close

Вход

Забыли?

вход по аккаунту

?

Microwave spectra of certain benaldehydes and the dynamics of twofold internal rotors

код для вставкиСкачать
This thesis, having been approved by the
special Faculty Committee, is accepted
by the Graduate School of the
University of Wyoming,
in partial fulfillment of the requirements
for the degr
'
"
Dean of the Graduate School
Date--Augus_t-_2^-197_Q
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MICROWAVE SPECTRA OF CERTAIN BEHZALDEHYDES
AND THE DYNAMICS OF TWOFOLD
INTERNAL ROTORS
by
CIBRARTR
O F THE
Ramesh K. Kakar
UNIVERSITY OF WYOMING
L ARAMIE
A Thesis
Submitted to the Department
o f Physics and the Graduate School
o f the U niversity o f Wyoming in P artial
F u lfillm e n t o f Requirements fo r the Degree of
Doctor o f Philosophy
U niversity o f Wyoming
Laramie, Wyoming
August, 1970
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI N um ber: D P 14800
INFORMATION TO USERS
The quality of this reproduction is dependent upon the quality of the copy
submitted. Broken or indistinct print, colored or poor quality illustrations and
photographs, print bleed-through, substandard margins, and im proper
alignm ent can adversely affect reproduction.
In the unlikely event that the author did not send a complete m anuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
®
UMI
UMI Microform DP14800
Copyright 2007 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Ml 48106-1346
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ACKNOWLEDGMENTS
I would lik e to express my deepest appreciation to Professor Edgar
A. Rinehart fo r his patien t supervision and guidance. The frie n d ly
atmosphere created by him in the Microwave Spectroscopy Laboratory made
the task o f th is work an enjoyable one, I am p a rtic u la rly gratefu l to
him fo r supporting me from his grant, throughout my graduate career.
To Dr. C. R. Quade I owe additional thanks. We had soma in terestin g
di scussions about th is work during the e a rly part of my graduate career.
I ad d itio n , I would lik e to acknowledge the help provided by the
Hewlett Packard Company, Palo A lto , C a lifo rn ia , by granting permission
to use th e ir K-band Microwave Spectrometer fo r part o f th is work.
F in a lly , my wife Sneh did more than her part in the completion of
th is work. She displayed exceptional patience and tolerance, especially
during the la s t few months o f th is work. To her I give my thanks and
unending love.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
. . . . . . .
LIST OF FIGURES
.
.
.
.
.
........................................................
.
.
LIST OF TABLES................................ ....
.
.
.
.
.
.
.
,
.
.
. .
. .
.
.
.
. .
. .
.
.
.
.
ii
v
.
v ii
CHAPTER
I.
II.
INTRODUCTION
. .
.
. .
MICROWAVE SPECTRA OF BENZALDEHYDES
Experimental
.
.
.
.
. . .
.
. .
.
.
Deuterated Benzaldehyde
III.
.
.
1
. ................................6
. .
Id e n tific a tio n o f Rotational Transitions .
Benzaldehyde
.
.
.
.
.
.
6
.
.
.
7
............................ 12
..................................... 31
Para Fluorobenzaldehyde
.
. .
Para Chiorobenzaldehyde
.
.
.
. ............................... 32
.
.
.
.
.
.
.
44
THE DYNAMICS OF THE TWOFOLD INTERNAL R O T O R ........................... 60
Potential E n e rg y .............................
63
Kinetic Energy and Coordinate Transformations
fo r IAM
...........................65
Hamiltonian Operator and the Basis o f Representation . 73
Results in the High B arrier Approximation
.
.
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.
75
Page
IV .
V.
STRUCTURE OF THE BENZALDEHYDE FRAMEWORK
.
.
. .7 8
A N A L Y SIS OF SPECTRA WITH R ES PEC T TO THE THEORY .
.
. . 96
Height o f the Potential B arrier
.
.
.
.
.
. . .
Calculation o f the Parameters fo r Analysis .
.
Further Developments o f Theory ........................
.
. 96
. . 100
.
.1 0 7
APPENDIX
A.
ASYMMETRIC ROTOR ENERGY LEVELS IN RIGID-ROTOR
APPROXIMATION
B.
. . .
.........................
.
.
.
.
.
109
COMPUTER PROGRAM FOR EVALUATION OF STRUCTURAL
PARAMETERS................................................................................. 115
REFERENCES................................................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
118
LIST OF FIGURES
Figure
Page
1
Low resolution fast-scan spectrum o f CgH^ CHO
2
High resolution spectrum o f a typical roational
tra n s itio n o f CgH^ CHO
.
.
..1 0
.............................
3
High resolution J = 19^-18 band o f Cl^C^H^ CKO .
4
Structure o f the monosubstituted benzene .
5
Structure of the aldehyde group .
.
.
29
.
.
.
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
. 53
.
86
92
LIST OF TABLES
Table
I
Page
Ground and f i r s t excited torsional state tran sitio n s
o f C6H5 'CH0
II
...........................................
14
Second and th ird torsional state tran sitio n s o f
C6H5 CHO
............................................................ 20
III
Ground state Q-branch tran sitio n s o f CgHg CHO •
*
IV
Rotational constants o f CgHg CHO in the ground and
*
f i r s t three excited torsional states ................................
V
22
Moments o f in e rtia and in e rtia defects of CgHg CHO
in the ground and f i r s t three torsional states
VI
*
*
*
•
•
*
«
•
•
•
•
•
•
•
•
33
Second and th ird torsional state tran sitio n s o f
CgHgCDO
IX
30
Ground and f i r s t excited torsional state tran sitio n s
o f CgH5 CDO •
V III
25
R elative in te n s itie s o f selected ground and f i r s t
torsional state tran sitio n s .................................................
V II
21
.............................
40
Rotational constants of CgHg CDO in the ground and
f i r s t three excited torsional states .
.
.
.
.
.
41
X
Moments o f in e rtia and in e rtia defects o f CgHg CDO
.
XI
R elative in te n s itie s o f selected ground and f i r s t -
»
torsional state tran sitio n s .................................... . . . .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
43
Table
Page
X II
Ground state tran sitio n s of FCgH4 CHO
.
X III
Rotational constants andmoments of in e rtia of
Ground state tran sitio n s o f Cl
C6H4 CHO
XV
.
.
.
.
.
.
.
. . .
45
...............................49
FCgH4 CHO in the ground state
XIV
.
■35
CgH4 CHO and CV
.
.
.
.
.
.
37
.
.
54
Rotational constants, moments o f in e rtia and quadrupole coupling constants o f Cl
35
CgH4 CHO and Cl
37
C6H4 C H O .............................
XVI
58
Comparison of calculated and observed ro tatio n al con­
stants A and B fo r the ground state of fluorobenzene
and c h lo ro b e n z e n e ................................................ ......
90
XVII
Structural parameters o f the aldehyde group
93
X V III
Comparison of the calculated and observed ground
. . .
state principal moments o f in e rtia for the fiv e mole­
cules
XIX
. . .
.
94
Parameters fo r analysis o f the spectra of CgHg CHO
and CgH5 CDO...................................................................................101
XX
Comparison o f prin cipal moments o f in e rtia in the
ground and f i r s t three excited torsional states o f
CcH, CHO and CcHc CDO............................ ........
6 5
6 5
XXI
Comparison o f observed and calculated parameters
o f analysis fo r CgHg CHO .
XXII
103
.
.
.
.
.
.
.
.
.
104
Comparison o f observed and calculated parameters o f
analysis fo r CgHg CDO .
,
.
,
.
.
.
.
.
,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.
105
CHAPTER I
INTRODUCTION
With it s inherent accuracy and high resolution, microwave spectro­
scopy is very well suited fo r the study o f molecular structures and cer­
ta in intram olecular interactions o f polar molecules.
This technique ap­
pears to be a useful tool fo r studying the effects of substitution in the
benzene rin g .
However, due to comparatively higher principal moments of
in e r t ia , the microwave spectra obtained fo r such molecules are rich and
d i f f i c u l t to analyze, and consequently few such molecules have been a t­
tempted.
In the present work, structural information (fo r benzaldehyde frame­
work) was obtained, and the effec ts o f torsional motion o f the aldehyde
group in benzaldehyde were studied.
This study was undertaken as a step
towards a b e tte r understanding o f the dynamics o f molecules with tv/of old
b arriers to internal ro ta tio n .
Molecules possessing internal rotors with rotational properties o f a
symmetrical top are easier to tr e a t than those possessing twofold in te r ­
nal ro to rs , and have been studied extensively.
A review a r t ic le by Lin
and Swalen^ discusses various th eo retical approaches with respect to the
problem o f the former class o f molecules.
In such molecules, the moments
o f in e rtia do not depend e x p lic itly upon the angle o f ro tation o f the
internal ro to r.
The dynamics o f internal rotation is quite well under­
stood fo r these molecules.
However, since these molecules are a ll o f a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
£
benzaldehyde spectrum
using an assumed structure fo r the molecule.
In
order to minimise errors* the structure fo r the benzaldehyde framework
was determined in the present work.
The study of the microwave ro ta tio n ­
al spectrum o f CgHg CHO, CgHg CDO, p - chlorobenzaldehyde and p - flu o ro benzaldehyde
helped in the determination of the structure.
These mole-,
cules are sim ilar in the sense th a t each has an aldehyde group attached
to one side of the benzene rin g .
obenzaldehyde and g
The chlorine and flu o rin e , in p - chlor­
- fluorobenzaldehyde respectively, in the position
para to the aldehyde group, i t is assumed, do not change the enviornment
of the aldehyde group from th at which exists in benzaldehyde.
Substitu­
tions w ill c e rta in ly produce disto rtio n s in the regular hexagonal struc­
ture of benzene.
However, i t is noticed th at such distortions produce
neg lig ib le e ffe c t on the moments of in e r tia .
These are therefore ignor­
ed and a regular hexagonal structure fo r the benzene skeleton, in ben­
zaldehyde, is assumed.
Attempts to analyze the microwave rotational spectrum o f benzalde8
hyde have been made e a r lie r .
In th is previous attempt, approximately
90 1 ines of the a-typ e, R-brarich tran sitio n s were reported.
A character­
is t ic o f the assigned lin es was the occurence of two lin e s , o rig in a lly
interpreted to be doublets, fo r each tra n s itio n .
In these measurements
i t was not possible to ascertain re la tiv e in te n s itie s within a pair of
lin es because of poor signal-to-noise ra tio and a rich spectrum with much
interference from Stark lobes of other lin e s .
At one time i t was f e l t
th at the orig in of a pair was an internal rotation doublet v/ith d i f f e r ­
ent rotational co efficie n ts fo r each torsional substate.
However, th is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3
single type, v/here the internal rotor and hindering potential have a
threefold symmetry a x is , th e ir study has not helped in pinpointing the
orig in o f the potential barriers hindering internal ro ta tio n .
I t is hop­
ed th at an understanding of the dynamical behaviour o f molecules with
twofold potential barriers to internal ro tation w ill provide a wider ba­
sis of attack toward understanding the o rig in of these potential b a rr i2
ers .
34
Burkhard and Irwin s * approach fo r solving the wave equation fo r
%
an asymmetric molecule with an asymmetric top rotor takes on a compli­
cated form and the results do not lend themselves e a s ily to the analysis
of the spectrum and the determination o f the b a rrie r o f a sp ecific molecule.
Quade and Lin's
treatment of the internal ro tation in completely
asymmetric molecules on the other hand, contains a minimum of complica­
ting features and can re a d ily be applied.
Their theory is general in
the sense th at i t is applicable to a ll molecules in which the internal
rotor is not a symmetric top.
The assumptions used in deriving the Ham­
ilto n ia n operator fo r the internal ro tation in th is treatment are ( 1) the
molecule is rig id except fo r the degree of freedom fo r internal ro tatio n
( 2) the top ( i . e . internal ro to r) and the framework ( i . e . remainder of
the molecule) each posses planes o f symmetry.
Quade
tightened the sec­
ond of the above assumptions and imposed the condition of a twofold axis
o f symmetry on one of the rig id portions o f the molecule.
This theory
fo r molecules with twofold potential barriers has been applied to the
analysis of the present data.
E a r lie r , Quade's theory was applied to the analysis of the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
was not consistent with the expected high b a rrie r to internal ro tatio n
and large reduced torsional in e r tia .
Therefore, the o rig in o f the p air
o f lin es remained uncertain fo r a number o f years.
B arriers to internal rotation of the aldehyde group in benzaldehyde
have been determined using other methods also, v iz . using nuclear mag­
netic resonance (NMR) techniques0 and from the infared spectrum*0 .
Bar­
r ie r heights, Vg, obtained in both cases were higher than those obtained
in the present microwave work.
These being 7.9 kcal/mole by the NMR
method and 6.4 kcal/mole from the d ire c t measurement o f the torsional
frequency fo r liq u id benzaldehyde in the infared region of the e le c tro ­
magnetic spectrum.
In the case of the NMR technique, b a rrie r heights in
the range of 5 to 20 kcal/mole may be determined by measuring the widths
of resonance lin es as functions of temperature.
However the time scale
is such th a t, i f the b a rrie r height is less th at about 5 kcal/m ole, the
internal rotation appears to be free ^ .
This e ffe c t is possibly respon­
sib le in introducing a large source of error in the measured values near
the 5 kcal/mole mark.
Fately e t . a l d e t e r m i n e d the torsional frequency
in the infrared region fo r benzaldehyde both in the liq u id and in the
gaseous s tate .
Their results emphasized th at i f one is to deduce a to r ­
sional b a rrie r which is due only to forces within the molecule, i t is
absolutely necessary to make observations on gaseous samples.
S trik in g ly
d iffe re n t results were obtained in 1iquids and in vapors due to the in ­
fluence o f neighboring molecules.
133 c m "\
Thus they obtained the results
= 6,69 kcal/mole in the liq u id phase and v
4.92 kcal/mole fo r the gas phase of
CHO.
*
= 111 cm""*,
These investigators also
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
=
5
report
= 104.5 c n f \ Vg. =. 5,13 kcal/mole fo r CgH^ CDO in the gas phase.
A more recent work
12
by th is group includes the d ire c t measurement of
torsional frequencies fo r p - chiorobenzaldehyde
and p - fluorobenzalde-
hyde in the in fra - red region.
Hanyu, B r itt and Boggs
trum of nitrosobenzene.
13
have analyzed the microwave ro tatio n al spec­
They reported a-type R-branch tran sitio n s fo r
the molecule in several torsional states.
From the torsional dependence
o f the in e rtia defect and r e la tiv e in te n s ity measurements, they were able
to estimate the b a rrie r to internal ro ta tio n , Vg = 1350 cm~^.
In many
respects the spectra o f benzaldehyde and nitrosobenzene are s im ila r.
This arise s, o f course, because both molecules are benzene d e riv a tiv e s ,
with comparable barriers to internal rotation and large reduced torsion­
al moments o f in e r tia .
Another sim ilar molecule on which extensive studies have been made
5 14
15
is phenol *
and several o f it s isotopic species .
Although phenol
has been found to have a high b a rrie r to internal ro ta tio n , the reduced
in e rtia fo r the torsional mode is small (in contrast to benzaldehyde).
The tunneling frequency appears d ire c tly in the fin e structure o f the
microwave rotational spectrum and th is has le d >to an accurate determina­
tio n of the b a rrie r to internal ro tation o f Vg = 1175 cm” *.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER
II
MICROWAVE SPECTRA OF BENZALDEHYES
•Experimental
Commercially available samples o f benzaldehyde, p - chlorobenzaldehyde and p - f 1uorobenzaldehyde were used.
The deuterated benzaldehyde
(CgHg CDO) sample was prepared by Merck, Sharp and Dohme of Canada and
was supplied with 98% D.
A ll the samples, except p . - chiorobenzaldehyde,
were vacuum transferred once, p - chiorobenzaldehyde occurs in the solid
state a t room temperature but has s u ffic ie n t vapor pressure and was used
in the solid state .
A ll measurements were made a t room temperature with the sample pres­
sure in the 3-30 mtcrr range.
A ll the data, except th at fo r p - chioro­
benzal dehyde, was taken on the Wyoming Hewlett-Packard Model RE05-8400
B Microwave Spectrometer.
Data fo r p - chiorobenzaldehyde in the range
18.0 - 26.5 GHz only was taken on a sim ilar spectrometer situated in
Palo Alto a t the Hewlett Packard Company.
The microwave source fo r th is spectrometer consists of a backwardwave o s c illa to r which is always frequency s tab ilized by phase-locking
techniques and can be swept over any part or a ll o f a frequency band
(26.5 - 40.0 GHz in the case o f the Wyoming spectrometer).
The frequency
can be read d ire c tly on a counter as the source is always locked to the
same harmonic of a reference o s c illa to r .
The spectrometer employs 33.333
kHz square wave Stark modulation^5. The Stark - modulated signals are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7
demodulated by a broad-band crystal detector.
The demodulated 33.333
kHz signal is am plified, phase detected and can be displayed on a meter,
recorder or oscilloscope.
In the present case, chart display was used
fo r observing tra n s itio n frequencies.
With th is arrangement i t was pos­
sib le to measure frequencies with an accuracy o f b etter than ± 0.05 MHz
with this spectrometer, i f care was taken to ensure th at no d isto rtio n
resulted from sweep speeds.
The spectrometer also has a c a lib ra tio n arm
fo r more accurate re la tiv e in ten sity measurements.
This arm provides a
calibrated signal th at is compared to the signal from the tra n s itio n
in the molecule.
This is accomplished by passing microwave power through
a microwave modulator, which modulates the power a t a 33.333 kHz ra te .
This small amount of modulated microwave power, which can be adjusted
in phase and amplitude, is added back into the Stark c e ll.
The signal
from the Signal C alibrator is then compared to the signal from the sam­
p le.
Ident i fic a tio n of Rotational Transitions
The microwave rotational spectrum fo r each of the molecules stud­
ied in th is work, consists of a large number o f closely spaced high J
lin es whose weakness prevents th e ir id e n tific a tio n by means o f the Stark
e ffe c t.
Therefore fo r the purpose of assigning and id en tifyin g the
lin es and to obtain an idea of how the spectra of these benzaldehydes
would appear an approximate t r ia l structure was assumed fo r each one o f
them.
C-F anc C-Cl bond distances used in th is structure fo r p - flu o r -
obenzaldehyde and p - chiorobenzaldehyde were calculated from the B ro tation al spectra o f fluorobenzene
17
and chlorobenzene
18
respectively.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R
These were calculated to be C~F = 1 .300
and C-Cl - 1.713
R.
Structur­
al parameters from other molecules such as benzene derivatives and acetaldehyde were the basis of choice of t r i a l structure fo r the benzalde­
hyde framework.
Thus the ring group was assumed symmetrical with the
bond distances C-C = 1.397
R
and C-H = 1.084
R.
The C-C bond connecting
the aldehyde group was assumed to be 1.480 R with C-0 = 1,211 R, C-H =
1.193
R,
/ CCH » 124° 6 ' , and / CCO = 123° 2 3 '.
This assumed, structure
was accurate enough to assist in the assignment of the spectra.
I t was now possible to predict general features of the rotational
spectrum fo r any one of these molecules.
The in e rtia tensor
19
with re ­
spect to the center of mass o f the molecule was diagonalised to y ie ld
approximate values fo r I . I , , I„ , the principal moments of in e r tia .
a
D
C
And
since the principal moments of in e rtia are inversely proportional to the
20
rotational constants ' , an approximate set of rotational constants (ex­
pressed in MHz) As B, and C respectively was thus obtained.
Using the
expression fo r the energy levels of an asymmetric rotor (see Appendix A)
i.e .
ECA.B.C)
=
^(A+C)J( J + l)
+
!s(A~C')E(K)
(1)
and the appropriate selection rules fo r rotational tran sitio n s an approx­
imate spectrum fo r the molecule was calculated.
This calculation fo r a ll these benzaldehydes predicted th at they
are near prolate symmetric top molecules and from th e ir configuration
i t was observed that each has a large component of dipole moment along
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
9
the "a" axis.
These approximate calculations fu rther predicted th at a-
typ'e R-branch transitions should consist of small regions with a high
concentration of lin es separated by large regions with a few lin es in
them.
