STUDIES IN THE CENTRIFUGAL DISTORIION THEORY OF
' TRIANGULAR TRIAIOMIC MOLECULES

By

Azam Niroomand-Rad

A DISSERIAIION

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

DOCTOR OF PHILOSOPHY
Department of Physics

1978

W

AGO}

ABSTRACT

STUDIES IN THE CENTRIFUGAL DISTORTION THEORY OF
TRIANGULAR TRIATOMIC MOLECULES

by

Azam Niroomand-Rad

In this dissertation the molecular vibration-rotation
Hamiltonian of Darling and Dennison is used in the expanded form of
Nielsen, Amat, and Goldsmith. Expressions for four linearly inde-
pendent linear combinations of the ten sextic centrifugal distortion
coefficients of triangular triatomic molecules are presented. These
combinations are formed in such a way that the resulting expressions
depend only on the equilibrium geometry and the harmonic force field
of the molecule. These expressions appear to be potentially useful
as a set of planarity—constraints on the ten sextic coefficients ob-
tained in the expansion of the Darling-Dennison Hamiltonian and can
be utilized to effect a Watson-type reduction of the sextic portion
of the Hamiltonian. Also in this dissertation, the quartic and
sextic centrifugal distortion coefficients of ozone have been cal-
culated and the results compared with experiment. The agreement is

found to be quite satisfactory.

Dedicated to
Gholamhossein Gharehgozloo Hamedani, my husband
and

Mahmoud Niroomand-Rad, my father

ii

ACKNOWLEDGMENTS

I am most grateful to Professor Paul M. Parker for suggesting
these studies and for his kind help and extensive guidance as well as
for many valuable discussions throughout this work.

I am particularly indebted to my husband and my children for
their patience and understanding. I would like to thank Ms. Zohreh
for her help in taking care of my children during my study at Michigan
State University. I would also like to thank Arya-Mehr University of
Technology, Tehran, Iran and the Ministry of Sciences and Higher
Education of Iran for their partial support of my studies at Michigan

State University.

iii

TABLE OF-CONTENTS

List of Tables

List of Figures

Chapter

1

2

Introduction
Molecular Vibration-Rotation Hamiltonian

1 The Darling-Dennison Hamiltonian

2 Watson's Simplification of the Darling-
Dennison Hamiltonian

2.3 Expansion of the Hamiltonian

2.
2.

The Successive Vibrational Contact Transformation
Technique

3.1 First Contact Transformation
3.2 Second Contact Transformation
3.3 Third (and Higher) Contact Transformations

The Normal Modes Problem for Triangular Triatomic
Molecules

4 l Equilibrium Geometry of the XYz-type Molecule
4 2 Normal Coordinates of the XYz-type Molecule

4.3 Molecular Parameters of the xyz—type Molecule
4 4 Molecular Parameters of the xxx-type Molecule

Centrifugal Distortion Coefficients for Triangular
Triatomic Molecules

5.1 Development of the Hamiltonian
5.2 Second-Order Centrifugal Distortion Constants
5.3 Fourth-Order Centrifugal Distortion Constants

Rotational Contact Transformation Technique and
Reduction of the Hamiltonian

6.1 Reduced Hamiltonian and Determinable
Combinations of Coefficients

iv

Page
vi

viii

23
23
3O
35
38
38
41
46
51
57
57

63
67

72

72

Chapter

6 (continued)
6.2 Rotational Contact Transformation
6.3 Determination of Quartic and Sextic
Determinable Combinations of Coefficients
7 Calculation of Asymmetric-Rotator Centrifugal
Distortion Coefficients by the Aliev-Watson
Method
7.1 Direct Perturbation Treatment for the
Calculation of the Sextic Centrifugal
Distortion Coefficients
7.2 Reordered Perturbation Treatment for the
Calculation of the Sextic Centrifugal
Distortion Coefficients
8 Centrifugal Distortion Sum Rules
8.1 Second-Order Centrifugal Distortion Sum
Rules
8.2 Fourth-Order Centrifugal Distortion Sum
Rules
8.3 Constrained Empirical Constants
8.4 Cylindrical Tensor Form Hamiltonian
9 The Centrifugal Distortion Coefficients of Ozone
9.1 The Fourth-Order Centrifugal Distortion
Coefficients for an XYXFType Molecule
9.2 Fundamental Molecular Constants of Ozone
9.3 Calculated Centrifugal Distortion Constants
9.4 Reduced Hamiltonian
9.5 Comparison of the Quartic Distortion
Coefficients
9.6 Comparison of the Sextic Distortion
Coefficients
10 Conclusion

List of References

Page

74

82

86

86

104

111

111
114
122
125
128
128
133
138
139
145
149
159

161

Table

(2-1)

(2-2)

(2-3)

(4-1)

(5-1)

(5-2)

(6-1)

(6-2)

(7-1)

(7-2)

(7-3)

(7-4)

(7-5)

(7-6)

(7-7)

LIST OF TABLES

Rotational Operators R of the Expanded
Hamiltonian Terms

The B Coefficients of the Expanded Hamiltonian
Terms

Terms of the Expanded Vibration-Rotation
Hamiltonian

The Coefficients th Introduced in Eqs. (4-82)-
(4-85) 1

Abbreviated Notation for Taeyd Coefficients

The Fourth-Order Centrifugal Distortion Co-
efficients of XYZ as Calculated by Sumberg-Parker

The Number and Species of Terms in the Standard
Form of the Hamiltonian (6-2)

The Number and Species of Terms in the Standard
Form (6-6) of SZm-l

S Operators for the Vibrational Contact Trans-
formation

Sextic Centrifugal Distortion Hamiltonian of an
Arbitrary Molecule

Asymmetric-Rotator Point Groups

The Fourth-Order Centrifugal Distortion Coefficients
of.XYZ as Calculated by Aliev—Watson

Contribution to the Distortion Constants Resulting
From the Elimination of the Non-Orthorhombic Terms

The Complete Expressions for the Fourth-Order
Centrifugal Distortion Coefficients of XYZ as
Calculated by Sumberg-Parker

The Differences Between Sumberg-Parker and Aliev-
Watson Sextic Centrifugal Distortion Coefficients

vi

Page

18

19

20

49

64

70

76

78

91

92

94

96

98

99

103

Table

(8-1)

(8-2)

(8-3)

(8-4)

(9—1)

(9-2)

(9—3)

(9-4)

(9—5)

(9-6)

Page
The Functions FiP, FiP, Pip, and Pip Defined
by Eqs. (8-16)-(8-l9) for the XYX(CZV) Molecule 118
The Functions Fiw, Fgw, Fgw, and Ffiw Defined
by Eqs. (8-16)-(8-19) for the XYX(C2V) Molecule 119
The Functions FiP, Pip, Pip, and PEP Defined
by Eqs. (8-16)-(8-l9) for the XXZ(CS) Molecule 120
The Functions Ff”, Fgw, Fgw, and FEW Defined
by Eqs. (8-16)-(8-l9) for the XYZ(CS) Molecule 121
The Fourth-Order Centrifugal Distortion
Coefficients for an XYXFtype Molecule 129
Calculated Sumberg-Parker (SP) and AlieveWatson
(AW) Sextic Distortion Coefficientsof 16O3 in cm.1 140
Calculated Second-Order Distortion Coefficients
Ti of 1603 in cm.1 141
Observed and Calculated Second-Order Distortion
Coefficients of 1603 in cm"1 148
Observed and Calculated Fourth-Order Distortion
Coefficients of 1603 in cm“1 151

Observed and Calculated Fourth-Order Invariants
16 -1 156

vii

LIST OF FIGURES

Figure Page
(4-1) Equilibrium Configuration of the Triangular
Triatomic Molecule 38
(4—2) Normal Modes of the X32 Molecule 46
(4-3) Equilibrium Configuration of the Nonlinear XYX
'Molecule 51
(9-1) Comparison of the Experimental and Theoretical
~ 152
Values of Hiof Ozone
(9-2) Comparions of the Experimental and Theoretical
Values of the Invariant Ii for Ozone 157

viii

1. INTRODUCTION

One of the principal tasks of the molecular spectrosc0pist
is to determine and to interpret the vibration-rotation energy level
structure of the molecule under study. The analysis of the infrared
spectra of molecules in the gas phase gives precise information about
the vibration-rotation energies, and knowledge of these energies can
lead to the accurate determination of molecular constants such as
bond distances, bond angles, vibrational frequencies, force contants,
centrifugal distortion constants, etc. An understanding of these
quantities is relevant in the determination of the detailed structure
and prOperties of the molecule and helps to better interpret the
physical and chemical prOperties of bulk matter.

The vibration-rotation Hamiltonian of diatomic molecules has
been treated to fourth and even higher orders of approximation many
years ago, and also the case of linear triatomic molecules (COz-type
molecules) has been studied extensively. It appears generally to be
quite impossible to find exact analytic expressions for the energy
levels of polyatomic molecules. For this reason, assumptions must be
made which one hopes are valid in practice, and the general theoretical
expression for the energies of polyatomic molecules must be treated by
an expansion formalism in successive orders of approximation. For
instance, assuming the validity of the Born-Oppenheimer approximation

allows separation of the vibration-rotation motion of the nuclei from

‘1

the electronic motion to a high degree of accuracy. Similarly, it has
been found that one can often safely ignore the energy contribution of
the nuclear spins to a high order of approximation.

During the last decade, the field of high resolution molecular
spectroscopy has experienced many impressive developments in instru-
mentation, resulting in particular from the availability of on-line
computing facilities, sensitive detectors, the progress of interfero-
metric methods, and the use of laser sources and detection methods.
For many polyatomic molecules, high quality rotation and rotation-
vibration spectra are being obtained both in the infrared and the
microwave regions. The complete interpretation of such spectra re-
quires the use of very accurate formulas for the frequencies of the
spectral lines expressed in terms of quantum numbers and molecular
parameters. To this end, it is necessary to compute rotation-vibra-
tion energies to a high order of accuracy which means that fourth, and
even higher, orders of perturbation theory need to be considered.

In cases such as the triangular XYXrtype (HfJ-type) molecule, fourth-
order centrifugal distortion coefficients are required in order to
account for the observed results to the experimental accuracy attain-
able, in some cases even for the low values of the angular momentum
quantum number J1, Therefore, one needs to consider a fourth-order
Hamiltonian to get a satisfactory and theoretically meaningful fit to
the spectral data.

The general vibration-rotation Hamiltonian for asymmetric
rotator molecules has been developed by Chung and Parker in the

2
Nielsen-Amat Goldsmith expansion of the Darling-Dennison vibration—

rotation Hamiltonian through the consideration of symmetry restric-
tions imposed by the applicable asymmetric rotator point group. Later
on, Watson3succeeded in developing a form of the Darling-Dennison
Hamiltonian which greatly simplified subsequent calculations. However,
even with this simplification, the general formulation of Amat-Nielsen—
Goldsmith”.5 is unnecessarily complicated when applied to the case of
the asymmetric rotator, principally because of its inclusion of de-
generate normal modes of vibration which are absent in asymmetric
rotator molecules. Therefore, instead of working with the general
formulation, Chan and Parkeréstarted with the Darling-Dennison vibra-
tionrrotation Hamiltonian in Watson's simplified form for the XYZ~type
molecule. Then the Hamiltonian was expanded and subjected to two
successive contact transformations of the Van Vleck type. The re-
sulting Hamiltonian for a given vibrational state has the form of a
power series in the angular momentum components. By using an extended
version of Watson's theoryi this Hamiltonian can be related to experi-
mental results in such a manner that meaningful fits to high-resolu-
tion experimental data are possible, at least in principle. Calcula-
tion of a complete set of fourth-order centrifugal distortion co-
efficients for X32 and XXthype triangular molecules was carried out
by Sumberg and Parkef7in a form which exhibits extensive cyclic and
algebraic regularities and which could readily be used in the present
work to determine a number of constraints on the sextic coefficients
due to the planarity of the nuclear framework configuration of the

molecule.

For the general case of asymmetric rotator molecules, Watson
has shown that the number of independent sextic coefficients (or
independent'linear combination of these) is seven. The constraint
due to planarity- reduces this number by one, to a total of six. The
principal aim of this thesis work was to find and to calculate, for
triangular triatomic molecules, four linearly independent linear
combinations of the ten sextic centrifugal distortion coefficients
which would represent the complete specification of the constraint
that reduces the ten original coefficients to six independent ones.
This attempt was successful, since the linear combinations that were
determined depend only on the equilibrium geometry and the harmonic
force field parameters of the molecule, quantities which are ordinarily
known to much better precision than either the calculated or the
empiriCal values of the sextic coefficients.

The triangular XYX—type molecule is considered explicitly as
a special case which leads to simplified expressions due to the higher
symmetry. Finally, the complete set of centrifugal distortion co-
efficients of ozone has been calculated and the results were compared
with the available experimental data. The agreement was found to be

quite satisfactory.

2. MOLECULAR VIBRATION-ROTATION HAMILTONIAN

2.1 The DarlingrDennison Hamiltonian

 

For any theoretical calculation of the energies of a mole-
cule, it is necessary to formulate a suitable quantum mechanical
Hamiltonian. The total Hamiltonian of a molecule would have to in—
clude an electronic part as well as a vibration-rotation part. The
electronic energy is not of interest here, since we wish to consider
vibration—rotation transitions during which the molecule remains in
its electronic ground state configuration. For such a case, Born and
Oppenheimerehave shown that it is allowable to separate the electronic
motion from the nuclear motion to a very good degree of approximation.
Since the electrons are moving much faster than the nuclei and con-
sequently the wavefunction of the electronic state is almost in-
dependent of the change in the internuclear distances, the Born-
Oppenheimer approximation is valid in many cases, especially when the
electronic state is one of zero total electronic angular momentum. To
the accuracy of the approximation, the total wavefunction can be
written as the product of an electronic and a vibration-rotational
wavefunction. For the present work, only the vibration-rotation

Hamiltonian is of direct interest.

The Schrodinger equation

(H - E)? = 0 (2-1)

of a rotating and vibrating polyatomic molecule was first discussed
by Wilson and Howard? and somewhat later by Darling and Dennisod? The
second formulation is now known to be equivalent to that of the former
authors and proves more convenient for the present discussion.
Essentially, the derivation is based on a classical develOpment of

the vibrational and rotational energies, subsequently transcribed into
the preper quantum mechanical operator form. We therefore begin with

the Darling-Dennison Hamiltonian for a polyatomic molecule which

reads:
k -h h
H bu ZG,B(PG - pa)ua8n (PB - p8)u

b

+ 5 1‘X * ' * h + V (2 2)
u 5 psp psu °

The lower-case Greek indices a and B (and lower-case Greek

indices in general) range over x, y, and z, the principal axes
directions of the equilibrium inertia tensor of the molecule. The
(x,y,z) coordinate system is fixed with respect to the equilibrium
configuration of the molecule ("body-fixed") and its origin is taken

to coincide with the instantaneous center of mass. Let the equilibrium
and instantaneous positions of the i-th nuclei in (x,y,z) be denoted

by “oi. and a1, respectively. Then the displacement of the i-th

nucleus from its position of equilibrium is:

“-3 -I{
1 1 ‘
01

(2-3)

.+
a

In Eq. (2-2), Pu is the a-th component of the total angular
momentum referred to the body-fixed axes and can be expressed solely

in terms of the Euler angles and the time derivatives with respect

to these angles. Thus the Euler angles can be taken as the rotational
coordinates of the problem. The vibrational coordinates to be used in
connection with Eq. (2-2) are the normal coordinates QS. To trans-
form to the normal coordinates, a set of transformation coefficients
1:8 is introduced which transforms mass-weighted cartesian to normal

coordinates as follows:

/m'a' = 2 2a

1 i s ist ' (2-4)

Here m1 is the mass of the i-th nucleus. For asymmetric rotor

molecules, no index enumerating essential degenerate modes of vibra-

11
tion is required as these do not occur. The vibrational momentum

*
ps conjugate to the normal coordinate QS is defined as:

p: = -ifi——— . (2-5)

The internal angular momentum pa occurring in Eq. (2-2) can then

be defined as:

(1*- * 26
Ru 3 ZSASPS ‘ stsvcsstsva 9 < ' )

where

a 8 £7 - 8 Y ' -
Cs's zi<£is'£ is 215213,), a,8,y cyclic (2 7)

AG

3

BY__ Y
Zs'ZiU'is'g'is 218113,)QS '

2 svcgtstv 9 G’ng CYC11C° (2‘8)

The ;:,8 are the Coriolis coupling coefficients. It is clear from

their definition that ;:,S - -§:S. and 5:3 - O.

The effective moments and products

are defined, respectively, by:

, B _ a 2
Ida Ida zs<As) ’

a 8

where

2 2
Ian ' Zim1(81 + Yi)’

IaB = ‘Zimiaisi’

of inertia, I' and I'
do a

B

(2-9)
a # 8 (2-10)
a # 8 H Y (2-11)
a # 8 (2-12)

are the instantaneous moments and products of inertia. The effective

inertia tensor and its determinant are defined by:

( I' -I' _I! ‘
XX

 

 

xy xz
-1
3 -I' I' -I' = I' , 2-13
In] yx yy yz I ] ( )
4;... 5.
L 4
[u] = [I'J‘l , (2-14)
a det[ ] = ——L—— (2-15)
u u detII'] ’
a I! I! + I 0 ’ 2-
Has u( YY a8 IayIBy) a # B # Y ( l6)
2
a 9 V _ 9 _
uaa MIBBIYY IBY)’ a f 8 # Y - (2 17)

The components of the effective rotational angular momentum, Pa — p ,

and the components of the angular velocity ma

[P - p] = [I'JIw]

a

are related by

(2-18)

Finally, V is the effective vibrational potential energy
which is usually written as a positive definite power series in the
normal coordinate QS, with the harmonic portion, quadratic in the
Qs’ the leading and lowest power terms. All quantities occurring in

the Hamiltonian (2-2) have now been defined.

2.2 Watson's Simplification of the DarlingrDennison Hamiltonian

If one commutes out n from the first two terms of Eq. (2—2),

the Darling-Dennison Hamiltonian becomes:

*2
= - - + .—
H sza 8(5 pa)“a8(PB p8) szsps + U + V (2 l9)

9

where

U = -52
a

9

I. _ A: a _ 1. 4.
Bu (Pa pa)ua8u [p8,u ] 82a Bu [pa,u ]
_ a * 4. * 1.
x uaB(PB pa) + kzsu Psu [P3,u ]
5 * “h * _
+ 122811 [Psw 1pS . <2 20)

To obtain this result, one uses the fact that Pa operates only on

the Euler angles and thus commutes with all quantities in H except

P and P .
B Y

In applications of Eq. (2-19), it has been customary to start

by introducing the power series eXpansion of “a and u in terms

8
of the normal coordinates, and to evaluate U from these expansions

' 3
to the desired degree of approximation. However, Watson has been able
to show that it is much simpler to use the commutation relations and

the prOperties of the ”08 tensor to evaluate U directly, without

first expanding it. After lengthy calculation, Watson finds that:

10

l 2
U - 8 M Sc uaa. (2-21)

This amazingly simple result constitutes Watson's simplification of
the Darling-Dennison vibration-rotation Hamiltonian which can thus be

written as:

‘*
) + kisfi: -'% M22 u + V .

8 i - —
H éza,B(Pa pa)uaB(PB p a do

8
(2-22)
This form of the Hamiltonian will be taken as the starting point for

the expansion in the normal coordinates in the next section.

2.3 Expansion of the Hamiltonian
In this section, a review of the expansion of the Watson form
of the fourth-order of approximation in the energy is given, and
terms standing in the various orders of approximation are identified
12 13

and written out explicitly in Amat-Nielsen-Tarrago and also in Watson

notation. We start with Eq. (2-22) and write it as:

H ' x12c.8“aepaps ‘ 122c.8(pauae + “aspa)P8 + IzaBpauaBpB

12 *2
8 M zauaa + 523 pS + v . (2-23)

The terms represent, in succession, the pure rotational energy, the
Coriolis coupling energy, a correction to the Coriolis energy,
another correction to the Coriolis energy, the vibrational kinetic
energy, and the potential energy of vibration.

It is not possible to find directly the eigenvalues of the
Hamiltonian (2-23). We therefore seek an expansion in orders of

magnitude of the general form:

11

2 3 4
H = H0 + 1H1 + 1 H2 + A H3 + A H4 +... (2-24)

where A is the expansion parameter. This form will allow us to
apply perturbation theory. We begin the expansion by writing the
effective moments and products of inertia in terms of the normal
coordinates. Substituting the appropriate form of Eq. (2-3) for

B

a and Yi into Eqs. (2-11) and (2-12), and using Eq. (2-4) to

i’ i

introduce the normal coordinates gives:

IaB Iaasas + 23 a8 QS + 23’s, Ass'Qst' a, B x,y,z (2 25)
where
I° = X m (82 + Y2 ) a # 8 ¥ Y (2-26)
on i i' O O. ’
i 1
1° = -£ m.a B , (2-27)
as i 1 Oi Oi
no a /- B Y _
as 22?1 mi (801$is + YOizis)’ a 7‘ B 3‘ Y (2 28)
a“8=-z/E(a 9.8 +3 2“) aa‘Ba‘Y (2-29)
3 i 1 O1 is 01 is ’
aa a B 'B Y Y _
Ass. 21(2182’18' + g‘iszisl) 9 a # B * Y (2 3O)
as a _ a B _
Ass' 21 kislis' , a ¥ 8 . (2 31)

Substitution of Eqs. (2-25) and (2-8) into Eqs. (2-9) and (2-10)

gives:

I a o 0-8 0&8 t _
IaB IaaaaB + is as Qs + Zs,s'(Ass') Qst' (2 32)

where

as . - a8 _ a 8 _
(A88!) A85! XS" CSSHCS' H ° (2 33)

S

12

The equilibrium products of inertia I° , a # B, are zero in the

a8
principal axes system (x,y,z).

Let us now continue with the expansion of the kinetic energy
part, H - V, of the Hamiltonian (2-23). To accomplish this, “68 and
non as defined by Eqs. (2-16) and (2-17) are written as a power
series in the normal coordinates. This is permissible, because “as
and “on are functions of the components of the instantaneous inertia
tensor, and therefore functions of the normal coordinates only. The
normal coordinates, in turn, are linear combinations of the in-
stantaneous cartesian displacement coordinates, assumed to be small

compared to the equilibrium separations. The u take the following

a8
general form:

V :- 8 o o 0-8 (18
“as “8a (MIMIBBHMO) + 23 9(1)s QS
a8 a8
+ 23,3! Q<2>SS'QSQS' + ZS,S',S" 9(3)SS'S"QS Q3 '0
+2 9(4 4)“ QSSQ QS QS +. 1 <2-34)
S,S',S",S"' SSISHSHV V H I” "

where the various Q are the coefficients obtained in the expansion.

The most compact and convenient form of these coefficients appears to
14
be the one given by Rothman and Clough which is the following:

9(0)“5 = IZBGdB, (2-35)
9(1)“B - 21° cl(nigs(:;§)“ 12:8 , (2-36)
52(2):B . 21° “1&6A :2 ')3 2 R::., (2—37)
n(3)“3. = 21° 1° Boiéi—l:§:)* 3Ra8, ", (2—38)

33' s" K6 as s

l3

 

9(4)“ =21° 1° (ASAS'AS"AS"')Y 4Ra8 <2-39)
SS'SHS"' ca 88 “8 SS'SHSH' 9
with
A‘YB
l 0.8 s _ _s_ _
Rs 21. , (2 40)
(10.
2Rue, =- - -3- ): (LRGY AY§ + 111°”! AYB) , (2-41)
33 8 s s s s
33““, .. - --2- z ( 2R°‘Y AY§+ ZRO‘Y" AY‘? + 2R°‘Y .. AYB).
as s 9 ss 3 s s
(2.42)
4 a8 = --2— Y8 +3 “Y YB
88181.8"! 32 Z (3 Rgsisfl A SH! +RSS'S"! A3"
+ 3R°Y 8.. AYB + 312°“: .. AYB ).
S S S 8
(2-43)
and
aa8 “2
A?- -I-—§ (— ) <2-44)
88

In these equations, the As appear in the harmonic potential func-

tion as:
1
V ='§ 2 A Q , (2-45)

i.e., they are proportional to the squares of the corresponding
normal frequencies. The potential energy function can be expanded

as the Taylor series:

.. av . _1__ av ,
V V0 + Z s[aT .] as + . 2 . , . use
3 o

3
3 V ,
3! Zs,s',s" Bagaa;.3a;n S
J o

2
‘ kzsxsqs + Z t n KSSISHQSQSUQSH

35?.25

 

 

+ zSiS':S"_<_S"' KSS'SHSH'QS QS 'Qs "Q3 "1 + co. (2_46)

14

Since this Taylor series expansion of V is taken about the equilibrium

positions of the nuclei, the total force at equilibrium in the

3V
3Q3 0
stant V0 has no physical significance and may be set equal to zero,

s-th normal mode must be equal to zero, hence ( = O. The con-
and V then can be rewritten as a function of the normal coordinates,

as shown above, with
A5 = 2wc m , (2'47)
3 s

where ms is the s-th normal frequency. The various sets of co-
efficients K are the force constants of the molecular force field
in the various orders of the expansion.

