ANALYSIS OF THE SPECTRA 0F PLANAR ASYMMEWC
MGLECULES‘, WITH APPUCATEOH T0 HYDROGEN
TELLURiDE

Thesis for the Degree of Ph D.
MECHIGAN STATE UNIVERSITY
N. KENT MONCUR
1967

     

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Ev -
‘ Iv - 72'

  
     
    

I‘H'c‘s;s

EJnhnns?

   

This is to certify that the
thesis entitled
ANALYSIS OF THE SPECTRA OF PLANAR ASYMMETRIC
MOLECULES, WITH APPLICATION TO
HYDROGEN TELLURIDE

presented by

N. Kent Moncur

has been accepted towards fulfillment
of the requirements for

Ph. D

. degree in Physics

WM4¢

' 'Major professor

Date July 28, 1967

 

0-169.

‘kk.

ABSTRACT

ANALYSIS OF THE SPECTRA OF PLANAR ASYMMETRIC
MOLECULES, WITH APPLICATION TO
HYDROGEN TELLURIDE

by N. Kent Moncur

The develOpment of the vibration-rotation Hamiltonian
for a planar asymmetric non-linear XYX molecule is out—
lined, and energy expressions are given by which energy
levels and other quantities useful in an analysis of
the Spectra of such a molecule may be calculated. Since
the upper vibrational states of some of the bands may
interact through a Coriolis type resonance, a modified
Hamiltonian is given which includes the necessary inter-
action terms for a description of these states.

A method is outlined by which the theoretical energy
expressions can be used in an analysis of spectra, and
is applied in the analysis of the absorption spectra of

l l

H Te near 2900 cm- and 4050 cm- .

2
Ground state combination differences from the in-
frared absorption bands vl+v2, v2+v3, 2vl, and vl+v3 are
used in a least squares analysis to obtain ground state
constants A, B, C, taus, and HK for the molecular species
H 130T 128T

126 125 124
2 2 e, 2 Te, H2 Te, and H2 Te.

upper states of the interacting bands vl+v2 and v2+v3 are

e, H H The

simultaneously analyzed to obtain upper state constants

N. KENT MONCUR
v0, A, B, C, taus, H and the perturbation coefficients

KI
. 130 128 126
G2 and ny, for the speCIes H2 Te, H2 Te, and H2 Te.

The upper states of 2vl and vl+v3 are treated in a similar

manner for the species H2130Te, H2128Te, H2126Te, H2125Te,

and H2124Te.
Using prOper combinations of molecular constants
from the different vibrational states analyzed, the equi-

librium constants Ae, Be’ and Ce are obtained from which

the equilibrium structure (re,6) of HzTe is calculated.

ANALYSIS OF THE SPECTRA OF PLANAR ASYMMETRIC
MOLECULES, WITH APPLICATION TO
HYDROGEN TELLURIDE

.9.“ By
C6

N? Kent Moncur

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1967

<7 ‘7 @197
/ 9

TO MY WIFE

ELAINE

ACKNOWLEDGMENT

I wish to thank Dr. T. H. Edwards for his help and
encouragement in the direction of my research. His ideas
and suggestions have been very useful and greatly appre-
ciated. I also wish to thank Dr. P. M. Parker and Dr.

C. D. Hause for their help and advice. I would like to
thank my fellow graduate students Dr. Thomas Barnett,
Lewis Snyder, Melvin Olman, and Don Keck for their many
helpful suggestions and advice. I would like to acknow-
ledge that the theoretical development of the interaction
Hamiltonian, which was used in some of the analysis of
this work, was capably carried out by Lewis Snyder. I
want to thank the M.S.U. Computer Laboratory for the use
of the 3600 computer in carrying out the numerical ana-
lysis of the data. I wish to express my gratitude to the
Air Force Cambridge Research Laboratories for their supe
port of our research.

I am indebted to my wife for her encouragement and
help during the course of this work, and I thank her for

her patience and understanding.

iii

TABLE OF

ACKNOWLEDGMENT . . . .

LIST OF

LIST OF FIGURES . . . . . . . . .
LIST OF APPENDICES . . . . . . .
INTRODUCTION . . . . . . . . .
CHAPTER

I. THEORY OF THE ASYMMETRIC MOLECULE

II.

III.

TABLES . . . .

Non-linear XYX Molecule
Asymmetric Rotor Wave Functions
Energy Expressions

Higher Order Distortion Terms

[Selection Rules
Intensities .
METHOD OF ANALYSIS

Line Fits . .

CONTENTS

Ground State Combination Difference

Fits . . .

Computer Programs

PERTURBATIONS IN EXCITED VIBRATIONAL

STATES . . .

Coriolis Interaction

nd

2 Order Distortion Interaction

Treatment of Two Interacting States

Computer Programs

iv

Page
iii
vi
vii

viii

13
15
16
19
21

24

25

26

30
30
31
32

39

CHAPTER

IV. THE INFRARED SPECTRA OF H Te . . .

2

Preparation of H Te . . . . .

2
Experimental Conditions . . .
Infrared Absorption Bands of H2Te

Isotopic Species of HzTe . . .

V. ANALYSIS OF THE INFRARED ABSORPTION

VI.
VII.
REFERENCES

APPENDICIES

SPECTRA OF HZTe . . . . . .
The 2900 cm-1 Region . . . .
The 4050 cm-1 Region . . . .
The 4900 cm.1 Region . . . .
Ground State Analysis of H
Reanalysis of v2 . . . . . .
Vibrational Analysis . . . .

STRUCTURE CALCULATIONS . . .

CONCLUSION . . . . . . . .

Page
41
43
45
46
51

53
53
59
67
68
73
73
76
79
81
84

Table

II.
III.

IV.

VI.

VII.

VIII.

Ix.

XI.

XII.

XIII.

XIV.

XV.

XVI.

LIST OF TABLES
Page

Classification of the Submatricies

E+, E', 0*, o' . . . . . . . . . . 12

Selection Rules by Parity Change of K_1and K+1 18

Experimental Conditions . . . . . . . 50
Molecular Constants of H2130Te for the
States v +v and v +v . . . . . . . 56
1 2 2 3
Molecular Constants of H 128Te for the
States vl+v2 and v2+v3 . . . . . . . 57
Molecular Constants of H 126Te for the
States v +v and v +v . . . . . . . 58
l 2 2 3
Molecular Constants of H2130Te for the
States 2V and V +V o o o o o o o 62
1 1 3
Molecular Constants of H2128Te for the
States 2v and v +v . . . . . . . . 63
l 1 3
Molecular Constants of H2126Te for the
States 2V and V +V o o o o o o o o 64
1 1 3
Molecular Constants of H2125Te for the
States 2v and v +v . . . . . . . . 65
1 1 3
Molecular Constants of H2124Te for the
States 2v and v +v . . . . . . . . 66
l l 3
Ground State Molecular Constants of H2130Te . 70
Ground State Molecular Constants of
H2128Te and H2126Te . . . . . . . . 71
Ground State Molecular Constants of
125 124
H2 Te and H2 Te . . . . . . . . 72
Molecular Constants of H2Te for the
State v2 0 O O O O O O O O O O O 74
Equilibrium Structure of HzTe . . . . . 78

vi

Figure

LIST OF FIGURES

Energy Level Diagram . . . . .
Elements of the Submatrix (E: + E
Elements of the Submatrix (0: + 0
Geometry of HZTe . . . . . . .
Normal Modes of HzTe . . . . .

Apparatus for the Production of HzTe

Absorption Spectra of H2Te near 2900 cm-
Absorption Spectra of H2Te near 4050 cm-

Absorption Spectra of H2Te near 4900 cm-

IsotOpic Absorption Lines of HzTe .

vii

l
1
1

Page
11
34
35
42
42
44
47
48
49
52

Appendix

I.

II.

III.

IV.

VI.

LIST OF APPENDICES

Nonvanishing Matrix elements of the
Hamiltonian Operators . . . . . . .

Assigned Lines and Observed Frequencies of

H2130Te for the States vl+v2, 02+v3, 201,

and v1+v3 . . . . . . . . . . .
Ground State Combination Differences

of H2130Te . . . . . . . . . .

Calculated Ground State Energy Levels . .

Calculated Upper State Energy Levels for
the Bigges vl+v2, v2+v3, 2vland vl+v3
of H Te

2
Listing of Program Spec-Fit 1 . . . .

viii

Page

84

91

110

115

126

151

INTRODUCTION

The analysis of the infrared spectra of polyatom-
ic molecules leads to precise information about their
rotational and vibrational energy levels which in turn
give information about inertial constants, centrifugal
distortion constants, force constants, anharmonic con-
stants, bond angles, bond lengths, and other data con-
cerning their structure.

Since a polyatomic molecule is a many body prob-
lem, it is impossible to find exact expressions for the
energy levels. A convenient approach to the problem
is to treat the general theoretical eXpression for the
Hamiltonian of polyatomic molecules by an expansion
formulation in successive orders of approximation, and
then apply it to the Specific case involved, making
simplifications whenever possible. The Born-Oppenheimer
approximation is used to separate the electronic problem,
which will not be considered here, from the vibration-
rotation problem.

Molecules may be grouped into three general classes
depending on their principal moments of inertia. Mole-
cules, with all three principal moments of inertia equal

(Ia=Ib=Ic), are called spherical top molecules and have

a spherical moment of inertia ellipsoid. Examples of

these include CH CC14, SiH4. Molecules with two

4'

moments of inertia equal have an ellipsoid which is an
ellipse of revolution and are called axially symmetric

molecules. Examples of these are the prolate (Ia<Ib=IC)

molecules CH3Cl and CH3D, and oblate (Ia=Ib<Ic) molecules

CD3H and BF3. These molecules generally have spectra
with regular features. Asymmetric molecules have the

most general moment of inertia ellipsoid with Ia<Ib<Ic.
Examples of these are the non-planar molecules CH2D2,

CH2C12, etc., and the planar molecules H CO, CZH H O,

2 4’ 2

H S, HZSe, and HzTe. The spectra of asymmetric molecules

2
are generally very irregular and identification of the
absorption lines is one of the major problems in an
analysis.

In the following, the development of the vibration-
rotation Hamiltonian for a planar asymmetric non-linear
XYX molecule will be outlined, and energy expressions
will be given which may be used in an analysis of the
vibration-rotation spectra. These expressions will be
used in an analysis of the infrared absorption spectra

of H Te.

2

CHAPTER I
THEORY OF THE ASYMMETRIC MOLECULE

A number of investigators have attacked the problem
of finding the Hamiltonian of a semi-rigid, rotating,
polyatomic molecule. A convenient formulation has been

given by Darling and Dennison:

Zul/4(Pa-pa)uasu-l/2(PB-p8)ul/4 <1)
8

1/4 -1/2p 1/4

kn +V .

The subscripts a and B have the range x,y,z (the prin-
cipal axes of inertia of the molecule), Pa and pa are
components of total and internal angular momentum respec-
tively, and “as are the cofactors of a determinant

which contains the moments of inertia. The pk are the
momenta conjugate to the normal coordinates qk. In gen-
eral, the eigenvalue problem using the Hamiltonian in

the form of Eq. (1) is impossible to solve, so the usual

procedure is to make an expansion in orders of magnitude,

H=HO+H+H2+... (2)

l

and apply perturbation theory to obtain the eigenvalues

and eigenfunctions. Such an expansion is given by Gold-

smith, Amat and Nielsen.2 The Hamiltonian is then cast

into a more convenient form by applying a contact trans-

formation:

H' = THT"l = H' + AH' + AZH' +
o l 2

iAS -l -iAS
e - e

... (3)

where S is a function of vibration and rotation operators.
Substituting the expanded Hamiltonian of Eq. (2) into
Eq. (3) and equating the coefficients of like powers of

A, one finds

'—
HO — HO (4)
Hi = H1 + 1[S,Ho]

I... i I
H2 — H2 + 2[s,(Hl+Hl)] .

S is then chosen such that all off diagonal H1 terms are
removed to Hi leaving Hi with only diagonal vibrational
matrix elements, which simplifies the perturbation calcu-
lation. Herman and Shaffer3 give a suitable S for trans-
forming H.

For an asymmetric molecule the situation is somewhat
simplified since S may be chosen such that Hi = 0. The
energy is then obtained by the perturbation method of
considering only the diagonal vibrational elements of

H' =Hé + Hi. This approach is valid provided, no res-

onant denominators appear during the transformation,

and all off diagonal vibrational matrix elements of Hi

which have been ignored are small. These problems will
arise later when the possibility of a Coriolis interac-
tion is discussed. Some of the individual terms in the
eXpanded Hamiltonian are considerably simplified for a

non-linear XYX molecule and will be given in detail by

Snyder4 for both the untransformed and transformed Ham-
iltonian.

There are 3! ways the principal axes (a,b,c) of a
molecule may be attached to a cartesian coordinate system
(x,y,z). Since the molecules of interest here are near
the oblate limit, where the angular momentum Operator Pc
becomes diagonal, the IIIr system will be used, where
a = x, b = y, c = 2.

nd

To 2 order of approximation, the transformed asy-

mmetric tOp Hamiltonian as given by Chung and Parker5 is

H' = hv + AP: + 8P: + CP: + itaaaaP: + irbbbbpé (5)
+ %TCCCCP: + Arbbcc(P:P: + ngg) + ATaacc(P:P: + Pipi)
+ %Taabb(P:P§ + Pipi’ + %Tbcbc(Pch + Pch)2
+ %Tacac(PaPc + PcPa)2+ iTabab<Pan + PbPa)2°
The Pa' Pb' and PC are the components of angular momentum

along the a, b, and c axes, and the taus are the centri-

fugal distortion constants.

Non-linear XYX Molecule

 

For a non-linear tri-atomic molecule in the ab plane

6
Dowling,6 and Oka and Morino7 have shown that the follow-

ing relations exists between the taus:

Tbcbc = Tacac = 0 (6)
_ e e 4 e 2 2
chcc _ (Ia/1c) Taaaa + 2[Ia I b/(Ic ) ] aabb

+ (Ib/Ic) Tbbbb

e e 2
Tbbcc ‘ (lb/Io) Tbbbb + (Ia/1c) Taabb

_ ' e e 2 e e 2
Taacc — (Ia/Io) Taaaa + (lb/Io) Taabb

where 1:, IE, and I: are the equilibrium moments of in-

ertia. Using Eqs. (6) and the commutation relation

2 _ 2P 2 2P 2 2 _ 2 -
(Pan+ PbPa) — 2(PaP b + PbP a) + 5Pc 2P (7)
_ 2 2 2
where P - Pa + Pb + Pc ,

the Hamiltonian Eq. (5) simplifies to

2 2 2
' _
H ‘ hv + APa + BPb + CPc + Taaaaoaaaa + Tbbbbobbbb (8)
+ Taabboaabb + Tababoabab '
where
_ 1 4 2 4 2 2 2 2
Oaaaa -- Z[Pa + r PC + r(PaP C + PCP 3)] (9)
_ 1 4 2 4 2 2 2 2
Obbbb ‘ 4[Pb + 5 Pc + 5(PbP c + PCP b)]
1 4 2 2 2 2 2 2 2
Oaabb a z[2rsPC + (PaP b + PbPa 2) + s(PaP c + PcP a)
2 2 2 P2
+ r(PbP c + PCP b)]
_ P2 2 P2 2 2
Oabab — M[2(P b + PbP a) - 2P + SPC] .

In the absence of any resonating states, the vib-

rationally dependent molecular constants A, B, and C of

Eq. (8) are defined as

_ _ A l
A - Ae zai(vi + 2) (10)
_ B l
B ‘ Be ‘ Zai<vi + 2)
l
_ C l
l
Ae, Be’ and Ce are the equilibrium molecular constants

and are inversely prOportional to the equilibrium moments

of inertia. Expressed in cm"1 they are defined as

_ 2 e _ 2 e _ 2 e
Ae — h/8n CIa, Be — h/8n ch, Ce — h/8n CIC, (11)

where h is Planck's constant and c is the speed of light.
Detailed expressions for the Oi will be given by Snyder.4

In Eq. (8), h is a pure vibrational term and is a

V
constant within a vibrational state.

The r and s are defined by

r = C /A (12)

/B

s = C

(I’M (ON
0N (DN

Use of the planarity condition

e _ e e
Ic - Ia + Ib
gives
1 _ l l
E'A+B'

e e e
which can be used to express r and s in terms of only

two of the equilibrium constants. Eliminating Ce from

Eqs. (12) we have

_ 2
r — B /(Ae+Be) (13)

s=A

('DNCDN

2
/<Ae+Be) .

In practice, when equilibrium constants are not available,
ground state constants A and B are used in place of Ae
and Be to calculate r and s. We chose to eliminate Ce
rather than Ae or Be from the definition of r and 5 since
for the type of molecules of interest, such as H28,8
HZSe,9 etc., it happens that Ce differs from the ground
state C more than Ae and Be differ from the ground state
A and B. Thus, calculations of r and s using A and B
are closer to the actual values than those using C.
The Hamiltonian of Eq. (8) differs from that used

by Hill and Edwards10 in that the portion of the centri-

fugal distortion of the last term

2 l 2 1

_ __ P _ _ 3 2
— 2Tabab a 2Tabab

2 2
+ 5pc) Pb + ZTabach

l
Tabab4(-2P

was included by Hill in AP: + BPg + CPi, thus, his de-

. . . . 1 l
finition of A, B, and C include -§Tabab, -§Tabab, and

3 .
+41abab respectively.

For a non-linear XYX molecule, Chung and Parker11
have calculated the centrifugal distortion constants.

In cm"1 they are

3 . 2 2 2 2
Taaaa ~16Ae(31n y/wl + cos y/wz) (l4)

_ 3 2 2 . 2 2
Tbbbb — -l6Be(COS y/wl + Sin y/wz)

3 2 2 2 .
Taabb = -l6(AeBe) / (l/wl - l/w2)51nycosy

_ 2
Tabab _ -16AeBeCe/w3 ‘

The mi are the normal frequencies in cm-l, and y, a dimen-
sionless quantity, is a function of the harmonic force

constants.

Asymmetric Rotor Wave Functions

 

A perturbation treatment requires use of only the
diagonal vibrational elements of the transformed Hamil-
tonian (except when vibrational resonances occur such as
in a Coriolis interaction, then off diagonal elements
must be considered), therefore, the rotational Hamiltonian
is all that must be diagonalized. When forming the rota-
tional matrix elements of the Hamiltonian it is conven-
ient to use the combinations of the symmetric rotor wave

12 for a basis set. These

functions suggested by Wilson
combinations break each J submatrix, where J is the total
angular momentum quantum number, into four smaller sub-

+, and 0’. Thus, the

matrices, designated E+, E-, 0
size of the matrices that must be diagonalized is reduced.

The basis set of wave functions used are

q’(JIKIY)

/%[¢(J,K) + (-1)Yw(J.K)], K ¢ 0 (15)

W(J,0,y) w(J,O) K = 0,

where K is the projection of J along the axis which be-

comes unique in the nearest symmetric top limit of the

10

moment of inertia ellipsoid, so that KSJ. The w(J,K) are
the symmetric rotor basis functions, and y is an odd or
even integer (except for K = 0 where only even y exists).

The symbols E+, E-, O+

, and 0- identifying the submatri-
ces refer to the evenness (E,+) or oddness (O,—) of the
(K,y) values in the matrix. The matrix elements of the
operators in the basis set W(J,K,y) are given in Appen-

dix I.

A convenient parameter K, indicating the asymmetry

of a molecule, has been defined by Ray,13'14 where
= 225%29 and has the range -1<K<l. K = -1 for a

prolate molecule (B=C), and K = +1 for an oblate molecule
(A=B) .

The energy level of a given vibration-rotation state
may be identified by the three vibration quantum numbers
(vl,v2,v3), the total angular momentum quantum number J,
and both values of K to which the level corresponds in
the prolate (K = -l) and oblate (K = +1) symmetric top
limits. Thus, a rotational level for a particular vib-

rational state is designated by J In another

K-1K+1'
method of labeling energy levels, the pseudo quantum num-
ber r is used, where T = K_1-K+1. Thus, a level is
labled as JT. In the IIIr representation, K_1+K+1 is
equal to J for levels with y even and J+1 for those with
7 odd. Fig. 1 gives an energy level diagram for J = l,

2, and 3. The classification of the submatrices in terms

of the parities of K-1 and K+1 is given in Table I.

K

11

Fig. 1. Energy Level Diagram

 

J K.1 . JT JK_IK+1, K+1 J
. . o
__3____.=========725-—“"' 31 ‘
32.1 3
3 3 2

 

 

 

 

 

 

 

 

l E o M101
I I 1 l
-l.0 -0.5 0.0 +0.5 +130
prolate oblate

12

sec 0 o

 

 

 

 

 

 

 

Gw>w H m
«\Aa+nv ~\n o O o 0 .0
«\AH+nv «\n o m o o +o
~\Aflunv mxu m m m 0 nm
mxfia+nv H+ Nxs m o 0 m +m
b OOO e c0>m L+m Hum H+x Hum
XHHflMEfldm m0 ONHm b 660 h. Gm>w XflHuwMEnaflm

 

 

 

 

 

coflumpammmnmmm. HHH

.H

m .+m mmowuumEQSm may no GOHUMOHMHmmmHU .H OHQOB

 

13

The wave functions A(J,T) which diagonalize the
rotational matrix are linear combinations of the basis
functions:

JT
A(J,I) = 2 SK W(J,K,y) , (16)
K
where the sum is over only those wave functions W which
- +

. + - .
form the same submatrix E , E , O , or O , 1.e., even K

even 7, even K odd y, odd K even y, or odd K odd y.

Energy Expressions

 

The energy of a given vibration rotation state may

be evaluated from

E(V,J,I) = G(vl,v2,v3) + W(A,B,C) (17)

2 2 2
<V1V2V3|hVIV1V2V3> + A<Pa> + B<Pb> + C<Pc>

+ Taaaa‘oaaaa> + Tbbbb‘obbbb> + Taabb‘oaabb>
+ Tabab‘oabab> '
where
<V1V2V3th|V1V2V3> = G(vl,v2,v3) (18)

G0 + 291(Vi+2’ + Z Exik(vi+%)(vk+%)°
1 ls

G(vl,v2,v3) is the pure vibrational contribution to the

energy.15 The xik are the vibrational anharmonicity

constants, and G0 is a constant term. W(A,B,C) is the

rotational energy where the average values of the rota-

tional operators are taken in the representation A(J,T)

which diagonalize the rotational Hamiltonian. The cons-

14

tants A, B, and C are dependent on the vibrational state
involved. Although the definition of the taus given in
Eqs. (14) show them to be independent of the vibrational
states, the empirical taus determined in an analysis may
show a vibrational dependence because they may try to
adjust for the effect of other centrifugal distortion
terms from a higher order of the Hamiltonian expansion
which have not been included here.

The difference between the ground and excited state

energies involved in a transition is given by

v = G'(vi,vé,v§) - G"(vi,v3,vg) + W'(A',B',C') (l9)_

_ W" (All’Bll ,C") .

(G'-G") is the pure vibrational contribution which is
called the band center or orgin, and is denoted by v0.
(W'-W") is the difference between the excited and ground
state rotational energy levels.

A second order vibrational resonance, which exists
between states of the type (vl,v2,v3) and (Vl-Z,V2,V3+2),
was first recognized by Darling and Dennisonl in H20 and

later by Allen and Plyler16 in H28. The vibrational
energies of two such resonating states are given by

_l ..
Gp — 2[Gl(vl,V2,v3) + G2(vl 2,v2,v3+2)] (20)

1 22 12
*il‘GI‘Gz’ +Y vl(vl-l)(v3+l)(v3+2)] / .

where Gp represents the energy of a perturbed level, and

15

G1 and G2 that of the unperturbed levels as defined in

Eq. (18). The interaction matrix element is represented

by Y-

Higher Order Distortion Terms

 

When higher order terms in the rotational Hamiltonian
are considered, many P6 distortion terms appear. Pierce,
DiCianni and Jackson17 have suggested that for nearly
symmetric rotors it may be a sufficiently good approx-
imation to retain only those P6 terms which do not vanish
in the symmetric rotor limit. This results in the ad-

dition of the following four terms to the Hamiltonian:

6 4 2 2 4 6
HJP + HJKP PC + HKJP PC + HKPc . (21)

These H'S derived from an analysis of the spectrum have
only empirical significance.

Chung and Parkers'18

have explicitly derived all

the terms in the asymmetric rotor Hamiltonian that cont-
ribute to the energy in the fourth order of approximation.
Using these results and the angular momentum commutation
relations, Kneizys, Freedman, and Clough19 reduced the
number of rotational coefficients appearing in the Ham-
iltonian. When their results are examined, the same four
terms that are in Eq. (21) are found among their P6 terms,
which lends more justification for their addition to the
Hamiltonian.

20

Watson, in a recent work, has shown that the ro-

16

tational Hamiltonian of an asymmetric molecule in general
form may be transformed, by means of a unitary transforma-
tion, to a reduced Hamiltonian which has the same eigen-
values but fewer parameters. This procedure removes any
possible indeterminacy which may exist in the coefficients.»
This reduced Hamiltonian, complete through the sextic
terms, has been given in a form which is suitable for
fitting to observed energies. For reasons that are not
clear, our efforts to fit the observed ground state com-
bination differences of H2Te using Eq. (37) of Ref. (20)
were not successful, since the constants would not con-
verge to stable values. It may yet be possible to fit
these combination differences with this equation by al-
lowing only certain combinations of the constants to vary
at one time in a fit, or by using a different form of the
equation, however, this formulation of the Hamiltonian

and energy expressions will not be considered further.

