FOURTH -ORDER CENTRIFUGAL
DISTORTION COEFFICIENTS FOR .
NONLINEAR TRMTOMIC MOLECULES .

Thesis for the Degree of Ph. D”
MICHIGAN STATE UNIVERSITY I
‘ DAVID ALLAN SUMBERG
1972

.4— ._

  
   
    

1..er ”"11

Michigan State
University

This is to certify that the

thesis entitled

FOURTH-ORDER CENTRIFUGAL DISTORTION COEFFICIENTS

FOR NONLINEAR TRIATOMIC MOLECULES
presented by
DAVID A. SUMBERG

has been accepted towards fulfillment
of the requirements for

Ph .D; degree in PHYS IQS

mmGfiW

Major professor

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Datew
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C'SDNS
QIJIIIMK BlNllERY INC.

IRAIY BINDERS

 

ABSTRACT

FOURTH-ORDER CENTRIFUGAL DISTORTION COEFFICIENTS
FOR NONLINEAR TRIATOMIC MOLECULES

BY

David A. Sumberg

In this dissertation the molecular vibration—rotation Hamilton-
ian of Darling and Dennison is expanded in the formulation of Nielsen,
Amat, and Goldsmith and specialized to the case of the most general
(XYZ-type) nonlinear triatomic molecule. Expressions are derived for
the ten fourth-order centrifugal distortion coefficients in terms of
the full set of cubic anharmonic potential constants and those
fundamental molecular constants which specify the equilibrium geometry
and the normal modes of vibration. These expressions are specialized
to the case of the XYx-type molecule and are found to be consistent

with previously published work.

FOURTH-ORDER CENTRIFUGAL DISTORTION COEFFICIENTS

FOR NONLINEAR TRIATOMIC MOLECULES

BY

David Allan Sumberg

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1972

6—73” $2.0

In mama/Ly 06
my motile/L

mm Wu Sumng

ii

ACKNOWLEDGMENTS

I would like to express my sincere appreciation to Dr. Paul M.
Parker for suggesting this problem, for his guidance, for his patience,
and, above all, for his warm friendship during the years I have spent
at Michigan State University. I would also like to thank my wife,
Lois, for her constant encouragement and her endless patience during
the period in which this work was undertaken, and for helping to

maintain the grammatical integrity of this thesis.

iii

TABLE OF CONTENTS

LIST OF TABLES AND FIGURES

CHAPTER

III.

IV.

VI.

VII.

VIII.

INTRODUCTION
DEVELOPMENT OF THE GENERAL VIBRATION-ROTATION HAMILTONIAN

II.1 Darling-Dennison Vibration-Rotation Hamiltonian

II.2 Watson's Simplification of the Vibration-Rotation
Hamiltonian

II.3 Development of the Hamiltonian

EQUILIBRIUM GEOMETRY, NORMAL COORDINATES, AND MOLECULAR
PARAMETERS

III.l Equilibrium Geometry and Normal Coordinates
III.2 Molecular Parameters

GENERALITIES OF THE CONTACT TRANSFORMATION TECHNIQUE
FOR FOURTH-ORDER CALCULATIONS

THE FIRST CONTACT TRANSFORMATION FOR THE XYZ-TYPE
MOLECULE

SECOND CONTACT TRANSFORMATION FOR THE XYZ-TYPE
MOLECULE

THE FOURTH-ORDER CENTRIFUGAL DISTORTION COEFFICIENTS
VII.l General Expressions for the XYZ-Type Molecule
VII.2 Specialization to the Case of XYX-Type Molecules

VII.3 Determinable Combinations of Constants

SUMMARY

LIST OF REFERENCES

iv

Page

vi

20

20

28

34

40

49
62
62
68
71
72

74

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

1.

THE FOURTH-ORDER CENTRIFUGAL DISTORTION CONSTANTS
¢i FOR XYZ MOLECULES

THE COEFFICIENTS (:fxss, APPEARING IN H2 AND IN
O R

1 TH OUGH @10
COEFFICIENTS OF THE S-FUNCTION APPEARING IN $1
THROUGH $10

THE COEFFICIENTS a APPEARING IN i[S, HO]R AND
I ¢

N 1 THROUGH @10

THE COEFFICIENTS Y APPEARING IN i[S, H0]v AND

IN $1 THROUGH @10

Page

76

84

85

86

88

LIST OF TABLES AND FIGURES

Table Page

8

l. The Coefficients t: Introduced in Eqs. (III-85)-(III-87) 30

2. Abbreviated Notation for the Distinct Fourth-Order

Centrifugal Distortion Constants O 54
aBYGEn
.. .aBa a8.
3. The Nonvanishing Coefficrents aS , Css" and (ASS,)
for XYX 69
Figure
l. Equilibrium Geometry of the Nonlinear XYZ-Type Molecule 21

vi

I . INTRODUCTION

In recent years considerable improvements in experimental
techniques and the high resolution obtainable in infrared spectroscopy
have necessitated taking into account terms in the vibration-rotation
Hamiltonian that are of higher order of approximation in the energy
than the second. In cases such as the nonlinear XYX-type molecule,
in order to account for experimentally observed results in a
satisfactory manner, it is necessary to include such terms as fourth-
order centrifugal distortion coefficients even for low values of the
total angular momentum quantum number J.1 The vibration-rotation
Hamiltonian of diatomic molecules has long been treated to fourth and
even higher orders, and Amat and his co-workers2 have considered this
problem extensively for the case of the linear triatomic molecule. It
is natural, then, to proceed to nonlinear triatomic molecules of the

HDO-type as well as of the H O-type as the next feasible case for

2
complete compilation of fourth-order vibration-rotation coefficients.
The general vibration-rotation Hamiltonian of asymmetric rota-
tor molecules has been developed by Chung and Parker3 in the Nielsen-
Amat-Goldsmith4 formulation of the Darling-Dennison Hamiltonian

through the consideration of symmetry restrictions imposed by the

asymmetric rotator point group. Recently, a form of the

Darling-Dennison Hamiltonian has been given by Watson5 that greatly
simplifies the expansion of this Hamiltonian. However, even with this
simplification, the general formulation of Amat-Nielsen-Goldsmith6-9
is unnecessarily complicated to apply in the case of the asymmetric
rotator, principally because of its inclusion of degenerate normal

modes which are absent in molecules of this type. Therefore, instead

of working with the general formulation, we start with the Darling-
Dennison vibration-rotation Hamiltonian in Watson's simplified form,

but specifically written for the XYZ-type molecule. This Hamiltonian

is then expanded and subjected to two successive contact transformations
of the Van Vleck type. Within the framework of the work by Chung and
Parker3 the Hamiltonian is further simplified through extensive rear-
rangements based on angular momentum commutation relations. The re-
sulting Hamiltonian is, for a given vibrational state, a power series

in the angular momentum components. By using an extended version of
Watson's theorys, the Hamiltonian can then be related to experimental
results in a manner which allows one to obtain meaningful fits to high—
resolution experimental data.

The principal aim of this thesis was the calculation of expres-
sions for the ten fourth-order centrifugal distortion coefficients of
the XYZ-type molecule. These were obtained in a form which exhibits
extensive cyclic and algebraic regularities. They were reduced to and
compared with the known expressions for the XYX-type moleculelo' 11

and found to be consistent with this previous work.

II. DEVELOPMENT OF THE GENERAL

VIBRATION-ROTATION HAMILTONIAN

II.l DarlingrDennison Vibration-Rotation Hamiltonian

For any theoretical calculation of vibration-rotation energy
levels of a molecule it is necessary to have a suitable quantum me-
chanical Hamiltonian as a starting point. The Hamiltonian of a mol-
ecule may be considered to be composed of an electronic part as well
as a vibration-rotation part. The Born-Oppenheimer approximation
allows separation of the electronic motion from the nuclear motion to
a high degree of accuracy, and therefore affords the possibility of
writing the total wave function as a product of an electronic and a
vibration-rotation wave function. Since we are not concerned with
the electronic level structure, only the vibration-rotation Hamilton-
ian is of interest.

The general vibration-rotation Hamiltonian of a molecule has
been studied extensively.12 Essentially, the derivation is based on
classical considerations of the vibrational and rotational kinetic
energies transcribed to the proper quantum mechanical operator form.

In the following discussion we begin with the Darling-

Dennison Hamiltonian for a polyatomic molecule:

1
-/2

1 1
H = kuéfa 8(Pa - paluasu (PB - p8)u4

p -y p
+ Hu‘is pgu “pgu4 + V - (II’l)

The symbols a,B take the values x, y, z and refer to a set of body-
fixed coordinates attached to the equilibrium configuration of the
molecule with the origin at the center of mass of the molecule. The
equilibrium and instantaneous positions of the i-th nucleus along the
axes x, y, and z in this coordinate system are denoted by aoi and oi ,
respectively. The displacement of the i-th nucleus from its position

of equilibrium is denoted by:

+' - I I (II 2)

In Eq. (II-I) Pa is the a-th component of the total angular
momentum referred to the body-fixed axes and can be expressed solely
in terms of the Euler angles and derivatives with respect to these
angles. Thus, the Euler angles are the rotational coordinates of the
problem. The vibrational coordinates to be used in this equation are
the normal coordinates Qs' which have been substituted in favor of the

oi. The transformation between the oi and the QS is given by:

a
Jfiiai = 2521393 ' (II-3)

. . a . .
where m1 is the mass of the l-th nucleus, 218 are coeffic1ents of the
transformation, and Q8 is the normal coordinate of the s-th normal

mode. The vibrational momentum p; conjugate to the normal coordinate

QS is defined as:

_ -- §_. _
p; - 1M3Qs. (II 4)

The internal angular momentum component pa is defined by:

pa = ZsAzp; a ISIS-anst-P; ’ (II-5)

where

a _ B Y _ B Y a _ a .
Cs's Zi<£is'£is £15213.) Css' I GIBIY cyclic
(II-6)
a - 8 Y _ B y
AS - ZS'Ziu'is'p'is giszis')Qs'
a O
- ZSICSISng I GIBIY CYCllC. (II 7)

From Eq. (II-6) it can be seen that the Coriolis coupling coefficients,

a I
C85,. are zero when s = s .

One now defines effective moments and products of inertia, Ida

and I& , respectively, as follows:

 

 

B
I = _ a 2 II-
aa Iaa 2isms) ’ ( 8)
a B
' = -I + A A II-
IaB as 2s s s ' a # B ( 9)
where
= 2 2 _
Ida Zimi(si + vi) . a a B i V (II 10)
Ice = ' imiaiBi ' a I B . (II-ll)
Ida and IaB are instantaneous moments and products of inertia.
The reciprocal effective inertia tensor and its determinant are defined:
r II _II _II I
xx xy xz
-1
u = _II I “I. II I (II-12)
I J W W W I I
'I I "I l I
‘ zx zy 22}
-1
(H) = (1') . (II-l3)

p = det (u) = ——1—-——- ' (II‘l4)
det[I')
“as = uII+YI&B + Iéyl§8) , a ¢ 8 # Y (II-15)
= I I _ l2 _
uaa M1881YY IBY) , a # B # Y- (II 16)

The tensor (1') is called the effective inertia tensor. If the compo-
nents of the effective rotational angular momentum, Pa - pa , and the

components of the angular velocity, ma , are written as column vectors,

then:

(P - p) = (mm .

Finally, V is the vibrational potential energy, and it is a

function only of the normal coordinates QS.

II.2 Watson's Simplification of the Vibration-Rotation Hamiltonian

 

By commuting out n from the first two terms of Eq. (11-1)

the Darling-Dennison Hamiltonian takes the form:

:11
ll

Iza,B(Pa - pa)uaB(PB - p8) + gZSPEZ + U +'v (II-l7)

where

C!
II

1/ -1/
1 4 2
- P -

1
‘4

P
P + 7 ‘ *
X ua8( ) 62811 P811

8 “ PB
1 1/4 -1/4
+ 42.514 [pg , u 1p; . (II-18)

Here, use has been made of the fact that Pa operates only on the Euler
angles and thus commutes with all quantities in H except PB and Py'

It has been customary in applications of Eq. (II-l7), for
instance in Goldsmith et al's treatment6-9 of higher-order contribu-
tions to the molecular vibration-rotation energy, to start by intro-

ducing the power series expansion of pa and u in terms of the normal

6
coordinates,and from these expansions to evaluate U to the desired

. . 5 . . .
degree of approximation. However, Watson has shown that it is simpler
in the long run to use the commutation relations and the properties of

the “a tensor to evaluate U directly, without expansion. This proce-

B
dure yields the simple result that:

l 2
= -— . II- 9
U 8 M 2a pad ( l )
With this, Eq. (II-l7) takes the form:

= 1 _ _ 1 _.l_ 2
H 62a,B(Pa pa)uoB(PB p8) + 625p; 8 M Zoned + V '

(II-20)

This constitutes Watson's simplification of the Darling-Dennison
Hamiltonian and will be the starting point for our expansion in

the normal coordinates.

II.3 Development of the Hamiltonian

 

In this section the Watson form of the molecular vibration-
rotation Hamiltonian will be expanded to fourth-order of approximation
in the energy and terms of various orders identified and written out

explicitly. As a first step in this derivation we write Eq. (II-20) as:

= 1 - 1 7
H éZaIBuaBPaPB éXoIB(pauaB + uoBPo)PB + éiaBpauaBpB
l
- g hzfauaa + 9223 p;2 + V - (II-21)

The terms of Eq. (II-21) represent, in succession, the pure rotational
energy, the Coriolis coupling energy, the first correction to the
Coriolis energy, the second correction to the Coriolis energy, the
vibrational kinetic energy, and the potential energy of vibration.

We begin the expansion by writing the effective moments and
products of inertia in terms of normal coordinates. Substituting the
appropriate form of Eq. (II-2) for oi , Bi , Yi into Eqs. (II-10) and

(II-ll), and using Eq. (II-3) to introduce the normal coordinates gives:

_ a a8 a8 =

IaB - Iaaéae + 2s as Qs + 23,5' Ass'Qst' a, B x,y,z

(II-22)
where
2 2

13a = Xi mimei + Y01" a # s a y (II-23)

13.8 = -Zi miaoiBoi , (II-24)

aa _ B Y

a8 "‘ ZZi/{n-i(Boi£iS + YOi'Q'iS) I a # B 5‘ Y (II-25)

GB _ _ B a _

as - Zi/fii‘aoizis + 8012.15) , a 7‘ B 7‘ Y (II 26)

10

“a = B B y y _
Ass' Zi(£is£is' + zisgis') ' a I B I Y (11 27)

GB _ _ a B _
Ass' - Xi £15 is' ' a f B . (II 28)

Substitution of Eqs. (II-22) and (II-7) into Eqs. (II-8) and (II-9)

gives:
I = o as as I
IaB IooaaB + 2s as Qs + Zs,s'(Ass') Qst' (II-29)
where
a8 , _ a8 _ a B _
(Asst) - ASS. ZS" CSSIICSISII ' (II 30)

In the above, a,B:Y = x,y,z have been taken as the axes of the coordi-
nate system in which the equilibrium inertia tensor is in diagonal form

(principal axis system), and I; = O for a f B.

B

We continue with the expansion of Eq. (II—21) by writing “a8

as a power series in the normal coordinates. This is permissible,

because the “a depend on the components of the effective inertia

B
tensor which, as we have just shown, are functions of the normal coor-

dinates. Furthermore, displacements in the normal coordinates are

assumed to be small. The “a then take the form:

8

— _ o o 08 a8
uaB - uBa - (l/IoaIBB)[Q(O) + is C(l)s Qs
a8 08
+ Zs,s' 9(2)ss'Qst' + Zs,s',s" 9(3)ss's"Qst'Qs"

a8

+
ss's"s

9(4) II IQSQSIQSIIQSHI + ...]

XS’SI'SH'SHI

(II-31)
where the various quantities Q are the expansion coefficients. With

the help of Eq. (II-31) the terms of Eq. (II-21) can be expanded as

ll

follows:

y _ 1 GB 0 o
22a,8 uaBPaP8 - 428,8[Q(0) /Ia HIBBlp P8

1 (18 o o o
+ 428’8Zs[9(1)s /I 88 1881QM P P8 + %X8 828 S.[Q(2>; B./I 88188]

X

Q8Q8.P8PB + IZG,BZS,S. 8.[9(3>:: Su/Igalg81Q8Q8.quP8PB

1 (181°
+ 620823 S. S" SnI[Q(4)sSISuSnI/I aa I881QS QS 'QS "Q5 "'P aPB
+ ..... (II-32)
_.g(pauas + Msp )P8 = -. m2 [9(0)“ B/I°8 I° 81p 8P8

kid 82$ [9(1):B/I°a 1° 1 [p8Qs + QspalP

88 B

I
SC

08 o o .
Za,828,s'[m2)ss'/IoaIBB] [ponQs' + Qst'palps

1/ (18 o o
--2XQBXSS.S.IQ<3> . "/18 ”I 81 [p Q HQ .98. + QSQS.QS.p81P8

SS S

+ ooooo (II-33)

= e28IQ(OI“B/I°8 I° lpap

1
62a,8 pa“asps 88 PB

+ 428,828[Q(1):B/1°H188Ip Q SP8

1 a8 0 O O... —
+ éXa,BZS,S'[9(2)SS'/IaaIBB]paQSQS'pB + (II 34)

_.l. 2 = _ l. 2 G“ o 2 _ l. 2 0a
8 M {a uaa 8 M 28 9(0) /‘Iaa’ 8 M 2828(e(1>s /

12

o 2 _ i. 2 “a o 2 .. _
(Ida) JQS 8 M Zazs,s'[9(2)ss'/(Iaa) JQst' + (II 35)

gig sz = gig p;2 , (II-36)

Finally, the potential energy can be expanded as the Taylor series

_ av , 1 32v a'a',
V ‘ V° + Zs[3aé] “s + les,s'[3aéaaé,] S S

 

1 33v
+— " ' one.
3125,5',s"[3a'8a',3a'"] asas'as" +
s s s o

a 1/ 2 ‘
22$ Ast + Zsss'ss"'Kss's"Qst'Qs"

+ EssslssllésllI KSSIS'ISIIIQSQSIQSHQS"| + .... (II-37)

The zero order (denoted hereafter by 0(0) ) term in Eq. (II-32) is the
rotational kinetic energy for the rigid rotator with the nuclear frame-
work of the molecule in the equilibrium position. Appealing to exper-
iment, the leading term of Eq. (II-33) is known to be of 0(1); also,

the leading terms of Eq. (II-34) and (II-35) are of 0(2). The first
term of Eq. (II-35) is constant and may be discarded, as it shifts all
energies by an equal amount. Equation (II-36) is the vibrational
energy which is of 0(0). The Taylor series expansion of V, Eq. (II-37),
is taken about the equilibrium positions of the nuclei. The total

force at the equilibrium position, [532} , must be zero, and the
S O

l3

constant V° has no physical significance and may be set equal to zero;
V is then rewritten as a function of the normal coordinates with AS
being the square of the s-th normal frequency, and the sets of coeffi-
cients K are the force constants in the various orders of the expansion.
Equation (II-21) appears impossible to evaluate in closed form.

