MECROWAVE SPECTRA, MOLECULAR

STRUCTURE, QUADRUPOLE COUPLENG,
DEPOLE MOMEN'F, AND RING IENDING
WBRAUCN OF 1, 3, 2:DIOX.ABOROLAME

Thesis for the Degree of DH. D.
MECKESEN STATE UNEVERSETY
James Henry Hand
1965

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THESIS

      

This is to certify that the

thesis entitled
MICROWAVE SPECTRA, MOLECULAR

STRUCTURE, QUADRUPOLE COUPLING,
DIPOLE MOMENT, AND RING BENDING
VIBRATION OF 1,3,2-DIOXABOROLANE

presented by

James Henry Hand

has been accepted towards fulfillment
of the requirements for ,

M— degree in wry

t / J» 9/01“» 5‘4““01u-
V I

Major professor

Date November 9: 1965

0-169

v____*7,

ABSTRACT

MICROWAVE SPECTRA, MOLECULAR STRUCTURE, QUADRUPOLE COUPLING,
DIPOLE MOMENT, AND RING BENDING VIBRATION OF 1,3,2-DIOXABOROLANE

by James Henry Hand

A brief history of microwave spectroscopy is presented. The theory
of rotational Spectra, with emphasis on the method of molecular struc-
ture determination and on vibration-rotation interaction, is discussed.
The microwave Spectrometer is described.

The microwave Spectra of molecules in three vibrational states

of OCHZCHZOBH, in three states of OCDZCDZOBH, in two states of
013CH2CHZOBH, and in the ground state of OCHZCHZOloBH have been ex-
g E

amined. Rotational constants and quadrupole coupling constants have
been deduced for each spectrum. The dipole moment has been measured
to be 2.28 Debye and lies solely along the symmetry axis. Structural
parameters consistent with the experimental second moments of inertia
are: c-c = 1.5b1 R, c-o = 1.b38 8, 8-0 = 1.368 R, C-H = 1.093 E,
ZHCH = 1090 9', Z cco = Iobo 514', _/ BOC = 1070 18', and [080 = 11140 12'.
The ring is slightly non-planar, with a C2 symmetry axis passing
through the H and B atoms and the midpoint of the C-C bond. The C-C
bond forms an angle of 7° 8' with the 080 plane.

The observed excited vibrational states have been assigned to
the first two excited states of a symmetric ring bending vibration.
The excited vibrational energy levels are estimated to lie 53 and 88 cm-1

above the ground state for ocnzcnzoen and 22 and 66 cm'1 above the

James Henry Hand

ground state for OCDZCDZOBH. The vibrational effects which have been
L_______J

observed are consistent with a double minimum potential with a small
central barrier.

A suggested method for calculating the energies of centrifugal

distortion is included.

MICROWAVE SPECTRA, MOLECULAR STRUCTURE, QUADRUPOLE COUPLING,
DIPOLE MOMENT, AND RING BENDING VIBRATION OF 1,3,2-DIOXABOROLANE

By

James Henry Hand

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1965

DEDICATION

To the future for David and Kristin

ii

AC KN ON LE DGIVIEI‘J TS

It is a sincere pleasure to thank Dr. Richard H. Schwendeman
for the generous guidance and intellectual stimulation which he
provided during the course of this research.

Financial aid in the form of fellowships from the National
Science Foundation and Petroleum Research Fund is gratefully
acknowledged.

Dr. Sheldon G. Shore deserves thankful recognition for his
interest in and initial suggestion of research on this compound,
for his gift of some of the chemical samples, and for his aid in

preparing the remaining samples.

TABLE OF CONTENTS

I HISTORICAL BACKGROUND AND AN INTRODUCTION .

II THEORETICAL BASIS OF ROTATIONAL SPECTROSCOPY

III

IV

2.1

2.2

2.3

2.11
2.5

Introduction . . . . . . . . . . . . . . . . . .
Rigid Rotor Model

A) Moments of Inertia and Principal Axes . .
B) Quantum Mechanical Treatment of Rigid Rotation

Perturbations to the Rigid Rotor . . . . . .

A) Stark Effect .
8) Nuclear Quadrupole Coupling

Determining a Molecular Structure . . . . . . .

The Coupling of Vibration and Rotation . .

DESCRIPTION OF THE MICROWAVE SPECTROMETER .

3.1
3.2
3.3
311
3.5
3.6

Introduction .

Klystron Oscillators .

Waveguide Absorption Cell . . . . . . .
Square Wave Stark Modulation .

Detection System . . . . . . . . . . . . . . .

Frequency Measurements .

MICROWAVE STUDIES ON 1,3,2-DIOXABOROLANE

11.1
11.2
11.3
11.11

Introduction .
Preparation of The Chemical Samples
Examination of the Rotational Spectra

Discussion of the Bond Characters . . . . . . .

iv

Page

\1

2O

2O
21

2b
27
36
36
36
ho
ho
111
. ’42
ho
b6
b6
52
oh

TABLE OF CONTENTS (Cont.)
Page

h.S Determination of the Molecular Symmetry and
Structure . . . . . . . . . . . . . . . . . . . 70

A) Relative Intensities of Absorption Transitions 71
B) Intensity Measurements . . . . . . . . . . . 73
C) Determination of the Dipole Moment . . . . . . 78

D) Boron Atom Coordinates and a Molecular
Structure . . . . . . . . . . . . . . . . . 83
b.6 Interpretation of the Ring Bending Vibration . . . 89
h.7 Discussion . . . . . . . . . . . . . . . . . . . . 96
V CENTRIFUGAL DISTORTION CALCULATIONS . . . . . . . . . . 99
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . 106
APPENDICES . . . . . . . . . . . . . . . . . . . . . . 109
A COMPUTER PROGRAM "EIGVALS": . . . . . . . . . . 110

B KRAITCHMAN-TYPE EQUATIONS FOR THE h-D
SUBSTITUTION . . . . . . . . . . . . . . . . 112

TABLE

II

III

IV

VII

VIII

IX

XI

XII

XIII

LIST OF TABLES

Klystrons, frequency meters, and crystal detectors'
now in use at Michigan State University . . . . . .

Initial structural parameters assumed for
1,3,2-dioxaborolane . . . . . . . . . . . . . . . .

Observed and calculated frequency splitting of
quadrupole hyperfine components for 1,3, 2—dioxa-
borolane: parent Species . . . . . . . . . . . .

Experimental hypothetical unsplit frequencies assigned
for 1,3,2-dioxaborolane: parent Species . . . .

Observed and calculated frequency Splittings of
quadrupole hyperfine components for l,3,2-dioxa~

borolane: h,h,S,S-tetradeutero - species . . . . .~

Experimental hypothetical unsplit frequencies assigned
for 1,3,2-dioxaborolane: b,h,S,5-tetradeutero—
species . . . . . . . . . . . . . . . . . . . . .

Observed and calculated frequency splittings of quadru-
pole hyperfine components for 1,3,2-dioxaborolane:
13C Sp€Ci€S O O O O O O O O O O O O O O O O O C O 0

Experimental hypothetical unsplit frequencies assigned
for 1,3,2-dioxaborolane: l3C Species . . . . . . . .

Observed and calculated frequency splitting of quadru—
pole hyperfine components and experimental hypo-
thetical unSplit frequencies assigned for 1,3,2-
dioxaborolane: loB species . . . . . . . . . . .

Rotational constants, principal moments of inertia,
and second moments of inertia for 1,3,2-dioxa- '
borOlane I O O O O O O O O O O O O O O O C O O O

Quadrupole coupling constants for 1,3,2—dioxaborolane .

Relative intensity of the (212 —> 303) to the
(202 -> 313) transitions in 1,3,2-dioxaborolane . .

Relative intensiqy of the (303 -> bl4) to the

(313 —> hoe) transitions in tetradeutero-
1,3, 2-dioxaborolane . . . . . . . . . . . . . . .

vi

Page

39

53

56

57

61

63

63

. ‘65

66
67

LIST OF TABLES (Cont.)

TABLE

XIV

XV

XVI

XVII

XVIII

XIX

Relative intensities of rotational transitions and
estimated vibrational energies . . . . . . . .

Data used in the dipole moment calculations

Calculated dipole moment components . . . . . . . . .

Principal axis coordinates of the atoms in OCHZCHZOBH
L______J

Structural parameters calculated for 1,3,2-dioxa-
borOlane O O O O O O O O O O O O O O O O O O 0

Values of <'O > determined from the model for the
vibrational motion .

vii

Page

77
81
82
86

87

9S

LIST OF ILLUSTRATIONS

Figure Page
1 Block diagram of the microwave Spectrometer . . . . . 37
2 Stark waveguide cross section . . . . . . . . . . . . b0
3 Detection preamplifier input circuit . . . . . . . . be

h Projection of 1,2,3-dioxaborolane in its principal
ab and ac planes . . . . . . . . . . . . . . . . . 88

viii

I. HISTORICAL BACKGROUND AND AN INTRODUCTION

The microwave region of the electromagnetic Spectrum lies between
the far-infrared and radio-frequency regions. It includes the wave-
lengths between .OS and 30 centimeters, corresponding to energies of
20 to .033 cm-1. The energies involved in microwave spectroscopy
are generally expressed in megacycles/Second (Mc/Sec.), with l Mc/Sec.
= 106 cycles/Sec., approximately 3 x 10—5 cm-l.

The first Spectroscopic experiments in the microwave region were
performed by Cleeton and Williams (1) in 1933 on the ammonia molecule.
This intense and profuse spectrum arises from the inversion motion
of NH3. However, in subsequent experiments on other molecules the
observed spectra were of rotational transitions, and today the micro-
wave region is considered the domain of rotational Spectroscopy.

Other experimentation at that time was limited by the spectrometer
sensitivity and associated technological problems. These were greatly
abated by advances made in developing the 1.25 centimeter radar system.
After the Second World War much surplus radar equipment became avail-
able to universities, stimulating a wide range of experimentation.
Advances were then made in microwave Spectroscopy which may be classi-
fied as instrumental advances to high resolution, high precision, and
high sensitivity, and theoretical and conceptual advances.

The first high-resolution experiments were performed by Bleaney
and Penrose (2), on the MRS spectrum, and by Coles and Good (3), also
on the NH3 Spectrum. The latter workers observed hyperfine splittings

which were later demonstrated to be caused by (and prove the existence

2
of) the 14N nuclear quadrupole moment (h). A second type of hyperfine
effect is due to internal rotation in which case the observed Split—
ting provides a sensitive method for calculating the height of the
potential barrier to internal rotation. The first of many barrier
height determinations was made by Burkhard and Dennison (5) on methyl
alcohol. With the treatments of Herschbach and others (6) micro-
wave spectroscopy is now the most accurate method for determining the
potential energy constants for small barriers. High barriers and
conformational isomers are also easily studied, since each excited
torsional state and each distinct conformation gwes rise to a separate
rotational spectrum which can be resolved and separately studied.

With present spectrometers, resolution of splittings of 0.5
Mc/Secu 'is almost routine. Therefore observation of hyperfine ef-
fects is generally the rule rather than the exception.

The high precision possible for microwave measurements was demon-
strated by Dakin, Good, and Coles (7) in 19h7. They measured the
J = l —> 2 transition of OCS at 2h325 Mc/Sec. with an uncertainty of
r 0.02 Mc/Sec., less than 1 part per million. This precision is now
generally attained, and may be surpassed in special cases.

The sensitivity of a microwave Spectrometer was greatly increased
as the result of a suggestion by Hughes and Wilson (8). They showed
that appLying a periodic electric field to the absorption cell would
produce a molecular Stark effect, modulating the absorption transitions.
This permitted the use of tuned and lock-in amplifiers in the detec-
tion circuit. Transitions with absorption coefficients as low as
lO_10/cm. have been observed, but lO-g/cm. is a morecxmmon lower

limit of detectability.

3

The Stark effect also provides a convenient method for measuring
molecular dipole moments. Debye (9) predicted that in an electric
field, splitting of rotational transitions should occur and Should
be proportional to the square of the dipole moment and the square of
the electric field strength. Dakin, Good, and Coles first demonstrated
this to be true for OCS. Precise measurements of the dipole moment
of OCS (10) have provided a secondary standard on which dipole
moment measurements may be based.

The increase in sensitivity of the Spectrometer resulting from
use of the Stark effect permitted experiments to be performed on
molecules with less intense Spectra. Many linear and symmetric top
molecules were then studied, due to the simplicity of their Spectra.
Asymmetric top molecules were considered later, when improved methods
for calculating their energy levels had been developed, and when the
quantum-mechanical treatments had been extended to permit calculation
of perturbation and hyperfine effects, (and when there were no more
Simple symmetric tops to study).

The determination of molecular structures is the most widely
recognized use of microwave Spectroscopy. The internuclear (bond)
distances and angles were at one time determined by varying them sys-
tematically to reproduce the moments of inertia calculated from the
rotational constants. Later, least squares fittings to the moments
of intertia of several isotopic species were made. Kraitchman (11)
suggested an improved method for determining the nuclear coordinates
from differences between the moments of inertia of the normal species

and those of isotopically—substituted species. Costain (l2) later

ii
showed that better results, giving less variation with the isotopic
Species used, were obtained by this "substitution" method. This
method is rather insensitive to atoms located near a principal iner-
tial plane but is highly sensitive to the effects of vibrations of
these atoms. No Simple relationship between substitution coordin-
ates obtained by the Kraitchman-Costain method and equilibrium values
has yet been established. Presently a prohibitively large amount
of eXperimental data is reeded to obtain an equilibrium structure.
These include knowledge of the vibration—rotation coupling constants
for each singly excited vibrational state for each isotopic Species,
or complete knowledge of the quartic and cubic potential constants
for each vibrational mode. It is therefore not surprising that the
determination of a substitution structure is accepted as a pragmatistic
experimental goal.

Direct observations of vibrational effects on rotational spectra
have also been made, in which pure rotational transitions for molecules
in excited vibrational states have been observed. One example of this
is the study by Chan, Zinn, Fernandez, and Gwinn (13) on the four—
membered ring trimethylene oxide. They observed rotational spectra
at room temperature for the ground state and four excited vibrational
states of a ring puckering vibration. From intensity measurements
and far-infrared data they estimated the first vibrational state to
be about 70 cm.1 above the ground state. They also found that the A
and C rotational constants increased in a zig—zag pattern with in-
creasing vibrational energy. This non-linear increase could be ex-

plained only by a double minimum potential function for an out-of—plane

5

vibration. A further discussion of their work, and Similar results
for other ring compounds is included with the results of the present
study.

The experimental work reported in this thesis deals with the
microwave Spectrum of 1,3,2-dioxaborolane. The existence of this
compound was first reported by Schlesinger and Burg (1h) in l9h2.
Recently Rose and Shore (15) at The Ohio State University have refined
the preparation procedure and have extensively investigated its physical
and chemical properties. They report the vapor phase to be essentially
monomeric and highly stable at -78°. The disproportionation reaction

in the vapor phase

 

l 1
has an equilibrium constant at 250 of K25 = 25 i 3. At room tempera-

Eq

ture the condensed phase is a tacky or viscous liquid which has a
great tendency to change to a non—volatile glassy solid. The liquid
may be regenerated by careful warming to about LOO.

Dr. R. H. Schwendeman of Michigan State University began the study
of the microwave Spectrum of 1,3,2—dioxaborolane for several reasons.
One was that the value of the dipole moment was of great interest to
the Ohio State chemists; also there was his general interest in micro-
wave studies of small ring compounds, with the added stimulus of the
uniqueness of this small boron heterocycle having sufficient vapor
pressure at -78° to permit its gas phase spectrum to be examined.

Dr. Schwendeman found that 1,3,2-dioxaborolane was a suitable
and interesting subject for microwave spectroscopic experimentation.

The Spectrum was fairly intense and he estimated the dipole moment to

6

be about 2 Debyes. He observed rotational transitions for the ground
vibrational state and similar transitions for an excited vibrational
state. The relative intensity of this second set of transitions in-
dicated that the vibrational level involved must be quite low to be sohigI--
1y populated at dry ice temperature. It also seemed that the effects
of nuclear Spin statistics on the relative intensities were present.
His hypothesis was that the effects of a ring deformation vibration
were being observed, this being the most reasonable type of low-fre-
quency vibration which might be expected.

The present study was then undertaken to learn more about
1,3,2-dioxaborolane in general and about the vibrational effects in
particular.

In presenting the methods, results, and conclusions of this
study, a general theoretical background to rotational spectroscopy
will first be discussed. Next the Spectrometer which was used will
be described. Then the methods used to analyze the rotational Spectrum
and the basic results of the analysis will be presented. Finally,

a model system for the vibrational effects will be considered, and
the observed effects and their implications will be discussed in light
of results on similar molecules.

The final chapter in this thesis presents the results of a
study concerning the problem of the calculation of the effects of

centrifugal distortion on rotational spectra.

II. THEORETICAL BASIS OF ROTATIONAL SPECTROSCOPY

2.1 Introduction

 

Molecular rotational spectrosc0py is most conveniently considered
in terms of the angular momentum properties of a model rigid rotor.
This approximate description of a molecule predicts the major features
of its rotational Spectrum. The effects of an electric field (Stark
effect), a quadrupolar nucleus, and vibrational motions, can then be
treated as perturbations to the rigid rotor results.

The theory of the rigid rotor is presented first, including a
discussion of moments of inertia and principal axes, and a quantum
mechanical treatment for symmetric and then for asymmetric top mole—
cules. The second section introduces the perturbation results for
the Stark effect and for nuclear quadrupole coupling. The third
section is a discussion of how a molecular structure may be determined
from the frequencies of rotational transitions. Finally, a theoretical
approach to vibration-rotation coupling is presented, along with some
implications of this coupling for the determination of molecular co-
ordinates. The consideration of vibrational effects is continued in
Chapter IV in order that the theory for each of several perturbations
might be presented at the time the vibrational effects which were

observed in the rotational Spectrum are reported.

2.2 Rigid Rotor Model

 

A) Moments of Inertia and Principal Axes

 

In the rigid rotor model a molecule is considered to be a

7

8

rigid body composed of n point—masses with the entire mass of each
atom concentrated at its nuclear position. If an arbitrary Cartesian
. I O O ' O Q I 0 '
coordinate syStem lS eStabl1shed, each mass m1 at a dIStance ri from
i i l

the origin has coordinates Xi’ yi, 2,. The moment of inertia about

the c axis is then defined as

n
10.0.: 21 m1<Biz+Y:)) G. +B+Y=X':y'sz'° (2-1)

The product of inertia IGP for a i B is defined as

n
Ia, = _ Iii midi ,. (2-2)

To shift the arbitrary coordinate system to a center of mass
system, the parallel axis theorem may be applied. To apply this
theorem, the coordinates of the center of mass in the original system;
xo,yo,zo, must be established by applying the center of mass condi-
tions to obtain

n
0.0 = —1- 2 mini etc., where M = 2 mi. (2-3)
i=1 i

3

Then by the parallel axis theorem

2 2
la = Ia - Weo + v0), and

, <21)
IGB - IGP - M(aOB0).

