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3 7 ?
/J Sid /V<7, 2 6 0 9
A COMPREHENSIVE MODEL FOR THE ROTATIONAL SPECTRA*OF PROPYNE
CH3CCH IN THE GROUND "AND Vx0=1,2,3,4,5 VIBRATIONAL STATES
DISSERTATION
Presented to the Graduate Council of the
North Texas State University in Partial
Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Won Myung Rhee, B.S., M.S.
Denton, Texas
December, 1986
hK
Won Myung, Rhee, A Comprehensive Model for the Rotational
Spectra of Propyne CH^CCH in the Ground and Vn n=l> 2,3,4 and
5 Vibrational States. Doctor of Philosophy (Physics),
December, 1986, 104 pp., 14 tables, 6 illustrations,
bibliography, 60 titles.
The energy states of C 3 V symmetric top polyatomic
molecules were studied. Both classical and quantum
mechanical methods have been used to introduce the energy
states of polyatomic molecules. Also, it is shown that the
vibration-rotation spectra of polyatomic molecules in the
ground and excited vibrational states can be predicted by
group theory.
A comprehensive model for predicting rotational frequency
components in various v 1 0 vibrational levels of propyne was
developed by using perturbation theory and those results
were compared with other formulas for C 3 V symmetric top
molecules. The v10=l,2,3 and ground rotational spectra of
propyne in the frequency range 17-70 GHz have been
reassigned by using the derived comprehensive model. The
v 1 0= 3 and v 1 0= 4 rotational spectra of propyne have been
investigated in the 70 GHz, and 17 to 52 GHz regions,
respectively, and these spectral components assigned using
the comprehensive model. Molecular constants for these
vibrationally excited states have been determined from more
than 100 observed rotational transitions.
From these experimentally observed components and a model
based upon first principles for C 3 V symmetry molecules,
rotational constants have been expressed in a form which
enables one to predict rotational components for vibrational
levels for propyne up to v 1 0= 5. This comprehensive model
also appears to be useful in predicting rotational
components in more highly excited vibrational levels but
data were not available for comparison with the theory.
Several techniques of assignment of rotational spectra
for each excited vibrational state are discussed. To get
good agreement between theory and experiment, an additional
term 0.762(J+1) needed to be added to Kjt=l states in v 1 0= 3.
No satisfactory theoretical explanation of this term has
been found.
Experimentally measured frequencies for rotational
components for J-»(J+1)=+1 (01 J <.3) in each vibration v 1 0 = n
(Oi.ni.4) are presented and compared with those calculated
using the results of basic perturbation theory.
The v9= 2 rotational spectrum of the propyne molecule was
introduced in Appendix A to compare the rotational spectra
of the same molecule in different vibrational levels v9 and
v. 1 0
TABLE OF CONTENTS
Page
LIST OF TABLES . . vi
LIST OF ILLUSTRATION viii
Chapter
I. INTRODUCTION 1
Rigid Rotor Distortable Rotor The Effects of Vibration
II. HAMILTONIAN FOR TREATING THE VIBRATION-ROTATION INTERACTION OF POLYATOMIC MOLECULES 20
Derivation of Hamiltonian Power Series of the Hamiltonian Twice Contact Transformation Non-zero Matrix Element of the Twice
Transformed Hamiltonian for Molecules of C 3 V Symmetry
III. DERIVATION OF A COMPREHENSIVE MODEL 33
Perturbation Theory Accidental Degeneracy
IV. EXPERIMENTAL APPARATUS 45
IV
TABLE OF CONTENTS-- Continued
V. ANALYSIS OF EXPERIMENTAL RESULTS 55
The Ground State Spectrum The v 1 0= 1 State Spectrum The v 1 0= 2 State Spectrum The v 1 0= 3 State Spectrum The v 1 0= 4 State Spectrum The General Form of the Rotational Constants The v 1 0= 5 State Spectrum
VI. CONCLUSIONS . . 75
APPENDICES 78
A. The v?=2 Rotational Spectra of Propyne B. The Linear Molecule and Symmetric Top
Molecules as a Rigid Rotor C. Distortable Rotor For Prolate Symmetric
Top Molecules D. Rotational Energy of Symmetric Top Molecules
in the Excited Vibrational State E. Accidental Degeneracy F. Program To Assign v 1 0= n Rotational Constants G. Program To Calculate v 1 0 = n Rotational
Transition Frequencies
REFERENCES 99
LIST OF TABLES
Table
I. Classes of Molecules 3
II. Diagonal Matrix Element of the Twice Transformed Hamiltonian H+ 31
III. Non-Diagonal Matrix Element of the Twice Transformed Hamiltonian H+ 32
IV. The Comprehensive Model of Propyne and C 3 V Symmetry Molecules 44
V. Rotational Transition Frequencies for CH3CCH in the Ground State 56
VI. Rotational Transition Frequencies for CH3CCH in the v 1 0 = 1 Excited Vibrational State 59
VII. Rotational Transition Frequencies for CH3CCH in the v 1 0= 2 Excited Vibrational State 62
VIII. Rotational Transition Frequencies for CH3CCH in the v 1 0= 3 Excited Vibrational State 66
IX. Rotational Transition Frequencies for CH3CCH in the v 1 0= 4 Excited Vibrational State 69
VI
LIST OF TABLES-- Continued
Table
X. Rotational Constants for CH3CCH in the
v1Q Vibrational States 71
XI. General Forms of Rotational Constants . 72
XII. Rotational Transition Frequencies for CH3CCH in the v 1 0= 5 Excited Vibrational State 73
XIII. Rotational Constants for Propyne in the v?= 2 State 79
XIV. Rotational Transition Frequencies for CH3CCH in the v?= 2 Excited Vibrational State 80
VI1
LIST OF ILLUSTRATIONS
Figure Page
1. The Symmetric Top Molecule, Propyne 6
2. Symmetric Top with Space-fixed (XYZ) and Molecule-fixed axes (xyz) 8
3. Vibrational Modes of a Triatomic Linear Molecule 17
4. Coriolis Force in a Linear XYZ Triatomic Molecule
5. Spectrometer for Measuring Rotational Spectrum of Propyne 46
6. A Typical Chart Recorder Scan of Rotational Components of Propyne 51
v m
CHAPTER I
INTRODUCTION
D. M. Dennisoni and H. H. Nielsen2 laid the early
foundations for the theory of vibration-rotation energy of
polyatomic molecules in 1931 and 1951, respectively. From
1950 to 1965 many scientists, Nielsen3, Anderson et al.4,
Goldsmith et al.s, three papers by G. Amat et al.6,7,s, s .
Maes', and Grenier Besson10 made major contributions to set
up the vibration-rotation energy and frequency formula for
polyatomic molecules. In 1965, Tarragon developed a theory
and applied it to some polyatomic molecules which have C 3 V
symmetry. Also she made specific rotational energy and
frequency formulas for C 3 v symmetric top molecules in the
ground state and vibrationally excited states v 1 0= 1, 2 and
3. These main contributions by Nielsen, Amat and Nielsen,
and Tarrago are examined with varying degrees of detail in
the following chapter.
Although a number of papers have appeared for specific
cases of C 3 V symmetry molecules, no one has set up a general
form for the vibration-rotation energy and frequency formula
for polyatomic molecules which have the same symmetry.
Therefore, all the energy and frequency formulas which have
appeared in the literature before appear in different forms.
This work has four major objectives.
1. Derivation of a comprehensive model for vibration-
rotation energy and frequency for polyatomic mole-
cules which have C 3 V symmetry.
2. Reassignment of all the structural patterns of the
propyne molecule using the comprehensive model and
then make a comparison between the comprehensive
model and previous models and formulas.
3. To set up the general forms of rotational constants
for the propyne molecule.
4. To make additional measurements of rotational spectra
of the propyne molecule in the v 1 0= 3 and v 1 0= 4
states.
Generally, polyatomic molecules can be classified into
four different catagories depending upon their symmetry12.
The four classifications are designated as linear molecule,
symmetric top molecule, spherical top molecule and
asymmetric top molecule, whose classification depends upon
the principal moments of inertia of each molecule. Table I
gives a summary of these classification. Also, Table I
shows two different symmetric top molecules; prolate top and
oblate top. A prolate symmetric top is one in which the two
equal moments of inertia are greater than the other moment
of inertia. An oblate symmetric top is one in which the
cn w
D U w
o s t l o w w cn w < J u
H
w 1-3 CQ <C Eh
CO CD
rH 3 O <D
rH 0 a
•P c <D g
O a-
N
rd
rd H -P
04 U H (1) O C C H
5H m a . o
<u c a)
rH >1 -p
0 u <
o o o
II H I! N X >1
M H H
> ! N N M H H I! II II
X >i X H H H
n : u u
m as u
CJ co
a: o
CO
U tt
0 C rd
. c •P <D a
<D *0 >i
x : a)
T3 rH cd g u o u*
N N N M H H
N N % A V H
II M Th
>1 > i >1 > i >1 H H H H M II II II II
X X X X X H M H H M
a a a , eu a 0 0 0 o 0 E* EH Eh EH Eh
O 0 CD rH •H
•H -P <D fd W fd •P 0 «P <D n 4-> rH rd •H <1> W rd a) 0 rH u CO a) g u & <D £ rd a g a* o & >i
iH •H >i Ou CA u w <
CO •H X fd
u -P CD
>i cn
cn fd
N
(D g
3 w w <
degenerate moment of inertia is less than the other one. By
this definition propyne is a prolate symmetric top molecule.
Most of the symmetric top molecules observed in the
microwave region are prolate.
Group theory is a powerful tool in dealing with the
symmetry of molecules. It is useful to recall that there are
32 point groups in a crystal whose symmetry elements are
rotation, inversion, reflection, plane of symmetry and
center of symmetry.i3 Also, recall that a point group means
that there is always one point in the group that is
unaffected by all the group symmetry operations: rotations,
reflections, and inversions. An important point group of
symmetric top molecules is the group of molecules that has
C 3 V symmetry. The propyne molecule in this study has C 3 V
symmetry and an explanation of that property follows.i4
C n v - consists of C n and a v and any other elements to
have closure(E).
* closure; the "product" of any two elements in the
set must also belong to the set,i.e.,
the set is closed.
C - Rotations are labeled by C„ where the rotations are n n
2ft/n about an axis through the origin. The axis of
highest n is called principle axis.
* C3 is a 2K /3 rotation.
<r - reflection in a vertical plane that contains the
principle axis.(mirror image)
Figure 1 shows a diagram of symmetric top molecule CH3-
CCH(commonly called propyne and methyl acetylene) showing
bond lengths and bond angle. These structural parameters are
taken from a paper by Duncan et al.15 and by Dubrulle et
al.*6. Propyne is a molecule of interest to many areas of
science, especially for astrophysics. Its existence has been
confirmed in the atmosphere of Titan which is the seventh
moon of saturn as well as in the Taurus dark clouds.17
It is the rotational motion of molecules that is
primarily responsible for the spectra observed in microwave
spectroscopy.!8 There are several ways to get the formula
for the rotational spectra of molecules. The simplest
method to calculate the rotational energy of polyatomic
molecules is by using the rigid rotor assumption.19
1. Rigid Rotor
A. linear molecules - The rotational energy for linear
polyatomic molecule in the rigid rotor assumption is given
by20(see Appendix B)
Er(J,K) = hBJ(J+l) (1- l)
B. Symmetric top molecule - Figure 2 shows a symmetric top
molecule along with a space-fixed coordinate system XYZ. The
1.060 A'
1.2066
1.4586 A 0
c
H f 1.096 A 0
108 17 ' H H
FIGURE 1. The Symmetric TOD Molecule, PROPYNE.
coordinate system which rotates with the molecule is labeled
xyz. The rotational energy of the rigid rotor is given in
terms of the principal moments of inertia I , I and I . x y z
Classically2i, the pure rotational energy is
A pz E = + — ^ + (i- 2) r 21 21 21 K }
x y z
where P x IXWX, Py IyWy, PZ-IZW z
The center of mass is assumed as origin of the
principal axis.
For the Quantum Mechanical approach22, Pv/ P„, p„ and Ev A y Z IT
become operators
P2
X P2
+ -3L P2
+ Z
21 21 21 H r = — +-J- + — (1-3)
21 21 x y
and
P2 = px2+ Py2+ pz2 (1_ 4 )
P : total angular momentum operator
In a symmetric top, one of the principal axes of inertia
must lie along the molecular axis of symmetry. Here z is
chosen as the symmetric axis of the top. For the prolate
symmetric top molecule, the Hamiltonian operator of Eq.(l.