Selection Rules

 

In order that a transition from state n"+n' occur,

the electric dipole matrix element
Iw;.uwn.dv (22)

must be non—zero. The selection rules for vibration-ro-
tation transitions come from the vanishing or nonvanishing

of this matrix element and have been given and discussed

14

by Cross,.Hainer and King, Hill,21 and many others. The

l7

selection rules for J in the asymmetric rotor are the

same as in the symmetric rotor, AJ = -l,.0, +1 corres-
ponding to P, Q, and R branches respectively. The sel-
ection rules of the parity change of K_1 and K+1 are given
in Table II.

If the change in the electric moment during a trans-
ition lies along the axis of least moment of inertia
(a axis), the resulting band is called a type A band,
and we see that the parity of K_1 does not change and
that of K+1 does, i.e.; AK_1 = *0, *2, *4, ... and
AK+1 = *1, *3,.... .

For a change in the electric moment along the great-
est.moment of inertia (c axis), a type C band results and
the parity of K_1 does change while that of K+1 does
not, AK_1 = *l, *3, ... and AK+1 = *0, *2, *4, ... .

For a Change in the electric moment along the inter-
mediate moment of inertia (b axis) a type B band results
and the parity of both K-1 and K+1 change, AK-1 = *1,

*3, ... and AK+1 = *1, *3, ... .

The above selection rules are subject to the restri-
ctions K_1 + K+1 = J for even y and K-1 + K+1 = J+1 for
odd 7 for both initial and final states. Transitions in-
volving changes in K_1 and K+1 of 0 and *l are all we
normally need consider since changes greater than *1
generally involve transitions with intensities an order
of magnitude weaker.22 A type C band is not possible for

a planar molecule in the ab plane since there can be no

18

 

 

 

 

 

HI.
M 0 0A).? 0 O O U
o o o m
1:... J. o 0 MM . . a m
0 0 ++ O O
~+
M O 0 ++ 0 O m an
:H Oman cmEo
ma mwamnu H+M “Ix +4 L+M ”Ix wwowwo 0” make
wuwnmm one macauflmcmua CT3OHH4 Hmaamumm maxd pawn

 

 

 

 

~+M paw HIM mo mmcmno muwumm

an mmasm coauomamm

.HH OHQMB

 

19

out of plane vibrations. For a non-linear XYX molecule,
any band which includes an odd multiple of 03 is a type
A band, all others are type B bands.

We have found it convenient to use the familiar
symmetric top notation for designating transitions. The

terminology is AK

AJK(J) where J and K refer to the ground
state quantum numbers, K being the K in the nearest sym-
metric tOp limit (K_1 for prolate and K+1 for oblate).
Since the degeneracy in K is removed in an asymmetric
molecule, there are two components to each transition,

an even component which orginates from a symmetric energy
level and an odd component orginating from an antisym-
metric level. As usual the symbols P, Q, and R indicate
changes in J and K of -l, 0, and +1 respectively. Thus,

the possible transitions for A and B type bands include

RRK(J), PPK<J>, R0K<J), P01,01), RPK<J>, and PRK(J>.

Intensities
The relative intensities of transitions within a

band may be calculated with the expression

2 e(-W"hc/kT)

n.,n. , (23>

I S 9|u(v<)|

(-W"hc/kT) 2

is the Boltzman factor and |u(K)ln" n'
I

where e

the line strength. The line strength includes the factor
(2J+1) which arises because of the (2J+1) fold degenercy
of the total angular momentum in the absence of an exter—

nal field. The g is the statistical weight factor of the

20

ground state level which arises when two of the nuclei

in a molecule such as H2Te are identical. For HZTe the

ratio of g for symmetric levels (ee, 00) and antisymmetric

levels (eo, oe) is 1:3.

CHAPTER II

METHOD OF ANALYSIS

The rotational energy, including the four terms of
Eq. (21) and written in a form which is convenient for

the analysis of spectra, is

W = “A + BB + YC + xlTaaaa + XZTbbbb + x3Taabb (24’

+ X4Tabab + xSHJ + XGHJK + x7HKJ + XBHK '
where

a = <P§> (25)
B = <P§>
y = <P§>

x1 = <0aaaa>

x2 = ‘Obbbb’

x3 = <Oaabb>

X4 = <oabab>

X5 = <P6> = J3(J+1)3

x6 = <P4P§> = J2(J+l)2<P:>

X7 = <P2P:> = J(J+l)<P:>

x8 = <Pg> .

The average values of the operators are calculated from

the eigenvectors of the diagonalized Hamiltonian.

21

22

The average value of any operator F may be expres-

sed as
Jr * JT
<JTIFIJT> = 2 Z,(SK ) SK, FKK, (26)
KK
where
*
FKK. = II (J,K.y)Fw(J.K:Y)dv
are the matrix elements of F in the original basis set.
For simplification of the calculation it should be noted
*
that since FKK' = FK'K for a hermitian operator, then
(27)
J1 * JT JT 2 Jr * Jr
2 2 (S ) s , F . = 2|s | F + 2 2 2Re[(S ) s ,F .1.
K K, K K KK K K KK K<K, K K KK

Re denotes the real part of the quantity enclosed with
brackets.

The method of analysis involves calculating the av-
erage values of the operators starting from the best
available values of the constants A, B, and C, from pre-
vious work or from an approximate symmetric tOp analysis.
Hill and Edwards10 have given a method of obtaining ap-
proximate values of the rOtational constants (A+B) and C
by using a rigid symmetric rotor approximation to the
combination differences.* Neglecting centrifugal dis-
tortion, the symmetric rotor approximation to the rota-

tional energy is
w = [(A+B)/2]J(J+l) + [c-(A+E)/2]K2 . (28)

Using this expression, the ground state combination differ-

ences formed from the zero series (RR and PP lines for

23

which K = J) are given by the simple expression

P

R ' II II II
RJ(J) PJ+2(J+2) = A + B + 4C (J+1) , (29)

where the double prime indicates ground state. Thus, if

a plot of the observed combination differences vrs (J+1)
is made, (A"+B") and 4C" may be obtained from the inter-
cept and slope of the straight line connecting these
points. Although this may also be done with other series,
the zero series is preferred because it usually contains
the strongest lines on each side of the band center and
has the lines least affected by the asymmetry and there-
fore the easiest to identify. In a similar manner the up-
per state combination differences formed from the zero

series are given by

R P _ u l I
RJ(J) - PJ(J) -- A + B + 4C J (30)

where the single prime indicates the upper state. A plot
of the observed combination differences vrs. J gives
(A'+B') as the intercept and 4C' as the slope of the
straight line connecting the points.

The initial A, B, and C are used in a least squares
fit of the assigned lines of the spectrum to get revised
values of A, B, C, the taus and H's, and to help assign
more lines. The process is repeated, this time calculat—
ing the average values of the operators using the revised
A, B, C, taus, and H's, and including all newly assigned

lines in the fit. This is continued until a stable set

24

of constants is obtained and all possible lines are as-
signed. It is usually necessary to refit more than once,
since the average values of the operators change somewhat
when small changes in the constants are made, and new lines

are often assigned.

Line Fits

 

To fit the energy levels to the spectral lines, the
best available constants are used to calculate the aver-
age values of the operators for each energy level of the
ground and upper state, and to predict a spectrum using
the expression

vcalc = V0 + wl (A',B',C') _ w" (A",B",C") ’ (31)

where the double prime indicates ground state and single
prime upper state.- The differences between the observed
frequencies and the calculated frequencies are then fitv

to the expression

v = Avo + a'AA' + B'AB' + y'AC' (32)

- v
obs calc

L. I I 8 I V II I!
+ Z xiAIi + Z xiAHi - a AA
1:1 l=5

_ BnABu _ YnAC" _ 2
i

II II
X'ATi

u
_ 1

1

_ u u
2 xiAHi °
1= 5

The Avo,AA, AB, etc., obtained from the fit, are the ch-

anges in the initial constants v0, A, B, etc., i.e.:

*

*
Avo = v0 - vo where v0 is the revised constant. These

25

changes in the constants are made and the process repeated
as explained above.

If, initially, only lines have been identified which
are not greatly affected by the asymmetry, it may be

desirable to use the-combinations of the constants figfi

and againstead of A and B. This was done by Hill 32 2£;10
and may be accomplished by replacing aAA + BAB in Eq. (32)
by £A(é;2) + ”A(§2§)’ where £= n+8, and n= a-B . The.
advantage of using these combinations of the constants
is that (5&2), which determines the asymmetry, may be
held constant in a fit until some lines which depend more

on the asymmetry are identified.

Ground State Combination Difference Fits

In a similar manner, the ground state energy levels
may be fit to the observed ground state combination dif-
ferences. The average values of the operators for each
ground state energy level are calculated and a set of

combination differences calculated from the expression

A = [W"(A",B",C")]l _ [W"(A",B",C")]2 ' (33)

v
calc

where this indicates level 1 - level 2. The difference
between the observed and calculated combination differ-
ences are then fit to the expression

-A = (GI-ag)AA"+(BiU-_Bfl)AB"+(Y3l-_Y'ZI)AC"

Avobs vcalc

(34)

1+ 8
+ g=l[(Xi)1-(Xi)2]ATi + §=5[(Xi)l-(xi)2]AHi .

26

Again, it may initially be desirable to write the above
formula in terms of A(§;§) and A(§i§) instead of AA and
AB.

It is generally advantageous to obtain the ground
state constants from a fit with Eq. (34) as Opposed to
a line fit, since the upper state energy levels, which
may be perturbed, are not involved, and the ground state
combination differences from all bands may be fit simul-
taneously.

The upper state of a single band may also be analyzed
by fitting its upper state combination differences in a
similar manner. Since this method involves fitting dif-
ferences in energy levels instead of the energy levels
themselves and does not allow v0 to be determined, we
chose to analyze the upper state by fitting the spectrum
to Eq. (32), holding the ground state constants fixed
and varying v0 and the upper state constants. In either

case,'the possibility of perturbations in the upper state

may hamper the analysis.

Computer Programs

 

Several computer programs written in Fortran were
used on Michigan State University's 3600 CDC computer.
The main subroutines used to diagonalize the Hamiltonian
and Perform least squares fits are respectively:

a. SUBROUTINE JHERMX

This subroutine, obtained from N. w. Naugle,23 is

27

the adaption of the Threshold Jacobi Method for
evaluating the eigenvalues and optionally, the
eigenvectors of a hermitian matrix.

b. SUBROUTINE REGRESS
This subroutine is a stepwise multiple regression

24 and is used in all

routine written by Efroymson,
least square fits in this work. An explanation
of the mathematical method including computer flow

charts is given in Ref. (24).

The following gives the name and description of the

computer programs used.

1.

ENG-CALC

A, B, C, taus, and H'S are input and the computer
calculates and prints out ground state energy levels
and the level identification J K-1K+1. Upper state
rotational energy levels may be calculated by inputing
upper state constants and inserting cards in the deck
that calculate r and s with the ground state A and B,
otherwise the program calculates r and s with the A
and B that are input.
SPEC-CALC

Ground and upper state constants, and band center
of the spectrum to be calculated are input. The computer
calculates rotational energy levels for both ground
and upper states and stores them in memory. Following
the program is a deck of IBM cards which the computer

reads. Each card contains the quantum numbers for an

28

allowed transition and the line strengths for this
transition tabulated in steps of 0.1 in K. The quantum
numbers designate the location of the energy levels
in memory and are output along with the frequency cal-
culated with Eq. (31), and the intensity of the tran-
sition calculated with Eq.(23) and scaled such that
the largest intensity is 10. The line strength is ob-
tained by interpolating between the line strengths on
the transition card.
CD-FIT

Ground state molecular constants are input along
with all the observed ground state combination differ-
ences Avobs,and the quantum numbers (J K_1K+1)l-(J
K_1K+1)2 that identify them. The quantities (Avob -

A ) are formed by the computer and fit by least

vcalc
squares to Eq. (34) to determine the revised constants.
Any combination or all of the constants may be varied'
simultaneously.

Upper state combination differences may be fit
in the same manner by inserting a card in the program
which calculates r and s with the ground state A and B.
SPEC-FIT

Ground and upper state molecular constants and
band center are input along with the observed frequen-
cies of all the assigned transitions and the quantum-
numbers (J' KilKll- J" KfllKil) that identify them.

The quantities (Vob ) are formed by the computer

- v
s calc

29

and fit to Eq. (32) to determine the revised constants.
Any or all of the ground and upper state constants may
be simultaneously varied.
CD-CALC

All the frequencies, quantum numbers, and weights
of the assigned transitions within a band are input.
The computer forms and prints out all possible ground
and/or upper state combination differences that can.
be obtained from the input transitions. When the same
combination difference appears more than once within
the same band, a weighted average is formed and printed

out .

CHAPTER III

PERTURBATIONS IN EXCITED VIBRATIONAL STATES

25 and Herman and Shaffer3 have shown that in

Nielsen,
the event of an accidental degeneracy, some terms which
have been transformed into Hi from Hl_by S Eq. (4), may
have resOnance denominators and become very large. If
this occurs, a modified function 8* must be used in place
of S so that terms with resonant denominators do not ap-
The modified

25 and

pear in the transformed Hamiltonian Hi.
function 8* is discussed and listed by Nielsen

Herman and Shaffer.3

Coriolis Interaction

 

For a non-linear triatomic molecule such as H2Te,
where ”l is very nearly equal to w3, a Coriolis resonance
does occur between states (vl,v2,v3) and (vl*l,v2,v3*l),
which leads to resonant denominator terms in H'.

The modified function 8* used in place of S in Eq.(4)

now leaves the transformed Hamiltonian with Hi non-zero
and equal to
Gz‘q1P3 ' q3pl)Pz ' <35)

and changes the molecular constant C [see Eq. (10)] to C* :

C* 1
*= - -
C Ce Eai(vi+2) (36)

30

31

where
“I* I “1
.g* .. .g
ag* # cg .

The explicit definitions of ai and a; will be given by

Snyder.4 The other terms in the transformed Hamiltonian

are the same as before. In cm.l G2 is

G2 = -;13(w1+w3)Ce/(w1w3)l/2 , (37)

where :13, the Coriolis coupling coefficient for a non-

linear XYX molecule is

:13 = [(Ce/Ae)l/2cosy - (Ce/Be)l/Zsiny 1 . (33)

nd

2 Order Distortion Interaction

 

Upon examination of the transformed Hamiltonian, as

suggested by Clough and Kneizys,26 another interaction
term
nyq1q3(PxPy - PyPX) . . (39)

is found-in Hi which has non-zero vibrational matrix
elements <vl,v2,v3lvl*l,v2,v3¥l>, connecting the same
states as the Coriolis term, and is large enough that it

cannot be ignored.. This is a term which orginated in the

untransformed H2 Eq. (2). In cm.1 G is

Ky
(40)

2 .
SlnY] .

ny = -6(l/N1Q3)l/ZAeBe[(Ce/Ae)l/zcosy+(Ce/Be)1/

32

The above two interaction terms also have the non-
zero matrix elements <vl,v2,v3lv1*l,v2,v3*l>, but as
noted by Wilson,12 since the energies of the interacting
states must be reasonably close together for any effect
to be observed, the interaction is seen only between the
states (v1,v2,v3) and (v1*l,v2,v3*l).

For HZTe, some of the states that satisfy the re-
quirements for these interactions to be observed are,
v1 and v3, v1+vz and 02+v3, 2vl and vl+v3, vl+v3 and

Zv +2v . Note

v2 3
1 and 2V3, thus, an

3, 2v1+v2 and vl+vz+v3, vl+vz+v3 and

that v1+03 interacts with both Zv

analysis Of any of these states should include all three,
even though one may be too weak to be observed. The

same situation-is true of the three bands 201+v2, v1+02+-

v3, and v2+2v3.

The interaction terms which must be added to the

Hamiltonian when analyzing these interacting bands is

HI = Gz(q1p3-q3pl)Pz + G q1q3(P P -P P ) . (41)

XY XYYX

The matrix elements of these Operators are given in Ap-

pendix I.

Treatment of Two Interacting States

and v +v will

1 2 2 3
be considered in more detail. As mentioned earlier, each

The two interacting states v +v

J block of the matrix elements of the Hamiltonian without

the interaction terms may be factored into four submatrices

33

+ - + - + - + -
for each state, E1, E1' 01, 01’ and E3, E3, 03, 03.

The subscripts l and 3 refer to states vl+v2 and vz+v3
respectively. Similarly, each J block of the matrix ele-
ments of the Hamiltonian including the interaction terms
may be factored into the four submatrices (E: + E3),-

(E; + E1), (0: + OS), and (0; + 01). The matrix elements
of the interaction terms connect the submatrices of the
states 1 and 3. The general form of (E: + E3) is shown

in Fig. 2 and that of (01 + 0;) in Fig. 3. The E1

KK' are
the matrix elements of the Hamiltonian in the basis set

Y(J,K,y) for the vibrational state i, and ELL' are the
matrix elements of the interaction terms of Eq. (41)
connecting states i and j. The matrices (E; + E1) and
(0; + 0;) are Obtained by interchanging the superscripts

1 and 3 on ERK' and Eii, and the subscripts l and 3 on

Wi-

The wave functions A(V,J,T) which diagonalize the
Hamiltonian with the interaction are now combinations of
not only the rotational basis set W(J,K,y), but also of

the vibrational wave functions wv of the two states in-

volved. The basis set of wave functions is

4.

l va(JIKIY)

where v = l or 3. The A(V,J,T) are linear combinations of

the 0.
l

A(V,J,I) = Z SZJT¢i , (42)
l

34

 

 

 

 

 

m

o . .
. . . emu em «mm . v.mfimm m.mfimm .o .
. . . sea as New o «.mfimm m.~fimm «.mfimm o

. 311. :11. :11.
. . . . . . . . o .
. . . m+qmwm ~+qmw o . emu 4mm mmm o
. . 311 311. . .1. .1. .1. .1.
. . . o ~+qmm H+qmmm o ems «mm mmm emu
. . . . H+qmwm . o emu «Mm own
9 . . m+qo ~+q H+qe . «a me No He

10.x.s11m1 u z
AOsmshv&MS "m+q
Aesfishv&ma HN+A
AQsM~hv9H9 " A

Amsmshvifla V

Amsfishv?H3 m

A0~Nsbv9H9 N

Aw~05hv9H9 H

e

e

e

e

m + was xnuumsnsm we» no mucmsmam .N .mnm

35

O

m+q.m
m1
m+q.~
m1

~+g.m
ma
m+q.~
m1
~+q.a
m1

H+q.m
m1
H+q.1
m1

 

 

 

 

m+q

N+A

H+A

O

o o 0
mm mm am
an as Hm
mm mm am
am Hm Hm
m1 m1 H1
as am an
me me He

+

10.x.hvsms n 20

1O.m.nvsms um+qe

A0~m~hv5ma "N+A0
AOsHshv&m5 "H+QO

10.m.nvsas m

II
o

Amsm~hv9H9 N

I)
0

II
o

Amsflsbv9H9 H

HOV xwuumfinsm mzu mo mucmemam .m .mHm

36

where the sum is over only those wave functions of the
two interacting submatrices involved.

The energy of a given vibration—rotation state is
the expectation value of the Hamiltonian in the represen-

tation A(V,J,T):

*
E(V,J,I) = <VJTIH|VJT> = Z Z (SYJT> SYJTH,. , (43)
1 1 J 13
where
* * * I I
Hij = feiH0jdv = (wv w (J,K,y)H0V,w(J ,K ,7 )dV .

It is convenient to separate the sum in Eq. (43) into

three parts:

* VJT 1

E(V,J.I) = E §(Si ) Sj Hij (44)
VJT * VJT 3
+ S. S. H..
g? 1 ) J 13
VJT * VJT 13
+ E §(Si ) Sj Hij I
where Hij are the matrix elements of the Hamiltonian in

state 1 (01+v2), H?. are the matrix elements of the Ham-

13
O O I 13
iltonian in state 3 (02+v3), and Hij

are the interaction
matrix elements connecting states 1 and 3.
The Hamiltonian may be broken up into its various

components such that Eq. (44) becomes

(45)
E(V,J,I) = X {( (sZJT)*s¥:T[vO (1)+ A(1)(P: )i jy+B”)(P ).
i j
+ C *(1)(P:)i (1) (O ) + (1) (O ).

xxxx xxxx ij yyyy yyyy ij

fii) (O ). .+ (1) (0 ) ]
yy xxyy ij xyxy xyxy ij

2 (SVJT)*SVJT “’33)

(3) (3)
i j +A (Px)ij+B (P; ).

13

+
P-M

37

*(3) 2 (3) (3)
+ C (Pz)i'+1xxxx(oxxxx)ij+ Tyyyy(oyyyy)ij
+ (3) (O ) (3) (O ) ]

xxyy xxyy ij xyxy xyxy ij

2 (SYJT)SSVJT[Gz(q1P3 9391)i'(Pz)

13 ij

+
)mbn
(.1

+ ny(qlq3)i j(PxPy Pny)ij] ’

The three sums in both Eqs. (44) and (45) are not each
summed over all i and j. The first is summed over-only
the i and j of state 1, the second over only the i and j
of state 3, and the third over only the i and j which
connect states 1 and 3.

To aid in the calculations, we can use the relation

for any hermitian Operator F,

2 2 (SVJT)*SYJTP.. = 2 ISYJTIZF.. (46)
1 J 13 . 1 11
1 j 1
VJT * VJT
+ 2Re S. ) S. F.. .
1% . . 1

Consideration of higher order P6 type distortion
terms results in the addition of the four terms of Eq. (21)
to both the first and second sums of Eq. (45).

The energy including these four terms may be expressed

£1)vo(1)+a(1)A(l)+B(1)B(l)+y(l)C*(1) (47)

(l) (l) (l) (l) (l) (1) (l) (l)
xl xxxx+ 2 Tyyyy+x3 Txxyy+x4 Txyxy

E(VJT) = x

+

+
+ x(3)v (3)+a(3)A(3)+8(3)3(3)+Y(3)C*(3)

O O

The superscript 1 is for state (v1+v2) and 3 for (v2+v

The a, BI

Operators

gr»

(II)

a

(n)

B

Y(n)

X(n)

PM PM PM PM PM PM PM PM P-M P-M P-M P-M P-M

U-M UM UM UM UM UM UM U-M U-M U-M UoM UM U-M

(3) (3)

T

(3)H (3)

HJ

+ x9Gz + xloG

Y: and Xi

var * VJt
(si ) sj
(5‘1”?) sxg’JWuij
*
(SXJ‘) s¥3‘(93)ij
*
(sgar) 5‘3.”prij
VJ? VJT'
(Si ) *Sj (Oxxxx)ij
*
(SXJ‘) s‘j’JWoyyyy)ij
*
(SXJT) S¥J1(Oxxyy)ij
*
(ng‘) s-‘J."“(oxyxy)ij
(SXJT) *SVJTJ 3(J+1)3
3
* ,
(SXJ‘) SgJT(P:)ijJ2(J+l)2
(sVJ‘) *ngT(p:)ijJ(J+1)
(SXJI)*S¥JT(Pg)ij
(SVJT)*S VJT

l

+X((3) $3)

38

(3) (3) (3)

xxxx 2

Tyyyy x3

(3)3

H1(+"7

xY

represent the average values of the

and are calculated with the expressions

xxyy x4

(qlp3- q3pl)-- P

39

VJT * VJT
x = {(S. )S. (qq)..(PP-PP)..
10 E j 1 j l 3 13 x y y x l]
where n may be 1 or 3 and the sums of “(n)’ 8(n), Y(n)'
and xén) through xén) are over only those i and j of the

state n, while the sums of x9 and x10 are over only those

i and j which connect states 1 and 3.

Computer Programs

 

Several more programs were written to aid in the

analysis of the two interacting bands vl+v2 and v +v

2 3’
l. SPEC-FIT 1
Ground state constants, upper state constants

and band center for both bands, perturbation cons-
tants G2 and ny, and all the frequencies of the
assigned transitions with their quantum numbers are
input. Using the constants that are input, the
computer calculates a spectrum and the average
values of all the Operators for the upper state
levels, and then fits the differences between the
observed and calculated frequencies to Eq. (49)

to obtain revised values of the upper state con-

stants of both bands and G2 and ny.

= z [x(n)Av(n)+a(n)AA(n)+B(n)AB(n)
o

-\)
obs calc n o

u
(n)AC(n) + X
i=1

+ Y xin’ATi“) <49)

8

+ 2 xin)AHin) ] + X9AGZ +xlOAny
1:1

In this equation n is equal to l and 3.

40

A listing of program SPEC-FIT l is in Appendix VI.
CORIOLIS

Ground state constants, upper state constants
and band center for both bands are input along with
the perturbation constants G2 and ny. The energies
of all the upper state levels of both bands with
and without the perturbation terms are calculated
by the computer. The perturbed calculation, un-
perturbed calculation and their difference are pr-
inted out for each level. This calculation allows
one to see the effect the perturbation has on the
individual levels. Optionally, the matrix elements
of the Hamiltonian before diagonalization and the
eigenvectors of the diagonalized Hamiltonian may be

printed out.