. .. . .. . -> . . . .
Alternat1veLy, the locat1on veCtor ri of each atom 1n the center-

of-mass system is related to its original vector ?? by the vector

—> —> ‘->1 —> -> 7> 7> ._> .
r0 as r1 = r1 - ro where r0 = x01 + yoJ + zok. Then the

moment of inertia tensor or dyadic I may be written in a dyadic nota-

tion for the center-of-mass system as

I = 2 min: i> - “E; - F?) (2-5)
1

where I) is the unit dyadic.

A tensor P, the second moment tensor, is defined by

PXX ny PXZI

P = , Pyx Pyy Pyz , (2-6)
P P P
L ZX zy ZZ

 

 

with elements of the form

2; ma; ,
1

"U
u

1

(2-7)

p
xy

Pyx = 2%: mixiyi .
.. .. . . -1/ .. .
If p01nts proportioned to the values of I 2 are p10tted 1n
a Cartesian coordinate system which has its origin at the center of
.. . .. . . . . -1/ .
mass and its axes 1dent1cal to those about which I 2 was evaluated,
the surface which is described is designated the momental ellipsoid.
The principal axes of this ellipsoid are the "principal axes of inertia"
or simply principal axes. In practice the principal axes of a mole-
cule are located by rotating an arbitrary center-of-mass axis system
until the products of inertia vanish.
The moment of inertia tensor may be regarded formally as a sym-
metric matrix. Therefore diagonalizing I' by the similarity trans-
formation SI'S = I is equivalent to rotating the arbitrary axis

system to the principal axis system. The same transformation

lO
diagonalizes P, since P commutes with I. The eigenvalues of I and

P are related by the equations

+ P , etc., (2-8)

I .
and P = §II$ + IY - I I, etc. (2‘9)

The procedures for locating the principal axes may be simplified
for molecules which possess elements of symmetry. The following rules
may be used to determine principal axes by inspection:

1) A plane of symmetry contains two of the principal axes and

is perpendicular to the third.

2) A two-fold axis of symmetry is a principal axis.

3) A three—fold or higher axis of symmetry is a principal

axis and the other two axes are any two mutually perpendicu-
lar axes which pass through the center of mass and are per-
pendicular to the symmetry axis.

The utility of principal axes and the dependence of a rotational
spectrum on the principal moments of inertia are discussed in the
following section. The determination of molecular coordinates from

observed moments of inertia is discussed in section 2.b.
B) Quantum Mechanical Treatment of Rigid Rotation

The quantum mechanical treatment of rigid rotation utilizes the

Hamiltonian operator,

H = E [T— + I + _] , (2‘10)
x y z

where Pi, P;, P: are operators corresponding to the square of the

ll
projection of the total angular momentum along the x, y, 2 molecule-
fixed principal axes, and Ix’ Iy’ Iz are the principal moments of
inertia.

The angular momentum operatmxsare defined by the commutation

relations,
PXPz - PZPX'E (Px’ P2) = -1Pz
(Py, P2) = -1PX (2-11)
(Pz, PX) = -1Py

From (2-11) it may be shown that the operator for the square of the
total angular momentum P2 commutes with the square of each of the

three components; that is, if

2 .
P2 = Px + P; + P: , then (2-12)

(P2, Pi) = (P2, P;) = (P2, P2) = 0. (2-13)

Combining the convention for designating the eXperimentally
observed moments of inertia as Ia, Ib, IC with the requirement that

I f Ib _<_ I a molecule may be classified as shown below.

 

 

a c’
Name Requirement
Spher1cal top Ia = Ib = IC
Symmetric top
L1near Ia = O, Ib = IC
Prolate Ia + Ib = IC
Oblate Ia = Ib + IC

Asymmetric top Ia + Ib + IC

12
In discussing symmetric and asymmetric tops the question of
how to identify the x, y, z axes with the a, b, c principal axes

arises. The six possibilities are

 

 

Ir 11 IIr 111 IIIr III1
x <—> b c c a a b
y <+> c b a c b a
z <e> a a b b c c

 

The following discussion will assume an oblate top and axis
identification 1111; to obtain the corresponding expressions for
a prolate top it is necessary only to interchange a and c and
also interchange K+land K41’ This will correspond to axis identifica-

. . I‘
t1on I .

Symmetric Tgp.- For the case with two equal moments of inertia

 

in (2-10) the SchrBdinger wave equation may be solved in closed form.
The resulting eigenfunctions and eigenvalues are labeled by three
quantum numbers, J, K, M,

J

0,1,2, .......
where J(J + l)h2 is the square of the total angular
momentum of the molecule,

K = 0,: 1,: 2, ......, i J

where Kh is the projection of the total angular
momentum along the molecule-fixed z axis, and

M = 0,: 1,: 2, ......., i J

where Mh is the projection of the total angular mo-

mentum along the Space fixed Z axis.

13

The eigenfunctions qJJKM satisfy the operator equations

PZWJKM = J(J + 1) hZLIJJKM ,
PZL‘JJKM = K h LVJKM , and (2-1h)
IEVnm = MPW.MM'

Matrix elements in thelP JKM basis set may be constructed for

the operators in equations(2-ll). The non—zero elements are

<K‘PXIK i 1> = - <K|PyIK i l> = ‘ éifi [J(J + 1) ' K(K i 1)]1/2:
< KIPZIK > = hK,
(2-15)
< KlPiiK > = < K‘PZIK > = 162[J(J + 1) ‘ K2];
Y

< K‘P:|K : 1> = - < K|P51K i l>

' ‘ I62[{J(J+1)-K(Kt1)}{J(J+1)-(Kr1)(Kt2)liL/Z,
< K|P:|K > = hZKZ,

where

< KIPX|K> = <J,K,MlPX|J',K',M'> for J = J', M = M',

and

<.J,K,M‘PX|JI’K13M1> = ij/j,K,M Px UJJ',K',M' d7.

The arbitrary phase factor has been chosen to make Py real positive
and Px imaginary.

The eigenvalues for free rotation correSponding to the LP JKM
eigenfunctions depend onLy on J and K, and the rotational constants

B and C as

wJK = h[B(J)(J + 1) + (c - B)K2] (2-16)

where B = 11 , C = h

8n21 81121C .

 

 

lb

Asymmetric Top.- The rigid asymmetric rotor problem has no

 

general closed-form solution. It is best treated in a matrix formu-
lation; fortunately the matrices encountered may be factored into
direct sums of finite matrices. Consequently, the numerical problem
of diagonalization is well-suited to high Speed digital computers.

Since the symmetric top eigenfunctions form a complete orthogonal
set when an infinite number are taken, a suitable combination of
them may be used as a basis set to solve the rigid asymmetric rotor
problem to as high a degree of accuracy as desired. The total
angular momentum operator P2 commutes with the asymmetric rotor
Hamiltonian, as does PZ , but PZ does not. Thus J and M are
still "good" quantum numbers but K no longer is.

The Hamiltonian in matrix form may be represented as

H=AP£+BP§+CP§ (2-17)
ax”- as”
where P: , etc. indicate the matrices whose elements are given in

M!

(2-15). The desired eigenvalues are the elements of the matrix W,
obtained by reducing H to diagonal form (diagonalizing H). This

is accomplished by the similarity transformation
I H I = w (2-18)

The task of diagonalizing the whole Hamiltonian is greatly
simplified by three properties of H,.

(1) Since there are no non-zero matrix elements listed in (2-15)
which connect levels of different J or M, the entire matrix is
diagonal in J and may be separated into a block for each J and M.

Each block will be of dimension 2J + 1 corresponding to the possible

15

values of K. Note that this also permits a finite number of symmetric
rotor eigenfunctions to be used as the basis set for asymmetric
rotator functions of a given J.

(2) Since H_ has elements only of the type <K|K> and <KlK i 2>,
none of the elements of odd K are connected to elements of even K
and the blocks may be further factored. A unitary transformation due
to Wang (l6), X:H X = H), accomplishes this. Here H} is the
Hamiltonian in the ”Wang symmetric rotor basis". A block of X, has

the form

— —

+1 O O O +1
O +1 O
_£_ f7

O O O O
r2

O +1 O

 

L+l O O O -l

 

This transformation is equivalent to taking linear combinations of

the basis functions as

91,0 = W J,O ,

Bil
ll

1
+ —(LIJ +l+j _ )3
JK fl J,K J,K

1
QJ,K- = _ (LPJ,K ' )1) J,-K-)’

J7

The Wang transformation factors each J block into four sub-blocks
labeled E+, 0+, 0-, E- since each contains only elements of odd or
even K and the symmetric (+) or antisymmetric (-) combination of

the wave functions.

l6
(3) The Hamiltonian may be recast to a "reduced energy" form
by assuming a general linear transformation of the variables A, B, C

as
A' = 6A + P,

B' - GB + p, (2-19)
C' = CC + p.
(Note that the operators Pg, P;, P: do not depend on this scaling
of A, B, C.) The rigid rotor Hamiltonian in terms of the primed
constants is
H (A', B', c') = A' P: + B' P; + c' P: .

Substituting from (2—19)
H(A',B',C') = [(0A + p)P: + (GB + P)P§ + (0C + p)P:]. (2-20)
Collecting terms in o and P

H(A',B',C')

2 2 2 2 2 2
o(APa + BPb + CPC)+ p[Pa + Pb + PC]

0 H(A,B,C) + 9 P2. (2-21)

Thus the original Hamiltonian H (A,B,C) may be written as

H = H'/o - p/b P2. (2-22)
Of the many possible choices for o, p two very useful ones are

those of Ray (17) and Wang

 

Ray Wang
_ 2 O = 2
° A—C 'T2CIBTK7
p - “:2 p it?
A-B
1 = 1 _
A 1 A m 130
23-1-0 ” B-A
' = = ' = —— = _
B A-C *’ B 2C-B-A b0

c' = -1 c' = 1 .

17

In both cases it is seen that H(A',B',C') is a function of one
variable, determined only by the relative Sizes of A, B, C. Thus
the eigenvalues of H' are "reduced" energies, functions of the
asymmetry parameter )( or b0.

The matrix result of this transformation assuming either of
the symmetric rotator bases described above is that 9/6 J(J + l)E,
a constant times the unit matrix, is subtracted from the diagonal
of each J block and a second constant, 0, is factored from the
remaining matrix block.

The total matrix result may be written as

W=TH(A£fifl
~~ w p 1~~ p~~
= T X [ =._ - P2] X T =.— T X H X T- T X P2 X T (2-23)
~~ o o~ ~~ o~~-~~ o ~~~~~
and since I'X P2 x_I = J(J + 1)E, w_ is diagonal when T'X H} X I.

is diagonal.

The rigid rotor energy may be written aS

(2-2b)
wr/h= A+C TJ(J+ 1) +A_§_ CEJ()(), or w /h = fig? J(J+ 1) + 29.,_,§-'-A-1.1J(bo)

where EJ (§() and wJ (be) are the reduced energy eigenvalues of
T T

H' = P2 - P2 + «P2 and
a C , b (2-25)
H' = P: - bo [Pb - P2] reSpectively, and are

calculated by numerical methods.

The original utility of the reduced energy transformation was
that tables of reduced energies needed to be prepared at intervals
of only one parameter, the asymmetry parameter. Entries for an arbi-

trary value of the asymmetry parameter could be obtained by interpolation.

18
The availability of computers has eliminated the use of tables for
most work, but the reduced energy method still has advantages for
these calculations.
Partial derivatives of the energy levels with reSpect to the

rotational constants, for equations such as

1 [air 9 Wr a wr

dw - mdA +WdB +Tdc], (2—26)

r — B
may be calculated at the same time that the energy levels are com-

puted. These derivatives are just the diagonal values of 2:, 5;,

E: calculated in the same basis set as the energy, i.e. the diagonal

R'E: E I_etc. where I, was the matrix which diagonalized

?“N

values of
the Hamiltonian. These derivatives are used, as will be shown later,
in calculating the nuclear quadrupole coupling energies and in deduc-
ing rotational constants from an observed spectrum.

For the asymmetric rotor, K is not a proper quantum number so
it is necessary to label the eigenfunctions and energy levels with
another quantum number. The easiest way is to number the energy
levels in each J block in order of increasing energy; the symbol 7
is used and assumes the integer values from —J to +J . A second,
more useful way depends on the energy limits which are approached as
the asymmetry tends toward each of the symmetric rotor limits. As
§Q4> -l, the prolate symmetric rotor limit, the calculated energies
approach those of the prolate symmetric rotor. Similarly as X—> +1,
the oblate symmetric rotOr limit, the calculated energies approach
those of the oblate symmetric rotor. Thus each asymmetric rotor

energy level may be labeled with the K values of the symmetric rotor

19
1
levels which it "connects" as J ; K being the prolate value
K419K+1 ‘1
and K+1 the oblate value.

The following relations exist between T and K_1, K+13

T = K-l 'K+l )
K - J t T + { O } (to make K11 integral).
:1 ‘ 2 l/2 (2-2?)

The frequency of a transition in the microwave spectrum from an
initial level of energy Wl to one with energy WZ is given by the

Bohr condition v = W2 ' W1 . (2—28)
h

A transition is allowed only if it may have a non-zero intensity,
This restricts AJ to values of -l, O, 1 (called P, Q, R branch transi-
tions) and AK_1, AK+1 to values listed below depending on the
principal axis orientation of the permanent electric dipole moment u.

Component of u

along AK‘l AK+1
a axis (a type transitions) even odd
b axis (b type transitions) odd odd
c axis (c type transitions) odd even

The dipole moment for an asymmetric top molecule does not
necessarily lie along one principal axis (as it must for a symmetric
top), but may have components along any or all of the axes. Thus the
a, b, and c type transitions are not in general mutually exclusive.
Symmetry within a molecule, however, may limit the orientation of u
and allow components along only one or two of the principal axes.

The symmetry of the rigid asymmetric rotor wave functions has
been examined by King, Hainer, and Cross (18). The Hamiltonian

a
belongs to the Four-group, V(a,b,c), which has the operations B, C2,

20
C2, C2. The character table for V(a,b,c) is shown below, with the
final column listing the symmetry class for a rotational level of
JK_1K+1' It will be noted that the symmetry depends only on the

relative odd and even nature of the K's.

Character Table and Symmetry Classification for the Group V(a,b,c)

Species representation Opergtion b c Symmetry of
E c2 c2 c2 K-1K+1
A +1 +1 +1 +1 2 e
Ba +1 +1 -1 -l e o
Bb +1 -1 +1 -1 o 0
BC +1 -1 -1 +1 0 e

e = even, 0': odd

2.3 Perturbations to the Rigid Rotor

A) Stark Effect

 

The rigid rotor energy levels are (2J + l)—fold degenerate in
the spatial quantum number M in the absence of a unique space-
fixed axis. The application of an electric field establishes a
unique (field) direction, producing a molecular Stark effect through
the interaction between the field and the molecular electric dipole
moment.

The Stark effect eigenvalues are obtained by adding a potential
energy term Jfi>° E>(E?is the molecular electric dipole moment vector
and E? the electric field vector) to the rigid rotor Hamiltonian.

E? is then transformed from its molecule-fixed axis system to the

Space—fixed axis system by the direction cosine transformation,

21

”P =§ QFgug , (2'29)

F X, Y, Z, the space fixed axes,

g = x, y, z, the molecule fixed axes.

The solutions for the non-degenerate asymmetric rotor take the form (19)

w = 62 2
s (1: (A0. + Bab/12mg, a = a, b, c, (2-30)

where Aa, Ba are calculated from second order perturbation terms

 

of the type
~~ 2
l i I
[< J,1-,M|T X Em X Tl J ,T ,M >] (2_31)
)
wJ,T‘_ WJ',T'
where 6 is a direction cosine matrix. The solutions are functions

Fa

of the quantum numbers and molecular parameters.

The three major applications of the Stark effect are: (l) to
modulate the absorption signals, thereby greatly increasing instru-
mental sensitivity; (2) to aid in assigning quantum numbers to an
observed transition by the characteristic pattern of the Stark ef-
fect; and (3) to determine the molecular dipole moment components
from the change in the perturbation energy with a change in the

electric field strength.

B). Nuclear Quadrupole Coupling

 

The measurement of nuclear quadrupole hyperfine effects in rota-
tional spectroscopy is of interest because the quadrupole coupling
constants obtained provide information about the electronic configura-

tion (bonding) in the molecule.

22

Quadrupole effects arise from the coupling of the nuclear spin
angular momentum of an atom possessing a nuclear quadrupole moment
with the rotational angular momentum. This coupling is the electro«
static interaction between the non-spherical (quadrupolar) nuclear
charge and the entire extra-nuclear charge distribution. The inter-
action energy is different for each orientation of the nucleus with
respect to the entire molecule.

The method of calculating the perturbation energies produced by
the presence of one quadrupolar nucleus in an asymmetric rotor will
be briefly outlined, utilizing the results of Casimir (20) and Bragg
and Golden (21). The energy is a function of the nuclear quadrupole
moment Q, the nuclear spin I, and the gradient of the electric field,
if e, at the quadrupolar nucleus. The field gradient is equal to
fins? V , where V is the electrostatic potential at the nucleus due
to the entirecndra-nuclear charge distribution.

The perturbation Hamiltonian for the interacting nuclear quadru-
pole is

HQ = eQ<eEL

[ 3(3>.f>)2 + 3/2(T>-f>) - 1212 J
a 22 Av.

21(21 - 1) I (21 - 1)

 

(2-32)

where e is the electronic charge,

J is the total angular momentum for the rigid rotor molecule,

 

and 32V . . . . . . . . .
is the average field gradient at the nucleus in the
922 AV.
2 direction, the direction of J.
The numerator of the term in square brackets has eigenvalues
derived by Casimir to be 3/h c(c + 1) - 1(1 + l)J(J + 1),
where C = F (F + l) - I(I + l) - J(J + l), and (2_33)

F=J+1,J+1-1,-~,|J-I|.

23
The total angular momentum quantum number F is a constant of the
motion and hence a good quantum number. The levels are normally
labeled with F, J, T, where J and T are the quantum numbers of
the level in the limit of zero coupling. Therefore the effect of a
quadrupole coupling is to split each pure rotational level in 21 + l

(or 2J + l, whichever is smaller) levels.

2
Bragg and Golden evaluated<g§zgy using first order

 

Av.
perturbation to be
W w w
2 6-1' d.r' d.r
M23 + 370 + l) [ qaa 7‘ + qbb m + (ICC _3_C ], (2-3b)
2
V
where qaa etc. are the field gradients 5—3 in the direction of the
a

principal axes,
sir
31V

By defining several new terms the quadrupole interaction energy

and etc. have been defined in (2-26).

may be more simply expressed. The quadrupole coupling constants are
Iaa = eQ qaa’
Ibb = 80 qbb’ and (2'35)

ICC

and within the validity of the Laplace equation (6 {7 v = o),

80 qcc’

:Xéa +jyibb +:3(cc = 0. Thus there are only 2 independent coupling
. . . . . _ . . ,
conStants, which may be taken to bejyfaa and (:(bb :Kicc)' Ca51m1r 3

function Y(F,I,J) is

_ 31h C(C + 1) - 1(1 + l)J(J + 1)
“Film " 21(21 - 17(21 - l)(2J + 3) (2‘36)

and has been tabulated for values of I and J.