3) can be expressed as
2 2 P Pz ( 1 1 \
Hr + ~ ( ) (1-5) 21 2 I I
x z x
Eigenvalues for the angular momentum operators and energy
operator for the symmetric top rotor are
(J,K,MlP2|J,K,M)= *2 J(J+1) (1- 6)
(J,K,M|Pz|J,K,M)= fi K (1- 7)
(J,K,M|PZ |J,K,M)= fi M (1-8)
where
J = 0, 1, 2, 3, ... . (1-9)
K = 0 , ±1, ±2 , ±3 ,±J (1-10)
M = 0, ±1, ±2 , ±3 ±J (1-11)
By equations (1-4),(1-5),(1-6) and (1-7), the final form
of the rotational energy of a symmetric top molecule with
10
rigid rotor assumption is given by23 (see Appendix B for
details)
Er(J,K)= h [BJ(J+l)+(A-B)K2] (1-12)
where J is the total angular momentum quantum number, K
is the quantum number for the projection of the total
angular momentum along the symmetry axis, M is the quantum
number for the projection of the total angular momentum
along the Z space-fixed axis (see Figure 2). The rotational
constants used in Eqs.(l-l) and (1-7) are defined by
B = 8 * 2 l V l , ( 1 - 1 3 )
x y
A = 1 ^ 7 " ( 1 - 1 4 )
z
It is necessary that any permanent dipole moment must lie
along the symmetry axis in a symmetric top molecule. All
matrix elements of this dipole moment resolved along a
space-fixed axis vanish except those between states
corresponding to J •> J or J+l, K * K.2* Th e selection rules
for the absorption spectra23 are
AJ= +1 , AK= 0 (1-15)
11
Application of these rules to the rotational energy formulas
for linear and symmetric top molecules gives the formula for
the absorption frequencies.
v r = 2B(J+1) (1-16)
Under the rigid rotor assumption, linear and symmetric top
molecules have the same absorption rotational spectra
formula.
2. Distortable rotor
When the prolate top molecule propyne is rotating about
its center of mass, bond lengths and bond angle are
changing. These distortions are due to the centrifugal
forces at play in the molecule. Centrifugal stretching is
treated as a perturbation on the eigenstates of a rigid
rotor. A complete procedure for derivation of rotational
energy of a distortable prolate symmetric top molecule is in
Appendix C. The expression for the rotational energy of the
distortable symmetric top molecule to the first order
perturbation treatment is26
E (J,K) = h [BJ(J+1)+(A-B)K2-D_J2(J+l)2 (1-17) 3T u
-DJKJ(J+1)K2-DKK4]
12
With the selection rules for absorption spectra, this
energy equation gives the rotational frequency as
v r(J,K) = 2B(J+l) - 4DJ(J+1)3 - 2D j k(J+1)K2 (1-18)
If one includes higher order centrifugal distortion
effects to the sixth power in the angular momentum
components P 6 to the rotational energy, the following result
is obtained.27
E (J,K) = h [BJ(J+1)+(A-B)K2-D J2(J+1)2
J
'DJKJ(J+1)K2"DKK4+IVJ3(J+1)3 (1-19)
+H J 2(J+l)2R2 +H J(J+1)K 4+H K<* ] «J & KJ K
By applying the same absorption selection rules and
considering the centrifugal distortion to sextic order, the
rotational frequency formula for the symmetric top molecule
becomes
v (J,K) = 2B(J+1)-4D (J+l)3-2D (J+1)K2 (1.20) "*• JK
+H (J+l) 3[(J+2) 3-J 3]+4H K 2(J+l) 3
J JK
+2H (J+1)K4
KJ
13
3. The effects of vibration
Only pure rotational frequencies and energy states of
linear and symmetric top molecules both under the rigid and
distortable rotor assumption were discussed so far. But
other major departures from those assumptions occur for
molecules in excited vibrational states. The effects of
vibration of molecule on the rotational spectra are as
follows.
A. Changing the rotational constants
For symmetric top molecules in excited vibrational
states, the effective rotational constants differ from those
of the ground state. One obvious reason for that is the
principal moments of inertia of a molecule are altered by
the vibration of molecules. If a symmetric top molecule
bends,the moment of inertia about an axis perpendicular to
the symmetry axis will decrease and the rotational constant
B will increase. If a symmetric top molecule stretches, the
moment of inertia about an axis perpendicular to the
symmetry axis will increase and B will decrease. This is
opposite to the behavior of the rotational constant A in
prolate symmetric top molecules. In 1980, Claude Meyer
et.al.28 expressed the rotational constants in excited
vibrational states as
u
BV=Bo- a v- yv2 (1-21)
A v=A 0- Bv- 6v2 (1-22)
Where B v and A v are the rotational constants in the
excited vibrational states, B 0 and A0 are the rotational
constants in the ground state, and <x,v,& and 6 are called
constants of anharmonicity.
From equation (1-18), the rotational frequency of either
a prolate or oblate symmetric top can be expressed by
v (J,K)= 2B (J+l)- 4D^ (J+l)3- 2D^ (J+l)K2 (1-23) j- V J J K
The new values of all the rotational constants in excited
vibrational states are expected to have systematic variation
with the vibrational quantum number.
15
B. Degenerate Coriolis splittings (z-type doubling)
The best approach to consider the effect of a degenerate
vibrational mode on the rotational spectrum of a symmetric
top molecule is to begin by considering a simple triatomic
linear molecule. A linear molecule with n atoms has (3n-5)
modes of vibrations,29 however, not all of these modes have
different frequencies. The bending modes are always doubly
degenerate, i.e., there are two bending modes of vibration
both of which have the same frequency. So, a non-symmetric
triatomic linear molecule has two nondegenerate parallel
modes vx and v3 and a doubly degenerate bending mode v2.
Figure 3 illustrates these three vibrational modes of
triatomic linear molecule.30 Two parallel modes correspond
to the stretching vibrations of the two bonds XY and YZ.
The bending modes v2 is doubly degenerate because the
molecule is free to bend in two orthogonal planes. It is
obvious that these stretching and bending modes cause a
change in the averaged or effective molecular parameters;
bond lengths and bond angles. This leads to a slight change
in the moment of inertia and a consequent altering of the
rotational spectra.
In addition to these effects, there are other effects of
interaction of rotation with the degenerate bending
vibrational modes. If z is chosen along the molecular axis,
16
the degenerate vibrations of the non-rotating molecules can
be considered as orthogonal bending motions of the molecule
in the xz and yz planes. In the rotating molecule this
degeneracy is lifted by the interaction between vibration
and rotation. This interaction is due to the Coriolis force
in the vibrating-rotating molecules.3i It has a value F =
2mvx o, where w is angular velocity of rotation of the
molecule and v is the linear velocity of the vibrating atom.
A Coriolis force occurs in all cases of the rotating and
vibrating polyatomic molecule. Figure 4 shows the effect of
Coriolis forces on a rotating linear triatomic molecule.32
If the molecule is rotating about the x axis, then
bending in the xz plane is not quite equivalent to bending
in yz plane. This non-equivalency is only because of a
Coriolis force in the vibrating-rotating molecule. The
direction of the bending in the xz plane and that of the
angular momentum J of the molecule is perpendicular to each
other, so this vibrational mode is excited by the Coriolis
force. But the bending in the yz plane and angular velocity
to have the same direction, so this mode is not changed by
the Coriolis force. Therefore, the effective moment of
inertia about each axis of rotation is different for the two
cases cited above. As result of this vibration-rotation
interaction, the two degenerate energy levels are slightly
split; this splitting is called £-type doubling.
17
V 1
v > 1 I 1
V 3
Fig. 3. Vibrational Modes of a Triatomic Linear Molecule.(Bending mode V_ is doubly dege-nerate) ^
V i •
v 2 L . - j ^ L i *
Fig. 4. Coriolis Force in a Linear XYZ Molecule.
Curved Arrow : Direction of Rotation. Solid Arrow : Normal Motion of the Vibrational Mode Dashed Arrow : Coriolis Force.
18
For symmetric top molecules such as CH3CCH which have a
linear group along the symmetry axis, there are degenerate
bending vibrational modes which give rise to I -type doubling
of the rotational levels similar to that described for
linear molecules. There is a component of this vibrational
angular momentum along the figure axis which adds to, or
subtracts from, the pure rotational angular momentum about
the symmetry axis. This component has the value Zfi and
with the Coriolis interaction term we can write
P - fMi (1-24)
I = v, v-2, - - - -, -v (1-25)
Where f is the Coriolis coupling constant and v is the
vibrational c[.iiantum number. When the vibrating motions are
perpendicular to the symmetric axis, f = 1 and the
vibrational component along z(symmetric axis) is Zfx, just as
in the linear molecules. In the symmetric top ,however, this
component along the symmetry axis ,in general, will be less
than Ifi, or 0< f <1.
By considering the &-type doubling contribution to the
rotational energy of the symmetric top molecule, the new
rotational energy of symmetric top molecules in the excited
vibrational states has a form33,34(see Appendix D for
derivation)
19
E (J,K,M= h[BvJ(J+l) + (A-B)vK2- 2fS-KAv] (1-26)
This gives a splitting in the energy levels of 4f£KA
between the levels with K£ positive and those with K&
negative. These splittings by £-type doubling are easily
detected in propyne when this molecule is excited to v 1 0=l
and v 1 0=3 vibrational states.
The basic frequency and energy equations and symmetry
properties for the propyne molecule have been outlined in
this chapter. Further details will be given as the need
arises.
The reminder of this paper is divided into five chapters.
Theory of vibration-rotation interaction in the symmetric
top molecule is discussed in Chapter II. Derivation of the
comprehensive model developed from perturbation theory is
found in Chapter III. The experimental equipment and
procedure used in this study are presented in Chapter IV.
The measurements made in this work and data analysis using
the comprehensive model are presented in Chapter V.
Conclusions and a brief summary of the entire work are given
in Chapter VI.
CHAPTER II
HAMILTONIAN FOR TREATING THE VIBRATION-ROTATION INTERACTION OF POLYATOMIC MOLECULES
Derivation of Hamiltonian
The Hamiltonian for a rotating vibrating polyatomic
molecule was discussed first by E. B. Wilson and J. B.
Howard.3s T o derive the exact Hamiltonian, Wilson and Howard
derived the classical Hamiltonian and then used the method
of Podolsky36 to obtain the quantum mechanical Hamiltonian.
J. E. Wollrab37 also explained this process well by
expressing the kinetic energy in terms of angular momentum.
An equivalent but more useful Hamiltonian was formulated in
1940 by B. T. Darling and D. M. Dennison.38 I n 1 9 5 1 H > H
Nielsen2 developed a useful Hamiltonian for calculating the
vibration-rotation energies of molecules. The Hamiltonian,
as expressed by Darling and Dennison is
H = ( 2 - D
V,. * ^ w + y p y ] + v A <r AV- ;4<r
20
21
Where « and £ range over x,y and z which are the
principal axes of inertia of the molecule. P and p are a
components of the total and internal angular momentum^
respectively, p^^is momentum conjugate to the normal
coordinate ^and V is the potential function of the
molecular force field. The equation for is given as
p* = - — ) (2- 2) ><3., ( 2 >
A complete procedure to derive the Hamiltonian and matrix
elements of the Hamiltonian for the vibrating rotating
symmetric top molecule is not the main topic of this
dissertation, so this chapter introduces only the steps to
get the matrix elements of the Hamiltonian for a symmetric
top molecule that can be used to derive a comprehensive
model for propyne. For completeness, a summary is now given
for the details needed to derive the comprehensive model.
The inertial parameters in eq.(2- 1) are given by
V r " s e V f " 'by'" < 2" 3 )
II = ( T * -Lt'T'\ ^ ot6 yy a6 aY yB
22
and
I -I -I xx xy xz
-I/ I / -I' xy yy xy z
-I -I xz yz zz
^ = Iv«/+£ a^QA«. + n A , Q n , . «« etd ir <4«r >fl(r *r,, , >a cr' -a er'
t' _Te ^ _ /«f-> *P> <*P ~fo-a^" -40- 9 'ir/ ATp'tr
(2- 5)
(2 -6)
Where Ij^ and ie (ocfts) are the equilibrium values of
the moments and products of inertia, respectively. By
introducing the Coriolis coupling coefficient we have
W * P =21 H e Q p
a »»- i'T' MA'V ^ A'rr' (2- 8)
and equation (2- 1) becomes
~P )u (P -p )1 2 o<^ " a a MotB B 6
(2- 9)
,1 r r * -ic * + 2 ^ 5 j ^ + vi [p_y 2 (p_ y -) ]}+ v at -Ocr
Also, rearranging Equation (2- 9) by the power of P P a' B'
the following result is obtained.
