3 7 ?

/J Sid /V<7, 2 6 0 9

A COMPREHENSIVE MODEL FOR THE ROTATIONAL SPECTRA*OF PROPYNE

CH3CCH IN THE GROUND "AND Vx0=1,2,3,4,5 VIBRATIONAL STATES

DISSERTATION

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

By

Won Myung Rhee, B.S., M.S.

Denton, Texas

December, 1986

hK

Won Myung, Rhee, A Comprehensive Model for the Rotational

Spectra of Propyne CH^CCH in the Ground and Vn n=l> 2,3,4 and

5 Vibrational States. Doctor of Philosophy (Physics),

December, 1986, 104 pp., 14 tables, 6 illustrations,

bibliography, 60 titles.

The energy states of C 3 V symmetric top polyatomic

molecules were studied. Both classical and quantum

mechanical methods have been used to introduce the energy

states of polyatomic molecules. Also, it is shown that the

vibration-rotation spectra of polyatomic molecules in the

ground and excited vibrational states can be predicted by

group theory.

A comprehensive model for predicting rotational frequency

components in various v 1 0 vibrational levels of propyne was

developed by using perturbation theory and those results

were compared with other formulas for C 3 V symmetric top

molecules. The v10=l,2,3 and ground rotational spectra of

propyne in the frequency range 17-70 GHz have been

reassigned by using the derived comprehensive model. The

v 1 0= 3 and v 1 0= 4 rotational spectra of propyne have been

investigated in the 70 GHz, and 17 to 52 GHz regions,

respectively, and these spectral components assigned using

the comprehensive model. Molecular constants for these

vibrationally excited states have been determined from more

than 100 observed rotational transitions.

From these experimentally observed components and a model

based upon first principles for C 3 V symmetry molecules,

rotational constants have been expressed in a form which

enables one to predict rotational components for vibrational

levels for propyne up to v 1 0= 5. This comprehensive model

also appears to be useful in predicting rotational

components in more highly excited vibrational levels but

data were not available for comparison with the theory.

Several techniques of assignment of rotational spectra

for each excited vibrational state are discussed. To get

good agreement between theory and experiment, an additional

term 0.762(J+1) needed to be added to Kjt=l states in v 1 0= 3.

No satisfactory theoretical explanation of this term has

been found.

Experimentally measured frequencies for rotational

components for J-»(J+1)=+1 (01 J <.3) in each vibration v 1 0 = n

(Oi.ni.4) are presented and compared with those calculated

using the results of basic perturbation theory.

The v9= 2 rotational spectrum of the propyne molecule was

introduced in Appendix A to compare the rotational spectra

of the same molecule in different vibrational levels v9 and

v. 1 0

m

TABLE OF CONTENTS

Page

LIST OF TABLES . . vi

LIST OF ILLUSTRATION viii

Chapter

I. INTRODUCTION 1

Rigid Rotor Distortable Rotor The Effects of Vibration

II. HAMILTONIAN FOR TREATING THE VIBRATION-ROTATION INTERACTION OF POLYATOMIC MOLECULES 20

Derivation of Hamiltonian Power Series of the Hamiltonian Twice Contact Transformation Non-zero Matrix Element of the Twice

Transformed Hamiltonian for Molecules of C 3 V Symmetry

III. DERIVATION OF A COMPREHENSIVE MODEL 33

Perturbation Theory Accidental Degeneracy

IV. EXPERIMENTAL APPARATUS 45

IV

TABLE OF CONTENTS-- Continued

V. ANALYSIS OF EXPERIMENTAL RESULTS 55

The Ground State Spectrum The v 1 0= 1 State Spectrum The v 1 0= 2 State Spectrum The v 1 0= 3 State Spectrum The v 1 0= 4 State Spectrum The General Form of the Rotational Constants The v 1 0= 5 State Spectrum

VI. CONCLUSIONS . . 75

APPENDICES 78

A. The v?=2 Rotational Spectra of Propyne B. The Linear Molecule and Symmetric Top

Molecules as a Rigid Rotor C. Distortable Rotor For Prolate Symmetric

Top Molecules D. Rotational Energy of Symmetric Top Molecules

in the Excited Vibrational State E. Accidental Degeneracy F. Program To Assign v 1 0= n Rotational Constants G. Program To Calculate v 1 0 = n Rotational

Transition Frequencies

REFERENCES 99

LIST OF TABLES

Table

I. Classes of Molecules 3

II. Diagonal Matrix Element of the Twice Transformed Hamiltonian H+ 31

III. Non-Diagonal Matrix Element of the Twice Transformed Hamiltonian H+ 32

IV. The Comprehensive Model of Propyne and C 3 V Symmetry Molecules 44

V. Rotational Transition Frequencies for CH3CCH in the Ground State 56

VI. Rotational Transition Frequencies for CH3CCH in the v 1 0 = 1 Excited Vibrational State 59

VII. Rotational Transition Frequencies for CH3CCH in the v 1 0= 2 Excited Vibrational State 62

VIII. Rotational Transition Frequencies for CH3CCH in the v 1 0= 3 Excited Vibrational State 66

IX. Rotational Transition Frequencies for CH3CCH in the v 1 0= 4 Excited Vibrational State 69

VI

LIST OF TABLES-- Continued

Table

X. Rotational Constants for CH3CCH in the

v1Q Vibrational States 71

XI. General Forms of Rotational Constants . 72

XII. Rotational Transition Frequencies for CH3CCH in the v 1 0= 5 Excited Vibrational State 73

XIII. Rotational Constants for Propyne in the v?= 2 State 79

XIV. Rotational Transition Frequencies for CH3CCH in the v?= 2 Excited Vibrational State 80

VI1

LIST OF ILLUSTRATIONS

Figure Page

1. The Symmetric Top Molecule, Propyne 6

2. Symmetric Top with Space-fixed (XYZ) and Molecule-fixed axes (xyz) 8

3. Vibrational Modes of a Triatomic Linear Molecule 17

4. Coriolis Force in a Linear XYZ Triatomic Molecule

5. Spectrometer for Measuring Rotational Spectrum of Propyne 46

6. A Typical Chart Recorder Scan of Rotational Components of Propyne 51

v m

CHAPTER I

INTRODUCTION

D. M. Dennisoni and H. H. Nielsen2 laid the early

foundations for the theory of vibration-rotation energy of

polyatomic molecules in 1931 and 1951, respectively. From

1950 to 1965 many scientists, Nielsen3, Anderson et al.4,

Goldsmith et al.s, three papers by G. Amat et al.6,7,s, s .

Maes', and Grenier Besson10 made major contributions to set

up the vibration-rotation energy and frequency formula for

polyatomic molecules. In 1965, Tarragon developed a theory

and applied it to some polyatomic molecules which have C 3 V

symmetry. Also she made specific rotational energy and

frequency formulas for C 3 v symmetric top molecules in the

ground state and vibrationally excited states v 1 0= 1, 2 and

3. These main contributions by Nielsen, Amat and Nielsen,

and Tarrago are examined with varying degrees of detail in

the following chapter.

Although a number of papers have appeared for specific

cases of C 3 V symmetry molecules, no one has set up a general

form for the vibration-rotation energy and frequency formula

for polyatomic molecules which have the same symmetry.

Therefore, all the energy and frequency formulas which have

appeared in the literature before appear in different forms.

This work has four major objectives.

1. Derivation of a comprehensive model for vibration-

rotation energy and frequency for polyatomic mole-

cules which have C 3 V symmetry.

2. Reassignment of all the structural patterns of the

propyne molecule using the comprehensive model and

then make a comparison between the comprehensive

model and previous models and formulas.

3. To set up the general forms of rotational constants

for the propyne molecule.

4. To make additional measurements of rotational spectra

of the propyne molecule in the v 1 0= 3 and v 1 0= 4

states.

Generally, polyatomic molecules can be classified into

four different catagories depending upon their symmetry12.

The four classifications are designated as linear molecule,

symmetric top molecule, spherical top molecule and

asymmetric top molecule, whose classification depends upon

the principal moments of inertia of each molecule. Table I

gives a summary of these classification. Also, Table I

shows two different symmetric top molecules; prolate top and

oblate top. A prolate symmetric top is one in which the two

equal moments of inertia are greater than the other moment

of inertia. An oblate symmetric top is one in which the

cn w

D U w

o s t l o w w cn w < J u

H

w 1-3 CQ <C Eh

CO CD

rH 3 O <D

rH 0 a

•P c <D g

O a-

N

rd

rd H -P

04 U H (1) O C C H

5H m a . o

<u c a)

rH >1 -p

0 u <

o o o

II H I! N X >1

M H H

> ! N N M H H I! II II

X >i X H H H

n : u u

m as u

CJ co

a: o

CO

U tt

0 C rd

. c •P <D a

<D *0 >i

x : a)

T3 rH cd g u o u*

N N N M H H

N N % A V H

II M Th

>1 > i >1 > i >1 H H H H M II II II II

X X X X X H M H H M

a a a , eu a 0 0 0 o 0 E* EH Eh EH Eh

O 0 CD rH •H

•H -P <D fd W fd •P 0 «P <D n 4-> rH rd •H <1> W rd a) 0 rH u CO a) g u & <D £ rd a g a* o & >i

iH •H >i Ou CA u w <

CO •H X fd

u -P CD

>i cn

cn fd

N

(D g

3 w w <

degenerate moment of inertia is less than the other one. By

this definition propyne is a prolate symmetric top molecule.

Most of the symmetric top molecules observed in the

microwave region are prolate.

Group theory is a powerful tool in dealing with the

symmetry of molecules. It is useful to recall that there are

32 point groups in a crystal whose symmetry elements are

rotation, inversion, reflection, plane of symmetry and

center of symmetry.i3 Also, recall that a point group means

that there is always one point in the group that is

unaffected by all the group symmetry operations: rotations,

reflections, and inversions. An important point group of

symmetric top molecules is the group of molecules that has

C 3 V symmetry. The propyne molecule in this study has C 3 V

symmetry and an explanation of that property follows.i4

C n v - consists of C n and a v and any other elements to

have closure(E).

* closure; the "product" of any two elements in the

set must also belong to the set,i.e.,

the set is closed.

C - Rotations are labeled by C„ where the rotations are n n

2ft/n about an axis through the origin. The axis of

highest n is called principle axis.

* C3 is a 2K /3 rotation.

<r - reflection in a vertical plane that contains the

principle axis.(mirror image)

Figure 1 shows a diagram of symmetric top molecule CH3-

CCH(commonly called propyne and methyl acetylene) showing

bond lengths and bond angle. These structural parameters are

taken from a paper by Duncan et al.15 and by Dubrulle et

al.*6. Propyne is a molecule of interest to many areas of

science, especially for astrophysics. Its existence has been

confirmed in the atmosphere of Titan which is the seventh

moon of saturn as well as in the Taurus dark clouds.17

It is the rotational motion of molecules that is

primarily responsible for the spectra observed in microwave

spectroscopy.!8 There are several ways to get the formula

for the rotational spectra of molecules. The simplest

method to calculate the rotational energy of polyatomic

molecules is by using the rigid rotor assumption.19

1. Rigid Rotor

A. linear molecules - The rotational energy for linear

polyatomic molecule in the rigid rotor assumption is given

by20(see Appendix B)

Er(J,K) = hBJ(J+l) (1- l)

B. Symmetric top molecule - Figure 2 shows a symmetric top

molecule along with a space-fixed coordinate system XYZ. The

1.060 A'

1.2066

1.4586 A 0

c

H f 1.096 A 0

108 17 ' H H

FIGURE 1. The Symmetric TOD Molecule, PROPYNE.

coordinate system which rotates with the molecule is labeled

xyz. The rotational energy of the rigid rotor is given in

terms of the principal moments of inertia I , I and I . x y z

Classically2i, the pure rotational energy is

A pz E = + — ^ + (i- 2) r 21 21 21 K }

x y z

where P x IXWX, Py IyWy, PZ-IZW z

The center of mass is assumed as origin of the

principal axis.

For the Quantum Mechanical approach22, Pv/ P„, p„ and Ev A y Z IT

become operators

P2

X P2

+ -3L P2

+ Z

21 21 21 H r = — +-J- + — (1-3)

21 21 x y

and

P2 = px2+ Py2+ pz2 (1_ 4 )

P : total angular momentum operator

Fig. 2. Symmetric Top with Space-fixed(XYZ) and

Molecule-fixed(xyz) axes.

In a symmetric top, one of the principal axes of inertia

must lie along the molecular axis of symmetry. Here z is

chosen as the symmetric axis of the top. For the prolate

symmetric top molecule, the Hamiltonian operator of Eq.(l.

