1966

   

LLBRARY

Michigan State
University

  
     

 
 

THdblS

 

 

 

 

v ../. / /. ../ / d / ,x. / A. / / x. ,_ / ...

 

ABSTRACT

A STUDY
OF CORIOLIS RESONANCE IN POLYATOMIC

MOLECULES WITH SYMMETRY C3V

by Roger L. Dilling

A theoretical analysis of Coriolis interaction be—
tween a parallel vibration-rotation band and a perpendicular
vibration—rotation band in polyatomic molecules with symme—
try C3V is given° Since the results are to be applied to
molecules of the general type XYBZ, the study is restricted
to cases in which antisymmetric nondegenerate Vibrations
are excluded.

Computer techniques are used to explore the effects
of varying the molecular parameters. The character and
magnitude of the energy shifts are given for several physi—
cally interesting caseso

As an application of this study, a perturbation in
the v3 + VA band of CH3F is investigated. It is assumed
that the perturbation results from Coriolis interaction
between the perpendicular band v3 + v“ and a nearby parallel
band. The parallel band is taken to be Vl + v3 so that the
general accidental resonance case p = 2, pt = l of the
Tarrago—Amat classification is being considered. Since the

general features of the experimental data are reproduced by

i

\. K \;

‘.
\

K

. i
\ \
X ‘h

2
t

EX \. \

a
|

3‘ "K y'x. \ __ -
\\ '\ \ ‘\ _‘ \ ‘ _ _ -

Roger L. Dilling

the theoretical curves, it is possible to conclude that the
perturbations in v3 + vu do indeed result from interactions

which include Coriolis interaction.

 

A STUDY
OF CORIOLIS RESONANCE IN POLYATOMIC
MOLECULES WITH SYMMETRY C3V

By

Roger Lfifibilling

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

1966

 

To my

wife

 

ACKNOWLEDGMENTS

It is a privilege to acknowledge the help of Dr.

Paul M. Parker throughout the course of this work. His
encouragement and direction were greatly appreciated.

My special thanks are extended to Professor T.

Harvey Edwards for many discussions regarding the appli-
cation of the results to experimental data.

I wish to thank Professor Gilbert Amat, Faculté des
Sciences de Paris, for several consultations related to
this project.

For many valuable services, I wish to thank the staff
of the Computer Laboratory at Michigan State University.

I wish to express my gratitude to the National Science
Foundation for their financial support in the form of Co-
operative Graduate Fellowships throughout my entire graduate
program.

It is a pleasure to express my indebtedness to Dr.
Charles S. Morris, Professor Emeritus of Physics, Manchester
College, North Manchester, Indiana, for his significant role
in the development of my interest in physics.

Finally, but foremost, I thank my wife, Carole, for
her inspiration and understanding during my graduate studies

and research.

TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS . . . . . . . . . . . . iii
LIST OF TABLES . . . . . . . . . . . . v
LIST OF FIGURES . . . . . . . . . . . . vi
LIST OF APPENDICES . . . . . . . . . . . ix
Chapter
I. INTRODUCTION . . . . . . . . . . 1
II. THE VIBRATION-ROTATION HAMILTONIAN . . . A
‘III. ACCIDENTAL RESONANCES IN THE SPECTRA OF
POLYATOMIC MOLECULES WITH SYMMETRY C3v . 19
IV. CORIOLIS RESONANCE . . . . . . . . 36
V. RESULTS AND DISCUSSION OF COMPUTER
ANALYSIS . . . . . . . . . 56
VI. ANALYSIS OF A PERTURBATION IN THE v3 + VA
BAND OF CH3F. . . . . . . . 87
VII. CONCLUSION . . . . . . . . . . . 113
REFERENCES. . . . . . . . . . . 115
APPENDICES. . . . . . . . . . . . . . 117

 

LIST OF TABLES

Table Page
2.1 Matrix Elements of the Fundamental Operators . 15
3.1 Perturbation Coupling Case a. p a 3, pt = 1 . 23
3.2 Perturbation Coupling Case b. p a 2, pt = 1 . 2A
3. Perturbation Coupling Case 0. p 2, pt = 2 . 2A
3.A Perturbation Coupling Case d. p 2, pt = O . 25
3.5 Terms in the Hamiltonian. . . . . . . 27
3.6 Types of Accidental Resonance Coupling . . . 28
3.7 Possible Accidental Resonance Cases from r3P . 32
6.1 Two—parameter Fits of Analysis II. . . . . 101
6.2

Summary Table of Molecular Parameters for CH3F 103

 

LIST OF FIGURES

Figure
4.1 Energy Matrix for J = 2 . . . . . . .
4.2 Energy Matrix for J = 2 in Block Diagonal
Form. . . . . . . . . . . .
4.3 Summary Matrix for the Submatrices . . .

5.1 Typical Unperturbed Perpendicular Band of
a Molecule with Symmetry C3v' . . .

5.2 Energy Level Diagram for Coriolis Resonance

in the Kth Positive Subband . . . .
5.3 Energy Level Diagram for Coriolis Resonance

in the Kth Negative Subband . . . . .
5.4 Energy Shifts due to Coriolis Resonance,

Cross— —over Case, AaB = O . . . . . .
5.5 Variation of Energy Shifts with Cy,

AaB = 0.. . . . . . .
5.6 Variation of Energy Shifts with gy,

AaB = .075 . . . . . .

5.7 Variation of Energy Shifts with AaB,

C = l o a o n o o o u 0 o o q

5.8 Energy Shifts for "Quasi—vibrational”
Perturbation . . . . . . . . . .

5.9 Comparison of Energy Shifts in RQ0 for
Ad = O and AaB = .075 . . . . . .

5.10 Comparison of Energy Shifts in RQl for
Ad = o and AdB = .075 . . . . .

5.11 Comparison of Energy Shifts in RQ2 for
Au = o and AaB = .075 . . . . .

Vi

Page
44

46
54.

59

61

63

66

68

7O

72

75

77

78

79

 

 

Figure Page

5.12 Comparison of Energy Shifts in RQ3 for
Ad = 0 and AdB = .075 . . . . . 80

5.13 Comparison of Energy Shifts in HQ“ for

Ad = 0 and AaB = .075 . . . . . . 81
5.14 Comparison of Energy Shifts in RQ5 for

Ad = 0 and A08 = .075 . . . . . . . 82
5.15 Comparison of Energy Shifts in RQ6 for

Au = 0 and AaB = .075 . . . . . . . 83
5.16 Comparison of Energy Shifts in RQ7 for

Ad = 0 and AaB = .075 . . . . . . . 84
5.17 Comparison of Energy Shifts in RQ8 for

Ad = 0 and AaB = .075 . . . . . . . 85

5.18 Comparison of Energy Shifts in RQ9 for
Au = 0 and AaB = .075 . . . . . . 86

6.1 Experimental Energy Shifts in the RQl
Branch as Determined by Blass . . . . 89

6.2 Experimental Energy Shifts in the RQO

Branch as Determined by Blass . . . . 93
6.3 Energy Shifts for RQl and PQl Branches,

Analysis I. . . . . . . . . . . 95
6.4 Energy Shifts for RQO Branch, Analysis I . 96
6.5 Energy Shifts for RQl and PQl Branches,

Analysis II . . . . . . . . . . 98
6.6 Energy Shifts for R00 Branch, Analysis II . 99
6.7 Energy Shifts for R00 Branch, Analysis III. 100

R

6.8 Energy Shifts for Q1 and PQl Branches,
Analysis III . . . . . . . . . . 102

6.9 Energy Shifts for RQ2 and PQ2 Branches,
Analysis I. . . . . . . . . . . 105

6.10 Energy Shifts for R03 and P03 Branches,
Analysis I. . . . . . . . . . . 106

 

 

Figure

6.11

6.12

Energy Shifts
Analysis I.

Energy Shifts
Analysis I.

Energy Shifts
Analysis I.

Energy Shifts
Analysis I.

Energy Shifts
Analysis I.

Energy Shifts
Analysis I.

for

for

for

for

for

R
Q5

RQ7 and P
RQ8 and P

R P
Q9 and

viii

RQu and P
and P

RQ6 and P

Page
Branches,
. . . . . 107
Branches,
O O O C O 108
Branches,
. . . . . 109
Branches,
O O U 0 D 110
Branches,
. . . . . 111
Branches,
. . . . . 112

 

LIST OF APPENDICES

Appendix Page

I. Computer Program . . . . . . . . . 117

II. Printout of Analysis I . . . . . . . 123
ix

 

CHAPTER I

INTRODUCTION

Analysis of complicated physical systems requires
that both an experimental and a theoretical study be made.
Many times these studies complement each other. Experi—
mental results may suggest physical models in terms of
which detailed calculations can be made. Conversely, the
theoretical results may suggest additional predictions
which can be checked experimentally. Physical systems on
the atomic level must be treated from the quantum mechanical
point of View. Small molecules provide an excellent example
for the testing of the quantum mechanical formulation. They
are sufficiently simple so as to allow for an explicit treat-
ment of their motions, yet are complicated to a degree that
makes the problem interesting from a theoretical View point.

Historically, the analysis and interpretation of the
infrared spectra of polyatomic molecules followed that of
diatomic molecules. For the diatomic molecule the infrared
spectrum suggests that in many cases, at least to a first
approximation, a very simple molecular model can be chosen.
As a consequence of the simplicity of the model, exact
expressions are obtained for the energy eigenvalues. To

lowest order, the Hamiltonian of a diatomic molecule is the

 

sum of the simple one-dimensional harmonic oscillator
Hamiltonian and the rigid rotator Hamiltonian. For a poly—
atomic molecule the corresponding Hamiltonian is the sum of
the Hamiltonian of a set of uncoupled harmonic oscillators
and the rigid rotator Hamiltonian. In the expansion of a
more accurate molecular model, it is desirable that the
lowest~order terms be, in fact, the Hamiltonian mentioned
above, thus giving consistency with the gross features of
the experimentally observed data.

As experimental methods are refined, it is necessary
to consider higher~order expressions for the energies. More—
over, as more experiments are carried out, it becomes neces-
sary to consider in detail many of the special cases of the
general theory since these are likely to occur repeatedly as
more molecules are studied. When deviations from the lowest-
order expressions occur, it is possible in an ad hog manner
to add higher-order expressions to obtain results more con—
sistent with experiment. However, it is necessary to justify
these added terms by showing that they result from terms of
higher order in the expansion of the Hamiltonian of the mo-
lecular model. As a result, it is necessary to include any
other terms which occupy positions of equal importance in
the expansion as those which are used to obtain a better fit
to the experimental data.

The treatment of problems in molecular infrared spec—
troscopy is carried out to best advantage if the problems

are formulated for a general polyatomic molecule rather than

¥

 

 

 

 

 

 

 

for individual molecules, each considered as a special case.
Such an approach was first given, to the second order of
approximation, by Nielsen (l, 2). In the next chapter we

give a brief review of his formulation.

 

CHAPTER II
THE VIBRATION—ROTATION HAMILTONIAN

A molecule containing N nuclei and n electrons has
3(N+n) degrees of freedom, if the intrinsic spins are neg-
lected. To reduce the complexity of the problem several
approximations, which have been found to be valid to a
high degree of accuracy, must be made. With the exception
of certain cases, the electronic motion of the molecule can
be considered as uncoupled from the motions associated with
the other degrees of freedom. The validity of this approxi-
mation was established by Born and Oppenheimer (3). They

1/4

used a power series expansion in (m/M) of the terms in-
volving the coordinates of both the nuclei and the electrons,
where m and M are the mass of the electron and the average
nuclear mass, respectively. Thus in a calculation of the
vibration—rotation energy levels of a polyatomic molecule

one need consider only the 3N degrees of freedom of the N
nuclei. These 3N coordinates are the three coordinates of
each of the N nuclei in an inertial coordinate system. For
an isolated system it is always possible to separate out
three degrees of freedom corresponding to the three Cartesian

coordinates of the center—of—mass of the system. These are

the translational degrees of freedom. They are not of

 

immediate spectroscopic interest. Hence, exclusive of
translation, we must consider 3N—3 degrees of freedom for

the molecule. In general, it is not possible to separate
these 3N—3 degrees of freedom into rotational and vibrational
parts because there are interactions between vibration and
rotation. However, it is found experimentally, at least to

a first approximation, that a molecule may be considered to
be acting simultaneously as a rigid rotator and as a set of
uncoupled harmonic oscillators. In this approximation the
3N—3 degrees of freedom can be separated into three rotational
degrees of freedom and 3N—6 vibrational degrees of freedom.
For a linear molecule there is a separation into two ro—
tational degrees of freedom and 3N—5 vibrational degrees of
freedom since in this case two angles completely describe

the orientation of the system.

Consider first the rotation. It has been found to be
convenient to choose a coordinate system which takes ad-
vantage of the fact that the molecule is, at least to some
approximation, a semi—rigid body rotating in space. Since
the molecule is not a completely rigid body, it is not
possible to have a "body—fixed" coordinate system. Instead,
the moving coordinate system is attached to the equilibrium
configuration of the molecule. The non—linear molecule as
a rigid body requires three angular coordinates to completely
describe its orientation; the Euler angles may be used for
this description. Another choice would be the rotations

about the three coordinate axes of the Cartesian coordinate

 

 

system attached to the rigid molecule. In this latter
description the conjugate momentum components are the total
angular momentum components along the x, y, and z axes of

P

the molecule—fixed coordinate system, P and Pz’ re-

X, y)
spectively. These are denoted collectively as Pa‘ The
angular momentum P and the angular velocity g are not, in

general, in the same direction:

2:

H

=‘fl.

I is the inertia tensor of the rigid body, and the mole-
cule—fixed axes x, y, and z are, for convenience, chosen
such that they are the principal directions of ; evaluated
at equilibrium.

Next, let us consider the vibration. The molecule
considered as a set of uncoupled harmonic oscillators re—
quires 3N—6 vibrational degrees of freedom. The 3N Cartesian
components of the N nuclei cannot be used as coordinates
since they are not independent of each other. By taking into
consideration the six non-vibrational degrees of freedom, a
transformation is made from the 3N mass—adjusted Cartesian
components of the displacements from the equilibrium positions
of the N nuclei, Ali/mi, to the 3N—6 normal coordinates of the

problem, QSO:

a —1 -
§=x y z<£iso> (/MiAai)
3 3

 

 

or /MiAOti = 2 .9,
SC

The 13h nucleus has displacement Adi in the a direction.
The normal coordinates are labeled with two subscripts; s

181/2/2nc,

denotes the sth normal mode having frequency “5 =
and o enumerates the degree of degeneracy of mode 3. If

mode s is nondegenerate, 0 = 1 only, and consequently a is
omitted. If mode 5 is doubly degenerate, o = 1 and 2. If
mode s is triply degenerate, o = l, 2, and 3. The momentum
conjugate to the normal coordinate Qso is the linear momentum
p*,

80'

3
p * = -ih
so BQSO

 

Associated with the Vibrational motion is an internal angular
momentum p. The components of p are given by
0L
p=ZE C I'vav*a
a 805,0, sos 0 so s 0

(I

where
CSOS'O'

is the Coriolis coupling constant connecting
Vibrations so and s'o' with rotations about the a—axis of
the molecule-fixed coordinate system. These constants are

defined as

a N B Y Y B
Csos'o' = §=l(£iso£is'o' _ £isogis'o')’

With a, B, and y taking one of the three cyclic permutations

 

of x, y, and 2. Since P is the total angular momentum and
p is the angular momentum associated with vibration, P - p
is the angular momentum of the rotation of the molecule.
This angular momentum is related linearly to the angular

velocity of the molecule 3,
a = E'<E - a)

or w = EudB(Po — pa).
8

Thus g is the inverse of the inertia tensor when both pure
rotation and rotation due to Vibration are taken into con—
sideration.