We therefore arrange the Hamiltonian in orders of magnitude

= + + + + + ... I -
H H0 H1 H2 H3 H4 , ( I 38)

and apply the methods of perturbation theory. The leading term of a
particular expansion (II-32) - (II-37) is assigned to the order
indicated by the discussion following Eq. (II-37), and the subsequent
terms are assigned to successively higher orders of approximation. We

also introduce dimensionless operators ps and q5 which are given by

.0
II

p
(As/M2)"QS (II-39)

’0
ll

(”Z/X )%P* (II-40)
S S

and the following notation:

QB
(2(1)S
IaaIBB

E [as- s] (II-41)

a8
Q(2)35'

1° 1°
aa 88

a [a8; 55'] (II-42)

“(3)323"

1° 1°
aa 88

[a8; 33'5"] (II-43)

a8
83.328". E [aB;SS'S"S"'] . (II-44)
I I

aa 88

9(4)

 

14

As a matter of convenience in future calculations we introduce
symmetrization of the above coefficients through the following

definitions:

[aB; ss']' = ([dB; 53'] + [dB; s's])/(l + 65 ) (II-45)

SI
[08; SS'S"]' = ([a8; 58'8"] + [08; SS"S'] + [a8; 5'33"]
+ [a8; s's"s] + [a8; s"ss'] + [a8; s"s's])/

[(1 + 638, + 6 )-(l + 65 )] (II-46)

ss" '5"

[a8; SS'S"S"']'= ([aB; SS's"S"'] + [as; SS'S"'S"]
+ [a8; ss"s's"'] + [a8; ss"s"'s']
+ [a8; ss"'s's"] + [a8; ss"'s"s']
+ Ids; s's"ss"'] + [a8; s's"s"'s]
+ Ids; s's"'ss"] + [a8; s's"'s"s]
+ [a8; s"s"'ss'] + [a8; s"s"'s's]
+ Ids; s'ss"s"'] + [a8; s'ss"'s"]
+ Ids; s"ss's"'] + [d8; s"ss"‘s']
+ Ids; s"'ss's"] + [a8; s"'ss"s']
+ Ids; s"s'ss"'] + [a8; s"s's"ss]
+ [a8; s"'s'ss"] + [a8; s"'s's"s]

+ Ids; s"'s"ss'] + [a8; s"'s"SS'])/

15

)(1 + a + 6 ...)<1 + 63.3.”).

I SIS" S's

(1+6 +6 u slll
35 SS

(II-47)

According to the above scheme,the Hamiltonian, regrouped into terms of

the same order of magnitude, takes the form:

CB
II

b
Bfapg/I;a+ +% M23 A;(p§/M2 + q: ) (II-48>

_ a 3'
H1 - za, st (1)Xs qs P aPB + Eazs,s' (1)xs qsps'Pa

+ XS€S.<SH (1)Xss.su qsqs.qsu . (II-49)
where
“8x = 2 . _
(1)X s /(h /*s)%[a8'51 : (II 50)
a s'_ _ o . g a _
(1)Xs - (l/Iaa) (AS./As) css. , (II 51)
= 6 k = _
(1)XSS'S" (h />\SAS'}\S") KSS'S" thSS.'S" I (II 52)
H = 2 Z aBX q SSq .P up
2 a.B 5&5' (2) 58' B
ax s"
+ Zaisss'm" (2) Xss' 2(qsqs'Ps" + pansqs,)P
s" "I
+ 23$S' 33"33" I (2) x35. 41(q SSqS'P ups nu + pS "PS "qu qS u)
ESSS'SSHSSH' (2) xsslsusul q Sq s'qsuqsuu
_ _. 2 I0 _
8” 2a 1/Iaa (II 53)
where
OLBX = h [aB; ss'] , (II-54)

(2) ss' 2(Asls,)4

l6

 

 

 

 

A Z
a s" _ _ k s" l B . B . '
(2)xss' - M ZB[ASAS,] l + ass,[cs's"[a8'sl + Cssu[aB.s ]]
(II-55)
Z a a a a
Xs"s"' = Z 1 ASNASIII Cssncsnsnu + Cslsucssul
(2) ss' a I° A A , (1 + 5 ')(1 + 5 n n.)
dd 3 5 ss 8 s
(II-56)
”8 Z
(2)XSS'SHS"' = [ASAS'ASIIASH'] KSS'SHSH' = hC kSS'SHS"l
(II-57)
H = “5

3 Za,BZsss'ss" (3)xss's" qsqs'qs"PaPB
Z Z a Sfll l(
a S$S'\<S";S"' (3) 55.8" 2 qsqsuqsnpsnl
Sfllsll"
+ pslllqsqslqsll) Pa + ZS‘S"S";S""S"" (3) XSSIS"
xi(qqqp p +p p qqq)
2 3 SI S" sIll Sllll sill Sllll S SI 8"

Zs<sl‘<s"6slll$sll" (3)Xsslsllsfllsllfl qsqslqsllqslllqsll"

 

 

 

+ is (3)Xs qs ' (II-58)
where
1
a8 _ i M6 4 , I n _
(3) 55's" _ 2 [A A 'A'ZJ [afiy SS 3 ] , (II 59)
s s s
1
a s"! 2 AS." /4 Cgusnl [08; 53'].
X U II = — M ——
(3) ss 3 B AsAs'As" (l + 688" + 63's")
c§.S.. [as; ss“)‘ :28". Ice; s's"1'
+ +
(1 + 6 , + 6 . n) (l + 5 . + 5 n)
ss 3 8 ss 33

(11-60)

17

, A A Z
SIIISIIII _ /2 sIII sIIII
(3)XSSISII - h A A 'AS " [(1 + 65"'S III!) (1 + 6 SISII )
S S
-1 B
x (1 + ass.+ assu>1 2a B£<c§ H..c S".
a c8 ,. CB
+ CS'SH'CS Sun) [0'81 S ] +(c: SIII CSIISIIII

 

a B . a B
+ Cs"s"'CSSHH) [(18, S ] + (CSISIIICSIISIIII
+ c“ c8 )[as- s]} (II-61)
sllslll slsllll I I
M10 14
X = K
(3) SSISIISIIISIIII A )x ,A "A ",A u" SSISIISIIISIIII
S S S S 8
= he k . n u. n" . (II-62)
SS S S S
_ - l. 2 2 Z . _
(3)Xs ‘ 8 h 2a (K /As) [3“! S] . (II 63)

_ a8
H4 - ZGIBZSSS'ss'Ks'” (4)XSSISIISIII qsqslqsllqslll PGPB

a s" II

ESSS'§S"\<S'.';S"" (4) XSSI ISIISIII 2(qs qs lqs Ilqs II IPS II"

+ psunqsqs.qs"qsn.)Pa

SIIIISIIII I
+
ZssslssllésllI;SIIII\<SIIIII (4) SSISIISIII

l
x 2(qsqSIqsIIqsII IPSIIIIPSIIIII + pSIIIIpsIIIIIqsqslqsllqslll)

+ ESSS ISSIISSII 'SS""\<S""' (4) xss ISIISII Is" IISII II I

18

X qsqslqansqusnnqsnnI + ZSSS' (4)XSS' quS' I (II-64)

where

B 1 8 1“
a [ M ] [aB; ss's"s"']' (II-65)

(4)xss's"s"' 5-K A ,A "A
s s s s

1

a sun 3/22 ASH" 4 CES""[QB;S'S"S"']'
x I II III = M
<4) as s s 8 ASAS.AS..AS... <1 + 633.+ ass..+ 533-")

 

 

8 II III I
CSISIIII[0'B; SS 8 ]
(l + 6 I + 6 I II + 6 I III)
S S S S S S

 

B I III I
+ ;S"S"" {QB} 38 S ]

(1+6u+6lll+6"lll)
SS 38 SS

8
(1+6 III +6 III I+6 III II)
S S S S S S

IIII [a8; 88'8"] '

 

(II-66)

 

14
sIIIIsIIIII K[ASIIIIASIIIII ]. l

(4)XssISIISIII = A A 'A "A l + 6 u" n".
S S S S S S

a B (I 8 II III I
(CSSIIIICSISIIIII + CSISIIIICSSIIIII) [08; S S ]
+

x2a8(1+6,+6 a...)(1+5,,,+5
SS S SS SS

 

8.3...)

 

 

 

a B a B I II I
+ (CSSIIIIICSIIISHIII + CSIIISIIIICSSIIIII) [(1838 S 1
(I + 6 I + 6 II + 6 III)(1 + 6 III I + 6 III II)
SS S S SS 8 S S S
a B O. B . I III I
+ (Cssungsusuul + Csnsuncssuul) [08, S S -]
(l + 6 I + 6 II + 6 III) (1 + 6 II I + 6 II III)
83 SS SS S S S S
a B a 8 III I
+ (CSISIIIICSIISIIIII + Csllsflllgslslllll) [(18, SS ]
(1 + 6 I + 6 I II + 6 I III)(1 + 6 II + 6 II III)
S S S S S S S S S S

l9

 

 

a B a B . II I
+ (CBISIIIICSNISIIIII + CSIllsllllcslslllI) [0'81 SS ]
(1 + 6 I + 6 I u + 6 I u.) (1 + 6 III + 6 III II)
s s s s s s s s s s
a B a 8 I I
+ (CSIISIIIICSIIISIIIII + CSIIISIIIICSIISIIIII) [aB’ SS ] (II-67)
(l + 6 II + 5 II I + 6 II III) (1 + 5 III + 6 III I) ’
s s s s s s s s s s
x = (MN/A A A A A A )1"
(4) SSISIISIIISIIIISIIIII S SI SII SIII SIIII SIIIII
X K
SSISIISIIISIIIISIIIII
= he k I II III IIII IIIII I (II-68)
ss 3 s s s
_ l. 32 A A 'Z . ']' I
(4)Xss' — -8 M a ( s 5') [aa, 55 . (I -69)

In the above Hamiltonian Eqs. (II-5), (II-39), and (11-40) have

been combined to write pa in the following form:

_ X A A Z a II 70
pa - s,s' ( s'/ s) Z;ss' qsps' ' ( - )

0
Also, 9(0) has been taken equal to GaBIa6

body-fixed coordinate system have been taken to coincide with the

, because the axes of the

principal axes of the equilibrium moment of inertia tensor. The

. 7/
operators qs and p8 are dimensionless, conjugate variables; A82, which
is equal to ch s" is the normal frequency in radians per second. In

the cgs system of units all potential constants k are in cm-l.

III. EQUILIBRIUM GEOMETRY, NORMAL COORDINATES:

AND MOLECULAR PARAMETERS

III.1 Equilibrium Geometry and Normal Coordinates

 

The first definitive study of the vibration-rotation energies
of the bent XYZ molecule through the second order of approximation was
published in 1944 by Shaffer and Schuman.13 Their results were cast
by Nielsenl4 in 1951 into a form in which they appear as the special
case of a general expansion of the Darling-Dennison Hamiltonian. In
1953, Posener15 presented a detailed and careful reexamination and a
further development of the problem in his thesis. These and more

3,10,11,16-23

recent findings are incorporated in this presentation.

The equilibrium geometry of the molecule is shown in Fig. 1.

In the barred equilibrium coordinates in , yo. , zo. , i = 1,2,3, one

1. 1
has that
a = £01 — E03 , (III-l)
b = £02 - 3201 , (III-2)
c = 3701 - {r03 . (III-3)

For molecules obtained by isotopic substitution from XYZ, such as HDO
from H20, one may expect that a - b. However, since keeping a and b
distinct retains greater generality at very little cost in additional

complexity, we shall do so. With

20

21

masomaoz mmNBIwa umwcchoz may no hnumEomu Esflubflawswm

.H ousmwm

 

 

 

 

 

l>~"

m

X

22

M = ml + m2 + m3 , (III-4)

the equilibrium coordinates, with the origin at the center of mass, are

£01 a (m3a - m2b)/M , (III-5)
£02 = [m3a + (ml+ m2)b]/M , (III-6)
E03 = -[(ml + m2)a + mzbl/M , (III-7)
§°l a (m2 + m3)c/M , (III-8)
§02 = -m1c/M , (III-9)
§°3 = -mlc/M , (III-10)
E = E ' o . (III-11)

°1 °2 ' z°3 =

Transformation to the principal axes system of the equilibrium inertia

tensor is accomplished by taking

xOi = §°icos6 + §°isin6 , (III-12)
y°i = -x°isin6 + §°icose , (III-13)
zoi = 201 - o , (III-l4)

where the angle of rotation 6 is given through

tan 26 = T/Q , ' (III-15)
with
T = 2m1c(m3a — mzb) , (III—l6)
Q = m (m + m )a2 + m (m + m )b2
3 l 2 2. 1 3
2
- ml(m2 + m3)c + 2m2m3ab . (III 17)

The transformation gives for the equilibrium coordinates in the prin-

cipal axes system

x {(m a - mzb)cose + (m2 + m3)o-sin6}/M , (III-18)

°l 3

x c*sin6}/M , (III-19)

02 {[m3a + (ml + m3)b]cose - m

3

x = - {[(ml + m2)a + m2b1cose + m C'sin6}/M , (III-20)

°3 l

+ m3)c°cose}/M , (III-21)

Yol ' {(m3a - m2b)51n6 - (m2

23

= - + + ' + . -
Y°2 {[m3a (ml m3)b]sin6 mlc cose}/M , (III 22)

= + + . — o .-
y03 {[(ml m2)a m2b1s1n6 mlc cose}/M , (III 23)
z°l = 202 = 203 = O , (III-24)

o g _ o _ I _
Ixx 2(Izz I ) , (III 25)
1° = iao + I') (III-26)
yy 2 22 '
° = ° + I° = + + 2 -
Izz Ixx yy [9 2m1(m2 m3)c ]/M , (III 27)
with
2 2 2 1/2
I' = t[(T + 9 )/M ] . (III-28)
Denoting instantaneous position coordinates by xi , yi , and 21 , we
have that 21 a 22 = 23 a 0 because of the absence of out-of-plane
vibrations, and the nontrivial Eckart conditions24 are given by
mlxl + m2x2 + m3):3 = O , (III-29)
+ a .-
mlyl + mzy2 m3y3 O , (III 30)
{i mi(x°iyi - inxi) = o . (III-31)
Intermediate coordinates u, v, w are now introduced as follows:
u = x2 - x3 , (III-32)
v = y1 - (mzy2 + m3y3)/(m2 + m3) , (III-33)
w = x1 - (mzx2 + m3x3)/(m2 + m3) . (III-34)
In conjunction with the Eckart conditions, these give
xl - (u/ml)w , (III—35)
a: ' — -
x2 (I: /m2)u [ll/(m2 + m3)]w . (III 36)
_. _ I _ _
x3 — (u /m3)u [Ll/(m2 + m3)]w , (III 37)
Y1 = (u/ml)v (III-38)
:- .I II _
Y2 (u Y/m2)u + (ua/m2)v + (u /m2)w . (III 39)
a .. I _ _ ll _
Y3 (u Y/m3)u (HS/m3)V .(u /m3)w , (III 40)

24

with
u = [ml(m2 + m3)]/M .
u' = m2m3/(m2 + m3) ,
u" a mlyol/x23 ,
a = ' x13/"23 '
B = ' x12/x23 '
Y = y23/x23 = -tan6 ,
where
x12 = xol - x92 = - b cose + c sine ,
xl3 = xOl - x03 8 a cose + c sine ,
x = x - x = (a + b)cose ,

The kinetic energy of vibration

l .2 .2 .2
= — + ‘I'

can be expressed in terms of the intermediate coordinates as
1 - - t
T = 3' (m) (U) ((0) I
where (A) = (u v w), t denotes the tranSpose, and (u) is the 3

symmetric matrix with elements

“11 = u'(1 + yz) ,
“22 = u2[(l/m1) + (aZ/mz) + (82/m3)] .
u33 = u + (u"2/u')..
“12 = u21 = u"'Y .
“13 = u31 = u"Y r
u23 - U32 = u"u"'/u' ,
where
u"' = - mlx01/x23 = uu'[(a/m2) + (B/m3)]

(III-41)
(III-42)
(III-43)
(III-44)
(III-45)

(III-46)

(III-47)
(III-48)
(III-49)

(III-50)

(III-51)

(III-52)

X 3

(III-53)

(III-54)
(III-55)
(III-56)
(III-57)

(III-58)

(III-59)

25

The most general harmonic potential energy expressed in the intermediate
coordinates and invariant under the point group symmetry of the molecule

(Clh) is

v = (w)(k)(m)t , (III-60)

nus:

where (w) = (u v w) and (k) is the 3 x 3 symmetric matrix of potential

constants with elements kll' k , k , k = k

33 12 21' k I k "k = k

13 31 23 32°

The transformation from intermediate to normal coordinates Q5,

22

s = 1,2,3, is of the form

u = {5 nlst , s = 1,2,3, (III-61)

v = Z n Q s = 1,2,3, (III-62)
s Zs s ’

w = 28 n3st , s = 1,2,3, (III-63)

where the nS,S are obtained through solution of the secular equation
I A(u) - (k)| = o (III-64)
and each n , can be expressed as n , = N , /N where N , is the co-
s s s s s s s s 3
factor of the s'-th element of any row (e.g., the first one) of the
determinant, Eq. (III-64), with A a As , the s-th root of Eq. (III-64).
The quantity N8 is determined such that
- l. '2 '2 '2 -
T _ 2 (Q1 + Q2 + Q3) , (III 65)
Which requires that

