I

IIIIII

I

I
I
I

II

I

III

133
315

 

ROTATIONAL CONSTANTS OF
H238 AND 0232 FROM

THEIR INFRARED SPECTRA

Thesis for the Degree of M. S.
MECHIGAN STATE UNIVERSITY
John Tressler
1961

IMF—SIS

III III IIIIIIIIIIII IIIIIIIII IIIIII

312 301

 

L I B R A R Y
Michigan State
University

w“

202 .

 

4. PHYSICS [:I

H

J '1‘

4‘!

h
H

V

i i

II‘ll‘tl IllideI

ROTATIONAL CONSTANTS OF HZSe AND DZSe

FROM THEIR INFRARED SPECTRA

BY

John Tressler

AN ABSTRACT OF A THESIS

Submitted to
the College of Science and Arts,
Michigan State University
of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE IN PHYSICS

Department of Physics and Astronomy

1961

ABSTRACT

An analysis of the infrared spectra of HZSe and DZSe is made in
order to determine the rotational constants and the first order
stretching coefficients of these molecules. The analysis utilizes
the energy moment treatment of the quantum mechanical asym-

metric rotator developed by Parker and Brown.

ROTATIONAL CONSTANTS OF HZSe AND DZSe

FROM THEIR INFRARED SPECTRA

I
I ‘1‘}
.. (I.

111:“
John Tre s sler

A THESIS

Submitted to
the College of Science and Arts,
Michigan State University
of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE IN PHYSICS

Department of Physics and Astronomy

Acknowledgments

The author wishes to express gratitude to Dr. Paul M.
Parker for his interest in and helpful guidance with this work.
Thanks are due also to the National Science Foundation

for the fellowship which in part made this work possible.

ii

IV

Table of contents

Introduction
The or y

Analysis

Summary

iii

Page

 

16

36

List of table 3

 

 

Table Page
I Calculated parameters of the HZSe molecule 18
II Calculated parameters of the HZSe molecule 19
III Calculated parameters of the Dzse molecule 20
IV Calculated parameters of the DZSe molecule 21
V Values of Si/Pi for the HZSe molecule 22
VI Values of 51/ P1 for the HzSe molecule 22
VII Values of Si/ Pi for the DZSe molecule 23
VIII Values of 51/ Pi for the DZSe molecule 23

IX Calculated parameters of the HZSe and the
D286 molecules 31

X Stretching coefficients and centrifugal

correction term coefficients 33

XI SloPes and intercepts of Si/Pi vs. J(J+l) 34
XII Centrifugal correction term coefficients 37
XIII Rotational constants 38

iv

 

List of figure 3

Figure Page
1 Geometry of the HZSe and DZSe molecules 16
2 Plot of Si/Pi vs. J(J+l) for HZSe 25
3 Plot of Sz/Pz vs. J(J+l) for HZSe 26
4 Plot of S3/P3 vs. J(J+l) for ste 27
5 Plot of Si/Pi vs. J(J+l) for D256 28
6 Plot of Sz/PZ vs. J(J+l) for DZSe 29
7 Plot of S3/P3 vs. J(J+l) for DZSe 30

Introduction

In order to discuss the rotational energies of nonlinear
polyatomic molecules, it is both customary and convenient to
classify them according to the relative magnitudes of their prin-
cipal moments of inertia. The principal moments of inertia are
expressed in terms of a body-fixed coordinate system which has its
origin at the center of mass of the molecule and is so oriented that
the product of inertia -terms are all zero. Depending upon the
geometry of the molecule, the three moments of inertia may all
have the same value, or two may have the same value but differ
from the third, or all three may have different values. The mol-
ecules are then classified as spherical tops, symmetrical t0ps,
and asymmetrical tops respectively.

The vibration—rotation energies of asymmetrical t0ps can
be formulated in terms of a quantum mechanical Hamiltonian
function. 1 To a first approximation the vibrational energy of the
molecule is that of an ensemble of harmonic oscillators, and the
rotational part of the energy is that of a semi-rigid microphysical
rotator.