The Hamiltonian of the vibration-rotation motion of
the nuclear framework of the polyatomic molecule which is
used in this formulation is the so—called Darling—Dennison
Hamiltonian (4),

_ 1 1/4- -1/2 1/4
H - E“ g8(Pa - pa)u “aB(PB - pB)u

lul/uz p *u—1/2p *ul/4 + V

+ _
2 so so
so

where u = det (“ds)’

and V is the potential energy function associated with the
force field in which the nuclei vibrate. It is assumed that
this potential energy function can be expanded in a multi—
dimensional Taylor series about the equilibrium positions

Of the nuclei:

 

 

" "Ksos'o's"o"Qson'o'Qs"o" + "‘
The first term is of the form required by definition of the
normal coordinate approximation being used. The higher—order

terms are the anharmonic corrections. The K are

sos'o's"o"
the cubic potential constants.

The Schroedinger equation for the Darling—Dennison
Hamiltonian cannot be solved analytically. Thus it is neces—
sary to make approximations to obtain solutions which can be

correlated with the experimental results. The Hamiltonian is

expanded in the normal coordinates in a series such that suc—

cessive orders differ by approximately one order of magnitude.

This development has been carried out by Amat, Nielsen, and
Goldsmith in a series of papers (5 — 9). Mme. Tarrago (10)
has specialized these results for molecules with three—fold
rotational symmetry.

As previously mentioned, the lowest-order approximation
must have the following properties: (a) The equation should
be sufficiently simple for an analytical treatment to be
given and an appropriate representation to be obtained. (b)
The lowest—order results can be used to explain the gross
features of the experimental data. These conditions are
satisfied if the terms of the largest magnitude are those
corresponding to a rigid rotator plus a set of uncoupled
harmonic oscillators. Higher—order terms in the expansion

can then be treated in the rigid—rotator-harmonic-oscillator

 

.r.

i‘_\ M.

‘ iiifxifx

L ilk

\

4

_\_‘L

«7

 

10

representation. The expansion is carried out by writing
the quantities “dB in terms of the normal coordinates:

_ 1 (0)08 - (l)dB (2)d8
“o8 — I I [Q + 2 9so Qso + I 2. 'Qsos'o‘QSon'o' + "‘J'
o B so sos o

0!. .
The Qé§;,§, are functlons of the molecular parameters:
masses, normal coordinate transformation coefficients, force
constants, etc. The principal axis system is used for the

inertia tensor, the equilibrium moments of inertia being Ia;

in this case

Q(0)0LS = I 5 .
d dB

Since u = det (uaB)’ it follows that to lowest order

(0) _ ;L
“08 — Ia608°

This is a constant and commutes with all operators in the
Hamiltonian.

If H is expanded as

H = HO + H1 + H2 + ...,

(0)08

Q_____ 1‘ *2

I I PoPB + 22 pso + V
a B so

1
then H = _
O 208 O

_ i 2
Where VO — 2EOASQSO .

 

 

__—,3

11

This can be written in the expected form

 

l'Pd2
w1th HR = 521
(10,
— 1? *2 2
and HV _ 2:0{pso + Asto }

HR is the rigid rotator Hamiltonian, and HV is the Hamilton—

ian of a set of uncoupled harmonic oscillators. The first-

order part of H is found to be:

12 . Qso Z Q(0)08
H — A ——————Q P P — p P
1 20880 IdIB so a 8 GB IdIB d 8

+ l 2 z KSOS.O.SHO"Q Q

"on so S'O'QSHO'".

(1)08
So

The coefficients 0 are defined by

gum =

a
—a B.
so SO

The a2: are the first—order expansion coefficients of the

components of the inertia tensor in the normal coordinates,

i.e.,

 

12

There are no constant terms for the products of inertia
(o r 8) since the principal axis system is being used. It
can be shown that

N
d8 _ - O 8 O a
aso ' §=l/Mi(ai£iso + Biliso)’ (” I 8)

N
do _ O 8 O Y
a ‘ 2;: m_i(‘5i"isc + Yigiso)’
where a? is the a component of the equilibrium position of
the ith nucleus in the molecule—fixed coordinate system.
The first—order part of the Hamiltonian can then be written

as

+ 2 Z 2 KSOS'O's"o"QsoQS'o'Qs"o"'

The second-order part is

Q(2)0L8

sos'o'Q
;,o,§8 IdIB son'o'PoPB

 

 

l3

' Y
+ go; ' O 1 gnonsmomKSOS'O'S"O"S'"O"'QSOQS'O'QSHOHQSH'O'".
Expressions for H3 and H4 are given in the first paper of
the Amat, Nielsen, and Goldsmith series (5). There will be
no need for these expressions in the present work. The

first sum in H2 corresponds to a variation of the coefficients

of inertia with vibration. The second sum in H2 is a first

correction to the second sum of H1. The various higher-order

terms may be treated by the use of a perturbation theory

formalism.
Molecules with point—group symmetry C3V belong to the
general class of molecules with axial symmetry, IX = Iy # IZ.

In this case the Schroedinger equation

has known analytic solutions. The eigenfunctions are

denoted by
w = waV
where wR = I J K >
and wV = | V1 > IVS, > IVS is > .

The quantum numbers J, K, and is are angular momentum
quantum numbers, and vS is a vibrational energy quantum

number. In Table 2.1 the matrix elements of the rotational

 

l4

and.vibrational operators are given in this representation.

We have that

P2|J K> =J(J+:uh% JK m

PZI J K > = Khl J K >,

vav’
@ssz = ZsfiwV'

It should be noted that 6:2 is an angular momentum operator
in normal coordinate space so that téh must be transformed

into real space to obtain the actual angular momentum.

Since
= * _ *
622 ps1 Q32 ps2 Qs1
and p

_ U.
a _ gasls2agz’

the 2 component of the angular momentum in real space due

to vibrations of the sth mode which is two—fold degenerate

is

z _ z
ILs/ficsls2 _ ILs/fits'

It should be noted that in molecules with symmetry C all

3v
two—fold degenerate vibrations are in a plane perpendicular

 

 

15

TABLE 2.1——Matrix elements of the fundamental operatorsl (11).

 

 

< K I Px I K+1 > = < K+1 I Px I K >
= %h/(J_K)(J+K+l)
< K | Py | K+1 > = _< K+1 I Py I K >
= —%ih/——T(J_K)(J+K+l
< K I PZ | K > = Kn
< vr1 I qn I Vn+l > = < Vn+l I qn I vn >
= /(vn+1)/2
< vn I pn I Vn+l > = —< vn+1 I pn I Vn >
= —ihV(Vn+1)/2
< vt 2t I qtl I vt+l, ttil > = < vt+l, tttl | qt1 I vt 2t >
= ll/(v it +2)/2
2 t t
< vt 1t | qt2 | vt+l, ltil > = _< vt+1, tttl I qt2 I vt 2t >
= _li/Zv rt ¥25/2
2 t t
< Vt 2t I ptl I Vt+l’ itil > = -< Vt+l’ itil I ptl I Vt it >
_ +1 /(V t+
- ——2-l’h( 5?, t'I'2)/2
< Vt 2t I pt2 I Vt+l’ itil > = < Vt+l’ ttil I pt2 I vt it >
= —l"if/_(——rv *9. +2 /2
2 t t
1The vibrational operators are given in dimensionless
form
q = (A S/h MM” and p = (412 As )l/up *.

SO SO SO SO"

 

16

to the three—fold rotational symmetry axis. No three-fold
degenerate vibrations occur in molecules with symmetry C3v°
In the treatment of this problem where results accu-
rate to an approximation higher than first order are desired,
the method of the contact transformation (12) can be used.

The energy correct to first order is given by

1 0
Eé)=E£)+<nIH In>,

1

whereas to second order it is given by

2
Eé2) = Eél) + < n I H2 I n > + Z'|< n IHl I n1 >I
‘ v
1'1 n

 

This expression can be simplified by performing a similarity
transformation on H. This transformation will not change
the zero—order energy or the first—order matrix elements
Which are diagonal in the principal vibrational quantum
numbers {VS}. However, it will cause all the first—order
matrix elements not diagonal in the Vs’ here denoted by n

and n', of the transformed first—order part to vanish:

Let this transformation be denoted by T, and

-1
' = = ' ' '
H THT HO + H1 + H2 +

 

17

where
HO' = HO, < n I Hl' I n > = < n I Hl I n >,
and < n I Hl' I n' > = 0
Under these circumstances
Eé2) = Eél) + < n I H2' I n >.

Thus, at least formally, by use of the contact transfor—
mation, results accurate to second order are obtained with
no more difficulty than a first—order calculation, provided
such a transformation T can be found. For the calculation
of third— and fourth—order energies a second contact
transformation can be sought.

This procedure of performing contact transformations
to simplify the problem to a given order of approximation
is valid as long as the case of accidental near—degeneracies
does not occur. Such an accidental degeneracy will occur
when the physical parameters in the Hamiltonian take on
numerical values such that certain interacting energy levels
fortuitously have nearly equal values. This degeneracy is
in contrast to one which occurs because of basic symmetry
considerations only. In the case of an accidental near—
degeneracy in polyatomic molecules, Nielsen (13) has found

it possible to modify the contact transformation in such a

 

 

18

way that the terms not participating in the accidental
near—degeneracy can be removed. Of course, the degenerate

part still remains and must be treated by other methods.

 

CHAPTER III

ACCIDENTAL RESONANCES IN THE SPECTRA OF

POLYATOMIC MOLECULES WITH SYMMETRY C3V

The validity of the perturbation theory approach to
the calculation of vibration—rotation energies depends upon
the relationship between the diagonal and off—diagonal matrix
elements. To produce sufficiently rapid convergence, the
off—diagonal matrix elements must be small compared to the
diagonal matrix elements; in fact, for proper treatment of
the problem by conventional nondegenerate perturbation theory,
the off—diagonal matrix elements must be small compared to the
differences between the diagonal matrix elements.

When the experimental data agree with the theoretically
predicted spectrum over a wide range of energies but deviate
in certain regions, it is natural to assume that the devi—
ations result from the assumptions in the theory not being
valid in that region. Thus when pronounced perturbations
occur in a high resolution infrared spectrum of a molecule,
it is useful to explore the various possibilities which
could invalidate ordinary perturbation theory in order to
explain the deviations in the data.

In such cases there first must be a non—zero off—

diagonal matrix element coupling two states. Regardless of

19

 

20

the closeness to degeneracy of two levels, no deviation from
the results of ordinary perturbation theory can be expected
if the levels are not coupled through a non—zero matrix ele-
ment. Hence, it is necessary to tabulate the various matrix
elements which can be non—zero in order to investigate the
cause of possible perturbed states. Secondly, for those
levels which are coupled through a non—zero matrix element,
it is necessary to investigate the conditions under which
their energies are sufficiently close so that their energy
difference is so small as to preclude satisfactory con—
vergence in ordinary perturbation theory.

By assuming the existence of a non—zero matrix element
coupling two levels it is possible to define two types of
resonance, essential and accidental. Essential resonance
occurs when the interacting levels have the same set of
principal vibrational quantum numbers {VS}. Thus, this type
of resonance occurs between levels within the same vi—
brational state. The levels differ only in their secondary
Quantum numbers K and ts. The remaining two quantum numbers,
J and M, do not enter the discussion since all matrix ele—
ments of the Hamiltonian are diagonal in these quantum
numbers. This result occurs because the system is isolated;
thUS J cannot change, and there is no preferred direction in
inertial space, so M cannot change. Accidental resonance
occurs when the interacting levels are in different vi-
brational states, characterized by different sets of the

principal quantum numbers {vs}. The secondary quantum

 

21

numbers may or may not differ for these levels. Essential
resonance is due to inherent symmetry properties of the
molecule under discussion and may occur in any molecule with
the appropriate symmetry. On the other hand, accidental
resonance occurs when the numerical values of molecular con—
stants are such as to fortuitously produce near—coincidence
of two vibrational energy levels. In fact, if the pertur—
bation is localized within a small range, it is possible to
conclude that the vibrational energies alone will not coin—
cide. Rather it is the Vibration—rotation energy levels
which nearly coincide for a limited number of levels.

This work is concerned with accidental resonances.

Consider two vibrational states a and b. Each state
is characterized by a certain set of vibrational quantum
numbers {vs}. Since accidental resonance is assumed to be
present, the condition vsa # VSb must be satisfied for at
least two normal modes s. Let Avs = vsa — vsb. If s is a
two—fold degenerate mode the subscript t will be used in
place of s; similarly, the subscript n may be used for
nondegenerate modes.

Mme. Tarrago and Amat (14) have developed a classi—
fication scheme to enumerate the various types of accidental

resonance. This classification is based on two quantities p

and pt' These quantities are defined as

p = XIAVSI:
S

p = ZIAV I.
t t t

 

 

22

Thus, since molecules with symmetry C3v have only non-
degenerate and two—fold degenerate modes of vibration,
p = pt + ZIAVnI-
n

p denotes the number of quanta involved in the interaction,
pt is the number of degenerate quanta. As a simple example,
consider the interaction between Vn and Vt. In this example
Vsa = 0 for s # n, Vna = 1. Also va = 0 for s # t, vt = 1.
Therefore, Avn = +1 and Avt = -1, all other AvS = 0; hence
p = 2, pt = 1.

For accidental resonances p > 1 since there must be
at least two quanta involved; one to increase the energy of
one mode by one quantum; the other to decrease the energy of
another mode by one quantum. The matrix element coupling
the two states a and b has the general form rkPg, where r is
used to denote a vibrational coordinate QSO or pso*. All
SUbscripts are suppressed in this condensed notation. Now,
r has only first off—diagonal non—zero matrix elements.
Thus rk has kph, (k—2)th, (k—4)th, ... diagonal or first
off-diagonal matrix elements. Hence for accidental resonance
k > 1. Note k z p since possibly p = k — 2m, m being a posi-
tive integer.

The Tarrago—Amat tables, Tables 3.1, 3.2, 3.3, and 3.4,
list results according to the above classification. The
tables are constructed with the aid of Amat's Rule (15).

This rule may be stated as follows: For a molecule with a

 

 

23

.E 3.l--Perturbation coupling case a: p a 3, pt # 1,

 

 

Order of
Magnitude
Matrix Elements AK EARt Z Att, Cplg.
t t' p=3 p=LI p=5

 

 

 

 

 

 

 

 

 

 

 

 

(1,2,3) 0 0* 0 I l 2 3
'(3’u)/J(J+l)-K(Kil) il +2 0 II 2, 3,
(3) 0 :2 +2 I 3
'(5)/J(J+l)-K(Kil)
x/JIJ+l)-(Kr13(Kt2) :2 i2 0 IV 3,
(1.2.3) 0 i3 0 I 1 2 3
(3’u)/J(J+l)—K(Kil) i1 i1 0 II 2,3 3,
(3)K 0 :3 0 III 2
(3) 0 i1 i2 I 3
(5)/J(J+l)—K(Kil)
x/J(J+l)-(Ki1)(Ki2) 12 ll 0 IV 3,
9(-

(2’3) 0 0 0 I 2 3
(u)/J(J+l)—K(Kil) i1 :4 0 II 3,

0 i3 0 I 3

(3)

 

 

24

E 3.2——Perturbation coupling case b: p a 2, pt = l.

 

 

Order of
Magnitude
Matrix Elements AK Alt 2 Att, Cplg.
t' p=3 p=4 p=5

 

/J(J4i3—KZKtl) il il 0 II 1,2 2,3 3,
’3’“)

 

 

0 i3 0
’3) { 0 i .2 } I 2 3
—‘———7:—7—:'—
,5)/J(J+l K K 1)
/J(J+l)—(Kil)(Ki2) :2 ll 0 IV 2, 3.

 

E 3.3-—Perturbation coupling case c: p = 2, pt = 2.