These small regions of high concentration of lin es arise due to
the s lig h t asymmetry.
However, most o f these 2J+1 lin es c a lle d , the
high K_.j lin es in the case of a near prolate symmetric top, l i e close to
the band centers giving rise to these small regions with high concentra­
tion of lin e s .
The lines s ig n ific a n tly removed from these clumps are
the low K_-j lin e s .
In these near symmetric roto rs, the band centers are
approximately (B +C) MHz apart
MHz in
thecaseof
asymmetric
21
as compared to thedifference
top.
Therefore i f the
of 2 B
sum (B + C) c a l­
culated from the t r ia l structure is very much d iffe re n t from the true
sum, the t r ia l structure w ill calculate the location of band centers
which are fa r removed form the actual locations.
To avoid th is , fa s t-
scan low-resolution spectra were taken fo r each molecule.
This c le a rly
emphasized the near-symmetric top character of these molecules and clumps
were observed in each case th at were a constant number of MHz apart (See
Figure 1 ).
The difference between these band centers being approximately
equal to the sum of the rotational constants B "and C.
The i n i t i a l in d i­
vidual values of B and C were now obtained by using the fa c t that a ll
these molecules are planar in th e ir equilibrium configurations and there­
fore
1 + 1 -
A
B “
1
C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
3 3 4 6 0 MHz
3 6 2 4 0 M Hz
3 9 0 0 0 MHz
3 0 7 0 0 MHZ
V ll m r .
II
Figure 1.
Fast-scan low-resolution spectrum of CgHg CHO.
emphasizes the near symmetric-top nature o f the molecule.
I t c le a rly
The fr e ­
quency difference between band centers helped in the calculation o f
ground-state rotation al constants fo r the molecule.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The i n i t i a l value fo r A was assumed to he the same as th at calculated from
the t r ia l structure.
Intervals fo r the high
lin es which were quite
insensitive to the t r ia l structure, fa c ilita te d assignment of these lin e s .
As an example, i f the rotational constants A,B and C are s lig h tly o ff
from th e ir true values say by 15 MHz, 2 MHz and 2 MHz respectively the
calculated tran sitio n frequencies fo r 10g ^ 9g Q and 10^ 3
tions w ill be in error by an appreciable amount.
£ trans^‘
However, the frequency
interval between the two calculated values w ill be very close to the ac­
tual interval i . e . about 35 MHz fo r CgHg CHO (compare in Table I ) .
There­
fore the problem was reduced to f i t t i n g the lines observed near a band
center with the in tervals fo r tran sitio n s calculated on the basis o f the
t r ia l set of rotation al constants.
When three lin es were assigned i t
was possible to calculate an improved set of rotational constants by
using Eq. (1) and the selection rule fo r a-type R-branch tran sitio ns (see
Appendix A).
These improved rotational constants helped to assign more
intense low K_-j lin e s , which are considerably shifted from the high K_^
series and are more sensitive to the values of rotational constants.
These therefore helped to determine more accurate rotation al constants.
The rotational spectrum of a molecule was "considered id e n tifie d in
the ground state when the following conditions were s a tis fie d :
1.
The principal moments of in e rtia calculated from the ro -
rational constants yielded a near zero value fo r the in e rtia defect
A 3
I
C
- I
d
- I, .
0
This arises because each of these molecules is plan-
ar in its equilibrium configuration.
r
The in e rtia defect w ill be s lig h tly
d iffe re n t from zero because of the zero point vib ratio ns.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
2.
The rotation al constants calculated the frequencies fo r
other observed tran sitio n s .
These conditions were s a tis fie d and the ground state tran sitio ns
were id e n tifie d in the case of a ll these benzaldehydes.
The data and
characteristics of the rotational spectrum fo r each molecule are given
below.
Benzaldehyde (CgHg CHO)
In an e a rlie r attempt to analyze the microwave rotation al spectrum
of benzaldehyde, a ch aracteristic o f the assigned lin es was the occuren­
ce o f two lin e s , o rig in a lly interpreted to be doublets.
This was, how­
ever, not consistent with the expected high b a rrier to internal rotation
and large reduced torsional in e r tia .
The o rig in of the p a ir of lin es
remained uncertain fo r a number of years.
In the present work, the data taken with a more sensitive spectro­
meter, makes i t clear th at each member of a pair of lin es has a d if f e r ­
ent in te n s ity , with one lin e o rig in atin g in the ground and the other in
the f i r s t excited torsional state
20
.
Nearly two times the o rig inal num­
ber of 1ines have been assigned, including some* fo r the second and th ird
excited torsional states.
Rotational co efficien ts have been derived
from the data fo r each torsional state fo r which lin es have been observ­
ed.
I t has also been possible to obtain an estimate of the b a rrie r hin­
dering internal
ro tation from re la tiv e in te n s ity measurements.
The rotation al spectrum of benzaldehyde was observed in the fr e ­
quency range 26.5 - 40.0 GHz.
The spectrum was found to be rich and the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
nearly 225 assignedlin es a t best represent
the region covered.
25% of the lin es observed in
Although someQ-branch lin es were observed fo r the
ground s ta te , most o f the lin es id e n tifie d were a-type R-branch tra n s i­
tio n s , J -*• J + 1,
K .j were found.
= 0.
The lin es were r e la tiv e ly weak.
resolve Stark components;
mined.
For several values of J complete series in
I t was not possible to
hence, the dipole moment has not been deter­
The transistions observed in the lower part o f the frequency re ­
gion covered (low J tra n s itio n s ) were less intense than those observed
in the upper part (high J tra n s itio n s ).
r ie s ,
For a p a rtic u la r J -> J + 1 se­
the in ten sity increased with a decrease in the value of K_j.
fo r high J and low
Only
some rotation al lin es were assigned fo r th is mole­
cule in the second and th ird excited torsional states.
However, fo r the
ground and f i r s t excited torsional states complete series were observed.
This data is reported in Tables I and I I .
A search was made fo r selected b-type Q-branch and R-branch tran ­
sitions and several Q-branch lin es were id e n tifie d fo r the ground state
of the molecule.
These are lis te d
These assigned
in Table I I I .
Q-branch lin es assisted in deriving the rotational
constants fo r the ground state of benzaldehyde.- A ll the other ro ta tio n ­
al co efficie n ts fo r higher torsional states were derived using the as­
signed lin es lis te d in Tables I and I I .
These co efficie n ts are given in
Table IV fo r each torsional s ta te . Thecalculated values fo r tran sitio n s
frequencies were obtained by using a computer program w ritten by Beudet
22
.
This program is w ritten in Fortran and o rig in a lly compiled fo r an
IBM 7090 computer.
I t was s lig h tly modified to run on PHILCQ 2100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE I
Transition Frequencies (in MHZ) of CgH^CHO
fo r the Ground and F irs t Excited Torsional States
CgHgCHO
v=Q
Transition
, ^ 99 ,0
109 ,2^ 99 , 1
v-1
Observed
Calc-Obs
Observed
Calc-Ob!
27783.55
+ .18
27808.02
+ .22
27798.02
-.1 2
27822.33
+ .07
27818.66
+ .11
27843.25
.00
109 1
1° 8 , 2"<' 98,1
98 s2
, " 97,2
107 ,4- 97,3
107 3
o
1
+ .09
6 ,3*
27851.35
*
27875.93
, ~ 96 ,4
105 ,5* 95 , 4
105 ,6* 95 ,5
27908.40
10,
4,6
28048.83
-.0 2
28073.27
.00
27983.86
1
o
CO
10A
6 , 4A
28008.21
+ .03
28620.83
-.3 7
106 5
9. _
4,5
+ .03
+ .04
27906.50**
-.0 9
27932.86
-.0 8
27930.65**
29318.30
-.1 6
29337.74
+ .01
102,9^ 92,8
27079.55
-.01
27104.83
-.0 9
101»9- 91 ,8
28112.57
-.0 2
28131.90*
•
° , ^ 94,6
103 ,7X
' 93,6
103 ,8^ 93 ,7
102 ,8* 92 ,7
1 4 7
27906.50**
27930.65**
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
FABLE I (Continued)
C6H5CHO_
v = 0
Trans1tion
Observed
v = 1
Calc-Obs
Observed
Calc-Obs
10.1 ^10io ,o ^
30560.40
+ .41
30587.23**
30574.28
-.0 5
30601 .35
30593.17
-.0 6
30620.44**
10.2 ^1010,1 J
9.2 * 10 9,1
9.3 ^10 9,2
J
-.1 7
8.3 '*'10 8,2 ^
8.4 - 10 8,3
J
7.4 "‘1° 7,3 ^
7.5 - 10 7,4
J
+ .01
6.5 "*10 6,4
5,7 ^ 10 5,6
4,7 ^ 10 4,6
4,8 ^10 4,7
3,8 ^ 10 3,7
3,9 ^ 10 3,8
2,9 ^ 10 2,8
2 ,1 0 ^ ° 2,9
l , l < f 10 1,9
l , l l " 10 1,10
0 , 11^10. 0,10
-.0 6
-.21
30691 .67
30664.57
-.1 6
30742.32
-.0 3
30769.48
-.3 5
30736.24
-.3 2
30763.15
-.4 3
30950.66
-.0 7
30977.69
-.1 5
30824.22
-.1 0
30850.74
-.0 9
31717.83
-.3 5
31744.17**
30666.76
-.0 6
30692.80
-.04.
32212.09
-.1 8
32232.76
.00
29648.34
-.01
29676.23
+ .02
30564.90
-.0 7
30586.57**
27665.78
-.0 8
27701.68
-.01
27767.70
+.23
27801.68
-.0 6
6.6 “*10 6,5
5,6 ^ 10 5,5
30647.90
30620.44**
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-.3 9
16
TABLE I (Continued)
-CHO
C6H5
v = 0
Transition
Observed
12 5,7
5,6
12 5 . 8
5,7
12 4,8 ^
4,7
12 4,9 - 11 4,8
12 3,9 *'11 3,8
12 S.IO*'11 3,9
12 2,io *'11 2,9
12 2 , l l ‘'11 2,10
12 l . l i 4-11 1,10
lC1,12-*-11 1,11
12
lC0,12-*-11 0,11
12
13
12,1
v = 1
Calc-Obs
Observed
Calc-Obs
33590.40
+.29
33620.15
-.1 9
33576.35
-.01
33605.70
-.1 8
33894.75
-.1 2
33924.55
-.4 4
33666.55
-.11
33695.75
-.5 4
34841.48
-.1 8
34869.48
-.4 9
33401.38
+ .04
33429.60
+ .09
35028.02
-.0 4
35049.71
+ .37
32187.15
+ .13
32217.77
+ .23
32952.47
+ .18
32977.17
+ .82
30092.05
+ .35
30130.48
-.01
30157.45
+ .25
30195.11
.00
36114.41
+ .71
36146.28
+ .70
36126.83
+ .6 6
36158.84
+ .51
36143.28
+ .54
36175.06
+ .60
36165.57
+ .42
36197.13**
-*-12
12,0
1312,2 ^1212,1
1311,2 <'1211,1
1311,3 - 1211,2
13
10,3 ^1210,2
13
l,510,4 ^
io
^
13 9,4 * 12 9,3
13 9,5 ^
9,4
13 8,5 " 12 8,4
36197.13**
36228.90
13 8,6 ^ 12 8,5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
+ .12
17
TABLE I (Continued)
CcHcCH0
___
Transition
E L _2___
V=1
V=0
Observed
13 7,6 * 12 7 , S ' ,
13 7.7 * 12 7.6
J
Calc-Obs
Observed
+ .26
36243.09
Calc-Obs
+ .24
36274.88
+ .23
+ .20
13 6,7 *"12 6,6
36315. 71*
36347.50*
13 6,8 * 12 6,7
36456.75
+ .09
36488.70
-.1 5
36426.84
+ .08
36458.65
-.1 8
36890.90
-.0 2
36923.44
-.5 2
36505.71
+ .10
36536.95*
37960.23
-.1 8
37988.95
-.2 0
13 3,11^12 3,10
36106.17
+ .09
36136.68
+ .08
13 2 , n " 12 2,10
37754.81
+ .08
37777.60
+ .68
13 2,12^12 2,11
34698.46
+ .29
34731.91
+ .46
35302.17
+ .28
35330.28
+ .92
32511.75
+ .39
32553.50
+ .19
32553.24
+ .31
32594.63
+ .01
38891.34
+ .97
38925.68
+ .95
38903.36
+ .95
38937.90
+ .72
13 5 ,8 - 12 5,7
13 5 ,9 ^
5 ,8
13 4 ,9 - 12 4,8
13 4.10*'12 4,9
13 3,10^12 3,9
13 1 .1 2 *12 1,11
13 1 ,1 3 -12 1,12
13 0,13^12 0,12
14
14
13.1 " 1313,0 >
13.2 " 1313,1
J
14
12,2 +13i 2, n
14
12,3 ^ 1 2 , 2 J
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
TABLE I (Continued)
CgHgCHO
Transition
V = 0
Observed
**
T411,3 ^ 13l l , 2
^
1411,4 - 1311,3
J
v = 1
Calc-Obs
Observed
Calc-Obs
38918.91
+ .86
38953.35
+ .71
38939.60
+ .57
38973.83
+.62
38967.38
+ .51
39001.69
+ .46
39006.49
+ .49
39040.93
+ .27
39064.32
+ .49
39098.63
+ .26
1410,4 ^1310,3
1410,5 - 1310,4
i
14 9,5 " 13 9,4 ^
14 9,6 ^
9,5
J
14 8,6 ^ 13 8,5 ^
14 8,7 * 13 8,6
14 7,7
J
7,6 'T
14 7,8 " 13 7,7
J
14 6,8 * 13 6,7
39155.50*
39139.43*
14 6,9
^13 6,8
14 5,9
* 13 5,8
39344.92
+ .41
39379.42
14 5 , lO^13 5,9
39286.74
+ .12
39320.87*
14
4,10^13 4,9
39946.81
+.15
14
4,11^13 4,10
39336.33
14
3 , 12**13 3,11
.00
39981.91
+ .40
+ .14
39369.44
+ .08
38777.80
+ .29
38810.40
+ .45
14 2 ,1 3 ^ 3 2,12
37185.62
+ .39
37222.00
+ .62
14 1,13^13 1,12
37638.48
+ .41
37670.37
+ .99
14 1,14^13 1,13
34927.57
+ .48
34973.12
-.0 2
14 0,14^13 0,13
34953.50
+ .38
34998.58
+ .05
«
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
TABLE I (Continued)
Cr HcCHQ
Q—5 —
-
T ra n sitio n
Observed
Calc-Obs
Observed
Calc-Obs
39978.15
+ .58
40013.98
+1.17
37340.88
+ .53
37389.90
+ .08
37356.75
+ .57
37405.58
+ .11
39752.52
+ .62
39814.70
+ .11
39762.22
+ .61
39805.20
+ .05
152 , u ‘ 142,13
151.14 ^ 141,13
151.15 ^ 141,14
150.15 " 140,14
161.16 ‘ 15X,15
160.16 ^ 150,15
*
The lin e has a f l a t peak.
th is .
The reported frequency is the mid point of
* * More than one assigned lines of nearly same frequency.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
TABLE I I
C6H5CH0 Transition Frequencies (in MHz) fo r the second
and Third Excited Torsional States
va2
Transitions
Observed
+1.03
32254.01
- 1.12
32272.44
+ .55
27753.82
+ .80
27771.74
+ .54
27851.32
+ .24
27869.36
+ .92
33964.55
- .01
33984.78
-1 .6 3
33732.65
- .21
33753.75
-1 .7 4
34905.76
+ .04
34924.74
-2 .1 3
33464.98
- .63
33485.55*
35068.06
+ .29
35091.36
+1.41
32257.08
+ .34
32279.08
+ .60
30188.54
+ .41
30208.01
+ .11
38026.16
- 1.10
38044.12
+ .25
32613.96
-1.17
32636.01
+2.44
-f-13
1,13
35040.65
+2.28
35066.15
-1 .1 1
4-14
39745.71
- 1.21
39772.01
+ .60
124 ,8 ^ 4 , 7
124 , 9 ^ 4 , 8
123 i 10"113 ,9
122, 1 0 ^ 2 , 9
122,11^112,10
<-11
1 ,1 1
13, _rt<-12_ „
3,10
3,9
^31 ,1 3 ^ 21,12
15
1,14
2,14
Cal c-ObJ
28170.38
" o .u ^ o .io
14
Observed
+ .37
1 .8
" 2 . 9 - 1 0 2 .8
1 ,1 2
Calc-Obs
28148.65
’ ° 1 .9 *
12
v=3
2,13
*
* The reported frequency is the midpoint o f a f l a t peak
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
TABLE I I I
Q-branch, b-dipole Transition Frequencies
(in MHz) fo r C6H5CHQ in the ground s ta te .
145,9 <~144,1o
152 s14^151,15
34172.59
-.2 0
27753.79
34093.51
-.0 1
32691.21
+ .2 2
32537.48
-.5 7
175,13"174,14
34311.84
+.11
}85,1^,15
34630.88
-.0 1
194 ,1 6 *193,17
205,16^204,17
•
165 j>12'<'164,13
o
34126.32
1
35037.74
OO
, 35,9 “"134,10
+.45
o
134 ,1 0 **33,11
33726.75
I
125,8 ^124,9
Calc-Obs
•
125,7
Observed
C\J
•
1
Transition
33964.55*
35766.00
+.06
* The reported frequency is the midpoint of a f l a t peak.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE IV
Rotational Coefficients o f CgH^HO
fo r Ground and Excited Torsional States
A (MHz)
5234.28
5214.93
5156.84 .
5174.05
B (MHz)
1564.2411
1564.9066
1565.1363
1566.1259
C (MHz)
1204.6988
1206.4753
1209.4460
1210.1345
k
-0.82155
[%(B+C)]v + r [%(B+C)3,
-0.82116
-0.81979
-0.82038
1.2210
1.6002*
0.8391
* The discrepancy is due to non-harmonic behavior of v * 2. The average value o f [^(B+C)] +,
-[%(B+C)1 is 1.2201 which confirms harmonic behavior o f the other torsional states.
V
ro
i\ j
computer a t the University of Wyoming.
This program calculates a com­
plete microwave spectrum fo r the frequency region o f in te re s t from the
rotation al constants and includes the calculation of such special features
as quadrupole s p littin g s , Stark effects and in te n s itie s .
For calculation
of the rotation al levels the mathematical method used is exactly th a t of
King, Hainer and Cross
23
.
(See Appendix A)
As can be seen in Tables I and I I , the agreement between the obser­
ved and calculated term values is good fo r lower J
gradually becomes poor fo r higher J tra n s itio n s .
true fo r higher K_^ tra n s itio n s .
tran sitio n s but
This is p a rtic u la rly
I t therefore indicates th at c e n trifu ­
gal d is to ritio n effects become evident fo r higher values o f J.
The
groundstate spectrum of C,Hr CHO is a good means of studying th is e ffe c t
O0
in th is molecule.
For th is s ta te , (A-C) ana the asymmetry parameter
values were obtained from the observed Q-branch lin es and the individual
rotation al constants (see Appendix A) were determined to get the best
f i t to data, with the help of the J = ll-f-10, R-branch, a type tra n s i­
tio n s .
Thus the agreement between observed and calculated term values
is best fo r J = 11^-10 series.
Centrifugal d is to rtio n causes the observ­
ed tra n s itio n frequencies to be higher than the calculated frequencies
based on the rig id rotor approximation fo r lower
J tran sitio n s
and causes
them to be lower than the calculated value fo r higher than J = 11«-10
series.
This e ffe c t increases with the excited torsional state and was
observed to be the greatest fo r the second excited state of th is mole­
cule which does not follow the harmonic pattern of a ll the other torsion­
al states.
For an adequate f i t to the data, however, corrections due to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
centrifugal d isto rtio n do not appear to be necessary and no such correc­
tions were attempted.
There are a few isolated low J tran sitio n s fo r
which the agreement between observed and calculated frequencies is poor.
No reason fo r th is has been found.
These are apparently due to an ac­
cidental perturbation of the energy levels concerned.