When Eq. (2-34) is substituted into Eq. (2-23), the expanded

form of. H, Eq. (2-24) is obtained, with:

 

 

 

 

P2 P2
3 1 a h._§ 2 _
H0 52 “1° + HAS( 2 + qs)} (2 48)
s a on K
9(1)a8 2 2p P
l S H k a a
H=-ZZ{"T"‘?‘(“‘) qPP- . }+V (2-49)
1 28 a8 IaaIBB As 5 a B loo 1
9(2)“. 4
H2 8 2'2 Z ff;—_%§— (Aflk )h qsqs'PaPB
83' 08 do 88 s s'
2
:zu)‘YB 2 p
- —'§*—. . (“—Y‘W q + q .13)? +--? }+ V (2-50)
I I A a S S a B I 2
do 88 s 7 ca
no)“. .. 6
1 ss 8 M k
H ' - 2 Si: 0 o ( ) q q !q "P P
3 38'3" a8 IaaIBB AsAs'As" s s s a 8
9(2)“. 4
38 H h a
"' o o _ + P
IaaIBB (A818,) (paqsqs' qsqs'po) 8
M1)“8 2 2
___s_ L Y: L Y: -
+ Io I° (A ) PaquB + A(3)S( ) qs} ‘+ V3 (2 51)

aa 88 s As

15‘

 

 

 

 

8
52(4)“ . . 8
1 ss s"s" u
H2.— 2 Z{° ‘ ( )quqHQIHPP
4 2 ss's"s"' a8 IaaIBB AsAs'AsMsm s S S S a B
52(3)::usu M6 1‘
- IZGIEB (Aslsnlsn) (paqsqs'qs" + qsqs'qs"pa)PB
8
9(2)“ , 4 4
ss u a u z.
4- 130.133 (Asks') qusqsmB + A(4)ss'(>‘s)‘s') qsqs.} + v4. (2-52)

In these equations, qs is a dimensionless normal coordinate and p3

is its conjugate momentum defined by:
2
qs = (AS/u )YQS. (2-53)
2 *
p3 = (u mgp‘fias . (2-54)

1’ V2, V3 and V4 denote, respectively, the cubic,

quartic, quintic, and sextic portions of the anharmonic potential and

The symbols V

are defined through:

V1 3 he 2 ' H kSS'SuquS'qS" ’ (2'55)
$33.33

V2 a he 2 ' n kSS'SHS'” quS'qs"qs"' a (2‘56)
35§ 5s

V3 ' 1“: Z ' n kssvsnsnvsnn qsqslqansntqsmn (2‘57)
S<S <3

<8" ' < S" '

= h k , 7
V4 (18:8 ' <8" 88' S' S" v Snusml v ,quS ' qansntqsnnq'n: v

i-s'nvEvaS-Smn (2-58)

The quantities AK3)s and A(4)ss, which appear in H3 and H4

originate from the term U and are given by

2 9(1):”
A<3>s = --Z- i'z;:—;§ , (2-59)
on
u2 9(2)“:,
A(4) , = --- 2 (2-60)
33 4 o 2
(I )
no
The internal angular momenta pa may be expressed as:
= Z (A /A )h C“ q q - (2-61)
pa , s' 3 ss' 3 s'

3,3

It is convenient to denote the various terms in the expansion
of the vibration-rotation Hamiltonian in a systematic manner by
adopting Watson'suzotational scheme in which the various terms are
designated by Hmn’ where the first subscript is the degree in the
vibrational Operators (coordinates and momenta) and the second sub-
script is the degree in the components of the total angular momentum
vector J. To conform completely to Watson's notation, the following
modifications must be introduced:

qk,pk: dimensionless normal coordinates and dimensionless

momenta which correspond to Amat-Nielsen qk, 11 pk,
Luk: harmonic vibrational frequencies,
.1“: dimensionless angular momentum components which
correspond to Amat-Nielsen (Pa/“)-

The terms Hmn’ expressed in wavenumber units, can be related to

H0, H1, H2, H , and H4 as follows:

3

H0 - H02 + 320, (2-62)

H 3 H + H + H

1 12 21 30° (2'63)

= + «-
H2 H22 + H31 + H4O + H00 H40, (2 64)
* **
a + + , ..
H3 H32 + H41 + H50 H10 H50 (2 65)
* **
H = H + H + H + H + H . (2-66)

To give the various Hmn explicitly and in a convenient form, one
can define a set of rotational operators as listed in Table (2-1).
These definitions constitute an extension of Watson's definitions.

It will be seen that if an R operator has an upper index, it is
linear in Ja' and if it has no upper index it is quadratic in Ja'
Lower indices not separated by commas may be permuted, and inter-
changing an upper index with the corresponding lower index introduces
a factor: - (m upper/w lower).

It is also helpful to define the set of coefficients B
listed in Table (2-2). With the definitions just introduced, the
terms of the vibration-rotation Hamiltonian can be written as given
in Table (2-3).

It can be seen that in Table (2-3) all summations over vibra-
tional indices are unrestricted. As a consequence, the anharmonic
potential constants k' in the present scheme are not, in general,
equal to the anharmonic potential constants k used in the Nielsen-

Amat Scheme, Eqs. (2-SS)-(2-58). Rather one has:

kzzz 6k££2’ kllm 2k22m’ klmn kzmn (2 67)
' 3 ' I -

kllil 12k£££2 kzzzm 3k£££m (2 68)
t a ' a ..

klmmm 2k££mm’ kzzmn kzzmn’ (2 69)

18

Table (2-1). Rotational Operators R of the Expanded Hamiltonian
Terms

 

a8
= 2 B J J
Rk (1,8 It a B

a

2. k ‘1 a
Rk ' ‘(wz/“k)R2 ’ “2(w2/“k) : Ba Ckx J

, _.2 Ya YB YB Ya
sz Rzk 8 a S Y(Bk 32 + Bk B: )(JaJB/BY)
9 9

9. k ’1 0:88
Rk,m . -(w2/wk)R£,m ' “(mm/wk) z Bm Clia

0,8

6a

a a}. 50157
R R 42 (Ble+Bm

01,8
Y.€

2. k
Pigmn a Rk,nm :- -(wl/wk)R£,mn

. av 78 av YB '0
a : Y(Bm Bn + Bn Bm )Ckl(JB/BY)
’ 3

5Y YB
132 )BD (JdJB/BaBy)

II
I
oolw
A
if
\
KEV

a L so an to an M YB
32 Z (Bk Bl + BIL Bk )Bm Bn (JoJB/BYBEBn)
a.B.Y
e,n

9. 2.
Rk,mn,j - Rk,mn,j a -(wz/ulc)R1:,mn,j

_ l ‘1 (So 5y 60: (W Ys
4 (wk/wk) a Z (Bm Bn + Bn Btn )B (JB/BYBG)

Ca
kl
:89Y j
6

 

19

Table (2-2). The B Coefficients of the Expanded Hamiltonian Terms

 

l ’ o
' '5 [K “2“”an

3/2 Hi/2]( aaB/Io Io )

. -.l 3/2
B 2 [2. /(21c) “a BB

2., n . k,n-' a _ 2.,m
Bk m B: -(w£ /w k)B£,m (mu/wm)Bk,n

= (w;/wk)l/2(wn/mm)l/2 : B Ca C“

a k2 mn

32’“ . B“ 2 . -(w M/mk)3

k,m,j m,k,j = “(2 fl/Qm)B

l m.j k.n n.j

1/2 a8 a 8

1/2
= (wk/wk) (mu/mm) Z Bj Ckigmn

(198

1
Bk ‘ ‘ 4

£,n l ,n n,£
k.m.Jg k m.gj ”Em.k.js (“l/wk)3ln m.jg

. -(wn I” m)Bkm n.j8

l/2(wn/wm)l/2 z (BaYBYB + BaYBYB)(;“; a /BY)

.2
(ml/wk) a,8,y j g g 3 mn

B - B = - -—- 2 (BGBBaslBB )
o,8

 

Table

20

(2-3). Terms of the Expanded Vibration-Rotation Hamiltonian

 

02

20

12

21

30

22

31

4O

*1:
00

40

32

41

50

*1:
lO

50

42

=£BJ2
. do:
0.

R0

2 2
k(pk + qk)

Z w
k
quk

2.
2 QP
k’zkkkz

.1. .
6 Z kltSLmqquLQm
k, 2. ,m

.1.
2

who

>3 qq
k’szlkR.

9.
2 m Rk,m(qkp£qm + qmqkpl)

2.,n
F Bk,m qkpzqmpn

1 v
12 £.m.n kLdmn k lqm n

2 m Rlc£,mqkq2qm
2.
z n Rk,mn(qkp£q~mqn + qmqnqkpl)

Z 32’“ q P q qmp
k,£,m,n,j k,m,j k 2;] n

z quk

l
60 X k

1
cl q ‘1 q
k,l,m,n,j 1(2an k iq-m n j

12:2, Rl<2,m,nqkq£qmqn

mn

21

Table (2-3) (continued)

 

2’ .
H51 k,2,m',n,j Rk’m’j (qkppvqmqj + qmqnqjqkpl)

* 2 n
H = 2 B ’
60 k,2,m, k,m,jg qkp2qquqmpn
n:j:g

H - 2 B

q q
20 k2 k k 2

 

k!

k!

k!

k!

kl

k!

22222 a

222mm 3 6
22mmn E

222222 a
2222mm '
222mmm '

22mmnn a 2k

22222’

222mm’

22mmn’

222222’

2222mm’

222mmm’

22mmnn°

22

k!

12k

2222m a 2222m’
' a

k222mn 3k222mn’
' _

k22222m ' 30k22222m’
' I

k2222mm 6k2222mn’
' a

k222mmn 3k222mmn’

(2-70)

(2-71)

(2-72)

(2-73)

(2-74)

(2-75)

(2-76)

3. THE SUCCESSIVE-VIBRATIONAL CONTACT
TRANSFORMATION TECHNIQUE

3.1 First Contact Transformation

 

The energies of the system represented by the Hamiltonian
(2—24) can in principle be calculated in successive orders of approxi-
mation by the perturbation method. The zeroth-order energy would be
calculated only from the zeroth—order part of the Hamiltonian, H0.
The first-order correction to the energy, E1, is computed exclusively
from the diagonal matrix elements of H1. In the absence of de-
generacies, the off-diagonal elements of H1 will contribute only to
the second and higher-order corrections. The perturbation calculation
is thus principally complicated by the myriad of off-diagonal con-

tributions, especially those from H and H4, and it is therefore

3
highly desirable to transform the Hamiltonian to a more convenient
form. To attain such a form, Van Vleck. suggested the so-called con-
tact transformation technique. By a suitable unitary operator T, one

subjects the Hamiltonian H to a transformation and attempts to find

a Hamiltonian H',

1 = H + m' + 2211' +... , (3-1)

1 a '
H THT O 1 2

such that the zeroth-order term and the diagonal matrix elements of
the first-order term of the Hamiltonian remain unchanged while the
off-diagonal elements of the first-order term of the transformed

Hamiltonian H1, would vanish completely. The eigenfunctions of H
23

O

24

become eigenfunctions of H0 + Hi which is thus effectively a

zeroth—order term, if the zeroth—order energy is non-degenerate.
Since now there are no off-diagonal matrix elements in Hi, the
Hamiltonian H5 can be treated as a first-order perturbation term,
and the second-order corrections to the energy are obtained by taking

the expectation values of H5.

Thus, except in the case of accidental degeneracies, it is
advantageous to consider a partial diagonalization of the vibration—
rotation Hamiltonian in the vibrational quantum numbers by use of the
contact transformation. This can be done by determining the operator

T which leave HO of Eq. (2-48) unchanged and gives an Hi diagonal

16
in the vibrational Operators. The simplest method of obtaining the

suitable form of T is to set T = exp(iAs(1)) where the Hamiltonian
(1)

Operator 3 is called the Herman-Shaffer operator, and is chosen

such that the operator HO + AHi has only diagonal matrix elements

with respect to the vibrational quantum numbers vS in the repre-

sentation which diagonalizes HO.

To carry out the first contact transformation, we let

1 iAS(l) e-iAS(1)

H's-THT-ae H 2

= ' ' .-
H0 + 2H1 + A H2 +... (3 2)

To obtain the general expressions for the operators Hg, Eq. (3-2)

is expanded as

2 3
I - l v I 2
H H0+AH1+AH2+AH3+...
= (l + 123(1) - -l- AZS(1)2 - A 1233(1)3 +...)(H + AH + AZH +...)
2 6 o 1 2
(1 - 125(1) - % 225”” +365 1235(1)3 +...). (3-3)

25

Equating the coefficients of like powers of A, one obtains in general

that
. _ (l) (2)
an - Hn + i[s , Hn_1], (3-4)
where
<2)_ _1_ <1) <3) _
Hn-l Hn—l + 2 [S ’ Hn-ZJ’ (3 5)

and so on, to

(m) a _i_ <1) (n+1)
HI]. Hn + m [S a Hn-l ] 9 (3-6)
with Hgm) - H0 for all values of m. Writing out the first few

terms of H; explicitly, one obtains

H' ; H (3-7)

0 0
Hi = Hl + i[s(l), HO] (3-8)
Hé =- 32 + i[s(l),Hl] vfi- [5(1),[s(1),H0]] (3-9)
H5 - 33 + i[s(1),H2] “21- [5(1),[S(1),H11]

-% [5(1).[s(1).[s(1),H0]]] (MO)
11; =- H4 + i[s(l),H3] -% [5(1),[s(l).H2]]

-% [3(1).[s(1)[.s(1),H1]]]

+ é; Lsmds‘l’d 5(1).[s(1).H0]]]] (341)

26
In general,

n n-k
.££L___ {S(l)n-k

H; = .5. a.-.” , H19 <3-12>
where
(l)(0) -
{S , HR} : Hn (3'13)
(l)(1) - (1)
{S , Hn} = [S , Hn] (3-14)
{5(1)(2). H } s [5(1).[s(1), H 11 (3-15)
n n
k brackets
\—."____/
{ (1"k),nn} : [5(1).[5(1)...-[S(l).Hn]]--u] <3-16)

According to Eq. (3—8), the required transformed first-order

Hamiltonian H' is obtained if 5(1) is chosen such that

1
(l) a : _
i[s , H0] H1 + H1 (3 17)
where Hi is the portion of H1 which is diagonal in the vibra-
(1)

tional quantum numbers. This choice of S removes all but the
so-called first-order essential Coriolis term from the first order
Hamiltonian. For the case of asymmetric rotator molecules, it can be

shown that Hi - 0, because there is no symmetry-conditioned,

essential<zoriolis interaction for this type of rotator. Eq. (3-17)

thus becomes

i[s(l) (3-18)

’ Ho] = -111,

(1)

and constitutes the defining equation for the 8 function for the

case of the asymmetric rotator molecule.

27

(1)

Now the Hermitian operator S , and the partial Hamiltonian

Héfil) whose commutator appears in Eq. (3-6), are each made up of

combinations of the vibrational operators ps and qS and the rota-

tional Operators Pa. The two types of operator can be expressed

!
symbolically as 5(1) s 2k kS,a nd Hn ”is 2k k H with
k

t t
S - (kS)V(kS)R, and k H a (k I-I)V(k H)R where the subscripts V
and R denote the vibrational and rotational parts. For any ks

1
and k H, and suppressing the k—indices, one has that

(l) (l)

[8. , H1 = [s H] + [s‘l’H HIR (3-19)
where
[3(1), H] =[s(1), H v](s(l) + HRSél))/2 (3-20)
[5(1),n H]R = (3:1)av + H 5(1))[ 3(1) ,HR]/2 (3-21)
It can be shown that if an is of order n, then [8(1),H]v

[8(1)

is of order n+1, whereas ,H] is of order n+2. This comes

R
about since the commutators [Pa’ PB] occurring in» [5(1) ,HR] are
equivalent to -i )1 PY and are therefore of one order of magnitude
smaller than PaPB’ whereas the [p,q] occurring in [851), Hv] are
equivalent to -i )1 and are therefore of the same order of magnitude
as pq. For these reasons it is useful to allow for the possibility
of regrouping terms into orders of magnitude in accordance with the

above remarks, and the once-transformed Hamiltonian is therefore re-

written in the general form

3' a h' + h' +rh' +Hh' +-hfl +-

o 1 2 3 4 (3’22)

28

From Eq. (3-4), the transformed Hamiltonian has the form

(1) h(2)]v

.+ R[5(1) h(2)1R

h' ,n‘H +i[S
n

where one has, in analogy with Eqs. (3-5) and (3-6),

(2) a (1) h<3) <1) h<3)
hn Hn+%[5 1v1%SI . 2R1
and in general
(In) 1 (l) (n+1) _i_ (l) (n+1)
hn a Hn + m [S ’ hn-l 1v + m [S ’ hn-2 1R '

(3-23)

(3-24)

(3-25)

For the asymmetric-rotator case, the above equations give explicitly

that

hé . H0

hi - H1 + 1[s(1), Ho]v = 0

ha - H2 + i[S(l), HO]R+ %1IS(1). Hllv

hé - H3 +-% 1[s(1), H11R+ 6 [S 8(1).[S(1),H0]R]v

%1[s(1), 2hR + H2]v

ha . H4 + 1[s(1), H31v- Z 1[S(l).IS(l) h'+ M] 1v
+‘%§ 1[S(l)’[5(1)’[S(1)’H0]R]v]v _ %-[s(1),[s
+«% i[S(l),2hé + HRIR+ %IS(1).IS(1),HOIR]R

(l).HllR]v

(3-26)

(3-27)

(3-28)

(3-29)

(3-30)

Using Eqs. (3-20) and (3-21), and the commutation relations

17

as stated by Herman and Shaffer, it is possible to find the

S<1)

function defined by Eq. (3-27). It can be shown to have the general

form

29

 

 

 

 

 

 

(l) a a8 8 a
S za,st S PsPaPB + Zazs<s'( 833' quS.
+ “888' p p )P + 2 SS" 'qu q p + q q )
' s s' a sis';s" ss' 2 s s' 3" ps" 3 3'
88'3"
ZS<SI<SII pSpS'pS" : (3‘31)
where
aaB
- 2 .7. . (3-32)
2 O O
(K A ) I o£188
& A + A ca
a a l s ' ss' ,
Sss' (A A ,) A - A , I° ’ s * 3 (3-33)
8 an
5 a
. (A A .) c .
aSss =Rg§ s s is , S # S, (3_34)
A - A , I
h s s cc
A5 (A - A - A )
SS" 3 2wc 1 + R s" s" s s'
SS' " n < 688" + 58'8", G kSS'S'”
. ss's"
L (3-35)
(A A A ) 2
Sss's" 4nc s s' s" k
a v u , (3-36)
“3 G ' I. SS S
38 s
with
G . n = (A5 + A”. + A”..)(AL2 + A*. - A5")
33 s s s s s s s
B 5 5 k 5 5
(AS 18' AS")()‘S A8' + As") . (3‘37)

It may be verified that 633's" is invariant under all six possible
permutations of the indices s,s' ,3".

With the explicit knowledge of the S(l)-function, the first
contact transformation can be carried out according to the procedures

described by evaluating the required commutators. This has been

done by Amat, Nielsen and co-workers. The lengthy results are

30

12
collected in a reference book by Amat, Nielsen and Tarrago. The

explicit forms of h', h', and h" are

l 2
v a ..
h0 H0 (3 38)
v 3 ..
h1 0 <3 39)
hé - 2 (“BY5 Y)PGPBP PR + z 2 ("‘BY YS)pSPaPBP
am (2) Y may 8 (2) Y
a8 83' a8
+ Z X ( Y psps. + Yss.qsqs.)PaPR
a8 s,s' (2) (2)
sgs'
a 33's"
'I' Z Z ( Y )PSPS'pSHPO.
a s,s',s" (2)
833,18"
a s" l
+ Z Z ' n ( Yssy)2(qsqstl?sn + pSHquS')Pa
a 3,3 ,5 (2)
8:8"
H H'
l
+ 2 ( Y3? )—(q q 1? up In + P up nvq q t)
s,s',s",s"' (2) ss 2 s s s s s s s 3
8(8' ;s"<s" V
+ Z ( Y . n n.)q q .q "q nv' (3-40)
s,s',s",s"' (2) as s s s s s s
sis':SI'—<-SHV

The expressions for h3 and h; are also cited in entirety in the
Amat-Nielson-Tarrago book, as are the detailed forms of the above

coefficients (2)Y in terms of fundamental molecular parameters,

3.2 Second Contact Transformation
In order to cast the Hamiltonian into a form suitable for the
calculation of the vibration-rotation energies to the fourth-order

of approximation, it is necessary to perform a second contact

31

transformation on the once-transformed Hamiltonian H', such that the
twice-transformed Hamiltonian will be diagonal to second order in all

vibrational quantum numbers. We therefore let

H" a TH'T-l a eiAZS (2)HRe-1AZS(2)
. H" + AH" + AZH" + A3H" + (3-41)
0 l 2 3 °'°
where H3 + AH; + AZHS is required to be diagonal with respect to

the vibrational quantum numbers vs. On expansion of the exponentials

and equating like powers of A one obtains

Hg . hé (3-42)
H; 3 hi (3-43)
H3 . hé + i[s(2), ha] (3-44)
Hg . h; + i[s(2), hi} (3-45)
H2 = h; + i[s(2), h 21- %[ 5(2),[s(2), h 0]] (3-46)
H; = hé + 1[s(2), h 31- %[ (2),[s(2), h' lJJ (3—47)

.., . <2) S<2) <2) .
H6 h6+1[s,h'-J%s[ ,[s ,hRJJ

-6[s(2), [3(2),[s(2), h 01]] etc. (3-48)
Again allowing for the possibility of regrouping the terms of the

twice-transformed Hamiltonian by true orders of magnitude, it is

expedient to write H" as

H g H H II I! _
H h0+hl+h2+h3+... (3 49)

32

The above Hamiltonian can be used to calculate the vibration-
rotation energies to the fourth-order of approximation by including
only those matrix elements which are diagonal in all vibrational
quantum numbers vs, because ha, h; and h; are diagonal in all
vS by virtue of the contact transformations, and those matrix
elements of h; and hz which are off-diagonal in one or more
vibrational quantum numbers can contribute to the energies only in
orders of approximation higher than the fourth. The Hamiltonian H"
is diagonal to all orders in the rotational quantum numbers J
(total angular momentum quantum number) and M (magnetic quantum
number). However, it has matrix elements which are nondiagonal in
~the angular momentum projection quantum number k in a symmetric

rotator representation whose basis functions are rigid-symmetric-

to ei enfun tions .
p 8 c kaM
HISIIB
It was found that through the fourth-order of approximation,

the general Hamiltonian has terms which can be classified into the

following three categories:

(a) (0)2 r2, (2)2 r4, (4)2 r6;

(b) (0)2 P2, (2)2 P4, (4)2 P2, (4)2 P6;

(c) (1)2 rzP, (2)z r2P2, (3)2 r4P, (3)z r2P3, (4)2 r2P4,
(4)2 r4P2,

where r2 denotes any possible product of two vibrational operators
4

i.e., qsqs,, Psps" qsps,, or psqs,; r denotes any possible pro-

duct of four vibrational operators, etc. The symbols ‘Pn' stands

for any possible product of ‘Px,?Py and. Pz of total power n, and

33

and (m)Z stands, in general, for the coefficient of an Operator
appearing in the meth order of approximation.

The terms in (a) constitute pure vibrational operators in—
cluding anharmonicity corrections, the terms in (b) constitute pure
rotational operators including centrifugal distortion corrections, and
the terms in (c) represent vibration-rotation interaction contribu-
tions.

The pure vibrational energies will not concern us here since
we are interested principally in the rotational level structure built
upon a particular vibrational state rather than in the detailed
calculation of the pure vibrational level structure. In the absence
of vibrational degeneracy, it is found that the (1)2 r2 ‘P-type terms
have zero coefficients (1)2. Also, the (3)2 r4? and (3)2 rzPB-
type terms have no non-zero matrix elements diagonal in all v8 to
the fourth-order approximation.' Thus the odd order terms may be ex-
cluded from consideration unless accidental resonances occur which
would partially invalidate the order of approximation arrangement of

the Hamiltonian. The Hamiltonian of interest has then the general

schematic form:

" h"* h"* h"* 3 49
H 0 + 2 + 4 (- )
where
..* ,_ 2 _
"* 2 2 4
= z + P -
* 2 4 2 2 4 6

Z r P + Z r P + Z P . (3-52)

34

The asterisks indicate that terms of h; and hz which are of the

"* and h"* Also
2 4 °

- *
terms of hz not diagonal in all vs are to be omitted in hZ .

Regrouping terms in Eqs. (3-42)-(3-46) according to true

pure-vibrational type are to be omitted in h

orders of magnitude, the twice-transformed Hamiltonian takes the

following general form:

R3 hé , H0, (3-53)
h; = hi = 0, (3-54)
hg . hi + i[s(2), H0]v, (3-55)
h; = 115 + i[s(2),Ho]R + i[s(2), hi1v, (3-56)
.hZ - ha + i[s(2), hi]R + {215(2), bi + h'2'1v. (3-57)
The operator 8(2) is determined through Eq. (3-55) by requiring
the commutator -i[s(2), HO]v to be of a form such that if offsets
the vibrationally off-diagonal matrix elements of hi. This 5(2)
function is found to take the following form:
5(2) = za,B’st O‘BYSR qs PaPBPY
+ za,8 s,s' GB :' 2(qsps'+ ps'qs)PaPB
+ Zazsgstgs" asss's" qsqs'qs" Pa
+ Zazs;s[§s" “82's" 2(qsps'ps" + ps'ps"qs)Pa
+ Z s's"s"' 1

s;s':s"_gs"' S ‘2'(qSPSvPSnPSnv + Psvpsnpsnvqs)

SS" I
"I 88'3"

Eiqsqs.qsnpsnv+ p n q q .qsu). (3-58)

2
sgsljs";s s s s s

35

12
(2)are listed in the Amat-Nielsen-Tarrago
2 4 2
z Z
(4) P and (4) r P

terms from Eq. (3-58), since they constitute fourth-order corrections

where the coefficients 5

book. It may be allowable to omit the

to the second-order corrected rotational constants and as such their
effects are probably undetectably small. Upon substitution of 5(2)

into Eq. (3-56), it is found that the diagonal matrix elements of hg

vanish; the off-diagonal matrix elements of h; and hz contribute

to orders of approximation higher than the fourth. Extensive computa-

tion yields a twice-transformed Hamiltonian of the following form:

 

n a ‘ _
h0 Ho (3 59)
n a ..
h1 O (3 60)
h"'-2: “875 YPPPP
2 GBYG (2) a B y 6
2 a8 39' a8 1 sps'
+ z 2: (II Y + Y .A-(q q . + )P P
a8 38' (2) (2) ss 2 s s K2 a B
s-s'
+ pure-vibration terms. (3-61)

The very complicated explicit results for h", h; and hZ will be
12

found in the Amat-Nielsen-Tarrago book

3.3 Third (and Highgg) Contact Transformations

It is sometimes necessary to perform a third (or even higher)
contact transformation such that the three-times-transformed
Hamiltonian will be diagonal in all vibrational quantum numbers.

One then takes:

36

-1 1A3 (3)

3 (3)
H"! 3 Salve = e S Hue 1A 3

.2 33' a; + 2H; + fins: + 4H:

4 3
H! "I "I
where H0 + AH1 + A H4 + A 3

(3-62)

H"' are diagonal with respect to the

vibrational quantum numbers vs. In terms of the like powers of the

parameter A, Eq. (3-62) is equivalent to

H ' "
Ho ho

I" 8 I! (3) H
. H' . H 0 (3) H

II! a ll (3) H
HS h5 + i[S , h2]

In a n (3) n A (3) (3) 11
H6 h6 + i[S , h3] - 2[S ,[S , ho]] etc.

Eq. (3-66) is the defining equation for the operator
(3)

function S is constructed such that

(3) n _ _ u
1&3 ’ ho] ’ ha off diag.

8(3).

(3-63)

(3-64)

(3-65)

(3-66)

(3-67)

(3-68)

(3-69)

The

(3-70)

Again allowing for the possibility of regrouping the terms of the

three-times-transformed Hamiltonian into a "true" order of magnitude

arrangement, one has

H"' . ha' + hI' + hg' + hg' + hZ' +...

(3-71)

37

Recently Aliev and Watsonyhave calculated the sextic
centrifugal distortion constants of polyatomic molecules by a so-
called "reordered perturbation treatment", for which they had to
consider the three-times transformed Hamiltonian. We will discuss
this calculation in some detail in chapter 7.

,The procedure of successive contact transformations could
be applied as many times as required. The general theory of n-times-
transformed molecular Hamiltonians has been explored in some detail

19
by Aliev and Aleksanyan.