CHAPTER IV

THE INFRARED SPECTRA OF HZTe

Previous spectrosc0pic work on hydrogen telluride
has been quite limited. The far ultra-violet absorption
spectra of H2Te was reported on by Price, Teegan, and
Walsh (1950).27 The work of Rossmann and Straley28 pro-
duced a prism spectrum of v2 near 860 cm"1 and a prism-
grating spectrum of v1 and v3 which are overlapped near
2070 cm-1. From these they determined approximate values
for the molecular constants. In 1964 Hill, Edwards,
Rossmann, Rao, and Nielsen29 gave an analysis of a high-
resolution (=0.l cm-l) spectrum of v2 from which they ob-
tained values for both ground and upper state molecular
constants.

This work will consider the combination bands v1+v2,
v2+v3,l2v1, vl+v3, two bands near 4900 cm-1, and a re-
analysis of the fundamental v2.

Hydrogen telluride is an oblate planar asymmetric
molecule and is the most nearly symmetric member of a
series of asymmetric molecules which include H20, H28,
and ste. The geometry and normal modes of vibration

of H2Te are given in Figs. 4 and 5. As the figure shows,

it is a non-linear triatomic molecule with a bond angle

41

42

Fig. 4. Geometry of HZTe

/

 

 

Fig. 5, Normal Modes of H2Te

 

43

near 90°.

Preparation of H Te

2
The gas sample of hydrogen telluride was prepared

 

by reacting powdered aluminum telluride with distilled

30

water as discribed by Moser and Ertl. The reaction

is given by

AlZTe3 2 2

A diagram of the apparatus used in the production of the

+ 6H 0 + 3H Te + 2A1(OH)3 . (50)
gas is shown in Fig. 6. Small amounts of powdered alu-
minum telluride were allowed to fall on frozen distilled
water in the reaction chamber. The reaction went slowly
at first, but sped up when the heat of reaction'

began to melt the ice. The resulting gas was then pas-
sed through a drying tube and collected in the liquid
nitrogen cold traps. The system was kept under a vacuum
to draw the gas from the reaction chamber to the cold
traps. Since any H2Te that passed beyond the cold traps
would contaminate the oil of the vacuum pump, a 5 gallon
jar was evacuated and used as a vacuum in place of the
pump. A bypass around the first cold trap was provided
in the event the center tube plugged up. The most suc-

cessful run produced a 58% yield of 23 grams of H Te from

2

45 grams of AlzTe3 and 80 grams H 0. Care must be ex-

2
ercised when working with HZTe since it is a very poi-

sonous gas. It readily decomposes at room temperature

or in the presence of H20, 0 or ultraviolet light, so

2

44

 

SUOMNV I

 

 

 

 

mack. Boo

_

 

 

(H

 

J

 

 

 

O T.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

pom

a/I4 mcflmnflpm
. Oom \ 1< .325
\ Q T! a
mflma

 

 

 

 

 

mew: mo coau0260um How mopmummmd .o mam

45

the sample was kept as pure as possible and frozen at

liquid nitrogen temperatures until used.

Experimental Conditions

 

The infrared absorption spectrum of H Te in the

2
region 1.0 - 3.6u was obtained with the Michigan State
University high resolution infrared spectrometer. The
spectrometer, which has been discussed in detail by

31 and Keck,32 uses a Littrow-Pfund type nonochromator

Aubel
with 300 line/mm and 600 line/mm gratings mounted back to
back on a turntable. In the 1-3 micron region the source
used was a 300 watt zirconium arc and the detector a type N
lead sulfide photoconductor cooled by liquid nitrogen.
Beyond 3 u a type P lead sulfide detector was used

and the source was a zirconium arc box with a sapphire
window which transmits radiation with wavelength greater
than 3 microns. A small volume coolable multiple traverse

33'34 was used to hold the

cell of the J. U. White type
gas sample. All spectra were run with the cell maintained
at -50° C by circulating methyl alcohol, cooled by heat
exchange with dry ice and acetone, through the cooling
coils of the cell.

The frequencies of the infrared lines were obtained
by interpolating between fringes of constant frequency
spacing which were recorded simultaneously with the infra-

red spectra?5 These fringes were obtained from Edser-

Butler bands produced by a Fabry-Perot etalon of =3 mm

46

spacing. The fringes were calibrated by recording both
before and after the spectrum of interest, the absorption
lines of molecules which have been accurately measured

by Rank and co-workers.36'37r38

Infrared Absorption Bands of HZTe

 

Strong absorption bands of H2Te were found near

1 1

2900 cm- and 4050 cm-1, with weaker bands near 4900 cm- .
Slow speed slave recordings of these bands are shown in
Figs. 7, 8, and 9.

Throughout the 2900 cm-1 region may be seen strong
absorption lines of the (1-0) band of HCl which was used
as a calibration gas before the HzTe spectra and could
not be entirely removed from the cell. Fortunately the
HCl lines are widely spaced and seldom interfer with the

HZTe absorption lines.

The high resolution spectra were recorded on long

1 l

at 2900 cm“ ,

, and 0.06 cm"1 at 4900 cm-1.

charts with a resolution limit of 0.05 cm-
0.04 cm-1 at 4050 cm"1
Table III gives the eXperimental conditions under which
the spectra were recorded.

The procedure by which measurments of the lines on
the chart were actually made involves photographing the
charts and making the measurments on the film by means of
avadel semiautomatic digitized film reader connected to
an IBM 526 card punch. This procedure is given in some

39

detail by Barnett, and in Ref. 35. To help minimize

47

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. ._

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. ... ... ... ...... :.2...32.

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..

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HI

....3.... ...... ...... ...... 2...... ...

...... ...... .../...);

m mo muuommm coHumHOmQ¢ .h .mam

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O. i
1|

48

._.._... e, .../...)!‘144... .I 2......

.. .... .... 2...... ...... .3... ...?§£.’._.. (3.4.3.4.... . ...... ...... ...... ,- ......Ifl...

... . .. 4.11
...

...

 

 

 

 

 

_. 1.“....,..,....3:....._.H. age ... .44.... .1....1....... 95 121.114.... ......111544 .4413... .3... 3...}. ...)..111}...:..... *1...)..\/».,; \ II}, ...... 9.56.21.11.11... ...:I! 1.... 1.111.211... 1.1.)
J . .. __ ...! it .41: ._ .. .. . . .
_ . .

HIEo omov How: memm mo manommm coaumHOmnd .m .mflm

49

EU ommv

H 4 22
3.4:)... . .412... 512534.}. ..4.. 7.21. _
1:414 4.54.2.4 . .:::). .:::).3 421.4..fl_.).:: 4 4.4..4_:I.J.r:.:3.4..4 44:: .JJ..._4,.,.... 44 .... :..
. . . 4.42.2. : 4.445.. _: .. :.:.4 4: ... _
«231.444.... ...11.42;.34¢J;.4..4.)}...334 3.4.4441: : ::: :44 __::_4::. .:4 .:::: ::::: :::::::::: .:::: ::: :::::...2.2. :: 4:: ::: .
EU ommv
4.. .
EU oomv 22. 24.2.... 2.1.222}. 2
H1 \1:}ISJ._.\. 1...: 4.4 JJ 21.322145; ... ....
_ 3.21.344:2222...4_ : _.
1322.12... 4.4.
..rJ...:.J..:. 4412242434.. .3fi.:.:%_ 4.: 2.31.44.34.13—3; I _._ :.:—J}: $.12 :_ :.: :: :—
32}. .. .... : ._.._... 3.. .2: .4
2. :4 :
4
. .

N

o 002%..“
Ham: we m mo muuommm coflumHOmnd m
Eu oomv

H-

50

 

vmoo.o mmoo.o
AH‘o~Hvzom
AH.o.NvoNz
oo.o mo.o
mmao¢.o vmao«.o
com com
ma ma
Zumnm z-mnm

uum uumz com

m.m m.m
Uooml Uooml
mm mu
w v
oomv

 

hHoo.o

v0.0

mmHNm.o
com
ma

Zlmnm

Uooml

vvloa

vmoo.o
Acumvoo
Acumvaom
vo.o
nmamm.o
com
vH

Zlmnm

hNoo.o

vo.o

hmamm.o
oom
va

Zlmnm

cum uumz oom

m.m
Uooml
leoa
m

omov

w.m
Uooml
m

m

 

mmoo.o mmoo.o
AH~o.ovzum
Acufivaom
no.0 mo.o
mmhm~.o vmhom.o
com com
am om
mumnm mumnm
NOD OHM .UHflN
m.m w.o
ooomu ooom-
mu om
p m
comm

 

H

uflm cofiumunflamu
mo cofiUMH>wn cuqumum

mmmmo coaumunflamo

HIEo ‘cofiusHOmmm

IEo .COfiumummwm mmcflum
EE\m:H .mcflumnw

: .nuofizuflam Hmuuowmm
Houowuwa

monsom

a ‘Hamo :H sumcwq numm
musumuwmfiws mmw

EE .mHSmmmum mmw

.oz uumno

HIEo .coflwmm

 

mGOfluflwcoo Hmucmfifiuwmxm

.HHH manna

 

51

any experimental error, at least two recordings of the
spectrum in each region were made (three in the 4050 cm-1
region). Each of these charts were individually measured
and calibrated, and the frequencies obtained from charts

of the same regions were averaged.

Isotopic Species of HzTe

 

Tellurium in its natural state has the 6 isotopes

130Te, 128Te, 126Te, 125Te, 124Te, and 122

Te occuring with
percent abundance 34, 32, 19, 7, 5, and 2 respectively.

The absorption lines of the 6 molecular species of H Te

2
have been well resolved in all the bands recorded. A
small region near 4980 cm.1 which clearly shows this iso-

13oTe had

tope effect is reproduced in Fig. 10. Since H2
the strongest absorption lines, these were analyzed in
detail first. It was then a relatively simple matter to
analyze the other molecular species by working with the
apprOpriate lines just to the right of the corresponding
H2130Te line. Unfortunately, many of these lines, espec-
ially those of the less abundent species, were blended

or masked by other stronger transitions of another isotopic

species, and could not be used in an analysis.

52

Fig

 

Tellurium Isotope 130 128 126 124 122
125
Percent Abundance 34 32 19 5 2

 

 

 

CHAPTER V

ANALYSIS OF THE INFRARED ABSORPTION SPECTRA OF HZTe

The 2900 cm"1 Region

From Eq. (18) and the fact that v of H Te is

2 2
at 860 cm"1 and a survey of the fundamentals v1 and v3

1

indicate they are overlapped near 2070 cm- , it is expec-

ted that the combination bands vl+v2 and v2+v3 will both
be found in the 2900 cm"1 region. The differences betwe-
en the sums of the fundamental frequencies and the obse-
rved band centers of the combination bands is due to the
anharmonicities in the vibrations and any vibrational
resonances that may exist in the band under consideration.
Examination of the spectrum near 2900 cm-1 reveals several
regularly spaced series of lines proceeding out in both
directions from a group of strongly gathered lines near
the center of the absorption region. The regularly spa-
ced series were assigned as zero series (RRJ(J) and

P

PJ(J) ), first series (RRJ_1(J) and PP (J) ), second

J-l
series (RRJ_2(J) and PPJ_2(J) ), etc. lines and the gat-
hered lines assumed to be unresolved Q branch transitions
of a type A band. Attempts to identify the gathered

lines and the first few members of the series which are

Split because of the asymmetry, indicated that these

53

54

series and gathered lines were not those of an A band,
but those of a type B band where the gathered lines were
the RQO(J) transitions. After the band was assigned as
a B band (v1+v2) with band center at 2911.42 cm-l, most
of the stronger and many of the weak lines in this region
were identified and assigned to that band. After most of
the transitions of the B band had been identified, several
weaker series of unassigned lines were noted which even-
tually led to the identification of the type A band
(v2+v3) with band center at 2915.97 cm-l. This band was
found to be much weaker than v1+v2 and many lines were
masked, however, enough lines were identified to allow
an analysis.

The ground state combination differences formed from
the observed lines in these two bands were combined with

l and used in a

those formed from the bands near 4050 cm-
ground state analysis.

Although, as noted earlier, the upper states of
vl+v2 and v2+v3 perturb each other, the initial analyses
of both these upper states were made ignoring the inter-
action and the bands were fit separately to Eq. (32)
holding the ground state constants obtained from the
ground state analysis fixed and varying only the upper
state constants. Even though the fits by this method

were poor they allowed many assignments in both bands to

be made. From Eqs. (37) and (40), initial estimates of

55

G2 and ny were made which were used along with the upper

state constants from the single band fits as starting values

to simultaneously fit the bands 0 +v and v +v3 to Eq. (49).

l 2 2

The simultaneous analysis of the two bands including the
the interaction terms gave a very good fit and helped to
identify many more lines that were badly perturbed.

One of the most notably affected series of lines were the
1+V2° Although these lines

were strong and easily identified, they fit very poorly

lines of the PQl(J) series of v

in the single band fit in which the interaction was ig-
nored, the deviations being as large as 0.5 cm-l. When
these same lines were used in the perturbation analysis,
they very well, with most deviations no larger than 0.01

-1

cm . The results of the simultaneous analysis of v +v

l 2
and v2+v3 are given in Tables IV, V, and VI. The ranges
of the constants are for a simultaneous confidence inter-
val of 95% which is about 6 times the standard error in
these fits. The constants in the tables are given to

six numbers past the decimal for H2130Te so that future
calculations will agree with those given in the append-
icies. The significant integers may be determined from
the simultaneous confidence intervals. All fits were
made holding the ground state constants fixed at the values
obtained from the ground state analysis. The assigned
transitions of vl+v2 and v2+v3 for H2130Te are listed in
Appendix II, and Appendix V lists the calculated upper

State energy levels for these two bands both with and

56

130

 

 

 

Table IV. Molecular Constants of H Te for
-l
the States vl+v2 and v2 3, (cm )
State v1+v2 02+v3
Vo 2911.4157 i0.0074 2915.9697 10.019
A 6.347845 t0.0018 6.319069 10.0047
B 6.136936 10.0017 6.165087 10.0042
c* 2.960471 10.00035 2.975441 i0.0011
T -0.003776 10.00013 -0.004229 10.00083
aaaa
Tbbbb -0.003125 t0.00008 -0.003736 10.00042
T +0.003ll3 10.00006 +0.003769 10.00070
aabb
1 -0.000953 10.00006' -0.000471 10.00028
abab
HK -34x10"9 13x10-9 -l88x10-9 i96x10-9
K +0.876 +0.908
No. of pts. 199 47
G2 +0.025882 10.033
G -0.164l34 i0.0031
XY
Total No. 246
of pts.
Std. Dev. 0.007

 

 

 

 

 

57

128

 

 

 

 

Table V. Molecular Constants of H2 Te for
the States v1+02 and 02+v3.
State vl+vz v2+v3
V0 2911.583 10.010 2916.146 i0.027
A 6.3488 t0.0022 6.3176 10.0053
B 6.1369 10.0019 6.1659 10.0050
c* 2.96089 10.00048 2.9755 t0.0018
T -0.00376 10.00015 -0.00376 10.00097
aaaa
Tbbbb -0.00312 t0.00009 -0.00356 10.00046
T +0.00311 t0.00007 +0.00351 t0.00090
aabb
1 -0.00096 10.00006 -0.00063 t0.00036
abab
HK -37x10'9 15x10-9 -181x10'9:262x10‘9
K +0.875 +0.909
No. of pts. 167 35
G2 ' +0.033 10.036
G -0.1651 10.0034
XY
Total No. 202
of pts.

Std. Dev. 0.008

 

 

 

58

 

 

 

Table VI. Molecular Constants of H2126Te for
the Statesvl+v2 and 02+v3.
State vl+v2 v2+v3
vo 2911.752 10.008 2916.318 i0.037
A' 6.3506 10.0019 6.3216 10.0070
B 6.1378 10.0019 6.1662 i0.0052
*
C 2.96171 10.00048 2.9762 i0.0028
T -0.00382 10.00014 -0.0041? 10.00100
aaaa
Tbbbb -0.00309 10.00007 -0.00353 10.00050
Taabb +0.00311 i0.00007 +0.00342 t0.00160
T -0.00099 t0.00007 ' -0.00053 10.00081
abab
-9 -9 - -9
HK -42x10 t6x10 -118x10 t500x10
K +0.874 +0.90?
No. of pts. 139 20
G2 +0.04? 10.030
G -0.1665 10.0029
XY
Total No. 159
of pts.
Std. Dev. 0.006

 

 

 

 

59

without the interaction terms and the difference. This

allows one to see how much each level is perturbed beca-
use of the interaction of the two bands. It can be seen
for high J low K levels, the perturbation can contribute

as much as several cm-1 to the energy.

The 4050 cm"1 Region

 

It is expected that the bands 201, vl+v3, and 2v3
will be near the 4050 cm-1 region. Darling and Dennisonl
have shown that a vibrational resonance can exist between
states of the type (vl,v2,v3) and (v112,v2,v3*2) which
tends to push the two states apart. In analogy with

H25, where the bands 201 and 0 +03 were separated by

1
1 (203 was not observed)40, and HZSe21 where Zv
and vl+v3 were separated by 2.04 cm-l, the two bands of
130 1

H2 Te, 201 at 4062.89 cm‘1 and 01+03 at 4063.37 cm“ ,

were identified. The band 203 of H2Te was not observed.

The fact that the two bands were very close together and

2.24 cm_ 1

that the zero, first, second, etc. series within each
band were nearly coincident, combined with the isotope
effect to give the striking feature in this region of
strong, badly overlapped series of lines proceeding out
in both directions from the band center. Because of this,
many lines were blended or masked and could not be used
in the analysis. Between these large groups of lines
were found weaker lines that were identified as the Q

branches.‘

60

The ground state combination differences formed from
the observed transitions listed in Appendix II. were com—
bined with those formed from the bands near 2900 cm-1
and used in a ground state analysis.

As noted earlier, there is an interaction between
the upper states of 201 and vl+v3, and also between vl+v3
and 293. Because of this, an analysis of the upper states
of these bands should simultaneously include all three.
Unfortunately, 293 of H2Te was not observed and therefore
could not be used in any such analysis. It was expected

1)

and 203 was expected to be about 80 cm-‘1 away, the effect

since 201 and vl+v3 were so close together (20.48 cm-

of the interaction between 203 and vl+v3 would be much
smaller than that between 201 and vl+v3 and therefore the

effect of 203 could be ignored and the two bands 2v and

l
vl+v3 treated as two interacting bands.

The initial analysis of the upper states of 201 and
vl+v3 ignored all interactions and each band was fit
separately to Eq. (32), holding the ground state constants
fixed and varying only the upper state constants. By this
method many of the lines, such as the zero series and
first members of the Q branches which were least affected
by the interaction, were assigned and approximate values
of the upper state constants were obtained. These con-
stants were used as starting values along with the per-

turbation terms G2 and ny obtained from the analysis of

the 2900 cm.1 region in a fit of the spectrum of 2vl and

61

01+03 to Eq. (49). When the two bands were fit together
with the interaction term included, many lines that were
badly perturbed were identified in both bands. A few of
the upper state energy levels were extremely sensitive to
the values of the constants used, and a small change in
the value of a constant resulted in a large change in the
energy of that level. Any line that involved such a level
was not included in the fit until near the end of the
analysis when the constants used were almost the final
ones.. The results of the simultaneous analysis of 291
and vl+v3 are given in Tables VII , VIII , IX , X , and
XI. The ranges of the constants are for a simultaneous
confidence interval of 95% which for these fits is about

6 times the standard error. All fits were made with the
ground state constants held fixed at the values obtained
from the ground state analysis.

The definitions of G2 and ny (Eqs. (3?) and (40) ),
show they are not vibrationally dependent, so one would
expect to obtain the same values for these constants from
the analysis of vl+v2 and 02+v3, and from the analysis
of 291 and 01+03. Different values of these constants
were obtained from the bands in the two regions as can be
seen from Tables IV and VII. One reason for this may
be because the effect of the interaction of 203 with vl+v3

was ignored in the analysis of 4050 cm.1 region. Exam-

ination of the definition of Gx in Eq. (37) shows that

Y
its theoretical value is relatively insensitive to the

62

130

 

 

 

 

 

Table VII. Molecular Constants of H2 Te for
the States 201 and vl+v3r

State 2v v1+v3

Vo 4062.8902 i0.011 4063.3743 10.012

A 6.068978 10.0045 6.063802 10.004?

B 5.940074 t0.0046 5.946441 10.0046
3%

C 2.956935 10.00044 2.959565 i0.00052
T -0.003281 10.0001? -0.003351 t0.00028
aaaa

Tbbbb -0.002679 10.00018 -0.002805 i0.00014
r +0.002326 t0.00007 +0.002405 10.00013
aabb

1 -0.000508 10.000062 -0.000445 t0.00013
abab

HK -31xlo"9 t6x10-9 -22xlo‘9 t6xlo'9
K +0.91? +0.924

No. of pts. 214 187

G2 +0.011163 10.0059

G -0.229786 50.0018
XY

Total No. 401

of pts.

Std. Dev. 0.011

 

 

63

 

 

 

 

 

Table VIII. Molecular Constants of H2128Te for
the States Zvl and vl+v3.

State 2vl vl+v3
v0 4063.109 10.012 4063.594 10.014
A 6.0695 10.0056 6.0636 10.005?
B 5.9413 10.0056 5.9478 t0.0053
0* 2.95761 10.00059 2.96078 10.00076
Taaaa -0.00325 10.00022 -0.00312 10.00034
Tbbbb -0.00269 10.00021 -0.00282 10.00016
Taabb +0.00229 10.00009 +0.00226 t0.00016
Tabab -0.00048 10.00008 -0.00050 i0.00015
HK -31xlo'9 110x10'9 -26x10‘9 1‘20x10'9
K +0.9l8 +0.925
No. of pts. 179 144
G2 +0.0134 10.0066
ny -0.2303 10.0022
Total No. 323
of pts.
Std. Dev. 0.011

 

 

64

 

 

 

 

 

Table IX. Molecular Constants of H2126Te for
the States 201 and vl+v3.

State 2V1 vl+v3
Vo 4063.340 10.016 4063.832 t0.017
A 6.0685 10.0085 6.0630 10.0078
B 5.9448 10.0081 5.9503 i0.0075
c* 2.95846 10.00091 2.9599 10.0012
Taaaa -0.00323 10.00034 -0.00308 10.00044
Tbbbb -0.00283 10.00033 -0.002?9 10.00025
Taabb +0.00235 £0.00011 +0.00229 t0.00022
Tabab -0.00050 10.00010 -0.00064 10.00021
HK -45x10‘9 1255410"9 --30x10‘9 136x10‘9
K +0.920 +0.92?
No. of pts. 134 113
G2 +0.0166 i0.0077
ny -0.2317 10.0028
‘Total No. 247
of pts.
Std. Dev. 0.011

 

 

65

125

 

 

 

 

 

Table X. Molecular Constants of-H2 Te for
the States 201 and vl+v3.
State 20 v1+v3
“6. 4063.453 $0.031 4063.948 $0.011
A 6.065 $0.010 6.060 $0.010
B 5.950 $0.010 5.955 $0.010
C* 2.9592 $0.0020 2.9596 $0.0019
T -0.00307 $0.00052 -0.00305 $0.00064
aaaa
Tbbbb -0.00315 $0.00058 -0.00296 $0.00051
T +0.00252 $0.00046 +0.00234 $0.00029
aabb
T -0.00068 $0.00023 -0.00057 $0.00023
abab
HK -68xlo'9 1141x10'9 -20x10"9 1100x10'9
K +0.925 +0.931
No. of pts. 30 57
G2 +0.021 $0.011
XY
Total No. 87
of pts.
Std. Dev. 0.005

 

 

66

Table XI. Molecular Constants of H2124Te for

the States 291 and vl+v .

 

 

3
State 2vl v1+v3
vo 4063.588 $0.019 4064.068 $0.008
A 6.071 $0.012 6.068 $0.011
B 5.943 $0.011 5.948 $0.011
c* 2.9582 10.0007 2.9605 10.0010
Taaaa The distortion terms were held constant

at the ground state values in this fit.

 

 

Tbbbb

Taabb

Tabab

HK

K 0.919 0.924
No. of pts. 39 46

G2 +0.0150 $0.0085

ny -0.2305 $0.0050

Total No. 85

of pts.

Std. Dev. 0.006

 

 

 

 

67

value of y and therefore a fairly accurate estimate of
its size can be made by substituting reasonable values

of ”1' w3, A Be’ and C8 and most any value of y in

er
this equation. Doing this, we find theoretically, for a
y of 45°, ny z -0.11. In all cases the experimental.
value obtained in the least squares fit was much larger
than this by 50%-100%. The theoretical value of G2 is
quite sensitive to the value of y, so without knowing y
fairly accurately, a comparison between the theoretical

value and eXperimental value cannot be made.

The 4900 cm.1 Region

 

Consideration of the sums of the fundamental vibra-
tion frequencies and the Darling-Dennison Resonancel
between states of the type (v1,v2,v3) and (v1$2,v2,v312)
+v and v +v +0 to be

1 2 l 2 3
overlapped in the 4900 cm-1 region. Examination of the~

leads one to expect the bands 2v

absorption spectrum in this region revealed what appeared
to be a single band with the zero, first, second, etc.
series of the R and P branches distinct and well separated
and a few Q branches between these series.” Attempts to
form ground state combination differences from these ser-
ies which agreed with those combination differences formed
from the bands in the other regions failed. However, it
was possible by using ground state combination differences
to identify a few RQ lines from the assigned RR lines and

to identify a few PQ lines from the assigned PP lines.

