PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2/05 p:/C|RCIDateDue.indd-p.1

AB INITIO CONFIGURATION INTERACTION (CI) CALCULATION OF THE
CHARGE-DENSITY SUSCEPTIBILITY OF MOLECULAR HYDROGEN AND
HIGHER-ORDER VAN DER WAALS INTERACTIONS FROM
PERTURBATION THEORY

By

Ruth L. Jacobsen

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemistry

2006

ABSTRACT

AB INI T10 CONFIGURATION INTERACTION (CI) CALCULATION OF THE
CHARGE-DENSITY SUSCEPTIBILITY OF MOLECULAR HYDROGEN AND
HIGHER-ORDER VAN DER WAALS INTERACTIONS FROM
PERTURBATION THEORY

By

Ruth L. J acobsen

The charge-density susceptibility x(r, r’;w) of a molecule is defined as the
change in the w-frequency component 5pc (r, w) of the electronic charge-density
at a point 1‘ within a molecule, due to a perturbing potential ’08 (1", w) of frequency
w applied at a point r’ (within linear response). This work includes a derivation
of an ab initio expression for the charge-density susceptibility and its application
to calculate x(r, I"; w) of the H2 molecule as a function of r, r’, and w in the
aug-cc-pVDZ basis set using a configuration interaction wavefunction with single
and double excitations (CISD). Since CISD theory is equivalent to full configuration
interaction (CI) theory in a two-electron case, the results are exact within a given
basis set. Results of the calculations of X (r, r’; w) for the H2 molecule are analyzed,
with emphasis on the behavior of the function when the frequency w is close to a
molecular transition frequency from the ground electronic state.

In order to test the calculations of x(r, r’; w) for the H2 molecule, the result
for x(r, I"; (.0) has been used to calculate the frequency-dependent polarizabilities
an, (w), ayy (w), and an (w) of H2 as a function of w in the DZ, DZP and
aug-cc-PVDZ basis sets. Excellent agreement has been obtained between our results
for static polarizabilities and the corresponding finite-field polarizabilities obtained
with the MOLPRO quantum chemistry software package.

Following a review of known results for the second-order and third-order correc-
tions to the energy of interaction of two molecules in the polarization approximation,

complex contour integration is used to derive a new equation for the third-order

dispersion energy of two interacting molecules. The results for the second- and third—
order interaction energies are used to obtain approximations to these energies for
pairs and clusters of hydrogen fluoride molecules, in terms of the properties of the

individual molecules.

In Loving M emor‘y

0f
Juan M. Lafuente

June 27, 1940 - September 11, 2001

iv

ACKNOWLEDGEMENTS

There are several people that I would like to thank for helping me through graduate
school and with this work:
my advisors, Dr. Kathy Hunt and Dr. Piotr Piecuch, for their leadership, guidance,
patience, and for teaching me with endless energy and enthusiasm;
my former advisor, Dr. Warren Beck, for giving me a working knowledge of ongoing
research in biophysical chemistry;
my advisorial committee, indcluding Dr. Robert Cukier, Dr. James McCusker, and
Dr. Jetze Tepe, for facilitating my understanding of the role of my work in theoretical
chemistry as a whole;
Dr. James Harrison, for his encouragement and for help with MOLPRO;
Dr. D. J. Gearhart, Dr. Karol Kowalski, Dr. Mike McGuire, Dr. Ian Pimienta, Dr.
Marta Wloch, Dr. Armagan Kinal, Dr. Peng-Dong Fan, Jeff, Maricris, Anirban, and
Lisa Green, for their help and for their camraderie;
Bronwyn, Emily, and Lisa Taylor, for cycling, swimming, and running with me;
Chi Alpha Christian fellowship, especially Mark and Cheryl McKeel, Anne Grain, and
Janet Lewis, and also to Mary Frances and Victoria, for being as solid as bedrock;
my extended family and my in-laws, Sue, Jake, and Katy, for all of their love and
support;
my sisters Cynthia, Christine, and Catherine, for their unconditional love, acceptance,
encouragement, and for always reminding me to laugh;
my late father, for being my hero in so many ways;
my mother and Mike, for being much more than I can ever explain;
and finally, my Lord Jesus Christ, the Ultimate Mystery — I live and breathe in you,

my Hope and Strength forever.

TABLE OF CONTENTS

List of Tables ix
List of Figures xii
1 Ab Initio Configuration-Interaction (CI) Calculation of the Charge-

Density Susceptibility of H2 from Perturbation Theory 1
1.1 Introduction ................................ 1
1.2 x(r,r’;w) at C1 Level: Derivation .................... 8
1.3 Algorithm for x(r,r’;w) ......................... 12
1.4 Results of x(r, r’ ;w) Calculations .................... 12
1.5 Results of (10,300) Calculations ...................... 18
Higher-Order van der Waals Interactions from Perturbation Theory 23
2.1 Introduction ................................ 23
2.2 The Quantum Mechanical Theory of Intermolecular Interactions . . . 24
2.3 The Polarization Approximation ..................... 25
2.3.1 Introduction ............................ 25
2.3.2 The First-, Second-, and Third-Order Polarization Energies . . 28
2.3.3 The Convergence of the Polarization Expansion ........ 31
2.3.4 Summary ............................. 31
2.4 The Multipole Approximation ...................... 32
2.4.1 Introduction ............................ 32
2.4.2 The Cartesian and Spherical Formulations of the Intermolecular
Interaction Operator in the Multipole Approximation ..... 33
2.4.3 The van der Waals Constants .................. 35
2.4.4 Removing the Angular Dependence from the Multipole Expan-
sion for the Interaction Energy ................. 36
2.4.5 The Convergence of the Multipole Expansions of the Inter-
molecular Interaction Operator and Interaction Energy . . . . 40
2.4.6 The Multipole Approximation and Nonadditive Interactions . 42
2.4.7 Summary ............................. 42
2.5 The Bipolar Expansion .......................... 43
2.5.1 Introduction ............................ 43
2.5.2 The Bipolar Expansion of Buehler and Hirschfelder ...... 44
2.5.3 The Fourier Integral Formulation of the Bipolar Expansion . . 45
2.6 Symmetry-Adapted Perturbation Theory ................ 46
2.6.1 Introduction ............................ 46

2.6.2 Weak Symmetry Forcing: Symmetrized Perturbation Theories 48
2.6.3 Strong Symmetry Forcing: Hirschfelder-Silbey Perturbation The-
ory ................................. 49

vi

2.6.4 The First and Second-Order Energies in Symmetrized Rayleigh-
Schr6dinger Perturbation Theory ................

2.6.5 The Convergence of the Symmetrized Rayleigh-Schrodinger and
Hirschfelder-Silbey Perturbation Theories ............

2.7 Many-Body Perturbation Theory ....................
2. 7.1 Introduction ............................
2.7.2 The Electrostatic Energy in Many-Body Perturbation Theory
2.7.3 The First—Order Exchange Energy in Many-Body Perturbation
Theory ...............................

2.7.4 The Second-Order Induction Energy in Many-Body Perturba-
tion Theory ............................

2.7.5 The Second-Order Dispersion Energy in Many-Body Perturba-
tion Theory ............................

Molecules A and B

50
52
53
53
58
61
63

65

3 The Second-Order Correction to the Energy of Two Interacting

67

The Third-Order Correction to the Energy of Two Weakly Interact-

ing Molecules

4.1 Non-Zero Contributions to AEéi: ..................
4.2 Higher-Order Induction: Terms of First-Order in Both 11’“) and [13° .
4.3 Hyperpolarization: Terms of Third-Order in p30 or it“ ........

4.4 Induction-Dispersion: Terms of F irst-Order in 11’“) or p30 .......
4.5 Third-Order Dispersion: Terms of Order Zero in Both HA0 and p30

80

80
81
85
89
114

Numerical Estimates of the Second— and Third-Order Corrections
to the Interaction Energy of Two or Three Hydrogen Fluoride (HF)

Molecules for Various Geometries
5.1 Leading Contribution to the Second-Order Correction to the Interac-
tion Energy of Two Molecules ......................
5.1.1 The Second-Order Correction to the Interaction Energy of Two
Colinear Molecules ........................
5.1.2 The Second-Order Correction to the Interaction Energy of Two
Colinear HF Molecules ......................
5.1.3 The Second-Order Correction to the Interaction Energy of Two
Parallel Molecules .........................
5.1.4 The Second-Order Correction to the Interaction Energy of Two
Parallel HF molecules ......................
5.1.5 The Second-Order Correction to the Interaction Energy of Two
Perpendicular Molecules .....................

124

124

125

129

133

137

5.2 The 2nd -Order Correction to the Interaction Energy of Three Molecules 139

5.2.1 The Second-Order Correction to the Interaction Energy of Three
HF molecules, Arranged Colinearly ...............
5.2.2 The Second-Order Correction to the Interaction Energy of Three
Parallel HF molecules ......................

vii

140

5.3 The Third-Order Correction to the Energy of Interaction of A and B 143
5.3.1 The Third-Order Correction to the Interaction Energy of Two
Colinear Molecules ........................ 145

6 Summary, Concluding Remarks, and Future Perspectives 163
Appendix A. Fortran CISD Code for Calculating x(r, r’ ;w) and am; (w) 179
Appendix B. Tables 247
Appendix C. Figures 251

References 268

viii

List of Tables

Table 1. Non-zero terms in the third-order interaction energy of molecules A and B,

classified by order in the permanent dipoles of A and B. Note that we have let \I/gi) =
(0) _
WA) and \I’OB — '03). 248

Table 2. Comparison of an (w) , Orgy (w) , and azz(w) values calculated by inte-
grating x(r, I"; w) using the algorithm described here and by finite-field calculations
performed with the MOLPROl7 quantum chemistry software package. Polarizabilities
were calculated in the DZ, DZP, and aug-cc-pVDZ basis sets, and polarizabilities are
given in a.u. 249

ix

List of Figures

Fig. 1. The charge-density susceptibility of the H2 molecule at the CISD level with
w = 0 a.u., r’ = 0, 0,0,:r = 0, -3.25 S y S 3.25 a.u., —3.25 S 2 S 3.25
a.u., and Ag = A2 = 0.05 a.u. in the aug-cc-pVDZ basis set. For this calculation,
the internuclear axis of H; was oriented along the Z -axis of the laboratory frame.

251

Fig. 2. The charge-density susceptibility of the H2 molecule at the CISD level with
w = 0.3858668352248763 a.u., r’ = 0,0, 0,3: = 0, —3.25 S y S 3.25 a.u.,
—3.25 S 2 S 3.25 a.u., and Ag = A2 = 0.05 a.u. in the aug—cc-pVDZ basis
set. For this calculation, the internuclear axis of H2 was oriented along the Z -axis of
the laboratory frame. 252

Fig. 3. The charge-density susceptibility of the H2 molecule at the CISD level with
w = 0.4812104263202694 a.u., r’ = 0, 0, 0, :L' = 0, —3.25 S y S 3.25 a.u.,
-3.25 S 2 S 3.25 a.u., and Ag = A2 = 0.05 a.u. in the aug—cc—pVDZ basis
set. For this calculation, the internuclear axis of H2 was oriented along the z -axis of
the laboratory frame. 253

Fig. 4. The charge-density susceptibility of the H2 molecule at the CISD level with
w = 0.3858668352248763 a.u., r’ = 0,0, 07,513 = 0, —3.25 S y S 3.25
a.u., Ay = A2 = 0.05 a.u., and Ag = AZ = 0.05 a.u. in the aug-cc-pVDZ
basis set. For this calculation, the internuclear axis of Hz was oriented along the Z
-axis of the laboratory frame. 254

Fig. 5. The charge-density susceptibility of the H2 molecule at the CISD level with
w = 0.4648380650856789 a.u., r' = 0,0, 07,3: = 0, —3.25 S y S 3.25
a.u., —3.25 S 2 S 3.25 a.u., and Ag = A2 = 0.05 a.u. in the aug-cc—pVDZ
basis set. For this calculation, the internuclear axis of Hz was oriented along the z-axis
of the laboratory frame. 255

Fig. 6. The zz component of the frequency-dependent polarizability Ozzz (w) of H2 as
a function of frequency w at the CISD level in the DZ, DZP, and aug—cc-pVDZ basis
sets. 256

Fig. 7. The 22 component of the frequency-dependent polarizability Ozzz (w) of
H2 at the CISD level as a function of o.) in the DZ, DZP, and aug-cc-pVDZ basis sets.
Here, we show the data included in —200 S an (w) S 200 a.u. in Fig. 6.

X

257

Fig. 8. The xx component of the frequency-dependent polarizability an (w) of
H2 at the CISD level as a function of w in the DZP and aug-cc-pVDZ basis sets.
258

Fig. 9. The xx component of the frequency-dependent polarizability am (0}) of H2
at the CISD level as a function of w in the DZP basis set. 259

Fig. 10. The yy component of the frequency-dependent polarizability ayy (w) of H2
at the CISD level as a function of w in the aug-cc-pVDZ basis set. 260

Fig. 11. The semi-circular contour C of radius R1 in the upper complex half plane.
In this figure, 1m and Re denote the imaginary and real axes, respectively. Also 0 is
the angle between the real axis and R1, and S is the portion of C off the real axis.

261

Fig. 12. Colinear arrangement of molecules A and B. In this figure, as, y, and 2 denote
the axes of the laboratory frame, and :r’, y’, and ;’ denote the axes of the molecular
frames of A and B. Also, A1 and A2 are the nuclei of molecule A, 81 and 82 are the
nuclei of molecule B, COM A and COMB are the centers of mass of molecules A and
B, and RAB is the distance between the center of mass of A and the center of mass of
B. 262

Fig. 13. Parallel arrangement of molecules A and B. In this figure, :13, y, and 2 denote
the axes of the laboratory frame, and :r’, y’, and z’ denote the axes of the molecular
frames of A and B. Also, A1 and A2 are the nuclei of molecule A, Bl and 82 are the
nuclei of molecule B, COM ,4 and COMB are the centers of mass of molecules A and
B, and R AB is the distance between the center of mass of A and the center of mass of
B. 263

Fig. 14. Perpendicular arrangement of molecules A and B. In this figure, 2:, y,
and z denote the axes of the laboratory frame, 27’, y’, and 2’ denote the axes of the
molecular frame of A, and x”, y”, and 2" denote the axes of the molecular frame
of B. Also, A1 and A2 are the nuclei of molecule A, BI and B2 are the nuclei of
molecule B, COM ,4 and COMB are the centers of mass of molecules A and B, and
RAB is the distance between the center of mass of A and the center of mass of B.

264

xi

Fig. 15. Colinear arrangement of three HF molecules. In this figure, we have labeled
the three HF molecules HFI, HF2, and HF3 in order to distinguish between them.
Also, as, y, and z denote the axes of the laboratory frame, 33’, y’, and 2’ denote the
axes of the molecular frames of each HF molecule, H1, H2, and H3 are the hydrogen
nuclei in HFl, HF2, and HF 3, F1, F 2, and F3 are the fluorine nuclei in HFI, HF 2, and
HF3, and COMHFI, COMHF2, and COMHF3 are the centers of mass of HF], HF2,
and HF3. Finally, RHF1_HF2, RHF1_HF3, and RHF2_HF3 are the distances between
the centers of mass of HF1 and HF2, HF1 and HF3, and HF2 and HF3, respectively.

265

Fig. 16. Parallel arrangement of three HF molecules. In this figure, we have labeled
the three HF molecules HF1, HF2, and HF 3 in order to distinguish between them.
Also, :13, y, and 2 denote the axes of the laboratory frame, :r’, y’, and 2’ denote the
axes of the molecular frames of each HF molecule, H1, H2, and H3 are the hydrogen
nuclei in HF1, HF2, and HF3, F1, F 2, and F3 are the fluorine nuclei in HF 1, HF2, and
HF3, and COMle, COMHF2, and COMHF3 are the centers of mass of HE, HF2,
and HF 3. Finally, RHFl—HF2, RHF1_HF3, and Ryp2_Hp3 are the distances between
the centers of mass of HFl and HF2, HF 1 and HF3, and HF2 and HF3, respectively.

266

xii

1 Ab Initio Configuration-Interaction (CI) Calcula-
tion of the Charge-Density Susceptibility of H2
from Perturbation Theory

1.1 Introduction

The charge-density susceptibility x(r, r’; w) gives the change in the w-frequency
component 5pc (r, w) of the electronic charge density at point r within a molecule,
due to a perturbing potential ve(r’ , w) of frequency an applied at r’ (within linear
response),1

6pe(r,w) = fx(r,r';w) ve(r',w) dr’. (1)
Molecular properties, such as dipole and higher-order polarizabilities, Sternheimer
electric field shielding tensors,5 induction energies for interacting moleculesf"10 van
der Waals dispersion energies,11 infrared intensities,5 electronic reorganization terms

in vibrational force constants,6’7 and intramolecular dielectric functions4 depend on

the charge-density susceptibility x(r, r’; w).
The dipole polarizability 003(0)) of a molecule is the first moment of the
charge-density susceptibility x(r, r’; w) ,with respect to both r and r' ,

(1.30) = f 7.. r2. x<r, raw) dr dr', (2)

where Ta and 7"5 are the O and 5 components of r and r’ , respectively
( (Oz, 0 = x, y, Z)).3 Similarly, the quadrupole polarizability 005,75 (w) is related
to x(r,r’;w) by

Cagnfiw) = 3]; (3 T0, T5» — 60,3 r2) $(3 r1, "r3 — (5,5 7”) x(r, r’; w) dr dr'.
(3)

In Eq. (3), Ta and T5 are the O and 0 components of r, and I“; and T3 are the ’7

and 5 components of r’ .3

When a molecule is subject to an external electric field Fe , the net electric
field experienced by nucleus I within that molecule depends on the unperturbed
electric field within the molecule and the screening of the applied field resulting from
rearrangements in the molecular electronic charge distribution. If the applied field
Fe is both static and uniform, the shielding tensor 7!, 0 determines screening effects

1

which are linear in Fe . The shielding tensor 7C1, 5 depends on the static nonlocal
polarizability density a(r, r’) ,5

7,23 = —/dr dr' Tag(RI,I') a(r, r’). (4)

In Eq. (4), Ta5(RI, r) is the QB component of the dipole-propagator ten-
sor T(RI, r) with coordinates R1 of nucleus I . Also, the static nonlocal
polarizability density determines the static susceptibility,3

VV' : a(r, r') = —X(I‘,I"), (5)

where V and V’ are the gradients with respect to r and r’ .

When a pair of molecules interact at long range (where the overlap between the
charge clouds is weak), the permanent and fluctuating charge distributions of each
molecule give rise to a perturbing potential ve(r, an) that acts on the neighboring
molecule. Thus, both induction and dispersion energies depend on x(r, I"; w) .
Specifically, the dispersion energy of a pair of molecules A and B depends on the
charge-density susceptibility at imaginary frequencies."”3 Several researchers have de-
rived equations for the second-order dispersion energy of interaction between two
atoms or molecules that depend on the charge-density susceptibilities of the inter-
acting atoms or molecules. In 1963, McLachlan12 derived the first equation of this
type. Specifically, McLachlan derived an equation for the second-order dispersion
energ of interaction of two atoms that is given in terms of implicit forms of the
charge-density susceptibilities of the two molecules. Following the work of McLach-
lan, Longuet-Higgins derived the first equation for the second-order dispersion energy
of interaction of two molecules that has an explicit dependence on the charge-density
susceptibilities of the two molecules. Specifically, according to Longuet-Higgins’ work,

the second-order dispersion energy of interaction W between two molecules i513

W = —Z%/dr1/dr2/dr’1/dr'2

X 706150 (rlir2$2€) (I, (r’1,r’2,z€), (6)

In - 1"1lll‘2 - 1"2I

 

In Eq. (6), 0 (r1, r2, 2f) and Oz’ (1"1, r’g, 7.5) are the charge-density susceptibil-
ities of the first and second molecules, respectively. Note that we are using Longuet-

Higgins’ notation in Eq. (6); Oz and 0/ correspond to the function x(r, r’; w)

2

in this work. In Eq. (6), r1 and r2 are points in the first molecule, r’1 and
r'g are points in the second molecule, and 26 is an imaginary frequency. Longuet-
Higgins’ a susceptibility measures the response of the electronic charge-density at
r1 to an exponentially increasing perturbation at r2 with 8(3) time dependence.
The a’ susceptibility measures the corresponding response at r’ 1 to an exponentially
increasing perturbation at r’g with elf” time dependence.

Langhoffl4 has expressed the second-order Coulomb energy in terms of the Fourier
integral of the Coulomb potential and uses the contour integration techniques of
Casimir and Polder106 to simplify the energy denominators contained in this ex-
pression for the Coulomb energy. The resulting expression gives the second-order
Coulomb energy in terms of the response functions of the two systems, which are
essentially Fourier transforms of the charge-density susceptibilities of those systems.
Jacobi and Csanak used a similar procedure to write the second-order dispersion en-
ergy of interaction between two closed-shell atoms in terms of implicit forms of the
Fourier-transformed charge-density susceptibilities of those atoms.15 Langhoff’s work
is similar to Jacobi and Csanak’s work in that both involve transforming susceptibili-
ties from configuration to momentum space. Langhoff transforms the susceptibilities
in the second-order Coulomb energy from configuration to momentum space, and Ja-
cobi and Csanak transform the susceptibilities in the second-order dispersion energy
similarly. The momentum-space expressions for the second-order Coulomb and dis-
persion energies contain integrals over fewer variables than the configuration—space

expressions.

Following the work of Jacobi and Csanak, several researchers developed approxi-
mate methods for calculating the second-order dispersion energy of interaction of two
atoms or molecules from the charge-density susceptibilities of those systems. Koide18
developed the first approximation to the second-order dispersion energy of interac-
tion of two spherically symmetric atoms by deriving a convergent expansion for the

interaction energy where V is written in terms of spherical wave interactions. This

expansion for the second-order dispersion energy AE gives

til—23k” d3k’ €21"
47r—1—2 [€78

5; h/ du m; (k, k’;2U) aB(-k, -k'; m) (7)

AE

where we have used Koide’s notation for Eq. (7). In Eq. (7),
a A (k, k’; zu) and a B (—k, —k’; zu) are the polarizabilities of atoms A and B, re—
spectively. Also, k and k' are wavenumber vectors, and 221. is an imaginary frequency.
These polarizabilies correspond to Fourier transforms of charge-density susceptibili-
ties of atoms A and B, which depend on spatial coordinates r and r’, as well as on the
imaginary frequency zu . Koide derived the polarizability a (k, k'; m) in terms
of spherical harmonics and polarizabilities Oz; (k, k’; w) , for angular momentum

quantum numbers I = 0, 1, 2, ...; a; (k, k'; w) is defined as
’-
01(k’ka‘”)_hzw2_—w—2—
#0 p

X (0le (k ) |p><plQl”(/€')l0>, (8)

where we have let w = w . The states Ip) are the eigenstates of the unperturbed
atom (which is either atom A or B in this case), and I0) is the ground state of the
unperturbed atom. Again, we are using Koide’s notation in Eq. (8). Also, fiwp —
Ep—EO, where E'p and E0 are the energies of states | p) and I0). Finally, Q)” (It)
and Q)" (’6') are multipole moment operators which are defined in Koide’s work in
terms of spherical harmonics and Bessel functions. The polarizability a; (k, k’; w)
in Eq. (8) can be approximated using the variational method of Karplus and Kolker19
or other variational methods. After calculating Oz; (’9, k’; w) , one can use his
or her results for Oz; (k, k’; w) and the appropriate spherical harmonics to obtain
01A (k, k'; w) and a3 (—k, —k’; zu) . Finally, using the resulting expressions for
01A (k, k’; w) and Oz B (—k, —k'; W) and Eq. (7), one can obtain an approximate
dispersion energy of interaction of atoms A and B. References 1, 25—30 and 31 give

equations for second-order dispersion energies that depend on approximate charge-

4

density susceptibilities and include other derivations and calculations that are related

to these equations.

Recently, Kohn, Meir, and Makarov20 have used density functional theory (DFT)
to derive a seamless expression for the van der Waals interaction energy of two atomic
or molecular systems. The expression is seamless in the sense that it yields accurate
van der Waals energies at any intersystem distance. They began their derivation
by using either the local-density approximation (LDA) or the generalized-gradient
approximation (GGA) to describe the electron density 7) (r). Next, they separated
the Coulomb potential into short and long-range parts, and assumed that the van
der Waals energies could be completely attributed to the long-range interactions.
At this point, they used the adiabatic connection formula to write the long-range
interaction energy. After transforming this expression into the time domain, they
obtained the correct long-range limit for the van der Waals energy EvdW of two
spherically symmetric atoms A and B,

 

Co
EvdW = _E’ (9)
with the following expression for Cg :

3 °° °° (t) (t >

X2Z 1 X213Z 2
C = - dt dt . 10
6 71' f l/ 2 t1 + t2 ( )

0 0

In Eq. (9), R = IR A — RBI, where RA and R3 are the coordinates of the nuclei
of atoms A and B, respectively. In Eq. (10), X22 is the 2 component of the response
of the electron density to a perturbation applied in the 2 direction. In equation form,
this response is

X” = fdr1dr2X(r1,r2)ZIZ2a (11)

where X (r1, r2) is the static charge-density susceptibility.

Dobson and co—workers""1723 have developed a seamless density functional for cal-
culating the van der Waals interaction energy of atomic, molecular or other physical
systems. The functional is defined by four equations, and Dobson and Wang give spe—
cific forms of these equations for the interaction of two jellium metal slabs in reference

IGA(

21. The first equation gives the average ground-state electron density 77' z, z’) of

5

the interacting system. The second equation, which depends on fiIGA (Z, Z') and the
Kohn—Sham polarization response a??? of the system, gives the Kohn-Sham density-
density response function X KS (2, 2’, q”, 28) of the interacting system. The third
equation is the Dyson-like screening equation for the Kubo density-density response
function )0 (r, r’; w = 28) of the interacting system. When A = 1 in the Kubo
density-density response function, we obtain the frequency-dependent charge-density
susceptibility x(r, r’; w) defined in this work. The equation which gives the Kubo
density-density response function depends on the Kohn-Sham density-density re-
sponse function, the
exchange-correlation kernel fch (r, r’;w = 28) of the system, and a modified
electron—electron interaction AVCoul . Finally, the fourth equation for the van der
Waals density functional is the adiabatic connection fluctuation-dissipation (ACFD)
formula. This equation uses VCoula X» and X KS as input, and gives the correlation

1 notation to

energy of the system. In this work, we have used Dobson and Wang’s2
refer to all quantities contained in the four equations that yield their seamless density
functional for van der Waals interactions. Also, all quantities mentioned in this work
that appear in Dobson and Wang’s equations are defined in reference 21. Dobson and
Wang have carried out a related derivation (and calculations) in reference 24.
Equations for the second-order dispersion energy that depend on charge-density
susceptibilities are superior to the corresponding point-multipole expressions for the
second-order dispersion energy because the former equations account for charge-
overlap effects that are neglected in the latter expressions. Because charge-overlap
effects are accounted for, the dispersion energy as defined in terms of charge-density

susceptibilities is finite as R —> 0 , whereas the corresponding infinite series for the

point-multipole dispersion energy diverges.
The static charge—density susceptibility x(r, r’) determines the derivative of the
electronic charge density with respect to nuclear coordinates,5

5P8(r)_ A —1/ I I A r A-1
BRA —Z (47reo) dr x(r,r)V Ir R| , (12)

6

where Z A is the charge of nucleus A with coordinates RA , and VA denotes
the derivative with respect to RA . Eq. (12) holds because the electronic charge
density responds via the same susceptibility to an applied potential and to the change
in the nuclear Coulomb potential when a nucleus shifts.5 Changes in the electronic
dipole moment as a molecule vibrates depend on linear combinations of the deriva-
tives 8p8(r) /8RA ; thus the intensities of vibrational transitions are related to the
internal charge redistribution in the molecule by Eq. (12). Similarly, the electronic
charge redistribution term in harmonic force constants depends on x(r, r’) ; this
term corresponds to the induction energy energy of the molecule, due to changes in

the nuclear Coulomb field, as the molecule vibrates.7

A non-local intramolecular dielectric function ev‘1(r, r’; w) characterizes the
screening of an applied potential ve(r; w) by the electronic charge redistribution,
to give an effective potential veff(r, w) within the molecule

veff(r,w) = [651(r,r';w) ve(r',w) dr'. (13)

The dielectric function is related to x(r, I"; w) by4

60 e;1(r,r’;w) = 6(r — r') + (47reo) 1./r—-dr”| r”| 1)((r”,r';tu). (14)

The correlations of the spontaneous quantum mechanical fluctuations in charge
density are determined by the imaginary part of the charge—density susceptibility
x”(r, I"; w) via the fluctuation-dissipation theorem,57

$<5pe(r,w) 6pe(r’,w’) + 6pe(r’,w’) 5pe(r,w)> = (5;) x”(r, r’;w)
x6(w+w)coth (2%). (15)

For this reason, the total electronic energy of a molecule is determined by x(r, r’; w)
and the permanent molecular charge density,58

_ _ <p.<__r_)> 1 r r,<p.(r)><p <r'>>
25,32” ("r Ir—R_7I W/d d lr-r’l

 

 

(a) / . /M It? — (é)
(3) g 2;. R... / dr 04:31:}; r)“. (16)

In Eq. (16), 266 is an electronic self-energy. Additionally, the charge—density

 

susceptibility x(r, r’;w) is related to the softness kernel as defined in density
functional theory.”34

The static charge—density susceptibility x(r, I") has been calculated ab initio
using a finite field CI approach for water by Li, Ahuja, Hunt, and Harrison.32 Ando,35
Zangwill and Soven,36, Gross and Kohn,”:38 and van Gisbergen and coworkers39 have
developed techniques for calculating the charge-density susceptibility within density
functional theory (DF T). The charge-density susceptibililty can also be calculated ab
initio using time-dependent perturbation methods,

40’44’41743’45_5° or by quantum Monte Carlo methods.51-53 Fur—

pseudo—state techniques,
thermore, an ab initio expression for the charge-density susceptibility x(r,r’;w) can
be formulated by deriving the linear response function within coupled-cluster (CC)

theory. 54

1.2 x(r, r’ ;w) at CI Level: Derivation

Quantum mechanically,

X(1‘a 1";w) = (‘I’olpe(r) C(01) Pelr')|‘1’0>
+ (wfllpeh'l) G(—W)pe(r)|\110), (17)

where \Ilo is the ground-state wavefunction, pe(r) is the electronic charge-density

operator at r ,

N
r) = —Z (5(r — ri), (18)
i-l
pe (r’) is the charge-density operator at r’ ,and C(w ) is the reduced resolvent,

G(w) = (1— mm — Eo - n...) =2 E'waf’flw. (19>
K750

 

In Eq. (19), H is the electronic Hamiltonian for the unperturbed molecule, 90 is
the projection operator for the ground-state wavefunction, and W K is the Km
excited-state wavefunction. Also, E0 and E K denote the energies of the ground
and K th excited states, respectively. Thus

x<r,r';w> = Z00040011,.)(“WWW

 

 

K9“) EK - E0 — hw
((r‘l’olpe’ )l‘I’Kll‘I’KlpeO‘ )I‘I’0>
+ K220 EK- E0 + m . (20)

At CI level, the ground and Kth excited- state wavefunctions ‘1'0 and \I/ K have

|\IIO) = 0)|<I> +;c’a(K )Icrgl) + Zcfif
i>j.a>b
+ =le Olll‘DJ

J

26.11“ MM) (21)

the form

I‘I’K)

In Eq. (21), IQ) is the reference determinant for the ground state, VP?) is a singly-

excited determinant, and I933?) is a doubly-excited determinant. Additionally,
|<I>J) represents all possible determinants in the full CI expansion. The coefficients
CJ(K) are the CI coefficients, determined by solving the Schrédinger equation at
C1 level,

HI‘I’Kl = EKl‘I’Kl- (22)
From Eqs. (20) and (21),
x(r, r'; w) = A(r, r’; w) + A(r’, r; —w), (23)
Where

A(r,r’;w)= Z :c 0ch, Kc)J~(K)ch(0)

 

K750 J,,J' J” J’”
Ian/2400.1)<<I>J~Ipe<r>I<I>J~I> (24)
Ex — E0 _ hw .

Since pe(r) is a one-electron operator, the matrix elements in Eq. (24) vanish unless
J and J ' differ by at most one orbital occupancy. We separate A(r, I"; 02) into

9

four sets of terms, based on the relationship between the determinants [<13 J), |<I>JI),

ICIDJH) ,and |¢Jm> ,

A(l‘, I"; w) = Z AJ’J”(r, 1"; OJ) + Z AJ,J”#JW(I', I"; w)
'1le J": J,” J:JI~JH#JIN
+ Z: AJ¢J’,J”(r, I"; w)
J¢JI,JII=JIH
+ Z AJ¢JIJH¢JW(I', I"; w). (25)
J75J’,J"§£J'"

Terms in Eq. (24) with J = J’ and J” = J,” are included in AJ,J"(r, r’;w) ,
etc. In terms of the atomic spin orbitals (b1, , qfiq, (fit , and (Du and the expansion
coefficients dls that relate molecular orbital l to atomic orbital ((93 ,

AJ,J~(r.r';wI = Z ¢p(r)¢q(r)¢t(r’)¢u(r’)

pflqtu

x Z J): CJ(0)CJ(K)CJ”(K)CJ”(O)
K740 i( j(,J”) occ
dipdizdjtdju
EK— E0 — hw
= Z 83??wa (rr)¢q( )cMr’) 0.0“). (26)

pmqtu

 

In Eq. (26), the sum over i(J) runs over all molecular orbitals i that are occupied

in determinant J and similarly for the sum over j(J”) . The sum over p, q,
t , and It runs over all atomic orbitals. For J aé J ' ,we define l(J, J’) as
the molecular orbital occupied in J but not in J ', 7'(J, J ’) as the molecular
orbital that is occupied in J ' but not in J , d)”, J’)? is the coefficient of atomic
orbital p in molecular orbital l, and dr(J,J’)q is the coefficient of atomic orbital q
in molecular orbital 7‘ (and similarly for J ” 75 J m ). We obtain

AJ=JI’JII#JIII E AJJII¢JHI( r, 1‘" ,LIJ) —Z ¢p(r) 05t( I") gbu(r’)

pqtu
X Z Z CJ(0)CJ(K)CJ"(K)CJ”’(O)
K#Oi(J),occ
dipdiqdl(J/I,Jlll)tdr(JII,JIII)u
EK— E0 — hw

= Z BJJ"¢J'"(‘*’ “01M )¢q(r)¢t(r')¢u(r’) (27)

pmqtu

 

10

AJ¢J’,J”=J”’(rar,;w) = Z ¢p(1‘)¢q r) (M M75 N )

p, q,t, u
X 2 Z CJ(O )(CJ' KC)J"(K)CJ"(O)
K960j(J”) ,occ
dI(J,J’) pdr(J,J’)qdjtdju
EK— E0 — fiw
= 2 333:3,» J... an )¢q(r)¢z(r’)¢u(r’)(28)

pmqtu

 

AJ¢J’,J"#J’”(rIrI;w) = Z ¢p(r)¢q(r)¢t(r’)¢u(r’)

 

mam
d I dT‘ I d n m d7. II In
x Z l(J,J)p g,J)q It: ,Jfi); (J ,J )u
K760 K_ 0"

X CJ(0)CJ'(K)CJ"(K )ch(0)
: Z B3231J1195Jm(w LU)

pq,,tu

X ¢p(r)¢q(r W) WK )¢u( I")- (29)

The calculations of the charge-density susceptibility x(r, r’; w) in this work are
based on Eqs. (26) - (29).
To check the calculations of x(r, r’; w) , we have calculated a03(w) from

x(r, r’;w) and Eq. (25). The frequency-dependent polarizability 04030.0) is
given by

 

aaW) = :Wolflal‘l’mmwm

 

K #0 E K - E0 — hw
+ Z (‘I’olual‘I’KI (\I’KIHaI‘I’M (30)
K #0 EK— E0 + M
where ya and #3 denote the a and fl components of the dipole moment
Operator, given by [10: 2: 62-730, + £431 Z jRja where N IS the total number of
i=1 J:

electrons in the molecule and M 18 the total number of nuclei. Also, 62- is the charge

on the ith electron, ZJ - is the charge on the jth nucleus, Tia is the a component

th

of the vector from the origin to the z electron, and Rja is the a component

11

of the vector from the origin to the jth nucleus. Eq. (2) follows from Eqs. (20) and
(30). From Eqs. (25) - (29),

' t
OCH/3(a)) = Z B3?J3fJ",-]m(w) [liq ”:3”, (31)
J,JI,JII,JIH
where
Bqujzf‘lfls‘l’” (LU) = 333;}:(01) ($le 6JIIJIII + 33?;gaéJ/n (LU) 6JJI (1 — 6]”,1’”)
+ ngzjivjfl (w) (1 _ 6JJI) 6JIIJIII
t
+ B‘gzétjl,Jll¢JI/I(w) (1 — 6JJ’) (1 — 6J"J’”); (32)

uflq and [1.2" are dipole moment integrals in the atomic orbital basis, defined as

#2." = (Plfialcv and Mfg“ = (tlfialm -
1.3 Algorithm for x(r, r’ ,w)

We have used the General Atomic and Molecular Electronic Software System
(GAMESS)16 to calculate the one- and two-electron integrals, to find the molecu-
lar orbitals at restricted Hartree-Fock (RHF) level, and to transform the one- and
two-electron integrals from the atomic orbital basis to the molecular orbital basis.
Then we have solved the CISD (CI singles and doubles) equations to find the CISD
coefficients C J(K ) , the ground-state energy E0 , and the excited-state energies
EK . Using these quantities, the atomic orbitals, and the transformation coeffi-
cients from the atomic orbital to molecular orbital basis, we have calculated AJJH,
A Jaé J', J”, AJ,J”¢J’” and A J¢ Jl’ Jrr¢ Jm and then summed to obtain the charge-
density susceptibility x(r, r’ ;w) of the molecule.

To calculate 003(w), we have used dipole moment integrals f 7‘0, q§p(r) ¢q(r) dr
and f T’fi ¢t(r’) qfiu (r’) dr’ computed in the linear-response coupled-cluster pro-

gram written by Kondo and co—workers.55’56

1.4 Results of x(r, r’ ;w) Calculations

We have calculated the charge-density susceptibility of the H2 molecule at the CISD

level as a function of frequency w and points r and r’ using a program based on

12

the algorithm presented here. Since CISD is equivalent to full CI in a two-electron
case, our results are exact within the basis set that we used. Each plot of the charge-
density susceptibility that we will present here was generated by fixing w, r’ , and :17
,and calculating x(r,r’;w) for all y and 2 included in —3.25 S y S +3.25
and —3.25 S 2 S +3.25 a.u. with Ay = A2 = 0.05 . The aug-cc-pVDZ
basis set was used for the plots. Note that the internuclear axis was aligned with
the z -axis for all calculations, and that we used the equilibrium geometry of the H2

molecule (the equilibrium bond length of H2 is 1.40126 a.u.).

Before presenting the results of our calculations, let us discuss some important
pr0perties of the charge-density susceptibility that we will use to understand the
behavior of x(r, r’;w) . Recall Eq. (20) from Sect. 1.2,

x<r.r';wI = Z(WOIWN‘I’K)(‘I’Klpe(r’)l‘l'o>

 

 

KaéO EK — E0 — fiw
(\Polpe(rr’)l‘l’1<><\1’xlpe(We)
+ 1;)(EK — E0 + fiw . (33)

According to Eq. (33), x(r, r’; w) is singular for energies fiw that are equal
to :t (EK - E0) , if the corresponding terms in the numerator do not vanish.
We have verified this property of x(r, r’; w) by calculating x(r, r’; w) of H2
at r’ = 0,0,0, :13 = 0 , and w = E1 — E0 (data not shown). For these
conditions, x(r,r’;w) of H2 was approaching infinity at —3.25 S y S 3.25
and —3.25 g z s 3.25 with Ay = Az = 0.05 .

If W0 is a singlet state and \11 K is a triplet state, then the matrix elements
(‘I’olpe(r)l‘I’K> , (‘I’olpe(r’)l‘I’K>, (‘I’Klpe(r)l‘l’o>, and <‘1’Klpe(r’)l‘1’o> van-
ish. This holds because ,0.3 (r) and pg (1") are spin-independent operators, so that
it is impossible for pe(r) and pe(r’) to change the spin of \I1 K to a singlet.
Therefore, triplet states will not contribute to x(r, r’; w) , and x(r, r’; w) will
not be singular at energies which correspond to transitions to triplet states. This
will be true for all of our data, since W0 is a CI singlet ground state for all of our

calculations.

13

In the aug-cc-pVDZ basis, H2 has 18 spatial orbitals and 36 spin orbitals. Eigh—
teen of the spin orbitals have a spin functions, and the other eighteen have fl spin
functions. We label a spin orbitals with odd integers, and 5 spin orbitals with even
integers. In terms of spatial orbitals, the lowest energy spatial orbital (the lag
orbital) is doubly occupied in the ground-state configuration of H2. In terms of spin
orbitals, the lowest energy a and fl spin orbitals are occupied in the ground-
state configuration of H2. Recall from Eq. (21) that the ground- and excited-state
CISD wavefunctions are generated from a linear combination of all possible singly-
and doubly-excited determinants. Therefore, according to Eq. (21), the ground-state
wavefunction for H2 in the aug-cc—pVDZ basis is:

lilo) = 00(0)|<I>>+C§(0)|<1>l>+C%(0)|<1>l>
+ 03(0)|<I>3> + 03(0)|<P3> + + C§i(0)l¢l§> + C§§(0)l¢l3> +
= 00(0)|‘1>>+ 01(0) “‘1’?) - |<1>3>l + + C§3(0)l@l3> + (34)

where (1):), (13?, (1)4 , and <I>§ are singly-excited determinants which correspond
to exciting an electron from spin orbital 1 (1090,) to spin orbital 3 (10m) , from
spin orbital 1 (1090) to spin orbital 5 (2090,) , from spin orbital 2 (1095)
to spin orbital 4 (low) , and from spin orbital 2 (1095) to spin orbital 6
(2093) , respectively. Also, (1)333 and (Pig are doubly-excited determinants
corresponding to exciting electrons from spin orbitals 1 and 2 to spin orbitals 3 and
4, and from spin orbitals 1 and 2 to spin orbitals 3 and 6, respectively. In Eq. (34),
the quantity lib?) — ((1)3) is a singlet spin-adapted configuration state function.
This is because each determinant involved in the linear combination in either quantity
involves exciting an electron from the same lower energy spatial orbital to the same
higher energy spatial orbital, and the resulting determinantel configuration is an
eigenfunction of S2 and S = 0 . If we write configuration ICE?) — |<I>§> in
terms of spatial orbitals $5 and spin functions Oz and fl, where (i/I,|21Ij) = (525 and

(alfi) == 503 , we have:
I<I>iI — I<I>3I =

(1)0(2 - fi(2)a(1)l- (35)

According to Eq. (35), the spin part of FF?) — [@3) is antisymmetric with respect

to electron exchange. We can carry out a similar analysis on each excited-state \I’ K

14

in the aug—cc—pVDZ basis set in order to determine whether a particular W K is a
singlet or a triplet state.

Let us return to our discussion of the properties of x(r, I"; w) . To do this,
consider the symmetry of the H2 molecule. Since H2 belongs to the Dooh point group,
there are an infinite number of irreducible representations that can be used to classify
the symmetries of its orbitals and states, including 23', 2;, 2;, E; , Hg, Hu,
Ag , and Au . If I" = 0 and W0 is a 129 state, then matrix elements
(‘I’olpe(r)|‘I’K>, (‘I’olpe(l")|‘I’K>a (‘I’Klpe(l‘)|‘1’0> ,and (‘I’Klpe(r')|‘1’0> are
nonzero only for ‘11 K states that have 129 symmetry. Therefore, when I" = 0
, x(r, r’; (.0) will only be singular for energies fiw approaching the energy of
transition to 29 states. In order to understand why this is true, we need to consider
the composition of the charge-density susceptibility. According to Eqs. (25) - (29) in
Sect. 1.2, the charge-density susceptibility is essentially a sum of products of atomic
orbitals evaluated at r and I" which is weighted by CI coefficients, coefficients for
converting atomic orbitals to molecular orbitals, and energy denominators. Although
we have not done so here, we can also write the charge-density susceptibility as a
sum of products of molecular orbitals evaluated at r and r’ and weighted by C1
coefficients and energy denominators. Now, consider the molecular orbitals of H2 as
a function of r’ . The 0'g orbital is nonzero at Z, = 0 , and the au orbital
is zero when 2’ = 0 . The 7rg and 7ru orbitals are also zero when 2’ = 0 .
Therefore, the only orbitals and states which will contribute to X (r, I"; w) of H2
when r’ = (0,0,0) are Ug-type orbitals and 29 -type states. Note that this
property of x(r, I"; w) is also true when r = 0 .

If 1" lies along the molecular axis and W0 is 129 state, then matrix elements
(‘I’olpe(r)|‘1’K>, (‘I’olpe(r’)|‘1’K>, (‘I’Klpe(r)l‘1’o) ,and (‘I'KIpe(r’)l‘Po> will
only be nonzero for 129 and 12:1 states. Therefore, when r’ lies along the
molecular axis, the only singularities in x(r, r’; w) will occur at transitions to 129

or 12,, states. This property of x(r, r’; w) can also be explained in terms of

15

the molecular orbitals of H2. The 09(r’) and 0,,(r’) orbitals are nonzero for all
z’ except 2' = 0 . Therefore, these orbitals will contribute to x(r, r’; w) for all
r’ = (0,0, i2) except I" = (0,0,0) . Since the 7rg(r’) and 7n,(r’) orbitals
are zero for all 2" , they will not contribute to x(r, r’; w) when r’ is on the large
2’ axis. This property of x(r, I"; w) is also true for r = 0 . Note these results
hold when the molecular axis is along the z’ (or z ) axis, which applies for all of
our calculations.

If r’ is somewhere in the 5132: plane, then the matrix elements (\I’Olpe(r)|\IJK),
(ll/olpe(r')|\IJK), (\lelpe(r)|‘1’0), and (‘I’Klpe(r’)|‘1’0> will only be nonzero
for 1H,; and 1AxLyz-type states (of the states generated by the basis set used).
As a result, if r’ is in the 332 plane, x(r, r’; w) will only blow up for energies
which correspond to transitions to 111$ and 1A5,:2_y2 states in these calculations.

Figure 1 shows the charge-density susceptibility of the H2 molecule with w = 0
and r’ = (0,0, 0) . As expected, x(r,r’;w) does not become singular when
w = 0 , since we are not near any transition energies. Also, note that X (r, I"; (.0)
has the general shape of a 0'9 molecular orbital of H2. This is as expected, since
I" = (0, 0,0) , for which only 0’9 type orbitals and 29 type states contribute to
X(r, 1"; w) -

Figure 2 shows the charge-density susceptibility of the H2 molecule when r’ =
(0,0, 0) and w = 0.3858668352248763 a.u. Note that fiw is near (E1 — E0)
in the aug—cc-pVDZ basis set. As in Fig. 1, the susceptibility has the general shape
of a 0'9 molecular orbital of H2. Again, this is observed because I" = (0, 0, 0) ;
since only 0'9 - type orbitals and 29 «type states contribute to x(r, r’; w) for
I" = (0,0, 0). Also, although M is near (E1 — E0) , x(r,r’;w) of H2 does
not become singular at any r . This is because \111 is a triplet state, and, as we
have discussed, triplet states will not contribute to x(r, I"; w) at any frequency, r
or r’ value.

The charge-density susceptibility of the H2 molecule at r’ = (0,0, 0) and

16

w = 0.4812104263202694 a.u. is shown in Fig. 3. This value of fin) is
near (E4 — E0) for the aug-cc-pVDZ basis set. As was the case for w = 0
and w = 0.3858668352248763 a.u., x(r, r’; w) has the general shape of a 09
molecular orbital of H2 because r’ = (0,0, 0) . However, in contrast to x(r, I"; (.0)
at w = 0 and at w = 0.3858668352248763 a.u., x(r, r’;w) is singular near
w = 0.4812104263202694 a.u. This happens because \114 is a 129 state.
Figure 4 shows the charge-density susceptibility of the H2 molecule at r’ =
(0,0, +0.07) and w = 0.3858668352248763 a.u., which is near (E1 — E0)
. This is the same frequency that was used to calculate X (r, r’; w) as shown in Fig.
2. As was the case for x(r, I"; w) as shown in Fig. 2, x(r, r’; w) is not singular
at this frequency, since \111 is a triplet state. However, the shape of X (r, r’; w) in
Fig. 2 is different from the shape of x(r, r’;w) in Fig. 4. Whereas x(r, I"; w) as
shown in Fig. 2 has the general shape of a 09 molecular orbital of H2, X (r, r’; w)
as shown in Fig. 4 resembles a 0,, molecular orbital of H2. This results from the
fact that r’ lies on the molecular axis. Recall that when r’ lies on the molecular
axis, both 09 and au-type orbitals can contribute to x(r, r’; (.0). Therefore, both
0’9 and O'u-type orbitals contribute to x(r, r'; (.0) when r’ = (0,0, 0.7) , and the
overall shape of x(r, r’;w) depends on a sum of products of 09 and an orbitals.
The charge—density susceptibility of the H2 molecule at I" = (0, 0, 0.7) and
w = 0.4648380650856789 a.u., which is near (E3 — E0) , is shown in Fig. 5.
According to Fig. 5, x(r, r’; w) is singular near this frequency. This results from
the fact that \113 is a 12,, state, since x(r, r’;w) of H2 at r’ = (0,0, 0.7) is
singular for transitions to Eu or 29 states. Notice also that the charge-density
susceptibility of H2 in Fig. 5 resembles a 0,, molecular orbital of H2. Again, this
occurs because both Ug-type and (Ia-type orbitals contribute to x(r, r’; w) when

1" lies on the molecular axis, but I" 75 (0, 0, 0).

17

1.5 Results of 010,300) Calculations

In order to test our calculations of the charge-density susceptibility of H2, we have cal-
culated the 01,3012), ayy (w) and azz(w) components of the static and frequency-
dependent polarizabilities of the H2 molecule at the CISD level in the DZ, DZP and
aug-cc—pVDZ basis sets. Table 2 provides a comparison of the am (w), ayy (w) ,
and (122 (w) components of the frequency-dependent polarizability 0105 (w) ob-
tained by integration of x(r, r’; w) (using the algorithm described here) and by
finite-field calculations carried out with the MOLPRO17 quantum chemistry software
package. There is excellent agreement between the 01m (w), ayy (w) , and an (w)
values calculated here and the corresponding values calculated with MOLPRO”.

Figure 6 shows the azz(w) component of the polarizability of H2 in the DZ, DZP
and aug-cc-pVDZ basis sets as a function of frequency for various frequencies within
the range from 0 to 1.5 atomic units (a.u.). Note that for these calculations, the bond
length of H2 was set to 1.40126 a.u. (the equilibrium bond length of the H2 molecule).
According to Fig. 6, an (w) in the DZ basis set becomes singular at approximately
0.58 a.u., and azz(w) in the DZP basis has a singularity at approximately 0.57 a.u.
In the aug-cc-pVDZ basis set, (122(6)) is singular at frequencies of approximately
0.47 and 0.6 a.u. Figure 7 shows the same data as shown in Fig. 6, however, the
range of the (122(0)) values in Fig. 7 is restricted to :1: 200 a.u.

We will now explain the singularities of an (w), ayy (w) , and an (an) in the
DZ, DZP and aug-cc—pVDZ basis sets in terms of the spin states which contribute to
010300) of H2 and the symmetries of these states. According to Eq. (30), Crag (w)
can be written in terms of matrix elements (\I’OI/lal‘I/K) and (WKIugIWO) of
dipole moment operators #0 and 1L3 , respectively. Because pa and [Lg are
spin independent and ‘110 is a singlet state, matrix elements (\IJOIMQIWK) and
. (ll/Klpglqlo) will be nonzero only if the ‘11 K states are singlet states. Therefore,

only singlet \I/K’S will contribute to (rag (w) .
We can determine which singlet excited states contribute to 010304)) by analyzing

18

the symmetries of the matrix elements which contribue to 0403 (w) . To begin this
analysis, consider Eqs. (21) and (30). If Eq. (21) is substituted for [\110) and I‘I/K)
, then (103(00) becomes

aag(w) = P(a, flea) + P(,3, a, —-w) (36)

and

P(a,6,w) = Z Z CJ(0)CJ'"(0)0J"(K)CJ'(K)

 

K>0 J,J’,J”,J”’
(‘I’Jlfial‘I’J'>@fllfifll‘I’J'") (37)
Ex — E0 — fiw

where all quantities in Eq. (37) have been defined previously. According to group
theory, the matrix element (@JlfiQIQJI) will be nonzero only if the direct product
of the irreducible representations of (DJ , [La , and (by: equals or contains the
totally symmetric irreducible representation for the molecular point group. We will
determine the symmetries of (DJ, fro, and (by that make (@JIIZQICDJI) nonzero

for a=$,a=y,and 0:2.

Let us begin by determining the symmetries of (1).], (by, and [12 that make
((PJIfiZIQJI) nonzero when J = J'. Eqs. (21) and (37) indicate that determinants
(1)] are contained within the ground state CI wavefunction \I’O . Because all
determinants within a CI wavefunction must have the same symmetry as the overall
wavefunction, determinants (PJ must have the same symmetry as the wavefunction

\I’o. Therefore, since we have required that ‘110 has 23' symmetry (in the D00},

point group), (DJ must also have 2; symmetry. If we let 0;, = - (e 21 + e 22)
in <¢Jlfizl¢J> , we have
(q’Jlfizlq’Jl = “6(‘1’lerl‘I’Jl — 8 <¢JIZZI¢J>- (38)

In the Dooh point group, 2 transforms as Z: . Therefore, the direct product
corresponding to either (@JIleqJJ) or ((PJIZQICPJ) is 23' Z: 23' = 2:,
which does not equal or contain 23' . Therefore, matrix elements <¢J|fi2|¢Jl)
vanish for all J = J’ . Now, let us determine the symmetries of (DJ, (1),]: and [1,;
that make ((PJIfiZICPJr) nonzero when J 75 J' . If we let 112 = — (e z1+ e 22)
in <¢J|fi2|¢JI> , we have

(‘I’Jlfizl‘I’J'l = —6 (q’lellq’J') — 6 ((PJIZ2I‘I’J'l- (39)

19

Let us assume that (DJ: has 2;" symmetry. Since (DJ has 2'; symmetry and
21 has 2: symmetry, the direct product corresponding to either (‘13le1 ICDJI) or
(<1) J|ZQICI> Jr) is 23' 2: 2:" = 23' , which is the totally symmetric representation.
Therefore, ((PJI/lzICIJJI) will contribute to an (w) when (In: has 2,“: symmetry.
Since the excited states ‘11 K that contain (DJ: must have the same symmetry as
(PJI , we can also conclude that excited states with 2: symmetry will contribute
to an (0.2). If (DJ: has a non-zero projection of the angular momentum along the

z axis, (@JlfizICPJI) vanishes. Therefore, only states of 2.: symmetry contribute

to azz(w) .

At this point, let us determine the symmetries of (DJ, (1)], and fix that make
((PJI/lxICPJr) nonzero when J = J’. If we let [LI : —(e 5131 +ex2) in
(Clefixqu), we have

(q’Jlfixlq’Jl = -€<<PJ|$1|‘I’J) - 8 (<I>J|2:2|<I>J). (40)

In the Dooh point group, 1‘ transforms as Hu . Therefore, the direct product
corresponding to either (@JI$1|<I>J) or ((DJISL‘QICDJ) is 2; IL, 23' = Hu
, which does not equal or contain 23'. Therefore, matrix elements ((1) Jl fiIIQ J!)
vanish for all J = J, . Let us determine the symmetries of (DJ, (1),]: and fix
that make ((DJIfixICPJI) nonzero when J 75 J'. If we let fix 2 — (6 11:1 + am)
in <¢J|fix|¢JI>, we have

(‘PJlfixl‘I’J'l = -€<‘PJ|$1|‘I’J'> - 8 (‘I’Jl172lq’wl (41)

Let us assume that (DJ: has 11,, symmetry. Since (DJ has 2; symmetry and
1:1 has Hu symmetry, the direct product corresponding to either ((1) J|$1|<I> J')
or (CI) J|z2|<1> J!) is 2:11,, 11,, = 23' + 2; + Ag, which contains the totally
symmetric representation. The matrix element ((1) Jl 113M) Jr) will contribute to
on (w) when (DJ: has HUI symmetry. Since the excited states ‘I’K that contain
(1)]: must have the same symmetry as (PJI, we can also conclude that excited states
with Hug symmetry will contribute to an (w). Excited states of other symmetries
do not contribute. In the Dooh point group, y also transforms as Hu. Therefore,

matrix elements ((1) JI [1qu) J) will also vanish, matrix elements (@JlflyIQJI)

20

will also be nonzero when (DJ! has Hug symmetry, and excited states with Hug
symmetry will be the only ones to contribute to ayy (w) . In Fig. 7, we see that
an (w) in the DZ basis set is singular at two frequencies, approximately 0.58 and
1.47 a.u. We also see that an (w) in the DZP basis set is singular at 0.57 and at
1.43 a.u.

For the three basis sets, each of the frequencies where an (w) blows up corre-
sponds to a specific (EK — E0) value. In the DZ basis, 0.58 a. u. and 1.47 a. u.
correspond to (E2 — E0) and (E7 — E0), respectively. In the DZP basis, 0.57 a. u.
and 1.43 a. u. correspond to (E2 — E0) and (E7 — E0). Finally, 0.47 a.u. and 0.6 a.u.
correspond to (E3 — E0) and (E10 — E0) in the aug—cc-pVDZ basis set. According to
an analysis of the spins and symmetries of states 2 and 7 in the DZ and DZP basis
sets, state 2 corresponds to the 112: of H2, and state 7 corresponds to the 212:
state of H2. A similar analysis of states 3 and 10 in the aug—cc-pVDZ basis set shows
that these states correspond to the 112,: and 212: states of H2, respectively.

Figure 8 shows the an (w) component of the polarizability of the H2 molecule
in the DZP and aug-cc-pVDZ basis sets as as a function of (.0, where w varies from 0
to 1.75 a.u. Note that the am. (w) component of the polarizability of H2 in the DZ
basis set vanishes, since there are no p—type atomic orbitals on either of the H atoms
in the DZ basis set. According to Fig. 8, am (w) of H2 in the aug—cc-pVDZ basis
set is singular at approximately 0.57 a.u. Figure 9 shows the on (w) component of
the polarizability of H2 as a function of w in the DZP basis set, where w varies from
0 to 1.75 a.u. Note that these data were also shown in Fig. 8, however, in Fig. 8,
the an (w) scale is too large to show the behavior of an (w) in the DZP basis.
According to Fig. 9, the an (02) component of the polarizability of H2 in the DZP
basis is not singular within the 0 to 1.5 a.u. frequency range.

Only Hu-type states will contribute to the am (on) component of the polariz-
ability of H2. The energy ha) of the frequency at which an (w) is singular in the

aug-cc-pVDZ basis set corresponds to the degenerate energy differences (E3 — E0)

21

and (E9 — E0). According to an analysis of the spins and symmetries of \Ilg and ‘119,
these states are the 111-qu and Ill—lay states of H2, respectively. Transitions to
the 1111”: state account for the singularity of am (w).

The ayy (w) component of the frequency-dependent polarizability of H2 in
the DZP and aug—cc-pVDZ basis sets as a function of w , where 0 S w 3 1.75
a.u., is shown in Fig. 10. Note that the ayy (w) component of the frequency-
dependent polarizability of H2 in the DZ basis set vanishes. Fig. 10 shows us that
the ow (w) component of the polarizability of H2 in the aug-cc-pVDZ basis set
also has a singularity at w = 0.57 a.u.

Only I'Iu-type states will contribute to ayy (w) of the H2 molecule. As was the
case for the an, (w) , the energy hw corresponding to the frequency at which
ayy (ad) has a singularity corresponds to the degenerate energy differences (E8 — E0)
and (E9 -— E0). As mentioned above, lIlg and \Ilg in the aug—cc-pVDZ basis are the

III-lugc and Ill—lug states of H2, respectively.

22

2 Higher—Order van der Waals Interactions from
Perturbation Theory

2.1 Introduction

This chapter provides an introduction to intermolecular interaction phenomena and
a brief summary of the methods used to calculate these interactions, within pertur-
bation theory. In chemistry, the interaction energy Em of molecules A and B is
given by

Eint : EAB — EA _ EB (42)
where E A B is the total energy of the two interacting molecules. Also, in Eq. (42),
E A and E B are the energies of molecules A and B when they are separated from
one another.

In comparison to covalent bond energies, intermolecular interaction energies are
weak. Whereas covalent bond energies are on the order of one hundred kilocalories per
mole, intermolecular interaction energies range from fractions of kilocalories per mole
to kilocalories per mole. Despite the relatively small magnitudes of intermolecular
interaction energies, several phenomena are affected by these interactions. Some of
these phenomena include the structures and properties of intermolecular complexes,
molecular dynamics, solvation, and the behavior of bulk gases, liquids, and solids.
This chapter is organized as follows: In Sect. 2.1, we introduce the idea of an in-
termolecular interaction energy and briefly discuss the importance of this kind of
interaction in chemistry, biology, and physics. In Sect. 2.2, we present the basic
quantum mechanical theory of intermolecular interactions. In Sects. 2.3 and 2.4, we
present and discuss the polarization and the multipole approximations, respectively.
We use Sect. 2.5 to introduce various formulations of the bipolar expansion. Fi-
nally, we discuss symmetry-adapted pertubation theory and many-body perturbation
theory in Sects. 2.6 and 2.7.

