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l'flt‘s'cfi

 

This is to certify that the

thesis entitled

A Complete Fourth-Order
Vibration-Rotation Hamiltonian

of HZO-Type Molecules
presented by

Liek Wilardjo

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in PhYSiCS \l

'2

- Cm 727 . ”Kira;

Major professor

 

Date October '6; 1970

 

0-7 639

 

 

 

ABSTRACT

A COMPLETE FOURTH—ORDER VIBRATION-ROTATION

HAMILTONIAN OF HzO-TYPE MOLECULES

By

Liek Wilardjo

Starting with the Darling—Dennison Hamiltonian, a complete
fourth-order vibration—rotation Hamiltonian for HZO-type molecules
has been obtained. It is in the form of a power series in the com—
ponents of rotational angular momentum in the molecule-fixed frame.
The coefficients of this power series include the equilibrium ro—
tational constants, second—order and fourth—order centrifugal dis—
tortion constants and vibrational corrections to the rotational con-
stants and to the second—order centrifugal distortion constants.
Expressions for all the above coefficients in terms of the fundamental
molecular parameters have been obtained and their relation to exper-
imentally determinable quantities has been established. These funda-
mental parameters include the equilirium moments of inertia, the
normal frequencies, and the potential constants of vibration of the
molecule. The resulting Hamiltonian is appropriate for the calculation
of ground-state energies of vibrating and rotating H20—type molecules
to the fourth order of approximation. Such calculations may also be
possible for those excited vibration states which are not subject to

accidental vibrational and accidental vibration-rotation resonances.

A COMPLETE FOURTH-ORDER VIBRATION-ROTATION

HAMILTONIAN OF HZO-TYPE MOLECULES

By

 

Liek Wilardjo

 

A THESIS

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1970

 

ACKNOWLEDGMENTS

The author wishes to express his sincere gratitude to
Professor Paul M-.Parker for suggesting the problem, and for his
guidance and counsel throughout this work. He also wishes to thank
the U. S. Department of State for the Fulbright travel grant that

enabled him to come to the United States from Indonesia.

ii

 

t—l

N

TABLE OF CONTENTS

. INTRODUCTION

. GENERAL VIBRATION—ROTATION HAMILTONIAN

3. THE VIBRATION—ROTATION HAMILTONIAN 0F HQO—TYPE MOLECULES

b

5- TWICE-

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

. ONCE-TRANSFORMED HAMILTONIAN

TRANSFORMED HAMILTONIAN

I. Coefficients aéfi, Aégngv and Coriolis Coupling
Coefficients

II. The Expansion of}%u, yap

III. The Untransformed Hamiltonian
IV. Coefficients (2;de;a;-) in hé
V. Coefficients (2;d@;ab;-) in hi
VI. Coefficients (2;d@;—;ab) in hi

VII. Coefficients (2;d;abc;—) in hé and All
Possible Permutations

VIII. Coefficients (2;d;c;ab) in bi
IX. Coefficients (2;-;cd;ab) in hi

X. Coefficients (2;-;-;abcd) in hi and
All Possible Permutations

XI. Coefficients (3;xp36}—;a) in hé
XII. Coefficients (3;dfij;b;a) in hé
XIII. Coefficients (3;dfi;bc;a) in hé
XIV. Coefficients (3;dP;—;abc) in hi

XV. Coefficients (4;afi§?§ifi;-;-)

 

12

26

32

50
55
57
60
61

62

64

65

67

69

7O

78

89

112

 

 

APPENDIX. XVI. Coefficients (4 ”(mfg— ;ab)
APPENDIX XVII. Coefficients (4;«p7{{;ab;-)
APPENDIX XVIII. Coefficients (4!dfi3’,§£{J!—!-)
APPENDIX XIX. Coefficients (4!0((53/,$!—!ab) in h4@
APPENDIX XX. Coefficients (4!o(/9,’(5!-.!ab) in h4@

APPENDIX XXI. Coefficients (Moupyfhtab) in h4@

APPENDIX XXII. Coefficients (4!o(f53{$!abl—‘) in h4@
APPENDIX XXIII. Coefficients (4!X(3,3/5!ab:—) in h4@
APPENDIX XXIV. Coefficients (4!0(,(;’f{!ab!-) in h4@

BIBLIOGRAPHY

iv

114

117

123

125

126

127

128

129

130

131

 

Table

VI

VIII

IX

XI

XIII

XIV

XV

XVI

LIST OF TABLES

. . d _
The coefficients lin of the transformation
to normal coordinates

o(
Coriolis coupling coefficients {Pm

First-order coefficients afé, agj

dd 4 «a a
Second-order coefficients Aij’ Ai?» Aij , Aijf3
Character table of point group sz.and symmetry
species of the operators appearing in the
Hamiltonian

Coefficients ¢fl%*—*a) and ¢x*b*a) in the first-
order Hamiltonian H1

Coefficients €¥fi*-*ab) in the second—order
Hamiltonian H2

Coefficients GX*C*ab) in the second-order
Hamiltonian H2

Coefficients (—*cd*ab) in the second—order
Hamiltonian H2

Coefficients @flA*-*abc) in the third—order
Hamiltonian H3

Coefficients Gx*d*abc) in the third-order
Hamiltonian H3

Nonvanishing coefficients in the fourth—order
Hamiltonian H4

Coefficients [S;¥p;a;—] in the first contact
transformation

Coefficients [S;d;—;ab] and [S;q;ab;—] in the
first contact transformation

Coefficients [S;-;abc;-] in the first contact
transformation

Coefficients [S;-;c;ab] in the first contact
transformation

Page

l3

14

15

l6

16

21

21

22

22

22,23

24

24,25

27

27

28

28

 

Table

XVIII

XIX

XXI

Coefficients (2;JPXS;—;—) in the second—order
Hamiltonian hz'

Coefficients [d;dfiX;—;a] in.the second contact
transformation

Coefficients [6;dfigb;a] in the second contact
transformation

Coefficients B7;d;c;ab] in the second contact
transformation

Coefficients [6;u;—;abc] in the second contact
transformation

The constants Aabc and Bab,c
The coefficients $019an

The coefficients<bj

vi

Page

31

33

33

34

34
36
41

44,45

 

LIST OF FIGURES

Figure 2.1 11

Figure 3.1 Equilibrium Configuration of the HZO-type Molecules. 12

vii

I. INTRODUCTION

Among the microwave and infrared spectra of H20-type molecules,
experiment has revealed many instances in which it is necessary to in—
clude in the Hamiltonian fourth-order centrifugal distortion terms in
order to account for observed results to experimental accuracy. Molec—
ules for which such a situation has been found to exist include the
nonlinear triatomic molecules H20,l S02,2 and F20.3 Several workers
have included on an ad hoc basis one or more fourth—order centrifugal
distortion terms in their data analysis and thereby have obtained
closer agreement between theory and experiment. Therefore it seems de-
sirable to obtain fourth—order centrifugal distortion coefficients as
a function of fundamental molecular parameters. The taking into account
of the corresponding terms in the Hamiltonian should then give more
satisfactory and theoretically meaningful fits to spectral data. To
this end one can start from a general formalism of the Hamiltonian of
polyatomic molecules such as that of Wilson and Howard,4 or Darling
and Dennison.5 The two Hamiltonians are equivalent, but the latter is
in more convenient form for expansion into terms according to orders
of magnitude for perturbation calculations. Goldsmith, Amat, and Niel—
sen6 have done such an expansion and development of the Darling-
Dennison Hamiltonian up to the fourth order of approximation, but it
is complicated, and one cannot use their results with complete con-

fidence without duplicating the actual calculation. For this reason,

 

 

it seemed to us more advantageous to carry through the complete
development of the Hamiltonian for the H20-type molecular configur-
ation in particular rather than obtain the result as a special case
of the general theory.In this way also the serious complications of
the general theory due to degeneracy in the normal modes of vibrat-
ion are absent since such degeneracies do not occur in asymmetric-
top molecules and our calculation can avoid this complication from

7 showed that

the start. Furthermore, a very recent paper by Watson
the Darling-Dennison Hamiltonian can be written in a much simplified
form. This was not known to Goldsmith, Amat, and Nielsen and therefore
the details of the calculation in their expansion of the Hamiltonian
are much more complicated than need be.

Chung and Parker8 have studied the general vibration—rotation
Hamiltonian of asymmetric—top molecules in the Goldsmith-Amat—Nielsen
formulation. By subjecting the Hamiltonian to the symmetry restriction
of the asymmetric-top point group, it was shown that there are only
105 independent fourth-order coefficients out of the possible 729.
Watson's work9 further reduced the number of independent fourth—order
coefficients. This work also showed which parameters are determinable
from experimental analysis, and how they relate to the theory. For
these reasons, and since the number of independent fourth—order
coefficients can be still further reduced and their expressions simp—
lified by symmetry considerations, it was thought desirable to obtain
a complete, fourth-order vibration-rotation Hamiltonian of H20—type

molecules through a calculation specialized from the very start for

 

 

 

this configuration.

In this work we take as the starting point of our calculation
the Watson—simplified Darling-Dennison Hamiltonian, written specif-
ically for HZO-type molecules. This Hamiltonian is expanded and
subjected to two successive contact transformations of the Goldsmith—
Amat—Nielsen type.6 Pure vibrational terms of all orders are discarded
since our interest lies in the detailed rotational structure for a
given vibrational statee The resulting twice—transformed Hamiltonian
is then subjected to extensive regrouping of terms based on the angu—
lar momentum commutation relations to put it in the final form cited

10,11 For a given Vibrational state, it is

by Yallabandi and Parker.
in the form of a power series in the rotational angular momentum com-

ponents with coefficients that are functions of the fundamental mol-

ecular parameters.

 

 

 

II. GENERAL VIBRATION—ROTATION HAMILTONIAN

We wish to consider Vibration—rotation transitions during which
the molecule remains in its electronic configuration. Consequently the
part of the Hamiltonian coming directly from the electronic contribut—
ion to the total energy is of no interest here. To a very good degree

12 one may consider the electronic motion to be in-

of approximation
dependent of the nuclear motion, and hence separate them and disregard
the former in the formalism of the Hamiltonian. This procedure is

known as the Born—Oppenheimer approximation.12

Within this approximation the Darling—Dennison5 formulation of

the vibration—rotation Hamiltonian of polyatomic molecules is:

H = (1/2)[}Jl/40(ZB(PuL _ PflPng—l/pr _ 1}.)le +

where (l) and denote the principal axes of the equilibrium inertia

tensor x, y, z of the molecule ; in general, Greek indices
are assigned the range x,y,z except when they are used as
normal modes' degeneracy indices in vibrational normal co-
ordinates and momenta. Thus, for instance,(x and g in (2.1)
range over x,y and 2, whereas G’ranges over the degrees of
degeneracy.

(2) Ex and p, are the components along the principal axes of

inertia of the total and internal angular momentum, respect—

ively.

 

(3) p36 is the momentum conjugate to normal coordinate Q95

i.e.,

Pa. = dig/9056 (2.2)
(4) V is the potential energy of the molecular force field;

it is a function of Q56

(5) Mags/lull; N5 I’d/J‘- IWZ) (2.3)
Mm: 1“ £1st + I41 is) 0"”55 (2'4)

M‘1 = det. I);x —I):>/ -I;Z
—I)'(y I” -1);

l
"Ixz

(2.5)
_I_;IZ 1&2
In (2.3) to (2.5), Iéd and Iép are the instantaneous moments and

products of inertia, respectively. In terms of Q36 they can be

written as

aka = «+2; aSGQSGI+s§;‘x AsatrngQtt. (2 6)

s
J'fl =42: + \W aSGQsovi‘ ér Aso‘t'cQsoQtt (2. 7)

where a4: and A' .(d= @, 0’#{3) are constants for a given mol—

SO’S'O‘
ecule whose explicit evaluations are given in Appendix I. Since we
use the principal axes of inertia, we have

0 = o = Wm

Ivy/5 Sdeotd 9 LP {0,d#@ (2.8)
i.e., the equilibrium products of inertia I343 , CHE/5 vanish.
Using the defining formula of the internal vibrational angular mom-

q’
P0( = S; gr CsctrQscPE’c (2-9)

entum

d
where $:¢tt are Coriolis coupling coefficients defined in Appendix I,

the Hamiltonian (2.1) can be written as

 

-W:..:.. : .--'- 7' 7
-.—I~———"

 

whe]

 

 

 

H = (1/2);fi{[2¥ — pqwdrjpp _ pa] +Ml/4(H,/4,L(M’Cl/2(pfl)£-/4))}
+ <1/2>S;{p§% +jA1/4(p§6,4£1/2(p§oj&/4))} + v (2.10)
In (2.10) an operator in parentheses acts only on quantities in the
same parentheses, and not on the wavefunction.
The Hamiltonian (2.10) depends not only on/udp, but also on the
determinant/L. Since the Schroedinger equation
(H —E)\i/= 0 (2.11)
is apparently impossible to solve in closed form, the Hamiltonian
must be expanded in orders of magnitude to put it in a form suitable
for perturbation calculations. Thus we would have to expand both/A‘¥#
andjl as power series in Q56, which is quite a tedious calculation
to do. Fortunately,however, it is not necessary to expandjd, as it
has recently been shown by Watson7 that the Hamiltonian (2.10) can be

identically rewritten in the greatly simplified form:

H = (1/2)[°Z(B{;,L(PP°(P(, — [Pd/Mo‘fipfg + Papa/3P3] + RX/udppfl’}
+5; pg: - (fi2/4);}Am] + V (2'12)

If it is assumed that the displacements of the nuclei from their
equilibrium positions are small relative to internuclear distances,

we may expand/Mg“ and/Aqfi in (2.12) into McLaurin series in dim—

ensionless normal coordinates defined as

qsg = (“s/hZHMQSs (2.13a)

where5is is related to the normal frequencyaJS associated with q56

by

 

 

 

 

 

 

 

 

H = (1/2)°(:fi{[g( - quflquPp — p6] +Ml/MR‘1‘LNIUMPHMMD}
+ <1/2)sz;{p>§%r +)A1/4(p§6M1/2(p§51A}/4))} + v (2.10)
In (2.10) an operator in parentheses acts only on quantities in the
same parentheses, and not on the wavefunction.
The Hamiltonian (2.10) depends not only on/udp, but also on the
determinantji. Since the Schroedinger equation
.(H —E)\[/= 0 (2.11)
is apparently impossible to solve in closed form, the Hamiltonian
must be expanded in orders of magnitude to put it in a form suitable
for perturbation calculations. Thus we would have to expand both/AJ¥#
and)A as power series in Q56, which is quite a tedious calculation
to do. Fortunately,however, it is not necessary to expand/M, as it
has recently been shown by Watson7 that the Hamiltonian (2.10) can be

identically rewritten in the greatly simplified form:

H = (untggmpt Pp - [Payqppp + papa/5PM + Fix/Jappfé'}
+5; pg: - (fi2/4);/qu] + V (2.12)

If it is assumed that the displacements of the nuclei from their
equilibrium positions are small relative to internuclear distances,
we may expand/iLmtand/qu in (2.12) into McLaurin series in dim-

ensionless normal coordinates defined as
qso’ = (7Ls/h2)l/4Qs6' (2.13a)

whereiis is related to the normal frequencya)S associated with qsg

by

7LS = (21tcms)2 (2.13b)
Carrying through the procedure and collecting terms expected to be of
the same order of magnitude, we obtain a Hamiltonian in the form:
H=H0+H1+H2+H3+H4 (2.14)
where H0, H1, H2, H3 and H4 are the zeroth, first, second, third
and fourth order portions of the vibration-rotation Hamiltonian, res—

pectively. They are found to be of the form:

Ho =;;{(ot/5*—*—)I&Pp + (asl/z/Zfi)ps(27} + (Nagasl/quz
H1 =°( Z{@fi*'*SV)Qch<P/o + (0(*t'r*so’) (1/2){qs°—,Ptr}Pq/}
+ chthrstquqsfqu—
. H2 =0;92{¢‘{-’>*'*S°'t1)qso'thPdPF> + (<><*t1:*rpsw) (1/2){qrpCIsa~Pt¢}g<
+ (—*sc't'r*p‘r[rf) (1/2){qpuqr9,pscrpt1} - h2/4)(°‘°‘*"*')} +
.- Zucfizkprstqpfiqrquo/qtr
i H3 =°‘ZPZ{ (075*-*rPS¢tT)qrpqscqt1Po<P{5 +
’ (o(*t'r*pnrfSG’) (1/2){ qmqrqug.ptT}Pa +
i (—*so‘t't*owp1rrf) (1/2){ qwqpmqrfmsgpw} '
F _ (.12 /4) (“Lawn“), + ZECthopx-stqowqp'rtqrf qwqm:
H4 =;fi2{@p*-*pmrfsctr)qpmqrququg< Pp +
(°(*trt*oo)p1trf56')(1/2){qowqpnqrquo-dptrr}P°( +
(-*so*t1*n12’owprrf) (1/2){ qnflqowqpmqrj, .psypm} “
(h2/4)67(°‘*-"56t'c)‘-Isa"1t13 +

ZKCHELknoprstqn¥qowqpmqrfqsqqt1 (2.15)

where all summations are over all indices in the corresponding terms,

Par is the momentum conjugate to the dimensionless normal coordinate

 

 

qsG’ i.e.,

pso’ = -ih3/9qso. (2.16)

and the last term in each Hamiltonian represents the part of that
Hamiltonian that comes from the potential energy V, also expanded as
a McLaurin series in the dimensionless coordinates. The expansion of
)lw‘and)AdPis outlined in Appendix II, and the derivation of (2.15),
along with the definitions of the coefficients appearing therein,
are given in Appendix III.

Except in cases of accidental resonance interaction,l3 H1 in
(2.15) has nonvanishing diagonal matrix elements arising only in
the second term, which themselves vanish except for degenerate normal
modes of vibration. Hence the second term is the only one in H1 which
can actually contribute to the first order energy. Therefore, one may
partially diagonalize H by subjecting it to a contact transformation
T14 to get a transformed Hamiltonian H' of the form:

H' = THT-l = Ho' + H1' + H2' + 113' + H4' (2.17)

where T is so chosen that
H1! :kzs;¢x*sa-*so“) (1/2)1quc,p36}2,( (2.18)

Here (2.18) is the same as the second term in H1 with tt =56; i.e.,
it applies only to degenerate normal modes of vibration.

Choosing T to be of the exponential form:

T = exp(£lS) (2.19)

where T is a unitary operator which requires S to be a Hermitian

 

 

 

 

operator, andciis a smallness parameter, we have

H' = exp(iWS)Hexp(—ifls)
= ( 1 + ifls -fl?sZ/2 — ... )(HO +CLH1 +23H2 + ... ) X

( 1 - ias — 2252/2 + ... )
Equating terms of the same order of magnitude we get

Ho‘ = H0
H2' = H2 + (1/2)[is,(Hl + Hl‘)]

H3'

H3 + i[S,H2] - (1/3)[S,[S,(H1 + (1/2)Hl')]]
H4' = H4 + i[S,H3] — (l/4)[S,[S,(H2 + Hz')]] + (1/12) x

The proper Hermitian transformation function S has been found by
Herman and Shaffer15 and the treatment has been extended by
Goldsmith, Amat, and Nielsenl6 to apply to energy calculations
through the fourth order of approximation. If we perform the contact
transformation, however, a part of the Hamiltonian formally belonging
to Hn' becomes of the order of magnitude of Hull . Hence after con—
tact transforming the Hamiltonian we need to regroup the terms in true
orders of magnitude. This transformed and regrouped Hamiltonian is

H' = ho' + hl' + hz' + h3' + h4' (2.21)
where

hn<m> = an + (1/m)[is,hn£T+l>1V + (1/m)[is,h§‘3‘§1)]R (2.22)

 

10

and [13,11]V = (1/2)[isV,HV]{sR,HR}

[iS,H]R = (1/2){sV,HV}[isR,HR] (2.23)

The subscripts V and R denote the vibrational and rotational part,
respectively, of the corresponding operators. To be able to calculate
vibration—rotation energies to the fourth order, we need to subject

H' to a second contact transformation't such that ho', hl', and h2'
will be diagonal in the vibrational quantum numbers in the represent—
ation in which ho' is diagonal in the vibrational quantum numbers.
This contact transformation, too, will necessitate regrouping of terms
in true orders of magnitude. Thus H' is once again transformed and

once again regrouped to

H@ ="EH"E—'= exp(io’)H'exp(—icr) = h0@ + hl@ + h2@ + h3@

@
+ h4 (2.24)

where now h0@, hl@, and h2@ are diagonal in the vibrational quantum
numbers in the representation in which ho' is diagonal.The proper
Hermitian transformation function G‘has been found by Goldsmith,
Amat and Nielsen.17

For the asymmetric-rotator case, and for given quantum number J,
it has been found that through the fourth order of approximation the
Hamiltonian in matrix form in a symmetric-top basis can be cast into
the Wang forml8919 as shown in Figure 2.1.For the orthorhombic point
group symmetry to which HZO-type molecules belong, all matrix

elements outside the four diagonal blocks E+, E', 0+,and 0' along

 

11

the main diagonal vanish.20 Thus when we specialize our Hamiltonian
we need not retain terms that do not have matrix elements falling
inside the blocks E+, ET, 0+ and O", i.e., we shall take into account
only those terms that are of even-power in the rotational angular
momentum components Rx. We shall,moreover, not explicitly consider
pure vibrational terms as our interest lies-in the detailed rotat-
ional structure for given vibrational state. The third and fourth—
order terms nondiagonal in any of the vibrational quantum numbers can
be discarded as it is possible to show that these do not contribute

to the vibration-rotation energy up to the fourth order.21

 

 

 

 

E+ I2 112 III
12 E' 111 112
112 111 0+ 12
111 II2 I2 0'

 

 

 

 

 

 

Figure 2.1

III. THE VIBRATION-ROTATION HAMILTONIAN OF H20 -TYPE MOLECULES

Since any triatomic molecule is necessarily planar, we choose our
molecular frame so that the x-y plane is also the plane of the molecule
and the origin is at the instantaneous center of mass . We also hold
this frame fixed with respect to the equilibrium configuration of the

molecule. Thus we have:

 

 

z ' = z ' = z ' = 0 (3.1a)
M . 1 l 2 3
4% m(x2'+x3') + Mxl' = 0
, x (3.1b)
/:9/// c m \\\<3\> m(y2'+y3') + Myl' = 0
m m
3 I 2 mrosindb(y2'-y3') =
Figure 3.1 (1/2) rocosob(2xl'—x2'-X3') (3.1c)
Equilibrium Configuration
of the HZO-type
Molecule where r0,<xo,1n and M are shown in

Figure 3.1, and [lie the reduced mass

fl= 2mM/(2m + M) (3.2)

 

Equation (3.1a) is the condition of planarity, and equations (3.1b)
and (3.1c) are the Eckart conditions ( see Appendix I ).

From the geometry as shown in Figure 3.1 and equations (3.1b) we
have

x10 = -x20 = rosinxb

le = y20 = - rocosfib/Zm
X30 = 0 y30 = rocosab/M (3.3)
12

13

The transformation from the mass—adjusted coordinates mil/%xi'

( m1 = m2 = m, m3 = M ; O(= x,y,z ) to normal coordinates Qn

( n = 1,2,3 ) is given in Appendix I. The coefficients 1;; of this

transformation are listed in Table I.

Table I. The coefficients 1:;
of the transformation to normal coordinates

 

 

X
111

X
121

O

cos'zf/zl/2
—c;os’g{/21/2

0

—sin’5/21/2
sing/2U?—

WM) (M413) 1/2
-(u/2m) (tux/419W
”(fl/2‘“) (mw3)l/2

13

123

.Y
133

II

II

II

9A/M)1/Zsin7(
—1/2(/A/m)l/Zsin7(
—l/2(M/m)l/2sin’2(
(,u/M)l/2cos'g
-1/2(})/m)l/2cosg
-1/2 ()A/m)l/2cos')(

0

Wm) (In/[(13) l/ 2c011>(0
-9u/m) (m//M3>1/2cou><o

 

 

In Table I,)A3 and?( are constants defined in the following equations:

#3 =/M[l + ()A/2)cot20(3]

(3.4)

Sin/X = +<l -{Ak/[(Ak)2 + 4kl§]1/2})1/2/21/2

cos/)5 = +(1 - Sufi/pl” ;Ak = kn — kzz (3.5)

 

 

14

The Coriolis coupling coefficients g2: defined in Appendix I can now
be constructed from the coefficients lén. Those that are nonzero are
listed in Table II. Since the only nonzero Coriolis coupling coef—
ficients are those with superscript 2, this superscript is superfluous

and omitted henceforth.

o(
Table II. Coriolis coupling coefficients fin

 

 

§13 = —(31 = (Ix/IZ)l/2cosj - (Iy/Iz)l/zsin%
§23 = —§_:32 = (Ix/IZ)l/Zsin’z(— (Iy/Iz)l/2cos’)(

o( «X
§;n = — nm 0, o(= x, y ; m & n = 1, 2, 3

 

 

513 and §E3 are related by
5132 + $32 = 1 (3.6)
The equilibrium moments of inertia in Table II are:
Ix =/ur02coszo(0
I = 2mrozsingxo
12 = 1x + 1y (3.7)
With 1:; from Table I and «30, Bio, jio from (3.2). the nonzero coef-
ficients of expansion of the instantaneous moments and products of
inertia 5?“, a? , Aé?, Agg, A1?“ and Aigg are found to be as listed
in Tables III and IV.
The HZO—type molecular configuration belongs to the symmetry

point group 02v“ With equilibrium geometry as in Figure 3.1,

 

 

15

the components of the rotational angular momentum Px, Py and Pz, and
the components of the internal angular momentum px, py and pz belong to
irreducible representations of species B1, A2 and B2, respectively, as
given in Table V. Of the three components of the internal angular
momentum we need to consider only one, namely pz, since the vanishing
of :2; (cx = x,y ) means that px = py = O. The transformation (AI.6)
with the coefficients of transformation as listed in Table I, moreover,
is such that the normal coordinates ql, q2 and q3 belong to species
A1,A1 and B2, respectively. Table V gives the character table of point
group C2v- We include the operators qa, Rx and pz in that table show—

ing to which species each of them belongs.

Table III. First—order coefficients a:? 5:?
!

 

 

afx = Zle/zsinfi afix = Zle/zcosfi'
aIy = 2Iyl/2cosj 35y = -21y1/2sinl
aiz = afx + aZy = —ZIzl/2§Z3 agz = afix + agy = 2Izl/2313

agy = -2(Iny/Iz)l/2 = agx

 

 

Since the Hamiltonian must be invariant under all symmetry operations
of C2v: terms in the Hamiltonian that do not satisfy this condition
must vanish identically, i.e., their corresponding coefficients are
zero. Thus, for instance, all operator terms in which q3 appears in

odd power have -a priori— zero coefficients if Bx and/or pz appear in

 

 

16

a
Table IV. Second—order coefficients A13, Aig, Ai;u, Aigg

 

 

 

 

Aff = singj A11 = coszfi Afi = 1
Ali"; = coszlzs Ag; = sinz’d AEE = l
A§§ = Ix/Iz Ag = Iy/Iz Agg = 1
A’f = A33? = —(Ix/Iz)l/2cos’}( 235% = Ag’z‘ = —(IX/IZ)1/2sin1
A’fl = Ag = —(Iy/Iz)1/2sin')§ A§§ = Ag = -(Iy/IZ)1/2cos’y
Aiiz = f23 A222 = 301% A332 = 0
A132 = A212 = ' 13:23
ck .
Aiifl = 5?: ; CK = x,y 1 = 1,2,3
0! d .
A1119 = A1? ; 0(= x,y {3: x,y ij = 13,23,31,32

 

 

 

TableV. Character table of point group sz and symmetry species of
the operators appearing in the Hamiltonian

 

 

 

Group elements
Operators Species
E C2(y) o'(xy) 01372)
ql, q2 l l l 1 A1
q3, Pz, pz 1 -l l —l Bz
PX l —l —1 1 B1
Py l l -l —1 A2

 

 

 

 

 

 

 

 

17

those terms in even power. Using this symmetry consideration, we can
avoid evaluation of coefficients of expansion of [Map which ultimate-
ly would be found to vanish. One notes also, that in the potential
energy part of the Hamiltonian, ql and q2 appear symmetrically, but
not q3. This is due to the fact that ql and q2 both belong to the

totally symmetric species A1, while q3 belongs to species B2. Exp—

 

anding flow in (AII.2) and using low and lap from (AI.13), the coeff-

icients of the expansion can be shown to be as follows

zeroth order: B49*] = Sap/Lxg
first order : BXP*a] = —éj%/fxr,

second order: Bfl3*ab] = [éflézgafis + 5£Xa§:>/Is ' (Aab + Aha )1/

Zldlfifl + gab)
third order : [dp*abc] =%[ i -%8_ azgagaaia/Igla + (SZJAflifi'aggAgffl

 

IS}/6I°(I@] (3.8)

In the expression for Exfi*abc] 2% denotes summation over all distinct
permutations of a, b, and c.

Goldsmith et a1, 6 and Watson7 have obtained general expressions for

the coefficients of expansion of/Aap. Equations (3.8) above have been
obtained from the general coefficients of expansion ijdafi‘of Goldsmith
et al. The fourth—order coefficients Ffl3*abcd] which appear only in

the fourth-order Hamiltonian are not needed since they appear in the
final form of the Hamiltonian as corrections to the rotational constants
of the rigid rotator. Since the latter are of order zero, these fourth—

order corrections are negligible and hence discarded. The same

l8

argument also applies to other terms which originally stand in H4.