4. THE NORMAL MODES PROBLEM FOR
TRIANGULAR TRIATOMIC MOLECULES

In order to present and explore the details of the theory of
centrifugal distortion in triangular triatomic molecules, it is
necessary to review the normal-coordinate problem for this class of

molecules. The XYXstype molecule was first considered by Shaffer and

20
Nielsen, and discussions have since appeared in numerous publications,
21 22
e.g., those of Nielsen, or Chung and Parker. The XYZ-type molecule
. 23
was first studied by Shaffer and Schuman, and further details have

6
been presented by others, including Chan and Parker.
4.1 Equilibrium Geometry of the ngjtype Mblecule
Let the XYZ molecule lie in the E; plane, as shown in
Figure (4-1), with the origin of coordinates at the center of mass.
The X and Z atoms are located at the base vertices of the

y Y y

--

 

 

Figure (4-1). Equilibrium configuration of the triangular tri-

atomic molecule.
38

39

triangle and the Y atom is at the top vertex. They are numbered
3, 2 and 1, respectively, and X2 -is chosen parallel to the §

axis. In the barred equilibrium coordinates §,i,‘§;i and 201,

i = 1,2,3, one has that

a 3 gal - 1:93, (4-1)
b 8 x02 - x01, (4-2)
C a §°1 ' §03° (4-3)

The principal axes of the molecule, designated as x, y,
and 2 have the same origin as the E, § and E coordinates. The
z-axis coincides with the Z-axis and the x,y axes are in the plane
of the molecule and make an angle 6 with the E, 5 axis. The

equilibrium coordinates are found to be

ERR - (m3a - mRb)/M, (4-4)
22.2 = [m3a + (m1 + m2)b1/H. <4-5)
§,3 = -[(m1 + m2)a + m2b]/M, (4-6)
§°1 - (m2 + m3)c/M, (4-7)
§,2 = -mlc/M, (4-8)
§,3 . -mlc/M, (4-9)
2.1 . 2,2 = 2,3 = 0, (4-10)

with M being the total mass of the molecule, viz.,

M 8 ml + m2 + m3 . (4-11)

40

Transformation to the principal axes system of the equilibrium inertia

tensor is accomplished by taking

x°i = xoicose + yoisine, (4-12)
yoi = -x°isin6 + yoicose, (4-13)
201 3 E01 3 0’ (4-14)

where the angle 6 is given by

tan 26 = T/Q, (4-15)
with

T a 2m1c(m3a - mzb), (4-16)

9 = m (m + m )a2 + m (m + m )b2

3 l 2 2 l 3
2
- m1(m2 + m3)c + 2m2m3ab . (4-17)

For the equilibrium coordinates in the principal axes system the

transformation gives

x01 8 {(m3a - mzb)cose + (m2 + m3)c-sin6}/M, (4-18)
x°2 = {{m3a + (1111 + m3)b]cose - m3c-sin8}/M, (4-19)
x.3 = -{[(m1 + m2)a + m2b]cose + mlc-sin6}/M, (4-20)
yol a -{(m3a - mzb)sin6 - (m2 + m3)c-cose}/M, (4-21)
y°2 = -{[m3a + (m1 + m3)b]sin6 + mlc-cose}/M, (4-22)
y,3 8 {[(ml + m2)a + mzb]sine - mlc'cose}/M, (4-23)
2.1 = 2.2 = 2,3 = O . (4-24)

41

For the equilibrium principal moments of inertia one has

1:3 = -]2;(I;z - 1'), 2 (4-25)

;y - %(I;z + 1'), (4-26)

122 - IL; + 1;}, =- [(2 + 2m1(m2 + m3)c2]/M, (4-27)
with

1' - [(T2 + (3)/14215. (4-28)

The equilibrium coordinates and the equilibrium principal
moments of inertia of the HDO-type molecule can be obtained from the

above more general expressions by setting a = b.

4.2 Normal Coordinates of the XYZ—type Molecule

 

Denoting instantaneous position coordinates by xi, yi and

21, we have that z - 22 - 23 - 0 because of the absence of out-of-

1

2k
plane vibrations. The Eckart conditions can be written as

Zmi 0,130 (4-29)
1

sz xE)-o. (4-30)
1 i 01 i

The first condition keeps the origin at the center of mass at all
times while the second condition keeps the xyz coordinate system
attached to the nuclear equilibrium configuration. We now introduce

intermediate symmetry coordinates u,v,w as follows:

3’

v a y]. - (mzyz + m3Y3)/(m2 + “13), (4'32)

42
w = x1 - (mzx2 + m3x3)/(m2 + m3). (4-33)

Eqs. (4—3l)7(4-33) along with the Eckart conditions give for the

instantaneous cartesian position coordinates:

x1 = (u/ml)W. (4-34)
x2 = (u'/m2)u - [u/(m2 + m3)]w, (4-35)
x3 = -(u'/m3)u - [u/(m2 + m3)JW. (4-36)
Y1 = (u/ml)v (4-37)
Y2 = (u'Y/m2)u + (ua/m2)v + (u"/m2)W. (4-38)
Y3 = -(u'Y/m3)u - (uB/m3)v - (u"/m3)w, (4—39)
with
u = [m1(m2 + m3)]/M , (4-40)
u' = m2m3/(m2 + m3). (4-41)
u" - mly.1/x23. (4-42)
a - -x13/x23, (4-43)
3 - -x12/x23, (4-44)
7 = y23/x23 = -tane, (4—45)
where
x12 a x01 - x02 - —b cose + c sine, (4-46)
x13 = x°l - x03 = a c036 + c sine, (4-47)
x23 = x02 - x.3 = (a + b)cose, (4-48)
Y23 = Yoz - y03 - -(a + b)sine. (4-49)

43

The vibrational kinetic energy is

l . 2 . 2 -. 2 ..
a 2 (ullu + “22v + p33W + Zulzuv
+ 2u13uw + 2u23vw), (4-50)
which can be expressed in terms of the intermediate coordinates as
l . . t
T - §'(W)(u)(W) , (4-51)

where

w s (a v w), (4-52)

and t denotes the transpose; (u) is the 3 x 3 symmetric matrix

with elements

‘111 ‘ u'(1 + Y2). (4‘53)
2 2 2

“22 = u [(l/ml) + (a /m2) + (8 /m3)], (4‘54)

1133 = u + (11"2/u'), (4-55)

1112 I “21 a “H'Yy (4'56)

u13 ' U31 ' u"Y. ' (4-57)

1123 ' 1132 8 u"u"'/u', (4.58)

where

u = -mlxo1/X23 = uu'[(a/m2) + (B/m3)]. (4-59)

44

Now, the most general form for the harmonic potential energy

can be expressed in the intermediate coordinates as
l t
v - 3(4) (10(4) . W60)

where (w) a (u V'W) and (k) is the 3 x 3 symmetric matrix of

22’ k33’ 12 ' k21’ k13 ' k31'

k23 - k32. The potential is invariant under the group symmetry of

potential constants with elements kll’ k k

the molecule CLh and can be written explicitly as
l 2 2 2
V's 2 (kllu + k22v + k33w + 2k12uv
+ 2k13uw + 2k23vw) . (4-61)

The symmetry coordinates u,v,w are related to the normal

coordinates QS (s = 1,2,3) through a set of transformation coeffi-

cients n , ,
s s

u = 28 nls s’ s = 1,2,3, (4-62)
v = ZS nszS, s = 1,2,3, (4-63)
w = 28 n3S S, s = 1,2,3, (4-64)

where these coefficients can be obtained in general only by solving

the cubic secular equation
IA(u) - (k)| . o . (4-65)

Each ns's can be expressed as

n , - 3 3 , (4-66)

 

45
where Ns's is the cofactor of the s'-th element of any row of the
determinant, Eq. (4-65), and where 'A 2 is is the s—th root of Eq.

(4-65). The quantity NS is determined such that
l '2 '2 '2
T 3.? (Q1 + Q2 + Q3) , (4-67)

which requires that

2 2
Ns - 2:s'='l,3 [us's'Ns's + zs"#s' us's"Ns'st,"s]' (4-68)

Expressed in the normal coordinates, the harmonic portion of the

potential energy becomes
1 2 2 2
V a 2 (AlQl + AZQZ + A3Q3), (4-69)

where the associated normal frequencies, in radians per second, are
A: (s - 1,2,3). The normal frequencies A: could be specified in
closed form as the roots of the general cubic equation. These ex-
pressions are of limited practical use, as ordinarily it is the three
roots As for which numerical values are known and the potential
constants kij for which numerical values are sought. As there are
three As and six distinct potential constants kij’ the problem is
underdetermined and additional relations must be obtained through
intercomparison of isotOpically substituted species and through in-
formation derived via the second-order parameters of the Hamiltonian.
The normal vibrations problem has been presented above in
such a way that specializing the results of this and the following
section to the XYx-type molecule will reduce‘ these directly to the
results of Chung and ParkerizYallabandi and Parkeii and Chan, Wilardjo,

2
and Parker.

46

The normal mode 3 = l specifies the X-Y 'bond stretching
mode, 3 = 2 specifies the bending mode, and s = 3 the Y-Z bond
stretching mode. The three normal modes are sketched fin Figure

(4-2).

 

 

 

./ 12 X\ ,72 2% ‘W’;

X Y-X Bond Bending Mode v2 Y-Z Bond
Stretching Mode . Stretching mode v3
v
'1

Figure (4-2). Normal Modes of the XYZ Molecule.

4.3 Molecular Parameters of the XYZ-type Molecule

 

The coefficients 2:8, a = x,y,z of the transformation from
instantaneous position coordinates to normal coordinates are defined
by Eq. (2-4) and can be constructed for the xyz-type molecule with the
aid of Eqs. (4-34)-(4-39) and (4-62)-(4-64). In this manner one de-

termines that

21‘s - (u/m§)n38, (4-70)
2:8 ' (u'lmZMIS - [mug/(m2 + 1:13)]n38 (4-71)
2353 - -(u'/m‘3’>nls - [mfg/(m2 + m3>1n3s. <4-72)
231's - (u/m‘isz, (4-73)
gs - (u'v/m’pnls + (pa/m3)n28 + (4711391138, <4-74)
23's - -<u'y/m§)nls - (us/ugh“ - <u"/m‘3’)n3s, (4—75)

47
2 . z = 22 a 0 s = 1,2,3. (4-76)

Knowing the set of coefficients £38, one can construct the Coriolis

constants, cgs, from Eq. (2-7) as

3
- 8 Y _ Y B _
css' 1:1 (liszis' lislis')’ (4 77)

where n.8, and y denote x,y, and 2, respectively, or one of

their two cyclic permutations. One finds that

C v = C t a C = 0: (4‘78)

for any 3 and s'. The onlynon-Vanishing Coriolis constants are-

the following:

2' - _ z a _ _ uv _
Css' cs's “(n23n3s' n23'n3s) + u (nlanS' nls'n28)

+ u"(nlsn3s, - nls'n3s)’ 8 ¥ 3'. (4-79)

It is customary to suppress the upper index for 5:8,. There

are then three distinctnon-VaniShing‘30riolis constants, viz.,

C12 3 '521 C13 ' ’531 C23 3 "532’ (4’80)

4’28
and these constants obey the sum rule

2 2 2
(12 + :13 + :23 1. (4-81)

The coefficients of expansion of the instantaneous moments
and products of inertia introduced in Eqs. (2-28) and (2—29) take

the following form for the XYZ-type molecule:

(4-82)

48

yy , yy + yy yy _

aS t1 nls t2 1128 + t3 n38, (4 83)
22 _ 22 22 22 = xx yy _

as t1 nls + t2 n2 + t3 n3 a8 + a , (4 84)
XY . xv xy xy , yx _

as t1 “ls + t2 nzs + t3 n3s as ’ (4 85)

where the coefficients tie are summarized in Table (4-1). All
other a:8 vanish. The non-vanishing (A::.)' and (A::,)' are
given by

Wyayyv=yy=yy
(Ass') (As's) Ass' As's

a ' -
u nlsnls' + un3sn33., (A 86)
22 , . 2 22 , = 2 22 , a 2 _
22 , . _ zz , a ' zz , a _ _
(A12) C13‘23’ (A13) ‘12‘23’ (A23) ‘12‘13’ (4 88)

XVIayxgxygyx
(Ass') (A843) ASS. As's

, + p"'n n ' + u"n n

a _ '
(“ Y “lg“ ls 2s ls 3s'

ls

+ u n (4-89)

3sn23')°

In Eqs. (4—87) and (4-88), we have used that A::, - 538,.

Direct computation of (A::.)' yields the rather complicated

expression
2 2
xx,_lxx,-xx-xx=,2 ZLLL
(Ass') (As's) Ass' As's n Y nlsnls' + u (m1 + m2 +‘m3)n23n23’

2
H
E... "v
+ u' n38n33' + u Y(Illans' + nls'nZS)

H -
+ p 7(nlsn3s' + n1 .n3s) + E;%-— (n23n3s, + n23,n3s). (4 90)

S

49

Table (4-1). The Coefficients tc‘B Introduced in Eqs. (4-82)-(4-85)

 

 

 

 

 

 

 

 

i
2 ,

t - 2m2m3(a + b)sin 6 cyy a 2m2m3(a + b) cose
1 (m2 + m3)cose (m2 + m3)
txx - 2m1(m2 + m3)c tyy s O
2 M cose 2
xx:— yy:
t3 2m1yo1 tane t3 2mlxol
tzz - 2m2m3(a + b) a txx + tyy txy a 2m2m3(a + b) sine
1 (m2 + m3) cose l l .1 (m2 + m3)
zz . xx xy 8
t2 t2 t2 O
:22 - 2m1(m3a - mzb) a txx + tyy tgy , —2m1y91
3 M cose 3 3

x.l and y°l are as given by Eqs. (4-18) and (4-21)

 

50

28
Using the sum rule of Oka and Morino given for s = 3', namely
2 Aggg- 2, one can avoid the above result and write (A::)' for'the

a
XYZ-type molecule simply as

H‘s-W, = _
(Ass) 1 (ASS) , 3 1,2,3, (4 91)

12
with (AZZ)' given by Eq. (4-86). For 3 # s', Amat and Henry have

shown that
H's-W1 -
(Ass.> (Ass') . (4 92)
Combining Eqs. (4—91) and (4-92), we have

xx v YY v a _
(Ass') + (Ass') 688, . (4 93)
Amat and Henry have also shown that the following simple relations

exist between the age and the A::,:

as . =.l ay my 0 -
(Ass') 4 ZY as as,/Iyy, (4 94)

a8 . Ba 3 l ow BY av BY 0 _
(Ass') + (ASS,)' 4 Xy(aS as, + as,aS )/IYY . (4 95)

Another useful sum rule is given for 2 A:: which for X32 jyields

S
(A’l‘x 1)' + (A? 2)' + (A? 3)' -%+ (1° -I° )/21° (4-96)
yy 22
t v 1 2 o _ O o _
(A 3) + (A g) + (A g) - 2 + (Iyy Ixxm:zz . (4 97)

The equations, coefficients and sum rules given in this section can
be specialized to XYX-type molecules. This case will be discussed

in the following section.

51

4.4 Molecular Parameters of the Xngtype Molecule

The normal modes problem for the XYX-type molecule has been
discussed by Chung and Parker?2 The problem can be considered as a
special case of the XYZ develOpment just presented above. Let the

molecule be in the xy plane as shown in Figure (4-3) with the origin

at the center of mass. The X atoms are located at the base vertices

 

 

 

 

x
x 3 . 2 X
a a
, m m
Figure (4—3). Equilibrium configuration of the nonlinear XYX

molecule.

while the Y atom is at the top vertex of the triangle, and XX is

chosen parallel to the x axis. The equilibrium coordinates x

01’
yoi’ i - 1,2,3 are:
x01 = 0, (4-98)
x02 - r sina, (4-99)
x03 = -r sina, (4-100)
y01 - I)”; r cosa, (4-101)
yo2 a - gE-r cosa, (4-102)
y03 - --%;-r cosa, (4—103)

where

52

 

2m

11 - 2m + M (4-104)
The equilibrium principal moments of inertia are

I;x = urzcosza, (4-105)

0 2 2

I = 2m 1: sin 0:, (4-106)

YY

I° - I° + I° = 2m(u3)rzsin2a (4-107)

-22 xx yy D" ’
where

113 = “[1 + (p/Zm)cot2a]. (4-103)

Denoting instantaneous position coordinates by x1 and yi, we have

for the Eckart conditions

m(x2 + x3) +Mxl = 0, (4-109)
m(y2 + y3) + Myl = 0, (4—110)

m r sin (y2 - y3) =-% u r cosa(2xl - x2 - x3). (4—lll)

If we now introduce mass-adjusted symmetry coordinates u,v,w as

follows:

u - (tn/2)],(x2 - x3) , (4-112)
V 8 UHIYI " ‘é'UZ + 373) 3 (ll-113)
w 8 p3le -'%(x2 + x3)], (4-114)

then the vibrational kinetic energy becomes

2

T = %(a2 + (r + 6:2), (4-115)

S3
and the harmonic potential energy is

l 2 2 2
V = 2(kllu + k22v + k33w + 2k12uv). (4-116)

It is important to note that when the comparison is made be-
tween a molecule of the XXXétype and one of the XYZ-type, the con-
ventional definition of the intermediate coordinates of XYX, Eqs.
(4-112)-(4-ll4), is inconsistent with the corresponding definition
for xyz, Eqs. (4-31)—(4-33). Hence there arises an inconsistency in
the definition of the harmonic potential constants. This discrepancy
must be taken into account whenever expressions applying to XYZ are

specialized to XYX according to the replacement scheme:

kll(XYX') + % m2k11(XYX) , (4-117)
k22(XYX')A+ uk22(XYX), (4-118)
k33(XYX') + u3k33(XYX), (4-119)
k12(XYX') - (% umz)“k12(m>, <4-120>
k13(XYX') + 0, (4—121)
k23(XYX') + 0. (4-122)

Also, specializing to XYX, the angle 6 defined by Eqs. (4-15)-
(4—17) is equal to zero. The transformation from symmetry coordinates

u,v,w to normal coordinates Q1,Q2,Q3 can be taken as

u - Qlcosy - stiny, (4-123)
v - leiny + Q2cosy, (4-124)
w - Q3, (4-125)

with

54

siny - +(l//E)(l - {Ak/[(Ak)2 + 4k122]&})a, (4-126)
cosy = +(l - sinzy)5, 1 (4-127)
and with

Ak = (kll - k (4-128)

22)°

The normal coordinate transformation is such that Ql is the co-
ordinate of the symmetric bond stretching or "breathing mode", Q2
is the coordinate of the bending mode, and Q3 is the coordinate of
the asymmetric bond stretching mode. The normal frequencies A5

_A

19
3, A: (in radians per second) are given by
. l l 2 2 ., _
Al 2(kll + k22) + 2[(Ak) + 4k12 J , (4 129)
.l _l 2 2% -
. AZ 2(kll + k22) 2[(Ak) + 4k12 ] , (4 130)

The transformation between displacement coordinates and normal co-

ordinates, of the form

iai = : 2:8 QS a - x,y,z, (4-132)

can be develOped and the transformation coefficients 2:3 are found

to be the following ones.

211‘ . 0 any =- (u/M)”s1ny (4-133)
2le - cosy/)5 221V - -§(u/m.)fisin-y (4-134)
£31x = -cosY//2 £31? - -5(u/m)ysiny (4-135)

and

55

212x = 0 llzy = (u/M)&cosy (4—136)

222x = -siny//2 fizzy = -§(u/m)%cosy (4—137)

232x - sinyl/2 232y = —B(u/m)gcosv (4-138)
and

£133 a (ll/MHM/l-IB)’, any = 0 (4-139)

223x = -(u/2m) (In/113)], 123), = (u/Zm) (In/113)],cota0 (4-140)

1331‘ = -(u/2m) (In/113)}, 9.33}, = -(1.t/2In)(In/1J3)1’cotmO (4-141)
and

Z

2 = 0, i = 1,2,3; n = 1,2,3. (4-142)

in

a
is

constants tin from Eq. (4-77),

,Knowing the 2

Y_ BY
mn im ILin zin 2im )’

c x = c y . o, m = 1,2,3; n = 1,2,3;
mn mu
2 2
C12 ' ‘21 = 0’
z 2 H 8
:13 -§31 a (Ix/Iz) cosy - (Iy/Iz) siny,
c z a -c z - -(I /I )ksin - (I /I )Bcos
23 32 x 2 Y y z 7’

2
(£1332 + (4232) - 1.

Again the superscript z

Coriolis constants are those with superscript z.

coefficients, we can construct the Coriolis

(4-143)
This gives

(4-144)
(4-145)
(4-146)
(4-147)

(4-148)

can be omitted since the only non-zero

56

8

We also will need the coefficients of expansion a: and

(A::,)' of the instantaneous moments and products of inertia intro-

duced in Eqs. (2-28) and (2-29). The non-vanishing coefficients a?
are
xx 5 XX 5
81 ' 21x81ny a2 = ZIxcosy (4-149)
ayy - ZIhcosy ayy - -ZIysiny . (4-150)
1 y 2 Y
zz . xx yy 3 _ 5 22 8 xx yy 3 _ 5 _
a1 a1 + a1 2122;23 a2 a2 + a2 212:31 (4 151)
and
KY . _ 5 _
a3 2(Iny/Iz) (4 152)

The non-vanishin8(A::.Y and (A::,)' are given as

(A::)' - (A§§)' - sinzy (4-153)
(A::)' - (A{{)' = coszy (4-154)
(A§§)' - Ix/Iz (4-155)
(A§§)' - Iy/Iz (4-156)
}(A::)' - (A::)' - -(A{§)' - -(A§{)' = sinycosy (4-157)
and
(A§§)' - (A§:)' - -(Ix/Iz)5cosy (4—158)
(A?{)‘ = (A{§)' . -(Iy/Iz)§siny (4-159)
(A;§)' - (A§:)' - +(Ix/Iz)331ny (4-160)
(A§§)' - (A§§)' - -(Iy/Iz)5cosy (4-161)

These coefficients will be used in Chapter 9 for calculating the

centrifugal distortion coefficients of ozone.

5. CENTRIGUFAL DISTORTION COEFFICIENTS
FOR TRIANGULAR TRIATOMIC MOLECULES

In this chapter there will be described a number of schemes
of development and rearrangement of the Hamiltonian into a form in
which the effects of centrifugal distortion on the rotational energies

are explicitly apparent.

5.1 Development of the Hamiltonian

It was shown by Kneizys, Freedman, and Clouglfgthat the
vibration-rotation Hamiltonian for the XYZ-type molecule in general,
and for XYX in particular, could be given in a simplified form
through extensive rearrangement based on the angular momentum

commutation relations
[Pa, P8] = -1 u PY, a,B,Y cyclic. (5-1)

The form of the resulting Hamiltonian is, for a given vibrational
state, a power series in the angular momentum components which needs
for its specification, to fourth order of approximation, three co-
efficients A, B, and C, of terms of the second power in the body-
fixed angular momentum components; six coefficients T1 of fourth-
power angular momentum terms; and ten coefficients of sixth-power
angular momentum terms. For XYX-type molecules, the Hamiltonian of

29
a given vibrational state can eventually be written as

57

58

H = H2 + H4 + H6 , (5—2)
where
2
H = A P2 + HP + C P2, (5-3)
2 x y z
4 4 4 2 2 2 2 2 2
H4 TIPx + TZPy + T3Pz + T4(Psz + PZPy) + T5(Psz
2 2 2 2 2
+ PxPz) + T6933, + 9:13,), (5-4)
6 6 6 2 4 4 2 2 4
H6 . ¢1Px + ¢2Py + @BPZ + <I>4(PxPy + Pny) + ¢5(Pny
+ P4P2) + c (P2P4 +AP4P2) +.¢ (P2P4 +'P4P2)
x y 6 y z z y 7 z y y z
4 4 2 2 4 4 2
+ ¢8(P:Px + PxPz) + ¢9(PXPz + Psz)
2 2 2 2 2 2
. + ¢10(Px1>zpy + PszPx) , (5-5)

For the XYZ-type molecule, an additional contribution must be added

to (5—2), viz.,

1 3 3 1 3
H6a D(PxPy + Pny) + 4 r10(PXPy + Pny) + 4 111(Pny
3 1 2 2
+ PxPy) + 4 112(PXPZPy + PszPx)° (5-6)

18
This contribution was originally developed by Chung and Parker. The

coefficients 1 appear here with numerical subscripts. These take
18

a8y6'

of the terms quadratic in Pa are the effective rotational con-

the place of the more elaborate notation r The coefficients

stants, equal to the equilibrium rotational constants (l/ZIga) of
0(0) [order zero], plus second-order centrifugal distortion correc-
tions to the equilibrium Ti, plus second—order vibrational correc-

tions, a2. There are also terms of 0(4). More explicitly:

59

A - 1/(21;x) + 2231 01: (vs + 92:) -%u219 + 0(4), (5-7)

B - 1/(21;y) + 2231 (1: (vs + $4 - % uzrg + 0(4), (5-8)

C I 1/(2I22) + 2:31 a: (v +-%) +-%'M219 + 0(4), (5-9)
where

as . %;)xss + aaYs + “2(aayss). (5_10)

The coefficients

an (10 SS

(;)Xss’ Ys’ and y are given by Amat,

Nielsen and Tarrago}2'The other coefficients in Eqs. (S-3)-(S-6) are

given by:

D-23 (”x +ny +uzcxyyss)1(v +4)

831 (2) ss 33 s 2
1 2

- 4 M (110 + 111 — 2 112), (5-11)
I -l( + )+u2¢ (5-12)
1 4 T1 01 11’
T .lh + ) +1424 (5-13)
2 4 2 ‘32 12’
T --1-(1- +p)+u2¢ (5-14)
3 4 3 3 13’
T - l ( + *) + “2 4 (5-15)
4 4 T4 04 14’

1 * 2
T5 - 4 (rs + p5) + u 415, (5-16)

1 * * 2
T6 - 4 (1'6 + p6) + u 416, (5-17)

where the 1 are the centrifugal distortion coefficients of 0(2),

01

i

the r , and 4

0(4).