2b
The quadrupole perturbation energy, WQ, may now be written as

w - 2 Y(FIJ) [fly +3wrx+ aw CrDCC] (2-37)
Q‘J(J+i) H 3A" aa aa bb

 

and further simplified to

NO aq'x aa + Bq(ybb - ICC) ) (2-38)
by defining

Y(F I J) [2 DWr dwr dwr

Gq=m —‘—'- 43- . and

A aB 3c

 

Y(F 1 J) 3W_};_ 5%»

2J(J + l)[ 3B aC‘J'

Values for the coupling constants may be obtained from an analysis

 

of the hyperfine Splittings observed in the rotational Spectrum.

2.b Determining a Molecular Structure

 

In section 2.2 moments of inertia were calculated from the struc-
ture of a molecule. The inverse procedure of determining an unambiguous
structure from the observed moments of inertia is not so simple.

Three rotational constants are the maximum available data, but in
general there are 3n - 6 coordinates to be determined, although this
may be reduced by symmetry assumptions. Isotopic substitution,
therefore, is used to provide more information: an isotopic atom is
substituted for an atom in the original molecule and another set of
rotational constants is obtained from the spectrum. To gain informa-
tion in this way it is necessary to assume that the structure is un-
affected by the substitution. This is precisely valid only for the

equilibrium structure, since the rotational measurements are time-

2S
averaged over the ever—present vibrational motions, and these in turn
are altered by isotopic substitution. Therefore some of the high
precision of microwave measurements is sacrificed to vibrational ef-
fects in order to gain structural information.

In 1952 Kraitchman (ll) systematically described the details of
determining a molecular structure from isotopic substitution data,
and the following discussion is based on his work. It is necessary
to neglect vibrational effects at this point, but they will be
considered later.

For a molecule in its principal axis system
:5 m.a.
. 1 1
1
:E m.b.
. 1 1
1

O

3

n
O

, center of mass conditions (2-39)

2; "Hci = O ,
1
Pab ’ 2;: miaibi = O ,
Pac = o ’ principal axis conditions (2—hO)
pbc = O ’
= Z m.a.z
aa . 1 1 ’
1
P =

2
bb 2%: mibi ,

p
CC

2 miciz, and
i
M =22: mi , the molecular weight.
i

. . th . . . . .
If the mass of the 5 atom is increased by an amount Am, the

new principal axis system will occur with its center of mass shifted

by

26

AmaS AmbS Amc

s n . . .
MIKE , Milfi’ MIKE Irom the parent center of mass and
generally rotated with reSpect to the parent axes.

The elements of P for the substituted molecule, calculated in

the parent principal axis system, will be of the form

 

( :E:m' a.)2
P' = mea? ___1__i_1._, etc.,
aa 1 1 1 j; '
-:-m.
1 1
(:m'aHZm'b) ‘2‘“)
P,=Emab_iiiiii
ab i O. O

I
i 1 1
:Efm'
1 i

Since the only difference between these summations and those for the

parent is that m' = mS + Am, the equations reduce to
s

 

2
' (Am)2 aS
aa = paa + Amas - M + Am
-P + 2 et
- aa “as ’ C"
(Am)2 asbS
I = _ _
Pab AmasbS M + Am (2 A2)
= uasbS , etc.,
_ MAm
where H - M:Efi .

The experimental second moments (P28) for the substituted molecule

are the roots, P", of the determinant

2 _ n
Paa + l“L33 P LLaSbS uaSCS
2_ n = _
uaSbS Pbb + ubs P ubscS O (2 b3)
2 _ n
uaSCS lil‘bscfi PCC + “C3 P

 

 

27
When the determinant is expanded and the coefficients of powers

of P" are set equal to the respective coefficients for the polynomial

I I! I I II
+P)+P"(PP +PP +PP)-
CC aa

I I
"3 _ "2
P P (Paa + P bb aa ca bb cc

bb

I I I

P P P =o (2-AA)

aa bb ca
three equations in the three coordinates result. These may be solved

to give equations for the coordinates as

 

(P p )(Pbb-P )(p —p )nl/2

)a | = -1[ aa— aa aa' cc aa
5
(Pbb-Paa)(pcc-Paa)
(2—us)
P” -P P" -P 1
= {u 1(pn - p )(1 + 213—1112)“ + _C9_E£)} /2.
aa aa
P -P P -P
bb aa cc aa

Equations for )b3) and ‘Cs‘ may be obtained by cyclic permutations
The signs of the coordinates must be determined from other in—
formation, often chemical information such as the bonding skeleton
and plausible values for bond lengths and bond angles. Furthermore,
for complete substitution at all non—equivalent sites the final co-
ordinates with their signs must satisfy the center-of-mass conditions

and give reasonable values of Ia’ I and Ic'

b)
2.5 The Coupling of Vibration and Rotation
Some of the problems of vibrational origin which are encountered

in determining a molecular structure may be demonstrated by consider-

ing the perturbation of vibration-rotation coupling in some detail.

28

The following discussion will also form the basis for the discussion
of the vibration-rotation coupling observed for 1,3,2-dioxaborolane.

The effect of molecular vibrations on the rotational constants
is rather straightforward in principle but highly complex in prac-
tice. The evaluation of the effects of vibration-rotation coupling
is usually stymied by the lack of information concerning the normal
coordinates and the vibrational force constants. There are two
vibrational effects on the rotational energy levels, and thereby on
the rotational constants, which must be considered.

The first effect arises from Coriolis coupling. This coupling
is due to the fact that the vibrations are observed in a rotating
coordinate system. As a result an angular momentum term associated
with both the vibrations and the rotation must be included in the
total Hamiltonian. Generally the large contributions of angular
momentum arising from degenerate or near—degenerate vibrations are
absorbed into the vibrational Hamiltonian. The remaining contributions
are included in the rotational Hamiltonian, since a perturbation cal—
culation for them.yields terms which are quadratic in angular momentum,
and hence are indistinguishable from the terms arising from ordinary
rigid rotation. The result of this inclusion of Coriolis terms is
to alter the rotational constants, making them no longer strictly
proportional to the inverse of the moments of inertia at equilibrium.

The second effect arises from the dependence of the values of
the principal moments of inertia on the actual vibrational motions
of the atoms. This dependence may be elucidated by considering the

vibrational motion of each atom in the molecule.

29
To start, the Cartesian coordinants measuring the instantaneous
diSplacement of each atom from equilibrium may be written as a linear

combination of the 3N—6 normal coordinants QK’

N-6
1
6a. = m./2 1? Q
1 1 1K K’
K=1 (2-A6)
e inst
where 6 a. = a. - a. '
1 1 1

The six Eckart conditions (23), :g: miS’ai 0 etc.,

(2-b7)
2% «I: si + a: w

. 0 etc.
1

complete the transformation. These conditions assure maximum un-
coupling of the vibration and rotation, and establish a linear
dependence between the 3N-6 normal coordinates and 3N Siai.

The instantaneous moments of inertia may then be expressed in

terms of the equilibrium and displacement of coordinates as

inst. _ inst 2 inst 2

2 mi”? + Yiz] + 2:, "HHS: 5‘31 + Y: 5Y1] (2-A8)

i

+ . mi[(6Bi)2 +(5Y,)21,

1

and similarly for the products of inertia,

inst _ e e _ e e
ICLB ‘ ’ 2 mi CIi‘3i 2, [mi O'i 6‘31 + Bi Sci] (2419)
- Ziniwaikéai).
The first term in (2-h9) is zero by the definition of the equilibrium
principal axis system, and the second term is zero by the Eckart

condition.

30

These expansions demonstrate that the vibrations contribute both
diagonal and off—diagonal terms to the instantaneous moment of inertia
tensor. In order to express the effective moments of inertia, it is
necessary to average the instantaneous values over the vibrational
motions. The correct averaging is of the rotational constants over
the vibrational motions, since this is what actually occurs in an
experimental measurement. Therefore, it is necessary to invert the
instantaneous moment of-inertia tensor before taking the vibrational
average.

inst -1
)

The elements of the inverse instantaneous tensor (I “B may

be expressed as power series in the normal coordinates as

(11”; “LE- 2 a‘“3 0K -% % (A358, - % afiYafiY/ijmKoM -

1013
where glinSt ‘BZI lnSt
QB _ a QB _ a _
8K - (WE ) and AKM ‘ (FUR—39W). (2 50)

After averaging over the vibrational motions, the resulting expressions
are re—inverted to yield effective moment of-inertia terms.

1>]-1‘='IV=Ie (LB-+2 aCLB<QK>+:(AaB_ aaYaBYY/I)
Y

inst ‘
MI ) 0B aeg

<15
2
< OK > + --' . (2-51)

The cross terms of < QKQM > contribute to higher order than that
indicated for the expansion (2—51). Next the vibrational averages of
the Coriolis terms must be added. Herschbach and Laurie (2A) have

shown these to be

. ' a P 2
- )4 %2M [imeM (W ]< OK > , (2‘52)

31
a J:B . . . . . .
where of KM’ KM are the Coriol1s coupl1ng conStants,
and XK’ AM are the vibrational eigenvalues.

The final result is that the effective moments of inertia may be

written as an expansion in average values of powers of the normal

coordinates,
eff e K ‘ K 2
1 =I+ZC<Q>+§C<Q>+”‘ (2—53)
do a K 1 K K 2 K

The coefficients C1 etc. may be evaluated if sufficient data are
available. When this is not possible, they may be treated as
empirical parameters.

Two further simplifications are possible. The first is the
single mode approximation. In this scheme contributions to Ieff from
all vibrational modes except the one being considered (K = l) are
assumed to be separable. It is assumed that the interactions of the

other modes may be included in the coefficients Ci, c; etc. As a

result Ieff becomes

GP

I
Izgf = IE, 8&5 + c}' <:Ql > + c; <0? > + ...o (2-5A)

This approximation is generally used when there is one mode which
dominates the vibration—rotation interaction, and little or no data
is available for the other modes.

The second simplification results in the familiar vibration-

rotation coupling equations

BV =>Be - 2%: aK(vK + 1/2 dK), (2-55)

and the equivalent equation I = I + :E: 6 (v + 1/2 d ). These
v e K K K K

32
equations may be obtained from (2—53) by noting that the quantum
dependence of <10; > for an harmonic oscillator is (vK + 1/2 dK)'
Furthermore, the leading terms in a perturbation treatment of
<.QK > have this dependence also. Consequently the coefficients of
all the terms may be combined to form the GK of EK' Most of the
values of GK which have been determined are small positive numbers.

The preceding discussion indicates that in cases where the
vibration-rotation coupling is small, the rotational spectrum can
retain the features of a rigid rotor. The effective rotational con-
stants depend on the vibrational state, and there may be several
rotational spectra, each due to a different vibrational state. The
degree to which each spectrum fits a rigid rotor pattern is a measure
of the relative perturbation caused by the coupling of vibrations.

To examine the effect of vibrations on molecular structures
determined from isotOpic data, it is necessary to ascertain which
quantities change in value with the substitution.

The most obvious changes are in the vibrational frequencies of
normal modes which contain large motions of the substituted atom.
For diatomic molecules GK is proportional to (VK)-1’ so the de-
crease in the vibrational frequency which occurs upon substitution.

of a heavier isotope causes an increase in This means that the

EK -
effective moment of inertia for the substituted molecule is even
larger than would be predicted by the simple mass increase.

A second, less obvious, change upon isotopic substitution is in

the effective bond lengths. These in general decrease with the substi-

tution of a heavier atom. This is due to the fact that the energy

33

levels are now lower in the vibrational potential well, so the ampli-
tude of vibration is decreased, and the average value of the bond
length is smaller. This effective bond shortening has the greatest
primary effect when deuterium is substituted for hydrogen, but also
may be observed as a secondary effect on a bond A—B when C is the
substituted atom.

A third quantity which changes on isotopic substitution is the

inertial defect. For the equilibrium structure of a planar molecule

I: — I: - I: must be zero. A vibrating planar molecule has an
inertial defect A defined as A = IV - IV - IV . (2-56)
v v c a b

Actually, the inertial defect is a single term containing the effects
arising from vibration-rotation coupling for the totality of the
vibrational motions. It is a convenient parameter because it has
been determined for many planar and near-planar molecules, and good
methods for calculating its value have been devised, most notably in
the work of Oka and Morino (2S), and Laurie and Herschbach (26).

The fact that the inertial defect is a single term for all the
vibration-rotation coupling makes it convenient both for numerical
calculations and qualitative explanations. In addition the concept
of a quasi-inertial defect is a convenient one in examining the struc—
ture of molecules which are near-planar, or which are composed of a
large planar portion with a small out-of-plane component; for ex-
ample, a molecule like acetaldehyde in which only two of the methyl
hydrogens do not lie in the symmetry plane. In such a case the non-

zero 2 coordinates could be obtained from the equation

hm“ 2% = IX + 1% - 18 + A0 (2_57)

3h
if an estimate of A0 is available; or with substitution of deuterium

for the hydrogen nuclei from the equation

D H
h(mD - mH) z; =(IX + 1% - 18 D - (IX + 15 - 18)H + (A0 - A0) (2-58)

if the difference (AB - Ag) may be estimated or assumed zero.

The following statements are generally accepted as guides to the
problems and errors which are encountered in the determination of the
parameters of a molecular structure (12).

1) Parameters which are obtained by using experimental moments
of inertia in equations which are satisfied only by equilibrium
values, generally deviate from equilibrium values. A structure calcu-
lated in this fashion is called an rO structure.

2) Coordinates which are obtained by using only differences in
the moments of inertia of two isotopic Species also differ from the
equilibrium values, but are generally closer to the equilibrium values
than are the ro parameters. Furthermore, the coordinates are
generally smaller than the equilibrium values when the substitution
introduces a heavier isotope. A structure calculated solely from the
differences in moments of inertia of pairs of isotopic species is
called the rs structure.

3) A large uncertainty is expected in the location of a light
atom, particularly an H atom, if its r0 coordinates are determined.

A) A large uncertainty is expected in the coordinates of an atom
located near an inertial plane. In general the calculated distance
from the plane will be less than the equilibrium distance. Even if
this distance was determined from equilibrium values of the moments

of inertia, its uncertainty may be large, being proportional to the

35
uncertainty in I8 divided by the distance from the plane. A classic
example of this type of behavior is the linear molecule N20. Sub—
stitution of the heavier isotope 15N for 14N in 15N14N160 produces
a decrease in the moment of inertia. As a result an imaginary value
for the 14N coordinate is obtained. (27).

The general validity of these statements may be demonstrated by
some simple considerations based on the simplified results of vibra-
tion—rotation coupling which were presented above.

The greatest uncertainties in the structural parameters are due
to the vibrational motions which shift the effective moments of inertia
from their equilibrium values. For ro structures, all the vibra-
tion-rotation coupling is absorbed into the coordinates; necessarily
moving the atoms to compensate for the differences (Io-IQ). This is
a result of presuming that the values of I0 are compatible with the
equations in the equilibrium coordinates. Less uncertainty is present
in rS structure, since in this case the atoms are moved to compensate
only the differences ( 2E: (eé - 6K)). The non—equality of rS and

K

equilibrium coordinates which has been observed indicates that
( :E:(efi - EK)) is not necessarily small enough to be consistently

K
neglected. If a light atom must be moved to compensate the differ-
ence Io — Ie , the error in its coordinate will obviously be greater
than if a heavy atom had been moved. On the other hand, with the
substitution technique the error in the coordinates depends only on
the difference in mass between the isotopes, so a light atom may be

located with the same precision as a heavy atom.

III. DESCRIPTION OF THE MICROWAVE SPECTROMETER

3.1 Introduction

 

A conventional lOO kc/sec. Stark-modulated microwave spectrom-
eiLer was used in this work. It consists of the following elements:
a source of microwave radiation (klystron oscillator), a waveguide
absorption cell, a square-wave generator for Stark modulation, a
detection system, and a frequency measuring unit. Figure l is a

block diagram of the spectrometer.

3.2 Klystron oscillators

 

At Michigan State University the sources of microwave power are
reflex type klystron oscillators. In these devices a stream of
electrons is obtained from a heated cathode. They are drawn toward
an anode which is maintained at a positive bias, giving the electrons
a high velocity. Some of the electron beam passes through a hole in
the anode and is focussed to a gap which has been made part of a
resonant circuit. An external radio—frequency field established in
this circuit "bunches" the moving electrons by slowing or accelerating
them depending on which part of the field cycle they encounter as
they pass through the cavity. This series of electron packets then
is reflected back into the resonant cavity by a negatively charged
reflector. As they return through the cavity, the electrons induce
a large r-f field and lose energy in the process. If the radiant

energy emitted by the returning electrons is in the correct phase the

36

37

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

uowmaflwumo
7 .oom}: H
.ooposoupooam m>mzouome HmMMMWom ;/
onp Mo smnmmmp xooam .H madman m no>mooom poumflfimumo
ompmm mm>
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nopnooom nepmuocoo - nowmfiawumo
spooeuzmm umpfimm mm:
mmwQISOA \
homo: V— awmwxm x
bosom _ .
nonoopoa olkl.moxm.%irv - >Haadm
o>mpmmcom ooh nozom
. . . . oncomoHHmomo
.0 Q 0 < Emomnfimso mdflm mcHCSH
\/ p _ , E pepooflwom
nommmflasmoum o>m3 onmsqm coupqux
or 8H .oon\os 8H \
.IIIIIIu, 11A _
Houpmou .lri
dozens? A L:
o>m3 meadow uooompoo Hopmxuo.u, hobo: an coupqux
\Icooooooo ssoaom storm .1 5:8 83983.4 courage: _ -cosoosm 2m

 

 

 

 

r e

 

 

 

 

 

 

 

. 38
klystron resonates, producing an extremely monochromatic milliwatt
microwave output.

Tuning the klystron is accomplished by varying the size of the
cavity, thereby changing its resonant frequency. The tuning range
is normally about 15-20% of the central design frequency. Changing
the reflector voltage continuously between the values necessary to
maintain oscillation provides fine tuning and normally has a range
of 3o-uo Mc/sec. The klystrons used in this work are listed in
Table I with their frequency ranges. The frequency meter and crystal
detector used for each frequency region is also listed.

The klystron is powered by an F.X.R. type 2815B power supply
designed to provide beam voltages from 200-36OO volts, reflector
voltages from O to -lOOO volts, and grid voltages from O to -300
volts, all highly stabilized. An auxiliary supply provides a stable
D.C. heating current for the filament.

In order to use an oscilloscope to observe transitions, the
klystron and oscilloscope must sweep over a short frequency range
simultaneously. The simultaneous sweeping is accomplished by coupling
the output of the X-axis sweep voltage from the oscilloscope to the
reflector of the klystron through a D.C. isolation network. The
klystron frequency then varies uniformly as the beam sweeps across
the face of the oscillosc0pe. Thus the X-axis of the oscilloscope
is a linear display of the klystron frequency. A sweep width of 1-10

Me is normally used.

39

Table I. Klystrons, frequency meters, and crystal detectors now in
use at Michigan State University.

