MOLECULAR ASYMMETRIC-TOP-
VIBRATION-ROTATION HAMiLTONIANS

Thesis fer the Dam of Ph; D.
MECHiGAN STATE UNIVERSlTY
Kun-Mo Thomas Chung
1963'

'IHEsus

7 ‘ '- ‘32?
\ {v
C‘ 2 A1..,,_,;_- 1;: >

U :m'cz‘ Sit}!

 

MOLECULAR ASYMMETRIC-TOP
VIBRATION-ROTATION HAMILTONIANS

BY

Run-Mo Thomas Chung

A THESIS

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1963

ACKNOWLEDGMENTS

The author wishes to express his sincere thanks to
Dr. Paul M. Parker for his kind help and extensive
guidance throughout this work, and Dr. T. H. Edwards
for his help and interest in this work. It is a great
pleasant duty to thank Mr. Chi Whan Choi for enabling
me to come from Korea, and Dr. John A. Hannah, President
of Michigan State University, for granting me an all
university scholarship. This work was supported in
part by the U.S. Air Force Office of Scientific Research

under contract.

-11-

II.
III.
Iv.
v.
VI.

VII.
VIII.
IX.

TABLE OF CONTEETS

INTRODUCTION

THE GENERAL VIBRATION-ROTATION HANILTONIAN
DEVELOPMENT OF TFE HANILTONI‘?

ZERO ORDER ASYHHETRIC ROTATOR HAKILTONIAN
GEEERAL SYKHETRY COESIDERATIONS

ASYNRETRIC ROTATOR HAHILTOKIAN TO THE FOURTH
ORDER OF APPROXIMATION

PLANAR ROTATORS
FOURTH ORDER CEKTRIFUGAL DISTORTIOU ‘ERMS
CONCLUSION

APPENDICES

“BIBLIOGRAPHY

-iii-

as
68

75
86

88

100

LIST OF TABLES

Table Page
I 3*, E‘, 0*, 0‘ of asymmetric rigid rotator. 38

II Ranks and symmetry species of 3*, E“, 0*, 0-. 38
III Asymmetric rotator point groups. #5

IV Symmetry prOperties of coordinates and #7
angular momentum components.

V Tabulation of distinct second order centrifu- 51
gal distortion constants.

VI Symmetry preperties of PngPXPJ associated 52
with'rn.

VIII Corresponding quantities for the monoclinic 66
contributions to the Hamiltonian.

IX Planar asymmetric rotator point groups. 69

X Relationships among planar rotator distortion 7#
constants.

XI Coefficients of fourth order centrifugal dis- 79
. tortion Operators and relationships among ed'

-iv-

II
III

IV

LIST OF APPENDICES

Nonvanishing matrix elements of (VI-23).

Eigenvalues

Nonvanishin
and hc+.

The fourth
constant.

Cof (VI-23).

g matrix elements of ha+, hb+,

order centrifugal distortion

-v-

I. INTRODUCTION

Generally, the infrared absorption spectra of
molecules originate when a molecule is raised from one
vibration-rotation state to another state with higher
energy accompanied by the absorption of light. Therefore
one of the principal problems of molecular spectroscopists
has been the interpretation of the vibration-rotation
energy level structure of the molecule under study.

Just as in the diatomic molecule case, in which the
study of the infrared spectra gives precise information
about the vibration-rotation energies and these energies
lead to the accurate determination of the structure of
the diatomic molecule, we may obtain information about
bond distances, bond angles, vibrational frequencies,
force constants, dissociation energies, anharmonic con-
stants, centrifugal distortion constants, etc., by the
analysis of the infrared spectra of polyatomic molecules.
An understanding of these quantities leads to the deter-
mination of the detailed structure of the molecule and
should ultimately help us to better understand the
physico-chemical prOperties of matter in the aggregate.

In the case of polyatomic molecules the situation
is often very complicated, since we are considering
many-body problems. There are several internuclear dis-

tances, several force constants, several vibrational

-1-

-2-

frequencies, etc. Therefore it has been found con-
venient to formulate the general theoretical expression
for the energies of polyatomic molecules and then to
apply it in the specific case instead of trying to de-
duce~formulas for each specific case separately.

Hewever, in the case of polyatomic molecules it is
impossible to find exact general expressions for the
energy levels. For this reason some assumptions are
made which are valid in practice, and one treats the
general formulation by an expansion formalism in suc-
cessive orders of approximation. For instance, it is
possible in the study of infrared spectra of poly-
atomic molecules to assume the validity of the Born-
Oppenheimer approximation to separate the vibration-
rotation motion of the nuclei from the electronic motion,
and also one can safely ignore the energy contribution
of the nuclear spins until a certain high order of
approximation.

The general quantum mechanical Hamiltonian for the
polyatomic molecule was first formulated and studied by
Wilson and Howard.1 Then Darling and Dennison2 gave
their general Hamiltonian for the polyatomic molecule
which is of slightly different but equivalent form to
that of Wilson and Howard. The formulation by Darling
and Dennison proves somewhat more convenient for
deveIOpment. By use of the above-mentioned Hamiltonians

the vibration-rotation energy levels of polyatomic

-3-

molecules were calculated to the second order of ap-
proximation and it was found that the energy relations
calculated explained certain anomalies of the infrared
spectra of polyatomic molecules and gave the relations
between the energies and the parameters which charac-
terize the molecule and its dynamic behavior. This
success established calculation of vibration-rotation
energies from theoretical formulations.

In recent years improved experimental accuracy and
resolution in the infrared work in many cases necessi-
tated the taking into account of terms in the Hamiltonian
higher than the second order of approximation in order
to arrive at a satisfactory interpretation of experiment.
Vibrational effects, rotational effects and vibration-
rotation interaction effects higher than in the second
order had been observed in various experiments.

Recognizing this situation, Nielsen, Amat, and

Goldsmith3-6

extended the expansion of the Hamiltonian
to fourth order, and extensively regrouped the resulting
terms to obtain the expansion of the vibration-rotation
Hamiltonian in orders of approximation more closely
corresponding to experimenal evidence. This newly
formulated Hamiltonian gave satisfactory interpretations
of the more recent experiments. waever, this general

Hamiltonian to fourth order contains a very large number

of terms, many of them depending on the molecular para-

.4.

meters in a very complicated manner. Therefore it be-
comes important to review the expressions in the general
Hamiltonian and simplify them for solution of particular
eigenvalue problems. Such studies have been carried

out for symmetric and spherical rotators, principally
by Amat and his coworkers.7

Another large and important class of molecules is
of the asymmetric rotator type. In asymmetric molecules
the energy eigenvalue problem is more complicated than
in symmetric or spherical rotators in the zeroth order
of approximation; on the other hand, the Hamiltonian of
the asymmetric rotator is considerably simpler in the
vibrational and vibration-rotation interaction terms
since in the asymmetric molecule there are no essential
vibrational degeneracies such as occur in symmetric and
spherical rotators.

Since the symmetry prOperties of a given polyatomic
molecule qualitatively characterize its spectrum, it
should be feasible to distinguish the Hamiltonians for
each of the different symmetry groups of molecules.

We have found that the symmetry properties of a partic-
ular point group or point groups of molecules greatly
simplify the general Hamiltonian.

In this work we have studied the general vibration-
rotation Hamiltonian of asymmetric rotator molecules in

the Nielsen—Amat Goldsmith formulation by subjecting

-5-

this Hamiltonian to the symmetry restrictions of the
asymmetric rotator point groups. Particularly the second
and fourth order centrifugal distortion constants, which
will be defined later, and those terms of the Hamiltonian
which can be interpreted as vibrational corrections to
the rotational structure were of interest.

We shall first present a discussion of the general
vibration-rotation Hamiltonian, and we will subsequently
impose the symmetry restrictions of the asymmetric
rotator point groups. Finally, we will discuss some

prOperties of the symmetry restricted Hamiltonians.

II. THE GENERAL VIBRATION-ROTATION HAMILTONIAN

For the theoretical calculation of the energies of
a molecule it is necessary to formulate a suitable quan-
tum mechanical Hamiltonian. We shall reproduce the de-
rivation of such a general quantum mechanical Hamiltonian

for the vibrating-rotating molecule.

The total Hamiltonian of a molecule would have to
include a portion which represents the electronic con-
tribution to the total energy. This electronic energy
is not of interest here, since we wish to consider
vibration-rotation transition during which the molecule
remains in its electronic ground state configuration.

8 have shown that

For such a case, Born and Oppenheimer
it is allowable to separate the electronic motion from
the nuclear motion to a very good degree of approxima-
tion. Since the electrons are moving much faster than
the nuclei and consequently the wave function of the
electronic state is almost independent of the change in
internuclear distances, the Born-Oppenheimer approxima-
tion is valid in most cases. Nielsen9 has pointed out
that one could calculate the vibration-rotation energy
accurate to one part in 106 despite the Born-Oppenheimer
approximation. We will adopt the BorneOppenheimer

approximation for the formulation of the general vibra-

tion-rotation Hamiltonian, and hereby will not consider

-5-

-7-

directly the electronic motion any further. The
potential energy of nuclear vibration will, of course,
recognize indirectly the molecular electron configura-
tion in the time average over the rapid electronic

motions.

The classical kinetic energy of a molecular frame-

work of H nuclei is
(II-1)

where m1 is the mass of the i-th nucleus and Vi repre-
sents the velocity of the i-th nucleous in a space-fixed
coordinate system. Using the position vector R to the
origin of a moving coordinate system (whose manner of
‘ motion will be specified later), the angular velocity
of the moving system, ER and the position vector of the
particle in the moving system, 51(xi, yi, zi), the

‘- 1O

velocity Vi can be expressed as

V. : R + '17. + 3.5. (II-2)

Substituting Vi of (II-2) into (II-1) and utilizing the

rules of vector algebra, the kinetic energy is found

to be

DJ

, 1w 2 1—- ' 2‘. 1r—
- an? + “Fm. 2 2- .~' .+ e! .v.

A
Ev“ I»-
0

+ I'LL-iii}?5 + R°Z~émiITi + Zim—

i * l i

F!
H:
I...‘

-8-

where M is the total massjgmi, and viigi.

The position vector Fi is the vector sum of the constant
equilibrium position vector,'ai(xio, yio, 21°), and the
displacement vector from the equilibrium position,

710:1" yi” 31'),

V. : 73:1. (II-'5)

We now take the origin of the moving system at the cen—

ter of mass of the N—nuclei molecule. This is expressed

by the so-called first Eckart condition,11

gruff. = o , (II-6)
i l 1 ,

which also implies, because of (II-#), that

rniV- = O 0 (11"?)

S’ '3. =

i

1—» M

Since the molecule is semi-rigid and the nuclei remain
very close to their respective equilibrium positions,

it is meaningful to require the second Eckart condition,

2%;in = o (II—8)

i l
i.e., the moving system shall be "attached" to the
nuclear equilibrium configuration.

Eq. (II-8) also implies, again through (II-h), that

-9-

.Emi aix vi = O and;§miaix f1 = O . (II-9)

The second Eckart condition means that there is no rota-
tion of the system as a whole relative to the body—fixed
axes, since eq. (II—9) implies that the internal vibra-
tory motions do not produce any rotational angular
momentum of the molecule as a whole relative to the
moving system. However, the particles may still rotate
on infinitesimal orbits about their equilibrium positions.
The Eckart conditions are six linear constraints on the
moving system and the displacement vectors Ti, and yield
useful simplifications of the kinetic energy expression
(II-3).

The first term of (II-3) is the translational
kinetic energy, which is non-periodic and is related to
the temperature of the molecular ensemble and the
DOppler broadening effect in the spectrum. The trans-
lational energy is thus not of immediate interest in the
vibration-rotation problem and can be omitted from the
kinetic energy expression.

The second term of (II-3) is the rotational energy and

can be expressed as
“EV: Ix’x’ .022 + E. I; “l; “33.), (II-.10)

by defining the moments of inertia Im and products of

inertia Iq36x¢fi) as follows:

-10-

c J 2 ,,2
I“'::‘mi("i 4-}: ), (II-11)
I“a =-zimi®<ipi ("#5) . (II-12)

The indices a, fl, and d’are cyclic and each ranges over
x, y, and z of the "body-fixed" coordinates, i.e. over

x, y, and z of the moving system above.

Employing (II-h), we can express the moments and products

of inertia as

IO“ = IQ.“ +2;m.(f57.0.'.' +‘g'.0'. ') +Zm,( 7‘", 'd+«. '2)
~ 1 l '1 1 l i l 1 1
(II-13)
. = To - V m 0? I 2 0 a _-_ ,/z4,:
1—).17 .. _ . ,1...mi(_ i i +‘i .\i ) Aimi i ,
(II-14)

where 19“ and IS, are the equilibrium moments and
products of inertia, respectively.

The third term of (II-3) is the vibrational energy and
is equivalent to

J-f ' 2 ‘ I2 ' :2
2, m-(x.' +y. +z. ). (II-15)
i l l 1 1

It is convenient to assume a mass adjustment transforma-
tion on the displacement vector ?& and to express the

vibration energy by mass-adjusted coordinates, 51, as

3."T 2

in
r O

I? 2. SI 9
i=1

(II-16)

where

-11-

31+: {1112X2' , oooooo , S =‘.—E‘; z?!" . (II-17)

The fourth and fifth terms of (II-3) can be shown to
vanish by the first Eckart condition.