Therefore, although their detailed expressions have been obtained, we

shall merely list them in a simplified version of the notation used

in (2.15). The coefficients in (3.8) are not tabulated since what is

really needed are not these coefficients, but the coefficients of the

terms in the Hamiltonian (2.15). Of the latter, those that symmetry

allows to be nonvanishing are included below:

:11
II

(1/2)[(PX2/Ix + PyZ/Iy + 922/12) + (plzmll/z + p22/7L21/2 +

pgzmgl/anl + (111/2912 +l"21/2‘122 J'?‘3l/‘Z<132)/21’i (3'9”

[(XX*-*l)q1 + (XX*-*2)q2]Px2 + [(yy*-*l)q1 + (yy*-*2)q2]Py2 +
[(zz*—*l)ql + (zz*-*2)q2]Pz2 + (xy*—*3)q3(PxPy + Pny) +

3 2 2 2 3
2Kfic<klllql + k112q1 q2 + k122q1q2 + k133‘11‘13 + k222q2 +

2
k233q2q3 ) (3.9b)

[(xx*—*1l)q12 + (xx*—*12)qlq2 + (xx*—*22)q22 + (xx*-*33)q32]Px2+
[(yy*-*ll)q12 + (yy*-*12)q1q2 + (yy*-*22)q22 + (yy*-*33)q32]Py2+
[(zz*-*ll)ql2 + (zz*-*12)qlq2 + (zz*-*22)q22 + (zz*-*33)q32]Pzz+
[(xy*-*13)q1q3 + (xy*-*23)q2q3](PxPy + Pny) +
[(2*3*ll)(q12p3 + p3q12) + (2*3*12)(q1q2p3 + pgqlqz) +
(2*3*22)(q22P3 + p3q22) + (2*l*13)(q1q3p1 + p1q1q3) +
(Z*l*23)(q2q3p1 + P1q2q3) + (2*2*l3)(q1q3p2 + p2q1q3) +
(2*2*23)(q2q3p2 + p2q2q3)]Pz +

2 4
—(fi /8)(l/Ix + l/Iy + l/Iz) + 21thc(kllllql + k1112q13q2 +

k1122‘112322 + k1133q12q32 + k1222qlq23 + k1233q1‘12‘132 + -----

 

H3

19

4
-'- + k2222q2 + k2233(122‘132 + k3333q34) +
(-*33*11)( 2 2+ 2 2) + (—*33*12)( 2+ 2 ) +
Q1 P3 P3 Q1 QIqZP3 P3 qqu
(—*33*22)(q22p32+p32q22) + (-*l3*l3)(q1q3P1P3+P1P3q1q3) +
(-*23*l3)(QIQ3P2P3+P2P3Q1Q3) + (-*13*23)(q2q3plp3+p1p3q2q3) +
(-*23*23)(q2q3p2p3+p2p3q2q3) + (—*ll*33)(Q32P12+912q32) +

(-*12*33)(q32p1p2+plp2q32) + (-*22*33)(q32p22+p22q32) (3.9c)

[(xx*—*lll)ql3 + (xx*—*222)q23 + (xx*—*112)q12q2 + (xx*—*122)qlq22
+ (xx*-*l33)qlq32 + (xx*—*233)q2q32]PX2 + [(yy*—*111)q13 +
(yy*-*222)q23 + (yy*-*112)q12q2 + (yy*-*122)q1q22 +
(yy*—*133)qlq32 + (yy*—*233)q2q32]Py2 + [(zz*—*lll)ql3 +
(zz*-*222)q23 + (zz*-*112)q12q2 + (zz*—*122)qlq22 +
(zz*—*l33)qlq32 + (zz*—*233)q2q32]Pz2 + [(xy*—*ll3)q12q3 +
(xy*-*123)q1q2q3 + (xy*-*223)q22q3 + (XY*'*333)q33](PxPy + Pny) +
<-n2/4)[<<xx*—*1> + <yy*—*1) + (zz*-*1>>q1 +

((XX*-*2) + (yy*-*2) + (z2*-*2))q2] +

[<z*3*111)<q13p3+p3q13) + (z*3*112)(q12q2p3+p3q12q2> + <z*3*122>x
(q1q22p3+p3qlq22) + (2*3*222)(q23p3+p3q23) + (z*3*133)(q1q32P3+
p3q1q32) + (z*3*233)(q2q32p3+p3q2q32) + (z*1*113)(q12q3p1+p1q12q3)
+ (z*l*123)(qlq2q3p1+p1qlq2q3> + (z*l*223)(q22q3p1+qu22q3) +
(2*1*333)<q33p1+p1q33) + (z*2*113)(q12q3p2+p2q12q3) + (2*2*123)x
(qquQ3p2+P2q1q2q3) + (2*2*223)(q22q3p2+p2q22q3) + (2*2*333)X

4

4
(Q33P2+qu33)1Pz + 21“55U=<11111C115+k22222‘125“1<11112C11 q2+k12222qlq2

3

4 2 3 2
+k13333Q1Q34+k23333Q2q3 +k11122q1 q22+k11222q1 qz3+k11133q1 q3 +

k22233qz3Q32+k11233Q12Q2Q32+k12233q1Q22q321 + ......

 

 

20

. + [(—*33*111>(ql3p32+p32q13) + (-*33*112)(q12q2p32+p32q12q2> +
(-*33*122)(qlq22p32+p32q1q22) + (-*33*222)(q23p32+p3q23) +
(-*1l*l33)(qlq32p12+p1q1Q32) + (—*11*233)(q2q32p12+p12q2q32) +
(—*12*l33)(q1q32p1p2+p1p2qlq32) + (-*12*233)(q2q32P1P2+P1p2q2q§)
+ (-*22*133)(q1q32p22+p22q1q32) + (—*22*233)(q2q32p22+p22q2q32)
+ (-*l3*ll3)(q12q3p1p3+p1p3q12q3) + (-*l3*223)(q2q3P1P3+P1P3q2q3)
+ (—*13*123)(qlq2q3plp3+P1P3q1QZQ3)+(-*23*1l3)CQ§Q3P2P3+P2P3QEQ3)

+(—*23*123)(qlq2q3pzp3+P2p3q1q2q3)+(-*23*223)(q3q3p2p3+p2p3q§q3)]

(3.9d)
H4 = 0%53%\<C\<d(dp*_*ade) qaqchqux P(5+
Z
0% ;a$b\(c\<d( *e*abcd) (qaqchqdpe+ peqaqchqd)%( +
gif;§éhgc<d(-*ef*ade)(qaqchqdpepf + PePfanchqd> +
(—n2/4§§;E§ Gid*-*ab)qaqb +
Z.
zmdhaégfig(kabcdefqaqchqdqeqf (3.9e)
\ \

The detailed expressions of the coefficients in (3.9b) to (3.9d) are

listed in Tables VI to X. Table XI lists the nonzero coefficients in

the fourth—order Hamiltonian (3.9e)

Each of the coefficients in Tables VII to XI is equal to coefficient

or coefficients of the same type, that could be obtained by permuting

the indices of vibrational operators in the original one.

 

21

Table VI. Coefficients ¢Xfi*—*a) and (z*b*a) in the first-order
Hamiltonian H1

 

 

(xx*-*l) =
<yy*—*l) =
(zz*—*l) =

(xy*_*3) =

-(h2flll)l/4alxx/2Ix2 (xx*-*2) = —(fi2/7‘2)1/4a2xx/21x2
-<'52”\1>1/4a1yy/21y2 (yy*-*2) = —CHZAK2)l/4aZYY/2Iy2
-(-n2A?xl)1/4alZZ/2Iz2 (zz*—*2) = —(-h2mz)1/4azzz/2Iz2

—(n2/23)1/4a3XY/2Ix1y <yx*-*3> = -(n2/75)l/4a3XY/21x1y

 

(z*3*1)

(z*3*2)

= —(%3/31>1/4‘§3/Iz (z*l*3) = +c21/23)1/43:3/Iz
= -(%3fll2)l/4$;3/Iz (2*2*3) = +(72/“3)1/4§23/Iz

 

 

Table VII. Coefficients G“fi*-*ab) in the second—order Hamiltonian H2

 

 

(xx*-*ll)
(xx*—*12)
(yy*-*12)
(xx*—*22)
(xx*—*33)
(zz*—*12)
(Xy*-*l3)
(xy*-*23)

(yx*-*13)

= 3hsin2172IX2K11/2 (yy*-*ll) = 3ficosZfi/21y2flll/2
=4hsin’6COS/K/I§(7il7\2)l/4 (zz*—*ll) = 3fi§232/2122211/2
=—4hsin1cos'6/I§(lez}/4 (ZZ*—*22) = 313:32/2122fl21/2
= 3ficos2372lx2221/2 (yy*—*22) = 3fisin%fl21y2221/2
= 3h/2Izlx231/2 (yy*-*33) = 3n/2Izlyfl3l/2
=‘3h3135‘23/Iz20h7‘2)“4 (ZZ*-*33) = 0
=-3n[(Ix/12)1/2cosg+(1y/Iz)l/Zsing]/21X1y(2i33)1/4

= n[5(IX/Iz)l/Zsin1-3(Iy/Iz)1/2cosgq/21x1y61223)1/4

= (xyi<—*l3) (yx‘a'C—ft23) = (xy*—*23)

 

 

 

 

22

Table VIII. Coefficients €x*c*ab) in the second-order Hamiltonian H2

 

 

(z*3*ll) = 51/27131/4alzz Q3/2211/2122
(2.3.22) = hl/2g31/4azzz353/2n21/2122
(z*3*12) = hl/Zfi31/4(a1223;3 + azzz§13)/2(“{”2}/41§
(2*1*13) =—n1/2alzz§13/2231/4122
(z*l*23) :_fi1/lel/4a2zz 13/2(2233)1/4I§

(z*2*13) =ehl/2221/4alzz333/2(2123)1/41§

 

(z*2*23) =—hl/2aZZZS;3/2k3l/4Izz

 

 

Table IX. Coefficients (—*cd*ab) in the second—order Hamiltonian H2

 

 

 

(_*33*11)=(23/21)1/2$;32/4IZ (-*11*33)=(llflk3)l/23132/4Iz
(-*33*12)=(fi3zflhi22)l/4513§23/212 ('*12*33)=6ki225132)1/4513§23/212
(—*33*22)=GK3AA2)1/23332/412 (—*22*33)=(32/33)1/25232/412
(-*23*13)=—(22/21)1/4§13123/212 <-*13*23)=-(21(A2)1/4513523/212
(-*23*23>='3232/212 (—*l3*l3)=-5132/212

 

 

Table X. Coefficients @P*—*abc) in the third—order Hamiltonian H3

 

(xx*~*lll) = :fi3/Zsin33/Ix5/Zzi3/42

(xx*—*222) = :n3/2cos3z/Ix5/2fl23/42

(xx*-*112)
(xx*—*122)
(xx*—*133)
(xx*-*233)
(YY*“*111)
(YY*'*222)
(YY*‘*112)
(YY*'*122)
(yy*-*l33)
(yy*-*233)
(zz*—*lll)
(zz*-*222)
(zz*-*112)
(zz*-*122)

(xy*—*ll3)

(xy*—*123)

(xy*-*223)

(xy*-*333)
(yx*—*113)

(yx*-*123)

23

Table X (cont'd)

-llh3/25iancosX/6Ix5/zhll/Z%21/4
—llh3/zsin3hos2376Ix5/2A11/4fizl/2

ih3/2[23in/X+ (Ix/1y)1/2cos34/2IZIX3/2g11/4331/2
2h3/2[2cos,x+.(Ix/1y)1/2sin2a,2121x3/2321/4231/2
1fi3/2c053X/21y5/2213/4

-5h3/2sin33/61y5/2223/4
+11h3/2s1ngtos23761y5/22ll/2221/4
—1lh3/Zsin2vcos376Iy5/2gll/4221/2

253/2[2cos7{+ (Iy/Ix)1/231n%q/21y3/21£)11/4)31/2
+h3/2[(8/3)siIvX—-(Iy/Ix)1/2cogg]/21y3/ZIéZZl/423l/2
+fi3/25333/2125/2213/4

253/2gi33/2125/2223/4
—3fi3/2$135232/2125/%111/2221/4
+3fi3/25233132/2125/2311/4%21/2

+fi3/2 ' + I 1 1/2 2 + I I 1/2 '
211[§“i‘7§§:f‘/22§17éy’ 1 W W

x y z
+h3/2[3cos +7(I /I )1/2sin cos —8(I /I )l/zsin cos —
4sin23]/6:EIyIzX/2?%ikjk3)¥>4‘1 x y 3/ ’5

+fi3/2[sin COS’X+'(I gly)l/2c0322{+ (Iy/Ix)l/zsinzg]/

ZInyIzl 2231/22317
+h3/2/2(1x1y123)1/2%33/4
(xy*—*ll3), (yx*-*223) = (xy*—*223)
(xy*-*123), (yx*-*333) = (xy*—*333)

 

 

 

 

 

24

Table XI. Coefficients (i*d*abc) in the third-order Hamiltonian H3

 

 

(z*3*lll), (z*3*112), (z*3*122), (z*3*222), (z*3*l33), (z*3*233),
(z*1*113), (z*1*123), (z*l*223), (z*l*333), (z*2*113), (z*2*123),
(z*2*223), (z*2*333)

The detailed forms of these coefficients are not required for our

calculation.

 

 

Table XII. Nonvanishing coefficients in the fourth-order Hamiltonian
H4

 

 

(l) (7‘ *-*abcd):

 

CK? abcd
XX 1
yy ] 1111, 1112, 1122, 1133, 1222, 2222, 2233, 3333

zz ]
xy = yx 1113, 2223, 1123, 1223, 1333, 2333

 

 

f2) fd*e*abcd):

 

d. e
z 1 ]
2 11113, 2223, 1123, 1223, 1333, 2333

3 1111, 1112, 1222, 2222, 1122, 1133, 1233, 2233

 

 

 

 

25

Table XII(cont'd)

(3) (—*ef*abcd):

 

 

ef abcd
11 ]
12 = 21 ] 1133, 1233, 2233, 3333
22 ]
13 = 31 ]
23 = 32 ] 1113, 1123, 1223, 2223, 1333, 2333
33 1111, 1112, 1122, 1222, 2222, 1133, 1233, 2233

 

 

(4) (*x*-*ab)=

These coefficients have been listed in Table VII.

(5) kabcdef:

abcdef = 111111, 222222, 333333, 111112, 122222, 111122, 112222,
111133, 113333, 222233, 223333, 111222, 111233, 122233,

 

The detailed forms of all these coefficients are not required for our

calculation.

 

 

4. ONCE—TRANSFORMED HAMILTONIAN

The terms in H1 that truly contribute to the first—order energy
are absent in the case of asymmetric rotators. This is because there
are no degenerate modes of vibration. The S function of Herman and

Shaffer may expressed as16

S :23} %[S;d{5;a;-]pa§ 352% ([S;«;-;ab]qaqb + [Susabs—lpapbfld}
+ a$ <c[S;-abC;-]papbpc + C,Za$b[S;-c;ab](qaqbpcfi pcqaqb)/2
(4.1)
the coefficients [S;«p;a;—], [S;x;-;ab], etc. being given in Tables
XIII to XVI.We obtained H' of (2.21) from (2.20),(2.22),(2.23a) and

(2.23b) with the aid of the commutation relations
[qa,pb] = ihéab ; [Pd,PP] = —ifiP/x, cyclic

Throughout the calculations the vibrational operators were written in
symmetric form as (1/2)(qaqb...pnpm...+pnpm...qaqb...), but the rot-
ational operators are not symmetrized. They will be symmetrized in the
final stage of the calculations, and the symmetrized final results
will be Hermitian. Since in (2.23a) the anti—commutator {SR,Hfi} yields
rotational operators in symmetric form, we "desymmetrized" them by
subjecting the second terms to proper permutations of the indices.
This procedure leads to terms having coefficients that consist of two
parts, the second being a certain permutation of the first. Thus, for

instance, a symmetric form

26

 

 

27

[Y;xy;ll;12](l/2)(q1<I2P12 + P12q1q2)(PxPy + PyPX)
would be desymmetrized to

([sty;ll;12] + [Y§YX;11;12])(l/Z)(qquP12 + P12q1q2)PxPy

Table XIII. Coefficients [S;dfi;a;-] in the first contact transformation

 

 

[S;xx;l;—] = afx /21x2h3/29\13/4 [S;yy;l;-] = 3':Il>’>’/?-Iyzl‘i3/413/4
[S;xx;2;—] = azxx/21x2h3/2223/4 [S;yy523-1 = aZYY/ZI 2h3/2223/4
[S;zz;l;-] = alzz/2I22h3/2313/4 [S;xy;3;-] = a3xy/21x1y‘T‘I3/27‘33/4
[S;zz;2;-] = aZZZ/2122h3/27‘23/4 [S;yx;3;-] = a3xY/2leyfi3/2233/4

 

 

Table XIV. Coefficients [S;oQ-;ab] and [S;qfiab;-] in the first con—
tact transformation

 

 

3'13 (211mg) / (21-23) 12211/4231/4
3§3<flz+33n(712—13)12321/4a31/4

[S;z;-;l3] = [S;z;-;3l]

[S;z;-;23] = [S;Z;-;32]

 

[S;z;13;—] = [S;z;3l;—] = 2313311/4231/4/(21-23)15h2

[S;z;23;—] = [S;z;32;—] = 2525321/4231/4/622—75)Izh2

 

 

The once-transformed Hamiltonians, regrouped in true orders of

magnitude with the aid of (2.22), are found to be as in (2.21), where

 

 

28

Table XV. Coefficients [S;—;abc;-] in the first contact transformation

 

 

[S;-;lll;-] =
[S;-;112;-] =
[S;-;122;-] =
[S;-;l33;-] =

[S;-;233;-] =

—4mcklll/3fi3lll/2 [S;—;222;—] = -4Ick222/3fi3221/2

—4[ckllzfil/fi3fizl/2(431—22) = [s;—;121;-] = [s;—;211;-]

-4Hck12222/h3311/2(47b—fli) = [S;-;212;-] = [S;-;221;-]

—4nck133fl3/h37111/2(4713—21) = [s;—;313;-] = [s;—;331;-]

—4rck23323/n3221/2(423—22) = [s;-;323;—] = [s;-;332;~]

 

 

Table XVI. Coefficients [S;—;c;ab] in the first contact transformation

 

 

[S;-;l;ll]
[S;-;2;ll]
[S;-;l;12]
[S;-;2312]
[S;-;3;l3]
[S;-;l;22]
[S;-;3;23]
[S;-;l;33]

[S;-;2;33]

-2rccklll/mll/2 [s;—;2;22] = —27Lck222/‘fia21/2
-2tck112(221—22)/fiflzl/2<421-22)
—4chlléflll/2/fi(4fll-22) = [s;-;1;21]
—4Tck122221/2/fi(422-21) = [s;—;2;21]
—41[ck133231/2ffi(4513—711) = [S;-;3;31]

—2mck122(222421)/nAll/2<422—31)
~41ck233231/2/fi(4a3—22) = [s;—;3;32]
-2npkl33(233-21)/fiflll/2<47t4kl>
—27r.ck233 (223412) 5121/2 (433-32)

 

 

 

 

29
h ' = H (4.2)
H0 is given in (3.9a)

h1' = H1 + [iS,HO]V = 0 (4.3)

h2' = H2 + [15,111/2]V + [iS,H0]R =
dE-TrSQy/ayfi—rwopfipgpf +°(F%/Za(2;0(flg;a;—)pag(PP/,PZ, +
0%az<b[(2;w{5;ab;-)papb + (2;¥[3;-;ab)qaqb]g(% +

 

0(agbgc (2;u;abC;_)papbpcBX +
0,2,3“, (2;0(;<:;ab)(1/2)(qaqbpc + pcqaqb)%( +
aEbwédas-wdwb)(1/2)(qaqbp¢pd + pcpdqaqb) +

magma”?3'5ade)qaCIchqc1 (4.4)

Coefficients in the first sum are listed in Table XVII, and the rest

of the coefficients are listed in Appendixes IV to X.

 

h3' = H3 + [iS,(H2 + (2/3)[iS,H1/2]v + (l/2)[iS,Ho]R)]v +
[15,H1/2]R =
4&5; (3;qp’yf;-;a)anuPp 1’ng +
dgggxwgflbmfll/Zflqapb + pbqa)%(PgBj +
d%a;%\<c(3;o(p;bc;a) (1/2)(qapbpC + pbpcqamflg +

a(bgc(33“fl3-:abc)qaqchfi.39 + h3* (4.5)

The coefficients appearing in (4.5) above are listed in Appendixes XI
to XIV. h3* is the portion of h3' that is discarded henceforth; it
consists of terms of the types {qa,pbpcpd}g(, {qaqch,pd}g>< and pure
vibrational terms which do not contribute to the energy through the

fourth order of approximation.

 

30

114' = H4 + [15,013 + [iS/2,(H2 + [iS,Hl/4]V + [iSsH0/31RJV)]V +
[15,H1/31R)]V + [is,(H2 + [13,111/3]V + [iS’HO/ZJRHR
«%r4““°%53”21323323 + 333231} +
0(ng afifimafigfiflabmaqb + (4;dpg’{;ab;-)papb}g<%P1P§

+ h4’!‘ (4.6)

The coefficients appearing in (4.6) are listed in Appendixes XV to XVII.
h4* is the portion of h4' that is discarded henceforth; it consists of
pure vibrational terms, terms whose matrix elements in a symmetric—top
basis fall outside the diagonal blocks E+, E_, 0+, O_, and terms

quadratic in the components of rotational angular momentum QX' These

 

terms either do not contribute to the energy through the fourth order
of approximation, or represent negligible corrections to the terms re—
tained in the final form of the Hamiltonian.

The first sum in (4.6) is not desymmetrized in the components of rot-
ational angular momentum since h4' is already in the form in which it
will enter the final form of the Hamiltonian.0ne could also symmetrize
the second sum at this point,but instead of doing so now, we shall
symmetrize this sum later (Appendix XVI). We expect that identical co-
efficients in that appendix will automatically symmetrize the cor—

responding terms in this sum, and this is, in fact, the case.

31

Table XVII. Coefficients (2;Q@?63-;-) in the second—order

Hamiltonian h2

l

 

 

(2;xxyy;-;-)
(2syyzz;-;-)
(2;zzxx;—;-)
(2;nyy;—;-)
(2;xxxx;—;—)
(2;yyyy;-;-)
(2;zzzz;-;-)
(2;yyXX;-;-)
(2;zzyy;-;-

(2;xxzz;—;-)

(23XYYX3';—)

-[(a§x agy )441 + (agx agy )/22]/81x21y2

_[(alyya122)ztl + (azyyaZZZ)/az]/81y2122

_[(alZZalXX) /a1 + (aZZZaZXX) /Z2]/8Izzlx2

-(a3XY)2/81§
_[(alXX)2/fll
—[(alyy)2/fil
-[<alZZ>2/3l
(2;xxyy;-;-)
(2;yy22;-;-)

(2;zzxx;—;—)

(2;yxxy;-;—) =

13713
+ (a§X )2/421/813
+ (aZYY)2fi22]/8I§

+ (aZZZ)Zflxz]/81g

(2;yXyX;—;—) = (2;xyxy;-;-

 

 

 

 

 

5. TWICE-TRANSFORMED HAMILTONIAN

The second contact transformation that diagonalizes H in the vi—
brational quantum numbers through the second order of approximation
has been constructed by Amat and Nielsen16 from the basic Herman—
Shaffer functions. The portion of this function that is needed in
our case can be written as follows:

C7 1%3; [6;«M;-;a]qa%.Ff;P3 +°ég [0“;Orp;b;a](l/2)(qapb+pbqa)I;(Pfi

+0, c%,$1,[U;o(;ab;c](l/2) (qcpapb + papchfifi

ii on ._.
51%“,1 ,d, ,abCanqchI’.( (5.1)

The coefficients [dgdfia§—;a], [U;ufi;b;a], etc., are given in Tables
XVIII to XXI, with the constants Aabc and Babc appearing in BT;u;c;ab]
and [6;x;-;abc] as listed in Table XXII.

Expanding'f = exp.(i2zc) and’b-l = exp.(-i%?¢) in power series in
o’and collecting terms of the same order of magnitude in H@ =TII'1'1

we obtain:

@_
Ho ‘ ho
@_
H1 ' hl
H@=h'+i[ch']
2 2 ’ O
H@=h'+'[o’h']
3 1 ' 1
I V I
H4@= h. + new 1 —<1/2>[rr, [6,110 11 (5.2)
In (5.2) h' , h; , h; , etc., are the zeroth, first, second-order
portion, etc., of the once—transformed Hamiltonian regrouped to true

oreders of magnitude which we have obtained in Chapter 4 from the

original once—transformed Hamiltonian with the aid of (2.22).

32

 

33

Table XVIII. Coefficients [d;ufifi;-;a] in the second contact trans—
formation.

 

 

[6;ZZZ;-;3] = (2;zzz;3;—)/7\3l/2

BS;xxz;-;3] = (2;xxz;3;-)fla3l/2 [6;zxx;—;3] = (2;zxx;3;—)/7\31/2
BY;YYZ;'33] = (23YYZ;3;—)/fl31/2 [S;zyy;—;3] = (2;zyy;3;-)/23l/2
[O’sxzx;-;3] = (2;xzx;3;-)/'7\3l/2 [6;yzy;-;3] = (2syzy;3;-)/7\31/2

[6;xy2;-;j] = [<Y;ZYX;-;j] = (2;xyz:j;-)/7lj1/2

[0‘;xzy;-;j] = [6;yzxrsjl = (2sxzy;j;-)/?»jl/2

. . . 1 2
[o’;yx23-;J] = [O’szxy;-;J] = (2;YXZ;J;')/9)'j /

j = 1,2

 

 

Table XIX. Coefficients [Ozxfi;b;a] in the second contact transform-
ation.

 

 

[O’;o¢o<;j;j] =
[6;0<o<;2;l] =
[6”;°<°<;l;2] =
[5;xy;3;j] =
[0";xy;j;3] =

[0“;yX;3;J'] =

[(2;o<o<;jj;-) - (2;°<°<;-;jj)/h2]/2?tjl/2 ; °(= x,y.z
[(2;°‘°’;12;-)9\11/2 + (2;o(o<;-;12)7\21/2/‘h2]/(9\l —7\2)
[(2;ob(;12;—)7L21/2 + (2;oe(;—;12)211/2/n2]/(7\2 _’\l)
[(2;xy;j3;-)7\j1/2 + (2;xy;-;j3)?\31/2/‘r'12]/(9~j -7‘3)
[(2;xy;1'3;—)331/2 + (2;xy;—;j3)’>~jl/2/t2]/(9\3 —7\j)

[6;xys3;j] j = 1.2 [6;yX;j;3] = [6;xy;j;3]

 

 

 

 

Table XX. Coefficients K7kx;c;ab] in the second contact transform—

ation.

34

 

 

[6;zsjjs3]

[632;12;3]

Bf;z;l3;l]

[032;23;2]

[6;z;l3;2]

[632;23;l]

[032;21;3]

[032;32;j]

-(2; 2; 3;jj)A j3/fi2 + (2; z;j;j3)Bj -/fi2
—(2; Z;jj3; —)gj' 3,3(1'1'25j3) ; j = ji:§33

—(2: z;3;12)A12:/h2 + (2 z; 1; 23)B 3 2/52

-(2; z; 1 ,13)A113/h2 + (2, z; 1 ,13)Bll 3/n2
+2(2; z; 3; 11)}3l3 l/fi — 2(2; 2; 113 ,-)B13’l

—(2;z;2;23)A223/‘hz + (2;Z;2;23)322 3N2
+2(2;z;3;22)B23 2/n - 2(2;z;223;—)B23 2

—(2; z; 2 ,13)A 123/52 + (2; z, 1 ,23)B12 3/52
+(2; z;3;12)B23 l/n2 — (2; z; 123,-)B13 2

 

—(2; z; 1- ,23)A123/h2 + (2; z; 2 ,13)B2 3/h2
+(2; 2 3;12)Bl3 2/h2 - (2; 2; 1233'132m

[6:z;12;3] [632;3l;j] = [6;2;l3;j]

[6:Z;23;j] j 1,2

 

 

Table XXI. Coefficients BT;x;-;abc] in the second contact transform—

ation.

 

 

k7;z;-;1l3] = —(2;z;1;13)1313,l — (2;.v.;3;11)1311,3 4h2(2;z;113;-)All3

B7;z;—;223] = -(2;z;2;23)B23,2 - (2;z;3;22)B22’3 =h2(2;z;223;—)A223

[6;Z;—;123] = ”(2; Z, .1‘ ,23)B23 l - (2; Z, 2' ,13)B13 2 " (2;Z; 3;12)B12,3—
h2(2 z;123,—)A123

[Gfiz;-;333] = —(2;z;3;33)B33’3 — h2(2;z;333;—)A333

 

 

35

Here, too, it is advantageous to regroup H@ in true orders of magni-
tude. Upon subjecting H@ to.regrouping of terms according to true

orders of magnitude we have:
@=@= @. @ @ @ @
H h hO + hl + h2 + h3 + h4

where h0@ = H0
h1@ = o
h2@ = h2' + i[c;H0]V
h3@ = h3' + ikT,HO]R

h4@ = h4' + itcr,h2']V — (1/2) [0", [53h0'1v1v
The last two terms in h4' can be written as follows:

i[03h2']V - <1/2>[<r,[cr,ho']v1v = i[63(h2' + (i/2)[63h0']v)]v
=-i[d.(h2'/2 + <1/2)<h2' + i[03ho']v))]v

= iBT,(l/2)(h2' + h2@>1V = <1/2>[c,<h2' + h2@>1v

Moreover, it can be shown that i[03h2@]V does not contribute to the
energy through the fourth order of approximation except in cases of
accidental near—degeneracy.22 The same is true of h3@. Thus for the
calculation of energies through the fourth order of approximation we
only need to consider:

h0@ = H0

h2@ = h2' + i[CXH0]V

h4@ = h4' + i[o*,h2']V

h@ = h0@ + h2@ + h4@ (5.4)

 

36

Table XXII. The constants Aabc and Bab,C

 

 

 

 

 

Aaaa = —2fla1/2/3; a = 1,2,3
A112 = —2alza21/2(421 "%2) A122 = ‘232/“11/2(4g2 ' “1)
A113 = ‘211/7‘31/2Uml ‘ 7‘3) A133 = -2“3/7\11/2(4’7\3 -711)
A223 = '2“2fih31/2(4a2 ' Q3) A233 = "2%3/x21/2(4“3 ’6a2)
A123 = zékla275)l/2D123 ; ( and all permutations of indices. )
Baa,a = '(1/3)“a1/2 3 a = 1,2,3 B12,3 = 4x3l/2(“3 ‘9‘1 '5K2)D123
B13,2 = ‘azl/Zfikz ‘Afii "’23)D123 B23,1 = 'flll/zcxl -9\2 '7‘3)D123

B12,1 = ‘311/2/(4R1 -0x2) B12,2 = '“21/2/(4A2 -7‘1)

B13,3 = -131/2/(423 ‘9\1> B23,3 = “751/2/(4q3 ‘7‘2)

B13,1 = 'all/z/(4Zl ‘ ”3) B23,2 = ‘751/2/(479 ' “3)

B11,2 = CA2 - 221)/A21/2(421 -%2) B22,1 = (“1 — 27‘2)/7‘1l/2(47‘2 "m1)
B11,3 - (A3 - 221)/93l/2(4Q1 -%3) B33,1 = (”1 - 223)/a11/2(433 431)

B22,3 = 27b)/“3l/2(47& 'fl3) B33,2 = 612 - 233)fi“21/2<4”3 '“2)

b.)
I

 

In A , B and the corresponding constants obtained through
123 12,3

permutations of indices,

D123 = [gall/2+221/2+fi3l/2)Call/24321/2_A3l/2)(fill/2-221/2+Q31/3)

XCA11/2+%21/2-fl31/2)]_1 ; ( all permutations of indices )

 

 

37

Of the terms in H0 in (3.9a) only the first three will be kept in h0@
since the remaining terms are purely vibrational and of no interest

@

here. However, in the second term of h2 in (5.4) the pure-vibration-
al operator—terms in H0 must be included in the commutator [03H0]V

for it is precisely these terms that contribute to h2@, since for the

pure-rotational operator—terms in H0 (2—23b) yields

i[O',Ho]V = i[O‘V,HOV](l/2)(O‘RH0R + HORO’R)

Thus we have
h2@ = hz' + i[ch,HOV](1/2)(o§.1 + 1.0;.) = h2' + i[c7V,HOV10'R

and with 0 from (5.1) and the last six terms in (3.9a) for HOV, one

can show that

h2@ =0??? (2;uf52fX;—;—)g(r;936Pg +

0%; [fi2(2;o((5;aa;-) + (2;d{5;-;aa)](qa2 + paz/hz)g(% + 112*
(5.5)

i.e., except for h2*, h2@ is diagonal in all vibrational quantum

numbers; h2* consists of pure—vibrational operator—terms originating

from hQ'.Even though this portion of h2@

is not diagonal in the
vibrational quantum numbers, it can be disregarded as it can be shown
that the pure—vibrational operator-terms in h2@ would be diagonal if

the complete transformation function of Goldsmith et al had been used.