 

 

 

Frequency Crystal Detector
(ko/sec.) Klystron Frequency Meter (Microwave
Associates)
8.2 to 12.6 Varian X-l3 Narda 810 1N23
. . 1N23
12.h to 18.0 Var1an X-l2 Narda 809 1N78
18'0 to 22.5 EMI/US R9622 DeMornay-Bonardi 1N26
DBE—7lS-2 1N78

22.5 to 26.0 EMI/US R9602
26.0 to 30.0 Ratheon QK289
DeMornay-Bonardi

27.9 to 32.3 EMI/US R9518 DBD-715-2 1N26

32.0 to 37.5 EMI/US R9586

 

'Ao
3.3 Waveguide Absorption Cell

 

The waveguide absorption cell is a ten foot section of rectangu-
lar brass X—band waveguide with cross section and dimensions as shown
below. The inner surface is silver-lined for increased conductivity.
Teflon strips with a 0.010 inch deep groove running their length are
laid along the shorter walls to support and insulate the Stark septum,
a coin-silver strip which runs the length of the cell. An electrical
contact to the septum passes through a hole in the teflon and is
sealed with an N—type connector and an O-ring seal. The entire cell is

sealed at both ends with mica windows and O-ring seals and is kept at
high vacuum. An attached vacuum line serves as a sample introduction

and recovery system.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

sm.r—'Teflon Strip
Orhoou l RLT Stark Septum
l raL-losou
e———-——-—0.900” ’7

Teflon tape: 0.062" x 0.800" with 0.032" groove 0.010" deep.
Coin silver septum: 0.032" x 0.796".

Figure 2. Stark waveguide cross section.

3.h Square Wave Stark Modulation

 

Modulation of the absorption signal is provided by applying
a 100 kc/sec square wave to the Stark septum. A second signal from
the square wave generator is used as the reference signal for a phase-

sensitive detector. If the klystron is at an absorption frequency,

111
the square wave produces a Stark effect which shifts the absorption,
in essence turning the absorption at the given frequency on and off
100,000 times per second. As a resultiflwzmicrowave absorption sig—
nal is modulated at 100 kc/sec and may thus be detected by a phase-
sensitive'detector (or lock-in amplifier). A useful Stark effect
requires rather high fields and consequently the amplitude of the

square-wave must be variable up to at least 1000 volts.

3.5 Detection System

 

Two types of microwave power detectors are commonly used,
silicon rectifiers and bolometers. The bolometer is a temperature
sensitive impedance element. A constant current passed through it
gives a voltage change as the resistance changes with incident
microwave power. The silicon rectifier type of detector used at
Michigan State University is a diode circuit element. It consists
of a thin sheet of silicon in which a contact wire is imbedded. An
A.C. signal is partially rectified to give a D.C. output voltage
roughly proportional to the input A.C. power. The crystal is held
in a mount which is tunable to provide maximum sensitivity.

The output of the crystal is sent to the inductance-capacitance
circuit shown in Figure 3. The D.C. component passes through the
parallel circuit (:) to a milliammeter to indicate the power level.
The 100 kc/sec component is applied to the control grid of the first
preamplifier tube.

The output from the preamplifier is applied to the signal input

of a phase sensitive detector; a signal from the square wave generator

A2
is applied to the reference input. If the signal from the preampli-
fier has the same frequency and phase as the reference, the output
voltage of the phase-sensitive detector will be increased; if the

frequency or the phase is different, the output will be suppressed.

 

Input from L
crystal > W > T0
Preamplifier

 

 

 

 

2.. c2

 

 

 

 

 

To Ammeter

 

llr

/\ /\

Figure 3. Detection preamplifier input circuit.

3.6 Frequency Measurement

Rough frequency measurements, as in initially locating a transi-
tion, are made with one of a series of resonant cavity frequency
meters whose calibrations are supplied by the manufacturers. The
klystron oscillator is driven either manually or with a small worm-
drive motor. When an absorption peak is noted on the oscilloscope,
the frequency meter is tuned until the power trace on the oscillo-
scope shows an 'absorption' resulting from loss of power to the meter
at its resonant frequency. The scale reading of the frequency meter
is then converted to the microwave frequency with a precision of

0.03% at best.

'h3

Precise frequency measurements are made by determining with a
Collins 51J—b radio receiver the difference between the microwave
frequency and a generated standard frequency.

The corner-stone of the standard frequency generator is a Manson
Model RDlAO crystal-controlled l Mc/sec oscillator. The oscillator
frequency is maintained by adjusting it until its 10th harmonic and
the 10 Mc/hec carrier wave of National Bureau of Standards radio
station WWV have a beat frequency of less than 1/2 cycle/second. Over-
all stability of at least one part in 107 is obtained.

The 1 Mc/sec signal is input to a Gertsch AM-lA VHF interpolator
composed of three units. The first is a frequency multiplier (VHF)
with a selective stepped output of 19—38 Mc/sec. The second is a
1-2 Mc/sec low frequency oscillator (LFO) which may be adjusted to
fixed known frequency by tuning until a stable Lissajous figure is
obtained against the 1 Mc/sec oscillator signal. The sum of the multi-
plier and LED signals is used to control the frequency of the third
unit, a 20—Ao Mc/sec oscillator which acts as a filter and amplifier.

The l Mc/sec oscillator signal and the 20-A0 Mc/sec output of
Am-lA are input to a Gertsch FM-AA microwave frequency multiplier.
The FM-hA may be tuned to lock 10 Mc/sec above or below a harmonic of
the 20-h0 Mc/sec input. The output is an accurately produced signal
in the range 500-1000 Mc/sec plus harmonics.

The standard frequency generated in the microwave region may be

calculated from the equation

Vstandard = n["1("Li=o + vVHF) i 10]:
n,m = l,2,3,-", (3‘1)
vLm = l-> 2’
vVHF = 19,20, , 38

."AA

The klystron output and the standard frequency signals are multi-
plied and mixed in a tunable crystal detector. The beat frequencies
from this mixture pass through a 500 Mc/sec low-pass filter to the
Collins receiver which functions as a tunable, extremely narrow band-
pass filter and amplifier. The carrier wave is removed and the out—
put of the receiver is the noise from the beat signal. This signal
may be applied to the Y axis of one trace of the dual-beam oscillo-
scope. As the saw-tooth voltage causes the klystron frequency to beat
against the marker with the (radio) selected beat frequency, a 'pip'
appears on the oscilloscope. Both positive and negative beats are
possible. For example; if the klystron is swept from 20,980 Mc/sec
to 21,020 Mc/sec by the saw-tooth voltage, with the marker set at
21,000 (the 30th harmonic of a 700 Mc/Eec signal), the beat frequency
will vary from 20 to 0 and then 0 to 20 Mc/sec. If the receiver is
tuned to 12 Mc/sec a pip will appear on the oscillosc0pe when the
klystron frequency is 20,988 and when it is 21,012 Mc/sec. In the
more usual procedure either the positive or the negative beat is used,
the klystron range is set so that it does not straddle the marker
frequency, and only one beat is observed. For example, if the marker
is at 30,000 Mc/sec, the klystron swept from 30,010 to 30,0A0, and
the receiver set at 20 Mc/sec, one pip will be observed at 30,020 Mc/sec.

Since the marker and absorption signals have different amplifica-
tion times, the average of two frequency measurements made with oppo-
site sweep directions is used.

Frequency measurements are made using the oscilloscope display
whenever it is possible to observe the transition on the oscillo—

scope trace. When this is not possible because either the transition

145

is too weak or it is desired to scan a wide region of the Spectrum,
measurements may be made on a recording. To accomplish this the D.C.
output of the phase Sensitive detector is input to a recording potenti-
ometer while the klystron frequency is varied by changing the cavity
size with the motor. Marker pips are made on the Spectrum by noting
the points ("whistle-through") at which the klystron frequency and

the standard frequency differ by the frequency at which the radio re-

ceiver is set.

IV. MICROWAVE STUDIES ON 1,3,2-DIOXABOROLANE

8.1 Introduction

 

This chapter reportsixmzmicrowave spectroscopic experiments per-
formed on 1,3,2-dioxaborolane. It begins with a description of the
chemical method of preparation for each of the samples which were
studied. Next the transitions observed and the rigid rotor and quad—
rupole coupling constants deduced from the spectrum are presented.

Then a model for the molecular configuration is assumed; some of the
effects anticipated from this model are described; and the experimental
observations and conclusions supporting the assumed model are presented.
Following this the most plausible structure for the compound is deter-
mined. Finally, the vibrational aspects are considered in some de-
tail. A model for the low-frequency vibrational mode iS assumed and
the consistency of the many observed vibrational effects with this
model is demonstrated. The implications on the form of the vibra-

tional potential function are then discussed.

8.2 Preparation of Samples

 

Samples of 1,3,2-dioxaborolane (OCHZCHZOBH) were prepared by the
addition reaction of diborane (B2H6) and ethylene glycol (HOCHZCHZOH)
following the procedure of Rose and Shore (15) and with the aid of

Dr. Shore and his students, Gerald McAchran and Gerald Brubaker.

Lo

'87

Materials:

 

l.

Ethylene glycol was refluxed over sodium for 1 hour and
vacuum distilled,

Diborane was vacuum distilled at -l20° to remove contaminants,
N(CH3)3 was purchased from Olin Matheson and used without
further purification,
Ethyl ether was dried over LiAlH4 and vacuum distilled,

The following slush baths were used:

-120° Skelly solve F,

- 78° Dry ice/isopropanol,

63° Chloroform,

A50 Chlorobenzene,

35° 1,2-Dichloroethane.

Procedure: The preparation procedure may be summarized by the

three reactions:

 

(A) HOCHZCHZOH + 1/2 B2H6 > OCHZCHZOBH + H2?
Et 20 ~—-—-
(B) OCHZCHZOBH + N(CH3)3 -——> OCHZCHZOBHN(CH3)3

(c) OCHZCHZOBHN(CH3)3 + 1/2 B2H6 <—— > QCH2CH2°$H + HsBN(CH3)3

1,3,2-dioxaborolane is produced in step (A) but it is necessary to

react it with N(CH3)3 to form a low vapor pressure adduct in step

(B) to allow removal of the ether solvent. The free product is re-

generated in step (C) by taking advantage of the greater Lewis acid

strength of Bsz in forming an adduct with the N(CH3)3.

A8

In a 'dry box' 0.55 ml. (10 mmole) of ethylene glycol was trans-
ferred to a 100 m1. reaction flask containing a magnetic stirring bar
and closed by a stopcock. The flask was cooled to —196° and evacuated.
Approximately 35 m1. of ether was distilled into the flask. The mix-
ture was warmed until the ethylene glycol melted and was then cooled
to -l96° while stirring vigorously to disperse the slightly soluble
ethylene glycol. A previously measured 6.9 mmole amount of diborane
was distilled into the reaction flask and the mixture was warmed to
-78° with continuous stirring. A small amount of hydrogen was evolved
and was pumped off after trapping any unreacted BzHe at -l96°. The
trapped Bsz was distilled back into the flask. When the rate of
H2 evolution became negligible, the flask was allowed to warm to -A5°
at which temperature the majority of the Hz was evolved. The trapping
and pumping was repeated. When H2 evolution again became neglibible
the reaction was completed at room temperature.

To form the adduct, OcHZCHZOBHN(CH3)3, and remove unreacted B2H6,
1.5 ml. (approximately 17 mmoles) of trimethylamine was distilled into
the reaction flask and was allowed to react while stirring at -780.
The reaction was completed by stirring 2 hours at —A5°. The ether
was distilled from the reaction flask at -63°, and the last trace re-
moved by pumping on the system at —A5°.

The product was sublimed at room temperature into a 1 liter
pear-Shaped exchange flask equipped with a test-tube bottom to allow
immersion ina dewar flask. Then 12.5 mmoles 0f BzHe was distilled
into the exchange flask, and the mixture was warmed to 50°. After

two hours the contents of the exchange flask was passed very slowly

A9
through a train of cold traps at -35°, -l20°, and -l96°. The 1,3,2-
dioxaborolane was retained in the center trap. Unreacted 82He and
the trimethylamine adducts were distilled back into the exchange flask.
The exchange and fractional trapping was repeated 3 times. The
1,3,2-dioxaborolane in the ~120° trap was quickly warmed to A00 and
transferred to an evacuated, flame-dried, sample bulb. The bulb
was stored at -78° or -l96° until samples were needed.

A sample of A,A,S,5-tetradeutero-l,3,2-dixoaborolane was pre-
pared following the same procedure as for the parent species. The
deuterated ethylene glycol was purchased from Merck, Sharp, and Dohme
of Canada, as 98+ mole percent in methylene deuterium. It was vacuum
distilled over sodium before use.

A sample of (A)13C - 1,3,2—dioxaborolane was prepared following
the same procedure, starting with 13C ethylene glycol. The glycol
had been synthesized by the procedure summarized by the reactions:

NaOH

1) Kc*N + CH3I ———> CH3C*N > CH3C*OONa

 

2) CH3C*00Na + ¢-CH2C1 -——> CH3C*0C1

LAH

 

3) CH3C*0C1 > CH3C*H20H

e Al O -X- B -><-
A) CH3C9H20H -—3—3—-> CHZC H2 r2 > BrCHZC HzBr

 

HOH

5) BrCH 0*H Br > HOCH 0*H 0H
2 Z 2 Z

 

The starting material was 5.3 grams of KCN containing 19 mole%
13C. ‘Sodium acetate was prepared from the KCN following the modified
method of Cox, Turner, and Warne as reported in Murray and Williams,

Organic Syntheses with Isotopes (28). The KCN was placed in a flask

 

which had two male standard taper joints separated by a seal-off

50

constriction. Twelve m1. of water was added and the flask was attached
to a vacuum line, frozen, and evacuated. Seven and one—half ml. of
CH3I was distilled into the flask, which was then sealed between the
joints and Shaken overnight. The flask was cracked open and the
volatile contents vacuum distilled into 8 ml. of 20N NaOH. The hydrol-
ysis of the methyl cyanide was completed by heating at 78° under
partial vacuum for A.5 hours. The resulting solution of NaOOC*CH3

was rinsed into a 300 ml. flask with water, 30 ml. of 6N H2S04 and

99.8 g. AgZSO4 were added by mistake; 1.0 grams should have been

added. Nitrogen was slowly bubbled through the solution and the
mixture was distilled into a cooled beaker containing 0.9925N NaOH.
When the distillation was completed, the distillate was titrated to

pH 8.8 as measured on a glass electrode pH meter. The solution was
then evaporated to dryness. The yield was 90% of the theoretical
amount.

Acetyl chloride was prepared with 99% yield by distilling the
fused Na00C*CH3 with benzoyl chloride in a flame dried micro—appara-
tuS.

Ethanol was prepared in 98.5% yield by reducing the acetyl
chloride with lithium aluminum hydride following the method of Cox
and Turner (29). The hydride was placed in a flask equipped with
an addition funnel and stirring bar, and attached to the vacuum
line through a reflux trap with a cold finger. Diethylene glycol
diethyl ether was added Slowly through the funnel, while stirring and
cooling the flask. The system was warmed to room temperature and

pumped on for 1 minute to complete the hydrogen out-gassing. The

51

acetyl chloride was vacuum distilled in at 0° and the system was
frozen to -78° and re-evacuated. The reduction was completed by
stirring 2 hours at 0°. The desired ethanol was released by alcohol—
ysis with 2—phenoxy ethanol. This was accomplished by heating at

90° under reflux for 15 minutes and then at 1000 for 2 hours. Fin-
ally the ethanol was distilled from the mixture.

Ethylene bromide was then prepared from CH3C*H20H following the
method of Popplewell and Wilkings (30). The ethanol was slowly dis-
tilled up a 17" column containing A1203 (100-200 mesh) heated to
about 380°. At the top of the column was placed a cold finger and
then a bubble trap containing bromine and glass wool. The ethylene
formed by dehydration of CH3C*H20H added Brz, yielding ethylene
bromkkh The distillation took about A hours and was completed by
sweeping the system with N2. The excess bromine was vacuum distilled
from the less volatile product.

The final step of the synthesis was the hydrolysis of the
BrCH2C*H2Br to ethylene glycol. This was accomplished by reflux-
ing a mixture of ethylene bromide and 25% excess of 1 molar Na2C03,
The hydrolysis took about 12 hours and was continued 1 hour after
the mixture became homogeneous. The products were passed through an
ion exchange column to remove the sodium salts. Finally the water
was removed from the glycol by pumping on the vacuum line for 2 days
at 0°. The total yield of HOCHZC*H20H was about 2 ml.

The 13C enriched 1,3,2—dioxaborolane was prepared from the glycol
by a different procedure from that used for the parent and deuterated
compounds. This seemed to produce the product with less solvent con-

tamination at the eXpense of a lower yield.

52
In a "dry box" 0.2 ml. of ethylene glycol and 5 ml. of decalin

were placed in a reaction tube containing a magnetic stirring bar and

equipped with a stopcock and a side arm with a T joint ending in a

break—off seal. The tube was attached to the vacuum line, evacuated,

and about A mmoles of Bsz was distilled into it. Then the tube was

sealed below the stopcock and allowed to warm to room temperature.

The reaction was carried out by stirring the contents of the tube

for 20 hours. The excess Bsz and the product 1,3,2-dioxaborolane

were vacuum distilled from the tube at room temperature into a sample

tube. Finally, the diborane was distilled from the product at —120°.
A fourth isotopic species of 1,3,2-dioxaborolane, the l°B Species

was studied in its natural abundance (20%) in the parent sample.
A.3 Examination of the Rotational Spectra

Dr. R. H. Schwendeman began the analysis of the Spectrum of
1,3,2—dioxaborolane by assuming some "reasonable" values for the
molecular parameters (listed in Table II) by comparison with values
obtained for other molecules. Then he calculated the moments of
inertia and rotational constants for this structure, and used the
rotational constants to predict a rotational spectrum. He examined
the frequency region in the vicinity of several of the b—type low J
transitions and assigned quantum numbers to the observed transition
frequencies with the aid of the calculated Stark effects, quadrupole
splittings, and intensities.

The initial analysis was complicated by the existence of a

relatively intense second series of rotational transitions for the

53

 

 

Table II. Initial structural parameters assumed for 1,3,2-dioxa-
borolane
B-H 1.15 R Z 0130 111° 22'
B—O 1.38 R Z BOC 109° 19'
0-0 1.113 R 1 000 105°
c-c 1.511 R Z HCH 109° 30'
CH 1.10 R Z HCC 110° 3A'
1 HCO 110° 311'

The B,0, and C atoms were assumed to be coplanar.

 

SA
molecule in an excited vibrational state. The rotational assignments
were confirmed by fitting the observed frequencies to a rigid rotor
model. Afier the two series of transitions had been reduced to two
rigid rotor sets, the responsibility for further studies of the Spectra
was assumed by the author of this thesis. A third series of transi-
tions, similar to the first two, was later located and analyzed.

The work by both investigators was done with the absorption cell
cooled to -78° with dry ice and with the sample at the lowest pos—
sible pressure consistent with workable intensity. All frequency
measurements were made with oscilloscope diSplay, although recordings
were used to establish the presence of some of the small splittings
caused by the l°B nuclear quadrupole moment. An error of i .02 Mc/Sec.
is estimated for most of the measurements.

The values of the rotational constants and quadrupole coupling
constants were determined by a numerical least squares method. Start-
ing with approximate values of A, B, and C, the values of aq and Bq
(the partial derivatives of the energy with respect to :Mga and (:[bb
-'][EC) as shown in Equation (2-38)) were calculated. Then’liaa and
('ybb -)(CC) were fit to the observed splittings between the quad-
rupole hyperfine components in each transition using equations of the
form 50 = 5(Aaq)'jj;a + 6 (ABq)(:3(bb — :yfcc). Here A is used (A-l)
to indicate a difference between two energy levels and <5 is used to
indicate a difference between two frequencies, or between two energy-
level differences. With these constants the quadupole coupling
contribution to each component of a multiplet was calculated and was
then subtracted from the observed frequency of the component. The

average of these "hypothetical unSplit frequency" (H.U.F.) values was

taken as the rigid rotor frequency for the transition.