The last term of (II-3) is the Coriolis term and repre-
sents the vibration—rotation interaction. By the second

Eckart condition the Coriolis term can be written as

" (15m.‘.=...f’~.) . (II-18)

-'—¢

. 3: _ _ _
2P=QILT%?'*%;I.;'.+%i§;2+up0mqfivfi)
r. “,5 : ‘ i=1 i l. I
(II-19)

The general potential energy is composed of an
"internal" potential energy due to the time-averaged
electronic force field and an ”external" potential
energy. Assuming the overall molecular motion to
proceed in force-free inertial space, the potential
energy, V, is a function of the internal coordinates

s. only. Since the s-‘s are generally small, one can

1 l
expand the potential energy in a Taylor series about

the nuclear equilibrium positions,

V =‘V -+‘(3¥) si +-§. ( P~;) s. s +ifii (é é“fl
O P - K'- ’\ fig. — . . ’4‘ 5- .-
i=1")01 ij ~~L5~IO i j Cle *4 O 133 8k
+... . (11-20)

a \5
However, the force at the equilibrium positions, (§:)O,

must be zero, and the constant term V0 is of no sig-

-12...

nificance here, and can be set equal to zero.

we write the potential energy as
ratio simple harmonic terms, and

correction terms,

'._

4",“: f

V = 5' f..s ,
dijls

xi .3, s 8.8
ij 13 l J

1’ "w
..33.

where the force constants

r.. = —i:—e~‘ )
13k (ds,+i«' o’

1 9‘

etc.

i J k

f are defined

Therefore
the sum of the quad-

higher anharmonic

0.. ,
as

(II-22)

(II-23)

Let us introduce coefficients I;~which transform

the coordinates 51

into linear combinations of the

normal coordinates of vibration QS. Then in the asymmet-

ric molecule in which essential degenerate modes of

12

vibration are prevented due to insufficient symmetry

the mass-adjusted cartesian displacements will trans-

form as
n
[miei' =;ZQ;SQS, i=1, 2, ..., H (II-24)
S tizx, y or z ,
or also
n . .
53' Li 7:253 425- (II-25>

The index n is equal to the total number of vibrational

modes of the molecule.

efficients {Is

' ’5 1‘
f‘ -
Y

‘Ei'XiSLiS'

385'

Since the matrix of the co-

has the normalization property, we have

(II-26)

-13-

The sckart conditions constrain the /-matrix by the

following relations;13

:35, 23‘s _ o (II-27)
1

Z_§1(I.,. - ’.; ) = O . (II-28)
l 18 1 15

Taking the body-fixed cartesian coordinate axes as the
principal axes of the equilibrium inertia ellipsoid
and substituting (II—2h) into (ll-13) and (ll-1h), we
obtain expressions for the instantaneous moments and

products of inertia in terms of the normal coordinates:

_ O V‘ fwd ,1” .‘\O( . -'
INPx — 13;) +~2~ as- Q8 +- I'ASS'QSQS' , (ll-'29)
S SS
In 1 = E a”); Q +2: A" ”IQ Q I (’1'”'# V )3 (11-30)
~ 5 s 3 ss' 55 S S

where 12“ represents the principal equilibrium moments
of inertia, and the constants ad“ . a”? , AV”. and Am"
5 ' S SS 35

are defined by

‘\ -- r O " t,- O " Y
a ‘ = 2;.m (-. y; + f i. ) a
S l i 1 13 i is
al' :— '_ - ( O , ' + l’ O m )
= -¢.m. '3. j. I ’
s i l 1 18 i is (11-31)
\\ ___ _ 2 .1;
A l = z. ( y J + : '
33' i ‘is’is' LlS'iS' ’
AN 'p. = - >- ,: 1,.
ss' ‘E'is‘is'

Substituting (II-24) into (II-16) and utilizing (II-26),
the kinetic vibrational energy will be in normal coor-

dinates,

-14-

n
V 2 (II-32)

as
mogmljc‘iil =2: :: €25.93 'Qs (II-33)
1 7 ss
where the Coriolis coupling coefficients §:;. are de-
fined as
“ ,. 3 -’ n“ "’ ) (II-3h)

)ss' — i<iis‘is' ’is"is

Thus we can write the kinetic energ,

° .‘
= $7:qu ..2 + a; I 3.5+ my;
I S r}!

The potential energy of (II-21) becomes in terms of the

normal coordinates

: ~213— 2 §' -
V ¢¢s~w Q + L... 1{SS'S”QSQSSH ooo , (II 36)

S S SS'S"
whereféxtS is the square of the s-th normal frequency
and where the k's are the transformed force constants.

These are now functions of the f's, mi's and y:;'s.

From (II-35) and (II-36) one finds the conjugate

angular momenta, Ra, for each ul<and the linear momenta,

p *, conjugate to Q5

A

=git _ T . .
P“ "afla- ‘““‘*'+ Id3*t + Idza c ;: Jss'Qst'a
(II-37)
3 ' n
*E ““h , = +P(: 0 II- 8

-15-

Summing (II-37) multiplied by 4; ovcr~~ and adding
(II-38) multiplied by éq and summed over s, we obtain

twice the kinetic energy,
2T =? 9.»; + 2 p *b . (II-39)
S s 8
Substituting is from (11-38) into (II-39), we have
T = if; (B. - p. )«3: 105.2 (ll-1+0)
s

where p, is an internal angular momentum arising from
the vibretional motions whithin the "body-fixed" system,

and is eq*al to (;:.fiSS,QSp;.).
One can show by manipulating (II-37) and (II-38) that

sz " pf." = l:_ ' -. - 11:13] 1 - 1'.” , , (II-'41)
where 35' 2
= IN; as» (II—1+2)
1 r. f.“
1‘ =4 +?‘*” ‘:€* . ILA
'5? “P 2(é‘r‘s'3 Qst)(:,"s’5 Q3!) ( 3)

Eq. (II-41) can be written in vector form,

T5 - 1-). Z {/l-«u-‘l : (II-{+10
where

r 1-1

I“ = I, I! _II _I' \

t/au . xx * xz
f-I' I' -I'
: yx yy yz (II-M5)
'_II _I! I!
L zx zy 22 J.

 

The inverse equation to (II-4h) is,

-16-

:U =T/L, (5 - B) (II-H6)

where the elements of rth are

’3“.

/::= J‘(I}.13> + If 13.). < ¢.¢.) (11-n7>
52-: }:(Ii If, _ 1352) (II-H8)
.with _
(1 : d8t '31,!“ . (II—U9)

Combining equations (IT-36), (II-MO), and (II-#6) we
have an expression for the total energy in the classical

Hamiltonian form,

. 2 a
H = 9.“? -pg )(Pfj-p .) + tip; + irérgqse
+ :1 1 Q Q Q + ... (II—50)

55'5":SS'S" s 5' S"
Podolsky11+ considered the problem of obtaining the
quantum mechanical Hamiltonian corresponding to the
classical kinetic energy expressed in terms of momenta,

pi, conjugate to a set of generalized coordinates qi.

If the classical kinetic energy has the general form.

T = gsigijp,p. , (II-51)
ij 1 J
he showed that the proper quantum mechanical Hamiltonian
should be
1 i,. ij ‘% i so
H = as i.pig g pjg + V , (II-xé)

13
where g = det glJ} . Eq. (ll-52) is subject to the

requirement that its eigenfunctions should be normalized

in the configuration space qi's,

I ‘

1.4,,“ dq1dQ2ooodqn = 1. (11-53)

-17-

From (II-SO) the kinetic energy may be written in the
form

T = a: G15 pip. , (II-51+)
ij 3

by denoting

PM : p1* , P5 : pq* , .00 , P : pn*, (II-55)

Yeijj = {UL;; o , with G = detgci3}=/U, (ll-56)
\ o [I] ‘
where ill is the n x n identity matrix.
Then according to Podolsky the proper quantum mechanical

Hamiltonian would be
i i' _. l
H = % G42;Pie JG ~ch‘ + v. (II-57)

Hewever the conjugate coordinates to the momenta Pi do
not meet the requirement (II-S3). Thus we should trans-
form Pi into the preper form to satisfy (II-53).

After this unitary transfomation is performed, the

O
Hamiltonian (11-57) will become’

“41- “.- r" ...-21* -1- i ' -11; -11- 15' W ‘1-
H =-§G 4;;(3 ‘P182)G JG ~(s aP.s )§G* + v, (II—58)
ij 3
. o . , -1
with o = (Sins) .
Remembering that P3 is independent of the Eulerian
angle 9, which is the angle between the z-axis of the

body-fixed system and the Z-axis of the space-fixed

system, we finally obtain the prOper quantum mechanical

-18-

Hamiltonian as

'H

_ 1 .1_ __1_ g.
El: é}§#;a(pl-px)lés,1?(P,-p3)ifi +§$§Zps*fii2ps*yq
) , I S I

- ~ 2 s
+ £2 \SQS +83. '8" kSS'San QS 'QS" +°°’, (II-59)

where P" and Py here represent the modified angular

IL

A a -4 % l .%
momenta (sin :)‘Px(sin.9) ‘ and (sin63)‘Py(sin a) “,

respectively.

In order to calculate the vibration-rotation
energies of a molecule we should solve the Schroedinger
equation for the Hamiltonian Opera tor (II- 59).

Hewever this H Yctil tonian does by no means lend itself
to an exact solution of the Schroedinger equation.
Hence to make further progress, the Hamiltonian (II-59)
has to be developed in orders of approximation.

Since the desplacements«x£ are small relative to the
equilibrium coordinateScx:, the Iiamiltonian (II- 59)

can be expanded such that the zeroth order Hamiltonian
will be the equilibrium Hamiltonian. In the following

chapter we will perform this expansion.

III. DEVELOPMENT OF THE HAKILTONIAN

When we expand the Hamiltonian (II-59) we get

the following operator expression,

H = 453: ’_ R P3 - 3:2 (11‘ $2.3} ...flpx )23». 12»; pa lag? p‘
+ 55-1322 + *3? ’N- + V (III-1)
S
where
i ...}- J. _;~ -13 ’1
A ‘3” 359.1%, .u"<p; 414)) +.'~“‘:-(p’§/U (Pg/14)).

The terms of (111-1) represent the pure rotational
energy, the Coriolis coupling energy, the first correc-
tion to the Coriolis energy, the vibrational energy,
the second correction to the Coriolis energy and the
potential energy, in this order. *

Since }Q,and y are functions of the normal co-
ordinates which, in turn, are functions of the dis-
placement vectors from the equilibrium positions, it
is possible to develOp it; and u in power series of the
normal coordinates for the development of the Hamiltonian
in the way mentioned at the end of Chapter II.

Let us write then,

 

' ‘ r " '2 '\.1
ll = __.- '5 A « +\.). In"; + C; r. . + 1 - 9
1’ ,’ To IC) (d :EMS Q5 as :‘I'SS'QSQS' 000), (11¢ c.)
‘110 {1,317 o ' SS
L4 = ______l (*5 +5 (JQ +0.0), (111‘3)
’ IO IO IO * S S
7m (if; 52K
where
3"}: IO (III—1+:-

-19-

-20-

‘ {\‘R _ Cu;
J“LS s ' (III-4b)
( 01
fgrfi _ -Afl3 +§ €R {(3 +r: 5:,ag}
atss' ss' gh/ss”/s's" j; -—jtr-'. (III-Ho)
We also have that
x“'= 1 (III-5a)
I}. — _'Z a;
g 35 (III-5b)

A}.
-..”.

The index 5 ranges over x, y, and z, and 133 will
vanish for x¢fi if we take the axes of the body-fixed
system as the principal axes of the equilibrium moment
of inertia ellipsoid. Substituting the expansions
(III-“) and (III—3) into the terms of (III-1) each of
these terms will have a series expansion in the normal

coordinates. For example,
A = \‘+2 ”'Q -+2/

S

S s SS}‘é,QSQS,+... (Illao)

S

how regrouping terms by estimated orders of magnitude,

one gets the Hamiltonian in the series form

H = Ho + TA‘ H1 +fi‘21‘12 +131i3 + 000 (III-’7)

where fl\is a numerical parameter of smallness.

We expanded (Ill-1) such that the zeroth order

7

Hamiltonian db would represent the "rigid-harmonic"
Hamiltonian of the molecule, i.e., it represents the
energies of a rigid rotator with the nuclear framework

of the molecule in its equilibrium position plus the

-21-

energies of vibration from a potential which is a
quadratic form in the normal coordinates.

Replacing 31*4 by its value ISé , and Q5 and p* by
(h2/xsfiqc and (75/‘1‘12 )3 respectively, we obtain the

first order terms, H1, the second order term Hg, etc.,
C.

 

1
VV ‘ T 2
no ._- @2133. +(fi/2)Z)a Shag/112 + (13) (Ill-8a)
0" r./,)( S .
Ina: 2.1
PC '13! fl * . a D‘ Pa
111 = 'sZZWI-o—(j—Mf Pp -4 175'“
at n‘ .2 “ D "‘ Nu
l-‘ D “0‘ “er « 3‘-
+hc 2: K :qs qs :0 n (11" 0b)
" ss '5' is 1‘U
SS 3
to; . V. 2
<—' r- N»/ fig ‘4' I’LL)"
H : $4 i; ""‘T‘T‘ SET": 3 )‘*q qs pr. Pad-14.,
2 SS '"r: Inns Ifflz‘ ASAS' S 2-1, ii‘rX

-%:. s_lll:;(_r) {(p q w+q K)

 

S 4" ’ !
+ hcsszisnsmlisstsnsm qsqs.qs.,qs... (III-8C)
where
6 J.

th = kc ( fi )3 (III-0a)

SS'S" “S’s" ASASQ‘S" /
‘h8 % (III Ob)

th = ) t ‘/

 

k (
I t t . -. - _

and the internal angular momenta may be expressed as

- .l.
_ = <- ' )\ \ 4 -
p“ Zé"ss'( S./ s) qsps. (III 10)

Higher order terms of the Hamiltonian are given in the

reference.3

The energies of the system represented by the

-22-

Hamiltonian (III-7) can in principle be calculated in
successive orders of approximation by the perturbation
method. The zeroth order energy would be calculated
only from the zeroth order term HO. The first order
correction energies, En(1), are computed only from the
diagonal matrix elements of H1. In the absence of
degeneracies, the off-diagonal elements of H1 will
contribute to the second order rather than to the first
order correction energies. This means that the off—
‘diagonal matrix elements of H1 complicate the computa—
tion only of the second order correction energies, Eé2).
To calculate the energies to the second order it is
desirable to transform the Hamiltonian to a form more
Convenient for the perturbation calculation. Van Vleck

15

uggested the so-called contact transformation. By
‘a.suitable unitary transformation T, one attempts to

find a Hamiltonian H',

H' = THT'1 = Ho' +AH1' +2232! + (III-11)
such that the zeroth order term and the diagonal matrix
elements of the first order term of the Hamiltonian
remain unchanged while the off-diagonal elements of the
first order term of the transformed Hamiltonian would
Vanish completely. This unitary transformation would
leave the wave functions unchanged, and hence under
this transformation the eigenfunctions of HO would be—

come eigenfunctions of HO+H1', which is now "equivalent"

-23-

to a zeroth order term. Since there are no off-diagonal
matrix elements of H1', we can treat H2' as the first
perturbation term and get the second order correction

energy by taking the eXpectation values of H2'.

,
.’\

Thus, except in the case of accidental ae eneracies,

0'0.

it is advantageous to consider the partial diagonaliza-
tion of the Hamiltonian in the vibrational quan-
tum numbers by use of the contact transformation.

This is done by choosing a suitable operator T which
leaves HO of (III-8a) unchanged and gives an H1' inde-
pendent of the vibrational Operators. 16
The simplest method of obtaining the suitable form of
T is to set T = eiAs, where S is called the Herman—

Shaffer operator. Then,

H' = THT'1 = (1+1As-éx232+...)~(Ho+»H +22H2+...)

1

'(1-iks-ik2s2+...), (III-12)
OI‘
Ho' : Ho (III-13a)
H1' = H1 — 1(HOs-suo) (III-13b)
H2' 2 H2 + gils, (H1+H1')] an so on. (III-13c)

The requirements for a suitable contact transformation

are thus that we have for all v :

S
(...vs...ii(HoS-SHO)\...Vs...)=0 (III-1Ha)
(...vs...]i(HOs-SHO)I...vg...)= (III-1hb)

(...vs...!H1[...v§...) for vs#vg .

-2u_

The preper operators 8 were found by Herman and
Shaffer.17 However, if we perform the contact trans-
formation, a part of the Hamiltonian formally belonging
to Hm' becomes of the order of magnitude of Hmi1.