3) can be expressed as

2 2 P Pz ( 1 1 \

Hr + ~ ( ) (1-5) 21 2 I I

x z x

Eigenvalues for the angular momentum operators and energy

operator for the symmetric top rotor are

(J,K,MlP2|J,K,M)= *2 J(J+1) (1- 6)

(J,K,M|Pz|J,K,M)= fi K (1- 7)

(J,K,M|PZ |J,K,M)= fi M (1-8)

where

J = 0, 1, 2, 3, ... . (1-9)

K = 0 , ±1, ±2 , ±3 ,±J (1-10)

M = 0, ±1, ±2 , ±3 ±J (1-11)

By equations (1-4),(1-5),(1-6) and (1-7), the final form

of the rotational energy of a symmetric top molecule with

10

rigid rotor assumption is given by23 (see Appendix B for

details)

Er(J,K)= h [BJ(J+l)+(A-B)K2] (1-12)

where J is the total angular momentum quantum number, K

is the quantum number for the projection of the total

angular momentum along the symmetry axis, M is the quantum

number for the projection of the total angular momentum

along the Z space-fixed axis (see Figure 2). The rotational

constants used in Eqs.(l-l) and (1-7) are defined by

B = 8 * 2 l V l , ( 1 - 1 3 )

x y

A = 1 ^ 7 " ( 1 - 1 4 )

z

It is necessary that any permanent dipole moment must lie

along the symmetry axis in a symmetric top molecule. All

matrix elements of this dipole moment resolved along a

space-fixed axis vanish except those between states

corresponding to J •> J or J+l, K * K.2* Th e selection rules

for the absorption spectra23 are

AJ= +1 , AK= 0 (1-15)

11

Application of these rules to the rotational energy formulas

for linear and symmetric top molecules gives the formula for

the absorption frequencies.

v r = 2B(J+1) (1-16)

Under the rigid rotor assumption, linear and symmetric top

molecules have the same absorption rotational spectra

formula.

2. Distortable rotor

When the prolate top molecule propyne is rotating about

its center of mass, bond lengths and bond angle are

changing. These distortions are due to the centrifugal

forces at play in the molecule. Centrifugal stretching is

treated as a perturbation on the eigenstates of a rigid

rotor. A complete procedure for derivation of rotational

energy of a distortable prolate symmetric top molecule is in

Appendix C. The expression for the rotational energy of the

distortable symmetric top molecule to the first order

perturbation treatment is26

E (J,K) = h [BJ(J+1)+(A-B)K2-D_J2(J+l)2 (1-17) 3T u

-DJKJ(J+1)K2-DKK4]

12

With the selection rules for absorption spectra, this

energy equation gives the rotational frequency as

v r(J,K) = 2B(J+l) - 4DJ(J+1)3 - 2D j k(J+1)K2 (1-18)

If one includes higher order centrifugal distortion

effects to the sixth power in the angular momentum

components P 6 to the rotational energy, the following result

is obtained.27

E (J,K) = h [BJ(J+1)+(A-B)K2-D J2(J+1)2

J

'DJKJ(J+1)K2"DKK4+IVJ3(J+1)3 (1-19)

+H J 2(J+l)2R2 +H J(J+1)K 4+H K<* ] «J & KJ K

By applying the same absorption selection rules and

considering the centrifugal distortion to sextic order, the

rotational frequency formula for the symmetric top molecule

becomes

v (J,K) = 2B(J+1)-4D (J+l)3-2D (J+1)K2 (1.20) "*• JK

+H (J+l) 3[(J+2) 3-J 3]+4H K 2(J+l) 3

J JK

+2H (J+1)K4

KJ

13

3. The effects of vibration

Only pure rotational frequencies and energy states of

linear and symmetric top molecules both under the rigid and

distortable rotor assumption were discussed so far. But

other major departures from those assumptions occur for

molecules in excited vibrational states. The effects of

vibration of molecule on the rotational spectra are as

follows.

A. Changing the rotational constants

For symmetric top molecules in excited vibrational

states, the effective rotational constants differ from those

of the ground state. One obvious reason for that is the

principal moments of inertia of a molecule are altered by

the vibration of molecules. If a symmetric top molecule

bends,the moment of inertia about an axis perpendicular to

the symmetry axis will decrease and the rotational constant

B will increase. If a symmetric top molecule stretches, the

moment of inertia about an axis perpendicular to the

symmetry axis will increase and B will decrease. This is

opposite to the behavior of the rotational constant A in

prolate symmetric top molecules. In 1980, Claude Meyer

et.al.28 expressed the rotational constants in excited

vibrational states as

u

BV=Bo- a v- yv2 (1-21)

A v=A 0- Bv- 6v2 (1-22)

Where B v and A v are the rotational constants in the

excited vibrational states, B 0 and A0 are the rotational

constants in the ground state, and <x,v,& and 6 are called

constants of anharmonicity.

From equation (1-18), the rotational frequency of either

a prolate or oblate symmetric top can be expressed by

v (J,K)= 2B (J+l)- 4D^ (J+l)3- 2D^ (J+l)K2 (1-23) j- V J J K

The new values of all the rotational constants in excited

vibrational states are expected to have systematic variation

with the vibrational quantum number.

15

B. Degenerate Coriolis splittings (z-type doubling)

The best approach to consider the effect of a degenerate

vibrational mode on the rotational spectrum of a symmetric

top molecule is to begin by considering a simple triatomic

linear molecule. A linear molecule with n atoms has (3n-5)

modes of vibrations,29 however, not all of these modes have

different frequencies. The bending modes are always doubly

degenerate, i.e., there are two bending modes of vibration

both of which have the same frequency. So, a non-symmetric

triatomic linear molecule has two nondegenerate parallel

modes vx and v3 and a doubly degenerate bending mode v2.

Figure 3 illustrates these three vibrational modes of

triatomic linear molecule.30 Two parallel modes correspond

to the stretching vibrations of the two bonds XY and YZ.

The bending modes v2 is doubly degenerate because the

molecule is free to bend in two orthogonal planes. It is

obvious that these stretching and bending modes cause a

change in the averaged or effective molecular parameters;

bond lengths and bond angles. This leads to a slight change

in the moment of inertia and a consequent altering of the

rotational spectra.

In addition to these effects, there are other effects of

interaction of rotation with the degenerate bending

vibrational modes. If z is chosen along the molecular axis,

16

the degenerate vibrations of the non-rotating molecules can

be considered as orthogonal bending motions of the molecule

in the xz and yz planes. In the rotating molecule this

degeneracy is lifted by the interaction between vibration

and rotation. This interaction is due to the Coriolis force

in the vibrating-rotating molecules.3i It has a value F =

2mvx o, where w is angular velocity of rotation of the

molecule and v is the linear velocity of the vibrating atom.

A Coriolis force occurs in all cases of the rotating and

vibrating polyatomic molecule. Figure 4 shows the effect of

Coriolis forces on a rotating linear triatomic molecule.32

If the molecule is rotating about the x axis, then

bending in the xz plane is not quite equivalent to bending

in yz plane. This non-equivalency is only because of a

Coriolis force in the vibrating-rotating molecule. The

direction of the bending in the xz plane and that of the

angular momentum J of the molecule is perpendicular to each

other, so this vibrational mode is excited by the Coriolis

force. But the bending in the yz plane and angular velocity

to have the same direction, so this mode is not changed by

the Coriolis force. Therefore, the effective moment of

inertia about each axis of rotation is different for the two

cases cited above. As result of this vibration-rotation

interaction, the two degenerate energy levels are slightly

split; this splitting is called £-type doubling.

17

V 1

v > 1 I 1

V 3

Fig. 3. Vibrational Modes of a Triatomic Linear Molecule.(Bending mode V_ is doubly dege-nerate) ^

V i •

v 2 L . - j ^ L i *

Fig. 4. Coriolis Force in a Linear XYZ Molecule.

Curved Arrow : Direction of Rotation. Solid Arrow : Normal Motion of the Vibrational Mode Dashed Arrow : Coriolis Force.

18

For symmetric top molecules such as CH3CCH which have a

linear group along the symmetry axis, there are degenerate

bending vibrational modes which give rise to I -type doubling

of the rotational levels similar to that described for

linear molecules. There is a component of this vibrational

angular momentum along the figure axis which adds to, or

subtracts from, the pure rotational angular momentum about

the symmetry axis. This component has the value Zfi and

with the Coriolis interaction term we can write

P - fMi (1-24)

I = v, v-2, - - - -, -v (1-25)

Where f is the Coriolis coupling constant and v is the

vibrational c[.iiantum number. When the vibrating motions are

perpendicular to the symmetric axis, f = 1 and the

vibrational component along z(symmetric axis) is Zfx, just as

in the linear molecules. In the symmetric top ,however, this

component along the symmetry axis ,in general, will be less

than Ifi, or 0< f <1.

By considering the &-type doubling contribution to the

rotational energy of the symmetric top molecule, the new

rotational energy of symmetric top molecules in the excited

vibrational states has a form33,34(see Appendix D for

derivation)

19

E (J,K,M= h[BvJ(J+l) + (A-B)vK2- 2fS-KAv] (1-26)

This gives a splitting in the energy levels of 4f£KA

between the levels with K£ positive and those with K&

negative. These splittings by £-type doubling are easily

detected in propyne when this molecule is excited to v 1 0=l

and v 1 0=3 vibrational states.

The basic frequency and energy equations and symmetry

properties for the propyne molecule have been outlined in

this chapter. Further details will be given as the need

arises.

The reminder of this paper is divided into five chapters.

Theory of vibration-rotation interaction in the symmetric

top molecule is discussed in Chapter II. Derivation of the

comprehensive model developed from perturbation theory is

found in Chapter III. The experimental equipment and

procedure used in this study are presented in Chapter IV.

The measurements made in this work and data analysis using

the comprehensive model are presented in Chapter V.

Conclusions and a brief summary of the entire work are given

in Chapter VI.

CHAPTER II

HAMILTONIAN FOR TREATING THE VIBRATION-ROTATION INTERACTION OF POLYATOMIC MOLECULES

Derivation of Hamiltonian

The Hamiltonian for a rotating vibrating polyatomic

molecule was discussed first by E. B. Wilson and J. B.

Howard.3s T o derive the exact Hamiltonian, Wilson and Howard

derived the classical Hamiltonian and then used the method

of Podolsky36 to obtain the quantum mechanical Hamiltonian.

J. E. Wollrab37 also explained this process well by

expressing the kinetic energy in terms of angular momentum.

An equivalent but more useful Hamiltonian was formulated in

1940 by B. T. Darling and D. M. Dennison.38 I n 1 9 5 1 H > H

Nielsen2 developed a useful Hamiltonian for calculating the

vibration-rotation energies of molecules. The Hamiltonian,

as expressed by Darling and Dennison is

H = ( 2 - D

V,. * ^ w + y p y ] + v A <r AV- ;4<r

20

21

Where « and £ range over x,y and z which are the

principal axes of inertia of the molecule. P and p are a

components of the total and internal angular momentum^

respectively, p^^is momentum conjugate to the normal

coordinate ^and V is the potential function of the

molecular force field. The equation for is given as

p* = - — ) (2- 2) ><3., ( 2 >

A complete procedure to derive the Hamiltonian and matrix

elements of the Hamiltonian for the vibrating rotating

symmetric top molecule is not the main topic of this

dissertation, so this chapter introduces only the steps to

get the matrix elements of the Hamiltonian for a symmetric

top molecule that can be used to derive a comprehensive

model for propyne. For completeness, a summary is now given

for the details needed to derive the comprehensive model.

The inertial parameters in eq.(2- 1) are given by

V r " s e V f " 'by'" < 2" 3 )

II = ( T * -Lt'T'\ ^ ot6 yy a6 aY yB

22

and

I -I -I xx xy xz

-I/ I / -I' xy yy xy z

-I -I xz yz zz

^ = Iv«/+£ a^QA«. + n A , Q n , . «« etd ir <4«r >fl(r *r,, , >a cr' -a er'

t' _Te ^ _ /«f-> *P> <*P ~fo-a^" -40- 9 'ir/ ATp'tr

(2- 5)

(2 -6)

Where Ij^ and ie (ocfts) are the equilibrium values of

the moments and products of inertia, respectively. By

introducing the Coriolis coupling coefficient we have

W * P =21 H e Q p

a »»- i'T' MA'V ^ A'rr' (2- 8)

and equation (2- 1) becomes

~P )u (P -p )1 2 o<^ " a a MotB B 6

(2- 9)

,1 r r * -ic * + 2 ^ 5 j ^ + vi [p_y 2 (p_ y -) ]}+ v at -Ocr

Also, rearranging Equation (2- 9) by the power of P P a' B'

the following result is obtained.