1 11

The terms 4

11

t *
to 93, 94 to p6, of 0(4), are the vibrational corrections to

to 4 are rotational corrections to T of

16

to 4 and the p-coefficients are not the

16

60

subject of the present investigation, and their extremely complicated

*
forms will be omitted. The quantity T6 is defined as

*

Finally, the coefficients of the 1P6 terms are the fourth-order
centrifugal distortion coefficients, 41 to 410 in which we are
interested in this dissertation. These coefficients have been dis-
cussed extensively by Sumberg and Parker: and have been given in a
form which exhibits extensive cyclic and algebraic regularities.
The basic form of the transformed Hamiltonian given by Eqs.
(5—2)-(5—6) is referred to as the "4—form"?shAn alternate form of

the Hamiltonian which considerably reduces the computation of matrix

elements is expressed in powers of P2 and P2 (where P2 a P: +
Py + Pz). This is the so—called "H-form":
* 2 * 2 3 * 2
H2 (A -*D.6)Px +'(B - D6)?y + (C - 2‘06)Pz. (5—19)
* 4 * 2 2 * 4 * 2 2 2
H4=D1P +D2PPZ+D3PZ+D4P(Px-Py)
* 2 2 2 P2 2
+D Pz 2(P2 - P + P - P P2 +-D; P:
51 y) ( x y) 1 1( -Py)
l 4 2 2 4 2 2 S 2
- 2 (P - 2P Pz + P2) + h (P - 2 Pz)], (5-20)
6 4 2 2 4 6 4 2 2
H6 - Hl P + H2 P P2 + H3? Pz + H4Pz + H5 P (Px - Py)

2 2 2 2 l 6 4 2 2 4
+ H6[P (Px - Py) - 2 (P - 2? P2 + P P2)

2 4 2

+ a (P - «% P2P:)] + H7{P [P:(P: - P2) + (P:- P:)P:]}

+ 381P2< P: - P 3) + (P: W: 1 + a 9{[P: (P: 4:)2
-1-(P:-P)2P:22 ]- (P4P2- 2P2P:+P:)

+ 2 112 (P2 P: -: P:)} + “104?: - P333, (5-21)

where

with

and

haul- NPX- I-PX-

xfi’» £§j&

61

2 .
M1 + M (~2Hl - H2 - H6 + 3H9),

2
M2'+ M (12Hl + 6H + H - 20H9),

2 6

M + M2(-lOH - 5H

3 1 2 + 20H9),

M4,

+ 2H

2
M + K (2H5 + 4H7 10),

5

M + “2(4H1 + 2H - 6H9),

6 2

l

+'Z T

afloo

(Tl + T2) 6’

3 1
- 4 (T1 + T2) + (T4 + T5) -«5 T6,

1
- (T4 + T5) + T

(Tl + T2) + T 4

aflua

3 6’

(T

hflP‘

l - T2),

1 1
’ 4 (T1 ' T2) ' 2 (T4 ' T5)’

1
(T1+T2)-2T

cth

6’

l

-5
16(°1 + °2> + 8 (¢4 + 25”

15 3 3 ,
- 16 (¢l + ¢2) - 8 (454 + ¢5) +-—-(¢

4 7

1

+ P8)

(5—22)

(5-23)

(5-24)

(5-25)

(5-26)

(5-27)

(5-28)

(5-29)

(5-30)

(5-31)

(5-32)

(5-33)

(5-34)

(5-35)

62

H'_21(°+¢2)+% (¢4+¢569)+(<1>+<1>)

3
-—(<1>7+<1>8-) -1- (5-36)
24’10’
H=--S-(¢ +¢)+¢ -l(¢ +¢)-(¢ +<I>)
4 16 1 2 3 8 4 5 6 9
%7(<1> +¢S)+1<1> (5-37)
410’
H --g (¢ -‘¢ ) -'; (¢ - ¢ ) (5‘38)
5 8 1 2 4 4 5'
H--3-(<1> +¢)-l(¢ +4) (5-39)
6 8 1 2 4 4 5’
3 1 1
H7 =- --§ (<111 - <1>2)+z(<1>4 - 455) — 2(<17 - P8). (5-40)
3 1 1
H8 - 16 (41 - 42) - 8 (44 - 45) - 2 (¢6 - 49)
+l(¢ -<1>) (5-41)
2 7 8’
H9- -l§g(¢1+¢2%)+ (<114 +<115--)+1 (<b7 +48)
1
- 4 410, (5-42)
H =l(¢-¢)+-1-(¢-¢) (5-43)
10 8 1 2 4 4 5'

The equivalence of the H-form to the o—form can be verified
by substitution of Eqs. (5-22)-(5-43) into Eqs. (5-19)-(S-21) and by
rearranging the resulting expressions to the form specified by Eqs.

(5-3)-(5-5). Here the last term, viz., H has been omitted and

6a’

can be set equal to zero since the theoretical expressions for the

110, Ill 12 cannot be ififermined

from.experiment. This result has been obtained by Watson who showed

nonezero coefficients D, and t

that the transformed Hamiltonian can be reduced to a form in which all

63

indeterminacies inherent in fitting the complete Hamiltonian to

observed energy levels can be avoided.

5.2 Second-order Centrifugal Distortion Constants
From the information of Sections 4.3 and 4.4, the second-

order centrifugal distortion constants,

, -.£ 88 15 _
TOBYG 2 2 a a8 lIaIBIYIGAs’ (5 44)

S

can be constructed. These are the coefficients of the fourth-power
angular momentum terms of the vibration-rotation Hamiltonian of tri-
angular triatomic molecules. Taking account of the symmetry pro-

perties of the age for the case of planar molecules (i.e. azs - as“

s
and a:? - aZZ = 0), one obtains a total of thirteen distinct non-
zero coefficients T according to Eq. (5-44). These coefficients

aByd
are given in Table (5—1).
20
Shaffer and Nielsen have originally calculated these constants.
3393“ 35
They obey the relations of Dowling, and Oka and Morino which hold in

general for planar asymmetric-top molecules:

Tl - (122/1;x)215 - (I;y/I;x)216 (5-45)
0 O 2 O O 2

12 ' (Izz/Iyy) T4 - (Ixx/Iyy) T6 (5-46)
0 o 2 O G 2

r3 - (Iyy/Izz) 14 + (xxx/122) r5 (5-47)

T7 = 18 = O . (5-48)

Thus there are-four independent distortion constants from among T1

32
to t For the.XYZ-type molecule, Parker finds that additionally

9.

64

Table (5-1). Abbreviated Notation for Ta Coefficients

8Y6

 

(aBYG)

Tl : (arm)
12 : (yyyy)
(2222)

r4 : (yy22); (zzyy)
(xxzz); (zzxx)
16 : (xxyy); (YYXX)
r7 : (YZYZ); (ZYZY); (zyy2); (yzzy)
T8 : (xzxz); (zxzx); (zxxz); (xzzx)
19 : (nyy); (YXYX); (XYYX); (yxxy)
110: (xxxy); (yXXX)
111: (YYYX); (xyyy)
112: (xyzz); (ZZYX)

113: (xzzy); (yzzx)

 

65

T12 - (1;x/1;z)2410 + (I;y/I;z)2111 (5-49)
and

113 = 0, (5-50)
whereas for the XYX-type molecule 110 a 111 - 112 = 0.

The centrifugal distortion constants Tl through 16, and

19 with the molecule in the xy plane can be constructed with the
help of the information introduced intthapter 4 and this gives the

following result:

:1 . -<1/21:)[(c’l"‘)28ll + (ch)2822 + (c§X)Zo33 + (2cjxch)olz
. + (2tgxch)cz3 + (2t§xt:x)031], <5-51)
r . —(1/214)[<cyy)28 + (tyy)20 + (2tyytyy)o 1, (5-52)
2 y 1 11 3 33 3 1 31

:3 - -(l/ZI:)[(t:z)zoll + (tgz)2022 + (c§2)2833 + (2tiztgz)012

+ (Ztgztgz)023 + (Ztgztiz)031], (5-53)
14 - -(l/21§I§X(t{yt:z)oll + (tgytgz)o33 + (tiytzzw12

+ (tgztgy)oz3 + (tgytiz + tgztiy)o31], (5-54)
15 - -(l/21:I:)[(t:x1:z)oll + (tgxtgz)°22 + (tgxtgz)o33

+ (tixtgz + tiztgx)alz + (tgxtgz + tgztgx)023

xx 22 22 xx
+ (t3 t1 + t3 t1 )031], (5-55)

66

a 2 2 xx yy xx
T6 '(1/21x1y1(t1 t{33°11 + ( 3 3 M>°33 (t1 t2 )°12
(t xx tyy + tyyt xx

xx xv
+ (t 3 t1 3 1 )031]

t2 t3 )°23+ (5'56)

- _ xy2 xy 2 xyt xy _
19 (1/21:1:)[(tl ) 011+ (t3 ) 033 + (2t13)o31],(5 57)
where the coefficients tie are given in Table (4-1),
3 n n ,
ass, . z -J%%414¥ . (5-58)
i=1 i

The symbols x12, x13, x23, y23 are defined by Eqs. (4-46)-(4-49);

and

y12 yol - y02 = b sin 8 + c cos 6, (5-59)

yl3 yol - yo3 = -a sin 6 + c cos 6. (5-60)

Furthermore for brevity we have written all lac as Ia’ and the
latter designation will be used henceforth.

One can now write down the second-order centrifugal dis-
tortion constants for XYX-type molecules as a special case of the

XYZ molecule and the following results are obtained.

r1 . -(2/I:)ol, (5-61)
12 - -(2/I:)02, (5-62)
13 = -(2/I:)[Ixol + Iyoz + 2(Ix1y)*o31

- -<2/I:)[<c§3/11> + (ci3/12>1 <5-63)
:4 = -(2/1§1 W:)[I o + (Iny)5o3] (5-64)
:5 - -(2/I:I:)[Ixol + (Ixxy)*o3] (5-65)

67

a _ 2 2 5 _
T6 (2/Iny)(Iny) 03 (5 66)
19 = -(2/InyIzA3) (5-67)
with
sin2 cos2
0137—14-74) 0 (5-68)
1 2
co 21 sinzx
G‘2 ’ 18 + A > 0 - <5-69)
l 2
o = sin 7 cos 7 fl- - l-] < 0 (5-70)
3 A A '

l 2

Since Al > 12 from Eq. (4-129) and (4-130), always 03 < O.

5.3 Fourth-order Centrifugal Distortion Constants

 

' The principal information of interest obtainable from the
fourth-order centrifugal distortion coefficients concerns the cubic
potential constants kss's"’ which are the coefficients appearing in

the anharmonic portion V of the Taylor series expansion of the

3
potential energy in dimensionless normal coordinates. For XYZ-type

molecules, the expansion takes the form:

3 3 3 2

V ' hc (R111 q1 + k222 q2 + k333 q3 + k112 qlqz

3

2 2 2 2
+ k113 q1‘13 + k122 q1‘12 + k223 q2‘13 + k133 q1‘13

+ k (5-71)

2
233 q2‘13 + k123 q1q2q3)’

with the dimensionless normal coordinates qS defined by

qs = (AS/HZY‘QS . (5-72)

68

The potential constants kss's" are in cqu' when V3 is in ergs.

It should be noted from (5-71) that for XYZ there are ten distinct

cubic potential constants. Yet, only seven ¢i or combinations

thereof are determinable experimentally. Thus, not enough informa-
tion is available to determine the full set of cubic potential con-
stants from the fourth-order centrifugal distortion constants alone.
However, cubic potential constants also appear in the coefficients
a: which specify the second-order vibrational corrections to the
equilibrium rotational constants. Full use of both the a: and the

¢1 thus opens the possibility of obtaining a complete, consistent,

and accurate set of cubic potential constants. Furthermore,it is
7
. a

observed that the as do not contain those potential constants

kss's". for which 3 # s' # 3". Therefore, for XYZ the cubic

potential constant k is obtainable only through a determination

123

of the 61. For XYX molecules, there are only six cubic potential

constants, and in principle, the full set can be obtained either from

the ¢i alone, or from the a: alone.

By extensive regrouping and redefining of terms, Sumberg and

7

Parker have been able to express the ten ¢i coefficients in a

relatively compact form. Let us introduce the following definitions:

8:3 - aEB/Azl4 , (5-73)
3:8 . age/AS , (5-74)
(B‘)ae 7 Big C23 + 3:8 531 + 338 512 ’ (5‘75)

an _ _ _
I - (IY IB)/IGY a # B i Y and cyclic. (5 76)

69

Thus,

22 '
I = (Iy - Ix)/Iz , (5-77)

and recognizing that Ix + Iy = Iz one has that

XX

I . +1, 1yy a -1. (5-78)

Also let

we
N "———— - (5-79)
n1/2

In terms of the above definitons, the ¢i coefficients are listed
in Table (5-2).

Each of the ¢i has a set of terms linear in the cubic
potential constants kss's" with coefficients that are symmetrized
products Ofthe bze. In addition, there occur terms independent of
the kss's"° These terms depend only on the parameters of the harmonic
vibration problem and on the equilibrium geometry. Thus, the fourth-
order centrifugal distortion coefficients may be regarded as resulting
from the sum of two contributions: cubic anharmonic and harmonic.

The equilibrium geometry of a particular molecule and the
parameters of the normal-vibrations problem are usually known through
the zeroth and second-order part of the analysis of the high-resolu-
tion data, and therefore one may regard the only unknown parameters

occurring in the fourth-order centrifugal distortion constants to

be the full set of cubic potential constants of the molecule.

70

Table (5-2). The Fourth-order Centrifugal Distortion Coefficients
of XYZ as Calculated by Sumberngarker

 

6 = xx .3 xx xx xx ,
1x31 %N 2 2 2 (bsz:¥bs")kss.sn + 2 2 BS BS,(ASS.)

sjs'js" 8 S 3'
16¢ a AN 2 2 2 (byybYYbyZ)k , n +-3 2 2 Byy8y¥(Ayy,)'
y 2 4 s<s'<s" s s as s 8 s s' s 3 ss
6 1 22 22 zz' l 2 l 22 2
I243 ZN :<:.<:" (bs bs'bs")kss's" + 2(BC)zz 8 2 (BS )

12144 ='lN 2 2 2 (bxxbbeyZ + bxfbyybyfi + bxfbbeyy
x y 4 8 s<s'<s" s s s s s s s s s

+ 4byybebe + 4bbexybe + AbebXbey)k . u
S S S S S S S 8 S SSS

_ YY YY XX 1 XX YY xy U yy 1
+ 16 2 2' {BS BS,(ASS,) + 2(BS 33' + 238 3800188,)

xy yy xv . xv .
+ 488 BS,[(ASS,) + (As's) 1}

l zz A3 + As' xy yy xy yy
+'__ I Z 2' Css'[X—_:—l—71(Bs Bs' + Bs'Bs )

12 s<s S S

I4I2¢ same as 12144 with yy interchanged with xx and with
x y 5 x y 4

I22 replaced by _Izz

2 4 ‘ yy 22 zz yy 22 zz yy 22 22
IYIz¢6 lfiN :<:'<:n (b8 bs'bs" + bs'bs bs" + bs"bs'bs )kSS'S"

l_ _.l yy 22 Ԥ_ 22 zz yy ,
+ 2(BC)yy(BC)zz 8 2 B3 B3 + 16 g 2. B8 Bs'(Ass')

C v
1 ss :2 xy _ zz xy _
-T;j:-x;:][Bs BS,(AS, 2A8) + Bs'Bs (AS 218,)]

71

Table (5-2) (continued)

 

 

' 14124' - in 2 2 2 (byybYszfi + byybysz? + bbeyszz)k , n
y z s s s s s s s s 3 ss

7 8 sgsflgs" S
.i 2 _.£_ 22 2 _.l xy 2 .2 yy 22 yy .
+ 4(32;)yy 16 2 (BS ) 4 2 (33) + 8 2 2,33 33,6488.)
3 s s s
C t
-.l ___£EL___ X? Y? _ Ky 22 _
6 Z Z'[A _ A ,][Bs 38,0S 2A8.) + Bs'Bs (AS, 2A8)]
s<s s s
I4I2¢ same as 14I2¢ with re laced b xx thro hout and
x z 8 y z 7 YY P Y “8

change sign of last sum

1x1 69 same as I:I:¢6 with yy replaced by xx throughout and

change sign of last sum

1212126 s'lN 2 2 2 (bxxbbezfi + bxxbysz? + bxfib ¥ybzz
x y z 10 8 s<s'<s" s s s s s s s s s

Z 22 22
+ b:¥b:Yb:" + b§§b§y8:? + b§§b§st + 4b:yb:¥bsn

xy Ky 22 XY Ky 22 l. 2
+ 4bs bsnbs. + 4bsnbs.bS )kss.su + 2(Bc)xx(B<:)yy + (BC)xy

--l 2 [IxxBxx + Iyysyy + Izszz]2 -'l 2 BxxByy
4 s s s 8 s s s
l_ xy 2 §_ 22 xx yy , yy xx ,
+ 4 2 (Bs ) + 8 2 2' BS,[BS (Ass') + BS (A381) 1
s S S
1 22 xy xy 1 xy I
+ 4 2 2' B8.Bs [(Ass.) + (As.s) 1
s s
1 As' XX YY XY
- .2— Z 2' Css'R—T—‘T-X—MBS - BS )BS'
sis s 8
l 22 A8' - 3AS KY 22
_ Z I Z 2' gs's[-X-—,—:—X_—.]Bs BS .
sis s s

 

6. ROTATIONAL CONTACT TRANSFORMATION
TECHNIQUE AND THE REDUCTION OF THE HAMILTONIAN

6.1 Reduced Hamiltonian and Determinable Combinations of Coefficients

we have discussed earlier that in the absence of resonances,
the rotational energy levels of an asymmetric-top molecule in a given
vibrational state are the eigenvalues of a rotational Hamiltonian
which is obtained, in the form of a power series in the components of
the total angular momentum, from a perturbation treatment of the
vibration-rotation Hamiltonian. The coefficients of this power series
are the rotational and centrifugal distortion constants of the
vibrational state under consideration. Therefore, if the values of
these constants are known initially, at least in principle, it is
possible to compute the rotational energy levels from them. However,
the situation that arises experimentally is the reverse of this:
one wishes to determine the values of the rotational and centrifugal
constants from the observed rotational structure of the spectrum.
These coefficients can then,in conjunction with the theoretical
formulation,provide information about the structure and force field of
the molecule under study.

The difficulty which arises in this reverse calculation is
that some constants or combinations of constants truly contribute to
the observed energy levels while others can be arbitrarily assigned

36
without changing the energy eigenvalues. Watson has shown that

72

73

this problem can be approached by making use of the fact that the
eigenvalues of the Hamiltonian are-unaltered when the Hamiltonian

is similarity-transformed by an arbitrary unitary operator. If the
unitary operator is a power series in the angular momentum components,
then the transformed Hamiltonian will again be a power series in the
angular momentum components with_exactly the same eigenvalues

as the original Hamiltonian. Therefore on the basis of the experi-
mental results, the two Hamiltonians are indistinguishable. The
transformed Hamiltonian is called the "reduced Hamiltonian", and

the coefficients in this reduced Hamiltonian are, in general, func-
tions of the coefficients in the original Hamiltonian. Unitary trans-
formations whose effects are equivalent to merely changing the values
of the coefficients in the Hamiltonian are found to lead to arbitrary
contributions to some of the coefficients. Since the sets of co-
efficients before and after transformation are equally consistent with
the given set of eigenvalues, the arbitrary contributions are not
physically significant and can be removed through particular choices
of the coefficients of the originally arbitrary unitary transformation
Operator. This "reduction" is not unique and depends on the particular
way in which the removable terms are eliminated. However, certain
combinations of coefficients are unique and these are referred to as
"determinable combinations of coefficients." The only determinable
combinations are those which are obtained through elimination of the
coefficients of the unitary transformation. The number of deter-
minable combinations is therefore equal to the number of coefficients

which contribute independently to the Hamiltonian minus the number

74

of parameters which contribute independently to the unitary trans-
formation. Consideration of the orders of magnitude of the various
coefficients involved allows one to associate particular degrees of
freedom in the unitary transformation with particular terms in the
Hamiltonian. It is then fairly easy to find the number of deter-
minable combinations of the coefficients of the various types of
term. The arbitrary parameters specifying the unitary operator are
chosen such as to eliminate as many terms as possible from the trans-
formed Hamiltonian. Then the coefficients of the remaining terms are

of the maximum number that can be determined from experiment.

6.2 Rotational Contact Transformation

. Let us suppose that the usual vibrational perturbation treat-
ment has been performed far a general asymmetric-tap molecule so that
the calculation of the rotational levels of a particular vibrational
level has been reduced to finding the eigenvalues of a rotational
Hamiltonian whose coefficients are appropriate to the vibrational
state in question. Now the only remaining dynamical variables are
the components of the total angular momentum, and it is assumed that
the Hamiltonian is expressed as a power series in them. These com-
ponents, in units of M, are denoted by Jx’ Jy’ Jz; they satisfy

the commutation relations

[Jx’ Jy] = -iJz, cyclic, (6-1)

appropriate to the components in moving or "molecule-fixed" axes.

These commutation relations can obviously be used to alter any

75

expression involving the angular mementa in a way which is equi-
valent to changing the coefficients of the various terms. To see this,
let us consider the following general expression for the rotational

Hamiltonian:

H - 2 (JpJqJ + J :Jqu), (6-2)
p,q,r=0 hpqr x y 2

which contains one independent term for each combination of powers
of Jx’ Jy’ Jz. The expression in parentheses is chosen in the manner
shown because it is convenient that each term be Hermitian. The
hpqr are constant coefficients. If we now have a quantum product of
p factors Jx’ q factors Jy, and r factors Jz, in any order,
then because of commutation rules, it differs from
%(J:J;J: + JZJEJZ) by terms only of lower degree in the components
of J. These latter terms, in turn, differ from similar expressions
of (6-2) by terms of yet lower degree in J, and so on. By carrying
through this procedure, one can therefore express any term of the
quantum-mechanical Hamiltonian in terms of the form (6-2). It follows
that the rotational Hamiltonian may be assumed, without loss of
generality, to be in the form (6-2) which is referred to as the
"standard form".

The vibrational perturbation treatment can be performed so
as to preserve the Hermitian property of the Hamiltonian. Since the
expression in parentheses in (6-2) is Hermitian, it follows that the
coefficients hpqr in the standard form are all real. A second pro-
perty of the Hamiltonian, which should also be preserved in the

perturbation treatment, is its invariance under the operation of time

76

reversal, i.e., reversal of all momenta accompanied by complex con-
jugation of all coefficients. When applied to the standard form
(6-2), this means that the coefficients hpqr are real for even
values of n a p + q + r and purely imaginary for odd values of

n - p + q + r. It follows that the coefficients of terms with odd
values of n must vanish, and that the coefficients of terms with
even values of n are real. The number and species of terms in the
standard form of the Hamiltonian (6-2) is given in Table (6-1).

When the Hamiltonian (6-2) is subjected to a general unitary
transformation, all the coefficients in the Hamiltonian are changed
to some extent. However, the change is not significant if it is of
small magnitude relative to the coefficient itself, and in such
cases the coefficient is regarded as "determinable". On the other
hand, if.the change is of the same order of magnitude as the co-
efficient itself, the coefficient is "indeterminable". When the

Table (6-1). The Number and Species of Terms in the Standard
Form of the Hamiltonian (6-2)a

 

 

 

D2 Species p q r number of terms
A e e e % (m-l-l) (n+2)
Bx e o o -%-m(m+l)
By 0 e o %-m(m+l)
32 o o e %- m(m+1)
Total (2m+l)(m+l)

 

ap + q + r = 2m, for fixed m; e is even, 0 is odd.

77

coefficient is indeterminable in this way, the parameters in the
unitary transformation can be chosen to eliminate the corresponding
term from the Hamiltonian. However, all terms whose coefficients are
individually indeterminable cannot be eliminated simultaneously be-
cause certain coefficients of this type occur in determinable combina-
tions. The reduced Hamiltonian is obtained by choosing the unitary
transformation so as to leave only these determinable combinations
of coefficients.

If U is some unitary operation (U+ = U-l), then the trans-

~

formed Hamiltonian H can be written as

2 . u'luu . (6-3)

We suppose that a set of unitary transformations is applied succes-
sively, which is equivalent to expressing U as a product. Since
we want H to be purely a function of Jx’ Jy, Jz, we choose U to
be such a function. If U is also chosen to be invariant to time

reversal, then H will be invariant if H is invariant. The most

convenient form for a unitary operator is
U a exp(iS). 1 (6‘4)

The unitary condition requires S to be Hermitian. The invariance of
U under time reversal requires that S change sign. Together these
two requirements imply that, when S is expressed in a "standard
form" analogous to (6-2), it has real coefficients and contains terms

of odd u only.

78

On the basis of this result, we can introduce the factorized

form of U:

U a exp(iSl)exp(iS3)exp(iSS) ..., (6-5)
where Sn contains only terms with n = p + q + r:
s - 2 s ruquJ: + J:JqJ:), (6-6)
n=p+q+r Pq Y Y
with real coefficients Spqr' The number and species of terms in

the standard form (6-6) of S is listed in Table (6-2).

2m-l

Table (6-2). The Number and Species of Terms in the Standard

 

 

 

 

Form (6-6) of 82m_1.a
D2 Species p q r number of terms
A o o o %-m(m+l)
Bx o e e '%-m(m+l)
By e o e ‘%-m(m+l)
Bz e e o %-m(m+l)
Total m(2m+l)

 

ap+q+r = Zmrl, for fixed m; e is even, 0 is odd.
We now introduce the notation

H2m+2 - exp(-iS )Hzmexp(iS ), (6-7)

2m+l 2m+l

where m takes the values O,l,2,... and H0 is the Hamiltonian

79

B2 = exp(-iSl)HO exp(iSl), (6-8)
H4 = exp(-133)H2 exp(iSB), (6-9)
H - fi . (6-10)

When H m has been reduced to standard form, it can be written as:

2

32m - 2 h(2:) (JquJ: + J:JqJ:). (6-11)
p+q+r pq y y
even

It is found that Sl specifies that part of the complete
transformation which brings the rigid rotor Hamiltonian to principal
axes. Since this has already been anticipated in the expansion of

the Darling-Dennison Hamiltonian, one may take S a O, exp(iSl) 8 l,

l

and proceed to H of (6-9). This part of the Hamiltonian is

4
associated with the determinability of the quartic coefficeints.
From Table (6-1), there are fifteen independent quartic terms,

(2m = 4), in the Hamiltonian, while from Table (6-2), one sees that

there are ten independent terms in S Thus, if all these terms in

3.
83 would affect the Hamiltonian independently, then the number of
determinable combinations of the quartic coefficients obtained is
15 - lO - S.

The angular momentum commutation rules (6-1) are invariant

under the operations of the point group D2, and H2 is, of course,

totally symmetric. Therefore in the expansion of H4,

H = H + i[H2, S

4 2 +..., (6-12)

3]

the terms in i[H2, 33] are of the same symmetry species as the

80

corresponding terms in S3. One can therefore discuss the deter-
minability of the coefficients of the quartic terms of the different.
symmetry species separately.

We first consider the Bx species which is typical of the

three B species of D Putting m - 2 in Tables (6-1) and (6-2),

2.
we find that there are three quartic terms in H of species Bx, and

three terms in S3 of species Bx. The corresponding coefficients

are h031, h013, h211 and $120, 3102, S300. Deve10pment of (6-12)

leads to the relations

(4) , <2) _ _
(4) (2) . -

h013 ' 8013 + 2(C‘B)3102’ (6 14)
(4) ,, (2) _ _ _ _

h211 h211 + 6(C B)8300 + 4(A C)S120 + 4(B A)3102. (6 15)

. (4) (4) (4)
The three coefficients h03l’ h013, h211 are independent functions

of $120, $102, $300 and by a suitable choice of the latter, they
could be made to take any—arbitrary values, subject only to order-
of-magnitude restrictions. These coefficients are therefore in—

determinable, and the transformation should be chosen to eliminate

the corresponding terms from the reduced Hamiltonian, and thus

<4) _
031

(4) (4)

h 013 . h211 8 0. (6-16)

h

Similar results hold for the By and the 32 terms. It can there-
fore be concluded that the non-totally symmetric (non—orthorhombic)
quartic terms should always be omitted from the reduced Hamiltonian,

even when the particular point group symmetry of the molecule allows

them to be present.

81

We now consider the Arspecies terms. We see from Tables (6-1)
and (6-2) that there are six quartic terms in H and one correspond-
ing term in S3. The former have coefficients h400, h040’ h004’
h022’ h202’ h 220, and the latter has coefficient $111. It is found

4 4 . 2 2
that héog, hOZO’ héOZ differ insignificantly from héoa, 643,

hé§%, respectively, whereas for the remaining coefficients we have

h

“622 ' “622 + 2(C"B)Sul’ (6'17)
“232 h2§2 + 2(A-C)Slll’ (6'18)
11:33-11:23 + 2(B-A)Slll. (6-19)

(4) (4) (4)
Since each of h022’ h202’ h220

freedom in the unitary transformation, each of them is indetermin-

is affected by this degree of

able individually. However, we can eliminate the parameter Slll
from Eqs. (6-17)-(6-19) in two independent ways which yields two
determinable combinations of these coefficients. The most

symmetrical choice is

1‘64) h<4) h<4) ho2) + h(2) + 115:3 (Ho)
and
A802 + Bh(4) + Chégga Ahéggi+ Bh<2> + Chggg (6-21)

Therefore a set of five independent) determinable combinations of the

. h(2) (2) (2) (2) (2)
quartic coefficients is. h400’ hog ’hOO4’ h022 + h20 + h220 and

Anna + Bh<2> + Ch(§3, since none of these is affected significantly

by the similarity transformation.