68

It was concluded that the absorption lines observed
were actually due to two bands separated less than 0.2 cm_l,
one band at 4891.7 cm-1 having strong transitions with
AK=+l, and very weak AK=-l transitions, and the other at
4891.9 cm.1 having strong AK=-l transitions and very
weak AK=+1 transitions.

As noted earlier, there is an interaction between
the states 2v1+v2 and vl+vz+v3, and also between v +v +0

1 2 3

and 02+293. The two bands were assigned as 20 +v and

l 2
v1+v2+v3, and attempts made to analyze the two bands
together, as was done in the 4050 cm.1 region, using the

few identified lines were not successful, so the analysis

was not carried further.

Ground State Analysis of H Te

2
As mentioned before, whenever possible it is generally
better to analyze the ground state using ground state com-
bination difference fits to Eq. (34) instead of line fits.
An analysis by this method eliminates the upper state
energy levels, which may be perturbed, and the ground
state combination differences from all bands may be simul-
taneously used. The analysis of the ground state using
combination differences may proceed before or at the same
time the upper states are being analyzed, although final
values of the ground state constants are not obtained

until all possible assignments in all the bands have been

made and combination differences formed from them. The

69

ground state combination differences are very useful in
assigning new lines and verifying the assignments of others
already identified. Thus, the analysis consists of a series
of ground state combination difference fits and line fits,
each time identifying more lines and forming more combina-
tion differences from these lines until as many assignments
as possible are made. The ground state constants obtained
by Hill 33 21.29 were used for starting values. As many
ground state combination differences as possible were.
formed from the four bands analyzed, and each was given a
weight depending on the weights of the two lines used to
form it. If the same combination difference was formed

in more than one band, a weighted average was taken and
given a weight equal to the sum of the separate weights.

In this manner the ground state combination differences
were formed and fit to Eq. (34). Tables XII, XIII, and
XIV give the results of these fits. The ranges of the
constants are for a simultaneous confidence limit of 95%,
which is about 4 times the standard error in these fits.
The observed and calculated ground state combination

differences of H2130

Te are compared in Appendix III.

It was not necessary to include all four H terms to
get a good fit, but if at least one of them was allowed
to vary, the fit was improved. It was decided to include
only HK, although the choice was somewhat arbitrary as to

which H term to use because, for our data mainly lines

with K equal to or slightly less than J are observed,

70

130

Table XII. Ground State Molecular Constants of H2 Te

 

Taaaa

Tbbbb

Taabb

Tabab

No. of pts.

Std. Dev.

 

 

130

H2

6.247489

6.094834

3.035999

-0.003139

-0.002791

+0.002285

-0.000533

9

-20x10’

+0.905

218

0.006

Te

$0.00049

$0.00048

$0.00024

$0.000030

$0.000034

$0.000046

*0.000025

112x10"9

 

Values quoted by
Hill‘gglgl.29

6.2486

6.0970

3.0361 $0.0014

-0.0025 $0.0014

-0.0032 $0.0010

+0.00219 $0.00035

+0.906
118

0.028

 

 

Table XIII.

71

Ground State Molecular Constants of

 

 

Te and H2126Te
H2128Te H2126Te

A 6.24899 10.00053 6.25162 10.00073:
B 6.09460 10.00051 6.09470 10.00053.
c 3.03642 10.00025 3.03690 10.000213
Taaaa -0.003146 10.000034 -0.003273 10.000099
Tbbbb -0.002769 10.000034 -0.002748 10.000051
Taabb +0.002268 10.000047 +0.002312 10.000054
Tabab -0.000531 10.000026 —0.000555 10.000026
HK -21x10‘9 1:12:410‘9 --23xlo‘9 112x10'9
K +0.904 +0.902
No. of pts. 176 136
Std. Dev. 0.006 0.005

 

 

 

 

72

 

 

 

 

Table XIV. Ground State Molecular Constants of
H Te and H 124Te.
H2125Te H2124Te
A 6.2514 $0.0018 6.2533 $0.0021
B 6.0958 $0.0020 6.0943 $0.0018
C 3.03712 $0.00059 3.03722 $0.00014
Taaaa -0.00315 $0.00036 -0.0033l $0.00031
Tbbbb -0.00288 $0.0002? -0.00271 $0.00025
Taabb +0.00236 $0.00021 +0.00228 $0.00014
Tabab -0.00059 $0.00012 -0.00054 $0.00009
HK -41x10'9 142x10'9 -24x10"9 132x10“9
K +0.903 +0.901
No. of pts. 49 44
Std. Dev. 0.005 0.004

 

 

73

giving a high correlation between the quantum coefficients
of the H'S and therefore, the effect on the energy levels

of any of the H'S is about the same.

Reanalysis of 02

Because of limited computer programs available at
that time, the analysis of 02 by Hill gt 21°29 used only
transitions involving Jle even though quite a number of
lines were identified with J>12. For this reason it was
decided to reanalyze the upper state of 92 including all
the identified transitions. The ground state constants
obtained from the analyses of the other regions were
used and were not varied. The spectra had a resolution
limit $0.10 cm.1 and the various isotOpes were not resol-
ved on this record.’ 92 is a type B band and fortunately

the upper state is not perturbed so the spectrum was fit

to Eq. (32). The results are given in Table XV.

Vibrational Analysis

 

The band center of a particular state may be written
as a function of the normal frequencies mi and anharmonic
constants xik' From Eq. (18), in the absence of a vib-

rational resonance,

vo(vl,v2,v3) = G(vl,v2,v3) - G(0,0,0) (51)

l
E viwi + E g Xik[vivk + §(Vi+vk)] .

Expanding these sums for the bands that are available

74

Table XV. Molecular Constants of HzTe for the

 

State v2.

Values obtained Values quoted by

from this analysis Hill 22 3E.29
v0 ' 860.765 $0.022 860.79 $0.01
A 6.4296 $0.0050 . 6.4306
B 6.2259 $0.0048 6.2258
C 3.00555 $0.00052 3.0064 $0.001?
Taaaa -0.00379 $0.00056 -0.0034 $0.0016
Tbbbb -0.00318 $0.00040 -0.0029 $0.0013
Taabb +0.00314 $0.00016 +0.00257 $0.00046
Tabab -0.00095 $0.00014 -0.00088 $0.00036
HK -46x10‘9 110x10”9
K +0.880 +0.880
No of pts. 170 96
Std. Dev. 0.026 0.029

 

 

 

 

 

75

we have
00(0,1,0) = 02 + 2x22 + 2x12 + %x23 (52)
vo(l,1,0) = ml + 2xll + 02 + 2X22 + 2x12 + %x13 + %x23
00(0,1,1) = 02 + 2x22 + w3 + 2X33 + %x12 + %xl3 + 2x23
00(1,0,1) = wl + 2xll + m3 + 2x33 + %X12 + 2Xl3 + %X23 ,
and from Eq. (20) we have for 201,
00(2,0,0) = %[(2wl+6xll+X12+X13)+(2w3+6x33+X13+X23)]
- %[4y2+{(261+6x11+x12+x13)-(203+6x33+x13
+x23)}2 1/2

Unfortunately the above band centers that have been
observed are not enough to solve for the constants mi and
xik’ but if additional data is sometime obtained, it
may be possible to solve for some of these constants
individually or at least more useful combinations of them.
This additional data could come from a positive identifi-

l

cation and analysis of the bands near 4900 cm- , and from

an analysis of the fundamentals vl and v which have

3
been observed overlapped near 2070 cm-1. Besides yielding
valuable information for a vibrational analysis, v1 and v3
perturb each other through the interaction term Eq. (41)

and they could be simultaneously analyzed as two inter-

acting bands in the same manner as vl+v2 and 02+v3.

CHAPTER VI
STRUCTURE CALCULATIONS

It can be seen from the definition of A, B, and C
in Eq. :10) that by the analysis of the prOper set of

3
bands one can solve for the 0'

and consequently the
equilibrium constants Ae, Be' and Ce' With the equili-
brium constants available, the equilibrium structure may
easily be calculated.‘ The constants obtained from the
analysis of the ground state and the bands v2, vl+v2,

vz+v3 and 01+03 are sufficient to allow a least squares

analysis determination of the 0?, and a?.. In principle
the-0E cannot be determined from these bands since the
analysis of the upper states of the bands 01+02, 02+03,
and vl+v3 yields C* instead of C because of the Coriolis
interaction. However, since the Coriolis perturbation
coefficient Gz determined in the fits is so small, it may
well be possible that C*$ C for the states analyzed, then
the a? may be determined and Ce calculated. A test of
the validity of this would be a comparison of the calcul-
ated quantities Ce and AeBe/(Ae+Be) which, because of the
2+1:

From Eq. (10) we can determine the relations,

planarity condition I: = I should be equal.

1 A
Ae = A(0,0,0) + 52 6i (53)

76

77

and

+aA

A A
A(0,0,0)-A(vl,v2,v3) = a v1 + a 3v3 , (54)

1 2V2
with similar relations for B and C. The constants from
the ground state and the four bands mentioned were used

in a least squares fit to Eq. (54) to determine the 0?,

and similarly, a? and 0%

assuming C*= C. The results of
this fit are given in-Table XVI along with the equilibrium
constants Ae’ Be’ Ce calculated with Eq. (53) and the
equilibrium moments of inertia calculated from Ae, Be’ and
Ce through Eq. (11). The ranges given for the “i deter-
mined in the fit are the-standard errors. The equilibrium

structural parameters re and 0 [see Fig. 4.] listed in

Table XVI were calculated using the equations,

r = [(1/2m){1§ + [(2m/M) + 1 11:}11/2

e (55)

tan(§) = [IS/{1:[(2m/M) + 1]}11/2

where m and M stand for the masses of the hydrogen and
tellurium atoms respectively. In these calculations,

the constants from the 130 isotope of HZTe were used,
except for the constants from 02 where the isotOpes were
not resolved. This is valid since the 01 are approximately
equal for each isotOpe i.e.; iagz 1300?, iag= 1300?,

iaCz 130 C

J ajwhere i is equal to 128, 126, 125, 124, and 122.

Examination of the equilibrium constants obtained
1 . -1
shows thatCe and AeBe/(Ae+Be) differ by only 0.001 cm
which tends to suggest the validity of the assumption

1:
C 2 C.

78

 

 

Table XVI Equilibrium Structure of H2Te
a? +0.0787 .0.002 cm'1 a? +0.0885 10.0004 cm'1
A . B
02 -0.1793 0.002 62 . -0.1306 10.0003
A 1 B
03 +0.1064 0.003 03 +0.0602 10.0004
Std. Dev. 0.002 Std. Dev. 0.0004
-1 -l
A 6.2504 cm B 6.1038 cm
8 e
1: 4.4783 x 10’409 cm2 1: 4.5858 x 10‘409 cm2

 

*
Assuming C = C

 

0E +0.0454 10.0004 cm'1
0% +0.0302 10.0003
G? +0.0308 10.0004
Std. Dev. 0.0004
-1
c 3.0892 cm
6
I: 9.0608 x 10'409 cm2
. 513'
(re)H_Te = 1.646 A

0 = 90° 38'

 

CHAPTER VI I
CONCLUSION

The development of the vibration-rotation Hamiltonian
for a planar asymmetric non-linear XYX molecule has been
outlined and energy expressions by which energy levels.
and other quantities, useful in an analysis of the Spectra
of such a molecule, may be calculated have been given.
Since the upper vibrational states of some of the bands
may interact through a Coriolis type resonance, a modified
Hamiltonian was given which included the necessary inter-
action terms for a description of these states.

High resolution (10.05 cm-l) calibrated recordings
of the vibration-rotation absorption spectra of HZTe

1 1

were obtained near 2900 cm- , 4050 cm- 1

, and 4900 cm’ .
The frequencies of all individual absorption lines in
these regions were measured from these recordings.

A method was outlined by which the theoretical energy
expressions could be used to analyze spectra, and was
applied in the analysis of H2Te. Computer programs were
written which would calculate energy levels, line freq-
uencieslperform least squares fits of ground state comb-
ination differences or line frequencies and make a simul-

taneous least squares fit of the upper states of two bands

79

80

which perturb each other through a Coriolis like inter-
action.

The results of the ground state analysis yielded the
ground state constants A, B, and C, centrifugal distortion

constants (taus and HK) for the molecular species H2130Te,

2128Te, H2126Te, H2125T 124T

state analysis came the molecular constants, centrifugal

H e, and H2 e. From the upper

distortion constants, and band centers for the bands 02,
01+02, 02+03, 201, and 01+03,.and the perturbation coef-

ficients G2 and ny of the interacting states. The upper

states of the species H2130Te, H2128Te, H2126Te, H2125Te,

124
2

while the analysis of the upper states of vl+02 and v2+v3

included the species H2130Te, H2128Te, and H2126Te.

and H Te were all analyzed in the bands 201 and vl+v3,

Finally, by a proper combination of molecular constants
from the different vibrational states, the equilibrium
constants Ae, Be’ and Ce were obtained, from which the

equilibrium structure (re and e) of H2Te was calculated.

10.

11.

12.
13.

14.

15.

16.

REFERENCES
B. T. Darling and D. M. Dennison, Phys. Rev. 21,
128 (1940).

M. Goldsmith, G. Amat and H. H. Nielsen, J. Chem.
Phys. 24, 1178 (1956).

R. C. Herman and W. H. Shaffer, J. Chem. Phys. 16,
453 (1948).

L. E. Snyder, forthcoming thesis, Michigan State
University, (1967).

K. T. Chung and P. M. Parker, J. Chem. Phys. 28,
8 (1963).

J. M. Dowling, J. Mol. Spectroscopy 6,550 (1961).

T. Oka and Y. Morino, J. Phys. Soc. Japan 16, 1235
(1961).

T. H. Edwards, N. K. Moncur, and L. E. Snyder, J.
Chem. Phys. 46, 2139(1967).

R. A. Hill and T. H. Edwards, J. Chem. Phys. fig,
1391 (1965).

R. A. Hill and T. H. Edwards, J. Chem. Phys. 14,
203 (1964).

K. T. Chung and P. M. Parker, J. Chem. Phys. 43,
3896 (1965).

E. B. Wilson, Jr., J. Chem. Phys. 4, 313 (1936).
B. S. Ray, Z. Physik. 18, 74(1932).

G. W. King, R. M. Hainer, and P. C. Cross, J. Chem.‘
Phys. 11, 27 (1943).

H. H. Nielsen, The Vibration-rotation Energies of
Molecules and their Spectra in the Infra-red,

Handbuch der Ph sik, Vol. XXXVII/l (Springer-Verlag,
BerIin, 1959), p. 247.

H. C. Allen and E. K. Plyler, J. Chem. Phys. 35,
1132 (1956).

81

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.
34.

82
L. Pierce and N. DiCianni, J. Chem. Phys. 38,
730 (1963).

K. T. Chung and P. M. Parker, J. Chem. Phys. 43,
3865 (1965).

F. X. Kneizys, J. N. Freedman and S. A. Clough,
J. Chem. Phys. 44, 2552 (1966).

J. K. G. Watson, J. Chem. Phys. 46, 1935 (196?).
R. A. Hill, Thesis, Michigan State University (1963).

H. C. Allen, Jr. and P. C. Cross, Molecular Vib-

Rotors, John Wiley and Sons, New York, 1963, pgs.
I78 and 207.

 

N. W. Naugle, N.A.S.A., Manned Spacecraft Center,
Houston, Texas.

 

M. A. Efroymsen, Mathematical Methods for Digital
Computers, Ralston and Wilf (Eds.), New Yor ,' 0,
p 191.

 

H. H. Nielsen, Revs. Mod. Phys. 23, 90 (1951).

S. A. Clough and F. X. Kneizys, J. Chem. Phys. 11,
1855(1966).

W. G. Price, J. P. Teegan, and A. D. Walsh, Proc.
Roy. Soc. A201, 600 (1950).

K. Rossman and J. W. Straley, J. Chem. Phys. 34,
1276 (1956).

R. A. Hill, T. H. Edwards, K. Rossman, K. Narahare
Rao, and H. H. Nielsen, J. Mol. Spectry. 14, 221
(1964..

L. Moser and K. Ertl, Z. Anorg. Allgem. Chem. 118,
269 (1921).

J. L. Aubel, Thesis, Michigan State University,
(1964).

D. B. Keck, J. L. Aubel, T. H. Edwards, and C. D.
Hause, 1966 Symposium on Molecular Spectroscopy
and Structure, Columbus, Ohio.

J. U. White, J. Opt. Soc. Am. 32, 285 (1942).

T. H. Edwards, J. Opt. Soc. Am. 21, 98(1961).

35.

36.

37.

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83

K. N. Rao, C. J. Humphreys,.and D. H. Rank, Wave-
len th Standards in the Infrared, Academic Press,
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D. H. Rank, D. P. Eastman, B. S. Rao, ant T. A.
Wiggens, J. Opt. Soc. Am. 52, l (1962).

D. H. Rank, G. Skorinko, D. P. Eastman, and T. A.
Wiggins, J. Mol. Spectry. 4, 518 (1960).

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Wiggins, J. Opt. Soc. Am. 51, 929 (1961).

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1104 (1954).

APPENDIX I

NONVANISHING MATRIX ELEMENTS OF THE

HAMILTONIAN OPERATORS

The nonvanishing matrix elements of the Hamiltonian

operators in the symmetric t0p basis 0(J,K) are given

below.

f = J(J+1)

<KIPXIK$1>
<KIP IK$1>
Y

<KIPzIK> =

2 _ l
<K|PXIK>— -2-

<KIP§IK$2>
<K|P2IK> =
Y

<KIP2|K$2>
y

<KIP2|K>
Z

<KIP4|K>
x

<K|P4IK$2>
x

<KIP4|K$4>
x

$%[f-K(K$l)]1/2

§tf-K<K11)11/2

K
(f-KZ)

[f-(K$1XK*2)]l/2

= -%[f-K(K$l)]l/2

<KIP2|K>
x

= -<KIP2IK$2>
X

K2

2 2

%[3f2-2f-6fK +5K +3K4]

1/2

-%[2f—K2-(K$2)2]{[f-K(K$1)][f-(K$l)(K12)]}

§g{tf-K(K11)1tf-(K11>(K12)1[f-(K12)(K1311

[f-(K.3)(K14)]}1/2

84

85

<KIP4IK> = <K|P4IK>
y x

<KIP4IK$2>
Y

<K|P4|K14>
Y
<KIP:IK> = K

<KIP: P “+P PiIK> =

<K|P2P2+PZ P i|K$2>

<K|P2P2+P2P§|K> =

<K|P2P2+P2 zP2|K12>

<K|P2P2+P2 P2|K>
XY yx

<K|P2P§+P2 P2|K14>

<K|PxPy+PnyIK$2>

4
<K|Px

-<K|P4|K12>
X

IK$4>

K2(f-K2)

-%[K2+<K12)21{[f-K(K11)1[f-(K11)

<K12)1}1/2

<K|P2P2+P2P2|K>

—<K|P2P2+P2 P 2|K12>

2 2

[£2+2f-+2fK -5K +K4]

bIH

-%{[f-K(K11)1[f-<K11)<K12)1[f-<K12)
(K13)1[f-(K13)<K14)111/2

= *%[f-K(K*l)]l/2[f-(K*l)(K$2)]l/2

The non-zero matrix elements of the Hamiltonian

operators in the basis set W(J,K,y) are given below.

K ¢ 1, K #40,

<,%(4K11_K)|Pi|/%(4K11_K)> = %(f-K2)

K = 1,

1 2 1
<,§(1111_l)lPx|/§(wltw_l)>

K g 0,

l 2 l
'2'(f-K 2* If

1 2 1 _
WEEW’K’w-K)lPx'./2(“’K+2*“’-K-2)> ‘

-%{[f-K(K+l)][f-(K+l)(K+2)]}1/2

86

K = o,

K751,K7!0,

</%(wKtw-K)IP;I/%(WK2W_K)> = %(f-K2)

K = 1,

‘/%(¢1*W-1)|P§'/%‘¢1‘¢-1)’ = 2(2'K2)*%f
K g 0,

K = 0,

<¢0|Pi|/%(w2+w_2)> = ‘< P: >

</2(¢Ktw-KIP:I/%(¢K*¢_K)> = K2

K ¢ 0, K g 1, K g 2'

2 2

</%‘¢K*w-K)IP:I/%(¢K*w-x)> = %[3f2-2f-6fK +5K +3K4]

K =~0I

<wo|PilW0> = $[3f2-2f]

- %[(3f2-8f+8)12f(f-1)]

A
Ema
A
e-
[.1
H-
-e
I
H
v
’U
x.>
Ens
A
e-
[_I
5+
6-
I
l'-'
V
I

</%(Wz*w-2)IP:|/%(wz$w_2)> %[3f2-26f+68]$%3f(f-2)

K # 0, K g 1,

l 4 1
</2(wKtw-K)lle/2(WK+2tw_K-2)>

87

K 1 0,

./%(¢K1¢_K)|p:|/%(¢K+41¢_K_4)> = (1/16){[f-K(K+l)][f

-<K+1><K+2)1lf-(K+2)(K+3)1[f-<x+3)<x+4)1}1/2

K = o,
(wOlP:I/%(w4+¢_4)> = 8/%{f(f-2)(f-6)(f-12)}l/2
K#0.K;€1,K¢2,
</%(¢K1W_K) lP;l/%(¢Ktw-K)> = < P: >
K = o,

4 4
<¢0|Pylwo> = (WOIPXIwO>
K = 1,
< l(¢1*¢_1)lP;I/%(wl$w_l)> = %[3f2-8f+8$2f(f-1)]
K = 2,

4

<1/%(w2tw—2) IPy|é(W22W_2)> = < P: >

K110.K11.

1 4 1 _ 4
</§(¢Kiw-K) le|/§WK+2*¢_K_2)> "' "< PX >

K = o,

<wOIP;I/%(¢2+¢_2)> = -< P: >

88

K

1,

</%(¢l1¢_l)IP;|/%(w31w_3)> = %{(£-2)(f-6)}1/2[(2£-10)1%f]

K 1 0,

</l(wK$ w _K)|Py 4|/%(WK+4*¢-K—4)> = < P: 2
K = 0,

<¢0|P;I/%(w4+W_4)> = < P: 2

1 4 1 4
</2(¢Kiw-K)lel/2(2Ktw-K)> = K

K P 1,
</l(wK10_K )IP2P2+P2 P:|/%(1K11_K)> = K2(f-K2)
K = 1,
</%(w1$w _1)|P2P2+P22 P X”201,11 w _1)> = (f-l)1%f

</2(¢K*w-K)IP:P:+P:P:I/2(wK+2*w-K-2)’ = '%[K2+(K+2)2]{

[f-K(K+l)][f-(K+l)(K+2)]}1/2

K=0,

<w0|P2P2+P22 P x|/-(w2+w_ 2)> = -/2[f(f- 2)] 1/2

K 2 1,‘

1 2 2 2 2
</§(¢K1¢_ K)|Py P z+P 2P 2|/—(wK$ w _K)> = K (f—K )
K = 1,

</-(w1$ w _1)|P2 P 2+P 2P 2|/-(w1 w _1)> = (f-l)*%f

89

K ¢ 0,

</2("’K w -K)|PZPZ+Pz P yl/2(WK+2 2 -K- 2) = '< IPZPZ+PZPZI 2
=0,

..OIP2P2+P§ P jl,-<wz+w_ 2)> = -< IP2P2+P2 zP2| >

K g 0, K A 2,

</%(¢K1¢_K)|Pfipi+Pny|/2(¢K1 w _K)> = %[f2+2f-2fK2-5K2+K4]

K = 0,

<00|Pipi+Pipilwo> = %[f2+2f]

K = 2,

</%(¢21¢_ 2)|P2 Py 2+Py 2P:|/%(¢21¢_2)> = %(f2-6f-4)1%f(f-2)

K 1 1,

</%(wK1w_K)lPin+Pnyl/§(0K+2 1 _K_ 2)> = 0

K = 1,

</%(wl$w_l)lP2P§+P% Pil/%(w3iw_3)> = 1%f{(f-2)(f-6)}1/2

K P 0,

</%(¢K1¢_K)|PiP§+Pny|/-(4K+4$ 0_K_ 4)> = -§{[f-K(K+l)[

[f-(K+l)(K+2)][f-(K+2)(K+3)][f-(K+3)(K+4)]} 1/2
K = 0,

<¢0|P2 xPy+Py 2P2|,-<w4+w_ 4)> = 47-{f(f- -2)(f- 6)(f- 12)} 1/2
</2(2Ktw-K)lel/2(2wa-K)> = K

K = 1,

f

NI P-

1 1 _
</2Wltw- l)IPJKIPy'IJDnyl122-(212231)> _ t

90

K g o,
< 1(W 2w )IP P +P P I 1’(11: w )> =
/2 K -K x y y x /2 K+2‘F -K-2
' 1/2

+ 1tf-K(K+1)] 1/2

[f-(K+l)(K+2)]

K = o,

1 _ 1 _ 1/2
<w0|PxPy+Pny|/1(w1-w_1)> - 12[f(f 2)]

K > 2,

1 1 _
</§WK1‘2—K)IPxPy+Pny|/§(wK-2‘w-K+2)> _
' 1/2

-1[f-K(K-1>1 [f-(K-1)(K-2)]1/2

APPENDIX II

130

ASSIGNED LINES AND OBSERVED FREQUENCIES OF H2 Te FOR

THE STATES vl+v2, v2+v3, 2vl, AND v1+v3

The observed frequencies of H2130Te are compared

with the calculated frequencies, where the ground state
rotational energy levels are calculated with Eq. (24)

and the upper state energy levels with Eq. (47).