Substituting (4.6) for h4', (4.4) for hz' and (5.1) forC7 into the

 

 

38

last equation in (5.4), we obtain

@ _ . ._._
h4 -o((%%569 (IV,o({5’Jf£f), , )g‘?) 1361951381}, +
“g; agb [(IV;0(F/()f«*’;-;ab)qaqb + (IV;°(PX§;ab;-)papb]%lP@lé’P5
+ 114* (506)

where h4* includes terms of the type r4P2 and rkP6'k, k = 1,3,5.

 

Following the notation introduced by Amat and Nielsen,22 rn and Pm are
used to denote products of any n vibrational and any m rotational op—
erators, respectively; h4* is discarded henceforth since herein terms
of the first type represent fourth-order corrections to zeroth—order
rotational constants of rigid rotator, while terms of the second type

would contribute only to the energy in an order higher than the

 

21
fourth. The coefficients in h4@ can be shown to be the following:

<1v;«%‘£{>;—;—> = (Mggéqéfirr) + (4w;£§’e!—z—)

(1v;°(%5;—;ab) = (4;o(WS;—;ab) + (4!3(&a_’,§_!-!ab) +
(4!gré,g£!—!ab) + (4!o_(,&fl!-!ab)

(IV;o(/5’(rf;ab;-) = (4;oqz,ys;ab;-) + (4!%2!ab!-) +

(4!gé,@§!ab!—) + (4!g,M!ab!-) (5.7)

.(4;5Q,5&#3;—;—), (4;afi37;—;ab) and (4;dfi37;ab;-) are the coefficients
in h4', the underlining on the groups of indices indicating how they
can be constructed from the original coefficients to accomplish the
desymmetrization of the corresponding rotational operators. Thus, for

instance,

(4;z£,2gfaF;-;-) = (1/2)[(4;o(fl,gfaf>;-;-) + (4;6{’,0([&ga(;-;-)]

 

39

since the corresponding rotational operator—term
(1/2) (EngPfiPfPa PF + Bfff'Pa; 13(1)? ) is desymmetrized to
E‘fieBxggaafio.
The rest of the coefficients in (5.7), i.e., (4!QQZ;,§§fi!-!—), etc.,
belong to the terms in i[O§h2']V. Again, the underlines are used here
to show how they can be constructed from the original coefficients.
These original coefficients are given in Appendixes XVIII to XXIV.
The desymmetrization.procedure mentioned above was used merely to

@

facilitate writing the coefficients of h4 in (5.6) in general form.
In our actual calculation all terms were left in their original sym—
.metric—form since they are already in the correct form in which they

enter the final form of the Hamiltonian.

We now summarize the final result. From (5.4) we have
h@ = ho@ + h2@ + h4@ (5.8)

Excluding the pure Vibrational terms in H0, we have, from (5.4) and
(3.9a),

h0@ = (1/2)[ PXZ/IX + Pyz/Iy + Pzz/Iz 1 (5.9)

No further rearrangement is required in (5.9) since it is already in
the desired final form. One identifies the coefficients therein as

being proportional to the equilibrium rotational constants A0, B0 and
C0. The coefficients (23Xfi@§E—;—) in the first sum of (5.5) have been
listed in Table XVII. Following the notation of Chung and Parker23 we

write

<2;orr>z{f;-;-) = (1/4)Twpa’f

 

40

and in terms of the coefficients 1;¢3g3 the first sum of (5.5) and

Table XVII yield

(1/4)[q;’xxxxP4 x +‘fyyny Y4 + LzzzzP z 4x+rtxxyy(P 2Py 2 + Py 2P x2) +

2 2
Tyyzz(Py PZ + Pz 2P yZ) + TZZXXQZZPXZ + PXZPZZ) +
T + 2 2
xyxy(PxPnyPy PyPXPny + P xPy Px + Pny Py) ]

 

With the aid of the commutation relations [Ex’gé] = -ifiBa, cyclic,

the last term can be written as
(1/4)[ 2(Px2Py2 + PyZPXZ) + 352P22 — 2h2Px2 - 2n2Py2 1T;yxy
Thus we have, from the first sum in (5.5),

4 4 4
(1/4) [ ’L’lPx +"L’2Py + "E3Pz + ”(4(Pyzpzz + Pzzpyz) +

 

T(P2P2+P2P2)+'t>'<(P2P2+P2P2)+
5 z x X z 6 x y y x
2 2 2 2
Tgn (3Pz — 2Px — 2Py ) ]

The last term is of the second order but will now be grouped with h0@,

and the coefficients of sz, Py2 and Pz2 therein represent centrifugal
distortion corrections to the rotational constants A0, B0 and C0,

respectively. The coefficients'F “U etc., or their numerical—

XXXX’ YYYY’

subscript forms Tl, Ti, etc., are listed in Table XXIII.

The coefficients (2;dfl;aa;—) and (2;dfl;-;aa) in the second sum of
(5.5) are given in Appendixes V and VI, respectively, along with the
coefficients of the off—diagonal terms in h2'. We note that for a = b

(2;o((é;ab;-) and (2;0(fi;—ab) vanish if 0(74’9 . Hence the second sum of

(5.5) can be written as follows:

41
3 2 2 2 2 2
g 8%“ ‘5 (2;ob<;aa;-) + (2;ob<;-;aa) ]( qa + Pa f5 )Pix

Table XXIII. The coefficients "tom’s.

 

 

Txxxx = T1 = "[(alxx)2/)Ll + («392‘x 2/712]/2Ix4

Tyyyy = 1’“2 = ‘[(alyy)2/’/\1 + (815332/7‘21/213'4

Tam = 73 = -[<a1“>2/7x1 + <a§Z)2/9\21/2iz4
Tyyzz = Tzzyy = ’12. = -[(alyyalzz/’Al) + (agyazzzmzn/zlyzlzz
Tzzxx = Txxzz = T5 = -[(alzza1XX/’A1) + (afizazxx/Olzn/zlzzlxz
Txxyy = Tyyxx = ”E6 = _[(alxxalyy/Al) + (afixazyy/zzn/szlyz
Txyxy =Tyxyx = Txyyx =Tyxxy = T9 = -(a3xy)2/(21x21y29x3 )

76* =T6 + 2T9

 

 

If for a given vibrational state the eigenvalues of pa2 and qa2 are
written as
(pa2> = ( Va + l/2 )nz, <qa2> = ( va + 1/2)
a = 1,2,3
respectively, where va is the harmonic oscillator quantum number cor—
responding to normal mode a, we can write the above double sum as
3(1sz + U-(ZPyZ + 3(3P22

where 3
l = 521 (Va + 1/2)Eh2(2;XX333;-) + (2;xx;-;aa)]
”N2 = £21 (Va + 1/2)[fi2(2;yy;aa;—) + (2;yy;—;aa)]

M3 = 521 (Va + 1/2)[fi2(2;zz;aa;—) + (2;zz;-;aa)] (5.10)

 

 

42

SinceiHj ; j = 1,2,3 are the coefficients of terms quadratic in the
components of angular momentum, these terms, although of the second
order, are now also grouped into h0@. Therefore, they will be added
to ho@ along with the centrifugal distortion terms from the first sum
of (5.5). The3ij's represent the vibration-rotation coupling correct-

ions to the equilibrium rotational constants A0, B0, C0 .

 

The first sum in (5.6) contains terms sextic in the components
of rotational angular momentum, the coefficients of which are given

@ @

in (5.7). Since there are no sextic terms in ho or hz , no ambi-
guities will be introduced if the "IV” and "4" are omitted from the
coefficients. Thus we simplify the notation and write «*336?F)9

Gflgqfflaf) and Gfl63*5£f) for (IV;dfiQflEF;-;—), (4;dfl{6fi5fi) and

(4!agggia0!—!—), respectively. Comma and asterisk are used in the

 

second and third coefficients, respectively, to show which group of
indices originates from the transformation function, and which one
from the Hamiltonian. In our notation the group of indices preceding
the separating comma or asterisk is the one coming from the trans—
formation function. We use the comma when the function is S, and the
asterisk when the function is<7. The detailed expressions of the co—
efficients Gxfl;gz5f5 are given in Appendix XV. These are the coeffi-
cients of the sextic terms P6 in h4'. The coefficients GX@J%S£f) are
given in Appendix XVIII. One can see in these appendixes that the

operators PX, P and P2 always appear in even power in all the sextic

y
pure-rotational terms. To have the sextic terms in the form arrived

10

at by Yallabandi and Parker, the original form

43

(945:5?ng PdeBdP5,%l?) + PXIg-PFJP’DQPfi ) +

(dwsiepu .PxPfiPngPaPF +- PSIZLIEJ PXPp P'b’ )
was subjected to rearrangement based on the angular momentum commut—
ation relations. In carrying.out this rearrangement, Table II of
Kneizys, Freedman.and Cloughll was very useful. Entries 58 through 105
in this table were subjected to permutations of indices to obtain op-
erators with P2 symmetfically placed with respect to PX and Py. As a
result of extensive rearrangement we obtain sixteen new coefficients,
of which ten belong to P6—terms and the other six belong to P4—terms.
Three of the former stay in their original form; they are the coeff—
icients of PX6, Py6 and P26. The rest of the coefficients are found
to be certain linear combinations of the original coefficients. Act—
ually we also obtained terms quadratic in the components of angular
momentum, but these were discarded since they are negligible relative
to the coefficients A0, B0 and C0 in the Exz—terms. Following Yalla—
bandi and Parker,10 these sixteen coefficients are calledd)l t04716.
They are listed in Table XXIV. Where these coefficients stand in the
final form of the Hamiltonian will be shown later.
The coefficients of the r2P4—terms in (5.6) comprise eight diff-
erent coefficients, two coming from h4' and six from i[O:h2']V. The
notation in (5.7) will now be simplified as follows:

(4;o/pfl;-;ab) ——> («ea/fab) <4;o((53rf;ab;—> —-—> (abom’f)

“wags !—!ab) —9 («Micfiaw (410(ggf,§:ab2—) —> (abdfiW)

(4!o((5,fi£!—!ab) -> (0(f5*wab) (4!o(@,3ff!ab!—) -—> (abosw’i)

(4!o(,{§’6’5!—!ab) -> (0(*M5ab) (4!o(,fi2{5!ab!—) a (mam)

 

44

Table XXIV. Coefficients<bj

 

 

$1 =

<Ps=

—e}.~er
H c
0I II

(xx,xxxx)

(yy.yyyy)

— (zz,zzzz) + (zzz*zzz)

(yy.YYXX) + 2(yy.xyxy) + 4(xy,xyyy) + (1/2)(XX.yyyy)
(xx,xxyy) + 2(XX.XYXY) + 4(xy.XYXX) + (1/2)(yy.XXXX)

(zz,zzyy) + (zz,zyzy) + (l/2)(yy,zzzz) + 2(zzz*yyz) + (zzz*yzy)

‘ (yy,yy22) + (yy,zyzy) + (1/2)(22.yyyy) + 2(yy2*yy2) +

2(yyz*yzy) + (1/2)(YZY*YZY)

(xx—xxzz) + (xx,zxzx) + (l/2)(zz,xxxx) + 2(xxz*xxz) +
2(xxz*xzx) + (l/2)(xzx*xzx)

(zz,zzxx) + (zz,zxzx) + (l/2)(xx,zzzz) + 2(zzz*xxz) + (zzz*xzx)

_ 2(ztiXYY) + (XX,}7YZZ) + (yy,zzxx) + 2(ZZ:XYXY) + 4(xy,xyzz) +

(yy,zxzx) + (xx,zyzy) + 4(xy,zxzy) + 4(xxz*yyz) + 2(xxz*yzy) +
2(yyz*xzx) + (xzx*yzy) + 2(xzy*xzy) + 2(zxy*zxy) +4(xyz*xzy) +
4(xyz*zxy) + 4(xzy*zxy)

 

- —2(xx,xyxy) — 8(xy,xyxx) + 4(xy,xyzz) - (xx,zxzx) + (xx,zyzy) +

4(xy,zxzy) - 4(xxz*xxz) + 4(xxz*yyz) - 4(xxz*xzx) + 2(xxz*yzy) +
2(yyz*xzx) + (xzx*yzy) — (xzx*xzx) + 3(xzy*xzy) + 3(xyz*xyz) +
3(zxy*zxy) + 6(xyz*xzy) + 6(xyz*zxy) + 6(xzy*zxy)

" -2(yy.xyxy) - 8(xy.xyyy) + 4(xysxy22) + (yy.ZXZX) - (yy,zyzy) +

4(xy,zxzy) + 4(xxz*yyz) - 4(yyz*yyz) — 4(yyz*yzy) + 2(xxz*yzy) +
2(yyz*xzx) + (xzx*yzy) - (yzy*yzy) + 3(xzy*xzy) + 3(xyz*xyz) +
3(zxy*zxy) + 6(xyz*xzy) + 6(xyz*zxy) + 6(xzy*zxy)

(5/2)(zz,xyxy) + 6(xy,xyzz) — (zz,zxzx) — (zz,zyzy) +
6(xy,zxzy) — (zzz*xzx) - (zzz*yzy) + 3(xzy*xzy) + 3(xyz*xyz) +
3(zxy*zxy) + 6(xyz*xzy) + 6(xyz*zxy) + 6(xzy*zxy)

 

45

Table XXIV (cont'd)

($14 = 2(yy,-yxx) + 2(zz,zzxx) é 2(yy,yyzz) — 2(zz,zzyy) — 4(zz,xxyy)
-2(y-.ZZXX) + (ll/2)(yy.xyxy> - 5(22.’nyy) + 15(xy,xyyy) -
14(xy,xyzz) - (5/2)(yy,zxzx) + (5/2)(zz,zxzx) - (5/2)(yy,zyzy)
—(5/2)(zz,zyzy) — l4(xy,zxzy) + 3(zzz*xxz) — 3(zzz*yyz) -
(3/2)(zzz*xzx) + (3/2)(zzz*yzy) - 4(xxz*yyz) - 4(yyz*yyz) —
-3(yyz*xzx) — 5(yyz*yzy) —'(3/2)(xzx*yzy) - (3/2)(yzy*yzy) -
7(xzy*xzy) — 7(xyz*xyz) — 7(zxy*zxy) - l4(xyz*xzy) —
l4(xyz*zxy) — 14(xzy*zxy)

($15 = 2(xx,xxyy) - 2(xx,xxzz) — 2(zz,zzxx) + 2(zz,zzyy) - 4(zz,xxyy)
-2(xx,yyzz) + (ll/2)(xx,xyxy) — 5(zz,xyxy) + 15(xy,xyxx) -
14(xy,xyzz) -(5/2)(xx,zxzx) - (5/2)(zz,zxzx) — (5/2)(xx,zyzy)
+ (5/2)(zz,zyzy) — l4(xy,zxzy) -3(zzz*xxz) + 3(zzz*yyz) -
(3/2)(zzz*xzx) + (3/2)(zzz*yzy) — 4(xxz*xxz) - 4(xxz*yyz) —
5(xxz*xzx) — 3(xxz*yzy) — 2(yyz*xzx) - (3/2)(xzx*yzy) —
(3/2)(xzx*xzx) - 7(xzy*xzy) - 7(xyz*xyz) - 7(zxy*zxy) -
14(xyz*xzy) - l4(xyz*zxy) — 14(xzy*zxy)

 

4316 = -2(XX.xxyy) - 2(yy.y.VXX) - 2(xx,xxu) + 2(yy,yy22) + 8(zz,xxyy)
+2(xx,yyzz) + 2(yy,zzxx) - 5(xx,xyxy) - 5(yy,xyxy) + 8(zz,xyxy)
—8(xy,xyxx) — 8(xy,xyyy) + 12(xy,xyzz) + (5/2)(xx,zxzx) +
(3/2)(yy,ZXZX) + (3/2)(XX.Zyzy) + (5/2)(yyszyzy) + 8(Xy.zxzy) +
2(xxz*xxz) + 8(xxz*xzx) + 8(yyz*yyz) + 8(yyz*yzy) + 2(xzx*xzx)
+2(yzy*yzy) + (xzy*xzy) + (xyz*xyz) + (zxy*zxy) + 2(xyz*xzy) +
2(xyz*zxy) = 2(xzy*zxy)

 

Herein @Xfi;g§8f) and éipz%§bp) are as given in Appendixes XV and
XVIII, respectively.

 

 

In the simplified notation above, rotational operators' indices,
afigfi; either precede the indices of vibrational normal coordinates,
or follow those of the momenta conjugate to the normal coordinates.
Again, an asterisk is used to separate rotational operators' indices
coming from G‘and from Hi The two contact transformations, S and<5,
diagonalize H in the vibrational quantum numbers through second order

only. Thus we have both diagonal and off—diagonal elements in the

 

46

matrix representation of h4@. It can be shown,21 however, that the
terms that are off-diagonal in the vibrational quantum numbers va do
not contribute to the energy through the fourth order of approximat—
ion unless the vibrational states are spaced so closely that an accid—
ental nearedegeneracy results. For this reason we have not included

in Appendixes XVIII through XXIV the coefficients of terms off-diagon—
al in Va. The regrouping of the r2P4—terms into the final form of the
Hamiltonian can be accomplished either through commutator algebra

with the aid of the commutation relations [Fx,fi5] = _ifi25’ cyclic,

or by using the relations among the angular momentum operators of
Olson and Allen.24 This rearrangement has been done by Yallabandi and
Parker,10 and their results were usedo Following their notation, we

have, from the coefficients in Appendixes XVIII to XXIV,

P1 = Pxxxx = 2; (va + 1/2)[ (xxxxaa) + (aaxxxx)h2 + (xx*xxaa) +
(aaxx*xx)h2 ]

[’2 = Fyyyy = 2,, (Va + 1/2)[ (yyyyaa) + (aayyyyfi’l2 + (yy*yyaa) +
(aayy*yy)fi2 ]

f3 = fgzzz = 2; (Va + 1/2)[ (zzzzaa) + (aazzzz)‘h2 + (zzz*zaa) +
(aazzz*z)fi2 + (zz*zzaa) + (aazz*zz)‘h2 +
(z*zzzaa) + (aaz*zzz)‘h2 ]

fa = Pyyzz = 2; (Va + 1/2)[ (yyzzaa) + (aayyzz)h2 + (yyz*zaa) +
(aayyz*z)fi2 + (yy*zzaa) + (aayy*zz)‘fi2 +

(zz*yyaa) + (aazz*yy)‘h2 + (z*yyzaa) + (aaz*yyz)fi2]

=Pzzyy

 

 

47

r75 = fzzxx = 2.078 + 1/2)[ (xxzzaa) + (aaxxzz)‘l’i2 + (xxz*zaa) +
(aaxxz*z)l’i2 + (xx*zzaa) + (aaxx*zz)1't2 +
(zz*xxaa) + (aazz”‘xx)‘1‘i2 + (z*xxzaa) +(aaz*xxz)‘h2]
= Fxxzz
[)6 = {)xxyy = :3 (Va + 1/2)[ (xxyyaa) + (aaxxyy)'1’i2 + (xx*yyaa) +

(aaxx”<yy)'H2 + (yy*xxaa) + (aayy*xx)‘fi2 ]

 

= Pyyxx
F7 =f‘yzyz = :a' (Va + 1/2)[ (zyzyaa) + (aazyzy)1‘i2 + (yzy*zaa) +
(aayzy>"z)'h2 + (z*yzyaa) + (aaz""yzy)'1'12 ]

= szzy

{)8 =szzx = :8“ (Va + 1/2)[ (zxzxaa) + (aazxzxfi'f2 + (xzx*zaa) +

 

(aaxzx*z)h2 + (z*xzxaa) + (aaz*xzx)‘fi2 1
= [’xzxz
f9 = fxyxy = E, (va + l/2>[ <xyxyaa> + (azaxyxym2 + <xy*xyaa> +
(aaxy*xy)‘h2 ]
=Pyxyx
{910: {’zyyz = 2; (Va + 1/2)[ (zyyzaa) + (aazyyz)h2 + (zyy*zaa) +
(aazyy*z)‘fi2 + (z*yyzaa) + (aaz*yyz)‘h2 ]
P11= fyzzy = 23 (Va + 1/2)[ (aayzzy)‘h2 1
f12=fzxxz = % (Va + l/2)[ (2xxzaa) + (aazxxz)n2 + (zxx*zaa) +
(aazxx*z)'h2 + (2*xxzaa) + (aaz*xxz)h2 1
7313: fxzzx = % (Va + l/2)[ (aaxzzx)‘h2 ]
F14: fxyyx = g (Va + l/2)[ (xyyxaa) + (aaxyyx)‘1‘12 + (xy*yxaa) +
(aaxy*y><)fi2 ]

P15: Pyxxy = P14 (5.11)

48

After all these rearrangements, the Hamiltonian h@ in (5.4) can

be written as
h@ = h2 + h4 + h6 (5.12)

where

h2 = (A0 ~— Tgaz/z +3-(1)Px2 + (Bo — 9152/2 + 3<Z)Py2 +
(c0 + 3T9n2/2 + H3)PZZ (5.13)
m + M4 + h2¢111Px4 + [<12 + PM + “2431211374 +
[(T3 + {73)/4 + n2¢13]Pz4 + [(T4+ fZ)/4 +h2¢l4](P§;P§+P§P§)

[(1'5 +f§)/4 + n2¢151(PZZPx2 + PXZPZZ) +

h4

 

[(Tg + 123/4 + fizfpmMPxZPyZ + Pszxz) (5.14)
h6 =431Px6 +(192Py6 + ¢3Pz6 +¢4(Px2Py4 + Py4Px2) +

(P5(Py2Px4 + Px4Py2) + (P6(Py2Pz4 + Pz4Py2) +

 

437052215374 + Py4PZZ) +¢8(P22Px4 + PX4P22) +
¢9(Px2Pz4 + PZ4PXZ) +4710(PX2PZZPYZ + Pszzszz) (5.15)
In (5.14)
{34* = {>4 +[)7 + (l/2)f’lo + (1/2)f’11
{9* =fs + {’8 + (“Dru “1”)?13
I); = {)6 + [)9 + (l/2)Pl4 + (1/2){>15 (5.16)

h@ is a complete fourth—order vibration—rotation Hamiltonian for
H20-type molecules. The detailed expressions for the coefficients
therein have been obtained and are given in the tables and appendixes
referred to in this chapter. The expressions for all coefficientsf?j
and the expressions forCPll throughCFl6 are original with this in-
vestigation.

The fact that two Hamiltonians H and H have identical eigenvalues
if they are related by any arbitrary unitary transformation U has been

utilized by Watson9 along with an order—of—magnitude argument, to

 

49

deduce the form of a rotational Hamiltonian devoid of experimentally
indeterminable coefficients. This theory has been extended by

Yallabandi and Parker,10

who, in their order—of—magnitude argument,
included second—order corrections of the P2~type and fourth—order
corrections of the P4—type in the zeroth and second—order terms, re—
spectively, of the transformed Hamiltonian. In this way it was shown
that one can impose one constraint and thereby reduce the number of
independent coefficients of the quartic terms to five, and three further
constraints to reduce the number of independent coefficients of the
sextic terms to seven. Subject to certain restrictions25 one can choose
these constraints in several ways. For further information, the orig—
inal papersg’lo’25 should be consulted.

In this thesis, a complete Hamiltonian was developed through the

 

fourth order of approximation for the vibrating and rotating HZO—type
molecular configuration. Explicit expressions were calculated for all
coefficients occurring in the Hamiltonian in terms of fundamental mol-
ecular constants, viz., the equilibrium principal moments of inertia
(which are related to the equilibrium geometry of the molecule), the
normal frequencies of vibration, and the qubic and quartic potential
constants of the molecule. Since vibrational and vibration-rotational
accidental resonances have not been considered, the theory at the pre-
sent stage is applicable to the ground state, and to bands not

subject to such resonances.

 

 

APPENDICES

 

 

 

APPENDIX I

“B
Coefficients aég, A168,. and Coriolis Coupling Coefficients $2:

 

 

 

Z
To show how the constants a§§,
dol a
mi éflg, Ascs'o” and Aéfigvdv relate
,
2 Pi to the Hamiltonian (2.1) it
a i
Ri suffices to consider the class-
31
A ical Hamiltonian of a rotating
R Y
CV and vibrating polyatomic molec—
x
4443 ule--specifically the part of
O
X the Hamiltonian coming from the
Figure AI—l

kinetic energy.
If the iEE nucleus, whose mass is mi, is denoted by the radius vectors
E;(X,Y,Z) and fi(x,y,z) in the inertial or space—fixed and moving or
molecule-fixed frames, respectively, and if F1 is its displacement

relative to its equilibrium position 31 in the latter frame, we have

_,
Ri

i + ?i (AI-l)

I31 = 121- 31 (AI.2)

where E is the radius vector of the moving frame's originla’in the
inertial frame. In the moving frame 31 is a constant vector. If T is
the kinetic energy of the molecular framework,
2T = Zmfiij’i {mini =2 mifi/ + 3, + (Z'ox 31)]2 (AI.3)
l i i

—-,
where Vi is the velocity of the iEE nucleus in the inertial frame,

51

3; is the velocity of the iEE nucleus in the moving frame, V is the
velocity of the moving frame relative to the inertial frame andEB is
the angular velocity of the moving frame relative to the inertial
frame. Use.has been.made of the operator relation
_>
d/dtL = d/dt + 60x
inertial moving
(AI.3) can be written as
-9 _, '7
2T = VZM +§ mi((—;)X f1).(coX ri) +2 miviz + ZZmi(V.vi) +
I i
_. —)
22mi(c7;x ri.V) + 22mi(§’i.?ax ii),
i v
where M is the mass of the molecule. The rules of vector algebra
allow rearrangement to
2T = W2 + 21111912502 — (3.313))2] + Zmiviz + 2V.Z mivi +
i t i
—> -9
2V Xib.::mi?i + 20L2:mi?i X 31
g I

Choosing the origin of the moving frame to be at the center of mass
of the molecule makes the fourth and fifth terms vanish identically.

Thus
2: miri = 0 (AI.4)

T = (1/2)Mv2 + (l/2)§__mi[?:§.m2 - (Pi.'<15)2] + (1/2)Z_miviz +
I l

and

a).Zmi?i X 3i
.
The first term in the above is the kinetic energy of the overall
translation which we shall discard as it leads only to an additive
constant in the total energy, the second term represents the rotat—
ional kinetic energy Trot: the third term represents the vibrational
kinetic energy Tvib and the last term represents the Coriolis energy,

the energy of rotation—vibration interaction, T Equation (AI.4)

c.
is called the first Eckart condition.

The rotational kinetic energy can be written

 

 

52

=(l/2)Zmi["12'd)2 - 51.39%] = (1/2)§_ I“ we)», (AI.5)
«(e f5

where

_ . .0 .' 2 0 l 2

Ida - Imam. +5.) + (2)1 +31) ]
= 2' Ininai02 +1102) + Zimdfiiopi' +’Xi0’a’i') +
Ei‘miqain + r5112)

= 1‘9,x + 22mi(fgiopi' +XIO’JI') + 211153552 +Wi'2)’ cyclic
—;2mi°(i{31 + ij-(dioei' + (aiodi') — gmiO/i'fii'
-Ig@ + Zf‘mi(0(i0fii' + fiiodi') - Zimidi'fii', cyclic

_I°(

7D
ll

10rd and L4" are the instantaneous moments and products of inertia,
respectively, and 12(0( and Tap the corresponding equilibrium values;
di"fii' and 11' are the components of F1 and 0(10, #10 and 310 the
components of 31.

If normal coordinates QsG’ are introduced by the following:

mil/2di' = loJ'fSO'Qso' (AI.6)

where 0(1' are the x, y or z—component of I31 ( i = 1,2,......N) and
1:26- are the elements of the matrix of transformation to normal
coordinates, then 104% and qug can be written as

Ida = lad +;:« as Qso’ +§scs' 'G'q Add

IMP. 0 + sZo-'as Qso'+ 255—6-5329“ Astrsér" Qchs' o" (AIJ)
where

a:- ZZimil/Zqfiolgs +giolii's)

a? = {mil/Rafi. + 6101?. > ; “#5

(BSVS'O’ QSG'QS' 0'"

old E>
Asc's'o” = 25-(liso'lis 0" + liso‘lis' 5"
sq; 3a
ASG‘S'G' = Zlisc'lis' '0" = As'o" so“ 3 dffi
,Fhfgcyclic (AI.8)

o< 0M o(
This defines the constants 330" as?“ Aso‘s'cr' and Asfis'w"

 

 

 

53

Using the second Eckart condition
-v a
Zmi(ai x vi) = o (AI.9)
l
the kinetic energy of Coriolis interaction can be simplified to
.9
Tc = Elimipi X 31
1
Again, using the transformation to normal coordinates (AI.6) to ex—
.5
press f1 and V, in terms of normal coordinates, the<X—component of
the sum in Tc.can be written as
" -7
Zonipi X V123) =2 miqfi’b’i 39’2“)
l
’6
=§iUSo-' (lisolig' 'GJQSGQS '0” ' lis 'GJliS0Qscr'Qso)

Interchanging sc'and s'oJ in the second term, we have
a _ Z P 6 a 25 .
(mifig X Via. - is 3'0” (lisli s'G' _ lis '0"1iscaqchskj'
d O
= so's'Er' SGS'G'QSO'QS'O'
and the Coriolis interaction energy becomes
d 0
Tc =§°(_0%(S¢Sé_.qsogs.o,v (AI.10)
The coefficients

3068'

are called Coriolis coupling coefficients since the Coriolis term Tc

P) .
'0' = 231186113. .0,— lis. 27.11;) , cyclic (AI.11)
in the kinetic energy, and hence in the Hamiltonian, couples the
vibrational and rotational motions.
Now, if we define
o( d
‘I¢'>(o(= Iaa‘éggs' ”6’ '305 " 5"an 16, 'Q3 "0" n

Idp= I°(|%_ _Z”§:6::' 11¢ viincrnQ: '04an u (AI.12)

we then have, using (AI. 7),

W.
I' «a: 104x +3204?“ sachso‘ + $0.334“ Ason '0'Q30Qs2y'
. _ _ 9
1°@_ 0 ;- aso'Qsc' ES'SZ'ASQ’SH o" Qso-Qs '0" (AI'13)

where

se's'cJ scs'cr' s o‘ SO‘SO‘

= A (AI.14)

(X

«a, 0(O( Z °( go
A =A — nu unslvuu

3:

ASFé'G' SG‘S' ' _ u n
O‘ n n 80’s 0"

«ex
This defines the coefficients Aso's'o" and AZEé'OJ.

