MEASUREMENT OF HYPERFINE STRUCTURE INDEUTERATED ACETYLENES VIA MOLECULAR-

BEAM MICROWAVE SPECTROSCOPY.

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Authors TACK, LESLIE MARTIN.

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University Micr6films

International 300 N. Zeeb Road Ann Arbor, MI48106

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Tack, Leslie Martin

MEASUREMENT OF HYPERFINE STRUCfURE IN DEUTERA TED ACETYLENES VIA MOLECULAR-BEAM MICROWAVE SPECTROSCOPY

The University of Arizona

University Microfilms

International 300N. Zeeb Road, AnnArbor,MI48106

PH.D. 1982

MEASUREMENT OF HYPERFINE STRUCTURE IN

DEUTERATED ACETYLENES VIA MOLECULAR-BEAM

MICROWAVE SPECTROSCOPY

by

Leslie Martin Tack

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF CHEMISTRY

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

198 2

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have read

by Leslie Martin Tack the dissertation prepared

entitled Measurement of Hyperfine Structure in Deuterated Acetylenes

via Molecular-Beam Microwave Spectroscopy

and recommend that it be accepted as fulfilling the dissertation requirement

D f Doctor of Philosophy for the egree 0

( 2.--J>-8'2--Date

1'2---~-~ '--Date

/) - >J--f' 2-Date

, J2.-Y-!C Date

1::Z/fI/e2 Date •

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

Date

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

u?~, -ItA ""I SIGNED: __ Jph ____ ,_.,ICK-_______ _

ACKNOWLEDGMENT

I would first like to thank my research director, Professor

S.G. Kukolich, for his valuable assistance and guidance in the completion

of the projects contained in this work. I have met few scientists with

his professional competence and I will always appreciate the opportunity

I had under his direction at Arizona.

In addition, I would like to acknowledge the rest of my disserta­

tion committee: Professor Mike Barfield for his valuable discussions on

theoretical matters; Professors Robert Feltham and Robert Bates for their

careful editing of this dissertation; Professors Bill Hetherington and

Peter Bernath for complementing my dissertation committee on short

notice.

Many graduate students and research groups have helped me through

their generous loans of equipment and personal time. They are too

numerous to list here and I hope this will not understate my appreciation

and regard for each individual.

I would finally like to thank Jeannette Gerl for typing this

dissertation in the short amount of time alloted her.

LIST OF ILLUSTRATIONS.

LIST OF TABLES

ABSTRACT . . .

1. INTRODUCTION

TABLE OF CONTENTS

2. THEORY OF HYPERFINE STRUCTURE FOR ROTATING MOLECULES

Introduction ............. . Interaction of a Nucleus with Molecular

Electrostatic Interaction .... . Magnetic Interactions ..... .

Matrix Elements and Coupling Schemes .. Symmetry Considerations

3. EXPERIMENTAL DETAILS .......... .

Introduction. . . . . . . . . . . . . Stark-Modulated Microwave Spectrometer. Molecular-Beam Maser Spectometer ... Measurements on Individual Molecules.

Chloroacetylene-D .. Propyne- 03 . . . Cyanoacetylene-D

4. ANALYSIS OF DATA.

Introduction. . . . . . . . . .

Fields.

Data Analysis for Individual Molecules .. Chloroacetylene-D .. Cyanoacetylene-D .. Propyne-03.. ... .

5. DUSCUSSION OF RESULTS· ..

Introduction. . • . . . Indivudual Molecules.

Page

· vi

· vi i

.vi;;

· 1

.4

.4 · .. 5

. . . . . .5 .8 .10 .12

· .14

. . .. .14 .14 .16 .19 .19

. . . .21 · .25

.26

· . .26 .27 .27

. . . ·27 ·30 .31

· . .31

Chloroacetylene-D .. Cyanoacetylene-D . Propyne-D3 . . . . . . . . . . . . . .. ....

· .31 .31 .34 .36

Deuterium Quadrupole Coupling in Molecules. · .37

;v

TABLE OF CONTENTS--Continued

APPENDIX: PROGRAM FOR CALCULATION OF HYPERFINE STRUCTURE ..

REFERENCES. • . • . . . • • . . . . . . . . . . • . . . . . .

v

Page

. ·44

·53

vi

LIST OF ILLUSTRATIONS

Figure Page

1. Electrostatic interactions with nucleus in atoms

and molecules ....... . . 5

2. Block diagram of Stark-modulated spectrometer .15

3. Block diagram of beam-maser spectrometer. . . .17

4. Hyperfine components of J = 1 + 0 transition, C1CCD . .22

5. Derivative maser spectra of the hyperfine components of the J = 1 + 0 transitions, CD3CCH . . . . . 23

6. Hyperfine components of the J = 1 + 0 transition, NCCCD . 24

Table

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

LI ST OF TABLES

Measured and calculated frequencies for components of J = 1 + 0 transition of C1CCD ..

Molecular constants for C1CCD ........... .

Measured and calculated frequencies for components of J = 1 + 0 transition of NCCCD ..

Molecular constants for NCCCD.

Measured frequencies for J = 1 + 0 transition in CD3CCH.

Molecular coupling parameters and rotational constant for CD3CCH. . . . . . . . . . . • . • . . . . . .

Comparison of gas and liquid phase hyperfine components in NCCCD ................. .

Chemical shift tensor elements of 14N in NCCCD .. .

Comparison of calculated and experimental deuterium quadrupole coupling strengths for some selected molecules ................ .

Quadrupole coupling strengths along the C-D bond

Page

28

28

29

29

30

30

36

36

. 40

40

11. Quadrupole coupling constants and bond distances for the C-D moiety in several deuterated acetyl enes. . . . . . . . . . . . . . . . . . . . . . . 41

vii

ABSTRACT

This work describes the measurement of hyperfine structure in a

series of deuterated acetylenes via molecular-beam microwave spectroscopy.

Measurements of spin-rotation constants were used to calculate the

paramagnetic contribution to the chemical shielding of the concerned

nucleus. Where possible, comparisons with NMR measurements were made.

Measurements of the deuterium quadrupole coupling determined in this work

are compared with previous measurements on the same or similar systems.

A review of the theoretical work done in this area is presented as well as

a discussion of trends observed from high precision measurements of

deuterium quadrupole coupling. A computer program that calculates hyper­

fine structure for up to four coupled nuclei of arbitrary spin is

presented.

viii

CHAPTER 1

INTRODUCTION

Beam - MASER(microwave amplification by stimulated emission of

radiation) instruments have yielded some of the sharpest and most stable

resonances to be found in spectroscopy. Ultimately, a MASER device will

become the universal time standard with stability of the order one part

in 1014 .

For the molecular spectroscopist the MASER instrument is an

extremely accurate measuring device for both strong and weak interactions

in molecules. Resolution requirements are dictated by what interactions

are being investigated. Rotational energies can be routinely measured by

conventional microwave or even infrared techniques. The literature is

full of measurements of rotational energies made with good Stark-modulated

microwave spectrometers where the resolution was about one part in 104 or

even 105. The observation of hyperfine structure, which will perturb the

rotational transitions, may require higher resolution for full elucidation

of the effect. This is particularly true for hyperfine structure in

deuterated species. Resolution of the order on part in 107 is usually

needed; fortunately, the beam-maser instrument meets this requirement.

Hyperffne structure in rotational transitions arises when one

considers the finite extension of the nucleus. There are electric and

magnetic interactions that split the rotational line; electric interac­

tions dominate in diamagnetic molecules. Analysis of the data taken on

2

the beam-maser spectrometer will yield interaction strengths for electric

quadrupole, spin-rotation and spin-spin interactions. The deuterium quad­

rupole interaction is of special interest because the electric field

gradient at the deuterium nucleus arises primarily from external charges

in the molecule; thus, there is a probe for estimating the charge distri­

bution in the molecule. Electronic wave functions can be tested for their

power to preduct the quadrupole coupling at the deuterium site.

The spin-rotation and spin-spin interactions are magnetic effects.

The former results from the interaction of the magnetic moment of the

nucleus with the magnetic field produced by the rotation of the molecule. 1 Ramsey was first to point out the intimate relationship between the spin-

rotation constant and the paramagnetic contribution to the chemical

shielding (which is represented by a second-rank tensor). Ordinarily,

from a purely theoretical point of view, this would be a difficult contri­

bution to calculate; however, spin-rotation constants can be fairly

accurately measured (to within 10%) from the data taken from a beam

experiment. Where calculations are available for determining the diamag­

netic contribtuion (a much more tractable problem since it only depends on

knowledge of ground state wave functions) experimental and calculated

results can be combined to obtain the total chemical shielding. If NMR

data are available (where the total chemical shielding is measured) there

is a consistency check to use.

Spin-spin interactions are usually calculated from the molecular

geometry and nuclear moments to aid in the data analysis process.

3

Each of these interactions is discussed in more detail in

Chapter II. A computer program was written to calculate hyperfine struc­

ture for up to four coupled nuclei of arbitrary spin. A listing of this

program is provided in the Appendix. Chapter III gives a complete descrip­

tion of the microwave instrumentation used to make the measurements.

Analysis of the data along with the derived spectrscopic constants is

presented in Chapter IV; wherever possible these results are compared

with previous works- (usually involving less precision) in the same or

similar systems. In Chapter V is presented a discussion of the interpre­

tation of the measurements. Every attempt will be made to maximize the

information that can be obtained from each measurement. A review of the

theoretical work done in conjunction with these measurements will be

presented; problems in comparing theoretical and experimental results will

be discussed.

CHAPTER 2

THEORY OF HYPERFINE STRUCTURE FOR ROTATING MOLECULES

A. Introduction

The Hamiltonian used to analyze the data taken in this work is

as follows:

(1)

where H t is the rotational energy of the molecule which includes effects ro . due to centrifugal distortion and rotation-vibration interactions. These

effects are discussed in standard texts. 2,3

For the systems studied in this work, the rotational energy will

be much greater than the hyperfine interactions (approximately a factor

of 103). The hyperfine structure can then be viewed as a perturbation

of the rotational energy levels and the analysis will always include a

IIline center frequencyll which corresponds to the line we would see in the

absence of hyperfine sturcture.

The hyperfine Hamiltonian will include electric quadrupole, spin-

rotation, and spin-spin interactions. Each interaction will be discussed

and a scheme for calculating hyperfine structure for all interactions

will be presented.