 

 

 

 

 

 

 

 

 

Order of
Matrix Elements AK ZAzt Z Azt' Cplg. Magnitude
t t' p=2
)/J(J+l)-K(Kil) 11 $2 0 II 1,2
K 0 0* 0 III 1,2 ,
) x/
0 0* 0 ,f
) { _ } I 2 ./
0 i2 +2 .6
./
)/J J+l —K Kil) i//
fix";
/J(J+1)-(K11)(Ki2) :2 :2 0 IV 2 I
-———————————— ~,/
)/J(J+l)—K(Kil) - .
J(J+l)-(Kil)(Ki2) “A
J J+ -— i i t * O V 3
( l) (K 2IIK 3) 3 0 “fl/
.4

25

IBLEEL4-—Perturbation coupling case d: p = 2, pt = O

 

 

 

 

 

 

 

 

Order of
Matrix Elements AK 2 Att, Cplg. Magn1tude
t p=2
'(2)
. 2 .
(M)J(J+l) + W (4)K { 0 O } I 2
(u)/J(J+l)—K(Kil) $1 $2 II 3,
(6)/J(J+l)-K(Kil)
x/J(J+l)—(Ki1)(Ki2)
X/J(J+l)—(Ki2)(Ki3) i3 0 V 3,

 

._1

:ation or rotation-reflection axis of order N, the matrix

ement of a term in the Hamiltonian

< Vn’Vn"V .J,K,M

t’gt“.

vn+Avn,vn,+Avn,,v +Av 2 +A£ ..J,K+AK,M >

t t’ t t’°

don—zero only if

. 1
AAK + ZatAtt + ENZ'Avn, = 0,iN,i2N,...
t n
—1. if N is a rotation axis, A = %N-l if N is a rotation—
.ecnsion axis; a = l, 2, 3, ... according as the t-th

t

dixnensional irreducible representation is E E

1’ 2’ E3:

 

26

denotes those nondegenerate Vibrations which are anti—
'mmetric with respect to the operation correSponding to

tations of 2n/N about the symmetry axis. This formula

y be specialized for the case we are considering. For
e group C3v’ N = 3, A = -1, all two-dimensional repre-
ntations are of type E1, so at = l for all t. There are

B type nondegenerate vibrations for C so the last

3V’
rm on the left side of the equation is identically zero.
erefore, in this case,
-AK + EN, = O. .3, t6. .9,
t
so, all terms in the Hamiltonian must be totally symmetric,
e., the product of the symmetry species must contain the

irreducible representation. If T(rk) = E, then F(P£)

st also be of type E since

E x A = E, E x A2 = E, and E x E

A + A + E.

The terms can be grouped in the orders of expansion of
e Hamiltonian as given in Table 3.5.

Mme. Tarrago and Amat have considered only terms
rough third order so k < 6. Hence for l < p < 6, and since

< p+l, the following possibilities arise:

p = 5 pt = 5, 4, 3, 2, l. O
p = 4 pt = 4, 3, 2, l, 0

p = 3 pt - 3, 2, l, 0

p = 2 pt — 2, 1, 0

 

27

ABLE 3.5--Terms in the Hamiltonian.

 

I
|___

 

 

Order Terms

0 r2 P2

1 r3 r2P rP2

2 rLl r3P r2P2 rP3 PLI

3 r5 ruP r3P2 r2P3 rPLI r

4 r6 r5P ruP2 r3P3 r2Pu P6 r2
Le of these cases can be eliminated immediately: If pt = l,
Len F(r2) = E. In order to have p = 5, pt = 1 would require

term of the form rSP. This term is not found among terms
.rough third order. The remaining seventeen cases can be

ouped as follows:

Case a. p a 3 pt # 1
Case b. p a 2 pt = 1
Case 0. p = 2 pt = 2
Case d. p = 2 pt = 0

There are six types of coupling, classified according
the dependence of the off—diagonal matrix element W on
e rotational quantum numbers. These types are listed in

ble 3.6. (W0 is independent of the rotational quantum

nbers.)

 

28

ABLE 3.6—-Types of accidental resonance coupling.

 

 

 

 

 

Type Matrix Element W
I WO
1' WOJ(J+1) + wO'K2 + wo"
II WO/J(J+l) — K(Kil)
'II WOK
IV WO/J(J+l) — K(Kil)/J(J+l) - (Kil)(Ki2)
V WO/J(J+l)-K(Kil)/J(J+l)-(Kil)(Ki2)/J(J(J+l)—(Ki2)(Ki3)

 

the Tarrago-Amat tables the column labeled {Alt gives
e sum of the changes of the quantum numbers it for those
generate modes t satisfying Avt # 0. These modes t are
id to participate in the resonance. The column I Att,

’0'
ves the corresponding sum for those degenerate modes t'

ving Av 0. The modes t' do not participate in the

t!
sonance. If [At = 0 but {IAttI # 0, the appropriate

t
?O is labeled 0*. The matrix elements W are given sub-

?ipts W ). These subscripts denote the corre-

(a,b,c,..
>nding order of magnitude of the term in the case J = K s 1.
a order of magnitude column applies to the case where

=I< 2 30.

 

29

In order to look more closely at the construction of
le Tarrago—Amat tables, consider the term r3P which occurs
1 H2. Since k = 3, the only possible value for p is p = 3.
ane k = 3 and p = 3, it follows that E'Att, = 0. Now with

= 3, the possibilities are the following:

Lth pt = 3 Attl = i1, A£t2 = t1, A2t3 = i1
Altl = i1, A£t2 = t2
Alt = i3
Amt = 11
th pt = 2 Attl = i1, A£t2 = i1
Alt = 12
all Alt = 0
th pt = 1 Alt = i1

th pt = 0 all degenerate modes are of type t'

nSider the rotational part: If P = PZ then AK = 0 and
ere is Type—III coupling; if P = PX or Py then AK = il
1 there is Type—II coupling.

Amat's Rule must be satisfied:

 

—AK + EAzt = 0, i3, i6, i9, ...

30

.nce IXAztI < 4, the above is restricted to
t
—AK + {Mt = 0, 23.
t

Consider Type-III coupling, AK = 0, so

t t
gAzt = 0, we have
= 2 Attl = —A2t2 = i1, and all Alt = 0
= 0: all degenerate modes are of type t'

XAit = :3, all pt = 3 cases except the Afit = i1 case
t
cur, subject to the condition that all Al's are of the
us sign.

Consider Type—II coupling, AK = 11, so

 

 

. = 3 Altl Alt2 — 'Mt3 — :1
Altl = -2A2t2 = i2
Allt = 11

I = 1: Alt = 1l

T ZAlt = 12, we have

t

5 = 2: Aztl = A1132 = t1.

Alt = 1'2

1e r3P term thus gives the possible cases listed in Table
.7. Not all these possibilities occur in the Tarrago—Amat
ables. Consider the case p = 3, p1; = O in the list of
assibilities. Since pt = 0, only nondegenerate quanta are
lvolved. If only nondegenerate vibrations of type A1 are
)nsidered, F(r3) = A1. Now because AK = 0, then the P in
3

P must be PZ but F(PZ) = A2. Hence

3)xp(P)=A xA =A

“1“ l 2 2’

1d this case must be excluded. Furthermore, any of the
ises which have matrix elements with the same quantum number
Bpendence and quantum number changes as a lower—order term

?e not included in the tables. Consider p = 3, pt = 2,

< = 0 in the list of possibilities. This term has a matrix

 

I
:I

I
”I
/:
.2
.x‘
w”
.../1

‘x

‘3 _

4
u

i—

ii. i

.
K
I.
..

Li.

 

 

32

TABLE 3.7-—Possible accidental resonance cases from r3P.

 

) pt Case Matrix Element AK XAtt Z'Az

Coupling Order

 

t t t'
3 2 a W(3)K 0 0(*) 0 III
3 0 a W(3)K 0 0 0 III
3 3 a W(3)K 0 i3 0 III 2
3 3 a w(3)/m il 11 0 II 2,3
3 1 b Mam :1 il 0 II 2,3
3 2 a w(3)/WKTKTII :1 $2 0 II 2,3

element of the form W(3)K with changes AK = 0, ZARt = 0(*),
t

Akt. = 0 and Type—III coupling. The matrix element W(2)K
I

KI LVJ

in p = 2, pt = 2 has the same changes in the quantum numbers
and Type—III coupling. This term arises from r2P in the
?irst—order Hamiltonian. This lower—order term takes prece—
1ence over the p = 3, pt = 2, AK = O possibility. This
)mitted term from r3P would correspond to a first correction
20 the Coriolis term r2P due to vibration.

Now, the rules, which we have just discussed, for the
:oupling of levels through a non—zero matrix element should
>e applied to levels with nearly the same energy. If a
)erturbation in the spectrum results from transitions in—
rolving a particular set of levels, it is necessary to con-
sider nearby levels and their possible coupling schemes.

Thus, we see that in determining which level has the greatest

 

 

 

 

33

nmabflity of causing the resonance, it is necessary to
wmida=both factors simultaneously: coupling, as well
sunpmmurbed energy difference.

Lazm be the order of magnitude of the coupling term.
picalxflbration-rotation energies in zero order are

;, first order ~30 cm’l, second order ml cm_l, etc.

000 cm”
t the energy difference between two interacting levels be
- En'; we use the following formula to express it in terms

order of magnitude:
2 - alog (E — E )
3 10 n n' '

ng the perturbation formula

y.l<nIH|n'>I2
‘ E-E'
n

n' n
. guide, we define the order of magnitude of the term con—

uting to the energy as
2m — [2 — -2—log (E — E )J
3 10 n n' '

Holenl arises in determining this energy difference.
tlyr speaking, it is necessary to use the zero—order
rhic .frequencies in determining the energy difference.
inlerrbally, even for fundamental levels, the quantity
lend :is not a harmonic frequency but the fundamental
fitted. by all the measurable anharmonic contributions.

91‘, Since only orders of magnitude are considered, it

 

 

34

is possible to use experimentally obtained frequencies in
combinations to form the various overtone and combination
level energies. Following this procedure, one can obtain
an order of magnitude, in general not an integer, of the
possible contribution to the resonance. Since the loga—
fithmic function is slowly varying, it is only natural that
the order of magnitude is most dependent on the value of m.
We have made a survey, in the manner described above,
)f some of the perturbations in the high resolution infrared
spectra of the methyl halide and deuterated methane molecules
with symmetry C3V obtained by the infrared spectroscopy group
it Michigan State University. For each case of a perturbation
;t is possible to give in decreasing order of likelihood the
>ossibilities of the cause of the perturbation. In this way
.t has been found that the case p = 2, pt = 1 is a likely
:andidate for several of the perturbations. However, p = 2,
't = 1 does not uniquely specify a case, unless one restricts
Limself to the case of a dyad interaction. In this case, let

-t be the mode giving pt = 1, then the change in v must be

t
'rom zero to one and vice versa. Also if vn is the non—
egenerate mode participating, then vn changes from one to

ero and vice versa. If Vt > 1 or vn > 1, then in general

pOlyad would occur. If Vt = 2, then the interaction would
e between the triad levels (vt = 2, vn = 0), (vt = 1, Vn = 1),
nd (Vt = O, Vn = 2). Thus, restricting ourselves to the dyad

 

35

se, we see that the most general type of interaction we

sh to consider is between the two bands v + Z v ,v ,
n , n n
n #n
d Vt + 2' Vn'vn“
n sen

 

 

 

 

CHAPTER IV

CORIOLIS RESONANCE

In this chapter we wish to give a detailed description
the calculation of vibration-rotation energies of mole—

,les with first—order Coriolis interaction. This topic has
en studied extensively (2, 16—21). Special cases have
len considered, and approximation methods have been employed.
Iwever, the problem has not been subjected to an analysis
:ilizing numerical methods.

The molecules to be considered have point—group symme—
‘y 03v plus the added restriction that all nondegenerate

;brations are of species Al. This class of molecules in-

Ludes the methyl halides, CH3X, as well as molecules of the “/
eneral type XY3Z’ XY3ZW, XYBZWV. In considering first—order “/f
>riolis resonance there are two questions to be answered: I
1) What diagonal elements shall be included in the Hamilton— “//
in matrix? (b) What off—diagonal elements shall be included? '"/'
iturally, if an off—diagonal matrix element of a given order “/j
3 to be included, then it is certainly necessary to include
Ll diagonal and off—diagonal matrix elements of the same and ”/1
lwer orders. 'M/
It is convenient to work in dimensionless quantities. “”5
3r this reason qSo and pSO are defined as '/(
D/

36

 

 

 

SO 80'

A
<—§>1/“0
41

1e Coriolis operator to be considered has the form

A
s' 1/4
F) q S'C'}POL

1 a
H . . = — -—{
Coriolis gla 20;,O,Csos'o'( sop

lich is part of H1. Thus, a strictly first—order theory

>uld be developed by considering the Hamiltonian to be

>wever, if Coriolis resonance is to be considered, the

3

:her terms in H namely rP2 and r need not necessarily

l,
a considered. These terms have matrix elements off—

iagonal in vS only. They are important in cases other than

3 2

Driolis resonance: r causes Fermi resonance, while rP
ever causes an accidental resonance. Nielsen (13) has
iven a modified contact transformation TR* which removes

1e terms rP2 and r3 from Hl'. Thus, in a first—order theory
Or the case of Coriolis resonance the Hamiltonian can be

aken as

H0 + HCoriolis'
f we further restrict ourselves to the case described at
he end of Chapter III, viz., Coriolis resonance between
ands vn + 2 V

n'#n
onsider the entire Coriolis term H

v v + v n we need no
n' n' and t g'#nvn, n" the t
. . . R ther we c
Cor1011s a an

 

 

 

38

estrict ourselves to that part of H which involves

Coriolis

0th participating modes n and t. The relevant operator

hen is
H0 + (Hl)nt
here (H ) = -2-£{z 2 Cu (ii: l/4 }P
1 nt I , , sos'o' A qsops'o' c'
a a sos o s

[n;tl,t2]

he diagonal elements of (H1)nt are the usual first—order
egenerate Coriolis contribution -2CK£tc:lt2, which is
resent in all symmetric top molecules. The diagonal ele—
ents of HO are the zero—order energies. Including these
arms thus allows one to treat the problem to first order
a the case of Coriolis resonance between a parallel band
ad a perpendicular band. Within this general framework a
iestion arises: How much more can be included to allow
or a better fit to the experimental data? The answer is
hat nothing can be added if a rigorous treatment is de—
ired. However, it will be noted that the inclusion of
iditional terms in the diagonal matrix elements does not
Abstantially complicate the problem. Additional off—
iagonal elements, however, greatly alter the character

f the secular determinants obtained. Thus, in this work
e consider the case of the exact first-order theory of
ccidental Coriolis resonance, and we also consider cases

0 which one additional parameter is added to allow for a

 

39

tter fit of the experimental data. Possible additional
rms that could be included in second order are variation
the rotational constants with vibration, and centrifugal
stortion. The other possibilities come from terms which
e either off—diagonal in second order, such as essential
sonances, or from terms in the third— or fourth-order
miltonians, which require an additional contact transfor—
tion. The obvious choice is a diagonal term from second
der. It is true that for most molecules which this study
uld include the rotational constants do vary considerably
th the state of vibration, while at the same time centrifu—
,l distortion could be neglected. Therefore, we have in—
,uded in an empirical way the a terms of second order which
.low for variation of the rotational constants with vibration.
Le point should be made clear. To be consistent with an
:act second-order treatment, the added terms must all come
’om second order and not from higher orders. The term r2P2
1 second order will give a variation with vibration of the
>tational constants in the diagonal matrix elements. How-
rer, it will not correct rotational constants which appear
1 off—diagonal positions. Thus, one cannot simply replace

and C,

B , .1 .
B + Be + ;“s(vs + 228)

c l
c + 0e + gas(vs + Egg)

 

/

LA

40

1roughout the purely first-order program. Rather, ad-
-tional terms EGS(VS + égs) should be added only in the
;agonal elemenis. These terms can then be grouped with the
niilibrium rotational constants to give a vibrationally
>rrected rotational constant. To reduce the number of
irameters, an effective a is introduced which corresponds

> the difference between the total effect of the d terms

1 the two interacting bands. Thus, a AuB and a Aac are

lcluded in the diagonal elements of the 2t = 0 states; this

irresponds to a difference in the variation of the ro—
ational constants in the two levels being considered. For

v one has

+ Z :V 1

1 3
gaS(VS + Egs) = 5“ + at + I at! + I

or Vt + I Vn'vn' one has

n'#n

1 1
gas(vs + 2gs) = 2am + 2dt + E at, + E

a ,(V , + —).
n'#n n n 2

I¢t

B

hese terms are coefficients of [J(J+l)—K2]—terms for a ,

nd coefficients of K2—terms for do. Since all matrix ele—
.ents are diagonal in J, the value of J in the it = i1 levels

ust be identical with that in the it = 0 levels. To further
implify the problem the approximation Adc = O was used be—
ause the effect of interest on the energy levels comes from

the variation with B rather than C with vibration. Thus,

 

"\\

 

41

3e AdB which is included in the 2t = 0 levels corresponds to
B _ B B B 2 B 2
a J(J+1) - (an - at)J(J+1) - (dnKn - ath )
l B B B B l 2 2
+ {—d + d + 2 a , + 2 a ,(V ,+—)}(K - K ).
2 n t t'#t t n'fin n n 2 t n
ince Kt = Kn i l and K is small for the cases we are con-

idering, the dependence on K is dropped. We see then that
here is no simple exact interpretation for AaB, but ef-
ectively it corresponds to the difference in the rotational
onstant B for the two interacting bands.