The total energy expressed by the Hamiltonian can then

in this approximation be set down as the sum of two terms, a

 

1 H. H. Nielsen, Rev. Mod. Physics 23, 90 (1951)

1

2

vibrational and a rotational term. If the forces of attraction be-
tween the atoms of a molecule are considered strong enough to
hold the molecule almost rigid, it is found to a good approximat-
ion that the rotational energy can be separated from the vibration-
al energy. Thus, if one considers transitions for which the
electronic configuration remains unchanged and for which the
vibrational energy remains unchanged, then the case dealt with
is one of "pure rotation", and the rotational energies are added
to the energy of vibration of the molecule vibrating in one of its
normal modes. When this approximation is inadequate, a second-
order correction term involving both rotation and vibration is
necessary to account for the change in the average moment of
inertia due to vibration-rotation interaction, and depending on
the molecule and energy levels considered a correction term
may be needed to adjust for the change in the moment of inertia
caused by centrifugal stretching of the molecule.

In their literature, Parker and Brownz' 3’ 4 describe
a procedure which relates the energies of a stationary quantum
mechanical system to the physical constants appearing in the

Hamiltonian describing this system.

 

P. M. Parker and L.C. Brown, Amer. J. Phys. 21, 509 (1959)
P. M. Parker and L.C. Brown, J. Chem. Phys. 21, 1108, (1957)
P. M. Parker and L.C. Brown, J. Chem. Phys. Q, 909 (1959)

ADON

3

Utilization of this method makes it possible to compute
the effective rotational constants of a molecule from spectro-
scopic data. The method described by Parker and Brown deals
not only with the rigid rotator, 3’ 4 but is extended to include first
order centrifugal stretching effects. 5

The present work utilizes this energy moment treatment
of the quantum mechanical asymmetrical rotator developed by
Parker and Brown3’ 4' 5 for the analysis of the infrared spectral
data of DZSe and HZSe obtained by Palik and Oetjen. 6’ 7 The
effective rotational constants of DZSe and HZSe are calculated for
the rigid case; the stretching coefficients as well as the rotational
constants are determined for the non-rigid case.

An error analysis is made and the results are com-

pared with the rotational constants which Palik and Oetjené’ 7 ob-

tained by applying a different method of analysis to the same

 

data.

3 P. M. Parker and L.C. Brown, J. Chem. Phys. 31, 1108, (1957)
4 P. M. Parker and L.C. Brown, J. Chem. Phys. 39, 909 (1959)
2 P. M. Parker and L.C. Brown, J. Chem. Phys. 31, 1227, (1959)

E.D. Palik and R.A. Oetjen, J. Molecular Spectroscopy, l, 223,
(1959)
E.D. Palik and R. A. Oetjen, J. Molecular Spectroscopy, _3, 259,
(1959)

.4

11

Theory

(a) Rigid rotator analysis
The energies of a rigid quantum mechanical system can
be expressed by a Schroedinger equation,

i Wtk = PM k (2a.1)
where W is the appropriate Hamiltonian for the system, k enum-
erates the members of a supposedly complete set of energy
eigenfunctions / , and Fk their corresponding eigenvalues.

When it is possible to express the matrix elements
Wij of the Hamiltonian W in some convenient representation
Vk, the allowed energies of the system Fk are the roots of the
secular equation,

[wij - F5” = o, (2a.2)
where is the Kronecker delta symbol. This secular equation
can be expanded into a polynomial of degree n in the form,

F1 + chn'1 + can'2+.....+cn=O, (2a.3)
in which the n roots of F and the n coefficients form a complete
set of invariants of the matrix. The subsequent determination of
the physical constants of the system can in certain cases be effected
by equating values of Ci expressed in terms of experimentally de-

termined energy levels to the corresponding values of C]; expressed

in terms of the matrix elements W which contain the physical

parameters. Parker and Brown3 show that these constants Ci
can be expressed in terms of the first r moments of the energy
levels.
sr =27 (F )1‘ (2a. 4)
T = 1,2,3, ...,n.
and alternately by the equation,
51 = Tr(Wi) (2a. 5)
i = 1,2,3, ...,n .
where Sr represents the moments of the energy levels, and
Tr(Wi) are the traces of the submatrices of the Schroedinger
equation; their method is then develoPed in terms of moments of
the energy levels rather than in terms of the polynomial co-
efficients.
In applying this method to the rigid asymmetric ro-
tator problem, the Hamiltonian for the system is

w = H/hc = APXZ

2
+ BIPY2 + CPz . (2a. 6)
Here Px’ Py' and P2 are components of angular momentum of
the rotator in units of’h referred to the body-fixed coordinate
system which is defined by the diagonal inertia tensor; and
A - h/8FZI c B - h/87le c C - h/87TZI c (2a 7)
_ X i " y a " Z s .