82

The parameter S is now chosen so that only five in—

111
dependent quartic terms are left in-the reduced Hamiltonian. It

is convenient to make a choice for $111 which simplifies the
calculation of the eigenvalues of the reduced Hamiltonian. Once the
choice has been made, all the coefficients of 83 will have been

chosen in a definite way.

6.3 Determination of Quartic and Sextic Determinable Combinations
of Coefficients

For an orthorhombic molecule, the rotational Hamiltonian as
obtained from the vibrational perturbation treatment contains only
terms of species A in D2 because of molecular symmetry. For
molecules of lower symmetry there are, in general, also terms which
are non-totally symmetric in D2. In the latter case we assume that
thenon-totally symmetric terms in the Operators S1 and S3 have
been chosen such as to remove the non-totally symmetric quadratic and
quartic terms from the Hamiltonian, as described in the previous
section. Thus in either case, we are left with only the A-species
terms in the standard form (6-2). Therefore we can concentrate on
the one-parameter problem of the reduction of the A-species quartic
terms.

In the form of the Hamiltonian (6—2) apprOpriate to an
orthorhombic molecule, the only terms present have p, q and r

all even. Thus for this case, Eq. (6-2) becomes, up to sextic

terms ,

H - H + H4 + H6, (6-22)

83

where H2, H4 and H6 are defined by Eqs. (5-3)-(5—5). Here the
hpqr notation, which was useful for the general discussion, has been
dropped in favor of our previous notation.

We now proceed to consider the transformation of the

Hamiltonian (6-2) by the unitary operator

U - exp(is3)exp(185), (6-23)
where

538 W111(J J Jz + J zyJ J x), (6-24)
and

3
$5: 3311(JnyJz + J zyJ J3 x) + 8131(Jx Jsz + Jz Jy 3J x)
+ s (J J J3 + J3J J ) (6-25)
113 x y z z y x '

The four real coefficients 8qu are the parameters of the unitary
transformation (6-23).
Carrying out the transformation one obtains the transformed

Hamiltonian

(
I

a = u Inmza H2 + H4 + H6, (6-26)
where

H2 . H2, (6-27)

H4 - H4 + iIHZ, 83], (6-28)

” 1
H6 3 H6 + i[H4, $3] + i[H2, SS] --§[[H2, 83], S3]. (6-29)

After evaluation of the above commutators, H can be rearranged to
the form of Eqs. (5-2)-(5-5), with new coefficients that will be

distinguished by tildes, as follows:

84

A . A, B = B, 6 = G (6-30)
21 a T1, 22 a T2, i3 = T3 (6-31)
24 . T4 + 2(C-B)Slll, (6-32)
is - T5 + 2(ArC)Slll, (6-33)
.i6 - T6 + 2(B-A)Slll, (6-34)
51 . 21, $2 - @2, 53 - o3, (6-35)
54 = ¢4 + 2(B-A)Sl31 + 4(T2-T6)Slll + 4(A—B)Sill, (6-36)
55 - J5 + 2(B'A)5311 - 4(Tl-T6)Slll + 4(B-A)Sill, (6-37)
56 = ¢6 + 2(c-B)s113 + 4(T3-T4)Slll + 4(B-C)Sill, (6-38)
87 f J7 + 2(c-B)s131 - 4(T2-T4)s111 + 4(C-B)Sill, (6-39)
58 = ¢8 + 2(A—C)S3ll + 4(Tl-T5)Slll + 4(C-A)Sill, (6-40)
59 8 o9 + 2(A.-C)S113 - 4(T3-T5)S111 + 4(A—C)Sill, (6-41)
510 - J10 + 6[(c-B)s311 + (A-c)s131 + (B-A)8113]. (6-42)
It can be seen that the coefficients A, B, C, T1, T2, T3,
$1, Q2, and ¢3 are, by themselves, invariant and therefore de-
terminable. The other determinable combinations must be obtained by

elimination of the S-parameters. The most symmetrical choice of

transformationrinvariant functions appears to be:

T4 + T5 + T6, (6-43)
AT4 + BT5 + CT6, (6-44)
3(¢5 + o8 + ¢7 + ¢4 + @9 + @6) + ¢10, (6-45)

85
(A—C)<1>5 + (A-B)¢8 - 2(T1-T6)(T1-T5),
(B-C)¢4 + (B-A)¢7 — 2(T2-T65(r2-T4),

(C-B)<l>9 + (C-AM6 - 2(T3-T5)(T3-T4).

(6-46)

(6-47)

(6-48)

Any other choice of a set of determinable combinations can be ex-

pressed in terms of those given above. It is convenient in practice

to have all of the invariant quantities involving the ¢

with the

same dimensions. To this end, in place of (6-46)-(6-48), one may

divide each of these by (A23) to obtain:

2¢8 + (1-o)¢5 - 4(Tl-T6)(Tl-T5)/(A-B)
2¢7 + (1+o)<b4 + 4(T2-T6)(T2-T4)/(A—B)
'(a+1)¢9 + (0-1)¢6 - 4(T3-T5)(T3-T4)/(A-B)

where

o = (2C - A - B)/(A—B).

(6-49)

(6-50)

(6-51)

(6-52)

7. CALCULATION OF ASYMMETRIC-ROTATOR
CENTRIFUGAL DISTORTION COEFFICIENTS
BY THE ALIEV-WATSON METHOD

In 1976, Aliev and Watson13 presented a very direct and
efficient method for the calculation of the fourth—order centri-
fugal distortion constants of asymmetric, as well as symmetric and
spherical, rotator molecules. Their very compact results, when
applied to triatomic molecules, are consistent (but not identical)
with those of Sumberg and Parker7. It was deemed useful to give a
more extended discussion of the Aliev-Watson method in this chapter,
because the results are relevant to the present work, because the
method appears potentially useful for the calculation of higher-
order vibration-rotation interaction coefficients, and also because
the original paper gives only a very brief presentation whose verifica-
tion in detail required much effort. We also present in this chapter
a detailed comparison between the Aliev-Watson and the Sumberg-

Parker sextic centrifugal distortion coefficients and account for all

discrepancies between the two sets of results.

7.1 Direct Perturbation Treatment for the Calculation of the Sextic

Centrifugal Distortion Coefficients

 

Using Watson's notation as discussed in Section 2.3, we let

~

Hmn denote the various terms of the effective Hamiltonian resulting

from the perturbation treatment. The perturbation calculation of

86

87

the desired terms, viz., H06, can be carried out in two stages. In
the first stage, two successive vibrational contact transformations
are performed to eliminate terms off-diagonal in the vibrational
quantum numbers vs. This can be done by the technique described in
Chapter 3. We denote the result of this transformation by H62).

In the second stage the rotational Hamiltonian is subjected to a
rotational contact transformation to produce a reduced rotational
Hamiltonian containing only experimentally determinable coefficients.
This can be done by the technique described in Chapter 6. We denote

~(R)

the result of this transformation by HO6 . The final expression

for H06 can be written as

~ _ ~(V) ~(R)
H06 ' 06 + Hoe ° (7’1)

The vibrational contact transformations are carried out such that at
each stage only those terms that contribute to the desired order of
magnitude, klou§ib, and a degree of six or greater in Jo are re-
tained. The power of k indicates the order of magnitude of the
coefficients relative to the harmonic vibrational frequencies.38

To express the two requirements together, we introduce the notation
kmrn, where m indicates the order of the magnitude of the co-
efficients and.mnn is equal to the total power of Jo of the

associated operator. It is found that the terms required in the

calculation of H06 are:

0’0 2-0 3-1 4-2 2-1 1-0
H20<k >.H02<k ),H12(k >,H22(k ),H21<1< >.H3o<k )

(7-2)

88

where the explicit expressions for these terms are given in Table
(2-3). The contributions of these terms can be conveniently grouped
into the following three classes:

(a) Anharmonic: H H H H

30 12 12 12

(b) Harmonic: leleH22

(c) C°ri°lis‘ H12H12H21H21’ H12312321H02’ lenlznozfioz'

In the contact-transformation formulation, each of the above terms
denotes a product in which the individual factors can be either Sis)
functions or Hmn functions to be identified with the apprOpriate
commutator brackets of the transformed Hamiltonians. No other
combinations of terms from the Hamiltonian of Table (2-3) satisfy the

specified requirements. If we now express our initial Hamiltonian

in the form:

2

H = H0 + AHl + A H2, (7-6)
with

Ho - H20, (7-7)

H1 .. H02 + H12 + H21 + H30, (7-8)

H2 - 322, (7-9)

then the above contributions all appear in the fourth-order of

perturbation theory.

Now the first vibrational contact transformation can be

written as:

H6 = H0 (7-10)

(1)

v a ..
H1 H1 + i[S , H (7 ll)

89

. , <1) _ i <1) (1) _
H2 H2 + i[S , H1] 2[S ,[S , H01] (7 12)
3I0
. . <1) <1) (1)
33 //§ + its H21- %[8 ,[s . H11]
-%ts‘1’,[s‘1’,[s‘l’, H0111 (7-13)

.0 o
. (1) .l (1) (1)
34 - H 4 + i[S , J/g ] - 2[s ,[s , H21]

-%[s(l).ts(l).[s(l). H1111

(1)

its ,[s(1).[s(1).[s(l), H0111]. (7-14)

(1)

Eq. (7-11) defines the Operator S which is constructed such that

off-diagonal elements in the vibrational quantum numbers vS are

removed. Therefore Eq. (7-11) can be written as:

i[S(l), H0] 8 -H1(off diag.) (7-15)

where

1-0

H kz’o) + H k3’1) + H kz-l) + H (k ) . (7-16)

1 ' Hoz‘ 12‘ 21‘ 30

Considering vibrational commutators only, we can assign separate

(1)

functions of S to each term of H with the exception of the,

l

H term , and these functions are respectively denoted by

02
8(1)-S(1)(kH) + S(l)(kH ) + smacl'0 ). (7-17)
Now considering the second vibrational contact transformation, we
can write:
"- 1 -
H0 H0 (7 18)

n .1 v _
H1 H1 (7 l9)

90

(2)

H; = Hé + i[S , H6] (7-20)
II . I (2) I '
H3 H3 + i[S , H1] (7—21)
II _ I (2) I (2) (2) I
H4 - H4 + i[S , H2]- %[8 ,[S , H0]]. (7-22)
Using the fact that
(l) a _ . _ _
[S . Ho] 1(H1 H1). (7 23)

we can write Eq. (7-20) in the following way:

u , (1) . (2)
H2 H2 + %[S , (Hl + H1)] + i[S , H0] . (7-24)

This equation now serves as the defining equation for the operator

8(2). From Eq. (7- -22) it can be seen that the only 8(2 ) term con-

tributing to H is S(Z)(k5”2 ). Therefore in Eq. (7-24), the

(2)
13

diagonal elements, viz.,

06

operator S has to be chosen such that it removes the off-

[s(2). H01 = -1{-n2 --§[s(1’, (H1 + Hi)1} <7-2s>

off-diag.

where

~ ~

H' i H + H = H

1 02 21 1’ (7'26)

and H21 is the diagonal part of H21.

that the needed part Sig) of 313 must satisfy

It is seen from Eq. (7-25)

3(2)]-%[s<1> (2302 +'H .+ fi21)]1+- -[s(1)

12 ’ 21 21 ' H121' (7'27)

[H 0.

S(1)3(1) S(l)

(2)
The expressions for $12 , S21 , $30 , and 813 are given in Table

(7-1). On substitution of these into their respective defining

91

 

 

 

Table (7-1). 8 Operators for the Vibrational Contact Transformationa’b
<1) ,
S12 : RkPk/“k
(1) l k *
S --{2R(qq-pp)/(m +w)+2 k
21 2 k2 2 k 2 k 2 k 1 k1 R2(qkq£ + Pkp£)/(wk - w£)}
(1) _ 1 . 2 2 2
S30 6 Kin klmnnwzwmmn pRPmPn + 3wm(w2 - mm + wn)q£pmqn}/Q£mn
<2> . 1 k
313 41:12: qk {i(3wk + 5w£)R£R£/(wk + 1119‘)!»ka
- 2*(111 + w )R Rk/(w - w )w w + 41[ H ]/m2
2 k 2 z 1 k 2 k 2 Rk’ 02 k
a

- + + - - - .
Qzmn (ml mm wn)( ml + mm + wn)(w£ mm + wn)(w£ + mm mn)
The other notations are described in Chapter 2.

*
bln the Z sums, terms with zero denominators are omitted.

equations, it can be verified that these functions were chosen
correctly. Now using Eq. (7-25) along with the commutators of the
first vibrational contact transformation in Eq. (7-10)-(7-14), we

can write the fourth-order Hamiltonian in the form
~(4). - n s _ _i_ (l) (l) (1) _]_-, "
a 34 81s .13 .[s , (H1 + 3 H1>111
1 (l) (l) 1 (2) (2) _

From the various operators and commutators, the three classes of

term in fiéz) can be calculated by means of the equations
~(v) , .1 <1) <1) <1) <1)
H06 (anharmonic) - 8[S12 ,[S12 ,[S12 , H30] + [S30 , H12]]]
(7-29)
fi(v)(harmonic) - --l[s(1) [5(1) H 1] (7-30)
06 2 12 ’ 12 ’ 22

92

(1)

(1) (l)
12 , [S (4H

+ [S [321 ’ 12 ’

02 3H21 + 321)]]]}

(1) [5(1) (1) (2)

(2)
[s13 , H011.

(7-31)

8[S12

The commutators containing H in the above equations are evaluated

02

as rotational commutators; all the other commutators are vibrational

commutators. The diagonal Coriolis term fl is included for complete-

21
ness in order to be able to apply the results to symmetric and

spherical tops as well as asymmetric tops. For asymmetric rotators

fiZl 8 0. After considerable algebraic manipulation, the very compact

resulting expression for H62) is obtained as listed in Table (7-2).

Table (7-2). Sextic Centrifugal Distortion Hamiltonian of an
Arbitrary Moleculea

 

Egg) _ H(V)(harmonic) + H(V 6)(Coriolis) + H(V 6)(anharmonic)
ficv)(harmonic) = 5 HRkszRzlwk 2
fi(v)(Coriolis) = -‘E Z{£ Rsz/wlwk1/2+ i[Rk H ”21/3/2} 2

k 2

k k
RkR£R£ + * RkR£R£
(wk + “ H>wk 2 k2 (“k ' ”2)wk”2

 

 

%[£ 021

H(V 6) g .:£ '
(anharmonic) 6 inn kzmnRszRn/wzwmwn

 

aNotation as in Table (7—1).

(R 6)

Next, we proceed to calculate HO by subjecting the

rotational Hamiltonian

93
~ H(V) (V) ~(V)
Hmt a1102+ H04 + H06 +..., (7-32)

to a rotational contact transformation by a purely rotational

operator. This operator is taken in the form
U = exp(i Sos)exp(i 503). (7-33)

Then the transformed Hamiltonian can be written as:

ii = U lfi(v)U
rot rOtU
=- fioz + 1304 + {106 +..., (7-34)
with
1102 . 302, (7-35)
' {‘04 " 332" + 1L[503’H 02] ' Hg) + Héfi)’ (7‘36)
ii06 " 135? " %[So3’[303’ Hoz“ + i[Soa’Ho Hm] + 1L[5033302]
-~sz> ~32”,
(V)

where the second-order fourth-power (in J a) Hamiltonian H04

is defined as:

WaBYGJaJBJ J (7-38)

For convenience of reference, all possible asymmetric rotator

32
point groups are listed in Table (7-3). The asymmetric-top rigid-

is invariant under the operation of the

rotor Hamiltonian H02
orthorhombic group D Therefore the terms of H(v) H(v) S
2' 04 ’ H06 ’ 03’
and S can be classified according to the symmetry species of this

05

group. As mentioned in Chapter 6, even for non-orthorhombic molecules,

94

Table (7-3). Asymmetric-rotator Point Groups.

 

 

Crystallographic Group Group operations other than
nomenclature symbol identity operation
Triclinic Cl none
Ci 8 82 i
*
Monoclinic Cs 8 C1h 0
C2 C2
C2h C2,oh,i
0 h h bi C **
rt or om c 2v C2, two ov,
V 8 D2 three mutually 1 C2
Vh = D2h three mutually i C2, 1,

three mutually l o

 

*
XYZ-type molecule has point group symmetry CS.

**
XYX-type molecule has point group symmetry C2v'

the non-orthorhombic terms of 803 and S can be chosen to

05

eliminate all the hon-orthorhombic terms from I} and 1:1 so that

04 06

as orthorhombic operators.

~

it is sufficient to consider Q and H

04 06
Such operators contain only even powers of the individual components

Ja' The non-orthorhombic terms of 135:) and 803 in Eq. (7-36),

however, do contribute to orthorhombic terms in fourth order.

It is found that the operator form of the last two sums of

I.

n")

(
06
be completely removed by the choice of an appropriate term in 805'

(Coriolis) as given in Table (7-2) is such that these terms can

Therefore this part of the Coriolis contribution can be omitted.

95

The remaining terms of $03 and SOS are associated with the re-
duction of the Hamiltonian to avoid indeterminacies in the fitting
of experimental data, and since this reduction is not unique, there

is no unique choice for the remaining S-parameters in S and S .

03 05

The final expressions for the various sextic distortion
constants are obtained from fiéz) by:
(a) adding the second-order contributions resulting from
the elimination of the non-orthorhombic terms from
fi04’ and
(b) eliminating all the non-orthorhombic terms and the
orthorhombic contribution from the last term of fiéz)
(Coriolis).
The resulting expressions are presented in Table (7-4). In order to
effect a detailed comparison between the Aliev-Watson coefficients
and the Sumberg-Parker coefficients, the former have been trans-
scribed into Sumberg-Parker notation.
.30

The contributions to the complete coefficients ¢

1

which result from the elimination of the non-orthorhombic terms in
the second-order part of the Hamiltonian are listed in Table (7-5),
again in Sumberg-Parker notation. For triangular molecules, these
contributions arise only for the XYZ case. For XYX the oiR) all
vanish identically. The Sumberg-Parker calculation failed to in-

clude the non-orthorhombic terms for the XYZ case, and hence their

ER) as listed in

results should be augmented by including the ¢
Table (7-5). Table (7-6) presents the Sumberg-Parker coefficients

augmented by the non-orthorhombic contributions of Table (7-5).

96

Table (7-4). The Fourth-Order Centrifugal Distortion Coefficients of
XYZ as Calculated by Aliev-Watson

 

' I
¢l %N 2 Z.
S 8

xx §_ xx xx ,
i" (b:xb:?bsn)kss.sn + 3 i :' B:XBS.(ASS.)
.; _ X? X? x xx
+ 8[Ix/Iy(Iy Ix)]z 2' Asxs'Bs BS.BSXBS.
8,3

6 . _ yy yy yy .3 yy Y? Y? .
I ¢ 1N z z 2 (b8 bs'bs")kss' . + z 2 Bs Bs'(Ass')

y 2 4 sgs'gs" S 8 s s'

-.l - xy xy yy yy
8[Iy/Ix(Iy Ix)]Z z ASAS,BS 38,33 BS,

s,s'
6 , 22 22 22 l_ 2 l_ 22 2
Iz¢3 %N Z X. 2" (bs bs'bs")kss's" + 2(BC)zz — 8 2 (Es )
s_s _s s
1214¢' - AN 2 z z (bxxbbeyZ + bx¥byybyZ + bxifbyfiyy +4bybebeX
x y 4 8 s<s'<s" s s s s s s s s s s s s

X? + yy xy xy .2. YY YY xx v
+ 4bz¥§ibs" 4bs"bs'bs )kss's" + 16 i :' {Bs Bs'(Ass')

yy KY KY yy . KY XY XY .
+ 2(B:XBS, + 2138 33,)(ASS,) + 438 Bs,[(Ass,)

XV . _.2 - xv xv yy xx
+ (As.s> 1} 8[Iy/Ixay Ix)]: E'ASAS.BS 38,38 BS,

l. _ XY XY yy yy
+ 8[Ix/Iyuy Ix): :' ASA3,BS 38,38 BS,
’

Iiliog same as 1:134;4 with yy interchanged with xx
2 4 , 1V yy 22 22 yy 22 22 yy 22 22
I I ¢ -'—1 2 2 z (b b .b n + b .b b n + b "b .b )k . n
y z 6 8 s<s'<s" s s s s s s s s s as s
+-£(B;) (8;) --£ szszz +-3- 2 2 Bzszzmyy )'
2 yy 22 8 s s s 16 s s' s 3' 38'
l_ 22 xy _ zz xy
+ 4 2 2:.L;ss'(Bs Bs' Bs'Bs )
s<s
_‘l_ _ xy xy 22 22
l6[Iy/Ix(Iy Ix)]£ Z Asxs'Bs Bs'Bs Bs'

I
3,3

.l 2 xy Ky 22 yy _ xx
- 8[Iz/(Iy - Ix) ]: E'ASXS,BS 38,38 [(IyBs,/Ix) IyBS./Ix)]

97

Table (2-4) (continued)

 

I4Ich' - lN 2 2 2 (byybyybzz + byybyybzz + byybyybzzflc

y z 7 8 $£§,§§" s s' s" s s" s' s' s" 3 53’s"
+-%(B§)2-12(B:y)2 %2(Bxy)2+3 2 2 B’VB:Z(A:: )'
Yy16 S 8
s s s'
_l xyyy_yyyy
4 :<:,Css'(B Bs' B '38 )

- -[Iy /Ix (Iy - I x)12 2 A “A ByB’gByszz
S, 8'

--—[I2 /(Iy -I x) 22 2 A SA S,B:yB:yByy[(Iy B 3.)/Ix -(I xByy)/Iy 1
8"3

4 2

lo o8 same as IyIz<I>7 with yy replaced by xx throughout

I

NNflk
NHbNN

I Io o; same as 1:1:¢6 with yy replaced by xx throughout

121212¢' sin 2 2 2 (b:xbyybu s" + bxxbebzf + bxfby¥bzz
10 8 sis'<s' s s s s s s

+ bmszybgfi + bsz..s ”B2 s. + bxxb. YZB" + ABXYBXszfi
s s s s s 3
xy xy zz xy xy 22 .l
+ Abs b3..bs. + “be,"bs'bs )ks . .. + 2(Bc)xx(Bc)

88

+ (BC)2 -'l i[IxxBxx + IyyByy + IZZB:Z]Z
xy 48 S s

_.l yy .l xy 2 3 Ayy .
8 2 BszS + 4 2 (BS ) +- 8 2 2 B2 s,[B? (A 88,)

s s'
+BYY(A? s,) 1 +-% 2 2 3:23:V[(A:Y ) + (A:¥s)']
s s'
_‘l xy _ yy 22B 22
4 2 2' CSS,BS,[(Bs Bs ) + 218]
sfis

+--[Iz /(Iy - I x) 212 2 2828 BznyuIy ngB? .)/Ix
3"3

yy yy _ Y?
+ (IXBS BS.)/Iy (Iy/Ix + Ix/Iy)B:st.]

 

98

Table (7-5). Contribution to the Distortion Constants Resulting
from the Elimination of the Non-Orthorhombic Terms

 

I:¢iR) --—[Ix /Iy (Iy - I x)]2 2 AS AS B:yB:ny Bxx s,
s, s'
I3¢§R> same as I:¢§R) with xx and yy interchanged throughout
6 (R)
z¢3 0
I :1y 4¢(R)- --2[I /I (Iy ~ I x)]Z Z A SA xy xy yym
4 8 y x s s 3 RB B8 B8 B s'

+%[Ix/Iy(Iy - I x)12 2 AS AS ByByBWBW
S, S

Iy1”4¢(R) same as I :Iy¢(R) with xx and yy interchanged

throughout
I 21%“) = - Lu /I (Iy - I x)12 2 A SA S,B"3'B:"B:‘"'B:z
ya 26 16 y x ,
8,8
-—[Iz /(Iy - I x) 212 2 A SSA B:yB:yB:z [Ix BW/Iy -I ”B x”,‘x/I]
S, 8
14124310 = - l[Iy /Ix (Iy - I x)12 2 AS A Sz.B’;"B’£:3’BWB:2
y z 7 8
S, S
- %[Iz/(Iy - I x) 212 2 A8 A3 B:yB:yB:y[Iy BS x’fx/I -I xBW/Iy 1
S, 8'
I:I:¢§R) same as I:I:¢§R) with xx and yy interchanged
throughout
IiIi¢éR> same as IyI :¢éR) with xx and yy interchanged
throughout
1:133:39?me /(Iy -Ix) 21): 2 A “A ,B?B?[(Iy Bf‘Bf‘ ,x)/I

+ (I BWBWVI - (I /I + I /I )Bm‘BW
x s s y y x x y s s

 

99

Table (7-6). The Complete Expressions for the Fourth-Order
Centrifugal Distortion Coefficient of XYZ as Calculated
4§y SumbergéParker

1ch -$sz z"(b’:‘bs’fxb:x..)ks s"'+§£z B’D‘B’fi‘mna'
s 8 s' s 3 ss

s<s <8"

 

+—[Ix IIy (Iy - I x>12 2 AS As B:YB:VB’:‘B’°f
S, S

I 64> sin I z z (byybyyb? .s-Nc 8.8.. +-:- 2 z BWBZV(AW.')

yZ s<s _<_S" SS 8

+-[I /I (Ix - I )1: 2 AS AS ,B:VB:VBWBW
8 y x y s s'

2
16¢ -- in Z 2 z (bzzbzzbzfms S, .. + l(Bc)2 - -1- 2(Bzz)
z 3 4 s 2 22 8 s
s<s' <5" 8

=15sz z"(b’s°‘bWb:?I +bs,b:yb:.y .s)+b .bWbW
s<s' <3"

Iny¢4=

+ AbbebeX + 4bWb:yb:?’ + 4bWb’S‘yb 23% 8.8..

§_ yyyy XX. xxyy nyy W .
+ 16 : :.{Bs Bs'(Ass') + 2“33 BS' + 23$ Bs')(Ass')

xy yy xy 0 KY
+438 BS,[(ASS.) + (AS. S)']}

l 22 A3 + As' xy yy xy xy
+.l5 I :<:, z;ss'[)\s - A ,](Bs Bs' + Bs'Bs )

-—[Iy /IX (Iy - I x)1: 2 A3 A8 BzyayBan’,‘
S, S.