We have calculated the energy eigenvalues and the
ssociated eigenvectors of the Schroedinger equation using

he Hamiltonian

H = H + (Hl)nt’

O
= sp2 + (c B)P 2 - 2—l{2 Z “ <1§l>1/“ }p
- z I , ,Csos'o' A qsops'o' a
d a sos o s
[n;tl,t2]

onsider the Coriolis term (Hl)nt° The summation over

n;tl,t2] gives nine terms. Since by definition

a _ a
C Cs'o'so

sos'o'
’or all a; the ; "matrix" is antisymmetric. Thus only three

0L
ntl’

is non-zero

.ndependent C's are to be considered for a given a: c

G.

a a
nt2’ and g tlt2‘ Whether or not a given Csos'o'

s determined by the symmetry species of qsop . For

P
s'o' d

42
*os'o' to be non-zero, it is required that
P(qso) X F<psyov) X P(Pa) = A1 + ...,

.e., the product of the symmetry species of the operators
1 the Coriolis term must contain the totally symmetric

epresentation, Al' By using this requirement, it is found
iat since P(PZ) = A

2,

z _ z _
Cntl ‘ Cnt2 ‘ 0'

3r PX and Py the situation is less straightforward. One

inds (22), if n is of type Al:

x — _ y X = y =
Cnt2 ‘ Cntl and Cntl ‘nt2 0'

f on the other hand n is of type A

2,
X = y X = y =
Cnt2 Entl and Cntl Cnt2 - O'

n either case, all the off—diagonal matrix elements are
roportional to one constant, Cy = Cztl‘ Here we restrict
urselves to the case where all nondegenerate vibrations
re of type A1.

In the discussion that follows, the diagonal elements
nclude the effect of the a term, which allows for a vari—
tion of the rotational constant with vibration. The strictly

irst—order theory is obtained by-setting Ad = 0. Since all

he operators are diagonal in J, it is possible to consider

if
./
/
¢;
/

 

43

10h J value separately. For a given J, K takes on the
r+1 values -J 5 K 5 J. it takes on the three values

_ = 0, *1. 2t = O in the "parallel band" levels, and

LtI = l in the "perpendicular band" levels. Hence for a
Lven J the energy matrix has dimensionality 3(2J+l). The
atrix for J = 2 is given in Figure 4.1. Since we are re-

tricting ourselves to the Coriolis resonance case between

odes n and t, we write the relevant eigenfunctions as

I J K > I V v t a
n .

here vn = 0 or 1 only. If Vt = 0 then it = 0; if v = l

hen it = :1.

We see from this energy matrix for J = 2 that all the
on—zero off-diagonal matrix elements satisfy the condition
K = Azt. Thus, the change in K-zt is zero for the non—zero
ff—diagonal matrix elements. This condition is also satis—
ied for the diagonal matrix elements, By defining a new
uantum number R as R = K - it, we have for non—zero matrix
lements AR = 0. As a consequence of this fact, the rows
nd columns of the energy matrix can be ordered in such a
ay that the non—zero matrix elements are arranged in block
iagonal form. Each submatrix is characterized by a par-
icular value of R. Thus R is a "good" quantum number for

he Hamiltonian we are considering:

H = H + (H

O l)nt'

If
‘E/
./
¢/
./
_/,
.../I
/,
./’.‘

 

44

FIGURE 4.l——Energy matrix for J = 2.

-/2ABzy

E(1,o) dingy

—/3Ach

Jinan” E(l,l)

-IIAch

-/2AB;y

E(2,—1)

‘fiABCy

 

 

A

AA:

\
4x

it}

.32

AAA 2

 

 

45

r a fixed value of J we may abbreviate the notation for
e eigenfunctions as follows. Since it completely charac—

rizes the vibrational state, we may use I K, A >. The

t
= l, vt = 0, it = 0; the

- 0 state corresponds to vn

= 11 states correspond to vn = 0, V1; = 1, it = *1. The

ergy matrix for J = 2 written in the block diagonal form

shown in Figure 4.2. We see that for J = 2 the IRI = 3
bmatrices have dimension one; the IRI = 2 submatrices have
mension two; and the IRI = l and R = O submatrices have

mension three. In addition, the submatrices characterized
R and -R are identical if the diagonal elements satisfy

.e condition

E(K,tt) = E(—K,—tt).

lis condition is satisfied for all cases which we will con—

.der. The K2 degeneracy thus remains. The perturbation

t
a are considering does not remove this first—order de—
aneracy except in one case. The submatrix for R = 0 con—

acts I l, l >, | 0, 0 >, and | —l, —1 >. Thus the Kt

t
Bgeneracy for I 1, l > and I —l, -l > is removed by the
>riolis term.
We may generalize the results for J = 2 given above

5 follows. For a given J the energy matrix has dimension
:2J+l). This matrix can be written in block diagonal form
1th 2J+3 submatrices along the diagonal. The submatrices

1th IRI = J+l have dimension one; the two submatrices with

 

 

 

46

 

 

 

 

AH.Nuvm

 

Ao.muvm som<mxl

som<mxl AH.H-vm

 

AH-.muvm som<m\a
som<mxu Ao.Huvm

hmm<m I

AH.on

 

AH:.HIVm zom<m\u o

som<m I Ao.ovm som<m\-

o som<mxl

AH.HVm

 

Aau.ovm som<m\-
>Mm<m\|

O >Um<m\l

Ao.va som<w\l

o

AH.mvm

 

Aau.avm asm<m\u

s35- 8.3a

 

 

 

 

 

 

 

 

 

AH-.me

 

.H+.mu_

.o .m-_ .H+.H-_

.H1.m-_ .o

.H-_ .H+.o _

.HI.H-_ .o .o _ .H+

a

H+_

AH:.o _ Ao .H+

_

AH+.N+_ Aau.a+_ Ao .m+_ AH:

n

m+_

 

_Hu.m+v

 

47

H =.Ihave dimension two; and the 2J—l submatrices with

I <J have dimension three. The eigenvectors and eigen—
luesof the submatrices with dimensionality less than or
ualtm two are easily obtained analytically. It is possi-
eto solve the cubic equations analytically, but the situ—
ion is sufficiently complicated to call for a computer

>gram to solve for the eigenvectors and eigenvalues of the
Imatrices of dimension three. Using the computer program,
ch is described later in this chapter, we have been able
obtain the eigenvalues and eigenvectors of the Hamiltonian
the case of Coriolis resonance of the type we are con—
ering. In zero order the ”good" quantum numbers are:
itional—-J, K, M; vibrational——{vs}, {2t}. When’we in-

ie the first—order Coriolis term, the vibrations and

itions interact so that some of the "good" quantum numbers
must be combinations of rotational and vibrational quantum
ers. Previously we have seen that one such quantum number
, defined as R = K — 2t. Care must be taken in giving an
rpretation to R. K is the angular momentum quantum number
3 gives the projection of the total angular momentum along
synnnetry axis of the molecule K6. it gives an internal
Lar’lnomentum, Eth, but this angular momentum is in normal
liruate space. To obtain the equivalent angular momentum
>a1. space, one must transform by the usual orthogonal
x. The angular momentum is cgtth along the c-axis. c:
rr—zerm>only for a = z so the internal angular momentum

a1. space due to vibrations is citfih. Thus (K — tilt)h

 

48

5 the projection of the angular momentum due to pure rotation
f the molecule. R = K — it has no easily apparent physical
eaning since it is a combination of quantities in different
paces. We have lost both K and it as ”good" quantum numbers;
a have R as a "good" quantum number. There must exist an—
ther "good" quantum number for this Hamiltonian. It takes

he form of an index. This index I is used to order the

nergy levels for a given value of J and R. For IRI = J+l
here is only One level, T = 1. For IRI = J there are two
evels, T = l and 2. iFor IRI < J there are three levels,

=51, 2, and 3. The ET notation is used such that

The eigenvectors of the Hamiltonian HO + (H1)nt are
enoted by I J R r >. These eigenvectors are linear combi—

ations of the zero—order eigenvectors.”

I J R r > = Z ciETI J K it >.
K t t
’ t
he coefficients CEET are obtained in the computer solution
t
f the eigenvalue—eigenvector problem. CiET = 0 unless
t

-= K — it; it is also zero unless —J—l 5 R 5 J+l.

The eigenvalue equations are

HI J R T > = E | J R T >,

T

P2I

J R T > = J(J + l)h2I J R T >,

 

49

(PZ - Gtz)| J R T > = EhI J R r >.

12 is an angular momentum operator in normal coordinate

pace:
6tz = (qtlpt2 ' qt2ptl)

> C

hus G2 v 2 > = tthl vt z

zI t t t

Subtracting the same constant from all the diagonal
elements does not change the eigenvectors but does shift all
Lhe eigenvalues by this same constant. Hence, it is not
Lecessary to know the purely vibrational energies for the
;wo interacting levels; only their difference is important.
l1or this reason we have introduced a quantity Av which corre-
;ponds to this difference between the vibrational levels

:haracterized by 2t = 0 and those for it = 1l:

E (Vn=0’ Vt: , lt=*l) = constant
E (vn=1, v = , 2 =0) = constant + Av.

[n the calculation this constant is set equal to zero. Since
:he discussion of the energy levels will center around their
shifts from the unperturbed positions, the procedure of set—
:ing the constant equal to zero is valid. Therefore, the
:otal energy for the 2t = *1 levels is the pure rotational

:erm plus the vibration—rotation term; for the 2t = 0 levels

:he total energy is the rotational term plus the Av term.

 

 

50

ereis no vibration—rotation interaction term in it = 0
velssince the term is proportional to at.
Hm one—dimensional submatrices occur when IRI = J+l.

lsapmntly, K = iJ and 2t = 11 with the restriction

_= -J. Due to the previously mentioned Kit degeneracy,

4

ediagonal matrix element for R = J+l is equal to the one

'R = —J—l. The diagonal element is
E(IRI=J+1) = CJ(J + 2;:) + BJ.
eigenvectors are

I J=J, R=+J+l, I=1 > = 1'

 

J=J, K=+J, £t=-l >,

I J=J, R=—Jsl, T=1 > = 1-

 

J=J, K=-J, £t=+l >.

(D

levels are not shifted by the Coriolis interaction

Therefore, if lines corresponding to transitions to

3 levels occur, they will not be shifted.

For the two—dimensional submatrices, IRI = J. When
I the levels are K = J, 2t = 0 and K = J—l, it = —l.
2 = —J the levels are K = —J, it = O and K = —J+1,

l. The diagonal elements are given by

E(IRI=J,2t=0) = {B + AaB}J(J+l)

+ (c + Aac - {B + AuB})J2 + Av,

RI=J-1,I2tI=l) = BJ(J+1) + (c — B)(J—l)2 — 2Cc:(—J+l).

 

.4
2
./
v/
a/

‘\

\.

K

 

51

3e off-diagonal matrix element coupling theSe two levels is
At 1/4 An 1/4 y —
-{<;—) + <;—> 1B; /J.
n t

1e vibrational factor occurs frequently. It is denoted by A,

A A
A = {(;E>l/“ + (;E>l/“l.
n t

1e eigenfunctions are
| J, J, l > and | J, J, 2 > for R = +J,
| J, -J, 1 > and I J, —J, 2 > for R = -J.

For IRI < J there are three levels interacting. For

given R, the levels are K = R+l, 2t = 1; K = R, 2t = 0;
{d K = R—l, 2t = —1. The respective diagonal matrix ele—
‘nts are

E(R=K,t =+l) = C(R+l)(R + 1 — 2;:)

 

+ B[J(J+l) — (R+l)2], ./
c 2 ”A
E(R=K,£t=0) = {C + Ad }R
.,/
., 1"
+{B + AaB}[J(J+l) - R2] + Av, .
at
Z “J
E(R=K,tt=—l) = C(R-l)(R - 1 + 2Ct) I
«%
M

+ B[J(J+l) _ (R—1)2].

52

e off-diagonal matrix elements are

= -BA§y/(J—R)(J+R+l /2.

< 2 :0 I 2 =—1 > = < £t=—l I £t=0 >

= —BA;y/(3:E7T3:§:i77§s

e eigenfunctions are I J R l >, I J R 2 >, and I J R 3 >.

One special case must be considered. When R = O the

vels to be considered are K = l, 2t = l; K = O, 2t = 0;

d K = -l, 2t = —1. Because of the Kit degeneracy the

rst and third levels are degenerate. These levels are
upled, indirectly, through off—diagonal matrix elements

that the K2 degeneracy is removed by the first—order

t

riolis interaction. Since R = O,
tt=l I tt=0 > = < tt=0 | £t=-l > = —BAcy/J(J+l)/2 = w

is we may write the matrix for R = 0 as

E(R=0,t =+1) w 0
w E(R=0,£t=0) w
0 w E(R=0,2 =—l)

t

 

 

53

Fhis three—dimensional matrix gives a characteristic

{quation whose solutions are easily obtained analytically:

_ l _ _. _ _ .i / 2 2
El - §{E(R—0,£t——1) + E(R—0,£t—0)} +,2A I + 8w /A ,

D7]
ll

E(R=0,2t=il),

[‘11
II

I _ _ _ _ ;_ / 2 2
3 §{E(R-O,£t—il) + E(R-0,tt—0)} - 2A 1 + 8w /A ,
here A = E(R=0,tt=il) — E(R=0,£t=0);

e have assumed A > 0. The possibility of obtaining ana—

vtic solutions results directly from

E(R=0,£t=—l) = E(R=0,£t=+l)

)th being in the same three by three submatrix. It is not
result of all the non—zero off—diagonal matrix elements
sing equal. One eigenvector for this case is of particular

tterest, viz.,
| J 0 2 > = eél J l l > --%:| J -l —l >.
/2 /2

te that this eigenfunction does not mix it = 0 with it # 0.
We may summarize these results schematically by writing
Single six—dimensional matrix, Figure 4.3.