23
H = ^ ^ P s" l P* 6P«'pB < 2-">
^ " a S ^ 1 + ? } r £ + D + V
with replacements
U " Z "''"J (2-11)
" i u i ' S ^ - 1
It is not possible to find directly the eigenvalues of
the Hamiltonian operator in Eq.(2-10). The general procedure
to get the solution of the Hamiltonian for vibrating
rotating polyatomic molecule is as follows 3 9, 4 0,,*2,43 #
Power Series Generation of the Hamiltonian
1. Expand the Hamiltonian in a power series to the fourth
order with respect to normal coordinates to get
H = H0 + AHx+A2H2+X
3H3 +\"Hh (2-12)
According to the customary notation, x is 3- parameter of
smallness equal to unity, which denotes the order of
magnitude of the different terms in H. By detailed
calculation, comparison and substitution, each term of the
24
Hamiltonian can be shown to have the following forms.5 In
these equations, is a force constant of the harmonic
potential given by V =?Ex Q2 and q is a dimensionless "AV A a t AT
normal coordinate and p is its conjugate momentum which is
defined by Q , ^ = ~ ^ - § q • The final results for M«r
H are in the following form.
v t 1 E Act a
U t
. aoc
act
•+s >6(7
V * E E <6a ap> 1* I* aPtxP|i"
a/Y rvAaa ^ a + V, (2-13)
V I E I /xsb'a' aft
Q p> . . " * w r ^ \
T1 l1 ^ A«<* l[bp>
^ / fi \4, T* !*• U , j (f« 71" «« VP T«^ j
+ vrt
ri I E • /> G/$o'<i>"cr" (X
st>?A>'crl4>',Cr"
7 T p i l««
V °i/5O-Cl,C7-'Ci.0"CT"fB
0V«t> • • 4 i
Wa'cr' / -ft \ * Je j 4 VX^Xy/ «< '52f5'cr'+Cl-io-S'cr'
O05™^ <L i W r \X
— f~ V>h O A(3' a e. V > I °, w + rr~) q t>o* + V
25
Twice Contact Transformation
2. Perform the first contact transformation4o,*i in order
to partially diagonalize the Hamiltonian matrix. W. H.
Shaffer, H. H. Nielsen and L. H. Thomas 4 3 in 1939 and Van
Vleck and Kemble44 first formulated such a contact
transformation T to remove the off-diagonal element of the
first order terms in T may be expanded in terms of a
Hermitian matrix S42 as
m iAS /X A2 ~ T = e = I + iA S - j S2 + (2-14) / \
I : unit matrix
H - T H T = Hq+ AH' + A2H2+X3H3+X (2-15)
Ho= H0
Where the S function4s is chosen such that the first two
terms,HQ+^HJ, in the expansion of H are diagonal with
respect to the quantum number v.
3. Group the terms of the same true order of magnitude in
X, which means that one expands the transformed Hamiltonian
and arranges the terms in order of powers of x as
26
H - h0 + Xht +X h2 +X3h3 + x ' t h l f + - - - - (2-16)
By comparing Eq.(2-15) and Eq.(2-16)
h'0 = Ho = H,
hi - Hi + [ iS,H0] (2-17)
h' = H2+ [iS,h"] + [is, H0 ]
h' = H3 + [iS,]/] + [is, h'j ]
These equations can be expressed by a combinations of the
vibrational operators and q^r and of rotational operators
W V 4 6
4. Perform the second contact transformation5! using
^ * A S i it X = e = I + iX2S - ± S2 + (2-I8)
then
H = T H T = Ho + +^2H2 + - - - (2-19)
in such a manner that + AH* +\ZH^ will be diagonal with
respect to the vibrational quantum number v. In general,
these operators will not be diagonal in £ or K, where % s s
corresponds to the vibrational angular momentum associated
27
with the s-th vibrational mode and K is the rotational
quantum number corresponding to the projection of the total
angular momentum about the symmetry axis.
5. Group the terms, taking into account the true order of
magnitude of the terms in powers of x- The twice
transformed Hamiltonian H then can be expressed as
H + = hj +A h+ +x2h+ + (2-20)
By comparing the equations (2-17),(2-19M2-20), the
expressions for the operators H +/h
+ are as follows n n
Ht = h0
H+ = h\
H2 = h2 + 1 ^S,h/°] (2-21)
= h3 + i [S,h^]
K = h4 + 1 C S' h2 ] " jfc'CS.h'o]}
The regrouped components of the twice transformed Hamiltonian
are
ht = h„
h+ = h[
h2 = h2 + C i S' ho ]
28
ht = h3 + [iS,^] + [is, h'0 ] (2-22)
h4 = h4 + tiS'h2+;I + CiS,^]
where
^2 ~ ^2 + [iS,h0] = i(h^ + h* )
The explicit expression47 for regrouped components of the
doubly transformed Hamiltonian in terms of the vibrational
rotational operators are not presented in this work. Even
though the operators h^,h* will have terms that are
off-diagonal in v , these terms will contribute to the
energy in an order higher than the fourth and do not need to
be considered for the fourth order correction to the energy.
Furthermore, all of the operators are diagonal in J and M so
that only matrix elements of the form
<J,M,v1,v2, V g, | hJ+h++h++h++h"J;| J,M,vx ,v2 , > (2-23)
need to be calculated. For axially symmetric top molecules
such as CH3CCH, the only off-diagonal terms which are of
consequences to fourth order of approximation arise from h2
and are off-diagonal in K and I only.
6. Solve the secular equation; the eigenvalues are obtained
by arranging the matrix elements into submatrices of the
general energy matrix and solving the secular equation. This
29
equation is very complicated and lengthy to solve for the
polyatomic molecule. Symbolically the procedure is to solve
the following equation
det [ — , V S , |ht+h++h2++h++hj |
J,K',M, — , v , i ' »M ' ) s s
^KK' 51 l' fiM M7 EV.R1 0 (2-24) s s s s
Non-Zero Matrix Elements of the Twice Transformed
Hamiltonian for Molecules of C 3 v Symmetry
When calculating the energies of molecules with special
symmetry, there are basically two approaches which can be
used. The first approach is to solve the secular equation
of the general Hamiltonian matrix. This is the exact method.
The second approach which has been followed here is an
approximation method by the perturbation theory when matrix
elements are calculated. Matrix elements for molecules of
C3V symmetry will have the form
< J,K,M,Vn,Vt,£t | H + |
J,K+AK,M,V + A V ,V +AV ,I +A£. > (2-25) n n t t t t
30
Where and v_ are the quantum numbers corresponding to
a non—degenerate normal vibration and a degenerate normal
vibration, respectively. Unless accidental resonances occur
only those elements with Avr and Av equal to zero need to
be considered. In any case, the matrix elements of Eq.
(2-25) for symmetric top molecules which have C 3 V symmetry
are non-vanishing only if9,4s,4?
- AK + £ A£t = 3p (2-26)
where p is an arbitrary integer.
It is not necessary to present all of the details of the
derivation of the matrix elements because in 1975, A. Bauer,
G. Tarrago and A Remy published a paper5° giving all the
matrix elements of the Hamiltonian for molecules of Cjy
symmetry. All these diagonal and non-diagonal elements are
expressed in terms of all molecular constants and
parameters. Also, these matrix elements are written by
using the zeroth order functions |v,J,K,£> as basis and
including all the contributions originating with twice
contact transformed Hamiltonian to the fifth order. For
convenience, Table II and Table III show the diagonal and
non-diagonal matrix elements which can be used for the
vibration-rotation energy calculation by using the
perturbation theory in Chapter III.
31
EC
M-l O
CO -p c <D e <u r—I w
X •H
4J rd S
n3 C O
fd •H Q
H
w
PQ <
Eh
CN
X + o
X
w
II
A
b >
EC
o*
b >
V
CN
W r—i
C N
ro PQ I
ro < C C N
CN
CN
ro
ks <
+ b
C N o*
cn PQ +
C N
> C N
PQ +
> f—I
PC +
o PQ
PQ I CM CN >
< CN
+ >
PQ I
O PQ I o <
k_p < ^ CN + ^
> r H
H + ^ I-} k-P — ' <3 CN w b + o >
+
O h} Q
CN
k-P <
C N
I I
+ b
> rH b
Q +
o h3 Q
I
+ b ro
LD
+
<=>* ro
+ b
O K
EC +
rH +
b
b
O I-)
EC +
CN
CN
+ b
W CN b id
{3 +
CN
> > > b H fc<H H b H ^ v — O r £~ CN + + + i-}
O K O b O K
b O b
EC + ro
+ b
cr +
ro hD b o b
EC +
32
>
H » CN X I
+1 rH
<0
>
+1 CM
CN
T 5 + h )
X I r H + i r o + U* + 1 f d ffi ^ 5 •>»
CN m h + 1 CN —' r o 0 <d p—i
CN + 03 CN
• P CN r o c CN W H + i CD « CN MH s MH + CN (D + 1—' >
r-H x—s r H i—i •»
w r H + CN + 1
X + « X
X h ) ' — ' ^ r o • H ' *"D o * MH
M CN > + > - P CN b H »* + 1 r-H f d n (n MH U - ^ T q
r-H f d 1 4 CN s MH +
U - ^ T q + X + > « r H « r-H > CN h > o * + 1 I 'D
CN r-H h D > H ) C CN MH + i c n h D r o CN + i 0 MH + f d MH + 1 < D 1—' f d + 1 v o t r> + CN + X I CN f d f d o r-H CN r o VO r H k D • H t r MH C n MH m h MH Q 1 *-r-»
MH
G 0
II 11 II II II II II II
S A A A A A CN CN CN • + 1 1 + W * A A + 1 M - A H
H o * o * <£> o * H
H «k + 1 v V H CN r H CN r o r-H VD
W J + 1 + 1 + 1 + i + 1 + i + 1 W J W W J
s. CQ < l-D •"D h ) *"D h D CQ <
K. v. V E h > > > > > > > >
ffi <=>*
> V
—
*k. > V
ffi «
V •""D >
V
+ EC
«
*fc
h ) >
V
M
h ) >
V
J E o *
w
h ) >
V
- S . c>*
J *
h 3 V > V
EC
O ? *»
« *-D
K. > V
•$*
C + i
I
£ + i
+
I c
CN
i—I +
I >
I a
CN
+ 1 r H + o * +
>
C K II fiK II c c
II II o *
« > h D
c + 1 C + 1 CN
t o X
CD U a )
X ! £
CHAPTER III
DERIVATION OF A COMPREHENSIVE MODEL
Bauer et. al.so have published a paper with diagonal and
non-diagonal matrix elements of the Hamiltonian for a C 3 V
symmetry polyatomic molecule. The origin of that work was
from the formulation by Nielsen et. al.2,4,7,s of the
vibrational energy up to the fourth order in vibration and
the sixth order in rotation. Tarragon has produced a paper
along similar lines. In this chapter a similar model will
be derived from fundamental concepts.
To obtain the rotation-vibration energy, a perturbation
treatment is applied to the Hamiltonian to effect its
solution for finite vibration-rotation interaction. Similar
work has been done by Amat, Nielsen, Tarrago3? and used in
several other paperss1,s2,53,s4,ss. b u t e a c h t ± m e t h e
resulting energy equation has a different form and does not
predict results for vibrations greater than v10=4. Since
these formulas do not predict the energy or frequency for
v10=5 and higher excited vibrational states of the propyne
molecule, a general model has been developed to do so.
In this chapter a comprehensive model which can be used
for any vibrational state of a C 3 V symmetry molecule by
33
34-
substituting the proper v and i into a general energy
equation is derived. The derivation is as follows.
Perturbation Theory
The Hamiltonian is subdivided in perturbation treatment
as
H = Ho + Hi (3- i)
Where H° is an unperturbed Hamiltonian whose
eigenfunctions are known and Hi is a perturbed Hamiltonian
which is significantly smaller than the zeroth order terms
of H° in the absence of near degeneracies. The basis
functions are the product of y y where y is the basis r v r
function of the rotational part of H° and y is an
orthonormal function of the molecular normal coordinate. H°
is diagonal in the vibrational quantum numbers, although it
need not be diagonal in all of the rotational quantum
numbers. Hi is generally not diagonal in either the
rotational or vibrational quantum numbers. As was seen in
Chapter IX some off—diagonal terms are partially
diagonalized by performing two contact transformations on H.