4

5

B. Interaction of a Nucleus with Molecular Fields

1. Electrostatic Interaction.

The general electrostatic interaction between a charged nucleus of

finite size and the electrons and nuclei in the rest of the molecule is

given by:

He1 . = f f Pe(r:)pn~~)dTedTn (2)

Te Tn ~

where P e(re) is the charge density of the electrons and nuclei in the ... volume element dT at the position r relative to the center of the rele-e e -vant nucleus, P n(rn) is the nuclear charge density of the nucleus con-

.> cerned in the volume element dTn at the position rn relative to the cen-

ter of the nucleus, and r is the magnitude of the radius vector joining

dT and dTn as shown in Fig.1. p (~) is negative for electrons and posi-e e e tive for positive charges.

If electronic charges more distant than the radius R of the

nucleus are considered, l/r may be expressed, using the cosine law, as

follows: r 1=_1 +_n_p

r r 2 1 e re

where Po is the Legendre polynomial of cose ,so that, ]V en

Pl = cose en

1 2 P = - (3cosa -1) 2 2 en

(3)

(4a)

(4b)

The first term in the expansion corresponds to an electric monopole, the

second to the electric dipole, and the third to the electric quadrupole

moment.

.",.--......... ... , '\ ,,,.,,.,'

I

dr" i P" R

\ \ , ,

I I \ I \ I \ , , / " / ' ......... _ ---"filii'

Fig. 1. Electrostatic interactions with nucleus in atoms and

molecules.

6

7

The monopol e term wi 11 1 ead to no hyperfi ne structure and is there­

fore of no interest. The dipole term can be shown to vanish from parity

considerations; thus, the only important term is the quadrupole moment.

Higher moments are generally too small to be considered. 4

Ramsey has shown the electrostatic quadrupole interaction can be

written as:

2 _ e qJQ ~">2 3~-> 22

HQ - 21(21-1)J(2J-1) [3(I·J) +"2 I·J -I J ] (5 )

where Q is the arbitrary nuclear constant and is called the magnitude of

the electric quadrupole moment and is given by:

r ~ 2 2 eQ = J P (r) (3z - r ) dT (6)

n n m =1 n n n I

The integral is evaluated for m1=1 as indicated. eqJ is defined as the

electric field gradient coupling constant and is given by:

r ~ 2 1 eqJ = ~P \ r. ) (3cos e - 1)3 'dT e e =J ez r e

r > mJ e e

(7)

-.I-

where e is the angle between r and the z axis. As written above there ez e is an implicit J dependence on qJ' The quadrupole interaction can be

rewritten with explicit J dependence by utilizing the following relation­

ship:

(8)

8

where Ve is the potential from all charges external to the nucleus and Zo

is along the symmetry axis of the molecule. Substituting (8) into (5)

gives: 2 -eQ a V 2

_ . a Zo -l ~ 3 ..£. .,.1. 2 2 HQ - 2I(2I-l)(2J+3)(2J-l) {3(I'J) + 2 I·J - I J } (9)

The above expression is only applicable to linear molecules. Thaddeus,

Krisher, and Loubser5 give the quadrupole interaction in asymmetric rotors

for the Kth coupling nucleus as:

_ eqJQ ..t.. 2 3 ~ ..> HQ - 21 (2 -1)J(2J-l)x{3(IK' J ) + 2(I K'J)-I K(I K+l)J(J+l)} (10)

K K

where, for an asymmetric rotor:

. \' 2 qJ = 2L <J > q /J(J+l) g g gg (11 )

qgg is the electric field gradient tensor along the gth principal iner-

tial axis and <Jg2> are the average values

nents of J along the gth principal axis.

2. Magnetic Interactions

of the square of the compo-

There are two types of magnetic interactions. The first corres­

ponds to the interaction of the magnetic moment of the nucleus of interest

with the magnetic field produced by the rotation of the molecule and is

referred to as the spin-rotation interaction. The second corresponds to

the magnetic interaction of the two moments and will be referred to as the

, spin-spin interaction.

The spin-rotation interaction for nucleus K is given by Thaddeus,

Krisher, and Loubser5 as:

9

-J. ..> HSR = C(K)IK·J (12a)

C(K) = \ <J 2> N /J(J+l) L. 9 gg 9

where (12b)

N are the diagonal elements of the spin-rotation tensor in the prin­gg cipal axis system.

When there are two nuclei present with non-zero spin there may be

a measurable interaction between the moments. Classically, the dipole­

dipole interaction is given as: .

(13 )

where it is the vector connecting the two dipoles. Thaddeus, Krisher,

and Loubser5 write the spin-spin interaction between nuclei K and L as

the contraction of two symmetric dyadics:

HSS = S:R (14)

where

(15a)

(15b)

average R over symmetric wave function we obtain:

dJ 3 ~ ~>tr < R> = J(2J-l) I'2 JJ + (JJ) - J(J+l) 1] (16a)

where dJ = 2\ <J 2> R /(J+l)(2J+3) (16b) l. g gg g

Substituting (17) into (14) the tensor contraction can be written in

terms of scalar product operators

10

For symmetric tops dJ will simplify to:

_ -h2J 3K2 - ~in2~ dJ - (2J+3) {l - J(J+l) H--ir3i<---} (18 )

where ~ is the angle between the vector connecting the two nuclei and the

symmetry axis of the molecule. Thus, the spin-spin interaction strength

can be calculated from knowledge of the nuclear moments and the molecular

structure.

C. Matrix Elements and Coupling Schemes

As can be seen from examination of the hyperfine Hamiltonian, the

Hamiltonian matrix is determined by evaluation of matrix elements for the J,. -'- J.-1-

scalar product operators IK'J and I(Jl in an appropriate coupling scheme.

The molecules in this work represent the determination of hyper-

fine structure for several general cases. Chloroacetylene-D and

cyanoacetylene-D represents the case of two nuclei with great by unequal

couplings. Two papers have applied the method of irreducible tensor

operators for the most general case of N coupling nuclei: Thaddeus,

Krisher, and loubser in 1964; Cook and Delucia6 in 1971. The basic

differences between the two papers are that Cook and Delucia included

terms in the Hamiltonian matrix off diagonal in J; however, they omitted

the spin-spin interaction in the hyperfine Hamiltonian.

A program for calculating hyperfine structure up to four coupling

nuclei of arbitrary spin was written based on these two papers. The

coupling scheme for the general case of N coupling nuclei is as follows:

J + 11 = Fl

Fl + 12 = F2

Basis functions in this coupling scheme are as follows:

-I. ~ .J..":>

11

(19)

(20)

The matrix elements for the scalar product operators IL"J and IL"IK are

diagonal in the total angular momentum quantum number F" Matrix elements ....L. .-I.

for IL"J are given by: oJ. ~

<F'l """ FL_1JIL"JI Fl """ FL_l >

= (-1 )r[J(J+l) (2J+l )I L ((IL+l )(2I L+l)] 1/2

x fF L \ F L - l}L - 1 fF i -1 F i .~ (2Fi+1 )(2Fi+1) 1/2 1 FL_1I L 1-1 Fi Fi _l L-2

where r = (L-l) :i[l (Fi_l + Ii + Fi ) + (FL_l + IL + FL) ..l. ..lo.

Matrix elements for IL"IK for K<L are given by:

> >-<Fl """ FL_1IIL"IKIFK """ FL_l >

x

(21 a)

(21 b)

(22)

12

L-2 x n [(2F~+1)(2F.+l)]1/2

i=K 1 1

o. Symmetry Considerations

When dealing with molecules where there are equivalent coupling

nuclei, such as propyne-~, it is advantageous to make use of the

symmetry restrictions imposed on the spin functions to reduce the energy

matrix. This problem has been discussed in the literature over the last

thirty years?,S,g The program used to analyze all molecules in this work

did not impose initial symmetry restrictions on the spin functions. This

results in increased CPU time (due to the spin-spin interaction) but main­

tains the program's useful generality. In any event, the symmetry argu­

ments that are invoked to reduce the energy matrix will be outlined. More

detailed explanations are avaiable in the literature. 10

The deuteron has a spin of one, requiring the total wave function

to be symmetric with exchange of the two deuterons. Propane-03 has C3V symmetry and the total wave functions must transform as the irreducible

representations of the point group. The total wave function is written

as an inversion, rotational, and spin function:

(23)

(~rWv) will transform as the irreducible representation of C3V. For

K=O, (wrwv) will be either Al or A2 to give a product Al . The symmetry

of the spin function is determined by the total spin, IT = 101 + 102 + 103 .

The total deuterium spin and its associated symmetry are tabulated below

13

IT Symmetry

3 Al

2 E 1 A1 + E D A2

IT = 2 is symmetry forbidden because the cross product does not contain

Al . IT= Dis allowed but uninteresting because it does not yield any

hyperfine splitting.

Basis functions are written as IIT,J,F> and the coupling scheme

is IT + J = F. Kukolich and COgle)Ohave worked out the matrix elements

for three equivalent deuterons in CD3Br with the above coupled scheme.

CHAPTER 3

EXPERIMENTAL DETAILS

A. Introduction

Initial measurements were made on a Stark modulated microwave

spectrometer patterned after the instrument developed by Hughes and

Wilson. ll Hyperfine structure due to chlorine and nitrogen was easily

resolved but only partial resolution could ever be achieved due to deuter­

ium quadrupole coupling. Typical resolution was 80-100 kHz on the Stark

modulated spectrometer. Resolution could be increased by a factor of 10

or more by goi ng to the beam experiment where the spl i tti ng due to

deuterium could be fully resolved. Each instrument will be described as

well as the specific experimental conditions for all measurements.

B. Stark Modulated Microwave Spectrometer

A block diagram of the Stark modulated spectrometer is shown in

Fig. 2. The absorption cell consisted of a 4 meter section of C-band wave­

guide hermetically sealed by the compression of O-rings against Mylar

windows. Thus, transition pieces could be connected to the absorption

cell without breaking the vacuum. The system was evacuated by use of a

liquid nitrogen trap in series with an oil diffusion and mechanical pump.

The microwave radiation was generated by reflex klystron tubes.

Frequency stabilization was achieved through a phase-lock loop, using a

stable 10 MHz quartz crystal as reference. The tenth harmonic of the

14

Valve Valve

. . Klystron I Stark Cell

(oetector ~)

L..J \.