23

2.2 The Quantum Mechanical Theory of Intermolecular In-
teractions

In the Born-Oppenheimer approximation, the time-independent electronic Schrodinger
equation is

HI‘I’kl = Ekl‘I’kl- (43)

In Eq. (43), H is the Hamiltonian for the system, Nu.) is the exact electronic

wavefunction for the kth state of the system, and E], is the exact energy for the
kth state of the system. If the system consists of two interacting molecules A and
B, then Eq. (43) becomes

qulkAB> ”—— EkAqujkAB>3 (44)

where \IlkAB and Elena are the exact wavefunction and energy for the kth state

of the interacting system. In Eq. (44), the overall Hamiltonian H is
H=HA+HB+V, (45)

where H A and H B are the Hamiltonians for molecules A and B when they are

isolated from one another. The Hamiltonians H X are given by

H. = {am-:23:-

iEX iEX 06X
1

”Z. .Tz'j

2,36X,z<]

where X = A for molecule A, and X = B for molecule B. Indices i and j run
over all electrons in X, and Oz runs over all nuclei in X. Also, (i) V? is the kinetic
energy operator for the ith electron in molecule X, Z0 is the charge on nucleus
0 of molecule X, and Tia is the distance between the ith electron and nucleus (1 in
molecule X. Additionally, T25 is the distance between the ith and jth electrons in
molecule X. The Hamiltonians H A and H B satisfy the time-independent electronic

Schrédinger equation

Hxlq’zx) = szl‘llzx), (47)

with X = A for molecule A, and X = B for molecule B. In Eq. (47), [\Ilgx)
is the exact wavefunction for the lth state of molecule X , and E1 X is the
corresponding energy of that state. Finally, in Eq. (45), V is the intermolecular

24

 

interaction operator, which is

=;:::Z 36—2:—

 

76A 6E8 nEB 7€A1
2 25;— + 2 2 -—,. (48>
mEA (SEBT 7716.4 7268 mn

Indices ’7 and 5 run over all nuclei in A and B, respectively, and m and 77. run
over all electrons in A and B. Also, Z7 is the charge on nucleus 7 in molecule
A, and Z5 is the charge on nucleus 5 in molecule B. We also use 7'75 to denote

the distance between nucleus '7 in A and nucleus 5 in B, rm to denote the

th

distance between nucleus 7 in A and the n electron in B, and rm; to denote

the distance between nucleus 5 in B and the mth

th

electron in A. Here, Tmn is

th

the distance between the m electron in A and the n electron in B. When we

combine Eqs. (44) and (45), we have
(HA + H8 + V) I‘ljkAB) = EkABl‘IJk/IB)‘ (49)

Eq. (49) can only be solved exactly for the interaction between a hydrogen atom
and a proton. The energy of interaction between two larger systems can be obtained
by solving Eq. (49) approximately, using either perturbation theory or variational
theory.280 Although variational methods have successfully been used to calculate in-
termolecular interaction energies,280 these methods will not be discussed in this work.
For the remainder of this chapter, we will briefly discuss various perturbative schemes
for solving Eq. (49). For more detailed descriptions of each of the methods mentioned

- 9
here, we refer the reader to several rev1ews.7°_73w75

2.3 The Polarization Approximation
2.3.1 Introduction

In Rayleigh-Schrédinger perturbation theory, the Hamiltonian for any perturbed
atomic or molecular system is

H = H0 + v, (50)

where H 0 is the Hamiltonian for the unperturbed system, and V is the term that
describes the perturbation which is being applied to the system. If we assume that

25

the Hamiltonian in Eq. (49) has the form of Eq. (50), then H A + H B = H 0 in Eq.
(49) and we can solve Eq. (49) with perturbation theory. This specific partitioning of
the Hamiltonian in Eq. (49) is known as the polarization approximation (PA).69’76’70
In this formulation of Eq. (49), V is the potential of interaction between A and
B. When A and B are infinitely far apart, V = 0 and H = H 0 , since there
is no interaction between A and B when they are isolated from one another. Let us
consider only the ground—state Schrodinger equation for the interacting system. In
this case, Eq. (49) becomes

(HA + H8 + V) IqIOAB> : EOAquIOAB>' (51)

Now, let us derive expressions for the exact ground-state wavefunction and energy of
the interacting system. Let us introduce an ordering parameter A into the expression
for the Hamiltonian of the interacting system, so that H becomes

H =.- H0 + AV. (52)

Then, let us expand both ‘I’OAB and E0 A B in a power series in A . When we do
this, ‘I’OAB becomes

\IJOAB -_- 2 mg, (53)
where
(n) _1<9"‘I’0
‘11 11.0... 51—7” M
and, E0 A 8 becomes
E0... = 2 ”Eli; (55)

In Eq. (53), W83; is the nth-order correction to the exact wavefunction for

the interacting system, \IJOAB . Similarly, in Eq. (55), E6313 is the nth -order
correction to the exact energy for the interacting system, EOAB . Now, let us impose

the intermediate normalization condition, so that

011‘“ lWo.3>— —— 1 <56)

0.43

Then, substituting Eq. (53) )for I‘I’OAB) in Eq. (56), we obtain

Eva/f owl MAL) =0. (57)

n=1

26

Since A 75 0 , Eq. (57) simplifies to

00

201’ 323M219: 0. (58)

n=1

If we substitute Eqs. (53) and (55) into Eq. (51) and combine terms of the same
order in A , we obtain a system of equations of infinite order. The general form of

an equation in this system for a given n is

(HA + H8) “1101) > + VI‘IIUL 1)) : EéglJ‘Ifln) >E(1) |\Il(n 1)>+

0A3 0A3 0A8 EOAB DAB
1 0
+ E0123)I‘I’OA)B>+E0A)B|‘I’E)A)B>a (59)

(0)

where n = 1,2, ...00 . If we multiply each term in Eq. (59) by (\IIOABI and use

Eq. (54), we obtain

Efnl = (@(0)

0A8 0A8

WWW”). (60)

0A8

The nth -order energy E33; is known as the nth

-order polarization energy.
The ground state energy for the interacting system is the sum of the nth-order

polarization energies for all possible values of n , that is,

EOAB: ”2:0 E0203 (61)

Also, the n h -order polarization wavefunction (1,01) Bis calculated recursively from
(n) n 1) (MG ‘11(n—— k)
‘1’0A3=‘GV‘I’0AB +:Z:::EOBG \IIOAB , (62)

where G is the reduced resolvent given by Eq. (19) in Chap. 1. The 1‘“, 2nd and
3rd -order polarization energies have been studied extensively, and their physical
interpretations are well understood. In the next section of this chapter, we will
present the equations for each of these energies and briefly discuss their physical

interpretations.

27

2.3.2 The First-, Second-, and Third-Order Polarization Energies

The 13t-order polarization energy is obtained by letting n = 1 in Eq. (60),

E‘” =< ‘0) IVIWJL.) (63)

0A3 0A8

We approximate W82; by the product of the exact ground-state wavefunctions
\IJOA and ‘IJOB of A and B, so that

\Ilffim— _ \IJOAQIOB. (64)
Using Eq. (64) in Eq. (63), we obtain
EDA):B <WOA‘IIOBIVlw0A wOB)‘ (65)

The energy given by Eq. (65) is called the electrostatic energy. Classically, E323 is

interpreted as the Coulombic interaction energy between the charge distributions of
A and 8.7035 When the distance between A and B is large enough that the potential
energy of interaction of A and B is asymptotically approaching zero, the electrostatic
energy is the sum of the energies due to the interactions between the permanent
multipole moments of A and B. When A and B are relatively close together, the
electrostatic energy contains terms that can be attributed to the overlap of the charge
distributions of A and B. These terms are largely responsible for stabilizing van der
Waals complexes that consist of an atom and a diatomic molecule.77_80 Also, the
structures of dimers of polar molecules, especially of hydrogen-bonded dimers, are

largely determined by the electrostatic energies of these complexes.231—8594336—93

We obtain the 2nd-order polarization energy by letting n = 2 in Eq. (60),

2
E5). = < SEBIVI‘II3.B> (66)
Using Eqs. (64) and (62) in Eq. (66), we obtain
Elli); : —<\IJOA\IJOBIVGV|‘IIOA\IIOB>‘ (67)

If we allow excitations of A or B only in Eq. (67), we obtain the induction con-

tribution to E323 . The term that arises from exc1tations 1n A can be physrcally

28

interpreted as the polarization of molecule A due to the static electric field produced
by B. Similarly, the term that arises from excitations in B can be interpreted as
the polarization of molecule B due to the static electric field produced by A. It is
important to note that the induction energy does not account for intermonomer elec-
tron correlation, that is, the correlation of the motion of the electrons in A with the
motion of the electrons in B. When the distance between A and B is such that in-
teraction energy is asymptotically approaching zero, the induction energies of A and
B can be calculated using the permanent multipole moments and static multipole
polarizabilities of A and B. At shorter distances, where the charge distributions of
A and B overlap, the polarization propagatorsg‘l’97 of A and B are also needed to
calculate E833 . Equations that express the second-order induction energy in terms
of polarization propagators are given in references 95 and 187. Distributed multi-
pole moments and polarizabilities are often used in calculations of induction energies
generated by the interaction of two larger molecules.98“100 However, because the po—
larizabilities used in these calculations are non-unique, these calculations are often
inaccurate.100 Angyan et. al. have improved these types of calculations by defining
distributed multipole polarizabilities so that they are basis set independent, and have

also used these polarizabilities to calculate induction energies. 1‘”
We obtain the 3rd—order polarization energy by letting n = 3 in Eq. (60),

3 0 2
E5). = (\PSALIVI‘I’EALl (68)
Using Eqs. (64) and (62) in Eq. (68), We obtain
E631; = (‘I’OA‘I’OBIVGVGVI‘I’OA‘I’OBX (69)

where V = V - (\IIOAWOBIVIWOAWOB). We obtain the 3rd-order induction
energy in the same way that we obtain the 2nd-order induction energy. When we
allow excitations in states of A or B in Eq. (69), we obtain the 3rd-order induction
energy as a sum of four terms. The first two terms represent the polarization of

molecule B due to the static electric field produced by A and vice versa. The second

29

two terms represent the mutual polarization of A and B by the fields of B and
A, respectively. As was the case for the 2nd-order induction energy, the 3rd-order
induction energy can also be calculated using the permanent multipole moments and
static multipole polarizabilities at large intermonomer distances. At distances where
the charge-densities of A and B overlap, the static polarization propagators of A and
B are also needed to calculate the 3rd-order induction energy. Moszynski et. al. has
derived an expression for the 3’d-order induction energy that expresses this energy in
terms of the static polarization propagators of A and B,187 including the quadratic
polarization prOpagatorsQ7"1°2‘105 of A and B.

We have discussed the terms that arise in the expressions for E3333 and E323 when
we allow excitations in A or B only. If we allow simultaneous excitations of A and B
in the expressions for E52; and E323, we obtain the dispersion contributions to these
energies. The dispersion energy appearing in E533 can be physically interpreted as
a consequence of the correlation of the motion of the electrons in A with the motion of

the electrons in B. At large intermonomer distances, the 2nd

E(2,disp)

OAB

-order dispersion energy
can be calculated from the dynamic multipole polarizabilities of A and

B .106 At distances where charge overlap is important, the polarization propagators of

2,d' _. . .
A and B are also needed to calculate ESABZSP) .95'107 109 The 3rd -order polarization

energy contains terms corresponding to a combined induction-dispersion effect as well
as terms corresponding to a pure dispersion effect. Moszynski et. al. have derived an

. . . . ' ' 3a. d‘d
equation Wthh expresses the induction-dispersmn energy E6142” 23”)

in terms of
the electron densities and polarization propagators of A and 8.187 In this work, the

p)

. . 3,d" . .
dispersmn energy EéAst is expressed 1n terms of frequency-dependent monomer

susceptibilities for the first time. It should be noted, however, that our result is
‘correct only for large distances between A and B, where exchange can be neglected.
. ,d'

Prev10us attempts to express EéiBzSp)

by Chan et. 01.110

in terms of monomer properties were made

30

2.3.3 The Convergence of the Polarization Expansion

The convergence of the polarization expansion has been thoroughly studied.1“_125
The polarization expansions for the interaction energies of H-H+ and H-H systems
converge. According to studies performed by Chalanski et. al., Jeziorski et. al., and
others, the polarization expansion for the H-H+ interaction energy slowly converges
to the energy of the 1509 ground state of the system at large H-H+ distances. Addi-
tionally, at small intermonomer distances, calculations of the polarization expansion
for the interaction energy of two ground-state He atoms carried to high order show
that the series is convergent. However, in general, the polarization expansions for
the interaction energy of other many—electron systems either converge to energies of

unphysical states or diverge.
2.3.4 Summary

There are two major advantages to using the polarization approximation to study
intermolecular interactions. The first advantage is that the polarization approxima-
tion is conceptually simple, relative to other perturbative methods for calculating
intermolecular interactions.280 The second advantage, which is more important than
the first, is that the energetic expressions which result from these calculations have
physically meaningful interpretations, as discussed earlier in this chapter.280 However,
there are also several drawbacks to using the polarization approximation to study in-
termolecular interactions. One major problem with the polarization approximation is
that the unperturbed Hamiltonian H 0 has the wrong symmetry with respect to elec-
tron exchange. Although H 0 is symmetric with respect to the exchange of electrons
within molecule A or B, H 0 is not symmetric with respect to electron exchange
between A and B.72 As a result, the polarization approximation does not account
for exchange effects.280 The inability of the polarization approximation to account for
exchange effects inspired the development of symmetry-adapted perturbation theory,

which will be discussed later in this chapter.

31

2.4 The Multipole Approximation
2.4.1 Introduction

Another common method for calculating intermolecular interaction energies is known
as the multipole approximation. In the multipole approximation, we express both the
intermolecular interaction operator and interaction energy as infinite series in inverse
powers of RAB , where RAB is defined as the distance between the centers of mass
of A and B. The multipole expansion for the intermolecular interaction operator V

in an arbitrary space-fixed coordinate system is

00 Vn
V = —'fi, 70
":2; (RAB) ( )
where Vn is given by280
n—l
Vn = Z: Wm—l—l- (71)
[=0

In turn, V1,n_1_1 describes the interaction between the 21 instantaneous moment
on A with the 2714—1 instantaneous moment on B. When we express the inter-

molecular interaction operator V as a multipole expansion in powers of —1— , we

RAB
can write the interaction energy of A and B as282’284v283

(6.4.03.1?)
(12.43)"

In Eq. (72), C X (X = A for monomer A and X = B for monomer B) is the

 

Emt (RABa CA, (3,131) N i C"

n=1

(7?)

Euler angle that describes the rotation of a coordinate system fixed on X with respect
to the spaced-fixed coordinate system in which V and Eint (R A B, CA, (3, R) are
defined. Also, R = (0, ¢) are the polar angles that indicate the orientations of
the molecular axes of A and B with respect to the space-fixed coordinate system.
The functions 0,, (CA, (3, R) appearing in Eq. (72) contains coefficients that are
called van der Waals constants.

In the following sections, we will discuss various formulations of the intermolecular

interaction operator, van der Waals constants, and intermolecular interaction energy

32

in the multipole approximation. Finally, we will discuss the convergence prOperties

of many of the expansions that we will discuss.

2.4.2 The Cartesian and Spherical Formulations of the Intermolecular
Interaction Operator in the Multipole Approximation

We can write the multipole expansion of the intermolecular interaction operator as
described by Eqs. (70) and (71) in terms of irreducible sphericalm'136 or Cartesian

tensorslzil"145

of the multipole moments on monomers A and B. First, we will write
the multipole expansion of the intermolecular interaction operator in terms of the
irreducible spherical tensors. Let us begin by noting that we will refer to the operator
Vn,n_1-1 in the following presentations as a specific form of V1 A1 3 , with L4 = n
and l B = n — l — I. The quantities l A and l B refer to multipole moments on

monomers A and B, respectively. In the spherical tensor formalism, V; A) B is

IA+IB

VIAJB = XIA.IB(RAB)_1A_IB_1 Z (-1)m 1731301)

fll=—fA—IB

X [MIA ® MlBlZ-HB' (73)

where X1“ 8 is a constant, and the equation for this constant is

Xm. = (—1>’BIISH%. (74)
where S is given by
S = 2L4 + 2Z3 (75)
21,,

In Eq. (73), we also have that 011213 (R) are complex spherical harmonics

which are often replaced by real tesseral harmonics.144 We note that equations for

the real tesseral harmonics are given in reference 145. The quantities M1 A =
m m

{MIAA,mA = -lA, ..., +lA} and M13 = {MIBB,mB = —lB, ..., +l3} are

multipole moment tensors on A and B, respectively. The M 1:1" and M5328 com-

ponents of the multipole moment tensors M1,, and M13 on A and B are given
by

M133" = 2: 2,7?ch (fp) (76)

pEX

33

where X = A for monomer A and X = B for monomer B. In Eq. (76),
p runs over all nuclei and electrons in monomer X, Zp is the charge of the pth
particle in X ,and Clrzx (fp) is a spherical harmonic. Finally, the tensor product
[MIA ® MIBllAHB is

[M1A®M,B]}" :5: Z MgAMgBuA,mA;zB,mB|z,m), (77)

mA=—1A m3=—IB
where (l A, m A; lg, m Bll , m) is a Clebsch-Gordon coefficient, and where we have
let lA + l3 2 l.143
Now, let us write the multipole expansion of the intermolecular interaction oper-

ator as given in Eqs. (70) and (71) in terms of Cartesian multipole moment tensors

on A and B. In the Cartesian formalism, V; A 13 is

a l l ,3
V1,, 1...: Z Z M { ”($731M } (78)
{a} {5}
Here, the M [{Aa} and M123} components of the multipole moment tensors on A
and B are
M127} : Z Zprp.i1rp.72'-'Tpmxa (79)
pEX

where X = A and ”y = Oz for monomer A, and X = B and ’7 = B for
monomer B. As in Eq. (76), p runs over all nuclei and electrons in monomer X,

and Zp is the charge on the pth particle in X. Also, Tp 71,7}, ,2...Tp,.,,x are the

Cartesian coordinates of particle 19. In Eq. (78), the tensor T{[l 1:5; is given by

 

TllA-HBl _ (R )1A+lB+1(—1)IA (V V V V v v )
{a},{3} — AB lA!lB! 01 02'” 011A .31 .32'“ .318

x (51;) . (80)

Mulder et. al. give general expressions for TE: } Hg; for l A + I}; S 6 in reference
142, and Isnard and co-workers give specific expressions for T{[ AYES; for tetrahedral
and linear molecules in reference 246. One can convert between the spherical and
Cartesian formulas for V; A.) B using equations derived by Coope et. al.145-148 and

Stone 149,150

2.4.3 The van der Waals Constants

In this section, we will discuss various methods for calculating the van der Waals

functions 0,, CA, (3,11) . We obtain the first expression for 0,, (CA, (3, R
by solving the chrodinger equation for the energy of interaction of A and B in
Rayleigh-Schrodinger perturbation theory with the normalized Hamiltonian H N ,
which is

H” = H0 + V”. (81)

In Eq. (81), VN is a truncated form of the multipole expansion that is given by
N

Vn
V” ZZW’ ‘82)

with N = N A + N B , where N A and N B are the total number of electrons in

n=1

monomers A and B, respectively. The operator Vn appearing in Eq. (82) is defined
in Eq. ( 71). When the Schrodinger equation containing H N is solved, we obtain an
asymptotic expansion for the energies of the AB dimer which is expressed in powers

of Bi; and contains the van der Waals functions 0'” (CA, (B: R) .112 According

to Ahlrichs,160 one can calculate the van der Waals functions 0,, (CA, (3, R) re-
cursively. Jeziorski et. al. give the equations needed to perform these calculations in

reference 280.
We obtain another expression for the van der Waals coefficients by asymptotically

expanding the polarization energies E32; as

00 7(3) , ,‘
E12; (......) ~20 5:35?)

m=1

 

(83)
Then, each van der Waals coefficient Cm (CA, (3, R) is obtained from

Cm (CA, (3,11) = i1: 07)?) (CA, CB, R) a (84)
11:1

where M is an integer whose value depends on whether the interacting molecules A

and B are neutral or charged.”0

35

We can also use the standard Rayleigh-Schrodinger perturbation equations of the
polarization approximation to write expressions for the C1)? ) coefficients. One
obtains these expressions by substituting Eq. (82) for the intermolecular interaction
operator into the standard RSPT equations of the polarization approximation. In

this approximation, 05,1) (CA,CB,R) and C13,?) (CA,CB,R) are

0.),(CACB, R): <.‘1.IV.I\II.1.> (85>

and280
05.?) (CA,CB,R) = 201’ OABleGABVn- kl‘I’ow) (86)
kzx

where G A B is the reduced resolvent for the AB complex. The expression for G A B
is given by Eq. (19) with index K replaced by k, ‘11 K replaced by \I’kAB , EK
replaced by EkAB , and E0 replaced by EOAB' Finally, if the interaction energy
is known, then the van der Waals coefficients can be computed from

01 (no.3) = jigwR.BE.... (RAM/143.1%.) (87)
and
0. (4.. <3, R) = 1. £1300 (12.43)" (E... (RAB, (A, <3, R)
n-10.(c.,<3, R)

 

["1

(88)
:1 R(AB)

2.4.4 Removing the Angular Dependence from the Multipole Expansion
for the Interaction Energy

The irreducible spherical and Cartesian expressions for the interaction energy pre-
sented in Sect. 2.4.2, as well as the equations for the van der Waals constants pre—
sented in Sect. 2.4.3, are very useful. However, since all of these expressions depend
on the Euler and polar angles (A, (B, R , they need to be re—evaluated every time the
geometry of the AB dimer is changed. Fortunately, we can also write an expression
for the interaction energy in the multipole approximation which separates the depen-
dence on (A, (B, R from the rest of the expression for the interaction energy. In this

36

equation, the non-angular part of the interaction energy (which contains the van der
Waals constants) depends only on the distance R A B between the centers of mass of
A and B, so that this portion of the interaction energy only needs to be re—computed
if the distance between monomers is changed. In the polarization approximation, the

th

equation for the n -order correction to the energy of the interacting system which

separates the angular and radial components of the intermolecular interaction energy
is
A
E01132? }€(() )B(RAB) AM) (CA: (3.11) . (89)
{A}

In Eq. (89), {A}( 507: )B(RAB) is the radial expansion coefficient which depends on
the intermonomer distance R AB . Additionally, A{ A} (CA, CB, R) is a function
containing the angular dependence of E ( )3 .The equation for this function is

L

LA LB
A{A}(CA,CB,R) = Z Z Z SA'ILDJICfA,KA(CA)*

MAL=—LA MB=—L3 M=—L

x DMB, BKB(CB) CL (R), (90)
where
_ LA LB L
SML — (MA MB M) (91)

is a 3 J symbol,143 Cit/I (R) is a complex spherical harmonic, and Di}: ,KA ((A)*
and DiliKe (C3)“ are elements of the Wigner rotation matrix as functions of the
orientation angles of monomers A and B.143 References 247, 248, 133-135 and 249
also contain derivations of equations (89) and (90). To obtain this energy in the
multipole approximation, we replace the intermolecular interaction operator that ap-
pears in the equation for the radial expansion coefficients {Mama (R A B) with the
multipole expansion of the intermolecular operator, as given by Eqs. (70) and (71).
When we do this, we can write approximations for the radial expansion coefficients
{“5823 (R A B) that are determined by the irreducible spherical tensors of the po—

larizabilities and multipole moments of A and B33032 Van der Avoird et. al. have

37

reviewed these derivations in reference 75. We will present the equations for the elec-
trostatic, second-order induction, and second-order dispersion radial expansion coeffi-
cients in the multipole approximation. Note that we will denote these approximations

to the exact radial expansion coefficients {“583 (R A B) by {A}(-:0“ )A(R B).

The electrostatic radial expansion coefficient8 1n the multipole approximation is

 

 

{A} (1) _ _ LA (2LA+2LB+1)!
EOAB (RAB) — ( 1) 6LA+LBAL (211A)! (2L3)!
QK Ang
LALA+BB+1° (92)
(RAB)
In Eq. (92), f: and Q22: are the spherical portions of the 21’" and 2L3
multipole moments on A and B. The equations for Q5: and Q5: are
Q5): 5 (‘I’OAIMLAA I‘I’OA) (93)
and
QK§= — (‘I’OBIMKB B-l‘I’osl (94)

In Eqs. (93) and (94), MI}: and Mg: are multipole moment operators of A

and B, as defined in their respective molecular coordinate systems.

The second-order induction radial expansion coefficient in the multipole approxi-

mation i3130,132—135
{A}e( 0.2ind)A):(RB _l§:i§: $32131
21A=1l54=113=0 (RAB)
C

{A},
{A},md;B (95)

22,0, (RAB)

where {A} is a set of indices given by {A} = {L4, :4, [3, 1’3} , and

3?le

00
’3=0
00
ll 8:1

fi:‘M8
IIM8~

n=lA+lA+lB+l’B+2. (96)
Also, C { A} m d A and C { A} m d— B are long-range induction coefficients, where
0"“ is
{A}, ind— A

{A} L L L OK K
C{A},ind—A: €1Ai' 1:13, 00:11.)“ (0) [Q13 ® nglLZ- (97)

38

The long-range induction coefficient for B is given by Eq. (97) with A and B
K

interchanged. The irreducible product [Q13 ® Q13] L: is given by Eq. (77) with

MIA = Q18, M13 2 Ql’Ba m = KB , and 1: LB . The quantity ELALBL

 

1,151,313;
is a constant, given by
SLALBL = (_1)lA+l:q (21A + 2Z3 +1)!(2l’A+ 2l’B +1)! §lA
(111131;; (2b,)! (213)! (21:4)! (2123)! B
x [(2104 'l' 1) (2L3 + 1) (2L + 1)}%
x (IA + lB,O; l’A + l’B,0|L, 0). (98)
where
1A I; LA
I” = ’B 1’3 LB (99)

lA+lBliq+123 L

is a 9j symbol,143 and (l A + l3, 0; lg + liBi 0|L, 0) is a Clebsh-Gordan coefficient.
Finally, 052114) LA (0) is the irreducible spherical tensor portion of the frequency-
dependent polarizability on A,

2(E —E )

K H 0A

“(113mm (w) = :(E "E 2 2
”750 nA_ 0A) —w

... .. KA
x [(WOAIMIAI‘I’M®<‘I’nA|Mz:,|‘I’oA>]LA (100)

with the frequency w = 0. In this equation, WM and EM are the nth excited-state

wavefunction and energy of A, respectively. The irreducible product
.. ... KA

[(WOAIMIAIWnA) <8) (wnAlMlquIOAflL is given by Eq. (77) with MIA =
A

(\I/oAleAlqlnA), M13 = (\I’nAIMlquloA), m = KA, l = LA. We obtain the
equation for the irreducible spherical component of the frequency-dependent polariz-

ability on B by replacing A with B in Eq. (100).

The second-order dispersion radial expansion coefficient in the multipole approx-
130,132—135

{Me-:33?“ (RAB) = - Z Z Z Z —'

imation is

(101)

with

{A} _ 1 L L L K K
C{/\},disp — gngigzizg/aufmm (W)a(1§1;3)LB(W)dW- (102)
0

Van der Avoird and co-workers281

have reviewed the calculations of the electrostatic,
second-order induction, and second-order dispersion radial expansion coefficients.
Since the electrostatic and induction coefficients are determined only by the multipole
moments and static polarizabilities of the monomers, it is relatively straightforward
to calculate these coefficients.280 However, because the second-order radial dispersion
coefficients are determined by the polarizabilities of the monomers at imaginary fre-
quency, calculating these coefficients is relatively difficult. To date, these coefficients
have been calculated with several different methods, including many-body pertur-

236,251,237,238,240,239 the second-

bation theory, order polarization propagator approach

(SOPPA),250 and the multiconfigurational time-dependent Hartree-Fock (MCTDHF)

259—2‘” Other methods that have been used to calculate these coefficients

technique.
are the limited CI technique,244’252 the random-phase approximation (RPA), or the

time-dependent coupled Hartee—Fock (TDCHF) approach.2‘“343353354342

2.4.5 The Convergence of the Multipole Expansions of the Intermolecular
Interaction Operator and Interaction Energy

In general, the multipole expansion of the intermolecular interaction operator as de-
fined by Eqs. (70) and (71) is divergent. However, there is a small region of configura-
tion space where the multipole expansion of V is convergent. The exact specifications
of this region are given in reference 151.230

Like the multipole expansion of the intermolecular interaction operator V ,
the multipole expansion of the interaction energy is also divergent. Although the
divergence of this expansion has only been proven for the H; system,154_156 it is

expected that the expansion will also diverge for multi-electron systems. Additionally,

Damburg et. (11.158 and Cizek and co-workers159 have shown that Eq. (72) is not

40

summable for H"; using conventional summation techniques,157 so it is also expected
that the expansion will not be summable for any multi-electron complexes.

Some of the methods used to define and calculate the van der Waals constants also
have divergent expansions of the intermolecular interaction energy.280 For example,
the multipole expansion of the intermolecular interaction energy that is produced
when the Schrodinger equation is solved using RSPT with H N and VN (as
defined in Eqs. (81) and (82)) is divergent.280 This divergence results from the fact
that VN cannot be treated as a small perturbation, which is a direct consequence
of the fact that the spectrum of H N is continuous.

The convergence properties of the asymptotic expansion given by Eq. (72) are
not well understood. Although researchers have studied the convergence properties
of the expansions for the first- and second-order polarization energies, no one has
investigated the convergence properties of the expansions for higher-order polarization
energies. Jeziorski et. al. have studied the convergence prOperties of the asymptotic

209 and Berns and

expansion for the electrostatic energy E61; for the water dimer,
co-workers have studied the convergence properties of the same expansion for the
N2 dimer.262 The expansions of E3114; for both the water dimer and the N2 dimer
converge. However, neither expansion converges to the correct physical value of the
electrostatic energy for the appropriate system. Vigné—Maeder et. al have shown that
in general, the asymptotic expansion of the electrostatic energy is convergent for any
system when Gaussian functions are used to approximate the unperturbed charge
distributions of the monomers.263 Again, however, the expansion does not converge
to the physical ground-state electrostatic energy of the system of interest. Several
researchers‘“‘“"267 have studied the convergence properties of the asymptotic expansion
for the electrostatic energy for each of various large molecules using the distributed
multipole analysis. Stone has reviewed this method in reference 98. Dalgarno et.
01.268 have studied the convergence properties of the asymptotic expansion for the

2nd-order polarization energy of H; Specifically, they showed that the asymptotic

41

expansion for the 2nd-order induction energy of H; is divergent. Young269 studied
the convergence properties of the asymptotic expansion for the 2nd-order polarization
energy of the H2 molecule. The author showed that the asymptotic expansion for the
2nd-order dispersion energy of H2 is divergent. We should also mention that neither
Dalgarno’s final expression for the 2nd-order induction energy of H; nor Young’s final
expression for the 2nd-order dispersion energy of H2 are Borel or Pade summable.157

One might be able to use distributed polarizabilitiesgwm"272 to make these series

convergent. To date, however, no one has done these studies.

2.4.6 The Multipole Approximation and Nonadditive Interactions

Stogryn273,274

was the first to use the multipole approximation in the study of nonad-
ditive intermolecular interactions. Piecuch275 has derived equations in the spherical
tensor formalism for the interaction energy of M molecules to any order of pertu-
bation theory. These equations are based on Wormer’s perturbation equations in
the spherical tensor formalism for the energy of interaction of two molecules.130’132
Piecuch has also derived an expression for the anisotropic induction energy of M
molecules through third-order in perturbation theory.63 The author has also used his
equations to derive expressions for the isotropic interaction energy65 and anisotropic
dispersion energy64 of M molecules through third-order in perturbation theory. Fol-
lowing this work, the author derived expressions for the induction energies of M

molecules through fourth-order in pertubation theory.276’277

Finally, Piecuch used the
equations presented in reference 275 to calculate the nonadditive induction energies
of the Arg-HF and Ar2-HC1 systems. We refer the reader to reference 278 for a review

of these derivations and calculations.
2.4.7 Summary

There are significant advantages and disadvantages of using the multipole approxi-

mation to calculate intermolecular interaction energies. Probably the most significant

42

advantage of using the multipole approximation to calculate intermolecular interac-
tion energies is that it is easier to evaluate multipole interaction energies than it is
to evaluate the corresponding polarization energies. This is because the expression
for any particular polarization correction must be completely re—evaluated any time
the Euler or polar angles between monomers are changed, while only part of the cor-
responding expression in the multipole approximation needs to be re—evaluated when
these angles are changed. As discussed in Sect. 2.4.4, we can write the multipole
expansion of the intermolecular interaction energy as a product of radial expansion
coefficients and angular functions. When one changes the Euler or polar angles be-
tween monomers, only the angular functions need to be re-evaluated.

Although the multipole expansion of the interaction energy is easier to evaluate
than the corresponding energy in the polarization approximation, there are also signif-
icant disadvantages of using the multipole approximation to calculate intermolecular
interaction energies. The most significant disadvantage of using the multipole ap-
proximation to calculate intermolecular interaction energies is that by definition, the
multipole expansion does not account for charge-overlap effects. Charge-overlap ef-
fects are contributions to the intermolecular interaction energy that are caused by
the overlap of the electron density on A with the electron density on B, and these
effects are largest when the intermonomer distance is near or below the van der
Waals minimum for the dimer. There is another method for calculating intermolec-
ular interaction energies that has the many of the same advantages as the multipole
approximation, and also accounts for charge-overlap effects. This method is known

as the bipolar expansion, and this method is the subject of the next section.

2.5 The Bipolar Expansion
2.5.1 Introduction

Another method for calculating intermolecular interaction energies involves using the

bipolar expansion for the intermolecular interaction operator. There are several ad-

43

vantages to using this method to calculate intermolecular interaction energies. Like
the multipole expansion, the bipolar expansion of the intermolecular interaction en-
ergy can be separated into a radial component and an angular component. Also,
intermolecular interaction energies computed using the bipolar expansion of the in-

termolecular interaction operator include contributions from charge-overlap effects.

2.5.2 The Bipolar Expansion of Buehler and Hirschfelder

The exact bipolar expansion of the intermolecular interaction operator proposed by
Buehler and Hirschfelder is255v256

00 1<
—1— = Z Z .1)ng (r1,7‘2, RAB) Y1? (61, <51) Y,;’" (0342) - (103)

mg 1,1,113=0m=—-l<
In Eq. (103), T12 is the distance between particles 1 and 2, where particle 1 belongs
to monomer A, and particle 2 belongs to monomer B. Also, l< = l A if I A < l B
,and l< = l3 if l3 < (A . The quantities 7‘1, 51, $1 and T2, ég, (52

are polar coordinates of particles 1 and 2, respectively. Finally YIT (81,651) and

Ylgm (62, 432) are spherical harmonics defined with respect to particles 1 and 2,
respectively.

Buehler and Hirschfelder255’256 derived Eq. (103) after assigning coordinate sys—
tems to monomers A and B. The coordinate systems that they assigned to A and B
have their origins at the centers of mass of A and B, and their z axes are co—aligned.
Also, the a: and y axes of the system on A are parallel to the :1: and y axes of the
system on B. The :12, y , and z axes in both of these coordinate systems are parallel
to the corresponding axes of an arbitrarily selected spaced-fixed coordinate system.
Note that we have written Eq. (103) with the notation used by Meath and co—workers
in reference 257. Note also that although N g at. (11.257 use Jlljllg (7‘1,7‘2, RAB) in
place of the B IIZIIIB (7‘1,7‘2, R A B) quantities used by Buehler and Hirschfelder, these
functions are proportional to each other.

There are four different expressions for .7);le (7‘1,7‘2, RAB), and the form of

44

this function that we use in Eq. (103) depends on the intermonomer distances of
interest. Specifically, the form of J )3; (7‘1,7‘2, RAB) depends on whether we are
interested in calculating intermolecular interaction energies for R A B > 7‘1 + 7‘2 ,
T1> RAB + 7‘2, 7‘2 > RAB + T1, or l7‘1 — 7‘2' S RAB S |T1+ 7‘2'. For
RAB > 7‘1 + 7‘2, 7‘1 > RAB + T2, and 7‘2 > RAB + 7‘1, the equations for
JIIZL (7‘1,7‘2, RAB) are combinatorial expressions containing l A and [3. Each of
these combinatorial expressions also consists of a product of powers of R A 3, 7‘1, and
7‘2, and this product is also multiplied by m. For [7'1 — 7‘2] S RAB S In + ml,
the equation for 17):}; (7'1, T2, RAB) is a finite sum of different powers of R A B, 7‘1,
and 7‘2. The exact expressions for J 1 A1; (7‘1,’r2, RAB) are given in reference 257.
If we neglect the contribution of the JIIA1|B(7‘1,7'2,RAB) functions to i when
T1> RAB +7‘2, 7‘2 > RAB +7‘1, l7‘1 — 7'2] S RAB S [7'1 +T‘2I, then i:- reduces

to the multipole expansion.

2.5.3 The Fourier Integral Formulation of the Bipolar Expansion

We can also express the bipolar expansion of the intermolecular interaction opera-

tor in terms of a Fourier transform. In this formulation of the bipolar expansion,
1/7‘12 i8258

i = 1 d3 keik R e—ik-f‘leik-I‘z (104)
7'12 271'2 [(32
Kay and co-workers258 assigned the same coordinate systems to monomers A and B
as Buehler and Hirschfelder assigned to A and B. Kay et. al.258 also chose the same
laboratory frame as chosen by Buehler and Hirschfelder. In Eq. (104), eik'R, eikrl

, and 621°?“ , are given by

l A
“=2 2 2' %;)—"C“m (k) q;" (k, 1“), (105)

with f‘ = R for eik'R, f‘ = fl for 1‘3“"f1 ,and i" = T2 for (and:2 . Note that

the equation for eik'f is expressed in terms of the arbitrarily selected spaced-fixed

coordinate system. In Eq. (105), the unit vector k = % , and its orientation angles

45

are 6k and (bk . Also, Cl-‘m (R) is a Racah spherical harmonic, and it is given
byl43

 

4w %
Ci" (k)-—- (2, +1) Y. (61,0). (106)
Finally, the k-dependent multipole operator qlm (10,?) is
m .. 2l+1 .
q. (k, r>=-(-—— 2),.)01’" (r) 1020. (107)

The j) (197‘) quantity contained 1n Eq. (107) is a spherical Bessel function. If we
substitute Eq. (105) into the Fourier integral given by Eq. (104) and integrate over
91: and ¢k ,we obtain

1 . - .

_ = Z ZlA-lB-J (2] + 1) 21A+18+1

T
12 (MB j kAkB

lA!lB! kAL B A
x7r(2lA)! (2113!) JAB/41,13]- (CAaCBaR)

 

00
x fdkj,()kRAqu/1,(k r1)qu (1,12), (108)
0
where we have expressed :1; in terms of the coordinate systems centered on A and
B. Also
lA l3 '
JAB=(O 0 (3)) (109)
The quantity Am};- (C A (B, R) is given by Eq. (90) with

{A} = {lA, kA, l3, k3, j} . If we select a space-fixed coordinate system that
has R along 2 and replace V in Eq. (60) with Eq. (108), we obtain an expres-
sion for the nth-order correction to the energy of interaction of A and B where the

angular and radial components are completely separate from each other.

2.6 Symmetry-Adapted Perturbation Theory
2.6.1 Introduction

Although polarization theory lends itself to physically meaningful interpretations of
each term in the interaction energy of A and B, it cannot be used to obtain quan-

titatively correct interaction energies. This is because polarization theory does not

46

account for the exchange of electrons between monomers. This, in turn, is because the
. . . 0 .
zeroth-order approx1mat10n to the exact ground-state wavefunction W023 v101ates

the Pauli-exclusion principle. This problem can be overcome by using A‘Ilggg rather

O . . .
than @328 as the zeroth-order approx1mat10n to the exact ground-state wavefunct1on

of the AB complex, where .A is the antisymmetrizing operator for the system, defined
as

__ NA!NB!
— (NA '1” NB)!

In Eq. (110), AA and A3 are the antisymmetrizing Operators for monomers A

A

 

AAAB (1 +73). (110)

and B. Also, ’P represents the sum of all possible permutations which exchange an
electron between A and B, where the appropriate sign factors have been assigned to
all permutations. However, since A‘I’gg . is not an eigenfunction of H O = H A +
H B , this wavefunction cannot be used as the zeroth—order approximation to the
exact ground—state wavefunction in conventional Rayleigh-Schrodinger Perturbation
Theory. This problem has lead to the development of several new symmetry-adapted

perturbation theories which maintain the definition of H 0 used in polarization

(0)

0A8

theory and use A‘I/ as the zeroth—order approximation to the exact ground-
state wavefunction of the system.

The first formulation of symmetry-adapted perturbation theory
(SAPT) was published in 1930 by Eisenschitz and London.161 Other foundational
works in this field include those of Murrell, Randic, and Williams;162 Hirschfelder and
Sibley;163 Hirschfelder;76'164 van der Avoird;165—168 Murrell and Shaw;169 Musher and
Amos;170 Kirtman;172 and Carr.171 We will summarize the most important features
of these works here. We refer the reader to several reviews for thorough discussions
of these works.72’73’12011731174

In general, there are two classes of symmetry-adapted perturbation theories. The
first class of symmetry-adapted perturbation theories include those theories which

were developed using weak symmetry forcingm"173 The energy expressions resulting

from the development of these theories contain the antisymmetrizing operator A

47

However, the equations used to derive these energy expressions do not contain
the antisymmetrizing operator. These theories have been used to calculate interac-
tion energies between one-electron as well as many—electron monomers.280 The second
class of symmetry-adapted perturbation theories contains those theories which were
developed using strong symmetry forcing.175*173 In these theories, the equations used
to derive the energy of the system contain the antisymmetrizing operator A . These
theories have only been used to calculate interaction energies between one— or two—

electron monomers.280

2.6.2 Weak Symmetry Forcing: Symmetrized Perturbation Theories

The first kind of SAPT that employs weak symmetry forcing which we will discuss
is called symmetrized Rayleigh- Schrodinger (SRS) perturbation theory. 175 In SRS

 

 

theory, the exact ground-state energy of the interacting system E 5 R3 is the sum

of all nth-order SRS energies E011 5 RS) ,
ESRS (12,8 SR3)

Ema-1:: E0 (111)

n=0
e e Eéjj‘RS) is given by175’176
,we) (72 1))-1 (0)
((Jn,SRS) ___ (‘1’ OABIV'AI‘IIOABQ E0,“ SR)S (Howl/“ll 01(3)) (112)
AB (1,0)) (0)) AB (1,0)) (0)
(‘1’ oABIAI‘I’oAB) (‘1’ oABIAl‘I’oAQ

It can also be shown that the nt h-order SRS energy is the sum of the corresponding

nth-order polarization energy E6”; and an exchange term EOn’B SR5 cub) ,th at
is,
('n,SRS') (,——nSRS arch)
EOAB =on8 + EOA’B (113)

E(n, ,SRS—exch)

The exchange term 0A3

included in Eq. (113) represents the energy
resulting from intermonomer electron exchange. The energy resulting from intra-

monomer electron exchange is included in E62; .

The generalized Heitler-London method IS another weak symmetry forcing method

th

which expresses the 71 -order perturbation exchange energy in terms of the wave-

48

functions of polarization theory.193 Cwiok and co—workers176 have shown that the

generalized Heitler-London energies are equivalent to the SRS energies.
The next weak symmetry forcing symmetry-adapted method that we will discuss

is called Murrell-Shaw, Musher-Amos (MSMA).169170164 Jeziorski and co-workers17

have shown that the nt h-order correction to the exact ground-state energy Eon’B MSMA)

of the interacting system in MSMA theory can be written as

 

 

E53151...) : <wéZBIVAIwéZ;XSMA’>
(‘1’ 0A)B|A|‘I’0AB>
_ 53113634511114)(W3?B|¢:)lnggk'AISM/q)> (114)
k=1 (‘I’OABI-AI‘I’OAB)

The first- and second-order corrections to the exact ground-state energy of the in-
teracting system in MSMA theory are equivalent to the corresponding corrections in
SRS theory.280 Higher-order MSMA corrections are not equivalent to the correspond-

ing corrections in SRS theory) because the MSMA energies EéjfilSMA)

rather than
the polarization energies E6: Bare used 1n calculating ‘14)...
2.6.3 Strong Symmetry Forcing: Hirschfelder-Silbey Perturbation The-
ory
There are two categories of theories that employ strong symmetry forcing. The first
category contains what are known as one-state theories,161’16""1‘3‘3‘181’19“_198 and the
second category contains what are known as multi-state
theories.16‘°’*130’174'1”1199—204 In one-state theories, only one state is included in the
perturbation equations in order to account for intermonomer electron exchange and
that state is included via the antisymmetrizing Operator $1.280 In multi-state theories,
such as Hirschfelder-Silbey (HS) theory,163 all possible states that can be produced by
intermonomer electron exchange are included in the perturbation equations.280 Each
state is accounted for by the inclusion of the specific permutation operator that gen-
erates the state.280 Although one-state theories are significantly less complicated than

multi-state theories, one—state theories do not have the correct asymptotic behavior

49

in the intermonomer distance R A B- Speficially, several calculations carried out on
H; have shown that energies resulting from one-state calculations do not reduce to
the apprOpriate polarization energies at large R A 3.181117312021205fio"

2.6.4 The First and Second-Order Energies in Symmetrized Rayleigh-
Schrodinger Perturbation Theory

In this section, we will write the specific expressions for the first- and second-order
corrections to the exact ground-state energy of the interacting system in SRS per-
turbation theory. Then, we will discuss the physical origin of each term in these
expressions. If we let n = 1 in Eq. (112), we obtain

 

 

E(1,SRS) _ (‘1’ 0,31,)IVAI‘IzoAL). (115)
0A8 '—
(‘I’OAAIAI‘I’OAB>
Then, using Eq. (110) in Eq. (115), we can show that
(1, SRS) (1) (ISRS— erch).
EOA’B =E0AB +E0A’B (116)
In Eq. (116), the exchange term Eg::RS_euh) is given by
011‘” I (V — V) PIW‘O) >
E51,]? ,SRS— ezch)_ _ 0A3 0,43 (117)
(0) (0) ’
1+ (‘I’OABIPI‘I’OAB>
.(117) represents the largest component of the total exchange energy in SRS
theory. Specifically, at the van der Waals minimum, EOI’B SRS exCh) comprises

at least 90 percent of the total exchange energy for many molecules.280 Eq. (117)
corresponds to the expectation value of the Hamiltonian for the interacting system
over the wavefunction “(“1100)” , where AlligB is the wavefunction which accounts
for electron exchange between A and B.

Jeziorski and co-workers have developed a method for calculating exact values

f E(1, ,SRS— —exch) 211

DAB Additionally, by restricting the types of electron exchanges

to include single exchanges only, Jeziorski et. al. and Williams and co-workers have
(1, h
written an equation which approximates E0113 SR5 we ) .2112” Moszynski et. al.

1,SRS— h . .
have rewritten the approximation to E's/1,3 we ) 1n terms of the propert1es

50

of monomers A and 3.186 Specifically, they have rewritten the approximation to

E(l, ,SRS—exch)

0A3 in terms of the one- and two-particle density matrices of A and B.

E(1,B ,SRS— catch)

The resulting equations have been used to calculate for various

systems.186’73’80

If we let n = 2 in Eq. (112), we obtain the second-order correction to the exact
ground-state energy of the interacting system in SRS theory. The expression for this

 

 

energy is
(0 1
E31335): <wéAAIVAIw31AA> _ Balsam 12.1.4111}; (118)
0) A 0) '
““3 <véAAIAIwéAA> B <w3AAIAIwOAA>

(ZSRS)

. . . 2
We can separate E0 AB into the second-order polar1zation energy E828 and

(2 S RS I
an exchange component EOA’B we ”230

(2.5RS) (2, SRS—erch)
EOAB =E0j3+ EOAB (119)
‘ ~ (2 S RS ) . . 280
Also, the second-order exchange energy EDA n SRS theory 18 given by

(2,—SRS exch) (2,SRS—exch—ind)+ (2 SRS— erch— disp)
EOAB ZEOAB +EOAB

(120)

E(2, ,SRS—exch—ind)

0A8 is the induction contribution to the second-order ex-

E32, ,SRS—exch-disp)
AB

where

change energy, and is the dispersion contribution to the second-

order exchange energy.280 The exchange—induction contribution Eé2’SRS_exCh—ind)
to the second-order exchange energy in SRS theory corresponds to the induction
energy that arises when electrons are exchanged between monomers A and 8.280
Williams and co—workers have shown that the second-order induction-exchange en-
ergy in SRS theory is a significant portion of the overall induction energy, especially at
intermonomer distances R A 3 which are smaller than the van der Waals minimum.78

(2 535— h— (1
Due to the complexity of the expression for EOA’B we in )

, the second-
order exchange-induction energy in SRS theory is also difficult to evaluate. How-
ever, Chalasinski et. al. have derived an equation which is an approximation to the
second-order exchange-induction energy E023 SR3 ”Ch ind) .214 They obtained this

approximation by 1gnoring the energetic contributions of higher than single-electron

51

(,——2SRS exch— —2'nd)

exchanges to the second-order induction-exchange energy E043 .Chalasin—

ski and co-workers have shown that this approximation is valid at and around the
van der Waals minimum for the helium dimer.213 Similar calculations need to be per-
formed on larger systems in order to determine whether this approximation is valid

in a general sense.

(,—-2SRS exch— —d23p)

The exchange-dispersion contribution E023 to the second-order

exchange energy in SRS theory corresponds to the dispersion energy that arises
when electrons are exchanged between monomers A and B280 This energy, how-

ever, is not a significant portion of the overall dispersion energy. Specifically, the

E(2, S RS —exch—d2'sp)

magnitude of 0A8

comprises only about a few percent of the

. . . . (2 h— d
overall d1sper81on energy.280 L1ke the exact express1on for EOA’B SR3 8“ in )

E32, ,SRS—exch—disp)
AB

3

the exact expression for is also very complicated and diffi-

cult to evaluate. Chalasinski and Jeziorski have also derived an approximation to

E(2, ,-—SRS exch— dzsp) 214

DAR They obtained this expression by making the same assump-

2, S RS h— d
tion that they made when deriving the approximation to EéA'B we in ). In

other words, they ignored the energetic contribution of greater than single-electron

2, S RS h— d .
exchanges to E623 “C 1810) .Unfortunately, however, even the evaluat1on of

E(2, ,SRS—exch—disp)

0A3 is difficult. This is because the wave-

" - - . . 2, SRS— h-d '
functlon used 1n calculatmg the approx1mat10n to E6“ 6“ 13p )

the approximation to
must contain
charge-transfer terms, 15and it is also because the approximation cannot be simplified

so that it depends only on the properties of A and 8.280

2.6.5 The Convergence of the Symmetrized Rayleigh-Schrodinger and
Hirschfelder-Silbey Perturbation Theories

Several researchers have studied the convergence properties of the SR8 and HS
symmetry-adapted pertubation theories. In particular, these groups have studied the
convergence properties of SRS and HS expansions for several states and intermolecu-

lar distances of H22+ 311,112,175 H2,176’208 and H82.285 The calculations of the convergence

52

properties of H2 and H82 were performed with very large basis sets. Therefore, it is
very likely that the results of these calculations are close to the results that would be
obtained if an infinite basis set were used.

In general, both the SRS and HS perturbation expansions for H; , H2, and H82
converge rapidly.280 However, according to the results of the calculations performed
on H; and H2, the convergence of the HS expansion is better than the convergence
of the SRS expansion. Also, because the convergence radii for the HS expansions of

208,285

H2 and H82 are similar, it is likely that the HS convergence for H82 will also be

better than the corresponding SRS convergence for H82 when expansions are carried
out to high order in 72.280

The most important difference between the convergence properties of the SRS
and HS theories is that the two expansions converge to different energies.280 While
the HS expansion converges to the physically correct energy of the system, the SRS

expansion does not.280

If we subtract the energy that the SRS expansion converges
to from the correct physical energy of the system, we obtain what is known as the
residual exchange energy.‘“"*175 However, according to Jeziorski et. al, the residual

exchange energy is so small that it does not affect the accuracy of the SRS method.280

2.7 Many-Body Perturbation Theory
2.7.1 Introduction

The polarization approximation, the multipole expansion, and symmetry—adapted
perturbation theory are useful theories for calculating intermolecular interaction en-
ergies and determining the physical origins of each term in the equations for these
energies. If the full configuration interaction (FCI) wavefunctions of monomers A and
B are used in the polarization approximation expressions for the interaction energies,
the intramonomer correlation effects are accounted for in these energies. However,
these wavefunctions are rarely used when calculating these energies. This is because

the sizes of these wavefunctions increase very quickly as the corresponding system

53

sizes increase, so that these wavefunctions are impractical to work with for systems
in which monomers A and B have more than two electrons. Instead, the wavefunctions
of monomers A and B are usually approximated by their corresponding Hartree—Fock
determinants. However, if the HartreeFock determinants are used to calculate the in—
teraction energies of the system within the polarization approximation, the multipole
expansion, or SAPT, then the effects of intramonomer correlation are not accounted

for.

A new theory has been developed which accounts for intramonomer correlation.
This theory is known as many-body perturbation theory (MBPT) or double pertur-
bation theory?“218 In double perturbation theory, we assume that there are two
perturbations acting on monomers A and B. The first perturbation is the inter-
molecular interaction, and the second is the intramonomer correlation. We use the
intermolecular interaction operator V which is given by Eq. (48) in the double per-
turbation theory equations to represent the intermolecular interaction perturbation.
Additionally, we represent the intramolecular correlation perturbation with a second

perturbation operator W , where W is given by
W = WA + WB. (121)

In Eq. (121), WA represents the intramolecular correlation in A, and WB
represents the intramolecular correlation in B. In this formulation, the Hamiltonian

for unperturbed monomer X is183
HX = FX + WX, (122)

where Ex is the Fock Operator for monomer X . To obtain the equations for
H A and H B ,we let X = A and X = B in Eq. (122), respectively. Using
Eq. (122), with X = A and with X = B in Eq. (45), we obtain

H=F“+W“+FB+WB+V 0%)

In the polarization approximation, we partition Eq. (123) so that FA + F B = H 0
. Letting FA + FB = F and WA + W8 = W in Eq. (123), we have that

H=F+W+V (HQ

Now, let us introduce the ordering parameters C and A into the overall Hamiltonian
for the interacting system, which is given by Eq. (124). When we do this, H

54

becomes
H = F+<W+AV. (125)
Then, if we substitute Eq. (125) into Eq. (44), with k = 0 , the result is182
(F + CW + AV)|‘1’0AB> = EOABI‘I’OAel (126)

If we expand both ‘IJOAB and EOAB in powers of C and A and collect terms of
the same power in A , we obtain the following double perturbation theory expressions
for the nth -order corrections to the ground-state wavefunction and energy of the AB

complex in the polarization approximation,182

\IISAL— _Z \IIOIXB (127)

and

E532- — 2 E61113. (128)
220

In Eq. (127), @822 is the nth~order correction in V and the ith—order correction
in W to the exact ground-state wavefunction of the interacting system. Also, in Eq.
(128), E62: is the nth -order correction in V and the ith-order correction in W to
the exact ground-state wavefunction of the interacting system. Similarly, we obtain
the SRS energy of the interacting system by introducing the ordering parameter C
into Eq. (112), expanding this expression in powers of C and A, and collecting terms
of the same order in A. When this is done, we find that the nth-order correction in

V to the SRS energy is given by178

00
(n, SRS) (n2 SRS)
EOAB =2 EOAB . (129)
2:0
In Eq. (129), EéZZSRS) ° 1sthe nth -order correction in V and the ith-order correc-

tion in W to the exact ground— state energy of the interacting system in SRS theory.
SRS
Each E0213 m ) has a polarization contribution EA: m; and an exchange contribution

E0722, ,SRS— BexCh)S othat
(2 SRS) 2 ) n SRS— I).
E0; =on; + E03 ’ “Cl (130)

55

th

The total exchange energy for the 72 -order correction to the energy of the inter-

acting system is

0,43 0.48 ’

E(n,SRS—exch): _ZE E(m', SRS—exch) (131)

',SRS- h
w}... E33; 6“)

th- order in V and ith-order in W.

is the contribution to the overall exchange energy which is

71.2)

Reference 280 reviews the evaluation of the MBPT expressions for EOAB and
\I/(ni) in detail.280 The procedures for evaluating these expressions were originally

developed by Szalewicz and co-workers182 and also by Tachikawa et. al.219 The eval-

E(n2’, ,SRS-exch)

0A8 5 much more difficult than the

nation of the exchange energies

evaluation of the corresponding polarization energies because the MBPT expressions

' — h . .
for EészSRS we ) contam 1ntegrals formed by the overlap of nonorthogonal or-

bitals. Therefore, MBPT exchange energies E (m’SRS-—8uh)

0A8 have only been evalu-

ated under the assumption that intramonomer correlation has been neglected (that

. - . . ',S S— I
1s, 2 = 0 1n MBPT express10ns for E33; R ex“) .

10, S RS I
order exchange energy EéAB’ 8“ 1) 211220

(20, SRS— exch— —2nd) 214

Specifically, the first-

the second-order exchange—induction

energy EOAB and the second-order exchange-dispersion energy
2 ch— d7 . . . .
ESAZSRS ex 181)) 214 have all been computed 1n this approx1mat10n.

Hi)
OAB

If we write diagrammatic expressions for the MBPT equations for E( and

E(n2', ,SRS—exch)

0.43 , we see that these expressions include disconnected terms. As a

result, these expressions scale improperly with the size of the interacting system.

By manipulating these equations, one can eventually cancel the disconnected terms.

. . 10 IOB,SRS— h
Usmg thlS procedure, researchers have calculated E6118)?” E ( we )3“
EOAB E 0A8 ’ EOA ’ EOAB
Ema—5123 exch)214

0A8

and

It is difficult to manipulate the diagrammatic expressions for the MBPT equa-

(122') and E(n2, SRS—exch)

tlons for EOAB 0A8 in order to remove the disconnected terms that

these expressions contain. Therefore, in order to avoid having to perform these ma-

56

nipulations, several researchers have used CC theory““7’218’223“230 to rewrite the di-
agrammatic expressions for the MBPT formulations of the RS and SRS perturba-
tion equations.177_179’216 Specifically, Rybak and co—workers have written CC equa—
tions for the MBPT formulation of the RS perturbation equations in the polarization
approximation.177 Additionally, Moszynski et. al. have derived CC equations that
contain only connected diagrams for the first-order exchange energy in the MBPT
formulation of SRS theory.178 These authors used the connected CC expansion of the
expectation value232 and the single-exchange operator 'P1 to obtain the connected
CC expression for the first-order exchange energy. Also, when deriving these equa-
tions, they also ignored the contribution of higher than single electron exchanges to

the total exchange energym’zm’288 Finally, they used the CC equations for the MBPT
(lO,SRS—-e:rch) E(11,SRS—exch)

first-order exchange energy to calculate EOAB , 0A8 , and
12’ R _ h - - 1,SR — h
E ((321138 S we ) 2 and they used a CC approx1mat10n to calculate E6118 3 exc ).178

Other symmetry-adapted MBPT CC equations have been used to write orbital ex-
10) E611)

12
A3, A8 , and E6213) ,177 and a symmetry-adapted MBPT coupled-
(2adi3P) 135
OAB '

Many-body perturbation theory can also be used to derive equations for vari-

pressions for E6

cluster approximation has been used to derive a similar expression for E

ous components of the interaction energies that express these energies in terms of
the polarization propagators and electron densities of monomers A and B. Moszyn-
ski and co—workers184 and Moszynski et. al.77 have both derived MBPT equations
for the electrostatic energy of interaction in terms of the electron densities and po-
larization propagators of A and B. Also, Moszynski, Cybulski, and Chalasinski187
have written MBPT equations for the induction energy of interaction in terms of
the electron densities and polarization propagators of A and B, and Moszynski et.
al. have derived MBPT equations for the exchange energy in terms of the same
properties of the interacting monomers.186 These MBPT equations for the electro-
static energy, the induction energy, and the exchange energy can be formulated
in terms of Maller-Plesset expansions for the polarization propagators and electron

densities 232.287.96.97,102,192

If these components of the total interaction energy are for-
mulated in terms of the Moller-Plesset expansions for the polarization prOpagators

and interaction energies, then the resulting energetic expressions are called nonrelaxed

57

expansions.280 The MBPT electrostatic, induction, and exchange energies can also be
formulated in terms of what are known as relaxed expansions for the polarization
propagators and electron densities. The expressions for these energies are called re—
laxed expansions.280 In general, for a relaxed expansion of any particular polarization
component of the interaction energy in MBPT, Eq. (128) is

E(n A,resp) =: Eéjgresp) (132)

In Eq. (132), E(n’B Hes}, is the total nth-order relaxed correction in V to the

122,1‘63 .
p) 13 the nth-order

specified component of the interaction energy, and EOAB
relaxed correction in V and the 2t h-order correction in W to the specified component

of the interaction energy.
2.7.2 The Electrostatic Energy in Many-Body Perturbation Theory

The MBPT expressions for the electrostatic energy in the polarization approximation
and in SAPT contain contributions from the intramonomer correlation energies of
monomers A and B. Rybak, Jeziorski and Szalewicz have derived an equation for the
second-order intramonomer correlation correction E323) to the electrostatic energy
in the polarization approximation.177 Also, Moszynski et. al. have derived expres-
sions for the third-order and fourth-order intramonomer correlation corrections to the
electrostatic energy (given by E3343; and E83 , respectively) in the polarization
approximation," given in terms of the nonrelaxed expansions for the polarization
propagators and electron densities.96381974023921232 Moszynski and co—workers184 have

derived an equation for the second-order relaxed intramonomer correlation correction

l2, . . .
E321 Brew ) to the electrostat1c energy. Reference 77 also gives equat1ons for the re-

E(13, reap) and

laxed third- and fourth-order intramonomer correlation corrections OAB

E(14,resp)

0A8
derivations of the equations for Eéilga nd E31: r881) ) ,where k S 4 .280

to the electrostatic energy.77 Jeziorski and co-workers briefly review the

The same researchers who derived the equations for these intramonomer correla-

tion corrections to the electrostatic energy have also used these equations to calculate

58

numerical values of these energies for several different dimers. They have used the
results of their calculations to compare the magnitude of the intramonomer corre-
lation correction to the electrostatic energy of the dimer to the magnitudes of the
electrostatic energy and total interaction of the dimer. Rybak et. al. have used
their equation for E512; to calculate the second-order intramonomer correlation
correction to the electrostatic energy of the water dimer and of the hydrogen fluoride

dimer.177 The results of these calculations show that at or near the van der Waals

(12)

minima of the dimers, the second-order intramonomer correlation correction EOAB

to the electrostatic energy is up to 10 percent of the total interaction energy. This
indicates that for these dimers and at these distances, the intramonomer correlation
energy is significant, and that one cannot neglect intramonomer correlation when
trying to calculate the interaction energies of these dimers.