I f the energy levels of a rig id asymmetric roto r are expressed as
E(A, B, C)
= aA + $B + yC
where a , 3 and y are the derivatives o f energy defined as
6E
a
"
6k
’
._
B"
6E
6B *
Y
_ SE
" 6C
on
i t can be shown'
th at fo r a prolate symmetric top, a-type R-branch tran ­
sitions have a very s lig h t dependence on the A rotation al constant.
How­
ever b-type Q-branch tran sitio ns show an appriciable dependence on th is
ro tatio n al constant.
Thus a ll the three rotational constants were determined accurately
only fo r the ground state of benzaldehyde.
In a ll the excited torsional
states, B and C were determined more accurately than A since only the
a-type R-branch tran sitio ns were id e n tifie d .
The empirical rotation al co e ffic ie n ts w ere used to calculate the
e ffe c tiv e principal moments of in e rtia and the in e rtia defects which are
lis te d in Table V.
These quantities were then used to calculate the d i f -
frences \ , +] - 4 > e t c ., fo r successive torsional states.
The small,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE V
E ffe ctiv e Principal Moments o f In e r t ia * , In e rtia Defects, and Differences
fo r CgBgCHO in the Ground and Excited Torsional States
0
I A(amu-A2)
V=0
V=1
v-2
v=3
96.551
96.909
98.001
97.675
I B(amu»A2)
323.081
322.943
322.896
322.592
I c(amu*A2)
419.504
418.886
417.857
417.620
0
A(amu-A2)
-0.128
-0.966
-3.040
-2.747
• •
0.358
1.092
-0.326
0.375
0 •
-0.137
-0.047
-0.204
-0.129
•
•
-0.618
-1.029
-0.238
-0.628
•
•
-0.838
-2.074
+0.293
-0.873
*
^C^v+l“^C^v
A , ..-A
v+1
V
The NBS-NRC recommended value o f 505376 MHz-u-A2 fo r the conversion fa cto r has been used.
Average
*>
26
negative in e rtia defect fo r the ground state demonstrates th at benzaldehyde is planar in its equilibrium configuration.
The in e rtia defect can
be w ritte n as a sum of three contributions orig inating from v ib ra tio n a l,
centrifugal and electronic e ffe c ts .
A = A (v ib ) + A (cent) + A (e le c t)
The vib ratio nal contribution is usually the dominant term.
Morino
25
Oka and
have derived expressions fo r A fo r a planar molecule.
Assuming
th at any other vib ratio nal modes with which the torsional mode couple
are a t much higher frequency, Hanyu e t . a l .
13b
fo r the analysis o f the
microwave spectrum fo r nitrosobenzene showed th at
Av+i
or
~\ ~ "h/ 2u2c“t
A . - A
v+l
V
~
Ii>£
amu-ft2
Where u>t is the torsional frequency in cm"^.
Therefore using the value
-1
*
111 cm fo r the torsional frequency o f benzaldehyde obtained in the
in fra -re d
11
o2
work, Ay+-j -Ay is calculated to be -0.608 amu. A .
Thus the
more negative in e rtia defect fo r successive excited states confirms th at
the s a t e llit e lin es cbseryed do in fa c t arise from the torsional mode of
v ib ra tio n .
Also th is value is quite close to the value calculated by
Hanyu e t . a l . using the theory o f Oka and Mori no.
The r e la tiv e in te n s itie s of the ground and excited vib ratio nal
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
27
tran sitio ns (I- j/ I q) sre related by the Boltzman facto r i . e . to the r e la 1
tiv e populations in the two states. Thus
y io
=
( g ,/g 0)e sE/kT
The g's are the s ta tis tic a l weights, aE is the energy separation of the
vibrational state from the ground s ta te , k is Boltzman's constant and T
is the temperature a t which measurements are made.
ra tio of s ta tis tic a l weights is one.
In benzaldehyde the
Therefore a measurement of the r e l ­
a tiv e in te n s itie s between a rotation al lin e and its s a t e llit e gives the
torsional frequency.
Conversely, i f another vibratio nal mode has almost
the same energy displacement as one of the torsional states from the
ground s ta te , the v ib ra tio n -ro ta tio n interaction w ill give ris e to lin es
o f almost same in te n s ity in the two cases but displaced by d iffe re n t
amounts from the ground state rotation al lin e .
Since the b a rrie r to internal ro ta tio n , as determined by previous
investigators, has been found to be high, one would expect several to r­
sional states to lie below the top o f the potential w e ll.
In such a case
effects due to tunneling are n e g lig ib le , and aTh almost equal spacing fo r
the torsional energy levels is predicted.
V ib ratio n -ro tatio n interaction
then causes the tran sitio n s in excited torsional states (s a te llite s ) to
be displaced from the ground state tran sitio n s by almost equal amounts^.
The non-harmonic behaviour o f the second torsional state o f benzaldehyde
was detected because of a greater displacement of the s a te llite s than
predicted by harmonic behaviour.
The s h ift from the harmonic behavior
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
suggests a Fermi resonance
26
type o f interaction of th is state with
another vibratio nal mode o f the molecule.
For such an in teraction the
two vib ratio nal states should be nearly degenerate and have the same
symmetry.
Therefore the in te n s ity of the s a te llite lin es fo r th is "other
vibratio nal mode" should be comparable to th at fo r lin es of the second
excited torsional s tate .
There have been some Tines observed below the
a-type R-branch ground state lines which appear to be lin es o f th is
"other mode" (See Figure 2 ) .
However, a large number of lin es in these
regions with considerable interference from Stark lobes o f other lin es
makes id e n tific a tio n , using the in te n s ity o f the lin e as a gu ideline,
rather d i f f i c u l t .
Therefore, the existence o f these "other mode" lin es
has not been conclusively proved.
I t was possible to perform r e la tiv e in te n s ity measurements on a
few selected ground and f i r s t excited torsional state lin es.
These
chosen lin es were the ones th at were fa r removed from any neighboring
lin es and thus were lea st subject
to interferen ce.
Due to the ric h ­
ness of the spectrum, only a very few lin es f e l l in th is category.
resu lts o f these measurements are presented in Table V I.
The
S im ilar mea-
%
surements were not possible on the second and th ird torsional states due
to n o n a v a ila b ility o f lin es on which r e lia b le measurements could be made.
The C alibration arm of the spectometer allowed the determination o f the
torsional frequency within ± 1 , 0 cm~* fo r a given pair o f lin e s .
This
was checked by performing re la tiv e in te n s ity measurements on the p a ir a
number o f times.
saturation
28
27
The method of slope r a tio s ' using low pressures and
, and the method o f comparing unsaturated signals a t high
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
29
<u
t—
sz
o
•r-
</l
S=
R3
S■M
cn
*
<M
lO"
o
CM
CO
■a
c
n
o
5-
ra
■ &
with a number of unassigned
o
•r*
lin es
zsz
o
Lf>
DC
VO
O
*4—
O
c
3
01
0
<U
r*
>
5-
jC
o
CM
CM
UJIU
5°
. ..
nor
<U
3
•D>
■r*
s-
richness of spectrum
o
•r*
4-1
otl
is indicated
Z3
s.
4J
o
<y
cx
w
tc
F lirthPr reDroduction prohibited w ithout permission.
Reproduced with permission of the copyright owner. Further reprodu
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE VI
R elative in te n s ity measurements fo r
selected rotation al tran sitio ns o f CLH.CHO in the
O5
ground and f i r s t excited torsional states.
Transition
Rel. In t . (Ground=1.00)
Torsional Freq. (cm"^)
12, _ -1 1 , q
0.574
116.66
12i , i r 11i f i o
° ‘ 580
m - 39
13, Q -1 2 , 0
4,9
4,8
0.593
109.62
Z91Z
13, . , - 1 2 , . .
2fI l
*
0.595
~
108.92
M 3.12*” 3t 11
° - 578
1 U -94
U 0,14-130 ,13
° - 568
118-65
,50,15*140,1A
° - 585
112-64
160,16"150,15
° - 583
113- 45
CO
o
pressures were used.
31
Due to the low power available and the d if f ic u lt y
in saturating the tra n s itio n s , the results o f the la t t e r method are con­
sidered more re lia b le in th is case.
The value fo r torsional frequency
was determined to be 113.8 ± 5.0 cm"^.
The spread of approximately ±
-1
5.0 cm
is mostly due to the v a ria tio n o f r e la tiv e in te n s ity from one
p a ir of lin es to another, presumably due to interference from other lin e s .
Deuterated Benzaldehyde (C^Hg CDO)
The rotation al spectrum of deuterated benzldehyde was also observed
in the 25.5 - 40.0 GHz range.
The torsional mode corresponding to in te r ­
nal ro tatio n of the aldehyde (-CD0) group around the C-C bond, as fo r
normal benzaldehyde was expected to be the lowest frequency mode.
As
fo r normal benzaldehyde, most intense vibratio nal s a te llite s were observ­
ed a t frequencies somewhat above those o f the ground-state tra n s itio n s .
W ell-defined progressions, related both by e s s e n tia lly constant f r e ­
quency increments and by steady diminution of in te n s ity were observed.
From considerations o f in te n s ity and relationships discussed fo r normal
benzadlehyde, these fam ilies were assigned to successively excited
states o f the torsional mode.
U nlike, benzaldehyde, the second excited
state was observed a t the place predicted by the harmonic behaviour o f
the potential b a rrie r.
The close s im ila rity o f normal benzaldehyde and
th is molecule therefore stresses the fa c t th at in normal benzaldehyde
there happens to be present, another vibrational mode whose frequency is
quite close to the second torsional state frequency, and the in teraction
between these two vibrational states displaces the second torsional
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
state from the position dictated by harmonic behaviour.
Apart from th is minor d e tail the normal and deuterated benzaldehyde
spectra were found to be quite s im ila r.
Table V II gives the observed
tra n s itio n frequencies fo r the ground and f i r s t torsional states.
Where­
as second and th ird torsional state tran sitio n s are reported in Table
V III.
In th is case no Q-branch tran sitio n s were id e n tifie d .
Therefore
in a ll the rotation al constants derived from the data and reported in
Table IX the rotational constants B and C have been determined more
accurately than A.
ed in Table X.
The calculated principal moments of in e rtia are l i s t ­
I t was again possible to perform re la tiv e in te n s ity mea­
surements whose results are reported in Table X I.
Para fluorobenzaldehyde (FC H^.CHO)
This is another near prolate symmetric top benzaldehyde.
tatio n al spectrum was observed in the 26.5 - 31.0 GHz range.
I t ' s ro­
However,
in th is case the value fo r (B+C) is nearly 1797 MHz as compared to the
approximate value of 2769 MHz fo r the normal species o f benzaldehyde.
This means th a t, i f the lowest R-branch a-type series observed fo r nor­
mal benzaldehyde in the range o f frequencies cohered was J ~ 10
lowest series observed fo r p - fluorobenzaldehyde was J - 15
+14.
9 , the
This
fu rth e r means th at compared to 19 ( i . e . 2 J + l) tran sitio n s observed fo r
normal benzaldehyde in he lowest series, there were 29 tran sitio n s fo r
p - fluorobenzaldehyde in the lowest series.
Thus the p - fluoroben­
zaldehyde spectrum was a t le a s t, the ra tio of these two numbers times
( i . e . 29/19) as rich in lin e s as the normal benzaldehyde spectrum.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This
TABLE V II
Transition Frequencies (in MHz) of CgHgCDO
fo r the Ground and F ir s t Excited Torsional States
CgHgCDO
Transitio n
Observed
Calc-Obs
Observed
Calc-Obs
27340.40
+.39
27358.45
+ .45
27355.30
-.06
27372.52
-.22
27393.53
-.1 9
109 , l <i' 99 ,0
109,2^ 99 ,1
108 , 2^ 98 , 1
108,3'<' 98,2
1° 7 ,3^ 9 7,2
27376.65*
107 ,4 * 97 ,3
-.11
106,4^ 96 ,3
27409.55
106,5^ 96 ,4
105 ,5^ 9S ,4
27466.69
-.1 7
+ .48
105,6** 95 ,5
10A ,6- 94,5
10.
4 ,7
9. ,
4,6
103 , 7 - 93 ,6
103 ,8 * 93 ,7
, ^ 92 , 7
102 8
102,9^‘ 92 ,8
101,9^ 91 ,8
+ .19
27426.10
+ .13
27483.62
+ .18
27481.49
+ .45
27610.76
-.0 5
27626.66
+ .15
27542.56
-.20
27559.22
-.5 5
28191.35*
28205.22*
27459.88
+ .03
27476.66
-.21
28861.88
-.1 9
28872.88
-.6 2
26649.10
-.1 3
27649.51
-.16
26629.22*
27634.57
,00
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
TABLE V II (Continued)
C-.Hj.CD 0
6 5
T ra n sitio n
v = 0
Observed
v = 1
Calc-Obs
Observed
Calc-Obs
-.2 9
10,1 '",tl10,0 '
30073.58
+ .10
30092.80
30087.12
+ .14
30106.20**
10,2 "■1010,1 ■
9,2 ~10 9,1
9.3 ^
9,2 •
8,3 *10 8,2 '
30106.20**
30125.21
-.1 3
30153.06
+ .07
8,4 *10 8,3 7,4 - ,0 7,3 '
30134.60
-.0 7
7,5 "‘10 7,4 ■
+ .05
6,5 *10 6 ,4 '
30178.70
30257.87
5,7 ~10 5,6
30251.10*
4,6
30470.97
6,8 *10 4,7
-.1 4
30197.25
-.2 7
30276.12
-.3 4
30269.52
-.5 6
+.06
30488.47
-.0 8
30338.40
-.11
30356.22
-.1 8
3,8 *‘10 3,7
31245.60
-.4 8
31260.18
-.7 3
3,9 *10 3,8
30173.52
+ .03
30191.64
+ .20
2,9 ‘ 10 2,8
31702.72
-.01
31714.29
+ .08
2.10*10 2,9
29151.93
.00
29174.31
+ .06
30035.90
+ .34
30054.93
+ .42
27191.53
-.0 9
27222.35
.00
27285.90
+ .13
27316.52
-.0 9
V
5,6 *10 5,5
t
o
6,6 *10 6,5 -
-.0 8
1,10*10 1,9
1 .1 T 10 1,10
O.ll"10 0,10
- .2 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
TABLE V II (Continued)
CgHgCDO
T ra n sitio n
1211,1 ^
12
v = 0
Observed
Calc-Obs
«-1
1 .1 1,1 J
1210,2
10,1 ^
10,3
<-1
1 10,2 J
12 9,3
32806.32
+ .20
32826.96
+ .10
32819.32
+ .19
32839.90
+ .10
32836.94
+ .19
32857.44
+ .14
32882.16
+ .16
9,2
1
12 9 , 4 " ’
9,3
12 8 ,4
8,3
J
12
-<-1
' 8,5 1 8,4 J
12 7,5
Calc-Obs
11,0
11,2
12
Observed
32861.48**
+ .12
7,4 \
12 7,6 ^
J
12 6,6 ^
6,5 "1
12 6,7 ^
6,6
12 5 , 7 ^
5,6
12 5,8
5,7
12 4.8 ^
A ,7
12 4,9 ^
4,8
12 3,9 ^
3,8
12 3 ,10*^
3,9
12 2 , 10^
2,9
34464.40
12 2 , 11*"^
2,10
12 I ,!!**1
1,10
J
32898.54
+ .08
32918.69
+ .44
32955.96
-.1 0
+ .11
+ .29
32976.08
-.2 5
33062.97
-.0 2
33082.50
+ .02
33047.65
-.1 0
33067.25
-.0 8
33374.96
-.1 7
33393.47
-.2 2
33135.75
-.0 7
33154.93
-.0 5
34323,20
-.2 6
34337.90
-.3 7
32881.17
+ .02
+.17
34477.39
+ .22
31645.00
+.05
31670.01
+ .12
32377.41
+ .59
32398.89
+ .49
32861.48**
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
TABLE V II (Continued)
CgHgCDO
T ra n s itio n
v = 0
Observed
12 1, 1 2 ^ 1 1,11
12 o, ^ 11 0,11
v
Calc-Obs
-1
Observed
Calc-Obs
29574.33
.00
29608.32
+.11
29634.95
+ .17
29668.78
+ .09
35539.18
+ .21
35561.28
+ .37
35551.47
+ .44
35573.49
+ .63
35568.05
+ .39
35590.54**
1312,1 * 1212,0
1312,2 * 1212,1
1311,2 * 12n , i
1311,3 ‘ , 2 U , 2
1310,3 * 1210,2
1310,4 * 1210,3
13 9 ,4 * 12 9 ,3
35590.54**
35612.56
+ .39
13 9 ,5 * 12 9 ,4
13 8 ,5 * 12 8 ,4
35622.30
+ .21
35645.27*
13 3,6 '■’ 2 8,5
+ .23
13 7.6 ■"'2 7,5
35668.99
13 7,7 * 12 7,6
35690.75
+ .20
+ .79
13 6 ,7 * 12 6 ,6
35742.62
13 6 ,8 ^
+ .19
+ .23
6 .7
-.5 8
35765.11*
35886.52
-.0 2
35907.37
+ .08
13 5 ,9 * 12 5 ,8
35854.45
-.02
35875.49
-.0 2
13 4,9 * 12 4 ,8
36332.13
-.16
36351.56
-.21
35929.46
+ .03
35950.06
+ .08
13 5 ,8 ~ 12 5 ,7
13 4 , l < f 12 4,9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
TABLE V II (Continued)
C,HcCD0
6 5
Transition
13 3 ,1 0 * 12 3 ,9
13 3 ,i f 12 3,10
13 2 ,1 1 * ^ 2 2 ,1 0
13 2 ,1 2 * ^ 2 2 ,1 1
13 1 ,1 2 * 12 1 ,1 1
13 l , ^
12 1 ,1 2
13 0 ,1 3 ^ 12 0 ,1 2
v - 0
v = 1-
Observed
Calc-Obs
Observed
Calc-Obs
37391.85
-.1 2
37406.74
-.1 7
35518.37
-.1 8
35540.02
+ .25
37150.03
+ .52
37135.49*
34110.68
+ .30
34138.45
+ .40
34683.33
+ .52
34708.06
+ .65
31951,68
+ .16
31989.14
+ .01
31990.05
+ .06
32027.09
+ .12
38272.43
-.1 5
38296.50
+ .24
38284.25
+ .18
38309.00
+ .64
38299.70
+.38
38323.40
+ .54
38320.55
+ .18
38344.29
+ .23
38348.32
+ .48
38371.91
+ .57
38387.94
+ .43
38411.48
+ .42
141 3 ,1 " 131 3 ,0
141 3 ,2 ^ 31 3 ,1
141 2 ,2 4-131 2 ,1
141 2 ,3 4'13 1 2 ,2
141 1 ,3 4-131 1 ,2
141 1 ,4 4-131 1 ,3
141 0 ,4 4-131 0 ,3
141 0 ,5
1 4 9,5
<-13
1 0 ,4
<-13 n ,
9 ,4
14 9 ,6 <' 13 9 ,5
14 8 ,6 * 13 8 ,5
14 8 ,7 * 13 8 ,6
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
TABLE V II (Continued)
_______________C ^ C D O ____________________ _____
Transition
V =
Observed
146,8 - 136,7
146,9 - 136,8
145,9 +135,8
145,10^135,9
144,10^134,9
144,11^134,10
143,12*133,11
142,13^132,12
141,13*131,12
141,14^131,13
140,14-<"130,13
152,14^142,13
15l,1 4 * 14l f 13
151,15^141,14
l50,15^140,14
J
v
Calc-Obs
~1
Observed
+ .39
147,7 " 137,6 ^
147 , 8 ^ 137,7
0
38446.55
+ .29
Calc-Ob
+.34
38469.90
+.24
38540.62
+.37
38563.46
38537.87
-.0 9
38560.64*
38732.85
+ .16
38755.33
-.0 7
38692.84
+ .1 0
38670.35*
+ .48
39349.82
-.1 2
39369.79
-.0 4
38713.67
+ .12
38735.63
+ .26
38141.49
+ .42
38165.66
+ .41
36552.70
+ .40
36583.36
+ .50
36978.35
+ .63
37006.69*
34325.48
+ .20
34366.06
+ .18
34349.24
+ .16
34389.68
+ .13
38974.67
+ .64
39008.48
+.57
39278.38
+ .72
39310.35
+ .78
36696.84
+ .29
36740.71
+ .22
36711.36
+ .28
36755.15
+ .20
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
TABLE V II (Continued)
_____ JW______
Transition
v ~ 0
Observed
v =1
Calc-Obs
Observed
Calc-Obs
16, ,^ 1 5 , , t
1,16
1,15
39066.70
+.34
39113.70
+.41
16n ,,-«-15_
39075.46
+.37
39122.46
+.37
U 91 0
*
U , Xj
The lin e has a f l a t peak.
th is .
