MICROWAVE SPECFRA OF CBS
AND TRANQ ACETALDQXEME

Thesis For flan Degree of pit. D.
MECEEGEN STATE UNIVERSE“

Robert Stephen Rogowski
19-6-8:

 

 

THESIE

Date

0-169

——n:fl

L I B R A R Y
Michigan State

University

    
   

 

This is to certify that the

thesis entitled

MICROWAVE SPECTRA OF
CIS AND TRANS ACETALDOXIME

presented by

Robert Stephen Rogowski

has been accepted towards fulfillment
of the requirements for

__P_h.D_. degree in Chemi 5 try

1 ////z 4 «414444

[Major professor

February 21, 1968

 

 

 

ABSTRACT

MICROWAVE SPECTRA OF CIS AND TRANS ACETALDOXIME

by Robert Stephen Rogowski

A short historical account of the development of micro-
wave spectroscopy is presented. The theory of rotational
Spectra, including the effect of internal rotation on the
total angular momentum of the molecule, is discussed. The
microwave spectrometer is described.

The microwave spectra of cis and trans acetaldoxime
(CH3CHNOH) have been investigated in the region 8000 to
37000 MHz. Several transitions have been identified for
both Species and the rotational constants in MHz are as

follows: cis acetaldoxime, A = 17364.38, B = 6658.918,

C 4921.036; trans acetaldoxime, A = 44613, B = 4237.73,

C 3973.81. The transitions of cis and trans acetaldoxime
are split by tunneling of the methyl group through the
potential barrier to internal rotation. From these separa-
tions the barrier height has been determined to be 1830 cal/
mole for trans and 375 cal/mole for cis acetaldoxime.

The internal rotation splittings for cis acetaldoxime
were calculated using the high barrier approximation where
.Mflhieu functions form basis functions for the torsional
states of the methyl group. The calculations were also
carried out using free rotor functions to describe the

eigenstates for the internal rotation of the methyl group

(low barrier approximation) because the value of the barrier

Robert Stephen Rogowski
is near the limit where the high barrier approximation be-
comes inadequate. A comparison of the two methods of calcu-
lation, which yielded the same value for the threefold bar-
rier hindering the internal rotation, is included.
The quadrupole moment of the nitrogen nucleus contributes
a fine structure to several transitions and the components

of the quadrupole tensor have been determined. The quadru-

pole coupling constants are: cis acetaldoxime, Xaa =
—2.8 MHz, Xbb - ch = 0.6 MHz; trans acetaldOXime, Xaa =
4.07 MHz, Xbb - ch = -5.0 MHz.

The electric dipole moments of both Species have been
evaluated from measurements of the Stark effect. The total
dipole moment is 0.847 Debyes for cis acetaldoxime and

0.893 Debyes for trans acetaldoxime.

MICROWAVE SPECTRA OF CIS AND TRANS ACETALDOXIME

BY

Robert Stephen Rogowski

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1968

ACKNOWLEDGMENTS

I am indebted to Professor Richard H. Schwendeman
for his guidance and support throughout this study and
the preparation of this thesis.

Financial support from the Petroleum Research Fund
is gratefully acknowledged.

Appreciation is extended to my wife and parents for

their sedulous encouragement.

ii

II.

III.

IV.

TABLE OF CONTENTS

HISTORICAL BACKGROUND . . . . . . . . . . . .
THEORY OF ROTATIONAL SPECTRA . . . . . . . .
2.1 Introduction .-. . ... . . ... .-. . .
2.2a Model and Hamiltonian . . . . . . . . .
2.2b Rigid Rotor Hamiltonian . . . . . . .

2.3 Hindered Internal Rotation . . . . . .
2.4 Stark Effect in Rotational Spectra .

2.5 Quadrupole Hyperfine Structure . . . .

MICROWAVE SPECTROMETER . . . . . . . . . . .

3.1
3.2
3.3
3.4
3.5

3.6

Introduction . . . . . . . . . . . .

Reflex Klystron . . . . . . . . . . . .
Waveguide Absorption Cell . . . . . . .
Stark Modulation . . . . . . . . . . .
Detection and Amplification of Signals

Frequency Measurement . . . . . . . . .

MICROWAVE SPECTRUM OF TRANS ACETALDOXIME . .

4.1

4.2

Introduction . . . . . . . . . . . . .

Observed Spectrum and Assignment for
Trans Acetaldoxime . . . . . . . . .

Quadrupole Coupling Constants . . . . .

Barrier to Internal Rotation of Trans
Acetaldoxime . . . . . . . . . . . .

Dipole Moment of Trans Acetaldoxime .

iii

Page

(DQUIUH

15
24
27
32
32
32
35
38
39
41
46
46

47
50

55
59

TABLE or CONTENTS (Cont.)

Page
V. MICROWAVE SPECTRUM OF CIS ACETALDOXIME . . . 63
5.1 Observed Spectrum and Assignment . . . 63
5.2 Determination of the Height of the
Barrier Hindering Internal Rotation . 67
5.3 Quadrupole Coupling Constants of Cis
Acetaldoxime . . . . . . . . . . . . 74
5.4 Stark Effect and Dipole Moment of Cis
Acetaldoxime . . . . . . . . . . . . 77
VI. LOW BARRIER APPROXIMATION FOR CALCULATION OF
ENERGY LEVELS FOR INTERNAL ROTATION AND
OVERALL ROTATION . . . . . . . . . . . . . 81
6.1 Introduction . . . . . . . . . . . . . 81
6.2 Free Rotor Approximation . . . . . . . 83
6.3 Comparison of Results from Low Barrier
and High Barrier Approximations . . . 87
VII 0 DISCUSSION 0 O O O O O O O O O O O O O O O O 95

BIBLIOGRAPHY . . . . . . . . . . . . . . . . 106

iv

LIST OF TABLES

TABLE Page
I. List of klystrons now in use at Michigan
State University . . . . . . . . . . . . . 36
II. Frequency meters in use at Michigan State
University . . . . . . . . . . . . . . . . 42
III. Structure assumed for trans acetaldoxime . 47

IV. Frequencies of observed transitions for
trans acetaldoxime . . . . . . . . . . . . 51

V. Rotational constants, quadrupole coupling
constants and internal rotation parameters
for trans acetaldoxime . . . . . . . . . . 52

VI. Internal rotation splittings observed for
trans acetaldoxime . . . . . . . . . . . . 60

VII. Observed transition frequencies for the E

symmetry species of trans acetaldoxime . . 61
VIII. Stark effect of trans acetaldoxime . . . . 61
IX. Structure assumed for cis acetaldoxime . . 63

X. Experimental hypothetical unSplit frequencies
for cis acetaldoxime (A Species). . . . . . 66

XI. Comparison of observed and calculated transi-
tion frequencies for cis acetaldoxime . . . 70

XII. Rotational constants, quadrupole coupling
constants and internal rotation parameters
for cis acetaldoxime . . . . . . . . . . . 71

XIII. Internal rotation splittings for cis
acetaldoxime . . . . . . . . . . . . . . . 73

XIV. Quadrupole coupling constants for cis
acetaldoxime . . . . . . . . . . . . . . . 76

XV. Stark effect of cis acetaldoxime (A Species) 78

XVI. Results of fitting observed frequencies with
low barrier and high barrier approximation. 86

V

LIST or TABLES (Cont.)

TABLE

XVII.

XVIII.

XIX.

XXII.

XXIII.

Comparison of results from free rotor approx-
imation and Herschbach perturbation calcula-
tion for s = 16 . . 2 . . . . . . . . . .

Comparison of results from free rotor approx-
imation and Herschbach perturbation calcula-
tion for s = 12 . . . . . . . . . . . . .

Comparison of results from free rotor approx—
imation and Herschbach perturbation calcula-
tion for S = 10 O O O O O O O O O O O O I

Comparison of results from free rotor approx-
imation and Herschbach perturbation calcula-
tion for s = 8. . . . . . . . . . . . . .

Difference of calculated frequencies using
free rotor and perturbation calculation for
A and E species . . . . . . . . . . . .

Comparison of (VA-VE) values from free rotor
approximation and perturbation calculation.

Comparison of barriers to internal rotation
for cis and trans isomers . . . . . . . . .

vi

Page

88

89

90

91

93

94

96

Figure

1.

2.

3.

10.

11.

12.

13.

14.

15.

LIST OF FIGURES

Page
Block diagram of the microwave spectrometer . . 33
A circuit schematic for a typical klystron
oscillator . . . . . . . . . . . . . . . . . . 34
Waveguide absorption cell showing position of
septum . . . . . . . . . . . . . . . . . . . . 37
Trace of the 101 ——e 202 transition in cis
acetaldoxime after phase sensitive detection . 40
Reference frequency generator . . . . . . . . . 43

Projection of trans acetaldoxime in its plane
of symmetry . . . . . . . . . . . . . . . . . . 48

Quadrupole splitting for the 111-—-> 212 transi-
tion of trans acetaldoxime . . . . . . . . . . 54

Schematic diagram of J = 1-—> 2 transitions for
trans acetaldoxime Showing the splittings due to
internal rotation . . . . . . . . . . . . . . . 58

Projection of cis acetaldoxime in its plane of
symmetry . . . . . . . . . . . . . . . . . . . 64

Hyperfine pattern for the J = 000 —> 101 transi-
tion of cis acetaldoxime . . . . . . . . . . . 75

Observed and calculated Stark shifts for cis
acetaldoxime (E species) for the 101 -$ 202
transition . . . . . . . . . . . . . . . . . . 80

The torsional matrix . . . . . . . . . . . . . 82

The energy matrix in the free rotor representa-
tion . . . . . . . . . . . . . . . . . . . . . 82

-Structure of cis acetaldoxime showing overlap of

Van der Waal radii of oxygen and methyl hydrogen 98
Structure of trans acetaldoxime showing overlap

of Van der Waal radii of nitrogen and methyl
hydrogen . . . . . . . . . . . . . . . . . . . 100

vii

I. HISTORICAL BACKGROUND

The rapid development of radar technology during the
second World War provided reasonably good sources of electro-
magnetic radiation in the centimeter wavelength region.
These electronic sources of energy had advantages over the
heat sources used in infrared spectroscopy because the
radiation was confined to a small portion of the Spectrum.

A typical microwave source could provide milliwatts of
power at GHz frequencies in a frequency band less than one
MHz wide. To produce Similar output of power over such a
narrow frequency range a hot body would have to reach a
temperature of 1014 degrees C. In addition the microwave
radiation from a klystron could be made monochromatic to
better than one part in 107 and absorption spectra could

be studied by tuning the source rather than by selecting
frequencies from a broad source, as with a prism or grating.

The first experiment in the microwave region was per-
formed by Cleeton and Williams (1) who investigated the
absorption of ammonia vapor around a wavelength of 1.5 cm.
The inversion absorption frequency for ammonia was predicted
to lie in this region of the Spectrum. Using a magnetron
as their energy source, they obtained a broad absorption
band centered at a wavelength of 1.25 cm but could not re-
solve the fine structure because of the relatively high
pressure of the sample gas and the crude grating used to

resolve the spectrum.

2
The first high resolution work was performed by Bleaney
. and Penrose (2) using a klystron tube as a source, a reso-
nant cavity as an absorption cell, and a silicon-tungsten
crystal as detector. With this apparatus they were able to
resolve the various lines in the ammonia Spectrum which were
due to the effect of rotation on the inversion. At about
the same time Good (3) was studying the ammonia spectrum in
a different laboratory using a waveguide type of absorption
cell, 2 1/2 meters long. ‘With the longer path length avail-
able for absorption of the radiation and less difficulty
with saturation broadening he was able to obtain spectra of
ammonia at pressures below 10-2 mm Hg. The higher resolu-
tion possible at low pressure enabled him to observe quadru-
pole hyperfine structure in the spectrum.

During the first few years much effort was spent on
improving the instrumentation. The Spectrometer introduced
by Hughes and Wilson (4), in which the signal was modulated
by a periodic Stark field, greatly increased the sensitivity
of the microwave method. With the greater sensitivity avail-
able more complex Spectra were studied.

From analysis of the Stark splittings the dipole moment
of OCS was determined (5). The theory of the Stark effect
was extended to asymmetric-top molecules and now dipole
moments are determined for many molecules using the micro-
wave technique.

It soon became evident that a host of important informa-

tion could be obtained from energy transitions observed in

3
the microwave region. Examination of the spectra under
high resolution revealed a complex structure consisting of
many lines. If a nucleus with a quadrupole moment was pre-
sent in the molecule, additional hyperfine structure oc-
curred. Additional lines would occur in the spectrum if
internal rotation was present.

The problem of hindered internal rotation for symmetric-
top molecules was treated by Nielsen (6) and Koehler and
Dennison (7). Several years later the first experimental
microwave Spectra for a molecule possessing internal rota-
tion were reported by Hershberger and Turkevitch (8) who
published the spectrum of methyl alcohol. Herschbach (9)
has published an extensive treatment of the internal rota-
tion problem and has tabulated perturbation coefficients
which are very useful in calculating potential barriers to
internal rotation. E. B. Wilson (10) has reviewed the sub-
ject of hindered internal rotation discussing the methods
available for obtaining experimental values of the barrier
height and the theoretical interpretations of the data to
determine the physical origin of the barriers.

Probably the most important use of microwave spectros-
copy is the determination of molecular structure. The rota-
tional constants which are obtained from the Spectrum are
functions of the moments of inertia of the molecule. The
moments of inertia depend on the masses and positions of
the atoms in the molecule. By obtaining the moments of in-

ertia for several different isotopically substituted Species

4
of the molecule, the coordinates of the centers of mass of
the atoms can be found. An assumption required is that the
bond distances and angles suffer negligible change upon
isotOpic substitution. Kraitchman (11) developed equations
which are used for fitting differences in moments of inertia
of two isotopically different molecules to determine the
position coordinates of the substituted atom. Use of the
Kraitchman equations reduces the effects of changes in bond-
distances and angles upon isotopic substitution.

Because of vibration—rotation interaction it is dif-
ficult to relate the spectrosc0pically determined structure
to the equilibrium structure or the average structure of
the molecule (12). To establish the relations between the
various structures the anharmonic part of the vibrational
potential function must be determined accurately-~a process
which requires measurement of the rotational transitions
for a number of vibrational states and has been accomplished
for only a few polyatomic molecules. However Herschbach
and Laurie (13) have developed a method for obtaining, to
a good approximation, the average structure from the effec-

tive moments of inertia.