The rotational constants were then fit by using equations of

the form
3W WC W B, WC
vo-oc= MT? AC-HAO -A c,) + A(—-—- —-—-)(Bo- B C) + A(-3—wc1-)(Co-CC,) (Lt-2)
Cc

where the subscript 0 indicates observed values, and the subscript
c indicates calculated trial values.

After an initial quadrupole fit was obtained, the caluclated
hyperfine frequencies were corrected, where necessary, for Small
shifts (of the order of 0.05 Mc/Sec.) due to the presence of unresol-
ved hyperfine components.

The final values for A, B, C, :Xéa, (:)[bb - :yfcc) were obtained
by recycling the fitting procedure until a stability of better than
0.01 Mc/Sec. in each parameter was obtained. This iteration was neces-

B

sary because the partial derivatives used are all functions of AC, C,

CC.

The observed and calculated quadrupole splittings are compared
in Table III for the three vibrational states of 1,3,2—dioxaborolane.
The experimental hypothetical unsplit frequencies for the observed
transitions and the differences between these frequencies and those
calculated from the rotational constants are listed in Table IV.

The three observed states were tentatively assigned as the ground
and two excited vibrational states. The relative intensities de-
creased in order of increasing frequency of the 0 —> 111 transition.
The assignment of the first spectrum as the ground state was confirmed

by the fact that no similar transitions of greater intensity could be

found within any reasonable range of frequency. The second state was

56

.po>Homop poc

macabpaaam
m.

 

 

 

 

 

mp. - ac. - m m An\a A: “\av Am\a A: m\av sum A: mam
ca.a+ ma.a+ m m Am\a A1_o\av Am\m A: w\mv amm A: mam
cc. + pp. + m m Am\a A1.w\av Am\a Ai.m\av
cm. - ma. - am. - Ow. - mm. - Ow. - Aw\m A1.m\mv Ao\a A: “\av
m m mm.m+ mm.m+ Am\a A: m\av Am\m A: n\mv saw A: mom
m m No.a+ oo.H+ Am\a A..N\Hv Am\s A: w\av
Hm.a- mm.s- m cm.a- mm.a- Am\m A: o\mv Am\a A: s\av
m om. + am. + mm. + am. + Am\m A: m\mv Am\a A: m\mv Mad At Has
No. + me. + co. + am. + m Am\m A: m\av Am\s A: m\mv new A: ass
on. + B. + co. + ac. .. m a} A: QC 8} A: QB as A: a:
m oo.a- ma. - mo.a- oo.a- An\a A: m\mv Am\m hi w\mv ass A: o
m as. - as. - pa. - ma. - Am\a A: m\mv Am\m A: o\mv
oc. + mm. + cm. + am. + co. + am. + an\m A: “\mv Am\m A: w\mv
acappasam meappaaam oesopaaam meaopaaam acapomaam acaopaaam c as a
popmHSono po>uomoo pmpmHDUHmu po>nomno popmHSUHmo po>nomoo mpco o 00 co.pwmcmnb
oompm oopaoxm ocooom opmpm oopaoxm pmsam oumpm oesoso
A.qu\uz mo. n,mm nouns Hmucoemnmmxm,popmewpmo ocpv mowooam peonma "ocmflononmxowpnmwmwfl
now monocoaaoo ocwwauahz:ofioaaupm:d mo AcoM\ozv mmcwpbwfiam zucosqmum popmHSUHmu pom po>uomno

JHHH magma

57

 

 

 

 

Table IV. Experimental hypothetical unSplit frequencies (MC/Sec)
assigned for 1,3,2—dioxaborolane: parent species (the
estimated experimental error is i .03 Mc/Sec.)

Ground State First Second
Excited State Excited State
Transition Exp. Exp. — Exp. Exp. — Exp. Exp. -
Frequency Calc. Frequency Calc. Frequency Calc.
O -> 111 12,638.11 +.05 12,665.AO +.O3 12,670.7A +.06

110 —> 221 29,A09.59 +.01 29,AA6.18 -.03 29,AA9.91 -.05

111 -> 202 20,A03.38 +.05 20,A8A.28 +.01 20,503.30 +.11

101 —> 212 21,1A2.6A -.02 21,215.26 -.01 21,232.82 +.06

202 —> 211 10,518.89 -.02 10,A8A.20 +.03 10,A67.19 +.06

2,2 -> 221 l2,A00.35 -.03 12,3A6.A5 +.0A 12,325.82 +.02

202 —> 313 29,3SA.16 -.0A 29,A7A.96 +.01 29,505.18 +.01

212 -> 303 29,263.23 -002 29,385.511 “.06 29,b15096 -.06

312 ‘> 321 a a 9,873.29 -.06

 

a I ' O I
Tran51t10n n0t measured.

58
assigned a quantum number of l for a low frequency vibration. A
larger quantum number was not likely, due to the high population de-
duced from the intensity relative to the ground state. The third state
obviously was a still higher vibrational state, and was felt to be the
v = 2 level of the same normal mode. The fact that the three states
lay in a series of decreasing intensity, and that no other excited
states could be located, gave some foundation to the v = 2 assign—
ment for the third state. The possibility that it was the v = 1
state of a different normal mode could not be excluded at this time,
however.

The Spectra of three isotopically substituted species of 1,3,2-
dioxaborolane were also examined. These were the A,A,5,5-tetradeutero,
(referred to as A-D), the 13C, and the 1°B species. The rotational
spectrum was predicted for each of these by first performing the iso-
topic substitution in the assumed trial structure. From this the
second moments of inertia were calculated. The calculated second mo-
ments were improved by employing the observed second moments for the

parent molecule in equations of the form

Paa(isotope, predicted) = Paa(isotope, calculated) +
(Li-3)
[Paa(parent, observed) - Paa(parent, calculated)].
The rotational constants corresponding to these predicted second moments
were then used to compute a predicted rotational Spectrum. The analysis
of each Spectrum then followed the procedure outlined for the parent

species. These analyses were simpler than that for the parent since it

was possible to limit the range in which the spectral parameters could

59

be expected to be found. The experimental rotational constants for

each Species were generally found to be within 20 or 30 Mc/Sec. of

the predicted values. This meant that the desired transitions could

be located rather easily, close to the predicted frequencies. Com-
parisons with the parent transitions of Stark effects, quadrupole split-
tings, and line shapes were also of great help. Finally, the first
excited vibrational state served as an aid rather than a complication

in analyzing the substituted Spectra, since now at least two similar
transitions were expected in each region.

For the A-D species, rotational transitions for three vibrational
states were observed. The relative intensities decreased in order of
increasing frequency of the 0 —> 111 transition, as they had in the
parent. The states were tentatively assigned v = O, l, and 2 for the
low frequency vibrational mode. The possibility that the third state
had v = l for a different vibrational mode could not yet be excluded
in this case, as:fi;coukinoth1the parent. Some transitions which might
be due to a fourth vibrational state were observed, but it was not
possible to locate a sufficient number of them to confidently deter-
mine a set of rotational constants.

The observed and calculated quadrupole splittings are compared in
Table V for the three vibrational states of the A-D species. The ex-
perimental hypothetical unsplit frequencies and the differences between
these and the frequencies claculated from the rotational constants
are listed in Table VI.

An estimate of the magnitude of centrifugal distortion effects

may be made by comparing the frequencies of the O —> 111 and 211 —> 322

60

.po>Homou poc mcmppwaamm

 

om. + 3. + m 3. + am. + SE A: QB- - GR in NE Nam A: :m
cm.a+ cm.a+ am.a+ mm.a+ Na.a+ Na.a+ Am\a A: a\av - Ao\m Au w\mv add A: Nam
mm. + Hm. + me. + op. + mp. + me. + An\a A: w\av - Am\a A: m\av
ma. - ma. 1 ma. - ma. - ma. - cm. - Aa\m A: m\mv - Am\a A: @\av
m NN.H- am.a- cm.a- Hm.a- Am\m A: m\mv - Aa\a A: “\av sum A: mom
5. + om. + mm. + mm. + cm. + am. + a} A. QC - SE. in mg 2m. A: 2:
mg. + 4:. + m om. + om. + Am\m A: m\av - Am\a AI n\mv mom AI ass
m m ma.a- ma.a- Am\m A: m\sv - Am\m A: m\mv sum A: ass
m m am. - mm. - a} A: Q: - a} A: va
m 8. oc. .. cm. + B. .. a} A: NB - a} A: QB
do. - oo.a- mo.a- mo.a- Ho.a- no. - Am\a A: w\mv - Am\m A: m\mv ass A: 0
ma. - an. - ma. - ca. - ms. - a4. - Aw\a A: a\mv - Am\m A: @\mv
mm. + mm. .. 8. + am. + cm. + mm. .. a} in N3 - a} A: QC

 

acappaaam asappaaam mcaopaaam meapoaaam acappasam acappasam

monocomsoo compwmcmph
popmflsoflmo po>oomno popmHSono po>nomno popmHSUku po>pomno

 

opmpm oopaoxm ocooom upmpm oopaowm pupae oompm ocsouo

 

A.ooM\oz mo. H
mm nouns HmpcoawnoQXo popmewpmo ozpv momooam nonopsopmupopnmsmsqs: "ocmfiouonmxowpumwmsfl
pom monocoQEoo NCWMnmazn ofioasupmsq wo Aoom\ozv mmcappmflam zocosooum popmaoofimo pom pm>pomno .> canoe

61

Table VI. Experimental hypothetical unSplit frequencies (MC/Sec)
assigned for 1,3,2-dioxaborolane: A,A,5,5—tetradeutero-
Specied (the estimated experimental error is i .03 Mc/Sec.)

 

 

 

Ground State First Second
Excited State Excited State
Exp. EXp. - Exp. Exp. - Exp. Exp.-

Tran51t10n Frequency Calc. Frequency Calc. Frequency Calc.

 

0 4> 1,, 10,9OA.6O +.03 10,927.61 -.01 10,932.57 +.10
1,O -> 22, 25,076.57 -.02 25,105.83 -.03 25,110.02 -.03
1,, —> 202 18,2A3.16 +.06 18,31A.9S -.06 18,331.62 -.10
10, -> 2,2 18,581.68 -.01 18,60A.65 +.03 18,619.87 +.0A
202 —> 21, 8,981.59 +.01 8,958.18 +.03 8,981.56 +.08
2,, —> 22, 9,802.36 -.01 9,751.89 +.03 9,735.27 -.06
202 —> 3,3 26,086.31 +.02 26,152.71 +.01 26,178.92 +.03
2,, —> 303 26,026.70 -.05 26,138.21 .00 26,160.52 -.03
22, —> 3,2 31,780.96 —.03 31,878.96 —.OA 31,898.29 -.0h

a a

2,, -> 322 32,713.58 -.17 32,782.69 -.17 32.797.2A -.17

 

a I O I ‘ C O I o n a 0
not included 1n the r1g1d r0tor f1tt1ng.

62
transitions. For an undistorted rigid rotor .3V0 _>111 = V211->322.
The observed 2,,—>322 transitions are .2A to .A? Mc/Sec. lower than the
values of 3 VO_>111. This demonstrates that some centrifugal distortion
is present but that the effects are small enough to be ignored in the
present analysis.

For the 13C Species rotational transitions for two vibrational
states were observed, corresponding to the ground and lower excited
states found for the parent and A—D species. It was not possible to
locate transitions for the third vibrational state, nor to measure
the transitions as accurately as for the other two species, because
the 13C sample decomposed a few days after it was first examined in the
Spectrometer. Water or some other impurity had been admitted to the
sample tube either by leakage through the stopcock during the 1A months
that the sample had been stored, or directly from the spectrometer vacuum
system. Each time the sample was warmed to introduce it into the ab-
sorption cell, a substantial amount of non—condensable gas was released.
The sample was vacuum distilled in an attempt to eliminate the decom-
position, but this was only partially successful. It was possible to
measure enough transitions to determine the two sets of rotational
constants, but a greater error must be expected in these than in the
parent and A—D constants.

The observed and calculated quadrupole splittings are compared in
Table VII for both vibrational states of the 13C species. The experi—
mental hypothetical unsplit frequencies and the differences between
these and the frequencies calculated from the rotational constants are

listed in Table VIII.

63

Table VII. Observed and calculated frequency splittings (MC/Sec) of
quadrupole hyperfine components for 1,3,2—dioxaborolane:

13C Species. (The estimated experimental error is
i .10 Mc/Sec.)

 

 

 

 

Ground State First
Excited State
Transitior Co onents Obs. Calc. Obs. Calc.
‘ mp Split- Split- Split- Split-
ting ting ting ting
<3/2—>5/2)—(3/2—>3/2) + .57 + .57 + .55 + .60
O—>1ll (3/2—>5/2)-(3/2—>1/2) - .Liu - .115 a
(3/2—>3/2)—(3/2—>1/2) -1 01 -1.02 a
1,,—>202 (5/2—>7/2)-(1/2—>3/2) + .62 + .61 + .65 + .63
2,2_>22, (7/2—>7/2)-(5/2—>5/2) - .87 - .88 — .8A - .85
312">321 (9/é->9/2)-(7/?->7/?) - .60 - .59 - .66 - .61

 

a .... .
Splitting n0t resolved

Table VIII. Experimental hypothetical unsplit frequencies (Me/Sec) as-
signed for 1,3,2-dioxab0rolane: 13C species. (The estimated
experimental error is i .08 Mc/Sec.)

 

 

 

 

Ground State First Excited State

52:22:27“ Eiiie.‘ 5:13:22? 82:1..-

0 -> 1,, 12,878.99 + .12 12,501.91 + .08
1,0 —> 22, 29,025.AA - .05 29,061.85 - .06
1,, —> 202 20,169.85 - .2A 20,250.89 + .01
10, -> 2,2 20,873.97 - .02 20,985.73 - .2A
2,2 _> 22, 12,227.2A - .01 12,173.A1 + .10
202 —> 313 28,992.03 + .16 a
2,2 _> 303 a 29,029.83 + .15
3,2 —> 32, 9,8A2.7A + .07 9,813.2A - .03

 

a ... .
TranSition n0t measured.

6A

For the 1OB species rotational transitions for the ground vibrational
state were observed. The observed and calculated quadrupole splittings
and the experimental hypothetical unsplit frequencies and the differences
between these and the frequencies calculated from the rotational con-
stants, are listed in Table IX. The larger quadrupole moment of the
l°B nucleus caused greater splittings than for the other species, al-
though the values were still small.

The values of the rotational constants, principal moments of in-
ertia, and second moments of inertia for each of the species of 1,3,2-
dioxaborolane which was examined, are presented in Table X. The values

of the quadrupole coupling constants are listed in Table XI.

8.8 Discussion of the Bond Characters

 

Information on the electron distribution in 1,3,2—dioxaborolane
may be obtained from calculations based on the experimental values of
the quadrupole coupling constants. These calculations will provide an
estimate of the non—localized and polar characters of the B-0 and B-H
bonds. The results of the following calculations, which make use of
the ideas of Townes and Bailey (22), must be considered as no more than
estimates due to the number of simplifying assumptions which will be
invoked.

To start, the orbitals used by the boron atom to form its 0 bonds

are taken to be

‘1’.
Y2
W3

1/J'3 (s + yte Py) B-H bond,
1/QF3 (s + orb/2 PX — y’272 Py) B—0 bond, (A-A)

1/073 (5 - 0’6/2 PX - 9’2/2 Py) B-O bond.

65

Table IX. Observed and calculated frequency Splitting (MC/Sec) of
quadrupole hyperfine components, and experimental hypo-
thetical unsplit frequencies (MC/Sec) assigned for 1,3,2—
dioxaborolane: 1°B Species.

 

 

 

Transition Components Obs. Cale. EXp. -
SpliE- Split- Exp. HufC Calc.
ing ing
0 —> 1,, (3 —> A) - (3 —> 3) +1.23 +1.22 12,883.35 - .02

(3 —> A) - (3 -> 2) - .AO - .A3
(3 -> 3) ‘ (3 ‘> 2) -l.63 -l.6A

1,0 —> 22, (A —> S) — (3 -> A) +1.25 +1.2A 30,0A6.23 + .02
(A -5 5) - (2 _> 3) - .68 - .62
(3 —> A) - (2 -> 3) -1 93 -1.86

1,, —> 202 a 20,A96 0A - .05
10, —> 2,2 (A _> 5) — (3 _> A) +1.05 +1.06 21,A87.28 + .01
2,, —> 22, (5 —> 5) - (A —> A) -1.28 -1.36 12,838.38 - .03
202 —> 3,3 a 29,72A.10 + .0A
2,2 -> 303 a 29,569.69 + .02

312 ‘> 321 (6 -> 6)

(5 -> S) - .96 - .97 9,778.73 + .02

 

aNo splitting resolved.
bThe estimated experimental error is i .05 Mc/Sec.

c .. . I . . .
The eStimated experimental error 15 i .03 Mc/Sec.

66

.ooM\oz mo. H

mm zucmmpuooc: poemswpmmU

mm zpcmmppoocs pomewpmmo

.ooM\oz mo. H
.Amumv compmsqo an pubmHSUHmo

mm xpcwmppoocs popmewpmMU .oow\oz mo. H

.ooM\oz n NMIDE< HmmmOm nopomm cowmno>coom

 

 

 

 

o
cemam.m mwaam.mm msooo.mc cemam.aaa ammam.mc maaoa.mm oma.a6ma oao.0mas oma.ammw zmwaommomzwm
ammmc.m masoa.am aammm.mc Hamma.aaa cpawa.mc 6mmm6.Hc omo.mmma sad.occa cmp.aamw opoumomwwwoxm
aaooa.m mammo.am cappc.mc msaam.oma seapo.co aoamo.ac com.aaaa uca.aaca cam.msmw omwmwm mmmmxommossm
amaca.c aabgm.ac ammAH.ac ocmmm.amH camaa.as maaam.as osc.mawm oom.aa®c oas.©woa obmoWaWMWWoxm
mmmmw.c Hmmma.ac awaam.sc 4H06s.amfi cmama.aa monm.as oom.mmmm omm.aamc oma.awo~ oompmomwwwoxm
mmaac.c amams.ac amocc.ac cwawm.mma camam.aa NHNAm.Ha ocm.wawm. uaa.c6mc oao.cmo~ ommmww mmmmmmmmmw
agama.m gammm.cm aaoom.ac mocpo.waa mmmam.mc pmcmm.oo oao.apma omm.maAa oac.ammw oumoWCWMWWoxm
amcac.m aaaoc.cm maaao.ao msamN.paa whoam.mc maomm.oc oma.aama oaa.sgaa usa.oamw obmomomwwwoxm
meaam.m mmamm.cm cacma.ac maamm.msa ammam.mc cramm.oo o0m.mmma ocm.omaa ops.mmmw ommmww mmommommoo
com and one oH pH mH o m a

 

 

ocmaocoamxoao-m.m.a sow pama-pzav
mwpnocm Mo mucmEoE pcooom pcm amnwdubzdv mwpuocw mo mucoEoE Hmawocwna nAoom\ozv mpcmpmcoo Hmcowpmpom .x oHQMH

67

 

 

 

 

Table XI. Quadrupole coupling constants (MC/sec) for 1 3 2-dioxaborolane
3 3
i>faa :){bb CXEc

OCHZCHZOBH Ground State - .67a -2.A1a +3.08a
L_______J

First Excited State - .58a -2.23a +2.81a

Second Excited State - .50b -2.A0a +2.90a
000 CD OBH Ground State - .53a -2 2Aa +2.77a
‘ 2 2 I

First Excited State - .A2b -2.A1b +2.83b

Second Excited State - .37b -2.20b +2.57b
013CH2CH20BH Ground State — 72b -2 26b +2.98b

First Excited State - .60b -2.A1b +3.01b
OCHZCHzloBH - .90b -6.09a +6.99a
L________J

 

a .. . . . .
EStimated uncertainty

b .. . . . .
EStimated uncertainty

.10 Mc/Sec.