We have programmed the Control Data 3600 computer at

e Michigan State University Computer Laboratory to obtain

- a

x A, \ \_ \. \ xxx-«.-

 

    

514

.II a

..J J J. J. J. Jr

 

 

mmafilmvlaa+hkum +

ANUN + H-mvxa-moo

mxwa+mue~nx+ewxso<m u

 

mmhh+mue~wm+evxsu<m n

>< +
mmmlfia+hvhuflma< + mv+

mmfioo< + 0*

mmwa+x+e~nmne~xso<m .

 

 

Nunfi+m+h~Hmaeyxso<m :

 

mNAH+mVIAH+hkum +

AWUNuH+mVAH+mVo

 

 

 

H! I a Him I M

o I a

H+m_u x

Au-ws~+eola-eoo

somehxa

 

 

hum<bl

 

>< + hfims<+mv +

mwfloo<+ov

 

H n .o

a_ .Hne u _e_

e u _m1

 

A

u
N

am +

um+hvho

 

 

55

eigenvalues and eigenvectors associated with the energy
rices. In principle the energy levels could be obtained
all J values. In practice experimentally it is found
.t J values through thirty are usually adequate to identify
lines, at least for the cases which we shall consider.
.s in the program Jmax = 30. The program which was written
used is given in unformatted form in Appendix I. The
routine used to obtain the eigenvalues and eigenvectors
‘ the three—dimensional submatrices was obtained from the
' Computer Library, Co—op identification number F4 UCSD 1
EN. This subroutine was written to obtain the eigenvalues
. eigenvectors of a real symmetric matrix with dimension

ater than one and less than or equal twenty.

 

 

CHAPTER V
RESULTS AND DISCUSSION OF COMPUTER ANALYSIS

We now give some results which have been obtained
ith the method discussed in the previous chapter. Since
he problem is solved numerically, it is necessary to choose
numerical value for each parameter which appears in the
leory. In the general case there are nine parameters.
lese parameters are the equilibrium rotational constants
and C, the constants denoting variation of the rotational
)nstants with vibration AdB and Adc, the degenerate Coriolis
>upling constant IE, the vibrational energy difference Av,
{e harmonic frequencies on and wt in cm_1 which are used in
Le forms An = (2ncwn)2 and At = (2fith)2, and the accidental

riolis resonance coupling constant cy.

A and At always and only appear in combination as a

n
ctor
A A
A = Iq—t—W” + (YEW/”l.
n t
P an accidental resonance IAt — AnI << An, expanding the
ctor gives
A A
_ 1 t - n 2
A _ 2 + R'( A ) + OI.
n
56

, grim».

 

 

57

first order this factor is a constant, and thus two of
: parameters are eliminated from consideration.

The constants B, C, and I: for a given molecule are
lally known, at least to a good approximation, from an
Alysis of the unperturbed portions of the spectrum. These
cameters are thus fixed for the discussion of a given
lecule.

Four parameters Av, AdB, Adc, and cy remain. In
inciple, if the analysis has been performed on an unper—
rbed band, Av and the Ad's are known, and also Cy = 0.

r a band perturbed by an accidental Coriolis interaction

# 0. In our analyses that follow, the four parameters

ven above are chosen in such a way as to fit certain
Irtions of a perturbed spectrum. The cases of interest will
2 those in which a resonance occurs in a particular subband.
lis condition places a constraint on the four parameters. Av
a chosen so that for given values of the other parameters
le energy levels which are to interact strongly have nearly
ientical energies. Since we will focus our attention on the
ffect of the perturbation in a given subband, it is possible
0 set Aac = 0 because for a given K value the term which
orresponds to Aac is a constant and may be combined with Av.
km accidental Coriolis coupling constant Ey plays no role
rlthe choice of where the accidental resonance will occur;
.tis important in determining the character and magnitude

fl Um energy shifts for a given resonant case. We see then

:hm:the parameters B, 0, ti, AdB, and Av determine the

 

nu

_\ \ \X‘vé": I

 

 

58

position of the resonance, and Cy determines the energy
shifts for a given resonance.

To make a systematic study among all the possible
cases that can occur, it is necessary to determine reasonable
numerical ranges for the parameters. In general, for a
Coriolis coupling constant ICI 5 1. Inspection of Figure
4.3 shows that the eigenvalues are independent of the sign
of gy. In the expansion of the determinants, Cy appears in
the form (cy)2. We, therefore, may limit the variation of

;y to the range 0 5 cy'5 1. In writing cy =

Ey = ic. Our method does not provide for a determination of

c, we imply

the sign of :y. The physically interesting cases occur when
dB < B; we will limit ourselves to cases where AcB < B.
In Figure 5.1 we show a typical unperturbed perpen—

dicular band of a molecule with symmetry O The Q branches,

3V‘
which correspond to transitions in which there is no change
in the total angular momentum quantum number J, are not re—
solved. If the molecule had a greater variation of the

rotational constant with vibration, the individual transitions

for the various J values would be resolved. The pre—super—

 

scripts P and R denote respectively negative and positive

subbands, i.e. P denotes AK = —l and R denotes AK = +1. A

subscript is added to denote the K value in the lower level. ./
An as example, RQl denotes transitions AJ = 0 and K = l + K = 2, .3/
When dB is sufficiently large, it is possible to resolve the lei
individual lines for various values of J. For example, RQ1(J) «’I
corresponds to transitions K = 1 + K = 2 and AJ = 0, i.e. J + J° 4/

 

7/.
/’

FIGURE 5.1--Typica1 unperturbed perpendicular band of _/’
a molecule with symmetry C3v' ,J’

/J

 

 

 

 

 

 

60

The quantity of interest for our discussion is the shift
in the unperturbed line due to Coriolis interaction. To
obtain the spectral appearance in the presence of pertur-
bations, the shifts must be added to the corresponding un—
perturbed positions.
Once all the parameters except Av and cy have been

chosen, it is necessary to determine exactly what resonance
is to be considered. One selection rule for transitions is

AK = Azt. Transitions from the ground state (it = 0) to

the levels vt + Z v ,v . (At = :1) give a perpendicular
n'#n n n

band. The change in this case is Att = i1. If Att = +1,

then AK = +1 giving a positive subband. If, on the other

hand, Azt = —1 so that AK = —l, a negative subband is being

considered. When the strong resonance occurs in a positive

subband, Av is chosen in such a way that the following

equality between diagonal matrix elements is satisfied:

E<J,K,tt=+l) = E(J,K—l,£t=0).

If a purely first—order theory is being considered, the
above relation is independent of J and will be valid for
all J values of the subband. On the other hand, if AaB
is non—zero, the equation can be satisfied for only one
value of J. It should be noted that J need not be an
integer. It is usually necessary for the exact resonance
to be placed at a position between two integral J values
in order to represent the experimental data. Figure 5.2

shows the energy levels for the case of a resonance in a

i
I

u/
/.
J”
J/
p/
B/
/,

 

 

61

FIGURE 5.2—-Energy level diagram for Coriolis Resonance

in the Kth positive subband.

 

 

 

 

 

 

 

 

 

 

 

 

Perpendicular Band (it = *1) Parallel Band (it = 0)
K _ _ _ _ _ T_=l _ _ _ _ _ _ ._
Ktt>0 T K—l
I
I T=2
— — _ —| —I— —'— _ _— I__ — — _ — .-
| I
I I
l
K—l I |
I K—2
I
' I
(K—2)£f<0 | |
1. _‘_____T=3________
I I l
l ' l
I
I i I ,,
I .
PQ RQ --/
K-l K—l ,,
I I I
I ,-
SPECTRUM: I l I ./
l I I

 

 

 

 

62

positive subband. In this and the next figure all levels
have the same J value. If the strong Coriolis resonance

is to occur in a negative subband, the energy level diagram
appears as in Figure 5.3. In these two figures the dotted
lines denote the positions of the perturbed levels and
transitions. The intensities of the lines in the perturbed
spectrum will depend upon the expansion coefficients of the
perturbed eigenfunctions for the appropriate levels.

We have seen that no more than three levels are coupled
together through off—diagonal matrix elements. For cases of
interest, Av will be chosen so that two of the levels are
nearly if not exactly degenerate. The third level, if
present, will interact to a lesser degree. Consider three
interacting levels: Ki = K+1, it = +1; K” = K, 2t = 0;
and Ki = K—l, 2t = —1. Assume Av has been chosen so that

E(KL) = E(K”) or
IE(KL) — E(K”)I << IE(Ki) - E(KI)I.
In addition we require
E(KL) > E(K“) > E(Ki).

Hence, we are considering a prolate rotator (C — B > 0),

The perturbed energies and associated eigenvectors are:

3’
L?
.l/
V,
,2
./
,J/

 

 

 

(I
63
./
.J/
FIGURE 5.3—-Energy level diagram for Coriolis Resonance ‘
J”
in the Kfl negative subband.
V/
Perpendicular Band (it = 11) Parallel Band (at = 0) ‘1’.-
//
K+1 .//
__ __ __ __ T=1

(K+l)tt>o

—"I
I
I
l
I
|
l
|
l

 

fl
\

 

 

 

 

 

SPECTRUM:

 

 

_— __-_

 

 

E1: | J, R=K, l > = cifi+l| Kl, +l > + CKFIOI N, 0 >
+ 0%E1_1| Ki, —1 >,
E2: | J, R=K, 2 > = CKf5+ll KL, +l > + CKE50I K“, 0 >
+(%f,lIKL—l>,
E3: I J, R=K, 3 > = Giff+ll KL, +l > + Giffol ", 0 >
+ cif§_1| K', —1 >.
If ;y = 0, then
CKfi+l = CKEIO = CKff—l = +1'

All other C's are zero. If Cy # 0, then in general all the

C's are non—zero. For the resonant case being described

J

C $3 = 1; all the other C and CJK3 are small. The
KL’—l Ki,—l

energy shift for the level Ki = K—l, “t = —l is also small-

We Shall not consider it explicitly; however, the energy
shift of the third level is always implicitly given as the
negative of the algebraic sum of the shifts of the two
Strongly interacting levels with the same J and R values.
This is a consequence of the trace of the submatrix being
invariant under a similarity transformation. The curves

giving the energy shifts of interacting perturbed levels

 

—/

 

 

65

will be labeled by KI and K". K_L denotes the state for
wh1ch ICKL,+1| > ICK",O|' K” denotes the state for wh1ch
ICK”,0| > ICK1’+1I. In the event that both strongly 1nter—

acting levels are denoted by K1 or K“, we change K, to Kll

(or K” to KL) for the state with the smaller ICK (or

,,.II
ICKu,0I)- It may happen that even though E(KL) > E(K“),

the perturbed levels have KH above KL: These difficulties,
stated in the two previous sentences, arise because we wish
to describe our results using perturbation theory language;
however, perturbation theory is not strictly applicable for
these exceptions.

Let us first consider the consequences of a variation
of Av. For a given set of parameters, let Av(XR) denote that
value of Av for which the two strongly interacting levels
are exactly degenerate. We wish to consider cases where
Av = Av(XR) + v'. If v' < 0, then E(£t=0) is less than
E(£t=il); whereas if v' > 0, then E(£t=0) is greater than
E(£t=il). When v' = 0, the two degenerate levels are split
by the accidental Coriolis coupling terms. No problem arises
in this case because no crossing of levels can occur.

If v' m —{E(Kl) - E(Ki)}/50 and AdB = 0, then
Eczt=0) is N{E(KL) — E(Ki)}/50 cm‘1 below E(£t=tl) for all
J values. For small values of J the levels K1 have positive
energy shifts as expected due to "repulsion" of the two
levels; however, for larger J values there is a cross—over
and the K1 levels have negative energy shifts. This case is

shown in Figure 5.4. This behavior can be explained by

 

 

Energy Shift in units of .883 cm—1

66

FIGURE 5.4—-Energy shifts due to Coriolis Resonance,

B _
cross—over case, Ad — 0.

I\ \ '\.. \. \ \' "I

+1OI— I/J .
_ / V,x’

_- KL - 6 ///

//
J_ __ _...
K” 5 // /
/ v'
+5-

I L.

 

I
Ln
I
/
\

 

 

67

observing that the effect of the off—diagonal matrix ele—
ments is more important than the order of the energy levels.
For this case AuB = 0, so the energy difference is constant
and independent of J. However, the off—diagonal matrix ele-
ments are J—dependent. This case corresponds to a split Q-
branch. Neither the K1. nor the Ku level corresponds to a
single K value. There occur extra lines in the region of
the cross—over because each level results from a linear
combination of the K+1 and K states.

If v' is still small compared to the energy level
spacing but larger than for the case just described, the
general consequence is a decrease in the overall effect of
the perturbation as one may well expect. This situation,
therefore, becomes a less interesting case. An arbitrary
choice of v' might cause the resonance to appear in a differ—

ent subband. The general effect on the spectrum is not a
sensitive function of v'. Av cannot be determined precisely,
and hence the A's cannot be determined from this work.

Now we consider the effects of varying cy. The value

I" Cy is not important in determining the position of the
esonance. We consider the case where Av = Av(XR). Figure
,5 shows the variation of the energy shifts for two values

2; y in the case where AdB = 0. Note that the energy
ifts scale is adjusted so that the energy shift per unit
is a constant. The energy levels are shifted; thus, the
r-responding spectrum lines are shifted. This shift is

>roximately linear in J. Hence the Q—branch is spread

 

 

FIMME5.5-—Variation

+10

68

of energy shifts with gy, AaB

 

 

 

\ \. \ K \S "‘I

 

 

69

outdueto the Coriolis resonance. The Q—branch is also

smmadout due to the dB term occurring in second order.

Asa mmsequence, qualitatively the dB term of second order

andtheaccidental Coriolis term of first order enter into

thesmm effect. However, the Coriolis term gives a shift

Inwarin J, whereas the shift due to c3 is quadratic in J.

Whmiboth factors are present, the usual interpretation of

B mum:be modified to include the effect of the Cy term.

a
y are

0 case no pronounced variations with c

In the AoB =
The ty = 1

obtained; however, two results should be noted.

case has a stronger interaction with the third level, which

in this case lies below the two strongly interacting levels.

The effect of the repulsion is to push both of the cy = 1

levels upward. For Cy = .01 case the magnitudes of the

positive and negative energy shifts are approximately equal.
The implication here is that the third level is less important

and 11:8 effect could be more nearly neglected.
.A more interesting situation is obtained when AdB # 0.

In IfiigLure 5.6 the results for AuB = +.075 are given. Here,

A\)(XR) with the two strongly interacting levels being
both with J = 10. Note

b V =
= 5, it = +1 and K = 4, it = 0,
iai: jJ1 ‘these cases the curves have a completely different

Iadraxztexr as cy is varied. The scales have been adjusted

tliafl: 'the shift per unit Cy is constant for the four cases.

5 L18 c<>nsider these results using the perturbation

Dression

-.fi

\.\'\.\"\"

i
\.‘

 

 

a? J J J J J J J .J...

70
FIGURE 5.6--Variation of energy shifts with cy, AdB = +.075.

+l
+5

 

 

 

 

 

 

 

lO

   

I|<nIHl
n' En En' '

The energy difference En - En, = AaBJ(J+l). This is a second-

order quantity since the a term arises in second order. Now,
< n I Hl I n' > is approximately proportional to cy/J. Both
the numerator and the denominator are J-dependent; however,
Cy is, in general, a first—order quantity whereas AaB is of
second order. Thus, the /J would dominate and cause the
divergence as in the previous AaB = 0 case. As Cy is reduced
by an order of magnitude, the J(J+l) factor dominates for
large J and the asymptotic form is obtained.

An important contrast between the AdB = O and the
AoB # 0 cases is that the former case leads to curves which
are concave toward the J—axis, whereas the curves are convex
in the latter case. This provides a useful criterion in data
analysis.