Williams and co-workers have also calculated the 2nd-order intramonomer correla-
tion correction E533) to the electrostatic energy of the water and hydrogen fluoride
dimers.286 They have also calculated the third-order intramonomer correlation cor-
rection E652 to the electrostatic energy of the same dimers.286 Then, by adding
E5312; and E523 together, they obtained slightly more accurate estimates of the

total intramonomer correlation energy of each dimer. The results of these calculations

show that at intermonomer distances which are larger than the van der Waals min-

(12)

0A3 18 a much smaller

ima, the second-order intramonomer correlation correction E
component of the interaction energy for each dimer. Specifically, at the intermonomer
distances specified in their work, the total intramonomer correlation energy was only
two percent of the electrostatic energy of the water dimer, and five percent of the
electrostatic energy of the hydrogen fluoride dimer. Although the intramonomer cor-
relation energy of each dimer is a smaller portion of the corresponding total interaction
energy at larger distances, intramonomer correlation is still a significant part of the

total interaction energy. The authors of this work also computed the relaxed first-,

' - . . 12, 13,
th1rd-, and fourth-order intramonomer correlatlon correctlons EéABreSp), ((,ABreSp),

59

(14 ,)resp

and EOAB ,for the water and hydrogen fluoride dimers, and they provided an-

(12 ,resp)

other estimate of the total intramonomer correlation energy by adding EOAB ,

E(13, ,resp)

0A8 and EOABZTGSP) for each dimer. The results of these calculations were
very similar to the results of the calculations of the nonrelaxed energies.

Other groups performed similar calculations on the helium dimer and on (H2)2.
Specifically, Moszynski and co-workers calculated the second- order intramonomer cor-
relation correction Eéllj: and the corresponding relaxed correction E01: Twp) for
the (H2)2 dimer.184 They performed these calculations 1n order to determine the con-
vergence behavior of the nonrelaxed and relaxed MBPT expansions for the intra-
monomer correlation contributions to the electrostatic energy, which are given by
Eqs. (128) and (132). They determined how quickly these equations (with n = l
) converge by comparing the values of E623) and Eéizéresm with the electrostatic
energy of the dimer as calculated with the FCI wavefunction. According to their
calculations, the second-order intramonomer correlation corrections contain only be—
tween 50 percent and 70 percent of the FCI correlation energy. Therefore, higher-order
intramonomer correlation corrections are large enough that they cannot be ignored
in accurate calculations of the intramonomer correlation energy. Moszynski et. al.
have also calculated E833) and E011 ’pr for (H2)2.77 However, they have also

calculated higher-order intramonomer correlation corrections for this system, includ-

ing E033), E612, E833 “Sp dB“: “3”) .77 Additionally, they have calculated
Eéilgan dEéilzreSp) where n S 4 for the helium dimer. Then, for each dimer,

they computed the sum of Egg) for k S 4 and of EU: Twp) for k S 4 ,
where the former sum is given by

1 (12
63,33 (4): E0”) + E033) + E0143), (133)
and the latter sum is given by
63:21:31)) (4) : Eéliresm+ Eéiiiresp) + E014 resp) . (134)

In Eqs. (133) and (134), 6838 (4) and 6(l,resp) (4) denote the sums given in each

0.48
equation. In order to determine the convergence of both MBPT expansions for the

intramonomer correlation contribution to the electrostatic energy of each dimer, they

(1)

compared 60/13 (4) for each dimer to the corresponding total electrostatic-correlation
energy. They also compared 6811:8819 ) (4) for each dimer to the corresponding to-

tal relaxed electrostatic-correlation energy. The total electrostatic-correlation energy

60

6828 (F C I ) is given by

4,” (F01): —EO“’> (135)

EDA)B AB’

1 . . . . .
where E323 IS the electrostat1c energy of the dimer computed Wlth 1ts FCI wave-
. 1 . . . .
function, and E612 15 the electrostatlc energy of the dlmer 1n the Hartree—Fock (HF)

approximation. The total relaxed electrostatic-correlation energy 6(1’r83p) (F01)

DAB
is given by
6,(1 resp) E(l ,resp) (10.1‘esp)
60,13 (FCI)= EOAB —E0AB , (136)
where EOI’B resin) and E0“: Twp) are the same quantities as E323 and E83: ,

respectively, except that EDA contain relaxed expansions for

(,lresp) and E(:0,resp)
the polarization propagators and electron densities. When they compared the sum
through the fourth-order of the intramonomer correlation correction 61(11):; (4) to the
total electrostatic-correlation energy 681,33 (F C I ) for the dimer, they determined
that over 90 percent of the total intramonomer correlation energy is included in

6823 (4) . They obtained similar results when they compared 63228810) (4) to

6(1, resp)

€0.43 (FCI) for each dimer. These results indicate that the MBPT expansions

for the intramonomer correlation contributions to the electrostatic energies of (H2)2
1
and He,» converge quickly, and that for both dimers, 6012(4) an nd €(A’ gap) (4) are

very good approximations to the total electrostatic-correlation energies 6012(1'701)

and 68:28”) (FCI)77 .

2.7.3 The First-Order Exchange Energy in Many-Body Perturbation The-
ory
There are two methods for computing the intramonomer correlation contribution to

the first-order exchange energy in SRS theory.280 The first method involves using

E(l, ,SRS—exch)

the approximation to 0.48

developed in reference 73, which is given in
terms of the one- and two-particle density matrices of monomers A and B. In order
to account for the intramonomer correlation contribution to the first-order exchange

energy, the one- and two—particle density matrices originally included in the expression

61

for E(1,SRS—-e:rch)

0A8 in reference 73 are expanded in powers of (.186 The equations

for these expansions are given in reference 280.

The second method for calculating the intramonomer correlation contribution to
the first-order exchange energy in SRS theory is a coupled-cluster singles and dou-
bles (CCSD) technique which involves infinite-order summation methods that are

nonperturbative.178'235 Jeziorski et. al. briefly review this method in reference 280.

Moszynski and co—workers have used the expressions which give the approxima-
. (LSRS-exch)
tion for EOAB

in powers of /\ to calculate the first- and second-order intramonomer correlation

. 11,SRS— h 12,SRS— . h.
corrections EéAB we ) and E51,; 3 en ) to the first—order exchange en-

ergy of H82, (H2)2, He-HF, and Ar-Hg.186 Then, for each dimer, they added these two

g::RS_exCh) (2) . In order

to determine the convergence of the MBPT expansion for the intramonomer corre—

in terms of one- and two-particle density matrices expanded

corrections. We will denote the sum of these corrections 6

lation contribution to the lat-order exchange energy of each dimer, they compared

_ h . - '
(I’SRS 8x6 ) (2) for each dlmer to the correspondlng total exchange-correlation

0.48
energy eéthS-exch) (F C I ) , which is given by
(1,3RS—exch) (1,5RS—erch) (lO,SRS-e:l:ch)
60,3 (FCI) = E,” — E0” . (137)

1,3RS—erch)
AB

In Eq. (137), E5

is the total 13‘-order exchange energy of the dimer, as

10,SRS—e:rch)

computed with its FCI wavefunction, and E3143

is the exchange energy of
the dimer as calculated with its Hartree—Fock determinant. Moszynski, Jeziorski, and
Szalewicz have calculated the intramonomer correlation correction to the first-order
exchange energy in the CCSD approximation for H82, (H2)2, He—HF, and Ar-Hg.178
Jeziorski et. al. list and briefly discuss the results of these calculations.

According to the analysis of the convergence properties of the MBPT expan-
sion (for the intramonomer correlation contribution to the first-order exchange en-

ergy), the convergence of the MBPT expansion is relatively slow. For example, for

H62, the sum of the first- and second-order intramonomer correlation corrections

€(1,SRS—exch)

0A8 (2) to the first-order exchange energy constitutes only 50 percent of

. . l, —, .h
the corresponding total exchange-correlation energy 66.1sz e“ ) (FCI) .186 For

62

the H2 dimer, egigRS—exch) (2) is 25 percent larger than the corresponding total

(1,3RS-exch)
0A8

exchange-correlation energy 6 (F01) . However, for each dimer, the
intramonomer correlation contribution to the lst-order exchange energy computed
in the CCSD approximation is around 98 percent of the total exchange-correlation
energy 68::RS_exCh) (FCI) . Therefore, the CCSD-based method for comput-
ing the intramonomer correlation contribution to the first-order exchange energy is
much more accurate than the corresponding MBPT method for computing the same
quantity.

2.7.4 The Second-Order Induction Energy in Many-Body Perturbation
Theory

Moszynski and co—workers have derived an equation for the induction energy that in-
cludes an MBPT expansion for the intramonomer correlation energy and depends on
the polarization propagators and electron densities of the unperturbed monomers A
and 318712321287 This equation for the induction energy can also be written in terms of

the relaxed expansions of the polarization propagators and electron

(2,2nd)

densities.103‘1051188-1‘m’233’234 The induction-correlation energy 60/13 is given by
(2.2'nd) _ (2,2'nd) (20.17211)
60” — EOAB EOAB . (138)

In Eq. (138), Eggnd) is the exact second-order induction energy, and Eéii’md)

is the second-order induction energy in the Hartree—Fock approximation. Similarly,
(2,resp-z'nd)

the relaxed induction-correlation energy 60”

is given by

(2,resp—ind) _ (2,resp—ind) (20,7‘esp—ind)

0.3 — Ed... — EDA, , (139)

2, -' d

where EéAgeSP m )
(20,1'esp—ind)

EOAB

proximation. Sadlej has shown that each intramonomer correlation correction

E(2l,ind)

OAB
191

is the exact second-order relaxed induction energy, and

is the second-order relaxed induction energy in the Hartree-Fock ap-

(which is second-order in V and lth ~order in W ) can be written
as

E(2l,ind) : E(2l,ind—a) + Eé2l,ind—t). (140)

OAB 0A8 AB

In Eq. (140), EWnd")

DAB

2(,’ d 21,' d— . . .
EéABm ) . Also, E81432" a) is what 18 called the apparent intramonomer

is what is called the true intramonomer correlation por-

tion

63

. . 21," d .
correlation portion of EéABm. ) 191 One can calculate the apparent intramonomer

E(2l, ,—2'nd a)

(21 d) , '
OAB of EOAB m by usmg the random-phase approx}-

. . 2l,'nd
in the expresswn for E5“: )

correlation portion

280

mation (RPA) propagator . Jeziorski, Moszynski,

and Szalewicz (and references therein) discuss the apparent and true intramonomer

correlation contributions to the second-order induction energy in more detail.280 In

(2,resp-—ind)

summary of their discussion, the relaxed induction-correlation energy €0.43

is a more accurate representation of the second—order induction-correlation energy

(2,2'nd)

than the corresponding nonrelaxed induction-correlation energy 60.43

Moszynski, Cybulski, and Chalasinski as well as Moszynski et. al. have calculated
#227624“) and Eozzares” W” for the He—K+, He-F‘, (H20)2, and He—HCI

OAB

dimers. 187'“ Note that they did not calculate the first-order relaxed intramonomer
2(1 (1 . . . .
correlation energy EOAB reSp m ) because this contribution 18 equal to zero by the

Brillouin theorem.280 According to the results of these calculations, the magnitude

of E(22, ,resp—ind)

0.43 is very different for each of the different dimers. For dimers

. . . . 22, __ d
containing a rare gas atom and an ion, the magnitude of E ( "3317 m )

0A8 is relatively

E(22, ,resp— ind) .

large. For example, for the He-F‘ dimer, OAB 1S about 10 percent of the

20 d . . .
size of E6113 “Sp in ) at or near the van der Waals minimum of the dimer. For
22 d . .
dimers containing two polar molecules, Ell/13 res!) m ) is even larger, relative to the

E.(20, ,—resp ind)

(22 ,——resp ind)
OAB EOAB

size of .For the water dimer, is about 30 percent

20 d . . .
of the size of EéAB res!) in ) at or near the van der Waals minimum of the dimer.

Although the authors of references 74 and 93 did not calculate E02: ”Sp and)

(22, resp—
EOAB

and

ind . . .
) for any dimers containing a rare gas atom and a nonpolar molecule,

E(22, ,resp—ind)

Jeziorski et. al. mention that OAB

is so small for these types of dimers

- - . . 2, d
that it can usually be ignored in calculations of E64268!) m )

64

2.7.5 The Second-Order Dispersion Energy in Many-Body Perturbation
Theory

In general, there are two ways of writing MBPT expansions for the intramonomer
correlation contribution to the second-order dispersion energy in the polarization
approximation.280 Rybak and co-workers177 have written completely connected CC
equations for the first-order intramonomer correlation correction Eéill’gdiSp) to the
second-order dispersion energy. Additionally, Jaszunski et. al. have written the
original expression for the second-order dispersion energy in the polarization approx-
imation as an MBPT expansion in the intramonomer correlation.231 They did this by
replacing the polarization propagators in the original expression for the second-order
dispersion energy with MBPT expansions for these propagators.

One can also compute the second-order dispersion energy at large R A B if the
intermolecular interaction operator V is replaced with the multipole expansion
in the MBPT perturbation equations for the dispersion energy. Recall from Sect.
2.4.4 that in the multipole approximation, the second-order dispersion energy is de-
termined by the reciprocal of the distance RA B between monomers and the van
der Waals constants. In turn, the van der Waals constants are determined by the
frequency-dependent polarizabilities of the monomers in the multipole approxima-
tion. Jeziorski and co-workers obtained a MBPT expansion for the lth-order intra-

. . 2l,dis
monomer correlation correction E ( p)

0.48 to the second-order dispersion energy by

substituting the expressions for the polarizabilities with the Moller-Plesset expan-
sions for these quantities.280 Specifically, Wormer et. al.236_238’240'239 have derived
diagrammatic MBPT equations for correlated frequency-dependent polarizabilities,
and they have deveIOped another MBPT method for calculating correlated van der
Waals constants.

Moszynski, Jeziorski, and Szalewicz have developed a method for approximating
the long-range second-order correlation-dispersion energy which is known as the ring

approximation (RA).185 One obtains the second-order correlation-dispersion energy

65

(2 (1 RA . . . . . . .
EOA’B 23p ) in the ring approx1mat10n by replacmg the expressmns for the polariz-

abilities in the equation for the second-order dispersion energy with the expressions for
the polarizabilities in the random-phase approximation.24434312441242 If one writes the
equation for the dispersion energy in the ring approximation in diagrammatic form,
the resulting expression that he or she obtains will contain ring diagrams only.185

One can obtain the diagrammatic expression for the dispersion-correlation energy

0A8 in the ring approximation by summing the ring diagrams included

21, d '
( stp) over all l.185 Note that one can also obtain the

(2 ,disp— RA)

in each expression for EOA

second-order dispersion-correlation energy EOA’B in the ring approximation

E52Bld1319)

by replacing the polarization propagators in the expression for n the

polarization approximation with the expressions for the polarization propagators in

the random-phase approximation. 279345 Finally, one can use CC equations derived by

(2,d RA
Moszynski, Jeziorski, and Szalewicz to calculate E023 23p ) 185

Moszynski et. al. have used CC equations derived in the polarization approxima-

(20 dzsp) ,E(21 disp), and E(228 disp) fOI' H82, (H2)2, and (HF)2.185

0.48
Then, for each dimer, they used these results to calculate 653:1”) (2) , which is the

tion to calculate EOAB

sum of the intramonomer correlation contributions to the second-order dispersion

energy through second-order in W. They also calculated the E (22 (kw—RA) and

th 0A8
I -order intramonomer correlation corrections where l _>_ 3 to

the sum of all
the second-order dispersion energy in the ring approximation, which we will denote

' - A . . .
68:28}? R ) (3 -> 00). The equation for this sum is

6(2,disp—RA)(3 __) 00) _2 E021, ,disp— RA). (141)

0.48

According to the results of these calculations, the MBPT expansion for the intra-

monomer correlation contribution to the second-order dispersion energy converges rel-

(2, disp— RA)(

atively quickly. Specifically, for each dimer, the magnitude of 60 3 —+ 00)

(2, disp) (2)

is very small in comparison to the value of 60 , which indicates that most of
the intramonomer correlation energy in included in the first-, second-, and third-order

corrections to the total intramonomer correlation energy.

66

3 The Second-Order Correction to the Energy of
Two Interacting Molecules A and B

This chapter provides a review of known results for the second-order intermolecular
interaction energies in the polarization approximation, which is valid when molecules
A and B are separated by a distance R A B such that the overlap between their elec-
tronic charge distributions can be ignored. If the electronic overlap is non-negligable
but A and B interact noncovalently, then the interaction energy can be obtained from
exchange perturbation theory. However, the current work is limited to the polariza-
tion approximation. Then the second-order correction to the energy of interaction
between the two molecules is

AEm = (1139,;le G VAB (11(0) ), (142)

OAB OAB

where IWEBBB) denotes the ground-state wavefunction of the unperturbed system,
which we approximate by the product of the ground-state wavefilnctions W833 and

1118:) of molecules A and B,

0 0 O
1111‘ ’ > = 1111511115.) (143)

0A8

In Eq. (142), VA8 is the interaction potential, and G is the reduced resolvent
of H 0 = H A + H B , which was defined in Eq. (19) of Chap. 1. For the interaction

between molecules A and B, G is a sum of three components,

 

 

 

G = GA + G3 + 01423, (144)
where
(0) (0) (0) (0)
GA 2 Z ”’23. ‘I’oall‘l’ja ‘I’oel (145)
0 0 0 0 ’
#0 (E12) + E62) — ( 6A) + E8)
0 0 0 0
0 0 0 0 ’
r790 (E62 + E7852) _ (E62 + E613)
and
(0) (0) (0) (0)
l‘ij ‘Pra><‘1’j,, \Pral (147)

GAGBB ___ Z
0 0 0 0 '
1.1.10 (Egg + Egg) .. (Egg + Egg)

67

0 , .
M) denotes the Jth exc1ted state of molecule

A, and 4152,) denotes the Tth excited state of molecule B. Similarly, E39) , E53,) ,

In Eqs. (145), (146), and (147), \II(.

E53), and E62) correspond to the energies of molecule A in excited state j , B
in excited state 7‘ , A in its ground state, and B in its ground state. Eqs. (145),
(146), and (147) are the reduced resolvents constructed when considering excitations
in molecule A only, in B only, and in both A and B. In the following, we approximate
V in terms of the dipole-dipole interaction and thus we have

A AB ... A A B
V = —,ua Tag 113, (148)
The dipole propagator T03 is defined by

l 3 Ra R -- 50 R2
Tag = VQV3E = 612.5 8 .

In Eq. (149), Ra and R3 are the a and ,8 components of R, and Va and
V3 denote derivatives of R with respect to a and fl , respectively. The full

 

(149)

polarization approximation can be recovered by expressing V in terms of polarization
density Operators for the interacting molecules and a dipole propagator T (r, r’)
integrated over all space with respect to r and I" . Using Eq. (142), (143), and
Eqs. (144) - (147), we have

A

 

E552. = 4.511.. E (OAminé,OB'fl€lf’B>Wig,“(2‘:me
9'4““) (EL; + E08) - (EDA + E03)
1,11,, 2 <04|fi¢|0430€03mik~3>(mailer(gamma
rméo (E0, + Eng) — (E0, + E03)
—T76 Tax; 2: (OAllA‘filj/ifjfBlfiafBl(jAlligJO/i)(fofilfiglOBl’
jAJ‘BaéO (E), + Em) - (EDA + E03)

 

 

(150)

where we have let @823 = 0A, @823) = 03, ‘16-? = jA, and @533) = 7‘3,

and we have replaced the first and second interaction operators with VAB =

—ilf3 7175/1? and VAB = —fif Tap fig, respectively. Because (OAI/ZflOA)
. «B B .

#130, (OAlué‘IO/d = #240, (OBlfla I03) = #50, and (OBll‘gIOB) = 145° ,

A0
.7

where p , #240, 11,1530, and #50 are static dipole moments of molecules A and B,

68

we can write Eq. (150) as

(0 | A'l | Al0 >
2) A #1 JA) (JA #6 A
AEéAB— _ _TVA Tea" ”(1530”? Z< 7 (0)

3:19“) (Em jA —E0A)

(OBIfiB ITB> (TBIILB I03)
A0 2 6 ¢>
7‘37“) (ETB) — OB )

 

 

 

—T75T€¢
0 0 37‘ 0 7" 0
x Z ( All/74 7|JA)O() BlfiiJ)B:<j/llu(2:)l A)((:|fiB AI 3) (151)
JArA¢0 (EjA Eo A)+ (Era E0 3)

where we have also rearranged and simplified the denominators in Eq. (151). Now,
consider the first term in Eq. (151). We can write the first term in Eq. (151) as a
sum of two halves,

(0AMA IJA) (JAlfie|0A>_
_ T76 Ted) #601150 2 ’7 (0) _
JA9‘90 (E('()) —E0 ,4)

1 (oAmA IJA><JA|fiA|0A>
_ T... T... #5011130 (2) z 1,,
JA9£0 (EjA _EO 0))

1 (0AM1 IJA>(JA|/1A IDA)
— TAM/15011506) Z ( . (152)
mm (EA-A -EoA)

Interchanging the labels 6 with ’7 and (b with (5 in the second term on the
right-hand side of Eq. (152) gives

 

 

pf": <0A|fi7|JA> (JAMAIOA)
O 0
(12.2 A.)

1 <0A|11A IJA>(J'A|fiAl0A>
80 BO 2 7
j/fiéo j _

1 (DAIMIJAXJAlgA l0A>
- T T75AAgOAAc1530 (5):: ) 7
jA7é0 (EjA —E((lA))

l
= —T76 Tab M1530 #50 (5)

69

 

_ T76 Tab #60

 

 

 

(OAlfiflJAlUAlfiflofl
Z (A? — E35?)

(OAlfifileXjAl/‘y WA)
.20 (E13) — Eat?)

The bracketed quantity in Eq. (153) is the definition of the static polarizability a

X

 

(153)

of molecule A. Therefore, we can write the first term in Eq. (151) as

>:

     

(OAIfiA lJA)<JAlfi;4
_ T767101 BO #302 —T76 Ted)

‘1’ 0) 0
AA) (BA. - E62)
1
x 1135901150 (2)041. (154)

At this point, consider the second term in Eq. (151). We can also write the second
term in Eq. (151) as a sum of two halves

(OBI/1.113%} (TBIAB I03)
- T75Te¢ #70 If“): (0) A =
AAA.) (EAA— E5, ‘2)
1 (03M; |r3)(r3|[13|03)
A0 A0 ¢
— TASTE”? “6 (2) 2 Eu» Em)
T3¢0 ( TB — OB)

1 (OBllungBXTBlHB IOB>
— 796111113011?“ (2): (2A . (155)
T3750 (E718) _EOB))

If we interchange 5 with (b and ’7 with 6 in the second term on the right—hand
side of Eq. (155), then Eq. (155) becomes

 

 

(OBIfiB I'rB) half '08)

A0 ”A0 6 <15 _

- 7175716451”le E: (E(()) (0)) _
7'87“) TB — 03

1 (OBI/13 |7‘B>(7‘B|fi§ |03>
A0 A0 2 :
TB

1 (OBI/AB |:B>(7‘B|;§|OB>
_ T057176 #240 ”:10 (2) 2: ¢
7B7“) (E718) — E03))

70

 

 

1
= —T'76 T“? #240,130 (2)
X Z (OBI/lgerXTBl/lglOB)

Z (OBIAAAITBX'I'BIAAAWB)
men ( E _E(0))

The bracketed quantity in Eq. (156) is the static polarizability (1215322 of molecule B.

 

 

(156)

Therefore, we can write the second term in Eq. (151) as

(OBIHB ITB) >(7'Bl#¢ 103)
"'T75 Ted) AA‘YOAAG A0 2 = _T75 Tf‘f’ “£0 “:10

(14212)
x (é) a215,. (157)

Therefore, from Eqs. (151),(154), and (157), the second order correction to the
energy of two interacting molecules A and B is

2 l
4E11= — 2262211780115“ (2)

1
A0 A0 B
— '75 71¢ #7 :U’e (2) add)

— T76 Cred) 2:

JA 78750
(OAlfi7 IJA>(OB|#§|TB>(JAI#A |0A><r3|u§ ICE)

(0) (0) (0) (0)
(EJA) — EDA) + (E773 _ E03)
We can simplify Eq. (158) by replacing the denominator in the third term of Eq.
(158) with an integral,

 

 

(158)

 

1 )1 +00
(0) <0) (0) (0) = (17?) / Aw
(EJA -E0A)+ (EB —EOB) —oo
1 l

X (B)?- BJ’HW) + (£7212) —E(0)—zfiw)

 

 

71

1 1
(Eg) — E30) + 71%)) + (Egg) _ ((33) _ zfiw) . (159)

B

 

 

Before making this substitution, however, we will use complex contour integration to
prove that Eq. (159) is true. Consider the integral

+
foo du (a / (a2 + 212)) (b/ (b2 + 212)) . The integral over a semi-circular contour
“-CX)

C of radius R1 in the upper complex half plane is given by

 

 

b R‘ b
[du 0’ = /du 0’
a2+qu2+u2 a2+u2b2+u2
C R1

 

+ fdu a b (160)

a2+u2b2+u2’
S

where S is the portion of C that lies off the real axis, as illustrated by Fig. 11. We

take the limit as R1 approaches infinity. In this limit, the integral over S vanishes.
Along S, u = R1619 and du = 2R16'6d0, so that

7r

 

 

 

 

- a b ' 29
1111:1300 duo2 + u2 b2 + u2 _ 1111:1200 / lee
S 0
a b
x d6
0.2 + R1262“) b2 + R1262“)
= 0. (161)
Thus
b +°° b
a a
d = d . 162
[ruczz+u2b2+u2 / ”02+u2b2+u2 ( )
C —00

First, let us evaluate f du (a/ (a2 + 112)) (b/ (b2 + 112)) assuming that
C
a 75 b . The integrand of fdu (a/ (a2 + U2» (b/ (b2 + u2)) has 4 singu-
C

lar points, which are u = iaz and u = ibz . Because —a.z and —b2 are on
the negative imaginary axis, they are not located within the contour C in Fig. 11.
Therefore, according to the residue theorem,

b
fdu a = 27rz(K1+ K2), (163)
C

 

a2+u2b2+u2

72

where K1 and K2 are the 2 residues at the poles within the contour C. If we let
f(u) = (a/ (a2 + 13)) (b/ (b2 + 13)), p(u) = a - b, and
q(u) = (0.2 +212) (b2 +u2) , then f(u) = p(u) /q(u) . Also, q, (u) =
2a2u + 2b2u + 4113 , where q' (u) is the derivative of q (u) with respect to u
. We can determine whether at and b2 are simple poles by evaluating p (u),
q (u) , and q, (u) at each pole. Since q ((12) = 0,
q, (a2) = 2a3z + 2ab2z — 4a3z 75 O , and p(a2) = ab aé 0, a2 is a sim-
ple pole. Similarly, since q (b2) 2 0 , q, (bl) = 20.217; + 2b3z — 4b32 # 0 , and
p (bl) = ab 74 0 , b2 is also a simple pole. Since both a2 and bz are simple poles,
the residues of f (u) at cm and b2 are

 

 

 

 

 

 

 

 

 

 

 

 

K _ p (a2) _ b
1 — q'(az) — 2z(b+a)(b—a)
10 (IN) a
K = -—,—— = . 164
2 q(bz) 22(a+b)(a—b) ( )
Consequently,
a b _
dua2 + 11.2132 + U2 —- 27rz(K1+ K2)
C
— 2m b + a
— 22(b+a)(b—a) 22(a+b)(a—b)
1
— 7r [_(a + 5)] , (165)
and therefore
+00 b 1
a
[dua2 + U2 b2 + U2 _ 7r [W] . (166)
Ifwelet uzfiw ,then du=hdw,and
1 +00 b )1 +00 b
a CL
71'- / dua2 + u2 b2 + u2 _ (B) /dwa2 + Web? b2 + Wu)?
1
= . 167
(a + b) ( )
Now, we must manipulate Eq. (167) so that it resembles Eq. (159). Because
a 1 1
a2 + h2w2 _— [2 (a + zfiw) + 2 (a — 2%)] (168)

73

and

 

 

 

—b—— — 1 + 1 (169)
b2+112w2 — 2(b+zhw) 2(b—zhw)
we can write
h +00 b h +00
a
(B) /dwa2 + h2w2b2 + 7127.122 — (Z77) /dw

X lmmllr—mr-ml “70>

Letting a = (Egg) — E6?) and b 2 (Eg) — E8?) in Eq. (170) and using
Eq. (167), we have

(E) 7d“) (El-3) - E55?) (E13? — E3?)

2 2
-00 (Elm _ 530’) + 527.22 (E52) — Eff”) + 11%?

JA A B

 

 

 

= (1.)). 1 + 1
47F (BE? ——E§,‘j} +2710) (E)? —E§,‘jj final)

 

 

 

—00 .7
X 1 + 1
(El? — Egg) + zfiw) (El? — Egg); — 2m)
1
= , (171)
(Eé-f? — E32?) + (B? — E532)

which is essentially Eq. (159).
To complete the proof, we will evaluate f du (a/ (a2 + 712)) (b/ (b2 + 712))
C

for the case with a = b . Then f (u) is

f (u) = (a2 + u?) (a2 + u2)’

 

(172)

and the two singular points for f (u) are u = :taz . However, 0.7 is the only

singular point within contour C. Therefore, by the residue theorem, we have

 

a a
d =
/ u<a2 + U2) (a2 + 112) 27FZK1, (173)
C

74

where K1 is the residue at the pole 'u. = a2 . If we let p (u) = a2 and q (u) =
(a2 + U2) (a2 + U2) = a4 + 2a2u2 +114 , where f (u) = p (u) /q (u) , we can
determine whether a2 is a simple pole by evaluating p (u) ,
q(u) ,and q, (u) = 4a2u + 4u3 at az . Because p(az) = a2, q(az) = O
and q, (a2) = 0, a2 is not a simple pole. In order to determine the order of a2 ,
we need to rewrite f (u). Factoring and rewriting f (u), we have

 

 

a2
f (u) = (0.2 +112) (a2+u2)
2
= a 2, (174)

(u + az)2(u — (12)

which indicates that cm is a pole of order two. At this point, consider q” (u) .
Since q” (u) = 4a2 + 12212, q" (a2) = -8a2 76 0. Then, because 19 (a2) # 0,
q (a2) = 0, q, ((12) = 0 , and q” (a2) 74 0 , we can determine the residue K1 of

f (u) from
I 2 I”
‘1 (a?) 3 [q (672)]
From q" (u) = 4a2 + 12u2 ,we have that q", (u) = 2411. Using 19 (a2) = 0.2,
p, (u) = p, (at) = 0, q, (a2) = 0, g” (cm) -—- —8a2, and qm (a2) = 24m in Eq.
(175) gives K1 = —2/ 4a . Substituting the value of K1 into Eq. (173), we have

/du( a a = l. (176)

a2 + u?) (a2 + u?) 20.
C

 

 

 

Using Eqs. (162) with b = a and (176), we can write

+00

fdu a a 1. (177)

0.2+uza2+u2 2a

 

—OO

Letting u = fiw and du = hdw in Eq. (177) and rearranging, we have

+00
h a a 1
(7;) fdwa2 + M2 a2 + M2 — 2a° (178)

—00

 

75

Finally, recalling that a. = b , we can write

71 00 a b 1
(E) [dwa2+hw2b2 +111»? _ a+b’ (179)

—00

 

which completes our proof of Eq. (159).

Having completed the proof of Eq. (159), we are ready to continue developing the
second-order correction to the interaction energy AEéiB of A and B. Using Eqs.
(158) and (159), we have

1
AEéiB— — 4.5 2. BO #50 (-) as:

2
A0 A0 1 B
— T75 T635117 116 (—) 05¢

 

 

 

 

2
h +00
— T75 Tee 2 (a) /W<0Alfi¢IJA><0BIfi§ITB)
jATB¢0 _ap
>< (JAlfie |0A><TBIIL5 |0B>
X 1 + 1
(0) (0)_ (0) (0)
_(E, —E0 zfiw) (EM — EDA +271»)
1 1
x + . (180)
_ (E3? — E32) — zfiw) (E32) — E32) + zhw) 1

 

 

Let us consider the third term in Eq. (180). Rearranging the components of this
term allows us to write

+00
h A o A
_ 1,61... 23 (7;) / dw<oilu3m><oalu§lr3>

jAa7'87é0
- «A «B
X (JAM |0A>(7‘B|u¢ I08)

1 1
X +

L(B33) —E3°— ) m) (133? B—30)+zm)

 

 

1 1
+
(Em) — E32) — zfiw) (E32) — E3? + zhw) _

 

 

 

 

76

+00
’1
= _T,3:l}3 (47) /dwz

J'A7'50

(oimg‘lm'imfloa (orlnfiljmilmoi)
0 O 0 0
(E3) — E3) — zfiw) (EL) — E32 + my)

 

 

 

 

 

Z <08|fi§|TB><TBmgIOB> (OBlfifererglOB) (181)

.310 (E32) — E3? —zfiw) (E3?) -—E3‘;) +zfiw)

Since 0’74, fig, [1? ,and [15 are Hermitian and IOA), I03), IjA) and ITB) are

real, (OAlfliflJ'A) = (JAlllfiloA), (OAlfiflJ'A) = (JAIMIOA), (Oslflfer) =
(TBIfiflOB), and (Oglfiglrg) = (TBIfigIOB). Using these relationships in Eq.
(181) gives

+00
71 . . A
— T1572) 2 (E) /dw<0A|H§1|JA><OB|/J§ITB>

JAJBS‘O
. «A *3
X (JAIHeIOA><TBl/‘¢l08>

 

 

 

 

 

 

1 1
x +
(0) (0) (0) (O)
_(EjA — EDA — 271w) (EM — EDA +zhw)J
x 1 + 1
0 0
L (E7)? — E32) — zfiw) (E38) — E38) + zfiw) _

+00
h
: _T75Tw5 (Z7?) fab”:

-00 jxfiéo

(OAlfiflJ'AHJ'AIfifIOA) (OAlflfilJ'AHJAIMIO/i)
o o o 0
(E3) — E32 — me) (E( l — E33 + 2211.))

JA
2 (OBlfiferXTB 115103) (OBIfiEIT‘BXTBIfiaBIOB)
38710 (E32 — 3: — zfiw) (E32) — E3? + inw)

 

 

 

 

 

(182)

77

In order to complete our analysis of the second-order correction to the energy
of interaction between A and B, we need to derive expressions for the frequency—
dependent polarizabilities (1’74E (w) and 015, (w) of molecules A and B. According
to Orr and Ward,61 the first—order polarization P” of an isolated molecule in the
presence of an applied field of frequency w is

we:

(0|P|n><n|H'“’l0> (OIH'wlnManlO)
E30) — E30) — hw E30) — E30) + fiw

 

 

(183)

 

 

 

 

 

 

 

 

11790
if damping is neglected. In Eq. (183) In) is an excited (state and I0) is the
ground state of the unperturbed molecule, with energies E730) and E (0) .,Also
P is a polarization Operator, and H w is the perturbation due to the applied field
offrequency w. Ifwe let w— — w, H“) = H’AW = +2417?” and P— — [17,
then
<11 A>ee _ FA ,3, Z (DAV?4 IJ'A> (J‘Alile |0A> (DAM? IJ'A><J'A|/1§1 IDA]
7 _ (0) (0)
JAS‘é0 EjA) ”EDA -ZhLd EjA_ EDA) + Zhw
= FeA Warsaw-:17? zw afie (2(4)) ' (184)
In Eq. (184), we have also let I0): IDA), In): IjA),E =Eé?a ,
7(30) = E](O). Similarly, if we let w— —‘ w, H”: If)";W = O—figFf’nw and
p = I16 , then
(1153)“) : F333, Z (012%; ITBl <73|u§IOB> (Oelfiglre><rslfi§l03>]
(0)_ (0) (0)
TB¢0 ‘_'OBE ZBLU ETB - E08 '1‘ 2%
B, (178 B, B

with (0) = I03), In) = mg), E30) .—. E32} ,and E3“) = E32,) . Therefore,
according to Eqs. (184) and (185),

(CAI/LA IJA><JAIHel0A>
031001) = Z 07 (0)_
mm Eli) EDA h“

(OAl/ie le)<jAl#7 |0A>
E30)— E30) + zhw

 

 

 

(186)

78

and
(OBlfigerXTBl/lglOB)
r3750 EA? — E323) -— zhw
(OBlfifer><TB|/1§|OB>
EB _ Egg; + A. l '
Using Eqs. (186) and (187) in Eq. (182), we have

+00
’1 A . A
— TAT... Z (E)/dw<0A|/1§‘IJA><OBIB?ITB>

jAJ'BS‘éO

 

 

 

(187)

X (JAlllflOAXTBl/lgIOB)
r -
1 1

E(0) E(0) ’10.} + E(0) E(0) m
( JA — 0A —Z ) ( JA _ 0 +2 )J

A

 

 

 

 

 

 

+00
11
= _T’75 Tab (21;) [dwafie (w) (1333., (w) (188)

Finally, using Eqs. (180) and (188), we can write AEbiL
1

2
ABS/1);; = _ T76 Ted) #630 #50 (5) age

1
.. Te 1;, e30 17:10 (2) e38,
f}, +00
— T75 T€¢ (II—7;) / do.) 0;: (W) 05,) (2(4)) . (1.89)

The first two terms in AEéi; give the induction energy due to the polarization
of each molecule by the field of the permanent dipole of the other (within linear
response, and neglecting effects due to the non-uniformity of the field). The third

term gives the dispersion energy.

79

4 The Third-Order Correction to the Energy of
Two Weakly Interacting Molecules

In this chapter, the third-order interaction energy of a pair of moleculesis derived
within the polarization approximation. Results for the induction, hyperpolarization,
and induction-dispersion energies agree with earlier work, but the third-order disper-
sion energy is derived in a new form, as an integral of nonlinear response tensors over

imaginary frequencies.

4.1 Non-Zero Contributions to AEéi;

Again, we consider two molecules A and B, separated by a distance R such that the
overlap between their electronic charge distributions can be ignored. The third-order
correction to the emery of interaction between the two molecules is

magi; = (xi/(0)3 IVAB G V Bo VA3 130”) (190)

O .
where ”1828 denotes the ground-state wavefunction of the unperturbed system,

which we approximate by the product of the ground-state wavefunctions W823 and
A LAB
@823) of molecules A and B. In Eq. (190), VA8 is the interaction potential, V

is defined by

A

V B = VAB will IVABhI/(O) ) (191)
0A8 OAB’

and G is the reduced resolvent, given by Eq. (19) in Chap. 1. We split G into a sum

of three terms, one with excitation in molecule A only (GA), one with excitations in

molecule B only (GB), and one with excitations in both A and B (GA®B), as in Eqs.

(145) — (147) of Chap. 3. Here we consider only the dipole—dipole contribution to the

interaction potential, which was given in Eq. (148) of Chap. 3. We also have

LAB

A A O 0 A A 0 O
V 2 _HQ T03 H? + (‘I’hA‘I’ ) ‘I’cgglflg T03 #3 mg): @813) (192)
Furthermore, since
0 0 A
WWI/3:33.31 T 351%,. Web: T 3:3" 350, (193)
then . AB
V =—il€3 T33 #5 + T33 ufi” #50 (194)

80

In Eq. (194), #20 and [1.50 are the permanent dipole moments of molecules A
and B in the a and 5 directions, where the total permanent dipole moments [LA0
and ”BO of molecules A and B are

A0 A0“ A0“ A0“
[1 = #2: X ‘I’ fly 3’ ‘I' ”z Z
BO 30“ BO“ 80“
3 = 33X+3yy+3zz (195)
and , 5’, and 2 are unit vectors in the x,y and Z directions. If we define
-“—_«A_ A0 d;B_AB_ BO
#0 _' “a ”a an #6 - ”(B “3 a
‘A—AB A0
V = mTaflfig_ EilTaBI‘BO _ ”a T033113 (196)

Using Eqs. (144) - (147) from Chap. 3 for G and Eq. (196) for TA/JB , we transform
Eq. (190) into a sum of 27 terms, 15 of which are nonzero. Table 1 contains a list of

the non-zero terms and the order of u A0 and H BO in each of those terms.

4.2 Higher-Order Induction: Terms of First-Order in Both
”A0 and I180

The static polarizability 03:}; of molecule A is

 

 

A _ Z (0.33. I33><33I35AI03> +2 «333. 33333. I03

0.5 ‘ 0) 0) (0) (0 (197)
3.30 E( —E( 3.30 Eh —E0>

where IkA) = ‘I’S: is the kth excited state of molecule A. Similarly, for molecule
3,

 

 

B Z (OBIfivltB><tBIHBBIOB>+Z (OBIfi3BItBWBIlivBIOB)

- , (198)
0 (0 0
3330 33(3) E(B ) “#0 E(B > _E(B )

0‘76 —

where ItB) denotes the tth unperturbed excited state of B; then (12% GEE is

 

A B _
056 0’73 _ Z (EME(0)— E(O))1(Et(2)— Em)
0

kg, “3760

[(OAIlte IkA><kAI#6IA><OBIA1§ItB><tBIfigIOB>

(CAI/16 IkA><kAIfi5 IOAIIOBIH: BItB><tBII2§IOB)
(OAIfiElIkA (kAI/iAIOAHOBIfiB It3><t3|3§l033>
(OAIfi34IkAIIkAIMe IOAIIOBIMBB ItB><tBII~2§IOB>I - (199)

81

 

+++><

Now, let us evaluate each of the (1,1) terms listed in Table 1. From Eq. (148) for
the left-hand perturbation operator VAB, VAB = —[2A Tab [Lg , and Eqs. (145),

6

(146), and (147) from Chap. 3, the (1,1) terms 1, 4, 5, 7, 12, and 14 are
“ ;A ;B “
— (OAOBIVAB GA 1% T33 #3 GB VABIOAOB> =
- Z T03 T75 T63 #530 #240

jAJ'B9éO
(OAI/lfiIjA> (jAI/lgIOA> (OBIBSIM (TBIfigIOB>

(Elm — E323) (E592 — E3?) ’

JA

 

A

200)

— (OAOBIVAB GA #30 T05 fig GAGE VABIOAOB) =
" 22 T08 T75 THIS #30 #530

jAJ’B7é0
<03I3§Irs><rsl3§|03> (OAIfiélIJ'AXJ'AIflfloA)

0 0 0 0 0 0 ’
[(EJIA) — E33) + (E5...) — E33] (E12,) — E62)

— (OAOBIVAB GB 32:11.5 35 GA VABlvoB) =

A0 80
— X T03 T76 Ted) #7 #3
rBajA¢0

(OBIfl§ITB><TBIflEIOB)<0A|fi§IjA>UAIfi§I0A>
0 0 0 0 ’
(EIB) — 33;) (E): — E31)

A ;A A A
_ (OAOBIVAB GB 0 TaB #50 GAmB VABIOAOB> =
- X T33 T,5 Ted) #330 111,40

TBJA#0
<03|3§|33><33|3§|03><03l323|jA><jAl3fil03>

(32 — 5:) [(32 - 3:) + (Es-r — 301’

A ;.A A
— (0A033VAB 0333 3.. Tag 350 GB VAB|0A03) =
— X T333 511,, T... 35" 3f“

jAaTBfi‘éO

 

A

201)

 

(202)

(203)

 

82

(GAMMA)(Ml/130A)(OBIfigerWBIfifIOB)

 

 

a (204)
0 0 0 0
[(32 — 32) + <32 — 33>] (32> — 3)
and
A LB ‘
_ (OAOBIVAB GAG”? 33" T0333 0" VABIOAOBI =
_ X T03 T75 T605 #30 #31530
jAs'rB9é0
(OAIfifIJ'A)(J'AI/lfIOA)<08Ififlr3><r3lfi§l03> (205)

(0) (0) (0) (0) (0 (0) '
(E3 —B.,.,) [(133 ‘E03) + (3... " 03)I
Since a, H, ’y, (5, 6, and (15 are dummy variables, we can rename them. If we

convert Eto Oz, qb to fi,,3to’7,d to 03,7 to €,and a to 5,termlfrom
Eq. (200) becomes

— 0,103 VABG'AfiATagfiBGB VAB|0A03 =
a [3

— X T67 T60) T03 “(15:0 [1:140
jAaTB9éO

(OAIfiijA>(jAI/lglIOAXOBIfifITB)(TBIflgIOB>
(0) (0) (0) (0)
(EjA - EDA ) (E773 _ E03)

In term4 from Eq. (201), we convert (5 to (15,) to 6, 6 to (5 ,and 05 to ’7

 

(206)

to give
A L8 513 A
— (OAOBIVAB GA #20 Tag/13 GAVE VABIOAOB) =
- X T33 Tm T67 #330 #50

J'AJ‘B790
(OBIfigITBXTBI/lflOB)(OAIfifIJ'AlUAI/lisqloxi)

(0) (0) <0) (0) (0) (0) '
[(3, - E03) + (E... - 33)] (E3 - E03)

Similarly, in term 5 from Eq. (202), we convert ”y to a, 5 to 5, B to ’7, and

 

(207)

a to 5 which gives
A _“_A AB A
— (0A03|VAB GB 3,, Tag 35 GA VAB|0A03) =
- X T37 T33 T33 #530 #50

TB .jA7éO

83

<03|3§|r3><33|3§l03>(00334000332403)
0 0 0 0
(E33) — E32) (E32 — E32)

When we convert 7to,a, (Ito H, (b to ’7, B to (b, a to 6, and 6 to 5
in term 7 from Eq. (203) we have

 

(208)

_ (OAOBIVAB GB 4: Tafi “£330 GAEBB VABIOAOB> =
— Z T... T3 T3 350 3:30

TBJA¢0
<03I33I1~B><33I3§I03>0332403033303

0 0 0 0 0 0 '
<3: - 3:) [<32 — 3:) + (Es-2 — 32)]

In term 12 from Eq. (204) we convert 6 to a, (b to )8, 5 to 7, B to (b , 03
to 6 and ’7 to 5 ,so that this term becomes

 

(209)

A 4A A
_ (OAOBIVAB GAGBB #10 T03 #50 GB VABIOAOB> :

BO A0
- 2 71.5713 71.33., 3..
33.73750

(0033403)033003)<03|3§|r3><r3|35|03>
0 0 0 0 0 0 '
[(32 — 32) + <32 - 33)] <3: - 3:)

Finally, in term 14 from Eq. (205), interchanging (5 and 7 gives

(210)

 

_ (OAOBIVAB 0.4333 #20 T06 E133 GA VABIOAOB> =

- X T03 T57 Tea“) #20 #31230
jAfl‘BS‘O

(0333030333103<03|35I7~B><r333I03>
(0) (0) (0) (0) (0) (0) °
(E3 - E3) [(3. - E3) + (E... - 33)]

If we add terms 4 and 7 as specified by Eqs. (207) and (209), we have

 

(211)

A _"_B A
_ (OAOBIVAB GA #2140 T00 #6 GAEBB VAB|0A03>

A _,._A A
— (0A03|VAB G330, Tag 350 GA“ VABIOAOB) =
- X T03 T37 T63 #230 #50

3.343737“)

84

(OBIM3 ITB>(7‘BIM5 I0B><0AIM. IJ'A)<JA|MZ§1 I031)
0 0)
(E322 —E3 I) (E3AI —E3AI)

Also, adding terms 12 and 14 from Eqs. (210) and (211)) gives

 

(212)

A __»_A A
_ (OAOBIVAB GA$B ”a Tafi ”A1630 GB VABIOAOB>
A AA_A_B A
(OAOBIVAB GAQBM ACT #3 GA VABIOAOB> ___

A B
-Z T073 T677135 ”:0 [1:0
JA 73?“)

(OAIMa IJAXJAIMEA IOAIIOBIMB ITBW‘BIM3 I08)
(0) (0) ( ) (0)
(E.- ... >32 3)

B

 

(213)

If we replace [CA and t B with j A and 7‘3 in Eq. (199) and then use Eqs.
(199), (206), (208), (212), and (213), we can replace the sum of terms 1, 4, 5, 7, 12,
and 14 with

A0 B00 B
SI : _ 0123 T57 71¢ ”a ”’45 066 0’75 (214)

Physically, this sum represents a higher-order induction effect: The permanent dipole
of molecule A sets up a field that polarizes molecule B, producing a reaction field that
acts back on molecule A. Molecule A is polarized by the reaction field due to B; the

polarization of A creates a field acting on B, which alters the energy of the pair due

to the permanent dipole of B (and similarly, with the roles of A and B interchanged).

4.3 Hyperpolarization: Terms of Third-Order in #30 or 3’40

The static hyperpolarizability 787016 of molecule A is

 

7‘... Z Z (E‘OI—JEEI:I)1(1~73?—E(°I)

k. 1,.an
x [(0.1331 13.) (3.13233033. (0.)
(OAIMAIkA><kAIM7 IlA>(lAIM. IDA)
(OAIMaIkAIIkAIM. IlAIIIAIM7 IOA)
(DAIM7IkAIIkAIM. IIAIIIAIMCAIOAI

+++

85

     

A _"_A A

+ (0A AlkAl<kA|M7IlA><lA|M§|0A>
A ;A A

+ (0313241333313.11333133103],

(215)

where Wu) and HA) are the km and lth unperturbed excited states of

term 3 in Table 1. When we evaluate term 3, we have

If we convert

Converting ’7 to 6, 6 to ’7, 5 to ¢,and (bto 5 in Eq. (217) gives

If we convert

a

X

a

X

(OAOBIVAB GAMA T03 M3 OGA VABIOAOB> =

T03 T76 T73 M330 3230 35:
Z <0A|M7IJA><JA|MO|CIAWIA|MEIDA)

0 0 (0 0
3219/1760 (E154) —- E62) (EQA) — E512)

 

molecule A, and E19,) and E8) are the corresponding energies. Now, consider

(216)

to ’7, ”y to Oz, ,3 to 5 , and 5 to B in Eq. (216), we have

(OAOBIVAB GAZIA Tug/1500A VABIOAOB)=
T31 T03 713M?) 3%“ M5:
2 (OAIMQIJAXJAIM7 lqA><qA|Me IDA)

0 0
jAsqzfiéO (EjA) — E02) (EMA) — E32)

 

(0,403 IVAB GA -2 T 353 0GA VAB|0A03)=
T53 T03 T76 M50 M50 M21530

2 (OAlMé IJ'AXJAIME lqA>(qA|M7 IDA)
3.4.3130 (Eli - E11,.) (E33 - 130,.)

 

(217)

(218)

to ’7, ’7 to a, H to 5 , and 5 to B in Eq. (218), we have

(OAOBIVAB GAHA T75 T/LBO GA VABIOAOB) =
Tab T75 Tafi #50 MMO Mg:
2 <0A|M7 IJA><JA|M6A lqA><qA|Ma|0A>

0 0
JAJIAiéO (EL ) _E(() 21)) (E911)- E023)

 

86

(219)

Converting e to 7, 7 to 6, 5 to 14,45 and (b to 5 in Eq. (219) gives

— <0AOB|VABGAfif €¢pfOGAVAB|OAOB)=
BO BO BO

— 76Te¢TaBl16 #4, W}
Z (014mg IJA><jAlfit7IQA>< l
0) 0) 0 ’
334324950 (EjA) — E62) (ECSAL —E(() A?)

also, converting a to ’7, ”y to 01, HA to 5 and 5 to fl in Eq. (220),we have

—(0A03 IVAB GAfiA an? 00" VAB|0A03)=
— Tag 71¢ 22",, u?” #530 #530
X Z (OM/12“ le><jAlfiQIQA><QAlfi7lOA>
(Es? - 352) (E3? — 3?)

If we multiply Eqs. (216) - (221) by 1/6, sum the results, and use Eq. (215), the
result is

     

         

 

(220)

 

(221)

1
—<0AOB|VAB GAH: Tag #3 OGA VABIOAOB>— _ 80—67105 T75 Tap X

#5 Ougoufofifae, (222)
where we have replaced IkA), HA), Ella) and ES) in (215) with HA), I‘M),

E39) and Egg) . Physically, Eq. (222) represents the change in energy due to the
hyperpolarization of molecule A by the field from the permanent dipole of molecule B.
A similar analysis allows us to write term 8 in terms of the static hyperpolarizability

553W, of molecule B. The static hyperpolarizability fig”) of molecule B is

 

B _

563(2) “ $21837“) (E§2)_E 30:1) E( 0)_E E(0))
X [<OBlfifltBHtBlli: luB><uBlN¢ '08)
cal/251m)«dimmed/25103)
mangle)<talfiflua><usmflog>
(Oalfifltal(talfifluBHuBlfiEIOw
<oBm§ItB><tBIfifluB><u3mglog>

mangle)<tBIfi§IuB><uBm§IoB>l . (223)

+++++

87

In Eq. (223), (153) and (21,13) are the tth‘ and um unperturbed excited states

of molecule B with energies Egg) and E1303) . When term 8 is evaluated, the result
rs
— (voglV/‘B GB #30 To), if? GB VABIOAOB) =
- afi T76 Teeuéoflfo #50
Z <oBm§IrB><rB|fi§|sB><sB|25|oB>
rmsméo (E53,) _ E3?) (Egg) — E5?)
When we convert 5 to B, ,3 to (5, a to 'y, and ’7 to (1, Eq. (224) becomes
— (OAOBIVAB GB #30 T,5 fif GB VAB|0A03) :-
- T76 T as Tax» #30 #20 It?”

 

(224)

 

A LB A
Z (OBIMEIT‘BXTBIMISBXSBIHEIOBl
0 o 0 0
73.83350 (EE'B) — E613) (EgB) _ E812)

Converting a5 to 5, 5 to qb, 6 to ’7, and 7to 6 in Eq. (225) gives

A _A_B A
— (OAOBIVAB GB #er T“), [1,), GB VABIOAOB) =

A0 A0 A0
_ 60511037175 “6 ”a [1,),

Z <03I2§Ir3>(relfifIsBstmfloB)
r3,83¢0 (Egg) " Ed?) (Egg) - E3?)
If we set B to 6, 6 to B, a to 7, and 7 to a in Eq. (226), we have
- (OAOBIVAB GB #50 T6,, fif GB VABIOAOB) =
— Teas T75 T05 #410 #340 #30
Z <03I263Im><rglfiflsB><sBm§IoB>
resaaéo (13(2) — Eli?) (El? - E3?) ,
and, converting 45 to 6, 6 to (b, e to 7, and 7 to e in Eq. (227) gives
— (OAOBWAB GB “:30 T,5 fif GB VAB|OAOB) =
— T,5 Teas Tag #340 #240 #20

(225)

 

(226)

(227)

 

 

A ;B A
Z (OBlflngB> (TBIW |33><SBIHEIOB>
who (Eli? — E62?) (E933 — E3?!)

88

(228)

Finally, if we convert B to 5, 5 to B, a to 'y, and ’7 to a in Eq. (228), the result is

A _“_B A
— (0A03|VAEGE 23071,)”, GEVAE|0A03)=
_ TaBTe¢T75II£O “:10 “£0

 

x Z (OBI/(2|?)<T£)|I1'§|81(90))<SBIM(§)IOB) (229)
”3,33% (Era E0 B)(Ese —E03)
If we multiply Eqs. (224) - (229) by 1/6 and sum, the result is
— (0A03|VAE GE” gj‘oTag 2,, BVGE A|0AOB)=
— g as T76 21,”, 2, ”#2405532, (230)

where we have replaced t3, uB, E52) and E1)? in Eq. (223) with 7‘3, 83,

E52.) and El? . Eq. (230) gives the energy change due to the hyperpolarization of

molecule B by the field from the permanent dipole of molecule A.