.20 Mc/sec.

68

The gradient of the field at the B nucleus is then assumed to

be eXpressible in equations of the form,

9

a = [U0 - 1/'2(UB + UY)]qo , with (A-S)

c:
11

number of P electrons in the valence Po orbital

(a, B, Y are mutually perpendicular),

and qo contribution of one valence P electron to qa.

It is found that assuming non—polar bonds, with or without n
bonding leads to the constraint qX = q . This is contrary to the

Y
eXperimental result that

= = - “ /
quX (BCaa 0.67 Mc sec.,
quy = érbb = -2.A1 Mc/Sec.,
quZ =IjCcc = +3.08 Mc/Sec.

Therefore it is necessary to assume a polar character to some of the
bonds to boron. The electronegativity values (B = 2.0, 0 = 3.5, H = 2.2)
suggest assuming that the B—0 bonds are polar, with the oxygen more
electronegative than the boron.

Therefore nP electrons per B-O bond are assumed to have been
transferred from the boron orbitals to the oxygens. Of these electrons,
1/6 per bond are Py and 1/2 per bond are PX. The contributions to UX
are then

+ 1/2 electron from each of H} 2 and \Bs, and

-2(:;) electrons "lost" to the oxygens.
The contributions to Uy are .
+ 2/3 electron from Ayl, '
+ 1/6 electron from each of [V2 and lys, and

n
-2(?;) electron lost to the oxygen.

69

The only contribution to Uz is from n? electrons contributed to the

boron PZ orbital from the occupied PZ orbitals on the oxygens. There—

fore,
UX = l - nP ,
Uy = 1 - 1/3 1113, (A-é)
U = n ,
2 TT
and
quX = (1/2 - 5/6 n1D — 1/2 nn)qu0 = - .67,
Qqu = (1/2 + 1/6 nP - 1/2 n”)quO = -2.A1, (A-7)

quZ (—1 + 2/3 nP + nn)quO = +3.08.

Solving these gives

nP -4u7A/quO,

n
TT

1 + A.2A/quo.

If qu0 is assumed to be -5.3 (31), these become
Up = .33,
n1T = .20.

The number of "boron electrons" from the B—0 bond wave function,

180 = [LWB + X‘Wo][1 + 121-1/2, is 2/(1 + XZ)/ ‘This is equal to 1 - nP,
so X2 = 2.0 and the ionic character of the B-0 bond, (A? — l)/()\2 + l),
is 0.33. The value of n1T indicates that the n bonding system has about
a 10% contribution to the electronic structure.

Some polar character to the B—H bond may be assumed, with H more
negative than B. For a B-H bond of 10% ionic character, the B-0 bond

ionic character changes from 33% to A3% and the n bonding contribution

decreases to 5%

70
The stability and monomeric nature of 1,3,2—dioxaborolane has
been attributed in a large part to the formation of B-O n bonds (32).
To retain some n bonding in the above scheme, it is necessary to

deny that there is much ionic character in the B-H bond.

A.5 Determination of the Molecular Symmetry and Structure

 

Once the rigid rotor and quadrupole analyses had been completed
the molecular structure and the vibrational effects could be examined
in detail. The three most likely configurations of the heavy atoms
were a planar ring with C2v symmetry, a symmetrically staggered ring
with C2 symmetry, and a boat or bowl form with oh (Cih E Cs group)
symmetry.

Relative intensity measurements were expected to help discrim-
inate between the types of symmetry, since the C2V and C2 symmetries
could produce effects of nuclear spin statistks while the oh sym-
metry could produce no such effect. Furthermore, the relative in-
tensities of the same rotational transition in different vibrational
states would provide an estimate of differences in vibrational energy
of these states.

A second test of the symmetry was the principal axis orientation
of the dipole moment. In all three cases the major component of ”D
was eXpected to be along the b "symmetry" axis with a zero component
along the a axis, across the ring. The c component (perpendicu-
lar to the ring) would vanish for sz and C2 symmetries but should

be non—zero for the oh symmetry.

71
A third test of the symmetry was the geometrical coordinates of
the B atoms. The a coordinate should be zero in all three cases.
The c coordinate must also be zero for the C2v and C2 symmetries,
but Should be finite for oh symmetry.
The indicated experiments were then performed in order to estab-

lish the symmetry of the molecular configuration.

a) Relative Intensities of Absorption Transitions

 

The absolute intensity of a rotational transition is a function
of many factors. The relative transition intensities, however, for
a given sample at constant temperature, pressure, microwave power and
for reasonably close frequencies may be expressed far more simply,
as
I, = 8r f, 0n, ' (A-8)
where Sr is the relative rotational line strength and may be cal-

culated once the rotatiOnal constants and dipole moment are known,

fV is the fractional population of the vibrational state concerned

. _ -E /kT
1V ~ gve V

and gn is the relative nuclear spin statistical weight.

The nuclear spin statistical weight is necessarily included only
if the molecule is symmetric to a 1800 rotation about a principal
axis. The appr0priate gn is calculated by determining the number of
nuclear spin wave functions which combine with the vibration-rotation
wave functions to produce the correct symmetry to the C2 Operations.
For an interchange of n pairs of identical nuclei of Spin Ii the

number of symmetric Spin functions which may be constructed is (31)

72
n
“symm = 1/2 [131 (21i + 1)][i (21 + 1) + 1], (A—9)
and the number of antisymmetric functions is

[3 (21i + 1)] 1H? (21. + 1) - 1]. (A-io)
1:, —

nantisymm = l/?[
For 1,3,2-dioxaborolane a rotation about the b axis may inter-
change 2 pairs of hydrogens of Spin 1/2 (also 1 pair of carbons and 1
pair of oxygens, but their spins of zero do not contribute). The
ratio of the number of symmetric to antisymmetric functions in this
case is 5:3. When A deuteriums (I = l) are substituted for the
hydrogens, the symmetric to antisymmetric ratio becomes 5:A. In both
cases the total wave function must be symmetric to the interchange.
The symmetry of a given vibration-rotational level is the product
of the rotational symmetry and the vibrational symmetry. Wilson,
Decius, and Cross (33) demonstrate the following statements for vibra-
tional wave functions:
1) The ground vibrational state is totally symmetric.
2) An excited state of a non-degenerate normal mode is totally
symmetric for even values of the quantum number.
3) An excited state of a non—degenerate normal mode has the
same symmetry as the normal coordinate for odd values of
the quantum number.
A) Excited states of degenrate vibrations have a combination
of wave functions, with more than one symmetry Species

represented.

73

The symmetry of a vibrational state with respect to the C2
operation may be determined by measuring the relative intensities
of two rotational transitions which have initial rotational levels
of opposite symmetry. The relative statistical weights are then equal
to the relative intensities corrected by the rotational line strength
factor. Finally, by recognizing which statistical weight corresponds
to the symmetric initial rotational level, the vibrational symmetry

may be deduced.

b) Intensity Measurements

 

The accurate measurement of intensities is still a major problem
in microwave spectroscopy. Data on vibrational frequencies, barriers
to internal rotation, conformational isomers, quantitative analysis
etc., and the analysis of spectra in general could be much improved
by more precise methods. Even with the relatively large uncertainties
presently involved, however, it is possible to obtain useful informa-
tion from relative intensity measurements.

The most important single difficulty probably is caused by re-
flections of the microwave radiation within the system. These can
lead to an effective sample path length which is both unknown and
highly frequency dependent. Reflections also cause standing waves
which can make the detection process highly frequency dependent. To
minimize reflections use is made of tapered transitions between sec-
tions of the waveguide, thin mica windows on the absorption cell,
attenuators, and isolation units (in which attenuation in one direc-

tion is many times that in the opposite direction). Other problems

7A
encountered are ground loops, pickup, non—linear detection and ampli-
fication, variations in microwave power, and interference from nearby
transitions.

In measuring relative intensities for transitions of 1,3,2-dioxa-
borolane and the A-D species it was hoped that the problems listed
aboVe could be minimized if the following procedure was used. Compar-
able measurements were made at constant pressure with the waveguide
cell at dry ice temperature, with the microwave power level held
constant, with the crystal mount adjusted for maximum sensitivity,
with the klystron source tuned until the desired frequency occurred
at a flat portion of the mode, with constant sweep width, and with
the Stark modulation voltage for comparable transitions adjusted to
similar values consistent with the least interference from other
transitions. The measurements on which the estimates of the vibra-
tional energy levels are based were made with a ferrite isolator
between the absorption cell and the detector. The absolute intensi-
ties were found to decrease with increasing length of time after
introducing the sample into the cell. To minimize the effect of this
variation of intensity with time, the reported measurements were made
after the sample had been in the cell 2 or more hours, and an average
was taken over a series of measurements, alternating between the lines
being compared.

The transitions measured include both symmetric and antisym—
metric initial rotational levels for each vibrational level. The
choice of transitions was made on the basis of proximity of frequency

and likelihood of oscilloscope display. Tables XII and XIII list

75

Table XII. Relative intensity of the (212 —> 303) to the (202 — > 3,3)
transitions in 1,3,2-dioxaborolane.

 

 

 

Ground State First Second
Excited State Excited State

Measurement (1) .55 .89 .76
(2) .68 .7A .60

Average <.62> <.8l> <.68>

Expected

.598 .598 .598

1.672 1.672 1.672

c .997 .997 .997

 

a . .. . . . . . b
For Vibrational State symmetric to C2 .
b . .. . . .. . . . b
For Vibrational State antisymmetric to C2 .
c b .
For no C2 symmetry.

Table XIII. Relative intensity of the (303 —> A14) to the (3,3 —> A04)
transitions in tetradeutero—l,3,2-dioxaborolane.

 

 

Ground State First
Excited State

 

Measurement (1) .7A .7A

(2) .85 .7A

Average <.79> <.7A>
EXpected

.80 .80

1.25 1.25

c - 1.00 1.00

 

a o to u I o g I b
For Vibrational State symmetric to C2 .
b o no a. 00 - o u b
For Vibrational State antisymmetric to C2 .

CFor no Czb symmetry.

76

relative intensities for transitions of the two symmetries to Cab
within each vibrational state. The 303 —> A14 and 3,3 —> A04
transitions for the A—D Species are particularly convincing, since
in each vibrational state they were separated by less than 1 Mc/sec.
and were measured simultaneously on the oscilloscope. It was un—
fortunate that the same transitions for the second excited state
occurred in a frequency region which was inaccessible because of
klystron failure.

In each case the rotational level which is symmetric to 02b
is stronger than the level antisymmetric to C2b by an amount similar
to that predicted. The conclusion is that 1,3,2—dioxaborolane pos-
sesses at least a C2b axis of symmetry. Furthermore all three Vibra—
tional states have a wave function symmetric to this rotation. Other
evidence to support these conclusions will be derived from the dipole
moment orientation and the effective structure.

In order to estimate the difference in vibrational energy between
the ground and excited states, the relative intensities of several
corresponding rotational transitions in each vibrational state were
measured. The intensities are listed in Table XIV relative to the

ground state transition. The vibrational energy differences were

calculated with the relation

I -(E -E )/AT
T! = e V O , with kT/hc = 135.5 cm“1. (A-ll)
0

These measurements demonstrate clearly that both the first and

.. .. . .. ,. .. b
second exc1ted States' Vibrational wave IunCtions are symmetric to C2

0

77

They further show that the vibrational mode has quite a low frequency,

and that the frequency is markedly decreased on substituting deuterium

for the methylene hydrogen atoms.

Table XIV. Relative intensities of rotational transitions and esti—

mated vibrational energies (cm-l).

 

 

 

 

 

 

 

 

Ground First Excited State Second Excited State
State a k _ b
... R81. R81. El-Bo El-EoU Rel. EZ-an EZ‘EO
TranSIEIO“ Int. Int. Int.
1,3,2-Dioxaborolane
3,2 —> 32, (l) 1.0 .68 52 -18 .53 86 16
(2) 1.0 .6A 60 —10 .51 91 22
202 —> 2,, 1.0 .6A 60 131 .A9 96 167
000 _> 1,, 1.0 .73 A2 111 .56 78 1A6
Average (EV-E0) 53:15 88:20
A-D-l,3,2-Dioxaborolane
000 —> 1,, (1) 1.0 .87 19 A8 .67 5A 83
(2) 1.0 .8A 2A 5A .6A 60 91
Average (EV-E0) 22:10 66:20

 

aEnergy difference (cm—1) calculated assuming vibration symmetric to C2 .

b

bEnergy difference (cm'l) calculated assuming vibration antisymmetric

to C21).

78

C) Determination of the Dipole Moment

 

The dipole moment components were determined for the parent species
by measuring the frequency Shift of some absorption transitions at
each of several values of the applied Stark field. This was accomp—
lished by applying a high D.C. voltage through a calibrated voltage
divider to bias the square wave generator output above ground poten-
tial. The voltage settings were converted to values of the electric
field by calibration with the known Stark shift of the OCS 0 -> 1
transition at 12,162.99 Mc/sec.

There is a problem concerning the value of the OCS dipole moment
to be used in the equation for the known Stark effect. The best value
for “D was obtained by Marshall and Weber (3A). They used sufficiently
large field strengths to observe the second and fourth order field
[effects of pg and us and the effect of the second order polarizability
perturbation. Their "true" value of ”D was 0.712A i .0002 D. Pre-
viously Shulman and Townes (10) had measured the Stark effect at lower
field strengths and obtained an effective dipole moment of 0.7085 D.
The only concern here with the value of “D is that it is presently
necessary to use it to calculate the OCS Stark effect. The problem
could be circumvented by recognizing that OCS is Simply a convenient
laboratory standard, which is being employed to determine the effec-
tive electrode Spacing in the waveguide cell which was used for the
present work. Therefore, a table of frequency Shifts versus known
values of applied electric field, in a range of fields commonly
used, would be more useful and less subject to debate than the deter-

mination of true or effective values for the OCS dipole moment.

79
Consideration was given to preparing a table of Av versus E2 from
the data of Marshall and Weber and of Shulman and Townes, but the
former did not report A0 Versus E2 directly and both pairs of workers
obtained their Stark fields in a different manner from that used
here. In order to decide which value of “D is appropriate for the
calibration, the contributions to the Stark shift from the polariz-
ability and from the term in us were calculated, using the equations
and experimental values reported by Marshall and Weber. This was
done for a field strength of 6 statvolts/bm., which is approximately
the maximum field strength which was achieved in the present work.
The contribution due to the polarizability was entirely negligible;
the contribution due to the “B term was .016 Mc/sec., which is com—
parable to the estimated uncertainty (.02 Mc/Sec.) for the measured
frequencies, and which is 0.3% of the Stark shift calculated from
the term in us. It was therefore decided to use ”D = 0.712A as the
effective OCS dipole moment, since this is the most accurate value
available, and since no effects from the higher order terms could
have been observed at the field strengths used here.

The values of the OCS Stark shift were measured here with the
un—calibrated cell, plotted against the square of the voltage set—
tings, and fit to a straight line by least squares. The theoretical
lepe of this line divided by the experimental slope yielded a cali-
Ibration constant k . The least squares slopes of similar plots for

1,3,2—dioxaborolane were then multiplied by k to obtain values of

6 Au
2(5
inilues of Au from about 1.0 Mc/Sec. to 8.0 Mc/Sec. were measured with

 

2)‘ The experimental error is believed to be less than 1%, since

zan estimated error in each measurement of less than i.OA Mc/Sec.

80
An equation for d A0/§j(E2) may be derived by taking the differ-
ence in the Stark energies (Equation 2-30) for the energy levels in
the transition, inserting the proper value for M(AM = 0 for the pres-

ent experimental arrangement), differentiating with reSpect to E2,

and noting that (AWS) = vstark_ V0 is the Stark Shift. Thus
__ (2) (l) _ 2 2 2
AWS — ws -_wS — E [:§:(AAG + ABGM )“o ] , (A—12)
,- 2 = 2 2 2 _
A (ANS)/3 (E ) Caua + Cle-b + CCLLC 3 (A 13)

C = AA + AB M2, etc.
a a a

The coefficients Ca’ etc. were calculated by the computer pro-
gram EIGVALS. The components of ”D were then determined by solving
the linear system (or least squares system where there was a redund-
ancy) for the Stark shifts of several transitions. Table XV lists
the experimental values of ciAV/ 3(E2) and the Ca's which were used.

Since it was of great interest to determine the c axis com-
ponent of uD, the 202 —> 3,3 and 2,2 —> 303 transitions were measured.
Both of these have large contributions to CC from the near degeneracy
of the 202 and 2,2, and the 3,3 and 303 pairs of levels. The com-
ponents of ”D were calculated in four ways, all of which assumed “a = 0.
(l) The linear system derived for transitions (A) and (B) was solved
for ubz and ucz, (2) ub2 and ”c2 were fit by least squares for the
transitions listed as (A), (B), and (c), (3) AC2 was assumed zero,
and each of (A), (B), (C), and (D) was solved for the remaining ubz,
(A) ucz was assumed = 0.0001, and each of (A), (B), (C), and (D) was

solved for the remaining ubz. The results of these four calculations

are listed in Table XVI.

Table XV. Data used

81

in the dipole moment calculations.

 

 

 

 

 

 

Transition Sfifigéa au/AEZb CbC ccC
Ground State
(A) 202 —> 3,3 )M) = 1 + 2.0AoA 0.17600 +0.0339A01 +10A.57881
(B) 2,, -> 303 |M| = 1 + 1.2156 0.10A85 +0.0199771 -10A.55836
(c) 10, _> 2,, |M| = 0 + 6.917 0.5966A +0.11A3908 + 0.0337893
(D) 10, -> 2,2 )MI = 1 +A0.SA 3.A96 +0.6578236 + 16.937205
First Excited State
(A') 202 —> 3,3 |M| = + 2.05A8 0.1772A +0.0339338 +107.03608
(B') 2,2 —> 303 |M| = + 1.21A8 0 10A79 +0.0200211 -107.01565
Second Excited State
(A") 202 .> 3,3 |M| = + 2.0235 0.17A5A +0.0339675 +107.298A9
(B") 2,2 —> 303 |m| = + 1.1730 0.10118 +0.0200377 -107.27806

 

a .. ..
Practical units
b

104 (Me/sec)/(statvo1t/'cm)2

C104 «MC/Sec)/(statvolt/bm)2)/Debye2

82

Table XVI. Calculated dipole moment components (Debyez)

 

 

 

 

Ground State First Excited State Second Excited State
”b2 “c2 “b2 AC2 ubz AC2
(1) 5.2090 -7.6x10‘6 5.2271 -1.2x10'6 5.105A +10.Ax10‘6
(2) 5.2151 —8.0x10‘6
(3) (A) 5°185° °'°* 5-2231 0-0* 5.1385 0.0*
(B) 5'2A87 0-0* 5.2337 0.0* 5.0A9A 0.0*
(c) 5.2159 0.0*
(D) S 315 0.0*
(A) (A) 8'8775 0-0001* A.9077 0.0001* A.8226 0.0001*
(B) 5-7719 0-0001* 5.7685 0.0001* 5.58A9 0 0001*

(0) 5.2158 0.0001*

(D) 5.312 0.0001*

 

(1) Calculated by solving (A) and (B).
(2) Calculated by least squares fit of (A),(B), and (C).
(3) Calculated assuming uc = 0.0.