 

APPENDIX II

The Expansion of )ga, )kfla

The determinant in (2.5) can be expanded into

2-I'I'2—I'I'2—2I'I'I']

- I' 1'
xx yz yy xz zz xy xy yz xz

V V I
[Ixxlyylzz

.. 2 2 —l
)u _ [Ixxlyylaz - Ixxlyz _ Iyylxz " Iézlé§ _ 2Ixylyzlxz]
(AII.1)
Substituting (AII.1) into (2.3) and (2.4) we get
= I l __ I2 I I I _ I I2_ I I2_ I I2
)iW (IMIW 19,6)/(Ixxlyylzz Ixxlyz InyXZ IzzIxy

— 2léyl§zléz ) ; cg @,q’= x, y, z cyclic

1“? = (Il'leo'qb - 58%? )/(l,gxl§yI;z — 1.269% — gym; — 1121,25
- ZIxnyzIxz ) ; cc fi,g’= x, y, z cyclic (AII.2)

It has been shown in Appendix I that ng‘and ng3(w, fl = x, y, z)

are functions of the normal coordinates qu. Hence “we and udfl.are
also functions of st- Since the nuclei are assumed to move about
their equilibrium positions with small displacements F1, the Qsc's

are small relative to internuclear distances. Therefore, we may expand
Jamx and “a? into McLaurin series in Q56,

Scs<s
X QSo-Qs'o" + (IL/“$50.! (931MB/2QSC;)QS'SDQSIIdIMQS‘TQéQSIGL'
s
+..... ;o(=f;a:do(fl9 (AII.3)
or, in terms of dimensionless normal coordinates
qso.= (XS/n2)1/4QSU (AII.4)
fag: [aggk] +§G[o(fl*s<r]qsc. +ES_GSZ_¢'[O(fl*sfis'o-J]qscqsvo_; + .. (AII.5)

55

56

where [MW] = (144,)0
[4*va = - (51/2/10 do“? Mano/7131M
[o({b*sc5'S'c"] =EPCh/2!) (azuap/astQs Ian/@9301”
[VP*s€s'o~'s"<>-”] =§<n3/2/32> amp/2cm.seesaw
91315013514
etc., (All.6)
In (AII.6) 51,-: sum over all distinct permutations of s, s', s", etc.,
and in (AII.5) all summations are restricted, i.e., subject to

sgskgs" ... etc.

 

 

 

APPENDIX III

The Untransformed Hamiltonian

It will be understood that throughout this appendix there is a

degeneracy index associated with each normal mode. The Watson—simpli—

fied Darling-Dennison Hamiltonian in (2.12) has the following terms:

(1) (U2); )1“de p,
d?

(

(

(

(

2

3

4

5

)

v

V

v

_(l/2% [pdpxg + )andePg
(“2% ruffle,
(1/2)2; p32

‘ fiz/B) if.“
d

(6) V

We shall call this "term—1”; this
term contributes to H0 through H4.
We shall call this "term-2"; this
term contributes to H1 through H4.
We shall call this ”term—3"; this
term contributes to H2, H3 and H4.
We shall call this "term-4"; this
term contributes to H0.

We shall call this "term—5"; this
term contributes to H2, H3 and H4.
We shall call this "term-6"; this
potential-energy term contributes

to H0 through H4.

In general we shall use the notation Hm(n), n = l, 2, ...., 6;

m = l, 2, 3, 4, to denote the mEE—order Hamiltonian coming from

term—n

H

H

H

0

1

m

T

hus we have:
H0(1) 1 H0(4) + a0<6>

H1<1) + Hl<2) + Rice)

= Hm”) + Hm”) + Hm(3) + Hm(5) + Hm(6), m = 2, 3, 4

57

(AIII.1)

 

58

Using the expansion of‘pqfi,(see Appendix II, equation (AII.5)), one

has 1
HO()

140(4)
How)
Hlu)
H1<2>
111(6)
H2<1>
HZ<2)
112(3)
112(5)
H2(6)
H3cl)
H3<2)
113(3)
113(5)
33(6)
H4<1>
34(2)
H4<3>
H4<5)

H4(6)

(1/2>Z[o(p*11;P,
OI
(l/Zh) gag/213,2
V0 = <°rr/2>Z.9L.1/2<t.2

2:72);%;é E48*a]qagxn;

dab <><*b*a>(1/2){qa,pb}1>°<

(AIII.2)

2Trhcabc kabcqaqch (AIII.3)

(“Dd/53 Exffi *ab]qaqu°(P‘g

1¥ggb e**c*ab)(l/2){anb:Pc}%(

aggd (-*Cd*ab)(1/2){qaqb’Pcpd}
—(h2/8>Z [dew]
q’
kabcdqaqchqd (AIII.4)
5({37 *abc]qaqchP°,PP

zmhcabcd

am 2

«fiabc

= Qgcd (o(*d*abC)(1/2){qaqchspd}%(

= abcde (_*de*abc)(1/2){qaqch’Pdpe}

—Ch2/8) é; Fkx*a]qa

Meagan kabcdeqaqchqdqe (AIII.5)
(1/2)o(@abcd [dfi*abcd]qaqchqdl;(PB
Exabcde ex*e*ade)(1/2){QaQbQCQd»Pe EX
aggéef (_*ef*ade)(1/2){fiaqchqd’pepf}
«Wang [<>(o<*ab]qaqb

Zlficaéécef kabcdefqaqchqdquf (AIII.6)

In equations (AIII.2) through (AIII.6) the "bracket coefficients" are

those directly arising from the expansion Of}£¥5’ with the factor of

conversion from normal coordinates and momenta QS, ps* to dimension-

less coordinates and momenta qs, ps absorbed into them. They are

 

 

59

defined in Appendix II.
The "parentheses coefficients" are related to the corresponding "brac-
ket coefficients” by a. normal—mode depending factor that arises from
expressing the components of the internal vibrational momentum Rx in
terms of normal coordinates and momenta.

Substituting (AIII.2) into the first equation in (AIII.1) we
obtain the H0 in (2.15). Similarly, we obtain the H1, H2, H3 and H4
in (2.15) by substituting (AIII.3), (AIII.4), (AIII.5) and (AIII.6)
into the second, third, fourth and fifth equation in (AIII.1),

respectively.

 

APPENDIX IV

Coefficients (2;o(f3’g;a;-) in hi

Herein are given the coefficients (2;dpa§a;—) in the second sum of
(4.4). Throughout this appendix the.condensed notation Omega) will be

used to denote (2;qp3;a;—), and j = 1, 2.

(zzzB) = -(231/4/2123n1/2)(a§zq3 + @2323) [(31 + 9&2 —27\3)/
(31 - 713)”? 713) — <31 +32>/2%1%21
(xxz3) = -(231/4/8lleznl/2)(a’l‘XEZ3/21 + .ng/zz) +
a§y(Iz - Ix)/4IX21yI;233/4n3/2 — (231/4/4Ilein1/2) x
[a’l‘xgflal ‘33) + 32" 23/02 ’75)]
(zxx3) = (xxz3)
(yyzB) = —(R31/4/8Iy212n1/2)(afyflyzl + a§Y§2°3/>I2) +
a§y(ly - Iz)/4Iny21z 33/4n3/2 _ (231/4/4Iy212h1/2) x
[Elly 13/(9‘1 'q3) + aZyQ3/(9xz "9‘91
(zyy3) = (yyz3)
(xzx3) = a’3‘y(Iz - Ix)/ZIXZIyI£q33/4h3/2
(yzy3) = a§y(Iy - Iz)/ZIny21gq33/4h3/2
(xyzj) = (Ajl/4agycg3/4Ixiylznl/2)[(fij - sky/2230;, —7I3)] +
[agixay — Iz)/Ix + a§z(Ix — Iy)/Iz]/4leylgaj3/4h3/2
(zij) = (xij)
(xzyj) = [aIX(Iy — Iz)/IX + a§y(lz - Ix)/Iy]/4leylz‘fij3/‘*fi3/2
(yzxj) = (XZYj)
<zxyi) = (yxzj) = <3j1/4a'3‘ytg73/4leyiztl/2)[cfij — sap/223(23- #3)] +

[all’Yuz — Ix)/Iy + aJZZ(Ix - Iy)/Iz]/4leylz'7lj3/4n3/2

6O

 

APPENDIX V

Coefficients (2;qfi;ab;-) in hi

Herein are given the coefficients (2;dF;ab;-) of the first term in

the third sum of (4.4). Throughout this appendix the condensed

notation (xpab) will be used to denote (2k(P;ab;-), and 0(= x, y.

(04X 11)

GkKIZ)

@4x22)

@flx33)

(zzll)

(zz12)

(z222)

(zz33)

(Xyl3)

(xy23)

(xy3l)

=- (“c/10(253/2n2‘f‘ klll/9113/4 +7'1a’2Xk112/5'23/4w7'1 -7\2)]
= (06(21) = (21rc/Iof-n3/2)[223/4a’g"k122/211/2(422 —9I1) +
CRIB/aajakllz/azl/2(4al ' 75)]
= (WC/Iogfi3/2)[a§<k222/a23/4 +Q253xk122/213/4(4a2 ‘531)]
= (“ca3/chzfi3/2)[afk133/a13/W49‘3 ‘9‘1) + 52'1‘233/223/“45‘3'7‘21
= (WC/Izzh3/2)[3IzklllM13/4 +0‘151521<112/>\23/l'(“I "9‘2” '
all/zglg/Izzfial ' 9‘3)
= (2221) = (ztc/Izzn3/2)[213/4afzk112/221/2(4%1 —512) +
7\23/4a22k122/111/2(4a2 ‘ 31)] '
(Al/AZfl/Z'al +[A2 ' 29'3)$I3§23/Iz2fi(9'1-/3)(AZ—9‘3)
= (WC/Izzfiwz)[322k222/223/4 +aazalzk122/xl3/4(4“2 'C'l) —
azl/Zgg/Izzmaz ‘%3)
= (re/IZZn3/2)[afzkl33/fll3/4(4?3 —.Ql) +agzk233/223/4(423_%2)]_
(431/2/Izzmli‘lg/(21 —7\3> + 52%, —7\3>1
= 2r6X33/4agyk133/leyn3/2211/2(423 -§41) +
<4143>U4G§<Iy - I.>/I.2Iy2<31 43>
= 2Ic233/4afiyk233/leyn3/2X21/2(423 —-%2) +
(393)1/452‘3Jy _ Ix)/Iley2(22 _ 23)

= (yx13) = (yx3l) = (xyl3) ; (xy32) = (YX23) (yx32) = (xy23)

61

 

APPENDIX VI

Coefficients (2;qfi;—;ab) in hi

Herein are given the coefficients (23aP;-;ab) of the second term in

the third sum of (4.4). Throughout this appendix (23qflg-;ab) will be

written in the condensed form anab).

(xxll)

(xx22)

(xx33)

(xx12)

(yyll)

(yy22)

(yy33)

(yy12)

(zzll)

3fisin33721x2211/2 +

(mdhl/Z/Ixz)[Zakalll/fil3/4 + a§Xk112(321—22)/Q23/4(4%l—%2)1
3ncos23721x2211/2 +

(Kohl/Z/Ixz) [2«'=1§"1<222/9\23/4 + ai‘xklzz (542-41) />\13/“<47\2-%1)1
3n/21x12951/2 + (teal/z/Ixz)[aka133(395-73)/%13/4(4%3-2q) +
a§xk233(3a3-42>/Az3/4(4a3-%2)1

(xx21) = 3nsin3cos3721x262122)1/4 + (Hchl/Z/Ixz) x
[alxkllz<5alaaz)/al3/4(4al’a2'+82xk122(522'al>/223/4(4az'a1)1
3ncosZX/2Iy2211/2 +

(tcnl/Z/Iyz)[zeiyklll/Rf/4 + agyk112(321-%2)/%23/4(4%1-22)]
3nsin33/2Iy2221/2 +

Oman/13:2)[2a2yk222/123/4 + aiyklzz(392-“1>/213/4(4a2-“1)1
3fi/21yliq3l/2 + (“gal/z/Iyz)[afykl33(3fi3x%1)/A13/4(4234x1) +
a§yk233(375“32)/az3/4(4R3422)1

(yy21) = —3nsin3cos3721y2(2i%Q)l/4 + (flcfil/Z/Iyz) x
[aka112(5a1'%2)/al3/4(4A14x2)+a§yk122(5%2;%1)/a23/4(422‘%l)1
(n/2122951/2)[3§Z§ + Si§<21+%3)/(31423)1 +

(TDCfil/z/IZZ) [Zaizklll/Ql3/4 + 332k112(39\1-g2) /223/4(47\1-7\2) I

62

(zz22)

(zz33)

(zz12)

(xy13)

(xy23)

63

(‘fi/ZIZZ9\2V2 )[33‘13 + $33()2+23)/0I2-9\3)] +
(maul/Z/IZZ)[2a32k222/223/4 + aizk122(332<21)/213/4(4A2—fll)]
<n/2122231/2>ISZ§<fil+fis>/<“1-%3> + 33§<32+23>/<22423>I +
(‘njc‘hl/Z/Izz) [aizk133(323—7\1)/9\13/4(4’A3-7Il) +

322193 (32342) ”23M ($342)]
(2221) = (nSf3Q3/2I22211/4A21/4n-3 + 2(91+9\3)(911+22-223)/
éal-R3>(32-%3>I + (teal/2/122>Iaizkllz(591422>fi413/4<421422>

+ a22k122(5a2441)4223/4(422431)1
(yx13) = (xysl) = (yx3l) =‘chhl/zkl33(59t3-7\1)/Iny’)\§/4(47\3-7Il)
-(n/4Ix2Iy2?\11/4931/4) [3(Ix/Iz)1/2cos’o’+ 3(Iy/Iz)1/2sin’gf -
- zéfgcly-Ix>63I+23>/<%1-%3>1

(yx23) = (xy32) = (yx32)-_-1I:ch1/2k233(593-12)/Iny9\33/4(49I3-7\2)
+(n/4IX2Iy2221/4A31/4)[5(Ix/IZ)1/2sin3’- (Iy/Iz)1/2cosfi+

+ 2 23<Iy-Ix>(22+%3>/<%2-%3>1

APPENDIX VII

Coefficients (2;d;abc;—) in hi and All Possible Permutations

The following is the list of the coefficients (2; ;abc;-), written in

the condensed form exabc), appearing in the fourth sum of (4.4) :

(2113) = (2"67‘31/4/12‘52H 131‘111/9‘13/4 - 2€3k133a31/2/7'11/4(49'3'21)
+ g3k112ql/223/4C49'1’7'2) 1
(2223) = (Z'EC9‘31/"/:Lz"‘2)[52631‘222/7'23/4 ‘ 2~{2’3k233a31/2/a21/4("95:42)
+ 4:31‘12222/413/4‘49'2‘7‘1”
(217-3) = (“Icg37'21/47‘31/4”2527\11/2)[klzza‘zllz/W‘z‘ l) ' k1333‘31/2/
(433-41)) + (415c§:39\11/4>\31/4/lzfi2)21/2)[k112311/2/(49xl—M)
- k235231/2/(49‘3-22H
(2333) = (2Ic’A35/4/Iz'h2)[5:3k133/7I13/4(47\3-7\1) + §3k233/Az3/4(49I3—7\2)]

64

APPENDIX VIII

Coefficients (2;o(;c;ab) in bi

Listed herein are the coefficients (2;o(;c;ab), written in the condensed

form (o(cab), appearing in the fifth sum of (4.4) :

(2311) = Ti1/2231/43I23I34211/2122 + (Ida3l/4/Iz)[k111313(“3‘7al)/
g13/4(;5_ql) + kllzggg((221.22)/223/4(421:A2) —2221/4/(23—22))
' 2%31/25I3k1334all/4(498'21)1

(2322) = h1/29'31/4322523/9‘21/21z2 + (W0231/4/Iz>[k222333ca3-722)/
’223/4(23_7§) + k122§I3((222441)/al3/4(4)24Z1) -2211/4/(23221))
' 2231/2555k233/a21/4(495—72)1

'(2312) = (2321) =.fi1/2q3l/4(ai23:3 + a§z§:é)flgll/4fi21/4IZZ +
(2tefi31/4/IZ)[(23-921+222)Qll/4gi3k112/(4%1422)(23-21) +
(23-922.2q1y221/43§5k122/(422.21)(25_i2) _
fi3l/2$§3kl33/g21/4(4)3;Al) _ OBI/25:5k233/211/4(4q3_7b)1

(2333) = (ic’A31/4/iz)[3:3k133(2212+29(32—7217\3)mi” (433—711)(A3—ql) +
$§3k233(2X22+2A32—72223)/923/4(423J%2)(23JA2)]

(2113) = (2131) = -nl/2a§z l3/9\31/4Iz2 - (2tc/Iz)[2235/4(221+fi3)§:3 x
1‘133/(49'3'21)(aralfllw4 ' (21222)1/4§23k1124131/4(4ql-Q2> -
313k111/(21g3)l/41

(2223) = (2232) = Jhl/zangES/Q3l/4Izz - (zrc/Iz)[2235/4(222+fi3)$;3 x
1‘233/(49'3‘9‘2)Okrazw‘zw4 ‘ (22241)ll€§l3k1224k31/4(422441) ‘
$23k222/(327h)l/41

65

66

(2123) = (2132) = —n1/2a§zflll/4§:3/(fifiyl/fizz - (zinc/Iz)[(9\l’4_.’)l/4 x
q3k233a1+22 “97‘3”““3-7‘2) (Al-“3) + “31/452,31‘133 X
(29'3‘9'1)((2129‘2)l/4(49‘3‘7'1> ' “21/4f23k122m‘2'9‘lfl
(31223)1/4(422—'41) - 213/4523k112/9131/4(Ml-g2)1

(2213) = (2231) = —n1/2afzflzl/4§:3/(393)1/4122 - (21rc/Iz)[(42’h3)l/4x
$23kl33(22+2q1'9a3)“49'3"al)(22‘9'3) + “31/4fl'3k233 X
(23'3-92V(42231)“4C4'A3-22) fill/[‘stlleQ'l-fizfl
(22223)1/4(4’Al_22) - q23/43'13k122/7'31/4(‘92—’41)]

 

APPENDIX IX

Coefficients (2;-;cd;ab) in hé

Herein are given the coefficients (2;—;cd;ab) appearing in the sixth
sum of (4.4). Throughout this appendix (2;—;cd;ab) will be written in

the condensed form (cdab).

(1111) = (—4Ir2c2fh)[3k1§1/9111/2 +O‘ikiiz/flzl/2Wi-32N

(1112) = (1121) = (—8izc2k112/n)[klll/2ll/2 + klzjfil/221/2(4%1—22)]

(1122) = (—4n2c2/fi)[332k111k122/211/2(422-31) + k112k222/z21/2]

(1133) = (~4n2c2/n)[klllk133/211/2 + k112k23321/221/2(47h-7Q)] +
(Al/43)l/2§15/212

(1211) = (2111) = (—8n?c2k112/n)[3kllle/221/2(4Ql—22) +
k1227‘2/‘9‘11/2(49‘2-7'1)1

(1212) = (1221) = (2112) = (2121) = (—len?c2/n)[Alklfznzzl/2(421—fiz)
+7'21‘122/7‘ll/Z(49‘2‘9‘1)1

(1222) = (2122) = (-8n2c2k122/n)[k11221/221/2(421122) +
3k22292/a11/2(4“2-31)1

(1233) = (2133) = (-8n2c2/fi)[21k112k133/Q21/2(4%14X2) +
a2k122k233/‘7‘ll/2(49‘2‘7‘1)1 + (2122/332)1/45I3523/Iz

(2211) = (-4n?c2/n)[322k111k1222211/2(4224A1) + kllzkzzz/zzl/Z]

(2212) = (2221) = (-8n?c2k122/m)[kllgfizfiall/2(4fizlkl) + kzzz/fizl/Z]

(2222) = (—4n?c2/n)[22k132A211/2(4%2421) + 3k232/221/2]

(2233) = (-4ch2/n)[32k122k133/311/2(422131) + k222k233/221/2] +

2
(92/a3)l/2523/212

68

(3311) = (—4t2c2/fi)[323k111k133/211/2<47h‘31) +'23k112k233/
321/2(4’A3'9‘2)1 + (waif/233325

(3312) = (3321) = (—8r2c2ffi)[23k112k133fl411/2(433411) + 75k122k233/
‘321/2(423432)1 + (932/2122)1/I:13§23/Iz
(—4r2c2/n)[23k122k133/211/2(433431) + 37bk222k233/
221/2(4'A3—32)] + (Q3/32)1/2§'2§/21z

(—4t?c2/n)[Q3klg3/qll/2(4Og—Al) +323k2§32221/2(423422)1

(3322)

(3333)

(1313)

(1323)

(2313)

(2323)

(1331) = (3113) = (3131)
{34.1

(1332) = (3123) = (3132)
‘(al/22)1/43I3§23/Iz]
(2331) = (3213) = (3231)
-(22/Ql)l/ASI3§23/Iz]
(2332) = (3223) = (3232)

2
— 3/Iz

= [‘16“?Caaakl33k133/fiall/z(423491)
= ['16n20233kl33k233/fiflll/2(433431)
= [~16K2C223k133k233/fi7El/2(4X3izz)

= [‘16W2C233k233k233/nfizl/2(433‘32)

APPENDIX X

Coefficients (2;-;—;abcd) and A11 Possible Permutations

Listed herein are the coefficients (2;—;—;abcd) in the seventh sum of
(4.4). Throughout this appendix (2;—;-;abcd) will be written in the

condensed form (abcd).

(1111) = 2rhckllll — 212c2n[3k1§17211/2 + kl§2(zfilJAZ)/221/2(421-22)]

(3333) = zwnck3333 - 2m2c2nIkl§3(223-31)/211/2(47g—21) +
k2§3(223—7&)/%21/2(493422)]

(1122) = ztnckllz2 — 4n?c3n[k112k222(3%l—%Q)/221/2(421422) +
k111k122(37'2'7'1)/211/2(4f"2-7\1) + 2klizall/z/(‘IQ‘l-‘Afl +
2k122321/2/<432-7h)1

(1133) = ZIfickll33 — 2n9c25[k111k133(59g-221)/flll/2(423—21) +
k112k233((221412)/2221/2(47&—7b) + (223'92)/2321/2(433432))
+ 2k153a31/2/(4A3‘31)1

(2233) = 2Wfick2233 - 2n9c2‘h[k222k233(523—232)/7b1/2(475422) +
(k122k133/211/2><<2’42— l)/(4’Az—’/)l) + (223—21)/ (423—31)) +

(1112) = Z‘IHTCklllZ - 27L2C2’fi[klllk112(7ql—qZ)/fill/2(40l1-9I2) +
kllzklzz(I3Zffi2-2212-5222)/221/2(421422)(422421)]

(1222) = 2Kfick1222 - 2u9czn[k122k222(722—21)fik21/2(422431) +
k112k122(13211243215312)fill/2(47‘1422)(432—331)]

(1233) = 2nfick1233 — 4“?c25[k122k233((2)3422)/221/2(423422) + 221/2/
(432—21)) + k112k133<<223-21)/211/ 2423-31) +9111/2/<421—212)>
+2kl33k233731/2(Baa-al‘fiz>/<4a3‘21)(433422)]

69

APPENDIX XI

Coefficients (33dp36E—ga) in bi

Listed herein are the coefficients (3;dfizd:—;a) appearing in the first
sum of (4.5). They are expressed in terms of the auxiliary coefficients
(T;o(,(}yI;—;a) and (U;¢(5,’&’f;—;a) as follows:

(3;dfi')/7;-;a) = (Baha’i-:61) + (mgggL-m) (A XLl)
The detailed expressions for (T;agpgég—;a) and (U;afi4y£}—;a) are given
in this appendix following the list of (3;d@3$}-;a). Throughout this
appendix the condensed notations «(ya/(a), (o(,{53rfa) and («PQ’Ya are
used to represent (3;o({}a’[;—;a), (T;o(,{l;3'{;-;a) and (U;o(@,a’f;-;a),

respectively.

(1) Coefficients (x3763):

(xxxxl) = (xx,xxl) (xxxx2) = (xx,xx2)
(yyyyl) = (yy,yyl) (yyyyZ) = (yy.yy2)
(22221) = (z,zzzl) + (zz,zzl) (zzzz2) = (z,zzz2) = (zz,zz2)

(xxyyl) = (yyxxl) = [(xx.yyl) + (yy,xxl)]/2

(xxyyZ) = (yyxx2) = [(xx,yy2) + (yy.xx2)]/2

(yyzzl) = (zzyyl) = [(2.yyzl) + (yy,zzl) + (zz,yyl)]/2
(yyzzZ) = (zzyyZ) = [(2.yy22) + (yy.222) + (zz,yy2)]/2
(zzxxl) = (xxzzl) = [(z,xle) + (xx,zzl) + (zz,xx1)]/2
(zzxx2) = (xxzzZ) = [(z,xxz2) + (xx,zzZ) + (zz,xx2)]/2
(zxle) = (xzle) = (z,xle)/2

(zxzx2) = (xzsz) = (z,xzx2)/2

7O

71

(zyz-l) = (yzyzl) = (x,yzyl)/2

(zyzyZ) = (yzyzz) = (2.yzy2)/2

(xyxyl) = (xyyxl) = (yxxyl) = (yxyxl) = (xyacyl)

(xyxyZ) = (xyyxZ) = (yxxyZ) = (yxyx2) = (xy.xy2)

(xxxyB) = (xxyx3) = (xyxx3) = (yxxx3) = [(xx,xyB) + (xy.xx3)]/2
(yyxy3) = (yyyx3) = (xyyy3) = (yxyy3) = [(yy.xy3) + (xy.yy3)]/2
(zzxy3) = (zzyx3) = [(z,zxy3) + (zz,xy3) + (xy,zz3)]/2

(xyzz3) = (yxzzB) = [(z,xyzB) + (xy,z23) + (zz,xy3)]/2

(zxzy3) = (xzyz3) = zyzx3) = (yzxz3) = (z,xzy3)/2

(zyyz3) = (z,yyz3)/2 (zxxz3) = (z,xx23)/2

(2) Auxiliary coefficients (4,193;ng :
E51/23l3(al+9h)/3(>1‘7b)211/4Iz4](aIZ5I3 + 32a§23) X
[ (2 192-223) / (214(2) (AZ-33> - (Alma/211321
[PI/€523<qz+7b>/3(g2433Y)21/41241(alzfls + 322553) X
[ (21+22—223)/(21.7.2) (912-913) - (Map/2210121

(z,zxxl) =[ml/2313(%1+75)/12(xl—23)211/41x21221 x
[aIx-{l‘flfil-zfl/aflgl-zfi + 82x§3(3>|2->\3)/7\2 {32-713)}
-..}gyg‘galay/8(2,—23))(11/4a31x2122n1/2

(z,zxx2) =[ti/23:3c22+95)/12cx2-95x121/4Ix21221 x
-a§>'§§3<22+33>/802-23>9\21/43,ix2122al/2

(z,zyyl) =[fi1/23f3(7\1+fl3>/12(9Il-’>I3)9»11/4Iy21221 X
+3§Y§:;(Rl+)3)/8(114)3ykll/4231yzlzafil/2

<z.zyy2> =I'nl/ 2333612913)/1202—9I3)9£21/4Iy2122] x
+a3y523642+33>/8(a2"73)921/49s1y2122h1/2

 

(z,zzzl)

(z,zzzZ)

(z,xle)

(z,xxz2)

(Z.yyzl)

(z,yy22)

(z,xle)
(z,xzx2)
(z.yzyl)
(Z.yzy2)

(Z’WZ3)

(z,xzy3)

(z,zxy3)

72

_a§y 13(21423>/41x21z2(OHIZ3)7&l/423hl/2
'35y523(22+95)/4Ix2Iz2(124334221/41351/2
+a§Y5:;(3y+?b)/4Iy2122(21:75)?11/42351/2
+a§y5:3(22+23)/4Iy2I225'2“23ya21/43331/2

(z,zyx3) = [:nl/2a§¥§:§dllak3)/o(dez3xx3l/4IXIYIZZ] x
[(M—slngfigaz-fign + E-fil/ 2a§y§3§<32+23)/6(912-23> x
'231/41x1y122][(22—375)/223(AZ-93)1 + [six —a{Y(Ix-Iy)/
lz]I§T3611+23)/81xlylz?2f231/4(9h-23)fil/2] + [afiX—agyx
(Ix-1y) /Iz] [553 (4203) /81x1y12222331/ 4 (fiz—fisml/ 2]
(z,yzx3) = (a€V-a:*)§:3(ll+?5)/s l 31/4(21_23)Ix1ylgh1/2
(z,yxz3) = [sol/Zagy/lzfl35/4Ixiylzz1(513(21423) x
(’xfi—fip/(AI-fip +33%).)(2,-333>/<22-33>1 — Ia):y +
aizux—Iymzl £3<31+93>/8319sl/4<21-’43)Ix1y122nl/2 -
[aXY + a§z(IX—Iy)/Iz]533(22¥23)/33éA§/4IXIYI§(22—23)nl/2

 

(3) Auxiliary coefficients GXQQKfa) :