II. THEORY OF ROTATIONAL SPECTRA
2.1 INTRODUCTION

A molecule consists of nuclei held together rather
rigidly by the Coulomb forces between nuclei. and electrons.
In the Born-Oppenheimer approximation the total Hamiltonian
for the molecule can be separated into a nuclear part and
an electronic part. AS a consequence of the separation,
the electronic energy (including the energy of nuclear re-
pulsion) enters as the potential energy in the description
of the nuclear motion. The Schrodinger equation can be

written
(Tn + Ee)wn = Ewn (1-1)

where Be is the electronic energy as a function of
nuclear coordinates Xk’ Tn is the nuclear kinetic energy
Operator, E is the total energy of the molecule, and wn
is a wave function for nuclear motion.

The nuclear Hamiltonian, HT = Tn + Ee, includes the
translational, rotational and vibrational energy of the
nucleii. The translational kinetic energy is exactly
separable while vibrational and rotational energies are
only approximately separable. The interaction of vibra-
tion and rotation produces observable effects in both infra-
red and microwave spectra.

A Hamiltonian which is general enough to describe ef-

fects observed in the microwave region (1-300 GHz) would

5

include at least the terms

HT = HR + HRV + HS + HQ. (1-2)
In the above equation, HR is the energy of a rigid rotor,
HRV is the interaction energy between vibration and rota-
tion, H is the energy of interaction with an electric

S

field and HQ is the contribution to the energy from a
quadrupolar nucleus. Each of these effects will be con—

sidered separately in the sections that follow.
2.2a MODEL AND HAMILTONIAN

The model chosen to describe the molecule is a system
of point masses, which represent the nuclei , held together
by weightless springs. In order to describe the motion of
the system, suitable sets of coordinate axes are selected
which consist of a set of Cartesian axes fixed in Space
(X, Y, Z) and a set of rotating coordinate axes x, y, z
with origin at the center of mass of the molecule. In ad-
dition 3N—6 normal coordinates are used to describe the vibra—
tional motion in the moving system. The following treatment
is that of Wilson, Decius and Cross (14).

The position of the ch particle with respect to the
center of mass is given by the vector ra whose components

(w

in the moving coordinate system are xa. ya, za.

The center of mass of the system is located by the

vector R' from the origin of the space fixed axes. The

7
equilibrium position of the ath particle is given by the
vector ad which is fixed in the moving system. The dis-

Ml

placement vector pa is defined by the relation

p = r - a . (2-1)

"W’wfw

If at any instant the rotating axis system has the
angular velocity 9_ and if the vector Va is the velocity

of the ath particle in the moving system, the velocity of

the ch particle in space is given by

=.+ + , 2-2
“isfigxisiq. ()

where 3' is the time derivative of R; The kinetic energy

for the entire molecule is

2T = Z mfir2 = M R ° R + 2 m (w x

C1

+ 5- 2 m v ' v + 2R ° w x 2 m r

o o a “’ “’ o o

a W m a W
+2R;°2mv+2w°2mrxv. (2-3)

Gabe Gas. .2.

Since the origin is at the center of mass of the molecule

2 m r = 0. (2—4)

Three more conditions are necessary to fix the rotating
axes in the molecule.

A possible choice of conditions to fix the rotating
axes intjmrmolecule would be to have the angular momentum
vanish in the moving system. The angular momentum can be

defined as

g,= 2 mara x v (2-5)
a m f&

It is not convenient to set equation (2-5) equal to zero

and the condition actually employed is that due to Eckart (15).

2 m a x v = 0. (2-6)
a Q.J% 4J$

Eq. (2-6) states that the resultant angular momentum
relative to the rotating system will be zero in so far as
r can be replaced by :2: This choice of conditions to
locate the moving axes minimizes the rotation—vibration
interaction and is especially useful for small amplitude
vibrations.

Replacing :2, by :2.+ 32' in (2-3) and using condi-

tions (2-4) and (2-6) results in the following form for

the kinetic energy

- 2 + o + 2 + 0
2T Rima imQ(QX:%) (cexrg):mava 293mg“ X32)

'(2-7)
The first term in Eq. (2-7) is the translational energy,
the second is the rotational energy and the third term is
the vibrational kinetic energy. The last term in Eq. (2—7)
couples vibration and rotation and prevents complete separa-
tion of these forms of energy. This term is normally
neglected in approximate treatments. However, it produces
observable effects in the microwave spectrum of most mole-
cules. It becomes especially important in the case where
a molecule possesses low lying vibrational states or large

amplitude vibrations.

9

The procedure for handling Eq. (2-7) is to solve
the vibrational and rotational part separately and to treat
the coupling term by perturbation techniques. The transla-
tional energy does not contribute to the Spectrum and may
be ignored.

In order to obtain the quantum—mechanical Hamiltonian
from (2-7) the energy is written in terms of momenta. This
is not a straightforward process because the angular momenta
are not conjugate to any coordinates. The derivation of
the Hamiltonian can be found in Reference 14. For our
purposes we will consider first the rigid rotor part of the
quantum-mechanical Hamiltonian, which is derived from the
second term of Eq. (2-7). Then the effects of a quadrupolar
nucleus and internal rotation, which are not included in

Eq. (2-7), will be added as perturbations.
2.2b THE RIGID ROTOR HAMILTONIAN

The energy of rotation of a non-vibrating rotor is

given by
h2 P: ‘92 P:
HR - (‘2")('I_ + ix + T) (2‘8)
x y z

where Px' Py and P2 are components of angular momentum
along the x, y and z principal axes of the rotor, respec-
tively, and IX, Iy’ and I2 are its principal moments
of inertia.

The angular momentum is related to the angular velocity

by

P=I°gg (2-9)

where I! is the inertia dyadic and g’ is the angular

~

velocity vector. The components of I can be written in

~

matrix form

 

 

7 _
Ixx Ixy Ixz
I = I I I (2-10)
:3 XY YY YZ
Ixz Iyz Izz
where
(2-11)
= 2 + 2 =_ .-
IXX i ma(yQ 2Q) , Iyz i ma(yQ 2Q), 2 12

and the other elements are obtained by cyclic permutation

of the coordinates.

The tensor I’ is symmetric and can be diagonalized

by an orthogonal transformation, which is equivalent to a
rotation of the coordinate axes. The axes of the coordinate
system in which 1% is diagonal are known as the principal

axes of the molecule and the diagonal elements are called

the principal moments of inertia (Ix. Iy’ Iz in Eq. (2s8)).

The matrix elements of the angular momentum operators,

in the representation in which P2 = P: + P; + P: and P2

are diagonal, are given by the following equations:
<J,K|P2|J,K>= J(J + 1) (2413)
<J,K|Pz|J;K>= K (2-14)

1/2

%[(J I K>(J i K+1)1

<J,K i 1|Py|J,K>

= —i<J,K : 1|PX|J.K> (2-15)

where J = 0, i1, i2, ....; K = 0, i1, i2, .... and the
P's are in units of h/ZW.

The energy may also depend on the quantum number M,
the component of angular momentum along a Space fixed axis.
M is allowed the same values as K, but in the absence of
an external electric or magnetic field each state is
(2J + 1)-fold degenerate; i.e. the energy is independent
of M.

The above relations can be deduced from the commutation
rules for angular momentum operators which for the molecule-

fixed axes are

P P — P P = [P , P ] = -iP (2-16)
x y y x x y z
[Py,Pz] = -j.PX (2-17)
[PZ.PX]= —iPy. (2-18)

2

The matrix elements of Pi, Py and P: are obtained

by matrix multiplication and are used to write the matrix

for HR whose elements are given below.

1

<J,KIHR|J,K> = 71—6:— + ?)(J(J + 1)) + T{ ————— )K2
X Y ' Z y x

<J,K+2|HR]J,K> = <g;x|HR|J,K +2)

-62 1 1 1/

-§—[(f- _.f_) [(J—K)(J-K-1)(J+K+1)(J+K+2)] 2} (2-20)
y x

The principal axes are usually designated as a, b, c

with Ia < Ib

c axes can be identified with the x, y, z axes and cer-

< IC. There are six ways in which the a, b,

tain choices Simplify computation (16).

12

We can now define reciprocal moments of inertia

A _ h I B = h , C = h .

— 2 2 2
8w Ia 8W Ib 8w Ic

 

 

 

which are called the rotational constants of the molecule
with A > B > C. The Hamiltonian can be written in terms
of A, B, and C with P in units of h.

2

_ 2

+ CP:)' (2—22)

If x = b, y = c, z = a then the matrix elements of

(2-19), (2-20) become

<J.K|HR|J,K>/h = %-C-[J(J+1) - K2] + AK2 (2-23)

<J.K+2 IHRIJJO/h = '41-{(C-B)[ (J—K)(J-K-1) (J+K+1)(J+K+2)1 1/2}
(2-24)

In a symmetric-top molecule two of the rotational
constants are equal. A prolate-top has the two smaller
rotational constants equal (B = C) which results in a
diagonal matrix (the right side of Eq. (2-24) vanishes).
The energy of each J and K level is given by the value
of the diagonal element, Eq. (2—23). An oblate-top has
A = B and interchanging C and A in the above equations
also makes the matrix diagonal. The energy levels of a
symmetric-top are degenerate with respect to 1K, as can be
seen from Eq. (2—23).

The most general case is the asymmetric rotor where
A # B # C. The energy matrix is no longer diagonal and the

K degeneracy is Split. A change of variables prOposed by

13
Ray (17) is very convenient for determining the energy
levels. The Hamiltonian of Eq. (2-22) is written in terms
of parameters p and o by replacing
A by GA + p.
B by 08 + p, (2-25)
and C by CC + p.

Then

HR(GA+p. oB+p, OC+p)/h (oA+p)P: + (OB+p)P: + (oC+p)P:

 

 

 

o A,B,C
= HR h + pPz. (2‘26)
We choose
.. 2 _ _ A + C _

O ’ A - c ’ P ’ A - c (2 27)

so that 0A + p = 1, oC + p = -1,

_ 23 - A w C _ -

and GB + p - A _ C - K. (2-28)

The parameter K is Ray's asymmetry parameter. Employing

(2-27) and (2-28) yields

HR(1,K,-1)/h = A §~C HR(A,B,C)/h - 2 _ g P2 (2—29)

 

 

The energy of an asymmetric rotor becomes

"WR(A,B,C)/h = A——'—2*'—-9-J(J + 1) + A 5 C E(K), (2-30)

 

where the E(r) are eigenvalues of the matrix
HR(1,K,-1)/h.
The value of E(K) is the energy of a fictitious
rigid rotor with rotational constants 1,K,-1 and therefore

can be calculated from the matrix elements of the rigid

14

rotor by suitable substitution (18). Since E(K) de-
pends on only one parameter, it is suitable for tabulation.

The HR matrix is diagonal in J and can be factored
into blocks, each of dimension 2J + 1. Each J block can
be further factored into four submatrices if the Wang
transformation (19) is applied. The submatrices are labeled
according to the four symmetry species of the asymmetric
rotor wavefunctions. The eigenvalues of the submatrices
give the values of E(K) for the corresponding values of
J and the symmetry species.

To the eigenvlaues of E(K) are attached a superscript
J and a subscript T to identify the energy levels. For
asymmetric-top molecules K is no longer a good quantum
number; however, the energy levels may be specified by in-
dicating the limiting prolate-top value K =~K_1 and the

limiting oblate-top value K = K The subeuxript on K

+1‘

refers to the value of K which is -1 for prolate and

+1 for Oblate symmetric tOps. In this notation 'T = K_1-K+1

and K_ + K+1 = J or J + 1. The stationary states for

1

asymmetric molecules are designated by J .
K-1K+1

J
Tables of ET(K) are available for 0.: J.: 20 and
K = 0.000 to 0.999 in increments of 0.001 on r (20).

Values for negative K can be obtained from the relation

J ‘ i J 2 31
ET(K) - -E_T(-K)- ( )
By substituting the value of E:(K) into Eq. (2-30) the

energy for a given level can be calculated.

/
15
The microwave absorption frequencies are computed from

the Bohr relation

hv - WR - WR. (2-32)

The selection rule for J is AJ = 0, il. Transitions
are classified as a, b or c-type transitions according to
the direction of the dipole moment component which interacts
with the electric vector of the radiation. Selection rules

for these transitions are as follows:

a-type AK_1 = even AK+1 = odd
b-type AK_1 = odd AK+1 = odd
c-type AK_1 = odd AK+1 = even

Interpretation of the spectrum consists of assigning
the appropriate quantum number to at least three transitions,
the absorption frequencies of which depend on A, B, and C.
The absorption lines will have definite characteristics
such as Stark component patterns and hyperfine structure due
to interaction of a quadrupolar nucleus. These character—
istics facilitate the identification of quantum numbers with
transition frequencies. Once the constants A, B, and C
are determined the frequencies of the other spectral lines

can be predicted. ’

2.3 HINDERED INTERNAL ROTATION

Molecules are not rigid and the relative motion of the
atoms produces observable effects on the overall rotation

which are generally classed under the term "vibration-

16
rotation interaction." In the case of a linear molecule
the angular momentum of the vibrational bending modes may
alter the effective moments of inertia and cause a splitting
of the rotational energy levels, a phenomenon known as
l-type doubling (21). ‘Another form of internal motion is
called internal rotation or torsion in which one part of
a molecule rotates about a bond axis in relative motion to
the rest of the molecule. Internal rotation is observed
in acetaldoxime and the theory will be reviewed in this
section.

An example of a molecule with internal rotation is
ethane in which the two -CH3 groups can rotate relative to
each other about the bond joining them. The internal rota-
tion will usually not be free because of interaction of the
two groups, and during a 3600 rotation about the bond three
potential humps will be encountered. Each hump corresponds
to a position where the hydrogens on the methyl groups are
directly opposite each other.

The coupling of internal and overall rotation may pro-
duce observable effects in the pure rotational Spectrum and
is treated as a perturbation of the rigid rotor energy
levels. The complexity which this type of motion introduces
into the spectrum will depend on the height of the potential
barrier hindering the rotation and on the moments of inertia
of the rotating groups.

A useful model for describing the internal rotation in

many molecules consists of a rigid symmetric top, e.g., a

17
CH3 group, attached to a rigid frame which may be asym-
metric. Four degrees of freedom are considered, three for
over-all rotation and one for the internal motion.

In the method for treating internal rotation originated
by Crawford (22) and Wilson (23) the kinetic energy is
referred to the principal axes of the molecule as follows:

'2

2T = z I + 2 ' 2 x w + I . 3—1
ggwglaagggaa ()

Here mg is the 9 component of the angular velocity (9

= a, b, c identifies the principal axes), I9 is the moment
of inertia about the g-axis, IQ is the moment of inertia
of the internal top taken about its symmetry axis, d is
the angular velocity of the internal t0p relative to the
framework, and Ag is the direction cosine between the

axis of the top and the g principal axis.