There are two assumptions in the perturbation theory.56
One begins with the perturbed Hamiltonian Hi in some sense
being small compared to the unperturbed Hamiltonian H°
,i.e.,
35
n 1 < < E 0 M - ? nn n \o- Z)
The other assumption in the perturbation approach is that
the eigenstates and eigenenergies of the total Hamiltonian
, H, do not differ appreciably from those of the unperturbed
Hamiltonian H°.
The complete energy calculation is then
H» tpo = £0 tpo / .3 , X n n n J>
E n = <cf ° I H° | cpo > = H o ( 3 - 4 ) n n n nn
Where cp and E° are, respectively, the eigenstates
and eigenvalues of the unperturbed Hamiltonian. Also, by
some calculation and comparison of eigenstates and
eigenenergies57, the first order energy correction from
perturbation theory has a form
E1 = H 1 (3- 5) n nn '
and the second order energy correction has the form
H . E2 = Z 1 n i
i^n E°-E° n l
36
The total energy has a form given by
E = E » + E i + E 2 (7-7) n n n n [ J 7)
As can be seen from equations (3-4), (3-5) and (3-6), the
unperturbed energy and the first order energy correction are
calculated from the diagonal matrix elements of the
Hamiltonian. The second order energy correction is obtained
from the non-diagonal matrix elements of the Hamiltonian.
In order to apply this perturbation theory to our
vibration-rotation energy calculation by using Bauer's
diagonal and non-diagonal matrix elements of the doubly
transformed Hamiltonian ,H+, some observations must be made.
The unperturbed Hamiltonian H» in the perturbation theory
above is not the same as H0 given in Eq.(2-12), the power
series expression of the Hamiltonian. Also, the value of Hi
is not same as Hx. Because the twice contact transformed
Hamiltonian H+contains both unperturbed and perturbed terms,
rotation-vibration energy can be calculated by considering
diagonal contributions and non-diagonal contributions of H+.
ie,
E = E + E v.R Diag. Non-Diag. { '
37
The diagonal contribution to energy is given by:
EDiag= < V' J' K' £ I H + | V , J , K , £ > (3. 9 )
Then using Table IX with some new notations,
EDia^ V' J' K' £ ) = V X££<l2+ BvJ
( J +1) + yjl££2j(J+;L) (3-10)
+ (A-B)vK2+ (A3-B3n
2K2-2(Af ) vK£
- 2(Af)3KZ3- D J2(J+l)2 - D J(J+1)K2- D K
J JK K
+ "HjJC J+1)K£+ ilKK3£+ "HJJJ2 (J+l) 2Kj
+ 71JKJ(J+1)K3£+ 1 K KK
5^+ HjJMJ+l)3
+ H J J K J 2 ( J + 1 ) 2 K 2 + HJKK J ( J + 1 ) K l t + V 6
where Bv, Ay: Rotational constants.
Anharmonic constant.
{ : Coriolis coupling constant.
DJ' DJK , DK : First order centrifugal distortion constants.
HJ ' HJJK ' % K K ' % : H i 9 h e r order centrifugal distortion constant
B3 = y1%
% ' ^ J J ' ^ J K ' : Hi9her order diagonal matrix elements.
38
The non-diagonal contributions to energy is given by
H +. H +
E = I n i in N. D i4n T o ~0 (3-11)
Where
1*1 E~_ e Y
n i
<n | = < v , |
Ii> = |v,J,K±l,A?2>
Iv,J,K±2,l±2>
|v,J,K±2,£±4>
and where the quantum numbers v, J, K and & are the standard
set for vibration, rotation and Coriolis interaction.
In the energy expressions all the diagonal matrix
elements and three major non-diagonal matrix elements of the
following type were considered: <v,J,K,£ | H+| v,J,K±l,£+2>,
<v,J,K,& | H +| v, J,K±2, £±2> and <v,J,K,£ | H+|v,J,K+2,&±4>
which gives some contribution to the energy in the highly
excited vibrational state ,i.e., v 1 0 = 4 and higher. All
these non-diagonal elements are diagonalized about v and A .
The quantity q is the coefficient of <K,£ | H+| K±2,l±2>,
and q 1 2 and d 1 2 are the coefficients of <K,l | H+(K±l,£f2>,
f 2 4 is the coefficient of <K,i\ H+| K±2,l±4>. The
coefficients fj2, ^22 a nd -2\> i n the low excited
vibrational states, were small and ignored but were
considered with f 2 4 = 0.007 for the v 1 0 = 4 and higher
vibrations.
39
These non-diagonal contributions in terms of vibration
rotation quantum numbers and rotational constants are given
by:
_ tq12(2K+l)+d12(£-i)]2(a+b2)2 [q12(2K-l)+d10(£+l)]
2(a:bt)
N.P ~ ~ 3 + ^ L J _ HVJK-£~HVJK-U+2
HVJK£~HVJK+1£-2
( q va X ) 2
HVJK£~HVJK+2£+2
( f24 a2 b4 ) 2
HVJK£~HVJK+2£-4
(^y a2 b2 }
H + -w-*-vJK-C vJK-2£-2
( £24 a2 b4 )
HVJK£~HVJK-2£+4
(3-12)
where H^ V J K = [V JK | H«| VJK ] etc. . .
+ 1 ax » [J(J+l)-K(K±l)]2
± J- 1 a2 = [J (J+l)-K (K±l) ] 2 [j (j+l) _ (K±l,) (K±2) ]T
b? = [(v+£+l±l) (v-ZKLTl) ]T
fc>z = [ (v+£+l±l) (v-£+i+l) (v+£+l±3) (v-a+1+3)]T
(3-13)
To calculate the energy difference which appears in the
denominator of E N j y some small terms ,ie, A3, B3 =Ylj?,(A£) 3, D J R
' DK' DJ' nJ' nJj , njK' nKK' HJ' HJK' HKJ ' H k w e r e ignored,
since these terms are very small compared to A , B X v v ' 'IV
40
Therefore, the energy difference terms are simplified to
H i , V J K J T H * V J K + 1 1 £ - 2 = X J l 2 . I 2 - 2 " ( £ - 2 ) 2 ] + ( A - B ) [K 2-(K " 4 ) 2 ]
• 2 ( A < J [ K £ - ( K + 1 ) U - 2 ) ] = 4 X ^ ^ ( 4 - 1 ) - ( A - B ) ( 2 K + 1 J - 2 A C ( 2 K - 2 . + 2 )
By the same procedure,
H " " v J K £ " H * V J K - l £ + 2 = " 4 X ; . £ ( £ + 1 ) + ( A " B ) ( 2 K - 1 ) - 2 A C ( - 2 K - i + 2 )
H " v J K £ ' H " r * V J K + 2 £ + 2 = " 4 C X Z i ( " + 1 ) + ( A - B ) ( K + 1 J - A S ( K + i + 2) ]
H V J K £ - H * V J K - 2 , £ - 2 = 4 + ( A - B ) (K-L) -A; (K+Z-2) ] ( 3 - 1 4 )
H v J K £ _ H " " v j K + 2 , £ - 4 = 8 X Z i ( £ " 2 ) " 4 ( A " B ) ( K + D - 4 A C ( 2 K - i + 4 )
H v J K £ ~ H ' ' ' v j K - 2 , £ + 4 = ~ 8 X £ £ ( S , + 2 ) + 4 ( A - B ) ( K - l ) + 4 A £ ( 2 K - 2 - 4 )
Finally, the rotational-vibrational energy for C 3 V
symmetry molecules is given by
E(v, J,K, £) = E_ . + E n K 1 Diag. N.D (3-15)
By using the selection rule for absorption spectra
AJ - +1 , AK = 0 , AV = 0 , AA = 0 (3-16)
one can get the frequency
, _ . E ( v , J + 1 , K , £ ) - E ( V , J , K , £ ) v(v,J,K, ?_) = — [___ (3-17)
4 1
Hence, the rotational frequency becomes
(V,J,K,5-) - Aj + A + A 3 + A ( 3 - 1 8 )
Where
A 1 5
f 2 bv ( J + 1 ) + 2 y £ a £
2 ( J + 1 ) - 4 D j ( J + 1 ) 3 - 2 D j K K 2 ( J + 1 )
+ 2 r i j K& ( J + l ) + 4 r | j j KM J + l ) 3 + 2 r i j K K 3 J l ( J + l )
+ H J ( J + 1 ) j [ ( J + 2 ) 3 - J 3 ] + 4 H j j k ( J + l ) 3 + 2 H j k k K 4(J+l)
• { i (v-&) (v+l+2) ( J + l ) [ ( J + l ) 2 - ( K + l ) 2 1 c t 2
+ ( v + £ ) ( v -2 .+2) (J+l) R j + l ) 2 -
1 1 (&+1)+(A-B) (K+l ) -A? (K+£+2) x u U - l ) + (A-B) ( K - l ) - A C
( K - l ) 2 ] q 2
(K+2.-2)
A _ | 2 ( v + £ ) ( v - " + 2 ) ' ( J+ l ) [^12 ( 2 K + l ) + d 1 U - l ) 1
4 X U U - 1 ) - ( A - B ) ( 2 K + l ) - 2 A 5 ( 2 K - £ + 2 ) - 2 (v—j) (v+?,+2) (J+l)
4 X ^ ( 1 + 1 ) - ( A - B )
[ c ? l ? ( 2 K - l ) + d i o ( £ + l ) I 2
(2K-1) +2AC (-2K+2.+2) }
a 4 = f 2 4 ( v - i ) (v+£+2) (v+2,+4) (v-Z-2)
8 X U (*+2) - 4 (A-B) ( K - l ) -4AC (2K-H-4 ) V
£24 (v+J&) ( v - l + 2) 8 x » „ < J?-2 ) - 4 (A-B)
Jlit *
( v - ^ + 4 ) (v+g,-2)
(K+l) -4AC (2K-J.+4) Where
[ ( J + l ) ( J + 2 ) - K ( K ; i ) ] [ ( J + l ) ( J + 2 ) - ( K + l ) ( K + 2 ) ] - [ J ( J + l ) - K ( K + l ) ]
[ J ( J + l ) - ( K + l ) ( K + 2 ) ]
42
Accidental Degeneracy
Some quantum number sets had to have special attention.
Any combinations of quantum numbers which make K&=1 or K&=-2
cause accidental degeneracies. Accidental degeneracy makes
the energies in the denominators of the E N D terms same, and
ENft aPP e a r s be infinity. To avoid these undesired cases,
direct diagonalization was employed to solve the secular
equation for these special quantum number sets. In this
general model, quadrupole interaction, proton—proton
interaction, spin-rotation interaction etc..., which appear
not to be important for the propyne molecule from our
experimental results, were ignored.
The direct diagonalization method used is given byi
Hd^d-El^d d; diagonal
Hip = Atp
(ft- \l)ip = 0
/ \ / \ |
H- XI | = 0
By using the Table III in Chapter II, the determinant has
a form
v,J,K+2,1+2
v,J,K+2,1+2
v,J,K-2,H-2
E-X
+ , + qva2b2
% a 2 b 2
+ , + qva2b2
E-X
v,J,K-2 fZ-2
%a2h2
0
E-X
= 0
43
No interactions between (K+2,2+2) and (K-2,5,-2) were
assumed.
(E-X ) [ (E-X ) 2- ) 2 ] - (E-X) (qvajbj)2 = 0
X = e
X = E i q ^ U + b * ) 2 + (a'b")2]^
X = E±qv {[J(J+1)-K(K+l)] [J(J+l)-(K+l) (K+2)] (v+Jl+2)
(v-j,) + [j (j+i) - K (K-l) ] [J (J+l) - (K-l) (K-2) ] (v+J,) (v-£+2)}1/2
when
v=l if o —i / K——1 £=--l v 1 , K*. -1 | K = ! 1
X = E±qvJ(J+l)(v+1) (3-19)
Applying the absorption selection rule AJ=+1 to Eq.(3-19)
Av = ±2qy(J+l) (v+1) if k£=1 (3-20)
By the same procedure(see Appendix E), A4 becomes
Av = ±2f24(J+l)v(v+2) if Kl=-2 (3-21)
Finally, a comprehensive model for the rotational
frequency of C 3 V symmetry molecules has been obtained by
using the perturbation theory and partial diagonalization
method. The final form which gives the frequency of a
transition J -> J+l for any vibrational levels of the propyne
molecule in low J transitions is given in Table IV.