(A) Calculated assuming “c = 0.01.

”Assumed value.

83

These calculations show that for all three vibrational states
uc is certainly no larger than 0.01 D, and that it most probably is
zero. The best values for u = (ubz)l/2 are uv=o = 2.28 i .02 D,
uv=1 = 2.29 i .02 D, Hv=2 = 2.26 i .02 D. Thus, the dipole moments
in all three vibrational states are equal within the estimated range
of experimental error.

From the fact that no vanishes, the structure with Oh symmetry

is highly unlikely, confirming the conclusions from the relative

intensity measurements.

D) Boron Atom Coordinates and a Molecular Structure

The third evidence for C2 (or CZV) symmetry of the molecular
configuration was the location of the boron atom. The coordinates
of the boron atom were determined using the three Kraitchman equa—
tions (equation 2-A5), and the experimental second moments of the
1OB and parent ground state Species. The zero point vibration changed
sufficiently with the 10B substitution that the calculation of the a
and c coordinates resulted in imaginary numbers. This result is
precisely what is expected for an axial position of the substituted
atom, and must be taken as evidence for the on-axis location of the
B atom.

Thus, configuration with oh symmetry has been shown to be highly
linlikely, in the light of the observation of the effects due to
nuclear spin statistical weights, the dipole moment orientation, and
the axial location of the B atom. The existence of at least C2 sym-
metly'is therefore well established. This symmetry was therefore

invoked in determining the structure of 1,3,2-dioxaborolane. It

8A

requires that a, = —a2, bl = b2, cl = -c2 for the pair of oxygens,
the pair of carbons, and the two pairs of methylene hydrogens (Hc).

There is not sufficient isotopic data to completely determine
the molecular structure by the Kraitchman—Costain method. Only
the coordinates of the boron and the two carbon atoms can be cal—
culated from single substitution data. Once these coordinates are
known, however, their contributions to the second moments of inertia
and the center-of-mass equations can be subtracted. The B-H bond
length was assumed to be 1.20 8, so the contribution from.this hydrogen
can also be subtracted. Therefore, it remains to calculate the three
coordinates for the oxygen atom and six coordinates for the Hc's.
Three Kraitchman type equations have been derived (Appendix B) for
the A-D substitution, subject to the C2 symmetry condition. These
result in one equation involving the b coordinates of two non-equi-
valent Hc's, and two mixed equations involving their a and c

coordinates.

The b coordinates for the oxygen and for the two Hc's were
determined from the equation for Pbb of the parent, fromZmibi = 0
for the parent, and from the A-D Kraitchman-type equation for the
hydrogen b coordinates. The three equations could not be satisfied
Simultaneously, with real values of the coordinates, since they were
non-linear. A set of coordinates with minimal deviations did exist,
however, This set was then chosen as yielding the "best" values for
the three b coordinates, by adjusting them to satisfy both the A-D
substitution equation and the parent center of mass condition.

The a and c coordinates for the oxygen atom and for the
rmathylene hydrogen atoms were then determined from the two equations
for the parent Paa and PCC, the two Kraitchman-type equations in aH ,

. i
a , c , c , and the assumption that the methylene hydrogens are
H2 H1 H2 .

85

each equally distant from the bonded carbon and equally distant from
the non—bonded carbon. These two conditions are contained in the
quite plausible assumption that the methylene angle is bisected by
the O-C-C plane. It should be noted that a third constraint is im-
plied by this bisection assumption, namely that both hydrogens be
equally distant from the oxygen atom. This constraint was not used,
since it was felt that if it were invoked errors in the O and H b
coordinates would add large uncertainty to the a and c coordin-
ates calculated. Indeed, when the six a and c coordinates were
calculated, they did not satisfy this third constraint, and the
largest contributions to the inequality were the differences
(bO - le)2 and (b0 - sz)

The principal axis coordinates which were calculated by the
methods outlined above are summarized in Table XVII. From these,
bond distances and angles were computed by simple geometrical rela-
tions. These resulting molecular parameters are presented in Table
XVIII. Projections of 1,3,2—dioxaborolane in the principal ab
and ac planes are shown in Figure A.

Of these parameters, only C-C may be called an rs bond length.
The rest are rO parameters at best. The actualy C-C substitution bond
length may be slightly greater than the 1.5A1 A listed; as the c
coordinates are expected to be too small, due to the proximity to the
ab plane. The bond length is larger than the generally accepted
value 1.526 A for simple paraffin hydrocarbons (35), but is Slightly

smaller than the 1.5A9 8 reported for trimethylene oxide. The C-H

bond length is equal to the C—Ha distance (adjacent to the oxygen)

in trimethylene oxide, and the C-H distance in methane (36). The C—O

 

86

Table XVII. Principal axis system coordinates (A) of the atoms in

 

 

 

OCHZCHZOBH
. a b c
Atom Coordinate Coordinate Coordinate

B 0a +1.166996 0a
HB 0a +2.366996a 0a
0 il.1A7973 +0.A23731 10.0A9867
c : .7678AA -0.958A23 :0.062A2A
H, :1.111590 -1.A0A070 70.999106
H2 11.255828 -1.A63000 £0.775078

IZXP - Igalc = +0.15A95A AMU-fi2

IEXP - lgalc = +0.000012 Aim—R2

1?? — 1231C = +0.1A9397 AMU-Rz

,Emiaibi = Zimibici = O

:Ennaici = O.AOAAA3 AMU-R2

1

 

a
Assumed value.

Table XVIII. Structural parameters

87

calculated for 1,3,2-dioxaborolane

 

 

C-C
C-O
B-O
C—H

B-H

Z OBO
Z BOC
Z 000
_/_ HCH

Z HCC

H-

1.5A1
1.A38
1.368 r

H-

1.093 r
(1 20 R).

11A° 12'
1070 18'
10A° SA'
109° 9'

112° 31'

Dihedral Z'between CBC plane and

.003 A
.010 R
.010 R
.008 R

Assumed

20'

1+

t 30'
i 30'
t 20'
t AO'

0B0 plane = 7° 8' i 30'.

Dihedral 1 between 0,020 plane and czc,0' plane = 11° A8' 1 30'.

 

88

CD
Figure A. Projections of 1,3,2-

dioxaborolane in its principle
ab and ac planes.

v ' i

b a—>

 

 

 

 

 

 

 

 

89
bond length of 1.A38 R is less than the 1.AA9 8 determined for tri-
methylene oxide. The C-O bond length is in good agreement with the
C—0 bond length, 1.A3 i .03 R, determined by electron diffraction
for B(OCH3)3 (37). The B-O bond length of 1.368 A is in good agree-
ment with the value 1.38 i .02 determined for B(OCH3)3. The H—C-H
angle is quite close to the tetrahedral angle (109° 28') and is a
degree less than the angle Hd'“'Ca""HCI (110° 18') in trimethylene
oxide. The remaining bond angles (of the ring atoms) are determined
to a great extent by the skeletal structure. The average angle would
be 108° for a planar ring, but decreases as the ring becomes non—

planar.

A.6 Interppetation of the Ring Bending Vibration

 

Once the molecular configuration had been established, the ring
bending Vibrational motion and potential function could be explored
in greater detail. Any assumed model for the Vibrations should be
consistent with information already obtained, namely that:

1) there is either one low-frequency mode with two very low-
lying levels, or two low-frequency modes each with a single lowr
lying level;

2) the two excited states have vibrational wave functions sym-
metric to a two-fold rotation about the b axis;

3) the Vibrational frequencies decrease substantially on sub-
stituting deuteriums for the methylene hydrogens;

A) the rotational constants A, B and especially C increase with

increasing vibrational excitation, in a non-linear fashion; and

9O

5) the vibration—rotation interaction is Similar for the parent,
A-D, and 13C Species.

The first step was to examine the symmetry properties of the Vibra—
tional modes of 1,3,2-dioxaborolane. To Simplify this examination,
only the fiVe ring atoms were considered as a first approximation.

For the five-member ring with Czb symmetry, there are five modes which
are symmetric and four which are antisymmetric to C2. Of these only
two are likely to have a very low frequency of vibration. They are a
symmetric and an antisymmetric out—of—plane ring deformation mode.
These two modes are essentially angle bending modes, while the other
seven modes contain large amounts of bond stretching. Adding the

four methylene hydrogens to the ring increases the number of normal
modes by six symmetric and six antisymmetric modes. Completing the
structure by adding the boron hydrogen introduces one more symmetric
and two more antisymmetric modes. It was felt that the modes antici—
pated for the bare ring would not be altered appreciably by the addi-
tion of the hydrogens. This should be a good approximation, Since

in most cases the normal modes which are substantially hydrogen Vibra-
tions are of high frequency and have little effect on the motions of
the modes assumed in their absence.

Since (a) only one symmetric low-frequency Vibrational mode was
anticipated for the five-member ring model, (b) the two excited vibra-
tional states which were observed in the spectrum seemed to lie in
a series with the ground state, (0) PCC increased in the series, (d)

a plausible vibrational model can explain the two excited states, it
was decided that the correct assignment for the two excited states

was V'= l and v = 2 for a ring deformation Vibration.

91
The second step was to examine the qualitative nature of the
potential function for the ring deformation Vibration. Three facts
about the potential function may be deduced from the information al—
ready presented. The first is that it must be quite anharmonic, even
for low values of the vibrational quantum number. This is demon-
strated by the unequal Spacing between the energy levels which were

estimated from the measurements of relative intensities, (E, - E0.~

53 cm—l, E2 - E, -35 cm-l). The second fact is that the normal
coordinate for the vibration must be symmetric to Czb. This has

also been deduced from the relative intensities. The third fact is
that there is quite probably a potential barrier at the planar config-
uration. This is deduced from the zig-zag variation of the rotational
constants on increasing Vibrational excitation. In their work on
trimethylene oxide, Chan, Zinn, Fernandez, and Gwinn (13) demonstrated
that a zig-zag variation of the rotational constants with v could
be interpreted in terms of an equation for the Single-mode Vibration-
rotation interaction (as equation (2-53)) to be caused by a non-linear
dependence of <:QZ> on V. This zig-zag variation could be produced
only by a potential function which had two minima separated by a
potential barrier. The explanation for this was that superimposing

a barrier on a single-minimum potential function produced the least
perturbation on the energy levels whose wave functions have a node

at the barrier maximum, the levels with odd values of v. Levels
'with even values of V were much more strongly affected, being in-
creased in energy. At the limit of an infinite potential barrier

the V'= even levels had been made coincident with the next higher

'v = odd level, producing doubly degenerate excited states. Even if

92
the ground vibrational state was above the potential barrier, the
presence of the barrier had an observable effect on vibrational
energy levels. In fact, in the case of trimethylene oxide the bar-
rier was calculated to lie 11 cm-1 below the ground state.

In summary, then, all the effects which have been attributed to
vibration-rotation interaction for 1,3,2—dioxaborolane are consistent
with a double minimum potential function, and with the assignment of
v = 2 to the second vibrational state.

The third step was to devise a model to describe the motion of
each atom for the ring bending Vibration. An indication of the type
of motion present was obtained by considering the differences in the
carbon atom coordinates between the ground and first excited states.
The absolute values of the substitution coordinates calculated for
the first excited state showed an increase of 0.011 A in c, a de—
crease of 0.002 A in b, and a decrease of 0.005 A in a, compared
to the ground state. Although far from conclusive, these indicated
that the carbon atoms were moving out of the OBO plane, and that the
midpoint of the C-C bond was moving closer to the boron atom. Three
constraints were imposed on the motions in the model. First, only
angle deformations were allowed, Since the force constants for angle
deformations are generally much smaller than for bond stretching.
Second, the bond distances which had been calculated in section A.5
were employed, and were held constant. Third, in order to locate the
methylene hydrogens, they were oriented such that the HCH plane bi-
sected the OCC angle, and the OCC plane bisected the HCH angle. The

normal coordinate was taken to be 0, the angle between the two OCC

93
planes. A more convenient angle to work with was 0, the angle between
the CBC and 0B0 planes. It was examined over the range zero degrees
(a planar ring) to 30°. For each value of 0, it was possible to cal—
culate the molecular configuration and the second moments of inertia
for each of the isotOpic species which had been studied.

AS 0 increased from 0° to 30°, the angle BOC decreased from
107.58° t0 102.03°, and angle OCC decreased from 105.28° to 99.0l°.
The second moments of inertia, Paa and Pbb’ of each isotopic species
decreased continuously and the PCC increased continuously with 0;
the rotational constants A, B, C all increased with 0. There was no
point at which the change in A became negative (as it had experiment-
ally between the first and second excited vibrational states) so it
must be concluded that the observed decrease (A2 - A1 = -.77 Mc/Sec.)
was due to the averaging over the vibrations. The vibrational aver-
aging must also be responsible for the non-linear increase of B and
C with v, Since they varied smoothly with 0.

The model for the Vibrational motion was employed for two further
calculations. The first of these was an estimate of the effective
reduced mass for the vibration in both the normal and A-D substituted
molecule. The ratio of the reduced masses could be used to interpret
the differences between the normal and A—D Vibrational energies which
had been deduced from the measurements of the relative intensities.
The ratios of the normal and A—D vibrational energies had been esti-
rmated to be 53:22 = 1:0.A2 for the first excited states, and 88:66 =
1:0.75 for the second excited states. The reduced masses were com—

;m1ted as the sum of the mass of each atom times the square of the

9A

relative distance it had moved between two values of the normal co-
ordinate. The relative distances were normalized by dividing by the
change in 0 times its unit vector. The reduced masses calculated

for 0 = 7° to 0 = 7.5° were 20.A AMU for the normal molecule, and
30.0 AMU for the A—D substituted molecule, a ratio of 0.68:1. Since
the Vibrational frequency of a harmonic oscillator varies as the

- 1/2 power of the reduced mass, and the frequency of a quartic
oscillator varies as the -2/3 power, the estimated reduced masses

are qualitatively consistent with the observed decrease of Vibrational
frequency from the parent to the A-D species. Little information may
be gained from a quantitative comparison, due to the large uncertain—
ties in the estimated energy levels and in the reduced masses.

The second calculation based on the vibrational model was per—
formed to predict the expectation value <:0 > for the two excited
states. To accomplish this, the angle was determined at which the
calculated value of PCC was equal to the experimental value, and

similarly for P This was repeated for each vibrational state of

bb'

each isotopic species for which an experimental value had been ob-

tained. The results of these calculations are presented in Table XIX.
Although there is substantial variation in the absolute values

of <:0 > for each Vibrational state, the differences between states

is much more consistent. Using these differences and the ground state

zangle, <:0 > = 7.13A°, which was calculated in the determination of

the molecular structure, the corresponding <10 > for v = l is predicted

‘to be 9.9A 1 .80 and <:0 > for V = 2 is predicted to be 10.7 i 1°.

95

Table XIX. Values of < 0 > (degrees) determined from the model for
the vibrational motion

 

 

 

 

cc bb
OCHZCHZOBH v = 0 7 07A 8.65A
v = 1 10.632 11.109
v = 2 11.600 11.706
0CD2CD20BH v = 0 7.A79 10.735
v = 1 10.353 12.667
v = 2 11.21A 13.1A5
013CH2CH20BH v = 0 7.068 8.51A
v = 1 10.610 11.01A
OCHZCH201°BH v = 0 7.091 8.656

 

 

a . calc calc _ exp exp
Value of 0 for which PCC (Pbb ) — PCC (Pbb ).

96

A.7 Discussion

 

The evidence which has been presented leads to the conclusion
that the potential function for the ring deformation vibration has
two minima separated by a small barrier to inversion. The ground
Vibrational state must be higher in energy than the maximum in the
potential barrier, since none of the expected effects of a tunneling
inversion through the barrier were observed. Evidence for such tun-
neling would include anomalous Stark effects, similar to those con-
sidered for trimethylene oxide, and very large centrifugal distor—
tion or Vibration-rotation coupling perturbations, Similar to those
observed in cyclopentene (38). Furthermore, if the potential barrier
was high compared to the lowest vibrational energy levels, a degeneracy
might occur in the vibrational levels. For this case the effects of
nuclear spin statistical weights on the relative intensities could
not have been observed.

The interpretation of the molecular forces which produce the
double minimum potential function is quite straightforward. There
is obviously some strain energy present in the ring valence angles
due to ring closure. The strain-free valence angle for boron is
very nearly 120°, while the carbon and oxygen angles are expected to
.be close to the tetrahedral angle, 109° 28'. The average of these
‘ strain-free angles is 111° 3A', while the angles for a planar five-
member ring must average 108°. The minimum in the angle strain
enerpy’would occur at a planar configuration. Opposing the ring
tangle strain energy is the energy associated with the methylene hydro—

gen.cmientation. This energy is at a maximum for the eclipsed

97

orientation in a planar ring, and decreases with a torsion about the
C-C bond. Therefore, if the methylene groups are symmetrically located
with reSpect to the OCC planes, a small symmetric bending of the ring
will produce a substantial torsion about the C-C bond, and will de-
crease the methylene torsional energy. It must be concluded that
the potential function for the ring bending vibration is mainly due
to these two opposing forces, and that the barrier at the planar con-
figuration is not very large.

In recent years there have been a number of microwave studies
of four and five—member rings, with particular interest in the ques—
tions of ring planarity and low frequency modes of vibration. The
outstanding example of these studies is the thorough investigation
of trimethylene oxide, which has been extensively used for comparison
with the results reported here for 1,3,2-dioxaborolane. For a series
of four-member rings, a relation between the number of pairs of ad-
jacent methylene groups and the relative heights of the potential
barrier has been noted (39). Specifically, cyclobutane with four
methylene groups is decidedly bent (A0); trimethylene oxide with three
methylene groups has a barrier slightly lower than the ground vibra—
'tional state; and B-propiolactone with two methylene groups has no
(detectable barrier (39). Most of the five member rings which have
iwacently been studied are conjugated systems, such as furan (A1) and
;1,2,5-oxadiazole (A2) which have no methylene groups and are planar.
cywzlopentadiene is also thought to be planar (A3). Cyclopentene with
tluxze methylene groups and one double bond has a high central barrier,

aIKI it has been demonstrated that there is inversion by tunneling

98

through the barrier (38). 1,3,2-dioxaborolane has been shown here

to have a small barrier, most probably due to the two adjacent methyl—
ene groups. Since ethylene carbonate also has two adjacent methylene
groups in its five-member ring, but should have much less ring angle
strain (with its C=0 group replacing the B—H of 1,3,2-dioxaborolane),
it is predicted that a higher barrier will be present than that of
1,3,2-dioxaborolane, but lower than that of cyclopentene. The micro-
wave Spectrum of ethylene carbonate is being examined at the labora—
tory of Dr. James E. Boggs (AA), and it will be interesting to see

whether the prediction of the size of its potential barrier is correct.