The momenta used are

ET -
= = + I , 3—2
Pg 3_— I mg Ag G ( )

_aT .
= Z . 3-3
p do Io o + Id 9 x9 wé ( )

and the kinetic energy can be written in terms of the momenta,

I
2=ZP2 +1—z 2-£I°2. 3-4
T g g/Ig [ 9 Ag 19] a o ( )

The relative angular momentum of the top and frame—

work is defined as ,
p_a>=rxad (3-5)

2 .
= _ d
where r Id [1 29 Ag Io/Igllo is the reduce moment of

18
inertia for the relative rotation of the two groups. With
the use of Eqs. (3-2) and (3—3) we find from Eq. (3-5)

that @must be given by

9:: p1. -
g kg 9 a/Ig (3 6)

Then substitution of (3-5) into (3-4) yields

_. 2 _ 2 _ .
2T - 2:9 Pg/Ig + (p 0) /rIQ. (3 7).

For a symmetric internal top with a threefold axis of

symmetry (e.g., -CH3) the potential energy is
v = 1/2 v3(1 - cos 3s) + 1/2 v6(1 - cos 6a)+--x3-8)

where o is the angle of rotation about the symmetry axis.

Therefore, the Hamiltonian can be written as

 

H=AP2+BP2+CP2+F( -¢3)2+:§-(1-cos 3o)+
a b c p 2
V6
5— (1— cos 0,) + (349)
2 2 2
where A = fé-, B = -é—-, C = h and F =462/2rI .
Ia 21b 21¢ a

The first three terms comprise the usual rigid rotor
Hamiltonian, and except for the cross term -2Fp¢>the
Hamiltonian of Eq. (3-9) is separable into a rigid rotor
part and a torsional part. The Fflaz term contains even
powers of P9 and can be incorporated into the rigid rotor
terms. The V3 and higher terms in the potential are
usually too small to have an effect and the remaining tor-
sional part,

V3
sz + —2'—(1 - C05 30): (3’10)

19
which describes the motion of the internal top, can easily
be transformed into the form of Mathieu's equation. The
eigenvalues and eigenfunctions of Mathieu's equation have
been extensively tabulated (24) and are easily generated
with a high-Speed digital computer.

With a very high barrier the internal rotation reduces
to small oscillations and the torsional eigenfunctions can
be approximated by the wave functions for a harmonic oscil—
lator. In such a case the internal torsion will resemble
a typical vibration and the effect on the rotation will be
limited to an alteration of the rotational constants. For
intermediate barriers the effect on the Spectrum is more
complicated and will now be described.

The eigenfunctions of the torsional Hamiltonian (Eq.
(3-10)) must be invariant under transformations of the C3
group because of the symmetry of the internal top. There-
fore there will be two solutions, one corresponding to the
non-degenerate A Species (wavefunction is multiplied by
+1 upon rotation of 2w/3 about the internal axis) and one
corresponding to the doubly degenerate E Species (wave-

function is multiplied by e upon rotation by 2r/3

about the internal axis.) Thus each torsional level consists
of two sublevels labeled by quantum numbers YA (o = 0)
and vE (o = 11).

The torsional levels would normally be triply degenerate

because of the three equal potential minima. The splitting

20
of this degeneracy can be thought of as being due to tunnel-
ing through the barrier. The torsional motion would have
no effect on the spectrum except for the coupling term
-2FpQP in Eq. (3-9). This term can be treated as a pertur-
bation and is found to affect the A and E states dif—
ferently and hence produces doublets in the microwave Spec-
trum when the barrier is of intermediate height (200-2000
cal/mole) and one of the rotating groups has a small moment
of inertia.

If the matrix for HR is written using as basis func-
tions the product of the rigid rotor functions and the tor-
sional eigenfunctions, the matrix will be diagonal in J
and 0 but not in K or v. The elements non-diagonal in
K are those (equation (2-20)) of the rigid rotor and terms
involving PX and Py in the operator “3. The elements
non—diagonal in v come from the factor p in the coupling
term -2Fp Z9 .

Partial diagonalization in the quantum number v can
be achieved by applying the Van Vleck transformation (25)
which reduces the magnitude of the non-diagonal matrix ele-
ments of -2Fp so that they may be neglected. The
transformed matrix may then be factored into blocks, Hvo’
one for each torsional state.

As a result of the Van Vleck transformation, the rota-
tional matrix, Hvo' can now be written in the form

(n)

n
W Hr +F2wVG ,6) . (3-11)

H

21
The rigid rotor part, Hr’ is not affected by the trans—
formation. -The zero order term in (3—11) gives the pure
torsional energy which does not contribute to the rotational
Spectrum. Also, all W32) = 0 for n odd for A levels
(0 = 0).

The perturbation coefficients Wég) depend only on
the ratio of V3/F for a potential barrier of the form
(3-8) and can be,eva1uated by nth order perturbation
theory. However, Herschbach (9) has shown that the coef-
ficients W33) can be evaluated in the coaxial symmetric
top limit. The higher order coefficients can be evaluated
from linear combinations of zero order coefficients when
this method is used. The Wég) are tabulated in Reference
10 as a function of V3/F.

When the Hamiltonian matrix is written in the repre-
sentation in which Hr +-FW§§(,@P2 , the pseudo-rigid rotor
part of Hvo’ is diagonal, the odd order terms occur off
the diagonal. Usually the spacings of the pseudo rigid
rotor levels are large compared to the odd n terms which
connect them, so that the odd n terms may be neglected.
In this case the Hamiltonian can be given correct to second
order by

(2)

- 2 2 2 2
Hv0 APx + BPy + CPz + vao (aPX + spy + sz) (3—12)

where 2? has been rewritten from Eq. (3-6). Here

a = AaIa/Ia, 6 = AbIa/Ib and y = ACIa/Ic. (3—13)

22
Eq. {3-12) gives the Hamiltonian for a pseudo-rigid rotor

with rotational constants

A + ozrw”)

AVG = V0

13vG = B + 52mg) (34-14)
: 2 (2)

Cv0 C + yFWVO

The cross terms in (3-12) can be eliminated by a slight
rotation of axes but are usually simply neglected without
detectable error.

The effective rotational constants Avo’ Bv0 and Cvo
given by Eq. (3-14) will be different for A (o = 0) and
E (o = :1) levels. The selection rule is A0 = 0, so that
certain lines in the spectrum will be Split into doublets.
Each component of the doublet will be part of a rigid rotor
pattern with rotational constants slightly different from
those of the adjoining line. The difference in the rotational
constants for the two sets of lines is a sensitive measure

of the barrier height V3.

A. 4 .. = .2 4,433 - 45,3]. (3.15,

If AA, AB and AC are small, the frequency separa-

tion of the doublets is given by
5
Av = (3%) AA +13%)As + (3%)Ac (3—16)

where Bv/BA, Bv/BB and dv/BC are the differential change
in frequency of the transition with respect to a change in

A, B or C respectively and can be calculated from the E:(K)

23
values and their derivatives (18). In general, Av in-
creases with J because the derivatives increase with J
and the largest splittings between doublets are expected
for transitions Tinvolving states with a high J quantum
number. Once AA, AB and AC are known from the Spectrum
the barrier can be calculated by use of Herschbach's table
of values of W32) (10).

If the pseudo-rigid rotor levels are not widely separ-
ated, the odd order terms cannot be neglected. In this
case the first order term may make the largest contribution
to the perturbation. ‘Trans-acetaldoxime falls into this
category because it is a near—symmetric tOp with some closely
spaced energy levels.

The Hamiltonian to first order is

_ (1) . _
HvE — Hr + F va (on + BPY + sz) (3 17)

H H

vA Ar AP: + BP; + CP: (3-18)
where only the E levels will be affected Since n is odd.
The matrix of the Hamiltonian can be written in the symmetric
rotor representation (Eq. (2-19)-(2-20)) with the unique

axis taken as the axis of quantization (z axis). The energy
levels may be obtained by the method of continued fractions
(26) or by direct diagonalization. The A level transitions
will fit the pattern of a rigid rotor whereas the E levels
will not. Therefore the difference between the absorption

frequencies of the A and E levels is a measure of

aWéé) from which the barrier height can be computed.

24

The methods given above will no longer be adequate
when the height of the barrier falls below «:1000 cal/mole.
In this case the perturbation coefficients Wég) must be
included to fourth order in Eq. (3-11). The Hamiltonian
matrix of Hvo will contain terms off-diagonal in K up
to (KIK i 4). This type of calculation involving the higher
order terms is described in Chap. V. For compounds with
still lower potential barriers the internal rotor can be
treated as a free rotor (low barrier approximation (27)).
This approach is described in Chap. VI where the low and
high barrier approximations are compared.

An alternative method of treating the internal rotation
problem by making use of an internal axis system differing
from the principal axes has been developed by Nielsen (6)

and Kohler and Dennison (7).
2.4 STARK EFFECT IN ROTATIONAL SPECTRA

When a homogeneous electric field is applied to a
molecule possessing a permanent dipole moment u, the 2J+1
degenerate rotational levels are split by the perturbation.

The perturbation operator can be written as

H'=-EZ¢ -lEZZ¢

4-1
9 29 Lis! 2 ( )

zg ¢zh Pgh
Here E is the electric field directed along the space-
fixed z axis, ”g is the component of u along the g

principal axis of the molecule, Pgh is an element of the

polarizability tensor and $29 is the direction cosine

25
between the space-fixed z axis and the molecular-fixed
g axis. Polarizability effects are never observed in micro-
wave spectra unless the dipole moment is nearly zero. The
observable effects are due to the first term in Eq. (4—1)
which depends on the permanent dipole moment of the molecule.

The first order energy is given by the matrix element

AW = * E d 4-2
1 fw Zg ¢zg ”9 ¢ T ( l

and for symmetric tops is just E multiplied by the z
component of the dipole moment matrix element. The eXpres-
sion for AW1 becomes

AWI = -&(—_-J E? Ii) (4'3)
where J is the total angular momentum quantum number, K
represents the projection of J on the symmetry axis, and
M is the quantum number for the projection of J on the
space-fixed z axis. ‘Thus symmetric tops exhibit a "first
order" Stark effect, so named because of the power of E
involved in the expression.

Golden and Wilson (28) have carried out a perturbation
calculation for asymmetric top molecules using the Van
Vleck perturbation technique (25). They have shown that
the perturbation energy is a function of E2 and can be
written in the general form:

w( ) = 2 [a (J,T) + b (J,T)M2] u” 32 (4-4)

9 9 9 9
where a and b are constants which depend on the molecular

geometry.

26
The arrangement of the spectrometer in use at Michigan
State University is such that the Stark and microwave elec-
tric fields are parallel. The selection rule which governs
the transitions in this case is AM = 0. The Stark compo-
nents will be displaced from the zero field transition by

a frequency

MM = (wfilmc wfiQ/h =%— i [[ag(J',~r')- ago») +
{bgwum - bg(J,T)}leE2 U43] (4—5)

Since M can take on the values M = 0, i1, i2, ...,
iJ each transition will exhibit J+1 Stark components
where J is the lesser of the two J values involved in
the transition. Thus the value of J (lesser) for a given
transition can be determined by Simply counting the number
of Stark components, if they can be resolved. In addition
the M value of a given transition can be determined from
the relative intensity of the Stark components. 'The in—

tensity for transitions where AJ = 11 is proportional to

2
Jupper
tional to M2. Because of these characteristics the transi-

- M2 and for transitions where AJ = 0 is propor-

tions for low J values can be identified from their Stark
pattern.

The dipole moment components can be determined with
the use of Eq. (4-5). Once the rotational constants (A,B,C)
are known and the transitions assigned, the constants a

(J.T) and b (J,T) can be evaluated. Then a plot of

27

M

This means that the frequencies of three Stark components

v vs E2 yields a slope which is a function of the pa.

must be measured as functions of E2 to yield the three

independent relations necessary to solve for ug, u; and

2. For some transitions AvM may only depend upon one or

two of the ”g in which case evaluation of u is simpli-

Ll.

fied.
2.5 QUADRUPOLE HYPERFINE STRUCTURE

A molecule that includes a nucleus with an electric
quadrupole moment will exhibit a rotational spectrum in
which the transitions are split into several components.
The magnitude of the splittings will depend on the value of
the quadrupole moment, the magnitude and the symmetry of
the electron density near the nucleus, and the J levels
involved in the transition. When the quadrupole interac-
tion energy is small compared to the Spacing between rota-
tional levels, it can be treated as a perturbation.

The theory of the interaction of a nucleus with the
surrounding electrons was developed by Casimir (29). The
theory was later extended to include molecules and has been
developed for symmetric and asymmetric tOp molecules with
one or two nuclei» possessing a quadrupole moment (30-38).

The Hamiltonian for quadrupole coupling of a single

nucleus in a molecule is

- 62 3(J'I )2 +2(~'I I) _. J2I2
HQ - 69(5E¥)Av [2J(2J - 1)I(21 — 1) (5'1)

 

28’
where J is the total angular momentum of the molecule
except for nuclear spins, I is the nuclear spin quantum
number of the quadrupolar nucleus (I must be greater than
1/2 for an observable effect), e is the electronic
charge, Q is the quadrupole moment of the nucleus, and
(52V/522)Av. is the aVerage electric field gradient evalu-
ated at the nucleus in the direction of a z axis fixed
in space.

The Hamiltonian indicates that the quadrupole energy
is a result of coupling of nuclear spin angular momentum
with the angular momentum of the molecular rotation. The
coupling is a result of the electrostatic interaction of
the chamges within the quadrupolar nucleus with the extra-
nuclear charges. ’

Since the angular momentum of a molecule is constant,
the space-fixed z axis can be chosen parallel to the
angular-momentum vector J and labeled zJ. The problem
then is to relate (52v/522)Av, evaluated with respect to
the space-fixed axes, to field gradients referred to the
principal axes fixed in the rotating molecule.

The average electric field gradient in the direction

of an axis fixed in Space is given by Eq. (5—2).

52V ) ,3cos2 9 — 1

qJ E (637 Av : fPJJ‘ :3 M" .(5-2)

where is the charge density, r is the distance be-

pJJ
tween the nucleus and a volume element d1 containing

dT

PJJ ‘of charge, and 9 is the angle between r and the

29
zJ axis. The quantity pJJ could be determined from the
electronic wave functions if they were known.

The quantity qJ can be evaluated quite simply in
terms of the molecular axes in the case of linear and sym-
metric-top molecules (23). The expression for qJ for an
asymmetric top, though not simple, can be written in terms

of E:(K). The result from first-order perturbation theory

is (37)

qJ = { B:v [J(J + 1) + E(K) - (x + 1) é§é£l.1 (5-3)

+2373,- (BEIC'c ) +3-3- [J(J + 1) —E(Ic) + (1c - 1)BEK’C 1}

 

1
15 + 17(737 + 3K?!)

2
where 33%. is the electrostatic field gradient at the nu-

cleus due to extranuclear charges evaluated in the direc-
tion of the g principal axis (9 = a, b or c), and J is
the rotational angular momentum quantum number. Methods of
evaluating BE(K)/5K are given in Reference (18).

The part of Eq. (5—1) in square brackets has been

evaluated by Casimir using first order perturbation.

m...) = 3é3%‘2‘§f1%)2;&é‘i1)“ 1“ * “ <5-4)

Here c = FgF + 1) - 1(1 + 1) - J(J + 1), and F = J + I,
(J + I - 1% ----- ’(J — 1). Tables of Y are available for

J up to 20 and I up to 9/2 (39).

30

Eq. (5-3) can be put in another form by recognizing

that
SW}: 1) E‘Kz
EX_.= §{J(J + 1) + E(K) - (K + 1) K 1 . (5‘5)
aw
aw
5633.: g-[J(J + 1) - E(K) + (K - 1) BEKK ]:

where Wr is the rigid-rotor energy, and A, B, and C are

the rotational constants. Substituting Eqs. (5—5) into (5—3)

leads to
q = 52V BWr + 52V awr + 32V awr . (5-6)
J a 2 A ab2 53 ac2 C
2
If we let X = eQ 9—! , the quadrupole energy can
99 592

written as

w = Y(F,I,J)[Xaaf1(J,T,K)+( )f2(J,T,K)], (5-7)

Q Xbb ’ ch
where use has been made of Laplace's equation

Xaa + Xbb + ch = 0. The functions f1(J,r,K) and f2(J.r,K)
depend only on the quantum numbers J and T and.on the

asymmetry parameter K, and can be derived from Eq. (5-6).

 

 

f _ 2 awr l.5ch - l awr) 1 5 8

1a,...) ‘ “ants: " 2 5‘5— 2 ac 2J+3 - ('>
2 BWr awr 1

f2(J’T'K) = 373*5§_" BE’) 2J+3 ' (5’9)

For purposes of computation Eq. (5-7) is written in

the form,

31

W3 T: axaa + 6(Xbb ' ch) : (5-10)
where

a = Y(F,I,J) f1(J,$.K). (5-11)
and

B = Y(F,I,J) f2(J,T,K). (5_12)

The selection rule for F, the total angular momentum
of the system, is AF = 0, :1. The various components of
the hyperfine structure can be identified from the relative
intensities of the transitions which are given in published
tables (21). In acetaldoxime, where the spin of nitrogen
nucleus is one, there are three values of F for each value
of J (except for J equal to zero). Hence, each rota-
tional level is split into three levels by the perturbation.
From the frequency differences of the hyperfine Splittings
the values of Xaa and (Xbb - ch) can be determined.

The value of X parallel to a bond is a measure of
the electron distribution along the bond. Townes and Dailey
have shown that the main contribution to the electric field
gradient at the nucleus is due to p electrons in the va-
lence shell (40). Consequently, when an atom having a quad-
rupolar nucleus is bonded to another atom, the quadrupole
coupling constant depends on the hybridization of the orbit—
rals‘of‘the atom and on the bond polarity (41).

The value of Xbond also indicates the symmetry of
the charge distribution. For a cylindrical charge distri-

bution
Xzz = -2 Xxx : -2 ny’

where z is parallel to the bond direction.

III. MICROWAVE SPECTROMETER
3.1 INTRODUCTION

Spectroscopy in the microwave region has features com-
mon to Spectroscopy in other regions of the spectrum. The
klystron provides a source of electromagnetic radiation
which passes through an absorption cell, is detected, ampli-
fied and displayed on an oscilltrscope or recorder. In
addition, a method of measuring the frequency of the absorp-
tion signal is included. Improved sensitivity is obtained
by modulating the signal with a periodic electric field and
amplifying the modulated signal with an amplifier tuned to
the modulating frequency. This type of Stark—modulated
spectrometer, with a 100 kHz square—wave Stark field, was
introduced by Hughes and Wilson (4).

A block diagram of the instrument in use at Michigan
State University is shown in Fig. 1. A short description

of the main components of the instrument follows.
3.2 REFLEX KLYSTRON

A typical reflex klystron is schematically presented
in Fig. 2. The electrons are accelerated toward the reson-
ator grids and under oscillatory conditions will be velocity
modulated by the a.c. field of the resonator grids. The
velocity modulation will cause bunching of the electrons
in the beam. Beyond the resonance cavity the electrons are

32

33

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(Tube-Body)\\s p g
Cathode .)
Grid
Voltage
Beam Supply
Voltage
Su 1
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J_ Reflector
? 4‘ Voltage
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Filament _
Voltage J.
Figure 2.

 

 

 

T

Sawtooth Voltage
oscillator.

A circuit schematic for a typical klystron

35
reflected by the negative potential on the reflector and
sent back through the resonance grids in the opposite direc-
tion. The bunched electrons, on their return trip, pass
through the a.c. field during its retarding phase, giving
up energy to the cavity in the form of electromagnetic
radiation.

The frequency of the radiation can be changed by ad-
justing cavity dimensions or reflector voltages. If the
absorption signal is to be displayed on an oscilloscope,
the frequency of the klystron is changed by applying a saw-
tooth voltage to the reflector. For recorder presentation
of the signal the frequency of the klystron is changed
slowly by turning the klystron tuning screw which changes
the dimensions of the cavity.

At Michigan State University the klystrons are powered
by an FXR Type Z 815 B power supply manufactured by Elec-
tronics and x—ray Division, F. R. Machine Works, Inc. To
minimize klystron noise a separate, regulated, d.c. fila-
ment supply is used for the cathode heater.

In our laboratory klystrons are available to cover the

frequency range from 8 to 37 GHz and are listed in Table I.
3.3 WAVEGUIDE ABSORPTION CELL

A ten foot length of rectangular X band waveguide fitted
with flanges on both ends serves as the absorption cell.

The waveguide is made of brass with a silver coating on the

 

36

r

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A v
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A

A

 

 

 

 

3

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m come oncomsummm o.nm o.mm mm x m

N.mH ooom mD\H2m m.bm o.Nm mvmm m

N.NH ooom mD\Hzm m.Nm m.bm mama m

N.NH ooom mD\Hzm o.mN m.NN Noam m

w.HH coon Anva\H2m m.NN o.wH mmmm m

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wuflmum>wcb wumum cmmflnowz um mm: CH 30: mcouumhax mo umflq .H magma

37

inside faces. The sample is introduced into the waveguide

through a narrow slit cut in the broad face of the guide.

a /
? Teflon

YT Strip

0.050" I

0.900"
J

 

 

 

 

 

Stark Septum

 

 

 

 

 

 

+5-355"-a

 

;b

Teflon tape: 0.062" x 0.400" with 0.032" groove 0.010" deep.
Coin silver septum: 0.032" x 0.796"

Figure 3. Waveguide absorption cell showing position of
septum.

In order to provide Stark modulation a coin silver
septum is inserted in the middle of the waveguide parallel
to the broad face (ac faces in Fig. 3). The septum is in-
sulated from the top and bottom of the waveguide by two
strips of Teflon which also serve to hold it in place. A
square wave voltage of 100 kHz frequency is impressed on
the septum and serves as the modulating field. With the
septum placed in the position described, the microwave and
Stark fields are parallel when the waveguide is operating

in the dominant TElo mode.

38
The absorption cell is made vacuum tight by sealing
the ends with mica windows and rubber O-ring gaskets placed
between the flange on the absorption cell and the flange

on the connecting piece of waveguide.
3.4 STARK MODULATION

The signal-to-noise ratio of the spectrometer is greatly
improved by modulating the signal with a Stark voltage and
demodulating with a phase-sensitive detector. A square
wave generator designed by Hedrick* operating at 100 kHz
is the modulation source used at Michigan State University.
The amplitude of the square wave can be varied from 0—1200
volts. The base of the square wave is ordinarily at ground
potential but can be biased with a positive or negative
voltage.

As mentioned in Chapter II, the perturbation from the
electric field splits the M degeneracy of the rotational
levels and the magnitude of the splitting increases with
increasing electric field. In the presence of the modu-
lating square wave the detector receives a signal which
consists of the zero-field spectrum (unsplit) during one
half of a modulation cycle and the Stark spectrum (split)
during the other half cycle. Since the Stark spectrum is
180° out of phase with the unperturbed spectrum, the Stark

spectrum is inverted upon phase-sensitive detection.

 

*L. C. Hedrick, Rev. Sci. Instr. 22, 781 (1949).

39

An example of an absorption signal and its associated
Stark spectrum as it would appear on a recorder trace is
shown in Fig. 4 with the Stark transitions at each end of
the trace.

For the purpose of measuring dipole moments a d.c.
bias is applied to the square wave. The bias voltage can
be measured very accurately with a potentiometer and the
diSplacement of the Stark components can be measured as a
function of the bias voltage. Accurate values of the dipole

moments can be obtained using this method.
3.5 DETECTION AND AMPLIFICATION OF SIGNALS

A crystal rectifier is commonly used for detecting
radiation at microwave frequencies. This detector consists
of a metal whisker making a point contact with a semi-
conducting material (usually silicon). Because of the low
contact capacitance, these rectifiers can be used at high
frequencies. At Michigan State University the three types
of crystals used are the IN23, IN26, and the IN78. Usually
the sensitivity varies from crystal to crystal and a method
of trial and error is used to segregate usable crystals.

The output of the crystal detector is sent to a pre-
amplifier and then to the phase sensitive detector. The
phase sensitive detector amplifies signals which are in
phase with the 100 kHz reference signal which comes from
the square wave generator. Spurious signals whose frequen-

cies lie outside a very narrow band including 100 kHz are

4O

.GOAuomuwv m>auamcmm mmmzm Hmuwm
oEonoamumom mHo CH :oHuHmcmuu «ONAII.HoH on» mo momma

 

 

 

.v ousmfim

41
not amplified and therefore the amount of noise is greatly
decreased.
The amplified signal is then sent to an oscilloscope
or chart recorder where the signal received is plotted yg

frequency.

3.6 FREQUENCY MEASUREMENT

Measurement of the frequency of a transition is accom—
plished by aligning a frequency marker with the center of
an absorption signal displayed on a dual trace oscillosc0pe.
The signal is displayed on one trace and the marker on the
other. Both traces are driven by the sawtooth voltage
which is used to sweep the klystron frequency and hence
the two traces and the klystron frequency are synchronized.

One way of generating a frequency marker is by ab—
sorption of power in a wavemeter, which shows up as a dip
on the trace. The wavemeter contains a cavity whose di-
mensions are changed by a micrometer screw and the dimensions
of the cavity determine the wavelength of the microwave
radiation which is absorbed. A calibration chart fur-
nished by the manufacturer relates micrometer readings to
the frequency of the radiation. The accuracy of this
measurement is seldom better than 5 MHz. It serves the
purpose of a quick, approximate measurement when a new
transition of unknown frequency is found. A list of wave—
meters in use at Michigan State University is given in

Table II.

42

Table II. 'Frequency meters in use at Michigan State

 

 

 

University
Manufacturer Model Iiigzuiigi)
Narda(a) 810 8.2 - 12.4
Narda 809 12.4 - 18.0
DeMornay—Bonardi(b) DBE—715-2 18.0 — 28.5
DeMornay-Bonardi DBD-715-2 26.0 - 39.0

 

(ajThe Narda Microwave Corporation, 118-160 Herricks Road,
Mineola, New York.

(b)DeMornay-Bonardi Corporation, 780 S. Arroyo Parkway,
Pasadena, California.

Accurate frequency measurements are made by beating
a known frequency against the klystron frequency. The
known frequency is generated with the use of the instru-
ments shown in Fig. 5. The fundamental standard frequency
source is a Manson Laboratories RD-140 high stability, one
MHz oscillator, which varies in frequency only one part in
108 per day. The crystal-controlled oscillator is standard—
ized to better than one part in 107 by beating its tenth
harmonic against the 10 MHz carrier of station WWV from
the National Bureau of Standards. The frequency is ad-
justed to exactly one MHz by tuning the oscillator until
the beat frequency reduces to zero.

The one MHz output of the crystal oscillator is sent

to the Gertsch AMrlA VHF interpolator which produces

43

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rlll

 

 

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44

frequencies between 20 and 1000 MHz which comprise the input
to the Gertsch FM-4A. This instrument contains a 500—
1000 MHz variable tuned oscillator which may be locked to
a frequency which is :10 MHz from a harmonic in the signal
from the AMrlA. The result is generation of stable fre-
quencies between 500-1000 MHz known to one part in 107.

The generated frequency can be represented by the equa-

tion

= [m(k + v ) i 10]N MHz.

vs LFO
In this equation
m, k and N are integers
k = 19-38.
m, N = 1, 2, 3,...,

1—2, and

<
ll

k + VLFO = 20-40.

The frequencies of the absorption signals are then
determined by measuring the beat frequency obtained from
mixing a harmonic of the FMr4A generator and the unknown
klystron frequency. The beat frequency, which corresponds
to the difference between the known and unknown frequency,
is detected and measured with a Collins 51J-4* radio re-
ceiver. The frequencies of the Collins receiver can be
read directly to 0.001 MHz. The relation governing the

beat frequency is

 

*-
Collins Radio Company, Cedar Rapids, Iowa.

45

IVklystron — mvo| = vbeat,
where
= +
vklystron mvo ‘ vbeat,
and
mvo = the frequency of the mth harmonic of

the signal from the FM-4A.

The output from the Collins receiver is connected to
one trace of a dual beam oscillosc0pe. Since the receiver
is tuned to the difference frequency, it produces a signal
on the trace only when the klystron passes through a fre-
quency which will produce this beat.

The signal from the Collins receiver produces a pip on
the trace of the oscillosc0pe which serves as a frequency
marker. The pip can be centered on the absorption signal
which appears on the second trace by tuning the Collins
receiver. Since mvo is known the frequency Vklystron
of the absorption line is determined from the above relations.

The beat signal and the absorption signal pass through
different circuits, and therefore reach the oscilloscope
at slightly different times. To compensate for the time
delay, measurements are made sweeping first from low to
high frequencies and then from high to low frequencies.

The average of the two measurements is taken to be the fre-

quency of the transition.

 

IV. MICROWAVE SPECTRUM OF TRANS ACETALDOXIME
4.1 INTRODUCTION

Recent investigations of molecules exhibiting internal
rotation of a methyl group have been concerned with compar-
ing the heights of the potential barriers hindering internal
rotation in cis and trans configurations of the same mole—
cule. The principal difference in the two configurations
is the proximity of the interacting groups. In all cases
studied thus far the barrier height has been found to be
lower in the cis than in the trans conformation (42—47).

For example, Beaudet and Wilson (43) have found that the
barrier height for the methyl group rotation in cis-1-
fluoropropylene is 1057 cal/mole, whereas Siegel found a
barrier height of 2200 cal/mole in the transibrm of the same
molecule (44). The results have led to considerable specu-
lation concerning the physical origin of the barrier heights
and these speculations will be discussed in Chapter VII.

Acetaldoxime was studied in this work to determine the
heights of the barriers to internal rotation of the methyl
group in the cis and trans conformations.* The barrier is
found to be lower in the cis form, fitting the trend estab-
lished in previous studies. In addition, the quadrupole
coupling constants for the nitrogen nucleus were determined
from the hyperfine structure, and the dipole moment of each

configuration was determined from the Stark effect.

 

*-
Compare Fig. 6 and Fig. 9 for the definition of cis and
trans acetaldoxime. 46

47

4.2 OBSERVED SPECTRUM.AND ASSIGNMENT FOR TRANS ACETALDOXIME

The structure assumed for trans acetaldoxime is based
on the parameters for formaldoxime determined by Levine (47).
The bond angles and distances used are given in Table III.
A projection of the molecule in the plane of symmetry is
shown in Fig. 6. Levine found that the hydroxyl hydrogen
is located as shown in Fig. 6 so that the molecule possesses
a plane of symmetry. No evidence was found for the existence
of the other planar configuration of the molecule which is
obtained by rotating the hydroxyl hydrogen 1800 about the
N-O bond. The value of the dipole moment obtained for cis
and trans acetaldoxime from this work indicates that the

position of the O—H bond is as shown in Fig. 6 and Fig. 9.

Table III. Structure assumed for trans acetaldoxime

 

 

r(C2=N) 1.276 R A N-O-H 103°
r(N-H) 1.408 A C2=N-O 110°
r(O-H) .956 A C1-C2=N 116°
r(C2-H) 1.085 A H-C2=N 122°
r(C1-H) 1.100 A H-Cl-H 109.4o
r(C1-C2) 1.55

 

C1 = Methyl carbon.

For purposes of computation the coordinates of the atoms

are written in terms of Spherical polar coordinates fixed

48

x
I

:r:

I

O
“\

J

 

18

Figure 6. Projection of trans acetaldoxime in its plane
of symmetry.

49
in the molecule. With the aid of a program written for the
Control Data 3600 computer the moments of inertia and ro-
tational constants are then used to compute the energy levels,
transition frequencies, quadrupole splittings and Stark
coefficients using a computer program prepared by R. H.

Schwendeman and J. H. Hand.

 

A search of the Spectrum was made in the vicinity of
the J = 2 -> 3 a-type transitions with the waveguide sample
cell cooled to -10°C (acetaldoxime apparently slowly de—
composes atroom temperature in the sample cell.) No puri-
fication of the sample* other that trap to trap distilla-
tion in vacuo was necessary and no separation of cis and
trans isomers was attempted. The total pressure of the
sample in the waveguide was approximately 30 microns for
most observations. The J = 2-—> 3 transitions were
identified by their Stark effect, quadrupole hyperfine
structure, and internal rotation splittings.

After the J = 2-—> 3 lines were assigned, near 24.62
GHz, the J = 0 e> 1, 1 —> 2, and 3 -9 4 transitions cen-
tering near 8.21, 16.42 and 32.85 GHz were assigned without
difficulty. Doublets due to internal rotation of the
methyl group were observed for the K‘1= 1, J = 1,2,3 levels.
The A levels of the doublets fit a rigid rotor pattern and
the E levels did not. -The Splitting between‘ A and E

levels could be accounted for using the first order

 

*Acetaldoxime was purchased from Aldrich Chemical Co.

50

perturbation approximation described in Section 2.3. Table
IV lists the observed transition frequencies and includes

a comparison with the calculated values which include con-
tributions from the quadrupole coupling. The measured fre-
quencies for seven transitions were used in a least square
fit for the final determination of the rotational constants.
The parameters determined by this calculation are given in

Table V.
4.3 QUADRUPOLE COUPLING CONSTANTS

The transitions J = 111-—> 212 and J = 110-> 211
showed splittings of more than one MHz due to quadrupole
interaction with the nitrogen nucleus. Only two of the six
predicted components of the quadrupole multiplets were ob-
served on the oscilloscope; the others were too weak. The
observed components were the transitions F = 1 -¢ 2 and
2 —¢ 3 and these were identified from their relative in-
tensities. The frequency of the weak F = O-+> 1 compo-
nent for the 111-—> 212 transition was measured from a
recorded Spectrum.

With the use of Eq. (5—10) the values of the quadru-

pole coupling constants X and (Xbb — ch) could be

aa
calculated from the observed hyperfine splittings. Eq.
(5-10) expresses the quadrupole energy as a function of the

parameters a and 5 defined in Section 2.5.

JT
WQIF = axaa + B(xbb - XCC)

51

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.Amaw>wa <V wEHXOUHMOmUm mcmnu How mcoHuHmcmuu Um>ummno mo mmflucmdvmum .>H manms

52

Table V. Rotational constants, quadrupole coupling
constants, and internal rotation parameters for
trans acetaldoxime.

 

 

 

 

Rotational constants Quadrupole coupling constants
A = 44613 i 385 MHz Xaa = 4.07 + .11 MHz

B = 4237.73 i .03 MHz Xbb-ch = -5.0 i .8 MHz

C = 3973.81 i .03 MHz

Internal rotation parameters

 

IQ = 3.18 amu 82
F = 211.785 GHz
Aa = cos (19.50)
s = 40.4

v3 = 1835 1 10 cal/mole

 

53

To calculate the quadrupole contribution to the frequency

of the J = 111 ——> 212 transition we first Obtain
212 111
WQ‘ and W0] and take their difference.
F=3 F82
'111-9212 I212 I111 ( )111-5212
AW = w - w = m X +
Q F=2->3 Q F=3 Q F=2 2—>3 a“
111‘9212
(AB) (x - x )
2_>3 bb cc
. 111—2212 . . .
The quantity AWQI , 1f expressed in MHz, Will give
F=2—>3

the frequency correction which must be applied to the rigid
rotor transition frequency. Using the same procedure the
J: 111-2212

value of AWQ can be calculated.
F= 1—>2

The frequency difference between the hyperfine com-

ponents is then

111-9212 111_>212
-Aw| =6w.

AW |
Q F=1—>2 Q F=2—>3 Q

Fig. 7 is a schematic representation of the J = 11r—>212
transition and illustrates the use of Eq. (5-10). The fre-
quency difference was used in the calculation of the quadru-
pole coupling constants. A value of 0WQ was obtained
for each of the J = 111-9212 and J = 110_>211 (F = 2->3,

F = 1->2) transitions for the A and E level transitions
and the four values were used to obtain Xaa’ which makes

the major contribution to these splittings. The value of

Xbb - ch was calculated from the frequency difference

54

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mucosvmum “Hammcb .mw

55
between the transitions J = 11f“>212, F = 0—>1 and
J = 11f->212, F = 2-53 taken from the recorded spectrum.
The Splittings of these transitions are independent of the
value of rotational constants A, B and C.

The values found for trans-acetaldoxime are Xaa

4.07 i .11 MHz; - X = -5.00 i .8 MHz. More accurate

Xbb cc

values of the coupling constants could not be determined
because of the laEk of a sufficient number of transitions

with observable\quadrupole splittings.
4.4 BARRIER TO INTERNAL ROTATION OF TRANS-ACETALDOXIME

Trans-acetaldoxime can be classified as a slightly
asymmetric-top molecule because the rotational constants
B and C are nearly equal. As a result, pairs of energy
levels for certain K_1 values will be nearly degenerate.
In such cases ordinary perturbation techniques are incon-
venient and some other approach must be followed.

The procedure followed here is to set up the matrix

HVO in the symmetric rotor representation.

_ n (n)
HvomflR‘kiFE> Wvo
The matrix elements for HR’
n
in Section 2.2 and those for 6) are taken from Hersch—
bach's paper (10).
For the transitions observed only the odd order terms

give a measurable contribution and these occur only for E

the rigid rotor part, are given

 

1

56

symmetry levels. Therefore the A levels will fit the
energy pattern of a rigid rotor but the E levels will not.

Now, 9 = aPa + BPb + yPC with o. {-3, and y as defined
in Section 2.4. If acetaldoxime has a plane of symmetry,
then y = 0. Also, the axis of the methyl group lies
close to the a-principal axis (the angle between the two
axes is 19.50), Therefore Ab, the direction cosine between
the b—axis and the axis of the methyl group, will be small
so that B = AB Ia/Ib can be neglected (the fact that

> Ia further reduces the value of 5 compared to that

If the a—axis is chosen as the axis of quantization
(z-axis), then the matrix elements for Pa will occur on
the diagonal and the Pb and Pc elements will occur off

the diagonal. With the matrix set up in this fashion the

largest contribution to ZFQinég) will come from the term
n n (n)
ZnFa P wvo and the other terms can be neglected.
a

With these simplifications the matrix for the E

levels becomes (to third order)

(1)

3
+ a3 F P:.w()

HOE - HR + aFPaWOE OE
where HOE is the matrix for the v = 0 torsional state.
The matrix HOE factors into separate J blocks of order

2J + 1. Because the (KlKil) elements are neglected (HR
possesses (KIK) and (KIKiZ) elements only), each J block

will factor into two submatrices approximately of order

57
J + 1, one containing only odd K elements, the other
containing only even K elements. The eigenvalues may
be obtained by solving the secular equations by the method
of continued fractions.
The eigenvalues obtained for the K_1 = 1 levels are

as follows:

 

 

111 = A + 1/2(B+C) - 1/2 [(B-c)2 + 4x2]1/2
110 = A + 1/2(B+C) + 1/2 [(B-c)2 + 4x2]1/2
211 = A + 5/2(B+C) + 1/2 [9(B_c)2 + 4x211/2
212 = A + 5/2(B+C) - 1/2[9(B-c)2 + 4x21“2
312 = A + 1172(B+C) - 32§§7(§:g)2 + [9(B-C)2 + lel/z
313 = A + 11/2 (B+C)- 3gé§B£ggC)2 _ [9(B-c)2 + lel/z

1 ,
In these equations x = F (dWéE)+ a3Wé;)) and A, B, and

C are the rotational constants. Fig. 8 is a diagram of
the J = 1 -> 2 transitions showing the Splittings due to
internal rotation.

The rotational constants are known from the least-
squares fit on the A levels (Table V). The value of x
is then determined by the difference in frequencies of the
A and E level transitions. The value of x may be
used to calculate Wég) or Wéé) + Wé;)° The Wég) are
tabulated by Herschbach (10) as a function of the parameter
s. A plot of log W(1) y§_s is approximately linear as is

OE

1
a plot of log WéE) + Wég) XS S. The value of s correr

sponding to the value of W(1) or ‘W(1) + W(s) determined
OE OE OE

"l

58

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1|

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59

from x can be read off either one of these plots and the
value of V3 computed. Table VI gives barrier heights
determined using the first order term only and using first
and third order terms. Table VII lists the experimental
frequencies of the transitions for the E symmetry species.

The value of V3 depends on the structure of the mole-
cule through the parameters Ia and the angle formed be-
tween the methyl axis and the a-principal axis. Since
these are not known, but assumed, the true error in V3
cannot be determined. ‘The uncertainty reported in Table VI
is the mean deviation from the mean. Assuming an error in
the angle of i1° and an error in IQ of i.1 amu 22, the
propagated errors in V3 are $7 and 160 cal/mole reSpec-

tively.
4.5 DIPOLE MOMENT OF TRANS ACETALDOXIME

The dipole moment of a molecule can be determined with
the use of Eq. (4-5) provided that the frequency of the
Stark components can be measured as a function of the
electric field. The Stark coefficients a(J,T) and
b(J,T) of Eq. (4-5) are computed with the aid of the pro-
gram mentioned in Section 4.2.

The value of the electric field for a given voltage
impressed on the septum is determined by measuring the
frequency of the Stark component of the J = 1-—> 2 transi-

tion of OCS. Since the dipole moment of OCS is known*, the

 

= 0.7124 Debyes, S. A. Marshall and J. Weber, Phys.

*
u
OCS Rev., 105, 1502 (1957).

 

60

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P Jm

 

 

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61

Table VII. Observed transition frequencies for the E
symmetry species of trans acetaldoxime

 

 

 

 

 

 

 

 

 

 

a
J > J' F > F' VObS
111 -> 212 2 —> 3 16163.60
‘ 1 -9 2 16164.76
110 —> 211 2 —"'> 3 16682.00

1-——> 2 16683.35

211 —> 312 3 —-» 4 25028.35

212 ""> 313 3 —'> 4 24238.07
aMHz

Table VIII. Stark effect of trans acetaldoxime

a a b
2 2 '
Transition (av/6E )obs (av/3E )calc
101 —» 202 .M = o -3.86 x 10"6 —3.75 x 10'6
M = 1 2.98 x 10"6 3.05 x 10‘6
202 -9 303 M = o -0.631 x 10'6 -0.625 x 10'6
M = 1 -o.174 x 10‘6 -0.176 x 10"6
IMHz/volts/cm.
b

Calculated assuming u: = 0.7976 (Debyes)2, u; = u = 0.

\

62

proportionality constant, which converts the voltage im-
pressed on the septum into the value of the electric field
in the waveguide, can be calculated.

The frequencies of the Stark components of the
101 -> 202 and 202 -¢ 303 transitions were measured and
the values of ov/BE2 were obtained from a plot of v‘yg
E2. With the relations obtained from Eq. (4-5) the values
of u: and u; were calculated assuming .uc to be zero
because of the molecular plane of symmetry. The initial
calculation produced a small negative value for u; indi-
cating that it is also near zero. Therefore u; was
set equal to zero and the value of u: was found to be
0.7976 (Debyes)2. Consequently the total dipole moment is
u = ”a a 0.893 r .014 Debyes. The observed and calculated

values of ov/5E2 are compared in Table VIII.

V. MICROWAVE SPECTRUM OF CIS ACETALDQXIME
5 .1 OBSERVED SPECTRUM AND ASSIGNMENT

The spectrum of cis acetaldoxime was interpreted using
the same approach as that described in Chapter IV for the
trans compound. The structunfl.parameters were taken from
Levine's (47) values for formaldoxime except for the
CMe-C=N angle. This angle was assumed to be 126°, which
was a compromise between the value of 122° for the H-C=N
angle reported for formaldoxime and the value of 131° for
the CMe-C=N angle reported by Bierlein and Lingafelter (48)
for acetoxime. The assumed structure of cis acetaldoxime

is given in Table IX and Fig. 9 is a projection of the

molecule in the plane of symmetry.

Table IX. Structure assumed:fin:cis acetaldoxime

 

 

r (C: ) 1.276 R A N-O-H 103°

r (N—O) 1.408 A C-N-O 110

r (o-H) 0.956 A CH3-C=N 126

r :(C-H) 1.085 1 H-C-N 116

r (c-H) (Methyl) 1.100 A H-C-H 109.44
r(C-C) 1.50

 

63

64

 

 

 

a
AL
H
C
H
H
C
\
I'
/N
O
H
¥ l
12

 

Figure 9. Projection of cis acetaldoxime in its plane
of symmetry.

65
The moments of inertia of cis acetaldoxime are very
different from those of the trans form so that the spectra
of the two compounds were easily distinguished. The transi-
tions were observed with the sample cell cooled to -10°C
and the sample at 30u pressure. Effective rotational con—

stants were determined from the J = 1-> 2 transitions

of the A levels and are AA = 17364 i 15 MHz, BA =
6658.92 1 .05 MHz. CA = 4921.04 1 .05 MHz. Table X com-

pares the observed transition frequencies for the A
species with the calculated frequencies. The experimental
frequencies have been corrected for quadrupole hyperfine
splitting as described below. The comparison in Table X
shows considerable deviation from the rigid rotor frequencies
which is a result of the low barrier hindering the internal
rotation of the methyl group.

In addition to the departure of the spectrum from a
rigid rotor pattern, the value of the inertia defect, A,
indicated an effect of a low frequency vibration. The in-

ertia defect may be dflined for the present case as

A-I +I -Ia-.Ib'

It may be shown that A would be zero if acetaldoxime is
a rigid molecule with all atoms in the plane of symmetry
except the hydrogens on the methyl group. However, using
the effective moments of inertia obtained from the rota-

tional constants above the value of A is +0.83 amu 32, if

Ia is assumed to be 3.18 amu 22. If the molecule actually

66

 

 

 

 

Table X. Experimental hypothetical unSplit frequencies for
cis acetaldoxime (A Species).
.44;— I;
J > J' : vexp Vcalc Deviation
MHz MHz MHz
000-——> 101 11579.92 11579.95 -0.03
101 __¢,202 22965.02 22965.02 0.00
111 ——> 212 21422.03 21422.03 0.00
110 -—¢-211 24897.80 24897.80 0.00
202 ——¢ 303 33976.07 33976.07 0.00
220-——> 321 35502.76 35503.65 -0.89
212 ——§ 313 32017.09 32017.38 -0.29
221 -9 322 34737.86 34739.86 -2.00
313 -v 312 10413.05 10413.53 -0.48
414 -—¢-413 17281.2 17283.17 -1.97

 

aCorrected from observed frequencies using the quadrupole
parameters in Table XII.

bCalculated assuming a rigid rotator with A

B = 6658.918, C = 4921.036 MHz.

= 17364.379,

67
did not possess a plane of symmetry as assumed, then A
would be negative. A positive value for A usually results
from the effect of low frequency vibrations (49). On the
assumption that the anomalous value for A was caused by
the internal rotation of the methyl group, the value of
the potential barrier which would correct the moments of
inertia to the extent that A becomes zero was calculated.
An approximate value of the barrier was obtained by using
Eq. (3-14) (modified for the present case) which gives the
effective rotational constants as functions of the pertur-

bation coefficients from internal rotation.

2
+ 52 F éA)' C = C

= 2 (2) =
A A+a FW . B A 0'

A 0 0A A B0

0’ B0 and CO approximate the true rota-

tional constants in that they reflect more accurately the

The constants A

moments of inertia of the rigid molecule. With the approxi-
rnate value of V3 the frequencies of the E-levels could be
predicted. In addition, a calculation of the Stark effect
for the E level transitions indicated an unusual first order
effect for the Stark perturbation which was very helpful in
the identification of the transitions.

5.2 DETERMINATION OF THE HEIGHT OF THE BARRIER HINDERING

INTERNAL ROTATION
The low potential barrier hindering the rotation of

the methyl group in cis acetaldoxime produced large split-
tings between the A and E symmetry levels. To account

for the large splittings the matrix HOE was calculated

68
from Eq. (3—11) including the perturbation coefficients
(n) ‘
W0E to fourth order.

_ 4 (n) n
Hoxz:_Hr+Fn‘:1 W03 6) +HD

The extra term HD that appears in the above equation is

a correction term which results from the neglect of ro-
tational energy differences as compared with torsional
energy differences in the denominators of the perturbation
sums. The matrix elements for HOE were taken from Hersch-
bach (10) and were referred to the symmetric rotor basis

of D

a>h symmetry rather than to the Wang symmetric

rotor basis which is the conventional starting point for
asymmetric rotor calculation. The matrix factors into
blocks of dimension 2J + 1 for each value of J and was
diagonalized with the use of the Control Data 3600 computer
at the Michigan State University Computer Laboratory. The
energy levels are simply the eigenvalues of the matrix.
Values of the internal rotation parameters including
V3, the height of the three-fold barrier, were determined
from a least squares fit on 14 measured frequencies, seven
from the E level transitions and seven from the A levels.
Five parameters were fit without constraints; the five
parameters were the three rotational constants A0, B0 and
Co, the angle between the methyl axis and the a principal
axis, and V3. The molecule was assumed to have a plane
of symmetry so that the direction cosine between the methyl

axis and the c principal axis was zero and the direction

69
cosine of the angle between the methyl axis and the b
principal axis was the sine of the angle between the methyl
axis and the a axis. To perform the least squares fit
the frequencies were assumed to vary linearly with the

various parameters, p,

The derivatives (av/dpi) were determined numerically by
varying each parameter individually and noting the effect
on the v's. The results of the calculation are tabulated
in Tables XI and XII.

In an initial attempt at fitting, the parameter Ia
was included in the adjustment. The results are'also given
in Tables XI and XII. When Ia was allowed to vary without
bound the final value became Ia = 2.728 amu 82, which is
far from the putative value of 3.1 amu 22 which was as-
sumed for the five—parameter calculation. It is unlikely
that IQ is this small since for a CH bond length of
1.08 2, Id = 2.73 amu 22 would require an HCH angle of
100°. Other possibilities for the small value of Ia are
inadequacies in the perturbation treatment or in the original
Hamiltonian expression. The value of the barrier is near
the limit where the perturbation treatment becomes imprac-
tical if torsional eigenfunctions are employed as basis
functions, and the Hamiltonian contains no contributions

from interaction with the other vibrations.

Table XI.

Comparison of observed and calculated
frequencies for cis~acetaldoxime.

transition

 

 

A Species

 

 

 

 

 

Transition V:bs (Vobs-Vcalc)b (Vobs-Vcalc)C
MHz .MHz MHz
101 ——> 202 22965.02 -0.51 -1.11
111-——> 212 21422.03 -0.55 -2.30
110 -—> 211 24897.80 -0.12 -2.08
202 ——> 303 33967.07 -0.16 2.22
212 -—> 313 32017.09 -1.02 -2.86
221 ——> 322 34737.86 -0.76 -2.77
22o-——> 321 35502.76 -O.27 -7.61
I E Species
000 -> 101 11437.00 —1.43 -9.25
101 -—» 202 22745.78 -0.75 -9.58
211 -—> 212 22012.42 —0.96 3.47
110-——> 211 24173.35 2.26 9.47
202 -—> 303 33760.73 0.09 -1.07
212-——> 313 32244.30 2.81 4.98
221-—-> 322 35033.28 2.23 1.88
220-——> 321 34958.41 -2.51 6.70
aHypothetical unSplit frequencies corrected for quadrupole
splitting.
bvcalc gbggggegmfisfigg parameters in Table XII with IQ =
cvcalc obtained gsing parameters in Table XII with IQ =

3.1 amu.

 

71

Table XII. Rotational constants, quadrupole coupling con-
stants and internal rotation parameters for
cis acetaldoxime.

 

 

Effective rotational constants Quadrupole coupling constants

A = 17.36438 GHz X = -2.8 i.2 MHz
A aa

B 6.65892 GHz Xbb

4.92104 GHz

['17
ll

- X = 0.6 i .4 MHz
cc

0
ll

 

c
ignternal rotation parameters

IQ = 3.1 amu Rza 4 Id = 2.7275 amu 22b

A0 = 17.11130 GHz A0 = 17.21546 GHz

B0 = 6.625322 GHz B0 = 6.626483 GHz

c0 = 4.920999 GHz 00 = 4.920697 GHz

a = 0.066291 a = 0.056847

8 = 0.031493 3 = 0.028274

1a = cos(50.82°) 1a = Cos(52.26°)

Ab = cos(39.18°) Ab = cos(37.74°)

F = 174.65272 GHz F = 196.58085 GHz

v3 = 356.42 cal/mole v3 = 392.927 cal/mole

s = 9.510 s = 9.314

 

aParameters obtained from least square fit with I fixed
at 3.1 amu 82. a

bParameters obtained from least square fit with Ia varied.

72

In an effort to determine if the error resulted from
failure of the perturbation technique, the frequencies were
recalculated using a low barrier approximation in which free
rotor wavefunctions formed the basis set for the motion of
the methyl group. The results of the calculation gave

approximately the same results as the perturbation treatment.

 

A comparison of the two methods is given in Chapter VI.

Since the perturbation approach does not contribute a
significant error to the calculated transition frequencies
the low value obtained for Ia suggests an inadequacy in
the Hamiltonian expression. An attempt was made to add a
V6 term to the potential function of Eq. (3-10). The po-
tential would then take the form

V = §—-(1 — cos 3a) +'ir (a — cos 6a)

However, when the potential is put in this form and the V6
term added as a variable to be fit to the data, the V6 and
V3 term are strongly coupled and cannot be determined in—
dependently. No reasonable value of V6 would increase
the value of IQ' At present we have no explanation for
the anomalous value of IQ.

Another method of presenting the data from the internal
rotation Splitting is to calculate a value of the barrier
for each Splitting of the A and E symmetry levels. The
calculated frequency difference between and A and E
level transitions is plotted as a function of the barrier

height and the value of the barrier corresponding to the

73

observed frequency is read off the graph. The results are
presented in Table XIII and give an average value of 356
cal/mole for the barrier height.

The error in V3 will depend mostly on the error in
Ia and slightly on the error in the angle between the
methyl axis and the principal axes. An error of i0.1 amu 22
in IQ will produce an error of i10 cal/mole in V3, while .
an error of i1° in the angle between the methyl axis and
the principal axis will cause an error of i0.8 cal/mole.
In Spite of the apparent inadequacy of the theoretical treat-
ment for this low-barrier species, the values of the bar-
rier height in Tables XII and XIII are all comparable, so
that it is probably safe to conclude that V3 = 375 i 30

cal/mole.

Table XIII. Internal rotation splittings for cis acetald—

 

 

 

oxime.
J -—> J' V3(cal/mole)calc (wA-VE)obs
(MHZ)
000 ——> 101 350 +143.06
101 ——> 2,,2 352 +219.24
111 —> 212 352 -590.39
110 ——> 211 356 +724.44
202 -—> 303 356 -215.34
212 —> 313 351 -227.21
220 ——> 321 368 —544.35
221 ——> 322 350 ~295.42

 

‘Ave 354 i 6 cal/mole

 

74

5.3 QUADRUPOLE COUPLING CONSTANTS OF CIS ACETALDOXIME

The quadrupole coupling constants were determined from
splittings measured for several lines as described in Sec-
tion 4.3. The measurements were made from recorded Spectra
since all the splittings were too small to be resolved on
the oscilloscope trace. The Spectra were resolved using
the recorder by sweeping through the transition Slowly with
the pressure of the sample gas aS low as possible.

With the use of Eq. (5—10) and the calculated values
of a and B the values of Xaa and Xbb - ch were
determined from the frequency differences of the hyperfine
components. Three hyperfine lines were observed for each
of the rotational transitions examined and the frequency
differences between the extreme components were used in the
calculation. As an example, the hyperfine pattern for the
J = 000 “—5 101 transition is Shown in Fig. 10. The
largest splitting here (2.1 MHz) is between the F = 1 -> 0
and the F = 1 -> 1 transitions and this value was used
to calculate the value of Xaa directly, since this split-

ting doeS not depend on the value of Xbb - . In Similar

ch

fashion the value of Xaa could be determined directly from

the hyperfine splitting of the 101 -—9 202 and 221-——> 322

transitions. The value of Xbb - X was determined from
cc

the quadrupole splittings for the 111'-> 212 and the

110-—> 211 transitions. For each of these transitions

the term involving Xbb - ch made a contribution of

75

 

 

 

 

 

 

0.8MHz
1.3MHz
' F= I—>1 F= L—>2 F= L—>O

Figure 10. 'Hyperfine-pattern for the J = 000 ——¢.101
transition of cis acetaldoxime.

76
approximately 0.1 MHz to the Splitting. Since this is about
the accuracy of the measurement from a recorded spectrum,

Xbb - ch is obtained with a large percent error. Table

XIV lists the values of Xaa and Xbb - obtained from

XCC

the various transitions. The average value of Xaa is

-2.8i 0.2 MHZ; =0.6i 0.4 MHZ.

Xbb - ch

Table XIV. Quadrupole coupling constants for cis acetald-

 

 

 

oxime
I I a
J -—> J F -v F Av ) Xaa (Xbb - ch)
(MHZ (MHz) (MHz)
2.1 -2.8 b
1 —> 0
101 ""> 202 O _> 1
2.0 -2.7 b
2 -> 1
221 —> 322 3 ""> 4
1.0 -3.2 b
2 -> 3
1 8 -2.8 0 6
O --> 1
110 —> 211 1 ""> 2
1.6 -2.8 0.6
0 —> 1

Ave -2.8 i .2 MHz 0.6 i .4 MHz

 

aAv is the difference in frequency of the two hyperfine
components listed under F -> F.

bThe splittings do not depend on Xbb - ch'

-.k

77

5.4 -STARK EFFECT AND DIPOLE MOMENT OF CIS ACETALDOXIME

Confirmation of the assignment of quantum numbers to
all of the observed transitions was obtained by approximate
measurements of the Stark effect. The A Species transi-
tions exhibited the usual second order Stark effect and the
dipole moment components were determined from careful mea—
surements of the frequencies of Six Stark transitions as a
function of the electric field. The values of Bv/BE2 ob-
tained were fit by the method of least squares to a system
of equations obtained from Eq. (4-5). The measurements
were made in the manner described in.Section 4.5 and the
results are presented in Table XV.

The values of ug, u% and u: were obtained from the
least-squares adjustment. The initial calculation, in which
all three components were fit to the data, gave a small nega-
tive value for u: which indicated that ”c is near zero.
The final fit was obtained by setting uc equal to zero.
The values obtained for the components of the dipole mo-
ment were u: - 0.6368 (Debyes)2 and u; = 0.0801 (Debyes)2
which gives ”a = 0.798 i .004 and ”b = 0.283 i .050
Debyes. The total dipole moment is 0.847 i 0.02 Debyes.
Because the Stark Splitting was not greater than about
10 MHz for most of the transitions and the Splittings were
only Slightly dependent on uc, the value of uc could not
be accurately determined. However, the data indicate that

it is less than 0.07 Debyes.

'1‘

Table XV. Stark

78

effect for cis—acetaldoxime (A species).

 

 

 

Transition M (av/8E2):bs (av/5E2):agc
110 ——> 211 0 1.612 x 10’6 .1.631 x 10"
' 1 21.565 21.602
101 ——> 202 0 -2.120 -1.949
1 2.231 2.140
212 -—¢-313 1 0.813 0.797
2 3.736 3.660

 

aMHz/(voltS/cm)2.

b

The E level

observed to have

Calculated assuming u: - 0.6368, ug = 0.0801, ”c = 0.

transitions of cis—acetaldoxime were all

a first order Stark effect because of the

barrier dependent terms in the Hamiltonian matrix for the

E levels. To calculate the Stark effect for the E levels

the elements of the Hamiltonian matrix described in Section

5.2 were calculated with the Stark perturbation added. The

Hamiltonian takes the form

HOE = HR + F 2 WOE + H - 2 E u ¢z

where E is the

component of the

4 (n) n

D 9

n=1 9 9

electric field vector, “9 is the gth

dipole moment and ¢zg is the direction

cosine between the Space-fixed z axis and the 9th princi—

pal axis. The matrix elements for ¢zg may be found in

Strandbergs book (18).

79

The matrix of 7H is diagonal in J except for the

OE
term involving 029 which has nOn-zero (JIJ + 1) elements.
The energy levels for a given value of J, M and E are
obtained by diagonalizing the matrix. To obtain the transi-
tion frequencies the eigenvalues had to be determined for
a matrix of maximum J value one greater than the upper
value of J involved in the transition. For example to
obtain the values of the energy levels involved in a J =
2-—-> 3 transition a matrix containing elements up to
J = 4 must be diagonalized.

Careful Stark-effect measurements were made of the
101 -—> 292, E Species transition. This transition had
an unusual Stark effect in that one component was displaced
to the low frequency Side of the zero field transition fre-
quency at low field and then was displaced to the high
frequency side at higher fields. The comparison of the ob-
served and calculated displacements as a function of the
electric field is presented in Fig. 11. The values of ”a’
”b and ”c used in obtaining the calculated curves are
those obtained from the -A levels. The parameters used in
calculating the matrix elements are those given in Table XII
with Ia = 3.1 amu 82. The M = :1 Stark components of
the 101-——> 202, E species transition are very sensitive
to the value of “b' All the components are weakly dependent

on the value of the internal rotation parameters.

5.0

-"l’00

-500

-6.0

80

 

 

 

 

 

O
[A M=+l
O 43
A.
0
0A £3
' ~11
a_ .A
o M=-l
A
I] D
.A O A.
A 3 °
0
D
D
O
D
[13 M=O
D
0 Experimental Points
A [3 Calculated Points
1 L 1 I 1 1 1 1 1 1 1 1 1 1 J 1 l +
5 10 15

E x 10_2(volts/cm)

Figure 11. Observed and calculated Stark shifts for cis

acetaldoxime (E Species) for the 101-——> 202
transition.

VI. LOW BARRIER APPROXIMATION FOR CALCULATION OF ENERGY
LEVELS FOR INTERNAL ROTATION AND OVERALL ROTATION

6.1 INTRODUCTION

The perturbation treatment described in Section 2.3
uses a basis set of functions which are a product of sym-
metric rotor eigenfunctions and torsional eigenfunctions.
The torsional eigenfunctions are the solutions to the
Mathieu equation. Therefore, a typical basis function can

be written as

(9,¢) eiKX U(Q)

z”I = SJ.K,M v0

where S lKX

e represents the symmetric rotor

J,K,M.(e’¢)

wave function, J, K and M are the quantum numbers de-

scribed in Chapter II, and 6,¢,X are the Euler angles

that describe the motion of the molecule. Uv0(a) represents

the Mathieu eigenfunction,.which for a very high barrier

can be approximated by harmonic oscillator wavefunctions.
When the matrix for the Hamiltonian of Eq. (3-9) is

written using these basis functions the matrix takes the

form shown in Fig. 12. The Hamiltonian of Eq. (3-9) is

written

v
= 2 2 2 _ 2 ._i _ _
H APA + BPB + CPC + F (p Q?) +-2 (1 cos 3a) (6 1)

For convenience Eq. (6-1) is rewritten as

H = HR + HT + HIR (6—2)

where
81

82

/ //

Figure 12. The torsional matrix. Each v block contains
the matrix elements of HR and HT of Eq.(6—2) associated

with a given torsional state and the elements (VWH

 

 

 

 

\

 

 

 

 

 

 

IR|v).
~Each v block contains an infinite diagonal block matrix,
one diagonal block for each value of J from 0-—> oo.

 

m=-—9 “Va/Ll"

 

m=—6 -V3/4

 

 

m=- ~V3/H

 

 

m=0 -V3/H

 

 

 

 

m=6 -V3/4

 

 

m=9

 

 

 

 

Figure 13. ‘Energy matrix in the free rotor representation.
The dimension of each diagonal block is 2J + 1.

83

_ 2 2 2
HR — APa + BPb + cpc (6-3)
V3
HT + sz +'1f (1 - COS 3a). (6'4)
HIR = —2FpfiD . '(6—5)

HR and HT are diagonal in J, v, and o and will have

elements in the diagonal blocks only. H however, will

IR'
have elements non-diagonal in v and will occur off the
diagonal block in the Shaded area of Fig. 11. The elements
of -2Fp6J are treated as a perturbation.

The Van Vleck perturbation technique is convenient
when the separation between torsional energy levels (dif-
ference between energies of different blocks in Fig. 12)
is large compared with the separation between rotational
energy levels for a given value of v. However, as the
potential barrier hindering the internal rotation of the
methyl group decreases, the spacing between torsional energy
levels is decreased and the perturbation technique becomes

inconvenient. In this case the low barrier approximation

outlined below may be used.
6.2 FREE ROTOR APPROXIMATION

In the low barrier approximation the internal rotor
is treated in first approximation as a free rotor. In the
Hamiltonian of Eq. (6-1), p represents the angular momentum
of the internal top, which is an internal degree of freedom
described by the angle of rotation a about the symmetry axis

of the top (e.g. a methyl group). The angular momentum

84

momentum operator p can be represented by

p = -i (O/OQ)9’¢’X , (6-6)

and if the top is treated as a free rotor, the eigenfunc-

tions which represent the states of the internal top are

 

Simply Um = , where m = 0,il,i2, etc. The eigen—

values of p and p2 become m and m2 respectively,
consequently, m is the quantum number that represents
the angular momentum of the top. The matrix elements are

therefore
<mlp|m> = m and <m|pzlm> = m2.
The basis functions now take the form

iKX 1 e ima .
J20

Writing the matrix for the Hamiltonian of Eq. (6-1) in the

$11 = SJKM(9'¢) 6

new representation one finds that all the elements are di—
agonal in J and m except for the term -(V3/2)cos 3a

whose non-zero elements are
<m|('V3/2)COS 3alm i 3> = ‘V3/4,

and are non-diagonal in m only.

Because of the C3V symmetry of the methyl rotor the
matrix can be factored into three submatrices one of which
is represented schematically in Fig. 13. Each diagonal
block corresponding to a given m value contains the

matrix elements of the terms of Eq. (6-1) for a Single value

85
of J, and J has the same value for every m block. The
dimension of each diagonal block is 2J + 1. The only ele-
ments off the diagonal blocks are the elements -V3/4 which
occur along the diagonal in the blocks just off the diagonal.

The eigenvalues of the matrix in Fig. 13 represent the
energy levels for the A symmetry species (non-degenerate).
The energies for the E symmetry species are obtained from
a matrix in which the central block has m.= :1 and each
successive block in either direction has m increased or
decreased by 3.

The off diagonal elements -V3/4 may be treated by
perturbation when V3 is small. However, for purposes of
checking the original perturbation treatment the matrix of
Fig. 13 was diagonalized directly with the aid of the
CDC 3600 computer. For purposes of computation the matrix
to be diagonalized was chosen large enough so that ground
state transition frequencies were accurate to i0.01 MHz.

For a value of V3 of 350 cal/mole the matrix required for
the calculation of A species frequencies contained values
of m from —9 to +9 and the E species matrix contained
values of m from -8 to +10.

The transition frequencies observed for cis-acetaldox-
ime were used in a fit of the rotational constants A0, Bo,
Co and V3, Ia and the angle between the avprincipal axis
and the methyl axis. The values obtained for these parameters

are summarized in Table XVI where a comparison is made with

.SOHDMHSUHMU mwumsvmlummmH 0:“ SH omumdhom mum3 QUHSB

mumumfimumm 0:» mo £000 How SOHumH>mU oumocmum on» mum mmsHm> o Hmnuo was .uHm mGOHuHmcmHu

smmuusom map How mmHocmSUmum meMHsono was Um>nmm£o somBDmn GOHDMH>mU Unmocmum mH co

m

 

 

 

 

 

 

mmmo.o 5H.o mb.m vm.o mam.o Hv.mH hv.H >H
mmo.o va.o 8H.v wm.o ww.o Nv.om mm.H HHH
6 hm.o b¢.H HN.H mm.H Nm.>m vm.m HH
8 hm.o om.H mN.H o¢.H hm.bm om.m H
NoH mam m
..o 0 >0 00 mo mo mo
8
mumb.m ©N.Nm 8mm.mmm bmm.omm¢ mv.mmmm mv.mHmbH UmHHm> H .SOHDmQHsuHmm >H
o
hvmb.m wN.Nm moo.mmm mmm.ommv om.mmmm Hm.NHNhH omHHm> H .Houom mmum HHH
d
mm.om Nv.mmm mmm.ommv www.mmmo om.HHHbH H.m u H .SOHpmanuHmm HH
6
mm.om om.mmm mm¢.ommv www.mmmo b©.®OHhH H.m u H .Houom mosh H
Aux 580v Ammpv AmHoE\Hmov , .
6H 392 n> Anni o Anni m Ana: 4 69302
.soHumEHxOHmmm

HmHHHmn flmfln bum HOHHHMQ 30H fluHS mmHquDvam Um>umwno mCHDUHM mo muHSmwm .H>N OHQME

87
the values obtained from the perturbation calculation of
Chapter V. 'The results from both calculations are approxi-
mately the same, especially for the values of the three-
fold barrier height which differ by only 0.1 cal/mole.
6.3 COMPARISON OF RESULTS FROM LOW BARRIER AND HIGH
BARRIER APPROXIMATIONS

As the value of V3 decreases the perturbation treat-
ment (high barrier approximation) becomes more inadequate
and the free rotor approximation will improve. To deter-
mine the actual error involved in the perturbation calcu—
lation as the barrier is decreased, calculated frequencies
obtained using both approaches were compared at values of
s from 8 to 16 (299 < V3 < 599). In the comparison

the values of the frequencies obtained from

vfreerot’
the free rotor (low barrier) calculation, are assumed to be
exact since the free rotor basis set was chosen large enough
to eliminate error in the calculated frequencies. For ex-
ample for s = 16 (V3 = 599 cal/mole) the calculation of
the E Species energy levels included values of -14 < m < 16.
Using a greater number of basis functions had no effect on
the values obtained for transition frequencies although in
some cases the value of the energy level may have changed
:2 MHz.

Tables XVII to XX Show the differences between calcu-

lated frequencies resulting from the perturbation and the

free rotor calculation for Values of 8 < s < 16. The

88

 

.Nmz CH mmDHm> HH<m

 

 

oH.ou om.omm mo.ou Hm.ou sm.omvom Hum 111 omm
sv.o om.ovmu oo.ou HH.o- ow.ooowm «no 411 Hum
oH.o os.mm ov.H- HN.H- SH.ooHom «Hm 111.11m
mH.on ma.m1 oo.H av.H vo.owofim «Ho 41: mam
oo.o- mm.sm ow.o os.o oo.ofloom oom_4|u «om
oa.o om.muH oo.Hu om.ou so.Hmmvm Ham 11|.oHH
4H.o- om.nsu Ho.H sm.o mo.Hova «Hm 4|: HHS
mo.ou os.am mm.o Hm.o oS.oHomm «on 41: HoH
pummAm>I<>vl mmumA >1 >w uuwm>lmmum>, puma Immmm> mmuw> soauamsmne
009m1m>1¢>v m a m m «> < a . .

 

 

m.oH I m How SOHumHSUHmo SOHumnusuumm

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89

 

.Nmz Cfl WODHM> HHANM

 

 

AN

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SH.H sm.msmu mm.H- HH.o- so.oooom «um 111.1«m
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ov.ou mm.oou Ho.m mo.H ao.¢oonm «Hm 111.81m
om.o- mm.ooH mo.H mo.o om.Hvomm mom 4|: «on
sm.o oo.oo¢ mm.dn oo.o- mo.oomvm 11m 1||.oHH
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nomnnomumm

.NH n m

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paw SOHumEonummmm Houou mmum Eoum mUHSmmH mo somHHmmEoo

.HHH>X OHQMB

90

.Nmz SH mmsHm> HHmm

 

 

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so.H us.mmmu ms.Hu HH.o- mo.amsom «no 111.14m
ms.o 68.669 oo.m- Hm.H- mm.momsm «Ho 11: ”Ha
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mm.o- so.mmfi oH.H mm.o ms.soomm mom 11|.oom
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moumxm>ua>v m 4> m> h) ¢> a) «> . .