+8[Ix /Iy (Iy - I x)Jz 2 AS AS .B:VB:YBYVBYV
8,8
42 24

I I ¢ same as I I ¢ with yy interchanged with xx and with
X Y 5 x y 4

Css' replaced by -Css'

100

Table (7-6) (continued)

 

1212456 - in 2 2: 2 (bWb:zb:ff + by¥b:zb:,z. + bW’bs ‘2 b:z)kss
yz S<S <8"
__ .l yy 22 3 22 22 yy
+ 2(Bz;)yy(Bc)zz 8 2 BS 33+ 2 2 BS B (AS )'

l6 . s'
s s s

C 1
__l ss 22 xy _ zz xy _
6 :<:'[As - xs'][Bs BS'(AS' 2A3) + BS'BS (AS ZASI)]

- -11-6-[Iy/Ix(Iy- I X)Jz I A SA S,B:YB:¥B:ZB:Z
8, 8

--[I2 /(1y -I x) 2]): 2: AS A S,B:yB:yB:z[(Ix BW)/Iy _ (1y B s.)/Ix]

S, S
141‘?» =- 1N z 2 2: (bWbW'b? + bWbW'b:z + bWbWb:z)k , ..
yz 7 8 S S
s<s' <8"
_ _. l. W 2_ KY?- 3 W2 Y? .
+ 4(BB)yy 16 2(BS) “5 2 £38 ) +— 8 z 2 BS B8 20.85.)
S 8 S S
c
.1. 88 X? W _ xv yy
6 sfs'[A - A ,][Bs Bs'as st') + Bs'Bs (ls' ZAS)]

- in /I (Iy - I x)1: z A A .BxyBWBWB ?2
8
Y X s s s s s s s s

_l _ 2 xyxyyy xx
8[Iz/(Iy Ix) 152$? Asxs'Bs 35,88 [(IyBs')/Ix

YY .
‘ (IXBS')/Iy]

I Ii°8 same as I:I:¢7 above with yy replaced by xx throughout

and with Css' replaced by -Css'

Iil2¢9 same as 1:1:¢6 above with yy replaced by xx throughout

and with Css' replaced by -Css'

101

Table (7-6) (continued)

 

121212¢ . in Z 2: z (bxxbng’; + bS “‘1:be + b S,.bWb:z
x y z 10 8 s<s $5"

+ b’obebzz ,, + bmn‘bW bzz + bf‘bngz + AbWbszfi
S S S S S S S S S

1
+ 4b:yb:y .‘gb z+ 4b:zb:yb:z)ks 3.8.. + -2-(Bz;)xx(Bz;)

+ (BC)2 - l zu’mBm‘ + IWBW + Izszz]2
4
xy 3 s s

— -1- z Bx‘fisW +l 2 (BW)2 + 3 z 2 Bz?[B’°‘(AW.)'
8 s s s 4 s s 8 s s' s s 38

+ B?(A::,)'] +— 2 z B:ZB:Y[(AW )' + (AWS)']

s s'
1 Xs' xx 3?? XY
--5 Z 2. Css'fx_7_:—X;](BS - B )B '
sis s
l 22 A S' - 315 Ky 22
-"I Z X C v [S SSSJB BS
4 g 3 S A v
s¥s s
+3[I /(Iy - I x) 2]: 2 A A .BWBWHI B1 Emb/I
8 z s s s s y s s x

8,8

yyyy _ xxyy
+ (IXBS BS.)/Iy (Iy/Ix + Ix/Iy)BS 38.]

 

102

By comparing Tables (7-4) and (7-6), we can now trace the
remaining differences between the sextic centrifugal distortion co-
efficients of Aliev-Watson and Sumberg—Parker. These differences are
due exclusively to the Coriolis-type contributions which are handled
differently in the two formalisms, viz., in the Aliev-Watson procedure,
part of the Coriolis contribution is removed through the rotational
contact transformation, as described previously. These differences,

listed in Table (7-7), are defined as
A¢i = ¢i(Sumberg-Parker) - ¢i(Aliev-Watson). (7-39)

We have now fully accounted for the differences between the
Aliev-Watson results and the Sumberg-Parker results.
. For fitting to experimental data, the contact transformation
is, finally, fully specified and one may proceed to obtain a re-
duced rotational Hamiltonian in the manner described in Chapter 6.

The result can be expressed as:

2
x

3 ~ 6 ~ 6 ~ 6 ~ 4 2 4 ~ 4 2 2 4
H06 ¢1Jx + ¢2Jy + ¢3Jz + ¢4(JyJ + Jny) + <I>5(JXJy + Jny)

~ 4 2 2 4 ~ 4 2
+ ¢6(Jsz + Jsz) + ¢7(Jsz

+ J2J4) + 5 (J4J2 + J2J4)
z y 8 x z z x

~ 4 2 2 4 ~ 2 2 2 2 2 2
+ <I>9(JzJx + JxJz) + 610(JnyJz + JszJy). (7-40)

The relations between the coefficients 51 and a; are given in
Eqs. (6-40)-(6-47) with ¢i replaced by ¢i.

Table (7-7).

103

The Differences Between Sumberg-Parker and Aliev-
Watson Sextic Centrifugal Distortion Coefficients

 

 

0‘

I A¢

N
H

I A¢

NDO‘VO‘
N

IA ¢3 =

1214A¢=
x y 4

I2I4A¢
y x 5

IZIAA¢=
y z 6

1412A¢a
y z 7

a

4 2
Ix I ZA¢8

As + AS,
I2 "2 2 cs s3.(VT—w)as"??? + 3x313?)

S<S 8

same as I214A¢
X Y

1
12

4 with xx replaced by yy throughout

replaced by -Css'°

+gzaB:y)

and with Css'

A + As

1
ss'(l-

zz xy+
12 m)(B BS

2 z t
(8' S
A + A , x
S S )(BXYBYY + B YByy)
, s s s

-1— <
33' A - A
s

12 2 Z 5

s<s'

same as I:I:A¢7 with yy replaced by xx throughout

and with css' replaced by -Css'°

same as 1:1:A¢6 with yy replaced by xx throughout

replaced by -C .

and with C' ,
ss 33

A8 + A S, 22
H)BXYB

-XS.S

+l z z c .[BXWBY’I - Bx?) - Bxywyy - Bxxn

4 s<s' as s s s s' s s
A
.l . ______L__ X? yy xx
+ 2 Z 2' tss'(l- )Bs '(Bs -BS )
sis s

 

104

7.2 Reordered Perturbation Treatment for the Calculation of the

 

Sextic Centrifugal Distortion Coefficients

 

13
In an appendix of their paper, Aliev and Watson show that

the calculation of H82)

ing the orders of expansion of the terms in the perturbation treat-

may be simplified considerably by reassign-

ment. The terms are ordered according to the degree in the vibra-
tional Operators, except that, as usual, H20 is taken as the zero—

order Hamiltonian H0, while H 2 is placed in H2. Furthermore,

0
the initial Hamiltonian (7-6)-(7-9) can be broken up into three parts
to suit the three different classes of term as expressed by Eqs.
(7-3)-(7-5).

~(V)
In case of H06

initial Hamiltonian are:

(harmonic), the relevant terms from the

H a H0 + AHl + AZH2 (7-41)
with

Ho . Hzocko‘o) (7-42)

H1 -'312(k3'1) (7-43)

H2 - H22(k4'2) (7-44)

Subjecting the Hamiltonian (7-41) to the first contact transforma-

tion, one obtains

H6 - HO (7-45)
. - <1) -

H1 H1 + i[S , H0] (7 46)

a; - H2 + 113(1), 31] --%[s(1),[s(1), 30]] (7-47)

105

. a (l) _.l (1) (1)
H3 i[S , H J le ,[s , H11]

2
--§ts(1),[s(1’.[s‘l), H0111 <7-48)
a; . --§ts(1),[s(l’, n21] --§[ (l),[s(l),[s(l), H111]
+-%Z[s(l),s(1),[s(l),[3(1), H0]]]]- <7-49)

Eq. (7-46) now defines the 8(1) operator such that

‘1) (7-50)

i[S ’ Ho] = ”H1 off-diagonal ' ‘H12 off-diagonal'
In this case we have only one type of function for the Operator 8(1),
namely S§:)(k3-l). The harmonic contribution can then be obtained
from H4’ and gives, as before,
' ~<v> 1 (1) (1)
'. Sun— -
H4 06 (harmonic) 2[S12 ,[S12 , H22]] . (7 51)

Next we consider the anharmonic part. The initial Hamiltonian

can be written as:

H = H0 + AHl + A3H3 (7-52)
with

no . 320(k0‘0) (7-53)

nl - 312(k3‘1) (7-54)

H3 - H30(k1'°) . (7-55)

We now subject this Hamiltonian to the first contact transformation,

giving

106

H6 = HO <7-56)
Hi = Hl + i[S(l), HO] (7-57)
Hé = H2 + i[S(l), H1] --i[s(l),[s(l), H01] (7-58)
35 - H3 + i[S(1), H2]- %[s(1),[s(l), H11]

--§[s‘1),ts‘1’.[s(l’. H0111 <7-59)
a; - i[S(l), 331- %[s(1),[s(l), 32]]

_ %[S<1>,[S<1>,[S<1>, H111]

fits‘l),ts(l’.[s(l),[s(l’, H0111] <7-60)

Hg 3 _ _[S(1> [3 3(1), H3]]_ %[S(1),[S(1) [3 3(1), 321]]

2_4[S<1),[_.,,<1>,[SmJSm’ H1111]

+Tzo[5(l)’[s(l) [3 3(1),[s(1),[s(1), Holllll (7-61)
Hg = - %{s(1),[s(l),[s(l), H311]

Ta[s(l) [S 3(1),[s(1),[s(1), H2111]

Tzo[s(1)'[5(1)'[5(1)’[S(1)’[S(1)' “111111

-770[S(1),[S(1),[S(1).[S(1).[S(1).[S(l), 80111111. (7-62)

Again 3(1) consists of the single contribution Si:$(k3-1). An

examination of the terms shows that the anharmonic contribution to

fi(v )

I
H06 arises with H 6 only, viz.,

fi<v 6) , _ (1) (1) <1) _
(anharmonic) 6[S12 ,[S1 ,[S12 , H3011]. (7 63)

107

For the calculation of H56) (Coriolis) the appropriate terms are
H a H + AH + AZH I (7-64)
0 l 2
with
no 320(k0‘0) (7-65)
H1 - 312(k3‘1) (7-66)
H2 a H21(k2‘1) + 302(k2'0). (7-67)

Subjecting this Hamiltonian to the first contact transformation, we
have a set of equations identical to (7-56)-(7-62). However now H2
is given by (7-67). Proceeding to the second contact transformation,

we can write

H3 8 H6 (7-68)
H1 8 Hi 3 0 . (7-69)
Hg - Hi + i[S(2), H 0] (7-70)
33 - HS + i[S(2),H //f 1 - H' (7-71)
HZ - H; + i[S(2), H2] - %[s(2),[s(2), 30]] (7-72)
H'S' - H; + i[S(2), H5] - %[S(2)’ [3(2), %=?] <7-73)

.. . . (2) . _ (2) (2) .
Ho H6 + i[s ,H4] %[s ,[s , H21]

- %ts‘2’,[s(2’.ts(2’, H0111 . <7-74)

Eq. (7-70) defines 8(2) such that:

(2) (1) (l) (1)

off—diag.

(7-75)

108
NOW’the third contact transformation can be written as:

H"' . H" (7-76)

0 o

33' a H; = 0 (7-77)
33' - a; (7-78)
33' - H; + i[S(3), HO] (7-79)
HZ' . H2 + i[S(3), H2] (7-80)
Hg' . a; + i[S(3), Hg] (7-81)
Hg' a 3; + i[S(3), Hg] --%[s(3),[s(3), H0]] . (7-82)

Eq. (7-79) defines 8(3) such that
(3) n . _ .
i[S ’ Ho] H3off-diag. H30ff-diag.

- -{1[s‘1’, H21- §ts‘l’,[s(l’ H11]

(1) S(1) (1)
%[s [s ,[s , H0111}off_diag. <7-83)

(1)

Since S is linear in the vibrational operators, the second and

third commutator terms vanish and

(3) . _ <1) -
[S ’ H0] [S ’ HZloff-diag. (7 84)
The required part of 8(3) is 8(3), and it is given by
S(3) , _ -1 ‘2 _
$13 2 qk{2 RkRz (ml wk + i[Rk, 1102]!»k }. (7 85)

k

Now Eq. (7-82) can be written as

109
". a _ <2) (1) (1) _ ;_ <2) (2)
H 2[s ,[s ,[s , H2111 2[S ,[s , H211

- §Is(2’ [33(2),[s(1), H1111 +-{S(2), [s s‘z’.[s‘1’,[s‘1),H01111

.; <2) (2) S<2) _ <3) <1)
- 6[8 .[s [s . H0111 [s ,[s ,H211
- 2[S(3),[S(1),[S(1), H1111 + %ts‘3),[3(1),[s(1),[s(1’,H01111

Using Eqs. (7-75) and (7-83), we have:

u. , _.; <2) (1) <1) (2) <2) (2)
H 2[s ,[s ,[s , H2111 + §ts .[s ,[s , H2011]

(3>,[S<3), H

+2[3 2011- (7-87)

An examination of the terms shows that the only contribution to

fi(V)

H06 (Coriolis) is

(3)

Hue . H<v 6)(CorioliS) a %[S(3) l3 ’

6 13 ’[S

H (7-88)

20]]

Evaluation of Eq. (7-88) yields the result in Table (7-2) without the
last two sums of H(v)(Coriolis). These were the sums that were
removed by a rotational contact transformation in the previous pro-
cedure. In the present procedure, these terms do not arise in the
first place. Altogether, the result from Eq. (7-88) is the same as
that finally obtained in Section 7.1, but clearly Eq. (7-88) is

considerably simpler to employ than Eq. (7-31) because it is un-

(l)
21

tained by a much more direct procedure.

necessary to evaluate S , and the Coriolis contribution is ob-

110

From the three different initial Hamiltonians (7-41)-(7-44),
(7-52)-(7-55), and (7-64)-(7-67), for the harmonic, anharmonic and
Coriolis contributions,respectively, we can see that it is per-
missible to combine the harmonic and anharmonic Hamiltonians without
any change in the result. We could also combine the anharmonic and
Coriolis parts and get the same result. However, one cannot combine
the harmonic and Coriolis Hamiltonians since the H1 term has been

defined differently for each case.

8. CENTRIFUGAL DISTORTION SUM RULES

8.1 Second-Order Centrifugal Distortion Sum Rules

 

As mentioned in Section 5.2, the second-order centrifugal

distortion constants which are the coefficients of the fourth-

T1
power angular momentum operators of the rotational Hamiltonian can

be constructed from Eq. (5-44). Considering the fact that the co-

efficients Ti

covering operations of the point group of the molecule under con-

have to be totally symmetric with respect to the

sideration, one obtains a total of nine distinct non-zero coeffi-
cients for orthorhombic molecules (such as the XYX-type molecule),
and a total of thirteen distinctnon-zemacoefficients for monoclinic
molecules (such as the XYZ-type molecule), and a total of twenty one

distinct non-zero coefficients for triclinic molecules.

31+
For planar asymmetric t0p molecules, Oka and Morino and

2

3
Dowling have proved that among T to 19, the relationships

1
(5-4S)-(5-48) exist. To obtain these relationships one starts with

the planarity condition:

axx + azy

22
s as , for all s, (8-1)

if the molecule lies in the xy plane. This condition follows
immediately from the definition of the inertial derivatives ags,

Eqs. (4-82)-(4-85). Multiplying (8-1) by a:x and dividing by

111

llZ

I:23’ yields Eq. (5-45), viz.,

(a? as xx/I2 A S) + (azxa yy/I: A S) a (a:xa:z/I: A S) (8-2)

01':

2 xx xx 2 xxa 22

(IX as as /I: A S) + (12 yaS Hazy/I IyAS ) = (Iza s/Ix 12 ZA 8) . (8-3)

Subsequent use of the definition (5-44) then leads to the relation

T1 = (Iz/Ix)215 - (Iy/IZA2T6 . (8-4)

Similarly multiplying (8-1) by azy and dividing by I323’ gives

Eq. (5-46), viz.,
yy xx 2 yy yy 2 a yy 22 2 _
(asas /IyAS) + (a8 a8 /IyAS) (a8 a8 llyAS) (8 5)
or:

(12a yy a:x/I2 12 YA S) + (Iy 2ayyaYY/I: A S) a (12a yy 3:2/12 I2 A s), (8-6)
from which

12 = (Iz/Iy)214 - (Ix/Iy)216 . (8-7)

Also multiplying (8-1) by azz and dividing by IiAs, gives Eq.

(5-47), viz.,

(a agz a22/12 A s) + (a as 2293712 A S) = (a 22 3:2/12 A S) <8—8)
or:
(12a 22 axx/I: I: A 8) a2 2yy/I: I2 A S) = (12aS 22 azz/I2A ) (8-9)
x s s + (5's a s z s ’

from which

113

_ 2 2
I3 - (Iy/Iz) I4 + (Ix/12) I5 . (8—10)
Moreover, using the fact that a:2 = aZZ = O, the definition (5-44)

gives Eq. (5—48), viz.,

= 0, (8-11)

= I = 0. (8—12)

Therefore, in planar molecules, there are five constraints among

I1 to T9. This means that there are only four independent I's

among Il to I9 for orthorhombic molecules.

For the monoclinic space groups, it is easy to prove that
in addition to the above five constraints, there are two more con-
straints among I

through I Again multiplying (8-1) by

10 13°
azy and dividing by InyAS yields Eq. (5-49), viz.,

xy xx xy yy = xy zz _
(as as /InyAs) + (a8 a8 /InyAS) (a8 a8 /InyAS), (8 l3)

0].”:

2 xy xx 3 ny yy 3 . a 2 xy 22 2 _
(Ixas as /InyAS) +(Iyasas /InyAS) (Izas as /InyIzAS) (8 14)
from which
1: - (I- n )2: + (I /1 )2r <8-15)
12 x z 10 y z 11 '

Therefore there are altogether seven constraints among I1 through

T13. This gives us six independent I's for monoclinic space

groups.
30
However, as discussed in Chapter 6, Watson has shown that

the non-totally symmetric quartic tenns, i.e., non-orthorhombic

114

terms,can be omitted from the reduced second-order Hamiltonian. This
can be done irrespective of the symmetry of the molecule. The non-
totally symmetric terms do not contribute to the energy in second
order, and they can be transformed completely into the terms of
higher degree in the Hamiltonian. Therefore the effects of these
terms in second or higher order are indistinguishable from the
effects of higher degree terms in the Hamiltonian. Thus we can con-
clude that in general in the planar case there are three sum rules
and two zero I's, hence four independent determinable coefficients

for the second-order centrifugal distortion coefficients T1.

8.2 Fourth-order Centrifugal Distortion Sum Rules

In the fitting of molecular rotational energies with a
sixth-degree rotational Hamiltonian, some of the sextic distortion
constants are frequently found to have large experimental un-
certainties. This can limit the usefulness of these constants as
sources of information on the cubic anharmonic potential of the
molecule. In such circumstances, it is advantageous to apply as a
constraint any theoretical relation existing between the sextic dis-
tortion constants. Such a constraint will normally reduce the un-
certainity of the constants without interfering with the deduction
of potential constants from them. While a relation of this type is
unlikely to exist for a completely arbitrary molecule, Watson39
has shown that there exists a general relation for planar molecules.
This relation is analogous to a planarity condition for the quartic
distortion constants, and like these is only strictly valid for the

equilibrium constants.

115

30
For the general case of the asymmetric rotator, Watson

has shown that the number of independent sextic coefficients or in-
dependent linear combinations of these is seven. Since the
planarity relation (8-1) introduces an additional constraint,
planarity would be expected to reduce the number of independent
sextic coefficients by one, to a total of six. Therefore, there
should exist four linearly independent relations among the fourth-
order centrifugal distortion constants. The construction and
calculation of these four sextic equations of constraint forms a
part of the original work of this dissertation. The manner in which
these results might be useful in the analysis of high-resolution
vibration-rotation data will also be discussed.

By inapection of the anharmonic portions of the theoretical
expressions @i, and by repeated use of the conditions of planarity
(8-1), it is found that the following four linear combinations of

the sextic constants ¢ are independent of the cubic anharmonic

potential constants: i
F1 =-%I:¢3 - I:I:¢6 - 1:1:¢9, (8-16)
F2 = IiIzog - I:I:¢6 - I:I:¢8 + I3I:Q7, (8-17)
F3 - I:¢1 + IS¢2 + %I:¢3 - I:I:¢7 — 1:1:28’ (8—18)
F4 - Iial + Igoz - 12¢3 - ZI:I:¢4 - 21:I:¢5 + 22:2:Ii210' (8—19)

These expressions depend thus only on the equilibrium geometry and
the harmonic force field parameters of the molecule, quantities

which are ordinarily known to considerably better precision than

116

either the calculated or the empirical values of the sextic co-
efficients. These expressions therefore appear potentially useful
as planarity-conditioned constraints on the ten sextic centrifugal
distortion coefficients obtained in the expansion of the Darling-
Dennison Hamiltonian.

The indexing of 21 corresponds to the sextic part of the

Hamiltonian as follows:

6 6 6

4
”6 21px +I¢2py + ¢3pz

X

2 4 4 2 2 4 2
+ ¢4(Pxpy + Pny) + ¢5(PyP + PxPy)

2 4 4 2 2 4 4 2 2 4 4 2
+ 26(Pypz + PzPy) + 457032?y + Psz) + 28(P2Px + PXPZ)

4
z

2 2 2 2 2

2 4 2 2
+ ¢9(PXP + Psz) + ¢10(PXPZPy + PszPx)° (8-20)

In ourformulation, the molecular framework is located in the

xy plane. If it is desired to have the molecule located in the
yz or zx plane, the appropriate cyclic permutations of the
cartesian coordinates can easily be carried out. It should be re-
called from Chapter 7 that the theoretical expressions for the

- 13
sextic coefficients ¢' given by Aliev and Watson (Table 7-4))

1.
differ in part from those given by Sumberg and Parker (Table (7-6)),
in the manner described in Chapter 7. An independent calculation
carried out by Georghiou“o and using a different pertubation pro-
cedure gives the same results as the calculation of Aliev and Watson.
Which of the two available sets of coefficients gives better
agreement with experiment must ultimately be decided by experiment.

At the present time, neither set has been tested extensively. Aliev

13
and Watson find reasonably good agreement with experiment for

117

I40

502, and Georghiou for H28. Using both the Sumberg-Parker co-

efficients and the Aliev-Watson coefficients, we find reasonably
good agreement for the ozone molecule with either set of coefficients.
These results will be given in the next chapter.

The computation of the functions Fi given by Eqs. (8—16)-
(8-19) is mostly straightforwardnthough somewhat tedious. Frequent
use is made of the planarity condition, Eq. (8-1), in simplifying and
consolidating the expressions as the calculation proceeds. Tables
(8-1) and (8-2), respectively, list the results obtained on the
basis of the Sumberg-Parker coefficients and on the basis of the
Aliev-Watson coefficients for the XYX (CZV) case. The more complicated
results for the general triatomic XYZ (Cs) case are listed in
Tables (8—3) and (8—4). In all these tables, ABS denotes BZY - B:x.
For ease of comparison, we have transcribed the Aliev-Watson based
functions into the Sumberg-Parker notation and dimensions. As re-
quired by the respective formalisms, we verified that the Aliev-
Watson based functions FAW yield the corresponding Sumberg-Parker

1

based functions Fip except for the Coriolis-type contributions

which are different in the two formalisms.-
In the cgs system of units, the ¢1 are in g-Scm_los4 and
can be changed to wavenumber units (cmfl) by multiplying by (HS/Zuc).

The dimensions of the functions F are g cmzsa, and the coeffi-

i
cients 3:6 are in 85mm 82.

118

Table (8—1). The Functions PEP, FiP, F§P, and Pip Defined by

Eqs. (8-16)-(8-19) for the XYX (sz) Molecule)

 

 

F? = -71;(Bizc31 - 332:23)2 a -Izc§3c§1[(l/Al) - (l/Azfl2
2:? " ' %(B}IQC31 ' 32:222322 + %(B}12y231 " B222,223)2
+ 3232;2{u3113izu3 - 2A1)/_(A3 - Al)] - [:233‘22203 - 2A2)/(A3 - A2)]}
2:? a ‘ 2(2?231 ' B220222320322231 ‘ 322,523) + 2(3)?)2
+-%B§y{[c31ABl(A3 - 2A1)/(A3 - A1>1
- [2:231:3203 - 2A2)/(A3- *2)“
F2? ' 7(2)::231 ‘ B22212223M22122231 ' B2227223) + ‘2'“:2231 " 2:22:23)2
_ (B§Y)Z --%(Z IauBia)2 --%(Z IaaBga)2
a a
--%IZZB§Y;31[A31<A1 + A3) + 33:20l - 3A3>1/<A3 - A1)
+ %1223?;23[A32(A2 + A3) + 333202 - 3A3)]/(A3 - A2)

xy
- n3 A3{[§31ABl/(A3 - A1)1 - [c23ABZ/(A3 - A2)]}

 

119

Table (8-2). The Functions Fgw, Fgw, Fgw, and Fiw Defined by

Eqs. (8-16)-(8-19) for the XYX (C2?) Molecule

 

 

Replace terms with Coriolis-resonant denominators (A3 - Al)

and (A3 - A2) of Table (8-1).
SP . .1 xy zz _ 22

in F2 by. + 4 B3 (B1 :31 B2 :23)
SP . lxy ..

in F3 by. + 4 B3 (Anlg31 A32c23)

in FSP by: --1 Bxy(AB c - AB t ) + Izszy(322: - 322: )
4 2 3 1 31 2 23 3 1 31 2 23

 

120

SP SP SP SP

Table (8-3). The Functions Fl , F2 , F3 , and F4 Defined by

Eqs. (8-16)-(8—l9) for the XYZ (CS) Molecule

 

 

FSP . f-%(Bc):z- %2(B:2)2 +1%(I /Ix I y): 2 A SA S,B:YB:YB:ZB:?
S

SS

2

a .1 2 _'l _‘1 xx 2 ._ yy 2
F2 + 4(BI)xx 4(BI)yy 4:(Bs ) + 42038 )

l

__ _ zz xy _ zz xy _
+ 6:<:v[§ss'/(As As')][Bs Bs'us' 223) + Bs'Bs (As 22$”>]
- l —(I zx/I I y): z A SSA J:YB:YB:z .ABS

SS'

- -{Iz /(Iy - I x) 212 2 A “A .BZYB:YB:ZI(Iy B S./Ix ) - (I xByy/I y)1

S 8'
SP - .1 -.1 xxByy .1 xy 2
F3 + 2<Bc)xx(Bc)yy 2:35 5 + 2:<Bs )
1 _ xy _
.+.gz 2 [6 ./(AS - A S.)][B::Y ABS (AS , - 2A S)+ Bs ABs .(AS - 2A .)1
[ SS 3
s<s .
.1 - XY XY yy xx _ Y?
+ 8[I/(Iy I x)]: i A SA $.33 BS.[(IyBS Bs,/Ix) (Ingst,/Iy)]

+“[Iz /(Iy - I x) 2]: z A SSA B:YB:YABS [(Iy BS KII )- (IXBZY/Iy)]
83'

.1 2 _ xx yy 22 2
F +(Bc)xx<Bz;)yy 2(Br)zz :33 BS + h2<B ) + 203:):y

28

_ xy 2 _.1 an an 2 _ - KY -
2(Bs ) 22(21 Bs ) z 2' ;SS.BS,ABSAs./(AS. AS)
3 S a sfis
lz

_ xvz - ‘ _
212 2:22: Cs'sBs B 8,(As, 3As)/(AS, As)

l 22

_ _. XY XY -
61 :(g' ;SS.(Bs ABS, + BS,ABS)(AS +As,)/(AS As.)