The reported frequency is the mid point of
* * More than one assigned lines of nearly same frequency.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
40
TABLE V I I I
C^CDO Transition Frequencies (in MHz)
fo r the Second and Third Excited Torsional States
y-Z
Transition
111,11*'101,1Q
123,9 ^ 3 , 8
v=3
Observed
Calc-Obs
Observed
27252.88
+.39
27281.03
+.56
34350.80
+.07
34363.92
+.38
32919,01
+.41
+.59
32899.40*
15,1,15
.*<-14.1,14
..
CM
140 ,1 4 *130,13
29641.90
-.2 0
29673.68
+.14
29701.40
+.43
29733.06
+.55
35664.95
+.32
35685.37
-.2 3
35561.22
+.14
35581.38**
34406.10
1
141,14^131,13
32439.09
CO
CM
•
133,11*‘123,10
I
138,6 **128,5
32420.18
O.
^ O .ia ^ O .ll
Calc-01
34428.98
+.29
36784.06
36797.90
. 34443.43
+.63
34466.50
+.88
-.3 6
36824.65
+.37
+.14
36840.30
- 1.02
**
* The reported frequency is the midpoint o f a f l a t peak.
**The lin e is hurried under a much stronger Tine.
calculated tra n s itio n frequency.
The number is the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE IX
Rotational Coefficients of C6H5CD0
fo r Ground and Excited Torsional States
y=0
^1
v=2
v=3
A (MHz)
5106.54
5089.81
5073.58
5056.98
8 (MHz)
1540.6353
1540.7465
1540.8127
1540.8489
C (MHz)
1183.8888
1185.5251
1187.1244
1188.6762
k
£%(B+C)3v+1 ■[%(B+C)1
-0.81811
-0.81804
-0 .8 1 79S
-0.81792
-0.8738
-0.8328
-0.7940
4»
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE X
E ffective principal moments o f in e r t ia , in e rtia defects, and differences fo r
C6H5CDQ in the ground and excited torsional states.
v=G
vsl
A ® I C* I B” I A
v=2
v=*3
I. (arnu A2 )
A
0
98.966
99.292
99.609
99.936
I B(amu A2 )
328.031
328.007
327.993
327.985
426.878
426.289
425.714
425.159
-0.119
- 1.010
- 1.888
-2.762
0.325
0.318
0.327
-0.024
-0.014
-0.008
-0.589
-0.574
-0.556
-0.891
-0.878
-0.874
I c (amu
k)
%
A (amu A )
( I a W
A
v+ 1
( i a )v
“A
<
V
4*
ro
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE XI
R elative in te n s ity measurements fo r selected
rotation al tran sitio ns of C6'H5CD0 in the ground
and f i r s t excited torsional states
Transition
Rel. In t . (Ground»1.00)
Torsional Freq.(cm
” 2, 10* 102. » .
° - 60°
107- 27
12.
*1 1 . ,
4 ,8
4,7
0.599
107.56
12i , i f " i , n
° - 606
105- 35
° : 591
110- 59
°-537
111-80
0.585
112.59
° * 598
107‘ 95
0.609
104.13
144 , u i3 4 ,io
H*,lf«2.12
^ 0 ,1 4 * ,30,13
150,15^140,14
.
-1
)
CO
enhanced the problem of id en tifyin g the lin es form th at which existed in
the spectrum of benzaldehyde.
were id e n tifie d .
Consequently only the ground state lin es
However, enough tran sitio n s were observed to establish
the ground state rotational constants, though, the rotational constants
B, and C, again .were more accurate than A.
The id e n tifie d tran sitio n s
are reported in Table X II and the calculated roational constants and
principal moments of in e rtia are given in Table X I I I ,
Parachlorobenzaldehyde (C135C6H4 CHO and C l37 CgH4 CHO)
The natural isotopic abundance fo r Cl
is 24.47%.
Therefore fo r a
35
is 75.53% and th at fo r Cl
sample of parachlorobenzaldehyde introduced
into the c e ll o f the spectrometer the percentage composition o f Cl
CgH4 CHO and Cl
37
37
35
CgH4 CHO w ill be in the natural abundance ra tio o f the
two chlorine isotopes.
Since the in te n s ity of the rotation al lines de-
pends upon the number of molecules o f the species present
29
, the in ten­
s ity of the lines due to the two chiorobenzaldehydes would be less than
the in te n s ity when 100% of the e ith e r species is present.
The re la tiv e
in te n s ity fo r the lines due to same tran sitio n s should be and was found
to be nearly in the ra tio o f the natural abundance o f the two isotopes.
Nuclei possess a spin agular momentum f which is represented by an
integral or h a lf integral quantum number I .
I f a nucleus possesses a
spin angular momentum greater than ^ts* i t has a non spherical charge
d is trib u tio n
30
and therefore has a nonvanishing nuclear quadrupole moment.
In such nuclei the
spin angular momentum can couple with ro tatio n al an­
gular momentum to produce a hyperfine structure in he rotation al
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE X II
Ground State Rotational Transition Frequencies
(in MHz) of Parafluorobenzaldehyde
151 2 .3 " 141 2 ,2 ' I
26978.75
-0.44
26981.88
-0.25
27000.87
-0.60
27012.80
-0.24
27031.89
-
15i 2 , 4 - 141 2 , 3 J
151 1 .4 " 141 1 ,3 ^
1511.5 "1411,4
J
15 8 ,7 - 14 8 ,6 'I
15 8 ,8 " 14 8 ,7 J
15 7 ,8 * 14 7 ,7 ' I
15 7 ,9 ^
7 ,8 J
15 6 ,9 - 14 6 ,8 ")
0.1 0
15 6 , l ( f 14 6 ,9 J
15 5 ,1 Q "14 5 ,9 >
27065.05*
15 5 , l l " 14 5 , 1 0 j
-0.18
15 4 , 1 1 ^ 4 4 ,1 0
2 7 1 4 5 *69
15 M 2 "14 4,11
27111 .72
-0.17
15 3 ,1 2 " 14 3 ,1 1
27479.57
+0.02
15 3 ,1 3 ^ 14 3 ,1 2
27074.85
+0.04
15 2 ,1 3 ^ 14 2 ,1 2
27995.26
+0.19
15 1 , U - 14 1 ,1 3
2730S * 75
-
0.22
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
TABLE X II
1 6 ,, 0
14,2
%
14,1 ^
1614.3 ^ 5U ,2
J
1613.3 "1513,2 "t
16t13,4
j / "-15,0
13,3, }
J
1612.4 ‘ 1512,3 ^
1612.5 *1512>4
J
1611>5 *15U ,4 ^
1611.6 " 15i i ,5
J
28774.14
-0.32
28776.30
-
28779.55
-0.14
28783.63
-
0.01
0.20
1610.6 ^15i 0 ,5 ^
.28788.69
+0.04
28795.90
+0.03
28806.02
+0.04
28820.92
+0.07
28844.38
0.0
28885.27
+0.17
5,12**15 5 ,1 1
28833.17
+0.14
4 ,1 2 ^ 5 4,11
28989.14
+0.15
4 ,1 3 "15 4,12
28936.17
+0.10
3 ,1 3 ^ 5 3,12
29404.70
0.00
28872.30
-0.04
1610.7 ^1510,6
16
16
16
16
16
16
16
16
16
16
16
16
16
16
J
9 ,7 ^ 15 9 ,6
9 ,8 *15 9 ,7
8 ,8 *"15 8 ,7
8,9
15 8,8
7,9 " 15 7,8
7,1Q*<‘15
7»9
6 , h T 15 6 ,9
6,11^15 6,10
5 , i f ' 15 5 ,1 0
3 ,1 4 "15 3,13
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE X II (Continued)
T ra n sitio n
Frequency
163 ,1 4 '<' 153 ,1 3
162 ,1 4 ^ 152 ,1 3
162 ,1 5 ^ 152 ,1 4
Calc-Obs
28872.30
-0.04
29864.82
- 0.01
•28286.41
-0.05
28989.68
. +0.08
27115.26
-0.06
30574.59
-0.30
30577.15
+0.10
30581.13
-0.14
30586.18
-0.36
30592.26
-0.08
30600.83
- 0.01
30612.94
+0.04
30630.90
0.00
171 4 .3 ^ 161 4 ,2 'I
1?1 4 ,4 " 161 4 ,3 J
171 3 .4 * 1 6 1 3 , 3 "l
1 7 1 3 .5 ^ 161 3 ,4 J
171 2 .5 ^ 161 2 ,4 ^
1712.6 "*1612,5 J
1711.6 ^ 611,5 'I
U l l , 7 +1611,6
J
171Q,7 ^1510,6 ^
1710,8 ^ 610,7
J
17 9,8 ^
9,7 >|
17 9 , 9 ^
9,8 J
17 8,9 "16 8,8 1
17 8, 10^16 8,9
J
17 7 ,1 0 4“15 7 ,9 ' I
17 7 , l l * 16 7 ,1 0 J
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE X I! (Continued)
Transition
Frequency
76, l l ^ 166 ,io >
7
6,12
ic
6 , 11
7512^85 ,11
it
)
J
Calc-Obs
+0*05
30658.98
n
- ° - 07
30709.15
-0.02
75, 13*"165,12
30705.28
-0.32
74 i3*164
12
30842.48
+0.07
74 , 1 4 " 154 ,1 3
3 0 7 6 2 - 12
+ 0 «27
73 ,15^ 63,14
30663.92
-0.09
72 , 1 6 " 162 ,1 5
2" 9 5 - 58
- ° * 10
71
1 6 * 161 ,1 5
30648.27
+0.11
71
1 7 ^161 16
28665.78
+0.11
70 ,17*^0,16
28742.98
+0.07
81 , 1 8 " 171 ,1 7
3 0 3 1 4 - 10
- ° * 14
80 ,1 8 ^ 170 ,1 7
30372.90
+0.11
Not completely resolved.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE X II I
Rotational and In e r tia l Coefficients
o f Parafluorobenzaldehyde
A(MHz)
5097.760
B(MHz)
976.7452
C(MHz)
819.9251
k
-0.92668
I.(amu-A2)
99.137
0
I„(amu*A2)
517.408
I c(amu*Az )
616.368
A
a
A(amu-A2)
-0.177
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
spectrum i . e . t couples with the rotation al quantum number 3 to produce
a resu ltan t to ta l angular momentum
F - t + 3
I and J remain good quantum numbers whose
F = J + I, J + I - l ..........
vector is also quantized.
| J - I|
Thus each rotation al level is s p lit into 21*1 le v e ls .
This s p l it -
ting is proportional to eqQ, the nuclear quadrupole coupling constant
31
.
Here Q is the quadrupole moment, e the u n it electronic charge and q is
the e le c tric fie ld gradient a t the position of the nucleus along the
near symmetry axis of the extranuclear charge d is trib u tio n .
A ll these
quantities are expressed in units such th a t the units fo r eqQ is MHz.
The selection rule observed fo r the R-branch tran sitio n s is aF = +1 and
th at fo r Q-branch tran sitio n s is aF = 0.
Thus each ro tatio n al tra n s itio n
is also s p lit into 21+1 tra n s itio n s .
The f i r s t orderhyparfine struc­
ture is a lin e a r function of eqQ, and
the c o e ffic ie n t o f eQq is a lin e a r
function of the quadrupole asymmetry parameter n.
In other words, the
s p littin g due to the quadrupole hyperfine structure is o f the form
eQq (a + 8n)
where
aand
p are constants depending on the tra n s itio n involved.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The
51
s p littin g s of the lower
lin es are usually affected by the value of
the quadrupolar asymmetry paramenter
32
which is a measure o f the asymmetry
of the f ie ld a t the nucleus.
The introduction of the nuclear spin does not change the to ta l inten­
s ity of a ro tatio n al tra n s itio n and the sum of the in te n s itie s o f the re ­
solved components is ju s t the in te n s ity o f the unsplit lin e .
Both Cl
35
and Cl
37
isotopes have a nuclear spin o f 3 /2 .
th e o ra tic a lly , each ro tatio n al tra n s itio n fo r Cl
CHO is s p lit into four hyperfine lin e s .
35
Therefore
CgH4 CHO or Cl
37
CgH4
Thus in addition to a lower in ­
te n s ity due to isotopic abundance, the quadrupolar s p littin g makes i t
s t i l l lower.
In addition to th is the (B+C) values were found to be approximately
1301 MHz and 1273 MHz fo r Cl35 CgH4 CHO and Cl37 CgH4 CHO resp ectively.
Thus the J ** 21
+ 20 a-type
R-branch band occured around 27300 MHz fo r
Cl35 CgH4 CHO and around 25700 MHz for Cl3^ CgH4 CHO.
The former number
is ju s t above the 1ower end o f the frequency region covered by the Wyo­
ming spectrometer while the la te r is below th is lower end.
There w ill
be 41 ( i . e . 2J+1) ro tatio n al tran sitio n s corresponding to th is change o f
ro tatio n al angular momentum quantum number.
These tran sitio n s th e o ra ti­
c a lly , fu rth e r s p lit into four hyperfine lin es each.
This along with the
1ower in te n s ity gives an extremely rich spectrum o f low in te n s ity lin e s .
To make the situ atio n s lig h tly b e tte r, the rotation al spectrum
was observed in the 18.0 - 26.5 GHz region with the help o f a K-band
spectrometer a t Hewlett Packard Company in. Palo A lto , C a lifo rn ia .
The
sample pressure was also increased to around 30 m -torr to increase the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
population of the molecules.
I t was found th at the s p littin g s due to
the nuclear quadrupole effe c ts increase with increasing K_-j value.
quadrupole hyperfine s p littin g s o f the higher
The
lin es a re , fo r the
accuracy involved, dependent only on eQq. Therefore the value o f the
quadrupole coupling constant can be obtained to a high degree o f accura­
cy by observing the s p littin g s in the highest
lin e s .
However, things
are complicated due to a large amount o f overlap in the region o f high
lin e s and therefore there are a lo t o f unresolvable lin es in these
regions.
See Figure 3.
into a ll four components.
In fa c t none o f the tran sitio n s was resolved
Thus there was an ea rly indication th a t the
value o f quadrupole coupling constant was r e la tiv e ly low.
The value fo r
th is constant was calculated from lin es that were s u ffic ie n tly removed
from the overlapping high K_-j regions but yet had s p littin g th a t was re ­
solvable into two components.
This value is therefore quite approximate.
The id e n tifie d tran sitio n s are reported in Table XIV and the ro tatio n al
c o e ffic ie n ts , principal moments o f in e rtia and quadrupole coupling con­
stants are reported in Table XV.
ed fo r lin es with
Since almost no s p littin g s were observ­
< 3 i t has not been possible to measure n.
The
calculated spectrum including quadrupole s p littin g s , was again obtained
22
using Beudet’ s
computer program.
I t is of in te re s t to note th a t the quadrupole coupling constant de­
termined here fo r parachlorobenzaldehyde is quite close to the value
-71.10 MHz fo r C l^ and -56.10 MHz fo r C l^ obtained by Poynter^^0 in
his study o f the microwave spectrum o f chlorobenzene.
Townes and D a ile y ^
point out th a t, since the quadrupole coupling constant is a measure of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
*
*r
•
®
in
Figure 3.
°
^
High resolution J a 19+18 bend o f C l^ CgH^ CHO.
o
trt
O
u>
The region 24721 — 735 MHz
contains nine rotation al tran sitio n s each o f which is fu rth e r s p lit into four hyperfine lin e s .
The exceptionally ric h spectrum near the band centers therefore made id e n tific a tio n o f
individual components almost impossible.
u>
54
TABLE XIV
Ground State Rotational Transitions (in MHz)
h
Of C135C6H**CHO and
C137CJLCH0
6
(a ) C135C6H1|CH0
S p littin g s
Transition
^ ^ X O
18
19
19
4 ,1 4
7 .1 2
+-18
-<-18
19 7 ,13 18
Lov;er
Calc.
W
,l
. s j
33/2
35/2
31/2
37/2
4 ,1 3
Obs.
2* g ; « | )
429.05
4 3 i!q 0 )
« 0 .9 7
23499.56 >
499.61 )
499.85 A
499.90 J
AQQ „
499 *33
4gg gg
^ 37/2
3 5 /2
24747.13 \
i 39/2
7,12 J 33/2
747.89 \
747.91 ^
748*°°
24757.68^
757. 6 9 J
7„
737.23
7 ,1 1 1
19 ft
ft ,1io2'I1
6 .1- t3^ 18 6
35/2
3 7 /2
747J5 7
1 9 6 . 1 4 ^ 8 6 ,1 3 j
39/2
I t H V
20 9 ,XX* 19 9 ,1 0 ^
8972
26039.55)
29 9 . u * « 9 . n )
“
Calc.
..1 2 ^ 8 .1 1 1
20 8,13^19 8, 1 2 !
.8 * 8 .1 !)
»*
35/2
2”
}
045! I D
747
Obs.
•
0.34
0.36
0.78
0.79
0.57
0.55
0.89
0.89
?1
'4 / . 2 1
7Ao nn
757‘ 78
^
M ° - 75
044.74
045.63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
TABLE XIV (Continued)
Obs.
183 ,1 5 ‘ 173>14
2 3 6 3 9 ' 44
+ 0 ' 11
183,16<-173,15
2349K16
- ° ‘ 17
18, ,,* 1 7 ,
24032.63
+0.01
182 ,1 7 " 172 ,1 6
2 3 2 3 3 ' 33
' ° - 16
18, ,,+ 17, , ,
X)1 > Xj Xu
23802.47
+0.08
0.09
,*<-18-
24988.08
-0.22
0.21
19- ...<-18-
24796.80
+0.10
0.20
192 , 1 7 - 182 ,1 6
2 5 3 8 9 - 40
19, ,,* 1 8 , , ,
23602.50
-0.14
0.13
19, , o^18, , ,
25073.18
-0.63
0.07
19. 1Q«-18.
23684.08
-0.26
0.08
190 ,1 9 " 180 ,1 8
2 3 8 1 6 - 05
- ° - 15
20, ,,+ 1 9 , , ,
26342.75
-0.16
0.18
20. - fi«-19-'
26101.60
+0.04
0.17
“ 2 .1 S *192 . «
25768' 42
- 0 ' 02
20, ,.+ 19,
26333.38
-0.13
94911 • “
+° - ° 7
2 5 0 2 4 - 73
+ 0 ' 21
2 ,1 6
3 ,1 6
3 ,1 /
2 ,1 5
3 ,1 5
3 ,1 6
/> , 1(5
1 , lo
1 ,1 ?
3 ,1 /
3 ,lo
1 ,1 .7
,1/
1 ,1 /
l,lo
3 ,1 6
5 ,1 /
1 , lO
200 ,2 0 * 190 ,1 9
.
Calc.-Obs,
S p littin g s
(Calculated)
Transitions
0.23
0.23
0.15
0.14
0.13
211, 21*2° 1>20
26137-92
+0-02
21O,2r 20O,2O
26234' 22
+0- 10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.07
0.11
0.07
0.08
0.07
0.07
0.06
56
TABLE XIV (Continued)
(b) C137C H CHO
'
Transition
Lower
6
'
k
Calc.
24174.26
’Vs W -1 37/2
%% 2
Y4;z
3l)
174.35
35/2
175.49
’W
1
V * /
39/2
33/2
%%
fs%
175.58
)
37/2
35/2
39/2
33/2
24177.34
" 177.40
{ & * )
178.34
178.40 )
37/2
r 188 , i o V p
35/2
39/2
1 9 * , u r 1 8 s . i i j 33/2
-« »
24181.59
™
)
181.62
182.38
J88,8? )
182.41
»
M ^
19 s
V
8 ...