91

ASSIGNED LINES 0F H 13075 FOR

nu¢aaa~uo+

X
I
X
*0-

NDHW‘P‘RJFJOF‘AJHJDF‘RDPJDCDHiOW‘CDH‘DO‘C3HH3h.CDDHDP‘OM*CDH43hficHACDHWDF‘C’Hwatb

0ED\IO\IJMHEDNHJ

ICIICCOIIIIQOIIICCHCfllfllllllllitlllfllllllllIJIIIIC-‘II

L.

0WD\IOWJJBOHUF‘FMD

X
l
x

O

PNNNJ-‘NPNPNPNFMFPDHDFDPOPDPOHDHDHDHOMDFDPOFDHCPO

2

‘00D‘JO\3JHUIJF*H43

PmuwAP-M~
i)O‘iOJIJMHNH‘CI.CdNHACD0MD\JO\3OWdNH‘

u+bflh
hJHWHPMo

92

095

2920.733
2926.437
2926.610
29329283
2937.993
2943.338
2948.640
2953.793
2958.798
2963.646
2968.353
2972.907
2977.309
2981.571
2985.681
2989.640
2993.450
2997.119
2902.251
2896.018
2889.920
2883.534
2877.018
2870.350
2863.539
2856.585
2949.490
2842.252
2834.878
2827.378
2819.740
2811.970
2933.545
2938.981
2945.175
2950.865
2956.458
2961.882
2967.168
2972.301
2977.306
2982.121
2986.813
2991.349
3000.028
2889.780
2884.439

vl+v2

OHS-GALE

0.011
00.002
0.006
09009
00007
”0.001
90.001
90.000
0.002
10.004
90.003
I0.007
80.018
00.013
900013
90.012
'00003
00027
'900035
I00008
00007
.09002
01004
00005
09006
0.007
0.006
80.001
‘00009
009011
000021
90.036
09007
90.000
00002
90.003
00006
00003
00010
01011
00032
00008
00007
000006
00011
000010
100003

NY

0.00
1.00
1.00
0.40
0.40
1.00
1.00
0.40
1.00
1.00
1.00
0.40
0.40
1.00
0.40
0.40
0.40
0.40
0.00
0.01
0.40
1.00
0.01
0.40
0.40
1.00
1.00
0.40
0.01
0.40
0.40
0.00
1.00
0.40
0.40
1.00
1.00
1.00
1.00
1.00
0.00
1.00
1.00
0.00
0.01
0.40
1.00

03“.) 3.3.10

‘GIJLOJHOIGhbfiibfidlb‘CdJMNIIOLOCHNJGHDOINWH“OOH““OOH“GdNHDGdNJGFUGINMHOINW‘RDFWUF‘NDHJONIH

H».
ocn~aoxnauunapr.:rcun~aoxnauua:m

Ht‘h‘
rsawunla

(HhihfiHW‘abD‘OQDNIOWIlb.CHOHONSHwalDD‘HOKULdeNHU

QUICIIIIICCOIIICCIIICOCIC‘...IltfllflllliillllflIlilfilllt‘

HfiJF‘H
EDGHDU‘CWOID\IOWIOL‘

Fwd—5H
'OID\JOWIUMbILOJHRNDVDCVNCDULQCd

up.»
~60~Odl\lJnNH‘CD(DCHQ¢!OIUH30h&Cd

Glb‘flfitblb.Cd‘Il.Mlfitbfii.fil.filflfllfil“HUGH“GIMGlbuvflifl(IGHDGlflfi‘NHHKHUH‘NW‘NHJRDHIUHH‘“!

pAH-m- flvmws
awoktcnourvc>UI.caOI

”0‘.
-.(d0‘NU0HHOID‘JOWJIIOHHRDNW‘HHDE‘GP0¢D\JOWIJIOHHRDNHUF‘OHD\JOWIIIGHDF‘

93

2877.962
2878.029
2871.785
2865.424
2858.893
2352.238
2845.439
2838.508
2831.441
2824.238
2816.916
2946.840
2952.585
2957.989
2963.919
2969.410
2975.143
2980.938
2985.773
2990.872
3000.619
3005.776
3009.777
2872.919
2871.902
2866.126
2866.337
2860.180
2853.954
2847.595
2841.093
2834.450
2827.689
2820.759
2813.726
2959.433
2963.844
2965.809
2970.826
2971.303
2976.970
2977.n33
2982.785
2988.404
2993.882
2999.204
3004.418
3009.454
2865.448
2859.287
2861.666
2854.850
2854.361
2848.758

90.005
I'09006
09004
0.018
0.003
09007
0.006
01010
0.010
0.005
0.008
0.001
0.007
0.008
80.011
0.004
09003
00007
00003
09009
0.003
90.002
I0o020
I0.002
30.005
90.009
00.005
0.015
00.006
0.002
0.006
0.008
00026
0.006
0.009
'01006
'00029
00.001
30.009
I0I000
30.001
80.003
80.002
.00002
0.002
09001
0.038
0.042
'0.009
I0.003
0'003
l0.005
90.004
901004

DQQJ$JJUD~J

rbpoArsH
a.sasu«-J

HP‘

3JJAJ03¢.ANDH.r-.JO)V’@JVOJQ)J¢¢JOJ‘IJJOJQJJxI

\fl‘JHNH‘H‘ONIOMHRDHCDDWDGHidbO‘Q\JO\IOWJUH3UIO\IOHJJIUJLUM50I&\I&Jfllbm\M‘bm\lbwflédfllbfl

MNNMMNODDDDSHHHFF‘QCANHUN”30W‘UNROG&UNHPD‘3~OOVObaCdMNO-‘Vomb

CIIIIOI'ICO‘CIOIIIOQOIIIICUQCIICKIIIIOI......ICCHCIGCCI

9w.
O‘QOJIUHJOM‘CDOCD

9.5

(IOKULdehJOWIJIOHDF‘H50WD‘QCVNOD\JOHO¢D\JOHD‘OOD\IOJNF.GHO¢D\IOJD\I

I‘F‘PW‘F‘FW‘FW‘F‘FW‘CMDCDCHUF‘NH‘CDO‘HNH‘VIOJHRJHWD\JO\IOMHRJNH*‘HD‘JOWIOUURDHW‘CiO‘dOWfl

94

2842.852
2836.450
2830.099
2823.596
2972.665
2976.090
2979.156
2984.856
2983.594
2990.173
3001.626
3007.160
3012.545
3017.776
3022.864
2853.578
2850.582
2847.098
2843.580
2837.310
2831.520
2825.445
2819.237
2985.801
2988.083
2997.932
3003.021
3009.094
3014.761
2841.952
2839.571
2830.812
2826.431
3002.645
3005.946
3008.480
2830.459
2822.840
3016.938
3030.616
3044.n31
2908.173
2914.770
2915.209
2915.936
2916.951
2918.308
2920.029
2901.624
2902.679
2903.413
2904.222
2905.040
2902.825

90.008
.0.001

0.006
90.001
'00019
90.003
90.006

0.001
90.006
I00004
90.008
'00007
90.005
90.008
90.007

0.008

0.000
90.004
00.008
00.009
'0.008
00.004
.00001
90.003

0.006
'0.001

0.046

0.001
00.014

0.000
'.0.004
'0.005

0.001
.00016

0.011

0.014
00.003
00.006
00.020
90.025
30.049

0.005
90.000
90.012

0.001
00.000
80.001
00.007

0.001
00.002
l0.°03
90.007

90.001 ~

0.021

0.01
0.40
0.01
0.01
0.00
0.40
1.00
0.40
0.40
0.40
1.00
0.40
0.70
0.40
0.00
0.00
0.01
0.40
0.00
0.01
0.01
0.01
0.00
0.40
0.00
0.40
0.00
0.40
0.00
1.00
0.40
0.00
0.01
1.00
0.40
1.00
0.01
0.00
1.00
0.01
0.01
0.01
1.00
0.40
1.00
1.00
1.00
0.00
1.00
0.40
0.40
0.40
1.00
0.00

Vl’OJVIJJJOJDVI’JOD‘3’.)bOO‘DVVJbDH‘QJ’JJbfid)DflD’JbJQJ’J'de00‘!

P‘H\JUHHOIHWIJBOJONDOIIOKbJLNFOOHflUIOGdHW"IO\IJMJOIOJUOIP(IU050dOHDhioChOWflUfl‘JLNHU‘IU

OillL‘JLObbLflUUmklUBUHflfldOiaCdOIblbhbbibJI‘JURINFONDMWURJMFOOIaCdOIGCdONMF‘HW‘F‘HW‘F‘HWUND

‘QOV0¢D\6OJ3CPO‘D‘JOWfliDOHMOhvhfiiroab\FH\IJMD\WHIDOHUKIJB&(dGMO‘dOhOWflJHW‘QOKOJflUI‘(NAPO‘Q

1M!“lb$flflmh3\du\lbhafi.WIGCdOH‘Hh‘GIO\llflflhfiD\I.MNOHUQHFHUCD\IO\HlbuchH‘NIUW$1b00MF‘F‘OCD

UHJ\flUHM\flUh§Jbb45‘h‘JIOMbIbOHbOHHOWUCd0H30‘0”“OUHCdOMQGIGHURJNNDNHDAJNHUhJNHUflJNHURDHW‘

95

2905.770
2906.765
2921.389
2920.083
2922.846
2921.317
2923.601
2921.739
2924.184
2922.386
2895.319
2695.600
2896.797
2.97.077

2897.I03

2898.977
2900.276
2698.395
2927.981
2927.660
2927.688
2926.920
2928.110
2929.292
2928.520
2928.954
2930.134
2930.990
2688.576
2089.337
2491.772
2891.127
2092.133
2893.698
2893.164
2933.387
2933.967
2934.665
2935.280
2936.266
2936.518
2881.733
2882.647
2883.701
2884.642
2886.084
2887.260
2939.314
2940.000
2940.746
2941.661
2942.358
2674.742
2875.657

90.004
0.002
90.008
0.001
90.001
00.015
90.009
90.022
'0.005
'0.011
I0.001
0.003
0.001
0.003
l0.004
00.009
I0.019
0.012
0.006
00.009
0.002
90.008
l0.006
00.006
0.001
00.007
I0.007
0.013
.0.002
90.004
0.010
0.001
0.000
90.014
0.001
90.002
0.002
90.008
90.008
90.000
0.005
0.002
80.004
0.006
0.024
90.005
90.005
0.010
0.005
l0.005
80.014
80.012
I0o001
0.014

1.00
0.40
0.40
0.40
0.40
0.01
0.40
0.01
1.00
0.40
0.40
0.40
0.40
0.01
0.00
1.00
0.00
1.00
0.40
1.00
0.40
0.00
0.40
0.40
0.00
0.01
0.40
0.40
1.00
1.00
0.70
0.40
0.40
0.01
0.01
1.00
0.40
0.01
0.00

0.40

0.00

0.01
0.40
0.01
0.40
0.20
0.01
0.40
0.20

‘QJKOJD~C)\JJD0.D\|J-J(DD'thlabiDOJD\lfl.lhnD‘J\LfiJ|J.lJ-bddJID‘QJH)&IO>J‘QJIfi~lJlbad\DO‘Q

HW‘\lUHflGIHW’JIOHDROCDOIDJIONDEJO\RULOGIHW"IOWIJIUCIOHUOIHW¥UI.{H.H0hio(’OUW\IJD‘IDNFV\N

05hlhblb‘bbEflWHJUI“\ROIUCdOIqulhfilbOrblbfifivflDMFOAlMfiofl3M00OiquOIGGdOUUF‘HW‘P‘HW‘E‘HWONB

‘30“0‘3‘00JQCIOHD‘JOKUNO(D\JOWIJIQHOOD\FV\BJDGVN‘QOKOWflklJthdODCVVI’CFUL§Cd‘IOWF\DULD(dAPO‘H

‘MIO85‘00NDJ\CUKIOFGF.UHQOWGF‘HHQGbO\IOHBflJD\IOMflJLM€dHHUCI\ID\fllbW0flGU‘\flUN‘lbOIat‘H*DCD

UHJ\HUHM\IULbJL‘458h‘lbbublbbnflbflflfii9H3(dOUHCdOI“CdOHflfldGHHhDNHUFONHDhJNHUflJNHDfiJNHUBDHW*

95

2905.770
2906.765
2921.389
2920.383
2922.846
2921.317
2923.601
2921.739
2924.184
2922.386
2895.319
2395.800
2896.797
2097.077
2897.003
2898.977
2900.?76
2:93.395
2927.981
2927.060
2927.688
2928.520
2928.110
2929.292
2928.520
2928.954
2930oi34
2930.990
2680.576
2889.337
2491.?72
2891.i27
2892.133
2893.698
2893.164
2933.387
2933.967
2934.665
2935.280
2936.766
2936.518
2881.733
2982.547
2883.701
2884.042
2886.084
2887.260
2939.314
2940.000
2940.748
2941.561
2942.358
2874.742
2875.057

909004
0.002
.00008
0.001
50.001
90.015
90.009
90.022
'00005
”0.011
30.001
0.003
0.001
0.003
00.004
90.009
.00019
0.012
0.006
90.009
0.002
90.008
90.006
.01006
0.001
00.807
.00007
0.013
l0.002
00.004
0.010
0.001
0.000
"0.014
09001
00.002
00002
90.008
909008
90.000
0.005
0.002
80.084
0.006
0'024
'0.005
90.005
0.010
0.005
I0.005
309014
90.012

A-0o001

0.014

1.00
0.40
0.40
0.40
0.40
0.01
0.40
0.01
1.00
0.40
0.40
0.40
0.40
0.01
0.00
1.00
0.00
1.00
0.40
1.00
0.40
0.00
0.40
0.40
0.00
0.01
0.40
0.40
1.00
1.00
0.70
0.40
0.40
0.01
0.01
1.00
0.40
0.01
0.00
0.40
0.40
0.40
0.40
0.00
0.00
0.01
0.01
0.40
0.01
0.40
0.20
0.01
0.40
0.20

H.‘
._.._30

H.‘
fl’flljlflJ-JJJJ‘JOJNIJAJbC‘JOOODfljfl-GVIJ

94‘
CdOIOJU‘JOW’NflOLb\IflJH13(DOHNORUL§€thPWDRONDP4DC3Odohiuum€d

L»‘CdOFVF‘F‘HW‘f‘HlH*Dc3CDD<3C3GID€3<0JDO‘Q\IO‘Q‘JUHJ\IO*O{)

H.‘

p.

IF\flOMQ‘Q‘JOJI\flJLbIDCDCHI‘lOWBJbOHUCDi3€VOCD‘JGVH‘O\HDh‘GWO

'6‘

p.
NDDCHHL.‘QULfiGiflfluflw:(>D\JO\IJIGhOPWDH‘PW.:Dafi‘GHSHWDO‘h

\fl\fl‘h‘flflcflofllNHURJKH‘F3HW‘F‘HW‘F‘HHD\OODGHD‘QOMOCFOH’\QUVW

96

2878.362
2879.726
2881.146
2945.096
2945.899
2947.856
2867.618
2868.923
2950.674
2956.127
2957.066
2052.965
2961.416
2966.571
2952.739
2965.012
2977.949
2.91.059
3004.418
3017.975
3031.749
3045.715
3059.896
2957.836
2985.379
2987.493
2998.404
2994.762
3006.990
2025.481
3029.032
2995.582
3021.186
3013.099
3026.775

0.003
0.001
90.025
0.032
0.004
90.003
30.003
0.016
0.006
0.009
0.009
'0.000
0.001
0.009
90.002
00.003
'0.004
90.028
I0.015
00.017
90.010
90.020
80.04‘

90.001'

0.004
90.023
90.015
90.027
90.021

0.008

0.010

0.010
I0.013

0.006
.0.012

0.01
0.01
0.00
0.00
0.20
0.40
0.40
0.00
0.40
0.40
0.40
0.01
0.00
0.40
1.00
1.00
0.40
0.00
0.00
0.40
0.40
0.01
1.00
0.01
0.01
0.00
1.00
0.01
1.00
0.00
0.01
0.40
0.40
0.01
0.00

ASSIGNED anss or H213OTE FOR

94‘.

‘J:.V"D&COO\-'3“C.OCDV"J.u~&;‘vDCb-‘9.0~CJO¢.C‘J‘rfi'aJ OJ'.U.U~ (-

X
0
X
.-

9+:

"”HI‘FO‘VO‘OIOUIUNCOVOG&MMPDO\I.“HHOODOQ‘GMPODODU‘bCINN

. ‘ . . . » ~ - « p x n . - - d n. .

F‘HW‘
FHDF‘CHO‘JONIOHHIOHHOII\LbGINH‘t. 1.

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2906.970
2942.725
2440.270
2964.270
2969.379
2974.329
2906.I40
2000.780

2894.519'

2088.227
2081.013
2875.291
2060.634

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2033.620
29373580
2949.067
2909.603
2061.416
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2806.486
2002.647
2878.478
2070.221
2363.052
2057.350
2050.746
2044.004
2037.192
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2074.074
2000.208
2044.003
2058.789
2046.220
2039.750
2962.031

2909.019i~

2912.920
2913.554
2914.041

02+v3

OBS!CALC

0.007
90.035'
0.003
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2919.163
2919.311
2919.850
2006.257
2007.340
2909.138
2910.278
2911.542
2925.444
2000.010
2000.456
2901.527
2981.020
2093.346
2894.179
2988.679
2006.676
2007.591
2944.286
2970.339
2051.937

90.008
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0.029

0.002
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90.193
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4053.606
4047.397
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4027.471
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3998.763
3991.194
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3975.633
3967.624
3959.460
3951.153
3942.699
4003.295
4088.562
4088.966
4093.914
4093.969
4098.926
4103.748
4108.410
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4121.402
4125.425
4129.293
4132.948
4040.702
4034.249
4033.933

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4099.144
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4104.391
4109.104
4113.551
4117.682
4122.031
4125.986
4129.704
4133.429
4136.907
4140.222
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4013.706
4006.576
3999.142
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3451.355
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4052.084
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4051.200
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4070.454
4071.182
4009.500
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4043.211
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4076.004
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4072.285
4031.902
4029.949
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4005.173
4002.604
4079.604
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APPENDIX III

I
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.

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110

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12.147
24.378
37.679
25.514
61.683
49.557
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61.290
36.959
49.106
36.691
11.704
50.481
86.303
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25.770
75.087
48.061
24.275
49.382
37.595
86.346
36.787
73.621
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110.893

63.385
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111.446

59.643
23.887
61.778

110.113

85.410
61.184
97.967
37.173
98.785
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2

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111

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110.827
61.614
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60.867
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135.998

135.407
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134.464

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111.413
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109.519
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135.398
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72.994
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172.734
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160.414
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158.646
82.302

108.787

182.542
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145.930

158.836

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112

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96.

awn

p.
OOOOOOOOOOOOOGOODOOQOOG00OOOOOOVDNOQGNEOOOVVOVONQQOCDa“

OUPPNNO‘.VUI‘IGDPPNONG-Il‘uwb‘umm.OWONOGPPNbNPWQPPGNNO‘AUUWVW

...
OOOOOVOW‘OflPmmNOVOQO‘JWWO.VUIUI.PVVOOQNU’W0.0QO.VWMUWOO‘NN

113

171.502

71.962
109.838

61.472
171.700
110.044
158.653

60.613
208.127

74.071
146.335
208.480

86.759
110.231

48.368
197.440
134.054

35.968
110.439
110.680
121.722

23.471
109.308

10.867
232.639
121.834
121.901
231.946
195.969

60.639
122.103
158.305

48.263
122.351
146.088
122.616

35.853
133.793
122.922

23.351
121.398

10.715
254.733
220.134
133.867

72.673
182.387
170.286
158.102
134.751
135.117
145.825

23.222
133.468

I'08002
0.009
01002
0.002

'0.004

'0.003
0.004
01008

.08011

“0.010

“0.001

I'0.006

.08001

'0.003
0.001

.08005

.0400‘
0.001

.08006

'0.005

'0800‘

.08003

'0.004

.08008
04020
01006

.08008

'0.010

'0.020

'04001

.0000,

.09008

.0800;
01004

'00001
0.001

'0.002
0.002
0.005
0.005
0.000

01003

04021
01016
01004
'08002
"0.009
01003
'0.006
.08010
.0401.
'0.018
'0.006
.0800:

0.03
0.59
0.04
0.04
0.01
0.77
0.02
0.02
0.40
0.04
0.40
0.01
0.02
0.05
0.40
0.01
0.99
0.44
0.40
1.61
1.84
1.08
2.04
0.44
0.57
0.02
0.04
0.02
0.01
0.01
0.72
0.01
0.02
0.59
0.40
0.02
1.96
1.14
0.27
1.79
0.03
0.57
0.40
0.02
0.02
0.02
0.27
0.40
0.04
0.02
0.02
0.63

NOGHNQ‘OAO

11
10

10

12
10
13
13

10
10
10
11

10
12
11
13

NOUPNPANN

114

10.544
170.106
182.233
146.878
157.882
145.528
158.477
157.566
206.011

"0.009
.08004
'0.004
.0800‘
01002
08003
'0.014
01004
04010

0.02
0.04
0.02
0.33
0.04
0.97
0.02
0.02

APPENDIX IV

CALCULATED GROUND STATE ENERGY LEVELS

GRWU 0 STATE CQHSTANTS

A = 6.2474600000
Q 3 640749540000
C 33 3140159900000
TAAAA = -0.003139000
THWBH : -0.009701000
TAABH : 0.007285000
YAiAfi = -0.000533000
“J = -0.000000000000
HJ4 = -n.000000000000
HK! = -0.000000“00000
N< = -0.00“000‘20000
KA’PA = 009(1406

J K- 5+ th

1 1 0 12.3420

1 0 1 9.1290

1 1 1 9.2929

2 - 0 37.0200

2 0 2 - 74.4715

2 1 2 24.4934

2 1 1 33.6568

2 2 1 34.1104

3 6 0 74.0719

3 1 2 61.4379

3 2 2 61.4960

3 2 1 70.4160

3 0 3 45.8170

115

APPENDIX IV

CALCULATED GROUND STATE ENERGY LEVELS

GPWU 0 STATE CGNSTLNTS

A : 5,2474dnufi0n
Q 3 6.0943640NDn
C = 3.0159900fi0n
TAAAA : ~n.n0‘139”00
7H45@ : -n.n09/o1n00
TA\BH = fi.fl0?295"00
TA4AH = -n.FU“535fiUn
HJ = -n.rnnonor0coog
HJ< = -n.n0~onor09000
HK! = -n.nn"0nonnaona
“< = -U.PU“U“U’20000
KAJPA : 0.9"40é

J K- W+ tflb

1 1 n 12.3420

1 n 1 9.1290

1 1 1 9.2925

2 7 0 37.0700

2 n P . 94.4715

2 1 2 94.4334

2 1 1 33.6568

2 2 1 34.1104

3 A 0 74.0?19

3 1 2 ‘1.4379

3 2 2 ‘1.4968

3 2 1 70.4160

3 n 3 45.8170

115

UT A

UTU“:

U'l

U703 U30"!

O‘O‘O‘O‘

00‘

0‘0‘

00‘

.‘J

in K)..-

...L

3‘.)

fi

mwb

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(AH

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H

116

71.3755
“5.8177

125.5285
110.637?
76.2219

110.8133
75.2220

119.3991
95.1197

120.8942
95.1745

194.9126
171.9977
114.8791

172.4527
1‘4.d?96

180.5382
156.6816
156.6959

192.7706
186.6710
1"6.6889

256.7354
245.4309
298.6960
146.2152

246.2274
298.6986
146.2152

255.8226
2xo.3444
190.5965

236.8955
230.4321
1“U.5“65

0000

JMJ‘V

{05):}?