+-—[1/(Iy - I x)1): z A SAs ,B:YB:Y[(IY /I x)(23223:x - BYYBYY)
SS

yy yy _ xx xx
(Ix/Ty)(ZBS BS, Bs 38.)]
i[I z/(Iy I x) 2]: :8 A AS Bs B:YABs [(I xBYY/Iy )- (IyBs./Ix)]

%[Iz/Ix1y]z 2 A 8A S.B:YB:YB? BYY.
8.23

 

Table

121

(8—4). The Functions Fiw, Fgw, Fgw, and FZW Defined by

Eqs. (8-16)-(8-l9) for the XYZ (CS) Molecule

 

 

+~ch;)2 visa")2 +m1(1 /Ix I y): 2 AS AS BS xxB:YBz 823:?
~ 4 zz 4 s
S SS
1 2 1 2_I:m2I w2
+ 4(Ba)xx 4(Be)yy 42033 > + 4:(BS )

.1 KY 22 22 XY __. xy xy:

8
88'

-—[I2 /(Iy - I x)2 12 2 AS As B:YB:YB:Z[(Iy B s,/Ix ) - (Ix BYY/Iy )]

s s'
.1 _.1 xx yy .1 xy 2 1 xy‘
+ 2(BC)xx(BC)yy 2:33 Bs + 2:(Bs ) +'Z:<:.Css'(Bs ABs'

B:YAB s) + 8[1/(Iy - I x)1: z A $3A B:YB:Y[(Iy BYYB? ./Ix )
SS'

(IszxBZY/Iy)] +— %[Iz /(Iy — I x) 2]: z A HA BZYB:YABS [(Iy Bs m/I )
SS

YY
(IXBS.{IY)]

2 2 2
«Bowmanyy - -%(Bc)zz - 23:33? + inn?) + mac)xy
S S

_ xy 2 _.l on an 2 _ zz xy zz
2(Bs ) 22(2 I BS ) I 2 Z c 'Bs'Bs

s s a s¥s' S
m—i- 2;,38 (B’g‘YABS , - B:YAB 3) +—[Iz /Ix I y]: z A $3A B:YB:YB:" BYY
s' s s'
+8[1/(Iy - I x)JZ z A “A B:YB:Y[(Iy /I x)(2B:"B:’.‘ - BYYBYY)
S S'

Y? Y? xx
- (Ix/1y)(zBs Bs' - B:XBS.AJ

+Z[Iz /(Iy — I x) 212 z A “A B:YB:YABs [(Ix BYY/Iy ) - (Iy B s,/Ix )]
SS,

 

122

8.3 Constrained Empirical Constants

 

In order to utilize the sextic planarity relations in the

analysis of vibration-rotation data, the theoretical constants $1

must be related to the corresponding empirical constant 5 (Again,

1.
the tilde symbol is used consistently in this chapter to denote

empirical constants.) The relations existing between the theoretical

13
and the experimental constants have been studied by Watson. This

work was discussed in Chapter 6, and we shall use the results in the
25
form given by Yallabandi and Parker, Eqs. (24-26).
In general, the planarity relations among the empirical co-

efficients are obtained when the oi are replaced in Eqs. (8-16)-

(8-19) by the corresponding 5 with the aid of the set of Eqs. (26)

i

. 25
of Yallabandi and Parker. After this raplacement, however, the
resulting equations are more complicated and contain four S-para-
meters viz., $111, $113, 8131 and S311. A more promising approach
appears to be to let the planarity relations define a reduction of
the sextic Hamiltonian by using an empirical Hamiltonian which in-

cludes all ten 5 but with these planarity constrained by our

1’
Eqs. (8-21)-(8-24) below. This reduction is specified by taking

S - $131 - S - O which amounts to not applying the fourth-

113 311
order part of the similarity transformation of the Hamiltonian. The

advantage of this approach is that a reduction is obtained which has
a natural and straightforward relation to the theoretical formulation,
and that this relation can be explored without the necessity of

, and S which

determining non-zero numerical values for 8113, 311

S131
tend to be very poorly determined in practice.

123

Another advantage is the immediate one that the equations
relating the theoretical and empirical coefficients are much simplified
113 ‘ S131 ‘ S311
(8-16)-(8-19) that

by taking S = 0 in which case one finds from Eqs.

3 6~ 2 4~ 2 4 ~ 2 ~ ;
Fl 21253 - IyIz56 - 1x1259 - 4uslllI: [1: (T3 -T4) + Ix(T5 - I3)]
ZS 2
- 4K3 I4 [- I :(C- B) + I :(A-C)], (8-21)
1112
2 4~ 2 4~ 4 2~ 4 2~ 2 2 2
F2 IXIZ¢9 - IyIz<I>6 - IXIZ¢8 + IyIz¢7 - “551111z [IyIz(T3- T4)
2 2 ~ 4~ ~ 4 ; ~
+ Isz(T3 - T 5) + IX(Tl - T5) + Iy<l2 - T4)]
2 2 ~ ~ 2 2 2 ~ -
+ 4“ 2811112 [Ii (I: + Iz)(A-C) + Iy(Iy + Iz)(C-B)], (8-22)
6~ 6~ l 6~ 4 2~ 4 2~ 4 ~ ~
. F3 . 1x5l + Iy52 + 2I253 - IyIz¢7 - Isz58 - 4HSlllI:&Iy(T4 - T2)
4 ~ ~ 2 4 4 ~ ~
+ Ix(Tl - T5)]- 4hSlllIz [Iy(C- B)- Ix(A-C)], (8-23)

F - 165 + 165 - I65- 212x134?» - 2121255 + 2121212510
4 x l y 2 z 3 xyz
II2 2 2 2 ~ ~
-8H51111x1y[1y(T2-T6) + Ix(T6 - T1)]
232
-8K lllliy I2 [- -I: (B-A) + I :(B-A)]. (8-24)

Here A, H, and C are the experimentally determined effective

rotational constants and the T1 are the experimentally determined

coefficients of the quartic part of the Hamiltonian,

H4 = TIPx + TzPy + T3Pz + T4(Psz + PzPy)
~ 2 2 PP2 2 22
+ T5(Psz+ Px Pz) + T 6(P :Py + P :P i). (8-25)

124

The manner in which the reduction of H4 is effected determines the
value of the parameter. 8 When S - S = S = 0, then

111' 113 1311+1 311

the T1 are related to the corresponding Ti by

T1 . Ii + 0(4), 1 = 1,2,3, (8-26)
I4 - I4 + 2u(é-fi)slll + 0(4), (8-27)
T5 . I5 + 2“(AP&)Slll + 0(4), (8-28)
T6 = I6 + 2n(§-A)slll + 0(4), (8—29)

where 0(4) are contributions of order four which also depend on
8111. In practice one is forced to neglect these contributions at
the present time since the fourthporder contributions to the

theoretical coefficients T1 are very complicated and have not yet

been well developed. Consequently, the T need to be approximated

i

by the theoretical expressions for the corresponding equilibrium

quartic distortion constants T After a quartic reduction is

i.
specified, for example T6 - 0, one can proceed to obtain a value

for S111 from one or more of the Eqs. (8-27)-(8-29). Use of more

than one equation for the determination of $111 amounts to a test

for consistency, with the more reliable values of S 1 probably

11
those for which the difference in the rotational constants involved
is of the same order of magnitude as the rotational constants them-
selves. Whereas, in principle, it is possible to determine the
numerical value of 8111 strictly from theory, in practice both

experimental and theoretical input are required because the

theoretical development remains incomplete.

125

It appears very attractive to attempt to take S = 0, as

111

in that case the numerical values of all empirical constants would
correspond to the numerical evaluation of the associated theoretical

expressions, i.e.,
5 a 5 . (8-30)

For such a situation to apply it is, however, necessary to planarity-

constrain the quartic constants T1 in the data fit. These con-

straints, however, are known at the present time only for the
335

’

3
corresponding equilibrium coefficients 1 , and thus an error

i

of order four would be incurred in the data fit which may be accept-
13 42

able in special cases , but almost certainly not in general.

' Watson39 has given a general sextic planarity relation in-
volving only empirical rotational and distortion constants and valid
for any reduction of the quartic and sextic portions of the
Hamiltonian. The relation is, however, strictly valid only for
the equilibrium values of these constants. With this proviso, we
have confirmed that Watson's planarity relation is equivalent to

the first planarity relation Eq. (8-16). This equivalence holds for

both the XYX case and the XYZ case.

8.4 Cylindrical Tensor Form Hamiltonian

 

Cylindrical tensor forms of the Hamiltonian have been widely
and successfully used for data analysis. Starting with such a
Hamiltonian in the form cited by Yallabandi and Parkerzs, Eqs. (5-2),
(5-19)-(5-21), it is possible to develop the set of equations below.

These are again based on a full ten-term sextic Hamiltonian with

126

planarity contrained coefficients H Use of Eqs. (S-22)-(5-43)

1'

leads to the following set of equations:

491 + <92 3 2H1 + H6, (8-31)
<91 ' <32 " 2H5 + ZHIO’ (8-32)
53 - H1 + H2 + H3 + H4, (8-33)
5~ ~
~ ~ - 2 2 - -
<54 - <55 -H5 + 31110 + 1658111136 - 8K Slum-A), (8-35)
5 + 5 a 3E + zfi + a + 4us (fi + 25 ) - 45232 (26-A-fi) (8-36)
6 9 1 2 3 111 4 5 111 ’
- - - r - 2 2 - -
56 - 59 -H5 - 2117 - 2H8 + 4158111032 + 293) + 4H Sula-A), (8-37)
57 + 58 . 3H1 + H2 +-%a6 + H9 + 41131110)4 - 295)
+ 4n232 (26-A-fi) (8-38)
111 ' .
¢ - 5 . -25 - zfi + 4mg (5 - 5 > - auzsz (é-A) <8-39)
7 8 5 7 111 2 6 111 ’
~ ~ 3.. ~
510 3111 + H2 - 5H6 - 3H9. (8-40)

If in Eqs. (8-16)-(8-l9) the ¢i are replaced in accordance with
Eqs. (8-31)-(8-40) above, then the cylindrical tensor version of
the planarity relations for the empirical coefficients, Eqs. (8-21)-
(8-24), is obtained. If in these one takes 8111 a O and removes
the tilde symbols from the Hi, the cylindrical tensor version of
the planarity relations for the theoretical coefficients, Eqs.
(8-16)-(8-l9) is obtained.

An experimental test of the planarity relations requires that

the data fit be carried out in a particular way, viz., with the co-

efficients constrained as described. No such data fits have been

127

carried out to date, but it may be hoped that they will be attempted

at some future time.

9. THE CENTRIFUGAL DISTORTION
COEFFICIENTS 0F OZONE

A calculation of the centrifugal distortion constants of
the ozone molecule was carried out and will be described in this
chapter. For the sextic constants, both the Sumberg-Parker and the
Aliev-watson expressions were used. The calculated distortion con-
stants were compared to experimental results in the literature with
the aid of the reduced-Hamiltonian approach described in Chapter 6.
The agreement between theory and experiment was found to be gen-

erally quite satisfactory.

9.1 The Fourth-Order Centrifugal Distortion Coefficients for an

XYX-Type Molecule

 

The complete expressions for the ten sextic fourth-order
centrifugal distortion coefficients for an XYZ-type molecule as
calculated by Aliev-Watsonl3 and Sumberg-Parker7 are given in
Tables (7-4) and (7-6) respectively. To specialize to the XYX—type
molecule, we let m2 8 m3 - m, M1 - M, a 3 b, e - 0. The detailed
expressions for b:8 - (age/12,4), 3:8 - (age/As), (A:S,)' and
Css' are obtained from Section 4.4 in a straightforward manner,
and the expressions for the ten sextic centrifugal distortion co—
efficients for an XYX-type molecule can then be written out and

are given in Table (9-1).

128

129

Table (9-1). The Fourth-order Centrifugal Distortion Coefficients

 

for an XYX-type Mblecule

6 xx 3 xx 3 xx 2 xx
Ix¢l "%N[(b1 ) k111 + (b2 ) k222 + (b1 ) (b2 )k112

+ (bfx)(b§x)2k122] +-%(B:xsin y + ngcos y)2

2 - yy 3 yy 3 yy 2 yy
Iy‘pz %N[$b1 ) k111 + (b2 ) k222 + (b1 ) (b2 )k112

2 2
+ (biy)(b§y) k122] +-%(B{ycos y - Egysin y)

1:53 ’ %N[(b:z)3k111 + (b22)3k222 + (biz)2(bgz)k112
+ <b§z><b§z>2k1221 + %[B:zc23 + 33253112 - $05th + (3:2)21
IiI3¢4 a %N{3(b:x)(b{y)2klll + 3(ng)(b§y)2k222 + 4(biy)(b§y)2kl33
+ 4<b§5><b§Y>25233 + 1203:") (5312’) 532') + (a?) (591231512
+ [zcbiyxa’z‘xxag'b + (a’l‘x)<a’z’y)21k122}
+-%E(B{ysin y + Bgycos y)2 +-%(Iy/Iz)(B§y)2
+-%[B§x3{ycoszy + ngBgysinzy - sin ycos y(B:xB§y
+ B:xB{Y)] --%(I:/Iz)1/2B§ysin ycos YC§I --%;)
+‘I2 IzzB§y[(*1 + A3)‘13B{y/“1 ‘ *3)
+ (*2 + A3’C23Bgy/(A2 ‘ *3)]
-5%B§y(1ny/Iz)1/2(coszy/Al + sinzylxz)
I:I:¢5 "%N+3<biy)(b:x>2k111 + 3b2y(b§x)2k222 + 4(bixfibgyfikus

xx XYZ yy xx XX YY XXZ
+ 4(b2 )(b3 ) k233 + [2(b1 )(b1 )(b2 ) + (b2 )(bl ) lkllz

130

Table (9-1) (continued)

 

+ [20: ’°‘)<byy)(b’2°‘ ) + <5”) <b’°‘)21k122}

§__ xx _ xx . 2 §_ xy 2
+ 16(31 cos Y B2 Sln y) + 4(Ix/Iz)(B3 )

XX

+ %{(B:x)(B{y)sin2y + BxxByycoszy + sin ycos y(B{YBZ

2 2

+ ByyB1 )]- %(Ix 2/I z)l/szysin ycos 7(1/11 - 1/12)

_ 3 xy 1/2 2 2
533 (Iny/Iz) (sin y/Al + cos y/lz)

.l_ 22 xy xx
12I B3 [(*1 + A951331 /(*1 A3)

+ (*2 + A3)’;233}2{x/02 ‘ *3)]

12145 ---%N{3(b{y)(bzz)2k + 3(byy)(bzz)2k2

y z 6 11122

yy zza zz yy 22 2 yy 22 22
+ [2b1 b1a2+b2(b1) ]k112 + [2b2 b1 b2

22 2

YY yy zz yy 22 yy zz 2
+bl (b2 )]1<122}+8 —(B1 131 +132 32 )- E-[BI 131 :13
yy zz 2 yy 22 yy 22
+ B2 B2 ‘23 + C2341303132 + B2 B1 )1

L22 _zz 2_1xy _ 22
+ 16(Blcos Y B sin y) 633 [(A3 211)C13Bl /(A1

2

zz
+ (13 - 212)52332 /(12 - 13)]

‘ zz yy 2 zz yy 2
I 12° %N{3(b1 )(b1 ) k111 + 3(b2 )(b2 ) k222

+ [Zbyybyybzz + (b:Z)(b{y)2]k + [Zbiybgybgz

2 1 112

YYZ _.l KY 2 .§_ yy 2 yy 2
+ blz (b2 ) 1k122} 4(33 ) + 16[(Bl ) + (32 ) 1

-1 yy22 yyZ yyw
4[(31 ) 3+ (32 > ‘23 + 2B1 B2 423413]

13)

131

Table (9-1) (continued)

 

yszz + Byszz)]

-§ yy zz 2 yy 22 ' 2 _
+ 8[B B B sin Y sin ycos y(B1 2 2 1

l 1 cos y + 32 2

_ 1 xy _ yy _ _ yv _
3B3 [(A3 2X1)Bl ‘13/(A1 *3) + (*3 2A2’32 ‘23/(A2 ‘3’]

IiIZ¢8 "%N{3b:z(b1x)2k111 + 3b22(b§x)2k222 + [Zbixngbiz

+ b22<bf>21km + when? +b32<b2>21k122

- %<B§y>2 + fi-t 031‘“)2 + (3’2“)21 - %[<B’f‘>2c§3 + (312°52c33

+ Zfong;23;13] +-%[B:x3:zsin2Y + ngBgzcoszY

+ sin Ycos “3:ngz + B§x3:2)1

+'%B§y[(*3 ' 2A1>‘13B)1‘/(A1 ‘ A3) + (*3 ‘ 2A2)‘23B}2{x/“2 ' ‘3’]
12I2¢9 ' %N{3b:x(biz)2k111 + 3b:x(b:z)2k222 + [Zbixbizbgz '

xx zz 2 x zz 22 xx 22 2
+ b2 (b1 ) 1k112 + [szxbl b2 + b1 (b2 ) 1k122}

3 22 xx zz 1 x zz 2 x 22 2
+ 8(fo31 + B2 B2 ) ’ 2[le31 ‘13 + B2sz ‘23

xx 22 xx 22 3 . 2
+ c23;13(B1 B2 + B2 B1 )] + 16(3131n Y + Bzcos Y)

.;.xy _ zz _
+ 633 [(A3 221M133l /(A1 A3)
22
+ (*3 ’ 2A2)‘2332 /(*2 ’ A3)]

2 2 2 - yy 22 yy 22 x yy zz
InyIzolo %N[6b:xhl b1 klll + 6b§Xb2 b2 k222 + 2(blxb1 b2

22 yy 22 yy xx yy 22 yy 22 xx
+ bixbl b2 + b1 b1 b2 )k112 + 2(bfxb2 b2 + b1 b2 b2

22 yy xx zz xy 2 22 xy 2
+ b1 b2 b2 )k122 + 4b1 (b3 ) k133 + 4132 (b3 ) k233]

132

Table (9-1) (continued)

 

.3. W W vixy2_ yy2
+ 8(31x31 + 32x32 ) + 4(33 ) ”[331”;13

yy 2 yy x yy
+ 32x32 ‘23 + (31x32 + B2x31 )‘23‘131

BXX

%[BlzB l ZZB ZZ

cosZY + BxxB: zsinzY - sin Ycos Y(Bl B2

Bzz Bxx 22 xx 2 2
+3132 )] + 8[Bl Bl cos Y + 32 B22 sin Y

- sin ycos Y(Bl zzB 332+ Bzz Bxx)] + %[B:zBiysin2Y
+ 32 ngycoszY + sin Ycos Y(ByyB:z + Byszz)]
--%(Ixx3:x + Iyysiy + IZZB Biz) 2- %(Ixx3xx+ +1373};y
+ Izngz)2_233xy(Iz)1/2{[(1ny)1/2 + Izsin Ycos Y]/)\l
+ [(ley)l/2 — Izsin Ycos Y]/12} - %B§ycl3(8:x
’ B{KB/(*3 ‘1) 2Bgy‘23( 32x B§YM3HA3 k2)
. - :1 Izszyt(A1-3A3)B:zcl3/(Al A3)

22
+ (A2 - 313)§2332 /(>\2 - 13)]

 

133

9.2 Fundamental Mblecular Constants of Ozone

The calculation of the theOretical centrifugal distortion
constants requires the following input data: the equilibrium geometry
of the molecule, the harmonic force field (which determines the
normal frequencies), and in the case of the sextic constants, the
cubic anharmonic portion of the molecular force field.

At equilibrium, ozone 1603 is known to have the geometry of
an isosceles triangle. The equilibrium apex angle Zoe - ll6°47(2)'
and the bond lengths of the equal-length sides of the isosceles
triangle, re - l.27l7(2)A, are well established through microwave
spectroscopyka’uu, with the numbers in parenthesis representing the

quoted uncertainties in units of the last decimal place given. The

above values allow determination of the equilibrium moments of
16

inertia for 03 as
2 2 2 2 a -40 2 _
Ix ‘Smrecos ae 7.868(10) x 10 g cm , (9 l)
I = 2mrzsin2a = 6.233(4) X 10.39 g cmz, (9-2)
y e e
-39 2
I2 Ix + Iy 7.020(4) X 10 g cm . (9-3)

18

For 03, the above values scale by a factor of (18m/1

6111) = 1.12531.
The most reliable harmonic frequencies, harmonic force field,

and cubic potential constants available for 1603 and 1803 at the

as
present time are those determined by Barbe, Secroun, and Jouve,
obtained through an analysis of thirty-three band center positions,

and these values were used for the present work. For the harmonic

frequencies of 16O3, Barbe et a1. give

134

1

m1 = 1134.9(2) cm" , (9-4)
-1 _
”2 = 716.0(2) cm , -(9-5)
-1
w3 = 1089.2(2) cm . (9-6)
These are related to the corresponding As by
mg a Ai/Z/ch, s - 1,2,3. (9-7)

The fourth parameter needed for the complete specification of the

46
harmonic field was taken to be :31 as determined by Barbe et a1,
viz.,

:31 = 0.604(1). ‘ (9-3)
This value of :31 is more precise than, and consistent with pre—

44
vious determinations by Tanaka and Morino and also by Clough and
.+7

Kneizys who found

:31 - 0.60(l). (9-9)
Since

2 2

:23 + g31 = 1, (9-10)
we obtain

:23 - -c32 = -0.797(1). (9-11)

:31 = -§13 a +0.604(1). (9-12)

With these values, use of Eqs. (4-144)-(4-148) now allows calculation
of the angle Y associated with the normal coordinate transformation

and one obtains

135

sin y = 0.836(2), (9-13)
cos y = 0.549(2), (9-14)
Y = 56°43(13)'. (9-15)

The angle 7 as well as the Coriolis constants have the same

numerical values for 16O3 and 1803 under the assumption of

1+8
negligible nuclear size effects due to isotopic substitution.

The signs of the Coriolis coupling constants :23 and :31 as
given by (9-9) and (9-10) are consistent with the arbitrary choice
of placing Y in the first quadrant. Different choices of
quadrant for y correspond to other mutually consistent choices of
phase for the two totally symmetric modes v1 and v2.

. For the cubic potential constants, the set given by Barbe,

L. s .
et al. was adopted. However, care must be exercised because the

signs of the cubic potential constants depend, in general, on the
choices of phase for the normal coordinates. In order to obtain
signs consistent with those of our y, :23, and :31, the cubic

potential constants were recalculated from the rotation-vibration

bk
a
interaction constants as measured by Tanaka and Morino , and

1+5
those of Barbe et al. The theoretical expressions for the a:

are well-established and are reproduced in the paper by Tanaka and

an
Morino . As a sample calculation, we have:

 

 

 

XX XX
1 2 3/4 111 3/4 112 2 1/2
2I A A 4wc I A
x 1 2 x 1
Y? yy
(1y 261.2. {381 k + 32—. k } + 3“2c°92 (9_17)
l 2 3/4 111 3/4 112 2 1/2
21 A A 4ncI A
y l 2 y l

with

and

Thus we can write the following relations for

a: = (1.2451x 10'3)k

oi . (3.6673 x 10'5)k

Z
“1'

136

 

 

2

2 1/2 C13113

22
2 >‘3/4 k4'111A3/4 k112 2 1/2..-
21 41rchA1 ch A
z 1 2 z 1
F13 = A3/(A1 - A3) = 11.66977
K a -39
‘5;- 5.5987 x 10 g - cm
aix . 21: sin y = 4.690 x 10"20 g7 - cm
{y a 21: cos y = 8.669 x 10-20 g5 - cm
22 a _ B . ‘20 5 _
a1 212 :23 13.355 x 10 g
six . 21: cos Y= 3. 080 x 10 20 g5 - cm
aZy - -21;sin y = -13.200 x 10.20 g11 - cm
22 a 8 a _ ‘20 5 -
a2 212:13 10.121 x 10 g cm
A: a 2.1378 x 1014 radians/sec
A: a 1.3487 x 1014 radians/sec
A: a 2 0517 x 1014 radians/sec

111

111

+ (0.5439 x 10‘3)k

- (3.7146 x 10'5)k

112

112

X

i, and oz:

+ 4.67705 x 10'

(9-18)

(9-19)

(9-20)
(9-21)
(9-22)
(9-23)
(9-24)
(9-25)

(9-26)

(9-27)
(9-28)

(9-29)

(9-30)

+ 3.21395 x 10"4

(9-31)

137

z a -5 _ -5 -3
61 (4.4539 x 10 )klll (2.2453 x 10 )kll2 + 5.61252 x 10
(9-32)
44

Using Tanaka and Morino data, we have
a: . -3.037 x 10'3 cm‘1 (9-33)
ai = 2.540 x 10"3 cm'1 (9-34)
0: . 2.317 x 10'3 cm’1 (9-35)

while Barbe et al. data determined
a? a -2.981 x 10‘3 cm’1 (9-36)
oi - 2.554 x 10..3 cm.1 (9-37)
0: = 2.319 x 10‘3 cm'1 (9-38)

Thus these values are reasonably consistent. Using the newer data

45
by Barbe et al. , one obtains

(1.2451)klll + (0.5439)k112 = -43.79 (9-39)
(3.6673)1<111 + (-3.7146)k112 = -287.5 (9-40)
(4.4539)k111 + (-2.2453)k112 = -285.3 (9-41)
Solving for klll and R112 in the three possible ways gives
-1
klll -48.2, - 48.6, -49.8 cm , (9-42)
—1
k112 - +29.8, +30.7, +28.2 cm . (9-43)

The remaining cubic potential constants can be determined in a

similar manner. Using the average values of these constants and

 

138

attaching the appropriate uncertainties to these quantities, we

obtain
k = -48 1(7) cm"l (9-44)
111 ’
-1

4222 +19.2(6) cm (9-45)
-1

k112 = +29.7(10) cm (9-46)
-1

klzz - -25.5(30) cm (9-47)
-1

k133 = -225.8(30) cm (9-48)

k a +59.3(10) cm'l (9-49)

233

This complete set of cubic potential constants is the same as that

45
determined by Barbe et a1 , except that positive sign is

obtained for k k112’ and k

222’ 233'

For our calculations, the uncertainties limiting the pre-
cision of the final results are principally determined by those of
sin y and cos y above, and by those of the cubic potential con-

stants. When the cubic potential constants as well as the harmonic

frequencies of 1603 are scaled by factors of (16m/18m)3/4
(l6m/l8m), respectively, they are in satisfactory agreement with the

and

corresponding experimental values for 1803 as determined by Barbe

45 “9
et al. A recent study by Hennig and Strey also confirms the

1+5
anharmonic force field of Barbe et a1.

9.3 Calculated Centrifugal Distortion Constants
One is now ready to calculate the sextic centrifugal dis-

tortion coefficients 91. The coefficients ¢i will now be

139

separated into three parts: a harmonic part, an anharmonic part,

and a Coriolis part. The expressions for the 0 , as calculated by

i
Aliev-Watson and Sumberg-Parker for XYXEtype molecules, differ only
in part of the Coriolis portion. The calculated values for ozone of

the harmonic, anharmonic, Coriolis, and total 9 of Aliev-watson

i
and Sumberg-Parker are given in Table (9-2). The.Coriolis entries
list only the part of the Coriolis contribution which differs in the
two formulations. The matching parts have been included in the
harmonic contributions.