The constants A, B, and C are the reciprocals of the effective

principal moments of inertia Ix, IY’ and 12 apart from universal

 

3 P. M. Parker and L.C. Brown, J. Chem. Phys. 31, 1108, (1957)

constants.

2

It is shown that the first three moments are adequate

in the determination of A, B, and C. The value of the first
moment S1 is obtained from the experimental data by adding the
(ZJ + 1) levels F corresponding to a fixed value of J; the electronic
state is assumed to remain fixed throughout the procedure. The

first moments is given by the equation

S1 =Z( (2a.8)

FT)
’T’ =J, J-l, ...,-J.

The mean value of the energies FJ corresponding to a given J is

next found by dividing 51 by the number of terms in the summat—
ion:

FJ = 51/2J+1. (2a. 9)

Subtraction of P from each of the (2J+l) values of F yields

J '1'

13;]? = (1:1; 5}). (2a.10)
This modifying process shifts the zero level equally for all
energies considered, but does not Change any of the physical as-
pects of the problem. The second and third moments can then be
evaluated by squaring and cubing the sum of the (2J+l) modified
energies,

52:2. (F (2a.11)

o )2
T
53 =2: (F537,)3 (2a.12)

 

2 P. M. Parker and L.C. Brown, Amer. J. Phys. _2__7_, 509 (1959)

It can then be shown that3’ 4

51 e pl ,1 (2a.13)
s2 = p2Y2(1+p 2/3) (2a.14)
$3 -'- psY 3(1-p 2) ~ (2a.15)

The quantities p1, p2, and p3 are polynomials in J as follows:

p1 = 2J(J+1)(2J+1)/3[ (2a.16)
p2 = 2J(J+l)(ZJ-1)(2J+l)(2J+3)/3(5!) (2a. 17)
p3 = 2J(J+1)(2J-3)(2J-1)(2J+1)(2J+3)(2J+5)/3(71) (2a. 18)

The parametercfi , which is a measure of an equivalent spherical
rotator, now follows from (2a. 13). The parameter/B measures
the deviation from a symmetrical top, and has a value that lies
in the range ~1< fl < 0. Its value can be determined by em—
ploying (2a.14) and (2a. 15) simultaneously to give
1 ~ 8 = (1-p2)2/(1+F2/3)3 = p23532/p32523 . (2a.19)

where (3 is first determined from the experimental data and ,8 is
then obtainable from a tabulation of F in terms of (5' . Having
determined P , thenY , which shows the deviation of the top
from spherical symmetry, is given by (2a.14).

When a , /8 , and T are known, A, B, and C can

be found by solving the equations

 

3 P. M. Parker and L.C. Brown, J. Chem. Phys. 2_7, 1108, (1957)
4 P. M. Parker and L.C. Brown, J. Chem. Phys. 33, 909 (1959)

8

A = (1/3IL'cc-Y/2u-p13 (2a.20)
= (1/3)E<£ -"r'/2(1+p)j (2a.21)
= (1/3)|:cr. + ’7’j . (2a. 22)

The principal moments of inertia readily follow from equations
(2a. 7).
(b) Non-rigid rotator analysis

The basic scheme of vibration-rotation energies is ex-
pressed by the harmonic oscillator and rigid rotator formulas.
However, a precise quantitativeiadjustment to spectral data must
consider both the anharmonicity of the potential energy and the '
centrifugal stretching of the rotating molecule. Only the effects
of stretching are considered here.

To correct for non-rigidity, one may set down a
Hamiltonian of the form

W=WO+W (2b.1)

1'
_ 2 2 2

W0 - APx + BPy + CPz (2b. 2)
is the Hamiltonian of the rigid rotator, and W1 is the centrifugal

distortion term which can be written to a first order in the

following form5:

2
W = 01P:2+ GZP4+ 0 P:+ 04(PZP: +Pz

1 y 3
05(Piip +P§2P x)+o~ 6(PZPZ+yPyP:.)