(xx,xxl) = (3h1/2/21X49q1/4)[aixsin%fffil + agxsinvcosa72/ ] +

(ZIcafx/3le'fil3/4) Izaixklll/9L13/4 + agxk112(321_>'2)/
a23/4(431'32)] + dtcagx/3Ix4423/4)[aka112(531'22)/
9(13/4(4il_9«2) + .gxklzzwzz—ap023/4432-11)I

(xx,xx2) = (351/2/21x4221/4)[agxcongVQZ + agxsinacosaVZAl] +

(Z‘llficaaéx/31x4223/4)I:2a}2cxk222/’/'\23/4 + a}ka122(37\2-9\1)/
213/4(4fizexl)1 + (reafx/3IX4213/4)[akallz(5filéhz)/
a13/4(411‘22) + a2(xk122(59‘2"7‘l)”23/4(4’Az‘7\1)1

73

(xx,yyl) = (351/2/21x21y2a11/4)[afxcosfif/fll - agxsingcosz7232] +

(mcaffX/ 313:1323I/4) [zalyklll/X‘l + 32yk112(35‘l‘32)/

753/4(421432)1 + (rca§x/3lx21y%(23/4)[aiyk112(521412)/
3/4 _ _ 3/4 _

9‘1 (421 a2) + agyklzzwxz 99/32 (M2 21)]

(xx,yy2) = (3fi1/2/21x21y2221/4)[agxcos%3722 — aixsin3c0357221]-+

(xx,zzl)

(xx,zz2)

(xx,xy3) =

(”caJZm/ 31231372221”) [2332]},1‘222/ 7'2 + aiyklzzw‘z‘qlfl
‘213/4(422421)] + (tcafx/31x21y%xl3/4)[aiykllz(521-22)/
21344214,) + against—31>/2,3/‘*<422-21>I
(sol/Zafz/slxzizaall/4)[afxafz/Ql + agxagz/zfiz] +

[Til/2 l3al+X3>/3Ix21229\11/40‘1‘9‘3)1[313/31 + 2523 X
(31+512-2313H22 @2143” + (zxca’l‘x/lezlzzfif/l‘)[2aiz x.
XIII/31w 4 + a2zk112(37'1-9'2)/qz3/ 4“ml-7'2” + Wea’zm/
31x2122223/4)Ia§2k112(521-22)f313/4(421:22) + agzklzz x
(5912-21)/>123/4(422-7Il>1
(SKI/Zagz/MXZIZBZZI/l')[agxagzaz + ai‘xafz/zfil] +
“71/2523 (92+Q3V 31x2122X21/4az‘7‘3” [{2‘3/9‘2 + 2513"
(31+32-223)A1(31-913)] + (zuca’z‘x/3Ilez7qz3/4)[2a§z x
1‘222/323/4 + a‘1221‘122(3312'7'1)”‘13/4(‘*X2'7\l)1 + (IcaIx/
3Ix21z27'13/4)[aIzkllZ(5’41’9'2)al3/4(47'14\2) + a"2321(122 X
(522-711)/423/ 4(4712-711H

(xx,yx3) = (-351/2a’1‘x/891231/4lx41y2)[(Ix/lz)1/2cosa/+
(Iy/Iz)l/Zsin’X] + (fl/Za’Z‘X/SOIZ’A31/4Ix41y2)[5(Ix/Iz)1/2x
sinyf- (Iy/Iz)l/2cos 7] + (Ito/31x3ly9x33/4)[a’ka133 x
(5X3‘X1)/X13/4(49‘3'9‘1) + a'2mk233<5;'3'7\2)(423/4(M3412)1
Ih1/2(Iy-Ix>/8IX4I§231/4J[an§13631+?g>/31<314%3) +

a2x§302+a3>flz<a2913>1

74

(yy.xx1> = (3fi1/2/21x2Iy27111/4)[ai'ysinZX/Al + agysingcosa/mz] +
(21ccaXY/3Ileyfll3/4) [2a’kalll/7cl3/4 + a§Xk112(37\l-’)(2)/
‘323/4(421—9§)1 + (rcagy/31x21y3223/4)[aka112(521—22)/
0'13/4U‘a1'qz) + agxklzz(522'7‘1)/7)23/4(47‘2'Xl)1

(yy,xx2) = (3hl/2/ZIXZIy29I21/4)[agycosz’b/Az + aiysin'b/cosy/fil] +
(zuca5Y/3Ix21y2223/4)[2axxk222/223/4 + aka122(3224kl)/
9123/4(4%2—7\1)] + (roa{Y/3rxzry2213/4> lawman—22w
a13/4(49‘1‘7‘2) + a'2'1'1‘12263‘2‘ 1)/7‘23/4(47‘2'Xl)1

(yy,yyl) = (3n1/2/2Iy4211/4)[afycoszgifll — agysingcosgvzfiz] +
(zucaXY/3Iy3113/4)[2aXYk111/213/4 + a§Yk112(321422)/
7123/4(431-22)1 + mea§Y/3iy4323/4> [aiykllz (521-22>/
7'13/4(4a1'9‘2) + aiyklzz(59‘2‘7‘1V9‘23/Hflz’al)1

(yy.yy2) = (3h1/2/21y4221/4)[agysiay/flz — aiysinatOS/J/Zfil] +
(27Lca5y/31y4 23/4)[2a§Yk222/7Iz3/4 + afYklzzehz—le
0113/ “422-31” + (reap/31374213”) [a{yk112(521-22)/
7'13/4(4al'7‘2> + a32'Yk122(5’42—7\l)/o\23/4(47\2—21)]

(yy,zzl) = (3n1/2afz/8ly2123211/4)[afyafz/al + agya§Z/222] +
m“ 2 51°3(21+9\3>/3Iy21z1211/ 461—23>1[a?5§3/21 + 2a§y x
g, (31+12-2213H912 (22413)] + (21ECaXy/3Iy2122213/4) [2afzx
klll/9\l3/4 + a22k112(321'22V9'23/4MQ'1'X‘2H + (th3237
3Iy2122223/4)[aIzkllzwa'aflfll3/4(49'1'q2) + aizklzz X
(fig-’Ap/ 23/4<4’Az—31>I

(zz,xx1) = (351/2/21x2122211/4)[aizsinfiyzfil + agzsingcosavzflz] +
(zucafZ/3Ix2122213/4)[2akalllnal3/4 + agxk112(321432)/
7'23/4(‘*”'1‘32)1 + “cagz/3Ilezzaz3/4) [alxk112(59‘1'7\2)/
7t3/4(47&J7§) + a§Xk122(522421)[323/4(422421) 1

75

(yy,zz2) = (3T‘Il/Zazz/8Iy2 IZ§ZZ3221/4)[a§ya /;\2 + a1yaf z/2Q1] +
[“1/253/31172122221/402 >13)][5‘2y 23(22+23)/32 +
2a1’Y51’3011+A3) 01+)2-223)/21(?\->3)1 + (zucagy/sly 2Iz 2x
9‘23/4H2a22k222/7123/4 + alzklZZCEQZ 7‘1)/)'13/4(4>‘2'7\1)]+
mca1y/31y21227113/4Ha1z k112(511—32)/9\13/“(421—%2) +
322k122(57‘2">'1)/223/4(422 ‘31)]

(zz,xx2)= (3fil/Z/ZIXI 2 2222““)[azz cos 23422 + afzsinarcosg/zfifl +
(27[ca§z/3Ix2122’>I23/4)[2a§xk222/)I23/4 + aka122(3’>12-21)/
213/4(4%2—’A1)] + (mail/31121 22313/4)[a’1‘xk112(591—7Iz)/
7‘13/4(431-22) + a ’z‘zxk122(5%2—7I1)/223/4(42 41)]

(yy.xy3) = (yy.yx3) = (-3fr1/2a{y/81ley421'a31/4)[(Ix/Iz)1/2 cos1+
(ly/Iz)1/2sin'b(] + (ml/Zagy/stzIy4QQ3l/4)[5(Ix/Iz)1/2
x sin/64 (Iy/Iz)l/2cosg] + (no/3ley3233/4Ha1yk133 x
(5”‘3—Xl>//\13/4(49‘3—Xl) + a2yk233(5’X3-9\2)”\23/4U/A3-32)1
[fil/2(Iy-Ix) /81x21y") 31/41 em. (31%) Al (”41-23) +
.yyr3c22+23>m2<az-2,>1

(zz,yyl) = (3fi1/2/21y21 2221““)[a1z coszf/Xl - agzsin’a/cosa’nfiz] +
Quail/31221 327q13/4H23ka111/a13/4 + 82yk112(331-7\2)/
323/4(4'>\1—22)1 + (IcaEZ/3Izzly2223/4)[aka112(5ql-7Iz)/
f’) EMMA—7‘2) + 32yk122(57‘2'9‘1>/9'23/4(47‘2'9‘1)1

(zz,yy2)= (3111/2/21y 21 22;) 21/4)[a2z sin2X/22 — a1zsin3'cosf/221] +
(zucagz/nz 21y2223/4)[2a§Yk222/x23/4 + aiyk122(322—21)/
213/4(4312_’/11)] + (7cca1z/3IZ2 1112?, 13/4)[ayyk112(5’)\1—22)/
213/4421—32) + agyk122(532-21)/223/4(422-’/\1>I

76

(22.221) = (sol/2 822/2125 21am)“ .2); /21 + (a3 732/2221 + [1.11/2 x

5:3(al+33)/3Iziall/4(alaa3)1[alzsI34Xl + 2822533 X
(al+55-223)/X2(X2433>1 + (ZIbaIz/3IzXXI3/4)[Zaizklll/
9'13/4 + agzk112(3ql-22)/9\23/4(49\l-9\2)] + (Icagz/3Iz X
’223/4)[afzk112(591422)/>13/4(421’Az) + a 22k122(592‘91)/
’k 23/4(43 2431)]

(22,222): <3fi1/2a zz/zlz 52 21/411133212112 + (a1ZZ>2/zx11 + ffil/Zx
3.23/4 + a1 zk122(3>\2- 1913/“ (43241)] + (reef/312 4 x
3K13/4)[a12k112(5)1412)/213/ 4(421332) + a2 zk122(522‘ 1)/

filmed 241)]

(zz, xy3) = (zz,yx3) = (-3nl/2aZZ/81x211y21222'2;1/4)[(lx/IZ)1/2cos7{
+ (Iy/Iz)l/Zsinvi + cal/Zafiz/81x21y212222951/4) X
[5(Ix /I 2z)1/Zsin’gf— (I y/Iz)1/2cos'6’] + [El/2(Iy -I x)/

SIX2 1229‘ 321/4][a1z§1°3(21+7\3)/7I1(3I1-9\3) + azz 23x
€32¥;:>/32(324fi3>1 + (WC/BInyIZ 2233/4)[a12 k133 x
(533-31>/2\13/4<413-7«1> + afizk233(93-Xz)/X23/4(47‘3-7\2)]

(xy,xy1) = (xy.yxl) = (yx.xyl> = (yX.yxl) = (-3fil/2a§y/8Ix3ly323 X
9111/4)[(lx/IZ)1/2cosg+ (Iy/Iz)1/2sin7fl + [mang133 x
(5A3JZ1)/31x21y2 33/2(423J)1)] + Pal/2513(Iy-Ix) x
a§X(ale3)/8Ix3Iy333all/4631JA3)]

(xy,xy2) = (xy,yx2) = (yx,xy2) = (yx,yx2) = (ml/Zagy/81x3ly323 x
9.21/4)[5(Ix/Iz)1/2siny-(Iy/IZ)1/2cos3’] + (rca’gyk233 x
(523432)/3Ix21y2233/2(423432)1 + [51/2§§3(Iy-IX) x
a§Y (3295) /81x3ly39\39\%/4 (2243)]

77

(xy,xx3) = (YX,xx3) = (Bill/Zagy/ZIYIZIXZA35/4) + (‘Itca’3‘y/3lxaly X
“33”) [ aka133< 37‘3'31) ”'13/4 (433141) + a’2‘xk233 ($342”
Q23/4(47\3_’)12)]

(xy.yy3> = (yx.yy3) = (351/ Zap/21211152235“) + (lrca’g‘Y/31y31x X
9.33/4)[31’Yk133(32.3_9.1)fl13/4(49.3_9(1) + 32yk233(37‘3-’/\2)/

<xy.zza> = <yx.zzs> = (finely/41115122235“)[£32/(A1—33)+;z‘32/
(423.211) + agzkz33(323-’x\z>/9\23/4<4’A3-22>I

APPENDIX XII

Coefficients (3;dfl3§b;a) in hé

Expressed in terms of the auxiliary coefficients (T;dp@§b;a),
(U;°"F”K;b;a)’ (N;o(,@’b/;b;a), (X;o(€),'<y;b;a) and (G;o<fi’b’;b;a), the co-
efficients (3943qfib;a) appearing in the second sum of (4.5) are listed
in this appendix. The detailed expressions for the auxiliary coeff-
icients (T;dfiq§b;a), (Ugagpqfibga), etc., are given successively there-
after. For simplicity, the symbol "3" and all separating semicolons
will be omitted , and the symbols "T”, "U", etc., in the auxiliary

coefficients will be kept only in the list of (3tq933b;a).

 

(l) Coefficients (3gdflngia)

(dp’Oba) = (Iapgba) + (U21, ms) +
(Ng,%ba) + (Xgfi,lba) + (Gaqasza)

In the following first ten entries j= 1, 2 :

(zzz3j) = (Tzzsz) + (Uz,zsz) + (Nz,zz3j) + (Xzz,23j)

(zzzj3) = (Tzzzj3) + (Uz,zzj3) + (Nz,zzj3) + (Xzz,zj3)

(zxx3j) = (xxz3j) = (szx3j) + [(Uz,xx3j) + (Nz,xx3j) +
(Xxx,23j)]/2 + (szx3j)

(zxxj3) = (xxzj3) = (szxj3) + [(Uz,xxj3) + (Nz,xxj3) +
(Xxx,zj3)]/2 + (szxj3)

(zyy3j) = (yyz3j) = (szy3j) + [(Uz.yy3j) + (Nz.yy3j) +
(nyyz3j)]/2 + (Gzyy3j)

(zyyj3) = (yy233) = (szyj3) + [(Uz,yy13) + (Nz,yyj3) +

(ny-213)]/2 + (Gzyyj3)

78

79

(xzx3j) (szx3j) + (zex3j) (xzxj3) (szxj3) + (zexj3)

(yzy3j) <Tyzy3j) + (Gyzy3j) (yzyj3) (Tyzyj3) + (Gyzyj3)

In the following three entries i = 1,.2 ; j = l, 2 :

(zyxij) = (Txyzij) + [(Uz,xyij) + (Nz,xyij> +

(xyzij)

(Xxy,zij)]/2 + (nyzij)

(zxyij) - (yxzij) = (szyij) + [(Uz.xyij) + (Nz,xyij) +
(Xxy.zij)]/2.+H(szyij)

(yzxij) = (szyij) + (zeyij)

(xzyij)

These last three entries are for a = b = 3 :
(xyz33) = (zyx33) = (Txyz33) + [(Uz,xy33) + (Nz,xy33) +

(Xxy,z33)]/2 + (ny233)

(zxy33) = (yxz33) = (szy33) + [(Uz,xy33) + (Nz,xy33) +
(Xxy,z33)]/2 + (szy33)
(xzy33) = (yzx33) = (szy33)

(2) Auxiliary coefficients (neigfgbm 5 (egg; ba):

j = 1, 2

(zzz3j) = {-4rckj33fi33/4/3Iz3n1/2(423-95)][£2313 + £2533] x
I 011+32-2X3V 01-313) (3233) - (31%) mm 21

(zzzj3) = {-41rckj337x31/4(2713-21)/3123fi1/27\11/2<4’Xs-Xj>1[aiz§l3 +
85253][(/11+?I2_2Q3)/0(1_’\3) 02-33) - (Arm/231321

(xxz3j) = (zxx3j) = [- ckj33 33/4/3Ilezhl/2(4 3- 1)][afx 13 x
(3al‘a3)/ (AF/Afial + angg3(332.33)/(>~2.23)211 +
nckj33a§Y/2Ilez(433-31)9I31/4n3/2

(xxzj 3)

(YYZ3j)

(W213)

(xzx3j)
(xzxj 3)
(yzy3j)
(yzyj3)

(xyzll)

8O

(zxxj3) = I-Irckj33(233—11)’)\31/4/3IXZIZ’>\11/2(4%-fij)nl/2]

X [angIBOAfO‘BVal‘QBVfi + a§x£3(39\2-?\3)/(?\2—99)9\2]
+ Tlickj 3333‘y {2331—711 ) /2IxZIz/)'jl/2 (433.911 )9133/4113/2

<2yy3j> = ['IIij33 33/4/3Iy212f1/2c433-3pl[c.1813 x
(3XI‘X3V (XI-aflal + a23'g2'3(3">‘2")‘3)/ (9‘2'7‘3Vzl "

'IIij 33a§Y/21y2]:z (423;):1 )fi31/4L113/2

(zyyj 3) = [’TIij 33 (2%3-91131 >131/4/3Iy219j1/2 (475—7111 )‘Ifl/z]
X [318153 (321143) / (kl-afia'l + 32323 (3324‘s) / 024932]

_ 2 1/4 2

lrckj 33a};y (233141 ) /1X2129\jl/27133/4 (49.34).j 153/ 2
_-n—ck133a}3<y/1y 1931/4 (42 3‘23] 1.53/2

-7Ickj33a’3‘y(223-fij ) /1y21z91j 1/29133/4 (491341 )n3/2

(zyxu) = TL‘31‘1113‘36'5‘1'3 (21-3013) / 31,111,191“4 (41-7911“ 2 +

(ficklll/lelylqus/4fi3/2)[aiz(Ix-Iy)/Iz - a’f‘] +Itck112 x

clung/43:13: 3203 02—323) / 31x1y120‘2'7‘3) (49'1'7‘2p‘31’il/2 +

(ICk1127XLl/2/21xlylzfl23/A(47'1‘7'2fii3/2) [322(Ix'1y)/Iz'a§x]

(xyz22) = (zyx22) =chk222a§y 521012—323)/31nyIzQ121/4()2_ 3)'h‘l/2 +

7121/271 ll/4a§3’ $13 011-3713) /31nyIz Oil-33> (47‘2- 97‘3".1

(nck222/21x1y12325/453/2)[a§z(lx-Iy)/Iz - a’z‘x] +Itck122 X
1/2
+

(nck122’2121/2/2leyIQI13/4(49\2-7\1)fi3/2) [a1z (Ix-Iy)/Iz-a’1‘x]

(xyzZl) = (zyx21) =7Ick112a§y£3q11/4(27\1—9\2) (Al-39B)/3leylz X

321/2(47\1_')(2)(’)\1_7137fi1/2 + (“$11203 l-7\2) /21x1y129\21/2X

fil/Zaxy (ii-39‘ )/3III$\(91- )(4 -’)I)1I1/2+(-rrcx
2 3 3 2 3 3 2 3 1

xyz

k122/21x13112?‘2:L/4(l‘/)\2"’/.\1)1'i:3/2)[3'22(1x'1y>/Iz " a32cx1

81

(xyzm = <zyx12> = M122(222-21)2.21/‘*a2-323>a§y§3/31x1yiz x
[)‘11/29\3(X2-7\3>(4”\2-’)\l)‘1’il/2 + (Trek122(25'2-9\l>/21xlylz X
211/223/4(49\2—7\1)n3/2)Ia§Z(lx—Iy)/Iz - a>2<x1 +11ck112 x
213/453“ {521—913)/3leylz’)\3(22—’A3) (4911-22m1/2 + (71c x
1(112/2leylz ll/4(4>£l—>\2)‘fi3/2)[afz(Ix-Iy)/Iz — a’1‘x]

(zxyll) = (ylel) = Irck111a§Y§"13(fl1—3’)I3)/3IXIyIz’)I11/‘Q3(21—’A3)Ifil/2
+ (71ck111/zlxlylz’l15Ming/2)[afzaX—Iywlz + a1’y] + We x
k112’k11/2221/4a’3‘yff3(AZ—37x3) /3InyIz’/\3 (AZ—Q3) (421-32)?”
+ (Tck1122111/2/2InyIz’7123/4(4911—32m3/2)[a§Z(Ix—Iy)/Iz
+ a‘2’y]

(zxy22) = (yx222) = Ick222a§y g(912—3713)/3InyIz’)\21/4fl3(’>\2-'X3)'fil/2
+ (1rck222/2InyIz’/\25/‘m‘3/2)[afizax—IYVIz + afiy] +th X
k122321/29111/4a3‘yg3{21—323)/31nyIzg3()1—>I3)(422.9(1ml/2
+ (Irck122’121/2/2InyIzQ13/4(422—31m3/2) [aizux—Iynlz
+ afy]

(zxy21) = (yszl) = ‘n’ck112Q11/4(2’/)1—9Iz)01—fl3)a§y§:3/3leylzq3 x
9.21/2(7.1—23)(4’A1—712)n1/2 + (Trck112(2ql-f>\2)/ZInyIzO\21/2X
7113/4(421—712)r3/2)[aizux—Iywlz + aiy] + 71‘ck122’A11/4 x
Q21/2593“;(323,43)/31x1y12930‘2-f>‘3) (laza‘lfll/2 + (TIC X
k122/2InyIz’Azl/4(47\2—>\l)‘fi3/2)[a§z(Ix—Iy)/Iz + an

(zxy12) = (yxz12) =rr,ck122a’3‘y 53922-21)02-393)/3IXIYIZ’A11/223 x
OLZ-X3)(47\2-’/\l>fil/2 + (Treklzz(2712-31)/21nyIz’>‘ll/29\23/4X

(422—Z1m3/2) [a§-’-(Ix-Iy)/Iz + a§Y1 +nck1129x13/4a’3‘3’51’3x

(91'3q3)/3Ix1y1z?‘30'l"q3) (411-'>\2)'fil/2 + (TCkllqul/Z/

zlxlylz’/\23/4(4’A1-’/\2)fi3/2[afzax—Iywlz + a1”]

82

(xzyll) = (ylel) ='n’ck111(a{y - a§X)/4InyI£215/4fi3/2 +
71‘ck1127111/2(a§y - 3:2“)/21nyIz7Iz3/4(4A1-22)‘fi3/2

(xzy22) = (yzx22) = ck222(a§y - a§X)/4InyI£AZS/4fi3/2 +
'nck121321/2(a1y — afx)/2InyI;Z13/4(432131)h3/2

(xzy21) = (yzx21) =‘ch112(221—%2)(aiy — afx)/4InyI£AZl/2213/4 X

(421—22)n3/2 +-7rck122(ag’y - agx)/2leylé221/4(432—21)n3/2

 

(xzy12) = (yzx12) ='rck122(2X2—X1)(a§y - a§x)/4ley1gh1l/2A23/4 x
(4)2—A1)n3/2 + rck112(a1Y — a1X)/2lx1yl;h11/4(421—)zyn3/2

(xyz33) = (zyx33) = Irck133q31/2/n1/21x1y12(423421)1Ia§¥§13211/4 x
(31-3A3)/333(31—>3) + (afz(Ix-Iy)/Iz — a§X)/25213/4] +
[Ihk233431/2/fil/21x1ylz(4)3aA2)1[33y523321/4(Xz'323>/

3k3(k2—23) + (a§Z(lx-Iy)/Iz — a§X)/2hAZ3/4]

 

(zyx33) = (yxz33) = [Ickl3é13l/2/hl/lelylz(423-?fi)][a§y5:3qll/4 X
(A1—3k3)/3%3611- 3) + (afz(IX—Iy)/Iz + a1Y)/ZH}13/4] +
FI‘k233X31/2/“1/21xlylz(4X3JX2)1Ia§y§53221/4(az'333)/

323(22_k3) + (a52(lX—ly)/Iz + agy)/2h753/41

(xzy33) = (yzx33) = (Ic331/2/2leylih3/2)[(aiy — a’1‘x)k133/9113/4 x

(423-21) + (agy - a§X)k233/223/4(423-22)1

(3) Auxiliary coefficients (Uxx,py;b;a) e (a’ be) :
In the following,cx= x, y ; j = l, 2
(z. 13) = <-81rc7~31/4/3I 21,115”)[(313211/4/01—23»I§f1°(k111/21/’*
+ ag<kll2al/az3/4(4ala}2)1 + (3:3221/4/(225X3))[X§N X
k122 23/4/ 11/2(4 2' 1) + a1 k112 13/4/ 21/2(4 1' 2>11
(z, 23) = (—8Ic}31/4/3rleih5/2)[($31)21/4/(22—Z3))[aéak222/223/4
+ aIKk1227‘2/213/M422‘Z1H + (£3211/4/<9i1-23>>I§§°( x

k122223/4/211/2(422'al) + 51(kll2313/4/azl/2(4'l‘a2)1]

83

(2,2213) = (-8nc;\31/4/3Iz3fi5/2)[(313)11/4/(911-913D[afzk111/213/4
+ a2zk1127‘1/7‘23/4(47‘1‘7‘2)1 + ~('52'39‘21/4/(X2'X3D[5‘23
X1227‘23/4/311/2(4>‘2’7‘1) + 3221‘1127'13/4/221/2ml‘32) 1]
+ (8,3:3211/4231/4/312352)[211/2513/(91413fl + 7‘21/2 X
§§(21+22-223)/(21-9I3) (A [913%]

(2.2223) = (~81rc23 1’4/312 3~115/2>I<$"3 1’ 4/(22-3 ))[a2k222/912
+ a1 21(12232/113/4 (422-711)] + ((139)114/ “21-23mm? x
k1129'13/4 /X21/2(47‘1‘X2) + 3223/1‘1229‘24/Xll/2(47‘2'7‘l>11
+ (8{1"3221/4231/4/3133'52)[azl/X'S‘Zg/(221A3)2 +7I11/2 x
Q§<21+22-223>/<3\2-9~3>01-23%]

3/4

(42331) + 521(233/223/ 4(0)?in
(2.2233) = <-8Ic$13211/4231/4/3IZ3Q1-23m5/ 2>Iaizk133/<4%3-%1) x

9113/4 + agzk233/(4A3-22)9|23/4] + (8333211/4231/4/3123 x
(21- 3>fi2>I31§/(91-3\3) + rig/<22 -A3>I

(z,xyij) - (z,yxij)= [— 81Tca3 xyk133gg3qjl/4 M3/3IXI 121115/291 l”X
(2133x4331) + {13(13011219132)1/4clx-Iy>/2Ix21yzlzx
fi01-713M'31-713H . i = 1. 2 ; j = 1. 2

(z,yx33) = (z,xyll) + (z,xy22)

(z,xy33)

(4) Auxiliary coefficients (N5o(,&715b;a) :- (ogflfba) :
In the following,0(.= x, y ; j = 1, 2
(z,o(o(j3) = 63:3211/4/(’11-).3fi31/41221x + [41'3'1'391-11/49\31/4/31°(21z x
Til/205‘ 3)][351‘13309‘33‘9/313/4(4X3‘Xl) + an2(°(1<233 X
(333-32>/323/4<413-22>1

84

(Z’zzj3) = (8"h§§33jll4331/4/3Iz3<11-23rhl/2)[afzk133csfi3—Zl)/

(z,xx31)

(z,xx32)

(z,yy3l)

(z,yy32)

(z,xyll)

‘A13/4(4%3‘”1) + agzk233(37b‘z2)/%23/4(475'A2)1 + (333 X
Z1“ 4/<21-23)9\31/ 4123) [31921—29 + Egg/(22%)]
6§13231/4ein%1/(21—7g)211/41x212 + (sweizgfill/4Z3l/4/
3(21'fl3)1x21ifil/2)[zakalll/zl3/4 + a§xk112(37‘1-7‘2)/
(421—7§)9§3/4] + (41c£33221/4A31/4/3(A2—%3)Ileéfil/2)X
[(5}1"f)‘2)a‘}1‘xk112/(‘31‘7‘29‘13/4 + (59‘2‘7‘1)aimklzz/Azw4X
(422—Al)] + 33:3231/4sin3e0337(21423)221/4Ix212
65:5131/4cosaf7(22‘7§)x21/41x212 + <3TC§:3?El/4231/4/
3(22—'A3)ix212111/ 2) [za’Z‘XkZZZ/Rf/ 4 + a’l‘xklzzmz-zlw
(412-21)213/4] + (Afic§1§211/4231/4/3(21—13)Ixziifil/2)x
[(5%2"”‘1)a‘}2{xk122/(49‘2"9‘1flz3/4 + (521—22)a§xk112/213/4x
(421—fi2)] + 33:3231/4ein360s67(fi2—95)%11/41x212
63:3231/4eos337()1—Z3)211/41yziz + (8nc§:;A11/4231/4/
3(21—23)1y211n1/2)[zeiyk111/213/4 + a§Vk112(3)1:22)/
(421-7by123/4] + (41c§33121/4231/4/3(22;23)Iyzléfii/2)X
[(52132”?1‘112/(47‘1-7‘2)7‘13/4 + (522-21)a§yk122/z23/4X
(422421)] — 33:3231/4sin5eos37(g1-75)“21/41y212
6SZSQ31/4sinfiy/(32—33)}21/4Iyzlz + (swegg3fizl/Qfi3l/4/
3(32-R3)Iy21ih1/2)[2egyk222/223/4 + aXYk122(3§2421)/
(4;2_21)q13/41 + (amcgi3g11/4g31/4/3(214A3)1y21ifi1/21x
[(532—?1)a§Yk122/(422—91)%23/4 + (-r>”\1"’>‘2)<‘=‘ka112/9‘13/4X
(421422)] - 33:3951/4sinfieosg7(22—133211/41y212
(z,yxll) = [-3313/2(21afi3)1x21y2123/21[Ii/2 cosir+
Iyl/Zsinaq + 4we315211/4(5%34x1)k133/3(214%3)%31/2 x

(423-31>1x1ylifil/2 - 5:3(31+?3)(Ix—Iy)/2(%l—%3)ZIXZI§IZ

 

 

(z,xy22) =

(z,xy12) =

(z,xy21) =

(2,2231) =

(z,zz32) =

(z,xy33) =

85

(z,yx22) = [g3/2(22-’A3)Ix21y2123/2][sixl/Zsinar—
1y1/2coslg] + 4mQ3fizl/4(55\3-7\2)k233/3(AZ—913951” x
(413-791) InyIz‘fil/2_ 3330293) (Ix—1y)/2()12_913)2Ix21y212
(Z’YX12) = [giszll/4/g21/4(A1423)IXZIYZIZB/z][51x1/2 x
sin3’— Iyl/zeosg] + Ame5:3A11/4(523422)k233/3(21—23) x
(423-7‘29‘31/212213212‘51/2 - 3&37‘11/4afi7‘3) (Ix-Iy>/
Mai-“3) <“2—Q3921/41leyzlz