4.4 Induction-Dispersion: Terms of First-Order in MAO or (230

We are now ready to determine the contribution to AEéi: from terms listed in
Table 1. When term 2 in Table 1 is evaluated, the result is

—<0A03IVAE GAfzfi T022 BGAEEVAEIOAOB>=— a 71,, we?“
X Z (0AM,IJ:><JA|fla|qA><qufifl0A)
jAaQAJ'Bi‘éO

(OBlflg lTB> (Tel/1,), l08>
(E2‘3)E WK Em) E‘“)+ (Em Eé‘élll

We can replace 1/ (E53) — E6?) + (El? — E620] in Eq. (231) with an

integral over frequencies,

 

(231)

 

[(E(0)_((3))+1 (E(0)_ E320] : (g)

+00

X / d“ (59>- 531(3+zhw)

—OO

 

89

1 1
(E52) — E53) — zfiw) 5 (E732) — E52) + zhw)

 

 

1

0 0 ’
(E58) ‘ 33 " ml)

so that term 2 (as expressed in Eq. (231)) becomes

_ (OAOBIVAB GAfifi TaB/Jfi BGAEEB VABIOA OB):

— TseTttflztp?°(:— 7)]me

(0AM? lJ'A><J'AlMaICIA>(CMIJL12‘1 IDA)
0 0)
(1352—195?) (E5, — E53 +zhw)

+

 

(232)

 

JA QA TB¢0

(CAI/17 IJ'A>(JA|fis|qA><<JA|Mé4 IDA)
(E53) —- E50 l) (E5?-- E0 0) -—Zfiw)
(OBIMEITBl (TBIMEIOB) (OBIMEI’I‘Bl (TBIMfIOBl
(E52) — 5:) + m) (E5? — E5? — zfiw)
When term 9 is evaluated, the result is
— (OAOBIVAB GAEE fif Ta), fig GA VABIOAOB) =
- 2 Tea T76 Tap M50
J'A qA r3930
(OAIMA lJA>(JA|Ma|€1A><qA|MA |0A><OB|MB ITBW‘BIMgBIO B)
[(E23 E33))+ +(Er3 - E33 )l (E33) - E33)

From Eqs. (232) and (234), we have

 

 

(233)

.(234)

— (OAOBIVABGAEBBHQT Tag/L BG AVABIOAOB)=

— TQBT7671¢#50(4— 7rm):)-/:Od3‘1.Z:

90

JA (IA TB¢0

(OAIJ‘L7 |JA><JA|JLa |61J1><€1Alllié1 IDA)
(E5?-- E3”) (E(.0 --E(0 Hm)

(OM/11? IJA><JA|Ha|qA>(qA|fi€‘ IDA)
(Egg? —E(O)) (11*.O ) — E03 — m)

M

 

 

(OBlfiferXTBl/lglow (OBlfiflT‘BX’I‘BIfigWB)

 

 

 

 

(Big) _ E35: + m) (35.2) _ E33 _ 2m) ’ (235)
where we have replaced (1A with j A in Eq. (232). Term 13 is
_ <0A08|VAB GAeBfi-A “To ”500/1693 VABIOAOB>_
— Z Tag T76 Ta» #50
J'A.qA,r37é0
X (OAlfifIJA><JA|fi;:IqA>(qufifl0A>(OBIfi?ITB><rB|/1§|03>
" [(EJ —Esi:>) + (E? — 32)]
x 1 (236)

[(1953 E3°))+ (E53? -E33’)]'

We can use59

1/ { [(E‘fi-E Es") + (E52 - E326]
X [(E‘f? -Eéi?)+ +(E5f22— Emu-_-

e 72» 1 + 1
471' _00 (E£0)— E(0) + 2%) (Egg) _ Egg) _ 2m)
1
(133(0) EoA) +212») (Egg) —E0A +212»)

 

 

 

X

1
(Egon E“”— m) (E(0)— E53) flu)

 

+ (237)

91

to write Eq. (236) as

_ (OAOB IVAB 0A®B“A ”a To; 3/150 GAGEB VABIOAOB> ___

 

 

 

 

— 1.105717511053360
h +00
X (a?) / ‘3". Z
—oo JA QA TBJéO
(0A|H7 |JA>3JA|Ha lqA><qA|He IDAHOBIHJ3 |TB><TB|H§|03>
(35.?— E38 +zhw) (E3?- E33: + m) (E39,) — E33” + m)
(OAIHé1 IJA>3JA|Ha lqA><qA|He |0A><OB|H§3 |TB><TB|H§ IOB)
(E132) — E00) + zfiw)A (E333) —E(0)— zhw) (EQA) — EDA) —- zhw)
<0A|H7|JA><JA|Ha|qA>3€1A|He |0A><OBIH¢§3 |TB><TB|H§|03>
(E33,? —E33A’A— 22w) (EV-2-131632%” (E33) — E33) + zhw)
+ ()0 (0) (0)
(E332—13332—zfu) (EjAl— )zhw)— E33 — 3A find)
3

x <oAIu7IJA><JAIfi2|qA> (ma? Iowan? IrB><rB|u§ low] (238)

At this point, we will relabel Eqs. (233) and (238) so that the static dipole moment
#80 has the same index in Eqs. (233), (235), and (238). Converting 5 to 45, q)
to 5, 'y to 6, and 6 to ’7 in Eq. (233) and expanding gives

A LAT ;B A
_ (OAOBIVAB GA #0 T03 [:5 GAEEB VABIOAOB> :

— T0371¢T “yd/1(1) ( 7A,.)ZoodeZ

30A|He IJA>3JA|Ha(|;1A><CIA|H7 |0A>303|HE|TB><TB|H§|OB>
(15:33” —E30)) (E33) —E3A +zfiw) (E33; — E30 +zfiw)

JA (IA T3730

 

<0A|Hf IJA><JAIHa|qAXqA|H7 |0A><OB|HEITB><TBIH5 l03>
O O
(E E3.)— E32) (E E—33) E30) + zhw) (E33; — E33; —Zfiw)

 

92

(OAIHE |JA>3JA|Hn IqA>(qA|H7 |0A>303|H§|TB> <TB|H§3IOB>
(E3A — E33) (E32 — E33) —zhw) (E32,)— E30) +zfiw)
1
(E3.0 —E33A)) (E32) — E30) — m) (E732) — E332 — 2m)

JA

 

+

 

A . . ;A A A
x <07qu IJA><JAIu7IqA><qu7 IoA><oB|u§IrB><rBIu§IoB>] «239)

If we convert (15 to E, B to 05,01 to E and 6 to a in Eq. (238), the result
is

_ (OAOBIVAB GAEBBfi —A gTe (AugO GA$B VABIOAOB>—

_ 71¢T76Ta,3/350(4'h_ 72)me‘2

<0A|H7 IJA><JA|Hf|qA><qA|HA |0A><03|H§|m>(J‘BIHfi9 |03>
(Em) —E(0) +zth) (E30) —E0A) +zhw1(EqA) —E0A) +zfiw)

JA (IA 7‘33‘0

 

JA

><TB|H7133|03>

         

<0A|H7 |JA><JAIH24 IqA)<quHn |0A><

 

 

 

+ (E39, —E30) +zhw)A (E3O)E — E30 A—zhw) (E33 —E33A) —zhw)
<0A|H7 IJA><jA|fie IQA><QA|HQ(|0A><OB|/1§ IrB)<rBIH§ |03>
(1~37.AE30)—E(37O flan) (E33) — E3A°+zriw) (E33) —E33A)+ziiw)
1
+ (Eryn _ EM -777) (Egg _E03 -777) E59 -1339 -777)

x <0A|H7 |JA><JAIH24 IqA><quJIoA><oBmJ IrB><er§IoB>3 . (240)
Expanding Eq. (235) and adding this to Eqs. (239) and (240) gives

_ (OAOBIVAB GA HST Tafl Hf? GAoB VABIOAOB>

_ (OAOBIVAB GAeaB-AcA-A T03 Hg BGA VABIOAOB>

_ (OAOBIVAB GAeeB—A “a T (WAD GAEBB VABIOAOB)_

_ TaflT76R¢u<lp30(4— _7r-)/oodw'§:

93

JA (IA 73730

(OAIHA |JA><JAIHanA><qAIHA |0A><03IH§|TBI<TBIHAIOB>
(E3I— E33”) (E32I— E3°I + m) (E733;I — E08 33” + and)

3A

 

OAIHe IJA)3JAIHanA><qAIH7 I0A>3OBIH7§ITBXTBIH5 I03

 

OAIHA IJA><JAIHanA><qA|HA IDAHOBIHE ITB><TBIHB I03

 

< I
(E33II— E33II)(E E30I—E33’I A+zhw) (E733’7IE — E3°I —Zhw)
I
I

3
(EA E30)- E3°I)(E E33I— E30I— ind) (E33,;I — E33;I +5125

OAIHA IJA><JAIH5 IqA><qEA|HA I0A><OB|HE ITBIZ‘BIH5 I03

 

5*

<
(Es- «33)(E33-E E) (EE— 2

I
OAIHA IJAI<JA|H5 IqA><qA|HA IOAI<OBIH33 ITB><7“(BIH5 IOB

 

3 I
(E3BAI— E33”) (E30I—E E3°I + zhw) (E33?— E38) + zfiw)

OAIHA IJ'A>(JAIH:71 Iq AHQAIHA I0A><OBIfi§ITBI<TBIfig I03

 

< I
(E3AI—E 3°AIA)(E3.AIE E3°I —zhw) (E “I E3°I EB+zh/.u)

A
OAIHA IJAI<JAIH5 lqA><qA|H£1 IOA>(OB|H7’5B ITB><TBIH5 IOB

 

< I
(E3AI—E3 3A?) E3°I — E30I+zhw) (E733;I— 3(3—inn)

OAIH7 IJAXJAIH: IqA><qAIHe I0A>3OBIH§ ITB><TBIH5 I08)

 

E—IO) E8») Ej (0) _E(0)_ zfiw) (E52) _ E30) _ mu)

JA

(OAIHA (LJAIUAIfieIQAXQAIfiA I0A><OBIHAITB><TBIHB I03)

 

30AIH7 IJAI<JAIH5 ICIA>3€1AIHnI0AI<OBIH5 ITB>3TBIH5 I03)

 

E733I E3°I BA+zhw) (E33’I— E3AI— 5m) (E32I— 33II— 5m)

(OAIH7 IJAI<JAIH5 IqA><CIAIHn I0A>3OBIH§3ITB>3TBIH5 I03)

 

<
(EE
(my. gym) (5E 55,: +55) (En gum)
(°
(

E32I— E3°I— 5m) (E33) — E3AI +5555) (E33I — E33) + 52255)

94

1
(E32 — E33AIAI — mu) (E33II — E3°I zfiw) (E333,I — E3°I zfiw)

 

.3.
JA A— A—

A . . ;A A A A
X 30AIH’74IJAI3JAIH5ICIA>3QAIH§I0A>3OBIH5ITB>3TBIHEIOB>3o 3241)

Now we will show that Eq. (241) is equal to the product of the frequency-dependent
hyperpolarizability 33476 (—w; w, 0) for molecule A and the frequency-dependent
polarizability 013533 (w) for molecule B. We can use Eq. (187) from Chap. 3 with
45 converted to 78 to write the frequency-dependent polarizability 01% (w) of
molecule B as

 

QB (7w) : Z 3OBIH§3ITB>3TBIHEIOB>
55 Em) E3°I hw
T3750 TB — 03 '—Z
3OBIH5|TBI3TBIH§IOB>
E33;I — E33? + m

 

 

(242)

Orr and Ward61 have derived an expression for the second-order nonlinear polar-
ization Pw" of an isolated molecule due to applied fields of frequencies cal and
3412,

Pwa = K(—wa;w1,w2) 11.2 Z
m,n7é0

<0IPIm><mIHAIE><nIHAI0>

 

 

X
(E3?) -— E3°I — m3) (E33II — E30) — 5551)
+ <0IH'AIm><mIH7 ln><n|Pl0>
(E3?) — 3‘” + m2) (E3‘II — E33II + m3)
+ 3OIH’B2ImI3mI-1-BIH) 3HIH’MI3I)

 

(E333I — E3°I + M2) (E33II -— E30) — m1)
= K (—wa;w1,w2) (—h)’2 2

m,n7€0
30|1BI7TE>3mIH7 In>3nIH’w1|0>
(E3‘7II — E3°I — m3) (E33II — E33II — m1)

 

95

<OIPIm><mITE7AIn><nIHAIo>

 

 

 

 

+
(E(0)_ E(0)_ hug) (E310) _ E30) _ AM)
+ (OIH’AAImeIH’ |n><an|0>
(E333I— E3°I + fiw 7) (E33II — E30) + mg)
+ (OIH’B‘ImeIH |n><n|Pl0>
(E3?) — E30I + 5551) (E30) - E3°I + hug)
+ <0IH'AIm><mI‘PIn><nIH"‘" l0)
(E73,?I — E3°I + hug) (E30) — E3°I -— 5551)
+ (OIH’B’ lm><mlHIn><n|HB2I0> (243)

 

(E39) — E3°I + m1) (E33II — E3°I — m)

where wa = 0.)] + tag, and Im) is an unperturbed excited state of the molecule
with energy 137(2) . In Eq. (243), K (—w0;w1,w2) is determined by

K(—wa;w1,w2) = 2'" X D, (244)

where m is the difference between the number of polarization frequencies and field-
frequency labels in the set of frequencies w1,w2,..., excluding zero. Also, D is
the number of times that the field-frequency labels can be arranged distinguishably,
where +0) and —w are distinguishable for all w 75 0 . In addition, 11,2 indicates
that a second term should be produced from each term in 13”” by permuting cal

and (.02 , and 75 is defined by
H=P—mmm. 9%)

The quantities HA“ and HI”?- denote the perturbations due to the H11 and tag
frequency components of the applied field, and

A“) A w A w

H’ = H’ — (OlH’ I0>- (246)
If we let 13 = #2, col: 2w, L412 — 0 (therefore, wa— — 2w, —wa = w,) H'W— — IfI'AW
= #35314”, H’wz = H“ = —5;3FA° H 1 = H ‘ =41, AFAW, and H 2 =

-A—’A’0 A A A I .
= -EA FAA’O, where H A” is the perturbation when the field FAB" 18 applied to

96

molecule A and IT“) is the perturbation when the field F (‘4'0 is applied to molecule
A, then Eq. (243) becomes

A E _ FAA“ FAA” (OAIHEIJAXJAlHelqu<quH7|0A>
”(“940 (Eu) —E0 J—zfiw) (qu1) —E0A) —zfiw)

 

(OAIHE |JA><JA|H7 ICIA><€1A|HE |0A>
E<g> ..Ew) (E593 ..Egg>)

(OAIHE IJA><JA|H7 IqA><qAIH£ IDA)
E30) Egg>)(E<g> _Ego ME)
0

 

 

(CAI/‘7 le><jAlue lqA><QAlua IDA)
—E02 + m) (Egg) —E30)+ Em)

 

(DEIHSl IJ'A)<JAIHE IqrA><c1E|Hi;1 IDA)
E<)_ Ea») (Egg) _E(0)_ m)

<0A|H7 IJA><JA|HAE IQA><QAlfie IDA)
(E30)— E30) +zhw) (qu — E0?)
= (h) AFAOFAW A (—zw;zw,0). (247)

aye

 

(E333
(BE
(E93
(BE

 

Therefore, from Eq. (247), the frequency-dependent hyperpolarizability of molecule
A is

 

0 a 5 0
(_w; W 0) _ Z (of AIH’:()|)JA><J'AIHA IqE3)(qEIH,OI A)
jAaQAHO (EjA _ E0 _ Zhw) (EqA — EDA) — zhw)

A

5076

(OAllAgle>)<jAlH7IQA><QAlI124 IDA)
(E30) —E(3,3’j —zhw) (E30)— E53)

JA

 

(OAIHE IJA><JA|H7 |qE><qrA|H$l WA)
(0)
(E133) —E0?A )q)(E —E0f’A +zfiw)

 

97

<0A|H7 le><jAlfiel31A><3IAllAalOA>
(E33) _ E33: + 222:) (E3 3” — E33” + Em)

 

(OAIHQ4 IJA><JA|HE lqA><quH7 IDA)
(E33) --E33) (E33) —E33A) — Em)

<0A|H7 lJA><JA|HalqA>361A|He IDA)

(E3°)— E30) +zhw) (E33 — E33?)

Note that we have let K(—wa;w1,w2) = 1, Im) = IjA), In) = lqA),
and I0) = |0A>- Now, let us calculate the product of the frequency-dependent

 

 

(248)

polarizability 61% (w) of molecule B with the frequency-dependent hyperpolar-

izability £7: (—zw; 2w, 0) of molecule A. According to Eqs. (242) and (248),

afflw) a76(- zw;zw,0) is
0133 (w) :3” (—zw;zw,0) = 2:
JA GA 73750
(OAIHCA le><jAlHelQA><QAlfi7 IDAHOBIH BITB>(7'BIH§ IOB>
(E30) —E30)— 2m) (E39, —E30) flab) (E32) —E33;> ~2m)

(OAIHSIJAXJAIH7 IqA><qA|HéA |0A>(OB|HE ITBWBIH5|03>
0 0 0 0
(E3A ) — E3A) — zfiw) (E3)— 3A 30))(E3‘; -E3A ) —zfiw)

 

 

(DEIH:1 IJA><JAIH7|qA>(qE|HaA I0A><03|H3B ITB><TBIHEB IOB>
(E3°)— E30) (E33) — E30) +zfiw) (E33,; — E30) — Zhw)
(

 

<0A|H7 IJE1>(JEIH:l |<LE><QEIH31 IOA)<OB|H3B I'I‘BWBIHE IOB>
E30— E30)+zhw) (E30)— E3 Alarm) (E3; —E30 —zfiw)

 

(OAIHA IJA><JA|Ha|qA><qAIH7 IOAXOBIHSB ITB><TB|HB IOB>
E3.°)— E30) (E3? —E30 —zfiw) (E32) —E30)—zhw)

(DEIH7IJ'/1><J/1|HE|c1,4>(6114|H2‘1 |0A><OB|HEB IrE><rEIHE We)
0 O)
(E3A) — E3A + zfiw) (E3A — E3A) (E33; — E33; — 21w)

 

 

98

30A|Ha IJA>3JA|HE lqA>3qA|H7 IOAl3OB|HE ITB)37‘B|HI33IOB>
(E30)— E33?- 77w) (E39, —E30)— m) (E 3’) —E30)+zhw)

 

OAIHA IJA>3JA|H7|qA (qAIHA |0EA>3OB|HE ITB)<TBIHB Na)

)3
E(_o>_ E(0)_ 3733) (#0313332) (Em) _E(o> +zfiw)

3
(Eu
30A|HA |JA>3JA|H7 lqA)3qA|Ha 30A)3OB|HE ITB>3TB|HE NE)
(E 3’) —E33A’) (E3, E3” — E30 +zhw) (E32 —- E30) +zhw)
3
3
3

 

 

 

+
JA
30A|H7 IJA>3JA|H24 IqA>3qAIHA |0A><OBIHE IrB><rBIH§ IOB>
E3fll— E30l+zhw) (E3? —E3 (2+2hw) (E3°— 3°l+zhw)
+ OAIHA IJA><JA|Ha|qA>3qA|H7 IOA)<OB|HEITB>3TBIHB I03

 

>
E3°)— E(0))(EAA Em) _E(0)_ WA) (E30l— Em)+ +hw)
>

30AIH7IJA>3JA|HE7|qA>3€1A|H§1 IOA>3OB|HE ITB>3TB|H§A |OB
(E33? — E3A +zfiw) (E39, - E3A) (E33; —E30) + zhw)

 

. (249)

Let us return to Eq. (241). If we convert a to ’7, B to 5, ’7 to Oz, and 5 to B in
the first, second, sixth, eighth, tenth and twelfth terms in Eq. (241), the result is

— (OAOBIVAB GAEA TaE E3 BGAAB VABIOAOB)
— (OAOBIVAB GAABEA TOE E3 BGAV ”3|vo )
__ <0AOB|VAB (314633—"1/1’6 T (lb/ABC GAQBB VABIOAOB>_
— TaET7671¢H¢1730 (4h 71’) food“) 2
JA 731A 7‘87“)

30A|HA IJA>3JA|H7 lqA>3qA|Ha IOA>3OB|H¢1§3 ITB>3TBIHE |03)
(E30)— E33”) (E30l— E30) +zfiw) (E32) — E33” +zfiw)

 

30AIHA IJA)3JA|H7|qA>3qA|HA IOA)3OB|HB ITB><TB|H§ IOB>
(E(-0)-— E30)) (Eq E30) __ E03) +zhw) (E732) __ 30) —zfiw)

 

99

30A|HA IJA>3JAIHAIOqA><qAIHA |0A><OBIHEBITB><TBIHA I03)
(E30— E3°)E3)(3 —E33Al -zhw) (E3 E—30 +zfiw)

 

30A|HA IJA>3JA|HanA>3qAIHA I0A>3OBIHE ITB)37'BIHB I03)
(E30)_ E30)) (Eq EI—(O) E30) —zhw) (E(B 0)_ E30) _zm)

 

30A|HA IJEA>3JAIHE IqA>3qAIHA l0A><OBIHA ITB><TBIHE I03)
(E53)- WWW-E E3” WNW-A EA” +7777)
30A|HA IJA)3JAIHAI€1A)3CIAIHE I3)A)30BIHEA73I7'B)(THIHA3 I08)
(Ea??- E3A) (EA-3’- MMA? EA” +7777)

30AIH7 IJA()3JAIHaIOCIA)3(1AIHAI0A)3OBIHA3ITB)3TBIHEI03)
(Egg)—E‘ (33)) E30) E30)+ M)(E3g) E30) —zhw)

 

 

 

JA
OAIHA IJA)3JAIH7 ICIA)3QAIHA IOA)3OBIHE ITB)37"BIH353 I08)
E(0)_ Ea») EAGLES? _ 7w) (E(0)_ EBA-211w)
TB

JA

 

30A|HA IOJA}3J/dH24 IqA>3qAIHAI0A>303IHA ITB>3TBIHAIOB>
E(O)—-— (2+ zhw)A (E(O)— —E00) + zfiw) (E33) — (0) + zhw

 

30A|HA (:IJAXJAIHAICIAXCIAIHA I0A>303IHB ITB)<7‘BIHB I03)

3
(,7
< >
(1232— J+H><a WW? - -7777
< >
< m)

 

30AIH7 IJA)3JAIHE ICIA)3C1AIH: IOA)3OBIHA3 ITB)37‘BIHE I08)

 

E30)—-— E30— 7m) E3°)—15130l+zhw) E33 —E30)+zfiw

JA

.3.

 

E3 —E3°— m)(3Efj) —El30)— mat)(E33 —E3°l—
>< 30AIH27A|JA>(JAIH:1 IqA)3qA|H7 AIOAXOBIHAITBWBIHA IOB>3 (250)

If we assume that the matrix elements of [Ag and [1? are real, then

3OBIH¢BITB) = 3TBIH<SBIOB)
3TBIHgIOB) = 3OBIHEI7‘B) 3251)

100

Using these relationships in the first, fourth, fifth, eighth, ninth, and twelfth terms
in Eq. (250) gives
A LAT :3 A
_ 3OAOBIVAB GA :u’a Tafi ”fl GAEBB VABIOAOB>
— (voElvAB GAEBBE‘A TaE E3 BGA VAB|0AOB)
_ (OAOBIVAB GA€BB —: T ¢fl¢0 GAGBB VABIOAOB>Z

— TaETETAHEOQ-B— )fdwz

30A|HA IJA)3JAIHAI€IA)3(IAIHA IOA)3OBIHE ITB)37‘BIHB I08)
(E30I _ E30I) (E32I- E30) + zhw) (E33,? — E33;I + m)

JA

JA (1A 7‘37“)

 

30A|HA IJA> (JAIHA IqA> (qAIHA IDA) (OBIHAITB> 3TBIHAIOB>
(E30) — E33?) (E30) - E00) + zhw) (E732) — (3):) - zhw)

30A|HA IJA)3JAIHA IqA)3qAIHA IOA)3OBIHA ITBI3’I‘BIHAIOB)
(E33II _ E33?) (E32I - E30) — 21w) (E33,;I— —E3°I +zfiw)

30A|HA IJA>3JAIHAIqA>3qAIHA I0A>3OBIHA ITB>3TBIHA I03)
(E33II _ E3AI) (E333,I — E33II -- 2m) (E3? — E33,;I — 2m)

 

 

 

A
30A|HA IJA>3JAIHanA>3qA(IHA I0A>3OBIHAITB>3TBIHA I03)
(E33I E3) (E33II— E3A I+ 2m) (E333I— —E33II +2773»)

30A|HA IJA)3JAIHAIqA>3qAIHA IOA)3OBIHAI7‘B)3TBIHB I03)
(E3? —E3°I) (E30I — E3A) — 2m) (E33,? E—30I +zhw)

 

 

30A|HA IJA>3JAIHA IqA>3qAIHA IOA)3OBIHA ITB>3TBIHA I03)
(E333,I —E3°I) (E30 I —E3AI +zhw) (E33,;I — E33;I — 2m)

JA

 

30A|HA IJA)3JAIHAIC1A)3CIAIHA IOA)3OBIH5 ITB)37‘BIHAIOB)
(E33I— E3 I) (E33:I — E3AI — 2m) (E33,;I — E33,:I — m)

 

101

<0A|fi7 IJA) (JAM-3 lqA><qA|Ha |0A> (013W? ITBXTBlfifloB)

 

 

 

 

(E332, —E((,O)+zhw)A (E3ol—E33jj +3233) (E 3” E30)+zhw)
(DAMS IJA)(JA|#.3|CIA><CIA|#A |0A><03|fingBXTB|fiB IOB)
(E332- E33” BAHM) (E3.0 —E33]j —zhw) (E30—E E30) —zfiw)
(OAII::( IOJA> (JAM lqA><quMa lOA><OBl/~‘(153 ITB><TBII1§ |OB>
(E32- 32—31233) (E33)— E33: +3133) (E33) —E30) +3133)
1
+ (E3332— $32233) (E30— —E33:) —zhw) E30) -—E30)— 3223;)

>< (DA/153 le><JA|u3lqA><qufi7 I03)<03|fi§|r3><r3|fi3IOB>]. (252)

Each term in Eq. (252) is identical to a term in Eq. (249). Therefore, we can write
the sum of the (0,1) terms as

_ (OAOBIVAB CAI—1*: T033 fifi BOA/638 VABIOAOB>
_ (OAOBlVAB GAEBB/l: Tafl fig BGA VABIOAOB)
_ (OAOB IVAB GAEBB—A #6 T ”HOG/1633 VABIOAOB>—

— T33T73T333§°(4—h 7,) / mfm mm- 73330). (253)

Now that we have derived the final expression for the sum of the (O, 1) terms, let
us consider the (1,0) terms. When term 6 in Table 1 is evaluated, the result is

A LA LB 33
(0303|VAB GB 30, T33 333 GA“ VAB|0303) = —Ta3 71,3 713, 335,30

A ;B .
X Z (Oaluglraflralug ISBXSBWfIOBHOAlua |JA>3JAl/13 IDA)
JA3TB3SB#0
1

(33 — 32>) [(3:02 - 33>) + (33 — 3377

Using Eq. (232) with qA replaced by J A and TB replaced by SB in Eq. (254)
gives

 

x (254)

— (OAOBIVAB GB fif T33 71-? 013633 VAB|0303) =

102

h
- T06 T75 Ta; H7 0(4—A)

(OBIH5 ITBl (TBIHg ISBXSBIHfloBl
X d“ (0) <0) (0)
3333;33330 (E8 —E0 3) (E3322 — E08 + 31333)

(OBIHB lTB><TBlޤlSB><SBng '08)
(E33; — E32) (E332— E30)— 3133;)

(OAlHalJAXJAlH3 IDA) <0A|H3|JA><JA|H3 IDA)
A) (0) (A . (255)
(EAA — EDA +2133) (EAA - E3A — 3m)

 

 

When term 10 in Table 1 is evaluated, the result is

__ (OAOBIVAB GAEBBfi: Tafi “fl BGBV VAABIO O B____>
— Z Tafi T75 Te: ”:10

JA37‘B3339‘90

A .23 A
(OBIHcSBlTBXTBlflfl ISB>(SBIH§IOB)<0A|H7 lJA>(JA|Ha|0 A)
0) 0 0 0
[(E3 E‘ l) + (E53 -E5!)l (E33 $93)

From Eqs. (232) (with CIA replaced by jA) and (256),

.(256)

_AT
_ (OAOB IvAB GAEBBfi-a T:/3 #3 BGBV VABBIOAO >_

- TaHT75T6¢M240(4— :)Zoodcdtz

(OBlHa ITBl (7‘8ng ISB><SB|H5 We)
0 0
(E33)— E3 8)) (E32,) — E33}; + #33)

<03|H5 ITBXTBIHBISBXSBIHB '08)
(E33; — E38) (E32) — E332 — 3m)

<0A|H7 IJ'A) (JAIHa IDA) (OAIH7 IJA> (JAlHa IOA)
( 0) 0
(EAA -— E3A +2133) (E3A—l E30)— zfiw)

JA 7‘3 839390

 

 

 

 

103

= _ TaHTvéTerbeO('4_ 72)me2

(OBIH5 ITB>3TB|H5 |SB> (SBlH5 I03) 30A|H7 |JA> (JAIH3 IDA)
(E3?) —E3 ) (E - E3A +2253) (E33) E—30) +3533)

(OBIH5 |;B><TBIH§ |83><SB|H5 |03><OA|H174IJA>3JAIH3 IOA)

JA3 7'33 38750

 

 

 

+ (E33 E—(O) E3)(E32)— E33 +sz E30)— E33) —zfiw)
+ (03 |H5 ITBWBIH§|83><SBIH5 l03> 03lH¢|J3><JA|H3 IOA)

(E333) — E30)E)(B)E30) —E3:—1—EAAzh/.u)(E(.O) —E33’A +2533)
+ (E33? —E33A’A))(E E32— 3331—3533) (E33’l—E E30) —inw)

x <03m3317~3><5313315353133103><03|33 IJ3><33|3103>]<>257
Term 15 is

3 ;B 3
— (0303|VAB GAHB 330 T33 33 014393 VAB|03033 =3
_ Z T03 T75 Tab #30

 

JA3TB3SB#0
x 1
0 0 0 0 0 0 0 0
[(52 - 52> + <52 — 552)] [(5- > — 5:) + (5:: — 52>]
3 .;B A
X (OBIHflTBWBIHa |SB><83|H3§|013)(031|H7|J'3)<j3|3’l;‘1 IDA). (258)
Using59

1/{[(E3'?- 52)) + (52> — 52)]

3 [(139—335) (33-3373:

(3) _/m 3, 1 + 1
473 (E30— E33” +3533) (E33) —E33A) —zhw)

 

 

1
(2+ zfiw) (E32,) — E3? + zfiw)

104

1

+ (E33; — 33A) —zhw) (E32) — E33? — 3533)

 

3 (259)

we can write term 15 as

_ (OAOBIVAB CA€BBAA AOT 5H3 6034353 VABIOAOB> =

- T03T76T5¢H£0(4— JIM:

(OBIH5 |TB><TB|H5|88><SB|H§IOB)(0A|H7 lJAXJAlHOé: IDA)
(E3A — E33? + 3753;) (E33,? — E33; + 3533) (E33; — 3A + 31533)

JA3 TB 33750

 

3 .23 A
(OBIH5BITB><TB|H5 ISBXSBIHfIOB) (OAIH7 |JA><JAIH3 |05>

 

 

 

+ (33>- 353333) “(35- 3 -333) (33g -330>_333)
(OBIH5 ITB><TB|H5ISB>383|H§IOB><0A|H7 |JA><JA|HA IDA)
(E30)—E E<0)_ MM Em) _E(0) + m) (Em) _ E00) +2533)
+ (3.5-3 3533- 333x35 33-333) (3552. 333; 333)
>

X (OBIH5ITB><TBIH5ISB<83|H5l0:><0A|H7|J'A><J'A|H3|05>3 (260)

Now, we will relabel the indices in Eqs. (255) and (260) so that the component of
the static dipole moment of molecule A in Eqs. (255), (257), and (260) has the same
index in all three equations. Converting ’7 to 6, E to 7, ¢ to 5, andd to (f) in Eq.
(255) and expanding gives

,3 ;A _"_B A
— (0303 IVAB GB 50 T33 53 03335 VABIOAOB) =

_ TaBTech 763% 04(h 71..)Z00j2dw

(OBIH5 ITB> (TBIH5 |SB> (SB|H§IOB>(0A|H3 |JA> (JAIH7 |0A>
(E30)— E35?) (E33,? —E30) +3755) (E33 — E33? +3533)

JA3 TB 389'“)

 

105

OBI/133ITB><TB|HEISB><SBIH§|OBl(OAIHQle><jAle7l0A>
32>- -3332?” .33: + 33 33> 35,52 _ 333)

 

<
<03lfiflr3><r3IH§I83>(83513I03><03IH£ |JA><JA|H7|0A>
:,3.3,?(E—E31~333)(3,?—E33’—mw E3? —E33? +3133)

 

 

E3 E33?) (E30 —E38— 3133) (E33? — E33? — m)
X <08|fi3lTB><TBlfiglsB><SBlfi§ IOB>30A|H3§|JA>(JAIH’74IOA>] 3261)
Converting 6 to 01, a to 6, (Z) to H and ,3 to (25 in Eq. (260), we have

_ (OAOBIVAB GAeaB/L A0 T3 3H: GAeB VABIOAOB> :

- 71¢T75T05H340(4h7r-)Zodwz

(OBIH5 VB) (TBIH5 ISBXSBng IOB) 30A|H7 |JA> (JAlHa |0A>
(E33? — E3 ?+zm) (E33; — E3 33? B+zfiw) (E33; — E3 BMW)

JA3 T133889“)

 

(OBIH5 (ITBXT‘BIHMSBXSBIHE IOBXOA|H7 IJA><JA|H§ IDA)
E30) EDA )+ zfiw) B(E(B (0) — E32) — 27w) (E32) — ((0)_ zfiw)

JA

 

(OBIH5 ITB)<7‘B|H5 ISBHSBIHg I03) 30A|H7 IJA) (JAlHa IDA)
(E3 .—3? E33?— 3233) E33? — E33? + 3533) (E33; — E38 + 2733)

(E
E3” — E33? — m3) (E322— 33—2333) (E33? - 33? — 2m)
)

 

 

+

JA
X <08|H5 IT‘B)(7‘B|H5 I83 (83ng IOBB)(0A|H7 IJA><JA|H3 |0:)] (262)
Adding Eqs. (261), (257) and (262) produces
— (0,303|VAB GEE: 3333531438 VABIOAOB )
— (OAOBIVAB GAHB H: T033133 GB VA 310,303)
<0A03|VAB GAeaB “:40 T33 H: GAeaB VABIOAOB> :

106

TafiTeabT 76/33 04('h— 7r_)A:/:sz:dw

J3A 7‘33 33790

(OBI/15ITBI<TBIHBI38)(SBIflgIOBI(OAIfiéIjAI3jAIfi7IOAI

 

(E33? — E33”) (E33? — E33? + 3333) (E33’ — E33? + 32335)

(OBIH33 IT3I<TBIH3ISBI<SBIH5 I03I<OAIH£ IJAI<JAIH7 IOAI

 

(E33? — E33?) (E33? — E33? + 3333 (E33’E E33? —3r333)

A _’;_B
(OBIHfITBI 3TBIH3 ISBI<SBIH5 IOBI<0AIH5 IJAI3JAIH7 IOA

 

(E33’— —E33?) E33’— E33?— —3h33) (E33’— E33’ +3335)

3’BIH5 ITBI (TBIH3 IASBIiAASBIH5 IOBI 30AIH5A IJAI UAIH7 IOAI

 

I
E33’— E33’)(E33? ?—3h33)(33’ ——E33’ zhw)
I

3’ BIH5 BTI BAI< TBIH3I8BI:BIAH5 IOBI30AIH7 AIJAI<JAIH5A IOA

 

E3 E33?) (E33?— b33333) (E33? E—33’ + 37333)

OBIH5 I;BA7‘I< BI 3ISBI:BIH5IOBI<0AIHAIJAI<JAIH5I0AI

 

E3? E33?) (E33? —E33? + 3333) (E33’— E33? —3h33)

03|H5 IT3I< BIH3 ISBI<SBIH5I03><OAIH7 IJAIUAIHQIOAI

 

0BIH5 I1EB(><TBIfiAISB>:BIfi¢ IOBI<0AIH7 AIJAI<JAIH5A IOAI

 

E3? E33?) (E33?— ?—zh33) (E3A’ -E33’ — 31333
3

(OBIH5 AITBI TBIH5 ISB:<SBIfiEIOBI<OAI/17IjAI<jAIfiaIOAI

 

E30I— EDA OI+2hw) 8(E3g) — E03) +27%) (E58) — E00 A)+zhw)

(OBIH5 I7‘BI (TBIH5 ISBI (SBIHEIOBI30AIH7 IJAI (JAIH5 IOAI

 

3
(
3
(3
3
3°
3
(E33?— E33? ) (E33? ——E33? 3233 (E33’— E33’ +3733)
3
3°
(
(

E(OI_ E((AOI+ 2m) (E73333) _ E323) __ 2%) (E33) __ E03)

107

_ 33w)

308|H5 ITBI 3TB|H5 ISBI (SBIH3 |03I 3OA|H7 IJAI 3JA|H5 IOAI
(E33’— E33?— 3333) (E33’— E33? +3333) (E33’ E33? +3333)
1
(E33’ — E33? — 3333) (E33’— E33? — 3333) (E33’— E33? — 3533)

x @3133 33333353333133IoB><03I337 333333103] (263)

Now we will show that Eq. (263) is equal to the product of the frequency-dependent

 

 

hyperpolarizability 538633 (— -—;w w ,0) for molecule B and the frequency-dependent
polarizability CIA (w) for molecule A. We can use Eq. (186) from Chap. 3 with 'y
converted to Oz and 6 converted to ’y to write the frequency-dependent polarizability

aA(

0’7 w) of molecule A as

30A|HA IJAI3JAIHA IOAI
02732”) Z Z (0) (0)_
”#0 E-A —E0 zfiw

30A|H7 IJAI3JA|H33|0AI3 .

 

 

 

264
E33’—3’E +3333 ( )

At this point, we will use Eq. (243) to derive the frequency-dependent hyperpo-
larizability 3,5363 (—zw;zw,0) of molecule B. If we let P— — pg, (311: w,

(322:0 (therefore, wazzw, —wa=— w), H“1 =HIB’W =—[LBF5WB’ ,
,3 , A , ,3 _"..’W1 ;'B3w A B 6.23%
W = H3,” = —33 Ff”, H = H = :35 E33“, and H =
-A—,B’0 A B A I

H = —-]Z¢ F f ’0, where H BAA" is the perturbation when the field Ff’w

is applied to molecule B and H [B ’0 is the perturbation when the field F f ’0 is
applied to molecule B, then Eq. (243) becomes

<AB>W _ FB’O F333 2 3OBIH3 ITBI37‘BIH5ISBI3SBIH5 IOBI
B
TB,SB¢0 (E53) — E62) '_ Iliad) (E83) _ Egg) — 73%)
4B ,3
3OBIH33I7'BI37'BIH6 ISBI 38BIH§I0BI
(E33’— E33?— 3733;) (E33’ —E33?)

30BIH5I7‘BI37‘BIH3153ISBI35BIH3IOBI
(E30I__ E30I) (E30I_ E30) +73%)

 

 

 

108

(OBI/15B IT‘BXTBlfi-e |SB><SalueB|08>
(E3? — E3: +zh:) (E3? — E30) + m)

 

(OBIH5 |7°B>(7‘B|fig ISB> (SEW? IOB>
(E3? — E3: ) (E30) — E3? — zfiw

 

 

)

(OBlfiB ITBWBIIZelSBXSBlfiB |03)

(E3? — E3: + zhw) (E§?E — 3‘?)
= Ff’OFaB’wfi%¢(—w;w,0). (265)
Therefore, from Eq. (265), the frequency-dependent hyperpolarizability of molecule

Bis

(OBIMEB ITB><TB|#5ISB)<SB|#B IOB>
— E30 — 21w) (E3? — E3‘? — zhw)

 

[856(1) ("W3 W: 0) Z Z (EBB?)

T333750

(Ogle? |;B><TBII-‘6 ISB><SBIHE ICE)
(0

(E3? — 252:) (E3? -— E 38))

(OBI/15 IT::<7‘BIH5 ISB><SBlflg I03)

 

 

 

 

 

(E3?— E3”) (E“? —E30 +25%)

+ (OBI/25B (lTBXTBlMelSBXSBng NE)
(E“?— ‘?+ 271:) (E33 —E3‘; +zfiw)

+ (OB W5|TB)<TBIMEISB)<SB|I1§|08>
(E3‘?— E3?) (E3? — E38 — zfiw)

+ (OBll‘gerXTBll—ifiISB><SB|fig|OB> (266)
(E3? —E30 Hm) (E3 “’) —E3‘?)

Note that we have let Im) = |7‘B), In) = [33), and l0) = I03). Now,

let us calculate the product of the frequency-dependent polarizability 034,7 (w) of
molecule A with the frequency-dependent hyperpolarizability 311396 46 (—zw; 2w, 0) of

109

molecule B. According to Eqs. (264) and (266), (1A3 (2w) B55365) (—zw; 5w, 0) is

e27 <w>fl535<—w:wa0> = Z

JAJ‘B @3550

;B A
(OBlfie |TB><TB|I15 ISB><SBIII§IOB><0A|H5 IJA) (Ml/1’731 |0A>

 

(EB E(0)— E330 33—2111») (EB: — Em) — 275w) (E30) — E33) — zfiw)

OBI/3B |7‘B)(:B|#5ISB)<SB|#B IOB><0A|#5 |JA><JA|MA IDA)

<
(E‘03— E‘03— 233:)(E(5) ()0 Em) (E3323 Em) —zfiw)
<
(
(°

 

OBI/15 ITB><TBIM5 ISB><SB|11§|08> 30A|fi5|JA33JA|fi7 IDA)
E? —E3‘33 3) (E3? — E323 +5155) (E323 — E323 - zfiw)

 

(OBlfigerWBlm |SB> (SBlugloBXOAl/Ie lJA> <JA|fi7 IOA)
E? —E323 + 55:) (E3? — E323 +2552) (E‘23 — E323 — zhw)

 

(OBlfi5 ITBWBMEISBHSBW? IOEHOMMZ,4 |J,«a)(J/a|fli7‘1 IOA
(E323— E323) (E323 -— E323 -— zhw) (E323 — E323 - zhw
(

 

 

>

;B
03|fl5 |TB><TBIM5 lsB) <83|fl5 I03) (OAlfié‘ |J5><J5|£t’7‘1 IDA)
E323— 3‘33 E3+zhw) (E332 —-E3‘.23E323)( E3 -zfiw)

         

(
<03|fl5 ITB)< )(sBlfif? |03> <0A|fi7 |JA><JA|£L5 |0A>
(E3‘? —E3‘333— z m)(E323—E323—zm) (E323 —E3 23+sz)
(

 

05mg |T3><r3|fl5|83)<83|il5 IOB><0A|fl7 IJA><JA|fi£ WA)
0 0 O 0) 0

(E3?— E333 —z h:E3‘?)( —E333) (E‘3 —E333+zrw)

< )

08|fi5 ITB><

 

(SBIHE I03) (OM/17 IJA) <J5|fi5 lOA

          

 

E323— E323) (E323— E323 + 2255)E‘ E323 + zhw)

(OBI/15B ITB><TBlfl5 ISB> (SEW; ICE) 30Alfl7 IJAXJAIfie IDA)
(E3? —E33‘33 + 555;) (E323 — E323 + zfiw) (E323 — E323 + 255;)

 

110

         

(OB |H5ITB3< 3<sBIH5I053<0A|H7IJA3<JA|H5|0A3
+(E‘23 —E3‘333) (E323— E323— 555;) (E323— E32 3+sz
<

0B|H5 ITB) (TBIH5 |8B3 (SBIH5 |0B3<0A|H7 IJA3<JAIH5 |0A3
(E‘3 —E323 + 255) (E323 — E323) (E323 — E323 + 555;)

 

. (267)

Let us return to Eq. (263). If we convert a to ”y, ”y to O, B to 6, and 5 to ,5 in
the first, second, seventh, eighth, tenth and twelfth terms in Eq. (263), the result is

A LAT ;B A

_ (OAOBIVAB GB #0 Tfilfi “[3 GAQB VABIOAOB>

_ (OAOBIVAB GAGBB-lifi T053 fiflB BGB VABIOAOB>

_ <OAOBIVAB GA$BHJ A0 T (1)/13¢ BGAEBB VABIOAOB> ____

_ TafiTe¢T75 “:10('4_ 72>]:de

(OBIH5 ITB3<TBIH5 I833 (SBIH5 I033 (OAIH7 IJA3<JA|H5 I0A3
(E3? —E3‘333) (E3‘? —323E + 555;) (E‘23 — E3 23+sz)

5515512555155 ISB3<SBIH5 IOB3<0A|H7 IJA;<JAIH5 I0A3

(E323— )(35—53+hw)( 3—555)

(OBI/155ITB><TBIfiziISB3<SBIH3§ IOB3<0AIH5 )IJA33JAIH7 I0A3

JA 7‘3 83740

 

 

 

 

 

     

       

 

+ (53> 5,3»)(5 553-55 (553 533555
+ (OB IH5 ITB3<TBIH5 I8B3:BIH5 |0B3(0A|H5 IJA3E<JAIH7 I0A3
(EB?) —E3H(7?33)( —zfiw (Ejg) AI—zfiw)
+ (0B |H5 |;B3<TBIH5 ISB3<SB|H5 I0B3<0AIH7 IJA3<JA|H5|0A3
(E3?— E323) (E32313 — E323+ +255) (E‘23 —323 +555»)
(OBIH5 ITB3<TBIH5 ISB><SBIHgIOB3<O 3E<JAIH5 I0A3
(E323 —E3‘333) (E323 E32 3+ 271w) (E‘23 233—255;)

111

A LB 5
(OBIHBITB337‘BIH5 I833<8BIHBIOB3<0 A|H5 IJA3(JAIH7 I0A3
—E323) (E323 — E323 -— 5555) (E‘23 —E323 +5555)

 

+

OBIH5 I'rB3 (TBIH5 ISB3<SBIH§I0B3(0A|H5 IJA3 (JAIH7 |0A3
E__(O) (0))(E E;g)_ E(0)_3,,w) (E‘O3— E(0)_ 3,333)

 

(E333

(

(E35 5

(OBIH5 |TB3<TBIH5ISB3<SBIH5 IOB3<0AIH7IJA3(JAIH5I0A3
(E‘m— E323 + 555) 3(3E32 -E323 +5555) (E32 ‘33 — E323 + 555
(

(°

(

 

(OBIH5 ITB3 (TBIH5 I883 (SBIHB IOB3<0A|H5 IJA3(JA|H7 |0A3
E‘—23 E323+5m) B(E323—E E33‘3 33—5555) (E323 — 323 —Zhw)

 

(OBIHB I7‘B3(TBIH5ISB3<SBIH(5B BIOB3<0AIH7 IJA3<JAIHE I0A3
E‘3 E323— m5) E323— E3323+zhw) (E323— E32 I+zhw

 

JA

 

+

E‘23—E E(0)_ 3,333) (E‘03— Elm) _ 3,32,) (E323 _ Em) _ 3m)
x <05I5555551555555555 memo/11551555515,10,5] (268)
If we assume that the matrix elements of ya and [1,3 are real, then

(OAIHBIJA3 = <JAIHBI0A3

(JAIH5 I0A3 = (OAIH5 IJA3 (269)

Using these relationships in the second, third, sixth, seventh, tenth, and eleventh

terms of Eq. (268) gives
_ (OAOBlVAE GEE: T55 H5 BGAEBB VABIOAOB >
_ (OAOBIVAE GAeB—fu T5571? 303 VAE|0A08>
_ (OAOBIVAB (3751533324 ESGAeB VABIOAOB> :

" T557155T76H240(4— 7:)()/oo:w_2

(OBIHBIT‘B337‘BIH5ISB3<SBIH5 IOB3<0AIH7 IJA3<JAIH5 I0A3
(E323 — E323) (E323 — E03 “33 + zfiw) (E33 ‘33 —E323 +555)

JA TB 885“)

 

112

3OBIH5 I7‘B337‘BIH5 I383:BIH5BIOB330 AIH5 IJA33JAIH7 I0A3
(E7323 _ E603) (E303_ hLd(E(O)—l; E303 —zfiw)

3
3

 

(OBIHB I7°B3 3TBIH5 ISB33SBIHB IOB330AIH7 IJA33JAIH5 IOA
(555-5 5)(55g>_5 55m -55 (55.5? 5533 55

 

0B

3OBIHB ITB>3TBIHZEISB>3ESBIH§ IOB330A IH 5IJA33JAIH7 I0A3
(E (2)—E(0))(E323 —zfiw (EBB—E E30) —zfiw

 

0BIH5BI7"B337‘BIH5ISB33E8B:IH,153I0B330AIH7IJA33JAIH5I0A
E323— E323) (E323— E323 + 555 E‘23— (23+zhw

 

           

OBIHB ITB33TBIH5 ISB33 0B330AIH gIJA3-333JAIH7 IOA

3
3
3 3
3
3
(E323— E323) (E323—E E323 +2552 3—555)
3
3 3
3

 

 

3

OBIHB I;B(337‘BIH5ISB3:BIH 5BIOB330 AIH7 IJA33JAIH5 I0A3

E323— E3‘333)(E323— —zfiw (E‘23— E323 +555
3

30B IH5 ITB337'BIH5 I8B338BIH5 I0B330A IH5IJA33JAIH7 I0A

 

E5, “33 —E323) (E323 E323 — 5755; E“? — E323 — 555;
3

3OBIH5B I7'B3 7‘BIH5ISB33SBIHBIOB330AIH, IJA33JAIH§ I0A3

 

3A

E3) E30) + zfiw) 8(E3g) — E32) + 231w) (E323—E (2+ zfiw

3
3°
3OBIH5 I7'B3 3"‘BIH5 I833 3SBIH5BIOB330AIHBIJA3 3JAIH5 I0A3
(E303 —E323 +5552) B(E323 - E323 - 555) (E323E — E323 —zhw
3

3

 

JA

3OBIHB ITB3 3TBIH5 ISBXSBIEBB IOB330AIH5 IJA3 3JAIH7 I0A3
E‘3 —3—E32 5555) E323— El,‘,‘333+2m) (E ‘33 E323+m)

 

JA

+

 

3
E‘—‘33E3‘33—zh5e) (E323-131323—2m) (E323— 323-
3

JA
x 5515555555518555155IOB>55152555553105](270)

Each term in Eq. (270) is identical to a term in Eq. (267). Therefore, we can write

113

the sum of the (1,0) terms as

_ (OAOB “/1413 03%: Ta fiufi BGAEEB V:BI0AOB>
— (0,403 IVAB GA‘BBJ ya Tag 57315 BG'BV ABIOAOB)
_ (OAOBIVAB GAGBB I135 A0 n¢— #43) BGAQB VABIOAOB> =

+00
’1
'— T033 THH T75 ”:10 (In?) / dw 035117 (w) figd) (-290; w: 0) ' (271)

4.5 Third-Order Dispersion: Terms of Order Zero in Both
#540 and “BO

In this section, we derive a new expression for third-order dispersion energies as
integrals of nonlinear response tensors over imaginary frequencies. When we evaluate
term 11 in Table 1, the result is

—30AOBIVAB GAEBBfi: T30 ’13 BGAQEB VABIOA083—
" T03 T76 Tab :3

jAaQA57‘85337é0

1
[(5:2) - .55) + 55 - 55)) [(523 - A?) + (52> — 55))

X 30AIH7 IJA33JAIH5I€1A33QAIH5 I0A3
X (OBIfiaITB337'BIHBISB33SBIHL¢IOB3° 32723

 

Replacing q A with j A in Eq. (232) and using Eqs. (232) and (272), we have

_ (OAOBlvABGAG9B—QTB a Bufi BGAEBBvABIOAOB)_ _ —Ta,’3 T75 cred)

1
X Z
0 0 0
JA59A TB SB¢0 [(E3/1) _E( A))+ (Ea: -E(() 12)]

X 30AIH7 IJA33JAIH5 IqA33qAIH5 I0A33OBIH5 I7‘B337‘BIH5 I8B335BIH5 I0B3
+00

x (43) / dw 1 + 1
7‘ (E33-2) — E33) + zfiw) (E33-3) - E33) — zfiw)

—OO

 

 

 

114

1 1
+
(ESQ — 133? + zhw) (Efi? — E3?; — 2m)
Replacing T3 with 83 in Eq. (232) and using Eq. (232) in Eq. (273) gives
_ (OAOBIVAB GAGBBfia —AT Ta 3,1,3 BGAGBB VABIOA OB>_ _Taa T75 71¢
X Z (0AMIJ:>(JA|#a|qA)(qA|/1€ IDA)

jA quJ‘BaSBSéO

A ;B A
X (OBlflgerXTBl/IgISB>(SB|/1§IOB>

 

 

(273)

 

 

 

 

 

 

 

 

 

 

 

 

+m
I
x (1679) fdw ZOO dw
1 1
X (0) (0) + (0) (0)
_(EjA — EDA +zhw) (E3-A — EDA — zhw)‘
, -
1 1
X 0) (0) + (0) (0)
(EfiB — E08 + m) (Em — E08 — zhw)
F H
1 1
X 0) (0) + <0) (0)
(E131 — EDA +zfiw’) (EM — EDA — mm)
X 1 + 1
(Egg) — Egg) + my) (Egg? — Egg) — mm)
= —Tafi T76 Ted)

A . . ;A A
Z <OAIH£IJA><JAIHQIQA><QA|H24IOA>

J'A.QA.TB,SB¢0

A $.13 A
X (OB|#§|TB><TBIH;3 |SB><83|I£§|08>

 

1
X (16—7272)_Zo(1§dw E(,O)+zhw)(E§2)—Eé:)+zhw)

1
jo oodw(E(.°— E5 0)+zhw) (Efi2)—E3‘;)—zhw)

115

 

 

1
+OO_Z(E dw — E3?—m) (Efig)—E53)+zhw)

 

1
_Z(E dw —E00— m) (E72) —E(§:) —zfiw)

A

 

_fw dw’ 1
(E53 — .3) +zhw’) (ESQ — E33) +zhw’)

 

1
(__ZOdJO; (50) _E00 )+zfiw’) (E58) —E00)— zhw’)

1
+_/000:0E1dw,( ”7 —E<3— —zfiw’) (E 0) —E33)+zhw')

 

1
j. ...,
(Em) — E00) A—zhw’) (ES) — Egg) —- zhw’)

We can use the properties of the integrals in Eq. (274) to reduce the number of terms.

 

(274)

 

Inoo the integral
f:dw1/[(EJ(-?—Eég)+zfiw) (E£2)—Eé:)+zfiw)] , let w = —77 and
w=—m,
71.1 =
3+ zfiw) (E52 — E33) + zfiw)

 

JA

1
+/00 (Em) — E33) -— zhn) (E723 — E62) — zhn)

1
_Z( dw E30“ —E33)— 2m) (E72’—E33’—zhw)'

 

(275)

116

Similarly

7w 1 =
-00 (E313) — E53) + #1.») (E53) — E33) - 277w)

 

 

1
dw . (276)
.../o (El-3) - E63) - zfiw) (E72) — E33.) + W)

This means that we can write the w-integrals in Eq. (274) as

 

1
_Z(E dw - O)+ zfiw) (E12) -— E33) + zhw)

 

1
+ _Z( dw Egg)— E03) + zhw) (ESQ —E(§:) — zfiw)

 

+ / dw 1
(E3-3) —E(§3)— 2717.)) (Elm— E33)+zhw)

 

1
+ _/ 01d.»(E(3_ —E33)—zhw) (El37—E53Lzhw)

 

1
= 2_/00( dw EJlO)- 0)+ zhw) (E72) — E33) + zfiw)

 

1
+ 2 [(E dw . (277)
—E03’ + m) (E13) — E33) — zhw)

Similarly, we can write the sum of the w’ integrals as

777 1
(E(g)— Em) + zhw’) (ES. 0) —Eé:) + zfiw’)

 

 

1
(”_ZOOdwi (Em) — E0 (0) + zfiw’) (ES. 0) — E62) — zhw’)

117

1
+ _/00 dwl(E537— E537 flax) (E..— 07— 537mm)

 

+ Z dw' 1
(E537— E537 — 27w) (E537 — E537 — 27w)

 

 

1
= 2 f... 7,7

_00 (E53)E — 0)+ zhw’) (EQ— 3:) + zhw’)
+ 2 / dw' 1

-00 (E537 — E537 + my) (E537 — E537 — 27w)

Using Eqs. (274), (277), and (278), term 11 becomes

 

(278)

— (OAOBIVAB 07473 7‘2"T.,fi.3 074673 VABIOA 0.): —T.,T..T..

X E «147723 |7A><7A|7ta ICIA><QAlfie lOA>
jA,qA,TB.SB¢0

A 2.3 A
X (OBlltfsBlTBWBl/tg ISB><SB|77§|03>

 

1
(4:2) _ZO( dw 3507 _E537 +271.) (E537—E537+zm)

1
+ jo idw(E537- E537 + zhw) (E5. “77 —E537—zrw)

 

/ 77’ 1
_00 (E53) — E63) + zhw’) (EQ— E30) + zhw’)

 

/ 77’ 1
_oo (E537 — E537 + 271.7) (E537 — E537 — 27w)

+00

7 —-:_>7dw 77>:de

118

 

JA (IA 7‘3 83750

30A|M71 IJA)<JA|Ma|qA)<qA|M§A |0A>
(E337 — E337 + zfiw) (E337 — E337 + 27171;)

 

A ;.B A
(OBIMSBITBWBIMg I83) <88|M§|OB>

X
E337 — E337 + 27w) (E337 — E337 + 7777/)

 

<07|M74lj7>(7'71IMQ|7177><q71|71£1 I07)
E3) “E0: +zhw) (E(g)— E0: 3+2hw)

 

(OBIMB I7“B><7‘B|M73|5>‘B><7‘>‘7B|Mqli3 IOB>
E337 —E337 +2777) A33(E7 —E337 —zhw’)

 

30A|M7 IJAXJAIMQ lqA><qA|MEA IDA)

(
(
(
(E77 _ E73 + 2771;) (Ex; _ E77 .. m)
(
(

+

 

JA
A .*_B A
(OBIMéBITBWBIMB ISB><SB|M§IOB>
E337 — E337 + 27717) A(E337 — E337 + 27717)

 

(DAIMA le><jAlfia|9A><QAlfiAIOA>
E337 — E337 + 271w) (E337 — E337 — 2m)

 

(OBlMa ITB><TB|M3 ISB><SB|M§>3 I378)
(E337 — E337 + 77717) (E337 — E337 -— 27w)

 

(279)

When factored, Eq. (279) is

A LAT ;B 7 A
_ (OAOBIVAB GAEBB“ T10!3 ”B GAEBB VABIOAOB>
+w

< 2777 7°72

(OAIMA IJA><JA|Ma|qA><CIA|MA IDA)
(E337 — E337 + 2771;) (E337 — E337 +27w)

JA (1712750

 

119

 

A ;B A
X Z 3OBIM§3|TB73TB|MA ISB73SB|M5|037
73,3375 (E337 — E337 + zfiw) (E337 — E337 + 27717)

 

 

 

 

+ Z 3OBIMB |TB73TB|MgISB73SB|MB |037
(E72 — 72 + 77) (E2 — 72 W)
3037M};B ITB73TB|M73 ISB73SB|M5|OB7
+ 2 E30)_ E30)_ ho.) E307 E307 m7
TB,SB#0( 7'8 Z )( 33 — 03 +2 )
A LB A
+ Z 3OB|M§3|TB73TB|MAISB73SB|M5IOB7 (280)
rasB7o(E(g)-E3:)-zfiw)(79723-13323-77777’)
We define
O 0
b2“ —wa;w2,w1) = (0)371I776 IJA73JA|M7IQA73QA|Ma| A7
JA (171950 E _EOAL M)A(E(O QA A—EOAL 1

30A|M2A lJA73JA|Ma IQA73QA|M7 IOA7
E307— E307— M) E337—E3 ()0 _ 2

 

172,7 ( —wa;w1,w2) =

30A|MA lJA73JAlMe lqA7)3qA|EM(’7A O70A7

(7 777)
Z3 7777
:2772 72>+7)(72 - - 77)
253‘” 7
:73 7

7

 

37357 (Ml; _w0'9 w2) :

30A|Ma|JA73JA|M7 lqA73qA|M:A IOA7
00 go 0

 

17376 (7771; w2, —w0) =

30A|M7 |7A7<jA|MLA lqA73qA|Mal0A7

 

bfiga (W2; —w0’7 wl) :

 

A . _ 1
b7ae (7722,7721, —wa) — 7,2772% (E30) _ 1302+ m2) (E70) _ E772) + 77700
X 30A|M7 IJA><JAIIUQICIA>39A|H¢ 70717 (2817
Then
M2377(- -wa;w1,w2) = b277—( wa;w1,w2)+b§‘70(-wa;w2,w1)

120

+ b27377 (W1; —w0', LL12) + bayg (3‘01; W29 —LU0)
+ b17460 (7122; —wa, 7.721) + bf,“ (7172; w1, —w3) ,(282)

Similarly

M555 3—wa;w1,w27 = b37335 3—wo;w1,w27 + b5673(_w0;w27w1)
+ 175756771; —wa,w2) + b557, (7771; 7772, -wa)
+ ban (7172; —w0,w1) + b33333, (w2;w1, —wa) .(283)

I

If we let 77.71: 2712 —w and (712: w in Eq (282), then wa = wl +012 = w ,
and

 

b: (wwr_w _WI) 2 Z 30A|M7|JA73JA|Ma|qA7
QC ’ 2 0
7,472,770 (E777) — E77,, + zhw)

3QA|M2A |0A7
(E33) — E03) + zfiw')

 

(284)

Similarly, when we let 011 = w —zw and 7712:qu for molecule B then 7170 =
wl + (.722: w, and

 

AB

, 7 0 IABIT 737' I— IS 7

B . __ 3 B M5 8 B M B
b53777 (”Aw ”AA—Aw) — Z (0) <0)

r3.337é0 (ETB _ 08 +27%»)

33377325703)
(E337 ——E337+ zhw')

 

(285)

Comparison of bme (w; w), — 2w,— —zw’) with the A term in Eq. (280) indicates
that the (O, 0) term, as specified by Eq. (280), contains b33106 (w); w, - w, —w').
Also, examination of bggq, (w; w, — w, —w ) in Eq. (285) and the first B term
in Eq. (280) indicates that the (0,0) term contains bggé (2w; zed, — 2w, —zw') . If
we let wl 2 —2w' — w and tag = w in b533, (w2;w1, —wa), then wa = —zw'
and

 

AB

0 IABIT 737‘ l_ 75 7

B . , , _ 3 8M5 B 3M5 B
b57347 (“A—“‘3 _w’w) _ Z (0) <0)

r3,33¢0 (Era " E08 +2501)

121

x 3SB|M7€3|07 . (286)

(E337 — E32) — zhw')

 

Inspection of Eqs. (280) and (286) indicates that b53377 (2w; —zw’ — W, 27.12,) is the
second B term in Eq. (280). When we let an = 7712' + M and bag = —-zw in
b35367 (7172; w1, —w0), then wa = w, and

 

aB

’ 7 0 AB 7‘ T‘ — 3
777 (7272 +2.22) 2 E: < 77737 77071771 A
7.8958750 (ETB _ E08 _ 2’10!)

<83|M§|037
(E32,) - E32) + zhw')

 

(287)

Examination of Eqs. (280) and (287) indicates that ()5?ch (—zw; w), + w, —zw')

 

is the third B term in Eq. (280). When we let (.01 = w - 7,7,7), and 0.72 = -—zw in
b333, (w2;w1, —wa), then tag 2 —2w' and
A LB
3 7 7 303|M§3|TB73TB|M73 ISB7
A533" (_A‘A A” _ A” ’3’“) = Z (0) (0)
rasméO (Era - E03 — 7’53“”)
s O
3 BIMB l 7 3 (288)

 

(E337 -E337 — 271717)
which is the last B term in Eq. (280). Therefore, from Eq. (280),

_ (OAOBIVABGAQB-LATO fifiBGAGBBf/ABIOA OB) ___

— 7107371767171) (Al—h” 2—2)Zood3ad -ZOObA(dCUI 706 (W;W,—W,—Zw’)

x [b§5¢(w;zw— w—roouu)

b533, (2w; —zw — 7,117,270,)

+++

bffid, (—zw; 77.77, + 174—7717,)
B
37537 (

—w; w —— 2w], AAA/)7 . (289)

122

Consequently, using Eqs. (214), (222), (230), (253), (271), and (289), we have

3
MS;

+++

_ A0 30 all B
_ T03 T57 71¢ #0 ”go acts 07,3

1
6 T03 T75 72¢ MEG/160M?) 31015
1

6 T03 T75 Tap Ma 0M7 0M6 A0 508 3gb

T03T36Twp50(3_1h_ 7r.)Zodwa§3(zw) €a3(— 2w;zw,0)

’1
T03 T63, T35 #240 (_) ZOO dw a; (w) 3553, (—zw; zw, 0)

Tax? T76 Ted) (4: :) fodw Zo'dw b3a€( 2w; 2w, — tug—2412')

[b68603 (14mm — w, —w)
b53333 (w); —zwl — w, w’)
b35333 (—zw; w, + w, —zw’)

bgw —zw; 2w — w’, WIN . (290)

123

5 Numerical Estimates of the Second- and Third-
Order Corrections to the Interaction Energy of
Two or Three Hydrogen Fluoride (HF) Molecules
for Various Geometries

5.1 Leading Contribution to the Second-Order Correction to
the Interaction Energy of Two Molecules

According to Eq. (189) in Chap. 3, the second-order correction to the energy of
interaction ABS/24; of molecules A and B is

2 1
ABS/RB Z _ T75 71¢ M1530 #50 (5) a;

1
— T75 Tab M340 M240 (5) 05p

— Trim (5) 70cm a3: (Mama). (291)

Let us write Eq. (291) as

25533 = Mg; (0,2) + 231351 (2,0) + AEéij (0, 0), (292)
where

AE”) (02) — —T T. 30 301- A

0A3 a — 76 642/16 Iu’q) 2 036

0,13 3 — ’75 EQHV #5 2 aria)

h +00

2

AEéA)8(O,O) = —T3,5T€¢ (Z7?) [dwafiE (22100233 (w). (293)

Eq. (291) (or Eqs. (292) and (293)) can be used to compute AEéi; for any choice

of A and B and corresponding geometry.

124

5.1.1 The Second-Order Correction to the Interaction Energy of Two
Colinear Molecules

Let us derive AEéi; for a pair of colinear molecules whose internuclear axes are
oriented along the z axis of the x, y, z laboratory frame, as shown in Fig. 12. In Fig.
12 x, y and 2 denote the axes of the the laboratory frame, and :c’, y’ and 2’ denote
the axes of the molecular frames. The centers of mass of A and B are COM A and
COMB, respectively. Also, the nuclei of A are A1 and A2, and the nuclei of B are
BI and B2. Finally, R A B, is the center-of-mass to center—of-mass distance between A
and B.

It is important to mention that when we use the general formula for AEéiL
from Sect. 5.1 to derive AEéi; for a specific A — B geometry, we carry out the
derivation in the laboratory frame. The non-zero components of the dipole moments
and polarizabilities of A and B, however, are given in terms of the coordinates of
the molecular frame. Therefore, if the laboratory and molecular frames are not the
same, we need to transform the non-zero components of the dipole moments and
polarizabilities of A and B from the molecular frame to the laboratory frame. In
Fig. 12, :L", y’, and z’ of the molecular frame are aligned with 1:, y, and z of the
laboratory frame, so that the two coordinate systems are the same. Consequently,
there is no need to transform the non-zero components of the relevant prOperties of
A and B to the laboratory frame, and we can complete the derivation of [BESS/i; for
this particular geometry in the laboratory frame. All quantities and equations in the
remainder of this section will be given in terms of the coordinates of the laboratory

frame.

The center-of—mass to center—of-mass distance between molecules A and B in Fig.
12 is defined as RAB = RABIi + 12,4335! + RABzi, where RABI, 12,433 and
RAB, are the :r, y and .2 components of R A 3. For this geometry, R A Ba: = RABy =
0, so that RAB = RABzi- To begin our derivation of AEéiRB for A and B arranged

colinearly, we will determine the contributions of AEéiL (0, 2) and AE(2) (2, 0)

0A8

125

to AEéiL. Recall Eq. (149) in Chap. 3,

1 3R rag-5,, R2
T03=VOV5R= R5 3 ,

When we use Eq. (294) with R = RAB to evaluate Tag for all possible a and 3
where a and 3 can be I, y, or z, we find that

 

(294)

 

Txx = Tyy =

Tzz = (RAB)31 (295)

and T05— — 0 when a 7e 3. The expression for AEéi) B,(O 2) given in Eq. (293)

simplifies to

 

1
AEéi; (0’2): - (2)Tm: Trait/150 “Boa;

T Tyayufoufo 53y

NIH

BO BOO/12
$2: HTzzH Liza

'8

BO Boa A
nyl‘iE My Mata yx

EH

BO 800 Ay
nyyy ”y ”ya

Tyy 71223330335003 5.

T2 Txxugoflfoan A

BO BO A
TzzTypypz yazy

ml .... toll—I NIH [\DIH [\DIH NIH [\DIH
VVVVVVV
,3?

l
¢AAAAAA¢

)T (2..)