Numerically, the distortion constants range over more than
four orders of magnitude, with the harmonic and anharmonic contribu-
tions of comparable magnitudes. Generally, the harmonic contribu-
tions are better determined than the anharmonic contributions. The
quoted undertainties were arrived at by elementary standard methods
for propagating errors on the basis of the uncertainties specified
for the input data.

Table (9-3) lists the values calculated for the six non-

vanishing second-order centrifugal distortion constants Since

T1.
1

the smaller T are of the order of 10_6cm- and the largest 9

i i
is of the order of 10-8cm-l, the numerical separation by order of

approximation of the perturbation calculation seems adequate.

9.4 Reduced Hamiltonian

 

In order to effect a comparison of the calculated distor-

tion constants with the measured ground state constants of 160
50
recently determined by Maki from the

45

et al. from the v3

3.
v1 + v3 band and by Barbe

band, it is necessary to reconcile the

140

 

 

 

 

Hence x Aoovmm.m- Aamvmm.mn Amvm4.ou Amavao.fl+ Ammeom.~u Aomvm6.~- e
NHIoH x Aaquvus.o+ Ameavwm.o~- onom.m+ Am~v4~.-- Aaevem.a+ Aamv~a.o+ me
cause x Ammvmo.mu Amovos.ms- Amvam.~- Aoavma.eu Aamvsw.mu Aamvfle.mn we
NHIOH x Aofivqo.o+ Aoavmq.o+ AmvoMH.on Aqvm-.o+ AevoH.H+ Awflvmm.au he
NH1¢H x Amuvmo.ou Amflvmm.o+ Amvmsoo.os Aqvemm.o+ Amvmm.fl+ Amvam.au so
os-oH x Assovmo.mu AmeVOn.~H- o Aam~v44.q- AmmVoH.ou Amamvma.~- we
Hauoa x Ammvea.~- AoNVs~.ou o Asvom.~+ Aflvmo.os Am~v4~.~- 49
asses x Aqmv~6.on Aemv~o.ou o o AoHva.e+ Amevmfl.na me
«Huofl x Amavmm.o+ AmHVNm.o+ c o Amvc~.~+ Aaavuo.an Ne
muss x Anavmq.m+ Amsvmq.m+ o o Amvom.m+ Amvao.ou He
oomaowmmmou acoaowmmooo mHHoHuoo maaOfiuoo coausnauucoo .aoausnauucoo .mmooo
sauce 3< H6063 mm 24 mm 6466546: aficoaumna<
Hugo as mood
mo mucmfiuemoooo sensuoumfin uauxmm A3<v canons: >64H< can Ammv umxummumumnasm emuaaauauo .Amumv «Hams

141

Table (9-3). Calculated Second-Order Distortion Coefficients T

 

 

of 1603in cm.1 i
11 - -8.139(48) x 10’4 14 - -l.605(7) x 10"6
12 - -2.313(13) x 10'6 15 . +3.499(75) x 10'6
T3 = -1.221(4) x 10'6 r: - +0.23(13) x 10’6
To obtain the corresponding values for 1803, all entries should be

multiplied by (mm/18m)2 - 0.78969.

 

Hamiltonians used by these authors with the one used by us. As dis-
cussed in Chapter 6, one must take into account Watson's theory which
requires the Hamiltonian used to fit the data to be in a reduced form
such that no redundant coefficients, or redundant combinations of
coefficients, occur. The Hamiltonians used by Maki and by Barbe et al.
are identical in structure and very similar in notation. Inspection
of the Hamiltonian shows that the symmetric-top or H-formzs’51 is
used, and that the reduction is carried out by choosing one of the six
quartic coefficients and three of the ten sextic coefficients to be
equal to zero. Since the molecule is taken to be in the zx »p1ane,
one cyclic permutation is required to bring the molecule into the

xy plane. Carrying out the permutation and introducing the notation

25
of Yallabandi and Parker for the coefficients, it is determined that

H - Apz +.sz + 092, (9-50)
2 x y z
.444 -1422 ~*4 ~*22 2
H4 DIP + DZP Px + D3Px + D4? (Py — Pz)
~* 2 2 2 2 2 2
+ D5[Px(Py - Pz) + (Py - Pz)Px]’ (9-51)

142

and
H6 = filpe + H2P4P: + H3P2P: + fi4P: + H5P4(P: - Pi)
+ H7P2[P:(P: - Pi) + (P; - P:)P:1
+ H8[P:(P: - 9:) + (P: - P:)P:]. (9.52)

As we have mentioned in Chapter 6, we use the tilde with the ex-
perimentally determined constants of the Hamiltonian, whereas all
constants calculated on the basis of theory appear without the tilde.

The particular reduction used is specified by
-* - -
D6 - H - H - H = 0. (9-53)

The Hamiltonian, Eqs. (9-50)-(9-52), must be arranged into the form
specified by Eqs. (5-2)-(5-5) in order to develop the relations be-
tween the two sets of coefficients. This is done by expanding

(9-51) and (9-52), retaining all Hermitian groupings of operators,
and comparing coefficients. The listing of angular momentum operator
identities given by Kneizys, Freedman and Clough51 in their Table II
is quite helpful in carrying out this calculation expeditiously. We

find that

an

. A + n4<-sfil), <9—s4)

we

4 ~ - -
- B + u (-8H1 - 4112 + 4H5), (9-55)

- C + Mane}?l + 4fi

O:

2 - 4H5). (9-56)

Also:

143

~ ~* ~* ~*
Tl - + 02 + D3, (9-57)
'1' ~* + J:
2 D1 D4’ (9-58)
~ ~* ~*
T3 01 - 04, (9-59)
- 2* 2~
$4 01 - 4“ H1, (9-60)
T ~* ~* 1 ~* ~* 2 .” 2~ ~
5 (D1 - 05) + 2(1)2 - 04) + u (-4Hl - H2 + 2H5), (9-61)
- -* ~* 1 -* ~* 2 ~ - .
T6 - (D1 + 05) + 2(D2 + D4) +.u (+8Hl + 232 - 2H5). (9-62)
Furthermore,
01 - H1 + H2 + H3 + H4, (9-63)
02 - H1 + H5, (9-64)
03 - H1 - H5, (9-65)
.. 3.. ~ ~ ~
<14 - 51.11 + #2 + “5 + H7, (9-66)
5 - éfi + Q + ~ +31% + i + H (9-67)
5 2 1 2 2 3 2 5 7 8’
- 3~ 1~
06 "Efll - EHS’ (9-68)
.. 3... ~
37 EH1 +'%Hs’ (9'69)
- 3~ - - - - -
$8 Sal + Hz +‘2H3 "%H5 ' H7 ' Hs’ (9‘70)
- 3- - -
¢9 ' 2H1 +‘2H2 ‘ H5 ’ 37’ (9‘71)
010 - 3111 + H2. (9-72)

The reduction, Eqs. (9-53), is specified alternatively by taking

4. ~ 4. 2... ~ 2.. ~
T2 + T3 . 2T4 + 8“ H1 . 214 + 4H (02 + 93), (9-73)

.. .. 3 .. ..
96 + 07 2(02 + 03), (9-74)

Inverting Eqs. (9-57)-(9-62) and (9-63)-(9-72), and using Eqs.

(9-76) to eliminate T4, 56’ 57, and 5

and

U:

U:
wa-Na-Ha-

10 gives:
1 ..
2(T2 + T3),
- - 2~
-(T2 + T3) + (T5 + T6) - 4“ H1,

1 1’
1..
2(T2 ‘ T3)’
1 ~ ~ 1 ~ ~ 2 ~ ~ ~
- 4(12 - T3) - 2(15 - T6) - 2a (3111 + H2 - H5),

+ 63) + (5 4 + 99).

-%(6 + $3) - 20134 + $9) + (95 + 98).
+53) + 61+ (044-99) - (55+ 58)
-%(52 - 53>.
_.%(¢2 - 53) +'%(54 - 59),

~ ~

1

To relate the experimental to the theoretical constants, the

25

equations are the following :

(9-75)

(9-76)

(9-73)-

(9-77)

(9-78)

(9-79)

(9-80)

(9-81)

(9-82)

(9-83)

(9-84)

(9-85)

(9~86)

(9-87)

(9-88)

(9-89)

(9-90)

required

145

T1 = T1 + 0(4), 1 . 1,2,3, (9-91)
24 - 14 + 2nslll(0 - B) + 0(4), (9-92)
T5 = T5 + 2H3111(A - E) + 0(4), (9-93)
16 - i6 + zusmas - A) + 0(4), <9-94)
and
61 - 61, 1 . 1,2,3, (9-95)
¢4 . $4 + zusllB(i - A) + 4uslllciz 16) - 4nzsill(§ - A), (9-96)
¢5 . 5+ zus3ll(n - A) + 4uslll(i6 11) + 4uzsill(fi - A), (9-97)
66 - $6 + zusl3l(é - 6) + 4uslll(23 14) - 4uzsill(é - 3), (9-98)
¢7 . 57 + 2u5113(0 - 3) + ““5111(T4 -'12) + 4uzsill(é - B), (9-99)
68 - 68 + zus311(A - E) + 4151110:1 is) 4uzsill(1 - 6), (9-100)
69 . £59 + 211513101 - E) + 411511105 T3) + 4n zsmu'; - é), <9-101)
410 = $10 + 6u[(& - ii)s311 + (i - 1)5131 + (A - é)sll3]. (9-102)

As mentioned in Chapter 6, the coefficients S 1’ of second order

11

of approximation, and the coefficients $113, $131, and $311, of

fourth :rorder of approximation, are determined by the particular

reduction chosen.

9.5 Comparison of the Quartic Distortion Coefficients

~

we first examine the quartic distortion coefficients Di

given by Eqs. (9-77)-(9-82), with the T1 related to the Ti by

Eqs. (9-91)-(9-94). The T in turn, are given by

1’

Ti= '4H + 0(4), 1 . 1,2,3,4,5, (9-103)

146

T6 =

94h:

(116 + 2r9) + 0(4)

=2 + 0(4), ' (9-104)

kjh‘

in which the terms of 0(4) are not firmly established, other than
that they are very complicated. This necessitates approximating the
Ti by the corresponding T1, thereby incurring an error of 0(4).
Because of this, the terms of 0(4) in Eqs. (9—77)-(9-82) and (9-91)-
(9-94) should be dropped as well at this point, with the result that

to 0(2) we have that

-4 1

01 - -8-(12 + 13), (9-105)
-* 1 1 * - -

D2 - - 2(12 + :3) + 2(15 + 16) + znsulm - B) (9-106)

0* - 14 +-l(r + r ) --¥(T + 1*) - zus (é - 6) (9-107)
3 4 1 8 2 3 4 5 6 111 ’

-*. 1 '

D4 I. '§(T2 - T3), (9‘108)

0* l;<T T ) - l(r - 1:) + “S (2A - B - 0). (9-109)
5 16 2 3 8 5 111

In order to evaluate these, the value of S is needed. From

111
Eqs. (9-91)-(9-94) and again dropping terms of 0(4), we obtain

“S111 . (T4 - 4T4)/8(C - B) = (r5 - 4T5)/8(A - é)

* ~ .., ~
- (T6 - 4T6)/8(B - A). (9-110)'

-* -
Using Eqs. (9-57)-(9-62) and Maki's D1 to evaluate the T1, using

Maki's rotational constants A, 8, and C in place of A, B, and

C, and using the values of the T as given in Table (9-3), we find

i
from Eqs. (9-110) three independently determined values for the

dimensionless quantity u 3111, viz.,

147

6 6 6

, -0.465(6) x 10‘
(9-111)

us - ~0.525(19) x 10' , -0.459(4) x 10'

111

respectively. These values are reasonably consistent, although the
error ranges do not overlap completely which can be attributed to the
neglect of the terms of 0(4). To proceed, we chose to use the

average value and the maximum uncertainty, viz.,

6

KS - -0.483(l9) x 10' . (9-112)

111

The distortion constants D: can now be evaluated. The calculated
values are listed in Table (9-4) along with the two sets of observed
values. In order to provide a common basis of comparison for these,
the uncertainties shown with both data sets were taken to correspond
to twice the reported standard deviations. The ratio of calculated
to observed values are also given. These ratios are the "best" ones
obtainable in the sense that the uncertainties are used in such a way
as to give ratios as close to unity as possible. For two of the five
coefficients, 0*

i

an overlap of the error bars of the calculated and observed values.

, this ratio is equal to unity which corresponds to

For the remaining three coefficients the agreement is close, but not
complete. This can again be attributed to the neglect of terms of
0(4), and on the basis of this assumption, the agreement may be con-
sidered satisfactory. From Eqs. (9-105)-(9-109) it is seen that

-* -*
$111 is not needed to determine D1 and D4, Numerically, the

-*
111-term to the value of D3 is negligible,

~*
and it is less than 3% for D2. The contribution of the Sill-term to

-*
the value of D5 is, however, the dominant one and amounts to almost
902 of the total value of 0

contribution of the S

5.

148

.mcowumfi>mv vumvcmum woumswumo may oofi3u mum mooam> vw>ummno ecu nu“: cacao moauawmuuooca

.UNQU 00m .mGOHUQQHHUGOU HQGOfiUQHflHb. HON fiOuUQHHOUU

.oc monopommm ..omouuomam .Hoz .H ..Hm um mason .<n

.AmamHV 0.4 .mm..umouuooam ..62 .e ..xmz .o.<6

 

 

 

 

O
moo..+ ooo..+ a- Asq.v.em.mu A~Hv.a.~- Am.v.mm~.mn Aem~v4Hnm~.mu so- mm
o 4
ooo..+ Nam.o+ a- Aomvmem..- A.~vmam..u AHNVmammm..- Aoeevocmmam..- a..- «m
o m
«Ho..+ qua.o+ .1 AmHVmso.~n AmHVoma..- .m.v.eo...~- AHCNVmeeeHH.~n sq- «m
o N
mqo..+ ooo..+ on Aamvmow..+. Aumv.m..+ Amuveoem..+ AthVceeew.H+ ee<u «m
0
~ma.¢+ aha.o+ a- Aomva4.sn Ammvmon.eu A~.v.~4m.qn A»..vmm~em.eu n<u Hm
o n.m .mxm mommasoamu ocm>uomno vm>uomno mmm>uomno mass: mo .mwooo
Aumonv .mno\.oamo A coaumuoz
So :a m «o musmfioamwoou coauu0umfin quHOIvaoomm cmumaaoamo was vo>homno .Aelmv manna

0H

149

The third column of observed values of Table (9-4) gives the
values of the second-order distortion coefficients corrected for

~*
vibration. For this calculation we used the values of Di of the

45
(100) and (001) vibrational states given by Barbe et a1. , and

2* 52
the D1 of (010) obtained by Bellet and his group from microwave

*
3

improved, but in general, the agreement is not substantially better.

data. For the largest coefficient D , the agreement is thereby

*
1

represent the equilibrium second-order distortion constants, because

The vibrationally corrected values of the D still do not fully

there remains a vibration-independent fourth-order difference between

these and the vibrationally corrected quartic ground state coeffic-

25
ients .

9.6 Comparison of the Sextic Distortion Coefficients
Calculation of the seven non-zero sextic distortion coeffi-
cients H1 is based on Eqs. (9-83)-(9-90) with the 51 replaced by

the corresponding 0 of Table (9-1) according to the scheme des-

i
cribed by Eqs. (9—95)-(9-102). In Eqs. (9-95)-(9-102), the co-

efficient S111 is used as given by Eqs. (9-110), and the T can

i
be calculated from the experimental results with the aid of Eqs.
(9-57)-(9-62). Correlation effects among the T1 are not

considered in the present calculation. The coeffi-
cients 56’ 57, and 510 are eliminated from Eqs. (9-95)-(9-102)
through the use of applicable constraints, Eqs. (9-73)-(9-76).

According to Eq. (9-95), 5 and 53 in Eqs. (9-83)-(9-90) can

1’ 52,
be neplaced by 01, 9., and 03 respectively. The remaining seven

Eqs. (9-96)-(9-102) are used to determine the four combinations

150

(55;: 58) and (54;: 59) needed in Eqs. (9-83)-(9-90) along with
the three coefficients S113, S131, and $311. Carrying out the in-
dicated calculations, we find
us - +1 45(275) x 10‘12 (9-113)
113 ° ’
as = -0 19(200) x 10'12 (9-114)
131 ° ’
us - +2 30(415) x 10"10 (9-115)
311 ° . ’
- - -9 -1
05 + 08 a -2.15(15) X 10 cm , (9-116)
~ ~ -9 -1
¢5 - 98 a +3.62(530) x 10 cm , (9-117)
54 + 69 - -1.63(345) x 10'11 cm'l, (9-118)
54 - 59 = +3.36(335) x 10~11 cmfll. (9-119)

It is seen that, with the exception of (55 + 58), the above quantities
are very poorly determined. This is principally due to the occurrence
of near-cancellation of the calculated sextic distortion coefficients

in the determination of S and S which, in turn, leads to

113 131

the remaining large uncertainties. As a consequence, we obtain

~

poorly determined values for H2, H7, and H8. Since (55 + 58)
constitutes the principal contribution to H3, this coefficient is
much better determined, as are H1 to H5 which do not depend on any

of the Eqs. (9-113)-(9-119), and i for which the 6 and ($5 + 5

4 1

contributions dominate. The calculated values of the fii are listed

50
in Table (9-5) along with the experimental values of Haki and

8)

46
Barbe et al. which are in remarkably good agreement. In Fig. (9-1)

we compare the calculated values with the experimental values. The

 

.maoqumw>mw vumwcmum vmumawumm msu madam mum mm=Hm> co>uomno one Sufi: ezozm mowu:HMuuoo:=

.oc mosmumwwm ..omouuooem .Hoz .H ..Hm um onumm .<n

.uouum Hmoasamuwonxu mane vmaufimsoo mm: “an: .un .uommn m.fixmx :« am>fim
mm m: no: .m>onm em>Hw mm nu ma e: no newcoaxo was .Anmmavoae .mm. ..omouuooem .Hoz .H .fixmz .u.<o

 

151

 

mus. x Aeneme.o+ Amneee.o+ Aemvm-.o+ Ae.mVe~m~.o+ we am
..-e. x Aeomveo..- Anemone..- Acevee.ou Aoeev.o..o- ewe em
«Hue. x Aevmm.o+ Aevum.o+ AmNNmemm.o+ .mmevoHem.o+. we. mm
e-o. x ANVem.o+ Amvem.o+ Aeevmmmm.o+ Aeemvmemem.o+ we em
mno. x Amvee.e- A..vn~.o- xenoemm..on ANNeVeONm..ou a“: mm
..-o. x Aeeevme.on Awesome.en Amevmm.e- Aeemveen.ou a“: «m
Nelo. x Acevmm.o+ Acevmm.o+ Aeo.veem.o+ Aenevaem.o+ mm .m
Aoomumzl>owa<v Aumxummlwuonasmv mo>uomao po>ummno mez mo .mmooo
moundsoamu moumasoamu A m coaumuoz

 

 

also aw mood mo mucoaowmmoou soaquumHn Hocuolnuusom mommasoamo was vo>uomao .mwrav manna

152

 

 

 

 

 

 

Tau :12 x w ..c ...u .7.
3......” +IIIIII|I+-III-III11 a
e
m Tlolli
Z TIIOIll
a
3<fi|ll|'lllll'll|l.ll'l+ lllllll . lllllllllllll I.” N:
m TIIOII.
z TIoIIL
.5 .3 x . e e N o
H Md L. L P u
3... .mm TIII+IIli mm
m Tl
z Tflélll.
ES
.833» 123:. new 33 uoxummlmumnaom T I I l ..l I I ll
23 .Hm um onumm " 0 i
1. H
as 36: T . 4 . m
maouo no em wo mosam> Hoofiumuoone can Housmaauomxm one no somwummaoo .malav .wam

153

So OH x 05 . 9% 06 can Nun

mu 1

I

a . o N- .7
P -

3.2.1:unuIIIIIlallollalulnnnnlllnlai

.mmT IIIIIIIIIIIIIII dlllllllllllllllllln.
m _ _ mm

3<T|0I6|lll
mm TIIIIIOI'IIII. m
m T. m

2?...

 

~eoeeeueoev Aeuev .mem

154

agreement between observed and calculated values is quite good, with
the experimental values generally much more precise than the cal-
culated ones. The Sumberg-Parker and Aliev-Watson theories generally
give comparable results.

Another method of comparing calculated and observed results
is more indirect, but has the advantage that it does not require the
evaluation of 8113, S131, and S311. As a consequence, this method
leads to more precise calculated values and hence constitutes a more
exacting test of the theory. In this procedure, the constants S

113’

1, and 3 are first eliminated from the ten Eqs. (9-95)-

S
13 311
(9-102), leaving seven linearly independent equations relating the

~

Qi to the 01, valid for any reduction of H6 provided only that
the reduction exists. Such a set of seven reduction-invariant re-

lations I is

1
11: 61 = 01, 1 - 1,2,3, (9-120)
9 1 9 ~ 1~

I4: 2 (0i) +-3010 = 2 (01) + 3¢10’ (9-121)

1:4 1-4
2 ~ 2‘* + 1 ~ ”* 9 122
2 2‘* “* 9 123
I6. 68 + (1 - 6)¢5 = ¢8 + (1 - 6)05. ( - )
~ ~ 1 ”* ~ ”* 24
I7. (1 - 0)¢9 - (1 - 0)¢6 ( + 6)¢9 - (l - 0)¢6. (9'1 )

where

A - 3.553666 cm-l (9-125)
3 - 0.445283 cm‘l (9-126)
5 = 0.394752 cm'l (9-127)

6 = (20 - A - §)/(A - E) - -1.032513, (9-128)

 

 

155

and

«-

*=~ ~-~ - 22 ~‘~
¢4 ¢4 + 4h8111(T2 T6) 4“ 8111(B A), (9-129)

~

~*
with 05 through 09 given by similar expressions, easily deter-

mined by reference to Eqs. (9-95)-(9-102) and (9-120)-(9-124). The
51 were obtained from the experimental results with the aid of Eqs.
(9-63)-(9-72). Considering the left—hand sides of Eqs. (9-120)-
(9-124) as the calculated values and the right-hand sides as the ob-
served values, and carrying out the computations just described, one
obtains the results summarized in Table (9-6). The precision of the
calculated values is much improved over that in Table (9-5). How-
ever, since some manipulation of the experimental constants is re-
quired, the attendant accumulation of errors leads to a loss in pre-
cision of the experimental quantities. In Fig. (9-2), we compare
the experimental values of the Ii with the calculated values. The
algebraic signs of six invariants are reproduced correctly by the

theory. With the exception of II, all error bars overlap and the

agreement can be considered satisfactory.

 

.oe moawummom ..omouuomam .Hoz .H ..Hm um onumm .<n

.AmeaHVe.e .mm. ..eeouueeem ..o: .n ..xez .o.<m

 

156

 

«Hue. x Aemvmo.on Aomvmo.on Amemvmm~.os Aummvee~.ou 5.
sue. x ANMHme.m- AeeHVeH.m- Ameven6.ml A.~.Ve...m- 6.
«Hue. x Aeevee.o+ Amevee.o+ Aeemvwm...+ Ammmvoe...+ m.
ale. x Amevme..: Aeevoe.~- Aeevmmm..- Aeevemm..- 4.
m.-o. x Ashame.o- Aemv~e.ou Aewevew.o+ AeoNV.H.e+ m.
mesa. x Amevem.o+ AmHVem.o+ Ammevome.o+ Aeomvm.e.o+ NH
ale. x Amevme.m+ .mHVme.m+ Amevmee.m+ Aeeveee.m+ H.
Asomum3|>ofia<v AuuxummlwquEva ;
mommaooamo moundsoamu awo>uomno mom>ummno usmuum>cn

 

 

use a“ mm mo mucmaum>su umpuounuuaoh vmumaaoamo cam cm>uomno .mwrmv Manse

H 0H

157

 

 

 

 

 

 

 

 

 

Fig. (9-2). Comparison of the Experimental and TheoretiCal Values
of the Invariants Ii for Ozone
I E 3 __1' M
1 t—O—q B
l- ....... -. ........ ..SP, A‘q
T Y— 1 l _ “l
3.2 3.4 3.6 3.8Xl0 cm
he'li
I HM B
4 I... ________ _._... ________ .4 SP
F-—---—-—o— ----- _{AW
1 r F v _ _1
-4 -3 -2 -l x 10 cm
ha! M
B
I6 p.g
t_-___----. ........ ..sP
F- ------ " ----- -1 AW
f f r f _
-8 -6 -4 -2><10 9cm 1
I5 1 .31, 1 M
k e : B
1..-----—o-——--——l SP, AW
f‘ 7 i - _
0 1. 2 x 10 12cm 1

 

158

 

Fig. (9-2) (continued)

 

 

 

 

 

 

 

 

 

.LA = —4 B
p. ——————— -4 ------- ‘4 SP, AW
T 1 r '
x lO-l3cm-1
I3 :1 1 - 4M
1 3 i B
I_ ' ' ' ‘
-2 -1 0 1‘ ' 2
x 10-13cm-l
1 i —1‘ M
I7 .L = 1 B
P--—-O--‘---{ SP
1..-——-o—-—"‘i AW
t I ' I
-g 0 8
x 10-13cm-l

 

-16

 

10. CONCLUSION

The Darling-Dennison molecular vibration-rotation Hamiltonian
and an order-of magnitude expansion of this Hamiltonian appropriate
for asymmetric-top molecules were presented. After outlining the
contact transformation technique as applicable to the expanded
Hamiltonian, the calculation by this technique of the second-order
(quartic) and fourth-order (sextic) centrifugal distortion coeffi-
cients was described.

The results of the calculation of the theoretical expressions
for the sextic centrifugal distortion coefficients for triangular

triatomic molecules by Sumberg and Parker were compared with the
results of the more recent calculation by Aliev and watson. All
discrepancies between- the two calculations were determined and fully
accounted for.

The complete set of quartic and sextic distortion coefficients
was calculated for the ozone molecule and compared to experimental
determinations appearing in the recent literature. To carry through
this comparison, extensive use was made of Watson's theory of re-
duced Hamiltonians. Agreement between theory and experiment was

found to be quite satisfactory.

Also in this dissertation, four linearly independent linear
combinations of the ten sextic centrifugal distortion coefficients of

triangular traitomic molecules are developed. These are independent

159

 

160

of the cubic anharmonic potential constants and depend only on the
equilibrium geometry and the harmonic force field parameters of the
molecule. These expressions appear potentially useful as planarity-
conditioned constraints on the ten sextic centrifugal distortion
coefficients. Such sum-rule type constraints should normally reduce
the uncertainty of the experimental constants without interfering
with the deduction of the potential constants from.them and lead to
Hamiltonians devoid of indeterminable coefficients or combinations of
coefficients. The manner in which these original results might be
useful in the analysis of high-resolution vibration-rotation data

was discussed.

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10.

11.

12.

13.

14.

15.

16.

17.

18.

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2

HI“

3 1293 03169 00

lllllUNIi