23

H = ^ ^ P s" l P* 6P«'pB < 2-">

^ " a S ^ 1 + ? } r £ + D + V

with replacements

U " Z "''"J (2-11)

" i u i ' S ^ - 1

It is not possible to find directly the eigenvalues of

the Hamiltonian operator in Eq.(2-10). The general procedure

to get the solution of the Hamiltonian for vibrating

rotating polyatomic molecule is as follows 3 9, 4 0,,*2,43 #

Power Series Generation of the Hamiltonian

1. Expand the Hamiltonian in a power series to the fourth

order with respect to normal coordinates to get

H = H0 + AHx+A2H2+X

3H3 +\"Hh (2-12)

According to the customary notation, x is 3- parameter of

smallness equal to unity, which denotes the order of

magnitude of the different terms in H. By detailed

calculation, comparison and substitution, each term of the

24

Hamiltonian can be shown to have the following forms.5 In

these equations, is a force constant of the harmonic

potential given by V =?Ex Q2 and q is a dimensionless "AV A a t AT

normal coordinate and p is its conjugate momentum which is

defined by Q , ^ = ~ ^ - § q • The final results for M«r

H are in the following form.

v t 1 E Act a

U t

. aoc

act

•+s >6(7

V * E E <6a ap> 1* I* aPtxP|i"

a/Y rvAaa ^ a + V, (2-13)

V I E I /xsb'a' aft

Q p> . . " * w r ^ \

T1 l1 ^ A«<* l[bp>

^ / fi \4, T* !*• U , j (f« 71" «« VP T«^ j

+ vrt

ri I E • /> G/$o'<i>"cr" (X

st>?A>'crl4>',Cr"

7 T p i l««

V °i/5O-Cl,C7-'Ci.0"CT"fB

0V«t> • • 4 i

Wa'cr' / -ft \ * Je j 4 VX^Xy/ «< '52f5'cr'+Cl-io-S'cr'

O05™^ <L i W r \X

— f~ V>h O A(3' a e. V > I °, w + rr~) q t>o* + V

25

Twice Contact Transformation

2. Perform the first contact transformation4o,*i in order

to partially diagonalize the Hamiltonian matrix. W. H.

Shaffer, H. H. Nielsen and L. H. Thomas 4 3 in 1939 and Van

Vleck and Kemble44 first formulated such a contact

transformation T to remove the off-diagonal element of the

first order terms in T may be expanded in terms of a

Hermitian matrix S42 as

m iAS /X A2 ~ T = e = I + iA S - j S2 + (2-14) / \

I : unit matrix

H - T H T = Hq+ AH' + A2H2+X3H3+X (2-15)

Ho= H0

Where the S function4s is chosen such that the first two

terms,HQ+^HJ, in the expansion of H are diagonal with

respect to the quantum number v.

3. Group the terms of the same true order of magnitude in

X, which means that one expands the transformed Hamiltonian

and arranges the terms in order of powers of x as

26

H - h0 + Xht +X h2 +X3h3 + x ' t h l f + - - - - (2-16)

By comparing Eq.(2-15) and Eq.(2-16)

h'0 = Ho = H,

hi - Hi + [ iS,H0] (2-17)

h' = H2+ [iS,h"] + [is, H0 ]

h' = H3 + [iS,]/] + [is, h'j ]

These equations can be expressed by a combinations of the

vibrational operators and q^r and of rotational operators

W V 4 6

4. Perform the second contact transformation5! using

^ * A S i it X = e = I + iX2S - ± S2 + (2-I8)

then

H = T H T = Ho + +^2H2 + - - - (2-19)

in such a manner that + AH* +\ZH^ will be diagonal with

respect to the vibrational quantum number v. In general,

these operators will not be diagonal in £ or K, where % s s

corresponds to the vibrational angular momentum associated

27

with the s-th vibrational mode and K is the rotational

quantum number corresponding to the projection of the total

angular momentum about the symmetry axis.

5. Group the terms, taking into account the true order of

magnitude of the terms in powers of x- The twice

transformed Hamiltonian H then can be expressed as

H + = hj +A h+ +x2h+ + (2-20)

By comparing the equations (2-17),(2-19M2-20), the

expressions for the operators H +/h

+ are as follows n n

Ht = h0

H+ = h\

H2 = h2 + 1 ^S,h/°] (2-21)

= h3 + i [S,h^]

K = h4 + 1 C S' h2 ] " jfc'CS.h'o]}

The regrouped components of the twice transformed Hamiltonian

are

ht = h„

h+ = h[

h2 = h2 + C i S' ho ]

28

ht = h3 + [iS,^] + [is, h'0 ] (2-22)

h4 = h4 + tiS'h2+;I + CiS,^]

where

^2 ~ ^2 + [iS,h0] = i(h^ + h* )

The explicit expression47 for regrouped components of the

doubly transformed Hamiltonian in terms of the vibrational

rotational operators are not presented in this work. Even

though the operators h^,h* will have terms that are

off-diagonal in v , these terms will contribute to the

energy in an order higher than the fourth and do not need to

be considered for the fourth order correction to the energy.

Furthermore, all of the operators are diagonal in J and M so

that only matrix elements of the form

<J,M,v1,v2, V g, | hJ+h++h++h++h"J;| J,M,vx ,v2 , > (2-23)

need to be calculated. For axially symmetric top molecules

such as CH3CCH, the only off-diagonal terms which are of

consequences to fourth order of approximation arise from h2

and are off-diagonal in K and I only.

6. Solve the secular equation; the eigenvalues are obtained

by arranging the matrix elements into submatrices of the

general energy matrix and solving the secular equation. This

29

equation is very complicated and lengthy to solve for the

polyatomic molecule. Symbolically the procedure is to solve

the following equation

det [ — , V S , |ht+h++h2++h++hj |

J,K',M, — , v , i ' »M ' ) s s

^KK' 51 l' fiM M7 EV.R1 0 (2-24) s s s s

Non-Zero Matrix Elements of the Twice Transformed

Hamiltonian for Molecules of C 3 v Symmetry

When calculating the energies of molecules with special

symmetry, there are basically two approaches which can be

used. The first approach is to solve the secular equation

of the general Hamiltonian matrix. This is the exact method.

The second approach which has been followed here is an

approximation method by the perturbation theory when matrix

elements are calculated. Matrix elements for molecules of

C3V symmetry will have the form

< J,K,M,Vn,Vt,£t | H + |

J,K+AK,M,V + A V ,V +AV ,I +A£. > (2-25) n n t t t t

30

Where and v_ are the quantum numbers corresponding to

a non—degenerate normal vibration and a degenerate normal

vibration, respectively. Unless accidental resonances occur

only those elements with Avr and Av equal to zero need to

be considered. In any case, the matrix elements of Eq.

(2-25) for symmetric top molecules which have C 3 V symmetry

are non-vanishing only if9,4s,4?

- AK + £ A£t = 3p (2-26)

where p is an arbitrary integer.

It is not necessary to present all of the details of the

derivation of the matrix elements because in 1975, A. Bauer,

G. Tarrago and A Remy published a paper5° giving all the

matrix elements of the Hamiltonian for molecules of Cjy

symmetry. All these diagonal and non-diagonal elements are

expressed in terms of all molecular constants and

parameters. Also, these matrix elements are written by

using the zeroth order functions |v,J,K,£> as basis and

including all the contributions originating with twice

contact transformed Hamiltonian to the fifth order. For

convenience, Table II and Table III show the diagonal and

non-diagonal matrix elements which can be used for the

vibration-rotation energy calculation by using the

perturbation theory in Chapter III.

31

EC

M-l O

CO -p c <D e <u r—I w

X •H

4J rd S

n3 C O

fd •H Q

H

w

PQ <

Eh

CN

X + o

X

w

II

A

b >

EC

o*

b >

V

CN

W r—i

C N

ro PQ I

ro < C C N

CN

CN

ro

ks <

+ b

C N o*

cn PQ +

C N

> C N

PQ +

> f—I

PC +

o PQ

PQ I CM CN >

< CN

+ >

PQ I

O PQ I o <

k_p < ^ CN + ^

> r H

H + ^ I-} k-P — ' <3 CN w b + o >

+

O h} Q

CN

k-P <

C N

I I

+ b

> rH b

Q +

o h3 Q

I

+ b ro

LD

+

<=>* ro

+ b

O K

EC +

rH +

b

b

O I-)

EC +

CN

CN

+ b

W CN b id

{3 +

CN

> > > b H fc<H H b H ^ v — O r £~ CN + + + i-}

O K O b O K

b O b

EC + ro

+ b

cr +

ro hD b o b

EC +

32

>

H » CN X I

+1 rH

<0

>

+1 CM

CN

T 5 + h )

X I r H + i r o + U* + 1 f d ffi ^ 5 •>»

CN m h + 1 CN —' r o 0 <d p—i

CN + 03 CN

• P CN r o c CN W H + i CD « CN MH s MH + CN (D + 1—' >

r-H x—s r H i—i •»

w r H + CN + 1

X + « X

X h ) ' — ' ^ r o • H ' *"D o * MH

M CN > + > - P CN b H »* + 1 r-H f d n (n MH U - ^ T q

r-H f d 1 4 CN s MH +

U - ^ T q + X + > « r H « r-H > CN h > o * + 1 I 'D

CN r-H h D > H ) C CN MH + i c n h D r o CN + i 0 MH + f d MH + 1 < D 1—' f d + 1 v o t r> + CN + X I CN f d f d o r-H CN r o VO r H k D • H t r MH C n MH m h MH Q 1 *-r-»

MH

G 0

II 11 II II II II II II

S A A A A A CN CN CN • + 1 1 + W * A A + 1 M - A H

H o * o * <£> o * H

H «k + 1 v V H CN r H CN r o r-H VD

W J + 1 + 1 + 1 + i + 1 + i + 1 W J W W J

s. CQ < l-D •"D h ) *"D h D CQ <

K. v. V E h > > > > > > > >

ffi <=>*

> V

—

*k. > V

ffi «

V •""D >

V

+ EC

«

*fc

h ) >

V

M

h ) >

V

J E o *

w

h ) >

V

- S . c>*

J *

h 3 V > V

EC

O ? *»

« *-D

K. > V

•$*

C + i

I

£ + i

+

I c

CN

i—I +

I >

I a

CN

+ 1 r H + o * +

>

C K II fiK II c c

II II o *

« > h D

c + 1 C + 1 CN

t o X

CD U a )

X ! £

CHAPTER III

DERIVATION OF A COMPREHENSIVE MODEL

Bauer et. al.so have published a paper with diagonal and

non-diagonal matrix elements of the Hamiltonian for a C 3 V

symmetry polyatomic molecule. The origin of that work was

from the formulation by Nielsen et. al.2,4,7,s of the

vibrational energy up to the fourth order in vibration and

the sixth order in rotation. Tarragon has produced a paper

along similar lines. In this chapter a similar model will

be derived from fundamental concepts.

To obtain the rotation-vibration energy, a perturbation

treatment is applied to the Hamiltonian to effect its

solution for finite vibration-rotation interaction. Similar

work has been done by Amat, Nielsen, Tarrago3? and used in

several other paperss1,s2,53,s4,ss. b u t e a c h t ± m e t h e

resulting energy equation has a different form and does not

predict results for vibrations greater than v10=4. Since

these formulas do not predict the energy or frequency for

v10=5 and higher excited vibrational states of the propyne

molecule, a general model has been developed to do so.

In this chapter a comprehensive model which can be used

for any vibrational state of a C 3 V symmetry molecule by

33

34-

substituting the proper v and i into a general energy

equation is derived. The derivation is as follows.

Perturbation Theory

The Hamiltonian is subdivided in perturbation treatment

as

H = Ho + Hi (3- i)

Where H° is an unperturbed Hamiltonian whose

eigenfunctions are known and Hi is a perturbed Hamiltonian

which is significantly smaller than the zeroth order terms

of H° in the absence of near degeneracies. The basis

functions are the product of y y where y is the basis r v r

function of the rotational part of H° and y is an

orthonormal function of the molecular normal coordinate. H°

is diagonal in the vibrational quantum numbers, although it

need not be diagonal in all of the rotational quantum

numbers. Hi is generally not diagonal in either the

rotational or vibrational quantum numbers. As was seen in

Chapter IX some off—diagonal terms are partially

diagonalized by performing two contact transformations on H.