P (2b. 3)

2+
Y+

The coefficients 0‘ depend on the geometrical form and the

 

5 P. M. Parker and L.C. Brown, J. Chem. Phys. §_1_, 1227, (1959)

 

9
force constants of the molecule and their values can in principle
be determined from experimental data. The moment equations

of (2b. 1) are then shown to be5:

51/131?“ (cu-r113 (3/2)E011(3f—1) + d2(2f+1D; (210.4)

SZ/pz Y2 = (1+ 13 2/3) + 2(6f—5)7’1(1+’3F 1) - (2b.5)
4(£+5) 7.2““ [9,522) ;

s3/p3 T3 = (1- p21+ 9(f-2111+}a 2/311cL1 ”<3. 2) + (zb.6)

217’1(£—1)E(1- pz/3) —2 FF 1] _
42 7211(1— P2/3)— ZPP 2]

In these moment equations,

<9 1 =(15/2)’r)(1 1 =1, 2, 6, (2b.?)
¢1=()\1+)\Z+}\3). (2b.8)
C52=I7x4+7\5+)\6), (2b.9)
P1=(?\1-7\2)/(27\3-1\1-V\2). (2b.10)
52=('7\4-‘7\5)/(27\6-'“1\4-‘1~ 5). (212.11)
T1=115/14)(2\3 Wu A2). (210.12)
72 = (15/141(2~7\6 -\’\4 3h 5). (2b. 13)

f = J(J+1) (2b.14)

Application of the three moment equations to three different
values of J yields a set of nine equations which can be solved in
principle for the nine physical parameters cf, 73, 7’: (£1, fl 1,

"fl, (5,2, ,3 2, Y2. The stretching constants )1 can then

 

5 P. M. Parker and L.C. Brown, J. Chem Phys. 2, 1227, (1959)

10
be calculated by means of equations (2b. 8) through (2b. 13). The
constants 6‘ i then follow from (2b. 7).
(c) Non-rigid rotator analysis with ”initial conditions"
The procedure of calculating the values of the nine
physical parameters is simplified a great deal ide, P , and
Tare found by extrapolation to J = O as in a rigid rotator
analysis by employing the following method.
Equation (2b. 4) can be arranged in the form
51pr = at + 6/21“)” (<12 — 0C1) + <3/21TL'3c5l1 + 2&2] f.
(2c.l)
It is seen on a graph of 51 /p1 vs. f that (2c. 1) is the equation of
a straight line with intercept
ci+ (3/2)’)’(c1:.z - 6.1) (2c.2)
and with slope
(3/2)7/(3d:1 + zccz). (2c. 3)
Then if the "initial condition" is imposed that for J = 0
Sl/pl = d; , (2c.4)
it must follow that
(3/2)(<£2 .. 6/1) = 0, at, a (£2 (2c.5)
In imposing the "initial conditions” it is assumed that the Cf.
extrapolated to J = 0 obtained by the rigid analysis is the
"unstretched" or "true" value of 0):» . Using the values of C191

and (52 resulting from these conditions, the slope (Sl)1 of the

11
line in equation (2c. 1) is
(s1)1 = (15/2)‘Yc£1 . (2c.6)
Next, equation (2b. 5) can be arranged in the following form:
Sz/pz {(1+/82m - 107/1 - 10/9/9171 - 207’s - 20 Iii/9272172
+[12'Y‘1 + 12 1313173 - 4 ’13 — 4 P19172372 f, (2c.7)

again the equation of a straight line with the first bracketed term
equal to the intercept and the second bracketed term equal to the
slope of the line. Extrapolating to J = 0, it becomes
Sz/pz =7’Z[(1+fi2/3)-1o‘r1—1o [3,81]; zoppl 7’2]. (2c.8)
Imposing the condition that at J 3 0,
Sz/pz = 73(1 + [32/3) (2c.9)
requires that
71(1+pf1) +2'r2(1+,B/92) = o, (2c.lO)
and slope ($1 )2 then becomes
(SI)2 = ~28 TZTzIH PPz). 12c.11)
Now, sincecL 1 = dig, the third moments equation
arranged in sloPe-intercept form becomes
(S3/P3‘Y3 = 11- 189-2171 131- sz -2 PAD“
-42 T2 [(143 2/3) -2 Ffifl +21TIE1- P2/3 - 2 FPIII f .
(2c. 12)
This time'the condition imposed at J = 0 is
S3/p3 = 73(1- p2) , (2c.13)

requiring that

12

'Yl [(1-fi2/3 -2 HM] ~27; [(1- 52/3 —2pP2)] = o. (2c.14)