Now we consider the case when the parameter AdB is
varied with all other parameters except Av fixed. Av is
chosen so that in each case Av = Av(XR) for the same J value
(J = 10) in the fifth positive subband. Figure 5.7 shows
the results. These curves are similar to those for a vari—

ation of Cy. As would be expected, increasing AcB has the

same effect as decreasing Cy. However, the variation of

AdB has a much greater effect on the adjacent subbands. If

B . . . .
IAd I 15 large, a resonance in one subband 1mp11es a resonance

in the adjacent subband (at a different value of J, of course).

 

.,/’
“/1
«,f

\

\\

\

 

.meum IJ..J IJIIwII/ .J .J -

72

FIGURE 5.7-—Variation of energy shifts with AcB, ty
+10-

1

J

  

 

 

 

so Hes.a so noses on chasm smsoem

 

“
5
_

 

 

 

73

Thus, if experimentally the resonance occurs in only gag
subband, it is necessary that the theoretically predicted
resonance in the adjacent subband must occur at a J value
which is greater than can be experimentally observed. In
fact, this prediction might suggest that a search for such

a second resonance in adjacent subbands be undertaken. This
constraint as to the localization of the resonance to a given
subband or subbands thus allows an upper bound to be put on
the possible value of IAaBI in particular cases.

The methyl halide molecules are prolate symmetric
rotators, i.e., C > B. In this case, the term (C — B)K2 is
positive, and in a particular band the level (K+1) lies above
the (K—1) level for a given value of J. For the perturbation
to occur in the positive subband, it is the (K+1) level which
interacts strongly with the 2t = 0 level. If AdB is positive
and the resonance occurs for a particular J value, no other
resonance can occur in the RQK or PQK subbands since the K,
it = 0 level lies above the K+1 level for larger J values.
However, if AuB is negative, the K, 2t = 0 level lies below
the K+1 level for larger J values, and for some larger J
value there will be a resonance in the PQK subband. Thus,

a resonance in a positive subband with AdB negative implies
the existence of another resonance in the corresponding
neSative subband. Similarly, a resonance in a negative sub—
band with AdB positive implies the existence of another

resonance in the corresponding pOSitive Subband_ If only

one resonance is observed experimentally, aCCOrding to this

"‘I

\ \.. \ \ \ \\-I\.-

\

\,

A

I
._/

 

74

theory bounds may be placed on AdB so that any other implied

resonances occur in regions not observed in the particular
experiment. For these and another reason we have chosen to
investigate the case of AoB positive and primary resonances
in positive subbands. The other reason is that for positive
subbands the upper levels never occur in the one—dimen-
sional matrices. All the levels may be shifted. We are
thus not restricted to cases for which unshifted levels must
be fit into the description.

In conjunction with this last point an interesting
situation presents itself. It is obvious that a purely
vibrational operator in the Hamiltonian cannot cause a
rotational perturbation. However, one may ask whether a
vibration—rotation operator, such as r2P, could cause a
"quasi—vibrational" perturbation, i.e., cause shifts which

are independent of the quantum number J, for one particular

subband. As an example, consider the following case: Av is
taken to be Av(XR) for the levels J = 5, K = 5, 2t = +1 and
J = 5, K = 4, it = 0. The results for the subband of interest

is given in Figure 5.8. Of course, the adjacent subbands must
be investigated. The shifts for the "flat" case are —.75

cm—l. For the next lower subband the shifts for K1 = 4 are

1 for J < 9. For J = 4 the shift is .01 cm‘l.

For KL = 6 the shifts are less than .1 cm_1 also for J < 9.

less than .1 cm—

However, a resonance occurs in KL = 6 at J = 21. If the lines

are unresolved, the R0, branch is shifted in total by .75cm‘l

while for the adjacent subbands, at least for low J values,

 

 

 

I.III 1

. ... J. J / J. IJIIJIJIm ,,

 

r

 

If
M
J
J
a/
./
./

 

2O

5
7

10

 

 

 

FIGURE 5.8—-Energy shifts for "quasi-vibrational" perturbation.
+11—
+5--

IEo me. so when: on peach sateen

 

76

the shifts are smaller by a factor of at least seven. Of
course, the adjacent subbands will show a spreading, indi-
cating that it is in fact a rotational perturbation which
is being observed.

Finally, we wish to present a set of graphs for the
behavior of the various subbands under the influence of
Coriolis resonance. Both the purely first—order theory
(AoIB = 0) and AdB = +.O75 are included. These two cases
can be directly compared in a meaningful way since 4y = .2,
and the molecular parameters B, C, and CZ are the same in
both cases. Av was chosen for the AaB = +.O75 case so that
Av=Av(XR) forJ=lO,K=5,£t=+1andJ=lO,K=4,

2t = 0. The first—order case has Av = Av(XR) for the same

subband. These graphs are presented in Figures 5.9 - 5.18.

:\ x. \ \._ \ \ \. xxx-3

 

 

77

FIGURE 5.9-—Comparison of energy shifts in RQO for

Ad = 0 and AaB = .075.

+10

AdB = .075

---°-----Ac = 0

+
U1
i

 

 

Energy Shift in units of .587 cm—1

I
U1

I

l

 

—10 “

 

If
a/
J/
v;
J/
a/
.J"
“/,
/

 

 

78

FIGURE 5.10——Comparison of energy shifts in RQl for
B

AOL = 0 and Ad = .075.
+10Io
~~ —————— ME = .075
T “"—““‘ A0 = 0
+53—

 

 

 

Energy Shift in units of .587 cm—1

 

 

\ \. \. \'\"‘I

K
\‘.

V/
/I

 

 

79

FIGURE 5.ll——Comparison of energy shifts in RQ2 for

Ad = 0 and AaB = .075.

\ \. \. \ \J' N

+10?
J- —————————— AaB = .075
_ ———————— A0. = 0

—l
+
U1

_ \

 

 

 

 

Energy Shift in units of .587 cm

..I
a_ v//
,
—I— ’1'
'1,
I J
-I ax
/
.. -//
x
...5 _.I—
i//
a
,
.. ’1
,x

 

—10“

 

80

FIGURE 5.12—-Comparison of energy shifts in RQ3 for

.075.

0 and AcB =

Ad

.075

AaB =

+101I
-t

= 0

——-----Aa

 

+5-.

J. J J J J J JIJ/I J11? a

 

 

1 lI; .TI:-+II I I--TIIITIII+;IIrIII+III+IIIIIIIIIIIITII1
5 0
_

—1

HIEo sum. so none: on chasm swatch

 

81

FIGURE 5.13—-Comparison of energy shifts in RQM for

Au = O and AcB = .075.

+1 —————AcB = .075 I
_. __- __ _. Au = 0 If
/
//
/ Kn = 4
_F I

I
+54. /
I

_I

I

 

Energy Shift in units of .587 cm-1

 

I
\
—— \
\

_ 4~ \

5 \

\
__ \
\
\
\
—_ \
\
\
\ K =
__ \ 1 5
\
\
\
._ \
\
\

~10J— \

 

é;
./
V;
«x
/
«1”
V/
n/'
2

 

 

82

FIGURE 5.1M-—Comparison of energy shifts in RQ5 for

Au = o and AaB = .075.

+10T

 

Energy Shift in units of .587 cm‘1

 

  

 

 

 

83

FIGURE 5.15——Comparison of energy shifts in RQ6 for

AOL = O and AaB = .075,

+lGr-

7
j/
J
J
j
./

 

 

+5—
:—I
I
E _
o
L\
co __
Ln
CH _
o
w -— .
JJ /
~r—i
C1 .
:3 -r/
E
4.)
(H .
.,..| __ r/
g .
m ..
,/
is I "
$4 I
a) __ ._/
CI
LL] ’.
-5—
-_ /,

 

 

 

8”

Q7 for

R

FIGURE 5,16—-Comparison of energy shifts in

.075.

O and AuB =

Au

+10?-

.075

Ad

—-——--—Aa

H

 

 

 

_ — _ _ _ _ _ _
d _ A _

:50 same ho mung: cs phflem swsmcm

 

“
R,
_

 

 

85

FIGURE 5.17-—Comparison of energy shifts in RQ8 for

Au = 0 and AaB = .075.
+10
AOLB = .075
...... —Aa = O

U‘l
l
.

 

 

 

I Energy Shift in units of .587 cm.1 +

U7
1
l

 

 

 

86

Q9 for

R

FIGURE 5.18-—Comparison of energy shifts in

.075.

Ad = O and AaB =

AaB = .075

“"‘"“Aoc

 

 

 

_ _ _ . . . p .

 

 

..

+10]

_
_
5

+

H

_ _ _ _ _ q _ _
use smm. so muses 2H phflgm smmosm

_

_
5

_

CHAPTER VI
ANALYSIS OF A PERTURBATION IN THE

v3 + V4 BAND OF CH3F

The infrared spectroscopy group at Michigan State
University, directed by Professor T. H. Edwards, has studied
the methyl halide molecules for a number of years (23—26).
One study has particular relevance to the present work.
Blass (27) has obtained and studied localized perturbations
in the spectra of some axially symmetric molecules. In this
chapter we propose to give a quantitative explanation of a
particular perturbation in the v3 + v“ band of CH3F by as—
suming it to result from Coriolis resonance between the
perpendicular band v3 + VA and the parallel band v + v

1

3.
Thus, using the notation of the previous chapters, t = A

,
n = l, and n'#n is 3.

The fundamental frequencies of CH3F as tabulated by
Herzberg (28) were used in conjunction with the order of
magnitude equation to obtain an estimate of the expected
order of magnitude for the various terms in H1, H2('), and
H3('). The orders of magnitude 0.8, 0.9, 1.0 were obtained
respectively for v1 + v3, v3 + 2v5, and v2 + v3 + v5.
According to the Tarrago—Amat classification, these

87

if
Q;
4]
./
pf
/
p/
x"
/
-/.

 

 

88

correspond to p = 2, pt = l; p = 3, pt = 3; and p = 3, pt = 2-
The latter two possibilities are Case a. For this case the
purely Vibrational operators are the most likely candidates.
Since the data clearly show rotational perturbations, we may
disregard these possibilities. The remaining possibility is

p = 2, pt = l, i.e., the situation which was studied in detail
in the previous chapters.

Blass has tabulated the energy shifts in the transitions
for the various subbands as a function of J. These shifts,
however, cannot be compared directly with those theoretically
predicted. The experimental data and the theoretically pre—
dicted curves must be in compatible forms before they can be
compared. Blass' plot of the energy shifts for the RQl branch
is given in Figure 6.1. Note particularly that for three J
values (J = 10, II, and 12), two energy shifts are shown for
each J. Both shifts were measured from the same 2t = +1
level. The two levels for the same J giving the two shifts
are here assumed to be the two strongly interacting levels
in the Coriolis interaction. One of the energy shifts should

then be measured from the 2 = +1 level and the other must be

t
measured from the corresponding it = 0 level. If we require
that the energy shifts for J values in the region of the
resonance be nearly equal in magnitude, the positive energy
shift for J = 10 must be reduced, and the negative energy
Shifts for J = II and J = 12 must be increased. Therefore,
we conclude that the it = 0 levels cross the 2t = +1 levels
between J = 10 and J = II, and that AaB < O. The decrease

 

./’
/’
./
v/x

Xi

\

 

A

ll

 

 

I ‘

 

 

89

FIGURE 6.l--Experimental energy shifts in the RQl

branch as determined by Blass.

 

 

 

 

+lO‘r v’
- A
_ /
_ ./"
+5- A

H

I

E _

O

.:r

w --

H

I. . A

O A A

U) _ A A

AA

rs A A I. . I —I+ J

C ADAQAAAAio A 20 30

.,—| - ..

33 A A.

-:—I ..—

fl _

m /

>3 _ .

£10

£4

(1)

LE ‘ A

-5- /
~ A ./I
I I
-10 A

 

90

in the magnitudes of the energy shifts for J = 10, J = 11,
and J = 12 must be made consistent with the choice of AaB.
Let us consider AaB in more detail. Under the as—

sumptions of Chapter IV
AOLBJ(J + l) = (dnB — dtB)J(J + 1).

If Av = Av(XR:K,J), then the energy difference between the
two levels interacting strongly through Coriolis resonance

(J, K, £t=+lg and J, K—l, £t=0) is

AaBJ(J + l) + g

where E is chosen so that the entire expression vanishes for

the desired J value. If AaB < O, and we consider J values

below the resonant J, J(XR); then E(£t=0) > E(2t=+l). Thus

the energy shift for the E(2t=0) level is
The shift for the E(£t=+l) level is

E2 - E(2t=1).
Above J(XR), where E(2t=+i) > E(2t=o) the shifts are

El - E(2t=l),

E2 - E(2t=o).

 

 

 

91

These are the predicted energy shifts. However, using this

notation, the energy shifts given by Blass are

El - E<tt=+1),

To transform Blass' data we use
E(2t=o) = E(£t=l) + AaBJ(J+l) + g.

The positive energy shifts for levels below J(XR) are reduced

by

 

AaBJ(J + 1) + g, ’

whereas, above J(XR) this same quantity is subtracted from v//
the energy shifts corresponding to the E2 levels.
If the proposed explanation of the perturbation ob— /‘
served by Blass is correct, the theoretically predicted 1
I

curves for a given set of molecular parameters should fit ,$
all the experimental data points within their experimental
accuracy. However, if the theoretical treatment does not ‘
exactly correspond to the physical situation, it becomes .-I
necessary to compromise and obtain a "best" fit to all the
data. An alternative procedure would be to fit selected _ ,
portions of the data with a set of molecular parameters.
The set of parameters would be allowed to vary when con-

Sidering different portions of the data. A test of the

 

 

92

correctness of the theory presented in this work is to
determine if the variations of the molecular parameters are
within reasonably small intervals. We shall use this pro—
cedure in order to test our method's applicability to explain

Blass' CH3F data. Three analyses are considered:

Analysis I: Fit the curves to the J = 10 and J = 11
points in the RQl branch, in a self—consistent
manner.

Analysis II: Fit the curves to the points J = 10, J = 11,
and J = 12 in the Roi branch, in a self—

consistent manner.

Analysis III: Fit the RQO branch.

Here we take self—consistency to mean that the AaB chosen to
obtain the energy shifts from the E(£t=0) levels for the
experimental data is the same as that used to determine the
magnitudes of all the theoretically predicted energy shifts.

The data we wish to use in considering the above three
analyses are contained in the RQl and RQO branches. In the
RQl branch there is a well—defined perturbation showing a
mixing of the wavefunctions and, therefore, the apparent
doubling effect. The RQO branch can be treated analytically
Since the relevant levels appear in R = O submatrices. The
energy shifts for the RQO branch as determined experimentally
are given in Figure 6.2.

First we consider Analysis I. A self—consistent fit to

the data is obtained when AaB = —.0513 and 4y = .OAOIA. The

 

All\\ \\+.

'\

\

 

I. \li‘a

.'\“

 

   

 

93

FIGURE 6.2-—Experimental energy shifts in the RQO

branch as determined by Blass.
+10*

+5-—

_1
Energy Shift in units of 0103 cm

 

—10I ZS

 

   

QA

energy shifts for the RQl and PQl branches are given in
Figure 6.3. The corresponding energy shifts in the RQ0
branch are given in Figure 6.4. The relevant data points

of Blass have been included. The fit in the RQl branch is
excellent below the primary resonance, J < 10. However,

the agreement above the resonance is not as satisfactory.

A better fit to the experimental data could be obtained if
the magnitude of AaB were smaller. This would correspond to
a less localized resonance. The interesting behavior of the
energy shifts for the K” = 1 levels should be noted. This
situation corresponds exactly to one suggested in the previous
chapter. What appears to be a compound resonance, in actual—
ity, is the resultant of the inclusion of a single primary
resonance. The curve for KL = 0 giving the energy shifts for
the PQl branch is in qualitative agreement with the experi—
mental data. In fact, the experimental energy shifts for PQl
require that the K" = 1 levels behave qualitatively as they
do in the theoretical curve. Here again, a better fit could
be obtained if the resonance were less localized, i.e., if
IAaBI were smaller. Similarly, the curve for the shifts in
the RQO branch is in qualitative agreement, and a better fit
could also be obtained if |AaBI were smaller. Thus, we see
that these three curves give qualitative agreement with the
three Q—branches of the Blass data. More importantly, all
the discrepancies, RQl, J > 11; Png and RQO, can be reduced
if the magnitude of AaB is reduced. The parameter gy must

be readjusted in each case.