There are two assumptions in the perturbation theory.56

One begins with the perturbed Hamiltonian Hi in some sense

being small compared to the unperturbed Hamiltonian H°

,i.e.,

35

n 1 < < E 0 M - ? nn n \o- Z)

The other assumption in the perturbation approach is that

the eigenstates and eigenenergies of the total Hamiltonian

, H, do not differ appreciably from those of the unperturbed

Hamiltonian H°.

The complete energy calculation is then

H» tpo = £0 tpo / .3 , X n n n J>

E n = <cf ° I H° | cpo > = H o ( 3 - 4 ) n n n nn

Where cp and E° are, respectively, the eigenstates

and eigenvalues of the unperturbed Hamiltonian. Also, by

some calculation and comparison of eigenstates and

eigenenergies57, the first order energy correction from

perturbation theory has a form

E1 = H 1 (3- 5) n nn '

and the second order energy correction has the form

H . E2 = Z 1 n i

i^n E°-E° n l

36

The total energy has a form given by

E = E » + E i + E 2 (7-7) n n n n [ J 7)

As can be seen from equations (3-4), (3-5) and (3-6), the

unperturbed energy and the first order energy correction are

calculated from the diagonal matrix elements of the

Hamiltonian. The second order energy correction is obtained

from the non-diagonal matrix elements of the Hamiltonian.

In order to apply this perturbation theory to our

vibration-rotation energy calculation by using Bauer's

diagonal and non-diagonal matrix elements of the doubly

transformed Hamiltonian ,H+, some observations must be made.

The unperturbed Hamiltonian H» in the perturbation theory

above is not the same as H0 given in Eq.(2-12), the power

series expression of the Hamiltonian. Also, the value of Hi

is not same as Hx. Because the twice contact transformed

Hamiltonian H+contains both unperturbed and perturbed terms,

rotation-vibration energy can be calculated by considering

diagonal contributions and non-diagonal contributions of H+.

ie,

E = E + E v.R Diag. Non-Diag. { '

37

The diagonal contribution to energy is given by:

EDiag= < V' J' K' £ I H + | V , J , K , £ > (3. 9 )

Then using Table IX with some new notations,

EDia^ V' J' K' £ ) = V X££<l2+ BvJ

( J +1) + yjl££2j(J+;L) (3-10)

+ (A-B)vK2+ (A3-B3n

2K2-2(Af ) vK£

- 2(Af)3KZ3- D J2(J+l)2 - D J(J+1)K2- D K

J JK K

+ "HjJC J+1)K£+ ilKK3£+ "HJJJ2 (J+l) 2Kj

+ 71JKJ(J+1)K3£+ 1 K KK

5^+ HjJMJ+l)3

+ H J J K J 2 ( J + 1 ) 2 K 2 + HJKK J ( J + 1 ) K l t + V 6

where Bv, Ay: Rotational constants.

Anharmonic constant.

{ : Coriolis coupling constant.

DJ' DJK , DK : First order centrifugal distortion constants.

HJ ' HJJK ' % K K ' % : H i 9 h e r order centrifugal distortion constant

B3 = y1%

% ' ^ J J ' ^ J K ' : Hi9her order diagonal matrix elements.

38

The non-diagonal contributions to energy is given by

H +. H +

E = I n i in N. D i4n T o ~0 (3-11)

Where

1*1 E~_ e Y

n i

<n | = < v , |

Ii> = |v,J,K±l,A?2>

Iv,J,K±2,l±2>

|v,J,K±2,£±4>

and where the quantum numbers v, J, K and & are the standard

set for vibration, rotation and Coriolis interaction.

In the energy expressions all the diagonal matrix

elements and three major non-diagonal matrix elements of the

following type were considered: <v,J,K,£ | H+| v,J,K±l,£+2>,

<v,J,K,& | H +| v, J,K±2, £±2> and <v,J,K,£ | H+|v,J,K+2,&±4>

which gives some contribution to the energy in the highly

excited vibrational state ,i.e., v 1 0 = 4 and higher. All

these non-diagonal elements are diagonalized about v and A .

The quantity q is the coefficient of <K,£ | H+| K±2,l±2>,

and q 1 2 and d 1 2 are the coefficients of <K,l | H+(K±l,£f2>,

f 2 4 is the coefficient of <K,i\ H+| K±2,l±4>. The

coefficients fj2, ^22 a nd -2\> i n the low excited

vibrational states, were small and ignored but were

considered with f 2 4 = 0.007 for the v 1 0 = 4 and higher

vibrations.

39

These non-diagonal contributions in terms of vibration

rotation quantum numbers and rotational constants are given

by:

_ tq12(2K+l)+d12(£-i)]2(a+b2)2 [q12(2K-l)+d10(£+l)]

2(a:bt)

N.P ~ ~ 3 + ^ L J _ HVJK-£~HVJK-U+2

HVJK£~HVJK+1£-2

( q va X ) 2

HVJK£~HVJK+2£+2

( f24 a2 b4 ) 2

HVJK£~HVJK+2£-4

(^y a2 b2 }

H + -w-*-vJK-C vJK-2£-2

( £24 a2 b4 )

HVJK£~HVJK-2£+4

(3-12)

where H^ V J K = [V JK | H«| VJK ] etc. . .

+ 1 ax » [J(J+l)-K(K±l)]2

± J- 1 a2 = [J (J+l)-K (K±l) ] 2 [j (j+l) _ (K±l,) (K±2) ]T

b? = [(v+£+l±l) (v-ZKLTl) ]T

fc>z = [ (v+£+l±l) (v-£+i+l) (v+£+l±3) (v-a+1+3)]T

(3-13)

To calculate the energy difference which appears in the

denominator of E N j y some small terms ,ie, A3, B3 =Ylj?,(A£) 3, D J R

' DK' DJ' nJ' nJj , njK' nKK' HJ' HJK' HKJ ' H k w e r e ignored,

since these terms are very small compared to A , B X v v ' 'IV

40

Therefore, the energy difference terms are simplified to

H i , V J K J T H * V J K + 1 1 £ - 2 = X J l 2 . I 2 - 2 " ( £ - 2 ) 2 ] + ( A - B ) [K 2-(K " 4 ) 2 ]

• 2 ( A < J [ K £ - ( K + 1 ) U - 2 ) ] = 4 X ^ ^ ( 4 - 1 ) - ( A - B ) ( 2 K + 1 J - 2 A C ( 2 K - 2 . + 2 )

By the same procedure,

H " " v J K £ " H * V J K - l £ + 2 = " 4 X ; . £ ( £ + 1 ) + ( A " B ) ( 2 K - 1 ) - 2 A C ( - 2 K - i + 2 )

H " v J K £ ' H " r * V J K + 2 £ + 2 = " 4 C X Z i ( " + 1 ) + ( A - B ) ( K + 1 J - A S ( K + i + 2) ]

H V J K £ - H * V J K - 2 , £ - 2 = 4 + ( A - B ) (K-L) -A; (K+Z-2) ] ( 3 - 1 4 )

H v J K £ _ H " " v j K + 2 , £ - 4 = 8 X Z i ( £ " 2 ) " 4 ( A " B ) ( K + D - 4 A C ( 2 K - i + 4 )

H v J K £ ~ H ' ' ' v j K - 2 , £ + 4 = ~ 8 X £ £ ( S , + 2 ) + 4 ( A - B ) ( K - l ) + 4 A £ ( 2 K - 2 - 4 )

Finally, the rotational-vibrational energy for C 3 V

symmetry molecules is given by

E(v, J,K, £) = E_ . + E n K 1 Diag. N.D (3-15)

By using the selection rule for absorption spectra

AJ - +1 , AK = 0 , AV = 0 , AA = 0 (3-16)

one can get the frequency

, _ . E ( v , J + 1 , K , £ ) - E ( V , J , K , £ ) v(v,J,K, ?_) = — [___ (3-17)

4 1

Hence, the rotational frequency becomes

(V,J,K,5-) - Aj + A + A 3 + A ( 3 - 1 8 )

Where

A 1 5

f 2 bv ( J + 1 ) + 2 y £ a £

2 ( J + 1 ) - 4 D j ( J + 1 ) 3 - 2 D j K K 2 ( J + 1 )

+ 2 r i j K& ( J + l ) + 4 r | j j KM J + l ) 3 + 2 r i j K K 3 J l ( J + l )

+ H J ( J + 1 ) j [ ( J + 2 ) 3 - J 3 ] + 4 H j j k ( J + l ) 3 + 2 H j k k K 4(J+l)

• { i (v-&) (v+l+2) ( J + l ) [ ( J + l ) 2 - ( K + l ) 2 1 c t 2

+ ( v + £ ) ( v -2 .+2) (J+l) R j + l ) 2 -

1 1 (&+1)+(A-B) (K+l ) -A? (K+£+2) x u U - l ) + (A-B) ( K - l ) - A C

( K - l ) 2 ] q 2

(K+2.-2)

A _ | 2 ( v + £ ) ( v - " + 2 ) ' ( J+ l ) [^12 ( 2 K + l ) + d 1 U - l ) 1

4 X U U - 1 ) - ( A - B ) ( 2 K + l ) - 2 A 5 ( 2 K - £ + 2 ) - 2 (v—j) (v+?,+2) (J+l)

4 X ^ ( 1 + 1 ) - ( A - B )

[ c ? l ? ( 2 K - l ) + d i o ( £ + l ) I 2

(2K-1) +2AC (-2K+2.+2) }

a 4 = f 2 4 ( v - i ) (v+£+2) (v+2,+4) (v-Z-2)

8 X U (*+2) - 4 (A-B) ( K - l ) -4AC (2K-H-4 ) V

£24 (v+J&) ( v - l + 2) 8 x » „ < J?-2 ) - 4 (A-B)

Jlit *

( v - ^ + 4 ) (v+g,-2)

(K+l) -4AC (2K-J.+4) Where

[ ( J + l ) ( J + 2 ) - K ( K ; i ) ] [ ( J + l ) ( J + 2 ) - ( K + l ) ( K + 2 ) ] - [ J ( J + l ) - K ( K + l ) ]

[ J ( J + l ) - ( K + l ) ( K + 2 ) ]

42

Accidental Degeneracy

Some quantum number sets had to have special attention.

Any combinations of quantum numbers which make K&=1 or K&=-2

cause accidental degeneracies. Accidental degeneracy makes

the energies in the denominators of the E N D terms same, and

ENft aPP e a r s be infinity. To avoid these undesired cases,

direct diagonalization was employed to solve the secular

equation for these special quantum number sets. In this

general model, quadrupole interaction, proton—proton

interaction, spin-rotation interaction etc..., which appear

not to be important for the propyne molecule from our

experimental results, were ignored.

The direct diagonalization method used is given byi

Hd^d-El^d d; diagonal

Hip = Atp

(ft- \l)ip = 0

/ \ / \ |

H- XI | = 0

By using the Table III in Chapter II, the determinant has

a form

v,J,K+2,1+2

v,J,K+2,1+2

v,J,K-2,H-2

E-X

+ , + qva2b2

% a 2 b 2

+ , + qva2b2

E-X

v,J,K-2 fZ-2

%a2h2

0

E-X

= 0

43

No interactions between (K+2,2+2) and (K-2,5,-2) were

assumed.

(E-X ) [ (E-X ) 2- ) 2 ] - (E-X) (qvajbj)2 = 0

X = e

X = E i q ^ U + b * ) 2 + (a'b")2]^

X = E±qv {[J(J+1)-K(K+l)] [J(J+l)-(K+l) (K+2)] (v+Jl+2)

(v-j,) + [j (j+i) - K (K-l) ] [J (J+l) - (K-l) (K-2) ] (v+J,) (v-£+2)}1/2

when

v=l if o —i / K——1 £=--l v 1 , K*. -1 | K = ! 1

X = E±qvJ(J+l)(v+1) (3-19)

Applying the absorption selection rule AJ=+1 to Eq.(3-19)

Av = ±2qy(J+l) (v+1) if k£=1 (3-20)

By the same procedure(see Appendix E), A4 becomes

Av = ±2f24(J+l)v(v+2) if Kl=-2 (3-21)

Finally, a comprehensive model for the rotational

frequency of C 3 V symmetry molecules has been obtained by

using the perturbation theory and partial diagonalization

method. The final form which gives the frequency of a

transition J -> J+l for any vibrational levels of the propyne

molecule in low J transitions is given in Table IV.