The slope (Sl)3 is then

(Sl)3 =2173?fi(1-F2/3 ~2Pfi1). (2c.15)

If the value of 71 is found from equation (2c. 10) and
then substituted in (2c. 14), the following equation results after
some algebraic manipulation:

(pa-pl)(p2/3_3)=o. (2c.16)
As F f 0, it follows that p 1 = p 2; and if this fact is used in con-
junction with equation (2c.10) it is found also that 7’1 = -2 7’2.
Solving equation (2c. 11) for 7’2 yields

7;, = -(51)2/ 2872(1+fifl2). (2c.17)
Here it is seen that 72 is expressed, with the exception of ’82,
in terms of parameters that can be determined by a rigid
analysis. The value of P 2 can be found, however, in terms of
similar parameters by solving (2c. 11) and (2c.15) simultaneously,
and by using the additional information that p 1 = ,5 2 and T2 =

“VI/2-

The resulting expression for P 2 is

52 = 37’(51)z(1—fi2/3) — 2(s1)3 ,
2 p[(s1)3 + 3751);] (2c.18)

Substitution ofcc 1, p1, and Y1 in terms ofdtz,

 

)3 z, and ’I’z re8pectively in equations (2. b8) through (2b.13)

leads to the equations for the stretching constants xi:

13

\1 = 1/3 [5321 —(7/15)’V1<1-3)91)]. (2c.19)
)\z=1/3 Efii ~17/15)’)’1(1+3,31)j. (2c.20)
\3 =1/3 [3:1 +(14/15)’)’1], (2c.21)

\4 1/332 -(7/15)')’2(1-3Bz)], (2c.22)
X5 1/3ECZ ~(7/15)Yz(1+3#2).]. i (2c.23)

he 1/3ECZ +(14/15)9’2]. (2c.24)

If it is possible to obtainoc, )8, and 7’a1ong with (s1)1, (51);,

and ($1 )3 from an analysis of experimental data, (Ll can then be
found by (2c. 6), Y2 by (2c.17) and ,8 2 by (2c.18). All stretch-
ing constants, )\1, follow from equations (2c. 19) through
(2c. 24). Knowledge of these values then makes it possible to
determine the coefficients 0'1, of the centrifugal stretching
correction terms by employing equation (2b. 7).

(d) Intercept-slope method of analysis

If the rotating molecule were strictly a rigid rotator,

the effective values och , P , and Vwould be constants un-

affected by changes in angular momentum as shown by the equations:

Sl/pl =00, (2d.1)
Sz/Pz =7’Z(1+flz/3), (2.1.2)
S3/p3 =73(1-,BZ). (2d. 3)

Graphs of Si/Pl vs. J(J+l) are straight lines with zero slope show-
ing that the constants are unaffected by a change in the angular

momentum quantum number J.

V"

14
Analysis of experimental data shows that stretching
due to centrifugal force does occur, resulting in Changes in
these parameters. One may, however, attempt to find the
value of 51/13}. for J = 0 by fitting the effective values of Si/Pi

obtained from an analysis of experimental data to curves of

the form
Si/pl = ct. +791. (2d. 4)
s2/p_2 = 7211 +13 2/3) +247}. (2.1.5)
83/133 = 730 -,8 2) was. (2d.6)

where a straight line fit is obtained by applying the method of
least squares. 8 The,” i represent the slopes of the curves
and are to be interpreted as a measure of the deviation from
the zero-order rigid rotator theory; and in order for this
first order correction to be adequate for a given system, the
slopes must be very small.

Where this correction is adequate, extrapolation of
the Si/Pi curves to J = 0 enables one to determine the "true"
values of the principal moments of inertia.

The first order theory is approximate even for small
values of J; even for moleculesthat are not very "stretchy” the
approximation becomes progressively poorer at higher values

of J and higher order correction terms become necessary. In

 

8 See for example C. G. Lambe, Elements of Statistics, Chapter
VII, (Longmans Green and Co. , New York, 1952)

 

 

15

this case, the Si/Pi vs. f curves would no longer be straight

lines. 5

 

5 P. M. Parker and L.C. Brown, J. Chem Phys. 2_l_, 1227, (1959)

111

Analysis

Symmetric, nonlinear, triatomic molecules like HZSe
and DZSe have three fundamental vibration- rotation bands, 'V 1,
”V2, and 1/3. Associated with the fundamental modes of vibration
are the quantum numbers V1, V2, and V3; which serve to
describe the vibrational energy state. In the following procedure,
analyses of the rotational levels associated with a fixed mode of
vibration are carried out.

The geometry for both DZSe and HZSe is shown in Fig.
1, where it is seen that the principal axes are so oriented that
the Z axis is normal to the plane of the molecule. Both molecules
are asymmetric rotators with principal moments of inertia

Iz>Iy)Ix3 and rotational constants A>B>C.

z 2

Se x
H ,,/ ' . H s
/ I

H2 Se

 

 

 

 

 

 

Figure 1. Geometry of the HZSe and DZSe molecules.

16

of Palik and Oetjen

17

The data used in the analysis were taken from the works

6,7

as follows:

HZSe for the vibrational state V1 = V2 = V3 = O;

6

J O to J 7 from the first paper ,

7

J 8 to J 10 from the second paper .

HZSe for the vibrational state V1 = V3 = 0, V2 = 1;

J = l to J = 6 from the second paper7.

DZSe for the vibrational state V1 = V2 = V3 = O;

J = O to J = 6 from the second paper7.

DZSe for the vibrational state Vl = V3 = 0, V2 = l;

J = l to J = 3 from the second paper7.

Following the procedure discussed in part II (a), and

employing equations (2a. 8) through (2a. 21) a rigid analysis of the

preceding data was carried out. Five significant figures were re-

tained in all calculations and the physical parameters resulting

from the analysis are listed in Table I, Table II, Table III, and

Table IV. These are the effective values of the parameters.

Next, the values of Si/Pi corresponding to a given

value of J were determined from the data given in Tables I

through IV. The Si/Pi along with the J(J+l) for a given J are

tabulated in Tables V through VIII. Preliminary plots of Si/Pi vs.

 

6

E.D. Palik and R. A. Oetjen, J. Molecular SpectrosCOpy, _1_, 223,
(1959)
E.D. Palik and R. A. Oetjen, J. Molecular Spectroscopy, 3, 259,
(1959)

 

18

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Table V - 51/ Pi for the HZSe molecule in the vibrational state *

22

 

 

 

V1=V2=V3=0
J sl/Pl SZ/PZ -S3/P3
1 19.785 65.274 504.43
2 19.786 65.171 501.50
3 19.782 65.114 501.89
4 19.773 65.006 502.15
5 19.764 64.830 501.77
6 19.750 64.523 499.68
7 19.735 64.234 497.26
8 19.719 63.932 494.84
9 19.700 63.518 491.76
10 19.678 63.125 488.70

 

Table VI - Si/ Pi for the HzSe molecule in the vibrational state *

VI=V3=O;V2-l

 

 

 

J SI/pl Sz/Pz -S3/P3
1 19.670 91.602 843.23
2 20.167 74.163 608.11
3 20.159 73.810 604.73
4 20.150 73.865 602.52
5 20.139 72.273 603.26

 

units of cm” 3.

* 81/131 in finite of cm'l, Sz/pz in units of cm'z, S3/p3 in

Table VII - Si/Pi for the DZSe molecule in the vibrational state *

V1=V2=V3=0

 

 

J 51/P1 Sz/pz -S3/p3
1 10.045 17.055 64.397
2 10.048 17.018 64.395
3 10.048 17.015 64.352
4 10.048 17.019 64.343
5 10.048 17.019 64.345
6 10.048 17.011 64.220

 

Table VIII - Si/Pi for the DzSe molecule in the vibrational state *

 

V1=V3:O;VZ=1

.
:—__

 

 

1 10.135 17.633 66.388

2 10.182 18.559 74.083

3 10.184 18.661 71.121
* 81 /p1 in units of cm'l, Sz/pz in units of cm_2, S3/p3 in

units of cm" 3.

24
(J +1) indicated that the data for the vibrational state described by
the quantum numbers V1 = V3 2 0, V2 = l for both DZSe and HZSe
were not only too limited, but also very erratic, no further
analysis was attempted.

Further analysis of the data corresponding to the
vibrational ground state (Vl = V2 = V3 = O) was carried out for
both molecules, but here too, some values were discarded. Only
the values below the solid lines in the columns in Tables V and
VII were used in the subsequent analysis.

The method of least squares was then applied to the
data in Table V and Table VII in order to obtain the straight line
best fitted to these data. The six equations which resulted are:

HZSe(V1=V =v —0)

2 3 ‘
sllp1 = [19.795 - .0010595 J(J+1)j cm'1 (3.1)
sz/p2 = [65.431 - .021057 J(J+1)j cm'2 (3.2)
S3/p3 = [-506.55 + .16322 J(J+1)j cm‘3 (3. 3)
DZSe (v1 = v2 = v3 = 0)
sl/p1 = [10.048] cm‘1 (3.4)
32/132 = [17.019 - .00012621 J(J+1)] cm'2 (3.5)
S3/p3 .—. [-64.421'+ .004085 J(J+1)j cm'3 (3.6)

Graphs of these equations are shown in Figs. 2, 3, 4, 5, 6, and 7.
Extrapolation of these curves yielded the values of

Si/Pi correSponding to J = 0. A rigid analysis of these extrapolated

25
values gave the rotational constants of the molecules which are

listed in Table IX.

26

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32
Table IX - Parameters of HZSe and DZSe calculated from a rigid

analysis of extrapolated values (to J =0) of Si/ Pi

ste (V1 = V2 = V3 = O)

 

 

03 fl 7’ A B c

 

19.795 .17032 8.0501 8.1683 7.7115 3.9150

 

DZSe (v1 = v2 = v3 = 0)

d: ,8 7" A B c

 

10. 048 . 23758 4. 0871 4.1924 3. 8687 1 . 9870

 

cf. , .r.’ A, B, and C in units of cm'l; B dimensionless.

33

A non—rigid analysis of these data then followed. By
using the values of Cfl, ,8 , and Vobtained from the preceding
rigid analysis along with the slopes of the curves from equations
(3. 1) through (3. 6) it was possible to findCD 1, ’72, and P 2 from
(2c.6), (2c. 17) and (2c.18). The values ochZ, I91, and’rl then
followed from the relations: ($1.: ($2, B 1 = 82, and 7’1 =
-2 72. The x1 3 were next determined from equations (2c. 19)
through (2c. 24). Finally, theG‘i s were calculated by using
(2.b7). The values of the stretching coefficients Xi, and the re-
sulting coefficients 0‘ i of the centrifugal correction terms in the
Hamiltonian (2b. 3) are listed in Table X.

Error Analysis

An error analysis9 was performed in order to measure
the quality of the fit obtained by the method of least squares for
equations (3. 1) through (3. 6). As the final interest lies in the
rotational constants and in the coefficients of the centrifugal
correction terms, the probable error of the values used to calcu-
late them were found. These values were the probable errors in
the slopes and intercepts of equations (3.1) through (3. 6); results

are tabulated in Table XI.

 

9 F. A. Willers, Practical Analysis, Chapter—IV,
(Dover Publications, New York, 1948)

34

Table X - Stretching coefficients }\ i and

centrifugal correction term coefficients 6' i

HZSe (Vl = V

1
—_

+ 3. 0493 x10-5

3. 8924 x 10-5

.91317 x 10-5

2.4028 x10'"5

+1.0634x10'5

- .42125x10‘5

2

:V3:O)

I
__‘—:

o-1 +1.8413x10'3

2.3501 11:10"3

5'2 '-
53 - .55133x10'3
0'4 - 1.45073110-3

55 + .64505 3:10"3

.25433 1:10"3

.9

 

13286 (V1 = V2 = V3 = 0)

 

+ 2.1317 x 10"6

2.3160 x10"6

+.18433x10‘6
— 1.0658 x 10"6
+1.1580x10'6

. 09217 x 10“6

an +6.5344x10"5
<32 — 7.0994 31:10"5
6'3 + .56504x10"5
5'4 - 3.2671 x10"5
55 + 3.5497 x10"5

06 - .23252 3:10"5

 