-)
8 9 ,1 0 i
» M
V
V
2 ’ 8 7,11 1
. l f 18 7,12 '
37/2
1
^2
35/2
39/2
33/2
Obs.
S p littin g s
Calc.
. Obs.
’^ 98
1.32
1.40
1.06
1.06
0.82
0.78
0.62
0.65
1 7 5 - 38
177.00
178- 06
,81.40
182- 18
24187.77
8 7 l 7 S 187-60
187.78
188.38
188.25
188.39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE XIV (Continued)
Transitions
f
Obs.
Calc.-Obs
19. -,-*-183 16 3 j 15
19_3,17
. 7«-l8-3,16
24236.32
+0.04
24403.18
+0.30
19_ 17H8-
24794.75’
+0.23
192,18"182,17
23964'52
- ° ‘08
191 18^181 17
24523.58
+0.05
19,1 yJ.y
--*-18-1) lb
23179.92
+0.10
190,19"180,18
23318' 12
+°-°5
20,4.16
.,.1 9 ,4,15
„
25538.25
-0.09
20.4.17
,-^19.4 ,1,6,
25523.24
+0.03
20-
25723.75
-0.09
2°3 18^-193 17
25512.24
-0.04
20x ! l9 * 19l,1 8
26761 • "
- 0 - 11
201 , 2 0 ^ 1 , 1 9
24382' 55
- ° - 05
^11 /
3 .1 7
^ y lb
<-19- , ,
3 ,1 0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE XV
Rotational Constants, In e r tia l Coefficients and
Quadrupole Coupling Constants of Parachiorobenzaldehyde
Cl 35C6H1+CH0
Cl 37C6H1+CH0
A(MHz)
5058.250
5056.233
B(MHz)
691.9960
675.5286
C(MHz)
608.8328
596.0059
k
-0.96262
-0.96434
IA (amu-82)
99.911
99.951
k (arnu*A2)
730.316
748.119
Ic (amu’A2)
830.074
847.938
A (anui*A2)
-0.153
-0.132
eQq(MHz)
70.0
55.2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
the energy o f orien tation of the quadrupole nucleus r e la tiv e to the
e le c tric fie ld o f the e n tire molecule, i t is also a measure of the elec­
tron d is trib u tio n o f the bond to the atom in question.
Thus fo r a purely
covalent halogen, eqQ is nearly th at fo r the atomic halogen which has
one p electron missing from the valence s h e ll.
For 100% ionic halogens,
the p electron f i l l s the valence shell and the value of eqQ approaches
zero.
Therefore, values o f eqQ which are intermediate to the two ex­
tremes should be a measure of the amount o f ionic character in the bond.
Almost equal values of eqQ fo r chlorobenzene and parachlorobenzaldehyde
therefore indicate an almost equal ionic character fo r the C-Cl bond in
the two molecules.
In other words the presence of the aldehyde group in
one case has almost no e ffe c t in the other when no such group is present.
This
might ju s tif y to some extent the assumption made la te r in the
structural determination of the benzaldehyde framework.
This assumption
being th at the presence of chlorine or flu o rin e in a position para to the
aldehyde group in parafluoro and parachlorobenzaldehydes resp ectively,
does not a ffe c t the environment of the aldehyde group from th a t which
existed in normal benzaldehyde.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER I I I
THE DYNAMICS OF THE TWOFOLD INTERNAL ROTOR
Considerable work has been devoted to the studies of the microwave
spectra of molecules in which one group (called the top) may execute to r­
sional vibrations (also known as in ternal ro tatio n ) with respect to another
(called the frame).
te n tia l b a rrie r.
The free rotation of the group is prevented by a po­
Though the study of hindered internal rotation in mole­
cules has been a subject of in te re s t fo r nearly fo rty years and numerous
methods have been devised fo r investigating the phenomenon, i t is in te r ­
esting to note that the origin of the potential b a rrie r is not completely
understood.
There have been two general approaches to study the dynamics o f in ternal rotation in single top molecules.
< jr
Hershbach
a
, and others
34
,
/*
, the set o f principal axes of the whole molecule
is used as the coordinate system.
axis method or PAM.
In th at originated by Wilson
Nielsen
37
This is referred to as the principal
and Dennison
development o f the a lte rn a tiv e method.
33
are credited with the i n i t i a l
In this the axis about which the
top exectues internal ro tation is chosen as one of the coordinate axes.
The other two axes are fixed with respect to the framework and th e ir o r i­
entation is , in p rin c ip le , a r b itra ry , but the choice is usually determined
to some extent by the symmetry of the molecule.
the in ternal axis method or IAM.
This is referred to as
In th is representation the in teraction
terms between o v er-all and in ternal ro tatio n are considerably smaller than
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
those from PAM.
Thus these terms lend themselves more read ily to simple
perturbation treatment.
However, since the coordinate axes are, in gen­
e r a l, not principal axes, the ch ief disadvantage is th at the Hamiltonian
is complicated by the presence of the terms containing the products of
in e r tia .
In order to derive the Hamiltonian function a model can be form­
ulated from the symmetry of the molecule being studied.
The complexity
of treatment depends upon the degree of lack of symmetry o f the molecule.
A lack of symmetry in both the top and frame greatly complicates the
problem.
The molecular dynamics o f in tern al rotation is well understood fo r
molecules where the in ternal roto r has the rotation al properties of a sym­
metric top so th at the moments of in e rtia o f the e n tire molecule do not
depend e x p lic itly upon the angle o f internal ro ta tio n .
This work is sum­
marized in an excellent review a r tic le by Lin and Swalen^.
2
Quade and Lin have developed a theory fo r molecules where neither
the internal rotor nor the frame has the rotation al properties o f a sym­
metric top,
The assumptions used in the derivation of the Hamiltonian
were ( 1) the molecule is rig id except fo r the degree of freedom fo r in te r nal rotation i . e . other modes of in tern al vibration were neglected.
These
can become important i f the torsional and vibration modes have nearly the
same frequency,
( 2) the molecular z-axis was chosen to be p a ra lle l to the
axis of in ternal ro ta tio n , and (3) both the framework and top were assumed
to possess planes of symmetry.
In a la te r work Quade tightened assumption
(3) and worked out the solutions when only one of the rig id portions of
the molecule possessed a twofold axis o f symmetry.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
Using the PAM approach, i t was found that the moments of in e rtia were
no longer independent of the angle of internal ro ta tio n , the internal rotor
now being not a symmetric top.
Perturbation sums were more complicated
because of additional terms in the to rs io n a l-ro ta tio n a l in teractio n .
In the IAM approach i t was not possible to completely remove the cou­
pling between internal and o v e r-a ll angular momentum by a coordinate trans­
formation, as was possible in the case of three fold internal ro to rs .
transformation was made to reduce this coupling.
A
However, th is did not
prove very helpful because of the introduction of a-dependent terms into
the moments of in e rtia which were both periodic and nonperiodic in 2ir.
Therefore, solution to the internal ro tatio n problem using th is approach
was obtained only in the high b a rrie r approximation which is applicable
to cases where separations of the torsional energy levels are large com­
pared with the ro tatio n al energy separations.
5
Quade
compared the two approaches and showed th at PAM was best for
situations in which the potential b a rrie r is of intermediate height.
The
two methods compare favorably in the high b a rrie r approximation, but the
IAM approach is easier to use because the Fourier series fo r the a-dependence of the moments of in e rtia can be replaced by a power series.
Then
the torsional ro tatio n al in teraction sim p lifies and a straightforward
solution develops.
In benzaldehyde the separation of ro tatio n al levels is o f the order
o f -1% of the separation of torsional le v e ls , therefore, high b a rrie r ap­
proximation was used fo r the analysis of the spectra.
IAM or PAM w ill be equivalent in this approximation.
The results using
However the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
benzaldehyde data was analyzed using Quade's IAM approach, this being more
straightforw ard.
To do th is , f i r s t of a ll the form of the potential func­
tion is discussed and then the method fo r deriving the k in e tic energy in
order to obtain the Hamiltonian function for the problem is discussed.
The method of calculation of approximate eigenvalues and eigenfunctions
is covered in the succeeding sections.
Potential Energy
The origin of the potential b a rrie r not being c le a rly understood,
the only requirement th at can be imposed on the potential function is
th at i t be periodic in the re la tiv e angle a (called the angle o f internal
ro tatio n ) between the frame and the top.
I t is customary to assume the
potential b a rrie r to be of a sinusoidal shape appropriate to the f i r s t
term of a Fourier series expansion of the periodic p o te n tia l.
I f the mol­
ecule has N number of equivalent configurations within the interval of
a = 0 and
a=2-ni . e .
one complete internal revolution, the potential
function also repeats i t s e l f N times i . e . this function has N potential
minima.
The potential energy V(a) is then expanded in a Fourier series
as
V(a) =
I (a^, Cos Hkct +
Sin Hka)
k
I t is possible to place an angular reference so that a ll sine terms
are eliminated and the expansion becomes
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
V(a) s I a, Cos kfJa
k K
By a s h ift in the reference level of the potential energy, this may be
w ritten as
V
V(a) = j>
In benzaldehyde,
the
ring
(1 - Cos kNa)
group has a plane of symmetry, therefore N = 2
and V(a) becomes
, , v2 ,
V(a) = 2“ 0
“ CoS 2 a )
,
+
Vu
,
(1 “ CoS4a) + . . . . .
Experience has shown th at the above series converges rapidly and re ­
taining only the f i r s t term in the expansion serves as a very good approx­
imation.
This has the added advantage th at such a simple potential func­
tion leads to solutions fo r the torsional wave equation in terms of
Mathieu functions
39
.
For a higher accuracy, hijgher terms in the expansion
can be included and corrections to the energy levels can be calculated
using perturbation methods.
used.
In the present case only the f i r s t term was
Thus
V (a ) = jf v2 (1 - Cos 2a)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3 -1 )
65
where V2 is the b a rrie r height, intervening the potential minima.
I t is
not known from f i r s t principles and is a parameter to be determined from
the experimental data.
In the high b a rrie r approximation, the internal motion degenerates
into small o s c illa tio n s and the potential energy can be sim plified using
the harmonic o s c illa to r approximation.
Thus expanding the cosine potential
function
V(a) = V2 (a2 -'g- a1* .+
<x6 - ........ )
I f a is sm all, a ll terms beyond a2 can be dropped.
(3 -2 )
This shows th at i f
the b a rrie r is high, the lower torsional energy levels w ill be spaced at
nearly equal increments.
K inetic Energy and Coordinate Transformations fo r IAM
I n i t i a l l y , in the derivation of the k in e tic energy the molecular
axes are chosen to be fixed in the framework o f the molecule.
If ^ ,
ft. and a . are the position vectors of an atom if) the frame, an atom in
J
J
the top and the center o f mass o f the top respectively with respect to
the center of mass of the whole molecule as the o rig in , then the k in etic
energy of the system, trea tin g atoms as mass points can be w ritten as
2t .
.
+
j mj r *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
66
Using the relations
R. = a) X R. ,
1
1
and
tR. =oj X R.« + at X„ ->a .
vl
J
J
Where u> is the angular velo city of the overall rotation of the molecule
with respect to a set of axes
ve lo c ity
fixed in the molecule
of the toparound the axis
and o is the angular
of internal ro ta tio n ,
the expression
fo r k in e tic energy.becomes
The f i r s t two terms represent the k in e tic energy of the overall rig id
rotation of the molecule with angular ve lo c ity w, and can be w ritten as
a) •I«U
where I
is the in e rtia tensor fo r the e n tire molecule.
If I
a is
the moment o f in e rtia of the top then the th ird term can be expressed as
I a&2 .
The la s t term represents a coupling between internal and external
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
motions and on sim p lifica tio n
becomes 21; (» •& ).
a
Since z-axis is chosen
p a ra lle l to the internal rotation axis, this is equal to 2 Iaau)z .
The ex­
pression fo r k in e tic energy, therefore, becomes
2T = w+* 1 • to + 21 aa2 + 21aauiz
(3-3)
Apart from z-axis being chosen p a ra lle l to the axis of ro ta tio n , the
y-axis is chosen to be p a ra lle l to the plane of symmetry in the framework
(the ring group in this case) and x-axis is perpendicular to the plane of
symmetry with origin at the center of mass (CM) of the molecule.
The mo­
lec u la r axes remain p a ra lle l to th e ir orientation fo r a = 0 as a is allowed
to vary.
For a = 0 benzaldehyde is planar and
1°
xx
But fo r
af 0,
= 1°
+ 1°
Ay y
zz
the co efficien ts and products of in e rtia are
<V x x
3 'x x
( I CM>yy - &
-
S i" 2 «
+
51" 2 «
5
^Wzz * l zz
(Irm ).. . = " I
S in
' C M 'xy
a
a
COS
a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
* W y z “ “Xx
*W x z
* 0
(3 -4)
! a = ^CT^zz
where CT refers to the center o f top.
The e ffe c tiv e moments of in e rtia are related to these as
Ti j = ( I CM>ii
! ZZ =
^C M ^Z Z
"
ij
* ZZ
(3-5)
!a
The in e rtia tensor fo r a planar molecule with internal rotation in matrix
notation can be w ritte n as
I
o
3
i—*
^W xy
^W yz
*W zy
^Wzz
i
i
o
■ ('W y x
©
^CM^xx
The k in e tic energy according to (3 -3 ) can now be expressed as
2T = (Ip
t^ w wx
wv +' (ViCM'yy
V m) wu
LCM'xx
W jr z V z *
wy +'
(IrM),
'*CM'zz“z ~^2C
M 'x y V V
»Z
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
Components of to ta l and conjugate internal angular momentum can now be
calculated.
These are
Px = 9T/9“x = ^CM^xx “x " ^ W x y “y
Py * 3T/3a)y = ( I CM) yy wy - ( TcM^xy “ x " ^CM^yz“ z
Pz -
p
3T/a«2-
= 3T/3a
( I CM) ZZ «z - ( I CH) yz «y + ' I 0&
3 I
(o + 0)z )
I t is c le a rly seen that p couples to P as well as to P since I yz/ 0 .
J
fc
Jr fc
In the IAM fo r asymmetric molecules with a plane of symmetry p again
couples with P., and P .
y
z
these couplings.
However, Hecht and Dennison^ were able to remove
In th e ir
method, f i r s t a rotation is performed in order
to elim inate the Py p coupling and then a modified Nielsen transformation
37
is applied to remove the Pzp coupling.
In the present case, only one of the rig id portions has a plane of
syrimetry and therefore the problem is more complicated.
I t is only possi­
ble to reduce the coupling of p with P and P By applying two successive
y
^
coordinate transformations. The f i r s t transformation is a rotation through
an angle e about the x axis,
e s a tis fie s the condition th at i t is inde­
pendent o f a and is determined such that in the h ig h -b arrier lim it there
is no coupling of Py to p.
Under this transformation,
' 3 Su»
and
I ' = S IS -1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The condition th at coupling of Py and p vanish in zeroth order gives
Cos e - 1° N ,
yy
5 70
sin e = I N
x
where
H > ( U ° y ) 2 + ( I x) 2] -1 ^2
(3 -6 )
The coupling between p and P^ is removed in the zeroth - order by
transforming to a new set of molecular axes.
A suitable transformation
5
corresponds to a rotation about, the z axis in the manner
e" = e'
ip" =
<j>" =
tp‘ + r a ' ,
,
a" = a' ,
and
Cos ra + wy' Sin. ra
oj" = -to' Sin ra + toy' Cos ra
y
^
wz =
uz +
where
r = I a Cos e / I®2' = I a (NB)-i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3 -7)
71
(3-8)
B = 1° 1° - ( I ) 2
yy zz
v x'
and <j>, e,
\pare
the Euler angles which serve as the three external coor­
dinates *or fix in g the molecular axes in space.
The parameter r does the
same job as done by another parameter of sim ila r form in Nielsen tran sfo rmation, to remove the coupling o f p and
?z fo r
41
symmetrical molecules
or
in a modified Nielsen transformation in the case of asymmetric molecules
with a plane of symmetry
42
.
In this case i t is not possible
to remove a ll a-dependent coupling
o f P‘ , P ', P ', with p since r should be independent of a.
^
y
2
The k in e tic
energy a fte r the second transformation can be w ritten as
2T = u)"+ • J . w" + J d"2 + 2J
aa
aa d" • 2"
(3 -9 )
where
Ju
-
with Jaa generally referred to
+ Ju
«•>
<3- 10>
as the reduced moment of in e rtia fo r the
torsional motion given by
J°
aa - I a[1 - I aI® yyB '1]
and other 0?^ being given by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
'
(3-11)
'
The J. .(a ) contain a ll the a-dependent contributions to the moments
■J
o f in e rtia and a ll J . ^ a ) -> 0 in the lim it a -*■ 0. This can be checked
•J
from the expressions fo r J . .(a ) reported in Reference 5. These are not
*J
given here because they consist of Fourier series terms fo r the a-dependence of moment of in e r tia .
These are la te r replaced by a power series.
The power series expansion terms given la te r are more pertinent to the
present case.
A fte r these two coordinate transformations, the general a-dependence
of in e rtia coefficients is small but not completely removed.
This should
be compared to the case of asymmetric molecules possessing a plane o f
symmetry.
In these, Hecht and Dennison
40
, by applying the modified N ie l­
sen transformation were able to completely remove the a-dependence of
in e rtia c o e ffic ie n ts .
Therefore, in the present case the actual solution
of the problem is complicated by the introduction of these new a-dependent
terms some of which are not periodic in
2
tt.
Hence i t is not possible to
combine terms and sim p lify the Fourier expansions o f the moments of in ­
e r tia and ro tatio n al parameters.
I t is possible to combine them only in
the h ig h -b arrier approximation when the Fourier series fo r the a-dependence
of moments of in e rtia can be replaced by a power series.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
Hamiltonian Operator and Basis of Representation
The quantum mechanical Hamiltonian can be ea sily formulated from re ­
sults o f the previous section.
Development of the general theory has
been presented in d e ta il by Wilson e t a l . ^ fo r vibration - rotation in te r 5
action. This theory has been applied by Quade fo r the present case to
get the Hamiltonian of the general form
H = H» ♦ H» ♦ Hj ♦ Htr ,
where
HR and H° are the ro tatio n al and the torsional parts o f the Hamiltonian
resp ectively, which are independent of the a-dependent terms in the k in e tic
energy; whereas Hj contains a-dependent contributions to the k in e tic en­
ergy and H^r is the in teractio n of in tern al and overall ro ta tio n .
The
moments of in e rtia usually appear in the denominator and therefore y.j j
are introduced fo r convenience and include the Fourier series fo r the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
a-dependence of moments of in e rtia ,
If
g = iJ ^ jl
and
and
are related as follows:
y = 1 jI
t
then
g = g- y " 1 = g° +
example: yxx = y°x + y j ^
I
sin2 a
(-R)^ s i n ^ ^ a , e tc .
k=0
The convergence o f the Fourier series is determined by the ra tio
For solution of the problem using IAM, i t is most convenient to spec­
if y the ro tatio n al coordinates as the Euler angles 0, <|>,
<pand to
choose
the symmetric top representation as the basis fo r perturbation calcu latio n .
The form of the wave function is
* ^R( e ,<{>) exp (iP z^) X exp [ i ( l - r ) P za] <j>T(a)
where double primes have been dropped form the coordinates and dynamical
variables fo r convenience and exp [ i ( l - r ) P z a ] <j>j(a) is the torsional part
of the eignefunction.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Results in the H igh-Barrler Approximation
In the
pendence
h ig h -b arrier
approximation
of the moments of in e rtia may
be
the leading terms sim plify considerably.
indicated by the super-script on
(k)
the Fourier series fo r
thea-de­
replaced by apowerseries and
To the appropriate order in a
the terms
k
j(ct) then become on
expansion^
i 1- i . iw b- , k
=
->„<«)• 0 .
lN2lW-'a B'I]> V “J = °N2 *
4 2z ’
V * [" 2
J i 2a ’
*
ll
■ S ' x 8' *
■W “>
H
B‘ 2
.
’ 0
J Za < “ > ■
0-
<3- 1 3 >
and the d° elements are s t i l l given by Eqs. (3 -1 2 ).
•*
In the perturbation c a lc u la tio n , H° = HR + H° is the unperturbed
Hamiltonian and H* - Htr + H' + H®- is considered as the perturbation.