44
CD C >i CX 0
M-i 0
rH (D TJ 0 s
<D > •H W c <D JC (D U a g •
0 0) u a)
rH CD JC o 4-) Q)
rH 0
0 s MH
>1 fO rH 4-> 3 0 e u [| 0 >i 1*4
>1 > o m c U (1)
&H D 1 CD a) & u -P En o
a) ,c C EH
w
PQ <
Eh
CM O CJ
C M
CN^ T—* +
CO *
-5 Cr +
C M \ /
Q
+ "3
CN
+ >
* + —' > «*»•>
< ^
+ In
r» "O +
JN
CM r— O"
+
I ° I cr ^ CM
^ + pq c I £
""""'* r —
+ II P ^ I .
r-
+
C N
+ <=w
> CN
CO
+ wL
—
+ 3 > <;
£? P C O
C M 3
1 3 ^ 3" • I
C M O a*
C N +
_ cvr + r—
r -
+ + + > V
| < 1
T—
+ C M
1 3 C M
r " o r—
+ cr + — 5
+ C M I i
—» C r » > CD
+ G Q
1
JO G Q
1 * *
_,— _
i
< r -1
C N 1
+ + I! »
4- r—
> + « r—
1 —• 1
II
!
X *•—
CM t— "O +
»•-+
CN
CM T— CT
+ -5
C N
I >
X
CM
A-0 < CM
+ CM
> CD
I <
I
X
C N
!
I *
I
CN 1 _+ r—
+ + —3
CM |
—3 1
| < CN CN + 1
r— |
1 CN 1
_— _
> 1 CD i CN 1 + < -5
I P +
«— 3
"i r
l *
OJ +
CN
+
+ >
CN J-
+ >
I >
C N C N
C N +
+
+ *
C M +
*
1
CN +
+
I
C N
+ rr * l <5W
* .
C N p
* 5
i ? — >
* ^ — CN «—« > CN
CO I I C <r a -± JZ rt -
1 7 CN n/ +
^ Y '
C M
+
C N 4-
X oo rt=
+
* CN
< CN +
, > + ^
_ C M
C O +
< 5 * €
> C N 1
ii
+
1
2, K
> h
CN C N *•— X 0 0
<5w
J L.
CHAPTER IV
EXPERIMENTAL APPARATUS
A microwave spectrometer suitable for measuring the
rotational spectrum of CH3CCH in this work was available in
our laboratory. This spectrometer is similar to those
described by E. A. Rinehart et alss, j. a. R o b e r t s * a n d I.
C. Story et al^o. A block diagram of the spectrometer is
shown in Figure 5.
In a typical spectroscopic experiment, radiation is
supplied from a source and guided through an absorption cell
to be detected and displayed in some fashion. This whole
procedure can be divided into five areas of instrumentation
according to the function of each. The catagories chosen
here for discussion are listed below.
1. Radiation Source
2. Absorption Cell
3. Detection and Display
4. Sample Gas Handling
5. Generation of Standard Frequency
4-5
46
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d O D -J ^ Q . CO *
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Cn G •H
3 CO (0 CD 5
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LT)
CD u p tn •H pLi
4-7
Microwave Sources
Four klystrons were used in this experiment as radiation
source. A Varian X-12 and OKI 20V10 klystrons were used to
generate radiation in the 17 GHz region; an OKI 24V11 was
used to generate radiation in the 26 GHz region; an OKI
35V10 was used to generate radiation in the 34 GHz region as
fundamental frequencies. All of these klystrons provide a
stable, tunable source of monochromatic microwave radiation.
The detailed construction of the klystron is out of the
scope of this work and is given elsewhere in standard
microwave electronic books* i, <*2, 6 3, <s 4,6 s . I n s o m e c a s e s it
was necessary to generate harmonics to reach the higher
frequency transitions. A crossed waveguide multiplier66,67
was used to provide radiation in the 52 GHz and 68 GHz
regions by prpducing the second harmonics of the OKI 24V11
and 35V10 klystrons, respectively.
There are two different adjustments of a reflex klystron
to produce appropriate radiation. Coarse frequency
adjustment of the klystron was achieved by mechanically
altering the size of the klystron's resonant cavity by means
of a flexible diaphragm and a connected screw assembly. Fine
frequency adjustment was then achieved by the fine
adjustment of the repeller voltage. It is also possible to
sweep a range of frequencies by sweeping the repeller of the
4-8
klystron with a voltage ramp available from the sweep of the
oscilloscope used for signal display. The frequency response
of a klystron can be measured by applying a control voltage
to its repeller and measuring the frequency shift. A
typical value of klystron sensitivity indicates that one
volt change in the repeller voltage will result in one MHz
change in the output frequency of the klystron,
approximately. Each klystron is highly individual in this
respect. About ten megahertz frequency shift was achieved by
using the voltage signal from the time base of an
oscilloscope. The exact range was measured by applying the
ramp voltage and directly observing the frequency shift of
the klystron relative to the frequency markers which were
generated when the klystron frequency was swept relative to
a fixed standard reference source.
Absorption Cell
A K-band rectangular copper absorption cell which has a
silver coated inside wall to produce better conduction was
used. A thirty meter absorption cell was used for the strong
signals found in the ground, v 1 0= 1 and v 1 0=2 vibrational
states. To detect the weaker signals in the higher excited
vibrational states ,ie, v10=3,4, the absorption cell was
lengthened to 60 meter, thus giving improved sensitivity and
better signal-to-noise ratio.
4-9
The ends of the absorption cell were vaccum sealed with
Teflon windows to allow changing the transition sections and
detectors without altering the pressure in the absorption
cell. Breaking the conduction path of the radiation in this
manner gives rise to reflections which greatly enhanse the
standing wave pattern of the system. Various methods are
employed to reduce this effect. One technique is to employ
approximate quarter wavelength gaps at the end sections,
much like the effect of a quarter wavelength plate in
optics. This effect has been discussed in some detail by I.
R. Dagg et al68.
Detection and Display
Crystal diodes were used for primary detection. Germanium
1N26 diodes w^ere used to detect the frequency from the ten
gigahertz to forty gigahertz region. Germanium 1N53 diodes
were used for frequencies above forty gigahertz.
A dual-trace Hickok Electric oscilloscope (Model 1805A)
provided visual display for the absorption. Strong signals
in the low vibrational states were detected directly on the
video, but weak signals in highly excited vibrational states
were detected only by using a slow scan sweep with display
on the chart recorder and by optimizing the gain and fine
tuning of all parts of the equipment. In any case, the
50
spectral line profiles were displayed on a dual-trace
recorder manufactured by Rika-Denki (Model B201) with one
pen producing the frequency markers and another pen
producing the appropriate derivative of the spectral line
profile. Figure 6 shows one typical example of the line
profile and standard frequency markers.
Sample Gas Handling System
Both gas handling and high vacuum capabilities were
required for the apparatus. The absorption cell was
evacuated by using a Van Waters and Rogers HV-1 oil vapor
diffusion pump, cold-trapped with liquid nitrogen to remove
the pressure of the gas molecules in the absorption cell.
This was fore-pumped by a Welch Duo-Seal 1405 mechanical
pump.
Once the apparatus had been evacuated, high purity
propyne(99.6 percent) gas was admitted to the absorption
cell from a sample tank which had been pumped to remove
residual air and water vapor before admission of the sample
gas.
The peak line intensity of the absorption spectrum was
found to be very sensitive to the pressure of gas in the
cell. So, it was necessary to keep the appropriate low
pressure to get the maximum intensity of each line. The gas
51
I-6MHZH
/
LO 0) •
CO CO 5
Fig. 6. A typical chart recorder scan of rotational spectra(vg=l,2). B.G. is the background display.
52
pressure in the absorption cell was measured by MKS Baraton
Type 170M-27D pressure meter with accuracy to 0.0001 torr.
A gas pressure of about 20-30 mtorr of propyne was observed
to produce maximum peak absorption in this work.
Generation of Standard Frequency
In order to measure the rotational absorption frequency
which is displayed on the chart recorder, a well known
frequency interval must be recorded along with the
absorption profile. A method used in this work for accurate
measurement is to heterodyne the microwave frequency with a
precisely known standard frequency which lies within the
range of an available interpolation receiver. A
Well-Gardner BC-348 Q radio receiver was used. The setting
in the receiver was 1.5 MHz, so that as the klystron
frequency was swept through the range of interest, markers
appeared at 1.5 MHz above and 1.5 MHz below the center line
frequency, giving a total marker spread of 3 MHz. These
signals are beat notes between the fundamental of the
klystron and appropriate harmonics of the fixed standard.
All frequencies are ultimately related to the frequency
of a Hewlett-Packard 105B quartz oscillator which provides a
signal of 100 KHz that is good to one part in 10* °. A
General Radio 1112A, 1112B reference is phase locked to the
53
quartz oscillator and thus produces a highly stable 1 GHz
comb of signals. The beat notes are produced when this
Signal, a signal of less than 100 MHz, and the fundamental
of the klystron are combined in a mixer multiplier unit.
A beat note will be produced if
± mxlGHz ± nxf = f . . K v R (4- 1)
where as the klystron frequency, f R is radio
receiver setting(l.5 MHz in this work), and m and n
are integers•
The value of m can be easily determined by the use of a
wavemeter 9. The fundamental klystron's frequency can be
obtained roughly by reading the wavemeter with accuracy of
about 10 MHz. To determine n, a Tektronic 191 variable
oscillator was used. The output frequency of the variable
oscillator is measured with a Hewlett-Packard model 5383A
counter which has accuracy to 1 part in l0e. By assuming a
value of n and setting the variable oscillator at
f ' = n f v v -57* (4- 2)
for n' an integer, marker will be produced if the choice of
n is correct. The center frequency of the absorption profile
is easily determined by a method of interpolation.70
5U
The greatest experimental difficulty in this experiment
arises from the weak intensity of v 1 0=4 absorptions. If the
variation of the population of energy states in accordance
with the Boltzmann distribution is considered, the v 1 0=4
transitions will be less than one percent as intense as the
ground state transitions". Moreover, the v 1 0=4 transitions
are even weaker since a number of degeneracies have been
removed with respect to the quantum number producing further
redistribution of the spectral line intensity. So, some
weak components could not be detected.
All of the experimental frequency measurements of the
propyne molecule in the 17-70 GHz region from the ground to
v 1 0=4 vibrational state are tabulated and discussed in
Chapter V.
CHAPTER V
ANALYSIS OF EXPERIMENTAL RESULTS
A. The Ground State Spectrum
to assign the microwave spectra of propyne in the ground
state, Dubrulle et al derived the quartic equation'!
v (J.K) = 2B0(J+1) - 2Djk(J+1)K2 - 4 D < J + 1 )3
(5- 1)
+ 4 HJJK ( J + 1 ) 3 k 2 + 2 Hj k k (J+1)K
4+HT(J+1)3[(J>2)3_J.3 ]
Fro. our comprehensive model one can get the same formula by
substituting v=0 and ,=0. We have ignored the sextic order
centrifugal distortion constants to assign our ground state
rotational spectra, since the contributions of these terms
to the frequencies is less than J-ess than the experimental error in
the low J transitions studied.
our measurements were made using the video spectrometer
described in Chapter IV. The ground state lines were
exceptionally strong and therefore easily measured with
experimental accuracy „i t h i n ± 6 0 k H z . A s , ^ o f ^ ^
the result of Ware and Roberts the calculated values by '
Dubrulle et. al.'i a n d t h o s e f r o m OUJ; comprehensive model
are given in Table V for comparison.
55
56
TABLE V. Rotational Transition Frequencies for CH nr„ in the Ground State CH3CCH
^ ^ ^exp (MHZ)
a b c
b Vcalc
(MHZ)
1 0 34 183.349 1 11 34 182.727
2 0 51 274.947 2 11 51 273.980 2 ±2 51 271.000
3 0 68 366.298 3 ±1 68 364.990 3 +2 68 360.976 3 ±3 68 354.498
34 34
183.415 182.761
51 274.947 51 273.967 51 271.026
68 366.270 68 364.962 68 361.042 68 354.509
Roberts and Ware (72) Dubrulle et. al (16) Our Comprehensive Model
calc (MHZ)
0 0 17 091.718 17 091.743 17 091.743
34 183.415 34 182.761
51 274.947 51 273.967 51 271.025
68 366.269 68 364.962 68 361.040 68 354.503
(j/a.j/b)
(MH2)
-0.066 -0.034
0.000 +0.013 -0.026
+0.028 +0.028 -0.066 -0 .011
(pa.pc)
(MH2)
-0.025 -0.025
-0.066 -0.034
0.000 +0.013 -0.025
+0.029 +0.028 -0.064 -0.005
57
As can be seen from the table, the two calculated values
are nearly the sane. There are only l-2kHz difference even
with the ignorance of sextic order centrifugal distortion
constants. Both calculational values were obtained by using
the same constants set which was derived from the data of
Ware and Roberts72.