44

CD C >i CX 0

M-i 0

rH (D TJ 0 s

<D > •H W c <D JC (D U a g •

0 0) u a)

rH CD JC o 4-) Q)

rH 0

0 s MH

>1 fO rH 4-> 3 0 e u [| 0 >i 1*4

>1 > o m c U (1)

&H D 1 CD a) & u -P En o

a) ,c C EH

w

PQ <

Eh

CM O CJ

C M

CN^ T—* +

CO *

-5 Cr +

C M \ /

Q

+ "3

CN

+ >

* + —' > «*»•>

< ^

+ In

r» "O +

JN

CM r— O"

+

I ° I cr ^ CM

^ + pq c I £

""""'* r —

+ II P ^ I .

r-

+

C N

+ <=w

> CN

CO

+ wL

—

+ 3 > <;

£? P C O

C M 3

1 3 ^ 3" • I

C M O a*

C N +

_ cvr + r—

r -

+ + + > V

| < 1

T—

+ C M

1 3 C M

r " o r—

+ cr + — 5

+ C M I i

—» C r » > CD

+ G Q

1

JO G Q

1 * *

_,— _

i

< r -1

C N 1

+ + I! »

4- r—

> + « r—

1 —• 1

II

!

X *•—

CM t— "O +

»•-+

CN

CM T— CT

+ -5

C N

I >

X

CM

A-0 < CM

+ CM

> CD

I <

I

X

C N

!

I *

I

CN 1 _+ r—

+ + —3

CM |

—3 1

| < CN CN + 1

r— |

1 CN 1

_— _

> 1 CD i CN 1 + < -5

I P +

«— 3

"i r

l *

OJ +

CN

+

+ >

CN J-

+ >

I >

C N C N

C N +

+

+ *

C M +

*

1

CN +

+

I

C N

+ rr * l <5W

* .

C N p

* 5

i ? — >

* ^ — CN «—« > CN

CO I I C <r a -± JZ rt -

1 7 CN n/ +

^ Y '

C M

+

C N 4-

X oo rt=

+

* CN

< CN +

, > + ^

_ C M

C O +

< 5 * €

> C N 1

ii

+

1

2, K

> h

CN C N *•— X 0 0

<5w

J L.

CHAPTER IV

EXPERIMENTAL APPARATUS

A microwave spectrometer suitable for measuring the

rotational spectrum of CH3CCH in this work was available in

our laboratory. This spectrometer is similar to those

described by E. A. Rinehart et alss, j. a. R o b e r t s * a n d I.

C. Story et al^o. A block diagram of the spectrometer is

shown in Figure 5.

In a typical spectroscopic experiment, radiation is

supplied from a source and guided through an absorption cell

to be detected and displayed in some fashion. This whole

procedure can be divided into five areas of instrumentation

according to the function of each. The catagories chosen

here for discussion are listed below.

1. Radiation Source

2. Absorption Cell

3. Detection and Display

4. Sample Gas Handling

5. Generation of Standard Frequency

4-5

46

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Cn G •H

3 CO (0 CD 5

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LT)

CD u p tn •H pLi

4-7

Microwave Sources

Four klystrons were used in this experiment as radiation

source. A Varian X-12 and OKI 20V10 klystrons were used to

generate radiation in the 17 GHz region; an OKI 24V11 was

used to generate radiation in the 26 GHz region; an OKI

35V10 was used to generate radiation in the 34 GHz region as

fundamental frequencies. All of these klystrons provide a

stable, tunable source of monochromatic microwave radiation.

The detailed construction of the klystron is out of the

scope of this work and is given elsewhere in standard

microwave electronic books* i, <*2, 6 3, <s 4,6 s . I n s o m e c a s e s it

was necessary to generate harmonics to reach the higher

frequency transitions. A crossed waveguide multiplier66,67

was used to provide radiation in the 52 GHz and 68 GHz

regions by prpducing the second harmonics of the OKI 24V11

and 35V10 klystrons, respectively.

There are two different adjustments of a reflex klystron

to produce appropriate radiation. Coarse frequency

adjustment of the klystron was achieved by mechanically

altering the size of the klystron's resonant cavity by means

of a flexible diaphragm and a connected screw assembly. Fine

frequency adjustment was then achieved by the fine

adjustment of the repeller voltage. It is also possible to

sweep a range of frequencies by sweeping the repeller of the

4-8

klystron with a voltage ramp available from the sweep of the

oscilloscope used for signal display. The frequency response

of a klystron can be measured by applying a control voltage

to its repeller and measuring the frequency shift. A

typical value of klystron sensitivity indicates that one

volt change in the repeller voltage will result in one MHz

change in the output frequency of the klystron,

approximately. Each klystron is highly individual in this

respect. About ten megahertz frequency shift was achieved by

using the voltage signal from the time base of an

oscilloscope. The exact range was measured by applying the

ramp voltage and directly observing the frequency shift of

the klystron relative to the frequency markers which were

generated when the klystron frequency was swept relative to

a fixed standard reference source.

Absorption Cell

A K-band rectangular copper absorption cell which has a

silver coated inside wall to produce better conduction was

used. A thirty meter absorption cell was used for the strong

signals found in the ground, v 1 0= 1 and v 1 0=2 vibrational

states. To detect the weaker signals in the higher excited

vibrational states ,ie, v10=3,4, the absorption cell was

lengthened to 60 meter, thus giving improved sensitivity and

better signal-to-noise ratio.

4-9

The ends of the absorption cell were vaccum sealed with

Teflon windows to allow changing the transition sections and

detectors without altering the pressure in the absorption

cell. Breaking the conduction path of the radiation in this

manner gives rise to reflections which greatly enhanse the

standing wave pattern of the system. Various methods are

employed to reduce this effect. One technique is to employ

approximate quarter wavelength gaps at the end sections,

much like the effect of a quarter wavelength plate in

optics. This effect has been discussed in some detail by I.

R. Dagg et al68.

Detection and Display

Crystal diodes were used for primary detection. Germanium

1N26 diodes w^ere used to detect the frequency from the ten

gigahertz to forty gigahertz region. Germanium 1N53 diodes

were used for frequencies above forty gigahertz.

A dual-trace Hickok Electric oscilloscope (Model 1805A)

provided visual display for the absorption. Strong signals

in the low vibrational states were detected directly on the

video, but weak signals in highly excited vibrational states

were detected only by using a slow scan sweep with display

on the chart recorder and by optimizing the gain and fine

tuning of all parts of the equipment. In any case, the

50

spectral line profiles were displayed on a dual-trace

recorder manufactured by Rika-Denki (Model B201) with one

pen producing the frequency markers and another pen

producing the appropriate derivative of the spectral line

profile. Figure 6 shows one typical example of the line

profile and standard frequency markers.

Sample Gas Handling System

Both gas handling and high vacuum capabilities were

required for the apparatus. The absorption cell was

evacuated by using a Van Waters and Rogers HV-1 oil vapor

diffusion pump, cold-trapped with liquid nitrogen to remove

the pressure of the gas molecules in the absorption cell.

This was fore-pumped by a Welch Duo-Seal 1405 mechanical

pump.

Once the apparatus had been evacuated, high purity

propyne(99.6 percent) gas was admitted to the absorption

cell from a sample tank which had been pumped to remove

residual air and water vapor before admission of the sample

gas.

The peak line intensity of the absorption spectrum was

found to be very sensitive to the pressure of gas in the

cell. So, it was necessary to keep the appropriate low

pressure to get the maximum intensity of each line. The gas

51

I-6MHZH

/

LO 0) •

CO CO 5

Fig. 6. A typical chart recorder scan of rotational spectra(vg=l,2). B.G. is the background display.

52

pressure in the absorption cell was measured by MKS Baraton

Type 170M-27D pressure meter with accuracy to 0.0001 torr.

A gas pressure of about 20-30 mtorr of propyne was observed

to produce maximum peak absorption in this work.

Generation of Standard Frequency

In order to measure the rotational absorption frequency

which is displayed on the chart recorder, a well known

frequency interval must be recorded along with the

absorption profile. A method used in this work for accurate

measurement is to heterodyne the microwave frequency with a

precisely known standard frequency which lies within the

range of an available interpolation receiver. A

Well-Gardner BC-348 Q radio receiver was used. The setting

in the receiver was 1.5 MHz, so that as the klystron

frequency was swept through the range of interest, markers

appeared at 1.5 MHz above and 1.5 MHz below the center line

frequency, giving a total marker spread of 3 MHz. These

signals are beat notes between the fundamental of the

klystron and appropriate harmonics of the fixed standard.

All frequencies are ultimately related to the frequency

of a Hewlett-Packard 105B quartz oscillator which provides a

signal of 100 KHz that is good to one part in 10* °. A

General Radio 1112A, 1112B reference is phase locked to the

53

quartz oscillator and thus produces a highly stable 1 GHz

comb of signals. The beat notes are produced when this

Signal, a signal of less than 100 MHz, and the fundamental

of the klystron are combined in a mixer multiplier unit.

A beat note will be produced if

± mxlGHz ± nxf = f . . K v R (4- 1)

where as the klystron frequency, f R is radio

receiver setting(l.5 MHz in this work), and m and n

are integers•

The value of m can be easily determined by the use of a

wavemeter 9. The fundamental klystron's frequency can be

obtained roughly by reading the wavemeter with accuracy of

about 10 MHz. To determine n, a Tektronic 191 variable

oscillator was used. The output frequency of the variable

oscillator is measured with a Hewlett-Packard model 5383A

counter which has accuracy to 1 part in l0e. By assuming a

value of n and setting the variable oscillator at

f ' = n f v v -57* (4- 2)

for n' an integer, marker will be produced if the choice of

n is correct. The center frequency of the absorption profile

is easily determined by a method of interpolation.70

5U

The greatest experimental difficulty in this experiment

arises from the weak intensity of v 1 0=4 absorptions. If the

variation of the population of energy states in accordance

with the Boltzmann distribution is considered, the v 1 0=4

transitions will be less than one percent as intense as the

ground state transitions". Moreover, the v 1 0=4 transitions

are even weaker since a number of degeneracies have been

removed with respect to the quantum number producing further

redistribution of the spectral line intensity. So, some

weak components could not be detected.

All of the experimental frequency measurements of the

propyne molecule in the 17-70 GHz region from the ground to

v 1 0=4 vibrational state are tabulated and discussed in

Chapter V.

CHAPTER V

ANALYSIS OF EXPERIMENTAL RESULTS

A. The Ground State Spectrum

to assign the microwave spectra of propyne in the ground

state, Dubrulle et al derived the quartic equation'!

v (J.K) = 2B0(J+1) - 2Djk(J+1)K2 - 4 D < J + 1 )3

(5- 1)

+ 4 HJJK ( J + 1 ) 3 k 2 + 2 Hj k k (J+1)K

4+HT(J+1)3[(J>2)3_J.3 ]

Fro. our comprehensive model one can get the same formula by

substituting v=0 and ,=0. We have ignored the sextic order

centrifugal distortion constants to assign our ground state

rotational spectra, since the contributions of these terms

to the frequencies is less than J-ess than the experimental error in

the low J transitions studied.

our measurements were made using the video spectrometer

described in Chapter IV. The ground state lines were

exceptionally strong and therefore easily measured with

experimental accuracy „i t h i n ± 6 0 k H z . A s , ^ o f ^ ^

the result of Ware and Roberts the calculated values by '

Dubrulle et. al.'i a n d t h o s e f r o m OUJ; comprehensive model

are given in Table V for comparison.

55

56

TABLE V. Rotational Transition Frequencies for CH nr„ in the Ground State CH3CCH

^ ^ ^exp (MHZ)

a b c

b Vcalc

(MHZ)

1 0 34 183.349 1 11 34 182.727

2 0 51 274.947 2 11 51 273.980 2 ±2 51 271.000

3 0 68 366.298 3 ±1 68 364.990 3 +2 68 360.976 3 ±3 68 354.498

34 34

183.415 182.761

51 274.947 51 273.967 51 271.026

68 366.270 68 364.962 68 361.042 68 354.509

Roberts and Ware (72) Dubrulle et. al (16) Our Comprehensive Model

calc (MHZ)

0 0 17 091.718 17 091.743 17 091.743

34 183.415 34 182.761

51 274.947 51 273.967 51 271.025

68 366.269 68 364.962 68 361.040 68 354.503

(j/a.j/b)

(MH2)

-0.066 -0.034

0.000 +0.013 -0.026

+0.028 +0.028 -0.066 -0 .011

(pa.pc)

(MH2)

-0.025 -0.025

-0.066 -0.034

0.000 +0.013 -0.025

+0.029 +0.028 -0.064 -0.005

57

As can be seen from the table, the two calculated values

are nearly the sane. There are only l-2kHz difference even

with the ignorance of sextic order centrifugal distortion

constants. Both calculational values were obtained by using

the same constants set which was derived from the data of

Ware and Roberts72.