HCAJN 3

HLNJ‘V

NAU‘O‘

r-‘TIJJ'VC

V5303

O‘AMO

0301512

117

344.7436
330.8415
294.5394
232.3998

332.2417
294.5487
232.3898

339.1984
316.1115
266.6219
101.7971

343.1953
316.2543
266.6722
191.7971

442.8654
428.1297
392.4808
3R0.6413
243.4303

430.3918
392.5086
330.6414
243.4303

436.6210
443.8457
364.7120
290.2346

441.5825
414.1859
364.7132
290.2346

553.0095
537.1966
582.3450
440.8753
354.1154

540.6155
502.4169
440.8755
364.1154

118

9 V 1 546.0478
9 5 3 593.4181

9 4 5 474.7604

9 2 7 4fi0.6797
9 0 9 3n1,1n93
9 9 1 551.9555

9 7 3 594.0267

9 5 5 474.7646

9 3 7 490.6797

9 1 9 391.1093
10 1o 0 675.0‘59
10 8 2 657.9465
10 6 4 694.0063
10 4 6 562.9845
10 2 8 476.7308
10 n 10 364.8279
10 9 2 662.8421
10 7 4 624.1729
10 5 6 562.9953
10 3 8 476.7308
10 1 10 364.8779
10 9 1 667.4400
10 7 3 644.6777
10 5 5 596.6563
10 3 7 593.0965
10 1 9 474.0760
10 10 1 674.2911
10 8 3 645.7777
10 6 5 596.6690
10 4 7 523.0765
10 2 9 424.0760
11 11 0 806.9104
11 9 2 700.2689
11 7 4 757.3145
11 5 6 696.8479
11 3 8 611.1586
11 1 "0 499.9589
11 10 2 796.9925
11 8 4 757.6674
11 6 6 696.8507
11 4 8 611.1586
11 9 *0 499.9589
11 1n 1 890.7600
11 H 3 77/.4567
11 6 5 730.2730

11
11
11

11
11
11
11
11
11

12
12
12
12
12
12
12

12
12
12
12
12
12

12
12
12
12
12

12

12
12
12
12
12
12

13
13
13
13
13
13
13

13
13
13
13
13
13

n

HIAJ‘VCH

JinJ‘NOH

12
in

:v33‘

11

r-‘CAJ‘PV

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{boa-Mo

4.;
MC

MOmO‘bNO HOVU'IOJH HOVU'OJP MDOJO‘AN

AA

MOGObN

—A .-.l

119

657.1559
558.7973
434.5787

808.1¢b4
779.3087
730.3067
657.1560
558.7973
434.5787

954.4110
934.1357
9na.oa99
842.379“
757.2679
646.8408
510.3531

942.9795
902.7947
842.3384
767.2679
666.8408
510.3531

945.9658
921.5311
875.4645
802.9357
705.2634
5“1.9053

953.8144
974.5052
875.5457
842.9362
7n5.2634
591.9158

1111.4369
1U“9.4145
1058.1206
999.2779
914.9143
605.3318
669.8571

1100.7085
1059.4030
999.3n12
914.9144
805.3318
669.8571

120

13 12 1 1143.0016
13 10 3 1076.8“64
13 8 5 1032.0616
13 6 7 960.2190
13 4 9 845.3115
13 2 11 740.8220
13 0 13 592.1410
13 13 1 1110.930?
13 11 3 1081.2329
13 9 5 1012,2425
13 7 7 960.2207
13 5 9 863.3115
13 3 <1 740.8820
13 1 13 592.1410
14 14 0 1279,8686
14 12 2 1256.0314
14 10 4 1225.1637
14 h 6 1167.5199
14 6 8 1043.9390
14 4 40 975.2766
14 2 «2 840.9001
14 n «4 679.9310
14 13 2 1270.0793
14 11 4 1297.3954
14 9 6 1167.5779
14 7 8 1093,9394
14 5 «0 975.2766
14 3 «2 840.9001
14 1 14 679.9310
14 13 1 1271.7902
14 11 3 1243.2339
14 9 5 1199,8656
14 7 7 1128.8435
14 5 9 1032,7747
14 3 11 911.3531
14 1 13 763.8017
14 14 1 1279.430?
14 12 3 1249.330!
14 10 5 12P0.2420
14 R 7 1128.848?
14 6 9 «032.7747
14 4 «1 911.3531
14 2 13 763.8017
15 15 0 4459.610?
15 13 2 1433.8839
15 11 4 14h2.9587
15 9 6 1346,8589
15 7 8 1264.1475

NHL—I
\FU'IU‘

15
15
15
15
15
15

15
15
15
15
15
15
15
15

15
15
15
15
15
15
15
15

16
16
16
16
16
16
16
16
16

16
16
16
16
16
16
16
16

16
16
16
16
16
16

1n
12
14

WWH‘OVUWCN'H ANOODO~3N

UTCNr-‘OVUWCAH

P‘
U

A-JAA
0&NOCDO‘AN

337105“)

AVA 4. J
0.5ny

HOVWCAH

121

1156,5459
1U?3.3158
863,7;71

1450.988?
1496,6307
1346.9940
1264,1691
1156.5”59
10?3.3158

863.7771

1452.2328
1470,5732
1378,6411
1398.679?
1213.4817
1093,1516

946.8832

773.7997

1469,2273
1478,5988
1379,3787
1308,6435
1213,4818
1093,1516

946.893?

773,7n97

1650.604?
16?2.8194
1591,2524
1537,0467
1485.4n84
1348.8355
4216,9239
1058,8177

873.4620

1643,3303
1596,9729
1537.3624
1455,4131
1348.8355
1216.9239
1058,8177

873.4620

1644,2156
16n8,6982
1568.1496
1499.3752
1445,2454
1286,0954

122

16 3 «3 1141.266&
16 1 15 969.6195
16 16 1 1650.267?
16 14 3 1618,7963
16 12 5 1569,4759
16 18 7 1499,4119
16 8 9 1445.245;
16 6 11 1286,0954
16 4 13 1141.2"66
16 ? 15 969.6190
17 17 0 1852,8334
17 15 2 1622,6154
17 13 4 1789.8315
17 11 6 1737,8734
17 9 8 1657.4505
17 7 1n 1552,0652
17 5 12 1491,5296
17 5 14 1265.0114
17 1 16 1081.4611
17 16 2 1847,0031
17 14 4 1798,2818
17 12 6 1738.4856
17 10 8 1657,4637
17 8 10 1552.065!
17 6 12 1471.5294
17 4 14 1265,0114
17 2 16 1081,4611
17 16 1 1847,6196
17 14 3 1847,7466
17 12 5 1767,9494
17 10 7 17n0.8559
17 8 9 1607,8615
17 6 11 1489,9971
17 4 13 1346,5794
17 2 15 1176.688?
17 0 17 979.1705
17 17 1 1852,5412
17 15 3 1819.636?
17 13 5 1770.3439
17 11 7 1740,9434
17 9 9 1607,8628
17 7 11 1489,9871
17 5 13 1346,5794
17 3 «5 1176.688?
17 1 17 979.1705
18 18 0 9066,3186
18 16 2 9032,9950
18 14 4 1998.5424
18 12 6 1948,9556

123

18 19 8 1870,0998
18 8 10 1765,9773
18 6 12 16‘6.9216
18 4 14 1462,1123
18 2 16 1300,4776
18 n 18 1090,8156
18 1’ 2 9061,9663
18 15 4 2010,4128
18 16 6 1950,1886
18 11 8 1870,0944
18 9 10 1765,9777
18 7 ‘2 1636,9216
18 5 14 1482,1023
18 3 16 13“0,4776
18 1 18 1090,8156
18 17 1 9062,3283
18 15 3 7017,6965
18 13 5 1977,8140
18 11 7 1912.8130
18 9 9 1821,1070
18 7 11 1714,6130
18 5 13 1562,7928
18 3 15 1394,7134
18 1 17 1199,2352
18 18 1 2066,0738
13 16 3 9030,8035
18 14 5 1981,7735
18 12 7 1913,0113
18 10 9 1821.1196
18 9 11 1794,6130
18 6 13 1562,7926
18 4 15 1394,7154
18 2 17 1199.235?
19 1g 0 2291,0886
19 17 2 9253,6258
19 15 4 7217,2956
19 13 6 9169,9268
19 11 8 9092,9756
19 9 10 1990,3348
19 7 12 1862,8728
19 5 14 1709,8687
19 3 16 1530,2943
19 1 18 1592,9206
19 1“ 2 7297,9538
19 16 4 7235,2183
19 14 6 9172,1739
19 12 8 7093,0694
19 1” 10 1990.336?
19 H 12 1862,8729
19 6 14 1709,8687

19 4 16 1530,2943

124

19 7 ‘8 1572,9708
19 18 1 2288,2359
19 16 6 7238,5129
19 14 5 9197,3788
19 12 7 2134.9d60
19 1” 9 2044,7364
19 8 11 1999,7414
19 6 13 1789,0227
19 4 «5 1623.475?
19 2 17 1410,1665
19 U 19 1218,5751
19 19 1 7290,8946
19 17 3 2252.028)
19 15 5 92n5,5241
19 13 7 2135,3709
19 11 9 9044,7461
19 9 11 1929,7415
19 7 13 1789,6227
19 5 15 1693.475?
19 3 17 1430,1661
19 1 19 1298,3751
90 2" 0 a597.1449
20 18 2 9484,2997
20 16 4 9446,0290
20 14 6 9490,3434
90 1? 8 9625,8923
20 10 10 9224,8789
20 8 12 2099.137?
70 6 14 1948,0726
20 4 16 1770,6788
20 2 1a 1565,7328
?0 0 20 1531,8238
20 19 2 2525,0535
20 17 4 7466,5515
70 15 6 94n4.3232
20 13 8 2326,0936
?0 11 10 7294.883?
20 9 12 9099,1375
70 7 14 1948,0726
20 5 1 1770,6788
20 5 18 1565,7328
20 1 20 1531,8238
20 19 1 9525,2339
20 17 3 9470,2538
20 1S 5 9426,4n93
20 13 7 9366,8971
20 11 9 9278,4793
70 9 11 2155,1911
20 7 13 9026,8774
20 6 *5 16‘2.7387
20 J 17 1671,7338

?0

P0
20
9D
?0
20
20
20
70
20
20

21
18
16
14
1?
1n

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19

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11

15
17
1.9

125
1 452.4959

9527.013:
85.111.
9156/. 7421.)
9278,5167
9155,1215
7096,8774
18‘2,7387
1671,7338
1452,4959

APPENDIX V

CALCULATED UPPER STATE ENERGY LEVELS FOR THE

130
STATES vl+v2, v2+v3, Zvl, and vl+v3 of H2 Te

The output from PROGRAM CORIOLIS is given. The
upper state energy levels are calculated with Eq. (47)
both with and without the perturbation terms G2 and ny.
The perturbed calculation, unperturbed calculation and,

their difference are printed out for each level.

126

127

GRWUND STATE CONSTANYS

A a 6,2474890000

9 a 6.0945340000

C a 3.035999onun

TAAAA = 90.003139n00
78989 a 90.002791fi00
TAABR : n,n02285n00
TAAAR I gn,n00533noo

NJ 3 vflofiOODnDHOUOOO
“JR 8 v0.000000000000
HKJ I sfl.nUDOnOn00000
MK 8 —n,nononon20000

UPDER STATE CONSTANT?

A 3 6,347845000fl vl+v2
B I 6.1369360000

C : 2.9604710000

1131A s 90.003776000
TBQBR 8 90.n031?5000
TAABH I 0.003113000
TAQAB 3 909000953n00

HJ : encfi000fl0fi00000
NJ! I v0.000000fi00000
HKJ I -0.0000n0"00000
NK 3 909000000034000

BAHD CENTEq = Q911.415700

KADPA 3 098755

A s 6.3190690900 v2+v3
R : 6.1850870000

C a 2.9754410000

TAAAA s on.n04299noo

78088 : .n,n03736n00

TAABB = O.n03769n00

TAnAB = 90.non417noo

HJ 3 -n.nonononooooo
HJK : 90.nonononooono
HKJ : .n,nonon0000000

an a en.nonono188000
BAHD CENTER I 2915,969700
KADPA 3 “.9079

62 8 20.000110n

GXY = no.00000n

GZ 3 00°?5882

va : 90.164134

J K. K.
1 1 o
1 D 1
1 1 1
2 2 o
2 n 2
2 1 2
2 1 1
2 2 1
3 3 n
3 1 2
3 2 2
3 2 1
3 D 3
3 3 1
3 1 3

vl+v2
P FNG

2923.900

2920.510

2920.722

2948.875
2935.721

2935.734

2945.248

2945.880

2986.351
2973.092

2973.155

2982.320
2956.758

2983.559
2956.758

U9 ENG

2923.900

2920.512

2990.793

2948.866
2935.722

2935.739

2945.262

2945.892

2986.305
2973.093

2973.178

2982.363
2956.762

2983.616
2956.763

128

PBUP

09000

90.002

90.001

0.009
90.001

90.005

‘09014

009012

09047
90.002

90.023

90.043
009004

20.057
60.005

J K! K9
1 1 0
1 1 1
1 0 1
2 1 2
2 2 0
2 0 2
2 2 1
2 1 1
3 2 2
3 3 0
3 .1 2
3 3 1
3 1 3
3 2 1
3 0 3

vz+v3

P ENG

2928.454

2925.265

2925.110

2940.345

2953.418
2940.339

2950.390

2949.927

2977.754

2990.856
2977.721

2988.061
2961.451

2987.161
2961.450

UP ENG

2928.454

2925.263

2925.109

2940.354.

2953.414
2940.338

2950.375

2949.915

2977.799

2990.833
2977.721

2988.010
2961.454

2987.096
2961.453

P'UP

0.000

0.002

0.001

”00009

0.005
0.000

0.014

0.012

-0.045

0.023
0.000

0.051
90.004

0.065
”09003

b

b

b

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370

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910.5

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3036.340
3022.806
2983.703

3022.994
2983.703

3031.693
3006.605

3033.661
3006.611

3098.847
3084.767
3045.993

3085.199
3045.993

3093.330
3068.794
3016.561

3096.013
3068.815
3016.561

3173.858
3158.863
3120.580
3055.330

3036.203
3022.806
2983.709

3023.057
2983.709

3031.784
3006.619

3033.854
3006.627

3098.537
3084.766
3046.012

3085.342
3046.013

3093.492
3068.821
3016.569

3096.553
3068.852
3016.569

3173.266.

3158.869
3120.616
3055.341

129

0.137
90.000
90.007

80.064
90.007

90.092
90.014

90.194
l0.017

0.311
09001
w0.019

90.143
90.019

90.163
90.027
90.009

20.540
I0.037
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0.592
90.006
90.036
20.011

5.5

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3952.127
3826.140
3672.254

”0.237
'09135
"09060

0.003

”79760
’14397
'09928
900238
'04140
’09063

0.005

8.124
0.418
0.406
“09352
’09237
"09150
“04063
0.005

8.476

0.589
"0.613
'00307
'04185
’09100
'00029

14
14
14
14
14
14
14

15
15
15
15
15
15
15
15

15
15
15
15
15
15
15

15
15
15
15
15
15
15
15

OiP‘O‘JUMUtJ

n+4

AhRJDEDOLhNJD

“9;“.

4215.549
4165.305
4119.128
4045.545
3945.945
3819.632
3665.616

4399.621
4356.870
4322.691
4265.784
4180.484
4068.917
3930.399
3763.819

4367.491
4325.095
4265.878
4180.485
4068.917
3930.399
3763.819

4368.483
4339.172
4298.157
4226.408
4128.012
4003.090
3850.697
3889.583

4204.125
4171.298
4119.776
4045.848
8946.137
3819.752
3665.678

4385.213
4356.023
4323.156
4266.239
4180.797
4069.122
3930.530
3763.887

4375.541
4328.332
4288.459
4180.800
4089.122
3930.530
3783.807

4378.435
4340.890
4298.753
4226.798
4128.265
4003.255
3850.796
3669.622

137

11.523
95.993
90.548
90.303
90.192
90.121
00.062

14.403

0.847
90.465
40.455
$0.313
'0.205
'0.130
90.068

98.050
93.237
90.581
a0.315
40.205
90.130
90.068

97.952
91.718
20.596
90.390
90.253
90.165
90.098
90.039

14 13
14 11
14 9
14 7
14 5
14 3
14 1
15 14
15 12
15 10
15 8
15 6
15 4
15 2
15 15
15 13
15 11
15 9
15 7
15 5
15 3
15 1
15 15
15 13

9&‘(DN\IGH‘

n+4

1
3

4194.304
4172.640
4123.783
4051.074
3951.942
38269040
3672.225

4373.454
4332.350
4271.277
4188.204
4075.200
3937.042
3770.485

4399.879
4361.147
4328.445
4271.007
4186.280
4075.200
3937.042
3770.485

4399.721
4357.219

4202.203
4167.867
4124.147
4051.374
3952.127
3826.140
3672.254

4384.119
4334.138
4271.820
4186.585
4075.390
3937.147
3770.514

4389.449
4359.398
4327.040
4271.383
4186.577
4075.390
3937.147
3770.514

4388.956
4355.260

'79899

4.773
“00364
’09300
'09185
'00100
’0.029

0310.665

91.788
'09543
”0.300
.00190
"0.104
90.029

10.429
1.748
1.405

“00376

’06297

'00190

”04104

”0.029

10.764
1.959

15 15 1
15 13 3
15 11 5
15 9 7
15 7 9
15 5 11
15 3 13
15 1 15

4374.336
4353.412
4298.991
4226.421
4128.012
4003.090
3850.697
3669.583

43R4.965
4351.767
4299.956
4226.894
4128.265
4003.255
3850.796
3669.622

138

'109629

1.645
90.965
90.404
90.253
90.165
90.095
90.039

15
35
15
15
15
15

15
15
15
15
15
15
15
15

PUU‘V‘OH

4304.419
4231.982
4134.059
40090579
3857.411
3676.040

4399.746
4343.188
4303.213
4231.947
4134.058
4009.579
3857.411
3676.040

4305.108
4232.359
4134.299
4009.725
3857.477
3676.032

4385.042
4345.897
4303.496
4232.308
4134.299
4009.725
3857.477
3676.032

”09689
”00378
'0.241
"00145
'00066

0.007

14.704
"2.709
909283
'00361
“09240
"09145
'0Q066

0.007

 

_..- “_“h-_._

 

139

GROUND STATE CONSTANTS.

A a 6.2474890000
B 9 6.0948340000
C ! -,3.0359990000

.__IMU
1:898
TAABB
14948
HJ a
HJK I

1_HKJ 9.
HK I

I
I

II~I .-
3

.-H'010031390091u-
00.002791000
0.002285000
.00000533000

'0.000000000000 ,

'0.000000000000

90.000000000000._mr_1”_.hmhh_ -_m.

00.000000020000

UPPER STATE CONSTANTS

 

 

4196.0689780000“ __ “2.1“
a a 5.9400740000
c n 2.9969350000 1
#174444 9 r-o.oosza1ooo 1
79933 . -0.002679000
__TA498 a _nm01002326000__11-__fl”h_muh,-w“
TAaAa a -o.ooosoaooo
,_,HJ s.” 1-o.nooooooonnnon_
HJK - -o.oooooooooooo
_ HKJ_! _-n.nooonnonnnno__u_ _u1_fl_m
HK . -o.000000031ooo
"'5I05_EEN?Ei_3 4062.890200-
KAPPA - 7 0.9172 “fl”___—M”' 7"-
“A'i_ 6.0638020000h ” 7 ”””v1+¢3”‘
-_18 a M- 5.94644100001m-‘ n"-w__rmum___mu1_w1wmh -.
c a 2.9595690000
YAAAA . 7 -o.663351663_
13988 g _ 30100230500011 _11 111,11
7AAea . 0.002405000
.mTABABJs_ _-901000445000___mwn_WWNflhn 1
HJ g -o.oooooooooooo
__ HJK g 1 yn.nuounnoooonn___u .1 Nn_w
HKJ a -o.oooooooooooo
A an -1 -1" 1-n.nounnnnzznna_1mw.___m _
J”MBANn_nENtER”2111_140631§11111“11_1_111_1__1117.
_KAPPA g . 0.9244__ _ ._ _ _ .mm_1_u--un
611 9 .04000000
ny . '0.000000
Gi—H; “_31011133flh"7-“7 _' _
3x1 n ~01229286.__1 1,, @1111 -1

l. . lull . Ill . .
I'll-Ii it '3 i110]! fli‘l? «it‘liiu.

0..“11'11 . 1': '1! [3| I‘ll!!!" IIII

J K9 KO

201

P ENG

4074.899

4071.738

4071.863

4098.936
4086.710

4086.698

4095.436

4095.638

4135.039
4122.671

4122.614

4130.930
4107.477

4133.094
4107.476

U! ENG

4074.899

4071.796‘

4071.915

4098.909
4086.714

4044.724

4095.668

4096.052

4134.905
4122.444

4122.739

4131.467
4107.497

4132.830
4107.498

140

POUP

0.000

00,049

:09052

04027
90.009

10.029

0042337

.0'41‘

04139»
90.01,.

l09129

.0953,
003020

04864
I03021

J K9 Kt—

(d
FWH
..

vl+v3

P ENG

4075.384

4072.445

4072.331

4087.197
4099.421
9037.211
4096.757

4096.589

41234120

41354516
4123.188

4133.257
4107.999

4131.162

.4107.995

UP ENG-

4075.384

4072.397

4072.279

‘0574220‘

4099.396

4087.210

40964525

4096.175

4123.238

4135.392
4123.187

4132;688
41034007

4131.993

‘4108.007

....

0.000

0.048

0.052

’04023

0.025

0.001

09233

0.414

90.118

0.124
0.000

0,570

"°4°13

.04331
’00011

‘IIO.

‘4‘

OJFOIO'

(3&8.

HWH

awn

hi.

19h}.

.saru

C3hl.l’

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“Hi

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\IOH‘

\IGN‘

4153.227
4.74.497
4134.155

4170.362
4.34.155

4178.225
4185.403

4.41.772
4158.398

4243.437
4230.723
.....043

4229.427
4194.043

4237.314
4219.202
4166.736

4241.592
4215.183
4.44.734

4414.200
4302.444
4245.743
4205.219

4382.084
4170.567
4134.182

4170.722.

4134.182

4179.156
4155.471

4180.412
4155.475

4242.74:
4234.24.
4129.120

4230.645
4194.128

4238.699
4215.344
4166.772

4240.550
4215.360
4166.772

'4314.588'

4301.787
4265.933
4202.283

141

0.393
80.070
90.028

00.380
00.020

00.931
00.067

1.361
.0.076

0.085
0.035
00.085

I0c818
00.088

01.385f

00.142
00.035

2.042
00.178
00.033

1.970”

0.670
00.170
00.044

«‘85;

UHfl\fl

\IUHI

4°...

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H9“

8.6““

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“(‘9’

UHUF‘

4170.945

4134.698

4193.714
4171.110
4134.698

4151.917
4155.920

4178.482
4155.926

4229.993
4194.591

4244.058
4230.998
4194.592

4242.712
4215.707
4187.311

4237.599
4215.729
9397.381

4171.228
4134.711

4183.348
4171.076
4134.711

4180.850
4155.990

4179.706
4155.956

4231.157
4194.663

4243.227
4230.806
4194.662

4240.962
4215.881
4167.324

4239.277
4215.866

4667.324»

.00283
"0.013

0.366
0.035
.00013

1.067
.04070

“1.224
.00060

'10164
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0.827
0.147
00.071

1.750
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.14670
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OJ’O‘

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h3.lb

C3h3bdl

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OMQRIGD

Ohbfll

‘Q\IGH‘

4309.901

4265.760
4205.219

4308.183
4286.803
4238.575

4315.523

4238.575

4400.933
4355.995
4349.234
4286.995

4353.502
4349.224
4.3.,995

4390.817
4370.113
4322.195
4249.509

4302.468
4265.935
4205.263

4310.054
4287.053
4233....

4312.583
4287.100
4230.676

4398.122.
-4384.912
'4349.526

42...112

4346.144
4349.534
4289.112

4393.195
4370.513
43.2.3.7
424....1

142

01.567-

00.175
00.044

01.871
l0.249
00.100

2.940
00.359
00.100

2.911
1.083
00.292
00.117

.2.642'

I0.310

.0.117.

02.357
00,409
00.192
00.052

0450

OJ’O‘O

'V‘JN

'V‘4\FV

‘U\4N

QI’O‘

croso

-u~0~ru
.névu~0

0.043

91‘“

43h).lb

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IVJIO

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Cuba.

Cub“)

OLbIOC3

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"9“P

On§lgca

'utlaué

‘39“P

4301.034
4266.313

4205.031.

4316.562
4302.675
4266.316
4205.831

4315.415
4267.274
4239.156

4300.499
4287.335
4209.156

4333.575
4349.780
4299.614

4401.228
4366.215
4399.791
4209.614

6400.591
4370.531
4322.780
4250.254

4391.166
4370.666
4322.781

4302.988
4266.476
4205.844

4314.982
4302.297
4266.474
4205.844

4312.962
4287.627
4239.235

4310.660

4287.582-

4239.235

4386.674
4350.081
4289.700

4398.552
4365.455
4350.072
4289.700

4396.773
4371.161
4322.953
4250.267

’4393.808

4371.051
4322.952

”1.954
'0.163

"0.014

1.580
0.378
'00158
"0.014

2.653
“0.353
”0.079

0.2.162

'0.247
.00079

'3.099
“0.300
”0.087

2.677
0.760
’0.281
'0o087

3.819
”0.630
‘00172
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'2.692
.00384

"0.171

4330-. I l I. 1 4| .lfililv

)4... ‘ III; 011' .III 6010 .l" .II‘I).

CHDGIC’ IDOIOMDGD 'fl‘deV

.0000

rsoru~0

430.0n340

.saua~0

PWH\I\I

RD.GIO

t‘GHI‘IO

~uba.u»

\NUiIHH

OGIOJDGD

lbfllbdo

VH’OH‘

(Doi‘lUCD

4400.531
4869.990
4822.195
4149.599

4497.791
4481.320
4444.367
4864.493
4299.872

4477.566

4444.337‘

4364.493
4299.672

4485.197
4499.992
4417.507
4345.299

4.97.575
4444.747
4417.506
4845.299

4606.711
4888.332
4551.054
4491.614
4407.440

4396.434
4370.630
4322.388
4249.691

4493.448

‘4479.626

4444.821
43.4.7.9
4299.933

4491.420
4444.845
4384.789
4299.983

4488.011
4465.627

4417.825'

4345.432

‘4492.0i4

4465.876
4417.826

4345.482.

4600.466
4565....

4551.714

4491.959
4487.833

143

4.097
00.640
00.193
00.052

4.324
1.694
I09454
00.215
00.061

04.054

00.5085
00.215.
90.061.

62.81;
90.645

‘90l31.