 

 

\ \ cu \

\

.\
\

X.-

.‘ V I ,-
3 3 \. \
1 '\__\__

 

 

FIGURE

+lOr

‘—

I
.5.

—l
+
U1
I ‘I‘ ~I -~+-I

I

Energy Shift in units of 0.11 cm
T

I
U1

 

-101-

95

6.3—_Energy shifts for RQl and PQl branches,

Analysis I.

[X - RQl (experimental)

:7 - PQl (experimental)

  
  
   
 
  
 

K \ \. V. \ K" 'V

\‘.

‘J

 

 

“iii

.__.a

 

' R “(I
FIGURE 6.A-—Energy shifts for Q0 branch, Analysis I. J
I
a”;
+lO-- . ‘1
J
_. A - RQO (experimental)
-/
J1
/
v;

30
I+ J

Energy Shift in units of 0.11 cm

 

 

 

 

97

In the second analysis we consider the non-self—
consistent case for which the curves are fit to all the data
points in the overlap region, J = 10, J = 11, and J = 12.

The value of AdB

to fit the above three points is —.O2692.
The energy shift curves for RQl and PQl, and RQO are shown
in Figures 6.5 and 6.6, respectively. We may compare these
curves with those of Analysis I to see the effect of the de—
crease in the magnitude of AaB. The Kl = 2, J > 11 curve is
raised slightly, however still not enough to give a good
"average” fit. The curve for the PQl branch gives a much
better fit. Similarly, the curve for the RQO branch is
closer to a correct fit than Analysis I. The principal diffi—
culties of Analysis I still remain, however, all the dis—
crepancies have been reduced. There is an important reason

for considering Analysis II. The equation for transforming

 

the J = 10 and J = 11 data points contains two parameters,

AaB and 5. Since it is always possible to fit an equation «’
containing two parameters to two points, it would be de— ex’
sirable to have more points to check the consistency of the «”

two—parameter fits. Here this check can be made since three
points are available. In Table 6.1 we give the results of
the three possible two—parameter fits.

In Analysis III the molecular parameters are not
chosen to fit the perturbation in RQl but rather the ROD
shifts. Of course, we still require a resonant cross—over

at J = 10 in the RQl branch. Figure 6.7 shows the fit to

 

the experimental BQO branch. The two constants in this case

 

98

FIGURE 6.5—-Energy shifts for RQl and Pol branches,

+1qe

" Y7-— PQl (experimental)

Analysis II.

[X - RQl (experimental)

+5.

Energy Shift in units Of 0.11 cm'1

I
Lpi

 

-lg_

 

 

 

\,

\

 

 

 

99

FIGURE 6.6——Energy shifts for RQO branch, Analysis II.

+101—
-- Z: — RQO (experimental)

+
U'I
I

I Energy Shift in units of 0.11 cm"1

 

    

 

 

 

100

FIGURE 6.7—-Energy shifts for RQO branch, Analysis III.

+1
[X — RQO (experimental)

+
U1
I

-l

 

‘39-) J

 

Energy Shift in units of 0.11 cm
I

5
U7
I
I

 

 

-10 _I.._

 

 

 

101

TABLE 6.l—-Two—parameter fits of Analysis II.

 

 

 

Data Points Used AaB
J = 10, 11 —.0512
J = 11, 12 -.0493
J = 10, 12 _.0502
are AdB = _.021 and ty = .o3u1. It should be noted that for

the range 7 < J < 18 there is a systematic difference between
the data points and the curve. This difference could be re—
duced by a greater localization of the resonance. However,
if IAaBI is increased to localize the perturbation, the
curve will reach its maximum at a smaller J value, as in
Analysis I. We conclude that both the absence of the syste-
matic difference and a close fit near the maximum shift can—
not be obtained simultaneously in this theory if a resonance
is required in RQl at J = 10. Figure 6.8 shows the RQl and
PQl branch shifts for this case. The PQl fit is not quanti—
tatively satisfactory. Surprisingly, however, the RQl
shifts for J > J(XR) seem to give the best "average" fit of
the three analyses. Below J(XR) the agreement is not as
grafifying.

The results obtained for the three analyses considered
are summarized in Table 6.2. The appropriateness of this
theory in explaining the perturbation observed by Blass can

be judged from the parameters in the table. Since the

\ \.. \ \. '\ \ \. K ‘\ ‘IV

  

 

 

 

102

FIGURE 6.8—-Energy shifts for RQl and PQl branches,

Analysis III.

—1

 

+10?-
R .
__ [3* Q1 (experimental)
-' - PQ (experimental) K'
V 1 L
+5-
E —-—
O
H Ax
3 K1. = 2
~— A
E K — 1 ‘_le>
H A
3 I A
H 20
g V v.4" 'VA'A!_VNAVAV v.- _ +
‘ ‘ A‘A" “ _WV-V-VA' ‘ ‘ O
Q A. V‘— 3
.... ~1- A v
a: _ VV
'3 ‘* KI ' 2 K ,, l
m A
>, __
m
a
a) —-r—
53‘ a
...5 ——
' =
K; O
1_ KM
1101-

 

 

 

103

TABLE 6e2——Summary table of molecular parameters for CH3F1.

 

 

 

 

 

 

n n
_ -1
B — 0.8518 cm
0 = 5.081 cm“1
Z _
ct - 0.08
, 13= B_ B y=y
AnalySis Au (an at ) ; Qntl
I —0.0513 .0401“
II —0.02692 .033U9
III —0.021 .0341

 

lThe unperturbed values of B, C, and a: are those
given by Blass (27).

general features of the experimental data are reproduced by
the theoretical curves, we conclude that the perturbation in

v3 + v“, a perpendicular band of CH F, is indeed due to

3
Coriolis interaction with a nearby parallel band, assumed to
by v1 + v3. Of course, the agreement is not complete but it
appears that Coriolis resonance is the dominant factor.

Better agreement could possibly be obtained if other higher
order terms were included. In any case, the agreement appears
to be sufficiently good that this method could be used in

conjunction with a reanalysis of the data to determine the

aSSignments of the transitions with greater confidence.

 

\_ \ \'\ \‘Kt‘fi?

 

 

x 4X li‘igcxa a:

444‘;

R

 

10“

In conclusion we display the theoretical predictions
of Analysis I for all the transitions RQK, PQK, 3 < K < 10,
in Figures 6.9 — 6.16. Only K = l and K = 2 were plotted
when Analysis I was first discussed. Analysis I is the
self—consistent case. The energy shifts in these subbands
have been given by Blass. His data points are included.
The predicted and experimental values are in good qualitative
agreement for K < 5, at least for J < mlS. In some cases
this agreement holds for all values of J observed experi—
mentally. We note that for PQK (A < K < 10), which we have
not been able to fit, there is a systematic trend in the
experimentally obtained energy shifts. This character indi—
cates the likely presence of another perturbation since the
predicted energy shifts due to Coriolis resonance are small
in these cases. The differences between the predicted and
experimental values in the branches for K < 5 may be due to

assumptions we have made. In this connection, it may be re—

called that we set Aac = O and neglected the K dependence in
the expression for the energy difference between the 2t = O
and the lit! = 1 levels. These assumptions are completely

valid only for the study of the RQl and PQl branches. The
assumptions become increasingly less accurate for the QK

branches with increasing K. The complete printout of the
energy levels and transformation coefficients for Analysis

I are given as Appendix II.

 

.__¢-

\. \ \ \ \\”=“

 

 

 

105

-..- elf-12%

(f
./
J
V/
/
//

FIGURE 6.9——Energy shifts for RQ2 and PQ2 branches,

Analysis I°
+19—

_h Z§- RQ2 (experimental)

.1 :7- PQ2 (experimental)

—1

 

 

Energy Shift in units of 0.11 cm

 

 

 

 

 

FIGURE 6.10--Energy shifts for RQ3 and PQ3 branches,

”1

+
d‘

L
I

|
l

106

Analysis I.

[E - RQ3 (experimental)

:7 - PQ3 (experimental)

 

 

l I l
I I l

Energy Shift in units of 0.11 cm—1

I
k?

1
l

 

—1O-L

 

 

_ Ki = 2
v&3A&/\/<7§Z:7 ,+ J
V
VV Vvvv
u = 3

./
vr’
J,
/
/,

 

 

asses

   

107

, . R P t-t‘,‘
.E 6.ll——Energy shifts for QA and Q4 branches, y
Analysis I. V!

I" .
_ A- RQu (experimental) V/
P V/
._ v_ QM (experimental) V}.
/
//

 

 

 

 

   

108

.E 6.12—-Energy shifts for RQ5 and PQ5 branches,

Analysis I.

ZX' RQ5 (experimental)

. \. \. \. ‘; VJ'V

‘ §7- PQ5 (experimental)

 

 

 

 

 

 

 

.-'-‘-_"1d

.-

109

E 6.13—-Energy shifts for RQ6 and PQ6 branches, 3%
Analysis I. J
- J
_ [3- RQ6 (experimental) ‘7 v;
" V- PQ6 (experimental) V /
_ J
_ v ,‘
v/
i— v J
__ V .3
v ..
V WV

   

 

 

110

:Gmm 6.lu—-Energy shifts for RQ7 and PQ7 branches,

Analysis I.
R
[ [3 ‘ Q7 (experimental)

" V7 ' PQ7 (experimental)

 

 

 

 

 

 

F54].
" V
__ VV
V

" W VVK = K" = 7

A /\ AA‘L RA___AiA/\ jd
..- K;- = 6
5..

 

\. \. K K \ \ K K Kmfiifi

 

 

111

,

GURE 6.15--Energy shifts for RQ8 and PQ8 branches

Analysis I.

   

[3— RQ8 (experimental)

§7- PQ8 (experimental)

 

 

 

 

vV
V

V _
V K" 8 .’
A K, = 9 ’
VV A AAAA A I J /
K — 7 /
./’

 

 

 

112

NEURE 6.16—-Energy shifts for RQg and PQ9 branches,

Analysis I.

K K K K KKK

  

 

 

l _.
é]_ ZK— RQ9 (experimental)
..J/"
—“ :7- PQ9 (experimental) v/
_ V /..
__ V A
. W
+5—- V '
. V
5 I v
H V
t: "
O _- V
m A
o V K 0 AA f/
w ~- =
i if EXZKZ: Kn = 9 x/
C: V A A AA AA AA—zfi:+ J
5 A tr we /
C =
... ,_ K1 8 ,.
.p V
q_‘ I
H —- ./
S -r
m
a -~ y//
w
h u
(D ’/
c a-
m
-5—
ax”
4-
.- hfl/
_lOJ_ I”

 

 

 

CHAPTER VII
CONCLUSION

After briefly reviewing the development and expansion
‘ the Hamiltonian which describes the vibration—rotation
tion of polyatomic molecules, we surveyed, in a general
.nner, accidental resonances in molecules with symmetry
‘V. The case of Coriolis interaction between a parallel
.nd and a perpendicular band was studied in detail. The
.gnitude and character of the energy shifts were examined
Len the molecular parameters varied within physically
Lteresting limits. This type of analysis is an example of
.tuations which allow determination of certain parameters
1possible or difficult to evaluate in the absence of
rsonance effects.

This theory was then used in an attempt to explain

me perturbation observed by Blass in the v + v” band of

3
[3F. The analysis allowed us to conclude that Coriolis

Lteraction is very likely a dominant factor in this case,

.though it is clear that other interactions are also present.

Many anomalies are not observed in high resolution
lfrared spectra. Some of these are ascribed to accidental

esonances. Since accidental Coriolis resonance is the

113

\.. \ K '\ \ -\ KKK-

 

 

 

 

11“

west order accidental resonance giving rotational per—
.rbations, it is the most likely candidate when the spectrum
.ows such perturbations. This study is also of importance
th regard to compound resonances. If a good understanding
' Coriolis resonance is obtained, it should ultimately be

issible to understand the effect on the spectrum of the

 

,multaneous presence of Coriolis resonance and some other

,gher—order resonance.

i7
.5/
J
.../r
/
-/
«f
/
¢/
«*

 

 

 

REFERENCES

H. H. Nielsen, Revs. Mod. Phys. 2;, 90 (1951).

H. H. Nielsen, The Vibration-rotation Energies of
Molecules and their Spectra in the Infra-red, Handbuch
der Physik, Vol. XXXVII/l (Springer-Verlag, Berlin, 1959).
M. Born and R. Oppenheimer, Ann. Physik 83, 457 (1927).

B. T. Darling and D. M. Dennison, Phys. Rev. 51, 128
(1940).

M. Goldsmith, G. Amat, and H. H. Nielsen, J
g3, 1178 (1956).

Chem. Phys.

0

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

G. Amat and H. H. Nielsen, J. Chem. Phys. 8A5 (1957).
665 (1958).

Amat and H. H. Nielsen, J. Chem. Phys. 6, 1859 (1962).

__

31,
G. Amat and H. H. Nielsen, J. Chem. Phys. 22,
G.
G. Tarrago, Cahiers Phys. 19, 1A9 (1965).
W.

Shaffer, Revs. Mod. Phys. :L_6_, 245 (191414).

R. C. Herman and W. H. Shaffer, J. Chem. Phys. 16, 453
(1948).

H. H. Nielsen, Phys. Rev. 68, 181 (19H5).

G. Amat, Symposium on Molecular Structure and Spec—
troscopy, Columbus, Ohio, Invited paper (1962).

G. Amat, Compt. Rend. Acad. Sci. 250, 1M39 (1960).
H. H. Nielsen, Disc. Faraday Soc. 9, 85 (1950).

. S. Garing and H. H. Nielsen, Proc. Nat. Acad. Sci.
S. u_u, M67 (1958).

115

 

a.--

‘_‘__' .

KiK K K

 

'- .L\.. K K.

 

116

H. II. Nielsen, J. Chem. Phys. 22, 142 (1953).

J. S. Garing, H. H. Nielsen, and K. N. Rao, J. Mol.
Spectroscopy 3, 496 (1959).

J. M. Hoffman, H. H. Nielsen, and K. N. Rao,
Zeitschrift fur Elektrochemie. Berichte der

Bunsengesellschaft fur physicalische Chemie
g, 606 (1960).

G. Amat and H. H. Nielsen, Rotational Structure of
the Fundamental Band v of Methyl Cyanide, Molecular
Orbitals in Chemistry, Physics, and Biology Academic
Press, New York, 1964).

 

L. Henry and G. Amat, Cahiers Phys. 24, 230 (1960).

R. G. Brown and T. H. Edwards, J. Chem. Phys. 31,
1029 (1962).

R. G. Brown and T. H. Edwards, J. Chem. Phys. 37,
1035 (1962).

R. G. Brown and T. H Edwards, J. Chem. Phys. 28,
384 (1958)

T. H. Edwards, Symposium on Molecular Structure and
Spectroscopy, Columbus, Ohio, Invited paper (1962).

W. E. Blass, Thesis, Michigan State University, 1963.

G. Herzberg, Molecular Spectra and Molecular Structure

 

II. Infrared and Raman Spectra of Polvatomic Molecules

 

 

(D. Van Nostrand, New York, 1945).