In Eq. (296), we have omitted those terms that vanish because they contain Tag
with a 75 ,3. Recall from Chap. 4 (Eq. (195)) that the permanent dipole moments

126

MA0 and p80 of molecules A and B are
AO AOA AOA AOA

= 9.. Hit, HA. 2

BO BOA BOA BOA
u = uxx+uyy+uzz

(297)

For the colinear geometry of the dimer shown in Fig. 12, 113:0 = [1:0 = O and
p50 = #50 = 0, so that ”A0 = 112402 and #30 = pfoi. Therefore, Eq. (296)

reduces to

(2) _ 1 BO BO A
AEOAB (0, 2) — — (‘2‘) T22 T22 #2: Hz azz’
and similarly AEéi; (2, 0) reduces to

2 1
AEfLRB (210) = _ (E) T22 T22 #on #:1002132

When we substitute T22 = (Bi)? into Eq. (298), the result is
AB

 

2
AEéii. (o. 2) = —- (R36) ,3 2.8%.,

and substituting into (299) gives

2
A1233). <2, 0) = — (R3,) 2:0 wag.

 

(298)

(299)

(300)

(301)

Now, we need to derive an equation for AEéi; (0, O) in terms of R A B- From Eq.

(293),
h +00
AEéiL (0’ 0) Z _ (II—7F) / d“) [131"F TM? 0211‘ (201)053: (W)
+ Tm Tyy afiy (2w) afy (w) + Tm. T2,, a; (2(1))021.3
+ Tyy Tm 05:1: (W) 05:1: (W) + Tyy Tyy ayAy (W) 0‘
+ T33 T22 a; (w) a; (w) + Tzz Tm of; (w) a
+ Tzz Tyy a2, (w) (153 (w)
+ Tzz Tzz a: (W) 01.5.32 (20.1)] -

Let us consider the imaginary-frequency polarizability a3: (w) of molecule A,
which is given in Eq. (186) of Chap. 3.

0346 (w) = Z

(029M);4 IJA><JA|M£1 IDA)

 

 

 

j/fiéo
(OAIMf le><jAlfiyl0A>J . (303)
E30) Em) + the
In general, (13142.0) has nine components: (13:43 (no), 0:343 (2w), 03:42 (w),
0343 (w), 0343040), 0523(1)), (12430112), (12,/(w), and afz (2w). For any par-

ticular molecule, however, some of these components may vanish, depending on the
symmetry of the molecule. Let us determine which components of the imaginary-
frequency polarizibility vanish for molecule A, where A belongs to the C00,, point
group. Note that the components of the imaginary-frequency polarizability that van-
ish for molecule A will also vanish for any other molecule with the same symmetry
as A, and that the static and frequency-dependent polarizabilities have the same

symmetry properties.

Consider the product of matrix elements (0 Al )1 I ] A>< ] Al [1 IO A) contained in
N A I” A
Eq. (303). Using the fact that [1: = Z (32-Tia + Z: ZvRva, we can write the
i=1 v=1
product of matrix elements as

NA NA
(OAleIJ'AHJAIMfIOA) = ZZeme,,(0A|r,,,3|jA)0,4334%)
+ 22%” (OAlrm7|]A><JA|Rne|O/i>
"m:
+ ZZZZmen<0AIRmIJA>919.40..)
"Ii/737$:
+2222

m=1 n=1

>< (OAIRmIJ'A> (jAIRnelOA>- (304)

128

The matrix elements of Rm vanish between orthogonal electronic states. The po-
larizability components (13:13 (M), GA (2w), 0.31:3(ZLU), (152(201), azAx (w), and
a”: (w) vanish by inversion symmetry with respect to the J: and y axes, but a; (w),
CIA: (w), and (1A2 (w) are B,nonzero and a; (w) = 80343 (W). If molecule B also
has Com, symmetry, then a3 3(zw), 0:8 5(zw), and (12,041)) are also nonzero, and

(1535— — 0 if 5 74 (b. In this case, Eq. (302) reduces to

1'2

A1533), (0. 0) — (f5) / dw (T... T... a; (w) a5. (w)

+ Tny aA 00(zw)aB (w)

1‘19 yy yy
+ Tu Tu an (M) (15, (21.0)] . (305)
Finally, because aA x(zw)- — aAy (2w), 01515.3:r (w) = of; (w), with (295), the total

second-order correction to the energy of molecules A and B when they are aligned

colinearly is

magi; (0,2) + AEOle (2,0) + Ang} (0, 0) =

2 BO 800 A
#2 #20
—(RAB6 )

+00
_ (L) #AO MAOCYzz B _ (i) [dw [zafix(w)afx(w)
RAB6 z z 47r _ RA36

A
+ 4azz

 

 

zw OB 2w
(R:B6zz( )] . (306)

5.1.2 The Second-Order Correction to the Interaction Energy of Two
Colinear HF Molecules

Now, we will use the information in Sect. 5.1.1 to derive the second-order correction
to the energy of two interacting HF molecules. If we assume that A = B =HF in
Eqs. (300), (301), and (305) then we have

(2) _ 2 HF,O H1200 HF
AEOHF_HF(012) — — (m) Mz Mz 0122 , (307)
2
AEOJP HF (2,0) = — (m) MfF’WYOMfFO 2F, (308)

129

and

 

AEOH)F HF (0,0) : _(4—H) _Zmdw [20 gRHF)— H:: (W)

40/” (191)04sz F090]

 

+ ZZ

309
RHF—HF6 ( )

Maroulis62 has recently used coupled cluster theory at the CCSD(T) level and finite-

field perturbation theory to calculate the electric properties as well as the property
derivatives of the X12+ state of HF as a function of bond length. Accord-
ing to Maroulis, the most accurate values of these properties have been obtained
using CCSD(T) theory and a (16s 11p 8d 4 f / 10.9 6p 3d 2 f) basis set. At the ex-
perimentally measured value of the equilibrium bond length of HP, 7‘( H F)c =
1.7328 a.u., the CCSD(T) value of the permanent dipole moment pHF’O in the
(16311p8d4 f / 1036p3d2 f ) basis is ”HR“ = 0.7043 a. u., and the corresponding value
of the 22 component of the static polarizability of HF is aHF =6. 36 a. u. Recall
that for the geometry shown in Fig. 12, ,uA0 — —/.I.’zA 0Z Fand ”BO — —u§ OZ, so
pHFO— — ,uHFOZ. Substituting pH” = 0.7043 a. u. for pf 0and aHF— - 6. 36 a. u.
for 01sz into Eqs. (307) and (308), we obtain

6.31
AEOIBF— HF (0’ 2): AE0H)F— HF (21 0) = — (m) ° (310)

In order to obtain numerical estimates of A1303,” HF (0,0), we use the Unsiild ap-
proximation with the average excitation energy approximated by the ionization po-
tential Eff? of HF. Let us begin by considering the static polarizability of HF, as
specified by Eq. (303), with w = 0 and A =HF. If we assume that the matrix el-
ements in Eq. (303) are real, then (OHFI/lfyiFIjHF) = (ijlfigFIOHp) and
(OHFIMe HFIJHF): (JHFIMHFloHFl ,and EQ- (303) reduces to

 

(0 HF - AHFO
)_2 Z (HFlMy IJHF><JHFlMe lHFl.

(0) (0)
JHF7H0 EJHF _ OHF

(311)

We simplify Eq. (311) by using the Unséld approximation. This approximation is
suitable for numerical calculations if the transition energy from the ground electronic
state to the first excited state is similar in magnitude to the ionization energy, and
if high-lying continuum states do not contribute significantly to 076 (w). Then we

130

approximate

 

1 1
Em) Em) g W' (312)
jHF-— OHF 1P
Using Eq. (312) in Eq. (311) gives
(0 HF ' ' AHF 0
aHF N 2 Z (HFluy IJHF><JHFIHe I HF). (313)

 

EHF
JHF¢0 1P

When we solve Eq. (313) for the sum of the matrix elements of the polarizability

aHF

0176 of molecule HF, the result is

.HFlaHF

A o o A a
Z (OHFIIL‘S’FIJHF)<JHF|H§FIOHF> = 2

JHF¢0
We get the frequency-dependent polarizability of HF by letting A = HF in Eq. (303),

a?!" (w) = Z:

JHF¢0

(314)

(OHFIMHFIJHFXJHFIMG FIOHF>
E(0) —E(O) —zfiw

JHF OHF

 

 

 

(OHFIHFFUHF>(jHFlflgFIOHF>:| (315)
(0) °
EJHF_ E0H)F + Zhw
Setting (Ejn)p_ E62”) 91 Egg and using Eqs. (314) and (315) gives
EHF 2 HF
a??? (w) ( )a ale . (316)

 

: [(EIPF2) +h2w2]

Therefore, from Eq. (316) with ’y = 6 = as, ’7 = e = y, and ’y = 6 = z, we
have

 

 

 

2
0:27(uu) __ (13HF3052F
[(EIPF2) +h2w2]
2
aHF(zw) = (E11531?) a5?"
yy HF 2
[(EIP) +h2w2]
HF 2 HF
aifiw) = (E’Pg) 0‘“ - (317)
[(EEJF) +h2w2]

131

Using 033‘: Eq. (317), and Eq. (295) in Eq. (309) gives

ayHyF’

2 EHF 4 diff 2 h
M=— (12.1-1.6) (...)

XZOO dw
[(EHF)2 12712002]

4(15‘IEF“)(a§;F)2 )1 +00 1
— 1;:1F-HF6 (Z7?) /w[(EfiDF)2+h2w2]2. (318)

—00

 

 

 

 

In order to simplify Eq. (318), we need to evaluate the integral in this equation. If
we let Eff = a and flu = :13, then dw = dx/h and the integral is

 

 

_l d“ [(Efif)2l+ h2w2]2 = (H) _4 d1; (a2 +232)? (319)

Since the integrand on the right-hand side of Eq. (319) is integrable from ~00 to

+00 and it is an even function,

G?) de(a2 : 3:2)2 = (:3) (200de +1.22)? (320)

--00

 

 

According to the integral tables of Gradshteyn and Ryzhik,67

 

 

+00

1 (2n — 3)” 7r
/ ”Home" 2 <2n—2)z:a2n—v (321)
0

where (2n+l)ll=1-3---(2n+1) and (271)” = 2-4---(2n) . For
71:2,

 

 

+00
1 7r
[d1]? (a2 + $2)2 — 2—a3'. (322)
0
From Eqs. (320) and (322),
1 +00 1
7T
(5) / d$(a2 + 1:2)2 — “Th, (323)

132

and

 

 

+00
/ dw 21 2 = 11:3 . (324)
-00 [(Efif) + 7121122] [P h
Using Eq. (324) in Eq. (318),
131931)-“ (0, 0) = _ 355(03): _ EfivF(aZF):_
2(RHF—HF) (RHF-HF)
When we let of?" = 6.36 a.u., aflp = 0%!" = 5.22 a.u. (as calculated by Marouli862)
and EH! = 0.5896 a.u. in Eq. (325), the result is“8

 

 

(325)

 

 

31.9 . .
AE33F_HF (0, 0) = — a u 6. (326)

(RHF—HF)

- AF”) 2 0 AF”) 0 2 AE(2) 0 0 -
Addlng OHF—HF( , ), 0HF_HF( , ), and OHF—HF( , )g1ves the
total second-order correction to the interaction energy of two colinear HF molecules,
44.5 a.u.
AEH) = — . (327)
OHF—HF (RHF—HF)6

5.1.3 The Second-Order Correction to the Interaction Energy of Two
Parallel Molecules

At this point, we will derive an expression for the second-order correction to the

energy [SE/"((33

of two interacting molecules A and B whose internuclear axes are
parallel to each other and to the x axis of the 2:, y, z laboratory frame. This particular
arrangement of molecules A and B is shown in Fig. 13. In Fig. 13, x’, y’, and z’ are
the axes for the molecular frame (A and B have the same molecular frame), A1 and
A2 are the nuclei of molecule A, and 81 and B2 are the nuclei of molecule B. Also,
COM ,4 and COM 3 are the centers-of-mass of molecules A and B, respectively. We
have denoted the vector extending from the center of mass of A to the center of mass
of B by RAB.

Let us compare the geometry of A and B in Fig. 12 with the geometry of A and

B in Fig. 13. The internuclear axes of A and B are designated as the z’ axis of

133

the z’, y’, 2’ molecular frames. Whereas the :r’, y’, 2’ molecular and x, y, z laboratory
frames are the same in Fig. 12, these frames are not the same in Fig. 13. Rather, the
x’, y’, 2’ molecular frame in Fig. 13 is rotated 90° counterclockwise from its position
in Fig. 12. Consequently, we need to rotate the non-zero components of the relevent
properties of A and B for the geometry shown in Fig. 13 from the molecular frame
to the laboratory frame.

While deriving AEéi; for the colinear arrangement of A and B shown in Fig. 12
(see Sect. 5.1.1), we determined that the non-zero components of the dipole moments

and the static polarizabilities of A and B in the molecular frame are [1:10 , [1530 ,

A A A B B

2:232 yy) 221 arr, ayy’

01 01 Oz and 01:32 T.hese components have the same values 1n the

molecular frames of Fig. 13. We can rotate these components from the molecular

frame of Fig. 13 to the corresponding laboratory frame by interchanging the roles of

:1: and 2. Therefore, we have “214,0: #201 #280 — "MEO, afixr = 02:, 024,2, = 0133;)
(.153, = a2, and (132, = 02131 .

Recall Eq. (296) from Sect. 5.1.1. Because [.130 = #50 for the current geometry,
Eq. (296) reduces to

BO [1800! A
AEéi’. <0, 2) = — (g) T ”T. u. a... (328)
Similarly
2 A0 ”A00 B
AEHLRB (2’ O) = — (2) T1171? TILE #2: am (329)
In Sect. 5.1. 1, AE E033 (0, 0) is given by Eq. (305). Since we have (1:7,: C232 ,
01242 = (lg/m), 01m: 052;, and a2 = 051., for the current geometry, we can

also use Eq. (305) to write AEéiL (0, 0) for the geometry of A and B in Fig. 13.
When A and B are parallel to each other and to the :1: axis of the laboratory frame,
the total second-order correction to the interaction energy energy of A and B is

AE”) = AEOij, 2) +AEOi)B(2,0)+AE EH) (0, 0)

OAB 0A8

= _(;)TxxTxfoO/J;BBOQTA$- (2)TxxTrx/1£O%Aoafx

134

+00
5 A B
_ (47:) /dw[Tm Tm 02,2, (w) an, (2w)

Tyy Tyy 01A (W) QB (W)

yy yy
— Tu Tu a331, (2w) a331, (2w)] . (330)

5.1.4 The Second-Order Correction to the Interaction Energy of Two
Parallel HF molecules

Now, we will use the information in Sects. 5.1.1 - 5.1.3 to derive the second-order
correction to the energy of two parallel interacting HF molecules. If we assume that
A = B =HF in Eq. (330), the result is

 

 

2 2 2 2
AEéHL—HF : AEéJF-HF (0’ 2) + AEéI-BF-HF (2’ 0) + AEéH)F-HF (0’ O)
= _Txa: Tm: (#:IF)2 0gp
h +00
HF 2
_ (Z7?) fdw {Tu Tu [an (zw)]
2

_ Tyy Tyy [015? (21.0)]

_ T22 T22 [afle (W)]2} ° (331)
According to Eq. (295), Tm, Tyy and T22 for the colinear arrangement of A
and B are

—1
Tm: : T =
W (RAB)3
2
22 -——3, (332)
(RAB)
Using these equations with A = B =HF in Eq. (330), we have
2 2 2 2
AEéH)F—HF : AEéI-BF—HF (0’ 2) + AEéIBF—HF (2’ O) + AE(()H)F—HF (0’ 0)
1 HF 2 HF
: _ #27: am
(RHF—HF)6 ( )

 

— (Elf—i7?) (RHF1_HF)6_ZOdw [agp(w)]2

135

 

’ (In?) (Rnimfzdw [05? M2

 

h 1 +00
2
— — dw [ozHF (2111)] (333)
(7’) (RHF—HF) 22
-—00
where we have also distributed the integral over (.12. According to Eq. (317) in Sect.

w) aHF(zw), andafz F(zw) can

5. 1. 2, the frequency-dependent polarizabilities aHF ( , W

be approximated by

 

 

 

 

2
OzHF (2(1)) 2 7 (EIHF) Gigi
[(Efi; F2) + hsz]
2
aHF (2w) : (EHF) as)?
W [(Efigp)2 + hzwz]
aHF (W) : (EEDF)2O:IZF (334)
[(13552 + 1121.12] ’
From Eqs. (333) and (334), we have
AEOH)F— HF = AEOIBF— HF (0’ 2) + AEOHF- HF (2’ O) + AEéfB—F HF (0’ O)
1 HF 2 HF
: _ ”a: 0151'
(RHF—HF)6 6( )

 

 

(21%) (13;:)F((111;):HF)6)2_/;w[(EHF)21+h2w2]2
— (£7?) (13::)F% HF) i/wdw[(EHF) 21+ h2w2l2

_ (g) (1:11;:HF)6)2_/:dw[(EHF)1,302]? (335)

 

 

 

 

136

The integrals in Eq. (335) were evaluated in Sect. 5.1.2. According to Eq. (324) in
Sect. 5.1.2,

 

 

 

 

 

 

+00
1
/ d1.) 2 2 = ”:3 . (336)
-00 [(Efif) + M122] [P h
From Eqs. (336) and (335),
2 2 2 2
A3,)... = wt)... (02) +333),_,, (2,0) +AE6.’.-.. (0.0)
2 2
___ (Him) 33;; _ (E55) (05‘2")
(RHF-HF)6 4(RHF-HF)6
(E3333)? _ (E33335? (337,
4(RHF-—HF)6 (RHF—HF)6
Recall that in the calculation of AE3:)F_HF for two colinear HF molecules that

we took pi”: = 0.7043 a.u., Efif = 0.5896 a.u., 0gp = off = 5.22 a.u., and
off = 6.36 a.u. Because we have interchanged a: and z for the current geometry, we
have #:F = 0.7043 a.u., of: = agf = 5.22 a.u., and of: = 6.36 a.u.. Using these

values in Eq. (337), we obtain

 

(2) _ (2) (2) (2)
AEOHF—HF _ AEOHF—HF (0’ 2) + AEOHF—HF (2’ 0) + AEOHF-HF (0’ 0)
29.2 . .
= — a u 6. (338)
(RHF—HF)

5.1.5 The Second-Order Correction to the Interaction Energy of Two
Perpendicular Molecules

In Fig. 14, the internuclear axis of molecule A lies along the z axis of the 2:, y, z
laboratory frame, and the internuclear axis of molecule B lies along the x axis of the
1:,y, z laboratory frame, making these two molecules perpendicular to one another.
The nuclei of molecules A and B in Fig. 14 are labeled A1, A2 and BI, 32, respectively,
and the centers-of-mass of A and B are COM A and COM 3. Also, RAB denotes the

vector extending from COM A to COMB.

137

Let us call the molecular frame of molecule A in Fig. 14 the LL", y’, 2’ frame, and the
molecular frame of molecule B the :12”, y”, 2” frame. The internuclear axis of molecule
A lies along the z’ axis of its molecular frame, and the internuclear axis of molecule
B lies along the z” axis of its molecular frame. Because the internuclear axis of A
lies along 2 and z’, the molecular frame of A is the same as the laboratory frame.
Therefore, we do not have to rotate the non-zero components of the relevant properties
of A, and we can specify these properties in the laboratory frame. However, because
the internuclear axis of molecule B lies along 2” and x, we need to rotate the non-zero
components of the relevant properties of B. The :12”, y”, 2” frame is rotated 90° from
Ex ‘2 03%", and (15y 2’ ai = affix”. From
the properties of A and B and Eq. (293), we have

the 23,31, z frame. Thus #50 = p39, Oz

1
1319339 (0,2) = _ (5) Tm Tm (359)23fix, (339)
(2) 1 A0 2 B
AEOAB (2, 0) = — (‘2') T33 Tzz (Hz ) axtrxn, (340)
and
+00
(2) h A B
AEOAB(0,0) = — Tm TM (47?) [dwam(zw)azuzu(zw)
fi —+oo
— Tyy Tyy (21;) / d0) (lg/4y (2&2) (1513/11 (20.))
)1 _+00
_ TzZ Tzz (47f) /d¢uafz (3w) dag/3,4212). (341)

-00

Letting A = B =HF in Eqs. (339), (340), and (341) gives

2 1 HF, 2
AEéH)F—HF (0, 2) = — ('2’) TIIXI: T333 (#2,, 0) agF, (342)
1
AE3:)F_HF (2,0) = _ (5)71. Tu (#5502353 (343)

138

and

+00
2 h
AEéJF— up (010) = " Tara: Tan: (1;) /d“dafo (2w) 01:1”; (W)
-oo
h +00
— Tyy Tyy (4;) diuagF(w)afi,§~(zw)
h
— Tzz Tzz '—
(4.)
+00
x / dwagp (magi. (w). (344)

5.2 The 2'” -Order Correction to the Interaction Energy of
Three Molecules

In Sect. 5.1, we wrote the second-order correction AEéilA to the interaction energy
of molecules A and B (see Eq. 292) as

AEOAA— _ 3123340, 2) + AE” 8(2, 0) + AEO2)B (0,0). (345)

If we have three molecules, which we will denote A, B, and C, we can write the total
second-order correction to the energy of interaction AEéiLC of these molecules as
the sum of the second-order, two-body corrections due to the interactions of A and
B, A and C, and B and C,

AE(i:C= AEOAB + 435,336 + AE02) (345)

BC’

and an irreducible three-body energy) of second- order, AEOABZI' The second-order,
(2) is

two-body energy of interaction AEOA)B of A and B is given in Eq. (345), AE

given by Eq. (345) with B replaced by C, EOACS
Ang’fC: 435,33 (0,2) + AEoi) (2, 0) + 4353; (0, 0) , (347)
and AESQC is given by Eq. (347) with A replaced by B,
AEOBcz Ang; (0, 2) + AE(B 2’ (2,0) + AEOBC (0, 0) . (343)

139

The irreducible three-body energy A5133:

is a sum of three contributions, AEéi’B) ,
AE(2 B3), and AE(:’ 3) ,where

131332 3 - “03,3 Ta7(RAB) #50 Tfi6(RAC) 1U? 0- (349)
The full nonadditive second-order correction to the energy AEéigé
2, 3)
nEgAAC = _ a5, Tannin) n50 Tamra) a?”
- 055 Ta'7(RAB) #30 T36(RBC) #600
- c223 Ta7(RAC) #30 T36(RBC) 141530. (350)

Therefore, using Eqs. (346), (345), (347), and (348), we obtain an equation for
AEOA)BC’

ABM)” = 1:530:13 (0:2)+AE(:)i:e(2 0)
+ EoA:B(030) :E02)C (0 2)
+ :E0::C (2»O)+ AEoiLw ,0)
+ EOE},(O,2)+AEO2B)C (2,0)
+ EOB)C(O,O) AEOABC. (351)
We will use Eq. (351) and the results of the derivation of AEOHC_ CC in Sect. 5.1.2

to derive and equation for AEOJAL HF_ HA.

5.2.1 The Second-Order Correction to the Interaction Energy of Three
HF molecules, Arranged Colinearly

Fig. 15 shows three colinear HF molecules which have their internuclear axes on the z’
axis of the :c’ , y’, 2’ molecular frame. Their internuclear axes are also aligned with the
:r, y, z‘laboratory frame. Note that the molecular and laboratory coordinate systems
are the same, as in Fig. 12. In Fig. 15, we have labeled the HF molecules HP 1, HF2,
and HF3. Also, H1, H2 and H3 denote the hydrogen atoms in HFI, HF2, and HF3;
and F1, F2 and F3 denote the fluorine atoms in HE, HF2, and HF3, respectively.
The HR, HFg, and HR», molecules have centers-of-mass COMHF“ COMHFZ, and
COMHFa, respectively. We have let RHFl-Hth RHFl—HF31 and RHFz—HF3 be
the vectors between HFl and HFg, HFI and HF3, and HF2 and HF3. If we let A z

140

HF1,B = H172, and C = HF3 in Eq. (351), we have

AEéi)p,_na_ar3 = AEéi’CCHFA(0,2)+AE(§:)F,_HF (20)
+ AEéffCCHFC (0,0) +AE33.)F,_HF (0’2)
+ Angij HF, (20) +AE3?F,_HF, (040)
+ AE32,)CC_,,,A(0,2)+4311?” (240)
+ AEéijC an. (0,0) +AE3§:3_HF,-HF,

_ (2) (2) (2)
_ AEOHFl—HFQ + AEOHFl—HFg + AEOHF2—HF3
(2.3)
+ AEonF,_na_ur3- (352)

 

 

 

 

Recall Eq. (327) from Sect. 5.1.2, which gives AEé:)F_HF for two colinear HF
molecules
44.5 a.u.
AE<2) = . 353
OHF—HF (RHF—HF)6 ( )
Using Eqs. (352) and (353), we have
(2) 44.5 a.u. 44.5 a.u. 44.5 a.u.
AEOHFl-HFg—HF}; : — 6 — 6 — 6
(RHFi—HF2) (RHFr—HFe) (RHF2—HF3)
(2.3)
+ AEOHp1—HF2_HF3' (354)
From Eq. (350), the irreducible three-body energy for this geometry is
2,3 _
AEéAB; = _ a; Tzz(RAB) #23 Tzz (RAC) “S
'— CéZTLAJKAB)MfQQAIiBC)HS
— OS; T22(RAC) l1? Tzz(RBC) ”28- (355)
and, letting A =HF1, B =HF2, and C = HF3 in Eq. (355), we have
2,3
AE(() ) = — Gin TZZ(RHF1-HF2) [A513 TZZ(RHF1—HF3) [1st

HFl—HFg—HF3

_ 0.52% T22 (RHF1_HF2) ”EB T22 (RHFz-HFs) ”5”};
- 0‘th Tzz(RHF1—HF3) Him]
>< T22(RHF2-HF3) ”gm, (356)

141

5.2.2 The Second-Order Correction to the Interaction Energy of Three
Parallel HF molecules

Fig. 16 shows three parallel HF molecules which have their internuclear axes aligned
with the z’ axis of the :r’,y’, z’ molecular frame, and with the :1: axis of the any, z
laboratory frame. In Fig. 16, we have labeled the HF molecules HFl, HFg, and HF 3.
Also, H1, H2 and H3 denote the hydrogen atoms in HFI, HF2, and HF3; and F1, F 2 and
F3 denote the fluorine atoms in HF], HFg, and HF3, respectively. The HFI, HFg, and
HF3 molecules have centers-of-mass COM Hp” COMHF2, and COMHFC, respectively.

We have let RHFl—HFQ) RHFl—HFg , and RHFQ—HFg be the vectors between
HF} and HF2, HFI and HF3, and HF2 and HF3.

The relationship between the molecular and laboratory frames in Fig. 16 is iden-
tical to the relationship between these two coordinate systems in Fig. 13. Therefore,

we can use the equation for AE(2)

2 OHF—HF
Sect. 5.1.4 to write AEéA)C_,,C

Eq. (352) in sect. 5.2.1, AE”)

OHFl—HFQ—HF3

derived for two parallel HF molecules in
for each pair of molecules in Fig. 16. According to

is

AEéi).,_n.~,-nr, = AES:)F,-HF, (0, 2) + ”5%“, (2, 0)
+ AEéi)FA_HFA (0. 0) + AES:’.,-..~, (0, 2)
+ AE3:)C1_,,FC (2. 0) + AEéi)n,_nr, (0’ 0)
+ Milan... (0. 2) + 4E8; (10)
+ 4193122-“, (0. 0) + AEéi’if_....-nn,
= AEéi)Cl_HCA + AEéi).~,_nn, + AE3:)r,_nn,
+ AEéi’if-nr,_n.,- (357)

Recall Eq. (338) from Sect. 5.1.4, which gives AEé:)F_HF for two HF molecules

that are parallel to each other and to the x axis,
2) _ 29.2 a.u.

HF—HF — (RHF—HF)6.

 

4E5 (358)

142

Using Eqs. (357) and (358), we have

 

 

 

AE(2) _ 29.2 a.u. 29.2 a.u. 29.2 a.u.
0 — — --— ....
HFl- HF2- HF3 (RHFI— HF2)6 (RHFl—HF3)6 (RHF2—HF3)6
(2,3)
+ AEOHF1——HF2—HF°3 (359)
From Eq. (350), the irreducible three-body energy for this geometry is
2, 3
AEéABL— — _ O‘A' WEAR/(B) RAC)#C

Tm(
— Q§JIT12(RAB)3(RBC)H7I
— arz’ Tum/10) #2 M(RBC)B - (350)

and, letting A =HF1, B =HF2, and C =HF3 in Eq. (360), we have

__ HFI HF] HF3
1315613,- —HF2-HF3 — — 012,2, Txx(RHF1-HF2)#ZI Txa:(RHF1-HF3)MZI

- eff? sz(RHFr-HF2) 14521 Ramadan) #5172
— 0553 Tm(RHFr—HF3) #52
X Trr(RHF2—HF3)#:iF2. (361)

5.3 The Third-Order Correction to the Energy of Interaction
of A and B

According to Eq. (290) 1n Chap. 4, the third- order correction ABC? Bto the energy

of two interacting molecules A and B 1S

3 A
AESAL = — T05 T57 7161) #040 (15000.3 Aa'fi?
1
- gTea T75 Tee #50 #3 0145?” 3“...

1
_. -6—TO,A3 T76 71¢ ”00/170”er 36,30

_ T03T75Teaugo(zh- 7K)Zodwaw(w) fiCm(— w;zw,0)

h
_ T03 T64) T75 ”:10 (4;) [00 dw 0323, (w) @1354) (—zw; 2w, 0)
—00

143

— T03 T76 Te¢(%’ —:-)/oodw fmdw' 7C,C( (zw;zwI—w,-zw’)

X

[bfw (w; w — 2w, —zw)

+ 123,, (w; —zw' — w, 7412’)

+ bggd, (—ud; w, + w, —zw')

+

The individual terms in AE(3 )

AEB)

0A8

AE‘2’)

0A8

E02113

3
4E6).

AE(3)

0A8

AE(3)

0A8

[#53346 (—w; w — 741),, 74.12)] .

are

0.48
(1 1) Z _ 01/3715”) TE¢H£OH§§30 066055
1
(37 0) = —6 0113 T75 Tm) [1’20 “gm/1:10 186,133)
1
(0 3) = -6Tad T76 71314501430145” 33.
h
(0,1) = J11,3 71“,, TC, ago (47?) foo dang, (2w
-oo
x C3,, (—zw;2w,0)
h +00
(1’0) 2 “7111371357176”? (a) [@0127 (W

x 5535C» (—zw;zw, 0)

(0,0) = -TafiTyéTcd)(% —:)/oodeOdw'

A . I I
x me (2w,zw —- 2w, —sz0)
x [bfw (2w; 3w, — 2w, —zw')
+ 5%,, (M; —quI — 2w, 7.5),)

+ (253C, (—zw; w, + w, —zw’)

144

(362)

+ (253$ (—zw; 2w — w], 3111)] . (363)

We can use Eq. (362) or (363) to derive expressions for AEéi; as a function of

RAB for various geometries of A and B.

5.3.1 The Third-Order Correction to the Interaction Energy of Two Co-
linear Molecules

Let us derive an expression for AEéiZC as a function of R A B for the colinear geometry
of A and B in Fig. 12. To begin, let us derive an expression for the AEéiL (1, 1)

term in Eq. (363) as a function of RAB. Recall from the derivation of AEéi:
a function of RAB for this geometry that the molecular and laboratory axes are the
same, so there is no need to rotate the non-zero components of the properties of A and

B before deriving A1333; in terms of the laboratory coordinates. While deriving

AEéi; for this geometry, we also determined that Tm = Tyy 75 O, Tzz 75 0,
and T03 = 0 when a 31$ 3. In the colinear configuration of A and B shown in Fig.
12, pfio = 11on = 0 and pf” = pf” = 0. Therefore,

AEéi; (I, 1) = — Tzz me T22 #1240 ”280 Q2435 afz
— T... Ty, T... a?“ u?“ at; 0:52.
_ T22 Tzz Tzz #240 ӣ30 05:12 0282' (364)
Finally, since 0405 = 0 if a 75 ,3,
AEEEB (1,1) = —TZ,z Tu Ta 11:10 #230 or“; 05,. (365)
Recall Eq. (295) from Sect. 5.1.1,
—1
Tm: : T = _—
2
AB

Using Eq. (366) in Eq. (365) gives

 

A0 BO A B
(3) _ 8/4.. 14. area...
AEOAB (1,1) _ (RAB)9 . (367)

145

Now, let us derive an equation for AEéi; (0,3) for this geometry. According to
Eq. (363), AEéi’B (0,3) is
3 1
ABS). (0, 3) = 7311;) :13, T.) 35077530735?“ 34:... (368)

Since Tm = TW 79 0, T2; 74 0, and T03 = 0 when a # ,8, and ,ufo = #50 = 0,

1
AEéiL (0, 3) = 77;. T2. T... u§°u§°uf° 2‘; (369)
Using Eq. (366) in Eq. (369) gives

3
AE(3) 4015.30) fizz.
3(RAB)9

0.48

(0,3) = —

 

(370)

Similarly, according to Eq. (363), AEéi; (3,0) for A and B as arranged in Fig.
12 is

 

AEéiL(3,0) = —% asTslaTeiuéoufi°uf°fi£as (371)
and
AEéiL<30> = —%T..T..T.. (324°? .3 (372)
Using Eq. (366) in Eq. (372) gives
Mg; (3,0) = —4(”?0)3 :3 (373)
3(RAB)

At this point, we are ready to calculate AEéi; (0, 1) and AEéi; (1, O) for A
and B as arranged in Fig. 12. According to Eq. (363) in Sect. 5.3, AE(3) (0, 1)

0A8
15

+00
’5.
AEéiL (0,1) = —Tag T76 Tecb #50 (21;) / dw fig” (—zw; zw, 0)

X c7533 (2w) . (374)

146

Since a) T,m 2 TM 75 0, Tu 75 O, and T03 = 0 when a # B, b) pf,” = )1on = O and c)
070,3 = 0 when a 31$ B (for HF),

+00

AEéiL(0,1)= — T..T..T..u§°(4%) fdw 3...(— w w 0)
>< aim) “0°
_ T T Tzuz(4h 7K)Zodwfi,yy( —zwzw0)
x aijw)

+00
h
_ Tzz T22 T22 #50 (a) f d“) fiiz (_W; W) 0)

x (if, (w). (375)

In order to simplify Eq. (375), we need to determine which elements of the hy-
perpolarizability B22117 are nonzero. For a Coo” molecule with its axis in the a:
direction by reflection symmetry in the x2 and yz planes components of the [3 hy-
perpolarizability tensor that are odd in either subscript a: or subscript y vanish. By
rotational symmetry around the z axis, the non-zero components of the hyperpolar-

izability ,Bfm of molecule A (with C00 00,, symmetry) are Bxxz— "" '—

2:23: £112: _

551,2- - yAzy— — 22,3], and 324”. If we assume that molecule B also has Coo” sym-

metry, then the non-zero components of the hyperpolarizability of molecule B are

B _ B _ B _ B B
(811)832—— szx — 2.72.7) — yyz — yzy — zyy’ and 222' NOte that the non-

zero matrix elements of the static and frequency-dependent hyperpolarizabilities of

molecules A and B will be the same.

When we inspect Eq. (375), we see that it contains only non-zero matrix elements
of the hyperpolarizability of molecule A. Therefore, we cannot eliminate any more
terms from this equation for AEéiL (O, 1) . However, we can simplify the expres-

sion for A518; (0, 1) by simplifying the expressions for the frequency-dependent

hyperpolarizabilities contained in Eq. (375). Eq. (215) in Chap. 4 gives the static
hyperpolarizability H705 of molecule A. From Eq. (215) with ’7 and 6 interchanged,

147

we have

(33,: Z

1...)..70 (E,£E?— Eéf) (E :3 ES?)
x [(0117). IkA><kAmnzA><zAm710A>
(0.4163(3))(3.116311361163103
(OAIIIQ lkA><kAlfifllA> (lAlflAIOA)
(0.416.Alk.4>(k.4|fiflz.i)(1.4162(0))

)< > >
> )

 

_:_A
<0A|#7|kA kAlfls llA (lAlfialoA
;A
(01163373377311 (1.416? (0.4 J. (376)

As discussed in Sect. 5.1.2, we can use the Unséild approximation to write

1 1
(0) (0) g '7" (377)
EkA — EOA 1P

+++++

 

and

31(3) i E53) 2 3% (378)
When we use Eqs. (377) and (378) in Eq. (376), the result is

5.0.7 A) 2:: [< 0AM. lkA> (kAlflallAWAlH7 |0A>

(Erlp)2 1s,, 1,7730
(OAlltAlkAXkAl/te |1A><1AII17IOA>
(oilfiémmmflzix A163 (0.4)
(OAIfiAIk/i)(kAlfifllAXlAl/la IDA)
(0.4172333)(kAlfifizA><zAmAIoA)
(0.1)),(3A)<kA|71-ZIIA><IA|6;A (0.7)] . (379)

 

 

IIZ

+++++

Let us consider the sum over 16,; and IA of (OAlfi,E IkA) (kAlfijllA) (lAl/l7 IDA)-
We can write this sum as

A ;A A
Z (OAlflflkAXk/ilflallA><lA|M74|0A) =
kA,zA7eo

148

2:3

[$1790 1A

     

     

“(CAI/La |1A><1AIH7 IDA)

—(oAI6A (3.1331162 IoA)<oAI6A (0.7)] (330)

Because 2 IlA)(lA| = 1 , we can write Eq. (380) as

1A

+
X

Z (OAlfiAIkAXk/iluc,|1A><lAlfi7IDA)=
kAJA5£0

A LA A
Z [(OAIMEAIkAHkAlHaflAIOA)

kAaéO
(OAmA (aw/3.4162103(OAI6AIOA)]

Z [<0AI6AI3A><3AI6..6A IDA)

kA

(OAIMAIOAXOAIHQA 7|0A>

(OM/12A lkA)<kA|'/13|0A)(0A|fi7 |0A>

(OAlfiéA |0A)<0AI#QIOA>
>1

(OAlflAIOA (381)

Similarly, since 2 IkA)( [CA] 2 1,

[CA

+

X

Z (CAI/12A lkA><kA|fia|lA><lA|fi7 |0A>=
1s,, 1,4760

(OAlfle'flafl AIDA)

(0.1m? IDA)<OAI6;‘6AIOA>

(0.116621036366071)

((OAIIICIOA)

(0 (6210A)<0AI6A10A)) (332)

Recall from Sect. 4.1 that $0 = [1:3 — [1230. When we use this equation in Eq.

(382), the result is

A ;A A
Z (OAlfléAlkAXkAlua|1A><ZA|11A|0A> =
kAJAsaéO

149

(OM. #3617 A|06>
- (OM/12A |0A><0A|fiafiA|0A>

- (OAIAAflaloA)<0A|/1A|0A>
OAlfiA |0A><0A|HA|0A>(0A|#A |0A>
OAlflAIOAMA 0(0A|0A>(0A|/1A|0A>

(
<
= (0A Ail/AIDA)
<
<

     

0A llu’e loj><0AlflauA|0A>
OAII-le fiQIOA><OAIIJAA7 |0A>

A A A A A A
+ #6063067“ 13.06.7067”

= (OAlfléA'flafiAIOA>
— (OAIAA |0A><0AlflafiA|0A>
— (0AI6A62I06)<06I6A10A). (333)

Now, consider the sum over [CA and [A of (CAI/lg IkA) (kAlfifIIA) (lAlpA IOA) in
Eq. (379). We can use Eq. (383) with 6 and a interchanged to write this quantity

as
Z (OAlfiA

kAJAaéO
<06I6A6f6AloA)
— (oAI6AloA)<oAI6f6A(oA)
" (OAluaflsloAHOAluAIO/i) (384)

Again, using Eq. (383) with 6 replaced by a, (1 replaced by ’7, and ’7 replaced

     

_"_A A
Wail/3. llA><lA|uAIOA> =

by61n the sum over 6,, and 1A of (oAmglkA) (M6364) (lAlfiAlOA) gives
2 (OAlflalkA)<kA|#7 |lA><lAIHA |0A>— —
kAJAaéO
(OAlflAfiAfiA|0A>
- (oAI6AIoA><oAI6A6A(oA)
— (OAIfififi’AIOAHOAIfiAIOA) (335)

150

Using Eq. (383) with ’y and a interchanged in the sum over (CA and l A of
(OAIAA lkA) (kAlA7llA> (lAlAAl0A> yields

2 <0AI624I6A><6AI636A>«Ammo/1 =
1971,7720
(OAIAAA7 AAIOA)
'— (OAIAA |0A>(0A|AA;1§|OA)
- <0A|fifAAI0A><0AI6£I0A> (386)

We can also use Eq. (383) with 6 replaced by ’7, 7 replaced by a, and a
replaced by 6 to write

2 (OAIAAII‘CA)(kAlAAllA>(lA|AA|0A> =
6,4,1A9eo
<0A|A7 AA Aa AIDA)
- (OAIAAIOA><0A|AAA AIDA)
— (OAIAAAA IOAXOAIAAIOA) (387)

Finally, interchanging ’7 and 6 in Eq. (383) allows us to write the last product
of matrix elements (summed over all k A, l A 3A 0) in Eq. (379) as

Z <0Al6r,‘ 16A><6AI626A><1A6£ 61,4):
16,1, lAaéO
(OAIAAAAAAIOA)
- (OAIAAIOAXOAIAAAAWA)
— <0A|A7 AAIOAXOAIAA IDA) (388)

Therefore, we can use Eqs. (379), (383), (384), (385), (386), (387), and (388) to
write

 

LA A
><0Alfla MAMA)

     

A ;.AA A
m — A [<0AIA5AQAAIOA>-<OAIAA
(EIIP)2
ALA ,. A 4AA
- (OAIAE AAa|0A><0A|AA|0A> + (OAIAaAe AAA|0A>
(OAIAAIOAXOAIAAA A|0A>—<0A|AaAe AIOAXOAIAAKLA)

151

fl(Aory g

<0A|M0M7MA|0A>— (DAIMQIOAXOAIMZMA AIDA)
(OAlMaM7 |0A>(0A|MA |0A>+ +<0A|MA M7 AMAAIOA)
<0AIAA10A><0A<A£AA10A> — <0A|AA A3|0A><0AIAA10A>
<0A|A7AAAAI0A>—AA10A><0AIAAAA<0A>
<0A IAAAA“ |0A><0AIAA10A>+ +<0AIAAA2AAI0A>

<0AIA310A><0AIA2AAI0A>

       

 

     

<0 AIAAA2<0A><0A|AA I0A>] . <380)
Since the matrix elements in Eq. (389) are real, we can write Eq. (389) as
2 A ;A A A ;A A
(E? PA) [<0AIAAAAAA > -— <0AIAAI0A><0A<AAAA10A>

(GA IMfMa|0A><0A|M7|0A>+ +<0A|MaMfM74|0Al
<0AIAAI0A><0AIA3A710—A> <0AIAAAAAI0A><0AIA7 I0A>

< A3AAI0A>— <0AIAAI0A><0AIA7AAI0A>
<0AIAAA7I0A><0A|AA |0A>] (300)

         

Solving for the matrix elements in (390) gives

W) «A

 

2

(E

[<0AIAA AAAA10A>— <0AIAA I0A><0A<AAAA |0A>

<0A|MA Ma A|0A><0A|M7 |0A>+ +<0A|M§MA AM71|0A>
(OAlfifiloAXOAlfi-efi 7|0A>- (OAlMaMA|0A><0A|M7|0A>
<0A|M§M7MA|0A>- (OAIMA|0A>(0A|M7MA|0A>
<0AIAAA710A><0AIAA AA] (391)

_'~_A A
Expansion of the matrix elements in Eq. (391) using pa = [1:3 — Mg (and similarly

;A _A_A
for [LAY and [JG ) followed by comparison of terms shows that

<0AIMA MaM7 AIDA)

152

—<0AIAA I0A><0AIAAAAI0A><0AIAAA£I0A><0AIAAI0A>
= (OAIMAAMAM A—IOAl (OAlMaloAXOAIMAMAIOA)
'(0AIMAAMA IOA)<0A|M7|0A)
= <0A|M0M7MA IOA)— (DAIMAAIOAXOAIM7MA |0A>
— (OAlMaM7l0A)<0A|MA|0A) (392)

and therefore

M53703 Apz)
6

 

A ;AA
: [<0Al/J'? _IIQI’LAIOA)—<0AIH:1|OA><OA|/«LOM:?IOA>
- <0A|MA AMA|0A><OA|AA|0A)] . (393)

We can use Eq. (393) to rewrite the frequency-dependent hyperpolarizabilities of A

and B. We can use Eq. (248) with 0 replaced by 6, ’7 replaced by a, and 6
replaced by ’7 to write the frequency-dependent polarizability 5A 0:67 (- w; w ,0)
of molecule A as

_A_, Z (OAIMA IJAlUA|M7 ICIAXCIAIMAAIOA)
_ 0 0 0)
AAAAaeo (Eh ) — E3A) — WA) (EqA - E3A — WU)

(AOAIMA IJA><JA|MA|qA>(qAIM7 IDA)
E(.0— -E0A— zhw) (Eéol— E3 ‘33)

A
607

 

(—zw; 2w, 0)

 

<0A|M7 lJA) (JAlMa MA) (quMA MA)
E(.°— —E03) (EA? —E(O) + AAA)
Em

 

(OAlMA IJA><JA|M7 lqA)(qA|MA IDA)
’— E30l+ AAA) (EA fil—Eofl +AAAA)

 

<0A|M7 IJA><JA|MA lqA><qA|MA IDA)

AAA-AW WE Aw)

(OAIMM IJA)<JA|MA|CIA)<CIA|M7 WA)
0 o o 0
(E(. l — E( ’+ Am) (EA; -— E32)

 

(
(
(EA
(E9

 

(394)

3.4

153

In one version of the Unsold approximation,

0 0 0
EAA) — EA ’ = Egg) —— EAA’ = Efp, (395)

A

so that Eq. (394) is

(DAIMS1 IJA><JA|M7|9A><9A|MA IDA)
(EA — AA) (BA - AAA)

 

MAAA7(- MMO) ’-—‘-’ Z

JAqA¢0
(OAIMA IJA><JA|MA lqA>(qA|M7 i0A>
(EMP— zhw) EMF
<0A|M7 |JA><JA|MAl9A><qA|MA |0A>
EMF (EMF + zfiw)

(OAIMA IJA>(JA|M7 IqA>(qA|MA iOA>
(EAP + zhw) (EA [P + zhw)
(DAIMA IJA><JAIMA |9A>(qA|MA IDA)
Ej‘ p(EAp- 2AA)

(OAIMA IJA><JA|MA lqA><qA|M7 |OA>
(E1 IP + zhw) EMF

 

 

 

 

 

+ (396)

Using Eqs. (383) - (388) with [6,4 = jA and 1A = qA in Eq. (396) gives

A
507

(OAIMA M7 AMAIOA>— (OAIMA IOA)<0A|M7 AMA|0A>
(EJAP — ZMW) (BAP - zMW)
(OAIMAM 7|0A>(0A|MA|0A)

(EA —mA> (EA AA)

A A
(DAIMAAMAMAIOAl— (OAlMA IOAHOAIMAMAIOA)

(EIP zhw) EIP

(OAlMA MA|0A><0A|M7 |0A)
(EMPA — zhw) EMF

<0A|M7 MAMA A—IOA> <0A|M7 IDAHOAIMAMA IDA)
EMF (EIP + m“)

(- -zw;zw,0)

 

 

 

 

 

154

<0A|M7 MQIOAXOAIMEA IDA)
EEAP (EMF + 2h“)

ALAA ;AA
(OAlfiaM7MAIOA>- (DAIMCA |0A><0A|M7MA|0A>
(EAP +zhw) (E;4 ,P + zhw)

(OAIMQM AIOAXOAIMEA IDA)
(Ea) + W) (Ea) + m»)
A ;AA
(DAIMAME ”filo/1)"(OAIIA‘yIOAXOAI—fleA/AIAIOA)
EMP(E1P - 25w)
<0A|M7Me AIOAXOAIMA |0A>
E1 P(E;AP— 2h“)
(OAlMaMe AMA|0A)— (OAIMQ |0A><0A|Me AMAIOA)
(EAP + 2h”) EIP

(OAIMQME |0A><0A|M7 IDA)
(EAP + zhw) EIP

 

 

 

 

 

 

 

(397)

 

Since the matrix elements in Eq. (397) are real, we obtain

“2

A

son

A .LAA A ;AA
[<0AIMQM7 MAID/1) - <0A|M2|0A><0A|M7 MAID/1)
<0AI/22r1-13 |0A><0A|M2A Ion]

("w; w: 0)

 

 

A 1 1

(E1 IP + :fiw) (EiAP + 2h“) + (BAP _ zfiw) (EiAP " 27%)]
L<0A|Me MaM MA—|0A> (OAlMe |0A><0A|MaM 7|0A>
(OAIMe Ma |0A> <0A|M7A|0A>]

1 1
L(EAP “ m”) EfAP + EiAP (EMF + 2h”)

 

 

 

 

 

(OAlMAMe MAID/1)
A _2.A A
<0AméloA><oAIm u¢I0A>— (OM/13m low/alumna]

155

1 1
+
EiAP (EiAP - 2h“) (E3AI) + 2h“) EiAP
Then, from Eqs. (392) and (398),
N A LA A A _c.A A
2:7 (w; m0) = [<0AIuA uaufilon — <0Alqu0A><0A|uau¢I0A>

A 2A A
— <0A|ufiualoA><oAIufiloA>]

 

 

x . (398)

 

 

 

 

 

 

 

2 2
x +
EAP (BAP + 2m") EiAP (EIAP " Z’5‘")
1 1
+ . (399)
EfP -— zfiw) (EfP — 2%)]

+
A A

(EIP + zht") (EIP + 2’7‘") (
Using Eq. (393) in Eq. (399) and combining the frequency-dependent denominators
in Eq. (399),

A

can

MA)“ + <E¢p>2h2w2]
3(Ef‘P + 2m)2(E;1P — zruu)2 '

 

[BA (_w; 2w, 0) g

60:7

 

(400)

Let us return to the calculation of AEéiL (0, 1) . We will use Eq. (400) with
6=z,a=a: and ’y=:1:;with e=z,a=y and ’y=y;andwith

e=a=7=zinEq. (375)

h
AEéiL (0: 1) g _Tm Tm T22 ”:30 (—)

277
yaw/2A [3<EAP>“ + (Emwwfi

X P
3(EAP + zhw)A(Ef‘P ‘ zh’w)2

 

an; (W)

—00

h
_ T22 Tzz Tzz ”:30 (Z7?)

+00 A A 4 A 2 2 2

fizu 3 E + E hw

X /d‘*’ [(1,3) 2( ”3) Jam). (401)
3(Ef‘P+zhw) (EfP—zhw)

Then, using Eq. (317) from Sect. 5.1.2 (with HF replaced by B) in Eq. (401), we
find

 

“‘00

AB“) (0,1) g —TmeTzzpB°(£)

0’43 z 27r

156

ll?

HZ

£13.73

7:14»); 2” [3(E1P)A + (EAPMEAWAJ (EMP)AOtB
3(EAP + z7i‘*’)A(EiAp-Z MA) [(Ef3P)2 + hsz]

—00

T22 T22 TZZ/l’ 2804(h7r

(LAW (E1?) (EA?) ’12 2] (EP) 2
3(Ef‘P + 2%)2 (EfP — zhw)2 [(EIBP)A +h2w2]

"'00

h
—Tm TxxTzz M? 0(a) E214 (EIP) )A(EIBP)AO 0:ch
A

70dw [3E( EIPA'l’hAw )2]

)
_OO [( (E)f1P +h2w2 J2 [(EP )A+h2w2]

Tzz Tzz T22 MzB 0(—) 5A2(EIP)A (EIBP) 20::

[3(EIP)A + EA” 2]
/ dw 2
-00 [(EAP) 23112102] [( (Ego) +4thw2]

 

-sz Tm Tz zz AAA)AAA_(A)1AAA(EIP) 4(EIBP)Aaxa:

+00

 

/ A
A [(Efp) +h2w2] [(EFP) +5232]
3
Txx Txx T22 #280 (SA—71:) 32AM (Efp)2 (EIBP)2afx

+00

 

/ AAA 2 AAA 2
[(Efp) +W] [(EFP) +11%?)

—00

h
TA TA TA AP (4—) EAEAWM;

157

/ A“ i 2
_00 [(EA) )2+h23w2] [(EIBP) +h2w2]

_ Tzz Tzz Tzz ”:3 01h2( _) 18f22(EIP)2(EEP)aBazz

 

(2)2

/ dw 2 2 2
-00 [(Efp) +h2w2] [(EPP) +71%?)

 

. (402)

In order to simplify Eq. (402), we need to evaluate the integrals contained within
this equation. From the Mathematica software package,

 

+001 2
hid“) [( (Efp) +5222]? ((13,310? + W]
2h{[—3(E1P)2EFP + (E50)?
x (E)

Z
+ [3(Ei4P)2EIBP ’ (EIBP)3] 10g (E—fi;)

 

+ 2(EiqP)3

x (om—:3)— (E:— )l}

x 4(E2AP)3EPP[<EAP)2 — (1253))?

= D(E34PaEIBP)a (403)

where we have designated the result in Eq. (403) D (Efp, EIBP). At this point, we
will simplify the other integrals in Eq. (402), that is, those that have w2 in their
numerators. According to Gradshteyn and Ryzhik,67

332 ’IT

O/d$(:c2+a2)(a:2+b2)(2:2+c2) :2(a+b)(a+c)(b+c)' (404)

 

 

158

If we let :1: = fiw, d2: = hdw, a = b = Efp, and C = EFF in Eq. (404), then
we have

+oo

 

 

2
(m3 / dw 2“; =
0 [AAA + (EA) ] [AAA + <EA)2]
A A" B 2. (405)
4E1P(EIP + EIP)

The integrand in Eq. (405) is integrable from —oo to +00, and it is an even function.
Therefore,

 

WW <A> >2) [Am (E27

AZA “2 =
[h2w2+ +(EAP )2] [h2w2+ (EBP2)]

7T

 

 

A A B 2. (406)
EIP (EIP + EIP)

When we use Eqs. (403) and (406) in Eq. (402), the result is

h
AE(3) (0,1) g —Ta:x Txa: T22 ”2:30 ('27.?) 213EE4EPP205$D (EIAPa EIBP)

0A3

Txl‘ TIL‘IE T2 22 #223 O/Bzfgcxa ngI/‘P EIBP

12(E1p +EzBp)
— T22 T22 Tzz ”:30 (4h 71') fizzzE‘I AP:EIBP2azB;D (EEP’ EFF)

_ ngTzz Tull/£30 $201 aszIPEIP (407)
24(E,AP +E,B P)? ’

 

Now that we have derived AEéi) 3(0’ 1) for the colinear geometry A and B, let
us derive AE(3) (1,0) for this geometry. We can obtain AEOA)B (1,0) by

0A8

interchanging B and A in Eq. (407),

N 3h
AEéi)3(110): _ TM TN? TZZ/A’fo (16) fmeIBPa 51134230 (EPPvEIP)

159

Tam: Tana: Tzzflfo Erma arrE?P(EIP)2
12(E}3,, + Ef‘ P2)
" T22 T22 T22 ”:10 (%’62) fzzElBPasz(E1P1E1P)

_ T22 TzszZ#zOIB§zzasz1:3P(EAP2) . (408)
12(E,BP +E, P)2

These results are equivalent to those obtained by Piecuch at the level of the Unsold

 

 

approximation.63‘66

We are now ready to derive the expression for AEéi; (0, 0) for the colinear
geometry of A and B. According to Eq. (363), AE(3) (0, 0) is

0A8

(3) ’72
AEOAB(0,0) = —TQ3T,3T,,, 472

lb . I I
x [00th joobAdw ’706 (w,zw—uu,—zw)

X [23,, (7:;zw — 2w, —w’)

+ b333,, (2w; —zw’ — zw, 2(1),)

+ b§3¢(— —;zw zw +zw, —zw)

+ b§3¢(— zw; zw— 2w WWI):| (409)
w

Recall from Eq. (284) that bA ( flu) — 2w, —zw') is

706

, , (OM/14‘ le><jAlflA lqA)
bAae _ 7.— : 0
7 (2w ;zw w W) ”1&0 (Elm _E(0) + zhw)

JA

 

(QAlfi’e IDA)
(E53) — EDA) + zfiw')

 

(410)

We can use the Unsold approximation to simplify Eq. (410). Horn Eq. (395) and
(410),

bA (no no), — w, —zw’) =

706

160

A _AA
(OAlufflafiAW/O— (DAM? loAiloAlflaflAloAi
(Efip +zfiw) (E1P+zhw)

(OAIIJJZ :l04>(0A|#A IOA>
(EAP +zfiw) (EAP +2hw)

 

 

(411)

Recall Eq. (393),

185/1017 (EfP) 2
6

 

= (OAlue'flaflAAO/ii- (OAl/lflOAHOAI‘fla/lflClA)

- (OAlfif/lal04><OA|/1§‘IOA) (412)
Using Eq. (412) with 7 and e interchanged in Eq. (411) gives

bA (7142;qu —zw, —zw’)

vac
'yae(EIP)2
6 (El/113+ zfiw) (Efp + 2111/)“

 

(413)

At this point, let us simplify the terms associated with molecule B in Eq. (409).
Recall from Sect. 4.5 that bqu) (w2;w1, -wa) is

(4%), (w2;w1, -wa) =

A .23 A
Z (OBlfliITBXTBIfla ISB><SBIII§|0>
,3 3310 (El? WE?) + 12412) (El? -— E35? + 71%)

 

(414)

If we let (422 = to), col = w, — 7,0.) , and 0.20 = w, in Eq. (414), the result is
b§3¢ (2w; zw’ — 2w, —zw’) =

Z (OBIILB lTB><TBlޤ l88><83|ޤ ICE)
1'3 33760 (Ema) —E(()O) + 277-14)) (E83) _ E32,) + zm')

 

(415)

In order to simplify Eq. (415), we need an expression for molecule B which is

analagous to Eq. (388) for molecule A, specifically

A 2.3 A A .;BA
2 (OBI/igerXTBIHgISB><SBIM§IOB> = (Oalufmuflma)
1'3,835£0

A ;B A
- (Oalfi?IOB><OBIfi§fi§IOB>- (OBlflgflgIOBHOBl/Igl03)o (416)

161

We also need to write the Unsold approximation to b for molecule B. Using Eq.

(416) and the Unsold approximation for molecule B in Eq. (415) gives

bgw (w; w, — 2w, —-zwl) 2
AB A
(OBI/15 H5H§l08>— «)8le IOB><OBngH§loB>

(EBP +zhw) (EBP +zhw)

(OBIH? H§IOB>(OBIH§ |03>
(EIBP +zhw)(EBP +zhw')

 

 

(417)

Thus the results are equivalent to those obtained by Piecuch, in the Unsold

approximation.63’66

162

6 Summary, Concluding Remarks, and Future Per-
spectives

The charge-density susceptibility x(r, I"; w) of a molecule is defined as the change
in the Lil—frequency component 5,06 (r, w) of the electronic charge density at a
point r within a molecule, due to a perturbing potential ’08 (r’, w) of frequency
w applied at r’ (within linear response), as given by Eq. (1). Several molecu-
lar properties, including dipole- and higher-order polarizabilities, dielectric functions,
and other properties mentioned in Chap. 1, depend on the charge-density suscepti-
bility. We have derived an ab initio expression for the charge-density susceptibility
x(r, r’;w) in CISD theory, developed an algorithm for calculating x(r, r’;w)
which is based on this expression, and written a program which is based on this al-
gorithm. Finally, we have used our program to calculate x(r, r’; w) of H2 as a
function of r, r’ , and w in the aug-cc-pVDZ basis set and at the equilibrium bond
length of the molecule. Since CISD is equivalent to full CI in a two-electron case, our
results are exact within the aug-cc-pVDZ basis.

The charge-density susceptibility can be calculated using the method described
here, or it can be calculated using several other methods mentioned in Chap. 1. The
advantage of the method presented in this work is that one can easily obtain the exact
value of x(r, r’; w) for a molecule (in a given basis) at any frequency w. However,
using one of the methods mentioned in Chap. 1, one can only obtain the exact value
of x(r, I"; w) for a given molecule at limited number of frequencies.

According to Eq. (20), the charge-density susceptibility x(r, r’;w) of H2 is
finite at all energies hw 75 (E K — E0) . This is demonstrated in Fig. 1, which
shows x(r,r’;w) of H2 as a function of y and z with :1: = 0, w = 0, and
I" = (0,0, 0). In this case, he = 0, and x(r, r’; w) is small at all r and r’ .

Eq. (20) suggests that x(r, r’; w) is singular at energies fiw = (E K - E0)

. However, this is the case only for specific excited states \I'K. For example, triplet

163

excited states ‘11 K do not cause singularities in x(r, r’; w) at ha) = (E K — E0)
. This is because \110 for H2 is a 129 state (recall the H2 has D00), symmetry), and
[16(1‘) (fie (r’)) is spin-independent, so that matrix elements (\Ilolfie (r) Mix)
and (\Il Kl fie (r) [\Ilo) (and the corresponding matrix elements of fie (r’)) are non-
zero only for singlet excited states \I’ K . This is demonstrated, for example, by
Fig. 2, which shows x(r, r’;w) of H2 as a function of y and z with :1: = 0,
hw ~ (E1 — E0) ,and r’ = (O, 0, 0) . Although he) ~ (E1 — E0), x(r,r’;w)
of H2 is small because W1 is a triplet state.

In addition to the fact that x(r, I"; w) vanishes for triplet states, x(r, r’; w)
also vanishes for many singlet states when r and r’ take specific values. We have
determined which particular types of singlet states contribute to x(r, I"; w) for the
H2 molecule. We did this by writing x(r, r’; w) as a sum of products of orbitals
evaluated at r and r’ , and determining which types of orbitals are nonzero when
evaluated at various r and r’ values. The states that contribute to x(r, r’; w)
for particular r and r’ values are determined by the orbitals which are nonzero at
those r and r’ . When r = (0, 0,0) or I" = (0, O, 0) ,only 0'9 -type orbitals
and WK states with 129 symmetry contribute to x(r, r’;w) . This is because
09 orbitals are the only molecular orbitals of H2 that are nonzero at r = (0, 0,0)
or at I" = (0,0, 0) . For example, this is demonstrated in Fig. 3, which shows
x(r,r’;w) of H2 as a function of y and 2 with a: = 0, I" = (0,0, 0), and
flu ~ (E4 — E0) . The charge-density susceptibility of H2 is singular here because
\114 is a 129 state; x(r, r’; (.0) has the shape of a 09 orbital because 09
orbitals are the only nonzero orbitals at r = (0, 0, O) or I" = (0, 0,0) . If r or
1" lies along the molecular axis, 09- and (Tu-type orbitals and excited states ‘11 K
with 129 and 12,, symmetries contribute to x(r, r’; w) for H2. This is because
both the 09 and 0,) molecular orbitals of H2 are nonzero along the molecular axis.
This is demonstrated by Fig. 5, which shows the charge- density susceptibility of H2
as a function of y and 2 with :L‘ = 0, M N (E3 — E0) , and r’ = (0, 0, 0.7)

164

. The charge-density susceptibility is singular because W3 is a 12,, state, and
x(r, r’; w) has the shape of a an orbital because both 09 and 0,, orbitals
contribute to x(r,r’;w) when r’ = (0, 0, 0.7) . Finally, if r or r’ is in the
:L‘Z—plane, only 71'2 and 5x2_y2 orbitals and 1H,, and 1A332.“ states contribute to
x(r,r’;w) for H2.