(z,yx21) = [—35:3221/4/2A11/2(32123)1x21y2123/2][Ii/2x
cosX+ Iyl/Zsin’d] + 41e§23q21/4(5%3-fil)k133/3O(2-9\3> x
(423-215z31/21x1y12fi1/2 - §Z3§Zé>bl/4(>l+23)(Ix—1y)/

2 (21-)? (22-13fl11/4Ix21y212
[87rc5f3211/4A31/4/301—7x3)12351/2] [Zaizklll/Zl3/4 +
213212112(321-%2)/(49x1-'AZ)123/4] + 4 1§X31/4(9\1+>(3)/3IZ3X
111/4(}1—>3)2 + [Ame{E3221/4231/4/36X2—Z3)12351/2] X
[(59‘1‘9‘2Mfzk112/(431'7\2>7‘13/4 + (512J11)a§zk122/223/4x
(4%2—21)] + 4§13S;§)31/4(31+23)(R1¥>2-2fi3)/3(22—k3)2 x
xil/l‘al‘ganf + 33:3331/4(aiz)2/2421' 39‘11/4124 +
3ggix3l/4afzaEZ/4(224A3)211/4Iz4
1871C§3q21/4%31/4,3(Az_’>\3)123fi1/21[2afizk222023/4 +
13121212202241)/(47\2—7\1)7\13/4] + 432317231/4(’/\2+X3)/3iz4x
951/405—2972 + {41¢}:3211/4A31/4/3(21—'A3)12351/2] x
[(522—21)252k122/(422-21>223/4 + (521—%2>aizk112/%13/4x
(4h1—A2)] + 4§3§Z§A3U402+>x3>(21+?iz—2k3w3a1—9x3)? x
”21/4<22'%3>Iz3 + 3§35X31/4(a32)2/2(A2'fi3)321/4124+
33:3231/4352332/4(gl‘a3)%21/4Iz4

(z,yx33) = (z,xyll) + (z,xy22)

 

 

86
(5) fliliary o_oef£ioients (X;o(18 ,Qvf;b;a) e (figmba) :

In the following, o(= x, y, z

(”6231) = 3093:2131/4~(I3/Io<21227\15/4 + 33a(ai253 + 3525:39‘31/4/
zlqzlzlel/Mz + (2Tca°1(°(7‘31/4/31u212q13/4fi1/2)[kill x
\gBGB—7alflalB/4O‘33‘1) + k112g3((2)1‘22)/(49‘1‘32) X
7‘23/4 ' 2221/4/(23‘7‘2D “ 2k1333:931/2/(‘a3‘7‘lflll/41
+ (27coa§<°(7\31/4/3I°<2129\23/4hl/2) [k1125‘139x11/4(3\3-9’>\1+
2:12” (49‘1‘ flab—7‘1) + k122g3’A21/4O‘3"9%2+221)/ (47‘2"
gl>(7‘3-7‘2> - k133333’331/2/221/4(4)331) - k2335‘l3q31/2/
%“ 4(Ma-12>]

(222.232) = a°2‘%52%31/4§§3/IJI22225/4 + 3:“2332533 + aizfz’lflsm/
2It>(21229121/4711 + (211112;:91/31/4/311(2129123/4111/2)[k222 x
533(23'79‘2V22 3MO‘3'7‘2) + k122§$3((2>‘2'(>‘1)/(4IA2-9‘1) X
9‘13/4 ‘ 2211/4/(23—9‘1» ' 21‘23333931/2/(4%3'7‘2)“2l/41
+ (27rca:(°(7\3l/4/3I°<2127\13/4‘fil/2) [121223;3fi21/4fl3-9’Afi
2911)/(4A2-21) (7\3-'>\2) + k11zf‘1’3u1l/4a3—9l1+2xz)/(421-
712)(Q\3_’)11) " k2335:3’A31/2p‘11/4(03—32) ‘ k133(237‘31/2/
7x21/4(4A3—h1)]

(o(o<,zla) = (—a:(0(/ZIO(2‘hl/2;\13/4)[‘fil/zaizg3/Izz7i3l/4 + (47ft/3IZ)X
[2235/4(29‘1‘M3g’13k133/(42331)(75-7191?”4 ‘ 6‘11/2 X
9‘21/4fzfl3kllz/<‘*>‘l"'”2)”\3l/4 ‘ q3klll/zll/4p‘31/4“ + 7
(-af/ZIdzfil/zfizs/a) [fil/2615253311/4/122221/4231/4 +
(MFG/312 ) [fill/4231M(11+2712_97\3)§‘13k233/(423912) (21-
(A3) + 931/4(2233‘93‘2‘31‘133/(423'qlflll/29‘21/4 "A2“4 X
(232-3l)§3k122/ (4)2'7‘1fl11/2A31/4 - Q‘13/43‘23k112/
(421“229‘31/4

87

(cogz23) = (—a°1(°(/21d2h1/27\13/4) [151/2211212{2'39l21/4/1227l11/4A31/4 +
(47Cc/3Iz) [121/ 4231/ 4 (22+221—9’h3)52 3k133/ (42341) (2 _
7‘3) +q31/4(2;13_ 2)fl’3k233/(“23‘7‘29‘11/4221/4 ' fill/4X
(221‘7‘2)5‘121‘112/(l‘yl‘gzfibulo‘zl/2 ' 7‘23/4fl3k122/(422'
Q1)’A31/411 + (_a<>2«>(/210(2fil/2913/4)1111/2122”“(gang/Elm+
(”C/312)[2235/4(222+7\3)5:3k233/(49‘3'7‘2) (7L3-”\2);\23/4 -
' fill/[*Azl/Zggklzz/(4)2‘119‘31/442‘31‘222/>‘21/4A31/431

(xy,zll) = (yx,zll) = -a’3‘yafz§:3/2lxlylzz?i3 - (2ma§y/3leylz x
233/451/2)[2k133ff3235/4(2’A1+’A3)/(4%3—7\1)(A3—?\1)’>\13/4-
kllzgflll/Zazlmflf/a(42132) ‘ klllg3/211/4A‘31/4]

(xy,222) = (yx,222> = —a§ya§z§3/21X1yizzg3 — (27cca’3‘Y/31xiyiz x
g'33Mfil/2)[21(23332637‘35/4(2"210‘3)“47832)(7‘3-’>”2>”\23/4-
k122g3121/2A11/4/231/4(47‘2‘7‘1) ' k222g3/9‘21/4A31/4]

(xy,212) = (yx,z12) = -e’3‘ya§zfl‘39\ll/4/2lxlylzzfi3121/4 - (zirca’g‘y/
3InyIzq33/Z'fil/2) [k233—q3211/47‘31/4(7‘1”)‘2'99‘9“47‘3-
7902‘“? + k133g39‘3l/4Qq3'9‘1) 7‘11/27‘%M(4”\3‘R1) -
k122g37‘21/4(21231)fill/2951M("7‘2‘7\l) ‘ k112§E3 X
213/4M31/4(1Al_912)1

(xy,zZl) = (yx,zZl) = -a’3<ya§zQQ21/4/2lxlylzzfi37s1l/4 - (21roa>3<Y/
31nyIz’A33/4‘fil/2> [k133g39‘21/49L31/4(224411-975)/(4A3—
712)(21_7\3) + k233£3231/4(29‘3' 2)>\21/27‘ll/4(49‘3'7‘2) ‘
k112{163%1/4091—7‘2)”‘21/27‘31/4(4313‘? — k122§l3 X
9123/4/7131/4(4912_zl)1

88

(6) Auxiliary coefficients (G;o(9 ;,b;a) .5 (dfiba) :

In the following, c(= x, y ; i = 1, 2 ; j = 1’ 2

«xxzj3)
(0W 33' )
(dzoci 3)
(o(zo(3j )

(xyzij)

(xyz33)

(zxyij)

(zxy33)

(xzyij)

(20633) = [(_)dea§y/8Ix1§qj3/4Q31/4][£?{/LX2-2a§z/Izz]

2(2223j> = [(-)S“ya§y/8IXI§111/4333/4][5:x/Lx2— 232/1221

I(-f£*ya§y/4ley9fi1/4953/4][afiq/Iq? _ age/1221
(zyxij) = (aiZ/slzzki3/4Ajl/4)[agy/lyz — agx/lxz] +
(afix/8IXZQi3/4gjl/4)[agz/Izz _ aguy/11,21

(zyx33) = (a§Y)2/8Ix21y223

(yxzij) = (agy/81y2213/4211/4)[agx/ixz — agz/Izz] +
(312/8122213/47111/4) [agy/IyL aSSX/Ixz]

(yxz33) = -<xyz33)

<y221j> = <a§X/81X2Ai3/4211/4)[a32/Izz — a§Y/Iy21 +

(aXy/81y2fii3/4gjl/4)[a§X/Ix2 _ a§z/Iz2]

APPENDIX XIII

Coefficients (3;dfi;bc;a) in hé

The coefficients (3;dp;bc;a) appearing in the third sum of (4.5) have
the general form

(3;dp;bc;a) = (U;dp;bc;a) + (X;g3g5bc;a) + (P;dfl;bc;a) +

(T;o(['5;h,g;a) + (N;g¢;2.g;a) + (G;o([5;bc;a)

where (U;dfi;bc;a), (X;d,@;bc;a), etc“, are auxiliary coefficients to
be listed in details later in this appendix. The underlining of the
alphabetical indices means that the corresponding coefficients must
be constructed from the original ones as a consequence of the desym—
metrization of the rotational operators in the terms to which those
coefficients belong, The significance of the underlining of the nu-
merical indices is best illustrated in the example below:

(T;o<[5;£,g;a) = [(T;d(s;b,c;a) + (T;o<f>;c,b;a)]/(l +3“)
List (1) of this appendix shows specifically which ones of the above
auxiliary coefficients actually contribute to the nonzero (3;dfi;bc;a)—
coefficients. In this list the symbol "3" and all separating semi—
colons are omitted from the coefficients, with the understanding that
the alphabetical and numerical indices therein appear in their orig-
inal orders. The auxiliary coefficients (U;4P;bc;a), (X;d,p;bc;a),
etcc, are listed separately in lists (2) to (7). In these lists the
symbols "U”, "X", etc., are omitted since it is only when the coeffi—
cients appear together that one really needs to keep these distin—

guishing symbols. Again, all separating semicolons are omitted.

89

90

(l) Coefficients (3;fl%;bo;a) a jgpbca)
j fill_____

 

In the following,q’= x, y ; i = l, 2 ; j = l, 2

Gixiij)
(«0(le)
G**33j)
@exj33)
颥3j3)
(zziij)

(zlej)

(zzle)
(zz33j)

(zzj33)

(zz3j3)
(xyi3j)
(yxi3j>
(xy3ij)
(yx3ij)

(xy123)

(xyjj3)
(yxjj3)

(xy333)

(Umuiij) + (Pquiij) + (TQQi,ij)

@anlj) = (Uaule) + (Bxkle) + (wal,2j) + (wa2,lj)
wmflafi +(Eu3fi)

(Udaj33) + (waj33) + (Gauj33)

(dej33) + (Pea533) — (Geuj33)

(Uzziij) + (Xz,ziij) + (Pzziij) + (Tzzi,ij) + (szi,ij)
(Uzlej) + (Xz,lej) + (Pzlej) +

(Tzzl,2j) + (TzzZ,lj) + (szl,2j) + (szZ,1j)

(zz12j)

(Uzz33j) + (Pzz33j) + (T223,3j)

(Uzzj33) + (Xz,zj33) + (Pzzj33) + (Tzzj,33) + (Tzz3,j3)
(szj,33) + (sz3,j3)

(zzj33)

(nyi3j) + (nyi3j) + (Txyi,3j) + (Txy3,ij) + (nyi3j)
(xyi3j)

(nyi3j) + (nyi3 )‘+ (Txyi,3j) + (Txy3,ij) -(nyi3j)
(xy3ij)

(xy213) = (yX123) = (yx213) = (ny123) + (ny123) +
[(Xx,y123) + (xy,x123)]/2

(nyjj3) + (nyjj3)

(ijj3)

(yx333) = (ny333) + (ny333)

 

91

(2) Auxiliary coefficients (Ugezéz-bcia) 5 (dfibca) :

In the following, 0(= x, y, 2

W111) =

Qx&121) =

(doQZl) =

(-8m2o2k11é21/31d2n3/2221/2(421—a2»[£:dk112(521—xz)/
913/4(4’A1_9\2) + 32“12122(5%2—%1)/7\23/4(49\2-7\1)1 -(l6'fl.2c2X
klll/3szh3/2211/2)[2§? 1‘111/213/4 + £§Kk112(321_%2)/
223/4(421—99)1 - (lzmoklll/nlq2A1)[fistinZZ’+ ggycos§y1
41—80(2) (61tck1127\13/4/fi1d2323/ 4 (Ml-7x2» [<->1"$<Xsinz(x
0°33? + 6;z[‘3ICklll(aIz) /hIz321 ' 3nek112alzagéxl3/4/
2nlz3223/4(421—12)— snok111§:§(a1+h3)/3122521(A1—/3) -
‘8”°k112213/§§13§33(31+33)(21+32‘ZA3)/3122“%23/4(Al'7b)x
(Ry/13> (Ml-A2”

cmXle) = (-32t?c2kll§21/3lxzn3/2221/2(491—)9))[2&f‘k111/
Q13/4 + a§“k112(3A1-32)/223/4(4?1422)1 — (16n2o2k12222/
3Ld2h3/2211/2(422-21))[a:*k112(521-%2)/213/4(4%14k2) +
aoéwklzz(512-47‘1)flz‘o’mmxzfl‘l)1 ‘ (24nhk112A11/2/3Lx2 X
321/2(421_22))[S;xsin22(+‘oxyooSQKJ - (l-S;Z)(lznek122x
9‘23/ 4mx2213/ Roz-Q1» [<—>1'€Wsinycosg] — Scam-7m x
k112(alz)gll/2 ffilz3221/2(4“14A2> + 3nok122a§za§ZA23/4/
fiI23213/4(49‘2-9‘l) + 16n3k12iAQS/43IBXZ3(Al¥x3)(21+ZZ'
223)/31225213/4(Z1—A3)(22-23)(422421) + lonnkllilll/Zx
3:3(A1+23)/31225221/2(21—23)(4%1—Z2)]
('8H9C2k222/33x2h3/22ll/2>[5ixkllz(5ql‘“2)/Al3/4<4)l'12)
+ e§Xk122(5fl2-21)/223/4(4%2—Rl)] — (lon?o2k12222/3103 x
H3/2/r‘ll/2(‘*Q2‘9‘l))[2"=‘°l(°<klll/"l3/4 + agdkllz(321‘%2)/
Q23/4(431-%2>] - (lzncklzzfig/nlx221(4fiz— 1)[ngsin22{+
gxycosfif] - (l— dz)(ennkzzz/nrx2223/4flll/4)[(_)1'axxx...

92

(o<o<221), cont'd :
sing/cos’o’] - {dz[37fck112(aiz)2fl2/filz3fll(49\2-7\l) + an x
1‘2223:23?/21"Iz357{23/4Xll/4 + 8K¢k2223:35:3(ql+23)(Al+22
"213)/3Izqul/4223/4(Al‘”3)(32" 3) ‘ 8flfik12ikzgl§¢11+
23) Ms anil—13> Mao—11>]

(W112) = (-87L2c2k111/31q2fi3/27Lll/2)[a°{°(k112(521-22)/?\13/4(47\1-
7‘2) + af>§Xk122(57‘2-7‘1VA23/4wzz-7‘l)1 - (l6ngczklléxl/
3103n3/2221/2(421-22))[25§*k222/223/4 + aTVk122(3224A1)/
313/4(422—21)] - (lznbkllzfilthxZQQ(4x1422))[ngc0322f+
sensitize/1 - <1-§z><6vtckin/file2%i3’ 4X2“ bunks“ x
singoosgfi - 5&z[3nck112(a§2)221/fi123fi2(4al422) + 3[ck111X
aizagz/21123213/4221/4 ' 8n¢k1115:35;3(21+33)(21+“2-
213)/3Izzal3/4a21/4(Al'23)(“z-a3) ‘ 8WPk1122133§<22+A3)/
312222(22-23)(431412)]

«xx222) = (-lon?e2k222/3rx253/2221/2)[zééxk222/223/4 + efihklzz x
(312_A1)/113/4(422—x1)1 — (8n9o2k12232/31,?n3/2211/2 x
(4)2—zl))[d§xk122(5A2'%1)/A23/4(4A2'21) + ékallz(5al‘
22)/213/4(421—22)1— (lZflckzzz/fiLuzzz)[Schosz + axy x
sinzb’] - (1&2) (eukmfif/ 4m103213/ MHz-$1» [ (J‘érx
sinyeosg] — 5,2[3nek222ags2mz322 + mkllzafzagz x
223/4/fi123A13/4(4Al‘z2) + 8I9k22253§(22+23)/31225“2(12-
23) + 8W1227K23/4(22+23>(kHz-223)gags/312211711“4 X
(ll—A3) (kl—9x3) (42241)]

«£1122) = onlez) = (—32n2o2k121A2/3lq?n3/2211/2(4%2-11))[23? x
k222/223/4 + ekalzz(322—Al)/fl13/4(422-k1)1 — (l6m2c2 x

kllzzl/3lx233/Zazl/2(4z1—32))[5?“t112(521—%2)/Z13/4 x...

93

(06(122) , cont'd.

(W331) =

(0(O(332) =

(06(133) =

(ml—fig) + a‘ZXklzzohz-fll) /223/4<4%2-%1)1 —<241:ck122 x
“21/ szxzkll/ 2(422-A1» [axxcos;5/+ gysiéy] —(l—é°(z)x
(lznok112213/4/m2fl23/4(Ml-A2)) [(—)l_&xsin’o’cos’5] —
Suzwlmklzflagz 27‘21/2/‘mz3X11/2(Ml—X2) + 3thkllzaiz X
a32213/4/nlz3221/2(Ml—AZ) + 16mk1227\21/25§§(22+A3)/
312202—23) (47x2—21)7\11/2 — 167Lck112(21+13)(31+A2—223) x
7‘13/4 1:31:3/3122223/4ar913) (2243) (4131—79)]
(-l6752c2kl337l3/3Io(zfi3/ 2211/ 2 (Hg-A1» [2a°f‘klll /7\l3/ 4 +
aoédkllzwfil‘AZ)AZBMMRI’AZ)1 ' (W2C2k2337s/ 310(2n3/2x
951/2(4"\3'7‘2)) [acidklIZGQ‘i—az)/7\13/4(47‘1'7‘2) + ale§°<1<122X
(512-%1>/7tz3/4<422—11> 1 - (127021213323 fulfill (423-21) >x
[S;(xsinzfd’+ deoosfifl - (l—SW)(61cck2”Kg/13103123”4 x
7111/ 4(4’13—22»[<—)1'5°<xsing/cosz{1 — g°<z[311;c9l3aiz/fi1z3 x
7‘11“][aizkl33/7‘13/4MX3'7‘1) + a32X233/2323/4(47‘3'7‘2)1
(-87r,2c2k133 3/3kzn3/2fl11/2mA3—7ll» [a°f’(k112(55\1-7\2)/
9x13/4(421—>l2) + a°2<><k122(522-7\1)/9L23/ 4091—21)] — (1671:2X
C21I<233X3/3I<><2’X3/25X21/4“9‘34LG[2~‘2‘°26(k222/5‘*23/4 + 5‘3““an
(312-11)/9\13/4(4>\2->\1)1 - (121Ick2337L3/‘fi1w27lz(433—12))X
[gdxCOSZKO/i' Saysinzrgf] — (l-&2)(67cck1339i3/n10<2213/4 x
fl21/4<4A3->\1>[<—>1'5xx singcoszn — éXZI3rtck2337Lg(a§z)2/
nlz312(47\3-A2) + 3mk133fl3aizagz/2nlz 113/4K21/4(4A3-7L1)]
(o<o<313) = (46792812133913/3l°(?rn3/2211/2(log—Al)) [a°f’<k133x
(323‘Xl)/Xl3/4(423—9Ll> + a°<2°<1<233<3X3-7‘2>/223/4(4X3-7\2)1
-(247cok1339l31/2/n10(12111/2(423-11)) (l-S,(z)- 5;,ro x
k1339k31/2/3122fi211/2(4134(1)][fig/(X196) + 53/ (“z-13>]

 

94

@nx233) = ¢nX323) = (-lon?o2k23323/3l,?n3/2A21/2(4A3éx2))[éfi‘k133x
(333—11>Akl3/4(4A3—21) + §§<k233(3A3—22)/223/4(413—22)1
— (24nok235k31/2inrxigkzl/z(4A3—12))(1 — $Xz)—3Xz[l6np x
kzggxé/Z/alzznzzl/Z(423—22)1[$13/(21-R3> + 53§/<22-23>1

(xyll3) = (Yxll3) = (-8n¥c2(SQS—Xl)/3leyfi3/2a33/4)[klllkl33/(423—
X1)fl11/2 + k112k233911/2/(4x1—A2)(413412)] — (nc(Iy—Ix)/
IleyzfiXBl/A)[klll§:3(ql+23)/Al3/4(21_ 3) + k1123;39‘1)‘
(A2+x3)/AQ3/4(22—23)] + (3noklll/nix21yzal)[(lx/lz)1/2x
coszf+ (ly/Iz)l/zsinj] - (moklljkllfilx21y2223/Qfl31/4 x
(all—A2))[5(lx/IZ)1/2sin3/— (Iy/lz)1/2oosg]

(xy223) = (yx223) = (—8n?o2(513-xz)/3lxlyn3/2z33/4)[k222k233/(423-
)2>le/2 + k122kl33221/2/(422-7lfl(433-111)] - (IB(Iy-Ix)/
Ix21y25231/4>[k222333(22+X3>/7‘23/4(22-7‘3) + k12251322X
(Al+X3)/Xl3/4(Xl—23)1 + (313k1229‘2ffilleyz’tlwwbl/4 X
<4x2—21>)[(Ix/Iz>1/2cosa/+ (Iy/Iz>1/2sinafi - (nrkzzz/
nlx21y2123/4A31/4)[5(1x/lz)1/2singr— (ly/lz)1/2ooe1]

 

(xy123) = (yx123) = (xy213) = (yx213) = (—lon?o2/3lxlyfl33/4)[klzzx
k23322(533-7‘2)/211/2<47‘247‘1)(QB-7‘2) + k112k13391(523‘
X1)/221/2(421_22)(423—A1)] - (2ro<lx—ly)/lx21y25231/4)x
[k122533323/4(22+X3)/all/2(22'X3) + k112§:3213/4(Al+23)/
121/2(21_g3)] + (6kallzyl3/4/fi1x21y2221/2k31/4(4a1_

12))[(1x/Iz)1/2oosg’+ (Iy/Iz)l/Zsin6] - (nok122g23/4/n x
Iley2223/4A31/4(4914A2>)[5(Ix/Iz)1/2sin2{— (Iy/Iz)1/2x
c0531

(xyl3l) = (yx131) = (xy311) = (yx3ll) = -8m2o2kl§3231/4/3lx1§n3/2x
all/2(SA3IXI)(4X3J%1)2 ‘ TD3kl335:39‘33/40114'23)(Iy‘Ime

95

(xyl3l), cont'd.
1x21y2213/4(21—R3)(4234A1) + (6npkl33233/4/51x21y2(423-
21)fl13/4)[(Ix/Iz)1/2coe34+ (Iy/Iz)1/2sin31

(xy232) = (yx232) = (xy322) = (yx322) = —sngc2k233231/4/31x1yh3/2x
321/2(5%3-22><423—%2>2 — Irk233333233/4<22+Ao>(Iy— Ix)/
1x21y%223/4(A2_23)(493—22) ' (2W5k233333/4/X1x21y2(493'
22)A23/4)[5(Ix/IZ)1/2sin1{- (Iy/Iz)1/2coszfi

(xy333) = (yx333) = (xyl3l) + (xy232)

(xy132) = (yx132) = (xy312) = (yx312) = —8n2c2k133k233231/4/3Ixx
Iifi3/gzll/2(5X3-A2)_l(433—Z1)(433€A2)— ck133333233/4ca2+
23)(Iy—Ix)/Ix21y2211/2A21/4(22—7g)(4k3—21) - 2n3k133 x
X33/4/fi1x21y2211/ikzl/2(4%3-h1)>[sclx/Iz)l/2sin2(—
(Iy/Iz)1/2cosg]

(xy231) = (yx23l) = (xy321) = (yx321) = —8n?c2k133k233231/4/3Ix x
IyXS/zzzl/Z(523‘92)_1(4X3‘zl)(4X3'X2) ' X‘k233§:3(21+
93>233/4(Iy—IX)/Ix21y2221/2211/4(Al—A3)<4z3—22> + (622 x
k233233/4/h1xzxy2221/2211/4(423—x2>>[<Ix/Iz>1/2cos1r+

(Iy/IZ)l/Zsin6]

(3) Auxiliary coefficients (x;ocp;bc;a) = (ocfihggl_i

In the following, i = l, 2 ; j = l, 2 ; E.= [j+(—)(j+l)Sij]

<2.ziij> = {-4iegjg<zj+23>/3Izzhfljl/4caj-x3>1[3:3ckiii/ai3/4 -
2ki3é)3l/2/2i1/4(4A3'%i)> + 323k11331/353/4<4Xi'3g)1

(z,lej) = (z,zle) = [—8Wc§;3(fij+fi3)/3I;%h(fij—13)%jl/4231/4] x
[223/4231/4333k122/211/2(422"hl)‘g21/4233/€§33k133/
2ll/Z(47‘3'%l>+Xl3/4X31/4gakllz/le/z(4X14X2FXIU4 X
“33/43l3k233/X21/2(4%3JX2)1

96
(z,zj33)= (z, z3j3)= 2(z, zjjj) + (z, zjk__k)

(4) Miliary coefficients gdfibcm) a (o/fibca) :

 

In the following, o(= x, y, z, and in general (u/gbca) = (fiqcba).

66(111) = (—8113c2 a°{°</3Io(zh3/2213/4)[null/All” +91k112/221/2x
(fill—22)] - (87L2c2a°2(°(/3IO(253/2223/4)[klllkllz/le /2 +
7‘1k112k122/7‘21/2 (49192) 1

(W112) = (-81€c2a‘§“/3I°<Zh3/2223/4)[3221211112122/211/2(432—21) +
kllzkzzz/le/Zl ' (812c23lx/3lxzh3/22‘13/4) [klllkllZ/
211/2 +2lk112k122/221/2U'91'92)1

(OM22) = (-lom2c2a°‘°</3Idzh3/29\23/4)[Alk112k122M21/2(4?\1—7\2) +
3221212212222/911/2 (49291)] — (167gc2aix/31a2fi3/2313/4)X
[alkllz/gzl/ 2(Ml-A2) +9‘Zk122/q11/2U‘22-21H

(W121) = (~16Tec2aiU/3Iq2fi3/ 2213/ 4)[3211211112112/221/ 2(421-22) +
’Azk112k122/211/2(422—911)1 — (loqécza3<°(/3Ig(zh3/2223/4) x
[Xlkllz/le/ZMXl'Xz) + szlzzmll/ 2(47\2->\l>1

@4221) = (—87r,2c2a°‘°‘/319<2'H3/27\13/4)[fizklllklzz/Xll/zflaaal) +
kllzkzzz/Xzl /2] ' (878czagd/3quh3/ZX23/4)[fizkllzklzz/
p‘ll/2(‘*22 '21) + X122k222/le/2]

(M022) = (—87I2c2 aZX/3Iwzir3/2223/4)whiz/221”- +?\2k1%2/211/2 x
(22221)] - (8120.2 a‘f‘X/3I 2213/2213”)[1222212122/221/2 +
221‘1221‘112/“11/2(49‘2‘21) 1

(“(331) = (’8TD2CZaIX/3k2fi3/ 2913/ 4)[37‘3klllk133/le/ 2(l+’/\3-;\l)+ A3X
k112k233/9‘21/2(4)3‘9‘2)] - (815C235‘693lxzfi3/2323/4H9‘3 X
k112k133/9fll/2(423'Xl) + X3k122k233/9‘21/2fl‘9‘3'X2N +
(“cl/2332122 1 film 1’4)t§‘3°"‘/9«1+5§3a°§°‘/121

97

enX332) = (-8x?cza§{/3Iugfi3/2223/4)[A3k122k133/311/2(423-21) +
3231222212233/221/2(423-22)] - (876c2a°2(°‘/3Io(7-h3/2223/4) x
[23k112k133/211/ 2(423-911) + 23k122k233/221/2(433-32)] +
(231/2§3/21q212fi1/2221/4)[5382(Xa2 + 1351016021]

(«4133) = (~161gzc2aof/3Id2fi3/2gl3/4) [9312133/211/ Zak-21)] -
(lenéczagw93193h3/2223/4>[2321332233/211/2(423-21)1 -

2<x«' 2 ~ . car
1331 / 21a 12‘1'11/29113/4 ' 211/ 43352332 ”loczlzfil/zqz

«xxz33) = (-16n2c2a§“93la3h8/2223/4)L2§k2335221/2(4A3422)1 -
<16I2€23Tb732g253/2Xl3/4>[33k233k133/921/2(433122)1._
523,32” 210(212111/ 2223/ 4 ' 921/4535???“ / 21031211“ 29&1

(xyll3) = (-8E?cza§y/3Ix1§h3/%133/4)[k111k133flzll/2 + k112k233 x

’kl/gzl/2(4Al_gz)1 +:2llx2gigagy/21x11nl/2235/4

(xy223) = (-81?c2a§Y/3IXI§h3/2233/4)[k222k233L921/2 + k122k133 x
92A211/2(422421)1 +‘X21/ZSZ§a§y/21xlynl/%aa5/4

(xy123) = (-Sn?c2a§Y/3IXI§nS/2233/4)L21k112k133/321/2(491_;&) +
22k122k233/X11/2(432‘21)1 +‘211/4221/4}:3§Z3a§Y/zlxly X
12hl/2235/4

(xy333) = (~8I9c2agy/BIXIyH3/2233/4)[I3kl33k133fiflll/2(423;21) +
25k233k233/921/2(423'7\2)1

(xyiBj) = {-16flgczagy/31xI§n3/ZA33/4)[23k133kj33/gil/2(423-31)] _

211/4333§3a§y/21xlylzhl/221“4233/4 ; i = l, 2 ; j = 1,2

(5) Auxiliary coefficients (T;o(‘3;b,c;a) = gfibfia) :
In the following, Q/= x, y, ; i = l, 2 ; j = l, 2
(xy3,ij) = (yx3,ij) = 16n2c2k133kj33a§Y235/4/3Ix1§fi3/2211/2(423—
2i><423421> + 2nhk133211/iz33/4gj3cly—1x>/Ix2112<423-

21><Ri4h3>

98

(xyi,j3) = (yxi,j3) = 16¢?c2k133kj33agyfi33/4(2)3-X1)/3Ix1§n3/2 x
211/2211/zc423-21) (423-21) + zxck133313ay-Ix) (223-
91119131/4211/4/1x21y2311/2 (123411) (31313)

(xyi,3j) = (yxi, 3j) = 16n9cc2k111k333a§y233/4(29}‘21)l3lxlin3/2X
211/ 22 1/ 2(42 -2-1)<47l3-311) + 1612c2k111k133a3233/4
31XIYn3/2(421 -21)(423-A1) + 27cck1 1133231 /4211/4(221-
91)(Iy-Ix)leziyzajl/Zmzj-Ai){21-23) + 2Ick111 513(1),—
12)231/4213/4/1x2122(421-21)(hi-23> , j 7‘ i

(xyl,31) = (yxl,31) = 16E9c2k111k133a§y233l4/31infia/211(423:21) +
16n2c2k112k233a§Y233/€a11/2/3IXI§h3/2(421-22)(423-22)x
321/2 + chklllslaqsl/4<Iy-Ix)/Ix21y2311/4(al-X3) +
ZICkllz{23911/2321/4331/4(Iy-Ix)/Ix21y2(4ql-32)612493)

(xy2,32) = (yx2,32) = l6n§c2k222k233a§y933/4/BInyfi3/zfll(493412) +
16fl9c2k122k133a§Y233/@121/2/31xI§fi3/2(47b— 1)(413JA1)X
‘311/2 + 2W9k222523131/4(1y'1x)/Ix21y2321/4(32'X3> +

. 2I3k1223l3321/3311/3231/4(Iy-Ix)/Ix21y2(432-11)(”l-X3)

(oozl,ll) = (161L2c2 k1 11/31012113/22 11/2)[a°1°‘k 111/213/4+a2°(h1129\1/
2 23/4(421-22)] + (lon?c2k112211/2/3Ia3h3/2(421é22)[a2 x
k122323/ [all /2(432 31) + a1Wk112313/4/321/2(4911-312)1

ng2,11) = (16m?c2k112(221122>/312253/2(421432)221/2)[a<Xk111/
7113/4 + acgxkll2Xl/7‘23/4MXl-32H + (16fi982k122111/2/
31x253/2(4Xl'32))[32 Wk122323/4/211/2<432-21) + aTRX112X
213/4/221/2<421422>1

¢*11,12) = (16m2c2k122(2A2-21)/3szh3/%911/2(432-%1))[d§*k122 x
X23/4/311/2(432‘X1) + ikall2X13/4/le/2(431432)1 +

(167(2c2k1127\11/ 2/3103213/ 2(421—73» [23"12111/213/ 4 + ...