B. The v 1 0 - 1 Spectrum
Nielsens was the first to calculate the rotational
transition frequencies for symmetric top molecules in an
excited v I 0= 1 state. In 1951 Anderson et. al.« reported
calculations on the v,„=1 bending mode of trifluoromethyl
acetylene. Also, they reported some deviation of their
experimental values from their calculated values as high as
200 kHz, which is outside the range of their experimental
error. In 1962 Grenier-Besson and Amat'3 derived a frequency
formula for trifluoromethyl acetylene in the v,„=l state and
obtained a superior assignment. The formula used by
Grenier-Besson and Amat is given below.
Grenier-Besson and Amafs formula is given by:
jJ+l)-2DJK(j+l) (Kt-1,2.4D ( J + 1 )3 (5- 2)
+ 2p (J+l) (Ki-1)
t4qv(J+l) if K£ = 1
58
- 4<3y ( J + l ) 3
[ V V ( A C ) v ] (KJU1)
2
if K£#l
* 12r B — B ™D +n -f- .
v JK j BV-AV-2(A?)V
* 2cr2 8r^ P = ^ J " 2 0 ^ +
VV(A^v Bv-Av-2(A?)v
From our comprehensive model one can obtain the v 1 0=l
rotational frequency formula by putting v=l and £. ± i to get
( J , K ' £ ) 2 ( J + 1 ) (V YAA J l 2- Dj KK 2 +n jKA+Tl ; y KK
3A) (5- 3)
4°j (J+l) 3+4ri KJ, (J+i) 3+v J J non diagonal
The observed frequencies of Ware and Roberts'2, the
calculated values using Amafs formula and those using our
comprehensive model are tabulated in Table VI. close
agreement between the two calculated values again can be
seen from the table.
59
TABLE V I . ^ o t a t i o n a l ^ T r a n s i t i o n Frequencies f o r CH„CCH e x c i t e d V i b r a t i o n a l
{pa-V*>) (Va-Vc)
(MHZ) (MHZ)
0 0 ±1 17 139.588 17
1 -1 -1 34 246.269 34 1 + 1 ±1 34 277.184 34 1 0 ±1 34 279.161 34 1 1 1 34 313.523 34
2 -1 -1 51 369.214 51 2 +2 ±1 51 410.612 51 2 + 1 ±1 51 415.471 51 2 0 ±1 51 418.341 51 2 ±2 ±1 51 418.915 51 2 1 1 51 469.901 51
139.632
246.321 277.202 279.142 313.421
369.300 410.671 415.560 418.409 418.876 469.850
34 34 34 34
51 51 51 51 51 51
139.610 -0.044 -0.022
246.268 -0.052 +0.001 277.164 -0.018 +0.020 279.080 +0.019 +0.081 313.369 +0.102 +0.154
369.221 410.625 415.481 418.270 418.856 469.872
-0.086 -0.059 -0.089 -0.068 +0.039 +0.051
-0.007 -0.013 -0.010 +0.071 +0.059 +0.029
a Roberts and Ware (72) b Grenier-Besson and Amat (73) c Our Comprehensive Model
60
C. The v 1 0 - 2 Spectrum
m 1971. Bauer used a frequency formula, given below, for
v8 2 state to assign the rotational spectrum of
Acetonitrile7 .
v(J,K,0)=2 (J+l)Bv-4 (J+l) 3-2 (J+l) K2D T„+ U (J-M) [X (J+, , 2^„ |_, ,
P2K2-\2
K ' + 3 2 ( J + 1 ' 4K2t(<3,12)
2("-2p)+2oq'1,d"] +
. ,2
- + 32 (J+l) 4K2[(q',„)2fTr-2n^-i.5nrr.i2Ui._
_ u . . > 2 ,,t_2„2 2. 12 12' Tr(q' - d,^)2 / (4p2K2-7T2)
and
v (J,K,±2)=2 (J+l) [Bv+4A]_1]-4 (J+l) 3D -2(J+1)K2D
JK
+2 (J+l) K * n j + 4 (J+l) [4(J+1)2-(M-2)21 (a'. v2. f o r K Z * 4
V(Kl-4)+2X
/ for Kl*-2 -16(J+l)[q'12(Ki+l)+d^|]
2
p (Kl+2) -it
Jl6(J+l)[(j+l) 2-l] (qj )2
Q , for K Z-4
,/+ 16(J+l)fJ4
1 + 2 t d ^ ~ cj • 0 ]2
^ -16 (J+l) ft. i2 1 2 , for K£=-2
61
This theory of Bauer predicts weak splittings in the
Ki=-2 and Ki=4. We could not detect these predicted
splittings in the propyne molecule, although the resolution
of the spectrometer allows it. If these splittings in
Bauer's formula are ignored and v=2 and t-0, ±2 are put into
our comprehensive model, with the notation a s g i v e n
below, the formulas become identical and good agreement is
found wath observed experimental frequencies.
^ A - B - 2A f + X££
H = A - B - A }
ft - A — B + 4A f + 4x 11
P = A - B + 2A f
The contributions of the constants f 1 2. f „ , ^ t o t h e
frequency were found to be ignorable in the v10=2 spectrum.
Table VII shows the comparison between the observed values^
and those calculated from Bauer's formula and our
comprehensive model. Assignment of Bauer's new data, shared
with this laboratory by private communication, has also been
made by the comprehensive model.
62
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64
D. The v 1 0 - 3 Spectrum
Bauer, Godon and Maes~ have given a frequency
expression, as shown below, for v s =3 to assign the
microwave spectra of methyl isocyanide.
| £ | = 1 / = 2 By U+1) +• 2 j (J+l | _ 4 0 * u + 1 | 3 _ 2 0 * k ( j 4 „ k 2 + 2 7 j ( J + h k ?
32 q,2 (2Kg +1) 24 I q12 (2K«-1) + 2 d , |2 (J-hi + _
" p ( 2Kg + 1) _ 2n 1
_ 12 ( + qv 22 13U+l)2-Kt(Kt+2) | ) ( (j +|,2 _ (kS +1)2 , (J + „
M ( Ki - 1) + 2 X —
i / K S * , ? + 1 6 K 2 + "» /2
J2 ( 3 (J + „2 - Kg (Kg -2> [ } t U + 1)2 - ,KC_„2 , u + „
V (Kt-1) —
>{ Kt = \ •. t 8 qy (J +1) t 16 /2J2 (j +,,3
1*1=3 '* 2 B' ( J~V 8 »te«'"i-4 0j UM,3-2d;(Ji11k2 t 2, j U !„«
8 I q12 |2 Kg+ 3) + 6 d)2 |2
o (2 Kt - 9) — 6 rr
+ 9 Qv r "v <22 ' 27 'J^"2 ~ IKe~6) I > I 9 (J HI2 - (Kjt-3)2 | (j+1)
9 p (Kt -9) f 54 \
An identical formula can be obtained from our comprehensive
model by putting v=3. »-±1 a n d i = ± 3 i n o u r » o d e l a n d
tituting A, K• IT, p as linear combinations of A B f v 11
'1jjana V C h O S e n 2 e r o- T a b l e V I " shows large
differences between the two calculational values when the
r m S ^JJ a n d ^jk a r e included. The comprehensive model.
6$
which contains both i\ and n i« * JK' therefore seen to be
superior to the Bauer formula.
constant set in the v10=2 vibrational state was used
as a trial constant set to obtain approximate constants for
the vio=3 level. Good agreement with the experimental data
for all measurements was not found. Systematic departures
between observed values and calculated values were studied
to determine additional procedures to be used for refining
the calculations. Inclusion of the higher order diagonal
terms ^ r e m o v e d s y s t e n a t l c M a s ^ ^ s a t i s f a c t o r y
convergence values for 1^-0.00123 and 1JK=0.0022 which were
obtained from the experimental data. The J-type doubling
constant q„= 4.1778 was obtained from the data set of K£=l
quantum states. Prakash and Roberts- previously published
the rotational spectra of propyne in the 17-52 GHz region
for the vI0=3, vibrational state. This work has been
extended with new observations of the rotational components
of propyne for the J»4<-3 transition and good agreement has
been found between these data and the comprehensive model.
E. The v 1 0 = 4 Spectrum
In 1977, Bauer and Godon™ published a paper on the
microwave spectra of methyl isocyanide in the v„= 4 state.
To assign their observed rotational frequencies, they used
66
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68
an analytical formulation giving the frequency of an excited
egenerate state vs=4, which has been obtained by Tarrago
and shared by private communication with them.
There are three differences between Bauer et. al.'s
formula and our comprehensive model. First, they added one
more non-diagonal Hamiltonian matrix element to their energy
and frequency derivation than our comprehensive model; the
element <K., , H+|K,U6> whose coefficient is g.. secondly,
they ignored higher order diagonal terms „ , w h i c h l s
r e a S O n W h y t h e " difference between the
two calculated values in Table IX. The values given for n
=0.003 and -1JK=0.006 presented in the summary table were "
Obtained from the observed frequency values. Also, to get a
better fit, f24=0.007 was assigned in this highly excited
vibrational state.- The weak intensity due to the low
population in this state caused problems to detect all
frequencies in the 17-52 GHz region. The weak signals
produced larger experimental errors than were encountered in
the ground, v10=l,2 and 3 components.
F. The General Form of the Rotational Constants
All the rotational constants of the propyne molecule from
ground state to v10=4 state are shown in Table X. From
these constants, derived from experimentally observed
69
TABLE IX. Rotational Transition Frequencies for CH CCH inthe v 1 Q= 4 Excited Vibrational State 3
0 0 0
2 2 2 2 2
0 ±2 ±4
0 ±2
K
0 0 0
0 0
±1 ±2 - 1 +2 0 ±4
+ 1 ±4 ±1 ± 4 '
±1 0 + 1 - 2 ±2 ±2 ±1 ±4 * ±2 ±4
a Pexp
(MHZ)
b cal
(MHZ)
c Peal
(MHZ)
17 278.585 17 278.585 17 278.278 17 278.257 17 278.257 17 277.010 17 276.831 17 276.831
34 34 34 34 34 34 34
51 51 51 51 51
556.808 556.808 556.808 553.348 553.348 547.348 558.234
833.821 870.727 836.464 838.729 848.731
34 34 34 34 34 34 34
51 51 51 51 51
556.955 556.617 556.264 553.648 553.609 547.776 558.381
833.318 872.357 835.316 837.428 844.643
34 34 34 34 34 34 34
51 51 51 51 51
556.955 556.617 556.504 553.408 553.609 547.296 558.861
833.318 870.629 837.188 838.868 848.387
a Roberts and Rhee. (79) b Bauer and Godon. (78) c Our Comprehensive Model.
(pa.pb) (pa.pc)
(MHZ) (MH2)
+0.021 +0.179
-0.147 +0.191 +0.544 -0.300 -0.261 -0.428 -0.147
+0.503 -1.630 +1.148 +1.301 +4.088
+0.021 +0.179
-0.147 +0.191 +0.304 -0.060 -0.261 +0.052 -0.627
+0.503 +0.098 -0.724 -0.139 +0.344
70
frequencies. we can predict t h e g e n e r a l ^
a general power series to predict the constants for any
vibrational level. Sufficient information was obtained for
the following constants to predict the trend and express
each constant in a power series form. The other constants,
f.z- f2s- f24, fljK etc... could not be so expressed
with the data available. Table XI gives the general form of
a n rotational constants of the propyne molecule for which
sufficient data could be obtained to express them in a power
series form.
The constant set predicted for v10.5 from the general
form is reported in Table X.
G. The Vj 0 = 5 Spectrum
By putting the predicted constants set and v=5, t=±l,
*=±3, f ± 5 into the comprehensive model, one can predict the
rotational spectra of propyne in the v,„=5 state. Those
predicted calculational frequencies are shown in Table XII.
Due to the weak intensity and low population, we could
not detect experimental data with our conventional video
spectrometer for the frequency range accessible to us.