2: 2’ +
N5 Zs'=l,3 [us's'Ns's Zs"7€s' us's"Ns'st"s

The harmonic portion of the potential energy becomes
1 2 2 2
= — + '-
V 2 (AlQl 2292 + A3Q3), (III 67)
and the normal frequencies, in radians per second, are Al, A2, and A3

The As can be specified in closed form as the roots of the general

cubic equation. These expressions are, however, rather cumbersome and

26

of limited practical use,as ordinarily it is the three roots As for
which numerical values are known and the potential constants kij for
which numerical values are sought. As there are three AS and six

distinct potential constants ki , the problem is underdetermined and

3
additional information about the potential constants must be developed
through intercomparison of isotopically substituted species and through
information derived via the second-order parameters of the Hamiltonian.
Thus if it is the AS which are to be regarded as known, it appears that
the relations between the coefficients of the cubic equation and the
symmetric functions of its roots are potentially more useful than the
expressions for the roots because the former relations are considerably

simpler than the latter. Theory of equations shows that the three

roots As of the cubic equation

3 2 + + = -
C3X + C21 ClA CO 0 (III 68)

are related to the coefficients Cn through

— CZ/C3 = A1 + A2 + A3 , (III-69)
+ Cl/C3 = AlA2 + A2A3 + A3Al , (III-70)
- CO/C3 = A1A2A3 . (III-71)

For the problem under study we find that

C3 = det (u) , (III-72)
.. _ (2) _
C2 - ZsXs' Css' kss' ' (III 73)
_ (1) _
Cl - Zszs' Css' uss' ' (III 74)
CO = - det (k) , (III-75)

(2) (l)

where C , is the cofactor of p , of (u), C , is the cofactor of k ,
ss ss ss 55

27

of (k), and where the double sums are unrestricted.
Since a simple, closed form solution of the secular equation

cannot be written down, the transformation coefficients nS need to

I
be retained explicitly in what follows. The normal vibrations
problem is here set up in such a way (with s - l specifying the XY
bond stretching mode, 5 = 2 the bending mode, and s = 3 the YZ bond
stretching mode) that specializing the results of this and the _
following section to XYX will reduce these directly to the results

of Chung and Parkerlo, Yallabandi and Parkerll, and Chan, Wilardjo,

and Parker.25

28
III.2 Molecular Parameters

The coefficients 2:5, a = x, y, z, of the transformation from
instantaneous position to normal coordinates are defined by Eq. (II-3)
and can be constructed for the XYZ-type molecule with the aid of Eqs.
(III-35) - (111-40) and (III-61) - (III-63). In this manner one

determines that

2:3 = (u/mf)n3s , (III-76)
22s = (“'/m§)nls - [me/(mz + m3)]n3s (III-77)
2:8 = -(u'/m'§)nls - tum’j/(m2 + m3>1n3s : (“148’
£15 = (u/m?)n28 , (III-79)
22s = (u'Y/m§)nls + (ua/mEMZs + (u"/m§)n3s I (III-80)
figs = - (u'y/m§)nls - (“B/m§)n25 - (u"/m%)n3s I (III-81)
£2 = £2 = £2 = 0 (III-82)

Knowing the 2:8, one can construct the Coriolis constants, :23, , from
Eq. (II-6). These are all zero when a = x or y. The nonvanishing C:s'
with the upper index suppressed, are the following:

- n ) + u"'(n n

C = - c = -u(n

ss' s's 2sn33' Zs'nBS ls 25'
‘ nls'nZS) I u (nlsnBS' ‘ nls'n3s)'
s # s'. (III-83)

There are thus three distinct nonvanishing Coriolis constants, viz.,

:12 = - C21, :13 - - :31, and :23 = - :32, and these obey the sum

29

rule7,19,26

2 2 2 = -
O I a8 a8 0
For the XYZ-type molecule the quantities as and (ASS,)', introduced

in Eqs. (II-29) and (II-30) for the instantaneous moments and products

of inertia, take the following form:

xx xx xx xx
= + + -
as t]- nls t2 nZS t3 n3s , (III 85)
yy _ YY yy yy -
aS - tl nls + t2 n25 + t3 n3S , (III 86)
22 _ zz zz zz = xx yy _
aS - tl nls + t2 n25 + t3 n3S aS + as , (III 87)
XY _ KY KY XY = YX _
aS — t1 nls + t2 nZS + t3 n3s as (III 88)
with the coefficients t:8 as summarized in Table 1. All other a:8
vanish. The nonvanishing (A::,)'and (A::,)'are given by
YY I = YY .
(Ass') (As's)
= Ayy ' = All?
ss 5 s
= I + _
u nlsnls' 1m3sn3s' ' (III 89)
zz zz zz
(A11>' = (c23)2. (A22)' = (:13)2, (A33>' = (:12)2,
(III-90)
zz , _ _ zz , = zz , = _
(A12) ‘ C13C23’ (A13) C12:23' (A23) C12513'
(III-91)
xY . _ YX . = XY 3 YX
(Ass') (As's) Ass' As's
= _ I II I + I!
(n Y nlsnls' + u nlanS' u n1snss.'
+ . III- 2
u n3sn23') ( 9 )

In Eqs. (III-90) and (III—91) we have used A::, = 685,.

30

B

Table l. The Coefficients t: Introduced In Eqs. (III-85)-(III-87)

 

 

 

 

 

 

 

 

- 2
4.
xx 2m2m3(a b)s1n 6 yy 2m2m3(a + b) cose
t1 = (m + m )cose t1 = (m + m )
2 3 2 3
+
txx _ 2ml(m2 m3” tyy _ 0
2 M cose 2
xx _ _ yy =
t3 - 2mly°l tane t3 2mlx°l
tzz = 2m2m3(a + b) a txx + tyy txy = 2m2m3(a + b) Sine
1 (m2 + m3) cosB 1 l 1 (m2 + m3)
zz _ xx xy =
t2 - t2 t2 0
2m (m a - m b)
zz = 1 3 2 _ xx yy xy _ _
t3 M cosG t3 + t3 t3 2m1y°l

x and y01 are as given by Eqs. (III-18) and (III-l9)

 

31

Direct computation of (A::,)' yields the very complicated expression

 

xx , _ xx , _ xx = xx
(Ass') (As's) Ass' As's
2 2
= I 2 2 1 L... g.—
H Y nlsnls' + u m + m nZSnZS'
1 2 3
uII2
+ __... III
u' n3sn3s' p Y(nlans' nls'nZS)
+ u"Y(nl n3s' nls'nBS)
uIIuIII
+ . -
+ u, (nanBs' n28,n3s) (III 93)

For 5 = 3' use of this expression can be avoided by taking advantage
. l9 . ad .
of the sum rule of Oka and Morino , Viz., 2a Ass = 2, which upon

application to XYZ gives that
XX I _ _ YY I = _

For 3 # s' Amat and Henry27 have shown that

XX

(Ass'

)I = -(A::,)' , (III-95)
and thus Eqs. (III-94) and (III-95) can be combined to give

(A::.)' + (Ayy )' =

SS, 558, (III-96)

Amat and Henry have also shown that the following simple relations

exist between the a:8 and the (A::,):
ad , = 1_ cy ay 0 _
(Ass') 4 2y as aS'/IYY , (III 97)
a8 , Ba 1 ay BY ay By
'=— + o -
(ASS') + (ASS,) 4 fy<as as, as,aS )/IYY , (III 98)

Another sum rule that may prove useful is the one for is A:: which for

XYZ yields that19

32

xx , xx , xx ,
(A11) + (A22) + (A33)

nun:

+ o _ o o _
(Ixx Iyy)/2Izz ,(III 99)

3
(A{§)' + (A§§)' + (A§§)' 3-+ (I;y - I;x)/2I;z .(III-lOO)

a8

Knowledge of the constants as

B

and (A:S,)' allows one to
express the effective moments of inertia and the effective reciprocal

inertia tensor “as in terms of basic molecular parameters. Since

a = xz = yz = xz , = yz , =
IaB O for a # 8, and a8 a8 (Ass') (Ass') 0, from Eq.
(II-29) it follows that 1&2 = I§z = 0. Recalling that the expression

for the components of the effective reciprocal inertia tensor “a is a

8

power series in the normal coordinates, the coefficients of QS can now

be written as

9(0)“B/I;a1§8 = Gas/138 , (III-101)
[a8; 5] = - aZB/IaalgB , (III-102)
[aB; ss’] = (13a1§8)'1[-(A::,)' + 25 agaa:?/136],(III-103)
[cB; ss's"] = (ISaI88)-l{-25,e a26a2$a:E/(I§6I;€)

+ Za[a:6(A:?s")' + a:6(A:?s")']/I§5}

(III-104)

. . I I. = o o -1 a5 86 68 nn . o o .
[a8,ss s s ] (IaaIBB) {26,EIn aS as'as"as"'/(Idéleelnn)

a6 Be 62 nn 0 o o
26¢efn as as'as"as"'/(Iédleelnn)

a6 86 66 as 58 Ge

.. ._ ’020
261$ ‘as aS'(aS"aS"' as"as"')/(Idélee)

33

+ 26(AS: )'(A:§S n.) '/I°5 _ 28,5 3:833 386(A66 S" s". )I/(IO EIoé)

_ 86 as 86 , 86 as , o o _
Z€,6 as [asl (Asllslll) + asI (ASIISIII) /(IEEI66) }.(III 105)

Use of the sum rules (III-97), (III-98) allows one to write the follow-

ing alternate forms for [a8; ss'] and [a8; ss's"]:

 

,I_1 3 (18. Ba. _
[a8,ss ] 13 J88[1 + 53 s'][(ASSI) + (Ass,) J , (III 106)

1 Z l
IaaIgfi 6 I (1 +883' +839”l +Ss's")

 

[a8;ss's"] = -

a6 86 , 86 , a6 86 ,
X {as [(As's") + (AS. 5') ] + as'[(Ass")
86 , a6 86 , 86 ,
+ (A5"s) ] + as"[(Ass') + (As's) 1
86 A06 , a6 86 ,
+ as [(A s' s") + (As "5 ,)' ] + as '[(As:")
a6 a86

+ (Asus)'] + %[( :.)' + (A? s)']}.(III-107)

The coefficients [a8;ss's"s"'] are not included, because they turn out

to be unnecessary for the fourth-order calculation. We note that for

. . a8 a8 a8 08
the coeffiCients (I)xs' (2)xss' , (3)xss's" (4)xss's"s"' only

terms with a,8 equal to xx, yy, zz, and xy are nonzero; and for

, and

a SI a S" a SIII a sIIII

(l)xs ’ (2)Xss' ' (3) ssIs" r and (4)Xss's"s"' the nonzero terms are
- - 25 28 ss 25
those With a 2. Finally, (l)xs , (2)xss , (2)xss , (3)Xsss I and
as . _ 05X
(3)xsss vanish for S _ 1'2'3’ For our purpose, only the (2)X are
08”

needed, and these are given in Appendix 2 in terms of the aS and

<A“B.>'

SS

IV. GENERALITIES OF THE CONTACT TRANSFORMATION

TECHNIQUE FOR FOURTH-ORDER CALCULATIONS

and H as

The contributions to the energy from H 3, 4

IHIH

1 2

discussed in Chapter II may be evaluated by the usual methods of per-
turbation theory. The zero order energy is computed from the zero

order part of the Hamiltonian, H , while the first order correction to

O

the energy: E1' is computed from the diagonal matrix elements of H1

The off-diagonal elements of H contribute only to the second and high-

1

er order corrections. However, the perturbation is complicated by the

myriad of terms in H , H2, H , and H4, and it is therefore highly de-

l 3

sirable to transform the Hamiltonian to a more convenient form. Such
a form can be attained by subjecting the Hamiltonian to a contact

. 28
transformation ,

H' = THT-l = H6 + AHi + AZHé + "" (IV-1)

where A is a parameter of smallness.

With a suitably chosen unitary operator T the off-diagonal
elements of Hi can be made to vanish, while the zeroth order terms and
the diagonal matrix elements of the first order term of the Hamiltonian
remain intact. The zero order eigenfunctions are then the correct
eigenfunctions to first order when the zero order energy is non-
degenerate. Since there are no off-diagonal matrix elements of H' the

1!

34

35

evaluation of second order corrections to the energy is effectively
reduced to a first order perturbation calculation.

We shall let the unitary transformation operator be of the form
T = exp(iAS). The transformed Hamiltonian is then given by:

H' = THT-l = (1 + iAs - -:—-AZS2 - %-1A3S3 + ~-- )

2 3 .I.
x (H0 + AHl + A H2 + A H + )

3

x (1 - iAs - i—Azs2 + é-‘A3S3 + --). (IV-2)

Equating coefficients of like powers of A one obtains

H6 - H0 (IV-3)
Hi = Hl + i[S, HO] (IV-4)
. _ - _.£ _
H2 — H2 + 1[S, H1] 2[s, [S, H0]] (IV 5)
In general
n (i)n'k n-k
“I: = Lao W5 , Hk} (IV-6’
where
(0) _
{s , Hn} = Hn (IV-7)
(1) =
{S I Hn} — [SI Hn] (IV 8)
(2) z _
{S I Hn} — [SI [8. Hnll (IV 9)

The partial Hamiltonians which will be needed in subsequent calculations
are

= H (IV-10)

36

I= ' I-
H1 H1 + i[S, HO] ( v 11)
H' = H + l-i[s H + H'] (IV-12)

2 2 ' 1 1
H' = H + i[S H ] - 518 [s H + l-H']] (IV-13)
3 3 ’ 2 3 ' ' 1 2'.1

In the.above it should be noted that Eq. (IV-4) has been solved for
[8, HO] = i(H1 - Hi) and substituted into the expressions for H5 and Hi.

In general we have

(2)

Hn = Hn + i[S, Hn-ll' (IV-l4)
(2) _ I_. (3)
Hn_l - Hn-l + 2 i[S, Hn-2]' (IV 15)
' (m) 1 . (mm
= + — _

Hn Hn m i[S, Hn-l ] , (IV 16)
where Hém) = HO for all values of m. The required transformed first-
order Hamiltonian Hi is obtained if S is chosen such that

' = - * ° _

1[S, H0] (H1 + {a paPa/Iaa) (IV 17)
where

* a — _
pa Zszo<o' Cso,so'(qsopso' qso'pso) (IV 18)

with o as the index of degeneracy. This choice of S removes all but
the first-order Coriolis term from the first-order Hamiltonian. The
proper forms for the operator S have been considered by Herman and
Shaffer.29

Allowing for the possibility of regrouping terms into orders

of magnitude more appropriate to experience, the once-transformed Ham-

iltonian will have the general form

37

l= I | ' ' ' coo. -
H ho + h]- + h2 + h3 + h4 + . (IV 19)

The explicit forms of h', h', and h' will be given in Chapter V.

0 1 2
In order to calculate vibration-rotation energies to the fourth
order of approximation by the contact transformation technique, it is

necessary to carry out a second contact transformation T on the once-

transformed Hamiltonian H',

H" = TH'T = exp(il22)-H'-exp(-il22)

n u 2 n 3 u ... -
HO + AHl + A H2 + A H3 + (IV 20)

in such a manner that H8 + AH; + AzHg will be diagonal with respect to
the vibrational quantum numbers vs. In terms of the like powers of

the parameter A, Eq. (IV-20) is equivalent to

H3 a W) (IV-21)
H1. = hi (Iv-22)
H3 = hi + i[Z, ha] (IV-23)
H; = bi + i[Z, hi] (IV-24)
HZ = 1121+ i[E, hi] - is, [2, 115]] (IV-25)

Again allowing for the possibility of regrouping the terms of the twice-
transformed Hamiltonian, it is expedient to write H" as

H = hO + h1 + h2 + h3 + --- . (IV-26)

The Hamiltonian in the form (IV-26) can be used to calculate
the vibration-rotation energies to the fourth order of approximation,
because h", h", and h" are diagonal in all vs by virtue of the contact

0 1 2

transformation, and matrix elements of h; and h; off-diagonal in one or

38

more vs contribute to the energies only in orders of approximation
higher than the fourth. The Hamiltonian H" is diagonal to all orders
in the rotational quantum number J (total angular momentum quantum

number) and M ( magnetic quantum number). In a symmetric rotator rep-

resentation WJKM ,in which P2? = KM? , the Hamiltonian does, how-

JKM JKM

ever, have matrix elements which are nondiagonal in the quantum number

K, including second off-diagonal elements in H

It was found3'6'7'8'9 that to the fourth order of approximation

0.

the Hamiltonian has terms which can be classified into the following

three categories:

2 H 6
(a) (0)2 r , (2)2 r , (4)2 r ,
2 ‘+ 2 6
(b) (O)ZP , (2)ZP , (4)ZP , (4)ZP ,
2 2 2 1, 2 3 2 1+ L» 2

Here, r2 denotes any possible product of two vibrational operators
. H -

qsqs" Psps" qsps" psqs" r denotes any poss1ble product of four

vibrational operators, etc. The symbol Pn stands for any possible

product of Px' Py, and P2 of power n, and Z stands for the coeffi-

(m)
cient of an operator contributing to the m-th order of approximation.

The terms in (a) constitute pure vibrational operators includ-
ing anharmonicity corrections, the terms in (b) constitute pure rota-
tional operators including centrifugal distortion corrections, and the
terms in (c) represent the vibrational corrections to rotational and
centrifugal distortion corrections.

The pure vibrational energies will not concern us here, since

we are interested principally in the rotational level structure built

39

upon particular vibrational states rather than in the detailed calcula-
tion of the pure vibrational level structure. In the absence of vibra-

tional degeneracy one observes from Eq. (IV-18) that the Z r2P-type

(1)

terms vanish. It is also found that in the absence of vibrational

Z r4P and Z r2P3-type terms have no nonzero matrix

(3) (3)

elements diagonal in all vs. Thus, the odd-order terms may be omitted

degeneracy the

from consideration unless accidental resonances occur which invalidate
the order of approximation arrangement of the Hamiltonian. The Hamil-

tonian to be used can thus be written as

II II* II* II*
H — hO + h2 + h4 (Iv-27)
where
hII* — Z P2 IV 28
* 2 2 4
" = + I -
h2 (2)2 r P (2)2 P ( V 29)
* 2 4 2 2 4 6
h" = Z P + Z P + Z r P + Z P . IV-30
4 (4) (4) r (4) <4) ( ’

t
The asterisks denote that pure vibrational terms are omitted in ha ,

* *
h" , and h" . Also, terms of h"

2 4 4 not diagonal in all vs are omitted

- hn*
1n 4 .

V. THE FIRST CONTACT TRANSFORMATION FOR THE XYZ-TYPE MOLECULE

We have discussed the partial Hamiltonians Hn which consist of
combinations of vibrational operators ps and qs, and rotational operators
Pa ; and , similarly, the hermitian Operator S introduced in Eq. (IV-2)
consists of both rotational and vibrational operators. Both operators

k k . k k k
can be expressed as S = 2k S, and Hn = 2k H with S = (.S)v( S)R ,

and kH = (kH)v(kH)R where the subscripts v and R denote the vibrational

and rotational parts. Dropping the index k one discovers that

[SI H] = [Sr H]V + [SI H]R (V-l)
where

[S, H]v = [Sv, Hv](SRHR + HRSR)/2 (V-2)

[S, H]R = (SVHV + HVSV) [SR' HR]/2 (V-3)

It has been shown4 that if Hn is of order n, then [S, H]v
is of order n+1, while [8, HJR is of order n+2. Regrouping terms
which are truly of the same order of magnitude one finds from Eqs.

(IV-l4) and (IV-15) that

(2) (2)

h; = Hn + i[S, hn-llv + i[S, hn-2]R (V-4)
(2) _ . (3) . (3) _
hn - Hn + 2 i[S, hn-IJV + 1[S, hn-2]R (v 5)

4O

41

m) 1 (m

= H + -i[S, h 1 (m
m n-

+1) . +1)

+ —- S . -
l ]v m 1[ ' hn-2 ]R (V 6)
In the absence of degeneracy: the second term in Eq. (IV-l7),

viz., 2a nga, vanishes. Thus, using Eqs. (V-4) to (V-6), we have

I: -
hO HO (V 7)
h1 = Hl + 1[S, HOJV = O (V-8)
h' - H + '[s H ] + l-i[s H ] (v-9)
2 ' 2 1 ' o R 2 ' 1 v
h'=H +l-i[s H] +l-[s [5 H1]
3 3 2 ’ 1 R 6 ' ' o R v

1 . .
+ 3 l[S, 2h2 + Hzlv (v 10)

l
' = ' - — '
1'14 H4 + i[S, H3]V 4[5, [5, 1'12 + HZJVJV

+ —: 118,18, [8, H

l
l 1 ] ] -'3—[Sr [SI HllRlv

O R v v

1 . , 1_ _
+ 3 1[S, 2h2 + H2]R + 6[S, [S, H01R]R . (V ll)

With Eqs. (V-2) and (V-3), and using the commutation relations
as stated by Herman and Shafferzg, it is possible to find the S-function

which satisfies Eq. (V-8). It has the general form

_ as s a
S - Z01,st s psPaPB + Za£s<s'( Sss' qsqs'

S"

a ss' S
< I . II I
s s ,3 ss

1
+ S psps.)Pa + Z 2(qsqs.ps. + psuqsqs.>

+ ZS<SI<SH S pspslpsn I (V-lZ)

42

 

 

 

 

 

where
a8
a8 5 as
2(K As) IaaIBB
% A + A ca
a l s s' _ ss' , _
Sss' ‘ [A A ,J A - 1 , ' 1° ' s I S (V 14’
s s s 3 ad
4 a
a ss' 2 (Asxs') z;ss' , s # s'
S = -- o (V-lS)
2 A - A , I
K 8 aa
’4
S" 21": )‘su(}‘su - A - A I)
S I = - —(1 + 6 II + 6 I II) k I II '
53 H s s s G , " ss 5
ss 8
(v-16)
I II (A A IA II) 2
Sss s a 4Hc s s s k ' u ’ (V-l7)
3 G , " ss 3
h as s
with
1 1 1 1 1 1
G . . = <14 + 14, + 14.><x6 + 14, - 14")
ss s s s s s s s
1 1 1 1 1 1
~(14 - 14, - Aé")(A4 - 14, + 14") . (V-18)
s s s s s s
It can be seen that “ass a B“ss', xzss = yzss = 0, “sea, a “533 =0

2
for a f z, and 888 = 2S88 = 0. The nonzero terms of a8

Ss and 2S ,
as
for the XYZ-type molecule are compiled in Appendix 3. Other nonzero
terms are not needed for our fourth-order calculation.
With the explicit knowledge of the S-function the second and

third terms of hi in Eq. (V-9) can be evaluated and yields the following:

. - 087 s
i[S, HO]R - Za,8,y2s a psPaPBPy

43

 

 

 

 

 

 

GB a8 ss'
+ Za,BZs<s'( ss' qsqs' + a psps')P PB (V-l9)
where
aIY aIIY
aBYas l l ,as Ga"B as 6a'B)
- 1 o o \ o o
4<n2183)4 IBBIYY a'a' Ia"a"
BIY QBI BIIY 0'8"
6 6 6 +
+ l ,as GB" as B"Y _ as 03' a 65'Y\
o o \ Io Io I
“a Y-Y BIB! Bus"
I II
1 actY 68 u - azY 68 ,
+ o o < S l o ) I (V-20)
I I I° I
Ga 88 Y'Y' YuY'.
%
. a' a'
aBa ' g_ l A + As' Css' a"B _ css' a'B
I _ o o o 0
ss 2 ASAS, A8 A8, Ia'a'IBB Ia"a"IBB
BI B"
C I II c I6 I
—— :6 --——— 16 .
BIBI “a 8'18" “a
74 a! an BI
aBass' l_(AsAs') Css'aa"8 Css'Ga'B Css'fiaB"
- _ o o ‘ ‘1““17’ "?"‘—F‘
K A8 A8. IG'G'IBB IG"G"IBB IB'B'IGG
BII
1;ss' 08'
- E;———3;7- , s # s' (V-22)
BIIBII aa
. . zzz s
With a, a', a", etc. in cyclic order. It may be seen that a - 0,
I I
xzxas a 2 xxzas’ and yzyas = 2 yyzas; aBa ' = Baa .' aBass = Baass ,
ss ss
and yza . = xza ' = yzass . xzass = dad ' = aaass = xya
as ss ss 58
axyass - 0. The nontrivial a-terms necessary for our fourth-order

calculation are compiled in Appendix 4.

Evaluation of %-i [S,Hl]v gives the following result:

“BYGY Pap P P

1 . _
2 l [S'Hllv - Za,B,y,6 B Y 5

44

dBY S 08
+2”,st Y psvavgpgzmzsfsm Y... qsqs.

08 83' a 83.5"
+ Y Psps')PaP3 + 202853.55" y psps'ps"Pa

a s" 1
+ ZQZSES. 188. 2 (qsqs.ps. + psuqsqs.)Pa

SIISII I l

ZSES';S":S"' Yssl 5*qsqsopsnpsul + PsuPsnlqsqsn)

 

 

 

2553':S":S"' Ysslsusnl qsqslqsuqsnl I (v-23)
where
6 6
“BY Y = - is azsa; /(8ASI;aI§BI;YI§6 , (v-24)
_ Y dB
“BYYS - - 1 2 AS (AS 3A8.) :88. as!
- k s' A .(A . - A ) 1° 1° 1°
8“ s'#s s s s aa 88 yy
z;:s' a2?
+ W , / (v—25)
aa 88 YY
1 - -
a8 Hcflé Gss's" As"()‘s" As As')
YSS' - 2 Zsu(l + 63 n+ 63'8") 3/4
A G I° 1°
3" ss's" ad 88
A A - A2
as a 44 s s' s"
X as" ss's" + 2 28"(ASAS') (A _ A u) (A ' _ A n)
s s s s
a B a B
C II; I II + C I II; II
x as s s s s 33 (V-26)

 

(l + 688,) Igalge '

a8 ss'

a ss's"
Y

a S"
Yss.

45

 

 

 

 

 

 

 

 

(As A A1/2 >11
SI ASII a8
ZS..<1+6..+6...> I: a..k...
”3/25 ss s s ss's " IaaIBB 5 es s
l I A8 + As, - ZASH‘
__ 4
+ 2H Z5"”5’5" (As - As")(As, - As")
;: s.::. .8. +c:. S.c§s.
o o , “7-27)
(1 + 68 s)IaaIBB
L2 ’/2 /2’
(A A IA IIA III) 1 + 68 SIII + 6 I III
= 2Hc Z s s s" s s s
2 III 68
M S 88.8" l..+ SS" + 68.3"
1 1 1
c“. .. (xix .A .Aé..>1
. s s + Z s s s 3
10 8513" S". G ' n "3
aa 3 s s
l + 6 I II + 6 II III Ca III
. s s s s . ss.
1 + 5 ' ... 6 " Io Signs"!
3 s s cc
1 1 1
(A A4.A..A”"... )4 1+6 ...+6..
+ s s s' as s s
S". G II III 1 + 6 I + 6 II I
es 3 es s s
a
C I III
3%- SS..S... , <v-28>
ad '
a ’4.
"C ZS"'(1 + 6SSIII + 6SISIII) (CSIISIII) (ASHASH')
26 I III + (A II - A III) (A + A II - A III)
ss 5 s s s s s k
G I III(A II - A III) Io ss's".
as s s s cc
AS". 14
- TIC ZS"'( 1 + 688".)(1 + ass" + 68"SH') (r)

SI

46

 

 

 

 

1/2 a
A "(A II - A III - X ) C I III III 1
. S S S S . S S _ TTC 2 ( S )4
(1 + 6 ,) ,, 1° ss"'s" s'” A
58 SS 8 0101
(1 + 6 )(1 + 5 + 6 ) A1/2 (A A A )
. SISIII SIS" SIIISII . S" S" SI 8'”
1 + 6 I G I III II
SS 8 S S
C!
. Cssnn k (V-29)
Io SISIIISII I
0101

SIISII I a 4.".2c2

Z IIII(l + 6 IIII+ 6 )(1 + 6 II "n+6 III
S SS S S S

)

 

 

SSI M SISIIII sIIII
P
(ASIIASIIIASIIII) 2
G n u. n" ksSISIIIIkslISIIIsIIII I (v-30)
S S S
Y III III = r III III + r II I III + r III I II + r I II III
SS S S SS S S SS 8 S SS S S S S SS
+ P I III II + P II III I I (V-31)
S S SS S S SS
with
P I u "I = - "Czh Z an (1 + 5 nu + 6 I nu)(1 + 6 u an
SS S S S SS S S S S
+ (S III IIII)(1 + 6 II + 6 III + 5 I II + 6 I III
S S SS SS S S S S
Ag A A A
_1 SIIII( sIIII - SII - SIII)
+ 6 "6 I III)
SS S S G II III IIII
S S S
k I IIII k II III IIII ° (v-32)
SS S S S S

Appendix 5 contains the nonzero y necessary for calculating the fourth-
order centrifugal distortion coefficients.
Substitution of Eqs. (II-53), (V-19), and (V-23) into Eq. (V-9)

gives the once-transformed form of h' ,

47

GBYG aByYs

hé = Za,8.y,6 (2)Y PGPBPYPG + Xa,8.yzs (2) PsPaPBPY
a8 ss' a8
+ Za,BXsfis'( (2)Y psps' + (2)Yss'qsqs')PaPB
a ss's"
+ {afsfis.fis. (2)Y 9598.95. Pa

a S"

l
+ ZQZS:S';S" (2)YSS' 2(qsqslpsu + pansqsn) Pa

+ Z Ysusn'liq q p p + p p q q )
SES';S":S"' (2) ssl 2 8 SI SI! sIII SII sIII 8 SI

+ Zsfs'isufis'” (2)YSSISIISIII qsqslqsuqsnl ' (v-33)
where

aBy6

aByd
(2) Y

Y

I (V-34)

aByYs = aByys + aByas

(2) , (v-35)
(gfyss' = asyss' + aBass' ' (V-36)
(;§Yss' = (:fxss' + “3153- + “8°55- , (V-37)
(2?Yss's" = GYSS'S" ' (v-38)
(2?Y:;' (2?x:;' + GY:;' . (V-39)
(2)Y:;?"' = (2)X:;?"' + 1:2?"8 . (v-40)
(2)Yss's"s"' = (2)Xss's"s"' + YSS'SHSHI ° (V-4l)

Similarly, the once-transformed form of h; is given by

. _ 08Y5
h3 _ Za,8,y,dzs (3)Ys qs PaPBPypd

aBy s'll
+ Za'B'YZSIS' (3)Ys 2 (qsps' + ps'qs) PGPBPY

48

a8 SIS" i
+ ZQIBZSIS'ES" (3)YS 2 (qspslpsfl + Pslpsllqs) POPS

a8
+ Zaszsfs.fs. (3)Yss.s. qsqs.qs. PaPB
a SISIISII I1
+ Zazs;slfsllfsfll (3)Ys 2(qsp8IpsllpslIl+ PSIPSIIPSIIIqS)Pa

a sIII 1
+ XGZSESISSII;SIII(3)Ysslsll 2(quSIqSIIPSIII + PSIIIququSII)Pa

Z sISIISIIISIIII l(
S;S':S"§S"'ES"" (3) S 2 qustsIIpsIIIPSIIII

s" I 5" II

+ pSIpsIIpSIIIpsIIIIqS) + ZSESIESII;SIIIESIIII (3)¥SSISII

1
2 (qsq8.qs.ps..ps.. + ps..ps..qsqs.qs.)
ZSESIESIIESIIIESIIII (3)YSSISIISIIISIIII qsqslqansnlqsllu

+ {S mars qs. (v-42)

The coefficients (3)Y are not listed explicitly, because they will not

enter into the final expression for the fourth-order Hamiltonian.

VI. SECOND CONTACT TRANSFORMATION FOR THE XYZ-TYPE MOLECULE

In order to cast the Hamiltonian into a form suitable for
energy calculations to the fourth order of approximation, it is neces-
sary to perform a second contact transformation on the once-transformed
Hamiltonian H' so that the twice-transformed Hamiltonian will be diag-
onal to second order in all vibrational quantum numbers. Regrouping
terms in Eqs. (IV-21) to (IV-25) according to true orders of magnitude,

the twice-transformed Hamiltonian takes the following general form:

h5 = h6 = HO (VI-l)
hi = hi = o (VI-2)
h; = hé + i[Z, Ho]v (VI-3)
h; = hé + i[z, HO]R + i[E, hi]v (VI-4)
h; = h; + i[Z, hiJR + %-1[z, hé + hSJV . (VI-5)

The operator 2 is determined by requiring the commutator -i[£, HO]v
to be of a form such that the vibrationally off-diagonal matrix elements

of h' will be removed in (VI-3). This 2 function can be shown to take

2
. 29
the follow1ng form :

z = “BYE q P P P
S S a

20.8.st B Y

49

50

a8 3'

l
+ Z01,825,3' zs 3(qsps, + ps'qs) PaPB

a
+ Zazsfs'fs" zss's" qsqs'qs" Pa

 

 

 

a s's" 1
+ 20.28533" 25 3<qsps.Ps.. + 93.93..qs) Pa
SISIISIII 1
+ 23,338.55... 23 3<qsps.ps..ps... + PS.PS..PS...qs)
SIII
28:8.Ssu;sn' Esslsllz -1-(q ssq lqs "PS III + pSII Isqsqslqsll) '
(VI-6)
where
aBy _ -% aBYS -
2 - As (2)Y, (VI 7)
0823 ’2 aBY ss _ _
=l/(2A S)[(2)Y (2)Yss/h2 1, (VI 8)
a8 5' a k as ss' 2 2 as! _
Es [AS (2)Y + (As,/K ) (2)Y ss' Wl/(A A8,),
3 f s' (VI-9)
A1/2 (A - A , - A u)
a: _ s s s s aYs
ss's" G , n (l + 6 , + 6 u) (2) s's"
s s s 3 58
P
A”, (A , - A - A n) ,
+ s s s s aYs
G n , (l +»6 , -+ 6 , n) (2) ss"
ss 3 - s s s s
P
A2" ( II A - A I) II
+ s s s s ays
G I II (1 + 6 n + 6 u .) (2) SS.
38 s s s s s
1 I II
- [2H2(A A A ")6/G 1 “yss S , (VI-10)

(2)

SSISII

51

 

 

 

a s's" = _ % 2 5
ZS [2(ASAS'AS") /M Gssgsu] (2)0.YS'S"
1
l + 6 , Aé,(A , - A - A u) n
_ s s s s s s ays
l + 68's" M2 G n ' (2) s s
s as
1
1 + 5 u A f<A " - A - A .) .
_ s s a s
l + 6 n , M2 G , ,, (2) S s
s s s as
A30‘s - s' - As") a s's"s
+ (1 + 6 + 6 ) Y .
ss' ss" G (2)
s's"s
(VI-11)
It can be seen that “825 = BaZS , szs = yzzs = O, and xxx: = YYY:
s s s s s
= xxyz = xyyz = “zzz = Yzzz = zxzz - 2Y2: = o. It is allowable
s s s s s

to omit the (4)2P2 and (4)2 r”?2 terms from Eq. (VI-6), since they

constitute fourth-order corrections to the second-order corrected ro-
tational constants, and as such their effects are probably not directly
observable.3 Upon substituting 2 into Eq. (VI-4) we find that the
diagonal matrix elements of h" vanish; the off-diagonal matrix elements

3

of h; and h; contribute to orders of approximation higher than the

fourth.4 Taking these considerations into account, extensive computa-

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

h"

2 o _
0 2a Pa/(Z Ida) I (VI 12)

P P P + 2 “8255]

l
2 Zia,8,y,6 TaBydp a B Y 6 '_Ea, st[a (2)Y ss +M (2)

[qs2 + psz/hzl P P

a B , (VI-13)

h" = P P P P MP
4 ZGIBIYrérern GGBYGEU a P8 Y 6

52

P “P P P (VI-l4)

+ l-Z
4 a,B,Y,6 paBYG B Y 6
where
_ _ 3 a8 avé o o o o _
TGBY5 - 23:1 (asaS )/(2AS I MIBBI 15 5) , (VI 15)
Oa8y6en = [aBY; 6enl' + [(aB)(Y); 6en]'
+ [(aB)(y6); enl' , (VI-l6)
with
, 3 -’2 6 6
[aBY; 6en] = _ %.ZS=1A s/(aByY s +aByas)( enys + ends)
(VI-l7)
“2 ass 3 y
[(68) (y); 6enl' = - T6 23's. (1 + ass.){s u 833')
. (6enys + %_6ena s' ) + ( nss s)(y6eys'
+ §_Y6eas )1 + enss[(asfl )(Zy6Y s'
2_By6a s' aBy s'
+ 3 ) + ( 68 SS.)( Y
+§aeyas3n , (VI-18>
Z3. ass s 768 s'
[<ae)<y6>; enl' = 16 28 s.(1 + ass.){s [< )
en en 3.6n
(2 (2)xss' + Yss' + 3 ass')
ens s' y6 y6 g_y6
+ ( )(2 (2)xss' + Yss' + 3 0‘ss')]

an ass 3' y6 76
+ ”S [( )(2 (2)xss' + Yss'

y6s s' an + 08

y6o‘S
s') + ( (2)x ss' Yss'

)(2

GB
ass,)]} . (VI-l9)

nun) nun:

The paBYG have been studied by M. Y. Chan30 and are found to
be of the form is raBy6(vs + 1/2). However, because they are extremely

complex and not germane to our argument, their explicit form will be

53

omitted. A detailed discussion of the coefficients occurring in h;
has been given by Chung and Parker.3

From Eq. (VI-15), the point group symmetry of the molecule, and

the properties of a28 for the case of planar molecules (i.e., aza - aga

xz 3 ayz
s s

and a = 0), it follows that there are 13 possible nonzero

coefficients T In addition, there are 47 pasY6 and 729 O

aBy6'
3,33

aBy6en'

that 15 of the paBYG and 105 of the 0 have

It has been shown
a6y6en

matrix elements that lie inside the Wang's diagonal block531, custom-
arily denoted by E+, E-, 0+, 0-. It has been shown3 that these matrix
elements therefore contribute to the energy in the fourth order; the

remaining 32 paBY5 and 624 0 fall outside the Wang's blocks and

aBy6en

from this it can be shown3 that they contribute only to eighth order

in the energy. These latter terms can therefore be discarded. An

abbreviated notation for the 105 ea8y6en that are retained is given in
Table 2. In this notation O is abbreviated as e , G as O ,

xxxxxx 1 YYYYYY 2
etc.

Considerable simplifications in the Hamiltonian result from

application of the commutation relations [Pa’ P ] a - ihPY and a

B
judicious redefinition of coefficients.32 The transformed Hamiltonian
consists of three terms of the second power in P with coefficients A,
B, C, six terms of the fourth power in P with coefficients Ti' and ten
terms of the sixth power in P with coefficients 61. The 61 will be

calculated in the next chapter. The basic Hamiltonian to be used for

a given vibrational state can then finally be written a332
H = H2 + H4 + H6 + H6a , (VI-20)

where

54

 

 

 

Table 2. Abbreviated Notation for the Distinct Fourth-Order
Centrifugal Distortion Constants O
aBYGen

1 Oa8y6en l OaBY6en

l (xxxxxx) 41 (xxzzxx)

2 (YYYYYY) 42 (xzxxzx)

3 (222222) 43 (zxxxzx) = (xzxxxz)

4 (xxyyyy) = (yyyyxx) 44 (zxxxxz)

5 (yyxxyy) 45 (zxzxxx) = (xxxzxz)

6 (yxyyxy) 46 (xzxzxx) = (xxzxzx)

7 (xyyyxy) = (nyYYX) 47 (zxxzxx) - (xxzxxz)
8 (xyyyyx) 48 (xzzxxx) - (xxxzzx)

9 (xyxyyy) = (yyyxyx) 49 (xxzzzz) = (zzzzxx)
10 (yxyxyy) - (yyxyxy) 50 (zzxxzz)

11 (XYYXYY) = (YYXYYX) 51 (zxzzxz)

12 (yxxyyy) = (yyyxxy) 52 (xzzzxz) = (zxzzzx)
13 (yyxxxx) = (xxxxyy) 53 (xzzzzx)

14 (xxyyxx) 54 (xzxzzz) = (zzzxzx)
15 (xyxxyx) 55 (zxzxzz) = (zzxzxz)
16 (yxxxyx) - (xyxxxy) 56 (xzzxzz) = (zzxzzx)
17 (yxxxxy) 57 (zxxzzz) = (zzzxxz)
18 (yxyxxx) = (xxxyxy) 58 (xxyyzz) - (zzyyxx)
l9 (xyxyxx) = (xxnyX) 59 (yyzzxx) = (xxzzyy)
20 (yxxyxx) - (xxyxxy) 60 (zzxxyy) - (yyxxzz)
21 (xyyxxx) = (xxxyyx) 61 (xyyzzx) - (xzzyyx)
22 (yyzzzz) - (zzzzyy) 62 (yzzxxy) = (yxxzzy)
23 (zzyyzz) 63 (zxxyyz) - (zyyxxz)
24 (zzzzyz) 64 (yxxyzz) - (zzyxxy)
25 (yzzzyz) = (zyzzzy) 65 (xyxyzz) = (zzyxyx)
26 (yzzzzy) 66 (zyyzxx) - (xxzyyz)
27 (yzyzzz) = (zzzyzy) 67 (yzyzxx) = (xxzyzy)
28 (zyzyzz) = (zzyzyz) 68 (xzzxyy) . (yyxzzx)
29 (yzzyzz) = (zzyzzy) 69 (zxzxyy) - (yyxzxz)
30 (zyyzzz) = (zzzyyz) 7O (xyyxzz) = (zzxyyx)
31 (zzyyyy) - (yyyyzz) 71 (yxyxzz) - (zzxyxy)
32 (YYzzyy) 72 (yzzyxx) = (xxyzzy)
33 (YZYYZY) 73 (zyzyxx) = (xxyzyz)
34 (zyyyzy) - (yzyyyz) 74 (zxxzyy) - (yyzxxz)
35 (zyyyyz) 75 (xzxzyy) - (yyzxzx)
36 (zyzyyy) a (yyyzyz) 76 (yxyzzx) = (xzzyxy)
37 (yzyzyy) = (yyzyzy) 77 (xyxzzy) = (yzzxyx)
38 (zyyzyy) - (yyzyyz) 78 (zyzxxy) = (yxxzyz)
39 (yzzyyy) a (yyyzzy) 79 (yzyxxz) - (zxxyzy)
4O (zzxxxx) a (xxxxzz) 80 (xzxyyz) = (zyyxzx)

Table 2 (cont'd.)

55

 

 

 

 

1 eaBy6en l eaBy6en
81 (zxzyyx) =_(xyyzxz) 94 (xzyyxz) a (zxyyzx)
82 (yxyzxz) = (zxzyxy) 95 (yxzzyx) = (xyzzxy)
83 (zyzxyx) = (xyxzyz) 96 (zyxxzy) = (yzxxyz)
84 (xzxyzy) = (yzyxzx) 97 (zyxyzx) = (xzyxyz)
85 (xzyxzy) = (yzxyzx) 98 (yxzyzx) - (xzyzxy)
86 (yxzyxz) = (zxyzxy) 99 (zyxzxy) = (yxzxyz)
87 (zyxzyx) - (xyzxyz) 100 (yzxyxz) = (zxyxzy)
88 (yzxxzy) 101 (zxyzyx) = (xyzyxz)
89 (zxyyxz) 102 (xyzxzy) = (yzxzyx)
90 (xyzzyx) 103 (xyzyzx) = (xzyzyx)
91 (xzyyzx) 104 (yzxzxy) = (yxzxzy)
92 (yxzzxy) 105 (zxyxyz) = (zyxyxz)
93 (zyxxyz)
H = A P2 + B P2 + c P2 , (VI-20)
2 x y z
= 6 u u 2 2 2 2 2 2
H4 TlPx + TZPY + T3Pz + T4(Psz + PzPy) + T5(Psz
2 2 2 2 2 2 _
+ PxPz) + T6(PxPy + Pypx), (VI 22)
= 6 6 6 2 u u 2 2 6
H6 61px + 62py + 6392 + 64(Pxpy + pypx) + 65(Pypx
+ PHPZ) + 6 (Psz + P“P2) + 6 (P2P“ + PHPZ)
x y 6 y z z y 7 z y y z
+ 6 (P2P“ + PHPZ) + 6 (PZP“ + PHPZ)
8 z x x z 9 x z z x
2 2 2: 2 2 2 _
+ <I>].0(PxP2=Py + PszPx) , (VI 23)
H =D(PP +PP)+3‘-r (P3P +PP3)+lT (P3P
6a x y y x 4 10 x y y x 4 11 y x
3 l. 2 2 _
+ PxPy) + 4 T12(PxPzPy + PszPx). (VI 24)

The coefficients T now appear

with numerical subscripts. These merely

 

 

1'11! Ill-I'll]

56

take the place of the more cumbersome notation T The coefficients

3
aBy6'
of the P2 terms are the effective rotational constants, equal to
equilibrium rotational constants, l/(Zlga), of 0(0) plus second-order
centrifugal distortion corrections to the equilibrium constants, 1,

plus second-order vibrational corrections, 6. There are also terms of

0(4). More explicitly:

A = 1/(21;x) + {i=1 0,: (vs + %) - %M2r9 + 0(4), (VI-25)

B = 1/(21;y) + 22:1 0,: (vs + -:-) - §n219 + 0(4), (VI-26)

c = 1/(2122) + {i=1 a: (v + %) + 341219 + 0(4), (VI-27)
where

a: = (g?xss + Gays + n2(“°yss). (VI-28)

Other terms in Eqs. (VI-20) to (VI-22) are given by:

n = 23 [ xyx + ny + M2(xyyss)1(v§ + %p

s=l (2) ss 33
1 2

' 2'“ (T10 + Tll ' 2 T12) ' (VI-29)
l 2

T1 = K (1:1 + pl) + M (1511 , (VI-30)
_ 1 2

T2 - z-(IZ + p2) + M 012 , (VI-31)
l 2

T3 = Z-(T3 + p3) + M 013 , (VI-32)
l * 2

T4 = Z-(r4 + p4) + M 614 , (VI-33)
1 * 2

T5 = 2 (T5 + ps) + M 015 , (VI-34)
1 * * 2

T6 = Z‘(T6 + 96) + M Q16 , (VI-35)

57

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

* t
to p3, 64 to p6 are the vibrational corrections to T of 0(4), and 6

11
to 616 are the rotational corrections to T of 0(4). The terms pl to
* . 3
p6 are merely the pafiy6 redefined , and the terms 611 to $16 are not

the subject of the present investigation, and their forms will be

* * 'k *
omitted. The quantities T6, 64, ps, and p6 are defined as

T6 = T6 + 2 19 , (VI-36)
* = + + -]-‘- + ~1- (VI-37)
04 D4 D7 2 p10 2 p11 '

* = + + l- + l- (VI-38)
p5 D5 D8 2 p12 2 p13 '

* = + + i- + i- (VI-39)
D6 06 O9 2 p14 2 p15 '

Finally, the coefficients of the P6 terms are the fourth-order centrif—

ugal distortion coefficients, 6 to 6

l 10, which are the subject of this

dissertation. After an extensive rearrangement of terms the coeffi-

cients are found to be given by32:

03 = 83 , (VI-42)

¢ =6+§6 +§6+6 +%0+0 +6 +6 +6,

4 4 5 6 7 8 9 10 11 12
(VI-43)
l 1 1
¢5 = 013 + 2 614 + 2 615 + 016 + 2 017 + 918
+ O + O + G I (VI-44)

58

1 1 l
¢6 ’ 622 + 2 923 + 2 024 + 25 + 2 026 + 027
+ 028 + 029 + 030 , (VI-45)
1 l 1
$7 ’ 031 + 2 032 + 2 933 + 34 + 2 035 + 635
+ 637 + 038 + 039 , (VI-46)
1 l 1
¢8 ' 040 + 2 041 + 2 042 + 43 + 2 044 + 045
+ 046 + 047 + 048 , (VI-47)
1 1 1
¢9 ‘ 049 + 2 0so + 2 051 + 52 + 2 053 + 054
+ 055 + 056 + 057 , (VI-48)
¢lo = 058 + 059 + 060 + 661 + 662 + 063 + 064 + 065 + 666
+ 067 + 968 + 969 + 070 + 071 + 072 + 073 + 074 + 075
+ 076 + 077 + 078 + 079 + 980 + 981 + 982 + 983 + 084
1 l l 1
+ 085 + 986 + 087 + 2 688 + 2 089 + 2 090 + 2 691
+ 2-0 + l-G + 0 + 0 + 0 + 0 + 0 + 0
2 92 2 93 94 95 96 97 98 99
+ 0100 + 9101 + 0102 + 0103 + 0104 + 0105 ' (VI'49’
A full discussion of 61 to 610 is given in the following chapter.
An alternate form of the Hamiltonian called the "H-form"ll'32,

which considerably reduces the computation of matrix elements, is ex-

pressed in powers of P2 and P2 (where P2 a Px + Py

H2 =

A * P2 + B * P2
(-146)x (~M6)y+(c-

2 2 + Pi) and is given by:

3 * 2
2 M6 z , (VI-50)

59

* 4 * 2 * 4 * 2 2 2
H = M P + M P2P + M P + M P P - P
4 1 2 z 3 z 4 ( x y)
i: *
+ M [ P2(P2 - P2) + (P2 - P2)P2] + M [(P2 - P2)2
5 z x y x y z 6 x y
l 4 2 2 4 2 2 5 2
- 2 (P - 2P Pz + P2) + M (P 2 Pz)] , (VI—51)
6 4 2 2 4 6 4 2 2
H6 - H1 P + H2 P Pz + H3P Pz + H4 Pz H5 P (Px - Py)
2 2 2 2 1 6 4 2 2 4
+ - - —- - +
H6[P (Px Py) 2 (P 2P Pz P P2)
2 4 5 2 2 2 2 2 2 2 2 2
+ - - + - + -
h (P 2 P Pz)] H7{P [Pz(Px Py) (Px Py)Pz]}
4 2 2 2 2 4 2 2 2 2
+ H8[P2(Px - Py) + (Px Py)Pz] + H9{[PZ(Px - Py)
+ (P2 - P2)2P2] - (P4P2 - 2P2P4 + P6)
x y z z z z
2 2 2 5 4 2 2 3
+ - - + - -
2 M (P P2 2 Pz)} H10(Px Py) , (VI 52)
where
* 2 2 H + 3
M1 — Ml + M (- Hl - H2 - 6 H9), (VI-53)
* 2 2
M2 - M2 + M (.1 H1 + 6H2 + H6 - 20H9) , (VI-54)
* 2 2
M3 - M3 + M (- 10Hl - 5H2 + 0H9), (VI-55)
*
* 2
M5 - M5 + M ( 2H5 + 4H7 + 2H10), (VI-57)
* 2 4 2 6 I 58
M6 - M6 + M ( Hl + H2 - H9) , (V - )
with
M - 2-(T + T ) + l-T (VI-59)
1 8 1 2 4 6 '
M - - 2-(T + T ) + (T + T ) - l-T (VI-60)
2 ‘ 4 l 2 4 5 2 6 '

«H61

kflh‘

nbhd

5
l—6'(<I>1 + (132) +
15 3 3
- 16 (41 + 42) - 8 (44 + ¢5) + 4 (¢7 + 48)

1
+ —-¢
4

(T1 + T2) +

(T1 - T2) .

#JP'

(Tl + T2) -

(T

1

1

0

- T2)

I

T

3

kflh‘

GHPJ

60

- (T4 + T5) +

1
2'(T4 ' T5) '

(44 + ¢5) ,

l
4 T

I

15 3
16 (41 + P2) + 8 (44 + 45) + (46 + 49)

3
+ 4 ($7 + ¢8)
3
8 (¢1 ' ¢2) ‘
3
§'(¢1 + ¢2) -
3
' 8 (¢1 ‘ ¢2)
.41
16
1
+ 2 (¢7 ' ¢8)

3 l
2 (¢7 + P8) - 2 Q

+

hJP'

bJP'

10 '

GNP!

((1)—(D),

(44 + 45) ,

(VI-61)

(VI-62)

(VI-63)

(VI-64)

(VI-65)

(VI-66)

(VI-67)

(¢4 + ¢5) - (¢6 + ¢9)

(VI-68)

(VI-69)

(VI-70)

l 1
Z (<14 - <15) - 3 (<17 - <18) , (VI-71)

1 1
(¢1 ‘ ¢2) ' 8 (¢4 ' ¢5) ’ 2

I

(¢

6

_ ¢9)

(VI-72)

61

3 1 1
H9 - - 16 (41 + 42) + 8 (44 + 45) + 4 (47 + 48)
_ l. q) (VI-73)
4 10 ’
H =1“, -¢)+l<¢ -<I>) (VI-74)
10 8 1 2 4 4 5 °

The equivalence of the H-form of the Hamiltonian to the ¢-form
can be verified by direct substitution of Eqs. (VI-53) to (VI-74) into
Eqs. (VI-50) to (VI-52) and by rearranging the resulting expressions to
the form Specified by Eqs. (VI-21) to (VI-23). Here, the last term

H6a was omitted; the justification for this will be given later.

VII. THE FOURTH-ORDER CENTRIFUGAL DISTORTION COEFFICIENTS

VII.1 General Expressions for the XYZ-Type Molecule

In this chapter explicit expressions will be given for the ten

fourth-order centrifugal distortion coefficients 4 to ¢ As will

1 10'

be discussed in greater detail later, 4 , ¢ 43, and four combinations

1 2’

of the remaining seven ¢i can be determined from experiment. The
principal information of interest obtainable from these coefficients
concerns the cubic potential constants kss's"' which, it will be re-
called, are the coefficients appearing in the anharmonic portion V3 of

the Taylor series expansion of the potential energy in dimensionless

normal coordinates. For XYZ this takes the form:

3 3 3 2
V ‘ hc (k111 q1 + k222 q2 + k333 q3 + k112 qlqz
2 2 2 2
+ k113 q1‘13 + k122 q1‘12 + k223 q2q3 + k133 q1‘33
+ k 2 + k ) (VII-l)
233 q2q3 123 q1q2q3 '

with the dimensionless normal coordinates qS defined by

_ .214 -
qs- (As/M) QS , (VII 2)

and for V3 in ergs the kss's" are in cm-l. It should be noted from

(VII-1) that for XYZ there are ten cubic potential constants. Yet,

62

63

only seven 41 or combinations thereof are determinable experimentally.
Thus, even in principle, not enough information is available to deter-
mine the full set of cubic potential constants from the fourth-order
centrifugal distortion constants alone. However, cubic potential con-
stants also appear in the coefficients a: which specify the second-order
vibrational corrections to the equilibrium rotational constants. Full
use of both the a: and the 4i thus opens the possibility of obtaining

a complete, consistent, and accurate set of cubic potential constants.
Furthermore, it is observed23 that the a: do not contain those poten-
tial constants kss's" for which 5 # s' # 5". Therefore, for XYZ the

cubic potential constant k is obtainable only through a determination

123
of the 41. For XYx there are six cubic potential constants, and in
principle the full set can be obtained either from the 41 or from the a:.
The basic equations for the 4i are found by substituting the
apprOpriate Oi, as constructed from Eqs. (VI-16) to (VI-l9) and Table

2, into Eqs. (VI-40) to (VI-49). In this way one finds the expressions

for Q to P 0 given in Appendix 1. Next,it is necessary to substitute

1 l
. . as s z
from Appendixes 2, 3, 4, and S the expres31ons for the S , Sss' ,
GB aBY s as aBy s as . .
(2)Xss,, Y r Yss" a , and ass,. While this task appears

hopelessly complex, we found that by judicious and extensive regrouping
and redefining of terms, the ¢i can be expressed in suprisingly compact

form. Introducing the definitions

as = 08 3/4 -
bs aS /As , (VII 3)
a8 a8
B8 = as /As , (VII-4)
_ a8 a8 48 _
(BC)aB - B :23 + 32 g31 + 33 £12 , (VII 5)

64

the following expressions were obtained:

_ 1/2 06 xx
¢l - ("C/4M Ixx) 2855.53" (bybszsfl) kSS'S"
3_ xx xx ,
+ 8 is S, 3:338, (ASS,) , (VII- 6)
_ % .6 yy yy yy
4?2 - (no/4K Iyy) Zsfs'fs" (bs bs'bs") kss's"
2. yy yy yy . _
+ 8 2s,s' BS 88' (Ass') ' (VII 7)
k O6 22 22 22
¢3 _ (WC/4M Izz) Zst'fs" (be s'bs") kss's"
1 2 1 zz 2
+ 2 (BC)ZZ - 8 25 (BS ) . (VII-8)
_ 2 .2 .4 yy yy xx yy yy
44 — (He/8M Ixnyy) Xsfs'fs" (ngbs,bs" + bs,bs b8"
+ bxbeYbyy + 4byybx¥be + 4by¥bxybe + 4by¥bx¥bxy)
s s s s s s s s s s s s
. _§_ YY YY xx u xx YY XY xY
kss's" + 16 Zs,s'{Bs Bs'(Ass') + 2(Bs Bs' + 28s 35')
o _ o
o W l xy yy xy l xy 1 ._l M
(Ass') + 4Bs Bs'[(Ass') + (As's) ]} + 12 122
A8 + AS' xy yy xy yy
' is... Css- 7:77: (B. 38' + 35.3. >, (VII-9’
4 = (WC/8M%I°ZI°4) 2 (bYbexbxx + bYbexbxx
5 yy xx sis'fs" s s' s" s' s s"

+ bYXbxfbxx + 4bxxbx¥bx¥ + 4bx¥bebXX + 4bxfbx¥bxy)
S S S S S S S S S S 8 S

- k u + —3- ,{Bxxsxx(AYY,)' + 2(BYYBxx + zaxYBxY)
ss's 16 5,5 5 8' es s s' s s'

' Io _ IO
. xx 1 XY xx xY ! XY u 1 xx
(Ass') + 4Bs Bs'[(Ass') + (As's) ]} + 12 I22

xxx XXX
(BSYBS, + BS¥BS ), (VII-10)

1
(WC/8K6I°2 °
YY

I 4) Z , " (byybz‘szz + bYszzbzfi
22 $53 :3 S S S S S S

22 22

yy 1_ _ 1_ yy 22
bs"bs'bs ) kss's" + 2 (Bc)yy(Bc)zz 8 23 B5 B5
C
_2_ zz zz yy , _ 1_ SS!
16 28,8' Bs Bs'(Ass') 6 s<s' As - As'

22 xy _ 22 XY _ _
[Bs Bs'(As' 218) + Bs'Bs (As 218,)1, (VII 11)

4 .4 .2 yy yy 22 yy yy 22
(no/8M Inyzz) 2 (bs bs,bS" + bs bsubs,

yy yy 22 l_ 2 _ _l_ yy 2

ss's
C
1. XY 2. 2. YY 22 YY . _ l_ ____3§§L_
4 28(38') + 8 2‘ss' Bs Bs'(Ass') 6 Zs<s' As - As'

[3:YB:¥(AS - 215,) + B:¥B:Y(As, - 215)], (VII-12)

% .4 .2
(no/8h Ixszz) Z

(bxxbxszfi + bxxbxsz?
s s s s s S

$28.53"
xx 22 1 2 1 xx 2
b:¥b3"bs ) kss's" + 4 (B§)xx - 16 25(33 )
C
1_ xy 2 3. 22 xx , _ 1_ ss'
4 Zs<Bs ) + 8 2ss' B:st'(Ass') 6 Zs<s' AS - As'

[3:YB:T(AS - 215,) + B:¥B:x(xs, - 218)], (VII-l3)

% 2 04 x 22 22 x zz zz
(we/8n I;szz) 28:5.55. (bsxbs.bs. + bs’fbS bs"
x 22 22 1 xx 22
bsfibs'bs ) kss's" + 2 (B§)xx(B§)zz - 23 BS 5

1.
8
3 22 zz xx Css'

. 1. ________
Ig'zs,s' s Bs'(Ass') 6 Zs<s' As - As'

66

. zz xy _ zz xy _ _
[B8 BS,(1S, 213) + 38,38 (AS 213,)1, (VII 14)

k O2 O2 2
IO
10 (“C/8” Ixnyy zz) 2553'53"

'9'
ll

(bxxby¥bzf + bxbesz?
S S S S S S

+ bxfby¥bzz + 6"”bebez + bxfbyybz? + bxfbyfibzz
S S S S S S S S S S S S

+ 4bxybx¥bzf + 4bxybebz? + 4bx¥bx¥bzz) k
S S S S S S S S S

ss's"
l_ 2 1_ xy 2 _ 1 x yy
+ 2 (mummy), + (Roxy + 4 28(35) 8 is 3:38 2
I0 _ IO 0 _ 0 IO _ IO '
_ 1_2 22 xx Bxx + xx 22 Byy + xx xx B22
4 8 1° 3 I° s I° 8
xx yy 22
g, 22 xx yy . yy xx .
+ 8 Zss'Bs'lBs (Ass') + Bs (Ass') 1
§_ 22 XY XY . KY .
+ 4 Xss'Bs'Bs [(Ass') + (As's) ]
I° - 1° 1 , - 31
_ 1. xx 2 C s a. Bxszz
4 1° sfs' 38' A A , (s s'
22 s - s
1 1° - 1° 1’ ' xx
_ _. _EE_:__XX. Z ' C ' ____§___. BXYB '
2 I s#s ss A , - A s 5
xx 5 s
1 I§x - I22 1 A8' xy yy
- 2' I°~ - Zsrfs' Css' A '-- 1 Es Bs"<v;I-15)
yy s s

Each of the above 41 has a set of terms linear in the cubic
potential constants kss's" with coefficients that are symmetrized
products of the bzs. In addition, there occur terms independent of
the kss's"' These terms depend only on the parameters of the harmonic
vibration problem and on the equilibrium geometry. Thus, the fourth-

order centrifugal distortion coefficients may be regarded as resulting

from the sum of two contributions: a harmonic one, and a cubic

67

anharmonic one. The frequency resonance terms of the form

l/(As - 18,), which are introduced by the contact transformations and
profusely sprinkled throughout the coefficients S, y, and a, are found
to completely drOp out from the kss's" terms, whereas in the k-inde-
pendent terms they either drOp out or combine in a systematic fashion.

As was mentioned at the beginning of this chapter, the point of
view may be taken that the equilibrium geometry of a particular
molecule and the parameters of the normal vibrations problem are known
through the zero and second order part of the analysis of the high-
resolution data, and therefore the only unknown parameters occurring
in the fourth-order centrifugal distortion constants are the full set
of cubic potential constants of the molecule.

Upon inspection of the above results it becomes apparent that
there exist extensive regularities which have heretofore gone totally
unnoticed. The occurrence of these regularities gives one considerable
confidence in the correctness of the calculation. Why these regulars
ities appear only at the very last step of a very extended calculation
is hard to say. The possibility exists that the contact transformation
technique is not the most efficient way of dealing with the expanded
Darling-Dennison Hamiltonian, but this is conjecture. Whether the
coefficients pi and the remaining 4 through 4

(i.e., Pl 6' which

i 1 1
occur as part of the coefficients Ti) will similarly reduce to much
simpler expressions than those which emerge from the contact transfor-

mation is a question that remains to be answered through future work.

68

VII.2 Specialization to the Case of XYX-Type Molecules

 

The results of the previous section can be reduced to the

Special case of the XYX-type molecule and compared to earlier work.11

Taking z-x implies that the masses m2 and m3 are equal, hence we take

m2 = m3 = m, m1 = M, and also a = b, 6 = 0. Introduction of these
relations greatly simplifies the normal vibrations problem, which is
now soluble in closed form. The calculations are described in detail
by Chung and Parkerlo, and the labelling of the normal modes in their
work is such that s=l gives the symmetric bond-stretching mode, s=2 the
bending mode, and s=3 the asymmetric bond-stretching mode. This label—
ling is the customary one for XYX molecules.

To make a complete comparison of our XYZ results with those of

B

Yallabandi and Parker for XYX, the detailed expressions for b: =

3/4 a8 _ a8 a8 , as
s , Bs - as /As, (Ass') , and Css' are needed. The as and ass,

as

as /A
are tabulated in the paper of Chung and Parkerlo, and their results
are reproduced in Table 3. In particular, it is noted that 512 = O,

2
and (:23) + (:31)2 = 1. The (A::,

)', also listed in Table 3, can

be calculated from their definition (Eqs. (II-27), (II-28), and (II-30))
and from the values of the coefficients 2:8 for XYX which can be

found in Table III of Reference (10).

Since the expressions for the coefficients 4 through 4 for

1 10

XYX can be obtained in a simple and straightforward way from those for
XYZ, we shall not cite them separately. Upon comparison with the ex-
pressions given by Yallabandi and Parkerll, complete agreement was

found except for one discrepancy: in Table IV of Yallabandi and Parker,

69

8

GB

 

 

 

 

Table 3. The Nonvanishing Coefficients a: , §:§" and (Ass,)' for XYX
xx = 0% . xx 3 0%
al 21xx SinY a2 ZIxx cosY
yy 01/2 yy 01/2.
al - 21yy cosY a2 - ZIyy SlnY
zz = xx yy = _ 0% Z zz xx yy 0% 2
a1 a1 + a1 2I22 523 a2 a2 + a2 2122 C13
xv,” . . .’/2
a3 2(Ixnyy/Izz)
xx , = . 2 yy , . 2
(All) Sin Y (All) cos Y
xx , = 2 yy . = - 2
(A22) cos Y (A22) Sin Y
xxt=o o YYI=O 0
(A33) Ixx/Izz (A33) Iy Izz
xx . = - YY . = _ -
(A12) sinY cosY (A12) s1nY cosY
zz'gzz xY': yx'=-° 07/2
(A11) (:23) (A13) (A31) (Ixx/Izz) °°SY
zzlgzz XYU= yxn=_o 01/2°
(A22) (C13) (A31) (A13) (Iyy/Izz) einY
zz , = xy , = yx , = o o k .
(A33) 0 (A23) (A32) (Ixx/Izz) SlnY
zz . = 22 . = _ z 2 xy . _ yx . = _ . o 4
(A12) (A21) 513 C23 (A32) (A23) (Iyy/Izz) cosy
:2 = (I° I° )gcosY - (I° I° )gsiny = - :2
13 xx/ 22 yy/ 22 . 31
gz = -(I° I° )gsiny - (I° I° )gc05Y = - C2
23 xx/ 22 yy/ 22 32

 

70

which gives 4 0, the term

1

3/ZI°3/21
x yy zx xy

- (2ql sinY cosy)/(I; )

should be changed to read

°3/21°3/21 I

-.(2q1 SlnY cosy)/(Ixx yy zx yz .

It should be noted that the expressions contained in this thesis are
much to be preferred over those of Yallabandi and Parker, since they
were not cognizant at that time of the regularities in the 4i found
in the present work, and they therefore gave the expressions for the

4i in what must now be considered an extremely awkward form.

71

VII.3 Determinable Combinations of Constants

 

Watsonzz'34

has shown that not all constants occurring in the
vibration-rotation Hamiltonian can be determined from experiment. Some
constants are completely indeterminable and must be omitted from the
Hamiltonian. In our case this is true of the constants D, 110, Ill,

and 112, and hence the part of the Hamiltonian we have designated as

Hea.was dropped from further consideration as described in Chapter VI.
Concerning the fourth-order centrifugal distortion constants

4 through 4

l , Watson's work established that there exist among these

10
constants three relations, and as a consequence only seven determinable
combinations of these constants can be found from experiment. Intro-
duction of the apprOpriate constraints leads to what Watson has called
the reduced Hamiltonian. The reduction of the Hamiltonian, furthermore,
can be carried out in an infinite number of ways but not entirely
without restrictive conditions.

Watson's theory is based on the observation that two Hamilton-
ians H and H have identical eigenvalue spectra if they are related by
a unitary transformation U such that H = U-IHU. In general, the unitary
transformation is arbitrary. Watson's work demonstrates that choice of
a particular U corresponds to a particular reduction of the Hamiltonian.

22'34 should be con-

For the details of the theory the papers of Watson
sulted. The theory was also discussed by Yallabandi and Parker11 and
by Ford, Yallabandi, and Parker)35 Practical experience with this

theory in analyzing vibration-rotation data has been described by Ben-

edict, Clough, Frenkel, and Sullivan36, and by Kirchhoff.37

VIII. SUMMARY

As described in detail in this dissertation, the fourth order
of approximation of the vibration-rotation Hamiltonian of asymmetric

rotator molecules can be written in the form

H = 4 P6 + 4 P6 + 4 P6 + 4
X Z

6 1 2 y 3

2 4 + u 2
4(PxPy Pny)

+ 4 (P2P“ + PQPZ) + 4 (P2P1+ + P”P2) + 4 (P2P1+ + PHPZ)
5 y x x y 6 y z z y 7 z y y z

2 H k 2 2 4 4 2

+ 48(PzPx + PxPz) + 49(PXPz + Psz)

2 2 2 2 2 2
+ 410(PXFZPY + PszPx)

unless accidental resonance effects invalidate the treatment of the
expansion of the Darling-Dennison Hamiltonian by the contact transfor-
mation technique.

This dissertation presents a calculation of the expressions for
the fourth-order centrifugal distortion coefficients 41 through 410 in
terms of the fundamental molecular constants for the case of the bent
triatomic XYZ-type molecule. The fundamental molecular constants
comprise those that specify the equilibrium geometry and the normal
vibrations of the molecule, and the full set of cubic anharmonic poten-

tial constants. Even though the calculational details are massive,

the final expressions were found to have a relatively simple and

72

73

surprisingly compact form that exhibits many cyclic regularities. As
a consequence it is felt that these results can now be used in data
analysis with considerable confidence that they have been calculated
correctly.

through 4

The expressions for 4 for XYZ were specialized

l 10
to the XYX-type molecule, and the XYX expressions found in this way
agreed with those that were available in the literature.

It is hoped that the present work will ultimately contribute
toward the accurate and unambiguous determination of the full set of

the cubic potential constants of triatomic molecules, which has

turned out over the last forty years to be a very elusive task.

LIST OF REFERENCES

10.

11.

12.

13.

14.

15.

16.

17.

LIST OF REFERENCES

W. S. Benedict, Phys. Rev. 22, 1317 A (1949); P. M Parker and
L. C. Brown, J. Chem. Phys. 31, 1227 (1959).

For example, M. L. Grenier Besson, J. Physique Rad. 31, 555 (1960).
K. T. Chung and P. M. Parker, J. Chem. Phys. 38, 8 (1963).

G. Amat and H. H. Nielsen, J. Chem. Phys. 39, 1859 (1962).

J. K. G. Watson, Mol. Phys. 15, 479 (1968).

M. Goldsmith, G. Amat, and H. H. Nielsen, J. Chem. Phys. 24,
1178 (1956).

M. Goldsmith, G. Amat, and H. H. Nielsen, J. Chem. Phys. 21,
838 (1957).

G. Amat and H. H. Nielsen, J. Chem. Phys. 21, 845 (1957).

G. Amat and H. H. Nielsen, J. Chem. Phys. 32, 665 (1958).

K. T. Chung and P. M. Parker, J. Chem. Phys. 43, 3869 (1965).

K. K. Yallabandi and P. M. Parker, J. Chem. Phys. 42, 410 (1968).
E. B. Wilson, Jr. and J. B. Howard, J. Chem. Phys. 4, 262 (1936);
B. Podolsky, Phys. Rev. 22, 812 (1928); B. T. Darling and D. M.

Dennison, Phys. Rev. 51, 128 (1940).

W. H. Schaffer and R. P. Schuman, J. Chem. Phys. 13, 504 (1944);
59, 4124 (1969).

H. H. Nielsen, Rev. Mod. Phys. 33, 90 (1951).

D. W. Posener, Ph.D. Thesis, Massachusetts Institute of Technology,
Cambridge, MA, 1953.

D. Kivelson and E. B. Wilson, Jr., J. Chem. Phys. 39, 1575 (1952).

T. Oka and Y. Morino, J. Phys. Soc. Jap. 16, 1235 (1961).

74

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

75

J. M. Dowling, J. Mol. Spectrosc. g, 550 (1961).

T. Oka and Y. Morino, J. Mol. Spectrosc. g, 472 (1961).

P. M. Parker, J. Chem. Phys. 22, 1596 (1962).

R. A. Hill and T. H. Edwards, J. Mol. Spectrosc. 44, 203 (1964).
J. K. G. Watson, J. Chem. Phys. 44, 1935 (1967).

M. Y. Chen and P. M. Parker, J. Mol. Spectrosc. 43, 53 (1972).
C. Eckart, Phys. Rev. 41, 552 (1935).

M. Y. Chan, L. Wilardjo, and P. M. Parker, J. Mol. Spectrosc. 49,
473 (1971).

J. H. Meal and S. R. Polo, J. Chem. Phys. 44, 1119 (1956).

G. Amat and L. Henry, Cahiers Phys. 43, 273 (1958).

J. H. Van Vleck, Phys. Rev. 22, 467 (1929).

R. C. Herman and W. H. Schaffer, J. Chem. Phys. 44, 453 (1947).

M. Y. Chan, Ph.D. Thesis, The Ohio State University, Columbus, OH,
(1970).

S. C. Wang, Phys. Rev. 34, 243 (1929).

F. X. Kneizys, J. N. Freedman, and S. A. Clough, J. Chem. Phys.
44, 2552 (1966).

K. T. Chung and P. M. Parker, J. Chem. Phys. 43, 3865 (1965).
J. K. G. Watson, J. Chem. Phys. 44, 4517 (1968).

L. H. Ford, K. K. Yallabandi, and P. M. Parker, J. Mol. Spectrosc.
42, 241 (1969).

w. S. Benedict, S. A. Clough, L. Frankel, and T. E. Sullivan,
J. Chem. Phys. §§, 2565 (1970).

W. H. Kirchhoff, J. Mol. Spectrosc. 44, 333 (1972).

APPENDIXES

APPENDIX 1

THE FOURTH-ORDER CENTRIFUGAL DISTORTION CONSTANTS 4i FOR XYZ MOLECULES

2 xx 1 2 xx xx ) xxs3)2

-.l
4 — 2 u {( s ) (2(2)x11+Y11’ + ( xxs 2) 2(2(2)x 22+ y22 + (

xx xx xx 1 xx xx 1
-(2 x + )+(S)(xxzs)(2(2)x+l2 yl2)+(S)

(2) 33 Y33

xx xx 2 xx 3
(xxs 3>(2(2)x13 + y13) + ( s )( s )(2(2)x 23+ xxy23)} (1—1)

_ ;_ 2 yy 12 yy yy yy2 yy yy 3 2
4 — 2 K {( s ) (2(2)x11 + 711) + ( S ) 2(2(2)x 22+ y22) + ( s )

-(2YYx +YYY 3312)+(YYS)(YYS)(2YYX +W )+(YYsl)

(2) X33 (2) 12 Y12

yy yyx My yy2 yy3 yy yy _
( s 3)(2(2)x13+ 13) + ( s )( s )(2(2)x23 + v23)} (1 2)

_ _ 4_ 3 -% zzz s 2 _ 4_ 2 zzz 1 22 2 z zz 3
4 - 2 K 25:1 15 < y > 4 n {( H)[( s 2)( s ) + < $23)( 8 )1

+ (zzz 2)[(z s 2)(zzs1) + (2323)(zzs3)1 + (zzzy 3)[(z sl 3)(zzsl)

4_ 1,2222{(zzsl)2(2 + 22 ) + (225 2)2

)(z 25 2’1} + (2) x11 Y11

z
+ ( 823

2 (2(2fx22+ zzy22)+ (zzs3)2(2(2)x 33 + zzy32> + ("51Hz zs2)

zzx zz zz l zz
° (2(2)x 12 + Y12) + ( 5 )(z 25 3”20)" 13 + Y13)}

+ (2282Mzz s3 )(2(2)x 23 + zzv23) (193)

76

hJH

bdhl

+

77

122{(_yysl)2(2 xxX xxyll) + (yysz)z(2 xxX + xx

(2) 11 + (2) 22 Y22)

yy 3 2 xx xx yy 1 yy 2 xx xx
( S ) (2(2)X33 + Y33) + ( S )( S )(2(2)X12 + Y12)

YY 1 YY 3 XX XX yy 2 yy 3 xx xx
( S )( S )(2223X13 + Y13) + ( S )( S )(2(2)X23 + Y23)

xx 1 yy 1 xy 1 2 YY YY
2[( S )( S ) + 2( S ) ](2(2)X11 + Yll)

2[(xx82)(yy52) + 2(xysz)21(2 yyx + YY

(2) 22 Y22)

xx 3 yy 3 xy 3 2 yy yy
2[( S )( S ) + 2( S ) ](2(2)X33 + Y33)

[(xxs1)(yysz) + (xxs2)<yysl) + 4<xysl)(xysz)1(2 YYx + YY

(2) 12 712)

xxsl xxs3)<yysl) + 4<xysl)<xys3)1(2 yyx YY )

yy 3
)( S ) + ‘ (2) 13 + Y13

[(

[(xxsz)(yys3) + (xxs3)(yysz) + 4(“Ysz)(xys3)](2 yyx + YY

(2) 23 Y23)

8(xysl)(yysl)(2 xyx XYY11) + 8(xy52)(yy82)(2 xyx + XY

(2) 11 + (2) 22 Y22)

8(xys3)(yys3)(2 xyx + XY

(2) 33 Y33)

xy 1 yy 2 xy 2 yy 1 xY xY 3-xY
4[( s )( s ) + ( s )( s )](2(2)X12 + le + 3

xy 1 yy 3 XY 3 YY 1 xy xy 2_xy
4[( s )( s ) + ( s )( S )1(2(2)X13 + Y13 + 3 13

4I<xysz)<yys3> + (“Ys3><yys2)1<2 xYx + “Y

(2) 23 Y23 +

wlw

xy(123)} (1-4)

M2{(xxsl)2(2 yyx yy ) + (xxsz)2(2 yyx + yy

(2) 11 + Y11 (2) 22 Y22)

(““53)2(2 yyx yy ) + (xxsl)(xxsz)(2 yyx + YY

(2) 33 + Y33 (2) 12 Y12)

(”‘81) (”S31 (2 W

yy 3 + (xx82)(xxs3)(2 yyx + yy

(2) 23 Y23)

(2)X13 + Y13

78

2[(“xsl)(yysl> + 2(xysl)21<(§fxll + xxy13>

2[(xxSZHny2) + 2(x3732)2]( xxx + xx )

(2) 22 Y22
xx 3 yy 3 xy 3 2 xx xx
2[( S )( S ) + 2( S ) 1((2)X33 + y33)
xx 1 yy 2 xx 2 yy 1 xy 1 xy 2 xx xx
[( S )( S ) + ( S )( S ) + 4( S )( S )1(2(2)Xl2 + 712)
xx 1 yy 3 xx 3 yy 1 xy 1 xy 3 xx xx
[( s )( s ) + ( s )( s > + 4( s )( s )1 2(23x33 + yl3)
xx 2 yy 3 xx 3 yy 2 xy 2 xy 3 xx xx
[( S )( S ) + ( S )( S ) + 4( S )( S )1 2(2)X23 + y23)
8(xysl)(xxsl)(2 xyx + xy ) + 8(xysz)(xxsz)(2 xyx + xy )
(2) 11 Y11 (2) 22 Y22

xy 3 xx 3 xy xy
8( s >( s )<2(2)x33 + y >

33

4t<xys11<xxs2) + ("Yszum‘slmz xyx + “Y

2 xy
(2) 12 Y12 + 3 )

xy 1 xx 3 xy 3 xx 1 xy xy
4[( s )( s > + ( S )( S )1 2(2)X13 + Y13 +

xy 2 xx 3 xy 3 xx 2 xy xy 2 xy
4[( s )< s ) + ( s )( s )1 2(23x23 + y23 + 3- a23)} (1-5)

3 -% zzz s yyz s yyz s _ 1_ 2 zzz 1 z yy 2
M 25:1 AS < y )< y + 2 a > 2 M {( y )[( $12)( s >
z yy 3 zzz 2 z yy 1 z yy 3
( 813)( S )1 + ( y )[( $12)( 8 ) + ( 523)( S )1

zzz 3 z yy 1 z yy 2
( Y )[( 813)( S ) + ( S23)( S )1}

fi-M2{(szyl + g-sza1)[(zsl )(zzsz) + (251 )(zzs3)1

2 3

(YYZYZ + )(zz 3

S )1

(nhb

yyz 2 z 22 1 z
a )[( $12)( S ) + ( S23

(YYZY3 + g-sza3)[(zsl3)<zzsl) + (2323)(zzsz)1}

+

79

l_ 2 zz 1 2 yy yy zz 2 2 yy yy
4 K {( s ) (2(2)x11 + Yll) + ( s ) (2(2)x22 + yzz)

(zzs3>2(2 yyx + YY ) + (zzsl)(zzs2>(2 yyx + YY

(2) 33 Y33 (2) 12 112)

zz 2 22 3

(zzs1>(zzs3)<2 yyx + YY ) + ( s )( s )(2 yyx + YY )

(2) 13 Y13 (2) 23 Y23
2(YY51)(zle)(2(:2xll + zzyll) + 2(YYSZ)(ZZSZ)(2(:?X22 + zzYzz)
2(YYS3)(zzS3)(2(::X33 + zzy33)

[(YYSl)(zzsz) + (yy82)(zzsl)](2(::x12 + zzle)
[(YY81)(zzS3) + (yys3)(zzsl)](2(::xl3 + 22713)
[(yysz)(zzs3) + (YYs3)(zzsz>1(2 22x + 22 >} (1-6)

(2) 23 Y23

3 -% yyz s yyz s 2
K {8:1 AS [ Y + 2( a )1

$_ 2 yyz 1 2_yyz 1 z yy 2 z yy 3
4 M {I Y + 3( a )1[( 812)( S ) + ( $13)( S )1

yyz 2 é_yyz 2 z yy 1 z yy 3
+ [ Y + 3( a )1[(.512)( S ) + ( S23)( S )1

yyz 3 g_yyz 3 z yy 1 z yy 2
+ [ Y + 3( a )1[( $13)( S ) + ( $23)( S )1}

_:L1_222{(yysl)2(2 zzX + zzyll) + (yysZ)2(2 zzX + 22 3

(2) 11 (2) 22 Y22

22x + 22

(2) 12 112)

yy 3 2 22 zz yy 1 yy 2
( S ) (2(2)X33 + y23) + ( S )( S )(2

22
22x +

yy 1 yy 3 22 zz yy 2 yy 3
( S )( S )(2 X + 713) + ( S )( S )(2(2) 23 y23

(2) 13 ’

2(YYsl)(zzs1)<2 yyx + ylel) + 2(YY52><zzs2)(2 yyx + YY )

(2) 11 (2) 22 Y22

2(sz3)(zzs3)(2 yyx + YY

(2) 33 133)

80

yy 1 22 2 yy 2 22 l yy yy
[( S )( S ) + ( S )( S )](2(2)X12 + le)

[(YYs1)(zzs3) + (YYs3>(zzs1)1<2 yyx + YY

(2) 13 Y13)

[(yysz)(zzs3) +<sz3)(zzs2)1<2 yyx + YY

(2) 23 723)} (1-7)

8)]2

3 -% xxz s xxz
K 25:1 15 [ y + 2( a

fi-M2{txxzvl + g- )(xxs3)1

xxz 1 z xx 2 z
( a )1[( $12)( S ) + ( 813

+ [xxzvz + gxxxza2)1[(zsl2)(xxsl) + (2523)<xxs3>1

+ [xxzy3 + §4xxza3)1[(zsl3><xxsl> + (2823)<xxs2>1}

i’KZ {(xxsl)2(2 zzx + zzY ) + (xxSZ)

2 22 22
(2) 11 11 (2 x +

(2) 22 Y22)

xx 3 2 zz zz xx 1 xx 2 zz zz
< s ) (2(23x33 + v33) + ( s )( s )(2(2)x12 + v12)

zz xx 2 xx 3 22 22
) + ( S )( S )(2(2)X23 + y23

(xxsl)(xxs3)(2 zzx +

(2) 13 Y13 )

zsl)(2 xxx + xxYll) + 2(XXSZ)(ZZSZ)(2 xxx + xx

xx 1 2
S )( (2) 11 (2) 22 Y22

2( )

xxSB) (zzSB) (2 xxX xx )

2‘ (2) 33 + Y33

[(xxsl)(zzsz) + (xx52)(zzsl)](2 xxx + xx

(2) 12 Y12)

[(xxsl)(zzs3) + (xxs3)(zzsl)](2 xxx + xx

(2) 13 Y13)

283) + (xxs3)(zz82)](2 xxx + xx )}

xx 2 z
[( S )( (2) 23 Y23

(1-8)

3 -% zzz s xxz s xxz s
K {3:1 A3 ( v )< y + 2 a )

l 2 xxx 1 2 xx 2 z xx 3
g-n {( v )[( 512)< s ) + ( 513" s )1

+

+

+

+

+

+

+

+

+

+

(zzzY 2

)[( 251

zzz 3

(

1 2
4 M

22

(

(“81Hzz s 3)(2

2(

2(

[(

[(

[(xxs2)(z 283) + (xxs3><"sz)1(2

3
- h Zs=1 A

xxsl)(zz
xx83)(zz
xxsl)(zz

xxsl) (z

y)[( 231

1'M2{[xxzyl +

{(22 1 2

S3)2

(2 xxx

5 ) (2

S 3)(2

2)(

3)(

§_xxz

3(

+ 4 xxz 2

3(

xx 1

s ) + ( zs

xxsl) + (ZS

4 xxz 3

'4

(2)

+

(2) x33

(2)

S 1)(2

(2

(2

52) + (

253) + (

xx
x11+

XX
Y33

XX

x13 + Y

22
+

) X11

22
+

) X33
xxsz)(zz

xxs3)(zz

+ [2(‘3’z 5) + (xyzas>'12}

- fi-K2{tyyzvl

+ [yyzvz

23

23

1 z
a )1[( $12
a)1[( le

z
a )1[( $13

Y11

l3

Y33

+ §4YY232>11<ZS

8

)(

)(

)

2)

)

)

> + ("52Hzz s 3)(2

)

S 1)](2

S 1”Q

12

l

xx83)]

xx52)]

22 2

(

(22

(22

+

+ §4szal>lt<zs )<

13)(

(

) + (“slnzz s 2)(2

711) + 2‘

-1
é{2[yyzYs + 2(yyzasntxxzvS

xx 2

XX

S ) + ( 2s
31) + (25
$1) + (25

22 2 2

S ) (2

xx52)(zz

+

(2) X12

+

(2) X13

+

(2) X23

+

S )

81)

23

23

(2) x22

(2) x12+

( S

( S

l3)<zzs3)1

)(zzs3>1

)(zzs2)1}

XX

+ Y22)

XX

Y12)

XX )
(2) x23 Y23

22
+

$2 )(2(2)zx 22

)

(1-9)

l3)(""s3)1

23)(xxs3)1

+

+

”(xx

+[(

+[(xx

+

+

+

+

+

1 2
Z‘M {2[(

2(YYS1)(z

82

4 xxz l

yy 2 2
+3( a)][(zlS 2H S)+(S1

4 xxz a2

[ v +3( )1[(zlszuyys1) +("1s2

[ Y + éixxz

3 z yy 1 z
3 a)][(813)( S) +(S2

1

xyzY l+ _(
3

4[ "YZ a1>'211[(s

4[xyzY 2 + :(xyza 2) 11(281

3 3
4IXYZY+ +§<xyza

) m 215
xxs1)(Ws1) + 2(1‘Ys1)21(2

+ 2[(""s2)(yys2> + 2(1‘5’52 ) 21(2

(2) X22

+ 2[(“s3>(yys3) + 2(1‘Ys3>21(2

(2) X33

x"s1HYYs3> + <
$2“sz ) + ("x

XX
25 1)(2(2)X 11

XX

+ Y33

yy 3
2( S)(Z Z3S)(2(2)x33 )

[(yys1)(zzs2) + (YYs2)("s1)1(2

(2) X12+

[(YYs1)(zzs3) + (YYs3)(“s1)1(2

(2) X13+

[(yysz)(zzs3) + (YYs3><"sz)1(2

(2) X23+

xy 2 z
2)( S) + ($13
2)<st1> + (""s1

3)(st1) + (252

(2) x11

51“ny > + (""s2 )(sz1) + 4<"Ys1)("Ys2)1<2
xxs3)(Ws1) + 4("Ys1)("ys3)1<2
s32>(YYS) + 4(st23)<"¥$)1(2

v11) + 2(YYSZHz

§<YYza3)1[<zls3)(""s1) + (ZSZ3HXXSZH
yy 3
3H s )1

3) (ny 3)]

yy 2
3)( S )1

><xys3n
3) (”s3n
3) (W52) 1}

+ ZZY )

+ zzv )

22
4.