99

(96(1,12), cont'd.
+ ao29(1‘112911/223/4(4911—22)1
(442,12) = (1611;2c2k222/31012h3/2221/2)[a°§’<h122223/4/211/2(422—21)
+ a°f(k112')t13/4/7121/2(421—22)1 + (16L2c2k1229t21/2/31012x
113/2(4512‘31” ["105(L><1Xlll/9113/4 + 8°;k112fi1/223/4MQ11'912H
(221,21) = (16m2c2k112’211/2/3Ifn3/2(421—22))[a°§’<k222/9tz3/4 + aff‘x
k122g2A13/4ml’12'21” + (16 152211111/3103113/2311/2) [52%
k1229123/4/3111/2(432—31) + a1°l«’<1<112213/4/;121/2(43142)1
(o<o<2,21) = (l6fic2k122921/2/3Iu21r3/2(log—21))[a°2(°(k222/)\23/4 + a°1°‘x
k122’A2/fil3/4(422-911)] + (16792‘32k112(2911'7\2)/3Io<zfi3/2 X
7121/?"(431‘32D[302(k1220‘23/4/311/2(492"7‘l) + alxkllz X
9113/4/9121/2(4911_22)1
(2421,22) = (1671;2c2k122(2}2—%)/3&%3/2Q11/2(422—11))[agxkzzz/
7123/4 + aifl122q2/113/4Mqrfiin + (167ec2k112211/2/
310(21’13/2(421‘7‘2)) [392%1229‘23/4/211/2(491241) + aolWXllzx
313/ 4/321/ 2 (421-72) 1
W232) = (16702°2k222/3Iocz‘"3/2321/2)[21(2°(k222/7‘23/4 + a01("<1<1227‘2/
113/40.22-29] + (16m2c2k122221/2/3Io13h3/2(422—21)) x
[3?k122’X23/4/All/2(4912—21) + auk112313/4/9‘21/2(47‘l'
22)]
(423,330 = (161t2c2233/2/3Io(2h3/2(423—21))[afidk133/fi13/4wflg—kl) +
a“? k233/9~23/4(‘*;13'Xz)1
wj ,33) = (167c2c2k133(223—21)/3I°<2h3/2211/2(423-21))[a°1°‘k133/
9‘13/4(4713-Xl) + aéxk233/%3/4(4%-M>1

100

In the following, (b,ca) will be used to denote the expression

obtained

(221,11)

(z22,ll)

(zzl,12)

(z22,12)

(221,21)

(222,21)

(zz2,22)

(223,31)

(zz3,32)

(zzl,33)

(222,33)

when 2 is substituted for 0(in (*xb,ca) :

(1,11) — (encfifg/slzzh)[2k111313/(21—23) + k112(2H1—22) x
3:3311/4(ql+32-233>/321/4(431432>(31133>(32-33)1

(2,11) — (812515211/4/3122n(21423))[2k112(221422)§:3211/4/
121/ 2091-22) + k1229123/4~52‘3(31+7‘2-29‘3)“43231) (912—29 1
(1,12) - (8mc3f3/3lzzh(21-23))[k122(222—21)221/€§;3(A1+A2
-291)/211/4(422:21)(22-23) + 2211521 13/(421122)1

(2,12) - (8n2313211/4/3122h(21-23))[323(21+22—223)k222/(22
-21)221/4 + 2k122221/2211/4311/(422—21)1

(1,21) — (smc221/2333/31225(32—23))[2k112311/2323/(411-32)
klll¢13<£l+32’223)/211/4(31J33)321/41

(2,21) — (8Ic333/31225(22-A3))[2k12222322/(422421) + k112x
(231-22)311/4313621+32-233>/221/4(431-22)(Al-33)]

(2,22) — (schZ3/3122n(22-23))[2k222333 + k122211/4223/4 x
3:3(21+22—223) / (422-91) (9123)]

(3,31) - (lomck13523/3122h(423:21))[513/(21-23) + 3;§/<22-
913)]

(3,32) - (16m23323/3122mfl3—22)>[Q3/(21—23)+3‘2’5/(22-23>1
(1,33) - (lorck133(223411)211/2/3Izzh(423—21))[Kjg/(21—fi3)
+ 3:3/(22—913H

(2,33) - (12121233(222-92)231/2/31222(223—22>>[Sig/(21—23)
+ {21/011291

(6).Auxiliary coefficients(N;{,fi;b,c;a).

101

. («be .ca) :
‘+l

In the following, i = l, 2 ; j = 1, 2 ; m = 1, 2 3 S = ('13 +3

(2,21,11) = -2§3 j3qill4332/123111/ 2611-913)

(z,zmaij) =

.(z,zj,33) =

(2:23.33) =

(2:233j3) =

[811C §3gill4g31/4/
3122m21-23n [212133311‘3235/4 (22103) / (423-21) (9(3-21) x
313/4 - ksij323311/23211/4/331/4(4314‘s) ‘ 1111531
9131/4311/4]

451112121“ 4a1?Z/121‘h1/ 2 (2 In313) — [sire Q32m1/4231/4/3122x
15 (am-13) 1 [111/ 4231/ 4g: 3121 3 3 (213221-923) / (423-21) (7(1—
713) + 231/4313k133(223—9\1)/211/2211/4(4913-’A1) - 9&11/4X
(:3k111(2311-31)/211/4R31/4(47\1-21) ‘ X‘iS/l'gskiij/
231/4<431-21>1 , i 2 j

[8th {10321.1( 4231/ 931221101143) 1 [k133§13(2>(12+2232-
7317‘31/9‘13/4(433‘31) 013—711) + k233 3293022397132-

771223) 2123/ 4(423-2 2) (2342)]
(2231/2/12311/2311/4)12332211901143) + 3:3(2112113 +
eggs/(2823)] + [8&53111/4231/2/3122M31-23)] x
[12111(12(23-721)/213/4(%3-9\1> + k113§3<(221-23)/<421-
“9233/4 - 2281/4/(23-23» - 2213353231/2/211/4M9h-
21)] + [81Lc{8933181/4231/2/35215(28-23)][la-113313211” x
(23+2zS—9711)/(421-28) @341) + kjssgz3asl/4(9\3+221—
9713)/<428-9\1>(23->\s> - k133§§3231/2/9\s1/4<49\3—21) -
k.233 333331/2/3j1/4wfl3'3afl

(Z9219jl) + (z,zZ,j2)

102

(7) Auxiliary coefficients ((3419 ;bc;a) a (qubc_a_L:_
In the following, i = l, 2 ; j = l, 2 ;o(= x, y:
(«4133) = (00(3j3) = <-)§Y2§3211/4a§y/Ix1112251/2(21423)
(22131) = @2131) = —(xysij> = t€3211/4231/4/Igh1/2211/4(911-23)1

yy 2 xx 2
X [a1 /Iy - aj /IX]

APPENDIX XIV

Coefficients (333% ;—;abc) in hé

The coefficients (3;xp;—;abc) appearing in the fourth sum of (4.5)

can be written in general as

(3u@;-;abc) = (d?*-*abc) + (N;o(p;-;abc) +

*Z[(T;o<fi;-;rs ,t)+(U;o(,@;-3r.st)+(G;o(p;-:r.st)]
(rstsabc)

where GX@*-*abc) are the coefficients in H3 as listed in Table X,

(N;o(@;-;abc), (T;o</3;-;ab,c), (U;0(,(;;a,bc) and (G;o((3;-;a,bc) are auxi-

liary coefficients to be listed in Lists (2), (3), (4) and (5) of

this appendix, and *2:_ denotes the summation over all permut-
(rst=abc)

ations of the indices r, s and t, subject to the restriction that the

left of the two indices placed on the same side of the separating

comma is less than, or equal to, the right index.

The nonzero (3;x@;-;abc)-coefficients are listed in List (1), in

which the separating asterisks and commas are omitted from the coeff-

icients.

(l) Coefficients (3gfigj-gabc) s (Banbc):

 

(so/«111) = (o<o(lll) + (Ndoall) + (To(o<11,l)

(34422) = (26(222) + (M2222) + (M22,2)

(30(o(llZ) = (30(o(121) = (3.41211) = («02112) + (Nomlz) +
(1242113) + (M2124)

(322x122) = (3x2212) = (322221) = («4122) + (11222122) +

('Bxx12,2) + (To<o(22,l)

103

(3a8133)

(3&3233)

104

(322813) = (3ao331)

+ (un3,13)

(32u323) = (3««332)

+ (Gou3,23)

In the aboveo<= x, y

(322111)

(322222)

(322112)

(322122)

(322133)

(3zz233)

(3xy113)

(3xy223)

(3xy123)

(3xy333)

ll

(dHlB3) + (TdK13,3) + (Taw33,l)

(dfi233) + (TqN23,3) + (TdK33,l)

(22111) + (sz111) + (Tzzll,1) + (Uz,zl,1l)

(z2222) + (N22222) + (T2222,2) + (Uz,z2,22)

(zzllZ) + (Nz2112) + (Tzzll,2) + (Tzz12,l) +

(Uz,zl,12) + (Uz,z2,1l)

(322121) = (322211)

(22122) + (Nz2122) + (Tzz12,2) + (Tz222,l) +

(Uz,zl,22) + (Uz,22,12)

(Tz213,3) + (T2233,1) + (Uz,zl,33) + (Uz,z3,l3)

(322313) = (322331)

(T2223,3) + (Tzz33,2) + (Uz,22,33) + (Uz,z3,23)

(322323) + (32z332)

(xyllB) + (nyll3) + (Txyll,3)
(3xyl3l) = (3xy3ll) = (3yxll3)
(xy223) + (ny223) + (Txy22,3)
(3xy232) = (3xy322) = (3yx223)
(xy123) + (ny123) + (Txy12,3)

(nyl,23) + (ny2,13)

(3xyl32) (3xy213) = (3xy231)

(3yX123) etc, ...

(xy333) + (ny333) + (Txy33,3)

+

+

(Txy13,1) + (nyl,13)
(3yx131) = (3yx3ll)
(Txy23,2) + (ny2,23)
(3yx232) = (3yx322)

(Txyl3,2) +

(3xy312) = (3xy321)

105

(2) Auxiliary coefficients (Ngdggabc) a Gigabc) and all Bermutation:

In the following, o(= x, y, z

(«2111) = 4nchl/2af2kllll/lxzal3/4 — (8n2c2fil/2aTx/3lugfll3/4)[3 x
k121/211/2 + kl§2(221:22)/221/2(421-%2)] + rail/2:1;Ex
klllz/1x2223/4 - (2K202fi1/253“/31x2223/4)[klllk112(72l-
22)/211/2(421'22) + kllzklzz(132122—2212'52‘22)/221/2 X

(421—22)(4A 241)]

«22112) = znyh1/2$ k1122/112223/4 - (8m2c2fi1/2a éx/3192223/4) x
[kllzk222(32l'22>/221/2(421-22) + klllk122(322<21>/
2 ll/2(422‘21) + 2k122211/2/(47‘1-7‘2) + 2kl§2221/2/<422-
31)] + 3nhh1/2%Wk1112/I2Al3/4 - (2n;c2n2/2a$“71«2 x
213/4)[klllkllz(721'22)/211/2(42l‘22) + k112k122(132‘1 X
22—2A12-5222)/22 1/2(421-22)(422—21)1

(AXlZZ) = 31ch1/2af k1222/1X2a23/4 - (21,c2fil/2a2“/1«2223/4) x
[k122k222(222'21)/221/2(422'2l) + k112k122(132122‘2222'
5212)/211/2(43H-22)(42 —21)] + 2mcfil/2 a““k1122/1 2x
213/4-(813c2n1/2a3‘1“‘/3l,<2213/4)[kllzkzzzwfi :22>/<4xl—
22)2‘2l/2+ klll(322‘kl>k122/211/2(422221) + 2k1122‘11/2/
(422 —2 l) + 2klgzzzl/2/<421-Az)1

1/2a°2(°(k2222/I°(2223/4-(815c21'1'1/2aqx/3IX2223/4n3 x

k222/22l/2 + k122(222‘21)/211/2(47‘2 A1)] + mcfil/2a1dx

W222) = [Fitch

k1222/Io< 2213/4 ‘ (2n2c251/2$ /3LN 2213/4)[k222k122<722-
91)/221/2(4£2—>1) + klzzkllz(l32f22-2222—5312)/211/2 x
(422421)(421—%2)]

<xy113> = <yx113) = ziphl/2a§Yk1133/ixiyz33/4 — (anEcznl/Zagy/s x

1x1§233/4)[klllkl33(523‘22l)’211/2(423'21) + k112k233 X

106

(xyll3), cont'd.
<<2Q1-%2>/2221/2<421—%2> + (2%3-22)/2121/2<4%3e22>) +
2kl§323l/2/(423‘21)1

(xy123) = (yx123) = znefil/2a§yk1233/ley233/4 — (812e2n1/2a§y/3 x
I£37133”)[k122k233(<223-22)/9<21/2(423-522) +221/2 /<422-
21)) + k112k133<(2%3-11)/211/2<4fl3n21> + 211/2/<421422>)
+ 2k133k233231/2(823-21-22)/(423-21)(423-22)1

(xy223) = (yx223) = zmenl/2a§yk2233/1x1y233/4 _ (4flgczfi1/2agy/3 x
ley233/4>[k222k233(523-222)/221/2(423-22) + (k122k133/
211/2><<222-%1)/(422-%1> + (223-21>/<423421>) + 2k2§3 X
231/2/<423-%2)1

(xy333) = (yx333) = 4Hen1/2a§yk3333/IXI§A33/4 - (8m2c2nl/2a§Y/3 x

 

1x1§333/2)[kl§3(223421>/%ll/2<423-2h> + k2%3023‘22”
221/2(423-Az)]

(3) Auxiliary coefficients (Iigfi;ab,c) a («Qab,c) :

In the following, c1: x, y, z

(wx11,1) = (amen/Iq2flll/4)[2klll($xycos2y’- astin%()AAl3/4 +
(—)62X(l-$xz)kllz(221422)sin6eos37223/4(49n-22)1 —
8Mz(3WCHaiz/4Izgall/4)[2klllaiz/213/4 + kllza§z(221‘
q2)1123/4(121-gz)1 — (afoul/Zklll/slu2211’2>[2a$“%lll/
213/4 + £32k112(321‘22)/223/4(421'22)1 ' [4KCfil/2kllzx
(221'22)/3Lx2221/2(421'22)1[ékallz(521—22)/2l3/2(421‘
2'2) + aguklzz(522*2l)/223/4(422421>1

¢2x22,2) = (3nnn/la2221/4)[2k222(5;ysin26’— gdxcos%{)/q23/4 +
(—)gdx(l—E¥z)k122(222-21)singeos57213/4(422—21)1 —
gdz(3nCha§z/41232Zl/4)[2k222a32/223/4 + klzzafz x _._

(06(22 ,2) , cont

(06412,1) =

(«22,1) =

(0083.1) =

107

'd.

(2%2-11)/213/4<422~21>1 - (sien1/21222/3112221/2>[22§<x
1222/923/4 + a§xk122<3%2—Ql>/213/4<422—A1)1 — [4Icfil/2X
k122(222‘21)/3Id2211/2(422'21)1[£21k122(522'gl)/ 23/4X
(4A2411) + ékallz(52l422)/213/4<421-22)1
(6nufi/Lx2)[2k112(gxycosarLéfixsin%f)/(421-%2) + (—)gxxx
(1—SXZ)k1223in5co§37(422—21)] - SXZ(3nehafZ/2IZ3)[2a52x
kllz/(421—kz) + agzklzz/(422—Rl)] - [lenenl/2kllzfill/2/
32x'z(421‘22)][22>f1klll/213/4 + 8220(1‘112(321'7‘2V2‘23/4 X
(421—22)] — [sien1/2k1229h1/2/3lx2(422411)1[aka112(s%l
'22)/213/2(421‘22) + a§>:(2D(k122(522'2‘1)”123/(“42221)1
(snen/1X2211/4)[2k122(éxyeoefiy-Sstingg)(2224h1)/(4A2—
)l)%l3/4 + (—)32X(l-g;z)kzzzsinqfiosz¢ZZ3/4] — $Xz(3nex
'fiaiz/alz3all/4)[2k122aiz(2)2121)/fl13/4(422-Rl) + k222X
l/2k222/3I03221/2)[aixk112(521'92)/
213/4<421412> + 52‘k122<522-%l>/223/4<422-x1>1 — [encx
fil/zklzzaqz'gl)/3Io<2211/2(422'2‘l)1[2“=‘°f’<klll/Al3/4 +
22¢k112(321'>2)(223/4(421‘22)1
(3mth/Ia2211/4)[2kl33(gxycosszistinéf)(223—21)/(4R3—
Alyxl3/4 + (—) x(l—éxz)k233sin36os3«ihé—kz)/(498412)x
1/4

a32/223/4] - (anon

223/4] —$XZ(3wena§z/lz3ql )[2k133(2fl3—%l)a§z/Al3/4 x

(423—21) + k233<223—Rz>a32/%23/4(4R3—%2>1 - [snenl/Z x
k133cil3-kl>/3l«2911/2<4234x1>1[zafi‘klll/Al3/4 + 53* x
kllz(391—22)/223/4(421—%2)1 - [#mcfil/2k233(223-22)/3 x
Iagfizl/2(4fl3412)][$T(k112(521-22)f213/4(421—22) + 5g“x

1122(522-21>/923/4<422-fil>1

 

(ddll,2) =

«£112,2) =

¢2X33,2) =

¢£<i3,3> =

108

(swan/lafiqzl/4)[2k112(Sxysinzyzgxxeoséy)(221—22)/(421—
qQ7123” + klllsingfiong-)g;X(l-Syz)/213/4] - SXZ[3TCX
fihgz/4lz3221/4][2k112(221'22)a§z/223/2(421‘22) + klllX
afz/213/41 l2k112<2fil—?n)/3112221/2<4%11A2>1 x
[252xk222/5X23/4 + aDiwklzz(322421)”213/2(422—21)1 ' [4X
nehl/zklll/slaFfihl/Zl[51“k112(sfil-Az>/al3/4<4zl-Az> +
$1122 (5912—21) /$\23/ 4 (4911412) 1
(anon/10(2)[2k122(§1(ysin20/-ExeosZp/(Mz—Al) + 9&9:
(1—$1z)k112211/4sin3E0937221/2(421—22)] - sz(3mcna§z/
2123>12k122a;Z/<422—21> + kllzafqul/4/221/4(475532)1
1/2k112211/2/3QX2(421-22)1[31u2112(5A1'22)/(421
—A2)>113/4 + a°§°‘k122(522—)1)/9\23/ 4221—12)] - [l61tc x
'fil/2k122/3Lx2(422'21)1[Zégxkzzz/flzy4 + didk122(322'
hl>/213/4<4AZ-Al>1

(3nch/Id2fi21/4)[2k233(gxysin2Q{— axXcong)(ZR3—22)/(423
—223323/4 + (—) o(X(1%“)k133singéos’g/(223—Cil)/213/4 x
(423—)l)] - gxz(3mena§Z/4lz3221/4)[2agzk233(223- 2)/
923/4(423112> + aizk133<223éhl>/fl13/4<413:Al>1 — [afcx
f11/2k133(223-2l)/3Io12311/2<423-21>1[atmllzfifil-Ptzfl
213/4<4%14)2> + éé‘klzz(592-%l>1223/4<4a2-fil>1 — [encx
111/2k233(27‘3‘7'2)”2x222(423-22)1[2‘22”<1<222mz3/4 + 5TXX
klzz(3224A1>/a13/4(422—21>1

-(1— dz)[lsznkj33/Lxlz(4k3ékj)] — Egz[8nrhkj33/3Izz x
(4913—9\j>][3f3/ol—9\3> + fag/(£2291 — [létcfil/zkj33x
231/2/31°f(4’>13—9\j)1 [if‘k133023-91)/7\13/"(47§-$\1) +
a321233csfi3-22>/223/4<4fl3- 2)] ; j = 1, 2

-[&mn1

- [87Cc‘h

 

109

(xyll,3) = (yxll,3) = (3ionklll/21X2Iy2lzl/2fll3/4q31/4)[Ix1/2eos3/
1y1/2singq — n2e2fil/2klllkl33(sfl3—21)/3ley211/2233/4x
(423-21) +'mcfiklllff3(fll¥23)(Ix-Iy)/21x2Iy2213/4931/4X
(Al-A3) — rfenkllz(2214xz)/21x21y2121/%az3/4a31/4(421-
22)][51x1/2sing/— Iy1/2cosg1 — n2o2fi1/2k112k233czhl-
22)(523—22)/31x1y%21/2233/4<4%14>2)(4)34A2) +-icH5§3 x
k112(221—22)(Rz+23)(Ix—1y)/21x21y2223/4(421422)(hz—z3)x
231/4

(xy12,3) = (yx12,3) = [3nehkllgh11/4/1x2ly2121/2231/4(421422)1 x
[1x1/2cos/(+-Iyl/2singq - 27€e2h1/2k112k135211/2(3)3-
/l>/3Ix1y(421—22)233/4(423'21) + 13hk112§13621+23)(1x'
Iy)2ll/4/Ix21y2231/4C2143)(421-22) — fmcfik122221/4/
1x21y2121/2231/4(422411)][Slxl/Zsingi lyl/Zcosgj — [2x
mgc2fi1/2k12é221/2/3IXIY(429—21)][k233(SQ34%2)fl233/4 x

(495-22)] +-menk122§§3)21/4(22+95)(Ix—1y)/lx21y2231/4x

 

(422—Rl)()2423)

(xy22,3) = (yx22,3) = [3fnk122(222421)/21x21y2121/2231/9213/4(422

_Ql)][1Xl/2eosz{+ lyl/Zsing] — -€e2nl/2k122k133(222_
21)(523—21)/3Ix1yz33/4211/2(423—21)(422—21) + ncfiklzzx
{[3(21+%3>(2%2—%1>(Ix-1y)/2Ix21y%213/4Q31/4(422->1><21
—Q3).-.(ICfikzzz/ZIXZIYZIZl/Z223/4231/4)[SIXl/Zsin’f- ‘
lyl/2oosqfl — 13o2fil/2k222k233(57g-12)/3IXI§321/2233/4x
(423-22; i-mehk2223§3caz+%3)(Ix—1y)/2lx21y2A23/4Q31/4x
(22-23)

(xy33,3) = (yx33,3).= [axehkl33(223-21)/21x21y2121/2x31/4213/4 x

(4903-911) ] [le/Zcos ’{+ Iyl/Zsinfi] — “gczfil/2k133kl33x. .

llO

(xy33,3), cont'd.
22 a) 51-2 /31 I q 1/22 3/4 42-2 42 a

( 3 1 ( ’3 l) x y l 3 < 3 1>< 3 l) +'“9X
fikl33§12<21+33)(223‘21)(Ix—1y)/21x21y2213/4231/4(423'
fli)(21—23) - [nenk233(223—22)/2Ix2Iy2Izl/2fiz3/@fi3l/4 x
(4Q3—12)][51xl/Zsin2(— lyl/zcoszfl — figc2fi1/2k2%3(223—
22)(572'22)/3Ix1§221/2233/2(423‘22)2 + “3fi2233523(22+
23)(2R3—22)(IX—1y)/2Ix21y2223/4231/4(423—22)(fig-2B)

(xyj3,l) = (yxj3,1) = [3nenkj33231/4/Ix21y2121/2(423—25Y211/4] x

 

[IXl/ZCOS ’0"? Iyl/Zsinfi] - 873C2fil/2k133kj33(513-21)/
733/4(423421)(423-2j) +~mcfikj33313QBl/4(fll+q3)(Ix-1y)/
IXZIy2211/4(423-75)(21-23) ; j = 1, 2

 

(xy13,2> = <yxj3,2> = {—ncfikj33231/4/Ix21y2121/2221/4(423-23->1 x
[Sle/zsinzf— Iyl/Zcosfiq - 8Tgczhl/2k233kj33(523-22)/
g33/4Ua-3'Q2) + TCCfikj 33 §3Q3l/4(22+R3) (IX-1y)/Ix2Iy2 X

221/4(423'2j)(22‘23)

(4) Auxiliary coefficients (U;d,§;-;a,bc) a Qiga,bc) :

In the following, i = 1, 2 ; j = l, 2

(z,zi,ii> = —'fi3/2a§z§‘i3§3(/\1+%>/Iz3211/42j1/2<fli—23> - [ZTEC'H x
{1'3 (Ai+%3)/39\il/4<Ai-%3> 1 [kjjjgfifiyfljmf/“(K $3)
+ kiJ-j3Z3<(zfij-fiiwflB/Hflj—Zi) — 2211/4/<9\3-21>> —
2231/2fg3kj33/2jl/2(423-2j)1

(z,zj,12) = (z,zj,21) = 4h3/2§33(25+23)(afz$:3 + a325E5)/Iz3(fij—
913)91j1/4211/4221/4 — [Aneng3(aj+x3>/32j1/4<2j-23>1 x
[CZ3+222‘921)211/4313k112/(421'22>(23421) + (23+22l'
922)221/2§23k122/(422'21)(23‘22) ‘ 231/2522k133/(428'
2‘1)221/2 ' 2‘31/2g3k233/2‘11/4(42-3-22)1

(2,2j.33)

(2,23,13)

111

{—2tcfi§3(2j+k3>Alf/403.43)1[q3k133<2212 + 29132 -
72fh3)/213/4(423-21)(23-21) + 5§3k233(2222 + 2232 —
79223) /223/4(423-22) 03—22)]

(2,23,31) = 113/2a? 13(9103)/Iz39«3l/2211/4(21—23) +
[4123532109 /3122211/4131/4(21_23) 1 [2235/4<29«1+23>x
3131(133/(423‘21)(2329213/4 ‘ 2‘11/22‘21/l+(231<112/(421'
2293”4 - 3:3k111/211/2231/4] +'fi3/Zaiz 22(22+23>/
223231/2(22'2‘3) + [4"Cfig23(2)295)/3122221/4231/2(22'
23)][221/4331/4523k133(22+22 '923)/(423421)(22'23) +
231/4(223‘2213:3k233/211/2221/2(423-22) 'l211/4§:2 X
k112(221422)/221/2231/4(421‘22) ' 223/4§:3k122/231/4X

(4% $911) 1

(2,23,23) = (2,23,32) =-fi3/2a§z§:§(21+a3)/IZ3231/2221/4(>1na3) +

[41e56fg(21+23)/3Iz2111/49\31/4(7‘1-2\3)] [9111/4R31/4I13X
k233(21+222‘923)/(423—22)(21:23) *‘231/2(223'21)(23 X
k133/211/2221/4M23‘Ql) - 2121/4323k122(22‘2-9‘1>/9\11/2X
231/4(422—21) - 213/4323k112/fi31/4(421-92)1 +-n3/2a§Zx
323(22+23)/123221/4231/2(22—13) + [4ncn?§3(22+23)/3 x
I227‘21/4231/4(22-23)1[2235/ 23k233(222+%3>/(423-22)X
(23'2‘29‘23/4 - 211/4221/2313k122/931/4(422221) - 323 X

1/4 1/4
k222/92 7L3 1

(5) Auxiliary coefficients (G;o(B;-;a,bc) E (qga,bc) :

(Xyi 3 j 3)

(xx3.j3)

(yxi,j3) = [113/2:33<2j+23>/2izmj1/‘*231/4aj—23>1[aiy/
Iyzail/4 ‘ afix/Ixzhil/al ; i = l, 2 ; j = 1, 2
— <yy3,j3> = [413/2523(7\j+23)affy/leylszjl/4R3l/2(aj

"513) ;j=l,2

 

APPENDIX XV

Coefficients (4;q$;XXéf;—;—)

The coefficients (4;dP;XSEF;—;—) appearing in the first sum of (4.6)
can be written in the general form

(4dfiqufR—;-) = (Zfi/322:[S;xfl;m;-](3:262f;-;m)

m

where [S;dfl;m;-] are the coefficients of the first contact transform—
ation S, and (3;Z§2F;—;m) are the coefficients in hé; they have been
listed in Table XIII and Appendix XI, respectively. Written in the
condensed form GXPJQT8P), the coefficients (4;qp;3$£F;-;—) are listed

in List A and List B below.