Oscillator Stark Lock-in Synchro!lizer Modulator AmplIfier

Crystal Frequency Chart Oscillator Counter Recorder

Fig. 2. Block diagram of Stark-modulated microwave spectrometer.

-I

(J1

16

the crystal was displayed on a frequency counter. A sweep generator

would slowly change the crystal frequency, thereby causing the klystron

frequency to slowly change also.

One of the main features of the instrument is the Stark modula­

tor. 12 It applies a 0-1000 V square wave to the Stark electrode, which

was supported by grooves cut in the cell and insulated from it by split

Teflon tubing. References for lock-in detection also came from the Stark

modulator. The output from the crystal detector is filtered and amplified,

followed by lock-in detection at 10' kHz with a time constant of 3 seconds.

The output of the lock-in amplifier is displayed on a linear chart recor­

der along with frequency markers derived from the counter. Sample pres­

sure would be reduced to obtain the best resolution while maintaining

adequate signal to noise. The pressure was typically 1-3 microns.

c. Molecular-Beam Maser Spectrometer

The beam maser spectrometer was patterned after the one developed

by Gordon, Zeiger and Townes. 13 ,14 In a molecular beam microwave experi­

ment the electric radiation is perpendicular to

the velocity of the molecules; thus, the first order Doppler effect

vanishes. Pressure broadening effects are negligible as the beam chamber

pressure is maintained at 10-7 torr. These effects are inherently present

in a Stark cell experiment yielding a line width of 50 kHz or more.

Resolution of this magnitude is usually about a factor of 10 too low to

resolve deuterium hyperfine structure.

A block diagram of the beam-maser spectrometer is shown in Fig. 3.

The beam source is a stainless steel supersonic nozzle, a 3 inch cylinder

vacuum envelo e

focuser

MULTIPLIER PHASE CHAIN I--~ LOCKED )( 1000 KLYSTRON

TUNEABLE CRYST/~L 03CILLATOR

a DOUBLER

AMPLIFIER. PHASE SENSITIVE DETECTOR a 200 Hz OSCILLATOR

cavity

Fig. 3. Block diagram of the molecular beam maser

spectrometer system.

17

18

with a 0.15 mm hole at its head. Typically, a pressure of 100 Torr was

maintained behind the nozzle, causing the rotational temperature to be

very low (20-40 0 K). This is a favorable condition because hyperfine

structure was measured for the first t~otationa1 transition throughout this

work. After exiting the nozzle, the molecules pass through a section of

quarter-inch copper tubing in thermal contact with a liquid nitrogen trap.

The beam is collimated by a 6.3 mm hole in the baffle separating the

source and beam regions. The beam enters the focusser which consists of

4 stainless steel rods, 1/4" in diameter and 6" long. Alternate rods are

held at potentials of up to 20 kV. to produce a large, radial electric

field gradient. Molecules with positive Stark coeffcients~ usually upper

states, will be drawn to the axis of the focusser. Molecules with nega­

tive Stark coefficients are deflected away. Thus, use of the quadrupole

focusser allows spatial filtering of states which greatly increases the

sensitivity of the instrument.

The beam then enters the microwave cavity which is a cylindrical

section of copper or brass tubing and 19 cm long. For all experiments

TM010 mode cavities were used. For this mode the resonant frequency of

the cavity depends on its inside diameter and not on its length. Sections

of copper or brass tubing could be accurately machined to achieve the

required resonant frequency. Fine tuning of the cavity depended upon its

size. Small cavities could be temperature tuned by heating the cavity

electrically. This method is not practical for large cavities which were

fine tuned by inserting a quarter-inch brass rod (for higher frequencies)

or hylon rod (for lower frequencies) into the cavity.

19

Stimulating radiation was injected into the cavity via a 10 or

20 dB directional coupler. Emission was detected with a superheterodyne

microwave receiver. The stimulating and local oscillators were phase­

locked to a harmonic of the same stable 10 MHz crystal; however, the

intermediate reference frequencies are suitably chosen 50 that the

stimulating and local oscillators are always 29.4 MHz apart.

The stimulating power induces resonant emission. The microwave

radiation coupled out of the cavity is mixed with the local oscillator

in a balanced mixer to produce a beat frequency of 29.4 MHz. To increase

sensitivity the crystal oscillator is frequency modulated, so that after

filtering and amplification, the IF signal is demodulated and brought

through a lock-in detector referenced by the chopper at 225 Hz with a

time constant of 15 seconds.

The tenth harmonic of the stable 10 MHz crystal is displayed on a

frequency counter, referenced to the 60 kHz transmission of WWVB, Boulder,

CO. The output of the lock-in detector is displayed on a linear chart

recorder with automatic frequency markers derived from the counter.

D. Measurements on Individual Molecules

1. Ch10roacety1ene-D

Ch10roacety1ene rotational transitions were first measured by

Westenberg, Goldstein and Wi1son15 who obtained the rotational constant,

chlorine quadrupole coupling constant and dipole moment. Weiss and

Flygare16 have previously measured the deuterium quadrupole coupling

strength in a series of deuterated acetylenes using a Stark modulated

20

microwave spectrometer. In the case of chloroacetylene-D the splitting

due to deuterium was only partially resolved and line shape routines

were used to extract the coupling strength. We have made measurements

on this isotope using a beam maser spectrometer, enabling us to go beyond

the Doppler limited linewidht (50 kHz) and fully resolve the splitting due

to deuterium by obtaining linewidths of 3kHz.

A 20 cm cylindrical TM010 cavity was used. The resonant frequency

of the cavity could be adjusted by inserting a quarter-inch brass rod (for

higher frequencies) or nylon rod (for lower frequencies) into the cavity.

The beam was generated by use of a single hole (0.15 mm) supersonic nozzle.

A presssure of 100 torr was maintained behind the nozzle by cooling the

gas in a chloroform/N2 slush(-63°C). The stimulating signal was frequency

modwated at 200 Hz. Derivative spectra were recorded with lock-in detec­

tion at 200 Hz with a time constant of 10 seconds.

Chloroacetylene was prepared by the method of Bashford, Emelius

and Biscoe. l ? Eighty per cent deuteration was obtained by exchange with

alkaline D20 as confirmed by the IR spectrum.

An initial low resolution measurement was performed using the

Stark modulated spectrometer to check the frequencies reported for the

J=l+O transition. No resonances were observed at or near these frequen­

cies. Using the values for the rotational constant and chlorine quadru-

pole coupling strength reported by Westenberg we calculated the expected

frequencies for the hyperfine components of the first rotational transi­

tion. The calculated frequencies were 2 MHz above the frequencies reported

earlier. Resonances were observed very close to these calculated

21

frequencies. It should be mentioned that the spacings of the measured

resonances were close to those reported by Weiss and Flygare.

Derivative recorder traces of all hyperfine components are shown

in Fig. 4. The components are labeled for the J=l level since to first

order all hyperfine matrix elements vanish for J=O.

2. Propyne-D3

Initial measurements on propyne were made by GordylB, et ale

Previous work involving deuterium quadrupole effects was also reported

earlier,19,20 however, the spectra obtained were not well resolved. We

have remeasured the hyperfine components of this transition with a mole­

cular-beam spectrometer. Three hyperfine components were completely

resolved by obtaining an order of magnitude increase in resolution over

the previous work.

Propyne-D3 spectra were recorded using a single TM010 mode copper

cavity 10 cm long. The resonance of the cavity was coarse adjusted by

inserting a piece of brass shim inthe lengthwise slot of the cavity and

the inside diameter was changed by turning four screws which squeeze the

cavity. Fine tuning was achieved by electrically heating the cavity to

the optimum temperature.

Propyne-D3 was prepared by reacting dimenthylsulfate-D6 with

sodium acetyl ide in DMSO. The product was purified by fractional distilla­

tion on a vacuum line. A pressure of 100 torr was maintained behind the

nozzle source by cooling the gas in a pyridine/N2 slush (-42° C).

A recorder tracing of the strongest components of the J=140

transition for propyne-D3 is shown in Fig. 5.

22

F=5/2

(a)

__ -L __ ~ __ L-~ __ -L __ ~ __ L-~ __ ~ __ ~~~~~_~ +35 +28 +21 +14 +7 0 -7 -14 -21 -28 -35 -42

F=5/2

(b)

______ L-~ ____ ~ ____ _L ____ ~ __ ~ ____ ~~~~~~----~

o +9 +18 +27 +36 +45 -36 -27 -18 -9

(c)

kHz

-8 -4 o +4 +8 +12

Fig. 4. Hyperfine components of the J = 1 -+ 0 transition. eleen. (a) Recorder tracing of Fl = 3/2 components. Frequency relative to 10 358 017 kHz. {b) Fl = 5/2 components. Frequency relative to 10 377 921 kHz. (c) Fl = 1/2 component. Frequency relative to 10 393 892 kHz.

r- I I -- I I -40 -20 0 20 40

Fig. 5. Derivative maser spectra of the hyperfine components of the J = 0 to J = 1 transition, CD3CCH.

kHz N W

24

(a)

+44 +33 +22 +11 O' -u -22 -33 -44 -55

'-2 1'1 i81

-~-- V-----t- (b)

-+48 +40 +32 +24 +16 -+8 0 -a -16 -24 -32 -40 -48

(c)

-9 -6 -3 0 +3 -t6 +9

Fig. 6. Hyperfine components of the J = 1 ~ 0 transition. NCCCD. (a) F = 2 components. Frequencies relative to 8 4431386.0 kHz. (b) F, = 1 components. Frequencies relative to 8 442 092.0 kHz. (t) F, = 0 component. Frequency relative to 8 445 318.0 kHZ.

25

3. Cyanoacety1ene-D

Cyanoacety1ene is a molecule of strong astrophysical interest;

rotational lines of its various isotopes have been observed in inter­

stellar clouds since 1971. 21 The first microwave measurements were made

in 1950 by Westenberg and Wi1son. 22 Tyler and Sheridan23 looked at the

microwave spectrum of an excited vibrational state and determined the

structure. DeZafra 24 accurately measured the rotational constant and

hyperfine structure for the ground vibrational state on a beam-maser

spectrometer. A thorough review of previous measurements is available in

the 1iterature. 25 We have accurately measured the rotational constant and

hyperfine structure of the deuterated species via a molecular-beam micro­

wave,spectrometer, fully resolving the hyperfine structure due to 14N and

D.