Hopefully, we can obtain the observed frequencies in this
high vibrational state by increasing the sensitivity of the
detection system and by obtaining microwave oscillators
71
<D JZ -P
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in CO o CO cm CD CO
00
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h*
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72
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00 >
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TJ
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in o +
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73
TABLE XII. Calculated Rotational Transition Frequencies for CH CCH in the v = 5 E^ci?ed Vibrational State.3 1 0 excited
J K 2 J k e ^ (MHz) (MHJ
0 0 ±1 17 320.008 0 0 ±3 17 319.546 0 0 ±5 17 318.574
-1 -1 34 540.548 +1 +1 34 740.314 +1 34 638.016 0 ±1 34 639.636
~1 ±3 34 634.620 0 13 34 639.052
±1 ±3 34 642.257 71 ±5 34 630.248 0 ±5 34 637.080
±1 ±5 34 642.599
2 -1 - 1 51 810.713 2 + 1 +1 52 110.362 2 +2 +1 51 951.313 2 +1 ±1 51 956.199 2 0 .±1 51 958.506 2 ±2 ±1 51 960.574 2 +2+3 51 941.457 2 +1 ±3 51 951.337 2 0 ±3 51 958.482 2 ±1 ±3 51 963.804 2 ±2 ±3 51 964.433 2 +2 ±5 51 930.092 2 +1 ±5 51 944.307 2 0 ±5 51 955.453 2 ±1 ±5 51 964.627 2 ±2 ±5 51 972.968
CHAPTER VI
CONCLUSIONS
Some of the problems of the cure m f . f r°tataonal spectrum of a
Vlbrating-rotating polyatomic molecule has been
investigated. Both classical and guantum mechanical
approaches have been presented to explain the physical
significance of some of the gross features of the rotational
spectra of linear and symmetric top molecules. Vibrational
• ects to the rotational spectra have been studied in the
excited doubly degenerate vibrational states. The general
quantum mechanical Hamiltonian for vibrating-rotating
polyatomic molecule of arbitrary symmetry has been
discussed. A complete procedure to get the solution of the
fourth order series expansion of this Hamiltonian through the use of successive contact tran«f« *. •
act transformations has been outlined. By using Bauer's Hamiif •
S Hamj-ltonaan matrix elements which are derived for the r
3y symmetry molecules a
comprehensive model for predicting the rotational spectra of
polyatomic molecules of c J V symmetry has been developed from
asic perturbation theory. As test of this model
comparisons between this model and other formulas have been
made and very good agreement between the comprehensive model
r m 0 d e l S ' M h e" Edifications have been made in the non-general formulas, has been found.
75
76
A total of 86 rotational transitions for the ground,
V " 1 - V l 0 " 2 , v " = 3 a n d v i o M Vibrational states of propyne
molecule have been reassigned by using the comprehensive
model. of these 86 lines, the transition of j=3*4 in v10=3
and all the lines in the v , v i b r a t i o n a l states were ''
measured here for the first time. Also, a total of 34
rotational constants for structure of the propyne molecule
have been derived for the first- r. .. tame from the microwave data
of the ground state to v, A=4 vibran™*! 4- q vibrational state which were
measured as part of this work.
For the first time general forms of the 11 different
rotational constants of the propyne molecule have been
achieved by analyzing the 88 rotational constants which were
obtained from the observed spectra of the ground state,
V..-1, 2, 3 and 4 vibrational states. These general forms
of the rotational constants of the propyne molecule allowed
US to predict the rotational constants of the v10=5
Vibrational state of this molecule. From these instants,
rotational spectra of the propyne molecule in the v,„=5
state can be predicted.
Some exceptional phenomena have been detected in the
v10=3 rotational spectrum of propyne molecule. It has been
shown that the observed spectra in the K-±l and ^ s t a t e s
diverge systematically from the calculational values. The
divergence increase with J according to
77
v <J,±1,±1) = 0.762 (J+l) + „ (J,±1,±1) 3.
where va(J,±l,±i) l s th e frequency predicted by the
model.
This departure may be caused by the ignorance of the terms
fi2(J+l) and fjj(J+l) l n the non-diagonal matrix elements of
the Hamiltonian for the rotational energy calculation.
Similar exceptional phenomena were reported in the v,-l
vibrational state." No satisfactory theoretical
explanations as to why this exception arises for this
special quantum number set has been found.
This study of the microwave rotational spectra of highly
excited vibrational states has been greatly complicated by
the low intensity of some of these transitions. This
problem could be partially solved by increasing the
sensitivity of the detection system and by obtaining the
microwave oscillators which will gi v e u s access to the
70-125 GHz frequency range as fundamentals.
APPENDIX A
THE v.. 2 ROTATIONAL SPECTRA OF PROPYNE
In order to test the comprehensive model for other than
the v,0 vibrations, an experimental test was run on the v,
spectrum of propyne. Some rotational components in the v,-2
vibrational state of propyne have been observed in the
frequency range 17-72 GHz region... Molecular constants for
this vibrational^ excited state have been determined from
more than U observed rotational transitions. See Table
XIII below for a list of these constants. Good agreement
between the experimental frequencies and the calculated
frequencies obtained from a constant set for the v,-2
spectrum has been found.
Some transitions in v,.2 state are overlapped by the
strong lines of v,-l state at 34 223.401 and 51 334.949
MHz. (See Fig. 6 ) I h i s o v e r l a p ^ ,t i m p o s s . M e ^ a s s. 9 n
S O me C O m P°" e n t s P«="ely. The rotational components of the
propyne molecule which were obtained in the v»=2 state are
given in the Table XIV.
78
79
Table XIII. Rotational Constants for Propyne in the v,-2
V 9=2
B V
— 8 556.1996 MHz A V
= 137 392 MHz D J = 0.00296 MHz
D J K = 0.1672 MHz
"J =
0.4687 MHz ^JK = 0.0 MHz
11J J = 0.0 MHz
q v = 2.2613 MHz X
11 = 150 000 MHz *3L 2 = -10 MHz
d i 2 0.0 i = 0.94
y n
= 0.0 f22=f2 4= 0. 0
80
T a b l e X I V .
S '« s°-Propyne t n e V 9 ~ 2 V i b r a t i o n a l L e v e l o f
K l -exptMHz) i/cal(MHz) (,e j<p-„C3|< ; ) ( M H l ,
0 0
1 1 1 1 1
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3
0 0
+ 1
0 +1 0
+1
12 +1
0 +2 +1
0 +1 12
13 12 +1 0
*3 +2 +1 0
+1 12 13
0 0
0 0
12 12 12
0 0 0
12 12 12 12 12
0 +0 0 0
12 12 12 12 12 +2 12
34 227.592
51 322.344 51 330.025
51 340.980 51 343.854
68 447.100 68 448.875
68 427.917
68 448.633 68 454.518 68 457.956
17 112.387 17 112.285
34 223.992 | 34 224.672 \ (1)
34 220.070 34 224.508 (1) 34 227.492 +0.100
51 332.922 51 335.750)
(1) 51 336.781 I (1)
51 321.328 51 329.961 51 336.617 (1)
+ 1.016 +0.064
51 341.094 51 343.742
-0.114 +0.112
38 436.781 68 443.453 68 447.234 68 448.594 68 413.953
-0.134 +0.281
68 428.156 68 439.672
-0.239
68 448.547 68 454.516 68 458.047 68 458.812
+0.086 +0.002 -0.091
( 1 )
£ 2237n?PPHdC?V * e , e r y S , ' °n9 <"e
vg i at J4 223.401 and 51 334.949 MHz.
APPENDIX B
THE LINEAR MOLECULE AND SYMMETRIC TOP MOLECULES AS A RIGID ROTOR
sically the angular momentum and energy of rotation
of a rigid body is represented by
r 2 (B 1)
p = i" (B 2)
By choosing *,y,z as body fixed principal axis and the
and x - s as the corresponding angular momenta and foments of
inertia respectively, Eq.(B 1) becomes
1 / P^ P^ E = — / x y F r
i / x y p2 x
2 > T + T + T " ) (B 3 )
V T T -L
For the quantum mechanical treatment, E and the P's are
replaced by the Hamiltonian operator and corresponding
angular momentum operators. I h e m e c h a n i c a l
expression for a rigid rotor is given by
1 /P2 p2 p2
H = I ( x . v . P 1 ( X y ?! x r 2 ( — + _ . + ) ( B 4 )
X " 1 y z
81
82
where , p s are appropriate operators
The total angular momentum squared ,P2, is g i v e n b y
P Z Px + Fy + Pz (B 5)
ForQa linear molecule, with 2 assumed as the molecular axis, I = I z X y '
Eq.(B 4) becomes
X
H • — P2 / X r 21 ( B 7 )
X
By using the matrix elements for pz and H r from Eqs.(l-
« M 1 - 7) and the definitions of rotational constants from
Eq.(l-13), the energy of rotation for a linear molecule
under the rigid rotor assumption is easily shown to be
Er * h (B 8)
For a prolate symmetric top molecule, z assumed as the
symmetry axis and I, - Iy > V From Eqs.(B 4) and (B 5,
P2 P2 ,1 1 H = — , __z r 1 \ r 21 2L7~7) (B9)
x Z X
83
By using the same eigenvalue equations (1- 6 M l . 7 ) t h e
definitions of rotational constants from Eqs.(1-13),(1-14),
the rotational energy for a prolate symmetric top molecule
with the rigid rotor assumption is shown to be
Er(J,K) = h [ BJ(J+l) + (A-B)K2] (BIO)
APPENDIX c
DISTORTABLE ROTOR FOR PROLATE SYMMETRIC TOP MOLECULES
centrifugal stretching is treated as a perturbation on
the eigenstates of the rigid rotor. If Hr represents the
Hamiltonian of the rigid rotor anrf w g a rotor and Hd represents that of the
distortional energy, the total rotational Hanitoni, tan is
H * H + H . r d (CI)
Reference 10 has shown that the centrifugal distortional
Hamiltonian has the form
4
d T y., Sy.o*1 PS PY p5 (C 2) H, = Jl Z
"here represent the principal coordinate
axes of the moments of inertia and where each must
be summed over all three coordinate axes.
The first order perturbation treatment HJ has the form
H 1 = 1 V ' 2 , d 4 Taot08 P
a p g (C 3 )
84-
85
^ — Ar ' 4 - — r T p .
X X X X X
> 2 D 2 ^ ^ 2
" T [ T X X X X P V + T ' P 4 + T ' P 4 " J. ' xxxx X Y Y Y Y Y ^ Z Z 2 ZP Z + T X X Y Y
(PxPy + V x ' +Txx2z<PxPz + P?P?) z X
+ T' (P^P2 + p2p2\i yyzz v v z + P ^ P J J yyzz y z + P
ZPY>
Where x,y,z represent the principal axes and z
represents the symmetry axis of the top.
The definition of the coefficients in the distortional
Hani.iltonian is given by
T' = h V Tf h
vv w J"1 L # T xxxx Txxxx ' T X X Z Z ~ H ( T X X Z Z + 2 T ) , t * xxzz xzxz' - H T . R '
yyyy ^Tyyyy ' Txxyy (Txxyy + 2xxyxy) (c 4)
T z z z z " h T z z z z ' Tyy« = h ( Tyy" + 2Vzyz>
For a symmetric top molecule
r zzzz
Txxxx Tyyyy T;
W = Vyzz T x x y y = X x x x x _ 2 x ^ (c 5 )
xzxz ~ T y z y z
A general definition of the distortion coefficients has been
designated in the literature as
h4
D =- — x "J 4 xxxx
h4 ~2D — — (x +2T )
K J 2 x x z z xzxz (C 6)
h4 D = -D-D -il T
^ 4 zzzz
86
The first order energy correction in the perturbation due
to the centrifugal distortion is
Ed " < J- K iHfll J,K> { c ? )
By manipulating and rearranging the coefficients and using
the definitions for 0 j, D j r, in the Eqs.fc 3),(c 4).(c
)<{C 6) and (c 7), the resulting expression has the form
E d = " H [ D J 2 ^ J + 1 ) 2 + D J(J+1)K2+D K4] (C 8) a j J K ^ J o;
Addition of ,1 to the rigid rotor values of Eq.(B10) gives
the usual expression for the rotational energy of the
non-rigid prolate symmetric top molecule with centrifugal
distortion in the following form.
e\j,K)= h[BJ(j+l) + (a_b)K2-d J2(j+i)2 J
-D J(J+1)R2-D K«] ( R Q\ JK K (C 9)
APPENDIX D
ROTATIONALJNRAGY OF SYMMETRIC TOP MOLECULES N THE EXCITED VIBRATIONAL STATE
The first attempts to identify the splitting caused by
the Coriolis interaction in the rotational spectra was by
Teller , and Johnston and Dennison34.
They approached the problem along the following lines.
" *x.Py. and P2represent the over-all angular momentum
about principal axis including that caused by pure rotation
of the molecule and that caused by vibration, the pure
rotational Hamiltonian can be expressed as
« = l f £ f ^ 2 +
( V V 2 + < y p 2 >
2
2 I* (D 1)
where Px'Py.P2 represent the components of the angular
momentum which arise from the degenerate vibrational
motions.