 

A. For d’=@ = x, y, or z,
(ddfgfiaf) = (215/3)([S;oé(;l;-](3;XFZ€;-;l) + [s;«x;2;—1<3;3A’2f;-;2>>
The nonzero ones are:
(xx,XXXX), (yy,yyyy), (zz,zzzz), (xx,yyyy), (xx,ZZZZ), (yy,ZZZZ),
(yy,XXXX), (zz,xxxx), (22.yyyy), (xx,xxyy)=(XX.yVXX),
(xx,xxzz)=(xx,ZZXX), (yy,YYXX)=(yy.Xny), (yy.yy22)=(yy,zzyy)
(zz,zzxx)=(zz,xxzz), (zz,zzyy)=(zz,yyzz), (xx,yyzz)=(xx,zzyy)
(yy,zzxx)=(yy,xxzz), (zz,xxyy)=(zz,yyxx), (xx,xzxz)=(xx,zxzx)
(yy,x2x2)=(yy,zxzx), (zz,xzxz)=(zz,zx2x), (xx,yzyz)=(xx,zyzy)
(yy,yzy2)=(yy,zyzy), (zz,YZYZ)=(zz,zyzy),
(xx,xyxy)=(xx,xyyX)=(xx,YXXy)=(xx,yXVX)
(yy.xyxy)=(yysnyX)=(yy,YXXY)=(yy,YXYX)
(zz,XYXy)=(zz,xyyX)=(zz,yxxy)=(zz,yxyX)

B.%rz¢d#fi# m
(«g/flap : <2fi/3)[S;wp;3;—J<3;Jf8f;—;3>

The nonzero ones are:

112

113

List B, cont'd.

(xy,xxxy) = (yx,xxxy) = (xy,xxyx) = (yX.xxyX) = (XYsXYXX) =
(yx,xyXX) (xy,YXXX) = (yx,YXXX)

(xy,YYXy) (yx,yyxy) = (xy,yyyx} (yx,nyX) = (xy,xyyy)
(yx,xyyy) (xy,yxyy) = (yx,yxyy)

(xy,zxzy) (yx,zxzy) = (xy,xzyz) = (yx,xzyz) = (xy,zyzx) =
(yx,ZYZX) (xy,YZXZ) = (yX’yZXZ)

(xy,zzxy) - (yx,zzxy) = (xy,ZZYX) (yx,ZZYX)

(xy,xy22) (yXaXyZZ) = (xy,ZZYX) - (yX.ZZYX)

(xy,zxxz) (yx,zxxz)

(xy,zyy2) (yx,2yy2)

 

 

APPENDIX XVI

Coefficients (4;d&3§}-;ab)

e coe 1c1ents ;d ;—;a appearing in t e secon sum 0 .
Th ff' ' (4 X b) ' ' h d f (4 6)

can be written in general form

(4;dfi’Jv§-;ab) = (O;o<%Y;-;ab) + (R;z,fia€;-;g.2) + (Ssgga.fi;-;ab)

where (O;def}-;ab), (R;g, ;-;ab) and (S;gfi)zék-;ab) are auxiliary

coefficients to be listed in Lists (2) to (4) of this appendix. In

the condensed form G¥¢31ab), the coefficients (4;dfi2fiE—;ab) are listed

in List (l).0nly those with a = b actuall contribute to h .

(l) Coefficients (4;dsqéE-;ab) a (ngéab) :

(2

V

 

 

 

In the following, o(=x, y, z ;@=x, y, z ;dflp) ;i&j= 1. 2

(zddzij) = (zaezji) = [(l—Sdz)/2(l+%ij)][(Rz,2eo(i,j)+(Rz,2o(o(j,i)]

(o(o(o(a(ij) = (o(o(o(o<ji) = (Odddotij) + (S°(o(,o(o(ij) +S“z[(Rz,zzzi,j) +
(Rz,zzzj,i)]/(1+$ij)

(«imam = (Memos) + $¢Z(Rz,zzzz3,3)

(ddfspij) = (ddwji) = (pflouij) = (Wouji) = (OWf’ifiij) + (1/2) X
Motown) + (swxinl + [30(2/2(1+%ij)][(RZ,(9PZi,j)
+ (R2,Prgzj,i)]

(«464333) = (W403) = (owép33) + (&(z/2)(Rz,z(}»§3,3)

Auxiliary coefficients (O;dbg§3—;ab) E (dfigéab) '

 

In general,
(0;o(f32{{;-;ab) = (Hi/3% [S;-;m;ab](3;°<fi?)’f;-;m)
where [S;-;m;ab] are the coefficients of the first contact trans-

formation S, listed in Table XVI, and (3;d@36:—;m) are the

114

 

115

coefficients in hé’ listed in Appendix XI.
A. For a = b = l, 2 or 3, and for 3 # a $ b # 3,
(OHME-sab) = (zn/s)[[s;—;1;ab1<3;o<{>2(5;—;1> +

[S;-;2;ab](3;dP35}-;2)]

 

and the nonzero ones are:
oxaeuaa), onxxxlz) = (Anxle),(xxgpaa), enxpplz) = («29221),
(nyyaa) = (xyyxaa) = (yxxyaa) = (yxyxaa)
(xyxylZ) = (xyxyZl) = (xyyle) = (xyyle) =
(yxxylZ) = (yxxyZl) = (yxyle) = (yxyle)
B. For (ab) = (13), (31), (23) Or (32),

(0;dp){8-;ab) = <2h/3)[s;—;3;ab1<3;d&7(f;—;3>

 

There are a total of forty eight nonzero coefficients of this
type, seven of which are independent. They do not contribute

to the final Hamiltonian, and therefore are not listed here.

(3) Auxiliary coefficients (R;q,§3§3-;a,b) E (d,§3éa,b)
The form in which these auxiliary coefficients relate to the co—
efficients (4;°(fi3’X;-;ab) is (R31’M32’R): with the underlining
having the significance as explained in Appendix XIII.
In general,

(R;ot,ea’f;-;a,b) = <—2h/3>%[S;«;-;am]<3;ezf;m;b>

where [SflX;-;am] are the coefficients of the first contact trans—
formation S, as listed in Table XIV, and (3;p3$2m;b) are the co—
efficients in hé, already given in Appendix XII.
The nonzero (R;X,@35:—;a,b) that contribute to the final Hamilt—
onian are:

(z,zfiXiJ) = (z,o<o<zi,j) = (-2’fi/3)[S;Z;-;i3](3;z«o<;3;j)

(4

V

116

(zadzdid) =-(2'fi/3)[S;z;-;i3](3;°<Z°<;3;j)
(z,zo®0,3) =-(2h/3)[[S;2;-;l3](3;zwx;l;3) + [S;z;-;23](3;zXX;2;3)]

(z,dzd3,3) =—(2fi/3)[[S;z;—;13](3;dzx;l;3) + [S;dzd;23](3;dzx;2;3)]

In the expressions for the Gx,fi%éa,b)'s above, 0<= x, y, z

i = 1, 2 and j = l, 2

Auxiliary coefficients (S;d&,fiJE—;ab) E fixééxxab) :

In general,

 

(S;d@;X5}-;ab) = (Zfi/3)%%.[S;«@;m;-](3;333-;abm)(1+gam+gbm)

where [S;dfi;m;-] are the coefficients of the first contact trans-
formation S, as listed in Table XIII, and (3;fi§}—;abm) are the
coefficients in hé , as given in Appendix XIV.

In the following, o(= x, y, z ;9= x, y, z ;o( =€) and 475%

(«M,@fill) = (2h/3)[3[S;«d;l;-](3;fi@;-;lll)+[S;“W;2;-](3;fifi;-;112)]
GXK,9&12) = (4n/3)[[S;MM;1;-](3:9fi;-;121)+[S;d¢;2;—1(3;@g;-;122)]
(ww,@@21) = (aa,§filz)

(02,9122) = [[s;o<o<;1;—J<3;(343;-;221>+[s;oa;2;—1<3;fifi;-;222)1(zfi/s)

In addition to the above, there are:
(xy,xyjj) = (xy,Yij) = (yX.xyjj) = (yx,yxjj) =
(Zfi/3)[S;xy;3;-](3;xy;-;jj3)

(xy,xy12) = (xy,xy21) = (xy,yx12) = (xy,yx21) = (yX.Xy12) =

(yx,xy21) = (yx,yx12) = (yx,yx2l)
(2n/3)[S;xy;3;-](3;xy;-;123)

where j = l, 2, 3

 

APPENDIX XVII

Coefficients (4;dfix$;ab;-)

The coefficients (4;d913;ab;-) appearing in the third sum of (4.6)
have the general form

(4pr5;ab;-) = (O;d9’(5;ab;-) + (R;<_x_, 5;§.§;-) +

(S;1&,fi;ab;-) + (Uzdl’z’a’5;2.h;-)

where (O;dp($;ab;-), (R;d,p31;a,b;-), etc., are auxiliary coefficients
whose detailed expressions will be given later in this appendix. In
condensed notations, the nonvanishing (4;dfi%§;ab;-)'s that contribute
to the final Hamiltonian are listed in List (1), while Lists (2) to

(5) give the auxiliary coefficients appearing therein.

(1) Coefficients (4;¢gx$;ab;-) s (abd2fi5) :

In the following, °<= x, y, z ; fi= x, y ; i = j = 1, 2

(ijdddd) = (jiaexw) = (Oddddij) + (Sd«,wxij) + [bgz/(lfbij)] x
[(R2,zzzi,j) + (Rz,zzzj,i)]

(3344“) = (Oddd063) + (500‘ + bow) (Uama3,3) +%o(z(Rz’ZZZZ3'3)

(ijxxyy) = (ijyyXX) = (jixxyy) = (tiYXX) = (Oxxyyij> + (1/2) X
[(Sxx,yyij) + (Syy.xxij)] + [(Uxxyyi.j) + (Uxxyyj.i)]/
(1+bij)

(33xxyy) = (33yyxx) = (Oxxyy33) + (Uxxyy3,3)

(ijppzz) = (jifipzz) = (oppzzij) + [1/2(l+%ij)][(Rz,zppi,j) +
(R2,zpflj,i)] + (l/2)[(Sfip,zzij) + (Szz,pfiij)] +
[(Upfizzid) + (Ufipzzj,i)]/(l+‘oij)

(33ppzz) = (oppzz33) + (l/2)(Rz,zpp3,3) + (uppzz3,3)

(3322??) = (Ozzpp33) + (l/2)(Rz,zpp3,3) + (Uzzpfi3,3)

117

(2)

(ijzzpfi)

(ijxyxy)

(33XyXY)

(ijzpzfi)

(33ZFZP)
(ijzfifiz)

(334q32)
(ijpzzp)
(33PZZP)

118

(jizng) = (Ozngij) + [1/2(1+$ij)][(Rz,zHli,j) +
(Rz,z(a(aj,i)] + (1/2)[(5zz,’egij) + (392,22in +
[(Uzzggi,j) + (Uzzpfij,i)]/(l+gij)

(inYYX) = (ijyxxy) = (ijyXVX) = (jixyxy) = (jiXYYX) =
(jiyxxy) = (tiXYX) = (Oxyxyij) + (Sxy.xyij) +
[(nyxyi,i> + woman/(1613)

(33xyyX) = (33yxxy) = (33YXYX) = (0xyxy33) + (Sxyixy33)
+ (nyxy3,3)

(jizng) = (ozfezteij) + [1/2(l+SiJ-)][(Rz,gzpi,j) +
(Rz,{izgj,i)] + [(Uzpzpid) + (Uaezpj,i)]/(1+%ij)
(029223) + (l/2)(Rz, 2 3,3) + (U2 2 3,3)

(jizppz) = (1/2(1+%ij)][(Rz,zppi,J-) + (Rz,zggj,i)] +
[(Uzfipzij) + (Uzfipzj,i)]/(l+$ij)

(l/2)(Rz,2)853,3) + (1122,2233)

(jifizzp) = [(Ufizzfii,j) + (upzzfij,i)]/(l¢$ij)
(Upzzp3,3)

Auxiliary coefficients (O;Q’ ézab'—) E (d Jab) :

 

In general,

(O;xflqfizab;-) = (2fi/3)2% [S;-;abm;-](3:4fi353-;m)(1+sam+$bm)

where [S;—;abm;-] are the coefficients of the first contact trans—

formation S, listed in Table XV, and (3;N??$;—;m) the coefficients

in hg, listed in Appendix XI.

Listed herein are only those (O;«pa§}ab;-)'s that contribute to

the final Hamiltonian. They can be obtained from the general form—

ula above, with m running from I to 2 only.

In the following list, o(&@ = x,y,z 0(7‘fié; ’X= x,y ; 1&j = 1,2

v

119

(«exam = (dodei), (0479(063), (zfzfiij) = (z'dz’b’ji), (rm/33),
(«pa 11) = (W921i) = ((Wij) = (@Mji), (2W3) = (#463).
(xyxyij) = (nyxij) = (yxxyij) = (yxyxij) = (xyxyji) = (nyxji) =

(yxxyji) = (YXYin), (xyxy33) = (xyyx33) = (yxxy33) = (yxyx33)

Auxiliary coefficients (R;«J?35;a,b;—) E (M,fifi{a,b)
In general,
(R;«,W5;a.b;-) = (Z’ri/3)% [S;d;am;-](3;p?ff;b;m)
where [S;«;am;—] are the coefficients of the first contact trans—
formation S, listed in Table XV, and (3;dpg§fb;m) the coefficients
in hfi, listed in Appendix XII. Of the possible combinations of the
vibrational operators' indices, a and b, only the following ones
yield contributing auxiliary coefficients (d,fi3§a,b):
A. b E j = l, 2 ; a E i = l, 2 , for which the general formula
above reduces to
(Ream/8mm = <2fi/3>[s;o<;i3;-1(2915mm
B. a = b = 3, for which the above formula reduces to
(R;o(,pyg;3.3;-) = (2h/3)[[S;°(;l3;-](3;Mf;3;l)+ [S;«;23;-] X
(393$;3;2)]
For these two cases, Table XIV and Appendix XII yield:
(2,zqfii,j) = (z,ddzi,j), (z,zfix3,3) = (z,wxz3,3),
(z,dz«i,j), (z;X2X3,3),
where o(= x, y, 2 ; i & j = l, 2
Auxiliary coefficients (S;%§iz;;ab;-) a Qifiéfgab)
In general,
(s;o(g,'()’§;ab;-) = (Zfi/3)% [S;dfism;-](3;Xf;ab;m)

where [S;¢#;m;—] are the coefficients of the first contact

 

 

(5

V

120

transformation, listed in Table XIII, and (3;%$;ab;m) the coeffi-

cients in hé, listed in Appendix XIII. Of the nonvanishing

(S;u?,%$;ab;-)'s, only the following ones contribute to the final
Hamiltonian:
A. (S;au,f@;ij;-) = (S;dd.&@;ji;-) = (2h/3)[[S;xa;l;-](3;Ffi;ij;l)
+ [S;dd;2;—](3;pfi;ij;2)]
(S;o&.gg;33;—) = (2h/3)[[S;dd;1;-](3;PP;3;1) + [S;wd;2;-] X
(3;@p;33;2)] ;ol&fl = x,y,z ; i&j = 1,2
B. (33XY:XYSij;')=(S;Xy,yx§ij;')=(S;yX:Xy;ij;‘)=(S;yX:YX;ij;') =
(S;xy.xy;ji;-)=(S;xy.yX;ji;-)=(S;yx,xy;ji;-) =
(S;yx,yxzji;-)= (2fi/3)[S;xy;3;—](3;xy;ij;3)
(S;xy,xy;33;-)=(S;xy,yxs33;-)=(S;YX.xy;33;-)=(S;YX;yX;33;-) =

(2fi/3)[S;xy;3;-](3;xy;33;3) ; 1&3 = 1.2

Auxiliary coefficients (U;¢&3§;a,b;—) E GrfixSa,b)

In terms of the coefficients [S;d@;a;—] in Table XIII and the co—
efficients (2';dfifi;b;-) to be given at the end of this appendix,
the nonzero contributing (U;dP%f;a,b;—)'s are listed below. In
this list symbols and separating semicolons are omitted from the
corresponding coefficients, 1. e., [S;d@;a;—] and (2';dfi]fib;—) are

written in the condensed notations (Apa) and Gxfijb), respectively.

(xxxx3,3) = 2h(xy3)[2(zxx3) + (xzx3)]

(yyyy3,3) = -2fi(xy3)[2(zyy3) + (yzy3)]

(xxyy3.3) = (yyxx3.3) = -fi(xy3)[(xxz3) - (yyz3)]
(xxyyi,j) = (yyxxi,j) = fi(yyi)[(zxyj) + (xzyj)] -

fi(xxi)[(xyzj) + (xzyj)]

 

121

(xxzz3,3) = (zzxx3,3) = h(xy3)[(xzx3) + (zxx3) — (zzz3)]

(xxzzi,j) = (zzxxi,j) = h(xxi)[(xzyj) + (xyzj)] -
fi(zzi)[(yXZj) + (xyzj)]

(yyzz3,3) = (zzyy3,3) = h(xy3)[(yyz3) + (yzy3) - (zzz3)]

(yyzzi,j) = (zzyyi,j) = fi(zzi)[(xij) + (yXZj)] -
h(yyi)[(yXZj) + (Ysz)]

(xyxy3.3) = (yxyx3,3) = 'h(xy3)[(yzy3) - (xzx3)]

(xyxyi,j) = (yxyxi,j) = 2fi[(yyi)(xy21) - (xxi)(zxyj)]

(zxzx3,3) = (xzxz3,3) = h(xy3)[(zzz3) - 2(2xx3)]

(zxzxi,j) = (xzxzi,j) = 2h[(xxi)(zxyj) - (zzi)(xzyj)]

(zyzy3,3) = (yzy23,3) = h(xy3)[2(zyy3) - (zzz3)]

(zyzyi,j) = (yzyzi,j) = 2h[(zzi)(xzyj) - (yyi)(xy2j)]

mex3)=-flflmfi)@md>

(yxxyi,j) = 2h(yyi)[(xzyj) + (zxyj)]

(xyyx3,3) = 2fi(xy3)(zyy3)

(nyxi,j) = 4h(xxi)[(xzyj) + (XYZj)]

(xzzx3,3) = 2(xxzz3,3)

(xzzxi.j) = 2fi(xxi)[(XYZj) 4-(zij)]

(yzzy3.3) = 2fi(xy3)[(zyy3) + (yzy3) - (zzz3)]

(yzzyi,j) = 2(yyzz3.3)

(zxxzi,j) = -2h(zzi)[(xy2j) + (zxyj)]

(z-yzi,j) = 2fi(zzi)[(xy2j) + (zxyj)]

 

 

In the above list i & j = l, 2 ; not every one of the coefficients
is independent of the others.
The coefficients (2' ;dfi'ggbr) appearing in the preceding list are

closely related to the coefficients (2;XPJ§b;-) in Appendix IV. They

are as follows:

(2';zzz;3;-)
(2';ZXX;3;-)
(2';zyy;3;-)
(2';XZX;3;-)
(2';yzy;3;-)

(2';xzy;j;-)

(2';xy2;j;-)

(2';yx2;j;-)

122

(2/3)(2;zzz;3;-)
(2';XXZ;3;-) = (2/3)(2;zXX;3;->

(2';yy2;3;-) = (2/3)(2;zyy;3;-)

(1/2)(2;XZX;3;-)

 

(1/2)(2;yzy;3;-)

(2';yzx;j;-) = (1/2)(2;xzy;j;—); j = 1, 2

(2';zyx;j;—) = (2/3)(2;xyz;j;—) — Ia§Z(Ix—Iy) —
agxlz]/24InyIz2Aj3/4h3/2 ; j = 1, 2

(2';zxy;j;—) = <2/3)<2;yxz;j;-> - 1a§z<Ix-Iy) +

 

agylz]/24IXIyIz22j3/453/2 ; j = 1, 2

APPENDIX XVII I

Coefficients (41¢gy,§gf!—!—) in hi

The coefficients (4!o<fi’6,c§£f!—!—) in the second term of the first
equation in (5.7) can be written in general as
(41453Q52P!-!-) = (-n/2)2%-[oudp(;-;m](2;sip;m;-)

where [o’;o(F7’(;-;m] are the coefficients of the second contact trans—
formationo", listed in Table XVIII, and (2;:gf;m;—) the coefficients
in hi, listed in Appendix IV. The nonzero (4!o(€’J,ng!-!-)'s all con—
tribute to the final Hamiltonian; they are listed below in the con—
densed notation (dfi'fi‘géf) adopted in Chapter 5. Use has been made of

Table XVIII to express [6';o(@'(;-;m] in terms of (2;o<(31;m;—).

A. In the following, o(&(3 = x, y ; CHEF}
(222*222) = (45/293112)(2;zzz;3;—)2
(222*o(o(2) = (222*2o(o() = (><O(z*zzz) = (zdol*zzz) =

l/2)(2;ZZZ;3;-)(2;2o<o<;3;-)

(eh/223

(zzzwzw) = mama) = (.../223W)(z,zzz,3,_)(2,,(z,<,3;_)

¢4<z*d«2) = QwX2*2dd) = (zmx*WXz) = (zM«*zdd) =
(sh/223l/2)(2;z«d;3;-)2

(do<2*@@z) = new”) = (newt) = (zoeizgfi) = (Mam) =
(@fiz*zdoo = (223*222) = (25342030 =
(ffi/2431/2)(2;zdx;3;-)(2;zgfi;3;-)

(«9(2*0(zo<) = (ZOQX’XchoQ = (o(z0(*0<o<z) = (0(20(*de0 =
(.../223W)(z,zwx;3,_)(2,m;3,-)

(oex mag) = (Mdi‘fyzfi) = (pagicoaz) = (gzeizaoo =

l/2

(ffi/223 )(2;z«X;3;-)(2;§Zfi;3;->

123

 

QXZKNXZOO
(dzx*pzp)

. (xy2*XYZ)

(xzy*x2y)

(zxy*zxy)

(XYZ*Xzy)

(xy2*zxy)

(xzy*zxy)

124

(—h/2X31/2> <2;<xzo<;3;-)2

(Pzgndz,o = (sh/223l/2)(2;xax;3;-)(zzfizp;3;—>

(xy2*zyX) = (ZYX*XYZ) = (zyX*zyX) =(sh/2) X
[(2;xy2;l;-)2/9\11/2 + (2;xy2;2;-)2fl321/2]
(xzy*YZX) = (yZX*Xzy) = (YZX*YZX) =(=fi/2) X
[(2;xzy;l;-)‘2/9lll/2 + (2;xzy;2;-)2/221/2]
(ZXY*YXZ) = (yx2*zxy) = (YXZ*YXZ) = (=fi/2)X
[(2;zxy;l;-)2/9\ll/2 + (2;zxy;2;-)2/221/2]
(XYZ*YZX) = (zyX*xzy) = (ZYX*YZX) = (xzy*xy2)
(xzy*ZYX> = (y2X*XyZ) = (yZX*zyX) = (-fi/2) X

1/2 + (2;xyzz2;-) X

[(2;xyz;l;-)(2;xzy;l;—)/Xl
(2;xzy;2;-)/221/2]
(xy2*yx2) = (ZYX*ZXY) = (zyX*YXZ) = (zxy*xy2)
(zxy*zyX) = (yx2*xy2) = (yx2*ZYX) = (45/2) X
[(2;xy2;l;-)(2;zxy;l;-)/211/2 + (2;xy2;2;-) X
(2;zxy;2;-)/AZl/2]

(xzy*VXZ) = (y2X*zxy) = (YZX*YXZ) = (zxy*xzy)
(zxy*yzx) = (yxz*xzy) = (yxz*yzx) = (—n/2) x
[(2;xzy;l;-)(2;zxy;l;-)/'>\ll/2 + (2;xzy;2;-) X

(2;zxy;2;-)/XZl/2]

 

 

 

APPENDIX XIX

Coefficients (4!d ,{!-!ab) in h @
4

The coefficients (4!dpz,g!—!ab) appearing in the second term of the
second equation in (5.7) are of the following general form:
(4!d93§X!-!ab) = (sh/2);; [S;«fiafi—;m](2;$}m;ab)
where [5}2p7;-;m] are the coefficients of the second contact trans—
formation, and (2;§;m;ab) the coefficients in hz: they are listed in
Table XVIII and Appendix VIII, respectively. It was mentioned in
Appendix XVI, that , of the coefficients listed therein, only those
with equal indices of vibrational operator, a = b, actually contribute
to the final Hamiltonian. This holds true also of the coefficients
listed in Appendix XVII, which is why the condition i = j was imposed
in the beginning of the listing. As a matter of fact, this condition
of vibrational diagonality covers all quartic coefficients in the
fourth-order, twice transformed Hamiltonian h4@. Therefore, the coeff-
icients with a ¥ b will not be included in the list below. Again, the

condensed notation adopted in Chapter 5 will be used here.

(222*2jj) = (~‘r'1/2)[0";222;-;3](2;z;3;jj)
(20(0(*zjj) = (o(a(z*zjj) = (47/2)[°‘;zo<o<;-;3](2;2;3;jj)

(«29(*zjj) = (-fi/2)[5;o<zo<;-;3](2;Z;3;jj)

In all the coefficients listed above, j = l, 2 and O(= x, y

125

APPENDIX XX

Coefficients (4!dp;{5!—!ab) in h4@

The coefficients (4!dp,6§!-!ab) appearing in the third term of the

second equation in (5.7) has the following general form:
(4!°(P,a’{!-!ab) = (Ti/2):m [0“;dp;m;a](2;’){f;mb;—) (l +$bm)

where BS;«@;a;m] are the coefficients of the second contact trans-

formation, and (2;fi$}mb;-) the coefficients in hi; they are given in

Table XIX and Appendix V, respectively. Of the nonvanishing coeff—

icients (4!dfi,fig!-!ab), only those that really contribute to the

final Hamiltonian are given in the following list:

¢***&§ll) = (fi/Z)[2[O§d«;l;l](2;fip;-;ll)
+ [53MX;2;11(2;fl$;-;12)]
(“2*P522) = Ch/Z)[[Uidd;l;2](2;pfi;-;12) +
+ 2[03«d;2;2](2;99;-;22)]
(«#5232 = n1o;o<o<;3;3](2;pp;—;33)
(Xy*xyjj) = (xy*Yij) = (yX*xyjj> = (yX*ijj)
= Cfi/Z)[6;xy;3;j](2;xy;-;3j)
(xy*xy33) = (xy*yx33) = (yX*xy33) = (yX*yx33)
= (fi/Z)[[63xy;l;3](2;xy;-;l3) +

[62xy;2;3](2;xy;-;23)]

126

APPENDIX XXI

Coefficients (4!d,Pgéd—!ab) in h4@

The coefficients (4!d,@3§1-!ab) in the fourth term of the second eq-
uation in (5.7) can be expressed in general as:

(42«,p?{::—zab) = (41/2)? [o';o<;-;mab](2;pgf;n;—)(l + Sam +$bm)
where [6§q;—;mab] are the coefficients of the second contact trans—
formation, and (2;fi3§;m;—) the coefficients in bi; they are listed in
Table XXI and Appendix IV, respectively. The symbol X; denotes the
summation over m and over all permutations of (mab), subject to the
restriction: left index <_middle index <;right index.

Listed below are the coefficients (4!d,335!-!ab), written in the con-
densed notation GX*Qagab) introduced in Chapter 5, that actually con—

tribute to the final Hamiltonian.

<z*zzzii> = (eh/2)[o:z;—;jj31<2;zzz;3;—><1+28j3>

(2*Mkzjj) = (2*zdxjj)

(-‘h/2) [o’;z;—;jj3] (2;o(o<z;3;-) (1+2Sj3)

(znxaxii) = (=fi/2)[532;-;jj3](2;dzd;3;-)(1+25j3)

In the coefficients listed above, o(= x, y and j = l, 2

127

 

 

APPENDIX XXIII

Coefficients (4!qfig{5!ab!—) in h4@

The coefficients (4!dfig§5!ab!—) appearing in the third term of the

third equation in (5.7) are of the following general form:
«mayhem» = (41/2):m [6“;dp;a;m](2;gf;nb;—)(l+$bm)

where [5}d@;a;m] are the coefficients of the second contact transform-
ation, and (2;35;mb;—) the coefficients in hé; they are given in
Table XIX and Appendix V, respectively. Of the nonzero (4!dfi§{51ab!—)
coefficients, only those that actually contribute to the final Ham—
iltonian are included in the following list. The condensed notation

(ahxa*3é) of Chapter 5 is used here.

(11QH*2@) = (sh/2)[2[0§ofl;l;l](2;fifi;ll;-) + BT;MX;1;2](2;fi@;12;-)]
(223X*fifi) = (afi/2)[ET;WX;2;11(2;fifi;12;-) + Zlckdd;2;2](2;fifi;22;-)]
(33NX*@&) = =fi[632“;3;3](2;@fl;33;-)

(jjxy*xy) = (jjxy*YX) = (ijX*xy) = (ijX*YX)

= (-fi/2)[[Oixy;3;1](2;xy;l3;-) + [5;xy;3;2](2;xy;23;-)]

In the coefficients listed above, <X&(3 = x, y, z , j = 1, 2

129

APPENDIX XXIV

Coefficients (4!d, Slab——) in h @
{5 4

The coefficients (4!O(,fi’d’5!ab!—) in the fourth term of the third eq—

uation in (5.7) can be written in general as follows:
(4!d,935!ab!—) = (=fi/2)2% [C§x;ab;m](2;pfiflgm;—)

where [53dgab;m] are the coefficients of the second contact trans—
formation, and (2;fl(£}m;—) the coefficients in hé; they are listed in
Table XX and Appendix IV, respectively. Here are the nonzero, contrib—
uting (4!d,fi3g!ab!—)'s, written in the condensed notation (abd*fi3£)

adopted in Chapter 5:

(132*ZZZ) = (-fi/2)[ ;Z;jj;3](2;zzz;3;-)
(jj2*°<zo<) = (-‘fi/2)[ ;z;jj;3](2;o(z0<;3;-)

(jj2*XXZ) = (ij*zXX) = (sh/2)[UKZ;jj;3](2;ddz;3;-)

In the coefficients listed above, <X= x, y and j = 1, 2, 3

BIBLIOGRAPHY

 

l.

2.

10.

11.

12.

13.

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