A 20 cm cylindrical TM010 mode brass cavity was used. The cavity

was machined until the resonant frequency was slightly higher than the

desired frequency. The cavity could then be tuned down in frequency by

inserting a quarter-inch nylon rod into the cavity. The beam was gener­

ated by use of a single hole (0.15 mm) supersonic nozzle. A pressure of

85 torr was maintained behind the nozzle by cooling the sample in an ice

water bath.

Cyanoacety1ene-D was prepared by the method outlined by Mallinson

and Fayt. 26

Derivative spectra of all hyperfine components are shown in Fig. 5.

CHAPTER 4

ANALYSIS OF DATA

Introduction

A computer program was written to calculate hyperfine structure of

a rotational transition for up to four nuclei of arbitrary spin. The

general coupling scheme is as follows:

11 + J = Fl

12 + Fl + F2

I + F = F 3 3

(24)

Expressions for the matrix elements in this coupled scheme were taken

from the papers of Cook and DeLucia6 and Thaddeus, Krisher, and Loubser. 5

This program was checked against a previous coupled-scheme program and

against the uncoupled basis set calculation (graciously provided by Dr.

Sid Yeung). The case of hyperfine structure in the presence of three

equivalent spins has been previously discussed in detail.

The program input data consisted of the spin of each nucleus and

its associated hyperfine interaction strengths. The effective rotational

constant would be thrown in as a parameter so that we could fit to the

actual experimentnl frequencies. Good trial interaction strengths were

obtained from previous measurements on the same or similar systems. The

actual experimental frequencies were also read in to calculate the

26

27

standard deviation for the fit, defined as foll ows: \Ii - ri

) 1 /2 °fit = 0: obs. calc. (25)

i N-l

where N is the number of lines being fit. The interaction strengths and

rotational constant were varied to fit the experimental frequencies. The

error in each parameter was estimated by varying each one in turn until the

standard deviation changed significantly.

Data Analysis for Individual Molecules

Chloroacetylene-D

The hyperfine Hamiltonian consisted of the deuterium quadrupole

and chlorine quadrupole and spin-rotation interaction strengths. Second

order quadrupole energies were calculated for chlorine from the tables

given by Townes and Schawlow2 and added to the first order energies.

Table 1 lists the measured and calculated frequencies for each component.

The hyperfine constants are given in Table 2 and where possible compared

with those obtained from previous work.

Cyanoacetylene-D

The hyperfine Hamiltonian consisted of the nitrogen quadrupole,

nitrogen spin-rotation, and deuterium quadrupole interactions. The

calculated and observed frequency for each component is given in Table 3

as well as the standard deviation for the fit. Hyperfine interaction

strengths where possible are given in Table 4 and compared to previous

works.

Table 1. Measured and calculated frequencies for components of the J = 1 + 0 transition of C1CCD. Frequencies are given in kHz. The Standard deviation for the fit was Q~ 1 kHz.

Component (F 1 ; F) Measured

(1/2;1/2,3/2) 10 393 892.0

(3/2;1/2) 10 358 050.6

(3/2;3/2) 10 357 975.3

(3/2;5/2) 10 358 017.1

(5/2;3/2) 10 377 912.9

(5/2;5/2) 10 377 975.3

(5/2;7/2) 10 377 931.6

Table 2. Molecular Constants for C1CeD

eqQ(D)

eqQ(Cl)

This Work

208.5 + 1.5 kHz

-79 739.5 + 1 kHz

1.3 + .1 kHz

Calculated

10 393 892.1

10 358 050.6

10 357 975.2

10 358 017.1

10 377 912.9

10 377 975.5

10 377 931.6

Previous Work (Ref.)

225 + kHz

-79 670. + 100 kHz

-79 660 kHz

C (Cl )

B 5 186 973.9 + .1 kHz 5187 010. kHz

aRef . 16

bRef . 15

28

29

Table 3. Measured and calculated frequencies for components of the J=l to J=O transition of CNCCD. Frequencies are given in kHz.

Component (F1;F) Measured

(1;1) 8442051.56

(1 ; 2) 8442084.10

(1 ; 0) 8442130.42

(2; 1 ) 8443343.19

(2;3) 8443366.70

(2;2) 8443413.26

(0; 1) 8445318.05

Table 4. Molecular Constants for CNCCD

This Work

eqQ(D) 203.5 + 1.5 kHz

eqQ(N) -4318.0 + 1 kHz -C(N) 1.1 + 0.2 kHz -C(D) -.1 + 0.4 kHz

B 4221580.1 + 0.1 kHz

aMeasured from data taken on NCCCH (Ref. 23). b Ref. 23.

Calculated

8442051.51

8442084.11

8442130.58

8443343.39

8443366.93

8443413.39

8445318.03

Previous Work

-4318.8 + 1.2 kHz a

1.05 + 0.26 kHza

4221581.67 + 44 kHzb

30

Propyne-03

The hyperfine Hamiltonian consisted of the deuterium quadrupole,

spin-rotation, and mutual spin-spin interactions. The spin-spin inter­

action was calculated from the known nuclear moments and structure and was

not treated as a variable parameter. The measured and calculated frequen­

cies for the observed components are given in Table 5. Hyperfine inter­

action strengths are given in Table 6 and compared with previous work.

Table 5. Measured frequencies for J = 1 ~ 0 transitions in OC3CCH. Frequencies in kHz. F value is the total angular momentum in the J = 1 state.

F

1

2

3

Measured

14 711 522.6

14 711 530.7

14 711 565.9

Calculated

1411522.6

14711530.8

14711565.9

Table 6. Molecular coupling parameters and rotational constant for C03CCH.

Parameter

eqaaQ

eqzzQ

Co

0(0-0)

B

OJ

-55.0 + 0.5 kHz

174.0 + 6.0 kHz

0.06 + 0.08 kHz

-0.25a kHz

7 355 767.0 + 1.0 kHz

2.8b kHz

a, calculated for nuclear moments and structure, b from previous measure.

CHAPTER 5

DISCUSSION OF RESULTS

A. Introduction

Attention will now be turned to the interpretation of the measure­

ments made in this work. The results for each individual molecule will

be discussed followed by a general discussion on trends in measurements

of deuterium quadrupole coupling strengths. A review of the theoretical

work done in this area will be presented. Accurate spin-rotation con-14 35. stants were measured for Nand Cl ln cyanoacetylene-D and

chloroacetylene-D, respectively. A discussion of chemical shielding in

these isotopes will be presented and comparisons with NMR measurements

will be made. B. Individual Molecules

Chloroacetylene-D

By examination of Table II it is noted that the values of eqQ(D)

reported by this work and that of Weiss and Flygare16 agree within the

earlier limits of experimental error (i.e., ~ 18 kHz); however, the

measurement from this work has a much smaller uncertainty (~ 1.5 kHz). The

values of eqQ(Cl) do not agree within the limits of experimental error.

Since the frequency measurements in this work are more accurate by two

orders of magnitude and there is second order quadrupole effects are

included for chlorine in our analysis, there is more confidence in the

present results.

31

32

The analysis also yielded an accurate measurement for the spin­

rotation interaction strength for 35Cl . The spin-rotation interaction

strength for the kth nucleus is defined as:

C(K) = -I <J 2> M /J(J+l) (26) 9 9 gg

M are the spin-rotation tensor diagonal elements. For a linear molecule gg l. <Jg

2> is simply J(J + 1) and c(k) = -M = -M Calculation of these xx yy

tensor elements from spin-rotation measurements enables the estimation of

the paramagnetic and diamagnetic contributions to the total chemical

shielding. The diagonal elements of the shielding tensor for nucleus k

along the gth principal axis are given by:

o(K) = o(d) + (p) gg gg 0 gg (27)

O~g and O~g are, respectively, the diamagnetic and paramagnetic contribu­

tions to the shielding tensor elements and are calculated from the

following expressions: d e2 -3 2 2

o = (-2 ) < OII r . K (r. K - (r. K) ) 10> gg amc 1 1 1 9

p Ogg

e2 hCMgg -3 2 2 = (--2){ 2 egG - LZ, r i K [r £ K - ( r i K) g]}

amc )IN K gg

(28a)

(28b)

)IN is the nuclear magneton, gK is the nuclear 9 value, r£k is the vector

from nucleus k to the other £ nuclei, Ggg is the rotational constant along

th th . . 1 . e 9 prlnclpa aX1S. d 0gg cannot be found directly from experimental data; one must rely

on molecular orbital calculations. In contrast, the paramagnetic contri­

bution can be determined from measured spin-rotation constants and know­

ledge of the molecular geometry. This is very useful since calculation

of the paramagnetic term would require evaluation of matrix elements

33

coupling the ground state to excited states with nonzero orbital angular

momentum. The intimate relationship between the spin-rotation interaction

and chemical shielding was first pointed out by RamSey.l

Since chloroacetylene is a linear molecule, obtaining the compo­

nents of O~g is especially simple. Evaluating the right-hand side of

equation 26b and using our measured value of C(35Cl ) = 1.3 kHz, we obtain

op = op = a = -420 ppm The average value of oP is one-third the trace xx gg .

of the shielding tensor, -280 ppm.

The question now arises, what should be done with this result, or,

is there anything to compare it to? If NMR data were available, the

measurement of the paramagnetic part of the shielding could be combined

with a measured value of the total shielding. Note that the corresponding

components of each cannot be subtracted unless the total shielding aniso­

tropy was actually measured in the NMR experiment. Sometimes calculations

are available that give the anisotropy of the diamagnetic part of the

nuclear shielding. This data can be combined with the measured paramagne­

tic shielding and comparison with NMR measurements can be made. A good

example of this consistency check is the beam-maser work on CH3l3CN.