Here
H° » H j‘ + H°"
Where H®' gives that part of the torsional equation which gives a d iffe r e n tia l equation s im ila r to the extensively studied Mathieu's equation^
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
76
a fte r a change in variables
a =
.However,,
X
+
gir,
S = V2/y ° ,
b = E /y°
(3-14)
fo r S > 100..i.e . high b a rrie r approximation, theeigenfunctions
correspondingto
the Mathieu’s equation become the same as th at fo r
a
harmonic o s c illa to r .
5
Quade has calculated both the ro ta tio n al as well as the torsional
energy in th is case.
The torsional energy in the vth state is given by
Ev = Uj [(2V+DS1/ 2 -
I
(2v2+2v+l)] +
I
y |2 ) (2v2+2v+3)
(3-15)
where y° = 1/2 J °a » y |2^arises from the a-dependence in the torsional
k in e tic energy and S is called the reduced b a rrie r.
The e ffe c tiv e moments of in e r tia in the harmonic approximation were
calculated to be
( I e ff)t j =
+ 0 ^ > (2 v + l) / 2 S '/2
i i » xx, yy, zz;
(3-16)
i j * yz
and the in e r tia defect was calculated to be
Ay = 0 ^ } (2v + D / S 1/ 2
where
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3-17)
The e ffe c tiv e moments of in e rtia obtained above are not to be equated
with the e ffe c tiv e principal moments which have been derived from the data
fo r benzaldehyde.
A rotation of axes is applied to these moments fo r com­
parison with the empirical values.
A ll these ( r „ ) ,• „• and J° are to be calculated from the structure of
4 e f r 'ij
aa
the benzaldehyde framework.
P a rtic u la rly , small changes in the structure
of the aldehyde group can introduce errors in the calculated values.
That
is why i t is important th at a f a ir ly good structure fo r the benzaldehyde
framework must be known in order to compare the values o f moments of
in e rtia predicted by this one-de g re e -o f-1ntern a1-fre e dom model with the
values obtained from the data and thus determine the a p p lic a b ility of the
model.
In the next chapter the method used fo r obtaining structural
parameters is discussed and with its help a f a ir ly consistent structure
has been obtained.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER IV
STRUCTURE OF THE BENZALDEHYDE FRAMEWORK
•
For determination of the structure of a polar molecule in the vapor
phase, microwave spectroscopy is unmatched by any other technique in pre­
cis io n , power and c e rta in ty .
In p rin c ip a l, the method employed consists
of calculation of moments of in e r tia from the observed spectral frequen­
cies.
From the moments of in e r tia of several isotopic species, in te r ­
atomic distances and angles are deduced.
However, the moments o f in e rtia
and structural parameters in this case are the e ffe c tiv e or average
values fo r the vibrating molecule in the ground vib ratio nal s ta te .
Since
the structure of most asymmetric tops cannot be determined by only three
stru ctu ral parameters ( i . e . the three unequal principal moments of in e r­
t i a ) , i t is necessary to combine data from several isotopic species in
order to determine the structure of the molecule.
And since the d iffe re n t
isotopic species of the molecule w ill have d iffe re n t zero-point vib ratio n s,
the e ffe c tiv e stru ctu ral parameters w ill d iffe r , s lig h tly fo r the various
isotopic species.
I f i t were possible to observe the spectrum of the mol­
ecule in several successive excited states of each vib ratio n al mode, then
i t could have provided a means fo r extrapolating back to the hypothetical
non-vibrating state of the molecule.
However, this method is not p ra c ti­
cal f i r s t l y because the molecule has too many vib ratio nal modes and sec­
ondly due to the fa c t th at the population of molecules in upper vibratio nal
levels fo r high frequency vib ratio n s, would be fa r too small to be observed.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
Another approach would be to calculate the correction arising from the
v ib ra tio n -ro ta tio n in te ra c tio n .
Although the principles of this calcula­
tion are w ell known i t cannot be carried out in practice because the anharmonicity of the force fie ld s o f the atoms are not known.
Therefore,
the effects due to zero-point vibrations are usually ignored and the as­
sumption is made that the stru ctu ral parameters are id en tical fo r each
isotopic species.
Deviations that occur in the values o f structural para­
meters when d iffe re n t is o to p ic a lly substituted molecules are used in the
calculation can be attrib u ted mainly to these zero-point vib ratio nal
e ffe c ts .
Kraitchman
44
has developed exact solutions fo r the molecular coor­
dinates in terms of the equilibrium moments of in e r tia .
Thus in this
form ulation, the calculated stru ctu ral parameters r g correspond to posi­
tions the atoms would occupy i f they were at rest in the molecule.
This
means that fo r a planar molecule the in e rtia defect fo r the r s structure
w ill be zero.
Again i t is assumed th at the value of r g does not vary
with isotopic substitution and the coordinates of the substituted atom
are determined in the principal axis system of the parent molecule.
The other and more obvious method fo r obtaining the molecular structure is the solution of the moment equations
is known as the r
o
stru ctu re.
45
.
The structure so obtained
I t is assumed th at the value of r„ and
o
orientation of the principal axes is not affected by isotopic su bstitu tion .
I f th is is done, fiv e relationships can be imposed on 2 N atomic coordin­
ates of a planar molecule.
These include two diagonal moment equations,
one product o f in e rtia and two conditions on the center of mass o f the
molecule.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
In the present case, only one isotopic substitution was made i . e .
substitution of H by D in C H CDO.
6 5
Also due to presence of the isotopes
C l35 and C l37 in natural chlorine, i t was possible to obtain data on two
parachlorobenzaldehyde molecules.
A ll the molecules chosen in this study
are s im ila r in the sense that each has the aldehyde group on one side of
the ring and H, F, C l35 and C l37 are substituted in turn a t the position
para to the aldehyde group.
However, the length of the bond from its
origin on the ring to the substituted atom is appreciably d iffe re n t in
each case.
Therefore, the expressions of Kraitchman cannot be used fo r
determining the coordinates o f any two atoms in the principal axis system
o f one parent molecule.
Since the o rien tation of the principal axes is d iffe re n t fo r each
o f the molecules considered the r Q method also cannot be conveniently
used.
In the method actually used, expressions, fo r each molecule, which
were equal to the sum of principal moments of in e rtia in each case were
set up.
The structural parameters
obtained by this method are same as
those calculated by the r Q method because in each case the e ffe c tiv e prin ­
cipal moments of in e rtia obtained from the ground state rotation al spec­
trum are used in the expressions.
With only fiv e molecular species studies in this work, a ll the struc­
tu ral parameters fo r the benzaldehyde framework cannot be determined.
Therefore the following sim plifying assumptions were made.
(1) The ring forms a regular hexagon in monosubstituted benzenes
as well as in these molecules.
(2) The environment of the aldehyde group and its structure is not
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
affected by the presence of H, F, C l35 or C l37 in the position para to
this group.
(3)
aldehyde.
C-F bond length is same in fluorobenzene as in parafluorobenzS im ilarly C-Cl bond length is same 'in chlorobenzene and para-
ch1orobenza1dehyde.
Admittedly, the ring w ill be distorted by varying amounts by d if f e r ­
ent. substituents.
However, the f i r s t o f the above assumptions allows the
instantaneous determination of a large number of parameters.
I t is also
found th at the 'A' ro ta tio n al constants determined from the ground state
rotation al spectrum of a ll monosubstituted benzenes have very nearly the
same value.
The 'A' ro ta tio n al constant in these cases is proportional
to the moment o f in e rtia about the axis containing the C-X bond where X
is the substituted atom.
This shows th at the variatio n in d isto rtio n in
the ring structure from molecule to molecule can be ignored as a good
approximation.
J u s tific a tio n fo r the second assumption to some extent is provided
by the analysis o f the ground state ro tatio n al spectrum of parachioro­
benza 1dehy de carried out in the present work.
The values o f quadrupole
coupling constants obtained fo r the two molecular species in th is case
are almost id en tical with the values obtained fo r the corresponding chlorobenzenes.
The quadrupole coupling constant is a measure of the ionic
bond character o f C-Cl bond in this case
.
This therefore indicates
th at the C-Cl bond has the same percentage o f ionic character in the two
cases and whether the aldehyde group is present or not a t one end o f the
molecule, does not matter.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
A survey of molecular structure data derived from microwave studies
indicates th at the length of a bond of a given type is less subject to
variatio n from molecule to molecule than bond angles i f no unusual struc­
tural features are present.
This provides ju s tific a tio n fo r the th ird
assumption.
I f m^ is the mass o f the 1 ^ atom, r^ the distance of the 1 ^ atom
from the origin and i! the u n it dyadic, the in e rtia dyadic may be w ritte n
as
19
I = | m. (r?
The diagonal
T1 - r . ^ . )
(4 -1 )
elements of 2 are knownas the moments of in e r t ia .
For a
cartesian coordinate system fix ed a t the center of mass of the molecule,
the diagonal
elements of the in e rtia tensor are
! xx = |
Where I
( 4"2a)
are formed by a permutation of x,y and z .
*
diagonal elements called products of in e rtia are given by
yy
and I
'xy =
| m1 *1 y 1
The o f f-
(4 - 2b)
The in e rtia tensor is
symmetric with 1.^ = 1^. leaving only s ix indepen­
dent tensor elements.
For a planar molecule with x-axis perpendicular to
the plane, 'I
= I X2 *- 0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
In the matrix notation the in e rtia dyadic fo r a planar molecule can
be w ritten
0
XX
0
I
0
i
yy
yz
0
I
i
,
yz
zz
The orientation of the coordinate system used to define the in e r tia l
system w ill determine the values of the moments defined above.
I t is
always possible to rotate the coordinate axes so th at the matrix is diag­
onal.
In the language of matrix algebra such a diagonalization can be
obtained by means of a s im ila rity transformation.
The rotated axes are
called the principal axes, and the corresponding diagonal elements, I x ,
ly , I z> are known as the principal moments of in e r tia .
Spectroscopic
measurements only give information as to the values of the principal mo­
ments of in e rtia with respect to axes passing through the center of mass
of the molecule.
I f the origin of the coordinate axes is chosen a t the
center of mass of the molecule and Eq. (4 -2 ) is used to compute the e le ­
ments of the in e rtia tensor, the diagonalization procedure w ill automati*
c a lly give the principal moments of in e r tia with respect, to the center
of mass.
However, i t is often desirable to s ta rt with an a rb itra ry o r i­
gin fo r the coordinate system and s t i l l make the computations with re ­
spect to the center of mass.
the p a ra lle l axis theorem.
This can e a s ily be done by making use of
To fin d the desired relatio n s we substitute
in Eq. ( 4 - 1 ) , r . = R + r . represent the distance of th is atom from the
center of mass, and ft the distance of the center of mass from the o rig in .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
Upon substitution of the center of mass relation s v iz .
* =
I mi V I mi
and
I 'V ?
= 0
into Eq. (4 -1 ) the following experssion is obtained:
where 2 ° is the in e rtia dyadic with respect to the center of mass.
Typical
elements fo r a planar molecule have the form*
(I2 V } ) {I3 mi 2i )
___
.
(4 -3)
With an a rb itra ry origin fo r the coordinate axes the d ia g o n a liza ti on
procedure w ill autom atically give the principal moments o f in e r tia with
respect to the center o f mass i f the elements of the in e rtia dyadic are
computed from Eqs. (4 -3 ).
The C-C and C-H bond lengths in the symmetrical ring and C-F and C-Cl
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
85
bond lengths were calculated from the *B* rotational constants determined
in the study of microwave rotation al spectra of fluorobenzene^ and chlorobenzene
18
.
In both cases the o rig in of the coordinate system was chosen
at the point where the carbon halogen bond o rig in ates, with z-axis coin­
ciding with the carbon-halogen bond and the y-axis perpendicular to this
axis such that both axes are in the plane of the molecule.
Bond lengths
C-C, C-H, and C-X, where X is e ith e r F or C l, are put equal to a, b and
c respectively.
See Figure 4.
In th is orientation of the axes, z is
also a principal axis as the molecule is symmetrical about i t and the
product of in e rtia I y2 vanishes.
Therefore, i f Eq
(4 -3 ) is now used to
evaluate Iy . the quantity computed w i l l be the principal moment o f in e r­
t ia about an axis p a ra lle l to the chosen y-axis but passing through the
center of mass of the molecule i . e . another principal axis.
This p rin c i­
pal moment of in e rtia is inversely proportional to the ro tatio n al con­
stant B.
Thus we have
1°
= Xc2 + (9C+9H)a2 + 2Hb2 + 6Hab - R^ - j | F>[X2c2 + (6C+6H)2a2
+ H2b2 + 2H(6C+6H)ab - 2X(6C+6H)ac - 2XHbc]
= C.F./B
where X,C,H stand fo r the isotopic weights fo r the respective atoms in
atomic mass u n its.
'3* is the rotation al constant, Mol. Wt. denotes the
molecular weight of the molecular species being considered and C.F. is the
conversion facto r fo r converting the ro tatio n al constant expressed in MHz
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
oc
a
C-C = 1.399
o<
b = C-H = 1.088 ± .001
o<
c = C-C1= 1.713 ± .001
Figure 4.
C-F = 1.300 ± .001
Structure of the Monosubstituted Benzene
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o<c
c -
87
into the corresponding moments of in e rtia in amu-A2. The NBS-NRC recomo„
mended value o f 505376 MHz. p.A2 fo r the conversion facto r has been used.
This corresponds to C12 = 12 atomic-weight scale.
Using these values for
the constants and the value of 18* from the respective ground state ro ta ­
tional constants, the expressions fo r I ° fo r d iffe re n t molecules give
For C6H5C135
24.052 c2 + 62.687a2 + 2.007 b2 + 4.642 ab + 48.732 ac + 0.629 be
= 320.513 arnu-X2
For C H C l37
v
5
24.980 c2 + 63.640 a2 + 2.007 b2 + 4.667 ab + 50.613 ac + 0.653 be
= 329.710 amu-X2
For C H F
15.240 c2 + 53.644 a2 + 2.005 b2 + 4.409 ab + 30.879 ac' + 0.399 be'
= 196.595 amu-A2
The solution o f these equations fo r the determination of a ,b ,c and
c' is not straightforward because of the presence of cross terms.
In
these equations, co efficien ts of b2 , ab and be are small and nearly the
same.
For the expressions fo r the two chlorobenzenes these terms were
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
converted into constants by using values of a, b and c determined by
other methods and these constants were moved to the rig h t hand side of
the expressions.
This gave two equations in a and c with only the cross
product ac and the constant present.
The constants were eliminated from
the two equations and a single equation, quadratic in a/c was obtained.
This was solved to obtain the r a tio a /c .
The values of a in terms of c
and c in terns of a were plugged in one of the two orig in al equations to
obtain approximate individual values o f a and c.
plugged in one of the I®
vv
of b.
These values were then
expressions to obtain the approximate value
Thus this procedure provided the range of values fo r the bond
lengths a, b and c.
The true bond lengths were obtained with the help
of a computer program w ritten in Fortran and compiled fo r PHILCO 2100
computer.
This program is given in Appendix B.
In th is program, the a,
b and c distances are varied by increments of O.OOlR w ithin th e ir respec­
tiv e range of values.
The left-hand sides of the three I®
J -J
expressions
fo r the three molecules are then calculated fo r a ll permutations of a, b
and c values w ithin th e ir respective ranges.
Values fo r a, b and c and
the difference between the left-h and side and the right-hand side in the
three Ie x p r e s s io n s is printed out i f this difference is less than
± 0 .1 ,
simultaneously in a ll the three cases.
fo r a, b and c.
This
narrowed the range
Their values correct to the fourth decimal place in
9
0
tiv e ranges and using the computer program once again.
As a fin a l check
Angstroms were determined by giving increments of 0.0001A in th e ir respec
the value fo r the 'A' rotational constant was calculated and compared with
the values obtained from the ro tatio n al spectrum o f the molecules.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The
89
values fo r a, b and c and th at fo r the observed and calculated ro tatio n al
constants 'A' and *B' are shown in Table XVI.
The structural parameters fo r the aldehyde group were determined with
the help of ground state rotation al constants o f the fiv e molecules studied
in the present work.
In these benzaldehydes the orientation of the two
in-plane principal axes varies from one molecule to another.
Their orien­
ta tio n and the position of the center of mass in each molecule cannot be
found unless a ll structural parameters are known.
This is why individual
rotation al constants cannot be used fo r structure determination.
A well known theorem in m atrix-algebra is th at the trace of a matrix
is in varian t under a s im ila rity transformation.
Therefore i f the in e rtia
tensor with respect to the center o f mass of the molecule is diagonalized,
the sum of the principal moments of in e rtia thus obtained w ill be ju s t the
sum of the moments of in e r tia before diagonalization.
Since the ground
state rotation al spectrum provides the values fo r A, B, and C which are
inversely proportional to the principal moments of in e r tia , the following
re la tio n is obtained:
'° x + '?y + 'zz = C' F-
A ll these benzaldehydes are planar in th e ir equilibrium configurations.
•
‘ ’
Thus
t
°
xx
=
lyy zz
i°
+
T°
2 ( I° y + I ° z ) = 505376 ( J- + 1 + 1 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
TABLE XVI
a , b, c and Comparison of Calculated and Observed
Rotational Constants A and B fo r the Ground State
of Fluorobenzene and Chlorobenzene
W
C H C l35
C H C l37
6 j
a (A)
1.3985
1.3985
1.3994
b (A)
1.0872
1.0872
1.0875
0
c (A)
1.7129
1.7129
1.0875
6 5
CALC.'A' (MHz)
5672.64
5672.64
5665.73
OBS. 'A' (MHz)
5672.95
5672.53
5663.54
CALC.' B* (MHz)
1576.77
1532.90
2570.54
OBS. 'B' (MHz)
1576.77
1532.79
2570.64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The origin
point where
rin g .
of the coordinate system in each case was chosen at the
the C-C bond connecting the aldehyde group is attached to the
The z-axis was chosen to coincide with this bond and the y-axis was
chosen perpendicular to this bond such th at both the axes li e in the plane
of the molecule.
5.
The fiv e unknown structural quantities are shown in Fig.
The expressions fo r 1°
J
and 1°
are
J
! zz = 4Cf‘a s1n 60° ) 2 + 4H[( a+b) s1n 60° ]2 + 0d2 + H' e2
- KoTTTJtT
I®
[0d-H'e]2 .
and
» X( c-*-2a) 2 + C(2a)2 + 2 C (| a )2 + 2 H (i b + | a )2 + 2C(^- a )2
•3 J
+ 2H(£ a -
\ b )2 + Cg2 + Of2 + H'e2 -
tX(c+a)
+ C(2a) + 2C(^ a) + 2H(~ b + | a) + 2C(~ a) + 2H(1 a -
j
b)
-Cg-Of-H'e]2
%
Where X stands fo r the atomic mass of e ith e r C l, F or H depending on fo r
which molecular species the moments o f in e rtia are being calculated.
Only
fo r C H CDO, atomic mass of deuterium was used fo r atoms marked H‘ .
®5
2(1® +
yy
) was equated against the sum of principal moments of in e r tia fo r each
molecule to obtain fiv e equations in fiv e unknowns.
However, again the
solution o f these equations was not straightforward because of the presence
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
92
Figure 5.
Structure of the Aldehyde Group
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
TABLE XVII
Structural Parameters of the Aldehyde Group
d = z-0
=
1.003 ± 0.001 A
e « z-H
=
0.984 ± 0.001 A
f *
y-0
=
2.146 ± 0.001 A
g =
C-C
=
1.477 ± 0.001 A
h =
y-H
=
1.999 ± 0.001 A
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE X V III
Comparison of the Calculated (from the Structure) and Observed Ground
C l35C H CHO
6 *+
C137C H CHO
6 h
FCH.CHO
6* r
C H.CHO
65
CcHcCD0
65
CALC. I A
99.702
99.749
98.800
96.800
99.176
OBS. I A
99.911
99.951
99.137
96.551
98.966
CALC. I fi
730.371
748.187
517.569
322.701
327.950
OBS. I B
730.316
748.119
517.408
323.081
328.031
CALC. I c
830.073
847.936
616.369
419.503
427.126
OBS. I c
830.074
847.938
616.368
419.504
426.878
OBS. A
-0.153
-0.132
-0.177
-0.128
-0.119
1
of the cross terms.