To test our x(r, r’; w) calculations, we have used x(r, r’; w) to calculate
the CISD polarizabilities an (w), ayy (w), and an (w) for H2 at w = 0
and at several other frequencies in the DZ, DZP, and aug—cc—pVDZ basis sets. We
compared our static polarizabilities with the corresponding polarizabilities obtained
in the finite field approximation using the MOLPRO17 quantum chemistry software
package. The static polarizabilities calculated with the two methods are given in
Table 2. There is excellent agreement between the polarizabilities calculated with
the two methods. Note that as with the x(r,r’;w) calculations, all polarizability
calculations were carried out at the equilibrium bond length of the H2 molecule.

According to Eq. (30) in Chap. 1, the polarizability 0105 (w) is continuous
when fiw 75 (E K — E0). Eq. (30) also indicates that aag (w) is singular when
ha) = (EK — E0) . However, this is not the case for all excited states \11 K .
Specifically, as with x(r, r’;w) , triplet states do not contribute to 01030.12).
This is because \Ilo for H2 is a 129 state and ya and HH are spin-independent
operators, so that matrix elements (‘I’OlHal‘I’K>, (WKIMQIWO), (‘I’OIHfiI‘I’K>,
and (\leluglqlx) are nonzero only for singlet states \I/ K of H2.

The frequency-dependent polarizability of H2 is also finite at a number of singlet-
state transition energies. We determined which types of singlet states cause Clog (w)
to be singular at (11.0 = (E K — E0) by determining which symmetries of the com-
ponents of the matrix elements (@JIfiOICPJI) (and (CI) Jul [1ng J”’>) in Eq. (37)
make the direct product of those symmetries contain the totally symmetric represen-
tation of the D00), point group. Since ((PJI is a determinant in (WOI, (‘IIOI must

have the same symmetry as ((PJI . Since IQJI) is in I‘IIK), Ill/K) must have

165

the same symmetry as |<I>Jr) . Therefore, we determined the allowable symmetries
of (WM and I‘I’K) from MM] and [(1) J!) . Using this analysis, we determined
that an (ad) would be singular for singlet \IJ K states with 2: symmetry. Figs.
6 and 7 demonstrate this behavior. In Figs. 6 and 7, an (w) is singular at two
frequencies in the DZ, DZP, and aug—cc—pVDZ basis sets. In each basis set, these
two frequencies correspond to the energies of transitions to the 112,]L and 212:
states of H2. We also determined that 01er (w) is singular for singlet ‘1’ K states
with Hut symmetry and that am, (w) is singular for singlet \I’ K states with
Huy symmetry. Fig. 8 demonstrates this behavior for 01” (w) . According to
Fig. 8, Ozmy (w) is singular at one frequency in the aug—cc-pVDZ basis set, and
this frequency corresponds to the energy of a transition to the 111—In: state. Fig. 10
demonstrates this behavior for ayy (w) . According to Fig. 10, am, (w) is singular
at one frequency in the aug-cc-pVDZ basis set, and this frequency corresponds to the
energy of a transition to the 111-[uy state.

The second major topic of this work is intermolecular interactions. We began
our study of intermolecular interactions in Chap. 2, where we summarized the main
results of several of the major theoretical approaches to intermolecular interactions.
In Chap. 3, we reviewed known results for the second-order perturbation correction
to the intermolecular interaction energy of two molecules A and B in the polarization
approximation. This approximation is valid when the overlap between the electron
distributions of the two molecules can be ignored. First, we showed that the second-
order intermolecular interaction energy AEézAL is given by Eq. (158), where the first
two terms in this equation give the induction energy due to the polarization of each
molecule by the field of the permanent dipole of the other. Note that these results are
valid within linear response and when we neglect effects caused by the non-uniformity
of the field. Second, we used complex contour integration to prove Eq. (159), which
is the Casimir-Polder formula with A = E39) — E83) and B = E)? — E62)

from reference 106. Using Eq. (179), we showed that the denominator in the third

166

term of Eq. (158) is given by Eq. (159). At this point, we used the expressions for
the frequency-dependent polarizabilities of A and B as given by Eqs. (186) and (187)
to show that the third term in Eq. (158) is given by Eq. (188). Eq. (188) is the
2nd-order dispersion energy of interaction of A and B.

In Chap. 4, we derive the third-order perturbation correction to the intermolecular

. . 3
interaction energy AE( )

0A8 of two molecules A and B in the polarization approxima-

tion. In this derivation, we started with AEéiL as given by Eq. (190), with VAB
and la/AB given by Eqs. (148) and (196), and with G given by Eqs. (144) - (147),
respectively. We show that the third-order intermolecular interaction energy AEéiL
is a sum of six terms which are given by Eqs. (214), (222), (230), (253), (271), and
(289). Pg. (214), which is first-order in both ,1er and [130, describes higher-order
induction effects. The permanent dipole on molecule A produces a field, and this
field polarizes molecule B. This produces a reaction field at B that acts back on A.
Specifically, the reaction field at B polarizes A, which then gives rise to a field acting
on B and this alters the energy of the AB pair due to the permanent dipole of B. The
energy of the AB pair is also affected by the higher-order induction effect produced
by the same mechanism but with the roles of A and B interchanged. Eqs. (230)
and (222), which are third-order in [M0 and third-order in 1130, respectively, describe
hyperpolarization effects. The permanent dipole on molecule A produces a field that
hyperpolarizes molecule B, and this effect causes an energy change in the AB pair
that is given by Eq. (230). Similarly, the permanent dipole on molecule B produces
a field that hyperpolarizes molecule A, and this effect causes an energy change in
the AB pair that is given by Eq. (222). Eqs. (271) and (253), which are first-order
in [4’40 and first-order in #80 , respectively, describe induction-dispersion effects.
These effects result from the modification of the AB dispersion energy due to the
static fields from the permanent dipoles of A and B. The static field from the perma-
nent dipole of A, together with the field from the fluctuating dipole of A, polarizes B

nonlinearly, changing its energy. In addition, the field from the permanent dipole of A

167

alters the correlations of the spontaneous charge-density fluctuations on B, and this
affects the correlation energy (and similarly, with the roles of A and B interchanged).

The induction, hyperpolarization, and induction-dispersion energies given by Eqs.
(214), (230), (222), (271), and (253) agree with the results of earlier work. However,
Eq. (289), which is zeroth-order in both “A0 and [430 and corresponds to the
dispersion energy of the AB pair, is a new result. This is a pure dispersion effect,
associated with correlations in the fluctuating charge densities of A and B beyond
linear response. Molecule B is hyperpolarized by the fluctuating field from A; also,
the fluctuating field from A alters the correlations of the charge-density fluctuations
in B. Both mechanisms contribute to the third-order dispersion energy.

In Chap. 5, we use our results for A1363; and A138; obtained in Chaps. 3 and
4, the definitions of Tag, HA0, [130,01’4 (2w ) and 05155:) )asgiven by Eqs. (149),
(297), (186), and (187) to derive approximations to AEéi) and AE(3)B for specific
geometries of A and B. Eq. (306) gives the second-order correction A1302)” to the
energy of the AB complex when A and B are colinear, with each of their molecular
axes oriented along the z-axis of the labOoratory frame shown in Fig. 12. This equation
gives AEéiL in terms of #20 , ”£30 , 01:12, 05;, 01;: (w), 6!sz (w), and RAB
. Using Eq. (306) and letting A and B be hydrogen fluoride (HF) molecules, we also
derived an expression for the second-order correction to the energy of two colinear HF
molecules with their molecular axes oriented along the Z-axis of the laboratory frame.
This expression is given by Eqs. (307), (308), and (309), where the total second-order
correction to the energy AEOIBF- HF of two HF molecules is the sum of these three

0 HF

equations. These equations are given in terms of pH F , an , (1sz (2w), and

RH F H F We simplified our expression for AE62)
0

HF_ HP by realizing that for this

geometry, we can replace [1H F with pH F0’ in our equations for AE02)

HF— HF

. By assuming that the matrix elements in the equation for the static polarizability

of HF are real and by using the Unsold approximation, we have derived an equation

HF(

which gives a w) in terms of the ionization potential E jHPF of HF, the static

168

HF
76

HF

76 (w) is

polarizability a of HF, and frequency w. This expression for a

given by Eq. (316). We also simplified AEOH)F— HF by using Eq. (316) in Eq.
. . (2)
(309), g1v1ng Eq. (318) for the AEOHF— HF (O, 0) component of AEOJR HF. We

further simplified AEOH)—F HF (O, 0) by evaluating the integral over frequencies on
in Eq. (318), giving Eq. (325) for AEotin HP (0, 0). Finally, using Eqs. (307)
and (308) with 11"” 0) =n<HF0>_ — 0. 7043 a u and am": 6.36 a.
u., we obtained Eq. (310) for both AEOH)—F HF (2,0) and AEm (0, 2).

0HF- HF

Also, using Eq. (325) with EIPF = 0.5896 a. u., aHF— — 6.36 a u., and

22

Griff: Gig/F = 5.22 a. u., we obtained Eq. (326). Multiplying Eq. (310) by
two and adding the result to Eq. (326), we obtained Eq. (327) for AEOJfl HF .
Since Eq. (327) is of the form 06(Hp_Hp)/(RHF_HF)6 . where 06(HF—HF)
is a constant, then our results indicate that Cg( HF- H p) = —44.5 a. u. for two
colinear HF molecules.

We have also derived an expression for AEéiL when the internuclear axes
of A and B are parallel to each other and to the :r-axis of the laboratory frame
as shown in Fig. 13. In this geometry, the molecular and laboratory frames are
not the same. Therefore, before obtaining a final expression for A1363; for this
geometry, we rotated the nonzero components of the relevant properties of A and B
for the parallel geometry shown in Fig. 13 from the molecular frame to the laboratory
frame. We obtained the final expression for A5163; in this geometry by replacing
the components in the AEéi; expression for the colinear arrangement of A and
B shown in Fig. 12 (Eq. (306)) with the appropriate rotated components. The final
expression for AEéiL for parallel A and B as shown in Fig. 13 is given by Eq.
(330).

We have used Eq. (330) with A = B =HF for AEéiL when A and B are
arranged as shown in Fig. 13 to obtain an expression for the second-order correction
AEéi; to the energy of interaction between two parallel HF molecules. This expres-

sion is given by Eq. (331). We obtain this expression by letting A = B =HF in Eq.

169

(330). We simplified Eq. (331) using essentially the same procedure that we used to
simplify Eqs. (307), (308), and (309) for A1363: of two colinear HF molecules (re-

call that the sum of Eqs. (307), (308), and (309) gives the total 2nd-order correction

(2)

to the energy of interaction AE of two colinear HF molecules). After simpli-

UHF—HF
fying, we obtained Eq. (337) for AEéi)F_HF of two parallel HF molecules. Letting
#5“? = 0.7043 a. u., QZF = erg/F = 5.22 a. u., and 0521: = 6.36 a. u. in

Eq. (337), we obtained the final expression for AEéi; of two parallel HF molecules,
which is given by Eq. (338). Eq. (338) is of the form C6(HF—HF)/(RHF—HF)6 ,
where 06(HF—HF) is aconstant. According to Eq. (338), 06(HF—HF) = —29.2
a. u. for two parallel HF molecules.

We have also derived an expression for AEéi; when A and B are perpendicular
to each other, as shown in Fig. 14. In this geometry, the laboratory and molecular
frames are the same for A, but they are not the same for B. Therefore, before
obtaining an expression for AEéi; for this particular geometry, we rotated the
nonzero components of the relevant properties of molecule B from its molecular frame
to the laboratory frame. After rotating these components, we obtained an expression
for AEéiL in this geometry by replacing the components of molecule B in the

colinear expression for AEéiL as given by Eqs. (292) - (293) with the appropriate

rotated components. The expressions for the AEéi; (O, 2), AESjRB (2, 0), and
AEéi; (0, 0) components of [3133333 for this geometry are given by Eqs. (342),

(343), and (344).
We have also derived an expression for the second-order correction to the energy

of interaction AEm

0430 of three molecules A, B, and C. This energy is a sum of

the second-order, two-body corrections AE(2) AESZ) and AEéZ: due to the

OAB’ AC’

interactions between A and B, A and C, and B and C, and an irreducible three~body

energy of second-order 13136322,.
2,2)

ABC.

We denote the sum of the second-order two—body
corrections AEé Each second-order, two-body correction is a sum of three

terms, two induction terms and a dispersion term. For example, as shown in Chap.

170

3, AEéiL is the sum of the induction terms AEéiL (0,2) and AESQA) 3(2’ 0),an

the dispersion term AEOA)?2 (O, 0). Similarly, AEéi; is the sum of AESZL (0, 2),

AEéi)‘, (2, 0) , and AEOA)C (0, 0), as shown by Eq. (347). Also, as shown by Eq.

(348), AE(28)C is the sum of AEéf; (0,2), AE(2B)C (2,0), and AE0:)C (0, 0)
.The full nonadditive, 2nd-order correction to the energy AEé2’3)

(350).

is given by Eq.

ABC

Eq. (351) contains all contributions to the total second-order correction to the
energr of the interaction of molecules A, B, and C. Based on this equation, we have
obtained an equation for the total second-order correction to the energy of interaction

of three colinear hydrogen fluoride molecules. We call the three HF molecules that we

2nd

use in this derivation HF 1, HF2, and HF3, and we call the corresponding -order

a ' ' 2
correction to the energy of interaction of these three molecules AEéJFl [”2 HF
- " 3

. The three molecules are arranged so that their internuclear axes lie along the z-
axis of the laboratory frame, as shown in Fig. 15. We have denoted the distances
between the centers of mass of HF 1 and H172, HF 1 and HF 3, and HF 2 and HF3 by
RHFl—Hth R115- (H173 , and R3143- 111:3 , respectively. We obtain the desired

expression for AEQ) by replacing A, B, and C in Eq. (351) with
01-1171-

--—HF2 HF3

HF 1, HFg, and HF3, respectively. This expression for AEOHLF __sz —HF3 is given
(2 ,3)

”Fl—HFz—HF3

in Eq. (352) is given by Eq. (350) with molecules A, B, and C replaced by HP 1,
HF 2, and HF3. The second-order, two-body terms AEm AEOH F1

OHFI -HF2 ’

by Eq. (352). The irreducible three-body energy of interaction AEO

-HF3’

and AEéfiFTHFs in Eq. (352) are given by Eq. (306) with A, B, and C replaced
by HF 1, HF 2, and HF3, respectively. According to Eq. (327), the second-order

correction to the energy of interaction of two colinear HF molecules is AE(2)

OHF— HF‘ =

--44.5 a.u./(R1151- HF)6. We simplify Eq. (352) for AE0H)F1_M,2_W,3 by realizing
(2) (2) . .

that AEOW,1 —HF2 , AEOHL, _3HF , and AEOHF2-HF3 are all g1ven by Eq. (327) With

the appropriate substitutions of HF], HF2, or HF3 for HF. We also simplify Eq.

171

. (2,3)
(352) by replacmg AEOHFl—HFz—HFa

AE(2 3) which is given by Eq. (356).

0HF1—-HF2 -3HF’

in Eq. (352) with a geometry-specific form of

We have also derived an expression for the 2nd-order correction to the energy of

interaction AEOH)F1_ of HFl, HF2, and HF 3 when these three HF molecules

—HF2— HF3

ara parallel to each other. The three molecules are arranged so that their internuclear
axes are parallel the x-axis of the laboratory frame, as shown in Fig. 16. Although Eq.

(352) in our derivation of AE3:)F1_ ”F2 —HF3 applies to three colinear molecules, it is

general enough that we used it as our starting point for deriving AEOH F1” F2” F
3

for three parallel HF molecules. The irreducible three-body energy of interaction

AE‘z') ”3 in Eq. (352) for three parallel HF molecules is also given by Eq.

OHFI— --HF2 HF3
(350), with molecules A, B, and C replaced by HFI, HF2, and HF3. According to
Eq. (338), the second-order correction to the energy of interaction of two parallel HF
HF_ HF —- ——29.2 a.u./(RHF_HF)6 . We Simplify Eq). (352) for
this geometry by realizing that AE(2) AEm and AE02) are

OHF —2HF’ OHF -3aHF’ HFz- -HF3

molecules is AEé2)

all given by Eq. (338) with the appropriate substitutions of HF], HF2, and HF3 for
HP. We also simplify Eq. (352) by replacing AEéiib HF _HF in Eq. (352) with a
. 2 ,3
geometry spec1fic form of Eq. AEéfl’FL HF2- -3HF , which 18 given by Eq. (361).
Using the expression for AEOA; derived in Chap. 4, which is correct for any
geometry of the AB complex, we have also derived an expression for AEéi; when
A and B are colinear. This expression is given by Eq. (290) in Chap. 4, and again
by Eq.) (362) in Chap. 5. Recall that Eq. (362) consists of six terms, which are
(3) (3) (3) E(3)
E053 (1,1), AEOAB (3,0), AEOAB(O,3), AEOAB (1, 0),A B,(0 1), and
:EOA)B (O, 0). The equations for each of these six terms are listed in Eq. (363). We

derive the overall expression for AE33)B when A and B are colinear by) deriving

the expressions for AE$)B,(11), AEéi; (3,0), AEéiL (0,3), AEOiL (1,0),
AEOA)B (0, 1), and AE0:)B (0, 0) separately, and then adding all of the resulting
expressions together. Note that in this geometry, the internuclear axes of A and B

are oriented along the z-axis of the laboratory frame, as shown in Fig. 12.

172

First, we show that EOAB )(1, 1) in this geometry is given by Eq. (367).

GA B
022? C1227

RAB . Next, we show that AEé? 3(0’ 3) and AEGi); (3,0) (in this geometry)

In this equation, AE(3)B,(1 1) is given in terms of 11,0 , ӣ30 , and
are given by Eqs. (370) and (373),A respectively. Eq. (370) is given in terms of [1:30,
182221 and RAB , and Eq. (373) is given in terms of 11:10, 52 , and RAB-
Numerical values of an and fizzz are available for many molecules. Also, for this
geometry, ”:10 = ”A0 and #280 = ,uBO, and numerical values of [LAO and HBO
are also available for many molecules.

We then show that AEOA)B (0, 1) is given by Eq. (375) which contains Tm,
Tyy, Tu, [150, each diagonal component of the frequency-dependent polarizability
01A (w), and several components of ,BA (—zw; 2w, 0). We simplified Eq. (375)
by replacing Tm, Tyy , and T22 with the appropriate expressions for these

quantities given in terms of RAB , and by replacing 11,30 with ”BO. We also

simplified Eq. (375) by letting HF: A in Eq. (317), and using these expressions for

A A
m: yy

ities a? x(zw), 0:14,!(260), and 0124,0122) given by Eq. (317) (with HF: A) are given

14 .A
1131:, ayyia zzi

a (w), a (w), and 0A z'(zw) in Eq. (375). The expressions for the polarizabil-

in terms of the static polarizabilities a the ionization potential E 1 P
of A, and the frequency w . The static polarizabilities and ionization potential are

known for many molecules, so that it is easy to compute 03:42 (W), 03y (w), and
A

2, (w) for these molecules.

(1
To complete our simplification of Eq. (375), we needed to replace the various
components of 3A (—zw; 2w, 0) with expressions that contain quantities whose

numerical values are known. Using the expression for the frequency-dependent hy-

perpolarizabilityfi [3170— -—;zw 2w ,0) given by Eq. (394), the Unsold approximation,
and a few manipulations of Eq. (394), we showed that 22,7 (—zw; 2w, 0) is given

by Eq. (399). We also used the expression for the static polarizability fig?” given by

Eq. (376), the Unsold and closure approximations, and other manipulations to show

173

that the matrix elements in Eq. (399) for 3:37 (—zw; 2w, 0) are given by Eq. (393).
We obtained our final expression for £7 (-—zw; 2w, 0), given by Eq. (400), by
using Eq. (393) in Eq. (399), and by combining the denominators in Eq. (399). Eq.
(400) depends on 18.2017: Efp, and w. Since these quantities are known for several
molecules, one can easily compute fig?” (—zw; zw, 0) using Eq. (400). Equations
for specific components of ,3607 (— w; w, 0) can be obtained by replacing 6, Oz,
and ’y in Eq. (400) with the appr0priate Cartesian coordinates.
At this point, we obtained a simplified expression for AEéi; (0, 1) by replacing
fizm(-— 2w; zw ,,0) z@y(—w;zw,0), and £34,, (—zw;ua,0) in Eq. (375) with

Eq. (400), with the appropriate substitutions for 5, Oz, and ’7 . This expression
E(3)

EOAB

AE(3A) (0, 1) , given by Eq. (407), by evaluating the integrals in Eq. (402). We

OAB

for (0,1) is given by Eq. (402). We obtained the final expression for
derived the final expression for AEéi; (1,0) by interchanging A and B in Eq.
(407). We presented the final expression for AEéiL (1,0) in Eq. (408).

The last step in deriving an expression for AEéi; for the interaction between A
and B when they are colinear (with their internuclear axes oriented along the Z-axis of
the laboratory frame) involved the simplification of the expression for AE03)B (0, 0)
containedin Eq. (362). The form of AE(3)B (0, 0) inEq. (362) for AEéAL isgiven
in Eq. (363). To begin our simplification, we showed that (1’4
is given by Eq. (413). Here, Eq. (413) gives bA

7016 (w; 210' — 2w, —zw')

706 (w; zw — 2w, —zw’) in terms

of 3,405 , E3413 , and w . It is possible to give béficb (w; zw’ — w, —zw’) and

the other frequency-dependent (353.1 terms contained in AEéi; (0, 0) in terms of

5,534, , EB; , and w . Since numerical values of static polarizabilities and ionization

potentials are available for many 3molecules, it is possible to use our expression for

AE(3) (0, 0) to compute AEéi) 3(0’0) at various RAB and w.

0A3

There are several possibilities for future work on the charge-density susceptibility
x(r, l"; w) . Future applications of the charge-density susceptibility include calcu-

lating X (r, I"; w) for several other centrosymmetric diatomic molecules such as N2,

174

02, Clo, and F 2, and also for noncentrosymmetric diatomic molecules such HF, HCl,
and CO. We may also calculate X (r, r’; w) for small polyatomic atomic molecules
such as H20, and 002.

We may also calculate x(r, I"; w) for H2 in higher level basis sets and compare
x(r, r’; (.0) obtained in different basis sets to determine the relative contributions
of different orbital types to x(r, I"; w) of H2. We may also calculate x(r, r’; w)
of other molecules mentioned above as a function of basis set. This would allow us
to determine the relative contributions of different orbital types to x(r, I"; w) for
these molecules.

We plan to improve our results for x(r, r’; w) of H2 by calculating x(r, I"; w) of
H2 at all of the same conditions as those used in this work, with the exception of the
step sizes Ag and AZ between 3] and 2 data points (recall that we have
x = 0 ). Specifically, we plan on calculating x(r, r’ ;w) of H2 with step sizes
Ay and AZ that are smaller than the current step sizes. This will allow us to
determine if X (r, r’; w) for H2 calculated with smaller step sizes has features that
are unresolved in our results for X (r, r'; w) of H2. If we calculate x(r, r’; w) of
other molecules, we can also calculate x(r, I"; w) of these molecules as a function
of step size. From these calculations, we can determine if the shape of x(r, r’; w)
for these molecules is independent of Ag and A2, or if we resolve more features of
x(r, r’; w) as we decrease Ag and AZ.

Other future work may involve modifying the current program for calculating
x(r, r’; w) to improve its efficiency and speed. We could improve the efficiency
and speed of our program by explicitly removing triplet states and singlet states with
improper symmetries from the sum-over-states calculation of x(r, r’; w). Depending
on the results of x(r, I"; w) calculations for different molecules, r values, r’
values, and to values, it may also be possible to develop programs to approximate
X(r, 1"; w)-

Another very important future project involves the development of algorithms and

175

would be to develop algorithms and write computer programs that use x(r, r’; w) to
calculate other properties, including dipole and higher-order polarizabilities, induc—
tion and dispersion energies for interacting molecules, infrared intensities, and nonlo-
cal intramolecular dielectric functions. We have given an extensive list of molecular
properties related to x(r, r’; w) in Chap. 1.

Future work on intermolecular interactions will involve comparing the results of
the calculations presented here with the corresponding results of similar calculations
presented in the literature. For example, we plan to compare the values of C 6 obtained
here for AE3:)F_HF of two colinear and of two parallel HF molecules with the
corresponding values of Cs obtained by Meath and co—workers,289 and similarly for
two perpendicular HF molecules. We also plan to compare C9 values for three HF

molecules with the corresponding values of Meath et. al..289

176

Appendices

177

Appendix A. Fortran CISD Code for Calculating x(r, r’; w)
and flag (w)

178

00000000000

0000000000000000000000000000000000000

Appendix A. Fortran CISD Code for Calculating x(r, r’; w)

and (lag (w)

PROGRAM CHICALC

This program solves the configuration-interaction with singles
and doubles (CISD) eigenvalue problem for the specified
molecule and computes the charge-density susceptibility of
that molecule at coordinates r = x, y, z, and r’ = x’, y’, z’.
The program also computes the xx, yy, 22, and xy components

of the polarizability of the molecule.

Declare that all variables with names that begin with letters
A-H or 0-2 will be double-precision numbers.

IMPLICIT DOUBLE PRECISION (A-H, D-Z)
Description of all variables and/or arrays used in the program:

NRE = Integer array whose values correspond to the number of
irreducible representations in the C1 (lst element) point
group; the C2, CS, and CI point groups (2nd element); the C2V,
C2H, and D2 (3rd element) point groups; and the D2H point
group (4th element).

MC = Integer array which contains the irreducible represen-
tations of the C1, C2, C8, CI, C2V, C2H, D2 and D2H point

groups.

IG = Integer set to the number of irreducible representations
in the point group.

GRP = Point group of molecular system (character variable).

GRPREP = One-dimensional character array (with at most eight
elements) which is set to the list of (symbols for the) ir—
reducible representations for the point group of the
molecule.

SYMMl = One-dimensional array of integers. The value of

each element represents the symmetry of a spatial orbital in
the molecular system. The symmetries are arranged according
to the energies of their corresponding orbitals.

SYMM2 = One-dimensional array of integers. The value

of each element represents the symmetry of a spin orbital in
the molecular system. The symmetries are arranged according
to the energies of their corresponding orbitals.

IRREP = Two-dimensional integer array which contains the ir-

reducible representations of the symmetries of all spin
orbitals in the molecule.

179

0000000000000000000000000000000000000000000000000000000

NOCC1 = Number of occupied spatial orbitals.

NUNDCCI = Number of unoccupied spatial orbitals.

NTDTl = Total number of spatial orbitals.
NTDT2 = Total number of spin orbitals.
NELEC = Number of electrons in the molecular system.

SPIN2 = One-dimensional array of integers. The value of an
element with an odd index is +1, and the value of an element
with an even index is -1. The value of each element corres-
ponds to the spin that an electron would have if it were to
occupy that spin orbital (and if the value of the element
were multiplied by 0.5).

RSYM2 = Two-dimensional integer array which represents the
symmetry of each occupied orbital in the reference deter-
minant (each occupied orbital has between one and eight
values associated with it, depending on the point group of
the molecule. These associated values correspond to the
irreducible representation of the orbital’s symmetry).

RTOTSYM2 = One-dimensional integer array which represents
the overall symmetry of the reference (the array is the
irreducible representation which represents the overall
symmetry of the reference determinant).

C = Two-dimensional array of integers which is used to
construct all possible (but not necessarily spin or
symmetry allowed) determinants. The first index refers
to the determinant number (to count and distinguish
between determinants), and the second index refers to
a spin orbital. The element corresponding to a par-
ticular determinant and orbital will have a value of

1 or 0 if the orbital is occupied or unoccupied, res-
pectively.

NC = Integer variable which is used to keep track of the
total number of determinants (not necessarily spin or
symmetry allowed).

CSPIN = One-dimensional array whose index corresponds to
determinant number, and each of whose elements has a value
which (when multiplied by 0.5) corresponds to the total
spin of a specific determinant.

C2 = Two-dimensional array of integers which is used to
construct all possible spin-allowed determinants. The
first index refers to the determinant number (to count
and distinguish between determinants), and the second
index refers to a spin orbital. The element corresponding
to a particular determinant and orbital will have a value
of 1 or 0 if the orbital is occupied or unoccupied,

180

0000000000000000000000000000000000000000000000000000000

respectively.

NC2 = Integer variable which is used to keep track of the
total number of spin-allowed determinants.

CIRREP2 = Two-dimensional integer array which contains the
irreducible representations for the overall symmetries of
all possible spin-allowed determinants.

C3 = Two-dimensional array of integers which is used to
construct all possible spin- and symmetry-allowed
determinants.

NC3 = Integer variable which is used to keep track of the
total number of spin- and symmetry-allowed determinants.

H = CISD Hamiltonian matrix (consists of all possible mat-
rix elements formed from the Hamiltonian operator and two
determinants).

TDTDIFF = Two-dimensional array of integers used to keep

track of the total number of occupation differences bet-

ween any two spin- and symmetry-allowed determinants. For
example, if two electrons are in different orbitals in I

and J, then TOTDIFF(I,J) will be set to two.

E1SPAT = Array of one-electron integrals and indeces.
This array is read from the "*.int.out" input file, which
is generated by a GAMESS calculation. The integrals and
indeces in this array are in terms of spatial orbitals.

E2SPAT = Array of two-electron integrals and indeces.
This array is read from the "*.int.out" input file, which
is generated by a GAMESS calculation. The integrals and
indeces in this array are in terms of spatial orbitals.

VNN = Nuclear repulsion energy. Read from "*.int.out"
input file.

INPUT4 = Character variable (length 5) used when
reading one- and two-electron integral information
from the "*.int.out" input file.

CJ = Two-dimensional array of coefficients which are
produced when the CISD eigenvalue problem is solved.
Each CISD wavefunction is a linear combination of deter-
minants weighted by these coefficients. The first index
in this array corresponds to the determinant number, and
the second index corresponds to the state number.

E = One-dimensional array of energy eigenvalues produced
when the CISD eigenvalue problem is solved. The elements
of the array are arranged in order of increasing energy.

FFWDRK = One-dimensional array corresponding to the FWDRK

181

0000000000000000000000000000000000000000000000000000000

array in the rsm subroutine (see rsm subroutine for an
explanation of the FWORK array).

IIWORK = One-dimensional array corresponding to the IWORK
array in the rsm subroutine (see rsm subroutine for an
explanation of the IWDRK array).

CJSUM = One-dimensional array. Each element contains the
sum of the squares of all the CJ coefficients for a par-
ticular state.

NAO Total number of atomic orbitals.

NMO Total number of molecular orbitals.

OMEGA = Frequency variable, read from input file. This
variable indicates what frequency should be used to cal-
culate the charge-density susceptibility.

DOEFF = Two-dimensional array of coefficients for con-
verting atomic orbitals into molecular orbitals. Read
from the "mod..." input file.

CHI = Value of the charge-density susceptibility at the
r = x, y, z and r’ = x’, y’, z’ coordinates (shifted by
the nuclear coordintes) of interest.

AAMU = Three-dimensional array of dipole-moment integ-
rals. The first and second indeces in this array corres-
pond to atomic orbital number, and the third index, which
goes from 1 to 3, corresponds to x (1), y (2) or z (3).
This array is read from the "fort.250" input file.

POLXX = xx-component of the polarizability of the
molecule of interest.
POLYY = yy-component of the polarizability of the
molecule of interest.
POLZZ = zz-component of the polarizability of the
molecule of interest.
POLXY = xy-component of the polarizability of the

AAO = One-dimensional array of atomic orbitals evaluated
at r = x, y, z (shifted by the relevant nuclear coor-
dinates). These values are provided by the ADROUT sub
routine (the A0 array in the ADROUT subroutine is
returned to the main program as AAO).

ARPRIME = One-dimensional array of atomic orbitals
evaluated at r’ = x’, y’, z’ (shifted by the relevant
nuclear coordinates). These values are provided by the
AORDUT subroutine (the A0 array evaluated at r’ = x’,
y’, z’ (shifted by nuclear coordinates) in the AOROUT
subroutine is returned to the main program as ARPRIME).

AVEC = One-dimensional array used to facilitate reading

182

000000

of $VEC group from "mod..." GAMESS input file.

STUFF, STUFF2 = Character variables 80 characters in
length which are used to read lines from the "fort.250"
input file.

CHARACTER fileinp*80
INTEGER MC,NRE
DIMENSION MC(8,8,4),NRE(4)

CHARACTER GRP*3,GRPREP*3
DIMENSION GRPREP(8)

INTEGER SYMM1,SYMM2

DIMENSION SYMM1(100),SYMM2(200)
INTEGER IRREP

DIMENSION IRREP(200,8)

INTEGER SPIN2

DIMENSION SPIN2(200)

INTEGER RSYM2,RTOTSYM2
DIMENSION RSYM2(200,8),RTOTSYM2(8)

INTEGER C,C2,C3

DIMENSION C(10000,200),C2(10000,200)
INTEGER CSPIN

DIMENSION CSPIN(1000O)

INTEGER CIRREP2

DIMENSION CIRREP2(10000,8)

CHARACTER INPUT4*5,INPUT3*80
INTEGER TOTDIFF
DIMENSION TOTDIFF(10000,1000O)

COMMON /BLOCK1/ H(10000,10000)

COMMON /BLOCK2/ EISPAT(100,100)

COMMON /BLOCK3/ E2SPAT(100,100,100,100)
COMMON /BLOCK4/ C3(10000,200)

COMMON /BLOCK6/ NTOT2

COMMON /BLOCK12/ NC3

COMMON /BLOCK15/ NAO

DIMENSION AAO(100),ARPRIME(100)

DIMENSION E(10000)
DIMENSION CJ(10000,10000)
DIMENSION FFWORK(80000)
DIMENSION IIWORK(1000O)
DIMENSION CJSUM(1000O)

DIMENSION ABIG1(50,50,50,50)
DIMENSION ABIG2(50,50,50,50)

DIMENSION AVECCS)
DIMENSION DCOEFF(100,100)

183

0

2283
473
472

474

6009

6010
6008

6007

6006
6005

CHARACTER STUFF*80,STUFF2*80
DIMENSION AAMU(100,100,3)

DIMENSION B(10000,50,50)
DIMENSION BAOl(10000,50,50)
DIMENSION BAO(10000,50,50)
DIMENSION ILEFT(5000,5000)
DIMENSION IRIGHT(5000,5000)
DIMENSION ISIGNRHO(5000,5000)
DIMENSION IOK1(5000)
DIMENSION IOK2(5000)

Set all of the elements in each array equal to zero.

DO 472 II=1,100,1
DO 473 JJ=1,100,1
DCOEFF(II,JJ)=0.0d0
DO 2283 KK=1,3,1
AAMU(II,JJ,KK)=0.0d0
CONTINUE
CONTINUE
CONTINUE

DO 474 IK=1,5,1
AVECCIK)=0.0d0
CONTINUE

DO 6008 I=1,10000,1
CSPIN(I)=0
DO 6009 J=1,200,1
C(I.J)=O
C2(I,J)=O
CONTINUE
DO 6010 J=1,8,1
CIRREP2(I,J)=0
CONTINUE
CONTINUE

DO 6007 I=1,
RTOTSYM2(
CONTINUE

8,1
I)=0
DO 6005 I=1,200,1
SPIN2(I)=0
SYMM2(I)=0
DO 6006 J=1,8,1
RSYM2(I,J)=0
IRREPCI,J)=0
CONTINUE
CONTINUE

DO 6020 II=1,100,1
DO 6002 J=1,100,1
EISPAT(II,J)=0.0D0
DO 6003 K=1,100,1

184

C

0

6004
6003
6002

6020

6063
6062
6061
6060

6136
6135
6130

224

223
222

6000

DO 6004 L=1,100,1
E2SPAT(II,J,K,L)=0.0D0
CONTINUE
CONTINUE

CONTINUE

SYMM1(II)=0

AAOCII)=0.0D0

CONTINUE

DO 6060 IP=1,50,1
DO 6061 IQ=1,50,1
DO 6062 IT=1,50,1
DO 6063 IU=1,50,1
ABIGI(IP,IQ,IT,IU)=0.0d0
ABIG2(IP,IQ,IT,IU)=0.0d0
CONTINUE
CONTINUE
CONTINUE
CONTINUE

DO 6130 I=1,10000,1
DO 6135 M=1,50,1
DO 6136 N=1,50,1
B(I,M,N)=0.0d0
CONTINUE
CONTINUE
CONTINUE

DO 222 I=1,10000,1
E(I)=0.0DO
CJSUMCI)=0.0D0
IIWORK(I)=0
DO 224 J=1,200,1
C3(I,J)=0

CONTINUE

DO 223 J=1,10000,1
H(I,J)=0.0D0
TOTDIFF(I,J)=0
CJ(I,J)=0.0D0

CONTINUE

CONTINUE

DO 6000 I=1,80000,1
FFWORK(I)=0.0D0
CONTINUE
Use data groups to set the values in the NRE and MC arrays.

DATA NRE/1,2,4,8/
DATA MC/

00

0 00000

0

00

9998

000 0000

0000

00000

”RPEPWPRPK’RPK’

0,0,0,0,0,0,0,0, 0 0 0,0,0 0 0 0

1,1,1,1,0,0,0,0, 1,1,-1,-L 0,0,0, 0, 1, 1, 1, L 0, 0, 0, 0,
1,-1,-1,1,0,0,0,0, 0,0,0,0,0,0,0,0, 0,0 ,0,0,0, 0 ,0,0,
0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,

1,1,1,1,1,1,1,1, 1,1,1,1,-1,-1,-1,- 1, 1,1,-1,-1,1,1,-1,-1,
1,1,-1,-1,-1,-1,1,1, 1,-1,1,-1,1,-1,1,-1, 1,-1,1,-1,-1,1,-1,1,
1,-1,-1,1,1,-1,-1,1, 1,-1,-1,1,-1,1,1,-1/

Open the input file containing the basic molecular information.

call getenvC’INPUT’,fi1einp)
open(unit=20,file=fileinp,status=’unknown’,form=’formatted’)

Read the number of occupied orbitals, the number of unoccupied
orbitals, and the total number of orbitals from the input file
(all spatial orbitals).

READ(20,*)NOC01,NUNOCC1,NTOT1

Set the total number of spin orbitals.

NTOT2=NTOT1*2

Read the point group of the molecular system from the input file.

FORMATCAB)
READ(20,9998)GRP

Read the symmetries of the spatial orbitals into the SYMM1
array.

READ(20,*) (SYMM1(I), I=1,NTOT1)
Compute the number of electrons and set the result to NELEC.
NELEC=NOCCI*2

Read the frequency to be used for calculating the charge-density
susceptibility into OMEGA.

READ(20,*)OMEGA
WRITE(6,*)’Frequency:’,OMEGA
CALL FLUSHCG)

Set the symmetries of the spin orbitals (using the symmetries
of the spatial orbitals, which are contained in the SYMM1
array).

DO 26 I=1,NTOT1,1
SYMM2((2*I)-1)=SYMM1(I)
SYMM2(2*I)=SYMM1(I)

26 CONTINUE

186

0000000 0000 000000

00

000000

000000

28

30

34

33
32

Assign spin to each spin orbital in the system.

(Note: The spins are assigned to orbitals rather than electrons
to facilitate computation. Also, the spins are set to integers
rather than +1/2 or -1/2 for the same reason).

DO 28 I=1,NTOT1,1
SPIN2((2*I)-1)=1
SPIN2(2*I)=-1

CONTINUE

Set the total number of all possible determinants (not
necessarily spin and symmetry-allowed) to 1.

NC=1

Construct the reference determinant. If a spin orbital is
occupied in the reference determinant, the value of the
corresponding element will be set to 1. If a spin orbital
is not occupied, the value of the corresponding element
will be set to zero.

DO 30 I=1,NELEC,1
C(NC,I)=1
CONTINUE

Increment the total number of determinants.
NC=NC+1

Construct all possible (but not necessarily spin or sym-

metry allowed) singly-excited determinants. Increment the
total number of determinants each time a new determinant

is produced.

DO 32 I=1,NELEC,1
DO 33 J=(NELEC+1),NTOT2,1
C(NC,J)=1
DD 34 K=1,NELEC,1
IF (K.EQ.I) THEN
ELSE
C(NC,K)=1
ENDIF
CONTINUE
NC=NC+1
CONTINUE
CONTINUE

Construct all possible (but not necessarily spin or sym-
metry allowed) doubly-excited determinants. Increment the
total number of determinants each time a new determinant
is produced.

DO 55 I=1,(NELEC-1),1
DO 56 J=(I+1),NELEC,1

187

000000

0000

0000

0000000

59
58
57

56
55

68
67

74

73

DO 57 K=(NELEC+1),NTOT2,1
DO 58 L=(K+1),NTOT2,1
C(NC,K)=1
C(NC,L)=1
DO 59 M=1,NELEC,1
IF (M.EQ.I .OR. M.EQ.J) THEN
ELSE
C(NC,M)=1
ENDIF
CONTINUE
NC=NC+1
CONTINUE
CONTINUE
CONTINUE
CONTINUE

Decrease the total number of determinants by one to
prevent improper counting of the total number of
singly- and doubly-excited determinants (the previous
loop overcounts the total number of determinants).

NC=NC-1

Determine the total spin of each possible (but not
necessarily spin- or symmetry-allowed) determinant.

DO 67 I=1,NC,1
DO 68 J=1,NTOT2,1
IF (C(I,J).EQ.1) THEN
CSPIN(I)=CSPIN(I)+SPIN2(J)
ELSE
ENDIF
CONTINUE
CONTINUE

Set the total number of spin-allowed determinants to
one.

NC2=1

Determine which of the determinants in the C array are
spin-allowed. Add the spin-allowed determinants to the
C2 array. Increment the total number of spin-allowed
determinants each time a determinant in the C array is
added to the C2 array.

DO 73 I=1,NC,1
IF (CSPIN(I).EQ.CSPIN(1)) THEN
DO 74 J=1,NTOT2,1
C2(NC2,J)=C(I,J)
CONTINUE
NC2=NC2+1
ELSE
ENDIF
CONTINUE

188

000000

000000

Decrease the total number of spin-allowed determinants
by one to prevent improper counting of these deter-
minants (the previous loop overcounts the total number
of spin-allowed determinants).

NC2=NC2-1

According to the point group, set the irreducible rep-

‘resentations of the molecule. Point groups that can be

handled by this program are: C1, C2, CS, CI, C2V, C2H,
D2, and D2H.

IF (GRP.EQ.’C1’) THEN

IG=1
GRPREP(1)=’A’
ENDIF
IF (GRP.EQ.’CZ’) THEN
IG=2
GRPREP(1)=’A’
GRPREP(2)=’B’
ENDIF
IF (GRP.EQ.’CS’) THEN
IG=2
GRPREP(1)="A’"
GRPREP(2)="A”"
ENDIF
IF (GRP.EQ.’CI’) THEN
IG=2

GRPREP(1)=’Ag’
GRPREP(2)=’Au’
ENDIF
IF (GRP.EQ.’C2V’) THEN
IG=3
GRPREP(1)=’A1’
GRPREP(2)=’A2’
GRPREP(3)=’BI’
GRPREP(4)=’B2’
ENDIF
IF (GRP.EQ.’D2’) THEN
IG=3
GRPREP(1)=’A’
GRPREP(2)=’B1’
GRPREP(3)=’B2’
GRPREP(4)=’83’
ENDIF
IF (GRP.EQ.’C2H’) THEN
IG=3
GRPREP(1)=’Ag’
GRPREP(2)=’Au’
GRPREP(3)=’Bg’
GRPREP(4)=’Bu’
ENDIF
IF (GRP.EQ.’D2H’) THEN
IG=4

189

0000

GRPREP(1)=’Ag’
GRPREP(2)=’Au’
GRPREP(3)=’Blg’
GRPREP(4)=’Biu’
GRPREP(5)=’B2g’
GRPREP(6)=’B2u’
GRPREP(7)=’BBg’
GRPREP(8)=’BBu’
ENDIF

Set the irreducible representation for the symmetry of each
occupied and unoccupied spin orbital in the molecule.

DO 108 I=1,NTOT2,1
IF (SYMM2(I).EQ.1) THEN
DO 109 J=1,8,1
IRREP(I,J)=MC(1,J,IG)
109 CONTINUE
ENDIF
IF (SYMM2(I).EQ.2) THEN
DO 110 J=1,8,1
IRREP(I,J)=MC(2,J,IG)
110 CONTINUE
ENDIF
IF (SYMM2(I).EQ.3) THEN
DO 111 J=1,8,1
IRREP(I,J)=MC(3,J,IG)
111 CONTINUE
ENDIF
IF (SYMM2(I).EQ.4) THEN
DO 112 J=1,8,1
IRREP(I,J)=MC(4,J,IG)
112 CONTINUE
ENDIF
IF (SYMM2(I).EQ.5) THEN
DO 113 J=1,8,1
IRREP(I,J)=MC(5,J,IG)
113 CONTINUE
ENDIF
IF (SYMM2(I).EQ.6) THEN
DO 114 J=1,8,1
IRREP(I,J)=MC(6,J,IG)
114 CONTINUE
ENDIF
IF (SYMM2(I).EQ.7) THEN
DO 115 J=1,8,1
IRREP(I,J)=MC(7,J,IG)
115 CONTINUE
ENDIF
IF (SYMM2(I).EQ.8) THEN
DO 116 J=1,8,1
IRREP(I,J)=MC(8,J,IG)
116 CONTINUE
ENDIF
108 CONTINUE

190

0000 000000 00000 0000

0

0000

133
132

119

123
120

126
125

129

128
127

Set the irreducible representations of each occupied orbital
in the reference determinant.

DO 132 I=1,NELEC,1
DO 133 J=1,8,1
RSYM2(I,J)=IRREP(I,J)
CONTINUE
CONTINUE

Set (initialize) each element in the RTOTSYM2 array (the array
which will be set to the overall symmetry of the reference
determinant) to one.

DO 119 I=1,8,1
RTOTSYM2(I)=1
CONTINUE

Determine the overall symmetry of the reference determinant by
calculating the product of the irreducible representations (for
the symmetries of the orbitals) of all occupied orbitals in the
reference determinant.

DO 120 I=1,NELEC,1
DO 123 J=1,8,1
RTOTSYM2(J)=RTOTSYM2(J)*RSYM2(I,J)
CONTINUE
CONTINUE

Set (initialize) the value of each element in the CIRREP2 array
to one.

DO 125 I=1,10000,1
DO 126 J=1,8,1
CIRREP2(I,J)=1
CONTINUE
CONTINUE

Determine the overall symmetry of each spin-allowed determinant.

DO 127 I=1,NC2,1
D0 128 J=1,NTOT2,1
IF (C2(I,J).EQ.1) THEN
DO 129 K=1,8,1
CIRREP2(I,K)=CIRREP2(I,K)*IRREP(J,K)
CONTINUE
ELSE
ENDIF
CONTINUE
CONTINUE

Set the total number of spin- and symmetry-allowed determinants
to one.

NC3=1

191

00000

000000

00

0000

0000

139

140

138
137

621
620

Determine whether each spin-allowed determinant in the C2 array
is also symmetry-allowed. If a determinants in the C2 array is
symmetry-allowed, add it to the C3 array.

DO 137 I=1,NC2,1
DO 138 J=1,8,1
IF (CIRREP2(I,J).EQ.RTOTSYM2(J)) THEN
GO TO 139
ELSE
GO TO 137
ENDIF
IF (J.EQ.8) THEN
DO 140 K=1,NTOT2,1
C3(NC3,K)=C2(I,K)
CONTINUE
NC3=NC3+1
ELSE
ENDIF
CONTINUE
CONTINUE

Decrease the total number of spin- and symmetry-allowed deter-
minants by one to prevent improper counting of these deter-
minants (the previous loop overcounts the total number of
spin- and symmetry-allowed determinants).

NCB=NC3-1
Write all spin and symmetry allowed determinants.

WRITE(6,*)’Spin- and Symmetry-Allowed Determinantsz’
CALL FLUSH(6)

DO 620 I=1,NC3,1
WRITE(6,*)’Determinant: ’,I
CALL FLUSH(6)
WRITE(6,*)’Occupied Orbitalsz’
CALL FLUSH(6)

D0 621 J=1,NTOT2,1
IF (C3(I,J).EQ.1) THEN
WRITE(6.*)J
CALL FLUSH(6)
ENDIF
CONTINUE
CONTINUE

Close the input file containing the basic molecular
information.

CLOSE (20)

Open the input file containing the one- and two-electron
integrals.

192

00000

00

153

0000000

1507

0000

1509

0000000

0000000000

OPEN (UNIT=21 ,

&FILE=’h2cisd_MBS_C1.int.out’,
&STATUS=’OLD’)

Set the number of data-containing lines in the one-
and two-electron integral file (represented by the
variable NLINE) to zero.

NLINE=0
Read the first five characters in the input file into INPUT4.

FORMAT(A5)
READ(21,153)INPUT4

If the first five characters of the input file are not blank
spaces (meaning data is present), increment the number of

of data-containing lines, and read the first five charac-
ters in the next line. Continue to read and count lines
until there are no more data-containing lines.

IF (INPUT4.NE.’ ’) THEN
NLINE=NLINE+1
READ(21,153)INPUT4
GO TO 1507

ENDIF

Return to the beginning of the data-containing lines,
backing up one line at a time.

DO 1509 I=1,(NLINE+1),1
BACKSPACE(21)
CONTINUE

Read the first five characters of the current line (this
is a check to make sure that we are at the beginning of
the data -- if desired, print these characters to find
out if we are indeed at the beginning of the data). Then
return to the beginning of the current line.

READ(21,153)INPUT4
BACKSPACE(21)

Read the two- and one-electron integrals and indeces into
the E2SPAT and EISPAT arrays, respectively. Also, read the
nuclear repulsion energy into the variable VNN. Note: The
input file’s list of integrals does not include all possible
one- and two-electron integrals. Therefore, we will complete
the list while reading the integrals from the input file.
(The variable for the integral values is XXX, and the
variable(s) for the integral indeces are IP,IQ,IR, and IS).

DO 1510 I=1,NLINE,1

READ(21,*) XXX,IP,IQ,IR,IS
IF (IR.NE.0) THEN

193

0000

0000000000000000

1510

149

E2SPAT(IP,IQ,IR,IS)=XXX
E2SPAT(IQ,IP,IR,IS)=XXX
E2SPAT(IP,IQ,IS,IR)=XXX
E2SPAT(IQ,IP,IS,IR)=XXX

E2SPAT(IR,IS,IP,IQ)=XXX

E2SPAT(IS,IR,IP,IQ)=XXX

E2SPAT(IR,IS,IQ,IP)=XXX

E2SPAT(IS,IR,IQ,IP)=XXX
ELSE

IF (IP.NE.0 .AND. IQ.NE.0

.AND. IR.EQ.0 .AND. IS.EQ.0) THEN
E1SPAT(IP,IQ)=XXX
EISPAT(IQ,IP)=XXX

ELSE
IF (IP.EQ.0 .AND. IQ.EQ.0 .AND.
IR.EQ.0 .AND. IS.EQ.0) THEN

VNN=XXX
ENDIF
ENDIF
ENDIF
CONTINUE

Close the input file containing the one- and two-electron
integrals.

CLOSE(21)

For each pair of determinants (I,J) in which J is greater
than or equal to I, subtract the occupation number in J

from the occupation number in I for each spin orbital in the
molecule, sum these differences, and set the result to
TOTDIFF(I,J). Then, divide TOTDIFF(I,J) by two to determine
the total number of spin-orbital occupation differences
between the two determinants. If the two determinants

differ by more than two occupancies, set H(I,J) to zero.

If I and J differ by two occupancies, call subroutine (SLATER2)
for forming two-electron integrals from two determinants that
differ by two spin orbital occupancies. If I and J differ by
one occupancy, call subroutine (SLATERi) for forming one- and
two-electron integrals from two determinants that differ by
one spin orbital occupancy.

DO 9100 I=1,NC3,1
DO 9101 J=1,NC3,1
IF (J.GE.I) THEN
DO 149 K=1,NTOT2,1
TOTDIFF(I,J)=TOTDIFF(I,J)+

& ABS(C3(I,K)-C3(J,K))

CONTINUE

IF ((TOTDIFF(I,J)/2).GT.2) THEN
H(I,J)=0.0D0

ELSE

ENDIF

IF ((TOTDIFF(I,J)/2).EQ.2) THEN

194

00000000000000000000

CALL SLATER2(I,J)

ELSE

ENDIF

IF ((TOTDIFF(I,J)/2).EQ.1) THEN
CALL SLATER1(I,J)

ELSE

ENDIF

If I and J are the same, determine which spin orbitals are
occupied in both I and J. Convert each spin orbital K (if
occupied) in I and J to spatial orbital KS. Then, add the
one-electron integral in which KS is occupied in both I and
and J (E1SPAT(KS,KS)) to the Hamiltonian matrix element
H(I,J). Also, for each pair of occupied spin orbitals L and
M in I (and in J) in which M is greater than or equal to L,
convert L and M to spatial orbitals LS and MS, calculate

the spins of LS and MS, and set these spins to LSPIN and
MSPIN. Then, if the spins LSPIN and MSPIN are the same, add
the antisymmetrized two-electron integral for these orbitals
to the Hamiltonian matrix element H(I,J). The antisymmetrized
two-electron integral is E2SPAT(LS,LS,MS,MS) -
E2SPAT(LS,MS,MS,LS). If spins LSPIN and MSPIN are different,
add the two-electron integral EZSPAT(LS,LS,MS,MS) to H(I,J).
Note: For an explanation of the calculations of matrix
elements H(I,J), review the Slater rules for calculating
one- and two-electron matrix elements of the Hamiltonian.

IF ((TOTDIFF(I,J)/2).EQ.0) THEN
DO 160 K=1,NTOT2,1
IF (C3(I,K).EQ.1) THEN
IF (MOD(K,2).EQ.0) THEN
KS=(K/2)
ELSE
ENDIF
IF (MOD(K,2).EQ.1) THEN
KS=((K+1)/2)
ELSE
ENDIF
H(I,J)=H(I,J)+E1SPAT(KS,KS)
ELSE
ENDIF
160 CONTINUE
ELSE
ENDIF
IF ((TOTDIFF(I,J)/2).EQ.O) THEN
DO 161 L=1,NTOT2,1
DO 162 M=1,NTOT2,1
IF (M.GT.L) THEN
IF (C3(I,L).EQ.1 .AND.
& C3(I,M).EQ.1) THEN
IF (MOD(L,2).EQ.0) THEN
LS=(L/2)
LSPIN=-1
ELSE
ENDIF

195

000000

0000

IF (MOD(L,2).EQ.1) THEN
LS=((L+1)/2)
LSPIN=1

ELSE

ENDIF

IF (MOD(M,2).EQ.0) THEN
MS=(M/2)
MSPIN=-1

ELSE

ENDIF

IF (MOD(M,2).EQ.1) THEN
MS=((M+1)/2)
MSPIN=1

ELSE

ENDIF

IF (LSPIN.EQ.MSPIN) THEN
H(I,J)=H(I,J)+

& E2SPAT(LS,LS,MS,MS)-
& E2SPAT(LS,MS,MS,LS)
ELSE
ENDIF

IF (LSPIN.NE.MSPIN) THEN
H(I,J)=H(I,J)+

& E2SPAT(LS,LS,MS,MS)
ELSE
ENDIF
ELSE
ENDIF
ELSE
ENDIF
162 CONTINUE
161 CONTINUE
ELSE
ENDIF
ELSE
ENDIF

9101 CONTINUE
9100 CONTINUE

For each pair of determinants (I,J) with J greater than I, set

TOTDIFF(J,I) equal to TOTDIFF(I,J). TOTDIFF(I,J) was calculated
in the previous loop. This can be done because the Hamiltonian

matrix H(I,J) is Hermetian.

DO 2025 I=1,NC3,1
DO 2026 J=1,NC3,1
IF (J.GE.I) THEN
TOTDIFF(J,I)=TOTDIFF(I,J)
ENDIF
2026 CONTINUE
2025 CONTINUE

For each pair of determinants (I,J) with J greater than I, set

H(J,I) equal to H(I,J). H(I,J) was calculated in the previous
loop. This can be done because the Hamiltonian matrix H(I,J)

196

174
173

0000000000

2003

00000000

000000

00000

is Hermetian.

DO 173 IL=1,NC3,1
DO 174 JL=1,NC3,1
IF (JL.GE.IL) THEN
H(JL,IL)=H(IL,JL)
ENDIF
CONTINUE
CONTINUE

List the matrix elements of the Hamiltonian.

WRITE(6,*)’CISD Hamiltonian matrix elements:’
CALL FLUSH(6)

DO 2001 I=1,NC3,1
DO 2002 J=1,NC3,1
WRITE(6,*)I,J,H(I,J)
CALL FLUSH(6)
CONTINUE
CONTINUE

Set each diagonal matrix element within the set H(NC3+1,NC3+1)
...H(10000,10000) to 1000. This is done to facilitate the
operation of the rsm subroutine. This enables the main program
to provide rsm with a fixed-dimension Hamiltonian matrix H
without preventing rsm from solving variably-sized CISD eigen-
value problems.

DO 2003 NH=NC3+1,10000,1
H(NH,NH)=1000.0D0
CONTINUE

Set the error indicator IIERR (for the rsm subroutine) to 1,000
(This will ensure that if the value of IIERR is zero, then it
is zero because rsm set it to zero and not by default). Note:
IIERR in the main program corresponds to IERR in the rsm sub-
routine. Also, if IIERR is zero, the rsm subroutine ran without
generating any errors.

IIERR=1000

Call the subroutine for solving the CISD eigenvalue problem.
Note: To determine what each of the arguments in the rsm call
statement corresponds to, see the comments for this program
and the rsm subroutine.

CALL rsm(10000,10000,H,E,NC3,CJ,FFWORK,IIWORK,IIERR)
For each eigenvector, calculate the sum of the squares of the
coefficients for that eigenvector. Print the sum of the squares
of the coefficients, if desired.
DO 2023 LK=1,NCB,1

DO 2024 LL=1,NC3,1

197

2024
c

c &

2023

00000

8821

0000000000

CJSUM(LK)=CJSUM(LK)+(CJ(LL,LK)**2)

CONTINUE
WRITE(6,*)’Eigenvector:’,LK,’Sum of squares of coeffs.:’,
CJSUM(LK)
CONTINUE
For each state I, calculate and write the difference between
the energy of the Ith eigenvector and the 1st eigenvector
(i.e., calculate the resonances).

WRITE(6,*)’Resonances:’

CALL FLUSH(6)

DO 8821 I=1,NC3,1
WRITE(6,*)(E(I)-E(1))
CALL FLUSH(6)

CONTINUE

Write the CI coefficients.

WRITE(6,*)’Determinant ’,
’State ’,’CJ(Determinant, State)’
CALL FLUSH(6)
DO 2382 I=1,NC3,1
DO 2383 J=1,NC3,1
WRITE(6,*)I,J,CJ(I,J)
CALL FLUSH(6)

CONTINUE
CONTINUE
Open the "mod..." input file, which is the input file for
a GAMESS CISD energy calculation.

OPEN (UN IT=23 ,

&FILE=’mod_h2cisd_MBS_C1.inp’,
&STATUS=’OLD’, FORM=’FORMATTED’)

00000

6100

000000

6102

0000

Read the first line (actually, the first 80 characters of the
line, which is essentially the whole line) of the input file
into INPUTS.

FORMAT(A80)
READ(23,6100)INPUT3

If the first five characters of the first line in the input
file are NOT " $VEC", then read the next line of the input
file. Continue reading lines in the input file until the
first five characters of the line ARE " $VEC".

DO 6102 WHILE (INPUT3(1:5).NE.’ $VEC’)
READ(23,6100)INPUT3
CONTINUE

Set ICOUNT (an integer variable which will be used to keep
track of the number of lines in the " $VEC" group) to zero.

198

ICOUNT=0

If the first five characters of the current line ARE " $VEC",
read the next line in the input file and increment ICOUNT.
Continue to read lines and increment ICOUNT until a line
whose first five characters are " $END" is read.

000000

IF (INPUT3(1:5).EQ.’ $VEC’) THEN
DO 4446 WHILE (INPUT3(1:5).NE.’ $END’)
READ(23,6100)INPUT3
ICOUNT = ICOUNT + 1
4446 CONTINUE

ENDIF
: Set a new variable, JCOUNT, equal to ICOUNT plus one.
c JCOUNT = ICOUNT + 1
2 Return (by rewinding line by line) to the line whose
c first five characters are " $VEC".
c

DO 4447 IBACK = 1, JCOUNT
BACKSPACE (23)
4447 CONTINUE

Read the current line. It should be the " $VEC" line (you
have to print the contents of the line to make sure).
This is the beginning of the GAMESS " $VEC" group.

The actual data begins on the next line.

000000

READ(23,6100)INPUT3

Read the contents of the " $VEC" group into the
DCOEFF array. Note: The " $VEC" group always has

a very specific format -- the first number in every
line is an integer representing the molecular
orbital number, the second number is an integer
representing the current line number (each molecular
orbital has a certain number of lines of coefficients
in the " $VEC" group), and the last five numbers are
coeffients for converting the atomic orbitals into
molecule orbitals. The 4448 statement gives the
format for reading these lines.

0000000000000

4448 FORMAT (I2, 1X, I2, 5E15.8)

0

DO 6103 IVROW=1,ICOUNT-1
READ (23,4448) IMO,LIN,(AVEC(JVCOL),JVCOL=1,5)

DO 6104 IVCOL=1,5
IAO=(LIN-1)*5+IVCOL
DCOEFF(IAO,IMO)=AVEC(IVCOL)

6104 CONTINUE
6103 CONTINUE
c

199

Set the total number of molecular orbitals NMO equal

to the number of the molecular orbital IMO whose coefficients
were just read into DCOEFF. Set the total number of atomic
orbitals IAO equal to the total number of molecular orbitals.