There are problems with obtaining nuclear shielding constants for

chlorine (as well as the higher halogens). First, you must remember that

the shielding constant of a nucleus in a particular compound cannot be

directly measured in an NMR experiment; only the "chemical shift", which

is the different chemical environment is obtained. Chemical shifts of

chlorine, bromine and iodine have been mostly measured relative to the

corresponding halide ion in aqueous solution. Since the ion shifts are

themselves dependent on the nature of the counter-ion, salt concentration

34

and temperature, they are not ideal references. In addition, 35Cl or

37Cl NMR signals have considerable line widths (several hundred ppm) due

to quadrupolar relaxation, the chemical shift covering a range of about

1000 ppm. Thus, due to experimental difficulties only shielding data for

a few covalent chlorine compounds have been reported. The error bars

associated with these measurements are usually quite large, e.g., a(35Cl ) =

-340 ~ 100 ppm in chloroform. No chemical shift data for chloroacetylene

has been reported. The possibility of ever being able to sort out the

diamagnetic and paramagnetic contributions to the chemical shielding in 35 Cl compounds is not encouraging. The chemical shift in the gas phase

would have to be measured relative to a common reference such as aqueous

NaCl solution. This is a non-trivial problem.

Cyanoacetylene-D

It is noted from Table IV that the deuterium quadrupole coupling

strength along the C-D bond is accurately measured at 203.5 ~ 1.5 kHz.

Included are the hyperfine constants measured on CNCCH by Robert DeZafra24

with the molecular-beam apparatus at Columbia University. The hyperfine 14 constants on N are not expected to be very different for the two iso-

topic species and they are found to be the same within experimental error.

Although DeZafra was able to obtain high resolution in his measurements

(FWHM = 3 kHz) he was unable to resolve the splitting due to the spin­

rotation interaction on hydrogen. For the deuterated species the spin­

rotation interaction only causes a relative shift of the hyperfine lines

since no new quantum numbers are introduced when it is included in the

Hamiltonian. Its contribution os small with a correspondingly high

35

uncertainty. The spin-rotation interaction strength for hydrogen would

be expected to be about a factor of seven greater.

Vold,27 et al., have measured the 13C, D and 14N nuclear magnetic

relaxation of cyanoacetylene-D in toluene solution. Analysis of these

relaxation times gives liquid phase values for 14N and 0 quadrupole

coupling constants. Chemical shift anisotropies for 13C were also

measured. Table VII compares gas phase values with the liquid NMR results.

There is a noticeable difference (4%) between eqQ(14N) in the liquid and

the solid phase. The solvent undoubtedly plays a role in how it affects

cyanoacetylene-cyanoacetylene hydrogen bonding. The difference is less

than that found in solid vs. gas phase measurements (6-14%) of quadrupole

coupling constants.

Analysis of the data also yielded a value for the spin-rotation

interaction strength on 14N: C(14N) = 1.1 + 0.2 kHz. Proceeding in a

similar way for chloroacetylene, cr P (14N) = -593 ppm.

The diamagnetic contribution to the chemical shift can be com­

puted from ab initio calculations or more conveniently by Flygare's atom

dipole method. The former was chosen for its simplicity and the good

agreement it provides with ab initio calculations. The diamagnetic

and paramagnetic tensor elements for 14N in cyanoacetylene are given in

Table VIII. The diamagnetic and paramagnetic tensor elements can be

combined to obtain the components of the total shielding tensor. We find

that crAV = 18 ppm. Unfortunately, chemical shift data for cyanoacetylene

is not yet available (owing to referencing problems); however, the value

obtained is in reasonable agreement with that of 25 ppm given for CH3CH.

Note that this value is relative to the bare 14N nucleus.

36

Table 7. Comparison of gas and liquid phase hyperfine constants in CNCCD.

Parameter

eqQ( 14N)

eqQ(D)

C(N)

aThis work. b Ref. 27.

Gas Phasea

-4318.0 + 0.1 kHz

203.5 + 1.5 kHz

1. 1 + 0.2 kHz

Liquid Phaseb

-4 140 + 50 kHz

200 + 2 kHz

Table 8. Chemical shift tensor elements of 14N in NCCCD (all values in ppm).

Tensor element

xx

yy

zz

Average

Propyne-D3

d CJ

446

446

348

413

-593

-593

o

-395

CJ

-147

-147

348

18

By examination of Table 6 it is noted that the deuterium quadru­

pole coupling along the a-rotational axis is accurately determined at

-55.0 ~ 0.5 kHz; however, the coupling strength along the C-D bond direc­

tion (which is what we are really interested in has a larger uncertainty

(~ 6 kHz). This is so because when the error is propagated into eqQ along

the C-D bond direction the majority of the error comes from the uncertainty

in the molecular structure. This is an inherent problem with CD 3X type

molecules. With the assumption that the C-D bond is cylindrically

symmetric the value along the C-D bond is given as:

eq Q = egQ(C-D) (3cos 2e-l) aa z

where e is the C-C-H angle which has an uncertainty of ~ 0.5°. The

37

(29)

value of eqQ along the C-D bond of 174 kHz is in excellent agreement with

the previous value although the uncertainty is reduced by a factor of

three.

Deuterium Quadrupole Coupling in Molecules

It is useful at this point to make a connection with the theore­

tical work done in this area. High precision measurements of deuterium

quadrupole coupling have come out of this work; to predict the interaction

strengths obtained here would require equal precision in accounting for

the electronic distributions in these molecules. The problem is tractable

from a purely theoretical end for several reasons: (1) the deuteron contri­

butes almost insignificantly to the electric field gradient; (2) there

are no complications due to Sternheimer anti-shielding which is prominent

in heavy atoms.

The electric field gradient at the deuteron is the sum of an

electronic and nuclear contribution: 2 2

3z - rO * eq(D) = +IZn Dn5 'n - e<y I

rOn (30)

Z is the charge of nucleus n in units of e, the index n is the sum over n

the n nuclei, i is the index for the sum over the electrons in the mole-

cule. It should be remembered that what is measured in the lab is the

38

product eqQ and not eq. The measured quadrupole coupling can be converted

to the electric field gradient within the accuracy of the measurement

and the known value of Q for the deuteron. Reid and Vaida 29 have calcu--27 2 lated Q for the deuteron to be 2.860 x 10 cm based on an electric

field gradient computed from an 87 term electronic wave function. Their

analysis included effects for vibrational averaging.

The first term in eq. 28 is easily calculated from the molecular

geometry. Imagine at this point a molecule consisting of only bare nuclei;

now fill in the electron density in the molecule. This is the difficult

part of the problem.

There have been two approaches in the estimation of deuterium quad-30 rupole coupling in molecules. Snyder has computed electric field grad-

ients at molecular deuterons from~ initio wave functions in a double

zeta basis. Barfield,31 et al., have done the same calculations employing

semi-empirical molecular orbital theory (i.e., SCF-MO in the INDO approx­

imation). Both methods of calculation give consistently high values for

the deuterium quadrupole coupling. This state of affairs is probably due

to an underestimation of the electronic contribution. The ab initio

approach seems to work better when dealing with a C-D moiety. Table 9

compares the results between the calculated (by ab initio and semi­

empirical methods) with experimental measurements.

There are several qualitative and quantitative observations to be

made concerning the calculations and measurements: (1) eqQ ii positive for

a deuterium bonded to a first row atom. This is due to incomplete

shielding by bonding electrons of the nucleus to which the deuteron is

bonded. (2) As nuclei further away are considered from the deuteron and

39

its associated electronic distribution the electronic and nuclear

contributions effectively cancel one another. (3) The calculated quadru­

pole couplings are consistenly higher than the measured values (about 20%).

Snyder30 has responded to the third observation by inclusion of

p basis functions on the deuteron and d basis functions on the carbon atom

bonded to the deuteron. This has resulted in the lowering of calculated

values to within 8% of experimental measurements.

Observations (1) and (2) are well documented in Table 9 where

quadrupole coupling strengths along the C-D bond are listed for eight

molecules. The first four molecules involve quadrupole coupling on an

acetylenic deuterium; the final four are CD3X type molecules. All

measurements except for those on FCCD are high precision.

After examination of Table 10 two trends are immediately obvious:

(1) the deuterium quadrupole coupling is not very sensitive to sUbstitution

within a group where the deuterium is bonded to carbon of given hybridi­

zation (sp or sp3) (2) eqQ(sp) > eqQ( sp3) by about 20%.

Let us consider the first trend. We have available in this work

for the first time comparison of quadrupole coupling constants on acety­

lenic deuteriums where the measurements have been precise enough to show

real differences. It is interesting to ponder these fi~ite differences

with the available structural data. Table 11 gives the quadrupole coupling

for several acetylenic deuterons along with the bond distance of the C-D

moiety. Propyne-D l and cyanoacetylene-D have virtually the same bond

distance for the C-D fragment yet the dueterium quadrupole coupling is

significantly different (about 10%). Although qualitative statements must

40

Table 9. Comparison of calculated and experimental deuterium quadrupole coupling strengths for some selected molecules. All values are in kHz. DZ = double zeta basis, SE = semi-empirical, DZ + P = double zeta basis plus polarization wave functions.

Molecule OZ OZ + P SE expo

CH30 226 208 268.7 192a

DCN 248 213 314.9 194b

HOO 378 339 311.7 307.9c

NH20 308 310 304.8 291 d

a Ref. 32. b Ref. 33. c Ref. 34. d Ref. 35.

Table 10. Quadrupole coupling strengths along the c-o bond direction. All values are in kHz.

Molecule eqQ Ref.

Cl - C :: C - 0 208.5 + 1.5 This work -CN - C:: C - 0 203.5 + 1.5 This work F - C:: C - 0 212 + 10 16 -CH3 - C:: C - 0 228 + 2 36 -C03CN 168 + 4 37 -COll 166 + 5 38 -C03Br 175 + 3 39 -C03C :: C ., H 174 + 6 36 -

41

Table 11. Quadrupole coupling constants and bond distances for the C-D moiety in several deuterated acetylenes.

a b

c

Molecule

CH3C :: C - D

Cl - C :: C - D

NC - C :: C - D

Ref. 40.

Ref. 41-

Ref. 25.

eqQ(kHz)

228 + 2

208 + 2

204 + 2

C-D(A)

1.058 + 0.002a

1.0550 + 0.00005b

1.057 + O.OOlc

be taken with caution we might say the difference can be explained by more

effective shielding on the carbon nucleus bonded to deuterium in

cyanoacetylene. Flygare16 has pointed out that for acetylenes a 10%

change in the nuclear shielding of the carbon atom directly bonded to

deuterium will have the same effect as a 100% change for the next nearest

carbon atom. Since the bond distance of the C-D moiety is the same for

both molecules the significant difference in the deuterium quadruple

coupling points to the chemical shielding on the adjacent carbon atom.