Therefore, a procedure sim ila r to the one employed
fo r the determination of a , b and c was used to convert a ll the cross terms
in to constants.
Structural parameters obtained fo r the aldehyde group in
the acetaldehyde study
46
proved quite helpful as t r i a l values fo r d, e , f ,
g and h in the present case.
The approximate values fo r the fiv e parameters
were then determined by the usual method o f solving simultaneous equations.
Again, the computer program given in Appendix B was used to determine the
true values of d, e, f , g, and h.
in Table X V II.
These structural parameters are reported
With the determination of these fiv e parameters, the
structure fo r each one of the fiv e molecules was known.
I t was not possible
to calculate the moments and products of in e rtia so as to obtain the in er­
t ia tensor in each case.
Diagonalization of this tensor provided a set o f
calculated principal moments of in e r t ia , which are compared with the values
obtained from the ground state ro tatio n al spectrum, in Table X V III.
The
agreement between the calculated and observed values is quite good consid­
ering the sim plifying assumptions and zero-point vibrations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER V
ANALYSIS OF SPECTRA WITH RESPECT TO THE THEORY
Only fo r benzaldehyde and deuterated benzaldehyde, was i t possible
to id e n tify lin es upto the th ird excited torsional s ta te .
Therefore,
with the structure fo r the benzaldehyde framework obtained in the pre­
vious chapter, and the appropriate expressions fo r the effe c ts o f in te r ­
nal rotation fo r molecules with two-fold barriers derived on the basis
of a model with one degree of internal freedom, described in Chapter I I I ,
i t is now possible to analyze the observed spectrum fo r these two mole­
cules.
To do th is , we f i r s t of a ll determine the height V2 o f the po­
te n tia l b a rrie r hindering internal ro ta tio n .
Height o f the Potential B arrier
Since the b a rrie r to internal ro tation as determined by previous
in vestigators, fo r these two molecules has been found to be high, one
expects several torsional states to be below the top o f the potential
w e ll.
Physically th is means th at to go from one excited torsional state
to the next higher s ta te , the torsional angle increases by a small
amount.
Therefore, i t becomes ju s tifie d to reta in only the f i r s t term
in the expansion given in Eq. (3 -2 ) o f the potential function equation
(3 -1 ).
i.e .'
V(a) = V2a2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5 -1 )
97
This allows the use of the harmonic approximation fo r the evaluation o f
the b a rrie r height Vg.
The internal motion of the aldehyde group can now be compared to
the motion o f a torsional pendulam.
The torsional pendulam consists of
a body in the form of a disk suspended by a wire attached to it s center
of mass.
The other end of the wire is securely fixed to a solid support
such th at the disk can execute torsional motions about the axis fixed by
the w ire.
In both cases, fo r small tw is ts , the restoring torque is pro­
portional to the amount o f tw is t, or the angular displacement (Hooke's
law ), so th at
t
" -ka
(5 -2 )
the constant k in the above equation depends on the properties o f the
wire in the case of the torsional pendulam and is called the torsional
constant.
For the torsional motion of the aldehyde group it s value de­
pends on the height o f the potential b a rrie r hindering internal ro ta tio n .
Higher the potential b a rrie r, greater is the torque required to produce
a torsional tw is t through an angle a .
Eq. (5 -2 ) is known as the con­
d itio n fo r angular simple harmonic motion.
The potential energy V(a) a t
any instant is given by
(5 -3 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98
Comparing ( 5 - 1 ) .and (5 -3 ) we get
k
=
2V2
Therefore by analogy to the case of the torsional pendulam, the ex­
pression fo r angular torsional frequency becomes
v
«
t
(2 V ,/J
'
2
°)h
aa#
2
or
v2
3
“T
^5”4 ^
where y? = 1/2 J 0 and J 0 is the reduced moment o f in e rtia fo r the to rT
aa
sional motion.
aa
Once the structure o f the benzaldehyde framework is de­
termined, the value of
here,
can be calculated from Eq. (3 -1 1 ).
is expressed in inverse of amu.
u n it.
As given
I t can be expressed
in cm"* by the following expression
4
=
—
8ir2c Ja a
( 5- 5)
•
The torsional frequencies fo r CgHg CHO and CgHg CDO determined from
r e la tiv e in ten sity measurements as lis te d in Tables VI and XI are 113.8
± 5.0 and 108.4 ± 4.25 cm"*, resp ectively.
Using these values in the
expression (5 -4) the value o f Vg, the b a rrie r to internal ro ta tio n is
calculated to be 4.66 ± 0.41 kcal/mole fo r CgHg CHO and 5.06 ± 0.40 k c a l/
mole fo r CgHg CDO.
On the other hand the value of the potential b a rrie r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
99
determined fo r the internal motion of the same aldehyde group in
acetaldehydef^ (CH^CHO) is approximately 1.15 kcal/mole fo r several
isotopic species studied.
S im ila rly , b a rrie r heights fo r the internal
rotation o f the -OH group in phenol
OH)
14. 1
*
(CgHg OH) and methanol
are 3.36 kcal/mole and 1.07 kcal/mole respectively,
47
(CHg
Hanyu e t . a l .
1^
report a b a rrie r height of 3.86 kcal/mole fo r the torsional motion o f
the nitroso group in nitrosobenzene.
Thus in a ll these cases the usual
b a rrie r to ro tatio n around a single bond is g reatly increased by conju­
gation o f the ro ta tin g group with the aromatic rin g .
This indicates
th a t in aromatic molecules, the bond jo in in g the ring and the internal
roto r has a p a rtia l double-bond character.
The p a rtia l double-bond is
responsible fo r an increase in the value o f k in Eq. (5 -2 ) from the v a l­
ue i t would have i f the same rotor is attached to say, the methyl group.
This increase in k leads to higher potential barriers fo r aromatic mole­
cules.
I f the aldehyde group has a methyl group attached to i t in one case
and a benzene ring in another, then the length o f the C-C bond connect­
ing the two parts of the molecule should be smaller in the la te r case as
compared to the length o f the same bond in the'form er case.
This is due
to introduction o f some stra in - a cause o f p a rtia l double bond char­
a c te r, in the la t e r case.
This has been observed to be so.
The length
of th is bond in acetaldehyde^ is 1,501 R compared to the value 1.477
observed fo r benzaldehyde in the present study.
%
This shortening o f the
C-C bond therefore serves as a confirmatory te s t, to some exten t, fo r
the explanation given above fo r r e la tiv e ly higher observed potential
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
barriers in aromatic compounds.
Calculation of the Parameters fo r Analysis
The calculated value fo r the dimension!ess b a rrie r height parameter
S = V2/ y ° f is 819.63 fo r CgHg CHO and 1065.62 fo r CgHg CDO.
This ju s t­
if ie s the use o f h ig h-barrier approximation because fo r S > 100, th is
approximation is useful fo r calculating the various m atrix elements in
5
the Hamiltonian .
The e ffe c tiv e moments o f in e rtia in the harmonic approximation are
given by expressions (3 -1 6 ), where J.*? and J . ^
IJ
Ij
expressions (3 -1 2 ) and (3-13) resp ectively.
are expressed in terms of I
■j
J,
aa
can be calculated from
These equations fo r J.? and
•J
i f, i f ,
yy
44
I , I , N and B.
ot
x
The v a l-
ues fo r these la te r quantities can be determined d ire c tly from the c a l­
culated structure using expressions (3 -4 ), (3 -5 ), (3 -6 ) and ( 3 -8 ), and
the calculated values along with the values o f S and
CHO and CgHg CDO are reported in Table XIX.
fo r a ll torsional states.
However, the
The J . . values are same
2
JA
'
*J
(
fo r both CgHg
)
values have constant in -
crements fo r each successive torsional s ta te , i f i t is assumed^that S
has the same value in each of the torsional tra n s itio n s .
This assump­
tio n is same as saying th at the torsional energy levels are evenly spac­
ed i . e . the harmonic approximation is followed.
Only th is change in
(2)
7 values w ill be reflected in the values o f e ffe c tiv e moments o f in -
' J
e r tia calculated from Eqs. (3 -1 6 ).
The e ffe c tiv e moments o f in e rtia
are not to be equated with the e ffe c tiv e principal moments which have
been derived from data and given in Tables V and X fo r CgHg CHO and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
TABLE XIX
Calculated Values of the Parameters fo r Comparison
o f the Spectra o f CgHgCHO and CgHgCDO with the Theory
CgHgCHO
Q
I (amu*A2)
a
C6H5CD°
89.36
89.36
1° (amu*A2)
419.50
426.88
1° (amu*A2)
yy
315.59
321.67
1° (amu'A2
zz
103.91
105.21
I (amu*A2)
40.44
37.62
XX
X
N(amu*A2) -1
3.1 4x l0"3
3.09x10
31157.56
32427.64
J° (amu*A2)
aa
'
8.48
10.15
y^Ccm"1)
1.99
1.66
V ^ c n r1)
1629.00
1769.30
S
819.63
1065.62
B(amu*A2) 2
*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
CgHg CDO respectively.
A rotation of axes must be applied to the mo­
ments in Eq. (3-16) fo r comparison With the empirical values.
The com-
parison o f the observed and calculated principal moments o f in e rtia is
given in Table XX fo r both molecules.
The observed and calculated v a l­
ues fo r the differences in the e ffe c tiv e principal moments o f in e r t ia ,
energy and in e rtia defect Eq. (3-17) are compared in Tables XXI and
XXII fo r CgHg CHO and CgHg CDO respectively.
physical data, two approaches were used.
For comparison with the
Calculation I assumed th a t
values fo r the torsional frequency fo r these two molecules which were
measured by infrared techniques^.
The second calculation used the
torsional energy difference determined from the r e la tiv e in te n s ity data
in th is work.
Both the approaches show c le a rly th a t there is a large
difference between the observed and calculated values o f the differences
o f these dynamical q u an titie s.
work.
This is a very s ig n ific a n t re s u lt of th is
The lack o f agreement can be a ttrib u te d to the lim ite d applica-
b i1it y o f the assumed model fo r determining the dependence o f the e ffe c ­
tiv e moments of in e r tia .
The rig id top and rig id framework model with
degree o f freedom fo r only internal ro ta tio n ignores a ll interactions
between internal torsion and other vibrations o f the molecule.
This
n a tu ra lly sim p lifies the problem considerably and requires much less
data fo r evaluation of the fewer parameters needed.
The discrepancy
between observed and calculated behavior suggests the importance o f v i ­
bration in te rn a l-ro ta tio n in te ra c tio n .
I f the torsional frequency 1s
well below the frequencies o f the other vibrations only then i t is ju s t­
if ia b le not to consider other vib ratio nal modes.
However, in benzaldehyde
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE XX
Comparison of the Observed and Calculated Principal Moments of In e rtia fo r C H CHO and C H CDO in
6
5
6
5
the Ground and F irs t Three Excited Torsional States
v = 0
v - 1
v = 2
v = 3
Obs.
Calc.
Obs.
Calc.
Obs.
Calc.
Obs.
Calc.
I A(amu*$2)
96.551
96.484
96.909
96.558
98.001
96.632
97.675
96.705"
I B(amu*$2)
323.081
323.164
322.943
323.386
322.896
323.609
322.692
323.832;.
o
I c(amu*A2}
419.504
419.352
418.886
419.056
417.857
418.760
417.620
418.463,
-0.128
-0.296
-0.996
-0.888
-3.040
-1.481
-2.747
-2.073
IA(amu*A2)
98.966
98.886
99.292
98.942
99.609
98.998
99,936
99.054
I^Camu'A2)
328.031
328.149
328.007
328.404
327.993
328.659
327.985
328.915
I c (amu-I\2)
426.878
426.725
426.289
426.414
425.714
426.103
425.159
425.792
-0.119
-0.311
-1.010
-0.933
-1.888
-1.555
-2.762
-2.176
C,H,CH0
6 5
0
A(amu*A2)
c6h 5cdo
15
o
A(amu*A2)
'
104
TABLE XXI
This table contains a comparison of observed and calculated para­
meters fo r CJ-LCHO.
6 5
Calculation I assumes th at value V„2 = 1550 cm"*1
while Calculation IT assumes the value V
2
= 1629 cm"1 from the microwave
in te n s ity measurement.
j ( 2)
zz
= 1.44
j ( 2)
yy
= 1.99 cm"1
= 7.04
= 6.08
Observed
Calculation I
Calculation I I
(I
K .-d
)
xx v + l
xx v
-0.628 amu X2
-0.319
-0.296
(I
) ^ .-(1 )
yy v + l
yy v
-0.129
+0.242
+0.223
+0.375
+0.077
+0.073
(I
ZZ
V + l
A . - A
v+l
v
ZZ
)
V
-0.873
V2
* **•
s 1/ 2
*"*
vt
113.8 cm'1
-0.638
1550a cm"1
27.9
111 cm-1
-0.592
1629 cm-1
28.6
114 cm"1
aassumed value such th a t v corresponds to d ire c tly measured in fra -re d
frequency.
T
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
105
TABLE XXII
Same as TABLE XXI but fo r C H CDO.
6
5
Calculation I assumes the value
.
V0 = 1647 cm-1 whereas Calculation I I assumes the value V„ = 1769 cm"1
2
2
from microwave in te n s ity measurements.
i - 21
=
mu
J/
1.66 cm"1
Observed
(I
(I
) ,,-(1
XX
V + l
XX
)
V
) ^ -(I
)
yy v + l
yy v
( I ) .,-(1
v zz'v+1 '
A - - A
v+l
v
2
ZZ
)
V
T
3 5.68 amu X2
Calculation I
Calculation I I
-0.573 amu A2
-0.336
-0.311
-0.015
+0.280
+0.255
+0.323
+0.056
+0.056
-0.381
-0.672
-0.622
....
1647a cm"1
1769 cm"1
s1/ 2
v
= s -94 amu ° 2
31.5
108.4 cm” 1
104.6 cm"1
32.6
108.5 cm"1
aassumed value such th at v corresponds to d ire c tly measured in fra -re d
frequency.
T
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
106
a t le a s t, some evidence has been obtained in he present work about the
presence of a vibrational mode o f very nearly the same frequency as the
second torsional mode.
Hanyu e t . a l .
13
in th e ir analysis o f the micro­
wave spectrum of nitrosobenzene obtained a simple expression fo r the
change o f in e rtia defect fo r successive torsional state (Ay+j - Ay ) from
25
the complex expressions derived by Oka and Mori no , by considering only
the torsional mode and ignoring a ll other vib ratio nal modes.
The value
fo r the change o f in e rtia defect fo r successive torsional state calcu­
lated from th is simple expression was less negative than the actual ob­
served value.
They fu rth e r showed q u a lita tiv e ly th at i f the other v i­
brational modes are also considered the in teractio n between these other
modes and the torsional mode would tend to make the change Ay+.j - Ay more
negative than the value calculated by the simple expression.
The value
fo r Ay+-j - Ay fo r both CgHg CHO and CgHg CDO was calculated in Chapter I I
using th is simple expression.
These calculated values are quite close to
the values calculated from Eq. (3 -1 7 ).
Therefore i f the analysis o f
Hanyu e t . a l . is applied to the case o f benzaldehyde, i t becomes quite
cle ar th a t certain vibrational modes are present in th is molecule whose
frequency is not much d iffe re n t from the torsional frequencies and the
in teraction between vibration and internal rotation modes can not be ig ­
nored .
Internal ro tatio n can in te ra c t with the remaining vibrational
degrees o f freedom and
molecular ro tatio n s.
these, in tu rn , can in te ra c t with the o v e r-a l1
Thus, there is an in d ire c t coupling between in tern ­
al and o v e r-a ll ro tation which is due to n o n rig id ity o f the molecule.
A q u an tita tive understanding w ill have to await fu rth e r theoretical
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
developments.
Further Developments o f Theory
The effec ts of no n rig id ity o f the two halves o f the molecule were
f i r s t studied by Kivelson*8 from a semi-empirical point of view.
He ob­
tained an approximate formula fo r the ro tatio n al tran sitio n s o f symmetric
rotors and successfully applied th is formula to CHgSiHg and CHgSiDg.
these symmetric tops, a Van Vleck
49
transformation was applied to remove
the nondiagonality in the vib ratio nal quantum number.
tend th is treatment to asymmetric rotors
ber of adjustable parameters.
For
50
An attempt to ex-
produced an unmanageable num­
For th is reason, the semiempirical ap-
proach to asymmetric rotors was abondoned fo r some tim e.
Kirtman
51
re ­
examined, modified and extended Kivelson's o rig in al approach and was
able to discuss, in d e ta il, the o rig in o f various in teractio n terms.
Over-all and internal rotation were separated in zeroth order, by means
o f the Eckart
52
and Sayvetz
53
conditions.
Even with the comprehensive
treatment of Kirtman, i t was obvious th a t, in general, the microwave
absorption frequencies o f an asymmetric roto r w ill depend upon an un­
wieldy number of empirical parameters.
Only foY two group o f tra n s i­
tio n s , the frequency formula was found to be simple enough, provided the
high b a rrie r approximation was used.
sists o f the 0Q g
■*1Q -j*
For prolate rotors one group con-
the combination sum
h[Ij
\
q->2.j ^ + 1^ ^->-2^ 2]
and the 22 ^->-32 2 tra n s itio n s . The other group corresponds to tran sitio ns
*
*
54
between the two members o f an asymmetry doublet. Quade
has extended
Kirtman's treatment to include the in teractio n of normal vibrations
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
with the hindered internal rotation in molecules with asymmetric in te r nal ro to rs.
A Van Vleck
49
transformation was again used to is o la te an
i,L
e ffe c tiv e o v e r-a ll and internal ro tation Hamiltonian fo r the V
tional s ta te .
vib ra­
The coupling between o v e r-a ll ro tation and vib ratio n and
between internal ro tation and vib ratio n were treated as a perturbation
a fte r these couplings had been reduced by applying the Sayvetz-Eckart
conditions.
However, i t was possible to apply the theory of internal
ro ta tio n -v ib ra tio n interaction only to moleucles fo r which both the top
and the framework possessed planes o f symmetry.
S p e c ific a lly the theory
was applied to the microwave spectrum o f p a r tia lly deuterated acetaldehydes.
In view of lim ited experimental data availab le no conculsions
were possible about the a p p lic a b ility o f the theory, only an encourag­
ing trend was noticed.
In the present work s u ffic ie n t data has been
obtained to te s t any future developments o f the theory to include a ll
the vibratio nal modes in molecules in which only one part o f the com­
p le te ly asymmetric molecule has a twofold axis of symmetry.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX A
ASYMMETRIC ROTOR ENERGY LEVELS IN RIGID-ROTOR APPROXIMATION
The energy of a rig id asymmetric rotor is expressed as
E< v
v
v
^
+ w
(A- 1)
where Pa , Pb , and Pc are the components of the angular momentum along the
principal in e r t ia l axes a, b, and c of the roto r with I a~ I b * I c as the
corresponding moments of in e r tia .
A=T>2/ 2 I a ,
Defining the rotation al constants
B = t i 2/ 2 I b,
C=T>2/ 2 I c
(A -l) becomes
E(A,B,C) = (A P2 + B P2 + C P2) / t\2
a
D
C
The calculation of the energy levels is greatly fa c ilita te d by the change
53
of variables proposed by Ray .
With this change of variables he obtained
the expression
E(A,B,C) =
J}-
(A-C) E(k) +
\
(A+C) J(J-H)
(A-2)
Where J is the to ta l angular momentum quantum number and E(k) is called
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
no
the reduced energy.
E(k) = E ( i , k , - i )
i . e . E(k) corresponds to the energy of a hypothetical r ig id ro to r whose
A, B and C rotation al constants are 1, k
and -1 respectively where
k is
called Ray's asymmetry parameter and is given by
i. „ 2B - A - C
K
T rn r
I f B=A then k=l and we have the oblate symmetric top lim it and i f B-C,
then
k=-l corresponds to the prolate symmetric top lim it .
With the energy level expression given in the form (A-2) the problem
reduces to determining the value o f E(k) fo r a given set of quantum num­
bers, i f the energy level corresponding to these quantum numbers is desired to be calculated.
termining E(k) in d e ta il.