Hence we need to regroup the terms in true orders of
magnitude after the contact transformation. This

transformed and regrouped Hamiltonian is

H' 2 ho' + h1' + h2' + h3' + ... (III-15)
where
h ' = H (III-16a)
0 o
hl=_<' :91
1 fiinfc (III-16b)
, n ,
h2 “i%-; (2)Y? P3 P P-+ 2% (2)Yabcdqaqchqd
”(a abud
a4b<c<d
a fib
+2. 43(0’ pp+ 8 qu)pc.’p$
In3ab (2)Y a b (2)Y b a
aeb

r ,cd, ,
+aécd(2)[abchaqbpcpd+pcpdqaqb) + H2 *
a:b,eéd

(III-160)

where H2'* includes all terms of H2' non-diagonal in

one or more vibrational quantum numbers,

H2'* zis>>(:;;Ya)p P P P +:;7 (2)Yabc pa p bp ch

Iw a <xabc

~_bsc

+Z; b((2)Y:b)2 (q aq
“azb

bpc +pC q aqb)P . (III-17)

p+ of h1' , which is :4;;g, s:(qsrpsc-qs dpSC) in the

-25-

general case, vanishes for asymmetric molecules due to
the absence of degenerate modes of vibration, because
_§;:,S€= O in asymmetric molecules since the index 6'
which enumerates degenerate modes of vibration for
given s,:‘=cV= 1. The coefficients Y are complicated
functions of the molecular constants and detailed

1+-6

expressions for them are given in the literature.

In order to calculate the vibration-rotation
energy to the fourth order, it is necessary to apply

yet a second contact transformation j”to the Hamiltonian

H',
-0 2» -2
H+ =7‘H'3" zel MHe’“ f
_I+‘I+324+ 3’+ T-
— A0 +4H1 + \L2 +Atd3 + ... (ill 18)
in such a manner that HO+ +AH1+ +1?H2+ will now be

diagonal with respect to the vibrational quantum
numbers in the representation for which H0 is diagonal
in the vibrational quantum numbers. Again one observes
that the transformation will have the effect that cer-
tain terms of Hi which formally arise from the Operator
Hm+ contribute only to the order of magnitude of Hfi+T.
Hence it is also necessary to regroup the terms of the
twice transformed Hamiltonian by true orders of magni-
tude. After this regrouping the Hamiltonian is of the
form

H" =TH'3”= 110* + h1+ + 112* + h3+ +... (III-19)

-26..

By the requirements for the contact transformation,
ho+ is the zeroth order term H6 unchanged and h1+
vanishes for asymmetric molecules. The detailed ex-
pressions for h2+, h3+, and hh+ are given by Amat and

Nielsens’6 who have found the required transformation.

The Hamiltonian (III-19) can be used to calculate
vibration-rotation energies to the fourth order of
approx'mation by utilizing only those matrix elements
of (Ill-19) which are diagonal in all vibrational
quantum numbers vs, because hO+, h1+, and h2+ are al-
ready diagonal in all v3 and off-diagonal matrix
elements in any vS of h3+ and hg+ will contribute to
the energ es only in orders of approximation higher
than the fourth. The Hamiltonian (III-19) is diagonal
to all orders in the rotational quantum numbers J(total
angular momentum quantum number) and h<nagnetic quantum
number) but it is not diagonal in the quantum number K
in a symmetric rotator representation. In particular,
for asymmetric molecules ho+ is not diagonal in K, and
no closed form general transformation is known which
would bring the zero—order rigid asymmetric tOp
Hamiltonian to the diagonal form. This is, of course,
the essential fact which prevents one from obtaining an
analytical expression for the rotation-vibration
energies of asymmetric molecules in the general case

despite the perturbation formalism.

r‘,
- ,' -
‘—

heglecting components which are Off-diagonal in
the vibrational quantum numbers, we have the following
types of term to the fourth order of approximation in
the vibration-rotation Hamiltonian for the asymmetric

molecule:

1+ 6 2
a ’ Zr 2 7
()< > r ’(2) ’(u) r W)
, a, Z“ w). -0 r ,6 .,
(b), art. 49* ZP‘ AP (111—20)
RC) '(2) ’(to ’m
, 2 2 2 a h n 8 3 i 2 h r 470
mm ’(2) r ’< ) r ’(3) r WI) 1 P W)“
‘ ‘ 2

where we have use< the following notation: r stands

for products of any two vibrationa operators qaqb,

pqpb, Qapb’ paqb; r4 for any product of four vibration-

(A

al Operators, etc.; P3 for any product of PX, Py’ and
‘1 d8-'--',, ' ‘ ooo ‘
Hz; and (n); stands for 4a ’b ’ , which appear

+ (n) a,b,...
in H as the coefficients to the various Operators.

The subscript (n) Of (n)Z represents the order of ap—
proximation of the corresponding term. We also have
omitted the summation sign over the rotational and
vibrational indices. Detailed expressions Of (n)Z are
given in the references,536 and we will write down
explicitly only those (n)Z which will be needed in the
present work and as the need arises.

Terms (a) constitute pure vibrational Operators
including anharmonicity corrections, terms (b) constitute
pure rotational Operators including centrifugal distor-

tion corrections, and terms (0) may be interpreted as

-28..

vibration-rotation interaction terms and comprise such
contributions as the vibrational corrections to the
rotational and centrifugal distortion constants.
Generally the vibrational frequencies are 100 to 1000
times larger than the pure rotational frequencies.
Therefore, as discussed by Amat and Nielsen:8 the
subscript (n) of (n)Z indicates the order of magnitude
Of the contribution by the term correctly for Jz10 to
6

2-1I , ,
L+)Zr P and (“)5P

should more prcperly be regarded as contributions to

Jx30. If Jt1, then such terms as

A

the eighth and tenth orders of magnitude respectively
rather than to the fourth order indicated by the
subscripts. In any case, inclusion Of all of the above
types of term will be sufficient to fourth order for
all reasonable J, and terms such as (#)ZP6 need be
considered in the fourth order only for large values

of J. The relative importance Of the various types of
terms for given J can be ascertained from Table I of
the reference 18.

In cases Of resonances, in which two or more
energy levels are either closely spaced or actually
degenerate, our preceeding arguments must be modified,
since the contribution from the relevant Operators hn+
will more pronounced than in the non-degenerate case.
If two energy levels are very close, the off-diagonal

matrix elements Of hn+ will contribute to the energy

-29-

in a much lower order than the 2n—th. In the general
molecule there can occur two kinds of resonance. One

is the accidental resonance which is due to the proximi-
ty of two interacting levels with different vibrational
quantum numbers yet having nearly the same energies.
Coriolis resonance and Fermi resonance are typical
accidental resonances. The other kind of resonance is
the essential resonance between levels which have the
same vibrational quantum numbers but different internal
angular momentum quantum numbers, e.g. l, or different

h quantum numbers. In the asymmetric molecule essential
degeneracies are absent because of the low symmetry of
the molecule. Therefore in this work we need to con-
sider only the accidental resonances. In fact, since

it is very complicated to account for all possible acci-
dental deyeneracies, we will assume that our molecule

is free Of accidental resonances, or that, if they occur,
the energy levels involved in such resonances may be
excluded from consideration. Then the Hamiltonian
(III—19) is quite apprOpriate for the calculation Of
energies to the fourth order.

Terms (a) of (III-20) are associated with the pure
vibrational energies. We shall denote their total
diagonal contribution to the Hamiltonian by hv+*. This
hv+* gives the vibrational energies Ev to fourth order

and will not concern us further, since we are principally

-30-

interested in the rotational level structure built

upon particular vibrational states rather than in the
detailed calculation of the pure vibrational structure.
From the general vibration-rotation Hamiltonian we find
for asymmetric molecules that the (1)Zr2P-type terms
have zero coefficients (1)2, and that the (3)2r“P and
(3)2r2P3-type terms have no non-zero matrix elements

diagonal in all v Thus the odd order terms which are

S.
a source of considerable difficulties in symmetric and
spherical rotators may be excluded from consideration

in the asymmetric rotator case. Thus, to fourth order,

we have:
+ 4-"r +* r
H = hv+* + ho+ + h2 + h1+ , (III-21)
where
+ __ r. 2 _ y. 2 a 2 e 2
ho _ (0)4? _ ARK + BPy + cez , (III-22)
+* _ 2 2 r h
h2 _ (2)Zr p + (2);? , (III-23)
h +* = 2P2 + Zth2 + ZrZP” + 2P6 .
Lr (1+) (n) (M (1+)

(III-2H)

The asterisks denote that terms of hv+* are to be
omitted in h2+* and h4+* and also that terms of hu+ not

diagonal in all vS are to be omitted in hg+*.

IV. ZERO ORDER ASYLMETRIC RCTATOR HANILTONIAN

We return to the part of Hamiltonian (III—8a),
which remained unchanged under the two successive
contact transformation. The first summation term of
(III—8a) is the rigid rotator Hamiltonian and the
second term is the Hamiltonian of n uncoupled simple
harmonic oscillators.

The vibrational portion of (III-8a) was
1
2
Hov = gh§/\S?(ps2/fi2 + qs ). (IV-1)

With the aid of the vibrational matrix elements

(vsiQSZ’Ys) (vs+ %) (IV-2a)

(vslp52,vs) a2(vs+-g-), - (IV-2b)

we obtain the vibrational energy of the molecule to

zero order of approximation from (IV-1) as

i
*1 _ \- 2 .1 _
We can write (IV-3) as

Eov = hdZmé(Vs+ fi), (IV-h)
S

where pus are the normal frequencies of oscillation

expressed in cm'1.

The zero order rotational Hamiltonian Hbr can be

written as

1:11P2+1p2+1p2 IV-
1or 7(mx my thj-z-z) ( 5)

-31-

-32-

where x, y, and 2 represent the directions of the
principal axis of the inertia ellipsoid in the body

fixed coordinate system. With the appreviations,

1 1 1
A=—-u— B=—'o— c=—e— (IV-6)
2Ixx ’ . 2Iyy ’ 2Izz
Hor will be
_ 2 2 2
Hbr - APx + BPy + CPz , (IV-7)

and the definition of the asymmetric tOp molecule
implies
A i B i C .

We will assume A>B>C in this study. This ordering is
not always the conventional one, but this work can be
brought into agreement with any of the customary con-
ventions by prOper interchanges of A, B, and C.
The total angular momentum can be expressed as

P2 = PX2 + Py2 + P22 = PX2 + py2 + PZ2 (IV-8)
where X, Y, and Z are the axes of the space-fixed
Cartesian coordinate system. Each component PX, Py, and

Pz commutes with each of P}? Py, and PZ, and

[Pol ) P3] 'ifipi ’ d, fl , and X cycliC, (IV-9)

[PLL, F'i

{, +ihR , (Q, A, and a cyclic, (IV-10)

in which we, 3, and 3 represent the body-fixed system
coordinates and 51, fl, and fl represent space—fixed

system coordinates. The total angular momentum P2 and

-33..

P2 commute with H6 but PZ does not commute with H

r’ or
for the asymmetric molecule. Hence the eigenfunctions

of the asymmetric rotator will be designated by 43F

and a set of (2J+1) eigenfunctions 13V will be asso-
ciated with every possible pair of quantum numbers J

and M (1-1 42.1);

= fi2J(J+1) t (Iv-11)

P2 rm,

“f J14

P f, = ’H M W (IV-12)

z 1“n ".11.; °
The (2J+1) eigenfunctions associated with a given pair
of values J and M would have been identified by the
quantum number K in the symmetric rotator case, since
for the symmetric rotator we have that Pz¢5kM = thdKM’
i.e., K is a "good" quantum number.

Wang16 wrote de as a linear combination of the eigen—
functions of the symmetric rotator, JFP’
PM = 2Ilci’f‘ihcr; ’ (IV'13)

Substituting (IV-13) into the Schroedinger equation of
the asymmetric rotator we obtain

. _ 3.. 4‘ _
Hor éCK/PJILI-i ” Eor t" Ck‘m'J-i (IV 11*)

and from the condition for the existence of non-trivial
solutions of (IV-1h) we obtain the secular equation for

each pair of J and H as

Det lHKK' - Eoréghh'l = o (IV-15)
where
HKK, =jcp3fin Hbr th'M d1~. (IV-16)

In principle we can now calculate the energy levels of
the asymmetric rotator from eq. (IV-15).

In the symmetric rotator case we have

9 9*
“ t . .-.-. ‘ * (+1 - a
p rJ'II‘. 4’1 d<J+1) ‘VJKII (IV 16 )
__ L
Pé(thH "’fiL YJKK (IV-16b)
P I = ('1‘ ._ -
z 4731\1‘4 71K VJ“; . (IV 160)

Taking the phase angle for the angular momentum compo-

nents such that

. J-
(KIPX x+1)=(h+1lpxjr)=§n{(J-x)(J+L+1)J2 (IV-17a)

 

.1-
(klay[K+1)=—(K+1}pylk 2-‘%hf(J-K)(J+h+1)]z, (IV-17b)

2
y

2

O O . 2
the non-vanishing matrix elements of Px , P , and P2

are given by
(LIPXZIK)=(kgPy2§h)=%h2{J(J+1)-K3: (IV-18a)
(Llpx2gxi2)=-(L,Py2,Ki2) (IV-18b)
, , .1.
=e52[(J;t)(J;K+1><J:r+1)(Jix+2)32
(KIPZEIK)=h2K2 (IV-18c)

From these matrix elements we obtain the non-vanishing

matrix elements of [Hart] for given J and N as

-35-

(rtabrgh)=%(e+a)’J(J+1)-K23h2+cx9h2 (IV-19a)
(KjHor(L+2)=(L+2}Ior|K) (IV-19b)

-1-
=-%n2(A-B)[(J-k-1)(J-K)(J+X+1)(J+h+2)£“-

Eq. (IV-15) can be simplified by taking advantage of the
symmetry properties of the wave functions of the system.
Since the inertia ellipsoid must be invariant under
rotation hrough an angle W around any principal axis,
there exist symmetrized symmetric rotator wave functions
invariant under these symmetry Operations. According to
Hulliken17 the symmetrized wave functions belong to one

of four symmetry species, designated by A, Bx, B B

y, 23
which are associated with three rotations C2’ and the
identity Operation I. The relation between the symmetry

species and the rotational operations is as follow:

Symmetry operators

K’2 2 2
1'}. + + + +
Symmetry B- + - _ + (IV-20)
spec1es A
3” - + - +
J
u - - + +
z

where + and - designate whether the wave function is
symmetric or antisymmetric under a given operation.

. . . _, s ‘ .
Tne symmetrized wave functions ¢Lp of the symmetric

rotator are formed by

.p .2“ ‘ \ p=O or 1
(IV-21a)
,‘15 _ '..X 1__
1100 _\,o for k_o (IV-21b)
where 4%:‘¥§ru:('1)q5bJ~y and qzh when K2K or qzh when
\. uni; ' 1.;

K<M.

From (IV-13) we construct the symmetrized wave
functions of the asymmetric rotator using the symmetrized
wave functions of the symmetric rotator of (IV-21), and
get the corresponding secular equation to (IV—15) ex-
pressed by matrix elements in the symmetrized symmetric

rotator representation,

5 —-S-5
zzicvn: IV-22
\tJh x n TJKH ( )
Det [HfiK.p-E0r§?hh.‘ = o . (IV-23)

We find easily that the matrix [Hih'p] will split into

0 (LS ‘ ‘ S ‘ o o I S"
two submatrices iILK.OJ andiHLh'1j° DeSignating \Hhh'O]

.s,-. F 1: ’ F 1 ' °
and [HLL11] by L+; and 1-; respectively according to

the sign of (-1)p, the matrix [HEY'E will be of the form

 

 

s A,5
s’x'o fifl1
s , [+ 1 1
L0 (J+1)x(J+1) : O ,
S f’ T . ] , (Iv-2h)
¢K1 i_ O 3 JL; J

 

 

-37-

Wang20 has shown that [+] and I—] can be factorized into
two submatrices each, by collecting elements with even
K values and with odd K values. By performing this we

can transform LHKK') into a four-step matrix

 

E+

i Lg . ; (Iv—25)

 

 

f O-

 

 

 

where E and 0 indicate even K or odd K respectively,
whereas + and - signs indicate p=O or p=1. According

to (IV-25), the secular equation now factors into four

8
aM,p

associated with a given "step" belong to only one of

equations of lower orders and all eigenfunctions‘+

the four symmetry species. We give the form and the
ranks of E+, E“, 0*, 0' in Table 1, and the symmetry
species of their eigenfunctions in Table 2.

Although the orders of the factored secular equations
are lower, closed form solutions of the energy eigen-
values are possible only for a few small J values. In
most cases the calculation of the energy eigenvalues can
only be made by numerical methods except in cases of
small asymmetry for which case various approximation

methods are available.

-38-

Table I 3*, E“, 0*, 0' of asymmetric rigid
rotator.

 

 

 

 

+_r r7” a ._ 1
E _ Hboladog o o .l n -1 H22 Hgg o o .1
¥2H2O H22 H2H O . HgZ HRH Hgé O .?
‘0 IE2HM+%% . o Ebuflasfiw°i
7T i 1:, . ;
I o o ”6% H66 °i ; o 0 J86 H88 §
I . .

+ ’ - ~ -_ , ~
0 2.03 +J )IH3 o . 0._‘0n1-H41)1q3 o .‘

, 0 H53 H55 . 0 H53 H55 .5

 

 

IL 0 O O O O O O O

 

 

Table II Ranks and symmetry species of 3*, E ,
o , 0'.

 

 

 

 

 

 

 

 

rank symmetry species

step K P even J odd J even J odd J

 

3* even 0 %J+1 %(J+1) A B2
3' even 1 %J £(J-1) Bz A

+ i l

0 odd O ;J 3(J+1) By K
0' odd 1 3J §(J+1) B B

>4
0.4

-39..

’3
According to Ray‘1 the numerical evaluation of rigid
asymmetric rotator energies can be simplified by

introducing an asymmetry parameter Pi,

 

" _. (ZB-flJ‘C) _ a /< _r~./
h- (A—C) , 1ésa‘+1. (IV 40

Ifiaz-1 we have the case of the prolate symmetric
rotator and ifk<=1 we have the oblate symmetric rotator.
For H.=O we have the "most asymmetric” rotator. Denoting
Hor by its associated energies E(A,B,C), we find from
(IV—5) that

E(aA+b,aB+b,aC+b)

. 2 2 as 2 2
a(.—.Px +BPy +o.z ) + bP

aE(A,B,C) + bh2J(J+1). (Iv—27)

If we substitute

a = .2. , b = -.Aig , r«= aB+b (IV-28)
A-C A-C

and rearrange (IV-27), we have E(A,B,C) as
E(A,B,C) = 5392 s) + £§9J(J+1)h2 (IV-29)
where E(h) is E(1,n,-1).

E(A,B,C) for fixed J can have (2J+1) values, and
the (2J+1) values of E(K) as functions of H;associated
with the given J can be proved not to "intersect" when
s< is varied in the interval -1<¢a<1. Therefore we
can designate an index 'TKz-J, -J+1, ..., J) to identify

the (2J+1) energy levels associated with given J,

T .
E} =.§%§E§(K) + A§9J(J+1)fi2, (IV-30)

-MQ-

in such a way that

-J

. J J
b_J(g)<ZE_J+1(e)< ... <1EJ(M). (IV—31)
It can be shown that
aim = -E§,(a>. (IV-32>

Remembering that one can assign in the limiting
cases =+1 and ta=-1 to every E(A,B,C) absolute values
K_1 and K1 (which are the limiting symmetric top quantum
numbers K) we can identify the energy levels by K_1 and
K1, and ‘C will be found to be equal to h_1-K1. Since
we can relate the symmetry species of the rotational
wave function in the two limiting cases to the I quantum

numbers, we can also identify the symmetry Species of

an energy level by its '1 value in the following way:

 

T ‘ 4 J J'1 (1‘2 J-3 J_)_+ 000
T T
Symmetry even J i A Bx By B2 A ...
species , i
l odd J 1 B2 By Bx A 32 ... (IV—33)

 

In combination with Table 2 and (IV-33), we should be

. . - gJ' mJ lJ
able to find the eigenvalues “J, bJ_1, ”J_2’ ... from
+

the step matrices 3+, 0', 0+, 3', E , ... in this order.

Eq. (IV-29) is convenient for numerical calculations
of the energies of the asymmetric rotator, since we
could easily calculate the energies by obtaining E(h).
However, the evaluation of E(R) is by no means simple.

22

Hainer, Cross, and Ling give a review of the methods

-h1-

of obtaining E(#), and calculated23 3(a) for Ja12 for
M from O to 1 by steps of .1. Their evaluation of E(H)
was done by solving the secular equation of (IV-23) in
a continued fraction form. Later workers 2” have en-
larged and extended the Hainer-Cross-King eigenvalue
tables. Also, there exist methods which use Mathieu
functions, harmonic oscillator functions, or power
series expansions in the treatments of the secular de-
terminants. Such methods have been discussed by Hainer,

22 It should also be remarked that if

Cross, and King.
one considers higher order approzimations, the compi-
lation of eigenvaue tables is no longer practical nor
even feasible since entirely too many parameters are

involved.

Despite the discouraging aspect of the complexity
of the calculation of the zero order energy eigenvalues
of the asymmetric rotator, we found that one could
proceed to the consideration of higher order terms with-
out producing an undue amount of additional complexity.
In fact, as we shall show, some closed form solutions
of the energies including the centrifugal distortion
effects and vibration-rotation interactions can be given
up to the fourth order of approximation.

Presently obtainable resolution in the infrared
and microwave spectra of molecules requires that these

higher order approximations be considered if a satis-

-l|>2-

factory interpretation of the spectra is to be obtained.
We will undertake the consideration of higher order
terms in the following chapters. Since the general
symmetry prOperties of the Hamiltonian are important
for the further deveIOpment, we shall discuss these in

the next chapter.

V. GEfififiAL SYMNETRY CONSIDJRATIONS

The terms of the general Hamiltonian may either
remain unchanged or change sign under any of the rele—
vant symmetry Operations which are coordinate trans—
formations (reflectiorsor rotations) which will produce
an equilibrium configuration of the nuclei that is in-
distinguishable from the original one. Recognizing th
axial vector nature of the angular momentum components
and that the vibrational Operators qS and p5 must be
symmetric or antisymmetric under the point group Opera-
tions, it is found that for asymmetric molecule all
terms of the Hamiltonian are either symmetric or anti-
symmetric under any symmetry operation. Of course, for
higher symmetries more complicated situations arise; it
is true only for the asymmetric rotator point groups
that all irreducible representations of these groups are
one-dimensional irreducible repres itations.

The Hamiltonian of a vibrating rotrtor must be
invariant under all symmetry operations of the point
group to which the rotator belongs. Hence all terms oi
the Hamiltonian which are antisymmetric under one or
more symmetry Operations of the relevant point group
must be absent fr m the Hamiltonian for that group.

All {symmetric molecules must belong to one of eight

point groups. These eight point groups are contained

-h3-

-un-

‘

thirty-two possible crystallographic

Ff)

within the set 0
point groups, and hence asymmetric rotator point groups
can be referred to in the crystallographic language, if
one so desires. These asymmetric rotator point groups
an; their nomenclature are summarized in Table III, Where
we give the symmetry operations for the various point
groups in cust ma y notation. The orthorhombic point
groups have the highest symmetry, and hence one could
foresee that the vibration-rotation Hamiltonian will

have its simplest form for these point groups. The
monoclinic point groups have lower symmetry than the
orthorhombic point groups, but higher symmetry than

the triclinic point groups. As one would expect we

will see that the order of symmetry is closely related

to the degree of complexity of the Hamiltonian. There-
fore it is reasonable to discuss the vibration-rotation
Hamiltonian for each point group separately and we will
do this in the following chapters. ‘

In the asymmetric molecule every vibrational mode
is non—degenerate. For a given non-degenerate normal
vibration a symmetry operation can at most bring about
a simultaneous change of sign of all displacement co-
ordinates belonging to a given non-degenerate vibration.
Therefore a given symmetry operation will change the

the normal coordinates or it will leave them

}- .1'

Sign of sl

0‘.

all unchanged. Since a non-degenerate vibration can only

be symmetric or antisymmetric with respect to any symmetry

-45-

Table III Asymmetric rotator point groups.

 

 

 

Crystallographic Group Group operations
nomenclature symbol other than identity Operation
Triclinic C1 none

Cl:S2 l
honoclinic CS=C1h (j, Case(a)<7(xy)

Case(b) 6(yz)
Case(c)<3(zx)
C Cq, Case(a) 02(2)
Case(b) C2(x)
Case(c) C2(y)
02h CZ’Uh’i Case(a) C2(z),o(xy)
Case(b) 02(x),o(yz)
Case(c) C2(y),o(zx)

Orthornomblc C2v C2, two CV

VzD2 three mutually—1C2

thD2h tnree mutually.LC2, 1,

three mutuallyJ—J'

 

m...- *—

-h6-

Operation which is permitted by the symmetry of the
molecule, our statement about the symmetry of the
vibrational Operators in the first paragraph of this
chapter is valid. Now, from (III-20) we see that all
vibrational operators are present in our Hamiltonian
as even powers only. Since the vibrational operators
qS and pS are either symmetric or antisymmetric under
any symmetry operation for the asymmetric molecule, the
vibrational portion of any operator term will always
transform into itself. This means that all terms of
our Hamiltonian will be symmetric or antisymmetric
depending only upon the symmetry prOpcrty of the rota-
tional portion of the operators.

To find the symmetry prOperties of the rotational
Operators we have to consider the behavior of each
component of the angular m mentum under the possible
symmetry operations for the asymmetric molecule. The
symmetry properties of the coordinates and the angular
momentum components are given in Table IV, where + sign
stands for "symmetric” and - sign stands for "anti-
symmetric” behavior. Thus we can determine the symmetry
property of any rotational Operator by using Table IV.
For example, the operator Pxpypxpz is antisymmetric
under the symmetry Operation c(xy), but is symmetric
under the symmetry operation C2(x). It is interesting

to notice the symmetry Operators 6(Xy),<3(xz), and C(yz)

-h7-

behave equivalently to 02(2), 02(y), and 02(x),
respectively for the angular momentum components.
The time-reversal symmetry has not been considered

in this study, since we do not expect it to produce
further simplification in our problem.

Table IV Symmetry prOperties of the coordinates
and angular momentum components.

 

l —-
4 ‘~-_

 

 

Symmetry Angular momentum
Operation Coordinates components
x Y z Px Py pZ
I + + + + + +
1 - - - + + +
‘7(xy) + + - - - +
5(xz) + - + _ + _
UIYZ) - + + + _ _
C2(Z) - - + - - +
02(x) + - - + _ _

02(Y) ' + - ‘ + ’

 

VI. ASYKKETRIC ROTATOR HAMILTOIIAN TO THE

FOURTH ORDER OF APPROXIMATION

The Hamiltonian appropriate for a vibrating-rotating
asymmetric molecule was given to the fourth order of
approximation by (III-21) in an abbreviation form. In

greater detail (III-21) may be written as

H+ = (hv+*+ho+) + h2+* + hn+*a (Vi—1)
with
+* . at aa 2 MB 2
= 7.- ~'Y + ’Y p
h2 Qf‘a((2) pa {In aaqa )d~3
+2; CK¢“LYR.P;P,P, (v1-2)
(«bk/2" 2 I I 1‘
+* .. “‘3 -:‘ ”‘3 rbD 2..
h)+ - r23 (1+)ZID pt. + L ‘3‘ b(1+)éa“qa pb Papa
‘ ; ,
V“ ¢ 0» 2 aabb 2 2
+1 + Z p P
.Eééfb'a<b((”) aabbqa qb (4) p pb )“"
+ '«(°1’;z q + éaap 2)P P.P P
g‘ .a (h) aa a a P a I
+ EQI I°"k;,zR‘P,P,p R P,. (VI—3)

As mentioned at the end of Chapter III the pure vibra-
tional energy Ev from hv+* is not of immediate interest
in this study. The zero order rotational Hamiltonian
ho+(=HOr) was discussed in Chapter IV. Therefore we

+»

will focus our discussion in this chapter on h2 and

+*

h1+ o
The second term of hg* which is the second order

-u3_

all-31¢: Ev. -..-lain!

_%9_

centrifugal distortion term, and the last term of hg+*
which is the fourth order centrifugal distortion term,
are important for the analysis of the rotational struc-
ture. In most molecules the fourth order centrifugal
distortion term is very small except at very high J
values. But in some rare cases, notably for H2O, the
effect of this fourth order centrifugal distortion terms
is prominent for the lower J values too. This feature
was shown by Benedict,25 and by Parker and Brown.26
Since in most cases these terms give unobservably small
contributions, and since we want to discuss these terms
in Chapter VIII, we will exclude these fourth order
centrifugal terms in this chapter.

The second order centrifugal distortion terms can

be written as

71-: Lo“, 11.9413, P, (v1.4)

where thesecond order centrifugal distortion coefficients
’(u/ {I}. are

}

and where the factor % is introduced in the definition
(VI-5) to bring the 72,,0 into agreement with the
conventional definition of the second order centrifugal

distortion constants. The coefficients an,” are prOpor-

tional to

oat); ”a";
as as
O O O 0

where the molecular constants agf were defined by

 

z: (VI~6)
(II-31). We shall use for‘igog the alternative nota-
tion (air ) where convenient. Because of the non-
commutativity of the angular momentum components Pa,
(VI-M) shows that there could be a total of eighty-one
(1;? ). Because of (VI-6) many of these are equal to

each other, and one has in fact that

(“f/i)=(XZG%)=(BWz4)=(%SEV)
=(nogr)=(2> 3)=(a%t )=( <* ). (VI-7)
Application of this condition to (VI—4) shows that many
of the summation terms have common coefficients, and
one arrives, as is well known, at twenty-one distinct
(REVS). These are summarized in Table V and are further
classified into four sets.

Symmetry properties of the Operators associated
with these coefficients are given in Table VI. It will
be noticed that we have grouped the (xfiXS) on the basis
of their symmetry behavior. A further advantage of this
classification appears if one considers the position of
the matrix elements of the second order centrifugal
distortion terms in the total Hamiltonian matrix. In
the symmetric rotator P2, P2 diagonal representation,
which was employed for the zero order problem in Chapter

IV, the angular momentum operators whose (age ) belong

-51-

Table V Tabulation of distinct second order

centrifugal distortion constants.

”’(.n :1, 2’ 3, 0.0, 210

 

 

 

 

 

 

n (apaé) Set
1 (XXXX)

2 (yyyy)

3 (zzzz)

% (yyzz)=(zzyy)

5 (zzxx)=(xxzz) I1
6 (xxyy)=(yyxx)

7 (yzyz)=(zyzy)=(yzzy)=(zyyz)

8 (zxzx)=(xzxz)=(zxxz)=(xzzx)

9 (xyxy)=(yxyx)=(XYYX)=(yxxy)

1O (xxxy)=(xxyx)=(xyxx)=(yxxx)

11 (yyyX)=(yyxy)=(yxyy)=(xyyy) I
12 (xyzz)=(yxzz)=(zzxy)=(zzyx) 2
13 (xzzy)=(yzzx)=(zxyz)=(zyxz)

=(zxzy)=(zyzx)=(xzyz)=(yzxz)

1% (yyyz)=(yyzy)=(yzyy)=(zyyy)

15 (zzzy)=(zzyz)=(zyzz)=(yzzz)

16 (yzxx)=(zyxx)=(xxyz)=(xxzy) II1
17 (yxxz)=(zxxy)=(xyzx)=(xzyx)

=(xyxz)=(xzxy)=(yxzx)=(zxyx)

18 (xxxz)=(xxzx)=(xzxx)=(zxxx)

19 (222x):(zzxz)=(zxzz)=(xzzz)
2O (zxyy)=(xzyy)=(yyzx)=(yyxz) 112
21 (zyyx)=(xyyz)=(yzxy)=(yxzy)

=(yzyx)=(yxyz)=(zyxy)=(xyzy)

 

 

 

-52-

Table VI Symmetry properties of PgfigPlP,
associated with 'h.

 

Operator I i 6(xy) o(xz)<r(yz) C2(z) C2(X) C2(y)

n = 1 + + + + + + + +
2 + + + + + + + +
3 + + + + + + + +
1+ + + + + + + + +
5 + + 4 + + + + +
6 + + + + + + + +
7 + + + + + + + +
8 + + + + + + + +
9 + + + + + + + +
10 + + + — - + - -
11 + + + — - + - -
12 + + + — — + - -
13 + + + - - + - -
11+ + + .- - + - + -
15 + + - — + - + -
16 + + — — + - + —
17 + + - - + - + -
18 + + - + - — - +
19 + + - + — - - +
20 + + - + — -— - +

 

 

-53..

to sets I, or I2 may have nonvanishing matrix elements

of types (th), (K K12), (KIKiH) only, whereas sets II1

 

or II2 Operators may have nonvanishing matrix elements
of types (K1K11), (K!Ki3) only. Also, sets I1 and II2
matrix elements are real whereas sets I2 and II1 matrix
elements are pure imaginary. If we write the matrix
elements of (VI—h) in the symmetrized symmetric rotator
representation (which was discussed previously for the
zero order problem) we obtain a matrix which is Hermitian
and, in addition, symnetric about the "nonprincipal"
diagonal. After performing the wan; transformation,
matrix elements of the four sets of terms will stand in
the following positions in the transformed matrix for

given J:

 

 

 

 

 

 

 

 

 

11 I2 I12 II,
12 I1 ; II1 112
% (VI-8)
112 1 II1 I I1 1 12
II1 ; 112 i 12 1 I1

 

.. . . . . + +
If we add the transformed matrix of hO of (IV-25), we
have the matrix representation of the Hamiltonian of the
asymmetric rotator to the second order. To the four

submatrices in the diagonal positions of (VI-8) will now

-54-

be added 3*, E', 0*, 0', respectively, of the Wang
transformed matrix of the zero order Hamiltonian. In
other words, the matrix of the second order Hamiltonian
would be of the form (VI-9), where E+, E’, 0+, 0‘ now
stands for the sum of 3*, E", 0*, o’ of the rigid

rotator Hamiltonian, and the corresponding I,'s of

(VI-8):

 

 

 

 

 

f I I
+ I
. A L L
3 I E” i II ‘ II
, 1 . ..
L 2 F 2 J, (VJ-'9)
s z
+
i 112 111 0 12 !
II, , 112 1 12 0' i

 

If we consider a properly partitioned four step matrix

T which transforms (VI-9) to a basis in which only the
blocks on the diagonal of (VI-9) are completely diagonal,
we find that none of the I2 or II elements will fall
within the diagonal blocks because of the transformation.
Thus only “E through. I, (set 1,) are true second order

constants, whereas Tm through ,will contribute only

‘1
to the fourth order of approximation of the energies.
This argument is very important, since we shall use it

to good advantage for the fourth order terms.

Because of the invariance requirement of the

-55-

Hamiltonian, if any Ph-type term of (VI-#) changes sign
under one or more Operations of a given point group,

the corresponding (dirS) must be zero for that point
group. Using the symmetry prOperties of the angular
momentum components of Table IV, one finds that twenty-
one distinct ( ?"‘) will in general be nonvanishing

for the triclinic point groups, whereas only thirteen
distinct (d; 2) will be nonvanishing for the monoclinic
point groups, and only nine distinct (dtré) will be
nonvanishing for the orthorhombic point groups. Thus
Kivelson and Wilson's 27 second order centrifugal dis-
tortion treatment of (VI~%) will be applicable to second
order to any asymmetric top molecule except the planar
molecules, where we have linear dependences between
nonvanishing (48f ). The planar asymmetric tOp molecule
will be discussed later.

The remainder of h2+* and hg+* comprise vibration-
rotation interaction terms and additional corrections
due to the rotational motion of the molecule. These
terms are not, in general, negligible and will be

considered now.

The set I, of (a I») was nonvanishing for all

1’ II2

vanished completely for the orthorhombic point groups

asymmetric point groups, whereas the sets 12, II

and one of them was nonvanishing for the monoclinic

point groups. The distinction between I2, II 112

1,

-56-

elements is not very significant and depends on the
choice of which particular component Pd is taken
diagonal, or also on the particular way in which the
principal axes system is attached to the molecular
equilibrium configuration. The three situation which
can arise are shown in Table III as cases (a), (b), and
(c). The contributions of I1, I2, II, and I12 of the
Pu-type terms to the energy and their direct correlation
with the point groups give very useful hints for the
develOpment of the general fourth order Hamiltonian of
(VI—1).

We rearrange (VI-1) into the form

H+ = H+' + ha+ + hb+ + hc+ , (VI-10)
where H+' includes all terms which are symmetric under
the symmetry Operations for the orthorhombic point
groups 02v, v=D2 and Vh=D2h, and (ha++hb++hc+) repre-
sents the remainder of H+, which is antisymmetric under
the symmetry operations of the orthorhombic point groups.
Thus H+' is the Hamiltonian to the fourth order of
approximation of a molecule which belongs to the ortho-
rhombic point groups. For monoclinic point groups,
CS=C,h, C2, and C2h’ additional terms beyond H+' appear
in the Hamiltonian because of the lower symmetry. Which
extra terms have to be added to H+' depends on the

manner in which the body-fixe principal axes system is

attached to the molecule, and one can distinguish three

-57-

cases as in the case of (dfixé) in the way pointed out
in Table III. In eq. (VI-10) we denoted the additional
terms for the case (a) of monoclinic point groups by
ha+, the additional terms for the case (b) of monoclinic
point groups by hb+ and the additional terms for the
case (c) by hc+.

In the symmetrized basis, the entire orthorhombic
Hamiltonian HI' will have nonvanishing matrix elements

only within the four Wang blocks E+, a , 0+, 0’ and all

additional terms allowed by the lower symmetries have

 

non-zero matrix elements located only outside the diago-

nal blocks E+, E , 0+, 0'. This is ascertained either
by computing the matrix elements involved directly, or
by examining the even-or-odd character of the matrix
element integrands under the symmetry Operations of the
rotational inertia ellipsoid, 02(x), C2(y), 02(2), using
the symmetrized basis wave functions. In this manner

the following arrangement of non-zero matrix elements

in the Wang matrix is found:

( E+ Monoclinic honoclinic Monoclinic
Orthorhombic case(a) case(c) case(b)
Monoclinic E" Monoclinic Monoclinic I

case(a) Orthorhombic case(b) case(c) ‘
ionoclinic Monoclinic O+ honoclinic

case(c) case(b) Orthorhombic case(a) a
Honoclinic Monoclinic Monoclinic 0' !

case(b) case(c) case(a) Orthorhombic}

 

-58-

Now, as in the case of the Ph-type terms, if we consider
a properly partitioned four-step matrix T which trans-
forms the Hamiltonian to a basis in which only the Wang

+

h a- +
blocks a , n , O

, and O' are diagonal, then we find
that none of the matrix elements located outside the
Wang blocks will fall within the Wang blocks because of
the transformation. This means that monoclinic matrix
elements of h2+* will contribute to the fourth order of
approximation, whereas elements of hg+* will contribute
only to the eighth order. Therefore, to the fourth
order we need consider only the monoclinic matrix
elements of h2+*, but not those of hh+*’ From eq.(VI-2)
the terms of h2+* to be considered are (2)th and
(2)2r2P2. The additional contributions of (2)2r2P2
will be combined with those of the Ph-type terms. As

a result one obtains the additional terms for the
monoclinic point groups to the fourth order of approxi-
mation, and we will give these simplified’ha+, hb+, and

+
hc later.

Performing the symmetry Operations of the ortho-
rhombic point groups upon (VI-1) but without the last
term of h%+*’ we get H+' as

+* +
H+I : (hv +h0 ) + h2+*l + hh+*', (VI-11)

with

-59-

+*t 2 a? 2 2
he =:2;<?:) pa +<2)Yaaqa >Pa
+dlil ( (2)YP\PBPr P ) (VI-12)
+*I ._ S (“~r 2
hh ‘ :((h)‘R< )
bb 2 2 2
+§Ja:, b(1+)Zaaqa qb })d
2p b2 2
Z (O( 2Q b2+ OK248.8.bb ) P
+ abgagb (h) Zaabbqa (H) pap a
at”: aa 2
+,§‘;€(1:)Zaaqa2+ m7“ pa ”3 p Pipe"
‘ (VI-13)
where 2;' means the summation over only those P P,R,P;

which are symmetric under the symmetry Operations.
With the aid of the vibrational matrix elements
(IV-2) and the following additional vibrational matrix

elements
<v.,vb|qa2pb2lva,vb> = fi2<va+ r><vb+ e>

(va,vb1qa2qb2|va,vb) = (va+ %)(Vb+ %) (VI-1H)

/
(va,vbip32pb2|va,vb) fi7(va+ %)(vb+ %)

the vibration-rotation Hamiltonian H+' for the asymmetric

molecules belonging to the orthorhombic point groups is
H+' : 1%+* + (A+A[+AII+AHI)PX2 + (B+BI+BH+BHI)Py2

+(C+C'+C"+C”')Pz2 + t 25' (ILU +4f'r)2xagpfg .

«.r “5?5
«IKo
' (VI-15)
In (VI-15) we have that

+XXY

.. 2m -1.- 3';
- 2(fi (2) M(2) m><va+2)-‘a((2) )(V 8+1)

-60-

I _ 2 yy 33+ yy +1 I
B = Va 1
ZOE (”Y +V(2)Yaa)(a ,g') 221((2 )[ja )( + )

C' _ 2 22 aa+ zz _ 3 3
2m (2)1 +(2)Yaa)(va e) (“(2% >(v am,

and that

Thus

(VI-16)
11" = (:32 E (“W
B" = (332 E m6 '1 (VI-17>
C" = (532 E (1011" ’
A"'= ;:b ‘h2(h)zbb(va +£ )(vb +l)
1,1261% ((332 aabb+fi11<f§zaabb’(va +11% + 1
= ‘Z $((M§X:b)(va + é)(vb+2)
B"'= ;: 'n2 yybe(v a+%)(vb +3 :)
a, b (“>81
+aqiiagb ((igzaabb+fi fin yyzaabb)(va+%)(vb+%)
E §;b((u)3;')(va+2)(vb+2)
C"'= ;:g‘fi2(:§z “22(v + H)(vb+-)
+a ,3a b (($28 abb fihcfizaabbma +11(vb+11
r. Z.b((u)X“')(va+ é-)(vb+%) . (VI-18)

the effective rotational constants CZ, 68 , and

are to fourth order

a = A+A'+A"+AH'

-61-

A+Z((2)% ')(V a+~)+(hqa;"+-a2;w((h0 O<"')(va +9 §)(vb+;)

6 = B+Bi+BH+BHI
= B+“K B')(v +%)+ '“+2ib ( @‘”)(va +% )(vb+
:‘(2) a a (H)6 (H)m
C : C+C’+CH+CH'

C+Z((2)gra )(v 2f+2)+(1+)X"+aZb((1+)Kab)(va+:‘ )(vbn)
(1-19)

The various coefficients <X, Q, and X introduced here

are related to the coefficients Y and Z of references

5 and 6 through (VI-16), (VI-17), and (VI-18), and

depend on the molecular constans in a complicated manner.

Also in (VI-15) we have that

x»; __ dBmZ 2am X5 Zaa
% (“Mix “2‘ (1+)Za am m2 )(V a”)
.. 0/015
=%( (Lr)r8, r)(v a+%) = k [Do/f /( o (VI-20)

We have examined ’Qfxé of (VI-15) in the first part of
this chapter, and the detailed expression for fixgrg
reveals that fl£x5 = Ry/wx° We have reindexed fgfitg
in Table VII. With the aid of Tables V and VII we can
write the Hamiltonian for the orthorhombic point groups

to fourth order of approximation,

W! = hv+* +CLPX2 +fipy2 +CPZ2
+i(”‘+ )P h+i(7‘+F )P 1++('r‘+f’)P h
4 L1 F1 X 4 2 2 y K3 3 2

~ n. 2 2 D 2 2
+~i<fl++fl+>(Py2PZZ+PZ2Py2)+4(¢5+F5)(PZ PX ax P2)

-62—

Table VII Nonvanishing (L 5,.