00.334

5.561"
01.089-
00.321
00.134

6.242
_2.524
00.661.
60.345
00.151'

7

(DGDOMD

'O‘D‘TO

canrécum

600.04-

0

FWI\I‘I

“8.460

f‘GHQ‘J

7

IDOMbRB

(DahfiflWD

‘Q‘IGH‘

\QUHHF‘

CIOL‘IU

4250.254

4477.572
4444.895
4395.117
4300.575

4498.017
4481.504
4444.925
4995.117
4300.575

4497.604
4465.415
4488.096
4345.960

4485.580
4465.636
4418.097
4345.960

4502.974
4551.546
4492.242
4408.190

4606.869

4250.267

4402.161
4445.393
4365.302
4300.539

4493.859
4480.180
4445.370
4385.302
4300.589

4492.305‘

4466.410
4418.396
4346.055

4488.669
4466.172
4418.394
4346.055

4589.385

{4552.330

4492.556
4408.293

4600.812

’0.014

"4.588
”0.500
'0.195
’0.014

4.158
1.324
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”0.185
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5.299

~’o.994

’0.300
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”3.089
00.536
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“6.409
.0 .783
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6.058

«0101ro

O‘D‘HOID

10
18
10
10
10

10

00009

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6‘OND‘4O

t‘OUJ‘I.

truihéun

4’ID.ID

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OIDOM‘EMD

OMDOhbfo

4563.031
4590.944
4491.614
4407.460

4591.296
4.71.902
4524.408
4452.652
......32

4606.609
4.70.997
4524.404
4452.652
4356.032

4727.691
4706.945

4669.167'

4610.245
4526.666
4418.073

4.....32
4669.030
4610.244
4826.666
4410.073

4709.072

45......
4551.77.
4491.960
44.7.4.1

4594.512
4572.272
4524.894
4452.892
4356.102

4599.220
4572.757
4524.898
4452.892
4356.102

4719.01.
47.3.3.5
4470....
4610.75.
45......
441..151

4707.706
4670.232
4610.759
4526.980
4418.151

4712.630

144

95.900

150400

00.793'
00.9405
00.151.

03.216
00.371

90.989 _

00.239
00.069

7.389
01.760
00.494
60.239
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8.605
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00.264
00.070

07.873
01.201
00.515
90.264
00.078

03.558

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4508.461
9551.618
4492.242
4408.190

4606.610
4571.201
4524.996
4453.319
4356.789

4591.714
4572.152
4525.000
4453.319
4356.789

44.9.73.
4669.617
4410..73
4527.300
44......

4727.721
4706.999
4669.758

4410.874-

4927.380
4418.890

4586.366
4552.264

4492.556
4408.293

4599.458
4573.287
4525.468
4453.519
4356.803

4595.190
4572.825
4525.465

4453.519

4356.803

»4708.278

4670.782
4611.353
4527.595
4416.904

4719.305
4703.903
4670.629
4611.353

«4527.595

4418.904

2.095
90.646
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7.153
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90.472
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00.673
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00.014

.39556
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8.415
3.096
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10
10
10
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494.06.80

HNO\IUWIF5

4.69.682
9692.784
4671.550
4475.533

4727.564
4668.470
4642.773
4571.550
4475.533

4660.451
4637.067
4796.522
4740.260
4657.306
4549.449

4627.694
4798.316
4740.257
4657.388
4549.449

4638.465
4.19.205
4772.504
4701.868
45.6.52.
44.5.9.7

4690.311
4.43.4.6
4571.925
4475.701

4717.939
4691.177
4.43.4.6
4571.925
4475.761

4640.993
4832.162
4799.793
47......
4657.715
4549.634

4838.165

‘4600.099

1740.9.9
4.57.715
45.....4

4842.317
4819.586
4773.476
4702.414
4606.816

‘4486.073

145

90.429
90.702

00.375‘

.0.168

9.645
02.707
00.724
20.375
00.168

11.458

4.905
91.271
00.726
60.907
00.180

“10.271

01.783
90.732
60.407
00.185

03.851'

00.381
00.972
00.546
00.290

7
00.086

NL‘060C3

F‘QHJ‘O‘D

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4727.561
4688.627
4643.366
4572.219
4476.296

4709.526
4690.112
4643.377
4572.219
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APPENDIX VI
LISTING OF PROGRAM SPEC-FIT 1

This program is set up for the simultaneous analysis
of the two interacting bands v1+v2 and v2+v3. To set the
program up for the analysis of any other two interacting
bands (v1,v2,v3) and (vi,vé,v§), the only changes that
must be made are in the calculation of the vibrational

...—

matrix elements, PQ and'QQ, of the interaction terms.

I I I
PQ (VlIVZIVBIqlq3IV1rv21v3>

00 = <Vl'vz'v3lqlp3'q3pllvi'vé'vi>

For (v1,v2,v3) = (1,1,0) and (vi,vi,v5) = (0,1,1),

P0 = -l.0 and 00 = +0.5 as can be seen in the program.

It is necessary that the unprimed quantum numbers (v1,v2,v3)
represent the type B band, and the primed represent the

type A band.

151

HELJK

152

18 L LISlINb OF PROGRAM SPEc-VIT 1

JF4,151154,NKM, 5: ”OKCUH, SPEC-FIT 1
FT!)X’L

TYCJ’UTTCW

‘7

PROGRAM SPEC FIT 1
DIMENSION AV(14,E),AA(?).88(?).CC(2)IT1(2)IT2(2)IT3(2)IT4(2).H1(2)

1:H?<2).H5<2).H4(2>.voc2;.mstc?).PX(3).PY(6>.PZ(5).sz<3),Pvz(3),
2PXV(3).P>2(21.71>.PZ2<?1.21>.P1(21.21I.92<21.21).PS<21.21).P4<21.
dPl)IA(?1.21).SS(?1.21).PHXY(?1.21)IPRZ(21:21):JJU(800)IKKU(300).
4LLu<eoo>.JJG(500>.KKG<800).LLstaUO),oas<800).E~G(aoo>,wHT(2,aoo;.
bIFHSF(800).CAP(2)

COMMON JJU.AKU,LLU,JJG,KKG,LLG,OPS,ENG,NHT
COMPLEX A,Sb.FRXY,PbZ

THIS PROGRAM IS SET up FOR THE SIMULTANEOJS ANALYSIS OF THt BANDS
(1'110) AND (0:111)
THE VIBRATIONAL MATRIX ELEMENTS USED ARE (1:110 0103 091:1) a
(0.1.1 01C3 131.0) = 1./2.
(1'133 ONES-(“3‘1 0,1,1) 2 nit!
(0:1.1 0195'03P1 1:110) : *1*I

RENINU 51

PT?=SQRT(2.)

00: 0.5

PO .1 -1,0

MM=21

kHO:1,nE-Oo

IE°=0

PEAU 500. AA(1).F5(1),CC(1):Tl(1),T2(1)aTS‘l),T4(1),V0(1)3H1(1),

1H2I1),H5(1).H4¢1)

510 rovaT(6F1b,6/4F10.6.F10.5/4F20.12)

CA°(1)=(2.*65(1)*AA(1)'CC(1))/(AA(1)-CC(1))

AA(2)=AA(1) $ PB(2)=PQ(1) C CC(2):CC(1) $ 11(2)ST1(1)

T2f2)=T2(1) $ TS(2)=T3(1) fi T4(2)=T4(1) $H1(2)=H1(1) S H2(2)=H2(1)
H6!2)=H3(1) $ H4(2):H4(1) m VO(2) a VOC1)

PRINT 501.(AA(1).BB(1).CC(1).T1(1).T2(1).Té(1)oT4(1).Hl(1)oH2(1).

1H3(l):H4(1)ICAP(l))

R=Rd(1)t*2/(AA(1)+RB(1))*t2 i S=AA(1)**2/(AA(1)*BB(1))**2
mprscl):kPTS(2)=I=D

591 FOR“AT(*16RUUND STATE CONSTANTS*//* A: #715,10/t B = tF15,10/

1* C = «F15.10//t TAAAA a «F15.9/* T8898 I *‘15.9/* TAABB 8 «F15,9/
2 t TAEAB = «F1fi.9/,t HJ = tF20.1?/t HJK = *F20.12/* HKJ a tF20.12
3 / u HK a «F20.12//* KAPPA = *F10I4)

P0 12 J=1.2

D0 10 K=1.hOO

I=J+1

READ bU4IJJU(IngKU(I)ILLU(I),JJG(I)I(KG(I)oLLG(I)IUBS<I).NHY(1.
I).LASTDATA

5G4 FOPMAT(1X,bIS,F13.4,7X,F4,2,?9X,A8)

IFtLASTDATA,EU.8FENU DATA)GO 10 12
WHT(?II)=WHT(1.I)

ENG(I)=O.D

NPTS(J)8NPTS(J)*1

10 CONTINUE

153

12 1:1-1
NSHW = NFTS(1).NFT§(?)
DO 296 NTMS =1.2
IOPI=0
IFINTMS,FQ.1) CO 10 503
IOPT = 1
HEAD 502. (AA(I).bH(I).CC(I).TI(I).T?(!).To(1).T4(I)IVD(I).
1 Hj(1)3H2(I)!H3(I’DH4(I).I=1,2). GZ'GXY
5U? FORMAT(2(3F15.6/4F10.7.F10.5/4F20.12/).2F20.10)
CAP(1)=(2.*BH(1)-AA(1)9CC(1))/(AA(1)-CC(1))
CAP(2)=(2.IBB(9)-AA(2)-CC<2))/(AA(2)'CC(2))
PRINT 555.(AA(I),BB(I).CC(I).f1(I)aT2(I).T6(I):T4(I).H1(I).H2(l)n
1H3!IIIH4(I)IV0(I).CAP(!).I=1.2)IGZIGXV
565 FOPMAT(///t UPPER STATE CONSTANTS:2(/lt A = *F15.10/* B E *F15.1D/
1* C : *F15,10//t TAAAA = ¢F15,9/I T8838 3 0F15.9/t TAABB I «F15,9/
2 * TAHAB = IF15.9/.* HJ = *F20.12/w HJK = ‘F20.12/* HKJ I tF20.12
3 l r HK = «Fzfi.12//t RAND CFNTEF = *F15.b/*0KAPPA = «F10,4)lt-GZ
4 = vF15.6/* va = *Flb.6)
503 CONIINUE
DO 201 KKK=1.N9UV
2U1 IFHSE(KKK)=O
NHAVH :0
no 200 MM:1,NSHM
IF( [FUSE(MM),NE.0)00 TO 290
IF!VTMS,EQ.Z)GO T0 2
J=JJG<MM)
IFIJ.E0.0) GO To 290
IF(LLGtMM)-(LLG(VM)/2)*2)2n6.2fi4
234 J'C:
50:0.
GO TO 208
236 JEO=4
60:1.
915 IF(J,EO,KKG(MM)*LLG(MM))JEO=JEO-1
GO TO 3
2 J=JJU(MM)
IF (J,NE.O) Go TO 404
DC) 401 MAJ-31:14
401 AV(VN,1)=AV(NN,2)=0.0
N821
NLM=1
Hfizz
IF(AM,LE.NPIS(1)) 60 In 40?
NLM=2
N832
“9:1
432 IFtNBAVD.NE,(NnAkD/2)*?)NB=MB
AVI1,NLM)=1
EVGU=VO<NB)
WRITE TAPE blaJJU(WM),KKU(MM)ILLb(MM),JJG(MM)IKKG(MM)ILLG(MM)o
1 oas(MM).ENGU,ENG<MM).wHT(1.MM3.((AVIIIIJJ)oII=1o12).JJ:1.2),
2 AV(13,1),AV(14;1).MV
GO ID 290
4"4 CONTIVUE

154

IF:LLU(MM)-(LLH(VM)/7)~2)210.POQ
259 JEO=2
am To 211
710 JEU=A
211 IFtJ.t0.KKU(MM)+LLU(MW>)J&O=JFO'1
IFtfiM,LE.NPTS(1))GO TO 145
GO T0¢140.141.140.141).JE0
340 JEn=JEn+1
so To 142
141 JEO=J&O'1
14? COUTINUc
145 GO TOI146,147,146,147).JF0
146 IF(N6AMD.EQ.(WRAAUIZ).?)GO T0 3
GQ T0 148
147 IFIVRANU - (NPAhD/2)t?)3,148
14a ASV=AA(1)548v:28(1)£CSV=CC(1) i VUSV=VO<1)
TlQV=Tl(1) $ T28V3T2f1) $ T38V=T3(1)$T4SV:T4(1)
H19V=H1(1) $ H9SV=H2(1) ¢ H3SV=H3(1) fl H4SV=H4(1)
PI_):-PQ
AA(1)=AA(2) $ PB<1)=BB(2) I CC(1)=CC(2) $V0(1)=V0(2)
T1<1$=T1<2> f T2(l)=72(2) £T3(1)=T3(2) S T4(1)8T4(2)
H1(1)=H1(2) b w2(1)=H2(2) $ H3¢1)=H3(2) i H4(1)IH4(2)
AA(2)=ASV i HBIZ)=HSV $ CC(2)=CSV $V0(2)=VOSV
T1(2)=TISJ i Y?(2)=T2SV £T3(2)=T35V $Y4(2):T4SV -
“1(2)=H1$V S H9(2)=fl?SV $ H3(2)=H38v 5 H4(2)=H4SV
NBAVD : NdAND¢1
3 stttd+1)
IF(JEO-2)119,119.123
11Q NSZ=2*(J/2)*1
N=J/?+2
GO TO 124
125 NSI=2*((J+1)/2)
N:(J+1)/2
124 CONTINUE
no 4 1:1.VSZ
no 4 JJ=1.NSZ
4 A(I.JJ)= (0..0.)
no In I=1.2
IFIJEO-2)126.176.1?7
N:N-‘
CONTINUE
IFtN.&O.fl) GO TO 70
NN=D
IFKI.EQ.2)NN=(VSZ+1)/2
DO 7” L:1,V
IFtJhU-2)133.1K0.135
530 IF(I-2)16;15
15 K=Q~L
60 T0 17
16 K=2*L-?
GO TO 17
fljflj K=?~L-1
17 CONTINUE
M:L¢HN

«.1
~.\)
\19

155

PX9(H.H)=(F-h*V)/2.
PZ9(HIM)=K*K
px(1)=(5.*FiV-?.*F-6,9FtKrK+5,cKtKfisq‘.KgKaK)/d'
PY(1)=PX(1)
PZ(1)=KwK*K*K
Px7<1>=K*K*(F—KIK)
PY7(1)=PXZ(1)
pXV(j)=(FrF+2,¢F-2,tFtKwK-S,thK+KtK*(*K)/4.
IF<L.EU.N)HU 10 16
P¥2(M,M+1)=-SURT((F—K«(K+1))w(Fn(K+1)w(K02)))/4,
pZ?(HIM*1)=d.0
pX(23=-(2.*F'KtK-(K*2.)**2)*SQRT((F-Kt(K¢1))*(F-(K+1)*(K*2)))/8,
pv¢2>=-9X(2)
PZ(2)=PZ(S)=U,
PX?(?)=-(K*K*(K*?)**9)*SOHT((F-K*(K¢1))*(F-(K+1)*(K+2)))/4.
pY7(2)=-PXZ(2)
PXY(?)=O.
IF(L.EO.Na1)GO T0 15
PZB(P,M+?)=PX2(4,M+2)=0.0
PXCJl=SuRT((F-K*(K*l))*(Fu(K+1)t(K+2))t(F'(K*2)i(K*6))‘(Fv(K+3)*(K
1*41)‘/1b.
PYLS):PX(3)
PX7C3):U.
PY7(3)=0.
va(3)=-2,.px<x)
15 CONTlduE
IFth0-2)19.19,47
19 IF<K$25,20
?3 PX?{H,M.1):RT2rPX2(M,M+1)
PX(?)=DX(2)*RT?
PV(?)=-PX(2)
PXZ(2)=pr(Z)*PT2
FY7I?):-le(2)
PXCLS)=PX(3)IRT?
PYC53=PX(3)
PXY£3):~2,*PX(3)
GO TO 56
25 IFIK-2)53,30
50 IFtl-2)40.45
43 PX(1)=PX(1)+V*(F~2.)/16.
PY(13=PX(1)
va(1):PXY(1)-F.(F-2,)/8,
Go T0 55
45 PX(1)=PX(1)-F*(F-2.)/16,
PY(1)=PX(1)
PXY(1):PXY(1)+FI(F-2.)18,
GO TV 53
4] IF(K~1)56,43
4% XM: ,
IF(1.EQ.2)XM=‘1I
FX?(H,M)=PX2(M,M)-XM«FI4.
PXI1)=Px<1)-XM.Fw(F-1,9/4.
PY!1)=PY(1)+XM«F*(F01,)l4,
PX(2)=PX(2)+XMIF*SQRI((F-Z.)¢(F~6,))/16.

1‘I'iillllllil'!lll|1.lllll'l‘ll.llll"i Ill.

156

PY(2)=PY(?)*XM*F*SQRT((F-Z.)*(796,))/16,
FX7(1)=PXZ(1)~YM‘F/2.
PY7(1)=PYZ(13+xHaF/2,
PXV(?)=-XM*F*50HT((F-2.)*(Fw6.))/8.
53 C0N11NHE
035:3
IF(L,ED.N-1)NS=2
IF‘LIEQQN)EV8=1
DO 95 KK=IINS
JJ=M+KK'1
P1tW,JJ)=(PX(KK)+R*H*PZ(KK9¢RtPXZ(KK))l4,
P2tM,JJ)=(PY(KK)+$*S*P7(KK)¢S*PYZ(KK))I4.
P3(M,JJ)=(2,tR¢S*PZ(KK)+PXY(KK)tSchZ(KK)¢HrPYZ(AK))/4.
P4tM,JJ)=PXY(KK)/2.
IFtM.EQ.JJ)P4(M.M)=P4(MIM)*(5.&PZZ(M.M)w2.*F)/4.
A(H.JJ)=(AA(I)—RB(1))IPX2(W.JJ)+CC(I)tP22(MIJJ)*Tl(I)*91(MIJJ) +
1 T9<I)092(MIJJ)*75(l)t93(M.JJ)*T4(I)*P4(MIJJ)
IFIM,NF.JJ)GU TO 55
A(M,JJ)=A(MIJJ)+BB(I)t(F!KtK)+VO(I)+H1(1)tF*FwF¢H2(1)tFtFthK¢
1 H3<I)iF*KtiQ#H4(I)tK§t6
55 A(JJDM)=A(MIJJ)
70 CONTINUE
IF(NTM3,EO.1’GP T0 116
THE FOLLOWING SETS UP THE INTEPACTION MATRIX ELtMENTS
IF(JFO.GT,2 ) 60 T0 105
PHxY(1,N+2):00pSCNT(F¢(F-z,)/2,)t(0..1,)
A(1IN*?)=GXY*PRXY(1,N+2)
A(V+2,1):9A(1,u.2)
DO 115 M=1nN
K:?ifiw1
IISW
JJ34+V
IF(JFO,GT.2) 60 TO 110
K=K+1
II=II+1
JJ=JJ+1
113 CONTINUE
P87220.
P871:K
PBXlen.
DBIY2=SQRT((F-Kt(K+1))*(F-(K¢1)t(K*2)))/2.
IF(K~1)112.111.112 '
111 PB¥Y1=F/2.
112 9871=PQ*9611
P3¥V1=QJ*P5XV1
pR¥Y2=QQ*PBAYZ
ICNT a Il+1
JCNT = JJ+1
IF(fi,EQ.N) 60 TO 114
PBXYKIIIJCNT)=PBXY(JJ.ICNT)=PBXY2I(0.I1I)
A(IIIJCNT)=A(JJ.ICN11 =GXY.PB(Y(IIIJCNT)
A(JCNT,II) = A(ICNT.JJ) = CONJG(A(IIIJCNT))
114 pBl(II.JJ)=PH11*(0.;1.)
PB¥Y(IIIJJ)=PBYY1*(0..1.)

4
'J";

115
115
3C?

157

A(II,JJ)=GZ*PB7(II.JJ)+5XY.PHXY<IIIJJ)
A(JJ,II)= CONJG<¢III.JJ))
COerNut

COITINué

FORUAF(*O*)

CALL JHtRMX(A,N4.NSZ,109T,QHO,IER,SS)
MM1:MM+1

IF(NTMS.E0.2)GN T0 ZRR

00 T0(120,120,122.12?).JFO
NN:LLG(MM)/2 + 1
IFtJFO.E0,2)NN=NN¢J/?

00 T0 125

NN:(LLG(MM)*1)/2
IF(JFO.EU.4)NN=NN+(J+13/2
FNG<MW)= A<NNINN)

IFUSF<MM)=1

!F(WM.EO.NSUH)GO TO 290

Do 220 MMM=MN1,NSUW
IFchUQttMM4),ME.0)Gn Y0 226
IF(JJG(MMM>.NE.J)GO T0 226
1F(LLG(MMM)'(LLG(MMM)/9)*2)216,214
10:2

FfiFzfl,

30 T0 218

ID=4

EOF=1. .

IF<E0.NE.EOE)Gn T0 226
IF(J,ED.KKU(MMM)*LLG(MMM))ID=ID~1
no T0(920,220.v22.222>.10
NN=LLO(MMM)/?+1
IF(ID.EO.2)NN=NN+J/2

GO TO 225

NN=(LLG<MUM)*1s/?
IF(ID.FQ.4)NN=NV+(J+1)/2
FMG<MMM)=A(NN,NV)

IFUSF(MMM):1

CONTIVUE

GO Tn 290

CONTINUE

ID=JRU

IS=~1

IF(NM.GT.NPrS(1)) 10:10-Istwlo
GD T0 (230.230.234.234).10
NN=LLU(MM>/2*1
IFtID.FO.2)NN=MN+J/2

GO TO 236

NN=(LLU(MM)+1)/2
IFIID.E0.4)NN=NN+(J*1)/2
COM1INUE

DO 280 IJ=MMINSUV
IF(IJ.E0,MM)00 T0 249
IFIIFU§E(IJ).NC.0)GO T0 280
IFIJJUIIJ).NF,J)GO TO 780
IFILLUIIJ)-(LLU(IJ)/2)*2)239.238

?5d

769
?40

249
THE

950
349

121

152

150

158

10:2

80 T0 740

13:4
IFtJ.EQ.KKU(lJ)*LLU(IJ))YD=ID'1
IDR=ID

IF(IJ.LE.NPTS(1))GO TO 243

GO T0(?41.?42.741.242).ID
ID=IU+1

GO TO 243

ID=IP-1

CONTINUE

IFtID.NE.JEO)Gn T0 280

GO TU(946,240,948,248),IDR
MNzLLU(IJ)/2+1
IF(1DR,E0.2)NN=Nhtd/?

00 T0 949

NN=(LLO<IJ)*1)/2
IFIIUR,tQ,4)NN=NN+(J+1)/2
ENGU=A(NN,NN)

IFUsEIIJ)=1

FULLUNING CALCULATES THE AVERAGE VALUES OF THE OPtRATUHS
PO 169 1:1.14
Av(l]1):AV(112,=OQ
LSZ=INSZ+1)/?