King e t. a l .
23
have treated the problem fo r de­
In th e ir method, matrix elements o f the Hamil-
56
tonian are generated in the Wang
representation.
This consists of a
CJ
lin e a r combination of symmetric ro to r basis functions
%
,
\px (J,K,M)
i.e .
S(J,K,M, y ) = 2-1/2 [ /( J ,K ,M ) + H ) Y / ( J , - K , M ) ]
where y is odd or even, say 1 or 0.
For K=0 only
y
even (y = 0) exists
and
S (J,0 ,K ,0) = / ( J , 0 , M )
K and M being the projections of J on the molecule-fixed fig u re axis and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m
the space-fixed f ie ld axis respectively fo r a symmetric ro to r.
can take any of 2J + 1 possible values.
K and M
The matrix elements o f E(k) as
derived by King e t . a l. are
<JKMjE(k)|JKM> = FJ(J+1) + (G-F) k2
(A-3)
<J,K+2,M|E(k)JJKM> = H [f( J,K+1)3 x/ 2
(A-4)
where the f ( J , K+l) are given by
f( J ,n ) * f(J ,~ n ) = -J- [ J (J + l)-n (n + l)] [J (J + l) -n ( n -l) ]
(A-5)
Tabulated values of f(J ,n ) are given in Table I of Reference 23.
x ,y , and z axes re fe r to the prin cipal axes of the in e r t ia l dyadic.
I t is customary in molecular spectroscopy to designate these axes as a , b,
and c with the convention th at I fl< I b< I c.
There are n! or six ways in
which the a , b, c axes can be id e n tifie d with the x, y , z axes.
The values
of the co efficien ts F, G and H fo r the m atrix elements of E(k) depend on
the way a , b and c are id e n tifie d with x, y and z.
King e t . a l. showed
th at the rig h t handed representation, in which x is id e n tifie d with b, y
with c and z with a, in th e ir n o tatio n , is nearly diagonal fo r a near pro­
la te symmetric top molecule.
In this representations
F » 1 (k -1 )
(A-6a)
S = 1
(A-6b)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
112
H --
\
(k + l)
(A-6c)
For a given J the energy m atrix E(k) in any representation based upon
the if>x(J,K,M) can be obtained from expressions (A -3 ), (A -4 ), (A-5) and o f
type (A -6 ).
The expressions (A-6) w ill vary with the representation used
to id e n tify the two set of axes.
-J - K * J.
The matrix fo r E(k) is o f order 2J+1 since
I t may, however, be displayed as two submatrices whose indices
involve, resp ectively, only odd K's.
The transformation to a representa­
tion based upon S(J,K,M, y ) enables fu rth e r factoring of E(k) in to four
submatrices i . e .
X' E(k) X = E+ + E* + 0+ + 0"
X is of order 2J + 1 and is referred to as the Wang transformation
= X S
For the form of X, X ', E+ , E", 0+ and 0" see Reference 23.
With the matrix elements of E(k) known, calculation of the energy
levels of a rig id asymmetric rotor becomes a purely mathematical problem
involving the diagonalization o f the submatrices E+ , E", 0+ and 0".
of these submatrices is of the Jacobi an or tridiagonal form.
Each
These can be
e a s ily transformed to a form which allows the d ire c t application
of con­
tinued- fra c tio n technique; the technique f i r s t used by King e t. a l, to
di agonal ize these matrices.
However, nov/adays these can be diagonalized
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
113
to a high degree of approximation by high-speed d ig ita l computer techniques,
as done by Beaudet
22
in his computer program fo r example.
In th is computer
program, the mathematical method used is exactly that of King e t. a l.
The
matrix elements are generated fo r the Hamiltonian in the Wang representa*4* *■
tio n . Therefore, fo r each J value, the four independent blocks E , E , 0
and 0” are obtained, each one of which is diagonalized separately.
The
eigenvalues are then ordered w ithin each block by c a llin g another subroutine,
so that they can each be labeled properly.
As a check, the trace of the
diagonalized matrix is compared with a sum-rule.
In asymmetric ro to rs, the projection of the to ta l ro ta tio n al angular
momentum is no longer constant along any axis or direction fixed in
molecule, with the re s u lt th at K is
no longera good quantum number.
ever, the parameter K is kept to label the energy le v e ls .
labelled in the notation
where
the
How­
Each level is
is the value K would have in the
lim itin g case of an- oblate symmetric top, while K_1 is the value of K fo r
the lim itin g prolate symmetric, top.
values such th at -J s
t
I f K_1 - K = t then t takes on 20+1
- J.
Dennison58 was f i r s t to derive the symmetry selection rules fo r an
asymmetric ro to r.
For R-branch a-dipole tra n s itio n s , the selection rule
AJ = ± T, a K - 0 , ± 2 . . . and a K^ = ± 1, ± 3 . . . is followed.
The fr e ­
quency fo r a R-branch tra n s itio n calculated from (A -l) is given by
v = (A+C) (0+1)
+ 1 (A-C) [EJ+1 (k) - EJ (k)J
(A-7)
For Q-branch tra n s itio n s , A 0=0 and fo r b-dipole tran sitio ns a K_x = ± 1 ,± 3 ...
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
aK
= ±1, ±3 . . .
The frequency o f a Q-branch tra n s itio n is given by
-v =
\
(A-C) A E(k)
(A-8)
Therefore from Q-branch tra n s itio n s , only the values fo r (A-C) and
k can be determined.
These values when plugged in to the expression fo r re­
branch tra n s itio n s , the value fo r (A+C) is also obtained.
to obtain individual values of A and C and plugging
I t is
ro w
easy
these in the expression
fo r k the value of B can also be obtained.
A lte rn a tiv e ly , individual values o f A, B
R-branch transitions alone.
However, in this
and C can be obtained from
case, B and C w ill be c a l­
culated more accurately than A as explained in the te x t.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX B
COMPUTER PROGRAM FOR EVALUATION OF
STRUCTURAL PARAMETERS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
116
CV o
r-» c- T
o
ij** zz XT—o
o w*
o
<
►
» *
* • • *
CV *-? t*4o rv -v
II 11 tt It ir it II
H4
t. Jl
Ji —
» E
c c C r- L«. «< a
-X3 c* CO
r>* o X
o c• • •
jo
T“f »—
-j
c X o
It IS I*
if u it
X c X
c c u c a c
-r-e
—f
O
o
Z9
•
m
*■
* r-4
♦
r-i
r-» o
t-{ o
»~
X
AE
-I
X
r
r
t— O a
>—o i3» Hc.
<
<r
c.
o
r>
o
O
o
«
X
U. i
r.jwo *
u. O
• v-i *-i
• -5-4v-i
■f f—1) f 4- H 11 1 T
e: -1
C5 X X
-J
—s •
-J •
O cC a O a o II
w II
It
v-» II
X o X
X O' X
1
E 2 UJ:
L H c
X
*-*
a
:3
• ■o
o'
PRo ^AM
ASROTR
oc
o
cr
m
-C'
rn
tH
C
<3
ZZ
3
» in
•
-* H
«rf O
3
2> fJO
<w r~
tf~
J— ca o
h- a O
o
cr
cr
<S c
<
zz
o
o
C3
o
“*
_1T ▼.4■ V*
«► u_ -« % u. X
•
• <*1*t-,
• H
ri <-s
it t + 1—
f * t— II 1
:
C5 ”5 ->
zz
•
.J
-X »
j
,
T
a
£Z
E* n X X r^ if
tT» *51
r-4
•r-r
tf St
T-f a
w a
H
o X
wi. o X
U_ o/~ X
c
C. u a
c_ cr »— 5r
FOHTPAN
fi
'O N '. I J
3
n
M :0 T
J
I
? 3
r l :V -O
^
O •'v .3 ? O r l .V O -ST XV
CC C C C C O C C r'rlrJr'rt-'H rtririlV C V tV jaW N M
C
VC
VC
V
o
c? o
o
d
c= a . c : —- o
o
c
o
c . <= c= C5 c
-s: o
-4T. o
cr c r o
ct
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o
o
K
T'. 7T, "O •C>
o
c
c: o
o |
♦
-fa
>
•CD
v; *>
r~~
Cr
-»
>
cr
rc
O
4
CD
♦
?*«».
4
rv
rv
y^
r>
•
ca
4
X
V
o- -»
•
T\
.*v
<» CV CV •< r.
< *rH o < cv
•
*fa. ■fa r> c ♦
♦ Cl •o 4 O
•
< c ro o V.’ sT
o T-“. rv c
T—
? *
t —1 ♦ O
4 •
4 *o
• r - «T* * X
o* cro X
rv s \ • S s ra
>?• .%! * -z r>, »
■y*
r-n o
▼H
T-* O' cv ?o o t
I r~f o T*i
"O
rv o
4 •C y% > X
4 o*
r^ PC*
4
CV
cr a 4
• X- o
•
• ♦ LD
o
O
X 4 rc 'C
* rv rv -fr rv o ♦• w*
x> O
X- O
o
'J
«
>
■fa » x« ♦ i •c
.1 T-% ♦ r o
.-V.
T-S ro
-X
X
s— r - • s~ t j : ITe
X c*
C ■*■ X cr 4 •
*5 ♦ •f a <*r> X • rv ■x
X "ID
v^- ♦ X
Pfa ■+ a a 4
4 cv
<X ■c ♦ o ?>» 4 X cv
X
4
W
• X
•
• r- 4- • o -b
fs*
•r>
o
O
o
X
o
CV p«r. r- c: -* T*-’ jf*. X- •r-. X *
t-5
.
4
cv ^r
4 r"
+ T-4 «~n X 4 rv rr
• CD
*
• XN
« ■«c
£1- <C • cr c X X C rv r r o X*
■
w
T
*
-•.>•
4
♦
♦ ■
<
>
1 tr- nr < o . cr5 t cr
cr. f
• '♦
•
♦
4 «•
> 4
o •■nr
X X •
X nr*.
X rv
•s. fa < C*v -*- + rC 4 rv ■«r 4 4
X C 4 rv < .
I f •< ♦ fs. C
■» < > 4
~
♦ C\
♦ <
■*
O Cv o-- O
id1 <3. 5- Cv -*
y O*
o X n cr; 'C c r'
4
• CV cr • cr tH • e *c • cr rv
'.•“i 3 V"* «—
i •>«* r-f
> .M ra •4' > _
"O
cr V ”
rv T“i
sTt H
4 CV1 • 4 cv r4 + cv O' 4 y j o
• cr
•
• X*
•
rv
rr i: -<L zz » <c
< O »
* -*
v- 4
♦
K‘
<s I
4- ■X t Xi X f
t rD
-fa
4 *
•« <«
4
o. UJ .sJ •> UJ
UJ ■4 JfN UJ
*«- fa fa
4 «JL‘ < 4 U c -» U
n !*• (T j rv. 1.1 -*■ r\: u*. ■*• y - It l 4
r- fa fa
r f \ •* c c\ •* C
+
•r* -JK "C ■r*. •» x- ;Nfc 4
■c o CV
jw^.
v-» fa*. X r~l .V .'CX nT ;v
• ar- cv • V •v:
• X
C\ y .
V
y •X crT CC cc C cr crV
W C3 -< -H C J r**. r*i GN O’- —i :>
X
iZ , Cv
U
J
O- X
o cr11 • • II
• rv IK
• cv 11 •
•
*
•■
•
T-*
o
H
O
1— •C
o *'C
.V
•cr
4 4 u 4 4 t* 4 4
U 4 +
t—. .'V
-r-i Cv
*- .V
T - i Cv
rc
o
•
sf\
M
LlJ
cr
ac
o
il\
cv
ir»
u.
♦
u.
o
%
•
^
U.
TO
K
•
^
cr
O
•
i%
rv
po
U%
V4
*
cr
u.
*
v*4
UJ
la-
O
•
CV
Ul
fa
%
♦ - €>*
•
»
o
»4
fa
r i C
r
fa
c c c
•
J—
sr-
rr
CV O
*O
r
k:
if
'S .
< t <T <C
II
II II
*<
?
>—
^
• rlil z
cr m
:v -c<r r\ u. * x
»- I-
v4
s
fa
o
<r-f
it
cr-
fa
*■* »
-Y— --X. u_ •
C3
s«r
;I « C? CT
U. C/; (T, CT; CT •
cr T. :X - I * C3
c<cc
U
fa
*—»—*—
3*
ri
X
I—
o
X
c
-«
n
u_; U..3
r
Z
Z
n z> ~ 3
—
i w; H
ZZ 2
H H i
Z H
v— i— »— i— i—i— x x
Z Z Z - Z Z Z O T O Q
O O O O O O G * — Z
L
CL
X ! UJ UJ UJ 'x—
L
L C f
> C ^
-—I y—I
tJ
U. W U
fl >1 3
vH r l »~4
H
3
X
r
c
X
X
«
> S
c o c 5 c
CN u N £ > ’J
'3-LTi
cz z . o o o o c sr
C
5
•vr
=r- j \
in X \
ci- cr- cr c r
♦c* ?<r ■
2T
<
a
'O o ¥**.'
n X* X X
cr cr cr cr
c c n • c.
X
o
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
rv ~ o : 'Z TV X
X vC
X X X
o cr o cT rr
• *..
■r. • c;
REFERENCES
1.
C. C.
Lin and J. D. Swalen, Revs. Modern Phys, 31_, 841(1959).
2.
C. R.
Quade and C. C. L in, J. Chem. Phys, 3S3, 540(1963).
3.
D. G.
Burkhard, J. Chem Phys. 21_, 1541 (1953).
4.
D. G.
Burkhard and
5.
C. R. Quade, J.
6.
R. K. Kakar, E.
A. Rinehart, C. R. Quade and
Phys. 52, 3803 (1970).
7.
R. K. Kakar and E. A. Rinehart (to be published).
8.
T. Kojima, C. R. Quade, and C. C. Lin, B u ll. Am. Phys. Soc. 1, 44
(1962).
9.
F . A. L. Anet and M. Ahmad, J. Am. Chem. Soc. 85, 119 (1964).
10.
H. G. S ilv e r and 0. L. Wood, Trans. Faraday Soc. 6£, 5 (1964).
11.
W. G. F ately,
R.K. H arris, F.A. M ille r , and R.
Spectrochim. Acta 21_, 231 (1965).
12.
J. C. Irw in , J. Chem Phys. 23_,1355(1955).
Chem Phys, 47, 1073 (1967).
T. Kojima, J . Chem.
E.Witkowski,
F. A. M ille r , W. G. F ately, and R. E. Witkowski, Spectrochim. Acta
23A, 891 (1967).
13a
Y. Hanyu and
J,
E. Boggs, J. Chem. Phys. 43,
13b
Y. Hanyu, C.0. B r it t and J.
14a
T. Kojima, J .
Phys. Soc. Japan1_5, 284 (1964).
14b
H. Forest and
B.P.
3454 (1965).
E, Boggs, ib id . 45, 4725 (1966).
D ailey, J. Chem, Phys. 45, 1736(1966).
15.
T. Pedersen, N. W. Larsen, and L. Nygaard, J. Mol. S tru c t. 4 , 59
(1969).
16.
R. H. Hughes and E. B. Wilson, J r . , Phys, Rev., Vol. 71, 562L
(1947).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
119
17a
G.
Erlandsson, Arkiy Fysik 6, 477(1953);
7, 139 (1953).
17b
K.
E. McCulloh and G. F. Poll now,J. Chem
Phys. 2£, 1144 (1954).
18a
G.
Erlandsson, Arkiv Fysik 8 , 341(1954);
9,399(1955).
18b
H.
SeUn, Arkiv Fusik 13., 81 (1957).
18c
R. L. Poyenter, 0. Chem Phys. 39, 1962 (1963).
19.
H. G. Goldstein, Classical Mechanics (Addison-Wesley, 1950) Chapt.
5.
20.
C. 11. Townes and A. L. Schawlow, Microwave Spectroscopy (McGrawH ill Book Company, New York, 1955), p. 50.
21.
L. H. Scharpen, 24th Symposium on Mol. Struc. and S pect., Columbus,
Ohio, 1969, Paper 09.
22.
R. A. Beudet, th esis , Harvard U n iversity, Cambridge, Massachusetts,
1961.
23.
G. W. King, R. M. Hainer, and P. C. Cross, J. Chem. Phys. 11, 27
(1943).
~~
24.
H. C. A lle n , J r. and P. C. Cross, Molecular Vib-Rotors. (W iley,
New York, 1963), p. 188.
25.
T.
Oka and Y. Morino, J. Mol. Spectry.6,
26.
E.
Fermi, Z. Physik 71., 250 (1931).
472(1961).
27a
J. V. Dresback and E, A. Rinehart, 23rd Symposium on Molecular
Structure and Spectroscopy, Ohio State U niversity, Columbus,
Ohio, 1968, Paper N5.
27b
W. F. White, 24th Symposium on Molecular Structure and Spectroscopy,
Ohio State U n iv ersity , Columbus, Ohio, 1969, Paper 0 11,
28.
H. W. Harrington, J. Chem. Phys. 46, 3698 (1967).
29.
Ref. 20, p. 19.
30.
N.
F. Ramsey, Nuclear Moments (W iley, New
31.
C.
H. Townes and A. H. Schawlow, ib id . Chap. 6 and Appendix V I I .
32.
York,1954).
W. Gordy, W. V. Smith and R. F, Trambarulo, Microwave Spectroscopy
(Dover Publications, In c ., New York, 1966) p. 287.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
17_, 782, (1949).
120
33.
C. H. Townes and B. P. D ailey, J, Chem. Phys.
34.
E. B. Wilson, J r . , Chem, Revs,
35.
D. R. Herschbach, "Tables fo r the Internal Rotation Problem," Dept.
of Chemistry Harvard U niversity (1957).
36.
B. L. Crawford, J. Chem. Phys. 8 , 273 (1940).
37.
H. H. Nielsen, Phys. Rev. 40, 445 (1932).
38a
D. G. Burkhard and D. M. Dennison, Phys, Rev.
38b
J. S. Kohler and D. M. Dennison, Phys, Rev. 57, 1006 (1940).
39.
Tables Relating to Mathieu Functions, National Bureau of Standards
(Columbia U niversity Press, New York, 1951).
40.
K. T. Hecht and D. M. Dennison, J. Chem. Phys. 26, 31 (1957).
41.
Ref. 1 . p. 845.
42.
Ref. 1. p. 847.
43.
E. B. Wilson, J r . , J. C, Decius, and P. C. Cross, Molecular Vibra­
tions (McGraw-Hill Book Company, New York, 1955).
44.
J. Kraitchman, Am. J. Phys. 21_, 17 (1953).
45.
For example, J. E. Wollrab, Rotational Spectra and Molecular Struc­
ture (Academic Press, New York, 1967) p. 90.
46.
R. W. K ilb , C. C. L in , and E. W. Wilson, J r . , J. Chem. Phys. 26,
1695 (1957).
~~
47.
E. V. Ivash and D. M. Dennison, J. Chem. Phys. 21_, 1804 (1953).
27, 17 (1940).
84, 408(1951).
*
48.
D. Kivelson, J. Chem. Phys. 22, 1733 (1954).
49.
E. C. Kemble, Fundamental Principles of Quantum Mechanics (McGrawH ill Book Company, In c ., New York, 1937), p. 394,
50a
D. Kivelson, J. Chem Phys. 23, 2230 (1955).
50b
ib id . 23, 2236 (1955).
50c
ib id . 27, 980 (1957).
51.
B. Kirtman, J. Chem Phys. 37, 2516 (1962).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
121
52.
A. Sayvetz, 0. Chem. Phys. 7_, 338 (1939).
53.
C. Eckart, Phys. Rev. 47, 552 (1935).
54.
C. R. Quade, J. Chem Phys. 4£, 2512 (1956).
55.
B. S. Ray, Z. Physik 78, 74 (1932).
56.
S. C. Wang, Phys. Rev. 34, 243 (1929).
57.
See fo r example, L. Pauling and E. B. Wilson, Or . , Introduction to
Quantum Mechanics (McGraw-Hill Book Company, New York, 1935).
58.
D. M. Dennison, Revs. Mod. PHys. 3, 280 (1931).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
5 626 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа