Dmmmmmmé or- A0 FDR AXIALLY
symmmac MOLECULES

Thesis fer the Degree of Ph. D.
MICHIGAN STATE UNWERSHY
THOMAS L, BARNETT
1967

   
     

  

LIBRARY.

MiChigan stat!
University

THESIS

 

This is to certify that the
thesis entitled

Determination of A0 for Axially
Symmetric Molecules

presented by

Thomas L. Barnett

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Phxsics

Major professor

Date dilly/f; /7/7

0-169

ABSTRACT
DETERMINATION OF A0 FOR AXIALLY SYMMETRIC AOLECULES

by Thomas L. Barnett

A new method is deve10ped for determination of
accurate values of the molecular constant A0 for axially
symmetric molecules by simultaneously analyzing a degenerate
fundamental band and its first overtone. In particular, the

method is developed for a simultaneous fit of the Va and 2v”

A

o is

hands of a methyl halide. [A0 = h/(BanIoA), where I
principal moment of inertia of the molecule about its axis

of symmetry in the ground vibrational state. Accurate

values of IdA are necessary to determine the structures of
these molecules.]

The development of this new method begins from the
Amat—Nielsen generalized frequency exPression, listed here
complete through third order and containing many fourth-
order terms. This expression is then specialized to forms
apprOpriate to individual least squares fits of the v“ and
2v“ bands, and simultaneous fits of the Va and 2v“ hands.

This method of determining AC has been successfully
applied through least squares computer analyses to high-
resolution spectra of the methyl halides and similar mole-
cules. The excellent value of A0 = 5.1291 1 0.0009 cm“1
obtained for methyl bromide seems to clearly demonstrate

the superiority of this method over previous methods of

determining A0. Analyses of the other molecules of the

same type (methyl iodide, methyl chloride, methyl fluoride,
methyl cyanide, and singly-deuterated methane) led to less
precise values of A0, mostly because of perturbations
occuring in one or both of the bands involved.

"Ground state“ and "substitution" structures are
calculated for methyl bromide, making use of the excellent
value of Ao obtained here. The results are discussed in

light of theoretical predictions by Kraitchman and Costain.

DETERMINATION OF A0 FOR AXIALLYVSYMMETRIC MOLECULES

by

,iIE/
Thomas Lf;Barnett

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Physics

1967

TO MY WIFE

MARY

ii

ACKNOWLEDGMENT

I wish to thank Dr. T. H. Edwards for his encour-
agement and many helpful suggestions throughout my research.
Our many discussions have guided and shaped this work to a
large extent. I wish to also thank Dr. P. M. Parker for
various discussions, as well as for an excellent course on
the theoretical aspects of molecular spectrosc0py. I am
indebted to Dr. R. S. Schwendeman of the M. S. U. Department
of Chemistry for several very valuable discussions concerning
the problems involved in the determination of molecular
structures. I would like to acknowledge the useful sugges-
tions, help, advice, etc. throughout my years here from my
co-workers Kent Moncur, Lewis Snyder, Mel Olman, and Don
Keck. In particular, I wish to express a special appre-
ciation to Don Keck for the numerous times he has helped me
in aligning and adjusting the spectrometer, in which he often
did most of the work with whatever assistance I could offer.

I am very much indebted to the M. S. U. Computer
Center for the use throughout my research of their C. D. C.
3600 computer, without which none of the analyses would have
been possible. And finally, I wish to expressly thank the
National Science Foundation for their generous support of

our research.

iii

TABLE OF CONTENTS

ACKNOWLEDGEMENT - - - — _ _

LIST OF

LIST OF

LIST OF APPENDICES

TABLES - - - - — _

INTRODUCTION - - - — _ - _ _ _
CHAPTER I GENERAL FOURTH-ORDER HAMILTONIAN
CHAPTER II SYMMETRIC TOP ENERGIES AND GENERALI-
ZED FREQUENCY EXPRESSION - -
CHAPTER III SINGLE-BAND FREQUENCY EXPRESSION
CHAPTER IV SIMULTANEOUS FREQUENCY EXPRESSION
CHAPTER V LEAST SQUARES ANALYSIS OF THE DATA
CHAPTER VI METHODS OF OBTAINING AND TREATING
SPECTRA - - - - - _
CHAPTER VII ANALYSIS OF CfiaBr - - - -
CHAPTER VIII ANALYSIS OF CH3I - - — -
CHAPTER IX ANALYSIS OF CH3F - - - -
CHAPTER X ANALYSIS OF CH3CN - - - -
CHAPTER XI ANALYSIS OF CH3C1 - - - -
CHAPTER XII ANALYSIS OF CH3D - - - -
CHAPTER XIII STRUCTURAL CONSIDERATIONS - -
CHAPTER XIV CONCLUSION - — - - - -
LIST OF REFERENCES - - - - - _ -
APPENDICES - - - - - - - - -

iv

Page

iii

vii

viii

10

22
28
39

46
56
72
83
95
102
107
111
123
130
133

Table

II.

III.

IV.

VI.

VII.

VIII.

IX.
X.

XI.

XII.

XIII.

XIV.

XV.

XVI.

XVII.

XVIII.

XIX .

LIST OF TABLES

Definition of Symbols - - - -

Elements of Hamiltonian Matrix and
Corresponding Energy Terms - - -

Symmetric TOp Energy Expressions
through Fourth Order - - -

Classical Interpretation of Energy Terms

Amat-Nielsen Generalized Frequency
Expression - - - - - - _

Frequency Expression Suitable for
Single-Band Fit of v. or 2v“ - -

Frequency Expression for Simultaneous
Fit of v. and 2v. - - — - _

Frequency Expression for Simultaneous
Fit of H3 + v. and 2v“ - - _ -

Normal Equation Terms - - - -
Experimental Conditions (CHaBr) -

Coefficients of Simultaneous Fit of
CH3Br VA and 2v. - — - - -

Coefficients of Single-Band Pit of
cuasr v. - - — - - - -

Coefficients of Single-Band Fit of
CH3Br 29g - - - - - - -

Experimental Conditions (CH31) - -

Coefficients of Single-Band Fit of
CH31 v. - - - - - - -

Coefficients of Simultaneous Fit of
CH3I VA and 2Vu - — - — -

Experimental Conditions (CH3F) - -

Coefficients of Single-Band Fit of
CH3F ZVA - - - - -

Coefficients of Single-Band Fit of
CH3F v. - - - - - -

Page

11

13

14

16

25

29

39
42
57

67

68

69
73

78

81
84

89

9O

XX.

XXI .

XXII.

XXIII.

XXIV.

XXV.

XXVI.
XXVII.

XXVIII.

XXIX.

XXX.

Coefficients
CHaF V3 + VA

Experimental

Coefficients
CH3CN v5 -

Coefficients
CH3CN V5 and

Experimental

Coefficients
CH3C1 VA -

Experimental

of Simultaneous Fit of

and 2v. - -

Conditions (CH3CN)

of Single-Band Fit of

of Simultaneous Fit of

2V5 - - -

Conditions (CH3C1)

of Single-Band Pit of

Conditions (CH30)

ro Structural Parameters -

r8 Structural Parameters Derived from

A
rCBrls’ Io '

r8 Structural Parameters Derived from

A
rCBrls' Io
tion - -

rS Structural Parameters - Best Average

B
and I0 - -

vi

, and Center of Mass Equa-

92
96

99

101
103

106
108
115

123

124
125

LIST OF FIGURES

Figure page
1. Definition of Rotational Constants - 2
2. Methyl Halide Fundamental Vibration
Frequencies - - - - - - - 19
3. CH3X Normal Modes - - - - - - 20
4. Dynamic Pen Separations - - - - 48
5. Idealized Spectrum - - - - - - 50
6. Hydel Optics - - - - - - - 52
7. CfigBr Survey Spectra - VA and 2v. - - 59
8. R03(J) Section of CHgBr 2v. - - - 61
9. CH3I Survey Spectra - v4 and 2v. - - 75
10. RQ3(J) Section of CHgI 2v“ - - — - 79
11. CH3F Survey Spectra - Vu, 2V4, and V3+vu 86
12. R06(J) Section of CH3F Zvu - - - - 87
13. CH3CN Survey Spectra - v5 and 2v5 - - 98
14. CH3C1 Survey Spectra - v. and 2v. - - 104
15. CH3D Survey Spectra - vi and 2v. - - 109
16. CH38r Structural Parameters - - - 113
17. CH3Br Substitution Parameters - - - 121

vii

Appendix
I.

II.

III.

IV.

LIST OF APPENDICES

Alternate Methods of Obtaining A0

Listing of FALSTAF Program -

Listing of SCAN Program

Listing of Output Data from Simultaneous

Pit of CfiaBr v. and 2v“

viii

Page
133
136
147

152

INTRODUCTION

The methyl halides have been the subject of many
investigations in molecular spectrOSCOpy over the years.
Herzberg (1) summarizes the situation up to 1945, and many
papers on the methyl halides can be found in the literature
subsequent to this date. From the viewpoint of infrared
molecular spectroscopy these molecules have many attractive
features, being polyatomic (five atoms), but having a
three-fold axis of symmetry (see Fig. l). The presence of
this symmetry axis greatly simplifies the theory and to
some extent the experimental treatment, but introduces some
problems unique to such I'axially symmetric“ molecules.

One persistent problem has been the difficulty in
determining accurate values of the molecular parameter A0

A is the principal moment of inertia

[= h/(szcIoA), where ID
of the molecule about its symmetry axis in its ground
vibrational state]. It has long been known that a perpen-
dicular band of an axially symmetric molecule cannot be
solved for the value of A0 alone, but rather for only a
numerical value which represents A0 plus the coefficient of
a first-order vibration-rotation interaction term. Hereto-
fore, methods of determining A0 for axially symmetric
molecules have lacked precision, or directness, or both.

In this thesis a new method, permitting the
determination of A0 directly from infrared spectra, is
described and applied. The method consists of making a

simultaneous analysis of two or more suitably related

1

Fig. 1

3’
ll

 

 

Definition of Rotational Constants
I A
0
Ah
I
O x
I."
B
Io
C
H H
H
h h
__W m ' B0 = V W B
2 A 2
8n cIo 8n cIo
6 A
Ae ' zs=l as (Vs+gs/2)
6 B
Be - zs=l as (v3+gs/2)
A[v] for all v3 = 0
Blv] for all v8 = 0

 

vibration-rotation bands by means of least squares fitting
on a large, high speed digital computer. Chapter I contains
an outline of the procedure used by Nielsen, Amat, et a1.
(2) in obtaining the generalized fourth-order quantum
mechanical hamiltonian for a vibrating and rotating sym-
metric top molecule. In Chapter II the fourth-order sym-
metric top energies are tabulated along with the classical
interpretations of the various terms. The generalized fre-
quency expression, representing any symmetric top vibration-
rotation transition from the ground vibrational state to an
upper vibrational state, is also listed. In Chapter III the
generalized frequency expression is specialized to single-
band frequency expressions suitable for computer analyses

of the methyl halide vi and 2v. bands. The reasons for A0
not being obtainable from a single-band analysis are dis-
cussed in detail. The new method of obtaining A0 is des-
cribed in Chapter IV. Frequency expressions appropriate for
simultaneous least squares fits of v. and 2v“, and of V3+Vu
and 2v4, are given.

Chapter V contains a general discussion of the
least squares procedure as applied to our problems and com-
ments upon the statistical considerations, including the
use of simultaneous confidence intervals. The experimental
procedures used to obtain the spectra and extract the tran-
sition frequencies are the subject of Chapter VI. Chapters
VII, VIII, IX, and X contain the results of single-band and

simultaneous analyses of CHaBr, CH31, CH3F, and CH3CN re-

Spectively. Chapters XI and XII present the results of the
less complete analyses of CH3C1 and CH3D. A general dis-
cussion is given in Chapter XIII concerning the problems
involved in calculating structures from the measured values
of A0 and Bo for the methyl halides. Calculations of the
"ground state" and "substitution" structures for CH3Br are
reported and discussed. The Conclusion sums up the main
results and attempts to assess the value of this work and
the importance of obtaining accurate Values of A0 for the

methyl halides and other symmetric top molecules.

4. .

CHAPTER I

GENERAL FOURTH-ORDER HAMILTONIAN

.The general fourth-order hamiltonian for a mole-
cule undergoing vibration and rotation has been developed
by Nielsen, Amat, et a1. (3)° A very brief summary of
their procedure follows.

One begins with the Darling-Dennison quantum

mechanical hamiltonian (2)

.11 == 1/2[umzaswa-pumaeu’m(PB-pawl”
+ umZSUPSUWND1,2930%"m] + V,

where

a and 8 range over the principal axes x, y, z;

s ranges over the set of normal coordinates
necessary to represent the normal modes of the
molecule;

a ranges over only the components of doubly-
degenerate normal coordinate pairs (0 = l or 2);

P represents the a-component of the total angular

momentum of the molecule;

pa represents the c-component of the internal
angular momentum of the molecule;
930* represents the momentum conjugate to the normal
coordinate Qso’
V represents the potential energy of the molecule;
= I 0+ I I.
"GB (177 IUB IaY IYB )u.
3 I I_ l2
"an (188 IVY IBY )u:

 

 

‘1 = l _ I - I
u Ixx xy Ixz

'I l I I “I O

xY YY yz

.. I _ l I

Ixz Iyz I22
I ' = 9 +2 +2 A'°‘°‘ Qs 0Q
ca soa 300 so sos'o' sas'o' s 'O"

' = e _ a8 - 'GB

IaB IaB EBOaSOQSU 2sos'o'A sas'o'QsaQs'O

and since the principal axes of inertia are used as the

molecular base framework (I e = 6 6) then ace and
CB OBI on so
"256,0, are constants of the molecule.

The Darling-Dennison hamiltonian, g, is diagonal
in J (the total angular momentum is necessarily conserved)
but may, in general, contain terms off-diagonal in the
quantum numbers v3, 2t, and K (see Table I for definitions
of these symbols). Here, only axially symmetric molecules
will be considered.

This hamiltonian is not directly solvable, hence
a perturbation treatment of the problem is called for. It
is assumed that E can be expressed in a power series ex-
pansion, as a sum of terms of rapidly decreasing magnitude,
viz.,

£1. = H0+H1+H2+H3+HA+...
with Ho >> H1 >> H2 >> H3 >> HA >> ....

In the power series expansion of the hamiltonian,
g, in terms of the normal coordinates, the zero-order term,
H0, is the hamiltonian representing a harmonic oscillator

plus a rigid rotor. It is diagonal in all quantum numbers

Table I
vs -
2t -
J .—
K _

Definition of Symbols

vibrational quantum number representing s-th

vibrational mode

second vibrational quantum number for doubly-
degenerate normal modes; associated with internal

vibrational angular momentum

rotational quantum number associated with total

angular momentum

rotational quantum number for axially symmetric
molecules associated with component of total

angular momentum along symmetry axis

and is exactly solvable. Classically, the higher order
terms will represent corrections to the rigid rotor-harmonic
oscillator model, such as centrifugal distortion due to
rotation, anharmonicities in the Vibrations, and inter-
actions between vibration and rotation.

The perturbation treatment is carried out by
means of contact transformations on E, The first contact
transformation on g, yielding
' h' = Ho + h1' + h2' + h3' + ht' + ...,
is chosen such that H0 is left unchanged but H1 is diago-
nalized with respect to V3 in the representation which
diagonalizes Ho. This leaves the hamiltonian actually
diagonal through first order with respect to all the quan-
tum numbers (J, K, VB, and it). In the absence of any
accidental resonance, the energy of an axially symmetric
molecule through third order is obtained from the diagonal
elements of (Ho + h1' + hz' + ha'). The off-diagonal terms
from hz' will not contribute to the energy before fourth
order, and those off-diagonal in h3' will not contribute
before sixth order.

The second contact transformation, operating
on h', is chosen so as to leave Ho and h1' unchanged, while
diagonalizing hz' with respect to V3 in the zero-order
representation (the representation in which H0 is diagonal).
The twice-transformed hamiltonian,

h1‘ = Ho + hl' + h2+ + 113'r + h.+ + ...,

is diagonal in all quantum numbers through hl', and

diagonal with respect to V8 through h2+, but may have terms
<xff-diagonal with respect to K and it in h2+, and off-
<iiagonal with respect to vs, it, and K in h3+ and h.+.

It should be sufficient, in the absence of any

(accidental resonance, to obtain an "exact" representation
of'the energy to third order from the diagonal elements of

h+ through h3+, plus a partial fourth-order contribution to

the energy from the diagonal elements of hq+.

CHAPTER II
SYMMETRIC TOP ENERGIES AND GENERALIZED

FREQUENCY EXPRESSION

I. Energy Expression

 

In general, the symmetric top energy eigenvalues
are obtained by solving the secular determinant

I v
detl <J,K,...,vs,...,£t,...| h IJ,A',...,Vs',...,2t',...>

(GKK"°"'5V3V3'""'5£t£t""°) EVR J = 0'

To obtain the energies completely through third
order and partially through fourth order, one needs only
the diagonal elements of (Ho + hl' + h2+ + h3+ + hu+), viz.,

detl <J'K’eee’vspbcollthOOl(H0 + hl' + h2+ +

+

ha + hq+)|J'K’oee'VB’eee’£ > - E = 00

t,... VR]

The elements of the general twice-transformed
hamiltonian, as developed by Nielsen, Amat, et a1. (2), and
their contributions to the symmetric top energies are given
in Table II. [Note that in Table II and subsequent expres—
sions "s" runs over all the normal modes of vibration, "n"
runs over only non-degenerate modes, and "t" runs over only
degenerate mOdes.]

Table III contains the entire energy expression
for a symmetric tOp molecule, complete through third order
and containing the diagonal fourth order contributions.