B. The v 1 0 - 1 Spectrum

Nielsens was the first to calculate the rotational

transition frequencies for symmetric top molecules in an

excited v I 0= 1 state. In 1951 Anderson et. al.« reported

calculations on the v,„=1 bending mode of trifluoromethyl

acetylene. Also, they reported some deviation of their

experimental values from their calculated values as high as

200 kHz, which is outside the range of their experimental

error. In 1962 Grenier-Besson and Amat'3 derived a frequency

formula for trifluoromethyl acetylene in the v,„=l state and

obtained a superior assignment. The formula used by

Grenier-Besson and Amat is given below.

Grenier-Besson and Amafs formula is given by:

jJ+l)-2DJK(j+l) (Kt-1,2.4D ( J + 1 )3 (5- 2)

+ 2p (J+l) (Ki-1)

t4qv(J+l) if K£ = 1

58

- 4<3y ( J + l ) 3

[ V V ( A C ) v ] (KJU1)

2

if K£#l

* 12r B — B ™D +n -f- .

v JK j BV-AV-2(A?)V

* 2cr2 8r^ P = ^ J " 2 0 ^ +

VV(A^v Bv-Av-2(A?)v

From our comprehensive model one can obtain the v 1 0=l

rotational frequency formula by putting v=l and £. ± i to get

( J , K ' £ ) 2 ( J + 1 ) (V YAA J l 2- Dj KK 2 +n jKA+Tl ; y KK

3A) (5- 3)

4°j (J+l) 3+4ri KJ, (J+i) 3+v J J non diagonal

The observed frequencies of Ware and Roberts'2, the

calculated values using Amafs formula and those using our

comprehensive model are tabulated in Table VI. close

agreement between the two calculated values again can be

seen from the table.

59

TABLE V I . ^ o t a t i o n a l ^ T r a n s i t i o n Frequencies f o r CH„CCH e x c i t e d V i b r a t i o n a l

{pa-V*>) (Va-Vc)

(MHZ) (MHZ)

0 0 ±1 17 139.588 17

1 -1 -1 34 246.269 34 1 + 1 ±1 34 277.184 34 1 0 ±1 34 279.161 34 1 1 1 34 313.523 34

2 -1 -1 51 369.214 51 2 +2 ±1 51 410.612 51 2 + 1 ±1 51 415.471 51 2 0 ±1 51 418.341 51 2 ±2 ±1 51 418.915 51 2 1 1 51 469.901 51

139.632

246.321 277.202 279.142 313.421

369.300 410.671 415.560 418.409 418.876 469.850

34 34 34 34

51 51 51 51 51 51

139.610 -0.044 -0.022

246.268 -0.052 +0.001 277.164 -0.018 +0.020 279.080 +0.019 +0.081 313.369 +0.102 +0.154

369.221 410.625 415.481 418.270 418.856 469.872

-0.086 -0.059 -0.089 -0.068 +0.039 +0.051

-0.007 -0.013 -0.010 +0.071 +0.059 +0.029

a Roberts and Ware (72) b Grenier-Besson and Amat (73) c Our Comprehensive Model

60

C. The v 1 0 - 2 Spectrum

m 1971. Bauer used a frequency formula, given below, for

v8 2 state to assign the rotational spectrum of

Acetonitrile7 .

v(J,K,0)=2 (J+l)Bv-4 (J+l) 3-2 (J+l) K2D T„+ U (J-M) [X (J+, , 2^„ |_, ,

P2K2-\2

K ' + 3 2 ( J + 1 ' 4K2t(<3,12)

2("-2p)+2oq'1,d"] +

. ,2

- + 32 (J+l) 4K2[(q',„)2fTr-2n^-i.5nrr.i2Ui._

_ u . . > 2 ,,t_2„2 2. 12 12' Tr(q' - d,^)2 / (4p2K2-7T2)

and

v (J,K,±2)=2 (J+l) [Bv+4A]_1]-4 (J+l) 3D -2(J+1)K2D

JK

+2 (J+l) K * n j + 4 (J+l) [4(J+1)2-(M-2)21 (a'. v2. f o r K Z * 4

V(Kl-4)+2X

/ for Kl*-2 -16(J+l)[q'12(Ki+l)+d^|]

2

p (Kl+2) -it

Jl6(J+l)[(j+l) 2-l] (qj )2

Q , for K Z-4

,/+ 16(J+l)fJ4

1 + 2 t d ^ ~ cj • 0 ]2

^ -16 (J+l) ft. i2 1 2 , for K£=-2

61

This theory of Bauer predicts weak splittings in the

Ki=-2 and Ki=4. We could not detect these predicted

splittings in the propyne molecule, although the resolution

of the spectrometer allows it. If these splittings in

Bauer's formula are ignored and v=2 and t-0, ±2 are put into

our comprehensive model, with the notation a s g i v e n

below, the formulas become identical and good agreement is

found wath observed experimental frequencies.

^ A - B - 2A f + X££

H = A - B - A }

ft - A — B + 4A f + 4x 11

P = A - B + 2A f

The contributions of the constants f 1 2. f „ , ^ t o t h e

frequency were found to be ignorable in the v10=2 spectrum.

Table VII shows the comparison between the observed values^

and those calculated from Bauer's formula and our

comprehensive model. Assignment of Bauer's new data, shared

with this laboratory by private communication, has also been

made by the comprehensive model.

62

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64

D. The v 1 0 - 3 Spectrum

Bauer, Godon and Maes~ have given a frequency

expression, as shown below, for v s =3 to assign the

microwave spectra of methyl isocyanide.

| £ | = 1 / = 2 By U+1) +• 2 j (J+l | _ 4 0 * u + 1 | 3 _ 2 0 * k ( j 4 „ k 2 + 2 7 j ( J + h k ?

32 q,2 (2Kg +1) 24 I q12 (2K«-1) + 2 d , |2 (J-hi + _

" p ( 2Kg + 1) _ 2n 1

_ 12 ( + qv 22 13U+l)2-Kt(Kt+2) | ) ( (j +|,2 _ (kS +1)2 , (J + „

M ( Ki - 1) + 2 X —

i / K S * , ? + 1 6 K 2 + "» /2

J2 ( 3 (J + „2 - Kg (Kg -2> [ } t U + 1)2 - ,KC_„2 , u + „

V (Kt-1) —

>{ Kt = \ •. t 8 qy (J +1) t 16 /2J2 (j +,,3

1*1=3 '* 2 B' ( J~V 8 »te«'"i-4 0j UM,3-2d;(Ji11k2 t 2, j U !„«

8 I q12 |2 Kg+ 3) + 6 d)2 |2

o (2 Kt - 9) — 6 rr

+ 9 Qv r "v <22 ' 27 'J^"2 ~ IKe~6) I > I 9 (J HI2 - (Kjt-3)2 | (j+1)

9 p (Kt -9) f 54 \

An identical formula can be obtained from our comprehensive

model by putting v=3. »-±1 a n d i = ± 3 i n o u r » o d e l a n d

tituting A, K• IT, p as linear combinations of A B f v 11

'1jjana V C h O S e n 2 e r o- T a b l e V I " shows large

differences between the two calculational values when the

r m S ^JJ a n d ^jk a r e included. The comprehensive model.

6$

which contains both i\ and n i« * JK' therefore seen to be

superior to the Bauer formula.

constant set in the v10=2 vibrational state was used

as a trial constant set to obtain approximate constants for

the vio=3 level. Good agreement with the experimental data

for all measurements was not found. Systematic departures

between observed values and calculated values were studied

to determine additional procedures to be used for refining

the calculations. Inclusion of the higher order diagonal

terms ^ r e m o v e d s y s t e n a t l c M a s ^ ^ s a t i s f a c t o r y

convergence values for 1^-0.00123 and 1JK=0.0022 which were

obtained from the experimental data. The J-type doubling

constant q„= 4.1778 was obtained from the data set of K£=l

quantum states. Prakash and Roberts- previously published

the rotational spectra of propyne in the 17-52 GHz region

for the vI0=3, vibrational state. This work has been

extended with new observations of the rotational components

of propyne for the J»4<-3 transition and good agreement has

been found between these data and the comprehensive model.

E. The v 1 0 = 4 Spectrum

In 1977, Bauer and Godon™ published a paper on the

microwave spectra of methyl isocyanide in the v„= 4 state.

To assign their observed rotational frequencies, they used

66

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68

an analytical formulation giving the frequency of an excited

egenerate state vs=4, which has been obtained by Tarrago

and shared by private communication with them.

There are three differences between Bauer et. al.'s

formula and our comprehensive model. First, they added one

more non-diagonal Hamiltonian matrix element to their energy

and frequency derivation than our comprehensive model; the

element <K., , H+|K,U6> whose coefficient is g.. secondly,

they ignored higher order diagonal terms „ , w h i c h l s

r e a S O n W h y t h e " difference between the

two calculated values in Table IX. The values given for n

=0.003 and -1JK=0.006 presented in the summary table were "

Obtained from the observed frequency values. Also, to get a

better fit, f24=0.007 was assigned in this highly excited

vibrational state.- The weak intensity due to the low

population in this state caused problems to detect all

frequencies in the 17-52 GHz region. The weak signals

produced larger experimental errors than were encountered in

the ground, v10=l,2 and 3 components.

F. The General Form of the Rotational Constants

All the rotational constants of the propyne molecule from

ground state to v10=4 state are shown in Table X. From

these constants, derived from experimentally observed

69

TABLE IX. Rotational Transition Frequencies for CH CCH inthe v 1 Q= 4 Excited Vibrational State 3

0 0 0

2 2 2 2 2

0 ±2 ±4

0 ±2

K

0 0 0

0 0

±1 ±2 - 1 +2 0 ±4

+ 1 ±4 ±1 ± 4 '

±1 0 + 1 - 2 ±2 ±2 ±1 ±4 * ±2 ±4

a Pexp

(MHZ)

b cal

(MHZ)

c Peal

(MHZ)

17 278.585 17 278.585 17 278.278 17 278.257 17 278.257 17 277.010 17 276.831 17 276.831

34 34 34 34 34 34 34

51 51 51 51 51

556.808 556.808 556.808 553.348 553.348 547.348 558.234

833.821 870.727 836.464 838.729 848.731

34 34 34 34 34 34 34

51 51 51 51 51

556.955 556.617 556.264 553.648 553.609 547.776 558.381

833.318 872.357 835.316 837.428 844.643

34 34 34 34 34 34 34

51 51 51 51 51

556.955 556.617 556.504 553.408 553.609 547.296 558.861

833.318 870.629 837.188 838.868 848.387

a Roberts and Rhee. (79) b Bauer and Godon. (78) c Our Comprehensive Model.

(pa.pb) (pa.pc)

(MHZ) (MH2)

+0.021 +0.179

-0.147 +0.191 +0.544 -0.300 -0.261 -0.428 -0.147

+0.503 -1.630 +1.148 +1.301 +4.088

+0.021 +0.179

-0.147 +0.191 +0.304 -0.060 -0.261 +0.052 -0.627

+0.503 +0.098 -0.724 -0.139 +0.344

70

frequencies. we can predict t h e g e n e r a l ^

a general power series to predict the constants for any

vibrational level. Sufficient information was obtained for

the following constants to predict the trend and express

each constant in a power series form. The other constants,

f.z- f2s- f24, fljK etc... could not be so expressed

with the data available. Table XI gives the general form of

a n rotational constants of the propyne molecule for which

sufficient data could be obtained to express them in a power

series form.

The constant set predicted for v10.5 from the general

form is reported in Table X.

G. The Vj 0 = 5 Spectrum

By putting the predicted constants set and v=5, t=±l,

*=±3, f ± 5 into the comprehensive model, one can predict the

rotational spectra of propyne in the v,„=5 state. Those

predicted calculational frequencies are shown in Table XII.

Due to the weak intensity and low population, we could

not detect experimental data with our conventional video

spectrometer for the frequency range accessible to us.

Hopefully, we can obtain the observed frequencies in this

high vibrational state by increasing the sensitivity of the

detection system and by obtaining microwave oscillators

71

<D JZ -P

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00

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h*

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72

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00 >

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TJ

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rH II

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73

TABLE XII. Calculated Rotational Transition Frequencies for CH CCH in the v = 5 E^ci?ed Vibrational State.3 1 0 excited

J K 2 J k e ^ (MHz) (MHJ

0 0 ±1 17 320.008 0 0 ±3 17 319.546 0 0 ±5 17 318.574

-1 -1 34 540.548 +1 +1 34 740.314 +1 34 638.016 0 ±1 34 639.636

~1 ±3 34 634.620 0 13 34 639.052

±1 ±3 34 642.257 71 ±5 34 630.248 0 ±5 34 637.080

±1 ±5 34 642.599

2 -1 - 1 51 810.713 2 + 1 +1 52 110.362 2 +2 +1 51 951.313 2 +1 ±1 51 956.199 2 0 .±1 51 958.506 2 ±2 ±1 51 960.574 2 +2+3 51 941.457 2 +1 ±3 51 951.337 2 0 ±3 51 958.482 2 ±1 ±3 51 963.804 2 ±2 ±3 51 964.433 2 +2 ±5 51 930.092 2 +1 ±5 51 944.307 2 0 ±5 51 955.453 2 ±1 ±5 51 964.627 2 ±2 ±5 51 972.968

74

which Will give u s a c c e s s t o t h e 7 0 1 2 5 G H 2 f r e q u e n c y ^

as fundamentals.

CHAPTER VI

CONCLUSIONS

Some of the problems of the cure m f . f r°tataonal spectrum of a

Vlbrating-rotating polyatomic molecule has been

investigated. Both classical and guantum mechanical

approaches have been presented to explain the physical

significance of some of the gross features of the rotational

spectra of linear and symmetric top molecules. Vibrational

• ects to the rotational spectra have been studied in the

excited doubly degenerate vibrational states. The general

quantum mechanical Hamiltonian for vibrating-rotating

polyatomic molecule of arbitrary symmetry has been

discussed. A complete procedure to get the solution of the

fourth order series expansion of this Hamiltonian through the use of successive contact tran«f« *. •

act transformations has been outlined. By using Bauer's Hamiif •

S Hamj-ltonaan matrix elements which are derived for the r

3y symmetry molecules a

comprehensive model for predicting the rotational spectra of

polyatomic molecules of c J V symmetry has been developed from

asic perturbation theory. As test of this model

comparisons between this model and other formulas have been

made and very good agreement between the comprehensive model

r m 0 d e l S ' M h e" Edifications have been made in the non-general formulas, has been found.

75

76

A total of 86 rotational transitions for the ground,

V " 1 - V l 0 " 2 , v " = 3 a n d v i o M Vibrational states of propyne

molecule have been reassigned by using the comprehensive

model. of these 86 lines, the transition of j=3*4 in v10=3

and all the lines in the v , v i b r a t i o n a l states were ''

measured here for the first time. Also, a total of 34

rotational constants for structure of the propyne molecule

have been derived for the first- r. .. tame from the microwave data

of the ground state to v, A=4 vibran™*! 4- q vibrational state which were

measured as part of this work.

For the first time general forms of the 11 different

rotational constants of the propyne molecule have been

achieved by analyzing the 88 rotational constants which were

obtained from the observed spectra of the ground state,

V..-1, 2, 3 and 4 vibrational states. These general forms

of the rotational constants of the propyne molecule allowed

US to predict the rotational constants of the v10=5

Vibrational state of this molecule. From these instants,

rotational spectra of the propyne molecule in the v,„=5

state can be predicted.

Some exceptional phenomena have been detected in the

v10=3 rotational spectrum of propyne molecule. It has been

shown that the observed spectra in the K-±l and ^ s t a t e s

diverge systematically from the calculational values. The

divergence increase with J according to

77

v <J,±1,±1) = 0.762 (J+l) + „ (J,±1,±1) 3.

where va(J,±l,±i) l s th e frequency predicted by the

model.

This departure may be caused by the ignorance of the terms

fi2(J+l) and fjj(J+l) l n the non-diagonal matrix elements of

the Hamiltonian for the rotational energy calculation.

Similar exceptional phenomena were reported in the v,-l

vibrational state." No satisfactory theoretical

explanations as to why this exception arises for this

special quantum number set has been found.

This study of the microwave rotational spectra of highly

excited vibrational states has been greatly complicated by

the low intensity of some of these transitions. This

problem could be partially solved by increasing the

sensitivity of the detection system and by obtaining the

microwave oscillators which will gi v e u s access to the

70-125 GHz frequency range as fundamentals.

APPENDIX A

THE v.. 2 ROTATIONAL SPECTRA OF PROPYNE

In order to test the comprehensive model for other than

the v,0 vibrations, an experimental test was run on the v,

spectrum of propyne. Some rotational components in the v,-2

vibrational state of propyne have been observed in the

frequency range 17-72 GHz region... Molecular constants for

this vibrational^ excited state have been determined from

more than U observed rotational transitions. See Table

XIII below for a list of these constants. Good agreement

between the experimental frequencies and the calculated

frequencies obtained from a constant set for the v,-2

spectrum has been found.

Some transitions in v,.2 state are overlapped by the

strong lines of v,-l state at 34 223.401 and 51 334.949

MHz. (See Fig. 6 ) I h i s o v e r l a p ^ ,t i m p o s s . M e ^ a s s. 9 n

S O me C O m P°" e n t s P«="ely. The rotational components of the

propyne molecule which were obtained in the v»=2 state are

given in the Table XIV.

78

79

Table XIII. Rotational Constants for Propyne in the v,-2

V 9=2

B V

— 8 556.1996 MHz A V

= 137 392 MHz D J = 0.00296 MHz

D J K = 0.1672 MHz

"J =

0.4687 MHz ^JK = 0.0 MHz

11J J = 0.0 MHz

q v = 2.2613 MHz X

11 = 150 000 MHz *3L 2 = -10 MHz

d i 2 0.0 i = 0.94

y n

= 0.0 f22=f2 4= 0. 0

80

T a b l e X I V .

S '« s°-Propyne t n e V 9 ~ 2 V i b r a t i o n a l L e v e l o f

K l -exptMHz) i/cal(MHz) (,e j<p-„C3|< ; ) ( M H l ,

0 0

1 1 1 1 1

2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3

0 0

+ 1

0 +1 0

+1

12 +1

0 +2 +1

0 +1 12

13 12 +1 0

*3 +2 +1 0

+1 12 13

0 0

0 0

12 12 12

0 0 0

12 12 12 12 12

0 +0 0 0

12 12 12 12 12 +2 12

34 227.592

51 322.344 51 330.025

51 340.980 51 343.854

68 447.100 68 448.875

68 427.917

68 448.633 68 454.518 68 457.956

17 112.387 17 112.285

34 223.992 | 34 224.672 \ (1)

34 220.070 34 224.508 (1) 34 227.492 +0.100

51 332.922 51 335.750)

(1) 51 336.781 I (1)

51 321.328 51 329.961 51 336.617 (1)

+ 1.016 +0.064

51 341.094 51 343.742

-0.114 +0.112

38 436.781 68 443.453 68 447.234 68 448.594 68 413.953

-0.134 +0.281

68 428.156 68 439.672

-0.239

68 448.547 68 454.516 68 458.047 68 458.812

+0.086 +0.002 -0.091

( 1 )

£ 2237n?PPHdC?V * e , e r y S , ' °n9 <"e

vg i at J4 223.401 and 51 334.949 MHz.

APPENDIX B

THE LINEAR MOLECULE AND SYMMETRIC TOP MOLECULES AS A RIGID ROTOR

sically the angular momentum and energy of rotation

of a rigid body is represented by

r 2 (B 1)

p = i" (B 2)

By choosing *,y,z as body fixed principal axis and the

and x - s as the corresponding angular momenta and foments of

inertia respectively, Eq.(B 1) becomes

1 / P^ P^ E = — / x y F r

i / x y p2 x

2 > T + T + T " ) (B 3 )

V T T -L

For the quantum mechanical treatment, E and the P's are

replaced by the Hamiltonian operator and corresponding

angular momentum operators. I h e m e c h a n i c a l

expression for a rigid rotor is given by

1 /P2 p2 p2

H = I ( x . v . P 1 ( X y ?! x r 2 ( — + _ . + ) ( B 4 )

X " 1 y z

81

82

where , p s are appropriate operators

The total angular momentum squared ,P2, is g i v e n b y

P Z Px + Fy + Pz (B 5)

ForQa linear molecule, with 2 assumed as the molecular axis, I = I z X y '

Eq.(B 4) becomes

X

H • — P2 / X r 21 ( B 7 )

X

By using the matrix elements for pz and H r from Eqs.(l-

« M 1 - 7) and the definitions of rotational constants from

Eq.(l-13), the energy of rotation for a linear molecule

under the rigid rotor assumption is easily shown to be

Er * h (B 8)

For a prolate symmetric top molecule, z assumed as the

symmetry axis and I, - Iy > V From Eqs.(B 4) and (B 5,

P2 P2 ,1 1 H = — , __z r 1 \ r 21 2L7~7) (B9)

x Z X

83

By using the same eigenvalue equations (1- 6 M l . 7 ) t h e

definitions of rotational constants from Eqs.(1-13),(1-14),

the rotational energy for a prolate symmetric top molecule

with the rigid rotor assumption is shown to be

Er(J,K) = h [ BJ(J+l) + (A-B)K2] (BIO)

APPENDIX c

DISTORTABLE ROTOR FOR PROLATE SYMMETRIC TOP MOLECULES

centrifugal stretching is treated as a perturbation on

the eigenstates of the rigid rotor. If Hr represents the

Hamiltonian of the rigid rotor anrf w g a rotor and Hd represents that of the

distortional energy, the total rotational Hanitoni, tan is

H * H + H . r d (CI)

Reference 10 has shown that the centrifugal distortional

Hamiltonian has the form

4

d T y., Sy.o*1 PS PY p5 (C 2) H, = Jl Z

"here represent the principal coordinate

axes of the moments of inertia and where each must

be summed over all three coordinate axes.

The first order perturbation treatment HJ has the form

H 1 = 1 V ' 2 , d 4 Taot08 P

a p g (C 3 )

84-

85

^ — Ar ' 4 - — r T p .

X X X X X

> 2 D 2 ^ ^ 2

" T [ T X X X X P V + T ' P 4 + T ' P 4 " J. ' xxxx X Y Y Y Y Y ^ Z Z 2 ZP Z + T X X Y Y

(PxPy + V x ' +Txx2z<PxPz + P?P?) z X

+ T' (P^P2 + p2p2\i yyzz v v z + P ^ P J J yyzz y z + P

ZPY>

Where x,y,z represent the principal axes and z

represents the symmetry axis of the top.

The definition of the coefficients in the distortional

Hani.iltonian is given by

T' = h V Tf h

vv w J"1 L # T xxxx Txxxx ' T X X Z Z ~ H ( T X X Z Z + 2 T ) , t * xxzz xzxz' - H T . R '

yyyy ^Tyyyy ' Txxyy (Txxyy + 2xxyxy) (c 4)

T z z z z " h T z z z z ' Tyy« = h ( Tyy" + 2Vzyz>

For a symmetric top molecule

r zzzz

Txxxx Tyyyy T;

W = Vyzz T x x y y = X x x x x _ 2 x ^ (c 5 )

xzxz ~ T y z y z

A general definition of the distortion coefficients has been

designated in the literature as

h4

D =- — x "J 4 xxxx

h4 ~2D — — (x +2T )

K J 2 x x z z xzxz (C 6)

h4 D = -D-D -il T

^ 4 zzzz

86

The first order energy correction in the perturbation due

to the centrifugal distortion is

Ed " < J- K iHfll J,K> { c ? )

By manipulating and rearranging the coefficients and using

the definitions for 0 j, D j r, in the Eqs.fc 3),(c 4).(c

)<{C 6) and (c 7), the resulting expression has the form

E d = " H [ D J 2 ^ J + 1 ) 2 + D J(J+1)K2+D K4] (C 8) a j J K ^ J o;

Addition of ,1 to the rigid rotor values of Eq.(B10) gives

the usual expression for the rotational energy of the

non-rigid prolate symmetric top molecule with centrifugal

distortion in the following form.

e\j,K)= h[BJ(j+l) + (a_b)K2-d J2(j+i)2 J

-D J(J+1)R2-D K«] ( R Q\ JK K (C 9)

APPENDIX D

ROTATIONALJNRAGY OF SYMMETRIC TOP MOLECULES N THE EXCITED VIBRATIONAL STATE

The first attempts to identify the splitting caused by

the Coriolis interaction in the rotational spectra was by

Teller , and Johnston and Dennison34.

They approached the problem along the following lines.

" *x.Py. and P2represent the over-all angular momentum

about principal axis including that caused by pure rotation

of the molecule and that caused by vibration, the pure

rotational Hamiltonian can be expressed as

« = l f £ f ^ 2 +

( V V 2 + < y p 2 >

2

2 I* (D 1)

where Px'Py.P2 represent the components of the angular

momentum which arise from the degenerate vibrational

motions.