00000

NMO=IMO
NAO=NMO

Write (if desired) the coefficients for converting the
atomic orbitals to molecular orbitals.

WRITE(6,*)’Coefficients for converting a.o.s to m.o.s:’
CALL FLUSH(6)
WRITE(6,*)’ao mo coeff’
CALL FLUSH(6)
DO 8831 I=1,NAO,1
DO 8832 J=1,NAO,1

WRITE(6,*)I,J,DCOEFF(I,J)

CALL FLUSH(6)
C8832 CONTINUE
C8831 CONTINUE

000000000000

c
c Close the "mod..." input file.
c

CLOSE(23)
c
c Open the file containing the dipole moment integrals.
c

OPEN (UNIT=25, FILE=’fort.250’,

&STATUS= ’ OLD ’ , FORM= ’ FORMATTED ’ )
c
c Read the first line (actually, the first 80 characters
c of the first line) into STUFF, and read the second line
c (again, the first 80 characters of the second line) into
c STUFF2.
c
2277 FORMAT(A80)

READ(25,2277)STUFF

READ(25,2277)STUFF2
c
c Calculate the total number of lines of data in the
c file containing the dipole moment integrals and set
c this number equal to LIM. This number is obtained
c by multiplying the square of the number of atomic
c orbitals (NAO‘2) by three.
c

LIM=(NAO*NAO*3)
c
c Read the dipole moment integrals into the AAMU array.
c Also, write these integrals, if desired.
c
c WRITE(6,*)’IJ ’,’KJ ’,’LJ ’,’AAMU(IJ,KJ,LJ)’
c CALL FLUSH(6)

DO 2276 NL=1,LIM,1
READ(25,*)IJ,KJ,LJ,XXX

200

C
C

0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2276

902

904
903
901

AAMU(IJ,KJ,LJ)=XXX
WRITE(6,*)IJ,KJ,LJ,AAMU(IJ,KJ,LJ)
CALL FLUSH(6)

CONTINUE

Close the "fort.250" input file.
CLOSE(25)

Set each matrix element in the ILEFT, IRIGHT, ISIGNRHO,
B, BAOl and BAO arrays equal to zero.

DO 901 IKH=1,NC3,1
DO 902 JKH=1,NC3,1
ILEFT(IKH,JKH)=0
IRIGHT(IKH,JKH)=0
ISIGNRHO(IKH,JKH)=0
CONTINUE
DO 903 JKH=1,NTOT1,1
DO 904 KKH=1,NTOT1,1
B(IKH,JKH,KKH)=0.0d0
BA01(IKH,JKH,KKH)=0.0d0
BAO(IKH,JKH,KKH)=0.0d0
CONTINUE
CONTINUE
CONTINUE

***************************************************

At this point, we will begin to calculate the
matrix elements of the electronic charge-density
operator involving determinants that differ by
one spin orbital occupancy.

***************************************************

For each pair of configurations (II, JJ), where

JJ is less than II and II and JJ differ by

only one spin orbital occupancy, determine the
spin orbital which is occupied in II and vacant

in JJ and set this orbital equal to ILEFTS. Then,
convert spin orbital ILEFTS to a spatial orbital
and set the result equal to ILEFT(II,JJ). Similarly,
determine the spin orbital which is occupied in JJ
and vacant in II and set this orbital equal to
IRIGHTS. Then, convert spin orbital IRIGHTS to a
spatial orbital and set the result equal to
IRIGHT(II,JJ).

DO 7001 II=2,NC3,1
DO 7002 JJ=1,(II-1),1
IF ((TOTDIFF(II,JJ)/2).EQ.1) THEN
DO 410 KJ=1,NTOT2,1
IF ((C3(II,KJ)-C3(JJ,KJ)).EQ.1) THEN

201

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0

410

448

449

7002
7001

ILEFT(II,JJ)=((KJ+1)/2)
ILEFTS=KJ

ENDIF

IF ((C3(JJ,KJ)-C3(II,KJ)).EQ.1) THEN
IRIGHT(II,JJ)=((KJ+1)/2)
IRIGHTS=KJ

ENDIF

CONTINUE

Set ISUMLEFT (a variable which will be used to
determine the number of occupied spin orbitals
in II that are before ILEFTS) and ISUMRIGHT (a
variable which will be used to determine the
number of occupied spin orbitals in JJ that are
before IRIGHTS) equal to zero.

ISUMLEFT=0
ISUMRIGHT=0

Count the number of occupied spin orbitals in II
that occur before ILEFTS. For each of these
occupied orbitals, increment ISUMLEFT. Then,
count the number of occupied spin orbitals in JJ
that occur before IRIGHTS. For each of these
occupied orbitals, increment ISUMRIGHT.

DO 448 ML=1,ILEFTS,1
IF (C3(II,ML).EQ.1) THEN
ISUMLEFT=ISUMLEFT+1
ENDIF
CONTINUE
DO 449 MK=1,IRIGHTS,1
IF (C3(JJ,MK).EQ.1) THEN
ISUMRIGHT=ISUMRIGHT+1
ENDIF
CONTINUE

Now, we will determine the sign of the matrix element of the
electronic charge-density operator involving determinants II

and JJ, and set the result equal to ISIGNRHO(II,JJ). The default
value of ISIGNRHO is -1. If determinants II and JJ have the

same number of occupied spin orbitals before ILEFTS and IRIGHTS
(respectively), then set ISIGNRHO equal to 1.

ISIGNRHO(II,JJ)=-1
IF (ISUMLEFT.EQ.ISUMRIGHT) THEN
ISIGNRHO(II,JJ)=1
ENDIF
ENDIF
CONTINUE
CONTINUE

*********************************************************

202

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

415
7003

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

At this point, we are almost done calculating the matrix
elements of the electronic charge-density operator involving
determinants that differ by one spin orbital occupancy.
Before finishing this calculation, however, we will begin to
calculate the matrix elements of the electronic charge-
density operator involving identical determinants.

*********************************************************

For each pair of identical determinants (II, II), set the
IOK1(II) and IOK2(II) matrix elements equal to zero. Then,
find the first occupied spin orbital in determinant II,
convert this spin orbital to a spatial orbital, and set
the result to IOK1(II). Then, find the second occupied
spin orbital in determinant II, convert this spin orbital
to a spatial orbital, and set the result to IOK2(II).

DO 7003 II=1,NC3,1
IOK1(II)=0
IOK2(II)=0
DO 415 KJ=1,NTOT2,1
IF ((C3(II,KJ).EQ.1) .AND. (IOK1(II).NE.0)) THEN
IOK2(II)=((KJ+1)/2)
ENDIF
IF ((C3(II,KJ).EQ.1) .AND. (IOK1(II).EQ.0)) THEN
IOK1(II)=((KJ+1)/2)
ENDIF
CONTINUE
CONTINUE

*********************************************************

We are almost done calculating the matrix elements of the
electronic charge-density operator involving identical
determinants. Before finishing this calculation, however,
we will finish calculating the matrix elements of the
electronic charge-density operator involving determinants
that differ by one spin orbital occupancy.

We now have enough information to begin computing the CI
coefficients’ contribution to the charge-density
susceptibility.

*********************************************************

For each determinant II, where II is determinant 2 or
larger, set the CI coefficient for the II determinant
and state 1 equal (CJ(II,1)) to CJIND. Then, for each
pair of determinants (II,JJ), where JJ is less than II
and where the two determinants differ by only one spin
orbital occupancy, set ILEFT(II,JJ) (determined earlier
-- ILEFT(II,JJ) is the spatial orbital corresponding

to the spin orbital that is occupied in II but not in
JJ) equal to LL1. Also, set IRIGHT(II,JJ) (determined

203

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6997

7005
7004 CONTINUE

0 0 0 0 0

000000000

6998

7007
7006

earlier -- IRIGHT(II,JJ) is the spatial orbital corres-
ponding to the spin orbital that is occupied in JJ but
not in II) equal to MM1. At this point, we will assign
the sign of the matrix element of the electronic charge-
density operator involving determinants II and JJ
(ISIGHRHO(II,JJ) to the CI coefficients’ contribution

to the charge-density susceptibility for this matrix
element by multiplying ISIGNRHO(II,JJ) by CJIND, setting
the result equal to VAL, and multiplying VAL by CJ(JJ,K).
Then, we complete the calculation of the CI coefficents’
contribution to the charge-density susceptibility for
for these matrix elements by determining CJ(JJ,K)*VAL
for each excited state K, and setting the result equal
to B(K,LL1,MM1).

DO 7004 II=2,NC3,1
CJIND=CJ(II,1)
DO 7005 JJ=1,(II-1),1
IF ((TOTDIFF(II,JJ)/2).EQ.1) THEN
LL1=ILEFT(II,JJ)
MM1=IRIGHT(II,JJ)
VAL=(ISIGNRHO(II,JJ)*CJIND)
DO 6997 K=2,NC3,1
B(K,LLl,MM1)=B(K,LL1,MM1)+(CJ(JJ,K)*VAL)
CONTINUE
ENDIF
CONTINUE

Repeat the process described in the above paragraph for all
pairs of determinants (II, JJ), where II and JJ differ by
one spin orbital occupancy and where JJ is GREATER than II.

DO 7006 II=1,(NC3-1),1
CJIND=CJ(II,1)
DO 7007 JJ=(II+1),NC3,1
IF ((TOTDIFF(II,JJ)/2).EQ.1) THEN
LL1=ILEFT(JJ,II)
MM1=IRIGHT(JJ,II)
VAL=(ISIGNRHO(JJ,II)*CJIND)
DO 6998 K=2,NC3,1
B(K,LL1,MM1)=B(K,LL1,MM1)+(CJ(JJ,K)*VAL)
CONTINUE
ENDIF
CONTINUE
CONTINUE

********************************************************

We have finished calculating the CI coefficients’
contribution to the charge-density susceptibility

for matrix elements of the electronic charge-density
operator involving pairs of determinants (II, JJ) that
differ by one spin orbital occupancy.

204

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6999
7008

0 0 0 0 0 0 0 0 0 0 0 0 0 0

Now, we will calculate the CI coefficients’ contribution
to the charge-density susceptibility for matrix elements
of the electronic charge-density operator involving pairs
of identical determinants (II, II).

********************************************************

For each determinants II, set the spatial equivalent of
the first occupied spin orbital in II (i.e., IOK1(II))
equal to LL1, and set the spatial equivalent of the
second occupied spin orbital in II (i.e., IOK2(II))
equal to MM1. Then, set the CI coefficient for the II
determinant and the ground state (i.e., CJ(II,1))
equal to CJIND. At this point, we will split the
calculation of the CI coefficients’ contribution to
the charge-density susceptibility for matrix elements
of the electronic charge-density operator involving
identical determinants (II, II) into two parts: We

we calculate the contribution due to spatial orbitals
LL1 and MM1 separately. Specifically, for each excited
state K, we will calculate the CI coefficients’
contribution to the charge-density susceptibility

due to orbital LL1 by multiplying the coefficient

for determinant II and state K (i.e., CJ(II,K)) by
CJIND, and setting this result equal to B(K,LL1,LL1).
Similarly, we will calculate the same contribution
for each K due to orbital MM1 by multiplying

CJ(II,K) by CJIND and setting the result equal to
B(K,MM1,MM1).

DO 7008 II=1,NC3,1

LL1=IOK1(II)

MM1=IOK2(II)

CJIND=CJ(II,1)

DO 6999 K=2,NC3,1
B(K,LL1,LL1)=B(K,LL1,LL1)+(CJ(II,K)*CJIND)
B(K,MM1,MM1)=B(K,MM1,MM1)+(CJ(II,K)*CJIND)

CONTINUE

CONTINUE

********************************************************

We have finished calculating the CI coefficients’
contribution to the charge-density susceptibility.

Next, calculate the expansion coefficients’ (i.e., the
coefficients for converting atomic orbitals to molecular
orbitals) contribution to the susceptibility, and
combine this with the CI coefficients’ contribution

to the charge-density susceptibility.

********************************************************

DO 7014 K=2,NC3,1
DO 7011 LKH=1,NTOT1,1

205

0

0 0 0 0 0 0 0 0 0

7013
7012
7011

&
7016
7015
7010
7014

6036
6035
6034

DO 7012 JKH=1,NTOT1,1
DO 7013 MKH=1,NTOT1,1
BAO1(K,LKH,JKH)=BAOI(K,LKH,JKH)+
(B(K,LKH,MKH)*DCOEFF(JKH,MKH))
CONTINUE
CONTINUE
CONTINUE
DO 7010 IKH=1,NTOT1,1
DO 7015 JKH=1,NTOT1,1
DO 7016 LKH=1,NTOT1,1
BAO(K,IKH,JKH)=BAO(K,IKH,JKH)+
(BA01(K,LKH,JKH)*DCOEFF(IKH,LKH))
CONTINUE
CONTINUE
CONTINUE
CONTINUE

Write the contents of the BAO array, if desired.

WRITE(6,*)’K,KK,LL,BAO(K,KK,LL)’
CALL FLUSH(6)
DO 6034 K=2,NC3,1
DO 6035 KK=1,NAO,1
DO 6036 LL=1,NAO,1
WRITE(6,*)K,KK,LL,BAO(K,KK,LL)
CALL FLUSH(6)
CONTINUE
CONTINUE
CONTINUE

***************************************************************

Complete the calculation of the contribution of all coefficients
to the charge-density susceptibilty, and add the frequency
dependence to the susceptibility.

***************************************************************

DO 6040 NP=1,NAO,1
DO 6041 NQ=1,NAO,1
DO 6042 NT=1,NAO,1
DO 6043 NU=1,NAO,1
DO 6044 K=2,NC3,1
ABIG1(NP,NQ,NT,NU)=ABIGI(NP,NQ,NT,NU)

& +((BAO(K,NP,NQ)*BAO(K,NT,NU))/(E(K)-E(1)-OMEGA))

ABIG2(NP,NQ,NT,NU)=ABIG2(NP,NQ,NT,NU)

& +((BAO(K,NP,NQ)*BAO(K,NT,NU))/(E(K)-E(1)+OMEGA))

6044
6043
6042
6041
6040

CONTINUE
CONTINUE
CONTINUE
CONTINUE
CONTINUE

Write the ABIG arrays, if desired.

206

0 0 0 0 0 0

c
c6048
c6047
c6046

&

DO 6045 NP=1,NAO,1
DO 6046 NQ=1,NAO,1
DO 6047 NT=1,NAO,1
DO 6048 NU=1,NAO,1
WRITE(6,*)NP,NQ,NT,NU,
ABIG1(NP,NQ,NT,NU)
CALL FLUSH(6)
CONTINUE
CONTINUE
CONTINUE

C6045 CONTINUE

C

0 0 0 0 0 0

c
c6052
c6051
c6050

&

DO 6049 NP=1,NAO,1
DO 6050 NU=1,NAO,1
DO 6051 NT=1,NAO,1
DO 6052 NU=1,NAO,1
WRITE(6,*)NP,NQ,NT,NU,
ABIG2(NP,NQ,NT,NU)
CALL FLUSH(6)
CONTINUE
CONTINUE
CONTINUE

C6049 CONTINUE

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

*****************************************************

Calculate the charge-density susceptibility at each
specified x, y, z and x’, y’, 2’. For each x, y, z
and x’, y’, z’, calculate the contribution of the
atomic orbitals evaluated at these points to the
susceptibility, and combine this result with the
coefficients’ contribution to the susceptibility.

Also, calculate the xx, yy, zz, and xy components
of the polarizability. Unlike the susceptibility,
however, the polarizability components will only
be calculated once, since these quantities are
calculated from integrals over all x, y, z.

*****************************************************

Set the values of x’, y’ and z’ to XRP, YRP, and ZRP,
respectively.

XRP=0.0d0
YRP=0.0d0
ZRP=0.0d0

Call the subroutine (i. e., subroutine AOROUT) for
evaluating the atomic orbitals

at any x, y, z or x’, y’, 2’. Obtain the values of the
atomic orbitals evaluated at x’, y’, and z’ from
AOROUT. These values will be returned from AOROUT in
the AAO array.

207

00

00000

6369

6871

000000000000000000000000000

CALL AOROUT(XRP,YRP,ZRP,AAO)
Write x’, y’, and z’.

WRITE(6,*)’r-prime:’,XRP,YRP,ZRP
CALL FLUSH(6)

Set each AAO(N) equal an ARPRIME(N) (essentially,
create an array which is identical to AAO and name
it ARPRIME).

DO 6369 N=1,NAO,1
ARPRIME(N)=AAO(N)
CONTINUE

Write the ARPRIME array, if desired.

DO 6871 J=1,NAO,1
WRITE(80,*)ARPRIME(J)
CONTINUE

Use the three nested do loops that immediately follow
to choose the grid of x, y, 2 points at which to
calculate the spatial dependent part of the
susceptibility. Use the I, J, and K loops to set up
the x, y, and z coordinates, respectively. Each x
coordinate (XX) is selected by taking the value of I
(where I essentially corresponds to the number of

the x point), multiplying this by DELTAXX (which is
the x increment), and adding this to XXO (the initial
x). The y (YY) and z (22) coordinates are set up in
the same way.

When I, J, and K are all equal to one, set the
variables used for calculating the xx, yy, zz, and

xy components of the polarizability equal to zero.
Note: POLXX1, POLXX2, POLYY1, POLYY2, POLZZl, POLZZ2,
POLXY1, and POLXY2 are intermediate variables used

in calculating the xx (POLXX), yy (POLYY), zz (POLZZ),
and xy (POLXY) components of the polarizability.

DO 6361 I=1,1
DO 6362 J=1,1
DO 6363 K=-1,1

IF (I.EQ.1 .AND. J.EQ.1

& .AND. K.EQ.1) THEN

POLXX1=0.0d0
POLXX2=0.0d0
POLYY1=0.0d0
POLYY2=0.0d0
POLZZl=0.0d0
POLZZ2=0.0d0
POLXY1=0.0d0

208

000000

000000

POLXY2=0.0d0

POLXX=0.0d0
POLYY=0.0d0
POLZZ=0.0d0
POLXY=0.0d0

ENDIF

XXO=0.0d0
YYO=0.0d0
ZZO=0.0d0
DELTAXX=0.0d0
DELTAYY=0.0d0
DELTAZZ=0.05d0

XX=(XXO+I*DELTAXX)
YY=(YYO+J*DELTAYY)
ZZ=(ZZO+K*DELTAZZ)

Call the AOROUT subroutine for evaluating the atomic orbitals
at x, y, z. The atomic orbitals will be evaluated at XX, YY,

22 and their values will be returned to the main program in the
AAO array.

CALL AOROUT(XX,YY,ZZ,AAO)
Write the x (XX), y (yy), and z (22) coordinates, if desired.

WRITE(6,*)XX,YY,ZZ
CALL FLUSH(6)

A1SUM4=0.0d0
A2SUM4=0.0d0
DO 5365 IM=1,NAO,1
A1$UM3=0.0d0
A2SUM3=0.0d0
DO 5366 JM=1,NAO,1
A1$UM2=0.0d0
A2SUM2=0.0d0
DO 5367 KM=1,NAO,1
A1SUM1=0.0d0
A2SUM1=0.0d0
DO 5368 LM=1,NAO,1

A1SUM1=A1SUM1+
(ABIGl(IM,JM,KM,LM)*ARPRIME(LM))

IF (I.EQ.1 .AND. J.EQ.1 .AND.

& K.EQ.1) THEN

POLXX1=POLXX1+
(ABIG1(IM,JM,KM,LM)*AAMU(IM,JM,1)
*AAMU(KM,LM,1))

209

0

0000

POLXX2=POLXX2+

& (ABIG2(IM,JM,KM,LM)*AAMU(IM,JM,1)
& *AAMU(KM,LM,1))

POLYY1=POLYY1+
& (ABIGi(IM,JM,KM,LM)*AAMU(IM,JM,2)
& *AAMU(KM,LM,2))

POLYY2=POLYY2+
& (ABIG2(IM,JM,KM,LM)*AAMU(IM,JM,2)
& *AAMU(KM,LM,2))

POLZZl=POLZZI+
& (ABIGi(IM,JM,KM,LM)*AAMU(IM,JM,3)
& *AAMU(KM,LM,3))

POLZZ2=POLZZ2+
& (ABIG2(IM,JM,KM,LM)*AAMU(IM,JM,3)
& *AAMU(KM,LM,3))

POLXY1=POLXY1+
& (ABIGI(IM,JM,KM,LM)*AAMU(IM,JM,1)
& *AAMU(KM,LM,2))

POLXY2=POLXY2+
& (ABIG2(IM,JM,KM,LM)*AAMU(IM,JM,1)
& *AAMU(KM,LM,2))

ENDIF
A2SUM1=A2SUM1+
& (ABIG2(IM,JM,KM,LM)*AAO(LM))
5368 CONTINUE
AISUM2=AISUM2
& +(AISUM1*ARPRIME(KM))
A2SUM2=A2SUM2
& +(A2SUM1*AAO(KM))
5367 CONTINUE

A1$UM3=A1SUM3+(A13UM2*AAO(JM))
A2SUM3=A2SUM3+(A2SUM2
& *ARPRIME(JM))

5366 CONTINUE
A1SUM4=A1$UM4+(A1SUM3*AAO(IM))
A2SUM4=A2SUM4+(A2SUM3*ARPRIME(IM))

5365 CONTINUE

CHI=A1SUM4+A2SUM4

Write the value of the charge-density susceptibility
at x (XX), y (YY), z (22).

WRITE(6,*)CHI
CALL FLUSH(6)

210

Write the xx, yy, 22, and xy components of the
polarizability. Note: These will only be written once;
when I, J, and K equal one.

0000

IF (I.EQ.1 .AND. J.EQ.1
& .AND. K.EQ.1) THEN

POLXX=POLXX1+POLXX2
WRITE(6,*)’POLXX:’,POLXX
CALL FLUSH(6)

POLYY=POLYY1+POLYY2
WRITE(6,*)’POLYY:’,POLYY
CALL FLUSH(6)

POLZZ=POLZZi+POLZZ2
WRITE(6,*)’POLZZ:’,POLZZ
CALL FLUSH(6)

POLXY=POLXY1+POLXY2
WRITE(6,*)’POLXY:’,POLXY
CALL FLUSH(6)

ENDIF
C
6363 CONTINUE
6362 CONTINUE
6361 CONTINUE
C
END

0

SUBROUTINE SLATER1(II,JJ)

This subroutine evaluates the matrix elements of pairs of
determinants which differ in the location of one electron.

0000

IMPLICIT DOUBLE PRECISION (A-H,O-Z)

0

INTEGER C3

COMMON /BLOCK1/ H(10000,10000)

COMMON /BLOCK2/ E1SPAT(100,100)

COMMON /BLOCK3/ E2SPAT(100,100,100,100)
COMMON /BLOCK4/ C3(10000,200)

COMMON /BLOCK6/ NTOT2

COMMON /BLOCK12/ N03

COMMON /BLOCK15/ NAO

11A=0
I2A=0

Determine where (i.e., what spin orbitals) determinants
II and JJ differ, and set these orbitals to 11A and 12A.

0000

DO 151 L=1,NTOT2,1

211

00000000000

00

000000

0

000

00000

IF ((C3(II,L)-C3(JJ,L)).EQ.1
& .AND. I1A.EQ.0) THEN
IiA=L
ENDIF
IF (((C3(II,L)-C3(JJ,L)).EQ.(-1))
& .AND. I1A.NE.L .AND. I2A.EQ.0) THEN

I2A=L
ENDIF
151 CONTINUE
Determine the sign of the Hamiltonian matrix element which
is being formed from determinants II and JJ, where II and JJ
differ in the occupation of one spin orbital.
First, count the number of occupied orbitals in determinant
II that are lower in energy than the first orbital with
different occupancies in II and JJ, and set the total to
(Before beginning, set SUMI1A to zero).
SUMIlA=0.0D0
DO 448 ML=1,(IlA-1),1
IF (C3(II,ML).EQ.1) THEN
SUMI1A=SUMI1A+1.0D0
ENDIF
448 CONTINUE
Multiply SUMIlA by -1 and set the result to SIGNIiA.
SIGNIIA=((-1.0D0)**SUMIIA)
Count the number of occupied orbitals in determinant
JJ that are lower in energy than the second orbital with
different occupancies in II and JJ, and set the total to
SUMI2A.
(Before beginning, set SUMI2A to zero).
SUMI2A=0.0D0
DO 449 MK=1,(I2A-1),1
IF (CB(JJ,MK).EQ.1) THEN
SUMI2A=SUMI2A+1.0DO
ENDIF
449 CONTINUE
Multiply SUMI2A by -1 and set the result to SIGNI2A.
SIGNI2A=((-1.0D0)**SUMI2A)
Determine the overall sign of H(II,JJ)
by multiplying SIGNIIA by SIGNI2A and setting the result
equal to SIGNIA.

SIGNIA=(SIGNI1A*SIGNI2A)

212

000000000

00000

0000

Convert the spin orbitals 11A, I2A to spatial orbitals I1AS and
I2AS, respectively. Each unique pair of consecutively-numbered
spin orbitals is assigned to the same spatial orbital. Also,
set the spins of I1AS and I2AS. If the spin orbital is even,
set the spin of IlAS or I2AS to -1, and if the spin

orbital is odd, set the spin of IlAS or I2AS to +1.
(IlASPIN=spin of IlAS, I2ASPIN=spin of I2AS)

IF (MOD(I1A,2).EQ.0) THEN
IlAS=(11A/2)
I1ASPIN=-1
ELSE
IF (MOD(IlA,2).EQ.1) THEN
IIAS=((I1A+1)/2)
IlASPIN=1
ENDIF
ENDIF

IF (MOD(I2A,2).EQ.0) THEN
I2AS=(I2A/2)
I2ASPIN=-1
ELSE
IF (MOD(I2A,2).EQ.1) THEN
I2AS=((I2A+1)/2)
I2ASPIN=1
ENDIF
ENDIF

Determine if the one-electron integral contribution to the
Hamiltonian matrix element H(II,JJ) is spin-conserved. If
so, add this integral to H(II,JJ).

IF (IlASPIN.EQ.I2ASPIN) THEN
H(II,JJ)=H(II,JJ)+E1SPAT(IlAS,I2AS)
ENDIF

Convert the two-electron integral contribution(s) to H(II,JJ)
to (an) integra1(s) over spatial orbitals.

DO 2000 K=1,NTOT2,1
IF (K.NE.IlA .AND. K.NE.I2A
.AND. C3(II,K).EQ.1) THEN
IF (MOD(K,2).EQ.0) THEN
KS=(K/2)
KSPIN=-1
ELSE
IF (MOD(K,2).EQ.1) THEN
KS=((K+1)/2)
KSPIN=1
ENDIF
ENDIF
IF (I1ASPIN.EQ.I2ASPIN .AND.

& I2ASPIN.EQ.KSPIN) THEN

H(II,JJ)=H(II,JJ)+

213

& E2SPAT(KS,KS,11AS,I2AS)-
& E2SPAT(KS,I2AS,I1AS,KS)
ENDIF
IF (IiASPIN.EQ.I2ASPIN .AND. I2ASPIN.NE.KSPIN)
& THEN
H(II,JJ)=H(II,JJ)+
& E2SPAT(KS,KS,I1AS,I2AS)
ENDIF
ENDIF
2000 CONTINUE
C
H(II,JJ)=(H(II,JJ)*SIGNIA)
C
END

0

SUBROUTINE SLATER2(III,JJJ)

This subroutine formulates the two-electron integral for a pair
of determinants which differ in the locations of two electrons.

0000

IMPLICIT DOUBLE PRECISION (A-H,O-Z)

0

COMMON /BLOCK1/ H(10000,10000)

COMMON /BLOCK2/ E1SPAT(100,100)

COMMON /BLOCK3/ E2SPAT(100,100,100,100)
COMMON /BLOCK4/ C3(10000,200)

COMMON /BLOCK6/ NTOT2

COMMON /BLOCK12/ NC3

COMMON /BLOCK15/ NAO

0

INTEGER C3

Set the spin orbitals (in ascending order) which differ in
occupation number in determinants I and J to I1,I2,I3 and I4.

(Before beginning, set I1,I2,I3 and I4 to zero).

WRITE(6,*) ’slater 2 1st check’
CALL FLUSH(6)

000000000

Il=0
I2=0
I3=0
I4=0
DO 150 KK=1,NTOT2,1
IF ((C3(III,KK)-C3(JJJ,KK)).EQ.1)
& THEN
IF (I1.EQ.0) THEN
Il=KK
GO TO 150
ENDIF
IF (11.NE.0 .AND. I2.EQ.0) THEN
I2=KK
ENDIF
ENDIF

214

0000000000

000

00000000

000

0000

150 CONTINUE
DO 170 KK=1,NTOT2,1
IF ((C3(III,KK)-C3(JJJ,KK)).EQ.(-1))
& THEN
IF (I3.EQ.0) THEN
13=KK
GO TO 170
ENDIF
IF (I4.EQ.0) THEN
I4=KK
ENDIF
ENDIF
170 CONTINUE

Determine the sign of the matrix element H(III,JJJ).

First, count the number of occupied orbitals in
determinant III that are lower in energy than the ist
orbital in III which has a different occupation number
than in JJJ, and set the total to SUMI1.

(Before beginning, set SUMI1 to zero).

SUM11=0.0D0
DO 450 MM=1,(I1-1),1
IF (C3(III,MM).EQ.1) THEN
SUMI1=SUMI1+1.0D0
ENDIF
450 CONTINUE

Then, multiply SUMI1 by -1, and set the result to SIGNIl.
SIGNIl=((-1.0DO)**SUM11)

Count the number of occupied orbitals in
determinant III that are lower in energy than the 2nd
orbital in III which has a different occupation number
than in JJJ, and set the total to SUMI2.

(Before beginning, set SUMI2 to zero).

SUMI2=0.0D0
DO 451 MN=1,(I2-1),1
IF (C3(III,MN).EQ.1) THEN
SUMI2=SUMI2+1.0D0
ENDIF
451 CONTINUE

Multiply SUMI2 by -1, and set the result to SIGNI2.
SIGNI2=((-1.0D0)**SUMI2)
Count the number of occupied orbitals in

determinant JJJ that are lower in energy than the 1st
orbital in JJJ which has a different occupation number

215

0000

000

0 00000000

00

00000

0000000000

452

453

than in III, and set the total to SUMIB.
(Before beginning, set SUMI3 to zero).

SUMI3=0.0D0
DO 452 MO=1,(I3-1),1
IF (C3(JJJ,MO).EQ.1) THEN
SUMI3=SUMI3+1.0d0
ENDIF
CONTINUE

Multiply SUMIB by -1, and set the result to SIGNI3.
SIGN13=((-1.0D0)**SUM13)

Count the number of occupied orbitals in
determinant JJJ that are lower in energy than the 2nd
orbital in JJJ which has a different occupation number
than in III, and set the total to SUMI4.

(Before beginning, set SUMI4 to zero).

SUMI4=0.0D0
DO 453 MP=1,(I4-1),1
IF (C3(JJJ,MP).EQ.1) THEN
SUMI4=SUMI4+1.0d0
ENDIF
CONTINUE

Multiply SUMI4 by -1, and set the result to SIGNI4.
SIGNI4=((-1.0D0)**SUMI4)

Calculate the overall sign of the two-electron integral(s)
by multiplying the four signs together and setting the result
to SIGNI.

SIGNI=(SIGNI1*SIGNI2*SIGNI3*SIGNI4)

Convert the spin orbitals Il,I2,I3 and I4 to spatial orbitals
ISI,IS2,ISB and 184, respectively. Each unique pair of
consecutive spin orbitals is assigned to the same spatial
orbital. Set the spins of ISl,IS2,IS3 and 184. If the spin
orbital is odd-numbered, set the spin of the spatial orbital
to -1, and if the spin orbital is even, set the spin of the
spatial orbital to +1. The variables for the spins of I81,
182, ISB and IS4 are IISPIN, I2SPIN, IBSPIN, and I4SPIN.

IF (MOD(Il,2).EQ.0) THEN
IS1=(11/2)
IISPIN=-1
ELSE
IF (MOD(Il,2).EQ.1) THEN
IS1=((11+1)/2)
IlSPIN=1

216

0000000000000

ELSE
ENDIF
ENDIF

IF (MOD(I2,2).EQ.0) THEN
IS2=(I2/2)
I2SPIN=-1
ELSE
IF (MOD(I2,2).EQ.1) THEN
IS2=((I2+1)/2)
I2SPIN=1
ELSE
ENDIF
ENDIF

IF (MOD(13,2).EQ.0) THEN
IS3=(I3/2)
ISSPIN=-1
ELSE
IF (MOD(13,2).EQ.1) THEN
ISB=((13+1)/2)
IBSPIN=1
ELSE
ENDIF
ENDIF

IF (MOD(I4,2).EQ.0) THEN
IS4=(I4/2)
I4SPIN=-1
ELSE
IF (MOD(I4,2).EQ.1) THEN
IS4=((I4+1)/2)
I4SPIN=1
ELSE
ENDIF
ENDIF

WRITE(6,*) ’slater2, ok here’
CALL FLUSH(6)

Determine the two-electron integral contribution to
H(III,JJJ).

First, determine whether the two-electron integrals
formed by applying the Slater rules to determinants
III and JJJ are spin-allowed.

Add each spin-allowed two-electron integral to H(III,JJJ).

IF (I1SPIN.EQ.IBSPIN .AND.

&I2SPIN.EQ.I4SPIN .AND. ISSPIN.NE.I2SPIN) THEN

H(III,JJJ)=H(III,JJJ)+E2SPAT(IS1,ISB,IS2,IS4)
ENDIF

IF (I1SPIN.EQ.I4SPIN .AND.

217

00000000000000000000000000000000

&I2SPIN.EO.13SPIN .AND. IiSPIN.NE.IBSPIN) THEN
H(III,JJJ)=H(III,JJJ)-
& E2SPAT(ISi,IS4,IS2,IS3)
ENDIF

IF (I1SPIN.EQ.I3SPIN .AND.
&I2SPIN.EQ.I4SPIN .AND. IlSPIN.EQ.I4SPIN) THEN
H(III,JJJ)=H(III,JJJ)+
& E2SPAT(IS1,ISB,IS2,IS4)-
& E2SPAT(ISl,IS4,IS2,ISS)
ENDIF

Calculate the sign of H(III,JJJ).
H(III,JJJ)=(H(III,JJJ)*SIGNI)
END
subroutine rsm(nm,n,a,w,m,z,fwork,iwork,ierr)
integer n,nm,m,iwork(n),ierr
integer k1,k2,k3,k4,k5,k6,k7
double precision a(nm,n),w(n),z(nm,m),fwork(1)
this subroutine calls the recommended sequence of
subroutines from the eigensystem subroutine package (eispack)
to find all of the eigenvalues and some of the eigenvectors
of a real symmetric matrix.
on input
nm must be set to the row dimension of the two-dimensional
array parameters as declared in the calling program
dimension statement.
n is the order of the matrix a.
a contains the real symmetric matrix.
m the eigenvectors corresponding to the first m eigenvalues
are to be computed.
if m = 0 then no eigenvectors are computed.
if m = n then all of the eigenvectors are computed.
on output

w contains all n eigenvalues in ascending order.

2 contains the orthonormal eigenvectors associated with
the first m eigenvalues.

ierr is an integer output variable set equal to an error

completion code described in the documentation for tqlrat,
imtqlv and tinvit. the normal completion code is zero.

218

00000000000

000

o n n n n o o 0 0 0 n o n o

10

50

X

X

fwork is a temporary storage array of dimension 8*n.
iwork is an integer temporary storage array of dimension n.

questions and comments should be directed to burton s. garbow,
mathematics and computer science div, argonne national laboratory

this version dated august 1983.

 

 

 

ierr = 10 * n
if (n .gt. nm .or. m .gt. nm) go to 50

k1 = 1

k2 = k1 + n

k3 = k2 + n

k4 = k3 + n

k5 = k4 + n

k6 = k5 + n

k7 = k6 + n

k8 = k7 + n

if (m .gt. 0) go to 10

.......... find eigenvalues only ..........

call tred1(nm,n,a,w,fwork(k1),fwork(k2))

call tqlrat(n,w,fwork(k2),ierr)

go to 50

.......... find all eigenvalues and m eigenvectors ..........

call tred1(nm,n,a,fwork(k1),fwork(k2),fwork(k3))

call imtqlv(n,fwork(kl),fwork(k2),fwork(k3),w,iwork,
ierr,fwork(k4))

call tinvit(nm,n,fwork(k1),fwork(k2),fwork(k3),m,w,iwork,z,ierr,

fwork(k4),fwork(k5),fwork(k6),fwork(k7),fwork(k8))
call trbak1(nm,n,a,fwork(k2),m,z)
return
end

double precision function epslon (x)
double precision x

estimate unit roundoff in quantities of size x.
double precision a,b,c,eps

this program should function properly on all systems
satisfying the following two assumptions,
1. the base used in representing floating point
numbers is not a power of three.
2. the quantity a in statement 10 is represented to
the accuracy used in floating point variables
that are stored in memory.
the statement number 10 and the go to 10 are intended to
force optimizing compilers to generate code satisfying
assumption 2.
under these assumptions, it should be true that,

a is not exactly equal to four-thirds,

b has a zero for its last bit or digit,

219

00000000

000000000000000000000000000000000

10

c is not exactly equal to one,
eps measures the separation of 1.0 from
the next larger floating point number.
the developers of eispack would appreciate being informed
about any systems where these assumptions do not hold.

this version dated 4/6/83.

a = 4.0d0/3.0d0
b = a - 1.0d0
c = b + b + b

eps = dabs(c-1.0d0)

if (eps .eq. 0.0d0) go to 10

epslon = eps*dabs(x)

return

end

subroutine imtqlv(n,d,e,e2,w,ind,ierr,rv1)

integer i,j,k,1,m,n,ii,mml,tag,ierr

double precision d(n),e(n),e2(n),w(n),rv1(n)
double precision b,c,f,g,p,r,s,tst1,tst2,pythag
integer ind(n)

this subroutine is a variant of imtqll which is a translation of
algol procedure imtqll, num. math. 12, 377-383(1968) by martin and
wilkinson, as modified in num. math. 15, 450(1970) by dubrulle.
handbook for auto. comp., vol.ii-linear algebra, 241-248(1971).
this subroutine finds the eigenvalues of a symmetric tridiagonal
matrix by the implicit ql method and associates with them
their corresponding submatrix indices.
on input

n is the order of the matrix.

d contains the diagonal elements of the input matrix.

e contains the subdiagonal elements of the input matrix
in its last n-1 positions. e(1) is arbitrary.

e2 contains the squares of the corresponding elements of e.
e2(1) is arbitrary.

on output
d and e are unaltered.
elements of e2, corresponding to elements of e regarded
as negligible, have been replaced by zero causing the
matrix to split into a direct sum of submatrices.

e2(1) is also set to zero.

w contains the eigenvalues in ascending order. if an
error exit is made, the eigenvalues are correct and

220

000000000000000000000000

100

105

110
120
125
130

ordered for indices 1,2,...ierr-1, but may not be
the smallest eigenvalues.

ind contains the submatrix indices associated with the
corresponding eigenvalues in w -- 1 for eigenvalues
belonging to the first submatrix from the top,
2 for those belonging to the second submatrix, etc..

ierr is set to
zero for normal return,
j if the j-th eigenvalue has not been
determined after 30 iterations.
rv1 is a temporary storage array.

calls pythag for dsqrt(a*a + b*b)

questions and comments should be directed to burton s. garbow,
mathematics and computer science div, argonne national laboratory

this version dated august 1983.

 

 

ierr = 0
k = 0
tag = 0
do 100 i = 1, n
w(i) = d(i)
if (i .ne. 1) rv1(i-1) = e(i)
continue

e2(1) = 0.0d0
rv1(n) = 0.0d0

do 290 l = 1, n
i=0
.......... look for small sub-diagonal element ..........
do 110 m = l, n
if (m .eq. n) go to 120
tst1 = dabs(w(m)) + dabs(w(m+1))
tst2 = tst1 + dabs(rv1(m))
if (tst2 .eq. tstl) go to 120
.......... guard against underflowed element of e2 ..........
if (e2(m+1) .eq. 0.0d0) go to 125
continue

if (m .le. k) go to 130

if (m .ne. n) e2(m+1) = 0.0d0
k = m

tag = tag + 1

p = w(l)

if (m .eq. 1) go to 215
if (j .eq. 30) go to 1000

221

J'=J'+1
.......... form shift ..........
g = (w(l+1) - p) / (2.0d0 * rv1(1))
r = pythag(g,1.0d0)
g = w(m) - p + rv1(l) / (g + dsign(r,g))
s = .0d0
c = .OdO
= .OdO
mml m - 1
.......... for i=m-1 step -1 until 1 do -- ..........
do 200 ii = 1, mml

ll OHH

1 = m - ii

f = s * rv1(i)
b = c * rv1(i)
r = pythag(f,g)

rv1(i+1) = r

if (r .eq. 0.0d0) go to 210

s f / r

g / r

w(i+1) - p

(w(i) - g) * s + 2.0d0 * c * b
s * r

”+1) = g + p

— c * r - b

200 continue

H II II II II II

A

c
8
r
P
w
8

w(l) = w(l) - p
rv1(l)
rv1(m)
go to 105
.......... recover from underflow ..........
210 w(i+1) = w(i+1) - p
rv1(m) = 0.0d0
go to 105
.......... order eigenvalues ..........
215 if (1 .eq. 1) go to 250
.......... for i=1 step -1 until 2 do -- ..........
do 230 ii = 2, 1
i = l + 2 - ii
if (p .ge. w(i-1)) go to 270

8
0.0d0

w(i) = w(i-l)
ind(i) = ind(i-l)
230 continue
250 i = 1
270 w(i) = p
ind(i) = tag

290 continue

go to 1001
.......... set error -- no convergence to an
eigenvalue after 30 iterations ..........
1000 ierr = l
1001 return
end

222

000000000000000000000000000000000000000

100

subroutine tqlrat(n,d,e2,ierr)

integer i,j,l,m,n,ii,11,mm1,ierr
double precision d(n),e2(n)

double precision b,c,f,g,h,p,r,s,t,epslon,pythag

this subroutine is a translation of the algol procedure tqlrat,

algorithm 464, comm. acm 16, 689(1973) by rein

sch.

this subroutine finds the eigenvalues of a symmetric

tridiagonal matrix by the rational ql method.
on input

n is the order of the matrix.

d contains the diagonal elements of the input matrix.

e2 contains the squares of the subdiagonal
input matrix in its last n-1 positions.

on output

elements of the
e2(1) is arbitrary.

d contains the eigenvalues in ascending order. if an

error exit is made, the eigenvalues are c

orrect and

ordered for indices 1,2,...ierr-1, but may not be

the smallest eigenvalues.
e2 has been destroyed.
ierr is set to
zero for normal return,
j if the j-th eigenvalue has not
determined after 30 iterations

calls pythag for dsqrt(a*a + b*b)

been

questions and comments should be directed to burton s. garbow,

mathematics and computer science div, argonne

this version dated august 1983.

national laboratory

 

 

ierr = 0
if (n .eq. 1) go to 1001

do 100 i = 2, n
e2(i-1) = e2(i)

f = 0.0d0
t = 0.0d0
e2(n) = 0.0d0

do 290 l = 1, n

223

i=0

h = dabs(d(l)) + dsqrt(e2(l))
if (t .gt. h) go to 105

t = h

b = epslon(t)

c = b * b

.......... look for small squared sub-diagonal element ..........
105 do 110 m = 1, n
if (e2(m) .1e. c) go to 120
.......... e2(n) is always zero, so there is no exit
through the bottom of the loop ..........
110 continue

120 if (m .eq. 1) go to 210
130 if (j .eq. 30) go to 1000

J=j+1
.......... form shift . ...... .
11 = l + 1
s = dsqrt(e2(l))
g = d(l)
p = (d(ll) - g) / (2.0d0 * s)
r = pythag(p,1.0d0)
d(l) = s / (p + dsign(r,p))

h g - d(l)

do 140 i = 11, n
140 d(i) = d(i) - h

f = f + h
.......... rational ql transformation ..........
g = d(m)
if (g .eq. 0.0d0) g = b
h = g
s = 0.0d0
mml = m - 1

.......... for i=m-1 step -1 until 1 do -- ..........
do 200 ii = 1, mml

1 = m - ii
p=s*h

r = p + e2(i)
e2(i+1) = s * r
s = e2(i) / r

d(i+1) = h + s * (h + d(i))
g = d(i) - e2(i) / 3
if (g .eq. 0.0d0) g = b
h=g*p/r

200 continue

e2(1) = s * g
d(l) = h
.......... guard against underflow in convergence test ..........
if (h .eq. 0.0d0) go to 210
if (dabs(e2(l)) .le. dabs(c/h)) go to 210
e2(1) = h * e2(1)
if (e2(1) .ne. 0.0d0) go to 130

224

000000

210

230

250
270
290

1000
1001

10

20

p = d(l) + f
.......... order eigenvalues ..........
if (1 .eq. 1) go to 250
.......... for i=1 step -1 until 2 do -- ..........
do 230 ii = 2, l
i = l + 2 - ii
if (p .ge. d(i-1)) go to 270
d(i) = d(i—1)
continue
i = 1
d(i) = p
continue
go to 1001
.......... set error -- no convergence to an
eigenvalue after 30 iterations ..........
ierr = 1
return
end

double precision function pythag(a,b)
double precision a,b

finds dsqrt(a**2+b**2) without overflow or destructive underflow

double precision p,r,s,t,u
p = dmax1(dabs(a),dabs(b))
if (p .eq. 0.0d0) go to 20
r = (dmin1(dabs(a),dabs(b))/p)**2
continue
t = 4.0d0 + r
if (t .eq. 4.0d0) go to 20

s = r/t
u = 1.0d0 + 2.0d0*s
P = u*P
r = (s/u)**2 * r
go to 10
pythag = 9
return
end

subroutine tinvit(nm,n,d,e,e2,m,w,ind,z,

x ierr,rv1,rv2,rv3,rv4,rv6)

integer i,j,m,n,p,q,r,s,ii,ip,jj,nm,its,tag,ierr,group
double precision d(n),e(n),e2(n),w(m),z(nm,m),

x rv1(n),rv2(n),rv3(n),rv4(n),rv6(n)

double precision u,v,uk,xu,x0,x1,eps2,ep33,eps4,norm,order,epslon,

x pythag

integer ind(m)

this subroutine is a translation of the inverse iteration tech-
nique in the algol procedure tristurm by peters and wilkinson.
handbook for auto. comp., vol.ii-linear algebra, 418-439(1971).

this subroutine finds those eigenvectors of a tridiagonal

225

0000000000000000000000000000000000000000000000000000000

symmetric matrix corresponding to specified eigenvalues,
using inverse iteration.

on input

on

nm must be set to the row dimension of two-dimensional

array parameters as declared in the calling program
dimension statement.

is the order of the matrix.
contains the diagonal elements of the input matrix.

contains the subdiagonal elements of the input matrix
in its last n-1 positions. e(1) is arbitrary.

e2 contains the squares of the corresponding elements of e,

m

V

with zeros corresponding to negligible elements of e.

e(i) is considered negligible if it is not larger than
the product of the relative machine precision and the sum
of the magnitudes of d(i) and d(i-1). e2(1) must contain
0.0d0 if the eigenvalues are in ascending order, or 2.0d0
if the eigenvalues are in descending order. if bisect,
tridib, or imtqlv has been used to find the eigenvalues,
their output e2 array is exactly what is expected here.

is the number of specified eigenvalues.

contains the m eigenvalues in ascending or descending order.

ind contains in its first m positions the submatrix indices

associated with the corresponding eigenvalues in w --
1 for eigenvalues belonging to the first submatrix from
the top, 2 for those belonging to the second submatrix, etc.

output

all input arrays are unaltered.

2

contains the associated set of orthonormal eigenvectors.
any vector which fails to converge is set to zero.

ierr is set to

zero for normal return,
-r if the eigenvector corresponding to the r-th
eigenvalue fails to converge in 5 iterations.

rv1, rv2, rv3, rv4, and rv6 are temporary storage arrays.

calls pythag for dsqrt(a*a + b*b)

questions and comments should be directed to burton s. garbow,

mathematics and computer science div, argonne national laboratory

this version dated august 1983.

226

0

0000

ierr = 0
if (m .eq. 0) go to 1001
Mg 0
order = 1.0d0 - e2(1)
q=0
.......... establish and process next submatrix ..........
100 p = q + 1
do 120 q = p, n
if (q .eq. n) go to 140
if (e2(q+1) .eq. 0.0d0) go to 140
120 continue
.......... find vectors by inverse iteration ..........
140t 3g = tag + 1
= 0
do 920 r = 1, m
if (ind(r) .ne. tag) go to 920
its = 1
x1 = w(r)
if (s .ne. 0) go to 510
.......... check for isolated root ..........
xu = 1.0d0
if (p .ne. q) go to 490
rv6(p) = 1.0d0
go to 870
490 norm = dabs(d(p))
i=p p+1
do 500 i = ip, q
500 norm = dmax1(norm, dabs(d(i))+dabs(e(i)))
.......... eps2 is the criterion for grouping,
eps3 replaces zero pivots and equal
roots are modified by eps3,
eps4 is taken very small to avoid overflow ..........
eps2 = 1.0d-3 * norm
epsB = epslon(norm)
uk=q-p+1
eps4 = uk * epsS
uk = eps4 / dsqrt(uk)
s = p
505 group = 0
go to 520
.......... look for close or coincident roots ..........
510 if (dabs(xl-xO) .ge. eps2) go to 505
group = group + 1
if (order * (x1 - x0) .1e. 0.0d0) x1 = x0 + order * epsB
.......... elimination with interchanges and
initialization of vector ..........
520 = 0.0d0

do 580 i = p, q

227

rv6(i) = uk
if (1 .eq. p) go to 560
if (dabs(e(i)) .lt. dabs(u)) go to 540
.......... warning -- a divide check may occur here if
e2 array has not been specified correctly ..........
xu = u / e(i)
rv4(i) = xu

rv1(i-1) = e(i)
rv2(i-1) = d(i) - x1
rv3(i-1) = 0.0d0

if (i .ne. q) rv3(i-1) = e(i+1)
u = v - xu * rv2(i-1)
v = -xu * rv3(i-1)
go to 580
540 xu = e(i) / u
rv4(i) = xu
rv1(i-1) u
rv2(i-1) v
rv3(i-1) 0.0d0
560 u = d(i) x1 - xu * v
if (i .ne. q) v = e(i+1)
580 continue

if (u .eq. 0.0d0) u = epsS

rv1(q) = u
rv2(q) = 0.0d0
rv3(q) = 0.0d0

.......... back substitution
for i=q step -1 until p do -- ..........
600 do 620 ii = p, q
i = p + q - ii
rv6(i) = (rv6(i) - u * rv2(i) - v * rv3(i)) / rv1(i)

v = u
u = rv6(i)
620 continue

.......... orthogonalize with respect to previous
members of group ..........
if (group .eq. 0) go to 700

J - r
do 680 jj = 1, group
630 j = ' - 1
if (ind(j) .ne. tag) go to 630
xu = 0.0d0
do 640 i = p, q
640 xu = xu + rv6(i) * z(i,j)
do 660 i = p,
660 rv6(i) = rv6(i) - xu * z(i,j)
680 continue

700 norm = 0.0d0

228

do 720 i = p, q
720 norm = norm + dabs(rv6(i))

if (norm .ge. 1.0d0) go to 840
.......... forward substitution ..........
if (its .eq. 5) go to 830
if (norm .ne. 0.0d0) go to 740
rv6(s) = eps4
s = s + 1
if (s .gt. q) s = p
go to 780
740 xu = eps4 / norm

do 760 i = p, q
760 rv6(i) = rv6(i) * xu

.......... elimination operations on next vector

iterate ..........
780 do 820 i = ip, q
u = rv6(i)

.......... if rv1(i-1) .eq. e(i), a row interchange
was performed earlier in the
triangularization process ..........

if (rv1(i-1) .ne. e(i)) go to 800

u = rv6(i-1)

rv6(i-1) = rv6(i)
800 rv6(i) = u - rv4(i) * rv6(i-1)
820 continue

its = its + 1
go to 600
.......... set error -- non~converged eigenvector ..........
830 ierr = -r
xu = 0.0d0
go to 870
.......... normalize so that sum of squares is
1 and expand to full order ..........
840 u = 0.0d0

do 860 i = p, q
860 u = pythag(u,rv6(i))

xu = 1.0d0 / u

870 do 880 i = 1, n

880 z(i,r) = 0.0d0
do 900 i - p,

900 z(i,r) = rv6(i) * xu
x0 = x1

920 continue
if (q .lt. n) go to 100

1001 return
end

229

00000000000000000000000000000000000000

110

subroutine trbak1(nm,n,a,e,m,z)
integer i,j,k,l,m,n,nm
double precision a(nm,n),e(n),z(nm,m)
double precision 3
this subroutine is a translation of the algol procedure trbak1,
num. math. 11, 181-195(1968) by martin, reinsch, and wilkinson.
handbook for auto. comp., vol.ii-linear algebra, 212-226(1971).
this subroutine forms the eigenvectors of a real symmetric
matrix by back transforming those of the corresponding
symmetric tridiagonal matrix determined by tredl.
on input
nm must be set to the row dimension of two-dimensional
array parameters as declared in the calling program
dimension statement.
n is the order of the matrix.
a contains information about the orthogonal trans-
formations used in the reduction by tred1
in its strict lower triangle.

e contains the subdiagonal elements of the tridiagonal
matrix in its last n-1 positions. e(1) is arbitrary.

m is the number of eigenvectors to be back transformed.

2 contains the eigenvectors to be back transformed
in its first m columns.

note that trbak1 preserves vector euclidean norms.

questions and comments should be directed to burton s. garbow,
mathematics and computer science div, argonne national laboratory

this version dated august 1983.

 

if (m .eq. 0) go to 200
if (n .eq. 1) go to 200

do 140 i = 2, n
l = i - 1
if (e(i) .eq. 0.0d0) go to 140

do 130 j = 1, m
s = 0.0d0

do 110 k = 1, l
s = s + a(i,k) * z(k,j)

230

0000000000000000000000000000000000000

120
130
140
200

.......... divisor below is negative of h formed in tredl.
double division avoids possible underflow ..........

s = (s / a(i,1)) / e(i)

do 120 k = 1, 1
z(k,j) = z(k,j) + s * a(i,k)

continue
continue
return
end
subroutine tred1(nm,n,a,d,e,e2)
integer i,j,k,l,n,ii,nm,jp1
double precision a(nm,n),d(n),e(n),e2(n)
double precision f,g,h,scale
this subroutine is a translation of the algol procedure tred1,
num. math. 11, 181-195(1968) by martin, reinsch, and wilkinson.
handbook for auto. comp., vol.ii-linear algebra, 212-226(1971).
this subroutine reduces a real symmetric matrix
to a symmetric tridiagonal matrix using
orthogonal similarity transformations.
on input
nm must be set to the row dimension of two-dimensional
array parameters as declared in the calling program
dimension statement.

n is the order of the matrix.

a contains the real symmetric input matrix. only the
lower triangle of the matrix need be supplied.

on output
a contains information about the orthogonal trans-
formations used in the reduction in its strict lower
triangle. the full upper triangle of a is unaltered.

d contains the diagonal elements of the tridiagonal matrix.

e contains the subdiagonal elements of the tridiagonal
matrix in its last n-1 positions. e(1) is set to zero.

e2 contains the squares of the corresponding elements of e.
e2 may coincide with e if the squares are not needed.

questions and comments should be directed to burton s. garbow,
mathematics and computer science div, argonne national laboratory

231

0000

100

120

125
130

140

150

170

this version dated august 1983.

 

do 100 i = 1, n
d(i) = a(n,i)
a(n,i) = a(i.i)
continue
.......... for i=n step -1 until 1 do --
do 300 ii = 1, n

1 = n + 1 - ii
1 = i - 1
h = 0.0d0

scale = 0.0d0
if (1 .lt. 1) go to 130

.......... scale row (algol tol then not needed) ..........

do 120 k = 1, 1
scale = scale + dabs(d(k))

if (scale .ne. 0.0d0) go to 140

continue

e(i) = 0.0d0
e2(i) = 0.0d0
go to 300

do 150 k = 1, 1
d(k) = d(k) / scale
h = h + d(k) * d(k)
continue

e2(i) = scale * scale * h

f = d(l)

g = -dsign(dsqrt(h),f)

e(i) = scale * g

h = h - f * g

d(l) = f - g

if (1 .eq. 1) go to 285
.......... form a*u ..........

(j)

(j) + a(j,j) * f

j 1

k = jpl, l

g + a(k,j) * d(k)

232

 

0

0000000000000

200

220
240

245

250

260
280
285

290
300

e(k) = e(k) + a(k,j) * f

continue
e(j) =
continue
.......... form p ..........
f = 0.0d0

do 245 j = 1, l

e(j) = e(j) / h

f = f + e(j) * d(j)
continue

= f / (h + h)
.......... form q ..........
do 250 j = 1, 1
e(j) = e(j) - h * d(j)
.......... form reduced a ..........

do 280 j = 1,1
f = d(j)
g = e(j)
do 260 k = J, l
a(k,j) = a(k.j) - f * e(k) - g * d(k)
continue
do 290 j = 1, l
”d( )
d(j) = Ja(l,j)
a(l, j) = a(i,j)
a(i, j) = f * scale
continue
continue
return

end
SUBROUTINE AOROUT(XELEC,YELEC,ZELEC,AO)

This subroutine uses the input for a GAMESS CISD energy cal-
culation to compute the values of all atomic orbitals for a
particular atom. Specifically, each atomic orbital is computed
at a set of x, y, 2 electronic coordinates shifted by the X,
Y, and Z coordinates of the atomic nucleus on which the atomic
orbital is centered. Also, each atomic orbital is constructed
from a linear combination of primitive gaussians, as specified
by the GAMESS input.

Declare that all variables with names that begin with letters
A-H or 0-2 will be double-precision numbers.

IMPLICIT DOUBLE PRECISION (A-H, O-Z)

233

0000000000000000000000000000000000000000000000000000000

Description of all variables and/or arrays used in the sub-
routine:

XATOM = x-coordinate of nucleus of atom
YATOM = y-coordinate of nucleus of atom
ZATOM = z-coordinate of nucleus of atom

XELEC = x electronic coordinate
YELEC = y electronic coordinate
ZELEC = 2 electronic coordinate

XDIFF = difference between x electronic coord. and x-coord. of
nucleus of atom
YDIFF = difference between y electronic coord. and y-coord. of
nucleus of atom
ZDIFF = difference between 2 electronic coord. and z-coord. of
nucleus of atom.

XDSQRD = square of difference between x electronic coord. and
x-coord. of nucleus of atom
YDSQRD = square of difference between y electronic coord. and
y-coord. of nucleus of atom
ZDSQRD = square of difference between 2 electronic coord. and
z-coord. of nucleus of atom

R = Sum of XDSQRD,YDSQRD,ZDSQRD.

SNUMPG, PNUMPG, DNUMPG = Integer variables set to the number
of primitive gaussians in an S, P or D atomic orbital
(respectively).

SPGAUSS = 100 x 100 dimension array containing the exponents
and coefficients for the primitive gaussians used in a S-type
atomic orbital.

PPGAUSS = 100 x 100 dimension array containing the exponents
and coefficients for the primitive gaussians used in a P-type
atomic orbital.

DPGAUSS = 100 x 100 dimension array containing the exponents
and coefficients for the primitive gaussians used in a D-type
atomic orbital.

SCHIMU = numerical value of a S-type atomic orbital computed
at x-Z, y-Y, z-Z

PXCHIMU, PYCHIMU, PZCHIMU = numerical values of Px-type,
Py-type, and Pz-type atomic orbitals computed the x-Z, y-Y,
z-Z

DXY, DXZ, DYZ, DXSQRD, DYSQRD, DZSQRD = numerical values of
ny-type, sz-type, Dyz-type, stquared-type, Dysquared-type
and Dzsquared-type atomic orbitals computed at the x-X,y-Y,z-Z

PI = double-precision numerical constant for pi

INPUT1 = Character variables used to read the lines in the
GAMESS input file.

234

0000000000000000000000000000000000000000

SORBTOT, PORBTOT, DORBTOT = Total number of S, P and D basis
functions (3 x PORBTOT = total number of P orbitals,
6 x DORBTOT = total number of D orbitals).

PXORBTOT, PYORBTOT, PZORBTOT, DXYTOT, DXZTOT, DYZTOT, DX2TOT,
DY2TOT, DZ2TOT = variables representing the total number of Px,
Py, Pz, ny, sz, Dyz, stquared, Dysquared, Dzsquared orbitals,
respectively.

ATOM = character variable used to read the name of the atom on
which the molecular orbital is centered.

NUMELEC = Integer variable used to read the number of electrons
in the atom on which the molecular orbital is centered.

AORBTOT = Integer variable representing the total number of
atomic orbitals for an atom.

A0 = One-dimensional array which holds the values of all atomic
orbitals at x-X, y-Y, z-Z.

ATOMTOT = The total number of atoms in the molecule of interest.

SORB = Two-letter character variable used to read the letter
S denoting a S-type basis function from the GAMESS input file.
PORB = Two-letter character variable used to read the letter
P denoting a P-type basis function from the GAMESS input file.
DORB = Two-letter character variable used to read the letter
D denoting a D-type basis function from the GAMESS input file.

NUMATOM = One-dimensional array used to keep track of the

number of atomic orbitals for each atom. For example the element
corresponding to NUMATOM(1) is the number of atomic orbitals for
the first atom (as listed in the GAMESS input file) in the
molecule of interest.

NCOUNT = Integer variable used to keep track of the total number
of atomic orbitals for each atom.

DIMENSION SPGAUSS(100, 100), PPGAUSS(100, 100), DPGAUSS(100,100)
INTEGER SORBTOT, PORBTOT, DORBTOT, PXORBTOT, PYORBTOT, PZORBTOT
INTEGER DXYTOT, DX2TOT, DY2TOT, DX2TOT, DY2TOT, DZ2TOT, AORBTOT
CHARACTER INPUT*80, INPUT1*80, ATOM*15

REAL NUMELEC

INTEGER SNUMPG, PNUMPG, DNUMPG, ATOMTOT

CHARACTER SORB*2, PORB*2, DORB*2

DIMENSION NUMATOM(100)

DIMENSION AO(100)

COMMON /BLOCK15/ NAO
Set the value of the constant pi (PI).