Similar reasoning can be applied to explain the difference in quadrupole

coupling in C1CCD and NCCCD. The C-D bond distance is slightly smaller

for C1CCD leading to a higher nuclear contribution (Dositive) +0 the

quadrupole coupling. If the nuclear shielding on the adjacent carbons

were the same C1CCD would be expected to have a larger quadrupole coupling.

Note that these arguments are extremely qualitative and may have little

basis. The electronic contribution from chlorine has been completely

42

ignored. As a second row atom, computing its contribution to the

electric field gradient at the deuterium atom is a non-trivial problem.

Barfield3l has established distance criteria for the inclusion of

various contributions to the deuterium quadrupole coupling in propyne-Dl .

He evaluated two and three center integrals associated with atomic centers

in the molecule. The three center terms were found to make a contribution

of -53 kHz to the quadrupole coupling. The point of this is that signifi­

cant differences in quadrupole coupling may be directly related to these

terms and therefore transcend qualitative arguments.

The last point to be addressed is the effect of the hybridization

on the carbon atom to which the deuterium is bonded. Millett and Dailey42

measured deuterium quadrupole coupling strengths in the liquid phase on a

series of hydrocarbons and observed from the measurements that eqQ(sp» 2 3 eqQ(sp » eqQ(sp). This conclusion must be taken with caution on experi-

mental and theoretical grounds. When making comparisons of deuterium quad­

rupole coupling in different systems one must note the error bar associated

with the reported measurement. Many of the molecules they were comparing

had uncertainties of the order + 15 kHz. Absolute trends can truly be

established when measurements have been shown to differ within the limits

of experimental error. Furthermore, in support of their conclusion they

pointed to the work of Fung,43 et al., who studied the dependence of the

deuterium quadrupole coupling on carbon hybridization via a simple MO

description. They treated the C-D moiety as an isolated fragment insensi­

tive to substituent effects. This is an oversimplification and their

43

basis set was minimal. Clearly, the later treatments of snyder30 and

Barfield31 demonstrate the importance of not treating the c-o moiety as

an isolated fragment.

APPENDIX

PROGRAM FOR CALCULATION OF HYPERFINE STRUCTURE

The program KUPLD calculates how a rotational level splits due to

electric quadrupole, spin-rotation and spin-spin interactions. Hyperfine

structure can be calculated for the most general case of four coupling

nuclei of arbitrary spin. The order in which the nuclei are coupled is

immaterial as a full matrix diagonalization is always a part of the calcu­

lations. As a check the order of the coupling can be reversed even during

the same run since the program will accept more than one set of data.

The algorithm is simple enough to follow. The philosophy in which

the program was written was such that a person could calculate hyperfine

structure in rotational transitions even if their background in the area

was limited. One merely inputs the rotational level, spin of each nucleus

and their associated interaction strengths.

Note that the program can not differentiate between equivalent and

non-equivalent spins; no attempt was made to simplify the energy matrix

for special cases (e.g., CD3X type molecules).

44

PROGRAM KUPLDlINPUT,OUTPUT,TAPES-INPUT,TAPE6-0UTPUTI C THIS PROGRAM CALCULATES THE SPLITTING OF ROTATIONAL C LEVELS DUE TO QUAORUPOLE,SPIN-ROTATION,AND SPIN-SPIN C INTERACTIONS FOR THE CASE Of fOUR NUCLEI OF ARBITRARY SPIN. C THE BASIS STATES ARE WRITTEN INTHE COUPLED SCHEME AND ARE C OF THE FOR" :J,11,Fl,I2,FZ,I3,F3,1~,F).

REAL Hl3,31,Ul3,31,DJl61,CJl41,QI41,El31 REAL J,ll,IZ,13,14,J2,ISS,IJ COMMON/QSRSS/J,11,IZ,13,I~ COMMON/SOE/Q,BROT

C THE INPUT DATA ARE AS FOLLOWS:J IS THE ROTATIONAL LEVEL, CII IS THE SPIN OF NUCLEUS ONElTHE SAME FOR IZ,I3,I~I.

11 READlS,6001 J,Il,IZ,13,I~ 600 FORMATl5FIO.OI

WRITE16,7001 J,Il,IZ,13,I~ 700 FORMATlIX,'J-',FS.l,'ll-',FS.I,'IZ-',FS.l,

1·I3-"FS.l"I~-',F5.11 IFIJ.EQ.OI CALL PHOSTGP

C THE INTERACTION STRENGTHS ARE NOW READ IN. THE C OJ'S ARE THE SPIN-SPIN CONSTANTS BETWEEN THE fOU~ NUCLEI. C OJll' IS THE SPI~-SPIN INTERACTION STRENGTH BETWEEN C NUCLEI ONE AND TWO,OJIZI IS THE SSIS BETWEEN NUCLEI ONE C AND THREE,OJI31 IS THE SSIS BETWEEN NUCLEI ONE C AND fOUR,DJI~1 IS THE SS15 BETWEEN NUCLEI TWO AND THREE C OJISI IS THE SSIS BETWEEN NUCLEI TWO AND FOUR,OJlbl IS THE C SSIS BETWEEN NUCLEI THREE AND FOUR. THE CJ'S ARE C THE SPIN-ROTATION INE\TERACTION STRENGTHS. CJlll IS C THE SPIN-ROTATION INTERACTION STRENGTH FOR NUCLEUS ONE C lANALOGOUS FOR CJIZI ••• CJ(~I. THE Q'S ARE THE QUADRUPOLE C INTERACTION STRENGTHS FOR NUCLEI ONE THPOUGH FOUR.

REAOIS,661 BROT 66 FORMATIFIO.OI

WRITEI6,7b)BROT 76 FORMAT(IX,'THE ROTATICNAL CONSTANT IS',FI5.S)

REAOlS,BOOI IOJIII,I-l,bl REAOlS,BOO) lCJ(II,I-l,~) READ(S,BOO) (QII',1-1,4)

800 fORMATl6FIO.OI WRITCl6,9DOI

900 FORMATlIHO,'THE SPIN-SPIN CONSTAMTS ARE') WRITElb,9101 lOJ(II,I-I,bl

910 FORMATlIX,FIO.31 WRITElb,9Z01

9Z0 ~ORHATlIHO"THE SPIN-ROTATION CONSTANTS AREtl WRITElb,9101 lCJlII,I-I,41 WRITElb,9301

930 FORHAT(IHO,'THE QUADRUPOLE COUPLING CONSTANTS AREt) WRITElb,9101 lQlII,I-I'~) DO 31 NI- 1,3 DO 31 NZ- 1,3 H(NI,NZI- O.

31 CONTINUE C THE HAHILTONIAN MATRIX IS INDEXED:

Nl-l HZ-I

C IN THIS PART OF THE PROGRAM THE SPIN-SPIN

45

C INTERACTION IS CALCULATED DO 10 FI-ABSCJ-ll"AB5CJ+Ill DO 10 F2-ABSCFI-12"ABSIF1+IZ' DO 10 F3-ABSCF2-13"ABSCF2+13' DO 10 F-ABSCF3-I4"ABSCF3+14' DO 20 FIP-ABSCJ-Il"ABSCJ+Il' DO 20 F2P-ABSCFIP-I2',ABSCFIP+IZ' DO 20 F3P-ABSIF2P-13"ABSCFZP+13' DO 20 FP-ABSCF3P-141,ABSCF3P+141 IFIF.NE.FPI GO TO 30

C THE NEXT STATE~ENT ACTS AS SWITCH TO BYPASS CALCULATION C OF THE SPIN-SPIN INTERACTION IF ALL INTERACTION STRENGTHS C ARE SET EQUAL TO ZERO. THIS IS USE FULL AS CALCULATION C OF THE SPIN-SPIN PART USES UP QUITE A B[T OF ~XECUT[ON TIME.

IFCOJCll.EQ.0.AND.DJIZI.EQ.O.ANO.OJI31.EQ.0.ANO 1.DJI51.EQ.0.AND.DJlbl.EQ.01 GO TO 15

KOUNT-l DO 100 L-l,3 DO 100 lL-L+l,4 DO 200 FlPP-ABSIJ-[11,ABSIJ+I11 DO 200 F2PP-ABSIF1PP-IZI,ABSIFIPP+IZI DO ~OO F3PP-ABSIFZPP-I31,ABSIFZPP+[31 DO 200 FPP-ABSIF3PP-I41,ABSIF3PP+I41 IFIF.EQ.FPPI THEN HINl,NZI-HIN1,NZ'+ DJIKOUNTI.l1.5.IIJIFl,FZ,F3,F,FlPP,F2PP,

If3PP,FPP,LI.IJIFIPP,FZPP,F3PP.FPP,FIP,FZP,F3P,FP,LLI 2+IJIFl,FZ,F3,F,FlPP,FZPP,F3PP,FPP,lll·IJIFlPP,FZPP, 3F3PP,FPP,FIP,FZP,F3P,FP,LI,-IISSIF1,FZ,F3, 4F,FIPP,F2PP,F3PP,FPP,L,LLI·JZeF1PP,FZPP,F3PP,FPP, 5FIP,F2P,F3P,FPI I I

END IF 200 CONTINUE

KOUNT- KOUNT + 1 100 CONTINUE

15 DO 300 l-1,4 HINl,N21-HINl,NZI + CJILI.IJIFI,FZ,F3,F,FIP,F2P,F3P,FP,ll

1+QIll.VQIF1,F2,F3,F,FIP,FZP,F3P,FP,ll 300 CONTINUE

30 N2-NZ +1 20 CONTINUE

NlaNl+l NZ-l

10 CONTINUE IIRITECb,401

40 FOR~ATCIHO"UNDIAGNOL!ZEO HAMILTONIAN" M-3 CALL DUTCH,PU W-O. DO 50 1-1,3 w- II + ABSCHII,I"

50 CONTINUE FIN-IW/3.'·1.0E-12 CALL JACDBIH,U,3,3,1,FINI WRITEI6,41'

41 fORHATCIHO,'DIAGNOLIZED HAMILTONIAN" CAll oUTCH .. M' IIRITEC6,42'