Nbal

V1832

IFIJFC,NF.(Jt0/2)*2)GO TO 153
N822

“Bel

CUNTINUE

D0 150 11:1.N87
IF(Il.FQ.LSZ+1)NP=MB

KS7:]I+2
IF(II.F0.NSZ'1)<SZ=II+1
IF(I].FQ.NSZ)KQZ=II

DO 150 JJ=IIIK<Z

vsn a CUNJG(SS(1I.NN))¢SS(JJ.NN)
IFIII-JJ)121.151
AV(1,NP)=AV(1,NB)tVSO
§V(9,NR)=VQU*Fo*3 + Av¢9.VR)

AV(10.NB)=AV(10.h8)*VSQtFtt2*PZZ(11.11)
:v(13,ud)= AV(lle8)+V5J*F*PZZ(11,Il)t*2
IV(12,N3)=AV<12.NH)+str922(!Iall)tt3
AV(3,NR)=VSU*(F-PX?(II.II)-PZ?(II,II))¢AV($INH)

GO ID 152

VSO=2.IVSO
Av<3,NR)=AV(5,Nd)-Vsn.0x2(II.JJ)
AV(2’NR):AV(?,NB)*VSQth7(IIIJJ)
ch4,Nn)=AV(4,NB)+VS0.922(II.JJ)
AV(5,NH)=AV(5,MB)+VSOtP1(IIIJJ)
AV(6,NH)=AV(6,NB)*VSO+P2(II.JJ)
AV!7,NR)=AV(7,MB)¢VSQ*P3(IIIJJ)
AV(B,NR)=AV<6,NB)*VSO*P4(11,JJ)
CONTINUE

155

159

160
405

pun
290
996

159

YF(JE0,GT,2)GO T0 155

WP=LSZ+1
AV!14.1)=AV(14.1>+2.*C°NJG(SS(1:0N))*SS(NP,NN)*P6XY(1IVP)
C(TWTIFVUC

Do 160 M=1.N

II2W

JJ=M+N

IF(JFO,GT.2)UU TC 158

II=I]*1

JIJ=JIH1

CONTINUE

ICNT = lI+1

JCNT = JJ+1

IF(M,EO.N)UO TO 159
AV(13.1)BAV(13,1)*?,*CONJG(SS(IIINN))tSS(JCNTINN)*PBZ(IIIJCNT)*2.*
100NJ0(SS(JJ;NN))ISS(ICVTIMM)*PBZ(JJIICNT)
AV!14,j)=AV(14,1)+2.vCOVJG(SS(IIINN))OSS(JCNTINN)'°BXY‘IIIJCNT’*
1 2,.CONJG(SS<JJ.NN))«SS(ICNT,NN)*PRXY(JJ.ICNT)
AV(13.1)=AV(J$,1)¢2,*CONJ5(SS(IIINN)).SS(JJINN)oP8Z(IIIJJ)
AV(14.1)=AV(14,1>*2.rC0NJG(SS(IIINN))¢SS(JJINN)*pUXY(IIIJJ)
CONTINUE

WRITE TAPE 51,JJO(IJ>,KKU<TJ).LLO(IJ).JJG(IJ),KKG(IJ),LLG(IJ),
1 0FS¢IJ),ENGU,FNG<IJ),HHT(1:1J); (<AV<II.JJ)all=1.12).JJ=1o2).
2 Av<13,13,AvI14.1).1J '
CONTIHUt

CONTILUE

CONTINUE

CALL REGRESS(NRUV)

END

SURRDbTINE JHEWMX (A,NM,N,IOPT.RH0.IER.S>

DIMENSION A(21,21). S(?1.21)

COMPLEX A, 5. V1. v2, V3. cSNT. SNT, TEMP

10C = 0
IEF : n
FPIZ : 0.000000000001

FN = FLOATF(N)

DU 1 1:11N

[‘10 1 J=1,N

8(YIJ) = (0.",0.0)
JF(I.E0.J)S(1,J) =(1.n.0.0)
CONTINUE

IFIN.LE.1)00 TU 12

sorrn = 0,0

NM1 = Nvl

D0 2 I=1INM1

IJ = 1+1

PO 2 J:lJ,N

X1:A(I,J)

X22A(I,J)*(0-0.'1.9)

QOFFD = SDFFH * 2.0!(X1tX1+X2*K2>
IFISUFFU .LT. n.1E-10)GO To 12
THP : SQRTF<SOFF[)

FTHR = (RHO/thwTHR

IND = U

I . 4| Ilil .. . l l
, «ll. Illll‘ I.‘ II‘ I‘ 41' I‘Ii till I ‘ II! I] l I 1] III}! I l I! II I] I.] l 1 .‘ l I. Ill

.7
\)

4

16

10

160

TH? = THR / EN

00 10 L=2,N

LPH_ = L-J

DC) 1(I K=1_,LJ41

¥]=A(L.r\)

Y?=A(L|K)i(bo0,19(1)

IF¢SOPT(X1*X1 o x2ox2) ,LT, THQ) GU T3 10

1N0 = 1

V1 3 A(‘0K)

V2 = CONJG( A(L.K) )
V5 = ATLnL)

C a -AIMAG( A<L.K )

F = PtAL< A(L,L)

D PtAL( AIKIK)

THFT = 0,0

IFTAPSF(C).LT, FP12)GO T0 13

THET = 1.5707966

IF!AR5F(B).LT, EPI7)UU To 13

T a C / B

THFT = ATANF(T)

CUMTINUE

PH! c 0.78959818

E0 = ARSF(E-P)

lFttP .LT, tPI?)GO T0 15

7A0 t 2.0w<d*003F(THLT) + CIGIUF(THET)) / (t-D)

PHI :2 0,5 t A'TAVF(TAO)

CONTITNJE

FSMT =COSF(PHI) * (0.0.0.0)

CS : cnSF(THFT) t SINFTPHI)

83 = STNT(THFT1 . SIMF(PH])

SNT = C8 + (0.0.1.0) t 85

00 b 1:1,N

WEMP = A(I.K)*CSNT - AII.L)-COUJG(SNT)
A(T.L) = A(I.K)cSNT * A(I,L)tCSNT
A(I,N) : TEMP

IFTIUPT .E0. 0350 To a

TEMP = S(I.K)*PSKT - S(I.L)*CONJG(9NT)

8(111.) = 8(Ith)*SNT * S‘I'L)*C§NT

S(I,k) : TEMP

CONTINUE

U0 9 lzlaN

A<K.I) = CONJUI A(I.K) )

A(L.T) = CONJG( A<I.L) )

CONTINUE

A(V.h) = SNT*CONJU(SNT)0V3 t CSNTICSNT*VI'SNTtCSNTICONJG(V2)
.C'SNT«COI\JJL;(SNT)I:V2

A(L,L) = CSN1IESRTOV3 t COMJG(§NT)¢SNT*V1 * SNTOCSVT'

)
B = PEAL( A(La‘) )
)
)

II

2 C“NJE(V?) ¢ CONJG(5NT)tCSMT*V?

“(K'l’ = CSN‘*55“‘*V? - SNT'SNT*CONJG(V2) ¢ CSNTISNTIVI
'CSNTtSNTtV3

A(LIK) = CONJG( A(K,L) )

CONTINUE

IOC = 10C + 1

fiOOOQ'JOO

161

IF<Iru .be. 10n>60 To 21
IFIIND ,LE. DIGU T0 11
IN“ = n
GO TO 4
11 IFITHN-FTHR)12.12.3
12 RETUPN
21 IE? 2 1
RETUPN
END
SURdOUTINE RFGQESS(NODATA)
DIMENSION DATA<35), VFCTORI56.36I,AVE(35>.AVI26)I
1SIGMA<35>.CUEN¢55).SIGMCO(35).INDEX<3SI.CONSTIS5).NK<39)INTI 800)
2 .JJbIBUO)IKKU(800).LLU(800). JJGIBOOII KK0<800),LLG(800).
3 owscono>,tuh<900).Dv(800);NHT(2.aoo)
COMMON JJU,KFU,LLU,JJU.KKG.LLGIOBSIENGIWHT
TYPE DnUHLE VEFTOR, AVE. SIGMA. CotN. SIGMCU. SIGY
TOL:,un1 $ EIIN=EFOUT=,oan000101
IFNTleSTEP=IFAVE=IFCOFNzerRED:IFHEV:0
IFSTEle
100 READ 53o (NKIII. I=1.33).XDEVMAX.LNOUT,BDLNILAST
33 FORMATISSI2. F5.1. 1?, F5.1. 12)
XDFVNAX=XDEVNAYIXDEVHAX
NOPROB=INVAH:IFRAN:IFHFSD=IFCNST=1
05§N0=SUMNHT=0.
LNCNT : D
IFIBDLNI36.34
34 BDLN = 10,
36 CONTINUE
DU 38 1:1.55 $ IF(NK(I))37.39
37 INVAP=INVAR+1
38 CONTINUE
PEMIND 50
HENINU 51
IFUT = 1,THEN ALL NHTS = 1,0

IFSTIP = 1, DO NOT PRINT EACH STtp

IFQAW = 1 00 NOT PRINT QAN SUNS AND SQUARES
IFAVE = 1 DO NOT PRINT AVERAGES

Iruzsn = 1 Do NOT PRINT RESIDUAL SUNS SQUARES
IFCOFN = 1 Do NUT PRINT PARTIAL COEFFICIENTS
Irpngn : 1 Do NOT CALC PREDICTED VALUES
IFCNST = 1 00 NOT HAVE CONST TERM IV EQUATION

NOVAP = INVAH

NVDI = NOVAR + 1
495 CONTINUE

LNCNT : LNCNT*1

NDIN 3 U

VAQ z n U

K : 0
I"LEVEL. = 0.0
NOLjVI 2 0
NOHIN = 0
NOVA! : 0

3 0 DO 120

3 I 3 1: MVP}
.30 DU 1?0 J = in

NVPl

162

120 VECIGHII.J) = ”.0
IF(L.NCNT.GT.1)CU TO 506
519 D0 235M=1,NUDATA
REAO TAPE 51.J3.LL1.KK1.JJ.LL.KK.ORSFREQ.EN61.EGowT( M).
1 (¢V(II):II=1176):1J
FRFJ=EN61~EG
ENOIIJazFREO
OATAINOVAR)= OHSFREQ-FPEO
IF(dT( M))2L6,200.2P4
€F4 IFIAPSFIOBSPREO-FREO)~FOLN>264.264.205
?€b PRINT 218.J1.LL1.KK1.JJ.LL.KK.OOSFREO,FREO.IJ

718 FORMATI «THF HEIGHT OF THIS LINE Is BEING SET EQUAL T0 ZERO*//
1 ?(2x.313). 2F11,5.I11)
WHT(JIIJ)=09
wT(M)=O,
GO TO 206

2n4 SUPNhT=SUMNHT+wT( M)
OBSNO=OOSNO+J.
2C6 NU”=P
O0 220 I=1.Zb
IFINKII))223.2?O.210
210 NUM=HUM+1
UATAINUM):AV<I)
?20 CONTINUE
?KO wRITF TAP65O.JI.LL1.KAI.JJ.LL.KK.OBSFQEO.FREU.
(OATA(L), L=1.(OVAR), M,IJ
CONTINUE
CONTINUE
RENINU 50
AVENHT=SUM4HT/OOSNO
To 511 N=1,NOOATA
HEAD TAPEfin.JI.LL1.KKl.J.LL.KK.OBSFHE3.FREU. (DATAILIaL=1.NOVAR
1 ).NHUN
WGT=HT( N)/AV:NPT
IF!NOT)5$0.511
Sfifl DO 540 I = 1. NOVAR
H5O VECTOR (I.NOVAQ + 1) = VECTOR (I. NOVAR ¢ 1) + DATA (I) ' NGT
bOO DO 500 J = I. NOVAR
EAO VECTOR (l. J) = VECTOR (I. J) + DATA (I) - DATA (J) t NGT
ean VECTOR (NVPL. NVP1) = VECTOQ (NvPl. NV91) oust
€11 CONTINUE
COMPLETED SUNS OF SQUAQES AND CROSS PRODUCTS. THESE ARE IN
ISTORAOE IN LOCATION. VFCTOQ (I. J). THESE WILL OE PRINTED OUT ON
ETAPE o UNDER CONTROL OF STATEMENT 100
5M5 NOVA! = NOVAR . 1
93616 NIJ‘IPL = NOVAR t 1
568 PRINT 00. NOPHOB. NOOATA.NOVAR, VECTOR(NOVPL.NOVPL). tIIN. EFOUT
570 IF (IFOAU) 900, 550. 650
850 WRITE OUTPUT TAPFot. 15
500 WRITE OUTPUT TAPE61. 20.(I.VECTOR(I,N3VPL).I=1.NOVWI)
OOO kRITF OUTPUT TAPE61.25. VECTOR(NOVAR.VOVPL)
OIO wRITE OUTPUT TAPEél. 5O
(2O wRITF OUTPUT TAPEbl. $5. ((
OsO kRITE OUTPUT TAPEéi. 40. (I

b" n
:L L‘.‘
'3‘ Kl".
’ L

Ind.VECTOP(I.J):J=1.NDVWI),l=1,Novml)
.VECTOR(I.N0¢AR).I=1.NOVMI)

163

hen ORTTE OUTPUT TAPEOI. 48, VECTORIOOVAR,NOVAH)
GO TO 650
CALCULATION OF RESIDUAL SDMS OF SQUARES AND CROSS PRODUCTS
OuO IF(IFCNST) 9FO,651.135
n51 IFIVECTOR(NO»PL.NCVPL)) 652.652.655
652 PRINT 654
GO TO 910
(55 DO 660 I = 1. )OVAR
A10 DO 660 J = I. NOVAR
6h” VECTOR (I.J) = VECTOR (I.J) . (VECT0R(I.N0VPL) w VECTUH (J.NOVPL)
- / VECTOR (NOVPL: NOVPL))
FRO PO 690 I = 1. NOVAR
840 AVF(I) = VECTOR(I.NOVPL) / VECTOR(NOVPL.N0VPL)
7J0 3F (IFAVE) 9O0. 710, 735
710 WRITE OUTPUT TAPEbl. SO
720 NRITF OUTPUT TAPEbi. ?O. (I.AVE(I).I=1.NOVMI)
730 WRITE OUTPUT TAPEo1. ?5. AVEINOVAR)
755 IF (IFRESD) 9n0. 740, 780
740 VRITE OUTPUT TAPE61. 55
750 «RITE OUTPJT TAPEbi. 36,((I.J.VECTOR(I.J).J=1.NOVMI).I=1.NUVMI)
7~D «RITE OUTPUT TAPEOl. 40. (I.VECTOR(I.NOVAR).I:1.NUVMI)
770 ORITE OUTPUT TAPEoi. 46, VECTOR(N0VAR,NOVAR)
750 AOSTFP = -1
751 ASSIGN 1523 TO NUMRER
7n2 DEER = VECTODINUVPL.NOVPL) - 1.0
700 D3 830 I = 1.NOVAR
791 IFIVECTUR(I.I)) 792.794.510
792 PRINT 795. I
GO TO 910
7&5 FURNAT (31R ERROR pESIOUAL. SOUARE VARIABLE 14.51H IS NEGATIVE,PRDB
ELEM TERMINATEU )
704 NRITE OJTPUT TAPEbi. 795. I
790 SIGWAII) = 1.0
797 GO TO 200
7u5 FORMAT (IHOIOH VARIARLE 15.13% IS CONSTANT )
8:0 SIGHAII) :OSORT (VtCTOR (1.1))
800 VECTOR(I.I) = 1.0
820 D0 830 I = 1.NOVMI
Han 1P1. 3 1" 1
841 PO 830 J = 1P1, NOVAR
850 VECTOR(I.J) = VECTOR(1.J) /( SIGMA(I). SIGMAIJ))
n30 VECTORIJ.I) = VECTORII.J)
Rnu IF (IFCUEN) 900. 870. 1000
870 HRITE OUTPUT TAPEbl. 50
h74 NOJW? = NDVAI - 1
815 TO 885 I = 1. NOVM2
8‘10 1P1. = 1+ 1
885 wRITF OUTPUT TAPEo1. 3S.(I.J.VECT09(I.J).J=IP1.N0VMI)
570 VRITE OUTPUT TAPFOI. 4O.(I.VECTOR(I,N3VAR).181.NOVUI)
IOWD NOSTEP = NOSTEP o 1
1501 IF (VECTORI ADVAR.NONAR)) 1002.1002.1010
10m? ASTPM1'= NOSTEP - 1
PRINT 1004, NSTPMl
GO TO 1681

t .. 1 ‘TC-

1010
1015
1016
1017

1020
1050
1055
1040
1041
1042
1045
.1050
1050
1090
1100
1120
1150
1140
1150

004

1370
1100
1390

100
1310
IPLU
1280
1050
1230

”I3

1240
1250
1945
1245
1747
1950
1210
1990
1250
1340
16L0
1311
1312
1313
1314

1315
1320
1350
1340
1345

164

SITY = SIGMAINOVAR)
DEFR =OEFR-1.U

IF (DEFH ) 1017.1017,
PRINT 1019 .NOSTEP

GO TO 1381

VWIV = 0.0
VWAX : 0,0
NOII = 0
DD 1050 I = 1.N0VMI

IF (VECTORII.I)) 104?.
PRINT 1044. I. NOSTEP
GO TO 1681
IFIVECTORII.I)-TCL) 1050. 1000. 1050
VAR:VECTOQ(I.NOVAR)'VECTDR(NOVAQ.I)/VECTOR(I:I)

ODSQRT (VECTURIVOVARJNUVAR)/ DEF“)

1020

1050. [060

IFIVAHTIIOD. 1050. 1110
NOIJ = NUIN * 1
INOEXINOIN) = I

CUPNINOIN) . SIGMAINOVAH) / SIGMA (l)

«DSQQT (VECTOR(101))

VECTOR(I.NOVAR)
SIS'ICCHNOIN) = (510)! / SICMAHI)
IF (VMIN) 1160,1170.904

WRITE OUTPJT TAPEOI. 906

GO TO 910

VMIN = VAR

NOWIM = I

GO TO 1050

IFIVAR - VNIN)IOSO.1050.1170

IF (VAR - VWFX)10bO.1050.1210
VMAX = VAR ‘

NOMAX : I

CONTINOE

IF (NOIN) 903.1240.1?43

ARITF OUTPUT TAPEbi. 907

GO TO 910

WRITE OUTPUT TAPEbl.
GO TO 1350
IF (IFCNST)
CNST = 0.0
GO TO 1500
CNRT = AVE(NOVAR)
DO 1780 I = 1.NOIN
J a INDEXII)
CNST = CNST -
IF(IFSTEP) 900.1310.1520

IF (NOFNT) 1311.1311.1313

WRITE OUTPUT TAPEol. 91.N0§TFP,K
GO TO 1514

WRITE OUTPUT TAPE61.
WRITE OUTPUT TAPEblo

5.SIGY

900.1250.1246

(CCbNII) t AVEIJI)

9?.N08TEP.K
70.FLEVEL.SIGY.CNST.

1<INDEXIJ).COENIJ).SIOMCD(J).J=1.AOIN)

GO TO NUMBER. (1320:1580)

FLEVEL = VMIN t DtFR / VECTOR (NOVARgNOVAR)
IFIEFOUT + VLEVEL) 1350. 135”: 1340

K a NOMIN

NOFNT = 0

165

00 TO 1391

1350 FLEVEL = VMAX t DtFR / (VECTOR(NCVAR.VOVAR)~ VMAXI

1350 IF (FFIN - FLEVEL) 1370,1361.1380

1351 IF IFFIN) 1680,13bn.1370

1310 K=VOMAX

1390 NOEVT = K

1391 IFIK) 1302.139?.1400

1392 WRITE OUTPuT TAPE61.1305.NOSTEP

1394 GO TO 910

1430 30 1410 I = 1.NOVAR

1420 IF II-K) 1450,1410.1430

1430 00 1440 J = 1. NOVAR

1450 IF (J-K) 1460,1440,1460

1450 VEPTORII.J) = VECTORII.J) - (VECTORII.K) t VECTOR (K.J) / VECTOR
-<K.K))

1440 CONTINUE

1410 CONTIvue

1410 DO 1480 I = 1, AOVAR

1490 IF (I-K) 1500.1480.1500

1500 VECTDR (11K) 2 - VECTOR (I,K) / VECTOR (KoK)

1440 CONTINUE

1510 00 1520 J : 1. NOVAR

1550 IF (J-K) 1540,1520,1540

1540 VECTORIK.J) = VECTOR IK.J) / VECTOR (K.K)

1520 CONTINUE

1550 VECTORIKIK) = 1.0 / VFCTORIK.K)

1560 GO To 1000

1350 PRIVT 75. NOSTFP

1501 IF (IFSTEP) 900, 1550,1570

1570 ASSIGN 1500 TO NUMPEW

157] GO TO 1310

1550 kRIIE OUTPUT TAPEoi. 1586:(LaVECT0R(L,L)gL=1'NUVMI)

1551 IF! IFDRED) 900.1562,910

1552 PEUINU 50
PRINT 930.<NK(I).I=1.26)

330 FORMATIt-THE FOLLowING CONQTANTS WERE VARIEUtl/t UPPER STATEt//
1* RAND N0. 1*/ * V0 A B C TAAAA T8888 TAABB TABAB
2 HJ HJK PKJ FK t/13.3I4.4(3X,13.ZXI.215.216/taBAND N0, 2«/
3* V0 A B C TAAAA T8888 TAABB TABAB HJ HJK HKJ
4 MK */I3.SI4.4(3X.I5,9X),715.?16/*-PERTURBATION TERMS'I' GZ 'GXY
5‘/13.2Xa13)

1553 kRITE OUTPUT TAPF61. 85

1556 FOPMAT I24H0 DIAGONAI ELEMENTS //?0H VAR,NO. VALUE/l
1(1H I 7. F16.6))
[‘EV‘4AX'30|
SRFSQWT 3 00

1590 00 1660 N = 1. NODATA

1600 READ TAPE 50.J1. LL1.KK1. J. LL. KK. OPSFRtUa FREQ:
1 (DATAIL).L=1,MOVAR),NPUN,IJ

1010 YPRED = CNST

1520 00 163n I = 1,NCIN

1640 K : INUEXII)

1550 YPPED= YPRED ¢ COtNII) * DATAIK)

1650 FEV = EATA(NOVAR) n YPRED

166

UV{IJ)=UEV
81 FORMAT(2(2Xo5I3)o 3F10.5.2F11.3.F9.2.110)
SRESQNT = SRHSPWT* DEV¢t2th( N)
rEv:wT( N)t“tvcEEv
IF(DFVMAx-va)012.912.1660
912 bEVMAX=UFV
hHumx=NHUN
IJV=IJ
1660 CONTINUE
P0 1b64 h:1,NUPATA
FRFUPRD=DHS(N)-UV(N)
DIFF=OHS(N)-Ewn(n)
1564 PRINT R1.JJU(N).KKU(N).LLU(N).JJG(N),(KG(N)oLLG<N).FREUPRDa
1 063(N).th(N).LIFF.DV(N).NHT(1.N),N
SFANDARD DtvIATIUN FROM RESIDUALS
F2:ORS~O«5Ht82wT/((0&SNO-N01N)«SUMWHT)
1770 PRIVT 1775. S2
1775 FORMAT (//24H QUV SO. OF RFSIDUALS = , F14.8)
178C STDUEV = SURTF(S2)
1735 PRINT 1790. STDDEV
1790 FORMAT (39H an. DEV. CALCULATFD FROM RESIDUALS = o F10.8)
IF(LHCMT'LNUUT)1665. 910. 910
1555 IF<UFVMAx-xurvax)910,q1n,915
915 PRTVT o15,14v .
018 FCRMAT(//////w TFE wkIGHT 0F LINE N0, ‘13* IS BEING SET EQUAL T0
12EPU~)
rbsv0=nss~0a1.
SUMWHTSSUMNHT'NT( NWUNX)
th NRUKX)=0,
VHT('I:IJV)=Uo
so 10 495
910 IFtLAST)911o911o960
960 PO 901 N‘lnNFDATA
961 WhT(1oN)=WH1(2,N)
Go To 100
900 PRINT 905
911 CONTINUE
PRINT 3
6 FcumAT(1F1)
END...
5 FORMAT (3F10.5.315.1H 912,5X.F6.4)
10 FORMAT (6(F12.5))

15 ‘ORMAT (1H 49H SUM OF VARIABLES/l)
2U FORMAT (1H 11H SUM x< 12.3w) : F12.4.8H SUM x< IZoSH) :F12,4.
15H SUM x( 12.3H) =F12.4,8H SUM X(12,3H) =F12.4 )
25 FOQWAT (17H SUM Y =F12.4)
50 FORWAT(1HU 70H HAw SUM OF SQUARES A
1ND CROSS PRODUCTS/I )
55 FORMAT (1H 7H X(12.7H) VS x(12.3H) = Flb.6p
1 6H ¥(12.7H) VS ¥<12.3H) a F15.6:
2 6H x<I2.7H) vs x(12.3H) - F15.6 )
40 FORMAT (1H 7n X(12.12H) VS Y sF15.6o
1 6H x(12.12H) vs Y =F15,6.
2 6H y(12.1?H) vs v =F15,6 )

45 FOWMAT (1H 21H Y VS Y =F15.6)

59 FUWVAT (1H005H AVERAGE VALUE OF
- VARIAdLESI/ 3

:5 FowwAT<1F077F RtSIDUAL SUNS 0F SOUA
~FE9 AND CHUSR PRODUCTS//)

66 FUNWAT(1H069H PARTIAL CORRELATI
.nm COEFFICIFNTS/I)

65 FORMAT (?5h0 STANDARD ERROR 0F.Y = F12.6 )

7G FORMAT (11H F LEVEL F12,4/25H STANDARD ERROR 0F Y a F12.4/12H
1 CHNSTANT F13.5/56H VARIABLE COEFFICIENT STD ERR
aCR OF CUFF // (30H x-13,F18.11.F15.11))

75 FORMAT (10H COMPLETtD 15.2nH STEPS 0F REGRESSION)

8D FOHflAT (hH Fb.0.2H F12.S.3H F12.5a2H F12.5)

55 FOQMAT(//4x- UPPFH GHoqu PREDIcTcn 08$ CALC 0
lfis-CALC Dtv er.//)

on FCRWAT (22HlSTFPWISE PEGHESRION //1?H PROBLtM N0 110 //13H NO OF
1UATA = 15 //15u r0 UF VARIABLES a 110 //30H WEIGHTEU DEGREES 0F FR
ZFEDOM : F12.? //?8H F LEVEL TO ENTER VARIABLE a F12.9 //29H F LEVE
3L TO HFMOVE VAQIABLE = F13.9 Ill )

91 FODMAT (9HOSTEP no.18 /10H VARIABLE REMOVtU 18)

92 FQPMAT (9H05TtP no.15 /20H VARIABLE ENTERING lb)

93 FORMAT (5H RUN F6.0,3H F1n.5,3H r10.5.3H F10.5,6H F10.5.
15H F10.5/(14u F10.5.3H F10.5.3H F10-5:3H F10,
'5,3~ F10.b)) ‘

654 FoprT (31H ZLRQ NUMBFR 0F nATA. so LONG.)
U05 FODMAT (42H ERROR IN CONTROL CARD, PROBLEM TtRMINATtD)
996 FCWMAT (25H FRPOF; VMIN pLUS. SOLONG)
007 FORMAT (26H tPR(R,N01N MIN 8, SOLONG )
1304 FORMAT (1H067Hv SQUARE NON-POSITIVE.TERMTNATE STEP 1 b)
1919 FORMAT (1H029H NO MORE DFGREFS FREEDOW STEP I 5 )
1044 FCQMAT (1H010H SCUAHE x-IS,17H NEGATIVE. SOLUNG 15.6H STEPS)
139E FORMAT (12H h=0. STEP 16, 7H SOLONG)
END
PVN, 5, 2100

2
1
3