 

 

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nomnnomumm paw coHumEonummm

91

 

.Nmz CH mmsHm> HH¢0

 

 

 

 

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om.m 8.87 3.? oo.o.. 3.82% 2m All 3a
:4 3.on 3&1 3.7 8.3.5.2.. 2m 1| 2m
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uuwmAm>14>+1u0HmmumA >1 » unmm>lmmHM> pumm>lmmum uoummnm> coauamcmua
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m <
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nomnnomumm paw SOHumEonummm mouon mmum sown muHSmmH mo somHHmmEoo .xx mHnma

92
splitting between A and E Species is also included
since it is the value of VA-VE which is normally used to
determine the barrier. Tables XXI and XXII illustrate how
the various frequency differences decrease as the value of
5 increases (V3 increases) and the perturbation approxi-
mation becomes more accurate especially for higher values of
J. The important thing to note here is that the largest

difference in the tables would not affect the final deter-

mination of the barrier by more than i1 cal/mole.

 

93

 

 

 

 

 

 

Table XXI. Difference of calculated frequences using free
rotor and perturbation calculations for A and
E species.
(Véree-vgert)a
Transition 5 = 8 s = 10 s = 12 S = 16
101 ——> 202 0.22 0.23 0.22 0.21
111 -—> 212 0.91 0.95 0.94 0.87
110-——> 211 -0.92 -0.96 -0.95 -0.89
202-—-> 303 0.80 0.83 0.82 0.76
212-—-> 313 1.47 1.53 1.52 1.42
211 —-> 312 -1.25 —1.31 -1.30 -1.21
221 ——> 322 0.09 -0.11 -0.11 -0.11
220-—> 321 -0.87 70.92 —0.91 -0.81
vIfree-Vgert
101 ——> 202 0.27 0.32 0.27 0.23
111-——> 212 1.21 1.34 1.25 1.01
110 ——> 211 -1.28 -1.42 -1.32 -1.05
202 ——> 303 1.22 1.16 1.02 0.85
212-——> 313 2.72 2.35 2.01 1.60
211-——> 312 -2.36 -2.09 —1.79 -1.40
221 ——> 322 -3.35 -1.78 -1.28 -0.58
220-——> 321 0.58 -0.29 -0.38 -0.62

 

aA11 values in MHz.

Table XXII .

94

Comparison of (VA-VB) values from free rotor
approximation and perturbation calculation.

 

 

 

 

 

A E E a

(V -v )free - (VA-v )pert
Transition S = 8 s - 10 S = 12 = 16
101 _""'> 202 -0.05 -0.09 -0.05 -0.02
111 —> 212 -0.30 “0.39 ”O .31 "0.14
110 -—> 211 0.36 0.46 0.37 0.16
202-——> 303 —0.42 —0.33 -0.20 -0.09
212-——> 313 -1.25 -0.82 —0.49 -0.18
211-——> 312 1.11 0.78 0.49 0.19
221 '—'-> 322 3.26 1.67 1.17 0.47
220 ——> 321 “1.45 “0.63 '0.53 "O.19
aAll values in MHz.

 

VII. DISCUSSION

The large decrease in the barrier to internal rotation
from 1830 cal/mole in trans acetaldoxime to 350 cal/mole
in cis-acetaldoxime follows the trend found in other cis-
trans isomers. A comparison of barriers in cis and trans
forms of the molecules studied to date is Shown; in Table
XXIII. Beaudet and Wilson (43) have suggested that the
lowering of the barrier in the cis configuration of the
compounds is due to non—bonded interaction between the cis
substituent and the methyl hydrogen. When the distance
between the interacting atoms becomes less than the sum of
the van der Waals radiitthey should be strongly repulsed.
The non—bonded interaction potential should have the shape
of a Morse curve with the minimum occurring at rmin =
rw(1)-f rw(2) where rw(1) and rw(2) are the van der
Waals radii of the two interacting non-bonded atoms. At
distances less than rmin a steep repulsive slope is en-
countered and at greater distances than rmin a gentle
attractive slope takes over.

For example in cis-l-fluoropropylene the distance of
closest approach between a methyl hydrogen and the fluorine
atom occurs iniie equilibrium configuration and is 2.49 8,
whereas the sum of the van der Waal radii is 2.55 R = rmin'
Thus the atoms are interacting at distances where repulsive

forces are dominant and the increased potential energy in

95

96

Table XXIII. Comparison of barriers to internal rotation

for cis and trans isomers.

 

 

 

Compound Barrier (cal/mole) Reference
CH3CH=CH2 1978 61
trans CH3-CH=CHF 2150 62
cis CH3-CH=CHF 1057 .43
trans CH3-CH=CHC1 2170 45
cis CH3CH=CHC1 620 52
CH3CH=CF2 1252 53
trans CH3CH=CHCN 2100 63
cis CH3CH=CHCN 1400 64
,/0\
trans CH3CHr-CH-CH3 2600 66
cis CH3Cfi12:CH-CH3 1600 65
trans CH3CH=N-OH 1835 this work
cis CH3CH=N-OH 375 this work

 

97
the equilibrium conformation will raise the minimum of the
threefold potential barrier. As a result V3 will be de-
creased, since it is defined as the difference between
the maximum and minimum of the threefold barrier. The
equilibrium conformation of the methyl group is assumed to
have a C—H bond eclipsing the double bond, which is the
orientation found in all compounds for which experimental
data is available.

The cis substituent and the methyl hydrogen approach
each other at distances which are less than the sum of the
van der Waals radii in each of the cis isomers listed in
Table XXIII. In cis-l—chloropropylene there is a marked
overlap of van der Waals radii in the equilibrium form.

The distance, r(fiMe-Cl) is 2.4 R and rw(Cl) + rw(H) is
3.0 R. The strong repulsive forces at such close distances
are thought to account for the 1500 cal/mole decrease in
the barrier.

In cis-acetaldoxime, the methyl hydrogen interacts with
the oxygen atom as can be seen in Fig. 14. Using values of
rw(0) = 1.5 R and rw(H) = 1.2 R taken from Bondi (50)
the sum of the van der Waals radii is 2.7 2. An [_N=C-CH3
equal to 130° places the oxygen and hydrogen at a distance
r(H-0) = 2.30 R. The value of 130° for the N=C=C angle is
compatible with the value of 52° found for the angle between
the methyl symmetry axis and the A principal axis if all
the remaining coordinates are as listed in Table IX. Pre—

suming that the equilibrium conformation for the methyl

 

98

.cmmounha Hmnpme paw cmmmxo mo Haomu
manna H06 cm> mo mmauo>o mcfl3onm mEononuwom mflo mo onsuosuum

% H

 

m u mm.H

xv.H

/\

 

1
\

 

.vH muzmnm

99
group is as shown in Fig. 14 the repulsive force between
the hydrogen and oxygen atom will raise the minimum of the
threefold barrier and decrease V3.

In trans acetaldoxime the oxygen atom is far removed
from the methyl group but because of the smaller N=C-C angle
(116°) there is considerable overlap of van der waals
radii from the methyl hydrogen and the nitrogen atom. How-
ever, there is an important difference between the way the
electron clouds overlap. If Fig,14 is compared with Fig.

15 which shows the trans compound, it will be noted that
most of the overlap takes place on the bonding side of the
C-H and the C=N bond in the trans compound. It is generally
accepted that the electron distribution around the nucleii
is non-spherical with most of the electron cloud extending
out on the side further from the bond (51). If we con-
sidered the van der Waals volume for the nitrogen and hydro-
gen nucleii as hemispheres perpendicular to the C‘N and

C-H bond respectively, the interaction between the nucleii
would be all but eliminated in the trans compound. However,
if the C-C=N angles for cis and trans acetaldoxime are ap-
proximately equal, then the interaction between the nitrogen
and hydrogen will differ only slightly in the two isomers.

That the angle of interaction as well as the distance
of interaction is important is supported by the fact that
the barrier height for the methyl group in the gauche form

of propyl fluoride is approximately the same as in the trans

100

.Gmmouoms Hmnumfi 6cm Gmmouuflc mo fiawmu
mamm3 moo cm> mo mmauw>o msa30£m mEflxooamumom mcmuu mo muduosuum

 

 

.nH mudmflm

 

 

 

101

form, though the methyl hydrogen and the fluorine in the
gauche form are only 2.5 R apart. The reason is thought
to be due to the fact that the C-H and C-F bonds are di-
rected away from each other and therefore approach each
other on the bonding side, where the van der Waals radii
are rapidly collapsing to the magnitude of the covalent
radii (52)..

In an attempt to determine the change in the barrier
as a function of the proximity of non-bonded interacting
atoms Weiss, Peak and Flygare (53) have determined the
barrier in 1,1-difluoropropylene. They showed that the
distance between the methyl hydrogen and the cis fluorine
increases in 1,1-difluoropropene as compared to cis-l-fluoro-
propene. The closest approach of methyl hydrogen and
fluorine is 2.549 R in the former and 2.490 in the latter.
If the fluorine and methyl hydrogen are in the range of
repulsive interaction (less than rmin in the Morse curve),
then the barrier should be very sensitive to small changes
in r(H-F) and should actually increase in the difluoro
compound when compared with cis—l-fluorOprOpene. The bar-
rier in 1,1-difluoropropene is 1250 cal/mole which is 200
cal/mole higher than the barrier for cis-1-f1uoropropene.
The increase of 200 cal/mole is accounted for by the fluoro
substitution in the trans-position which produces an in-
crease of 175 cal/mole if the barriers for prOpene and
trans-l—fluoroprOpene are compared in Table XXIII. The

conclusion of Weiss et al, was that the barrier was

 

102

insenS1t1ve to small Hmethyl*F distances-and therefore
ruled out a dominant repulsive or attractive non—bonded
interaction for H -F. Either the interaction is non-

methyl
bonded and near the potential minimum or electrostatic in
nature with a low power 1/r dependence. A similar study
of 1,1-dichloropropylene Will be’more revealing if com-
pared to cis-l-chloropropylene because the barrier is

lowered much more in the chlorine compound and the van der

Waals radii overlap to a larger extend.

 

Scott and Scheraga (54), using semi-empirical methods,
have calculated barriers to internal rotation taking into
consideration non—bonded interactions and exchange inter-
actions of electrons in bonds adjacent to bonds which form
the axis of rotation. Following the suggestion of Pauling
(51) that the contribution of exchange electrons should be
constant for a class of compounds they have explained the
variation of barriers within a class asdue to a modified
Buckingham "6—exp" potential governing the interaction of
non-bonded nucleii. They choose the minimum of the poten—
tial as rmin = rw(1) + rw(2) where the values of the van
der Waals radii, rw, are from Bondi (50). In some cases
the non-bonded interaction is as much as 4 kcal/mole ac-
cording to their calculations. For example in CE3-CF3 the

contribution from the (6-exp) potential function is 1 kcal/

mole and in CF3-CC13 it is 4.8 kcal/mole.

103

In an effort to include the non-spherical nature of
the electron distribution about the interacting nucleii
they allowed rmin to increase for interactions whose line
of action lies at a small angle to the bond directions.
This approach produced no significant difference for the
cases studied.

In a radically new approach to the theory of internal
rotation Parr and co-workers (55—59) have based their study
on the integral Hellmann-Feynman theory. They have shown
that the barrier in ethane is due to proton-proton electro-
static repulsion and to a threefold electron density in
regions near the protons. In a semi-empirical extension
of the theory Lowe (60) has presented electrostatic models
of bare protons of the methyl rotor moving in the potential
field created by the non-rotating part of the molecule.
Applying the method to halo ethanes and prOpylenes Lowe
takes into account inductive and resonance effects due to
the presence of a Halogen. The model is not tested on cis-
halopropenes because not enough structural data is avail-
able and the model is very sensitive to structure. In any
case Lowe has stated that non-bonded repulsive interaction
between methyl hydrogen and the cis halogen is inconsistent
with the model.

The theories presented thus far fail to pass the test
of predicting the barriers to internal rotation for various
compound. The quantum mechanical problem is complex even

for ethane (43) and semi—empirical approaches are still

104

groping for an understanding of the origin of the barrier
in terms of established chemical concepts. From the in-
creasing amount of experimental evidence general trends
become obvious but subtle differences can produce large
variations and provide the incentive for a continuing
dialogue.

Returning to acetaldoxime, as was mentioned previously,
the largest error in the determination of V3 is caused
by the uncertainty in Ia' The angles between the methyl
axis and the principal axes produce smaller errors. A
different choice of IQ in the fitting of the internal
rotation parameters will result in a different value for
the angles. The [_N=C-CH3 (cis) of 130° obtained by vary-
ing only this coordinate in the structure can be compared
with the value of 131° found for this angle in acetoxime
(48) and 126° 1 1° found in_dimethylglyoxime (67). In-
creasing the C=N-O angle in cis acetaldoxime will tend to
rotate the principal axes also and produce the same angle
of intersection between the CH3 axis and the principal
axes. The actual situation is probably with the C=N-CH3
and C=N-O angles both increased slightly over the values
in Table IX. This would not change the distance r(H e-O)

M

a significant amount from the value used above.

105

Hamilton (67) also mentions that the methyl group is
somewhat distorted in the cis position of dimethylglyoxime,
which is not surprising in view of the steric interaction
with oxygen. Of course such distortion seriously affects
the value of IQ. Fitting Ia to the transitions for cis
acetaldoxime in Chapter V resulted in a value of Id of
2.728 amu 82. It is possible that Ia is actually this
small if the methyl group is severely distorted. The other
possibility is that the perturbation technique failed for
a low barrier of 375 cal/mole. However, this was investi—
gated by performing the calculations using a free rotor
approximation in Chapter VI and it was found that the same
values were obtained for the internal rotation parameters.
It was also found that changing the shape of the barrier
by including a V6 term would not lower 10' The third al-
ternative under consideration is that some other internal
motion is contributing to the Observed trhnsitions which
has not been included in the original Hamiltonian expres-
sion. This may be a low frequency vibration of another part
of the molecule or some high amplitude bending vibrations

of the C—H bonds in the methyl group itself.

8.

10.
11.

12.

14.

15.

16.

17.

18.

19.

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TY

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