42 FORMATCIHO,'TRANSFDRMATION MATRIX" CAlL DUTCU,M' WRITEC6,60'

60 FDRMATCIHO,"EIGENVALUES"'

46

C C C C C C C C C C C C C C C

C C

00 61 1-1,3 Etll-Htl,1) Erll -Elll + 2 •• BROT

61 CONTINUE CALL SORTCE,3) DO 63 1-1,3 WRITEr6,6ZIElII

63 CONHNUE 6Z fORMATIIX,F15.51

GO TO 11 END SUBROUTINE OUllX,MI DII'IENSION Xl3,31 DO ZO K-l,f1,10 WRITEI6,ZI

Z FORMATllHOI 00 20 J-l,M Ll-K

1

20 10

1

2 3 It

lU-K+9 IF ILU.GT.,..) lU-M WRITEr6,11 rXlJ,ll,l-Ll,LU) ~ORMATlIHO,10G13.71 IFIlU.EQ.,...AND.J.EC.MI GO TO 10 CONTINUE RETURN END SUBROUTINE JACOB I H, U, M, H, IFU, FIN) H IF THE ARRAY TO BE DIAGONALIZED. U IS THE UNITARY MATRIX USED FOR FORMATION OF THE EIGENVECTORS. M IS THE ALLOTTEe ORDER OF THE MATRICES H, U IN CALLING PROGRAM N IS THE ORDER OF H, lU) USED. N CANNOT BE GREATER THAN M IFU MUST BE SET EQUAL TO ONE IF EIGENVALUES AND EIGENVELTORS ARE TO BE COMPUTED. IFU MUST BE SET EQUAL TO TWO IF ONLY EIGENVALUES ARE TO BE COMPUTED. FIN IS THE INDICATOR FOR SHUT-OFF, THE FINAL LARGEST OFF DIAGONAL ELEMENT. THE SUBROUTINE OPERATES ONLY ON ThE ELEMENTS OF H THAT ARE TO THE RIGHT OF THE MAIN DIAGONAL. CAll ERRSET lZOe,0,Z5,1,01 CALL ERRSET l207,256,25,1,0) CALL TRAPSlO,20,201 DIMENSION HIM,MI, UlM,MI

~REPARATORY OPERATION GO TO rl,~), IFU DO 3 I - 1, N DO 2 J - 11 N utI,") - 0.0 Un,It - 1.0 IF eN .EQ. II GO TO 100 NSl - N - 1 SUM - 0.0 DO 6 I - 1, NSI IAl - I + 1 00 5 J - U1, N

5 SUM - SUM+Hrl,J).HlI,J) 6 CONTINUE

IF ISUM .LT. FIN.FINI GO TO 100 OFFMAX - SQRTeSUf1+SUM) HN - N

47

c C SCANNING FOR URGE OFF DIAGONAL ELEMENT C

C

7 OFFKAX • OFFI1AX/HN 8 MEKO • 1

00 17 I· 1, NS1 IA1 • I + 1 DO 17 J • I Al, N IF (ABSIHII,J» .LT. OFFMAX) GO TO 17 I1EI10 • 2

C TRANSFORMATION C

HII • HIld) HJJ • HIJ,J) HIJ • HII,J) TANG • SIGNIZ.O, IHII-HJJII'HIJ

1 II ABSIHII-HJJ) + SORTIIHII-HJJI"Z + HIJ"Z'~.OI) COSIN • 1.0/S0RTll.0 + TANG'*ZI SINE • TANG'COSIN 151 • I - 1 IF (lSI .EO. 0) GO TQ 10 DO 9 K· 1, lSI HKI • HIK,I) HIK,I) • HKI'COSIN + HIK,J)'SINE HIK,JI·. -HKI'SINE + HIK,JI'COSIN

9 CONTINUE 10 HII,I) • HII'COSIN"Z + HJJ*SINE'*Z + HIJ'SINE*COSIN*Z.C

HIJ,J) • HII'SINE"Z + HJJ'COSIN"Z - HIJ'SINE'COSIN'Z.O JS1 • J - 1 IF IIAI .GT. JSl) GO TO 12 DO 11 K· IA1, JSl HIK • HII,K) HII,K) • HIK'COSIN + HIK,JI'SINE HIK,J) • -HIK'SINf + ~IK,JI'COSIN

11 CONTINUE lZ HII"n • 0.0

IF I J • EO. N) GOT 0 ·8 JA1 • J + 1 DO 13 K. JAl, N HIK • HII,K) HII,K) • HIK'COSIN + HIJ,KI'SINE HIJ,K) • -HIK'SINE + HIJ,K)'COSIN

13 CONTINUE 35 GO TO 115,17), IFU 15 DO 16 K· 1, N

UK I .. UI K, IJ UIK,I) • UKI'COSIN + UIK,J)'SINE UIK,J) • -UKI'SINE + UIK,JI'COSIN

16 CONTINUE 17 CONTINUE

GO TO IIB,e" KEMO 18 IF 10FFI1AX .GT. FIN) GO TO 7

100 RETURN tND

C FUNCTION IJ CALCULATES MATRIX ELEMENTS FOR THE C 'StUAR PRODUCT OPERATOR UJ IN THE COUPLED SCHEME

fUNCTION IJIFl,F2,F3,F,F1P,FZP,F3P,FP,M) COMI10N/QSRSSI J,ll,I2,13,I~ REAL J,11,I2,I3,J4,IJ IFII1.EQ.l) THEN

48

IfCF1.NE.flP.OR.fZ.NE.FZP.OR.F3.NE. If3P.OR.F.NE.FP' THEN

JJ-O. ELSE IXI-J +Il+Fl IJ-C-l'··CIX1'·SQRTII1·CI1+1)*IZ.*Il+1.'*J*

lCZ •• J+l.'·IJ+l."·SlXJCJ,Il,Fl,Il,J,l.1 END IF ELSE IFCM.EQ.Z' THEN IFIFZ.NE.FZP.OR.F3.NE.f3P.OR.F.NE.FP) THEN I~-O. ELSE lxZ-Z.*FlP+IZ+FZ+J+Il+l. IJ-C-11**CIXZ'*SQRTIIZ*IZ.*lZ+1.,

1.CIZ+l.I.J·IZ.·J+1.1.IJ+1.I.IZ.·FlP+1.I.IZ.·Fl+1.11 Z.SIXJIF1,IZ,FZ,IZ,FIP,1.).SIXJIJ,Fl,Il,FlP,J,1.1 E~D If ELSE (F(M.EQ.3) THEN IFCF3.NE.F3P.OR.F.NE.FPI THEN 13-0. ELSE (X3 a Z.·FZP+Fl+FlP+13+F3+IZ+J+Il IJ-I-l'.·IIX31.SQRTC13·IZ.*13+1.I*I13+1.1.

lCZ.*FZP+l.I*Cz.*rZ+l.I*IZ.*FlP+l.'*CZ.*Fl+l.l. Z J*IZ.*J+l.)*IJ+l.11 *SlXJI~Z,I3,F3,13,F2P,1.1* 3SIXJCF1,FZ,IZ,FZP,FIP,1.I*SIXJIJ,Fl,Il,FlP,J,1.1

END IF ELSE (FCM.EQ.4' THE~ IFCF.NE.FPI THEN IJ-O. ELSE IX4-Z.*f3P+14+F+FZ+FZP+I3+Fl+FlP+IZ+Il+J+1. !J·C-11**IIx41*SQ~TII4*CZ.*I4+1.1*II4+1.1.

lCZ.*F3P+l.I*IZ.*F3+1.1*IZ.*FZP+l.I*CZ.*FZ+1.1 Z*IZ.*flP+l.'·IZ.*Fl+l.I*J*IZ.*J+l.'.IJ+l.',· 3SIXJIF3,I4,F,I4,F3P,1.I.SIXJCFZ,F3,I3,F3P,FZP,1.1 4.SIXJIF1,F2,IZ,FZ"FIP,1.I*SIXJIJ,Fl,Il,FlP,J,1.1

END IF END IF RETURN END fUNCTION ISSCF1,FZ,F3,F,FIP,FZP,F3P,FP,L,LLI COMMON/QSRSS/J,Il,IZ,13,I4 REAL J,Il,IZ,I3,I4,ISS (FIL.EQ.l.AND.LL.EQ.ZI THEN IFCFZ.NE.FZP.OR.F3.NE.F3P.OR.f.NE.FPI THEN ISS-O. ELSE IXS- J+ll+(Z+FlP+fl+FZ+l. ISS-l-l' •• lIXS'.SQRTIIZ •• IZ+l.'.IIZ+l.I*IZ*

lCZ.·ll+l.'.CI1+l.'.Il.IZ.·FlP+1.1*IZ.*Fl+1.11 Z.SIx~IFl,12,f2,I2,FlP,1.'·SIXJII1,Fl,J,FlP,Il,1.'

fND IF ELSE IFCL.EQ.l.AND.lL.EQ.3' THEN IFlF3.NE.F3P.DR.f.NE.FP' THEN ISS-O. ELSE IX6-Z •• FZ+Z.·FIP+I3+IZ+ll+F3+J ISS-l-11 •• CIX61.SQRTCCZ •• FZP+l.I.IZ •• FZ+l.,.

1(Z •• flP+l.'.CZ •• F1+1.'.13.IZ.~I3+1.'.II3+1.1. ZIZ •• Il+l.'.Cll+1.'.Il'.SIXJIFZ,I3,F3,13,FZP,1.1*

49

3SIXJ,Fl,fZ,IZ,FZP,FIP,I.'·SiXJ,Il,Fl,J,FlP,11,1.1 END IF ELse If,L.fQ.l.AND.LL.EO.4' THEN tf C F .NE .FP I THEN ISS-O. ELSE IX7-Z.·FIP+2 •• F3+F2P+F2+I4+I3+IZ+Il+J+1.+F ISS-'-I'··'IX7'.SORT"2.·F3P+1.'.'Z.·F3+1.'.

l'Z.·F2P+l.'·'Z •• FZ+l.I.'Z •• FIP+I.I·CZ •• FI+I.I· Z,2 •• 14+1.,.,14+1.1.14.,2 •• ll+l.I.'ll+l.'·I11 3.SIXJ,F3,I4,F,I4,F3P,1.'·SIXJ'FZ,F3,I3,F3P,FZP,I.I. 4SIXJ,Fl,FZ,I2,fZP,flP,I.I·SIXJ,11wFl,J,F1P,Il,I.1

fND IF ELse IF'L.EQ.2.AND.LL.EQ.31 THEN IF'f.NE.fP.OR.F3.NE.F3P.OR.Fl.NE.FlPI THEN ISS-O. ELSE !Xe-F3+FZP+FZ+F1+I3+IZ+1. ISS-'-II •• 'IXBI·SQRT'CZ.·I3+1.I.CI3+1.I·I3

1·'Z •• I2+1.1·IIZ+l.I.IZ.CZ •• fZP+l.J·IZ •• FZ+l.ll. 2SIXJCFZ,I3,F3,I3,FZP,1.J*SIXJ,12,F2,F1,FZP,IZ,1.1 ~ND IF fLSE IFCL.EQ.Z.AND.LL.EQ.41 THEN IFIF.NE.FP.OR.FI.NE.FlPI THEN ISS-O. ELSE IK9-F+Z.·F3+Z.·FZP+F1+14+13+IZ ISS-C-ll •• ,IX9J·SQRTIIZ.·F3P+1.1.IZ.·F3+1.1.