For a prolate symmetric top in a doubly degenerate
vibrational mod. with the symmetry axis taken as the z axis
1 = 1 = 1 , 1 = 1 x y b z a ' (D 2)
and
P = P = 0 . x Y (D 3)
87
88
Eq.fD 1) can be rewritten by using the E<js.(D 2) and (D 3)
as follows
p2 p2 (p -p )2 H = Z +. Z 2
2ib 2zb — «
= + 4 CT-4> p2 b 2 xa zb 2 ! a
m the last expression the tern, pz/2^ was neglected, which
represents pure vibrational energy and which does not change
with the rotational state.
By making some substitutions
P 2 = £2 J(J+1)
P = K and z
P„ = ? Z fi z
The final form of the rotational energy of a symmetric top
molecule in the doubly degenerate vibrational state can be
expressed as
E (J.K. I), h [E J(J+1) + ( J B , p . 2 f l K A ] ( D 5 )
V V v '
Where : rotational constants in the excited
vibrational state
? • Coriolis coupling constant.
APPENDIX E
ACCIDENTAL DEGENERACY
Quantum number sets K*~2 cause accidental degeneracies
which make the energy difference in of Eq.(3-ll) zero.
TO remove this undesired case, a direct diagonalization
method was employed for thes£> j. jt xwt tnese special quantum sets. This
procedure is given schematically as follows.
Hd^d E^d d : diagonal.
H jp= \rp
|H -Alj = 0
To solve this secular equation, Table III was used,
V'J'K'£ V,J,K+2,£-4 V,J,K-2,J£+4
v,J,K i
v,J,K+2,1-4
VfJ,K-2,£+4
E"A f24a2b4
f24a2b4
f24a2b4
f24a2b4
= 0
E-A
NO interactions between (K«,i-4) and (K-2,t+4) were
assumed.
89
90
(E-A) [(E-A)2-(f24a+b;)2]-(E-X) (f24a"b+)
2= 0
A = E
* - E±f24[(a2+b4-)
2 + (a-bj)2]3*
From the Table III
4 = [ J ( J + 1 ) - K ( K ± 1 ) ] ^ [ J ( J + l ) - { K + l ) ( K ± 2 ) ] ^
t>4=[(v+I+l±l) (v-ulfl) (v+il+1+3) (v-Jl+153) ]
A becomes
A = E ± f24{ [ ( J ( J + 1)"K(K+l)] [J(J+1)-(K+1) ( K + 2 ) J
(v+A)(v-£+2)(v+i-2) (V-£+4)+[(J(J+I ). K ( K_ 1 ) ]
[ (J+l) J- (K-l) (K-2) ] (v+£+2) (v-£) (v+A+4) (v-£-2)j
if Kl= -2 then K=l,*=-2 and K=-l,£=2.(see table IV)
(D 6) becomes
A = E ± f24 J ( J + 1) v(v+2)
Applying the selection rules for absorption spectra,
v ~ ± 2 ^24J J+^-) v (v+2) if K£=-2 .
APPENDIX F
PROGRAM TO ASSIGN v,„ - n ROTATIONAL CONSTANTS
The computer program developed for calculations in this
work is recorded here for convenience.
This program can be used to assign any rotational
constants in a n y vibrational states of propyne by putting
the proper v and the observed frequency values. This
program is an example to determine q 1 2, d 1 2 and x ^ for the
v,„= 4 rotational components of propyne. The machine
language of this program is FORTRAN IV.
-PROGRAM-
S N M ; , z-fYF ^ '
S S S S S S S S K P ? A £ ? W K j S 3 ' 3 4 ' S « ' " . S8. S5
DIMENSION VAL(45), EXPVAL(26) DATA EXPVAL/
v=4 .0 TOL=0.2 NUM=15
91
92
D l = •' 1 6 3 4 2 3 D 0 + V * 4 ? 4 6 2 D - 4 ^ 8 5 5 D ° * V * * 2 ~ " 0 0 2 1 8 3 3 3 D 0 * V * *3 S N l = . 3 3 3 5 D 0 + V * 9 . 1 0 5 8 D - 4 SN2=0.0 SN3=0.0
D2=. 00292D0+V*9 . 5 9 9 9 l D - 5 + V * * 2 * 2 00048D e, Q = 4 . 2 0 1 8 D 0 - 8 . 0 0 0 1 8 D - 3 * V ' 0 0 0 4 8 D 6
A = 1 5 8 8 5 3 . D O - 2 5 5 7 . 5 2 D 0 * V + 5 1 * v * * 2 P S O ; J 7 8 6 6 5 D 0 * V + • 111914D0»V« « 2 - 3 . 1 3 3 2 2 D - 2 « V " 3 + 3 . 083
F 2 = 0 . 0 F 3 = . 0 0 7
UPTTTTId 11\
FORMAT(IX, - B I - , 2 5 * . ' D 3 ' , 2 5 X , • * ! • , 3 5 X , 'FREQ' ,
DO 900 I Q = 1 , 5 0 , 1 Q l - I Q
901 Q l — Q l
DO 800 I D = 1 , 2 0 0 , 1 D3=ID
DO 700 I X = 1 4 0 0 0 0 , 1 7 5 0 0 0 , 1 0 0 X1= IX
IND=0
DO 600 I L = 1 , 5 , 2 L = I L - 1
601 L = - L
DO 500 I J = 1 , 3 J = I J - 1
DO 400 I K = 1 , I j K = I K - 1
401 K=-K
F 5 = 4 * ( J + l ) * ( Q + 2 * F 2 * ( J + l ) * * 2 ) — ( ^ + 1 ) * (B1 + G * L * * 2 - D 1 * K * * 2 + SN1 * tf * r , e M 5 icvic * «
S l = S l M * D 2 M J + l ) - 3 + 4 * S N 2 * K * L * u i l ) - 3 S N 3 K 3 L )
93
D6-4*X1* (L-l) - (A-Bl) * (2*K+1) -2*A*Z * (7*K m i 07=4*X1*(L+1) + (A-B1)*(2«K-1 -2*A*Z*\-£K+TJ>\
•* ,(f^' + JA"B1 > * <K+l >-A«Z* (K^L+2) * ^ ( *"1 ) + (A-Bl) * (K-l) -A*Z* (K+L-2)
C2=(V+L)*(V-L+2)*(J+l) C3=(V-L)*(V+L+2)*(J+l)
S3=C3*Q**2*((J+1)**2-(K+1)**2)/D8
IF ((D8.EQ.0.0) .AND. (V.EQ.1.0) ) S^--T?*
IF ((D8.EQ.0.0) .AND. (V.EQ.3.0) ) ffl-PpS
S4=C2*Q**2*((J+1)**2-(K-1)**2)/D9
IF ( (D9.EQ.0.0) .AND. (V.EQ.1.0) ) S4=F5 IF ((D9.EQ.0.0).AND. (V.EQ.3.0) ) S4=2*F5 S6=2*C2*(Q1*(2*K+1)+D3*(L-1))**2/D6
( 2* K _ 1 ) + D 3*
S8=S6-S7
IND=IND+1 VAL(IND)=S1+S5+S8
IF (K.LT.O) GOTO 401 400 CONTINUE 500 CONTINUE
IF (L.LT.O) GOTO 601 600 CONTINUE
NFREQ=0 DO 300 11=1,45 DO 200 JJ=1,26 IF ((VAL(II)-TOL).GT.EXPVAL(JJ)) GOTO 200 */((VAL(II)+TOL).LT. EXPVAL (J J) ) GOTO 300 NFREQ=NFREQ+1 w i u JUO GOTO 300
200 CONTINUE 300 CONTINUE
APPENDIX G
PROGRAM TO CALCULATE Vt- n ROTATIONAL
TRANSITION FREQUENCIES
This program can be used to calculate rotational transition
frequencies for any excited vibrational states of any c
molecule for which rotational constants have been derived!'
This program is an example for calculation of rotational spectra of
molecule in any excited vibrational states by changing v and L.
C COMPREHENSIVE MODEL FOR C
sJSf' G = S N 2 : £ 1 = D £ & - ^ = D ' Q ^ v ' Q 1 = q i 2 ' D 3 = di 2 ' 'SN2-i|jj,SN3=*|JirFl=fl2,F2=f22,F3=f*!
C c
REAL L REAL J REAL K
V=2 L=0
DOTBLE P R e S k n XI' F1DF2SP1' ? n 2 ; f 3 -02' 0' 21 - 03. A, Z DOUBLE S:«:S2!li!;6 T^ T?g ,S9 DS3 sS' s l
DOUBLE PRECISION S5,F9 ' ' ' ' '
COEFFICIENTS
i 4 5: 7 1 2 D 0 + 2 4 • 2046D0*V- . 11687D0*V**2 Bl+•0372304D0*V**3-.0101258D0*V**4
£=:163 9?23D0 0"M D46X:4 0 2 8 5 5 D O* V" 2-' 0 0 2 1 8 3 3 3 D O* V**3
3&0?L%2D0:^99.l0598D?45+V**2*2-0OC48D-6
SN2=0.0 SN3=0.0
95
96
C
Q-4.2018D0-8.00018D-3*V
' 9062D0*V+24 • 484D0*V**2-2 603R "?nn*v* * i D3=33.6602D0+50.1149D0*V-5.44498D0*V**2 A=158853•DO-2557.52D0*V+51*v**2 Z~ 1~ , 1 7 8 6 6 5 D 0* V +-111914D0 *V**?- '? „ ^ 5 « 6 4 - E 0 - 6 5 « - 1 2 D ° * V + l 5 4 0 0 2 M ' V » ? V 3 + 3' 0 8 32 d-3'V F1=0.0 F2=0.0 F3=0.0
DO 750 JJ=1,4
JK=2*J+1
DO 740 KK=1,JK K=KK-(J+l)
C DIAGONAL TERMS ^ * ( J + l ) * ( B 1 + G * L * * 2 - D 1 * K * * 2 + S N 1 * K * T J - c m i * V * * o * t \
T1=~4*D2*(J+l)**3+4*sn2*K*L*(J+ll**3 D9=T0+T1
C NON-DIAGONAL TERMS 1ST TERM
X 1 (l+1)+(a-B1)*(K+1)-A*Z*(K+L+2)
SMli-if/iii1 ' A N D' K' E Q'" 1' 0 0 1 0 350
s L ™ i E 9 ' 3 - A N D ' L - E S ' 1 • A N C - K • E Q • 1 ) G O T O 3 9 0 S1=SM1
G O T O 4 1 0
350 S1=-2*Q*(J+1)*(V+1) G O T O 4 1 0
390 Sl=.762 *(J+l)+SM1
2 N D T E R M
4 1 0 ?c:iy^T)*1
(T"^+2)*(j+1)*((J+l)**2-(K-l)**2)*Q**2
(L~l)+(A-Bl)*(K-l)-A*Z*((K+L-2)
IF (L.EQ.l .AND. K.EQ.l) G O T O 480
97
SM2=T4/T5
(V. EQ . 3 .AND. L.EQ. -1 AND Tf Fn i N r- ^ S2=SM2 -awu. K.EQ.-l) GOTO 520 S2=SM2
GOTO 540
480 S2=2*Q*(J+l)*(V+l) GOTO 540
520 S2=.762*(J+1)+SM2
c 3RD TERM
I ^ « / ^ < 1 " 1 ) " < A " B 1 ) ' * < 2 * k + 1 ) - 2 * A * Z M 2 « K - £ + 2 ) " S3=T6/T7
4TH TERM ^ (V-L) * (V+L+2 ) * (J+l) * (Ql* (2*K-1) +r> 1* r 4.1 \
' -(A-B1»* ' 2
5TH TERM
TT? Ir'fH'2 *AND* K-EQ-"1) GOTO 660 n( L - E Q - 2 *AND* K.EQ.l) GOTO 680
oO = U
GOTO 730 660 S5=2*F3*V*(V+2)*(J+l)
GOTO 730 680 S5=-2*F3*V*(V+2)*(J+l)
C FREQUENCY(F9) • 730 F9=D9+S1+S2+S3+S4+S5
J8=J+1 J7=J K7=K L7=L
732 WRITE (6,735)K7,L7,J7,J8,F9 735 FORMAT ( ' K=' 12 ' r - • TO > T • -r-, .
V K. , 12, L- ,12, J= ,12,' TO J= ' , 12, ' F=' ,F12 . 5
740 CONTINUE
WRITE(6,745)
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