 

‘-—--

 

(“5%) Fn'iflaxg
(XXXX) fl
(yyyy) F2
(2222) t3
(yyzz), (zzyy) f4
(zzxx), (xxzz) f5
(xxyy), (yyxx) F6
(yzyz), (zyzy) F7
(xzxz), (zxzx) Pg
(xyxy), (yxyx) F9
(zyyz) F10
(yzzy) V11
(zxxz) F12
(xzzx) F13
(XYYX) P14

 

-63-

J. 2 2 2 2
+‘(Yo+f%)(Px Py +PY PX )

+i7‘

. 2i 2 1_,f~ 2
L7(Psz+PzPy) +‘7é(PzPX+PxPz) + (P P,+P,P )

71‘9 x J J x

PZPy)+:fé(PZPXPZPx+PXPZPXPZ)

-10
+4|7(P P P P +PZPy

Y Z Y Z

+¢f§(PXPyPXPy+PyPXPyPX)

+%{’1O(P2Py2pz)+€ {J11(Psz‘PyHéF12(I>ZPXZPZ)

+%F13(PXP22PX)+%f1h(PXPy2PX)+&(35(Pny2Py). (VI-21)
From the angular momentum commutation relations Kivelson
and Wilson27 have established that

(qu

o c o 0
5+PBPR)“=2(Pd2Pgd+P3‘RX‘)+fi2 3P52-2Pd2-2P32),

(##flix3 and cyclic). (VI-22)
With the aid of these identities, (VI-21) can be

simplified somewhat to:

+1

+* ») 2 2 2
Ii 21% +an+fiPy+CPz
#414? W 24AM.” )p 2+-‘.<<r-+-° >9
1 1 x q ‘2 32 y 9 ‘
+iffh+2f7+)h)(Py2P22+P22Py2)
3.4- 4~ ;; 2 2 2 2
+4(‘5+2‘8+_5)(Pz PK +PK P2 )
+£(“ +2 +- )(P 2? 2+p 2P 2)
6 9 6 X y y x
-1.? " ‘ - I) '
+4 7(PszPyPZ+PZPyPZPy)+%.8(PXPZPxPZ+PZPxPZiX)

+4(c(PXPyPXPy+PyPXPyPX)

+iF1o(Pzpy2Pz)+iffi1(Pypz2py)+%?12(PZPX2pz)

-64-

+tf13<PXPZ2PX)+-é {’1 M PXPYZPX )+-zii F1 5< PyPXZPy).
(VI-23)

The relation (VI-22) has the effect of combining
the coefficients T7, T8, and T9 of (Pong +P5Pm)2 with
TL, 73, and‘fb of (R12P32+P92RX2). The grouping of
IL through '19 occurs as (TL+2Y% , (15+2Yé) and (7é+27§).
Thus, in general case only these combinations (in
addition to 1}, TE’ and-73) can be obtained by fitting
experimental frequencies. Hence for these cases only
six independent distortion coefficients can be found
and'Tg, “(5, “T6 cannot be separated from ”T7, ~18,
’f9, unless enough additional information is available
to compute some of the ifs involved directly from their
definition. Also, the expressions for C1, 43, and Q:
acquire the additional terms ifi2(37§-27§-2Té),
%n2(3Té-21§-21}), and £fi2(3T§-2Té-21§) respectively, as

a consequence of the application of (VI-22)

The nonvanishing matrix elements of the Hamiltonian
(VI-23) in the symmetric'rotator representation are
given in Appendix I. For a particular choice of J, the
diagonal blocks 8+, 3', 0+, 0" can be constructed easily
from these matrix elements of Appendix I. In Appendix

II we give those eigenvalues of (VI-23) which can be

established in closed form.

The additional contribution to the Hamiltonian for

-65-

the monoclinic point groups will be:

Case (a).

+ - l o 2 2 3
ha _ 4("0(PX Py+PX PyPX+PKy P PX +PyPX )

+é7;1(Py3Px+ PyZPxPy+ PyPX Py2+Pxy P 3)

i- 2 2 2 2
+4 12(PXPsz +PnyPz +Pz PxPy+Pz PyPX)

+£7‘3(P P 2P +P 1 2Px+P P P P +P P P P

x z y y z z x y z z y x z
+P zP P P +P P P 2? X+P xP zP P +P P P P )
x z y z y y z y z x Z
+£9(PXPy+Py PX), (VI-2H)
where
at? =Z((2)gé)(ve_+i-), (VI-26)
and
' = fyi“an2 xyy (VI-26)
(2)5e (2 > (2 ) 8a

. . . + . .
The nonvanishing matrix elements of ha in the symmetric

rotator representation are given in Appendix III.

Case (b). The additional contributions hb+ can be found
from (VI-24), (VI-25), and (VI-26) by making in these
the replacements indicated in Table VIII. Nonvznishing
matrix elements of hb+ in the symmetiic rotator repre-

sentation are given in Appendix III.

Case (0). The additional contribution he+ can be found
from (VI-2%), (VI-25), and (VI-26) by making in these

the replacements indicated in Table VIII. Nonvanishing

‘gn

-66—

Table VIII Corresponding quantities for the
monoclinic contributions to the
Hamiltonian.

 

 

 

ha+ hb+ hc+

T1o {1L1 <18

71 1 “C1 5 “£19

712 <16 <20

11 3 T17 T21

x y z

y z x

z X Y

«5* E (7
ny YZY ZXY
(2) aa (2) aa (2) aa
(2) (2) (2)

(2)3a (2)5a (2)»at

 

 

-57-

matrix elements of hc+ in the symmetric rotator repre-

sentation are given in Appendix III.

For the triclinic point groups, C1 and Cizs the

2’
Hamiltonian to the fourth order of approximation is

given by
H*' + ha* + hb+ + hc+ (VI-27)
where H+' is given by (VI-23) and ha+, hb+, and he+

are those of the monoclinic point groups.

VII. PLAKAB ROTATORS

When the equilibrium configuration of the molecule
is planar a number of "simplifying" features will apply.
Planar molecules must have at least one reflection
plane, viz., the reflection plane which lies in the
plane of the molecule. Thus, four of the eight asymmetric
rotator point groups from Table 111 need not be con-
sidered here. Furthermore, it can be shown by giving
specific examples that all asymmetric rotator point
groups which do have at least one reflection plane will

admit the planar condition, and

3’; = o , (VII-1)
where the {-axis is perpendicular to the plane in which
the molecule lies.

We refer to Table IX for the planar asymmetric
rotator point groups. It is noted that the triclinic
point groups do not admit of planar rotators. Hence,

the most general planar rotator Hamiltonian can contain

at most thirteen 1‘s, and to fourth order of approxima-

+

tion is composed of H+' and only one of ha , hb+, or

hc+, as explained in Chapter VI.

The condition for the planar molecule (VII-1)
implies

0
Ida + 130 = 13X , (VII-2)

-68-

 

 

m
Jmmo magma; .+
.H smaum>
..Tm
xxx amocflaucom >m
PH 0 canaoznompno
m, on . n: . an 0 one mm
/o w mum . + + + a e .+m
. am a o manna mammam m P m u
HH . HH . H no ono ocm PH :10
o «a. .m.
a a a mo mqo cam
wa nwocfifianoc + + + 2+m
NHH ..HH .mH mo oco one PH neoumo oanaaoono:
mamemxm cwfluopafiamm mo anon HoQHAm waspwaocoaoq
dam mnamw mo mpmm macho oangmnmoaawpmmno

 

 

.mQSOMm pcfiom nepMpom oahpmeszmw mmnwam NH manna

-70-

by the definition of the principal moments of inertia.
If we choose for convenience of discussion the x-y plane
as the plane in which the molecule lies, we have from

(VII-2) that

1/A + 1/B = 1/C (VII-3)
from the definition of A, B, C in (IV-6). One should
not confuse the equilibrium rotational constans A, B,

C with Cl, d?, C, of eq. (VI-19) which represent the
instantaneous rotational constants of the molecule.

Due to the inertia defect and non-rigidity one does not
have the same kind of relation (VII-3) among 62,68 ,(1 ,
and (VI-19) directly shows the relationship between the
equilibrium rotational constants and the instantaneous

rotational constants.

Since the molecule lies in the x-y plane we have

the following for eg3 and a:“ of (11-31) and by the

condition 2? = O;
agz = O
(VII-#)
2
ag = O ,
and
aix + aZy = 3:2 . (VIJ-5)

Also, in planar molecules we have the following prOperties
of the elements of the normal coordinate transformation

matrix 1:

1?s g 0, 1y

is ¢ 0: lfis = O for in-plane vibrations,

-71-

lfs = O, lis = O, lis ¢ 0 for out-of-plane
vibrations. (VII-6)

For the planar molecule Oka and Morino12 found the

relations,

aXX = agy = agz = O for out-of-plane vibrations,

s
(VII-7)
and
XX YY _ ZZ _ . , .
Ass + ASS — ASS — 1 for in-plane Vibrations,

AXX = AYY = 1 and A22 = o for out-of-plane
ss ss 55 ,
Vibrations. (VII-8)

a and A:: were defined by (II-31).

Similarly, one secures easily the following results
for the Coriolis coupling coefficient 5;; of (II—3H)
of the planar molecule by the relation (VII-6):

If a and b are both in-plane vibrations,

x _ _ z
ab _ o, ggb _ o, :fab 7! o. (VII-9a)

If a and b are both out-of-plane vibrations,

X Z
ab = o, ggb = o, gab .-. o. (VII-9b)

If only one of a or b is an in-plane vibration,

50;) a! o, 5gb 1! o, 1:2,) = o. (VII-9c)

Triatomic molecules, which are a special case of general
planar molecules, admit only in-plane vibrations. In

cases in which the plane of the molecule is the y—z plane

-72-

or the z-x plane, the appropriate equations corre-
Sponding to (VII-h) through (VII-9c) may be obtained
by permuting x, y, z cyclically.

When we apply (VII-H) to the definition of Tpr-,

5‘3 x.
7 — .1-?— a "' a.“
"16.x "" ..2 A S S

O O 0 O
SIV’11'3dIAKIN ,

 

we obtain after some rearrangement the Dowling28 rela-

.-‘ ,9.

tionships among 41 through 19:

/7 = ’8 = O ,

f1 = <A/C)225 - (A/B)2ié ,

‘r 2_ - 2,, (VII-10)
L2 = (B/C) 4 - (B/A) 6 ,

-- _ 2, - 2

if the molecule lies in the x-y plane. Thus there are

f

only four independent distortion constants among ;1
through “T9. Eq. (VII-7) reveals that the four inde-
pencent ~f's may be built up from contributions of the
in-plane vibrations only, so that the summation over
vibrational modes could be restricted to in-plane

29

vibrations only. Hill and Edwards gave the specific
expressions for the second order centrifugal-distortion
energy in planar asymmetric-top molecules in terms of

the four independent distortion constants.

multiplying (VII-5) by afiy, it is easy to prove a

relationship among the four additional T's which appear

-73-

in molecules belonging to one of the monoclinic point
groups. For the x—y planar molecule we find in this
way

(1/A)2’T‘ + (1/B)2-’q1 = (1/C)21” (VII-11)

1O 12 ’

and from (VII-h) we immediately secure that

~713 : o . (VII-12)
We summarize the conditions and relationships among the
”T's in Table X, in which we give the apprOpriate
equations for the y~z planar molecules and x—z planar
molecules corresponding to (VII-13), (VII-11), and
(VII-12).

The planar condition introduces also many simpli-
fications into the detailed definitions of the f's, but
no simple relationships similar to those holding for

the T"s appear to hold among the f's.

‘q

-7u-

Table X Relationships among planar rotator

distortion constants.

 

 

 

 

 

Plane of
molecule Relationships
x-y C7=f8= (13:0
7 _ 2,. _ 2r“
_ (A/C) a5 (A/B)26
1. _ 2f .0 ,fi
t2 - (B/C) tg - (u/A) ‘6
1 _ 1 2, ; 2e~
[3 — (C/h) ;5 + (C/B) 11+
(1/A)2730+(1/B)2(312(1/C)2 is (monoclinic)
710 = ii1 = (32 = O (orthorhombic)
’7‘ _ '7‘ _. ..
Y‘Z ‘8 " fig -117 - O
T} = (n/B)2fg + (A/c)2<5
a. _ 2%2 ‘ 24-
12 - (3/3) ‘6 - (B/C) 14
.e _ 2,. 2,-
‘3 "' (C/[&) \5 " (C/B) kl+ h .\
(1/B)27,g+(1/C)27;c=(1/A)‘716 (monoclinic)
7,4 = 1,5 = Ylé = O (orthorhombic)
z-x f7 =‘Té = {21 = O
_ 24~ 2,
Té = (B/C)27g + (B/A)21é
‘- _ 2x. _ 2,
(1/A)2C,E+(1/C)2f,9=(1/B)2’éo (monoclinic)

/‘

£18 2 119 = :20 = O (orthorhombic)

 

 

VIII. FOURTH ORDER CENTRIFUGAL DISTORTION TERMS.

In the previous chapter we neglected the fourth

6---type terms),

order centrifugal distortion terms (P
since their contribution to the energy is only signifi-
cant for J23O in general cases. However, as indicated

in the previous chapter, experimental results25 indicate
that the second order centrifugal distortion terms (Ph-

type terms) are inadequate to describe the stretching

ILNMA‘L

effects in some cases, such as for the planar triatomic
molecule H20, even for the lower J values. In these
cases it is necessary to consider the fourth order
centrifugal distortion effects to get a satisfactory
fit to the spectral data. For these reasons we present
in this chapter a survey of the fourth order centrifugal

distortion terms.

These fourth order centrifugal distortion terms

were given by the last term of eq. (VI-3) as

* = Z: 0‘3? "
II «3((511 (4)ZBPP P P Pqu , (VIII 1)
where V W(L$Z are the fourth order centrifugal distor-

tion constants (coefficients) and the detailed and very
lengthy expression of them in terms of molecular
constants is given in Appendix IV. We shall use the

““ K56, )‘Z where

alternative notation Exp r8? “2 for
convenient and further denote Legzlt'g_by 131, i=1,2,3,...

-75-

-76-

where the numbering of the 69's will be established
later.

Because of the noncommutativity of the angular
momentum components Rx, (VIII-1) shows that there is
a total of 36:729 coefficients[PBX156‘f. Hewever, if
we apply the symmetry Operations upon the P6-type
operators we find that only 183 terms are nonvanishing
for orthorhombic point groups and 365 terms are non-
vanishing for the monoclinic point groups. The P6-type
Operators, whose pr{§(vfi?s do not vanish for the ortho-
rhombic point groups are found to have real matrix
elements of type (KIK), (KIKiZ), (K‘Kih), (KIKi6) only.
Therefore, as in the case of the second order centrifu-
gal distortion terms, the entire matrix elements of the
183 nonvanishing P6-type terms for the orthorhombic
point groups will lie only within the four Wang blocks
E+, E’, 0+, 0- in the symmetrized symmetric rotator
representation. Further, all edgitional terms allowed
by the lower symmetries in the case of monoclinic or
triclinic point groups have nonvanishing matrix elements
located only outside the Wang blocks. This is ascer-
tained in the same manner as for the Ph-type terms.

u

Now, as in the case of the P -type terms, if we
consider a prOperly partitioned four step matrix T'
which transforms H* to a basis in which only the Wang

blocks are diagonal, we find that none of the matrix

-77..

elements located outside the Wang blocks will be brought

into the Wang blocks due to the transformation. This

means that we need to consider only the 183 P6-type

terms which are invariant under the symmetry Operations

of the orthorhombic point groups in the study Of the

fourth order centrifugal distortion effects not only Pt
for the orthorhombic point groups but also for the mono-

clinic and triclinic point groups, since the remaining

nonvanishing matrix elements will contribute to the

”than,

energies only in approximations higher than the fourth.

Close examination of the parameters in the ex-
pression Of Excxicyg in Appendix IV reveals the following

relations:

«58111 = easm (VIII-2a)
q'ngn -_- Pawn VIII-2b
(2) (2) ( )
“Han = min (VIII-2c)
0(8 _ 3"

(2)Umn _ (2)Umn (VIII-2d)

for asymmetric molecules. From these we are able to

obtain among thexwpzflkivj the very useful relation

I°<B¥86133=Mw81'51; . (VIII-3)

Relation (VIII-3) reduces the 183 coefficients to 105
independent fourth order centrifugal distortion co-

efficients.' The 105 independent coefficients may be

classified

Type
Type

Type

-78-

as
['XCX 01'0" 0" ’3. ‘. 3 coeffiCientS ,
[VQPVX1>PJand 5H coefficients,
permutations thereof, Ckfifi)

pot 8 f3 3’ Y] and 1t8 coefficients ,
permutations thereof, ngbfigj

Total 105 coefficients.
(VIII-u)

These 105 distinct coefficients are given in Table XI,

where they are grouped in the manner Of (VIII-h), and

indexed by i Of 61. The fourth order centrifugal dis-

tortion terms can now be expressed by the 105 81's and

their associated P6-type Operators with the aid of

Table XI and (VIII-1). This expression would be accurate

to the fourth order of approximation for the general

asymmetric-top molecule.

Since H20, one Of the molecules which show the

effect of Pé-type stretching terms even for low J values,

is triatomic, and further simplifications are feasible

for triatomic molecules, we considered the fourth order

centrifugal distortion effects in triatomic asymmetric-

tOp molecules as a special case.

In triatomic molecules simplifications arise due

to (a) planarity (any triatomic molecule is necessarily

a planar molecule), and (b) the absence of out-Of-plane

vibrations in these molecules. The planarity condition

-79-

Table XI Coefficients of fourth-order centrifugal
distortion Operators («ifib’oé 1]: 61 and
relationships among them.

 

 

I“

‘ Relationships* for triatomic molecules
i1 [eta {SE11} xy-piene 1 yz-plane 122-pien‘e

 

 

fl —Y

 

1

I 1
1 1 i 3 1
1 .
12www '

33 zzzzzz

l
L

1 hfxxyyyy=yyyyxx 6=7=8
1 1 .'
3 Sgyyxxyy 9=1o=11=12

 

12: O 6=7=8:O

 

10=7+9 11=12=O

 

i 61yxyyxy ‘9=10

- - -...- ...... -..-

; 71xyyyxy=YXYYYX.
f 8§xyyyyx

. --N .___ ..-... *4.— ---”--“-.-_-—-”_4

5 9 xyxyyy=YYYXYX1
101yxyxyy=yyxyxyi
111xyyxyy=yyxyyx i
123yxxyyy=yyyxxv f 1

A

T 7 7

113§yyxxxx=xxxxyy j15=16=17 1 15=16=17=0 121:0

1h xxyyxx S18=19=2o=211 2o=21=o 119=16+18
:15 xyxxyx 1 ;18=19
;16 yxxxyx=xyxxxy : ;
1171yxxxxy I
118'yxyxxx=xxxyxy 1
119 xyxyxx=xxyxyx
120yxxyxx=xxyxxy

 

 

”M“-.-_..p—.—.- ...- —-\ ... .— .n- — - .
.5

1211xyyxxx=xxxyyx 4-4
(continued)

-80-

 

 

 

 

 

l. AA xy—plane 1 yz-plane ? zx-plane
22 yyzzzz=zzzzyyfr2H=25=26=O 2h=25=26 130:0
123 zzyyzz +29=3o=o i 27=28=29=3o :28=25+27
2M zyzzyz 27:28 1 1
25 yzzzyz=zyzzzy 1 1
126 'yzzzzy 1 1 i
27- yzyzzz=zzzyzyl 1 2
28 'zyzyzz=zzyzyz_ 1 E
'29 yzzyzz=zzyzzyl ': f E
30 ‘zyyzzz=zzzyyzz E j
—1 1 ' 1 1
31 zzyyyy=yyyyzz 39=0 1 33=3#=35 1 33=3h=35=0 1
-32 1yyzzyy 37=3H+36 E 36=37=38=391 38=39=O f
'33 1yzyyzy E E 36=37 3
f3%‘ zyyyzy=yzyyyz 1
E35 zyyyyz . w .
136 zyzyyy=yyyzyz' : 1 1
'37 yzyzyy=yyzyzyl i 2
138 zyyzyy=yyzyyz Z 1
139 yzzyyy=yyyzzy1 : 1
’ c i’ ‘ “if ’%
Etc, zzxxxx=xxxxzz§ h1=0 : h2=h3=kh=0 é h2=h3=kk 1
EH1: xxzzxx 1 h6=H3+#5 ‘ b7=H8=O é h5=k6=h7=h8§
EMZ; xzxxzx i 1 h5=k6 1 1
1131 zxxxzx=xzxxng g 3 1
1h#i zxxxxz g l 1
3h5§ zxzxxx=xxxzxz§ E
' xzxzxx=xxzxle3

‘ ' (continued)I

-81-

! (Ky-plane)| (yz-plane) 7 (xz-plane)

 

 

 

 

1
szxzxx=xxzxxz 1 1
#8 xzzxxxzxxxzzx 1 1 f
#9 xxzzzzzzzzzxx l51=52=53=0 157:0 1 51:52:53
ISO zzxxzz 56=S7=O 55=52+5k 1 54=55=S6=57‘
151 zxzzxz '54255 1 1 ‘11
152 xzzzxz=zxzzzx 1 j
153 xzzzzx
15h xzxzzz=zzzxzx i
155 zxzxzzzzzxzxz : ‘ 1 6}
56 xzzxzzzzzxzzx 1 f 1 f
67 zxxzzzzzzzxxz j 1
7 —1 l
58 xxyyzzzzzyyxx - -
$59 yyzzxx=xxzzyy ‘
6O zzxxyyzyyxxzz
61 xyyzzxzxzzyyx 6126220 1 62=63=O E 63261=O
'62 yzzxxy=yxxzzy 1 . :
-63 zxxyyz=zyyxxz1
i: 1 1 #e
6% yxxyzzzzzyxxy ’6#=65 E 66:67 I 68:69
65 xyxyzzzzzyxyx; 68:0 ‘ 64:0 1 66:0
66 zyyzxx=xxzyy21 5
67 yzyzxxzxxzyzy?
:68 xzzxyy=yyxzzx§
zxzxyy=yyx2X21

69

I

 

+_(continued7A

1
7O
71
72
73
7M'
75*

xyyxzz=zzxyyx
yxyxzzzzzxyxy
yzzyxx=xxyzzy
zyzyxx=xxyzyz
zxxzyyzyyzxxz

xzxzyyzyyzxzx

-52-

xy—plane yz-plane 1 zx-plane 1

1'

. 1 I

70:71 T72=73 g7h=75
72:0 7k=O §70=O

57-73.

 

76‘
77
78

yxyzzx=xzzyxy
xyxzzy=yzzxyx
zyzxxyzyxxzyz

yzyxxz=zxxyzy

76=77=O 178:79=o 80=81=o
78=81=82=83 77=80=B3=84 76=79=8H=82§
80:75-69 76:71-65 78:73-67 ‘
.79=67-73 81:69-75 77:65-71

...,--.JL.

xzxyyz=zyyxzx'

yxyzxz=zxzyxy

zyzxyx=xyxzyz

zxzyyxzxyyzxz-

xzxyzy=yzyxzx'

xzyxzy=yzxyzx
yxzyxzzzxyzxy

zyxzyxzxnyyz

.yzxxzy
zxyyxz
xyzzyx

‘xzyyzx

yxzzxy

93 izyxxyz

l

1 T
i85:88=91 86:89:92 1 87:90:93
86=87+89-93 87:85+9o-911 85:86+88-92

(continued)

_83-

 

 

 

xxeplane [p yz—plane_J xz~plane#J
; if 11
9h xzyyxzzzxyyzx ,95=92= f96=9§= {94:91: 2
' 90+70-6%1 8 +72-661 89+74-68?
95 yxzzyxzxyzzxy ' +105-93 1 +103-91 1 +10%-92 '
' 1
96 zyxxzyzyzxxyz 1
971 xzyxyz=zyxyzx 197:96 198=9h _ 99:95
985 yxzyzx=xzyzxy 1oo=9u 11o1=/5 ; 102296
99% zyxzxy=yxzxyz .99=101= l97=102= ; 98:1002
? . 87+105-93 85+103-91; 86+104-92
100} yzxyxzzzxyxzy 3
E 102=103= :1ooz1ou= 1o1=1os=
101: zxyzyx=xyzyxz' 81+97 77*”:8 3 79+99
3 ; f 1
1021 xyzxzy=yzxzyx 98=1OH= .99z1052 972103:
' 77+101 79+1Q

1031 xyzyzxzxzyzxyi

10h: yzxzxy=yxzxzy§
1 i

i I
11051 zxyxyz=zyxyxz‘

81+1ooT

L 1 4__ I

t

*For the entries in these columns, EL

1

is replaced by i

in the interest of clarity of reproduction.

-au-

gives us, as shown in Chapter VII, the following

prOperties of a:3 of (II-31):

a:Q ¢ 0
(«#8) (VIII-5)
8dr = agar: O,
s s
if the molecule lies in the as-plane. Also, from . ,-

(VII-9a), which was proved in Chapter VII from the
prOperties of the normal coordinate transformation
matrix, we have the following prOperties of the Coriolis
coupling constants due to the absence of out-of—plane 4}

vibrations:

if the molecule lies in the cfiS-plane.
Applying (VIII-5) and (VIII-6) to the expressions

of the parameters appearing in Appendix IV, we have

found:
dem :uXSm zsxsm z’flsm : O (VIII-7)
cxsmn: Ssmnz O (VilI-S)
(Z3Umn= {Egumn= (gyUmn= (EgUmn= O (VIII"9)
:ZjUn = iigUn = O (VIII-10a)
?;:Un = igiUn = gggun = flégun = o (VIII-10b)
:giU“ = :::Un = iééun = éngn = Q (VIII-10c)
:égUn : f§§Un = :éiun = {éfvn = O (VIII-10d)

ox'd '
a c<n _ 33£<n : (rgpn _ (VIII-11a)

l
I
O

—'3 In "nal-nIL-rn'v.
I.
. m5, II.-.‘

agaxn : gag ,n = rot 251 = Xerxn = o (VIII-11b)
Po<n = 39fo<é<n = mallmdl : Bden = O (VIII‘11C)
o<.( a Y ,
30(1’1 = ldddin : V5¥0<n = «3!;an : O (VIII-11d)

The nonvanishing parameters are also further simplified
by application of (VIII-S) and (VIII-6) for the triatomic
case. Substituting (VIII-7) to (VIII-11d), one finds Fm
that 18 of the 105 distinct coefficients now vanish and .
that there exist 3% linear relationships among the non-
vanishing 91's. Therefore, we now have only 53 inde-
pendent distinct fourth order centrifugal distortion 9
constants for triatomic molecules. We summarized these

conditions in Table XI for xy—plane, xz-plane, and yz-

plane planar triatomic molecules.

The contribution of the fourth order centrifugal
distortion terms to the energy levels could be calcu-
lated, in principle, in terms of 105 9'5 for the general
case and in terms of 53 9'5 for the triatomic case by
evaluating the matrix elements of the relevant Pé-type
operators. For those energy levels listed in Appendix
II it would be possible to give closed form expressions
for the energy contributed by the Pé-type terms. However,
this has not been done because of the length of these
expressions, and therefore we have not listed the

matrix elements of these terms.

IX. CONCLUSION

By taking advantage of symmetry considerations,
the vast number of terms appearing in the general
vibration-rotation Hamiltonian for asymmetric molecules
to the fourth order of approximation can be reduced ».
considerably. In fact, by distinguishing three types
of Hamiltonians, for orthorhombic point groups, mono-

clinic point groups, and triclinic point groups, we

were able to rearrange the Hamiltonian into reasonably
tractable form.

For the orthorhombic point groups the Hamiltonian
takes the simplest form and closed form solutions for
the energy levels for a few low J values were obtained.
Although closed solutions for the energy levels are not
available for most higher J levels, we expect that
numerical analysis can now be invoked to higher accuracy
with the aid of computers. Therefore it will now be
possible to analyze centrifugal distortion effects and
vibration-rotation interactions in higher orders from
the deduced Hamiltonian.

For the monoclinic and triclinic point groups we
face even greater difficulty in getting closed form
solutions for the energy levels. However, as in higher
J cases of the orthorhombic point groups, it would be

ossible to emplo' numerical anal sis with the aid of
. Y

-86..

f. muFKq’aflNuuw
e

-87-

computer techniques, and thus our Hamiltonians should
be of practical interest.

Further simplification of the Hamiltonians can be
had in the case of molecules with planar equilibrium
configurations, and for triatomic planar molecules,
eSpecially, the analysis of fourth order centrifugal
distortion effects should become feasible.

Because fourth order effects are now in many cases
within the limit of eXperimental accuracy, this work
should be of value in obtaining still more satisfactory
analysis of the infrared and microwave spectra of

asymmetric molecules.

APPENDIX I

Herein are given the nonvanishing matrix elements
of (VI-23). The phase angle chosen for the angular
momentum components is such that

(xlpxjh+1) = (K+1 Pxfx) = %n[(J-K)(J+K+1)1%

1 , i
(R1? 3+1) =-(K+1!Py'K) =-%ih1(J-K)(J+K+1)]2

yl
(hlelK) = id:

« and f=J(J+1).

(I; H“; K) = Ev + '%h2(a+d3)(f-K2) + {125112

+ aa“<(3+123>1<‘*

+ lgh”<r1+12+2g+1+1‘9+ 111+ F2+2F6> (f-KZ)2

+ anhctg+15+2z7+2“8+fg+r§+f7+f8
+§130+gfg1+g{32+%{g3)(r-K2)K2

" tlfifihm +T2'2 @475" 51* 1””2'2 590219" P14“ 1 5)

xfif-K(K+1)flf-(h+1)(h+2f
+[r—K(h—1X[r-(h-1)(h-2H}

+ §hu(F11+F13)f

+ éfiH(-QP7-2fé-99-3fi1-3fi3+%f34+%f35)Ké

-88-

-89-.

(K1ai'1xia) = {a2<Cl-a3)

+ much-{2+ 91-92) [f-é—I-fi-aE-(Kizfil

+ éfih(-Ty+lB-ZT§+2Té-Pg+F5){K2+(Ki2)2]

+ i’fiu(-2f7+2f8- 1011+f13)(K:1)2

+ ihh(P1n-V15)]

x a—[r—Kum )1’5' [f-(K-r1)(1<t2)]%

4‘ A 7

(K\H“\ Kin) : $11<TI+TX2TKHT9+ P1+Fz'2 {2‘2 £9" F19“ F157.) 3

X \Lf'MKn)]%[f-(Ki1)(1<:2)]%

x [f-(Ki2)(K13)]%[f-(Kt3)(xthy]%

APPENDIX II

Herein are given those eigenvalues of (VI-23) which

can be established in closed form.’

E(OOO)
E(101)
an“)
E(11O)
21202)

3(212)

3(211)

E(221)

3(220)
M303)
n<313)
3(312)

_‘ ’3 _, .
Lv+h4(1+uJ+¢fih(11+

t1]

V

53+215+h:8+,1+_3+215+’1O+fH%)
)

E1362 (015+C)+-éfi)+(f2+ (3+2TLI’+1+T7+ P2+ 1734-2 91++ 012+ 915
)

~r+P+V+2P+:

a 2.. 11+ 1
uv+fi (LLH )+_4fi (11+E+2 L6+l+ 9 ,1 2 16 111+f13

EV+ +%:fi% -(312+h(,2)%j
Ev+h2(a+2-+h‘)
+fi’+(%f +13;'5‘2-1111'31271fi215-1.a -%+1+7;7+1H8+ 19
+%f3+4f)H+HF +2'H+2f5+2 6+ H7 8+ 9
110+?f11+f12+4f13+ilh+115)
Ev+h2(hd+&3+(3
+fi4(nf1+%Té+%i3+%fh+2:§+2I6+T7+h7g+h79
+ufy1+%f;+i13+afh+2fg+276+17+f8+yé
+P1O+P11+4412+'13+71u+4715)
Ev+h2G1+ME+C)
+hh(%?1+47 2+?f3+21h+%15+:26+hi7+C8+h7§
+4F1+ug +fi? 3+2"+2 5+2 f6+97+-8+i9
+3.1O+ f11 H112+f13+il1h+ #35)
+% a;1+(312+uy12)ii

+3%\ 2-(p2 2+4B’N2) ]

11‘v
E

ff;

V+
Ev+2 110(3"(33 :+l*/32)2J

E +17% _(" 1 2+4( 2);?)
v 2L h in h

-01-

I

E(322) Ev+khgox+i3+c)
+13%»: +11}: 2+L+r3+8 1+815+8r6+1617+1sz8+16<9
+8 F +L+ 1+8 f3+81+85+81 517+? 108+? 9’9
+l+1310+1+p11+1+f112+2+1&13+)+/01)++)+/015)

E(321) = E v+2k%2 4(3 22+4322)21
28331) = E:++:«3+<,a32+w32 >8;
13(330) = EJ811412 812$};
E(l+1)+) = E Vegas-(13 5 24115: )‘3
E(’+13)= E v-+2-[o/.6 -(f»6622+1+3’ )2.
30123) = Ef+2-18<7-(e:2+1+2r:2>f"

. _ 2 2 2
D(h32) - E v+§:d‘:+(35 +885 )

£104.31) = E v+%[m 6+(;: 624‘47562)2;
E(1+)+1) = E v+2Lm 7+( 7 2+1+dl72)i:

.. O 8- 2 2
E<52l+) - Ev+éi ((3 822+l+(8 )2_;.

)=Ev+2~'cx 0(84'LHV8)

N1 = %2(a+:5 +C)

1‘" f +‘ 'r 7" " “"
+h (4L1+”T'~+_3+284+2+5+216 +#.7+h.8+429
+l+ F1 +1+:f’ 2‘5'11-“3‘l'2 “+2 5+2 96+? 7+ '8+ 9
+r1o+r'11+r'12+5’13+r11+ '15)
1‘31 = 2132(34'4’3 -20
1+ «r r‘ 4' ’7" 7' 7' 'r
+h (2 {1+2f2-Li'1-3-2 ”...-2 “5+“. gé-l‘i' k7-]+‘-8+8 L9

+2P1+2F2-”f3'2? ’2€5+h€6_97-F8+2fé
#1014911-1’12*%F13*4'01‘++H15)

“Li-:5, ,'
1

-92-

>3 = (3)319’120- r”)

+(3Wfi(T1-r2-flufS-21'+2L8

] r‘

1.9 ___+, _1 _;
4-91-92“ Flt-...FS- 27+728 4311-9113 4914415
fi2<101+1+5+1oo

)
¥\2 =
+fil*(.g.l(1 +1+12+glr3+81+1 1 T5+8T6+16T7+22Z8+1679
+—L2i1-91+1+92+12fl93+8f71++1195+8f96+7 P7+7 58+]:9
+élf710+1+911+1+fi2+uP13+glpm+l+fl5)
32 = fi2(7a+6-8;‘)
1+ #2.. 1-__1_1_ _ 1»- «v-7, 4,-
3 . 1 .l ,. j 14 1;
+35 {1+ 82-20 F361 FT-395+-§ 166-2 1“ 7-2 98"”: : 9
,. 1 7
‘9! 916911391912" 913*52'911” r15)
X2 = -§(15)'5‘fi2(a—5)
+*(15)1§f14(5/c1-2Q- 5TL++5 {5-316-10 9-74.10/8-6 T9
9 p - Q _ p _ u? _ “
+5i1-2.2 Sui-595 3.96 Nf7+ {’8 3&9
"32*10'2 911+ 912+25’13‘ {HM-2915)
o<3 = fi2(1+1+10(8+1oc)
1+(8T+l+11'2+1+1T+22TL+16T5+16T6+hHTZ7+32Té+32T§
a J L) ,9 "
+89 +41fg+h193+22i4f1625+16=6+1”{7+1”rg+1999
+8€1O+8f911+11F12+89J13+8311++117'15)
(33 = {12 (914:7:3 ~80
+%h”(2f +357 2403-53-11154-1976-10’777—22’78+3829
+2 9185924093 5 u-11f5+19f6—1+97—1+98+11 (“9

+29 i? 2913+2 ‘11} +5 915)

X3 = -§-(15)‘i?fi:(d-t)
+£115>4fi (2 11-5 ' 2-51.35 15+3 '6-1o*7+w "8+6-c;
+2f5-5"2-5 1++5'5+36-1+ 7+h 8+3 9
'gf10'219114-33242F1342F1V1015)
o<1+ = 152(10 a+1063 +110
#:7511140” 11 +u112+st3+1 63141 6Z5+22€6+ 32 17+ 32758141111 '59
+111 f1+t11 42-18 03+16H1++1655+22 {6+111197+11+98+1l+€9
+8P1O+111011 +8f12 +11F13 +81111 +8141; >
911 = 1121251911145)
+hl+( 571 +512-11’r3-81‘LF-8 (5+1o'76-16 :7-16<8+2019
+5€1+5F2J+F3~81°h~8F5+1OF6-7P7-7 P8+8 {99
"“10“???” 412-3191349114'3'fis)
m = <15)%n2<a-e>
+115) 1114(51‘1 312- m+r5-217+2r8

(J'H

+2191 ~12 L145"“? 8
'4' “11 3+4 1114 15)

0‘5 = 1Ofi2(a+£}f +23)

*‘YILWBg-‘C' +£12+68T3+3214-3275+27f6+6l+f7+61+78+51+f9
+1+._1.2F1+’+.:l F2168134-321-132F5+27196+27397+27 8+27f9

+126F1O+221F11+161912+—2-ZF13+16P11++16P15)
{35 = 6fi2(a+i: -20
+11% 1 51—1 +1 wise-Ego 4564-60759
3 +3.’"8+21+13
+3911+gP13+—-F111+-2§1 F15)

+30 9 6+3?

+15€1+15fi2 ~60?) 7

9

War;

55 = (7)%h2(1rt)
~1-%(7)5fil*(5f1 - 5:2-1 014-1 0:15-20 7‘7+201'j8
+5 F1-5 .92-10 1131 OPS-9 K57+9 F8
4.191 0% :911+1+F12+§f13+gr11réfi15)
"‘6 = 132(20 cz+10d3 +1oc)
+fih(68f1+—2—72-1~2- ”1575-1-27 ”+32 5+32 5+5L1f7+6l+f8+61r9
+681’1-1~—-1+1229+—Lfé1—3F +27 ”W32 5-1-32 F6+27F7+2798+27F9
A 2
+16910+162P11+JFP12+1613+16K1M+EZP15)
V6 132(91-{3 ~80
+%fi”(9015-Sig-H013~HSTg-3515+3575~90“7-70"8+7o:9
~3§F1o-1‘+F11 ‘EESKC12‘1LH/13 *13V1lfi8r15)
= (”5913201110
1+ 7- ,3 ,- . ,. ., 3.
+(7)% £13. (10 K1-5132-5 l++5 5-5 6-1O\7+1O ~ -109
+10r1'572-51fi5‘75—5 1/6-)+ 7.)7+1+I/8-5f9
"3910-2 511*?)12”? 13"2 "111-3 ‘1 5)
0K7 = fi2(1o:;+2o:3+1o 3')
+fi’+(.li2lf1+687”2+lil ...-134324 )++27 5+32 6+6’+ 7451+ 78436141729
+—L*§1-F1+68KW~—2 #12"3+32 ”+27 V326 +27 7+27F8+27K9
+%Z€10+161’11+16p12+16p13"‘§21011++16.'15’)
(37 = fi2(-d+9Cf-8C)
+é’fib'(- 5': +901i2-110 ~35“? 4+5? +35f6-7o<5 -9o"8+7o”’9

3 1+ 5 7
- P- _ 9,, Q ,—,_ p- _ (a 3
5K3+9o.2 #OF3 3511+ us.s+35!6 36‘7 36 8+27 9

415.510-11191 1-32.5 912-1MK’13+8 F11++13 F15)

-95-

{7 = §(7)%h2(a~ )
+y}(7)‘%h‘+(5q-1o 12-57135ifs/6'4077+1018+1OT9
+5 Ki-1OF2-5fir5 f5+596_1+f27+1+ (23+; 59
"3910'2 F11*‘3?12*2 2131891132 £15)

(*8 =20h2(;’:1+/:+ C)

+fi)+( 68 7:, +681é+6873+82T1++82T5+82 gm 61+T17+1 61113 ”i
3 r 0 5/ 5' ‘ 5
+68 {1+68 r12"?68 €3+8211++82 » 5+82 6+771Q7+77 P8+771 9
+41 F1O+1+1 (’1 1+1+1 {0124,4191 3411 (31 1++1.31 {31 5+161+79)
138 = 6h2(a+1:3-2:) i f

+30 F1+3o 192-60 K‘3-30PLF-30P5-160 F6-27 87-27 38+51+' 9
21 3 21 , 1 1
‘15P1o‘2‘ 11'15V12"?‘F13*%‘P11+*52—F15)

X8 = 3(3)%fi2(Q-J)

1
- 1+ ’f‘ ’7 If "' — 7‘
+£(3Ph (20 -1-20 2-201320 ..5-ho\7+1+0 8

+20 F1 -20 3 2-20 fl++20 75-18 "7-1-18 ‘8

_ “ - ’3 £3 {1 AC _ [.3
8K10 9’11'*8'12'*9 13+‘1H '15)°

*The rotational energy levels are designated by E<J1~1_1K1)'

APPENDIX I I I

Herein are given the nonvanishing matrix elements

of ha+, hb+, and he+ introduced in Chapter VI.

(tha+lK:2) = liél\%fih(1‘ + r )r+zh””' <2at1)(2K:3)

-13
-ifih(’10+’11+2‘2l2) (10212192) + 112,15, ‘ n
x gr—K(Kt1)j2gf-(Kt1)(htg):% g
(tha+iKtH) = ;i(1/16)h“(*10-:31) 5
=4

r ...-L" -.--,—
x Lf-K(K:1);2Lf-(Ki1)(K:2);5 , 1

X Lf-(Ki2)(Kt3)3%Lf-(Ki3)(Kih):%

(thb+:Ki1) +i%(2L+1)1r-K(K+1)12

r K T I
x 3(1/8)fi (3‘14+716+2‘17)f

-(1/8m1+(3 711:2

1H+ ‘162

£311“ + 3 4. _¢

‘1’ ’1'
137111 15" 16'2 7)K

"' 4‘ -1 1+.” - “1
2C/’%hh~1h+‘fi “~16"?1 Z17)]
11(1/16)K“(€gK-136- i57)(2K:3)

(K:hb+iKi3)

.1. __ -1.
x;r-1<(Kz-1): 2 i‘f-(KLH ) (K12); % 1:20:12) (133132

           

1:3) (1/16)h“(f18-T2O-2 21)(2K+3)

XLf-me )‘j‘i' Lf-(Ki1 ) (K12)]%{f-(Ki2) (K1312

~96-

-97-

(thc+|x:1) = 1(2xt1)Lr—K<K:1)]%
x :(1/8)h1+(3118+'720+2121)f
-(1/8)h”(3<}8+7é0+2151)KZ

«111% 1223 8'71 ginzoi-K<“21 M

23" ...-J; Li's,” —l- )+/. .. 1+”?
”I -' ‘ 18“"h ~20 éh 221)].

“éT-‘u 2-,o1_1

APPENDIX IV

Here is the detailed expression of [Ff/’51: , ,.

on, ,.-._ ii 1a: m 5 52": 1151:
’ “"2”” ‘Bg’t S ( Snm(2)Un+” Smn(2)Un)

“1167111138876;

+ 'S< Smn(2)Un+ Smn<é1°n>1

 

 

F“
+15 (0' S11:1 XI : n35 1
‘8‘2.’ ( Sn (6’2’)Umn + ’nS (2)Umn ) ;
an 1’
i" 2 0’: 2 r
+ ’Sm(’x ’ns (2”)Umn -1-{8:”(21:Unm)f(1+9mn) 1
-fir 1 n1_/3‘-;7 :n 1,-1'1’1 -’
n
where
013
a 8m am
212‘, 1°11 *1“
m" _—_ gor’nn Pinyin); L
mn 3 _;1 ~-- -
(Am n)I°(J/I;};l

 

{/3 ( IX * r/‘s _31 (y: ‘11, .11". 4 1 ‘,
161121;“ I, I 1 2 (Arr/l) 2 n -...n J. ....
’ l¢n l

A
[\J
V
(is
I

" (

 

 

 

 

 

 

 

 

 

“8:11;” = 1m 1 3m gigga’fi + ai” gore" + £5211;
5% I. 1?; 1?: 133 15:13 11 113,113,
+ 61:16:51: _ 3315:9121 __ jfit’giii”
10.3151? ' 152118312); I; Iflqfll?!
_ 333’ av _ 3316’ 532?
1° 1° 1° 1311313,
d‘lU 1 “gag: + a”; afif)
(2)Umn‘81°_1°(1m3fi1nTi:1+£fm:) g 1?; _
.1

-98—

-99..

 

 

+ £4
81:. 1313(AmAn)t(1+5)
8' "A "i“ {D 1 A n+/\
x \ /_, (Eml/‘nl'i' Jml n1);—--—l(1+_ml)
1 l "l 3
1¢n
m X +
Cd " —m—1 1 +_nl
’3‘ (’ ml n1 ml nl)/\m'1(+ )
lfim
”H (/m+/\n)5mn ( 1 )
6(Am-%)(/Wm/n){10 1011/1 log/r ”7-! )
”’ min
and
_21- a; '1' 71‘" _ 1i" .1. 71- -
Chm,l- '- A 1%(\1—/‘m_ xin)(/\m +/’n2+”ld) ( md+'n2_ 1d) 1

\ % 1 é \ % -1 1 5 . é a % -1
(m'h“l) ('m'h""'l)
The primed Greek superscripts and subscripts (e.g. x,

cx; and nC’) are defined such that the order *\,:»',x’

presents a cyclic permutation of x, y, 2.

fix.

 

10.

11.
12.

13.

1h.
15.
16.

17.

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> In
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