1,Z •• FZP+l.I.,Z •• FZ+l.I.CZ.·I4+1.1.114+1.1.14. ZCZ •• IZ+1.1.'IZ+l.I·IZI.SIXJIF3,14,F,14,F3P,1.)· 3SIXJIFZ,F3,I3,F3P,FZP,1.I·SIXJIIZ,FZ,F1,F2P,12,1.,

END IF ELSE IFCL.EQ.3.AND.LL.EQ.41 THEN IF,F.NE.FP.OR.FZ.NE.FZP.OR.Fl.NE.FlPI THEN IS'5-0. ELSE IXI0-F3+F3P+FZ+F+I4+I3+1. ISS-,-II •• IIXIOJ.SQRT,CZ.·F3P+l.I.IZ •• F3+1.1.

lCZ •• I4+l.J.CI4+1.1·14.IZ •• I3+1el.,I3+1.I·I31. ZSIXJII3,F3,FZ,F3P,13,I.J.SIXJIF3,I4,F,14,F3P,1.1

E-ND IF END IF RETURN END

C FUNCTION VQ CALCULATES THE QUADRUPOLE INTERACTION C FOR THE K TH COUPLED NUCLEUS' UP TO FOUR NUCLEI C CAN BE COUPLEDJ

FUNCTION VOlfl.F2,r3,r,FlP,F2P,F3P,FP,MI COMHON/QSRSSI J,ll,I2,I3,I4 REAL J,ll,12,13,I4 IF , H • E Q • 11 THE N ,f'f.ne.fp.or.f3.ne.f3p.or.fZ.ne.fZp.or.fl.ne.flp.

lor.I1.lt.U then VO-O. else hU-J+ 11+f1 yq-'-11 •• CI_l11.sqrtC"Z •• J+l.I.'Z •• J+2.1.'2 •• J+3.J

IJ/,B •• J.,Z •• J-l.II'.sqrtCC'Z •• il+I.,.,2 •• il+2.'.'Z •• i1+3.11 Z/,B •• ll.'Z •• il-l.',I.sl-J,fl,il,j,Z.,J,11I end If ELSE IF'H.EQ.21 THEN

50

IFIF2.NE.F2P.OR.F3.NE.F3P.OR.F.NE.fP.or.i2.lt.11 THEN VQ-O. EL Sf IX12- 2 •• F1P+12+F2+J+l1 VQ-C-11 •• IIX121·SQRTI12.·FIP+l.I·12 •• Fl+1.11

1 .SQRTII12 •• 12+1.).12 •• 12+2.).IZ •• I2+3.)I/CIE •• IZI 2.12.·12-1.)I).SQRTCIC2 •• J+l.I.12.·J+2.1.IZ •• J+3.11 3/CC6 •• JJ.C2 •• J-1.'JJ·SIXJIF1,12,F2,12,F1P,2.) 4.S1XJCJ,Fl,11,F1P,J,Z.J

END IF ELSE IFCM.EQ.3J THEN IFIF3.NE.F3P.OR.F.NE.FP.or.i3.lt.1J THEN VQ-O. elSE IX13-Z.$FZP+13+F3+F1+IZ+J+Il+FlP VQ-C-1I •• IIX131·SQRTIIZ.·FZP+1.1.12 •• FZ+1.1.

liZ. Hl P +1. I. I Z. $f 1 + 1.1 I. S CRT II 12. *13+ 1. I. I Z •• I 3 + Z. I. 2IZ.·13+3.1'/118 •• 131*12.·I3-1.)11 3·SQRTII12.·J+l.I*12.·J+Z.I.12.·J+3.11/118 •• J).12 •• J-1.))) 4·SIXJIF2,13,F3,I3,FZP,2.I.SIXJIF1,FZ,I2,F2P,F1P,Z.I. 5SIXJIJ,F1,Il,FlP,J,2.1

END IF ELSE IFIM.EC.41 THEN IFCF.NE.FP.or.i4 •• t.11 THEN VQ-O. £LSE IXS-2.*f3P+14+F+F2+I3+Fl+IZ+FZP+J+Il+F1P VQ-I-ll··IIX81·SCRTII12 •• I4+1.1*IZ.·14+2.1*li •• I4+3.)1

1/1 18 •• I 4 1 • I 2 •• 14-1 • ) 1 1 *S QR TC 1 12 •• J + 2. I. 12 •• J+ 1. I. I 2 •• J + 3 • ) ) Z/IIS •• J).'2.*J-l.II)·SIXJIF3,14,F,14,F3P,2.) 3·SIXJIFZ,F3,I3,F3P,FZP,2.I.SIXJIF1,F2,IZ,F2P,FIP,2.)* 4SIXJIJ,Fl,Il,FIP,J,2.1 5. S QR TC 12 •• F3 P + 1. ) * I Z •• F 3 +1 • ) *1 Z •• F 2P + 1. )* I 2 •• F 2 + 1. ). 6IZ.·FIP+l.J*12.*F1+1.1)

END IF rNO IF RETURN END FUNCTION SIXJIJ1,J2,J3,Ll,L2,L31 REAL J1,JZ,J3,L1,L2,L3 IIMIN-INTIAMAX1IJI+JZ+J3,JI+L2+L3,Ll+JZ+L3,Ll+L2+J31+0.~I IIMAX-INTIAMIN1IJ1+J2+L1+L2,J2+J3+LZ+L3,J1+J3+Ll+L3)+O.5) W-O.O IFI"001IZMIN,2J.EC.11 GO TO 10 SlGN--1.0 GO TO 20

10 SIGN-1.0 20 DO 1 J-IZMIN,IIMAX

Z-J SIGN-C-l.)·SI6N A-Z-JI-J2-J3 B-Z-H-L2-L 3 C -Z-Ll-JZ-L3 0-Z-L1-L2-J3 e-Jl+J2+l1+L2-Z F-J2+J3+L2+L3-Z G-J1+J3+Ll+L3-Z IFIA.LT.-O.1.0R.B.LT.-D.1.0R.C.lT.-O.1.0R.D.LT.-D.1

1.0R.E.LT.-O.l.OR.F.LT.-O.l.OR.G.LT.-D.11 GO TO 1 V-W+SIGN.FACCZ+1.01/IFACIA).FACCBI.FACCCI.FACIOI.FACIE).

lfAC CF ).FAt C G) I

51

1 CO~TINUE SIXJ-DELTACJ1,J2,J3'.DELTAIJ1,LZ,L3'*DELTAIL1,JZ,L3'

1*DELTAIL1,LZ,J3).W RETURN END FUNCTION DELTAIA,B,C' IFIA+B-C.GE.-0.1.AND.-A+B+C.GE.-0.l.AND.A-B+C.GE.-0.l)

1GO TO 10 DELTA-O.O RETURN

10 DELTA-SQRTIFACIA+B-C)*FACIA-B+C'*FACI-A+B+C)/FACIA+B+C+1') RETURN END FUNCTION FAC ()() N-INTIX+0.5) HIN) ZO,30,10

20 WRITEI6,100)X 100 FOR~ATIIHO,'FACTORIAL',G16.B,'IS REQUIRED. UNITY IS ASSUM

leD' , FAC-1.0 RETURN

30 FAC-l.O RETURN

10 FAC-l.O DO 1 J-l,N FAC-FAC*J

1 CONTINUE R ETUR N END FUNCTION JZIF1,FZ,F3,F,FIP,FZP,F3P,FP) REAL JZ IFIF1.NE.F1P.OR.FZ.NE.F2P.OR.F3.NE.F3P.OR.F.NE.FP) THEN 8Z-0. ELS E JZ·Z. END IF RETURN eND SUBROUTINE SORTfA,B' INTEGER B DI~ENSION AlB' lFIB.EQ.l' RETURN L -B-1 00 10 l-l,L t(. B-1 00 ZO J-l,K IFIAIJ)-AIJ+1"ZO,ZO,30

30 TEHP-AIJ+l) AIJ+U-AIJ) AlJ)-TEHP

20 CONTINUE 10 CONTINUE

RETURN END

52

lIST OF REFERENCES

1. N.F. Ramsey, Phys. Rev. 86. 243-6 (1952).

2. C.H. Townes and A.l. Schawlow. Microwave Spectroscopy (Dover. NY. 1975) .

3. W. Gordy and R.l. Cook. Microwave Molecular Spectra (Interscience. NY, 1970) .

4. N.F. Ramsey. Molecular Beams (Oxford U.P., london, 1961).

5. P. Thaddeus, l.C. Krisher, and J.H.M. loubser, J. Chern. Phys. 40, 257-73 (1964).

6. R.L. Cook and F.C. Delucia, Am. J. Phys. 39, 1433-54 (1970).

7. R. J. Bersohn, J. Chern. Phys . .l§., 1124-5 (1950).

8. M. Mizushima and I. Ito, J. Chern. Phys. ~, 739-44 (1951).

9. J.W. Cederberg, Am. J. Phys. 40, 159-72 (1972).

10. S.G. Kukolich and C.D. Cogley, J. Chern. Phys. 76, 1685-91 (1982).

11. K.B. McAfee, R.H. Hughes, and E.B. Wilson, Rev. Sci. Inst. 20,821-6 (1949) .

12. S.G. Kukolich, Proc. IEEE 60, 136-8 (1972).

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