+22 )

(2) X12 Y12

+22 )

(2) x13 Y13

22

(2) X23 23

XX

zS 1 (2 Y22

(2) x22+

+1“)
Y12
XX)
Y13
XX

Y23)

83

2
2511(2 YY + yyyll) + 2(xxs )(zzsz)(2

xx 1 2
S )( (2')}(11

YY )

2( + y22

(2)X22

xx 3 22 3 yy yy
2( S )( S )(2(2)X33 + Y33)

282) + (xxs2)(zzs1>1(2 yyx + yyy 1

xx 1 z
1‘ S 1‘ (2) 12 12

[(xxs11(zzs3) + (xxs3>(zzs2)1(2 yyx + yy 1

(2) 13 Y13

xx 2 zz 3 xx 3 ~22 2 yy yy
11 s )1 s 1 + ( s >( s )](2(2)x23 + v23)

xy 1 22 1 xy xy xy 2 22 2 xy xy
8‘ S )1 S )(2(2)x11 + Yll) + 81 S )1 5 112(21x22 + 122)

xy 3 zz 3 xy xy
8( S )1 S )(2(2)x33 + 733)

1%

xy 1 zz 2 xy 2 zz 1 xy xy
411 s )1 s 1 + < s )1 s )1 (2(2)x12 +

out)

y +

12 12

$5

xy 1 22 3 xy 3 zz 1 xy xy
4[( S )( S ) + ( S )( S )1 (2(2))!l3 + Y13 +

ads)

13

$2

41<xys21(zzs3) + (“Ys3)(zzsz)1 (2 “Y xy +

(21X23 + Y23 )1

wlw

23

(1-10)

APPENDIX 2

 

 

 

01
THE COEFFICIENTS (2)8)(83' APPEARING IN H2 AND IN (bl THROUGH (I110

= xx , 2 o 2 xx _ xx , '4 o 2
(21311 3 ”(311) /12 11(1xx) 1 (21x12 ‘ 3 ”(312’ /(1112) (Ixx)

_ xx I 14 o 2 xx _ xx , 1/4 o 2
(gfxzz _ 3 M<A221 /12 12‘1xx’ 1 (2)x13 - 3 x<Al3> /<Alx3) (Ixx)
xx _ xx , 1/2 o 2 xx = xx , ’4 o 2
(2)X33 - 3 ”(A33) /[2 33(Ixx) 1 (2)X23 3 fi(A23) /(1213) (Ixx)
yy = yy . 2 o 2 yy g yy . 2 o 2
(2)x11 3 “(A111 /[2 31(Iyy) 1 (2)X12 3 ”(A12) /(A112) (Iyy)
YY = YY . 2 o 2 yy _ yy . 2 o 2
(2)322 3 ”(322’ /[2 12‘1yy) 1 (21X13 ' 3 ”(A131 /(1113’ (Iyy)
W = W ' 1/2 0 2 W I W I 14 o 2
(2)x33 3 M(A33) /[2 A3(Iyy) 1 (2)x23 3 h<A23) /(AZA3) (Iyy)
22x = 3 M( 2 13/12 A311° 121 22x - - 3M( 2 2 )/[<1 1 131°2
(2) 11 C23 ' ' 1 22 (2) 12 ‘ ' z:13123 - 1 2 22]
22x = 3 M( 2 12/12 A31I° 121 22x - - 3M( 3 z 1 [<1 1 131°21
(2) 22 113 2 22 (2) 13 ’ 112123 / 1 3 22
22x = 3 n( z )2/[2 13(I° )3] 22x = - 3M( Z z ) [(1 A 131°21
(2) 33 ‘12 3 zz (2) 23 512113 / 2 3 22
xy = X? . 2 o 0 xy 3 xy xy 2 o o
(21X11 3 M(A11) /[2 31 Ixxlyy] (2)x12 3M(A12+A21)/[2(1112) Ixnyy]
xy ~ g xy . 2 o 0 xy _ xy xy 2 o o
(2)322 3 ”(322) /[2 32 Ixnyy] (2)313 ‘ 331313+331)/12(1133’ 3xx1yy1
xy ~ = xy . 2 o 0 xy _ xy xy 2 o o
(2)333 3 ”(333) /[2 A3 Ixnyy] (21X23 ' 331323+A32)/12(1213) Ixnyy]

84

zz 1

APPENDIX 3

COEFFICIENTS OF THE S-FUNCTION APPEARING IN ¢ THROUGH ¢10

l/[2(M2Al)3/4(I;x)2] axx xxs2

/

1/[2<M213>3 4(I;x)21 a

2 3/4 0 2 yy yy 2
l/[2(M Al) (Iyy) 1 al 8

/4

2 3 O 2 yy
1/[2(M A3) (Iyy) J a3

/

1/[2<K2A1)3 4(I;z)2] azz zzsz

/

l/[2(M2A3)3 4(1;2)21 a

/4

1/[2(x211)3 I;xI° 1 axY xys2

yy 1

l/[2(M2A3)3/4I;xl;y] axy

'4 o

[1/(11A2)1 (muml + 12)/(Al -
l A A 2 ° A A A

1 /< 1 3)] (l/Izz)( 1 + 3m 1 -

2 o
[1/<1213)1 (mum2 + A3)/(12 -

85

1

1/[2<n212)3

1/12<2212)3

1/[2(M2A2)3

A ) z:12

2
A3) C13

A ) c2

/

/

/

4(I° )2]
XX

4 2
IO
( yy) 1

4 o 2
(122) ]

1/[2(M2A2)3/4I;x1;yl

m

APPENDIX 4

THE COEFFICIENTS a APPEARING IN i[S, HO]R AND IN $1 THROUGH ¢lO

xxz 1 _ o _ o 2 3 k 02 o 0 xy
a - (Ixx Izz)/[4(H Al ) 'IxnyyIzz] a1

XXZ 2 _ o _ o 2 3 1/4 02 o 0 XY
a (Ixx Izz)/[4(K A2 ) IxnyyIzz] a2

xxz 3 2 xy

_. O 0 2 3 1/4 0 O O
a — (xxx Izz)/[4(M 13 ) IxnyyIzz] a3

YY21= o _ o 2 314 o 020 XY
a (Izz 1yy)/[4(x1l ) Ixnyylzz] a1

yyz 2 = o _ o 2 3 a 0 O2 0 xy
a (Izz Iyy)/[4(M A2 ) Ixnyylzz] a.2

yyz 3 = o _ o 2 3 Z 0 02 0 xy
a (Izz Iyy)/[4(K A3 ) IxnyyIzz] a3

xyz 1 , = xyz 1 xzy l zxy 1 = 2 3 2 o o o
( a ) a + a + a {1/[2(M Al ) IxnyyIzz]}

.. o_o o YYo_o 0 220-0 0
{al (Izz Iyy)/Ixx + a1 (Ixx Izz)/Iyy + a1 (Iyy Ixx)/Izz}

1
(xyza2), = xyza2 + xzyaz + zxya2 = {1/[2(H2A 3)4 1° 1° 1° 1}
2 xx yy 22

22

XX 0 _ o o YY o _ o Io +
(122 Iyy)/Ixx + a2 (Ixx Izz)/ yy a2

o{az

(I;y- I;x)/I;z}

xyz3,=xyz3 xzy3 zxy3= 2374;00°
( a ) a + a + a {l/[2(M A3 ) IxnyyIzz]}

86

xY

xY

13

 

3

87

1° I° I°

xx yy 22

IO _ IO

IO 0 0
xx yy 22

_ o o
Izz)/Iyy + a

a C12

2

' :13

22

(I;y- IXX)/I;Z}

zzz 1
Y

zzz 2
Y

zzz 3
Y

APPENDIX 5

THE COEFFICIENTS y APPEARING IN

 

 

 

 

 

 

 

 

 

 

_ 1 A1 ‘A1 ‘ 3A2) C2
‘ ‘ 2 o 3 12
4“ (Izz) { A2H2 - A1)
'1 u(1 - 3A )
_ 1 2 2 1 C2
' 2 o 3 12
4” (Izz) L A1(A1 ‘ A2)
'1 5(1 - 31 )
_ 1 3 3 1 C2
‘ 2 o 3 13
4“'(Izz) _ A1M1 ' A3)
, 1
1 114(11 - 312)
= - 8M%(I° )ZI° A (1 A > C
xx 22\ 2 2 1
, 1
1 124(12 - 311)
= 2°20 ‘
an (xxx) IzzL 11(11 12)
,1
1 134(13 - 311)
= 202° ’=
8“ (Ixx) Izzk A10‘1 A3)
1
1 r11401 - 312)
8M’(I° ) I° 1 (1 - A )
yy 22L 2 2 1
r14
= 1 12 (12 - 311) c
2 o 2 o _
an (Iyy) 122‘ 11(11 12)

 

88

 

 

 

 

 

 

 

 

l
k A 3A
( 1 - 3) 2 a22
(1 _ A ) C13 3
3 3 1
z
(12 - 3A3) :2 a22
_ 23 3
3(A3 ’ A2)
Z 1 31 )
( 3 - 2“ C2 azz
23 2
2(A2 ' A3)
2
Al (A1 - 313) :2
13
A3(A3 - Al)
2
A2 (A2 313) :2
23
13(A3 - 12)
2
A3 (A3 - 312) z
+ £23
12(A2 - A3)
2
A1 (A1 - 3A3) z
+ 513
13(A3 - A1)
‘ -
A2 (A2 3A3) :2
23
A3(A3 - A2)

IN ¢ THROUGH ¢

10

 

 

 

Z 3
XY 2 1
Y

x z 2
y Y

3
Y

89

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7/4
1 13 (13 - 311) z yy A3 (13 312) z yy
2 2 ‘13 a1 ' 523 a2
8H’(I;y) 122 11(11 - 13) 12(12 - 13)
r 2 2 _
1 1 (A1 - 3A2) 2 xy A1 (A1 3A3) 2 xy
- -— 3- 2; a + r — g a
8K%(I° 1° 1° ) A (A - A ) 12 2 A (A - A ) 13 3
xx yy 22 L 2 2 1 3 3 1
r ”’4 _ 2 _
1 12 (12 311) z axy _ 12 (12 313) z axYW
8M%(I° 10 1° ) 1 (1 - 1 ) C12 1 1 (1 - 1 ) C23 3
xx yy 22 1 1 l 2 3 3 2
r 2 _ 2 _
1 13 (13 311) :2 axy + 13 (A3 312) CZ axy
2 o o o _ 13 1 _ 23 2
an (IxnyyIzz)L 11(11 13) 12(12 13)
A
1 1 xx _ xx -
wen”2 2a1 k111 (A2 3A1H2 k112 + (*3 3"1’533mk113
o 2 3/4 3/4 _ 3/4 _
(1xx) 1 ,Al 12 (12 411) A3 (13 411) J
r A
1 YY _ YY _ YY
my!”2 2a1 k111 (A2 3A1’az k112 + (*3 3"1W3 k113
o 2 3/4 3/4 _ 3/4 _
(xyy) 11 12 (12 411) 13 (13 411)
, zz _ zz _ zz
fiché 2a1 R111 (*2 3A1’32 k112 (A3 3A1’a3 k113
o 2 3/4 3/4 _ 3/4 _
(122) L A1 A2 (A2 4A1) A3 (A3 4A1) J
2 2
M (Al + 12)<;:2) (11 + 131(c:3>
+ +
o 2 2 2 _
2(122) A1 (A1 ’ A2) A1 (A1 *3)
f ‘
x x
flcflg 2a2xk222 + (*3 ' 3A2"“)30‘1‘223 + (*1 3A2’a1xk122
4 4 3 4
(I;x)2 123/ 133/ (13 - 412) 11 / (11 - 412)
r
1 YY - YY - yy
«0M6 2a2 k222 (*3 3}‘2”3 k223 + (*1 3"2W1 k122
4 4 3 4
(I;Y)2 123/ 133/ (13 - 412) 11 / (A1 - 412)
J
22 22 22
ncn2 2a2 k222 + (*3 ' 3A2’33 R223 (*1 " 3"2"“1 R122
0 2 3/4 3/4 _ 3/4 _
(122) 12 13 (13 412) 11 (11 412) J
2 2
M (12 + 11)(c:2> (12 + 13)<;:3)
+ +
o 2 2 _ 2 _
2(122) 12 (12 11) 12 (12 13)

 

 

 

 

9O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2
(1112) (11 - 13)<12

o 2
- A3) (122)

 

 

 

 

 

 

 

, r xx _ xx _ xx
"of2 2a3 k333 + (*1 3A3’a1 k133 + (*2 3A3’a2 k233W
o 2 3/4 3/4 . _ 3/4 _
(Ixx) 1 A3 11 .(11 413) 12 (12 413)
1r YY _ YY _ YY 1
«cf2 2a3 R333 (*1 3A3)al k133 + (*2 3A3)a2 k233
o 2 3/4 3/4 _ 3/4 _
(Iyy) 1 13 Al (11 413) 12 (12 413) J
,r zz _ zz _ zz 1
nché'zaB k333 (A1 3A3’a1 k133 + (*2 3A3’a2 k233
o 2 3/4 3/4 _ 3/4 _
(122) 1 13 11 (11 413) 12 (12 4A3) J
z 2 z 2
+ M (13 + Al)(§13) + (13 + A2)(§23)
2(1° )2 A 2(1 1 ) 1 2(1 - 1 )
22 3 3 l 3 3 2
1 _ x __ _ _
'rrcgri/2 (5A1 A2’a1xk112 (5A2 A1’a2‘xk122 G123+)‘3a3 A1 A2) xxk
' 02 3/4 A 1 + 1 3/4 1 1 + A 3/4 a3 123
Ixx A1 (4 1 ' 2) 2 (4 2 ‘ 1’ 2G123' 3
7 5 - - x + - -
«ché[( A1 A3’3’10‘1‘113 + (5A3 A1’a3xk133 + G123 A2(A2 A1 A3’axxk ]
02 3/4 _ 3/4 _ 3/4 2 123
Ixx 11 (411 A3) A3 (413 11) 2G123 12
7 .. - _ _
flcflé (5A2 A3’32“}‘223 (5A3~A2)a:xk233 G123+}‘10‘1 A2 A3) xxk
02 3/4 + 3/4 + 3/4 a1 123
1xx 12 (412 - 13) 13 (413 - 12) 23123 11
1 _ YY _ YY _ _
wcx3[(511 A2’31 k112 + (5*2 A1’a2 k122 + G123+)‘30‘3 A1 A2’ayyk ]
02 3/4 _ 3/4 _ 3/4 3 123
Iyy 11 (411 12) 12 (412 Al) 2G123 13
, _ yy _ yy _ _
flcM4[(511 A3’a1 k113 + (5*3 A1’a3 k133 + G123+)‘20‘2 A1 A3’ayyk ]
02 3/4 _ 3/4 _ 3/4 2 123
1yy 11 (411 13) 13 (413 11) 2G123 12
1 _ YY _ YY - ..
negé[(5*2 A3)a2 k223 + (513 A2)a3 k233 + 3123+11<11 12 131a k ]
02 3/4 _ 3/4 _ 3/4 1 123
IYY 12 (412 13) 13 (413 12) 2G123 11
1 _ zz _ zz - _
“C54[(511'12)a1 k112 + (5‘2 A1)a2 k122 + G123+A3M3 A1 A2’azzk ]
02 3/4 _ 3/4 _ 3/4 3 123
I22 A1 (4A1 12) A2 (4A2 A1) 2G123 A3
2 z z
“(Alxz ' A3 ) C13 ‘23

91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

, - zz _ zz _ .-
zz = Trcyt/2 (5A1 A3’31 k113+(SX3 A1’a3 k133+ G123+}‘20‘2 A1 A3’azzk
13 02 3/4 3/4 _ 3/4 2 123
I22 A1 (4A1 A3) A3 (4A3 A1) 2G123 A2
2 z z
_ M(1113 - 12 ) :12 C23
(A A )k (A - A )(A - A )(I° )2
1 3 l 2 3 2 22
1 - zz - zz ' _ _
zzY = myt/2 (5A2 A3’a2 k223+(5A3 A2’a3 k233+ G123+)‘10‘1 A2 A3’azzk
23 02 3/4 3/4 3/4 1 123
122 12 (4A2 A3) 13 (4A3 12) 25123 Al
2 z z
2 o 2
(A2A3) (A2 - Al)(A3 - A1)(Izz)
1 xY _ xy - xy
ny _ Trczm/2 2a1 k111 + (*2 3A1’a2 k112 + (A3 3A1’33 k113
11 ‘ o a 3/4 3/4 _ 3/4
Ixnyy 11 12 (12. 411) 13 (13 411)
1 xY _ xY _ xY
xyY a fiché 2a2 k222 + (*1 3)‘2)a‘1 k122 + (‘3 3A2233 k223
22 o 0 3/4 3/4 _ 3/4 _
Ixnyy 12 A1 (11 412) A3 (A3 412)
1 xy _ xy _ xy
ny = new2 2a3‘k333 + (*1 3A3’a1 k133 + (*2 3A3’a2 k233
33 o 0 3/4 3/4 _ 3/4 -
Ixnyy 13 11 (11 413) 12 (12 4A3)
1 _ XY _ xY _ _
xyy a "cflé‘ (5A1 A2)al k112 (5A2 A1)a2 k122+ 6123+A3(A3 Al A2)a k
12 o o 3/4 3/4 3/4 3 123
Ixnyy 11 (411 12) 12 (412 11) 2G123 13
2 - XY - XY - _
xy _ fiche (5A1 A3’a1 R113 (513 A1’33 k133+ G123+}‘20‘2 A1 A3’axyk
13 o o 3/4 _ 3/4 _ 3/4 2 123
In:yy 11 (411 A3) A3 (413 11) 23123 12
2 _ XY _ xY - _
ny = ngx’ (5A2 A3’32 k223+(5A3 A2’a3 k233+ G123+}‘1M1 A2 A3’axyk
23 o 0 3/4 _ 3/4 _ 3/4 1 123
Ixxlyy 12 (412 13) 13 (413 12) 2G123 11

   

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