35

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36
These values of Si/Pi along with their probable errors

were used in a rigid analysis once more to determined: , ,8, and

7. The worst possible combinations of these values and their
corresponding probable errors were used this time in order to
ascertain the largest error that might result in the present
calculations. These "worst" values ofCD, p, and Vwere next
used in the same manner to determine the rotational constants in
the way previously described. Results showed that A and B for

l

the HZSe molecule might be in error as much as i . 01 cm" and

the value of C by as much as i . 02 cm'l.
An error analysis of the stretching coefficients was
carried out in the same manner. Here, the slopes of the Si/Pi
curves along with their probable errors were used in con-
junction with the "worst" values of 03, B, and 7’to find the
greatest error likely to occur in the stretching coefficients 7‘1'
The x i s were in turn used to determine the greatest amount

by which the coefficients 6 i might be in error. The results of

the calculations are shown in the summary in Table XII.

   

 

 

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IV

Summary

The ”unstretched" values of the rotator constants A, B,
and C obtained by the energy moment method in the present work
are in very close agreement with the values which Palik and
Oetjen7 arrived at by employing a different method of analysis.

A comparison of the results is shown in Table Kill.

The coefficients of the centrifugal correction terms in
the Hamiltonian seem to be reasonably accurate for HZSe; Table
XII shows that the values calculated from these data might be in
error by as much as i. 04 x 10‘3cm'1.

Although the values of the rotational constants for
DZSe are quite satisfactory, the results obtained for the 0‘ i s
are rather poor;.the error may be as high as 28%. DZSe is not
a very “stretchy" molecule; the zero sloPe of equation (3. 4)’
and the very small slopes of the curves represented by equations
(3. 5) and (3.6) bear this out. As a consequence, a small error
in the calculated value of the slope shows up in the end results
as a very large relative error.

In this particular case it is believed that the data,

although sufficient to enable determination of the rotator con-

 

? E.D. Palik and R. A. Oetjen, J. Molecular Spectrosc0py, 2,

259. (1959)

37

 

38
Table XII - Coefficients of the centrifugal correction terms in
the Hamiltonian - equation (2b. 3)

ste (V1 = V2 '3 V3 = 0)

 

a
01=(+ 1.841 1.010)}:10'3
-3
63 -- (“-°5513i-°°3) x 10"3
_ ‘ -3
6'4 - (- 1.451 1: 000)x10
6‘5 — (+ .64503: .010) x 10‘3
_ _ -3
0'6 -( .25431.040)x 10

 

 

 

61=(+ 6.53 i1.84) x 10"5

4
62 = (- 7.099_+_1.79)x10'5

53 = (+ .5650: .05)x10‘5

_ -5
64— (- 3.267:1.10)x10

s = (+ 3.5501 .89) x 10’5

5
456 = (- .2825: .11)x10“5

 

39
Table XIII - Rotational constants of HZSe and DZSe for the

vibrational* state (VI = V2 = V3 = 0)

Present work Palik and Oetjen**

W

 

 

 

 

 

HZSe HZSe
A 8.163 8. 165
B 7.711 7. 712
C 3. 915 3. 915
DZSe DZSe
A 4.192 4.193
3.869 3.865
C 1. 987 l. 987

 

* All values expressed in units of cm'I.

** These values for the rotational constants are taken from the
works of Palik and Oetjen7 where it is stated that they may
be in error by as much as + . 02 cm’l.

7 E.D. Palik and R. A. OetjeE, J. Molecular Spectroscopy, _3,
259.(1959L

 

4O

stants with a fair degree of accuracy, is insufficient to obtain the
slope of the Si/Pi vs. J(J+l) curve accurately enough. Figs. 5,
6, and 7 show the graphs of these data for DZSe. Here it is seen
that although the data for the Si/Pi curve is satisfactory, the data
for the Sz/pz and S3/p3 are very limited and very erratic. In-
clusion of a value of 51/ Pi which may border on the point of
rejection, while it may not change the value of the intercept
appreciably, can result in a relatively large change in the sloPe
of the curve. The smaller the slope of the curve, the more
sensitive the values of O' i would be to such a change. Without
doubt, acceptable results could be obtained had more complet e
data been available for the DzSe molecule.

The data for the higher vibrational states is seen to
be far too limited and erratic to warrant any attempt at analysis
beyond the calculation of the effective values of the rotational

constants obtained by a rigid analysis.

 

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