V. CENTRIFUGAL DISTORTION CALCULATIONS

This chapter is concerned with centrifugal distortion, the name
applied to the changes in the rigid rotor energy levels when fourth
(or higher) powers of the angular momentum operators are included in
the rotational Hamiltonian. The chapter begins with a summary of
some well—known aSpects of the theory of centrifugal distortion. Fol-
lowing this, at equation (5—5), some possibilities are presented for
extending present methods for calculating and analyzing the effects
of centrifugal distortion in molecular rotational or Vibration-rota-
tional Spectra. It must be noted that the formulation of Kivelson
and Wilson (A5) is basic to this chapter.

The Hamiltonian for a centrifugally distorted or semi-rigid

rotor is
2 Z 2
H=AP +BP +CP +1/ r PPPP. -1
a b C AZZZZGBY, a,“ (5)
91375
The 81 TGBYS'S are the centrifugal distortion constants. Parker (A6)

has demonstrated that there are only 21 distinct T's, since in general

Tam = T7597 = T5976 Iowan

(5-2)

= T9957 — T6799 z T7599 = T5799 '

Further simplifications are possible in cases where the molecular
system under consideration has elements of symmetry. For each of CS,
C2, and Czh, a set of eight of the 217's must be zero, and for the

most widely studied cases of orthorhombic symmetry (CZV, V, Vh) there

are only nin non—zero T's.

99

100

To date, centrifugal distortion has been treated by perturbation
theory, usually to first order. To extend the accuracy, second—
order calculations have been performed; in addition, semi-empirical
P6 terms have been added to the Hamiltonian. It was felt that these
calculations might be improved by a different approach, both to ob—
tain greater accuracy and to extend the useful range of centrifugal
distortion analysis. The thought was that it would be possible to
exactly diagonalize the Hamiltonian matrix for at least the ortho-
rhombic case. These solutions would then form a convenient basis in
which to calculate the perturbation energies of systems with less
symmetry or with higher order Pn contributions. This diagonalization
would effectively reduce by one the order of the perturbation calcu—
lation. With this goal, the study of the orthorhombic case was under-
taken.

For a semi—rigid rotor with orthorhombic symmetry, there are nine

non—zero 7'3

three each of the types When the

’ Toaoa’ TooBB, TGBGB'

commutation relations (equation 2-11) are applied to the operators

. . ' . . . . . 2
which these T s multiply, terms of the type PaPB Pa _ and PGPBPaP
2P 2.

B

If similar operators are then grouped, and the three rigid rotor terms

B

may be reduced to a combination of terms only of the type Pd

included, the number of independent coefficients is reduced from 12
to 9. Kivelson and Wilson (A5) define the new coefficients (T') by

the equations

1 _ I _ ‘ _ ' 4
B X “ BX + l/L'OTYZYZ 2Tszz ”my? 8 i
_. _ _ 4 _
B'Y ‘ BY + l/MBTXZXZ 2T7272 2TXYXY) 4" , (5 3)
.. 4
8'2 ‘ '32 + l/L‘(3TX7XY ' 277272 ‘ ZTXZXZ) 5‘ i

f _
I ..e

101

I _
T 2222 ‘ 72222:

I ...
T - TXXYY + 2%???”

szz ‘ TXX22 + 2TX2X2:

+ 27

T' = T

7722 7722 7272'
Therefore, the Hamiltonian becomes
H=B'XPX2+B'YPY2+BZPZZ-'P+l/AITXXXX X 4+T'7777P74+
T'2222 P24 + T XXYY[PX2 P72 + P72px2 ] + T'XX22[PX2P22 + P2sz2 ]
1 flYYZZ[PY2PZZ + PZ 2P Y2]}. (5-A)

The Hamiltonian (5-A) is the one which is to be exactly diagonal-
ized. It is diagonal in J and M and has elements only of the type
< KKK >, < X|X : 2 > , and < KIK : A> . It could be diagonalized
directly, but some further manipulations will simplify the computation.

A new set of m's is defined here by the equations

II _ I

T XX 1/8 T XXXX
n = v

T 77 1/8 I 7777

I 22 ‘ 1/8 1'2222 (5-5)

= , _ I _ I
I"X7 1/A(+'XXYY 1/2 7 l/2 'r

n _ t _ I _ I
I XZ ‘ l/L'T XXZZ 1/2 T XXXX 1/5 T 2222)

n .__ I _ I _ '
T 72 1/L(T 7722 1/5 T 7777 1/8 I 2222)“

 

102

Then by noting that

2+P2+PZJ=P2P2

422 22_2 __
P + PX-PY + PX PZ — PX [PX Y Z X ) (S 6)

X
and similarly for y and z, equation 5—A may be re-written as

= y 2 I 2 l 2
H BX PX + BY PY + 82 P2 +

n 2 2 n 2 2 n 2 2
T XXP PX + T 77P P7 + T 22P P2 + (5-7)

P 2 + P 2P 2] +

2 2 2 n 2
+ P P J l I X 2 2 X

2
‘ PY Y X XZ

PX

THXY[ [P

2 2 2 2
[PY PZ + PZ PY J

T"72
This form of the Hamiltonian may be factored into five separate
parts to facilitate the diagonalization. The basis set in which the
elements are calculated has been assumed to be the same as was used
in section 2.2, including the Wang transformation, XHX. The
Hamiltonian is
H = H(O) + H(1) + H(2> + H'3) + H'4) , (5—8)

where
(0) = v I 2 —
H 1/2 (B + B )P (5 9)
X Y
is diagonal, and constant in the quantum number K;
(1) _ 7 2 2 _ 2 _
H - l/2[2BZ - BX' — BY'][PZ + b(PX PY )J, (5 10)

where

BXI __ BY'
QB 1 _ Bit _ v
z X BY

is in Wang's reduced energy form, and has elements < K K >

 

H0)
and < K)K i 2 >. Furthermore,

m7 = 027.1,, . We (s-m

103

is diagonal, and also constant in K;

(3)_ . 2 2 2 _ 2 _
H __ 1/213 [214122 _ THXX _ THYYJUDZ + fT (PX PY )] (S 12)

is a reduced distortion energy similar to H<l), with <K|K> and

< X\X : 2> elements, and with

H H

T XX ‘ T YY

T n _ n _ n
QT 22 T XX T YY

 

Finally,

P 2 + P 2P 2]

(4) _ n '2 2 2 2 2
H — T [P P + P PX ] + [PX Z Z X

Tn
XY X Y Y XZ

(5-13)

I! 22 22
+ T YZ[PY PZ + PZ PY ], and

has < K|K > , < XlX : 2 > , < K|K : b> elements.
The blocks of (5-8) in the Wang symmetric rotor basis are symmetric

across the principal diagonal and have the form

/ E00 «(7502 ”504 O 0
5+ = ¢r2502 E22 + E2.2 E24 B26 0

dr5504 E24 E44 E46 E66

 

 

 

+ "1
O = E13 + E1-3 533 E35 E37 0 (S b)
E15 E35 E55 E57 E77
\\ \ \ \. \ L;

\\ \\ \\ \\

_ t at
5+ Except/the first row and column are omitted and

U1
H

522 + E2_2 is replaced by E22 - 52-2.

0
M

0+ Except that the (1,1), (1,2), (2,1) terms are now

Ell - El-l) E13 ‘ 51-3) E13 ‘ 51—3 reSPBCtiVQIYo

10b

The K dependent elements of H' = H(1) + H<3> + H<4> are

2
< X|H'1X > =:V2 K2[2BZ' - BX} — 8Y1 + P (2T"ZZ ‘THXX — THYY)
+ T"XY {(P2 - K2)(P2 - K2 + 2) + 3K2}
n n 2_ 2
+2(TYZ+TXZ)(P KN)
<KW'Mfit2>==U%F‘iG%Kfi)EY'-BX'+TWY-flkX+ENwWZ—vwma

2
{K2 + (K + 2m + H‘XYHPZ — K02 - (P2 - [K 1.2102} 1, (5-15)

< KIH"K : L > = - 1/8 THX P : (P,K,O) F : (P,K,2) ,

Y

where F : (P,K,m) = [P2 - (X irn)(K t m t 1)].

In this form, the semi-rigid rotor should be relatively easy to
diagonalize. The detailed method to be used will depend on the con—
siderations of computer efficiency and timing, the size of the sub—
matrices to be diagonalized, and the ease of coding the procedure.
It should prove useful to take advantage of the continued fraction
program which is presently available (Appendix B). The < KIK i b >
elements could be treated by Givens' method (h?) to reduce the matrix
to tri-diagonal form, and the diagonalization completed with the
continued fraction method. This method would also allow computation
of the rigid rotor eigenvalues for comparison purposes.

Once the eigenvalues and eigenvectors of (5—8) have been ob—
tained, they may be used to calculate first and higher order pertur-
bation energy contributions from distortion constants admitted by

lower symmetry or from higher powers of PD.

105
A second use would be to check the accuracy of results which
have been obtained by perturbation theory using the rigid rotor
as the zero order problem. Such a comparison could be made, for

example, with the results of Pierce, DiCianni and Jackson for OFZ (h8).

\1

10.
11.
12.

13.

lb.
15.
16.

17.

BIBLIOGRAPHY

Cleeton, C. E. and N. H. Williams, Phys. Rev. 99, 23h (193b).
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b. Bleaney B. and R. P. Penrose, Phys. Rev. Z9, 775 (19h6).
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b. Good, W. E. Phys. Rev. 29, 213 (l9b6).

)

Dailey, B. P., R. L. Kyhl, M. W. P. Strandberg, J. H. Van Vleck
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Burkhard, D. G. and D. M. Dennison, Phys. Rev. 99, £08 (1951).
a. Herschbach, D. R., Thesis, Harvard University, 1958.

b. Herschbach, D. R., J. Chem. Phys. 21, 91 (1959).

c. Lin, C. C. and J. D. Swalen, Revs. Mod. Phys. 2;, 8&1 (1959).

Dakin, T. W., W. B. Good, and D. K. Coles, Phys. Rev. Z9,
560 (l9h6).

Hughes, R. H. and E. B. Wilson, Jr., Phys. Rev. 1;, 562 (1987).

Debye, P. J. W., Polar Molecules, Chemical Catalog Co.
(1929)-

New York,

3

 

Shulman, R. G. and C. H. Townes, Phys. Rev. 11, 500 (1950).

Kraitchman, 1., Am. J. Phys. 91, 17 (1953).

Costain, C. C., J. Chem. Phys. 99, 86h (1958).

Chan, S. 1., J. Zinn, J. Ferandez, and W. D. Gwinn, J. Chem.
Phys. 2;, 16b3 (1960); Chan, 5. 1., J. Zinn, and w. D.
Gwinn, J. Chem. Phys. 29, 1319 (1961).

Schlesinger, H. I. and A. B. Burg, Chem. Rev. 39, 1 (19h2).

Rose, S. H. and S. G. Shore, Inorg. Chem. 1, 7hb (1962).

Wang, s. C., Phys. Rev; 25, 213 (1929).!

Ray, B. s., Zeits. P. Physik.1§, 7b (1932).

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29.

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107

King. G. W. R. M. Hainer, and P. C. Cross, J. Chem. Phys. ll,

)

'27 (19b3).

Golden, S. and E. B. Wilson, Jr. J. Chem. Phys. 2, 869 (l9b8).

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Casimir, H. B. G., On the Interaction Between Atomic Nuclei and
Electrons, Teylers Tweeke Genootschxx E. F. Bohn, Haarlem

(19365.

 

Bragg, J. K. and 5. Golden, Phys. Rev. 15, 735 (19b9).

Townes, C. H. and B. P. Dailey, J. Chem. Phys. 11, 782 (19b9).
Eckart, C., Phys. Rev. 91, 552 (1935).

Herschback, D. R. and V. W. Laurie, J. Chem. Phys. 21, 1668 (1962).
Oka, T. and Y. Morino, J. Mol. Spec. 9, A72 (1961).
Herschbach, D. R. and V. W. Laurie, J. Chem. Phys. 99, 3182 (196A).
Douglas, A. E. and C. K. Mdller, J. Chem. Phys. 3g, 275 (1951).

a. Cox, J. D., H. S. Turner and R. J. Warne, J. Chem. Soc.,
3167 (1950); as reported (page 35) in,

b. Murry, A. and D. L. Williams, Organic Syntheses with Isotopes,
Interscience, New York, 1958.

 

Cox, J. D. and H. S. Turner, J. Chem. Soc., 3176 (1950); as
reported in reference 28-b, page 905.

Popplewell, D. S. and R. G. Wilkings, J. Chem. Soc., 2521 (1955);
as reported in reference 28-b, page A95.

Townes, C. H. and A. L. Schawlow, Microwave Spectroscopy, McGraw—
Hill, New York (1955).

 

Shore, S. G., private communication.

Wilson, E. B., Jr., J. C. Decius and P. C. Cross, Molecular
Vibrations, McGraw-Hill, New York (1955), page 1h7 ff,

 

Marshall, S. A. and J. Weber, Phys. Rev. 995, 1502 (1957).
Lide, D. R., Jr., J. Chem. Phys. 2;, 1519 (1960).
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NO.

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I?
\L’
6

Ah.

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50.

108

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b. Butcher, S. S. and C. C. Costain, J. Mol. Spec. 22, b0 (1965).

Boone D. W. C. O. Britt and J. E. Boggs, (to be published).

) 3

Dunitz, J. D, and V. Schomaker, J. Chem. Phys. 29, 1703 (1952).
Bak, B., D. Christensen, W. B. Dixon, L. Hansen-Nygaard, J.
Pastrup-Andersen, and M. Schottlander, J. M01. Spec. 9

12b (1962).

)

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Boggs, J. E., Private communication.
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APPENDICES

109

APPENDIX A

COMPUTER PROGRAM "EIGVALS"

This program calculates rotational energy eigenvalues, quadru-
pole coupling energies, transition frequencies, line strengths, and
Stark effect coefficients for a rigid asymmetric rotor. It was origin-
ally coded in FORTRAN-6O and has been modified to run in 3600 FORTRAN
for the Control Data 3600 computer at Michigan State University.

The heart of the program is the routine for diagonalizing the
reduced energy Hamiltonian. The reduced energy matrix is established
in the computer from the values of the rotational constants which are
read in as input data. The matrix elements are those appropriate to
the III1 or Ir axis identification, depending on whether the rotational
constants correspond more closely to an oblate or to a prolate rotor.
The diagonalization utilizes a continued fraction calculation with a
Newton-Raphson second—order iterative procedure as suggested by
Swalen and Pierce (h9). The "best development" algorithm of Swalen
and Pierce is also used, with good results. It has been found that
it is possible to speed convergence and eliminate the possibility of
convergence to an incorrect root, by employing an additional algorithm
to obtain the required initial estimate of each eigenvalue. The
linear and quadratic solutions for the low J matrix blocks are
calculated directly. It is then possible to extrapolate the eigen-
values to initially estimate higher J eigenvalues. The extrapola-
tion is approximately of the order (J — 2‘K|) in the successive dif—

ferences of the eigenvalues which have already been computed.

110

111
The extrapolation improves the initial estimates, especially for high
J, low K eigenvalues, which are preCisely the ones which have the
greatest deviation from the linear extrapolation which has often been
used previously. The final eigenvalue of each matrix block is esti—
mated from the criterion of invariance of the trace for the trans-

formation.

The eigenvectors are then used to compute the average values of

2
Z

P (i.e. < Pzz>) from which < PX2> and < Py2> are then calculated.
These have been shown to be also the partial derivatives of the energy
with respect to the rotational constants. The quadrupole coupling
constant coefficients aq and Bq (equation 2-38) are calculated from
the derivatives, and the energies WQ are obtained by multiplying by
the values of'Jfaa and (i):bb - :XTCC) which were read into the com—
puter as input data.

The final section of the program was written by Dr. Schwendeman.
The line strength for each possible transition is calculated by trans—
forming the apprOpriate direction cosine matrix element (50) with the
eigenvectors previously computed. Only the transitions which have
a frequency between 8,000 and L0,000 Mc/sec. and a relative line
strength greater than 0.1 are printed out. Finally the Stark coef-

ficients are computed by second-order perturbation from the direc-

tion cosine matrix elements and the energy levels.

APPENDIX B

KRAITCHMAN—TYPE EQUATIONS FOR THE h-D SUBSTITUTION

The substitution of four deuterium atoms for the four methylene
hydrogen atoms in 1,3,2-dioxaborolane produces the following changes

in the values of the second moments of inertia:

 

2
2 2 2 2 [Am(xl + X2 + X3 + X4)]

P2X = PXX + Am(xl + x2 + x3 + x4 ) - M + hAm etc.

Pl . = A171<X1yl + X2Y2 + X3}!3 + x4314) _

Ky ' (B—l)

Amz .
M r LAm[(Xl + X; + X3 + X4)(Y1 + Y2 + Y3 + Y4)] etc.
where

Am = mD - mH,

M = parent molecular weight, and
(x1, x2, ..., 24) are the coordinates of the hydrogens in
the parent axis system.
The elements in (B-l) may be simplified by satisfying the requirement

of Cgfsymmetry;

X1 2 'X3: X2 = “X4:
Xi = y3. Y2 = y4, (B-2)
Z1 = '23: 22 = ‘24'

These reduce the elements of (B—l) to

1 = 2 2
Pxx Pxx + 2Am(xl + X2 )9 (B-3)
_ 211m 2 2 2
P§y — Pyy + M_:_HZE[M(Y1 + y2 ) + 2Am(yl _ Y2) ],
Pr = P + 2Am(zlz + 222) ’

22 22

112

113

ny = O,

P = 0, and

yz

, ’ ‘ ‘ I! I! It ‘ ‘ "
The experimental second moments Pxx’ Pyy’ Pzz are the r00ts 01
the determinant lP'l = O. P' is already a root, therefore,
YY

P” - P = 21m [140 2 + y 2) + 2Am(y - y )2]. (EB-b)
W W m 1 2 1 2

PXx and P22 are the roots of the remaining 2 x 2 determinant

p1 _ pn p1
XX X2

=0 (8-5)

1 1 _ n
PXZ P22 P

EXpanding the determinant and equating the resulting coefficients of
H l H 0 ‘ H n H " . . . .
(P ) and (P ) to PXX + P22 and Pxxpzz’ leads to the two final

equations,

H _ = 2 2 2 2
PxxP'z'z Pxxpzz 2Am[Pzz(X1 + X2 ) + Pxx(Zl + 22 )

2
+ 2Am(x122 - x221) ] and (B-6)
) = 2Am[xl2 + x22 + 212 + 222].

(P” -P )+(P" -P
XX XX 22 22

These three equations (B—h), (B—5), (B-6),can not be reduced to
simple expressions in the coordinates of one atom. They were, however,
used in this form with other equations as reported in section A.5 to

determine the coordinates of the methylene hydrogens.

   

Millfll'llilllllslllllflllfll) 11111111111111)?!“
3 1193 01084 8349