For a prolate symmetric top in a doubly degenerate

vibrational mod. with the symmetry axis taken as the z axis

1 = 1 = 1 , 1 = 1 x y b z a ' (D 2)

and

P = P = 0 . x Y (D 3)

87

88

Eq.fD 1) can be rewritten by using the E<js.(D 2) and (D 3)

as follows

p2 p2 (p -p )2 H = Z +. Z 2

2ib 2zb — «

= + 4 CT-4> p2 b 2 xa zb 2 ! a

m the last expression the tern, pz/2^ was neglected, which

represents pure vibrational energy and which does not change

with the rotational state.

By making some substitutions

P 2 = £2 J(J+1)

P = K and z

P„ = ? Z fi z

The final form of the rotational energy of a symmetric top

molecule in the doubly degenerate vibrational state can be

expressed as

E (J.K. I), h [E J(J+1) + ( J B , p . 2 f l K A ] ( D 5 )

V V v '

Where : rotational constants in the excited

vibrational state

? • Coriolis coupling constant.

APPENDIX E

ACCIDENTAL DEGENERACY

Quantum number sets K*~2 cause accidental degeneracies

which make the energy difference in of Eq.(3-ll) zero.

TO remove this undesired case, a direct diagonalization

method was employed for thes£> j. jt xwt tnese special quantum sets. This

procedure is given schematically as follows.

Hd^d E^d d : diagonal.

H jp= \rp

|H -Alj = 0

To solve this secular equation, Table III was used,

V'J'K'£ V,J,K+2,£-4 V,J,K-2,J£+4

v,J,K i

v,J,K+2,1-4

VfJ,K-2,£+4

E"A f24a2b4

f24a2b4

f24a2b4

f24a2b4

= 0

E-A

NO interactions between (K«,i-4) and (K-2,t+4) were

assumed.

89

90

(E-A) [(E-A)2-(f24a+b;)2]-(E-X) (f24a"b+)

2= 0

A = E

* - E±f24[(a2+b4-)

2 + (a-bj)2]3*

From the Table III

4 = [ J ( J + 1 ) - K ( K ± 1 ) ] ^ [ J ( J + l ) - { K + l ) ( K ± 2 ) ] ^

t>4=[(v+I+l±l) (v-ulfl) (v+il+1+3) (v-Jl+153) ]

A becomes

A = E ± f24{ [ ( J ( J + 1)"K(K+l)] [J(J+1)-(K+1) ( K + 2 ) J

(v+A)(v-£+2)(v+i-2) (V-£+4)+[(J(J+I ). K ( K_ 1 ) ]

[ (J+l) J- (K-l) (K-2) ] (v+£+2) (v-£) (v+A+4) (v-£-2)j

if Kl= -2 then K=l,*=-2 and K=-l,£=2.(see table IV)

(D 6) becomes

A = E ± f24 J ( J + 1) v(v+2)

Applying the selection rules for absorption spectra,

v ~ ± 2 ^24J J+^-) v (v+2) if K£=-2 .

APPENDIX F

PROGRAM TO ASSIGN v,„ - n ROTATIONAL CONSTANTS

The computer program developed for calculations in this

work is recorded here for convenience.

This program can be used to assign any rotational

constants in a n y vibrational states of propyne by putting

the proper v and the observed frequency values. This

program is an example to determine q 1 2, d 1 2 and x ^ for the

v,„= 4 rotational components of propyne. The machine

language of this program is FORTRAN IV.

-PROGRAM-

S N M ; , z-fYF ^ '

S S S S S S S S K P ? A £ ? W K j S 3 ' 3 4 ' S « ' " . S8. S5

DIMENSION VAL(45), EXPVAL(26) DATA EXPVAL/

v=4 .0 TOL=0.2 NUM=15

91

92

D l = •' 1 6 3 4 2 3 D 0 + V * 4 ? 4 6 2 D - 4 ^ 8 5 5 D ° * V * * 2 ~ " 0 0 2 1 8 3 3 3 D 0 * V * *3 S N l = . 3 3 3 5 D 0 + V * 9 . 1 0 5 8 D - 4 SN2=0.0 SN3=0.0

D2=. 00292D0+V*9 . 5 9 9 9 l D - 5 + V * * 2 * 2 00048D e, Q = 4 . 2 0 1 8 D 0 - 8 . 0 0 0 1 8 D - 3 * V ' 0 0 0 4 8 D 6

A = 1 5 8 8 5 3 . D O - 2 5 5 7 . 5 2 D 0 * V + 5 1 * v * * 2 P S O ; J 7 8 6 6 5 D 0 * V + • 111914D0»V« « 2 - 3 . 1 3 3 2 2 D - 2 « V " 3 + 3 . 083

F 2 = 0 . 0 F 3 = . 0 0 7

UPTTTTId 11\

FORMAT(IX, - B I - , 2 5 * . ' D 3 ' , 2 5 X , • * ! • , 3 5 X , 'FREQ' ,

DO 900 I Q = 1 , 5 0 , 1 Q l - I Q

901 Q l — Q l

DO 800 I D = 1 , 2 0 0 , 1 D3=ID

DO 700 I X = 1 4 0 0 0 0 , 1 7 5 0 0 0 , 1 0 0 X1= IX

IND=0

DO 600 I L = 1 , 5 , 2 L = I L - 1

601 L = - L

DO 500 I J = 1 , 3 J = I J - 1

DO 400 I K = 1 , I j K = I K - 1

401 K=-K

F 5 = 4 * ( J + l ) * ( Q + 2 * F 2 * ( J + l ) * * 2 ) — ( ^ + 1 ) * (B1 + G * L * * 2 - D 1 * K * * 2 + SN1 * tf * r , e M 5 icvic * «

S l = S l M * D 2 M J + l ) - 3 + 4 * S N 2 * K * L * u i l ) - 3 S N 3 K 3 L )

93

D6-4*X1* (L-l) - (A-Bl) * (2*K+1) -2*A*Z * (7*K m i 07=4*X1*(L+1) + (A-B1)*(2«K-1 -2*A*Z*\-£K+TJ>\

•* ,(f^' + JA"B1 > * <K+l >-A«Z* (K^L+2) * ^ ( *"1 ) + (A-Bl) * (K-l) -A*Z* (K+L-2)

C2=(V+L)*(V-L+2)*(J+l) C3=(V-L)*(V+L+2)*(J+l)

S3=C3*Q**2*((J+1)**2-(K+1)**2)/D8

IF ((D8.EQ.0.0) .AND. (V.EQ.1.0) ) S^--T?*

IF ((D8.EQ.0.0) .AND. (V.EQ.3.0) ) ffl-PpS

S4=C2*Q**2*((J+1)**2-(K-1)**2)/D9

IF ( (D9.EQ.0.0) .AND. (V.EQ.1.0) ) S4=F5 IF ((D9.EQ.0.0).AND. (V.EQ.3.0) ) S4=2*F5 S6=2*C2*(Q1*(2*K+1)+D3*(L-1))**2/D6

( 2* K _ 1 ) + D 3*

S8=S6-S7

IND=IND+1 VAL(IND)=S1+S5+S8

IF (K.LT.O) GOTO 401 400 CONTINUE 500 CONTINUE

IF (L.LT.O) GOTO 601 600 CONTINUE

NFREQ=0 DO 300 11=1,45 DO 200 JJ=1,26 IF ((VAL(II)-TOL).GT.EXPVAL(JJ)) GOTO 200 */((VAL(II)+TOL).LT. EXPVAL (J J) ) GOTO 300 NFREQ=NFREQ+1 w i u JUO GOTO 300

200 CONTINUE 300 CONTINUE

94

1 W-W.Xl.NFREQ

700 CONTINUE

800 CONTINUE

I F (Ql'LT.0.0) GOTO 901 900 CONTINUE

STOP END

APPENDIX G

PROGRAM TO CALCULATE Vt- n ROTATIONAL

TRANSITION FREQUENCIES

This program can be used to calculate rotational transition

frequencies for any excited vibrational states of any c

molecule for which rotational constants have been derived!'

This program is an example for calculation of rotational spectra of

molecule in any excited vibrational states by changing v and L.

C COMPREHENSIVE MODEL FOR C

sJSf' G = S N 2 : £ 1 = D £ & - ^ = D ' Q ^ v ' Q 1 = q i 2 ' D 3 = di 2 ' 'SN2-i|jj,SN3=*|JirFl=fl2,F2=f22,F3=f*!

C c

REAL L REAL J REAL K

V=2 L=0

DOTBLE P R e S k n XI' F1DF2SP1' ? n 2 ; f 3 -02' 0' 21 - 03. A, Z DOUBLE S:«:S2!li!;6 T^ T?g ,S9 DS3 sS' s l

DOUBLE PRECISION S5,F9 ' ' ' ' '

COEFFICIENTS

i 4 5: 7 1 2 D 0 + 2 4 • 2046D0*V- . 11687D0*V**2 Bl+•0372304D0*V**3-.0101258D0*V**4

£=:163 9?23D0 0"M D46X:4 0 2 8 5 5 D O* V" 2-' 0 0 2 1 8 3 3 3 D O* V**3

3&0?L%2D0:^99.l0598D?45+V**2*2-0OC48D-6

SN2=0.0 SN3=0.0

95

96

C

Q-4.2018D0-8.00018D-3*V

' 9062D0*V+24 • 484D0*V**2-2 603R "?nn*v* * i D3=33.6602D0+50.1149D0*V-5.44498D0*V**2 A=158853•DO-2557.52D0*V+51*v**2 Z~ 1~ , 1 7 8 6 6 5 D 0* V +-111914D0 *V**?- '? „ ^ 5 « 6 4 - E 0 - 6 5 « - 1 2 D ° * V + l 5 4 0 0 2 M ' V » ? V 3 + 3' 0 8 32 d-3'V F1=0.0 F2=0.0 F3=0.0

DO 750 JJ=1,4

JK=2*J+1

DO 740 KK=1,JK K=KK-(J+l)

C DIAGONAL TERMS ^ * ( J + l ) * ( B 1 + G * L * * 2 - D 1 * K * * 2 + S N 1 * K * T J - c m i * V * * o * t \

T1=~4*D2*(J+l)**3+4*sn2*K*L*(J+ll**3 D9=T0+T1

C NON-DIAGONAL TERMS 1ST TERM

X 1 (l+1)+(a-B1)*(K+1)-A*Z*(K+L+2)

SMli-if/iii1 ' A N D' K' E Q'" 1' 0 0 1 0 350

s L ™ i E 9 ' 3 - A N D ' L - E S ' 1 • A N C - K • E Q • 1 ) G O T O 3 9 0 S1=SM1

G O T O 4 1 0

350 S1=-2*Q*(J+1)*(V+1) G O T O 4 1 0

390 Sl=.762 *(J+l)+SM1

2 N D T E R M

4 1 0 ?c:iy^T)*1

(T"^+2)*(j+1)*((J+l)**2-(K-l)**2)*Q**2

(L~l)+(A-Bl)*(K-l)-A*Z*((K+L-2)

IF (L.EQ.l .AND. K.EQ.l) G O T O 480

97

SM2=T4/T5

(V. EQ . 3 .AND. L.EQ. -1 AND Tf Fn i N r- ^ S2=SM2 -awu. K.EQ.-l) GOTO 520 S2=SM2

GOTO 540

480 S2=2*Q*(J+l)*(V+l) GOTO 540

520 S2=.762*(J+1)+SM2

c 3RD TERM

I ^ « / ^ < 1 " 1 ) " < A " B 1 ) ' * < 2 * k + 1 ) - 2 * A * Z M 2 « K - £ + 2 ) " S3=T6/T7

4TH TERM ^ (V-L) * (V+L+2 ) * (J+l) * (Ql* (2*K-1) +r> 1* r 4.1 \

' -(A-B1»* ' 2

5TH TERM

TT? Ir'fH'2 *AND* K-EQ-"1) GOTO 660 n( L - E Q - 2 *AND* K.EQ.l) GOTO 680

oO = U

GOTO 730 660 S5=2*F3*V*(V+2)*(J+l)

GOTO 730 680 S5=-2*F3*V*(V+2)*(J+l)

C FREQUENCY(F9) • 730 F9=D9+S1+S2+S3+S4+S5

J8=J+1 J7=J K7=K L7=L

732 WRITE (6,735)K7,L7,J7,J8,F9 735 FORMAT ( ' K=' 12 ' r - • TO > T • -r-, .

V K. , 12, L- ,12, J= ,12,' TO J= ' , 12, ' F=' ,F12 . 5

740 CONTINUE

WRITE(6,745)

98

745 FORMAT('0' 750 CONTINUE

STOP END

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