PI = DACOS(-1.D0)

235

00

0000

00

100

2222

00000000000 0000

00000

0000

0

Set the total number of atoms equal to zero.
ATOMTOT = 0

Set the total number of atomic orbitals for this atom
equal to zero.

AORBTOT = 0
Open the input file for a GAMESS CISD energy calculation.
OPEN (UNIT=24,

&FILE=’mod_h2cisd_MBS_C1.inp’,
&STATUS= ’ OLD ’ , FORM= ’ FORMATTED ’ )

FORMAT (A80)
Go back two lines in the input file.

BACKSPACE(24)
BACKSPACE(24)

Read the current line in the input file into INPUT
(actually, the first 80 characters of this line).

READ(24,100)INPUT

If the first five characters in the current line
are not " $con", then go to line 2222 in the
program. Note: " $con" belongs to the " $control"
line in the input file. Since " $control" is
always the first group in GAMESS input files,
reading " $con" indicates that we are at the
beginning of the file. Returning to line 2222
will allow us to keep reading until we reach

the beginning of the input file.

IF (INPUT(1:5).NE.’ $Con’) THEN
GO TO 2222
ENDIF

If we have read " $con", then go back one line
in the file (then, once again, we are at the
beginning of the file).

BACKSPACE(24)

Read the next two lines in the input file, one by
one, into INPUT.

READ(24,100)INPUT
READ(24,100)INPUT

If the first five characters of the current line
are not equal to " $dat" or " $DAT" (indicating

236

that we have reached the " $data"/" $DATA" group
in the GAMESS input file), the read the next line
in the input file into INPUT. Keep reading until
the first five characters ARE " $data" or " $DATA".

00000

101 IF (INPUT(1:5).NE.’ $dat’ .AND.
&INPUT(1:5).NE.’ $DAT’) THEN
READ(24,100)INPUT
GO TO 101
ELSE
ENDIF

Once we read the " $data"/" $DATA" group, read the
next three lines, one after another, into INPUT.

0000

102 FORMAT (A80)

IF (INPUT(1:5).EQ.’ $dat’ .OR. INPUT(1:5).EQ.’ $DAT’)

&THEN
READ(24,102)INPUT
READ(24,102)INPUT
READ(24,102)INPUT

ELSE

END IF

c Set NCOUNT equal to zero.
NCOUNT=0

c Read the next line in the input file into INPUT.

777 READ(24,102)INPUT
If the first five characters in the current line are
blank spaces, then read the next line into INPUT.

Continue reading until a line is read which DOES NOT
have blank spaces for its first five characters.

000000

DO 1999 WHILE (INPUT(1:5).EQ.’ ’)
READ(24,102)INPUT
1999 CONTINUE

c
c If the current line’s first five characters are
c " Send", indicating that we have reached the end of
c the data group in the GAMESS input file, then go to
c line 5000.
c

IF (INPUT(1:5).EQ.’ $end’) THEN

GO TO 5000
ELSE
END IF

0

Go back to the previous line.

BACKSPACE(24)

237

0

00

0000

00000 00000 00000

0000

0

103

Read the name of the atom, the number of electrons in the atom,
and the x, y, and z coordinates of the atom’s nucleus.

FORMAT (A15, F2.0, D20.10, D20.10, D20.10)
READ(24,103)ATOM, NUMELEC, XATOM, YATOM, ZATOM

Increment the total number of atoms.
ATOMTOT=ATOMTOT+1

Set the total number of s, p, d, px, py, pz, dxy, dxz, dyz,
dx2, dy2 and dz2 atomic orbitals equal to zero.

0
0
0

SORBTOT
PORBTOT
DORBTOT
PXORBTOT
PYORBTOT

0
0
0

U

N

M

H

O

F]
II II II II II II
OOOOOOII II II

DZ2TOT

Calculate the difference between the x, y, and 2 electronic
coordinates and the x, y, and z coordinates of the atom’s
nucleus.

XDIFF = XELEC - XATOM
YDIFF = YELEC - YATOM
ZDIFF = ZELEC - ZATOM

Square the differences between the x, y, and 2 electronic
coordinates and the x, y, and z coordinates of the atom’s
nucleus.

XDSQRD = XDIFF**2
YDSQRD = YDIFF**2
ZDSQRD = ZDIFF**2

Add the squares of the differences between the x, y, and 2
electronic coordinates and the x, y, and z coordinates of
the atom’s nucleus.

R = XDSQRD + YDSQRD + ZDSQRD

Set the INPUT1 character variable equal to the INPUT
variable.

INPUT1 = INPUT

9988 FORMAT (A5)

104 FORMAT (A80)

238

0000000000000

0000000 000000 000000

00000

200

201

105

If the first five Characters of INPUT1 are NOT all blanks,
read the next line in the input file into INPUT1. If the
first five characters of INPUT1 are " S ", then go to
line 201. If the first five characters of INPUT1 are

" P ", then go to line 202. If the first five Characters
of INPUT1 are " D ", then go to line 203. Note: " S "
indicates that we are about to read data for an

s-type orbital, " P " indicates that we are about to
read data for a p-type atomic orbital, and " D " indicates
that we are about to read the data for a d-type atomic
orbital.

DO 200 WHILE (INPUT1(1:5).NE.’ ’)

READ(24,104)INPUT1

IF (INPUT1(1:5).EQ.’ S ’) THEN

GO TO 201
ELSE IF (INPUT1(1:5).EQ.’ P ’) THEN
GO TO 202
ELSE IF (INPUT1(1:5).EQ.’ D ’) THEN
GO TO 203
ELSE

ENDIF

CONTINUE

If the first five characters of INPUT1 are blanks, then
go to line 4999. Note: If the first five Characters of
INPUT1 are blanks, this indicates that we have reached
the end of the "$data" group in the GAMESS input file.

IF (INPUT1(1:5).EQ.’ ’) THEN
GO TO 4999

ELSE

END IF

This is line 201, which is the line to go to if a
" S " was read into INPUT in the 200 DO loop.

Go back to the previous line.
BACKSPACE(24)

Read the "S" indicating data for an s-type orbital

and the number following the S (which is the number
of primitive gaussians in the s-type orbital) into

the SORB character variable and the SNUMPG integer

variable, respectively.

FORMAT (A2, I4)
READ(24,105)SORB, SNUMPG

Read the number of the primitive gaussian, the
exponent for the gaussian, and the coefficient
for the gaussian into the SPGAUSS array. Do this
for all primitive gaussians in the s-type atomic

239

00

00 00000

0000

0

000

000000

0000000

106

300

202

&

orbital.
READ(24,*) ((SPGAUSS(ISROW,ISCOL), ISCOL=1,3), ISROW=1,SNUMPG)
Use the primitive gaussians to calculate the value
of the s-type atomic orbital at R, and read the value of the
atomic orbital into SCHIMU.
DO 106 I = 1, SNUMPG, 1
SCHIMU = SCHIMU + DEXP(-1.0D0*((SPGAUSS(I,2))*R))*
(((2.0DO*(SPGAUSS(I,2)))/PI)**0.75D0)*(SPGAUSS(I,3))
CONTINUE
Increment the total number of atomic orbitals.
NCOUNT=NCOUNT+1

Read the value of the s-type atomic orbital into the
AO array.

AO(NCOUNT)=SCHIMU

Increment the total number of s-type orbitals.
SORBTOT = SORBTOT + 1

Clear the SCHIMU variable.

SCHIMU=0

Clear the SPGAUSS array.

DO 300 ISROW = 1, SNUMPG, 1

SPGAUSS(ISROW,1) = 0

SPGAUSS(ISROW,2) = 0

SPGAUSS(ISROW,3) = 0
CONTINUE

Go back to line 200.
GO TO 200

This is line 202, which is the line to go to if a
" P " was read into INPUT in the 200 D0 loop.

Go back to the previous line.
BACKSPACE(24)

Read the "P" indicating data for an p-type orbital
and the number following the P (which is the number
of primitive gaussians in the p-type orbital) into
the PORB character variable and the PNUMPG integer
variable, respectively.

240

0000000

000000

000

0000 000 0000

00

107

&

fifi’fi’

WWW

&

&

&
108

FORMAT (A2, I4)
READ(24,107)PORB, PNUMPG

Read the number of the primitive gaussian, the
exponent for the gaussian, and the coefficient
for the gaussian into the PPGAUSS array. Do this
for all primitive gaussians in the p-type atomic
orbital.

READ(24,*)
((PPGAUSS(IPROW,IPCOL),IPCOL=1,3), IPROW=1,PNUMPG)

Use the primitive gaussians to calculate the value
of the px-type, py-type, and pz-type atomic orbitals
at R, and read the values of these atomic orbitals
into PXCHIMU, PYCHIMU, and PZCHIMU.

DO 108 J = 1, PNUMPG, 1
PXCHIMU = PXCHIMU + DEXP(-1.0D0*((PPGAUSS(J,2))*
R))*XDIFF*(PPGAUSS(J,3))*
(((2.0D0**1.75D0)*
((PPGAUSS(J,2))**1.25D0))/(PI**.75D0))

PYCHIMU = PYCHIMU + DEXP(-1.0D0*((PPGAUSS(J,2))*
R))*YDIFF*(PPGAUSS(J,3))*

(((2.0D0**1.75D0)*
((PPGAUSS(J,2))**1.25D0))/(PI**.75D0))

PZCHIMU = PZCHIMU + DEXP(-1.0D0*((PPGAUSS(J,2))*
R))*ZDIFF*(PPGAUSS(J,3))*

(((2.0D0**1.75D0)*

((PPGAUSS(J,2))**1.25D0))/(PI**.75D0))
CONTINUE

Increment the total number of atomic orbitals.
NCOUNT=NCOUNT+1

Read the value of the px-type atomic orbital into
the AO array.

AO(NCOUNT)=PXCHIMU
Increment the total number of atomic orbitals.
NCOUNT=NCOUNT+1

Read the value of the py-type atomic orbital into
the AO array.

AO(NCOUNT)=PYCHIMU
Increment the total number of atomic orbitals.

NCOUNT=NCOUNT+1

241

0000

000

0 0000

0

000000 000

0000000

00000

325

203

109

Read the value of the pz-type atomic orbital into
the AO array.

AO(NCOUNT)=PZCHIMU
Increment the total number of p-type atomic orbitals.
PORBTOT = PORBTOT + 1

Increment the total number of px-type, py-type, and
pz-type atomic orbitals.

PXORBTOT = PXORBTOT + 1
PYORBTOT = PYORBTOT + 1
PZORBTOT = PZORBTOT + 1

Clear the PXCHIMU, PYCHIMU, and PZCHIMU variables.

PXCHIMU = 0
PYCHIMU = 0
PZCHIMU = 0

Clear the PPGAUSS array.
DO 325 IPROW = 1, PNUMPG, 1

PPGAUSS(IPROW,1) = 0

PPGAUSS(IPROW,2) = 0

PPGAUSS(IPROW,3) = 0
CONTINUE

Return to the 200 D0 loop.
GO TO 200

This is line 203, which is the line to go to if a
" D " was read into INPUT in the 200 DO loop.

Go back to the previous line.
BACKSPACE(24)

Read the "D" indicating data for an d-type orbital
and the number following the D (which is the number
of primitive gaussians in the d-type orbital) into
the DORB character variable and the DNUMPG integer
variable, respectively.

FORMAT (A2, I4)
READ(24,109)DORB, DNUMPG

Read the number of the primitive gaussian, the
exponent for the gaussian, and the coefficient
for the gaussian into the DPGAUSS array. Do this
for all primitive gaussians in the d-type atomic

242

00

0000000

orbital .

READ(24,*)((DPGAUSS(IDROW,IDCOL),

&IDCOL=1,3), IDROW=1,DNUMPG)

RPfi'fi’ RPWRP ”w” ”a“?

P”?

&
&
&

Use the primitive gaussians to calculate the value
of the dxy-type, dxz-type, dy2-type, dx2, dy2, and
dz2 atomic orbitals

at R, and read the values of these atomic orbitals
into PXCHIMU, PYCHIMU, and PZCHIMU.

DO 110 K = 1, DNUMPG, 1

DXY = DXY + DEXP(-1.0D0*((DPGAUSS(K,2))*R))*
XDIFF*YDIFF*(DPGAUSS(K,3))*
(((2.0D0**2.75D0)*
((DPGAUSS(K,2))**1.75D0))/(PI**.75D0))

DXZ = DXZ + DEXP(-1.0D0*((DPGAUSS(K,2))*R))*
XDIFF*ZDIFF*(DPGAUSS(K,3))*
(((2.0DO**2.75D0)*
((DPGAUSS(K,2))**1.75D0))/(PI**.75D0))

DYZ = DYZ + DEXP(-1.0D0*((DPGAUSS(K,2))*R))*
YDIFF*ZDIFF*(DPGAUSS(K,3))*
(((2.0D0**2.75DO)*
((DPGAUSS(K,2))**1.75DO))/(PI**.75D0))

DXSQRD = DXSQRD + DEXP(-1.0D0*((DPGAUSS(K,2))
*R))*XDIFF*XDIFF*(DPGAUSS(K,3))*
(((2.0D0**2.75D0)*

((DPGAUSS(K,2))**1.75D0))/((3.0D0**0.5D0)*(PI**.75D0)))

DYSQRD = DYSQRD + DEXP(-1.0D0*((DPGAUSS(K,2))
*R))*YDIFF*YDIFF*(DPGAUSS(K,3))*
(((2.0DO**2.75D0)*

((DPGAUSS(K,2))**1.75D0))/((3.0D0**0.5D0)*(PI**.75DO)))

DZSQRD = DZSQRD + DEXP(-1.0D0*((DPGAUSS(K,2))
*R))*ZDIFF*ZDIFF*(DPGAUSS(K,3))*
(((2.0D0**2.75D0)*

((DPGAUSS(K,2))**1.75D0))/((3.0D0**0.5D0)*(PI**.75D0)))

110 CONTINUE

Increment the total number of atomic orbitals.

NCOUNT=NCOUNT+1

Read the value of the dx2-atomic orbital into the AO array.

AO(NCOUNT)=DXSQRD
Increment the total number of atomic orbitals.

NCOUNT=NCOUNT+1

243

Read the value of the dy2-atomic orbital into the AO array.
AO(NCOUNT)=DYSQRD

Increment the total number of atomic orbitals.
NCOUNT=NCOUNT+1

Read the value of the dz2-atomic orbital into the AO array.
AO(NCOUNT)=DZSQRD

Increment the total number of atomic orbitals.
NCOUNT=NCOUNT+1

Read the value of the dxy-atomic orbital into the AO array.
AO(NCOUNT)=DXY

Increment the total number of atomic orbitals.
NCOUNT=NCOUNT+1

Read the value of the dxz-atomic orbital into the AO array.
AO(NCOUNT)=DXZ

Increment the total number of d-type atomic orbitals.
DORBTOT = DORBTOT + 1

Increment the total number of dxy-type atomic orbitals.
DXYTOT = DXYTOT + 1

Increment the total number of dxz—type atomic orbitals.
DX2TOT = DXZTOT + 1

Increment the total number of dy2-type atomic orbitals.
DYZTOT = DYZTOT + 1

Increment the total number of dx2-type atomic orbitals.
DX2TOT = DX2TOT + 1

Increment the total number of dy2-type atomic orbitals.
DY2TOT = DY2TOT + 1

Increment the total number of dz2-type atomic orbitals.

DZ2TOT = DZ2TOT + 1

244

0000

375

00000000

00000

4999

0000

00

00

5000

0000

0

Clear the dxy-type, dxz-type, dy2-type, dx2-type, dy2-type
and dz2-type atomic orbitals.

DXY 0
DXZ
DYZ
DXSQRD
DYSQRD
DZSQRD

0
0

0
O
0
Clear the DPGAUSS array.

DO 375 IDROW = 1, DNUMPG, 1

DPGAUSS(IDROW,1) = 0

DPGAUSS(IDROW,2) = 0

DPGAUSS(IDROW,3) = 0
CONTINUE

At this point, we have finished reading the information for

the d-type atomic orbitals and evaluating these orbitals at R.
We are ready to read the information for the next atomic
orbital, or (if all of the orbital information has already been
read) return the evaluated atomic orbitals to the main program.
Return to line 200, where we will decide what to do next.

GO TO 200

We are done reading the information for all atomic orbitals of
the atom and evaluating these orbitals at R. Now, we

compute the total number of atomic orbitals for the atom.

AORBTOT = SORBTOT + PXORBTOT + PYORBTOT + PZORBTOT + DXYTOT +
DX2TOT + DYZTOT + DX2TOT + DY2TOT + DZ2TOT

Read the total number of atomic orbitals for the current
atom into NUMATOM(ATOMTOT).

NUMATOM(ATOMTOT)=AORBTOT

Return to line 777.

GO TO 777

Close the GAMESS CISD input file.
CLOSE(20)

Return the values of the atomic orbitals (evaluated at R) to
the main program, and return to the main program.

RETURN
END

245

Appendix B. Tables

246

Appendix B. Tables

Table 1. Non-zero terms in the third-order interaction energy of molecules A and
B, 0Classified by order m) the permanent dipoles of A and B. Note that we have let

=|0A) and 111313—403).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Term ,1“ order 1130 Name
—(0,,03|VABGAH%T 0,3,1, BGBVABIOAOB) 1 1 1 — (1,1)
—(OAOB|VABGAfi:‘ TWO), GAQ’BVABlOAOB) 0 1 2 — (0,1)
‘(0AOBIVABGAMQ TaaflfaoGAVABIOAOB) 0 3 3 - (013)
-,1O(OAOB|VABGA :OT 3,18 BGAG’BVABIOAOB) 1 1 4— (1,1)
—(0,10,)VA’-"G’3fif,1 T0111, BGAVABIOAOB) 1 1 5 — (1,1)
—(oAOB|VABGBfif T11, BGAABVABmAOB) 1 0 6—(1,0)
—(0A03|VABGBfiaTawgoaAABVAB|0103) 1 1 7 — (1,1)
—<0101IVABGB112°T.1F§ CAI/“10103) 3 0 8 — (3, 0)
—(0 ,0,;|VAA‘GAéABfifi‘Tafifi;g GAVAB|0 103) 0 1 (0, 1)
—(01%|VABGAABEZTGBEEGBVAB|0103) 1 0 10 — (1, 0)
—(OAOB|VABGA@Bfi:T gfigGA®BVABIOAOB) 0 0 11 — (0,0)
—(0,,()B|VABGAAAAHZl T ,HgOGBVABonB) 1 1 12 — (1,1)
—(0,10,,IVA’E’GAAAB,1;l T ngOGAABVABonB) 0 1 -— (0,1)
(010,”VABGAAB,1A°T,,,,1,§3 GAVABloAOB) 1 1 14 — (1,1)
—(010,4VABGAABHgOTOBASG/AABVAB[0,03) 1 0 15 — (1,0)

 

247

 

Table 2. Comparison of an (w) , 01% (w) , and an (w) values calculated by in-
tegrating x(r, r’; w) using the algorithm described here and by finite-field calculations
performed with the MOLPRO17 quantum chemistry software package. Polarizabili-
ties were calculated in the DZ, DZP, and aug—cc—pVDZ basis sets, and polarizabilities

are given in a.u.

 

 

 

 

 

 

 

 

 

 

 

Basis Set q,t,,(w) ayy(w) ozzz (w)

DZ“ 0.00031451 0.00031451 5.74696714
DZ” 0 0 5.74696958
DZP“ 0.6753247 3 0.67532473 5.87133714
DZP” 0.67532474 0.67532474 5.87133956
aug-cc-pVDZ“ 4.35229766 4.35229763 6.54719224
aug-cc—pVDZb 4.35229593 4.35229593 6.54717965

 

a MOLPROl7 results.

b These results were obtained by integrating over x(r, r’;w) using the algorithm de-

scribed in this work.

248

 

Appendix C. Figures

249

Appendix C. Figures

 

 

 

 

 

 

 

0.1 —2 O
2
y (bohr)
0.15)
0|
—2 0 2

Fig. 1. The charge-density susceptibility of the H2 molecule at the CISD level with a) = 0
a.u., r' = 0,0,0, x = 0, -3.25 s y s 3.25 a.u., -3.25 s z s 3.25 a.u., and Ay = A2 = 0.05 a.u.
in the aug-cc-pVDZ basis set. For this calculation, the internuclear axis of H2 was
oriented along the z-axis of the laboratory frame.

250

 

 

0.05 C

 

 

 

 

 

 

 

,-
_0 05 1 4 L A 1 I I I I g;
0

Fig. 2. The charge-density susceptibility of the H2 molecule at the CISD level with to =
0.3858668352248763 a.u., r' = 0,0,0, x = 0, -3.25 s y s 3.25 a.u., -3.25 s z s 3.25 a.u.,
and Ay = A2 = 0.05 a.u. in the aug-cc-pVDZ basis set. For this calculation, the
internuclear axis of H2 was oriented along the z-axis of the laboratory frame.

251

 

13 ,
4X10 _2

> 0
2x103» 2 ~“/
: y mer) ‘
O i ' ‘ \._..

L
—2x 1013 L

 

 

 

 

 

 

l
—4x1013 ,

 

 

 

Fig. 3. The charge-density susceptibility of the H2 molecule at the CISD level with

a) = 0. 4812104263202694 a.u., r' = 0,0,0, x = 0, -3.25 s y s 3.25 a.u., -3.25 s z s 3.25
a.u., and Ay = A2 = 0.05 a.u. in the aug-cc-pVDZ basis set. For this calculation, the
internuclear axis of H2 was oriented along the z-axis of the laboratory frame.

252

 

 

 

 

 

 

 

l
0.6’ —2
i o 2!
0.2 ;
0;
-0.2: 1 g A g
—2 o 2
z (bohr)

Fig. 4. The charge-density susceptibility of the H2 molecule at the CISD level with a) =
0.3858668352248763 a.u., r' = 0,0,0.7, x = O, -3.25 s y s 3.25 a.u., -3.25 s z s 3.25 a.u.,
and Ay = A2 = 0.05 a.u. in the aug-cc-pVDZ basis set. For this calculation, the
internuclear axis of H2 was oriented along the z-axis of the laboratory frame.

253

 

 

2x1013[ ‘2

0 2 .. \
y (bohr)
l

-2 l A A ‘

' "f_“_“T"'_

 

0

 

 

 

 

 

—2 x 1013

 

A—A

0 2
z (bohr)

Fig. 5. The charge-density susceptibility of the H2 molecule at the CISD level with co =
0.4648380650856789 a.u., r' = 0,0,0.7, x = 0, -3.25 s y s 3.25 a.u., -3.25 s z s 3.25 a.u.,
and Ay = A2 = 0.05 a.u. in the aug-cc-pVDZ basis set. For this calculation, the
internuclear axis of H2 was oriented along the z-axis of the laboratory frame.

254

 

AAAAAALQL

 

0111(0))(au)

O
1

I“ y.
‘0
i
1
g:
I)
b

 

 

- 1

~5x1015 E u 7

. DZ 1

41:10“ A 3

i . DZP 1

l

-1 5x10“ ~ 0 aug-cc-pVDZ ,
o 25 o s o 75 1 1 25 1 5 1 15

Fig. 6. The 22 component of the frequency-dependent polarizability (13(0)) of H2 as a
function of frequency a) at the CISD level in the DZ, DZP, and aug-cc-pVDZ basis sets.

255

 

200

A L41 A

 

 

 

 

150 E . j
. ' i
. , 0 0 1
100 +5 .. . j
A
i '. . I
,7 50 _T c . . j
:1 1 ~
3 i o o o o o I . .‘j/ 4 ‘ l
A o r ‘..0000....‘ .— 0 ’ ‘
é ) .8‘ :
it! o. o . ,
5 "5° ) 3 1
: ° .
100 i : . . DZ 2
i ' . DZP i
o 1
-150 - . aug-cc-pVDZ ,
, . 1
. .

0.25 0.5 0.75 1 1 25 1 5

Fig. 7. The 22 component of the frequency-dependent polarizability 01,2 (0)) of H2 at the
CISD level as a function of a) in the DZ, DZP, and aug-cc-pVDZ basis sets. Here, we

show the data included in -2005 0122 (0)) £200 a.u. in Fig. 6.

256

.75

 

1x10 ...fiffifi.y,.fififi....fjfi ..r.+j,,..rf

 

 

 

 

t . 1
: 1
7.5x1015 : 1
* 1
5x10” 3 3
2.5x1015 i
A. :
=1 5 A
a o $ : : c : c : : —‘-— :x:::::: a :::--:-_::::::
I: : 7
8 : -
‘5 -2.5x1015; :
e i ‘
-5x1015 E . DZP 1
4.5.1111: E ' 111211-me A
i . 1
L x 1 1L 4 4 LL 4 kg 4 1 A L 1 #1_1 L + A 1 g1
0.25 0.5 0.75 1 1.25 1.5 1.75

Fig. 8. The xx component of the frequency-dependent polarizability (13(0)) of H2 at the
CISD level as a function of a) in the DZP and aug-cc-pVDZ basis sets.

257

 

an ((0) (a.u.)

r—r—v—w—r-j ‘T—H—V y—r-v—V-Or-j-w—v—r v—T—v—r v—T—r-r—v' w—v—r-T—r-r—r-T —1- y—‘v—v—
r . 4
O

0
O
O
O
O

 

Fig. 9. The xx component of the frequency-dependent polarizability 012,,(01) of H2 at the
CISD level as a function of (0 in the DZP basis set.

258

1111015
7.5111015
5111015

2.5111015

0
0
1

 

on
N
U11)

TV—V—r—T—‘H 'T' "I—V—rfl W'T—P‘Y—T—‘Tfifi—Y“.

(1..., ((1)) (a.u.)

-2.5><1015

y

—5><1015

4.5111015

ffi—Y—V—T—T‘Tfi—Tfiffifi—T

—1 x 1016
00 (a.u.)

Fig. 10. The yy component of the frequency-dependent polarizability (1),),(01) of H2 at the
CISDlevel as a fimction of 00 in the aug-cc-pVDZ basis set.

259

 

V

 

 

Fig. 1 1. The semi-circular contour C of radius R, in the upper complex half plane. In this
figure, 1m and Re denote the imaginary and real axes, respectively. Also, 0 is the angle
between the real axis and R1, and S is the portion of C off the real axis.

260

RAB

}...............q...-1,13........1

 

/\

 

 

 

Fig. 12. Colinear arrangement of molecules A and B. In this figure, x, y, and 2 denote the
axes of the laboratory flame, and x', y', and 2' denote the axes of the molecular flames of
A and B. Also, A, and A2 are the nuclei of molecule A, B, and B2 are the nuclei of
molecule B, COM, and COMB are the centers of mass of molecules A and B, and Rug is
the distance between the center of mass of A and the center of mass of B.

261

 

 
 
  

A2 ’1’, 32
l”’ Z
\ COM, COM, '
A1 31

 

Fig. 13. Parallel arrangement of molecules A and B. In this figure, x, y, and 2 denote the
axes of the laboratory flame, and x', y', and 2' denote the axes of the molecular flames of
A and B. Also, A, and A2 are the nuclei of molecule A, B, and B2 are the nuclei of
molecule B, COMA and COMB are the centers of mass of molecules A and B, and R4,, is
the distance between the center of mass of A and the center of mass of B.

262

 

A

  
 

RAB

 

Fig. 14. Perpendicular arrangement of molecules A and B. In this figure, x, y, and z denote
the axes of the laboratory flame, x’, y', and 2' denote the axes of the molecular flame of A,
and x", y", and 2" denote the axes of the molecular flame of B. Also, A, and A2 are the
nuclei of molecule A, B, and B2 are the nuclei of molecule B, COM, and COMB are the
centers of mass of molecules A and B, and RAB is the distance between the center of mass
of A and the center of mass of B.

263

RHFI- HF3

I.........................u

F2
H1

    
 

      
 

COMHFI

RHFI- HF2

COMHF2

RHFZ- HF3

 

  

  

COMHF3

Fig. 15. Colinear arrangement of three HF molecules. In this figure, we have labeled the
three HF molecules HF,, HF 2, and HF 3 in order to distinguish between them. Also, x, y,
and 2 denote the axes of the laboratory flame, x', y', and 2' denote the axes of the
molecular flames of each HF molecule, H,, H2, and H3 are the hydrogen nuclei in HF,,
HF2, and HF3, F,, F2 and F3 are the fluorine nuclei in HF,, HF2 and HF3, and COMHH,
COMHn, and COMHpg are the centers of mass of HF,, HF2, and HF3. Finally, RHF,-,,,:2,
Rama”, and Run-HF3 are the distances between the centers of mass of HF, and HP 2, HF,

and HF3, and HF2 and HF3, respectively.

264

 

 

       

H1 H2
1 " z
COMHFI COMHFZ I
F1 F2
*I...-.... ...-......-*..-.......‘..........'.|
RHFl- HF2 RHFz- HF3
. ...-........I..III.-I......-'...............I
RHFl-HF3
v

 

Fig. 16. Parallel arrangement of three HF molecules. In this figure, we have labeled the
three HF molecules HF,, HF2, and HF 3 in order to distinguish between them. Also, x, y,
and z denote the axes of the laboratory flame, x', y', and 2' denote the axes of the
molecular flames of each HF molecule, H,, H2, and H3 are the hydrogen nuclei in HF,,
HF2, and HF3, F,, F2 and F3 are the fluorine nuclei in HF ,, HF2 and HF3, and COMHH,
COMHn, and COMHB are the centers of mass of HF ,, HP 2, and HF3. Finally, RHF,-,,,:2,
RHF,-,,,:3, and RHnHm are the distances between the centers of mass of HF, and HF2, HF,
and HF3, and HF2 and HF3, respectively.

265

References

266

10

11

12

13

14

15

16

17

18

19

20

21

References

Linder, B.; Rabenold, D. A. Adv. Quantum Chem. 1972, 6, 203.
Linder, B. Adv. Chem. Phys. 1967, 12, 225.

Hunt, K. L. C. J. Chem. Phys. 1983, 78, 6149; 1984, 80, 393.
Jenkins, 0. 8.; Hunt, K. L. C. J. Chem. Phys. 2003, 119, 8250.
Hunt, K. L. C. J. Chem. Phys. 1989, 90, 4909.

Handler G. S.; March, N. H. J. Chem. Phys. 1975, 63, 438.
Hunt, K. L. C. J. Chem. Phys. 1995, 103, 3552.

Liang, Y. Q.; Hunt, K. L. C. J. Chem. Phys. 1991, 95, 2549.

Liang, Y. Q.; Hunt, K. L. C. J. Chem. Phys. 1993, 98, 4626.

Li, X.; Hunt, K. L. C. J. Chem. Phys. 1996, 105, 4076.

Hunt, K. L. C. J. Chem. Phys. 1990, 92, 1180.

McLachlan, A. D. Proc. R. Soc. London Ser. A 1963, 271, 387.
Longuet-Higgins, H. C. Discuss. Faraday Soc. 1965, 40, 7.

Langhoff, P. W. Chem. Phys. Lett. 1973, 20, 33.

Jacobi, N.; Csanak, G. Chem. Phys. Lett. 1975, 30, 367.

Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.;
Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J.; Windus, T. L.;
Dupuis, M.; Montgomery, J. A. J. Comput. Chem, 1993, 14, 1347.

Amos, R. D.; Bernhardsson, A.; Berning, A.; Celani, P.; Cooper, D. L.; Deegan,
M. J. 0.; Dobbyn, A. J.; Eckert, F.; Hampel, C.; Hetzer, G.; Knowles, P. J.;
Korona, T.; Lindh, R.; Lloyd, A. W.; McNicholas, S. J.; Manby, F. R.; Meyer,
W.; Mura, M. E.; Nicklass, A.; Palmieri, P.; Pitzer, R.; Rauhut, G.; Schiitz, M.;
Schumann, U.; Stoll, H.; Stone, A. J.; Tarroni, R.; Thorsteinsson, T.; Werner,
H.-J. MOLPRO, a package of ab initio programs designed by H.-J. Werner and P.
J. Knowles, version 2000.1.

Koide, A. J. Phys. B 1976, 9, 3173.

Karplus, M.; Kolker, H. J. J. Chem. Phys, 1963, 39, 2997.

Kohn, W.; Meir, Y.; Makarov, D. E. Phys. Rev. Lett., 1998, 80, 4153.

Dobson, J. F.; Wang, J. Phys. Rev. Lett., 1999, 82, 2123.

267

 

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

41

42

43

44

45

46

47

Dobson, J. F. In Topics in Condensed Matter Physics; Das, M. P., Ed.; Nova: New
York, 1994.

Dobson, J. F.; Dinte, B. P.; Wang, J. In Electronic Density Functional Theory:
Recent Progress and New Directions; Dobson, J. F., Vignale, G., and Das, M. P.,
Eds; Plenum: New York, 1998, p. 261.

Dobson, J. F.; Wang, J. Phys. Rev. B 2000, 62, 10038.

Koide, A.; Meath, W. J.; Allnatt, A. R. Chem. Phys, 1981, 58, 105.

Krauss, M.; Neumann, D. B. J. Chem. Phys, 1979, 71, 107.

Krauss, M.; Neumann, D. B.; Stevens, W. J. Chem. Phys. Lett., 1979, 66, 29.
Krauss, M.; Stevens, W. J.; Neumann, D. B. Chem. Phys. Lett., 1980, 71, 500.
Krauss, M.; Stevens, W. J. Chem. Phys. Lett., 1982, 85, 423.

Malinowski, P.; Tanner, A. C.; Lee, K. F.; Linder, B. Chem. Phys, 1981, 62, 423.
Linder, B.; Lee, K. F.; Malinowski, P.; Tanner, A. C. Chem. Phys, 1980, 52, 353.
Li, X.; Hunt, K. L. C.; Ahuja, C.; Harrison, J. F., unpublished.

Berkowitz, M.; Parr, R. G. J. Chem. Phys. 1988, 88, 2554.

Liu, P-H.; Hunt, K. L. C. J. Chem. Phys. 1995, 103, 10597.

Ando, T. Z. Phys. B 1977, 26, 263.

Zangwill, A.; Seven, P. Phys. Rev. A 1980, 21, 1561.

Gross, E. K. U.; Kohn, W. Phys. Rev. Lett. 1985, 55, 2850.

Gross, E. K. U.; Kohn, W. Phys. Rev. Lett. 1986, 57, 923.

van Gisbergen, S. J. A.; Snijders, J. G.; Baerends, E. J. Phys. Rev. Lett. 1997, 78,
3097.

Bishop, D. M.; Pipin, J. J. Chem. Phys. 1989, 91, 3549.

Bishop, D. M.; Pipin, J. J. Chem. Phys. 1992, 97, 3375.

Bishop, D. M.; Pipin, J. J. Chem. Phys. 1993, 99, 4875.

Bishop, D. M.; Pipin, J. J. Chem. Phys. 1995, 103, 5868.

Bishop, D. M.; Pipin, J. Int. J. Quantum Chem. 1993, 45, 349.
Bishop, D. M.; Pipin, J. Int. J. Quantum Chem. 1993, 47, 129.
Thakkar, A. J. J. Chem. Phys. 1981, 75, 4496.

Meath, W. J.; Kumar, A. Int. J. Quantum Chem. Symp. 1990, 24, 501.

268

48

49

50

51

52

53

54

55

56

57

58

59

6O

61

62

63

65

66

67

68

69

70

71

72

Kumar, A.; Meath, W. J.; Bundgen, P.; Thakkar, A. J. J. Chem. Phys. 1996, 105,
4927.

Bundgen, P.; Thakkar, A. J.; Kumar, A.; Meath, W. J. Mol. Phys. 1997, 90, 721.

Bundgen, P.; Thakkar, A. J .; Grein, F.; Ernzerhof, M.; Marian, C. M.; Nestmann,
B. Chem. Phys. Lett. 1996, 261, 625.

Ceperley, D. M. Phys. Rev. B 1978, 18, 3126.

Ceperley, D. M.; Alder, B. J. Phys. Rev. Lett. 1980, 45, 566.

Moroni, S.; Ceperley, D. M.; Senatore, G. Phys. Rev. Lett. 1995, 75, 689.
Piecuch, P. personal communication.

Kondo, A. E.; Piecuch, P.; Paldus, J. J. Chem. Phys. 1995, 102, 6511.
Kondo, A. E.; Piecuch, P.; Paldus, J. J. Chem. Phys. 1996, 104, 8566.
Callen, H. B.; Welton, T. A. Phys. Rev. 1951, 83, 34.

Hunt, K. L. C. J. Chem. Phys. 2002, 116, 5440.

Craig, D. P.; Thirunamachandran, T. Chem. Phys. Lett. 1981, 80, 14.
Buckingham, A. D. Adv. Chem. Phys. 1967, 12, 107.

Orr, B. J.; Ward, J. F. Mol. Phys. 1970, 20, 513.

Maroulis, G. J. Mol. Struct. (THEOCHEM) 2003, 633, 177.

Piecuch, P. Mol. Phys. 1986, 59, 1067.

Piecuch, P. Mol. Phys. 1986, 59, 1085.

Piecuch, P. Moi. Phys. 1986, 59, 1097.

Piecuch, P. Mol. Phys. 1989, 66, 805.

Gradshteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series, and Products; Academic
Press: San Diego, 2000, Ch. 3.

Lide, D. R., Ed.; CRC Handbook of Chemistry and Physics; CRC Press: Boca
Raton, 1999, p 10-180.

Certain, P. R.; Bruch, L. W. in Intern. Rev. Science, Phys. Chem; Butterworths:
London, 1972, Series 1, Vol. 1.

Claverie, P. In Intermolecular Interactions: From Diatomics to Biopolymers; Pull-
man, B., Ed.; Wiley: New York, 1978.

Jeziorski, B.; Kolos, W. In Molecular Interactions; Ratajczak, H., Orville—Thomas,
W. J., Eds; Wiley: New York, 1982; Vol. 3, p 1.

Arrighini, G. P. Intermolecular Forces and Their Evaluation by Perturbation The-
ory, Springer: New York, 1981.

269

73

74

75

76

77

78

79

80

81

82

83

85

86

87

88

89

91

92

93

94

95

Kaplan, I. G. Theory of Intermolecular Interactions; Elsevier: Amsterdam, 1987.
Hobza, P.; Zahradnik, R. Intermolecular Complexes; Elsevier: Amsterdam, 1988.

van der Avoird, A.; Wormer, P. E. S.; Mulder, F .; Berns, R. M. Top. Curr. Chem.
1980, 93, 1.

Hirschfelder, J. 0. Chem. Phys. Lett. 1967, 1, 325.

Moszynski, R.; Jeziorski, B.; Ratkiewicz, A.; Rybak, S. J. Chem. Phys. 1993, 99,
8856.

Williams, H. L.; Szalewicz, K.; Jeziorski, B.; Moszynski, R.; Rybak, S. J. Chem.
Phys. 1993, 98, 1279.

Moszynski, R.; Wormer, P. E. S.; Jeziorski, B.; van der Avoird, A. J. Chem. Phys.
1994, 101, 2811.

Moszynski, R.; Jeziorski, B.; Diercksen, G. H. F.; Vielhand, L. A. J. Chem. Phys.
1994, 101, 4697.

Buckingham, A. D.; Fowler, P. W.; Hutson, J. M. Chem. Rev. 1988, 88, 963.
Buckingham, A. D.; Fowler, P. W. J. Chem. Phys. 1983, 79, 6426.
Buckingham, A. D.; Fowler, P. W. Can. J. Chem. 1985, 63, 2018.

Buckingham, A. D.; Fowler, P. W.; Stone, A. J. Int. Rev. Phys. Chem. 1986, 5,
107.

Liu, S.-Y.; Dykstra, C. E. Chem. Phys. 1986, 107, 343.
Liu, S.-Y.; Dykstra, C. E. J. Phys. Chem., 1986, 90, 3097.

Liu, S.-Y.; Michael, D. W.; Dykstra, C. E.; Lisy, J. M. J. Chem. Phys, 1986, 84,
5032.

Liu, S.-Y.; Dykstra, C. E.; Kolenbrander, K.; Lisy, J. M. J. Chem. Phys, 1986,
85, 2077.

Dykstra, C. E.; Liu, S.-Y.; Malik, D. J. J. Mol. Struct. (Theochem), 1986, 135,
357.

Dykstra, C. E. J. Phys. Chem., 1987, 91, 6216.

Dykstra, C. E. Chem. Phys. Lett., 1987, I41, 159.

Dykstra, C. E. J. Comput. Chem., 1988, 9, 476.

Kolenbrander, K.; Dykstra, C. E.; Lisy, J. M. J. Chem. Phys, 1988, 88, 5995.
Dykstra, C. E. Acc. Chem. Res, 1988, 21, 355.

Magnasco, V.; McWeeny, R. In Theoretical Models of Chemical Bonding; Maksic,
Z. B., Ed.; Springer: New York, 1991; Vol. 4, p 133.

270

96 Jergensen, P.; Simons, J. Second Quantization-Based Methods in Quantum Chem-
istry, Academic: New York, 1981; p 142.

97 Oddershede, J. In Methods in Computational Molecular Physics; Diercksen, G. H.
E, Wilson, S., Eds; Reidel: Dordrecht, 1983, p 249.

98 Stone, A. J. In Theoretical Models of Chemical Bonding; Maksic, Z. 8., Ed.;
Springer: New York, 1991, Vol. 4, p 103.

99 Stone, A. J. Mol. Phys 1985, 56, 1065.
100 Le Sueur, C. R.; Stone, A. J. Mol. Phys. 1993, 78, 1267.

101 Angyan, J. G.; Jansen, G.; Loos, M.; Héittig, C.; Hess, B. A. Chem. Phys. Lett.
1994, 219, 267.

102 Olsen, J.; Jergensen, P. J. Chem. Phys. 1985, 82, 3235.

103 Salter, E. A.; Trucks, G. W.; Fitzgerald, G.; Bartlett, R. J. Chem. Phys. Lett.
1987, 141, 61.

10“ Trucks, G. W.; Salter, E. A.; Sosa, C.; Bartlett, R. J. Chem. Phys. Lett. 1988,
14 7, 359.

105 Trucks, G. W .; Salter, E. A.; Noga, J.; Bartlett, R. J. Chem. Phys. Lett. 1988,
I50, 37.

106 Casimir, H. B. G.; Polder, D. Phys. Rev. 1948, 73, 360.
107 Dmitriev, Y.; Peinel, G. Int. J. Quantum Chem. 1981, 19, 763.
108 McWeeny, R. Croat. Chem. Acta 1984, 57, 865.

109 Claverie, P. In Structure and Dynamics of Molecular Systems; Daudel, R., et. al.,
Eds; Reidel: Dordrecht, 1986.

“0 Chan, Y. M.; Dalgarno, A. Mol. Phys. 1968, 14, 101.

111 Chalasinski, G.; Jeziorski, 13.; Szalewicz, K. Int. J. Quantum Chem. 1977, 11,
247.

“2 Jeziorski, B.; Schwalm, W. A.; Szalewicz, K. J. Chem. Phys. 1980, 73, 6215.

“3 Cwiok, T.; Jeziorski, B.; Kolos, W.; Moszynski, R.; Rychlewski, J .; Szalewicz, K.
Chem. Phys. Lett. 1992, 195, 67.

“4 Ahlrichs, R. Chem. Phys. Lett. 1973, 18, 67.

“5 Claverie, P. Int. J. Quantum Chem. 1971, 5, 273.

“6 Schulman, J. M. J. Chem. Phys. 1972, 57, 4413.

“7 Certain, P. R.; Byers Brown, W. Int. J. Quantum Chem. 1972, 6, 131.
“8 Ahlrichs, R.; Claverie, P. Int. J. Quantum Chem. 1972, 6, 1001.

“9 Whitton, W. N.; Byers Brown, W. Int. J. Quantum Chem. 1976, 10, 71.

271

‘20 Kutzelnigg, W. J. Chem. Phys. 1980, 73, 343.

121 Adams, W. H. Int. J. Quantum Chem. S 1990, 24, 531.

122 Adams, W. H. Int. J. Quantum Chem. S 1991, 25, 165.

123 Adams, W. H. J. Math. Chem. 1992, 10, 1.

124 Tang, K. T.; Toennies, J. P. Chem. Phys. Lett. 1990, 175, 511.

125 Kutzelnigg, W. Chem. Phys. Lett. 1992, 195, 77.

126 Carlson, B. C.; Rushbrooke, G. S. Proc. Cambridge Phil. Soc. 1950, 46, 626.
127 Rose, M. E. J. Math. Phys. 1958, 37, 215.

128 Fontana, P. R. Phys. Rev. 1961, 123, 1865.

129 Steinborn, D.; Ruedenberg, K. Adv. Quantum Chem. 1973, 7, 1.

‘30 Wormer, P. E. S. Dissertation, University of Nijmegen, Nijmegen, 1975.
131 Gray, C. G. Can. J. Phys. 1976, 54, 505.

132 Wormer, P. E. S.; Mulder, F.; van der Avoird, A. Int. J. Quantum Chem. 1977 ,
11, 959.

133 Leavitt, R. P. J. Chem. Phys. 1980, 72, 3472.

13“ Leavitt, R. P. J. Chem. Phys. (E) 1980, 73, 2017.

135 Stone, A. J.; Tough, R. J. A. Chem. Phys. Lett. 1984, 110, 123.

136 Piecuch, P. J. Phys. A 1985, 18, L739.

‘37 Jansen, L. Phys. Rev. 1958, 110, 661.

138 Buckingham, A. D. Quart. Rev. Chem. Soc. (London) 1959, 13, 183.
‘39 Kielich, S. Physica (Utrecht) 1965, 31, 444.

”0 Kielich, S. Mol. Phys. 1965, 9, 549.

‘41 Kielich, S. Acta Phys. Polon. 1965, 28, 459.

142 Mulder, F.; Huiszoon, C. Mol. Phys. 1977, 34, 1215.

143 Brink, D. M.; Satchler, G. R. Angular Momentum; Clarendon: Oxford, 1975.

”4 Mulder, F.; van Hemert, M.; Wormer, P. E. S.; van der Avoird, A. Theor. Chim.
Acta 1977, 46, 39.

”5 Prather, J. L. Atomic Energy Levels in Crystals, NBS Monograph 19; National
Bureau of Standards: Washington, 1961.

14" Coope, J. A. R.; Snider, R. F.; McCourt, F. R. J. Chem. Phys. 1965, 43, 2269.

272

‘— iii Err

 

 

 

”7 Coope, J. A. R.; Snider, R. F. J. Math. Phys. 1970, 11, 1003.

148 Coope, J. A. R. J. Math. Phys. 1970, 11, 1591.

149 Stone, A. J. Mol. Phys. 1975, 29, 1461.

150 Stone, A. J. J. Phys. A 1976, 9, 485.

151 Stolarczyk, L. Z.; Piela, L. Int. J. Quantum Chem. 1979, 15, 701.

152 London, F. Z. Phys. Chem. (B), 1930, 11, 222.

153 London, F. Z. Phys, 1930, 63, 245.

154 Morgan, J. D., III; Simon, B. Int. J. Quantum Chem. 1980, 17, 1143.
155 Brezin, E.; Zinn-Justin, J. J. Phys. (Paris), Lett. 1979, 40, L511.

156 Cizek, J .; Clay, M. R.; Paldus, J. Phys. Rev. A 1980, 22, 793.

157 Bender, C. M.; Orszag, S. A. Advanced Mathematical Methods for Scientists and
Engineers; McGraw-Hill: New York, 1978.

158 Damburg, R. J.; Propin, R. K.; Graffi, S.; Grecchi, V.; Harrell, E. M., II; Cizek,
J.; Paldus, J.; Silverstone, H. J. Phys. Rev. Lett. 1984, 52, 1112.

159 Cizek, J.; Damburg, R. J.; Graffi, S.; Grecchi, V.; Hassells, E. M.; Harris, J. G.;
Nakai, S.; Paldus, J.; Propin, R. K.; Silverstone, H. J. Phys. Rev. A 1986, 33, 12.

‘60 Alrichs, R. Theor. Chim. Acta 1976, 41, 7.
"’1 Eisenschitz, R.; London, F. Z. Phys. 1930, 60, 491.

162 Murrell, J. M.; Randic, M.; Williams, D. R. Proc. R. Soc. (London) A 1965, 284,
566.

163 Hirschfelder, J. 0.; Silbey, R. J. Chem. Phys. 1966, 45, 2188.
164 Hirschfelder, J. 0. Chem. Phys. Lett. 1967, 1, 363.

“’5 van der Avoird, A. Chem. Phys. Lett. 1967, 1, 24.

166 van der Avoird, A. Chem. Phys. Lett. 1967, 1, 411.

167 van der Avoird, A. Chem. Phys. Lett. 1967, 1, 429.

168 van der Avoird, A. J. Chem. Phys. 1967, 47, 3649.

169 Murrell, J. N.; Shaw, G. J. Chem. Phys. 1967, 46, 46.

‘70 Musher, J. 1.; Amos, A. T. Phys. Rev. 1967, 164, 31.

171 Carr, W. J. Phys. Rev. 1963, 131, 1947.

172 Kirtman, B. Chem. Phys. Lett. 1968 1, 631.

”3 Jeziorski, B.; Kolos, W. Int. J. Quantum Chem. (Suppl. 1) 1977 12, 91.

273

174 Klein, D. J. Int. J. Quantum Chem. 1987 32, 377.

175 Jeziorski, B.; Chalasinski, G.; Szalewicz, K. Int. J. Quantum Chem. 1978, 14,
271.

176 Cwiok, T.; Jeziorski, B.; Kolos, W.; Moszynski, R.; Szalewicz, K. J. Chem. Phys.
1992, 97, 7555.

”7 Rybak, S.; Jeziorski, B.; Szalewicz, K. J. Chem. Phys. 1991, 95, 6576.
178 Moszynski, R.; Jeziorski, B.; Szalewicz, K. J. Chem. Phys. 1994, 100, 1312.

‘79 Rybak, S.; Szalewicz, K.; Jeziorski, B.; Jaszunski, M. J. Chem. Phys. 1987, 86,
5652.

180 Chipman, D. M.; Bowman, J. D.; Hirschfelder, J. O. J. Chem. Phys. 1973, 59,
2830.

181 Peierls, R. Proc. R. Soc. (London) A 1973, 333, 157.
182 Szalewicz, K.; Jeziorski, B. Mol. Phys. 1979, 38, 191.
183 Moller, C.; Plesset, M. S. Phys. Rev. 1934, 46, 618.

184 Moszynski, R.; Rybak, S.; Cybulski, S. M.; Chalasinski, G. Chem. Phys. Lett.
1990, 166, 609.

185 Moszynski, R.; Jeziorski, B.; Szalewicz, K. Int. J. Quantum Chem. 1993, 45, 409.

186 Moszynski, R.; Jeziorski, B.; Rybak, S.; Szalewicz, K.; Williams, H. L. J. Chem.
Phys. 1994, 100, 5080.

187 Moszynski, R.; Cybulski, S. M.; Chalasinski, G. J. Chem. Phys. 1994, 100, 4998.
188 Salter, E. A.; Trucks, G. W.; Bartlett, R. J. J. Chem. Phys. 1989, 90, 1752.
189 Salter, E. A.; Bartlett, R. J. J. Chem. Phys. 1989, 90, 1767.

190 Koch, H.; Jorgen, H.; Jensen, Aa.; Jergensen, P.; Helgaker, T.; Scuseria, G. E.;
Schaefer, H. F., III. J. Chem. Phys. 1990, .92, 4924.

191 Sadlej, A. J. Mol. Phys. 1980, 39, 1249.

192 Parkinson, W. A.; Sengelev, P. W.; Oddershede, J. Int. J. Quantum Chem. S
1990, 24, 487.

193 Tang, K. T.; Toennies, J. P. J. Chem. Phys. 1991, 95, 5918.

194 LOWdin, P.-O. Int. J. Quantum Chem. 1968, 2, 867.

195 Amos, A. T.; Musher, J. I. Chem. Phys. Lett. 1969, 3, 721.

196 Chipman, D. M. J. Chem. Phys. 1977, 66, 1830.

197 Polymeropoulos, E. B; Adams, W. H. Phys. Rev. A. 1978, 17, 11.
198 Polymeropoulos, E. E.; Adams, W. H. Phys. Rev. A. 1978, 17, 18.

274

199 Certain, P. R.; Hirschfelder, J. O. J. Chem. Phys. 1970, 52, 5977.

200 Certain, P. R.; Hirschfelder, J. O. J. Chem. Phys. 1970, 52, 5992.

201 Klein, D. J. Int. J. Quantum Chem. S 1971, 4, 271.

202 Chipman, D. M.; Hirschfelder, J. O. J. Chem. Phys. 1973, 59, 2838.

203 Polymeropoulos, E. B; Adams, W. H. Phys. Rev. A. 1978, 17, 24.

204 Jeziorska, M.; Jeziorski, B.; Cizek, J. Int. J. Quantum Chem. 1987, 32, 149.
205 Chalasinski, G.; Jeziorski, B. Int. J. Quantum Chem. 1973, 7, 63.

206 Kutzelnigg, W. Int. J. Quantum Chem. 1978, 14, 101.

207 Adams, W. H.; Clayton, M. M.; Polymeropoulos, E. E. Int. J. Quantum Chem.
S. 1984, 18, 393.

208 Cwiok, T.; Jeziorski, B.; Kolos, W.; Moszynski, R.; Szalewicz, K. J. Mol. Struct.
{Theochem) 1994, 307, 135.

209 Jeziorski, B.; van Hemert, M. C. Mol. Phys. 1976, 31, 713.

21° van Duijneveldt-van de Rijdt, J. G. C. M.; van Duijneveldt, F. B. Chem. Phys.
Lett. 1972, 17, 425.

2“ Jeziorski, B.; Bulski, M.; Piela, L. Int. J. Quantum Chem. 1976, 10, 281.

212 Williams, D. R.; Schaad, L. J.; Murrell, J. N. J. Chem. Phys. 1967, 47, 4916.
213 Chalasinski, G.; Jeziorski, B. Mol. Phys. 1976, 32, 81.

2” Chalasinski, G.; Jeziorski, B. Theor. Chim. Acta 1977, 46, 277.

215 Chalasinski, G.; Jeziorski, B.; Andzelm, J.; Szalewicz, K. Mol. Phys. 1977, 33,
971.

216 Jeziorski, B.; Moszynski, R.; Rybak, S.; Szalewicz, K. In Many-Body Methods in
Quantum Chemistry, Kaldor, U., Ed.; Lecture Notes in Chemistry 52; Springer:
New York, 1989; p 65.

217 Bartlett, R. J. Annu. Rev. Phys. Chem. 1981, 32, 359.

218 Bartlett, R. J. J. Phys. Chem. 1989, 93, 1697.

219 Tachikawa, M.; Suzuki, K.; Iguchi, K.; Miyazaki, T. J. Chem. Phys. 1994, 100,
1995.

220 van Hemert, M.; van der Avoird, A. J. Chem. Phys. 1979, 71 , 5310.

221 Tuan, O. F.; Epstein, S. T.; Hirschfelder, J. O. J. Chem. Phys. 1966, 44, 431.
222 Szalewicz, K.; Havlas, Z. Unpublished result quoted in reference 71, 1979.

223 Coester, F. Nucl. Phys. 1958, 7, 421.

275

224 Cizek, J. J. Chem. Phys. 1966, 45, 4256-

225 Cizek, J. Adv. Chem. Phys. 1969, 14, 35.

226 Cizek, J.; Paldus, J. Int. J. Quantum Chem. 1971, 5, 359.
”7 Paldus, J.; Cizek, J.; Shavitt, 1. Phys. Rev. A 1972, 5, 50.
228 Paldus, J.; Cizek, J. Adv. Quantum Chem. 1975, 9, 105.

”9 Paldus, J. In Methods in Computational Molecular Physics; Wilson, S., Diercksen,
G. H. F ., Eds; NATO ASI Series; Plenum: New York, 1992; p 309.

23° Paldus, J. In Relativistic and Electron Correlation Effects in Molecules and Solids;
Malli, G. L., Ed.; NATO ASI Series B, Vol. 318; Plenum: New York, 1994; p 207.

231 Jaszunski, M.; McWeeny, R. Mol. Phys. 1985, 55, 1275.

232 Jeziorski, B.; Moszynski, R. Int. J. Quantum Chem. 1993, 48, 161.

233 Rice, J. E.; Handy, N. C. J. Chem. Phys. 1991, 94, 4959.

23’“ Sasagane, K.; Aiga, F .; Itoh, R. J. Chem. Phys. 1993, 99, 3738.

235 Purvis, G. D., III.; Bartlett, R. J. J. Chem. Phys. 1982, 76, 1910.

236 Wormer, P. E. S.; Hettema, H. J. Chem. Phys. 1992, 97, 5592.

237 Rijks, W.; Wormer, P. E. S. J. Chem. Phys. 1988, 88, 5704.

238 Rijks, W.; Wormer, P. E. S. J. Chem. Phys. 1989, 90, 6507.

239 Rijks, W.; Wormer, P. E. S. J. Chem. Phys. (E) 1990, 92, 5754.

240 Wormer, P. E. S.; Hettema, H. Polcor Package, Nijmegen, 1992.

2‘“ Visser, F.; Wormer, P. E. S.; Stam, P. J. Chem. Phys. 1983, 79, 4973.

242 Visser, F.; Wormer, P. E. S.; Stam, P. J. Chem. Phys. (E) 1984, 81, 3755.
243 Visser, F .; Wormer, P. E. S. Mol. Phys. 1984, 52, 923.

244 Visser, F.; Wormer, P. E. S.; Jacobs, W. P. J. H. J. Chem. Phys. 1985, 82, 3753.
245 Cybulski, S. M. J. Chem. Phys. 1992, 96, 8225.

246 Isnard, P.; Robert, D.; Galatry, L. Mol. Phys. 1976, 31, 1789.

247 Piecuch, P. Chem. Phys. Lett. 1984, 106, 364.

248 Piecuch, P. Int. J. Quantum Chem. 1985, 28, 375.

249 Piecuch, P. Int. J. Quantum Chem. 1984, 25, 449.

25° Sauer, S. P. A.; Diercksen, G. H. F.; Oddershede, J. Int. J. Quantum Chem. 1991,
3.9, 667.

276

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

Wormer, P. E. S.; Rijks, W. Phys. Rev. A 1986, 33, 2928.

Rijks, W.; van Heeringen, M.; Wormer, P. E. S. J. Chem. Phys. 1989, 90, 6501.
Visser, F.; Wormer, P. E. S. Chem. Phys. 1985, 92, 129.

Hettema, H.; Wormer, P. E. S. J. Chem. Phys. 1990, 93, 3389.

Buehler, R. J.; Hirschfelder, J. 0. Phys. Rev. 1951, 83, 628.

Buehler, R. J.; Hirschfelder, J. 0. Phys. Rev. 1952, 85, 149.

Ng, K.-C.; Meath, W. J.; Allnatt, A. R. Mol. Phys. 1976, 32, 177.

Kay, K. G.; Todd, H. D.; Silverstone, H. J. J. Chem. Phys. 1969, 51, 2359.
Jaszunski, M.; McWeeny, R. Mol. Phys. 1982, 46, 483.

Jaszunski, M.; McWeeny, R. (E) Mol. Phys. 1986, 57, 1317.

Hettema, H.; Wormer, P. E. S.; Jergensen, P.; Jensen, H. J. A.; Helgaker, T. J.

Chem. Phys. 1994, 100, 1297.

Berns, R. M.; van der Avoird, A. J. Chem. Phys. 1980, 72, 6107.
Vigné—Maeder, F.; Claverie, P. J. Chem. Phys. 1988, 88, 4934.
Stone, A. J. Chem. Phys. Lett. 1981, 83, 233.

Stone, A. J.; Alderton, M. Mol. Phys. 1985, 56, 1047.

Sokalski, W. A.; Poirier, R. A. Chem. Phys. Lett. 1983, 98, 86.
Sokalski, W. A.; Sawaryn, A. J. Chem. Phys. 1987, 87, 526.
Dalgarno, A.; Lewis, J. T. Proc. R. Soc. (London) A 1955, 233, 70.
Young, R. H. Int. J. Quantum Chem. 1975, 9, 47.

Stone, A. J.; Fowler, P. W. J. Phys. Chem. 1987, 91, 509.

Stone, A. J. Chem. Phys. Lett. 1989, 155, 102.

Stone, A. J. Chem. Phys. Lett. 1989, 155, 111.

Stogryn, D. E. Mol. Phys. 1971, 22, 81.

Stogryn, D. E. Mol. Phys. 1972, 23, 897.

Piecuch, P. J. Math. Phys. 1986, 27, 2165.

Piecuch, P. Acta Phys. Polon. A 1988, 74, 563.

Piecuch, P. Acta Phys. Polon. A 1990, 77, 453.

Piecuch, P. In Molecules in Physics, Chemistry and Biology; Maruani, J., Ed.;

Kluwer: Dordrecht, 1988; V01. 2 (Physical Aspects of Molecular Systems), p 417.

277

279 Piecuch, P. Int. J. Quantum Chem. 1982, 22, 293.
280 Jeziorski, B.; Moszynski, R.; Szalewicz, K. Chem. Rev. 1994, 94, 1887.

281 van der Avoird, A. In Intermolecular Interactions; Pullman, B., Ed.; Reidel: Dor-

drecht, 1981; p 1.

282 Handy, N. C.; Amos, R. D.; Gaw, J. F.; Rice, J. E.; Simandiras, E. S. Chem.
Phys. Lett. 1985, 120, 151.

283 Harrison, R. J.; Fitzgerald, G.; Laidig, W. D.; Bartlett, R. J. Chem. Phys. Lett.
1985, 124, 291.

28“ Handy, N. C.; Amos, R. D.; Gaw, J. F.; Rice, J. E.; Simandiras, E. S.; Lee, T.
J.; Harrison, R. J.; Fitzgerald, G.; Laidig, W. D.; Bartlett, R. J. It Geometrical
Derivatives of Energy Surfaces and Molecular Properties; Jergensen, P., Simons,
J ., Eds; Reidel: Dordrecht, 1986.

285 Cwiok, T.; Jeziorski, B.; Moszynski, R. Reference 140 in reference 280.

286 Williams, H. L.; et. al. Reference 358 in reference 280.

287 Moszynski, R. Reference 366 in reference 280.

288 Bulski, M.; Chalasinski, G.; Jeziorski, B. Theor. Chim. Acta 1979, 52, 93.
289 Kumar, A.; Meath, W. J. Mol. Phys, 1985, 54, 823.

278

 

u _, v. v“, 5‘“,

 

   

ulglnllylngygguliy,I