“Ki

 

 

.2
-/3
w/
w;
L/
Vx’
V/T

 

 

M
2'

APPENDIX I

COMPUTER PROGRAM

In this appendix we give the computer program used
1 this study. It is written in Fortran for the 3600
antrol Data computer at the Michigan State University
amputing Laboratory. We have omitted the format state—
ants.
The following substitutions have been used:
A = AMBDAN

h

At = AMBDAT

Av = DELTANU

Z

Ct = ZETAZTT
y =
‘ntI ZETAYNT
AdB = DELALPB
0

Ad = DELALPC

 

R = IRR

PROGRAM CRLSRS
DIMENSION A(20,3), VALU(3)

READ AMBDAN, AMBDAT, DELTANU, B, c, ZETAZTT, "/
1 ZETAYNT, DELALPB, DELALPC

DO 107 J=l, 30

IBM = 2*J+3 ,
DO 106 IR = 1, IBM

W11 = B*J+C*J*(J+2*ZETAZTT) ./
W21 = (B+DELALPB)*J+(C+DELALPC)*J**2 "
1 +DELTANU

W22 = B*(3*J—1)+C*(J—l)*(J+2*ZETAZTT-1)

117 ...z

 

 

.K

118

W212 = B*((AMBDAT/AMBDAN)**(1/4)+(AMBDAN/AMBDAT)
l **(1/4))*SQRTF(FLOATE(J))*ZETAYNT

EGVll = W11

C11 = 1

EGV21 = (W21+W22)/2+SQRTF((W21—W22)*(W21—W22)/4
1 +W212*W212)

EGV22 = (W21+W22)/2—SQRTF((W21—W22)*(W21=W22)/4
1 +W212*W212)

C211 = ((W21—W22)/2+SQRTF((W21—W22)*(W21—W22)/4

K KKK—KKK

l +W212*W212))/SQRTF((W21—W22)*(W21—W22)/2
2 +2*W212*W212+(W21—W22)*SQRTF((W21—W22)
3 *(W21—W22)/4+W212*W212))

C212 = W212/SQRTF((W21—W22)*(W21—W22)/2+2*W212* v5
1 W212+(W21—W22)*SQRTF((W21—W22)*(W21—W22)/4+
2 W212*W212))

C221 = ~C212

C222 = C211

ITAUl = 1

ITAU2 = 2

ITAU3 = 3

IRR = IR—J—2

KlP = J

KlN = —J

K2Pl = J

K2P2 = J—l

K2Nl = —J

K2N2 = —J+1

LTP = l

LTO = 0

LTN = —1

IF(IR—2) 100, 101, 102

100 PRINT W11, KIN, LTP, J, IRR, ITAUl, EGVll, 011
GO T0 106

101 PRINT W21, K2N1, LTO, J, IRR, ITAU1,EGV21, 0211,
1 0212
PRINT W22, K2N2, LTP, J, IRR, ITAU2,EGV22, c221,

 

102 JAM = 2*J+2 .,~
IF(IR~JAM) 105,103, 104

103 PRINT W21, K2P1, LTO, J, IRR, ITAUl, EGV21, 0211, 12
1 0212
PRINT W22, K2P2, LTN, J, IRR, ITAU2, EGV22, 0221, ex
1 0222 1

00 TO 106 .-i
104 PRINT W11, KlP, LTN, J, IRR, ITAUl, EGVll, 011 '
00 T0 106 «2

105 A(1,l) = C*(IR—J—1)*(IR—J—1-2*ZETAZTT)+B*
1 (J*(J+1)—(IR—J—1)**2)
A(2,1) = B*((AMBDAT/AMBDAN)**(1/4)+(AMBDAN/AMBDAT)
1 **(l/4))*ZETAYNT*SQRTF(FLOATF((2*J+2—IR-l))/2.) .21

 

 

()0

106
107

119

A(3,l) 0
A(1,2) A(2,1)
A(2,2) (C+DELALPC)*(IR—J—2)*(IR—J—2)+
1 (B+DELALPB)*(J*(J+l)-(IR-J—2)*(IR—J—2))
2 +DELTANU
A(3,2) = B*((AMBDAT/AMBDAN)**(1/4)+(AMBDAN/AMBDAT)
1 **(1/4))*ZETAYNT* SQRTF(FLOATF((IR—2)*
2 *(2*J—IR+3))/2.)
0

>
A
[\J
\o
LA.)
v

H H H

A(3,2)
C*(IR—J—3)*(IR—J—3+2*ZETAZTT)+B*(J*(J+l)
l -(IR—J—3)*(IR-J-3))

A(1,1)

A(2,2)

A(3,3)

IRR+I

IRR

IRR-1

CALL EIGEN (A, VALU, 3, 1)

PRINT W31, K31, LTP, J, IRR, ITAUl, VALU(l),

E:
u)
U) u)
II II II II II ll

1 A(1,1), A(2.1), A<3.1)

PRINT W32, K32, LTO, J, IRR, ITAU2, VALU(2),
1 A(1,2), A(2,2), A(3,2)

PRINT W33, K33, LTN, J, IRR, ITAU3, VALU(3),
1 A(1,3), A(2,3), A(3,3)

CONTINUE

CONTINUE

END

SUBROUTINE EIGEN(A,VALU,N,M)
F4 UCSD EIGEN
EIGENVALUES AND EIGENVECTORS OF A REAL SYMMETRIC

MATRIX
DIMENSION A(20,20), B(20,20), VALU(20), DIAG(20),
x SUPERD(19), 0(19), VALL(20), s(19), C(19),
x D(20), IND(20), U(20)
CALCULATE NORM OF MATRIX
ANORM2=0.0
D0 6 I=1,N
DO 6 J=1,N

ANORM2=ANORM2+A(I,J)**2
ANORM=SQRTF(ANORM2)
GENERATE IDENTITY MATRIX
IF (M) 10, 45, 10

DO 40, I=1,N

DO 40, J=1,N

IF(I-J) 35, 25, 35
B(I,J)=l.0

GO TO no

B(I,J)=0.0

CONTINUE

PERFORM ROTATIONS To REDUCE MATRIX TO JACOBI FORM
IEXIT=1

 

 

 

OO

170

120

NN=N—2

IF (NN) 890, 170, 55

DO 160 I = 1,NN

II=I+2

D0 160 J =II,N

T1=A(I,I+l)

T2=A(I,J)

GO TO 900

DO 105 K=I,N

T2=COS*A(K,I+1)+SIN*A(K,J)

A(K,J)=COS*A(K,J)-SIN*A(K,I+1)

A(K,I+1)=T2

DO 125 K=I,N

T2=COS*A(I+1,K)+SIN*A(J,K)

A(J,K)=COS*A(J,K)—SIN*A(I+1,K)

A(I+1,K)=T2

IF (M) 130, 160, 130

D0 150 K=I,N

T2=COS*B(K,I+1)+SIN*B(K,J)

B(K,J)=COS*B(K,J)—SIN*B(K,I+1)

B(K,I+1)=T2

CONTINUE

MOVE JACOBI FORM ELEMENTS AND INITIALIZE
EIGENVALUE BOUNDS

DO 200, I=1,N

DIAG(I)=A(I,I)

VALU(I)=ANORM

VALL(I)=—ANORM

DO 230 I=2,N

SUPERD(I—l)=A(I—1,I)

Q(I—1)=(SUPERD(I—l))**2

DETERMINE SIGNS OF PRINCIPAL MINORS

TAU=0.0

I=1

MATCH=O

T2=0.0

T1 =1.0

DO A50 J=1,N

P=DIAG(J)—TAU

IF(T2) 300, 330, 300

IF(T1) 310, 370, 310

T=P*Tl-Q(J—1)*T2

GO TO 410

IF(Tl) 335. 350, 350
T1=-1.0

T=-P

GO TO 410

T1=1.0

T=P

GO TO 410

IF(Q(J-l)) 380. 350. 380

3’7
M
.17
g/
,2
./'
1/
./
J,
«’1

 

 

380
390

400
410

121

IF(T2) A00, 390, 390
=— .0
GO TO A10
T=1.0
COUNT AGREEMENTS IN SIGN
IF(Tl) A25, A20, A20
IF(T) AAO, A30, A30
IF(T) A30, 440, AAO
MATCH=MATCH+1
T2=T1
T1=T
ESTABLISH TIGHTER BOUNDS ON EIGEN VALUES
DO 530 K = 1,N
IF (K—MATCH) A70, A70, 520
IF(TAU~VALL(K)) 530, 530, A80
VALL(K)=TAU
GO TO 530
IF(TAU—VALU(K)) 525, 530, 530
VALU(K)=TAU
CONTINUE
IF(VLAU(I)—VALL(I)—5.0E-8) 570, 570, 550
IF(VALU(I)) 560, 580, 560
IF(ABSF(VALL(I)/VALU(I)-1.0)—5.0E-8) 570, 570, 580
I=I+l
IF(I-N) 5A0, 5A0, 590
TAU=(VALL(I)+VALU(I))/2.0
GO TO 260
JACOBI EIGENVECTORS BY ROTATIONAL TRIANGULARIZATION
IF (M) 593, 890, 593
IEXIT=2
DO 610 I=1,N
DO 610 J=1,N
A(I,J)=0.0
DO 850 I=1,N
IF (I—l) 625, 625, 621
IF (VALU(I-1)—VALU(I)-5.0E—7) 730, 730, 622
IF (VALU(I—l)) 623, 625, 623
IF (ABSF(VALU(I)/VALU(I—1)—l.0)—5.0E—7) 730, 730, 625
COS=1.0
SIN=0.0
DO 700 J=1,N
IF(J—I) 780, 780, 6A0
GO TO 900
S(J—1)=SIN
C(J-I)=COS
D(J—1)=T1*COS+T2*SIN
T1=(DIAG(J)—VALU(I))*COS—BETA*SIN
T2=SUPERD(J)
BETA=SUPERD(J)*COS
D(N)=T1
DO 725 J=1,N

J\ \K K. A. K '\ ‘K‘ K*:K'"Ki

 

122

IND(J)=O

SMALLD=ANORM

DO 780 J=1,N

IF (IND(J)-1) 750, 780,

IF (ABSF(SMALLD)— ABSF(D(J)))780, 780, 760

SMALLD= D(J)

NN=J

CONTINUE

IND(NN)=1

PRODS=1.0

IF (NN—l) 810, 850, 810

D0 8A0 K=2,NN

II=NN+1—K

A(II+1,I)=C(II)*PRODS

PRODS=—PRODS*S(II)

A(I,I)=PRODS

FORM MATRIX PRODUCT 0F ROTATION MATRIX WITH
JACOBI VECTOR MATRIX

DO 885 J=1,N

DO 865 K=I,N

U(K)=A(K,J)

DO 885 1:1, N

§7SAA(I, ,J)= 0. 0

DO 885 K=1 ,N

A(I, J)= B(I ,K)*U(K)+A(I,J)
RETURN

CALCULATE SINE AND COSINE OF ANGLE 0F ROTATION
IF (T2) 910, 9A0, 910
T=SQRTF(T1**2+T2**2)
COS=T1/T

SIN=T2/T

GO TO (90,650), IEXIT

GO TO (160,910), IEXIT
END

.2:
..F/
.2
w;
.2’
,2"
2"
22’
.3/

 

 

 

APPENDIX II
PRINTOUT OF ANALYSIS I

This appendix contains the output of one computer run.
The self—consistent Analysis I discussed in the text is used.
The numerical values of the molecular parameters are those
given in Table 6.2 for Analysis I.

The energies in cm_1 are given in exponential form;

for example, 1.234567+OO2 = 123.4567.

123

 

 

 

 

 

 

 

7 7 120 "I " N I. ’7 7" ' P .
1 11, “11111—11111 ,_ , 1 , ,__ _, 1 K/
E(J.KoLT) K LT J R TAU E(J.R.TAU) C(+1) C(0) C(31)
11 , 1 1 1 ,711111K_1#.1/’.
6.745760'0'000 £1 1 1 '2 1 607457609000 1.000
1 1 ,1 1 111 1111 1 1_11-./"‘
2.3153350001 91 0 1. ’1 1 2031535752001 10000 090.03
1.7030000000 131111 1 2112,10?O§§82*000 -0.003 1.000 ‘32
50119840*000 1 1 1 0 1 19587353*001 0.0105, 1.00.0 0.0051 1 /’
108872850001 0 0 1, 0 2 501198401'000 0.707 0.000 00,707
5.119050¢000 :1 21 1 0 3 5.1191690000 0.707 70.007, 0,707 ,2
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1.703600+000 0 *1 1 1 2 19703338201000 ”0,003 1.000

'6.745700¥000' 1’21 1 2 0.74570000007"12000W'V "WWWW' ’7 7 '

p

2.305352$001” .2 i 2 -3 1 2.365352-001 1.000

 

 

 

 

 

3.919005+001 .2 ”0"”?’ 12 1 3.9197172001 1.000 0.003
1.015290+001 21 1 2 ~2 2 1.0152600001 ~0.0007 1.000
5.110500+000 0 1 2 ~1 1 2.035750.00; 0.000 1.000110.01011111
2.035535+001 .1 0 2 21 2 2.0400110001 0.000 0.016 -1.000
12304919212911122171 2. 11 3 511!Ql592999 1.000 -0.006 v0.000
8.527040+000 i 1 2 0 1 2.2070920001 0.009 1.000.10.009, 1 3 ,,2
2,207.35.001 0 0 2 0 2 6.5270402000 0.707 0.000 ”0:707
8.527040+000 .1 21 2 o 3 0.524970«000 0.707 -0,012 0.707 1111_11/
1,12,..9‘110119021MLL 2 11 2 .1 11 2.1.6.3152258201011119 .016111 . 00 0 0 . 0 06 /
2.6355350001 1 0 2 1 2 2.0400110001 1.000 ~0.016 v0.000
5.110000+000 .1 1 3 5.1101400000 0.000 ~0.006, 1.000 7,474,,»
3.919685*001 2 0 2 2 1 3.919717o001 1.000 0.003 1 1 11 _ /
1.0152962001 1 -1 2 2 2 1.0152040001 «0.003 1.000
2.3053522001 2 -1 2 3 1 2.3053520001 1.000
5.072328+001 .3 i 3 ~4 1 5.0723200001 1.000 ’5 7 A 7’”
6.540235+001 a3 0 s as 1 6.540273+001 1.000 0.003 ’
11.?1§Z§fi§3109;,102 1 ,3 7.3 2122976399099},3Q19931101200111111111111___1,/
1.5203700001 .1 1 3 .2 1 4.5053090001 0.000 0.004 0,990 ,7,
4.399985+001 .2 0 3 .2 2 4,399309+001 0.005 0.998 90.064 ’
4.5045522001 .3 :1 3 -2 a 1.520295¢001 1.000 -0.005 00.000
111-022116108001. 11 £1 ,11 3 '1 1 3a11163832001111010091 1.00.01 11010211W_1 .2“
3.115035c001 .1 0 3 e1 2 2.550034«001 0.000 0.0 7-u1.000 '*
2.5512480001 .2 01 3 -1 3 1.0220200001 1.000 ~0.0oa 00.000

 

 

E‘JIKILT’ K
1.3037040001 1 1
21682185+001 0 0
19363784‘001 E1 '91
2.551240+001 2
50115835*001 1
1.022160~001 0 .
0 504562.001 3 1
4 3999§§10011112 0,
1. 5263760001 1 .1
6,5402350001 3 0
2.076432«001 2 .1
5.0723201001 10 91
0.7955040001 .4 1
1.017099.002 .4 0
5.7537680001 .3 1
3.557572.001 ’02’ 1
7.180635900l .3 0
0.1451301001 .4 .1
2.207a1e+001 .1 1
5.0403859001. :21 0
5.205992+001 .3 ~1
1.7035000001 0 1
3.756235+001 =1 0
3.232080+001 .2 .1
2.045224+0017 1 1
3.3281859001 0 0
2.045224o001 .1 .1
3.232000o001 2 1
3.75623590017771 ,0
1.703600*001 0 «1
5.205992+001 0 1
5.0403850001 2 0
2.207016+001 1 .1
8.1451300001W' 4 1
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