The classical interpretation of each term is noted in

Table IV. Actually, it is doubtful that the third and fourth

order terms can be assigned any classical significance.

10

Table II

11

Elements of Hamiltonian Matrix and Corresponding
Energy Terms

1/2ZaPa2/Ia + h/zf A1/2(p8 2/h2+q 2)

SO 3 SO

_ "2
BeJ(J+l) + (Ae Be)h + {Sms(vs+gs/2)

o b

1/2chzab (1)Ya (qapb+pbqa)Pa
z

'ZAeZtCt ttK

“875(2)YPGP P P

ZoByd B y 5

JK K

_ J 2 2 _ 2 _ u
DeJ (J+l) De K J(J+l) DeK

2ofizgybws(2)Yabpapb+as(2)Yabqaqb]PaPB
~28a2(vs+gS/2)J(J+1) -2 ”(a2 :)(vs+gB/2)K2

Xab ed [(2)Y§b11/2(qaqbpc Pd+Pc quaqb)
asb, csd

[(2)Y

+2abcd abcdlqaqchqd

asbscsd

Eggé'x“. .(v “+9 /2)(vS .+gs./2) + 2:3; xltlt stat,

aBy b .
Easyiabt mya ll/2(qapb+pbqa)PaPBPY

ZtniltJ(J+l)K

bcd
2 2g§°dd I“ (3)Ya 11/2(qapbpcpd+pbpcpdqa)Pa
Cid

+2 2

d [ (3)Yabc]1/2(qaqchpd+pdqaqch)Pa

aabgc

2tntztK3 + Ztlnt+zsnt,s(vs+gs/2H£tK

12

aByé aByd ab
[ (4)2abqaqb+ (4)2 papb]PaPBPYP5

Z“BY523?b

{ssg(vs+gS/2)J2(J+1)2 +{sng(vs +g S/2):<2J(J+1)

+{ss:(vs+gS/2)K“

a8 +08 abcd

P P P P ]
sctd a b c d

aaB
+Xabgdm (4)Z§b1/2(qa quc Pd+Pc qua qb)+“ 8(4)Z}PGPB

{zgs'.Yss'(vs+gs/2)(vs'+gs'/2)+Ztt' 7% 2 .2t2t°
gs tst' t t

+ABe}[J(J+1)-K2] + {X 7:3.(vs+gs/2)(vs.+qs./2)

Sié'

+2tt' *2 2 t t'+AAe }K2

te et' t t'

“BY5€“(4)ZPG P P P P enP

ZaBycSen B 6

JK KJ

1‘76

H 3J3(J+l)3 +Ho K2J2(J+l)2 +Ho K“J(J+1) +i1u

-K-
O

Zabcdefu‘“zabcdefqaqchqdqeqf

abcdef

+(4)Z P Hpbp PdPePf]

+Zabcdef[(4)z: bcd1/2(qa qch qdpepf+Pe Pfqa qch qd)

abcd

+(4)ze1/2(papbpcpdqeqf+qeqfpapbpcpd)1

ab
+ZgEb[(4)Zabqaqb+(4)Z papb]

233.3: yss.su(vs+gs/2)(vs.+gS./2)(vsu+gSn/2)
s£s' s"

+ Xstt' 'ysz 2 .(vs+gs/2)£t£t' + EsAws(Vs+gs/2)
tst' t t

13

Table III Symmetric T0p Energy Expression through Fourth
Order

EVR =

{S ”s (v SS+9 /2) + {S Ams (v s+g 8/2) +

Z

3?; xss'(vs +98 /2)(VS '+98 '/2) + Xtt' x2 2 +

tst- t t'
's"(vs+gs/2)(vs'+gs'/2)(vs"+gs"/2) +

ILt’Lt'
Z I n Y
g§s$ssw 88

XEEE: Ys£t2t'£t£tl(vs+gS/2) +

_ 2
AeK + Be[J(J+1) K ] +

[-ZAeXt Ctzzt + 2t{"t + 2s nt,s(vs+gs/2)} 2t] K +

DJK K

- DeJJ2(J+1)2 - K2J(J+1) - De K” +

- is asA(vs+gs/2)K2 - is asB(vs+gs/2)[J(J+1) -K2] +

K 3
nt 2 K +

zJ(J+1)K + it t

2t "tJ t

23 BSJ(Vs+gs/2)J2(J+1)2 + {s 3:K(vs+gs/2)K2J(J+l) +

23 38K(vs+gS/2)K“ +
2
[Egg;. Yss'("s +93 /2)(Vs '+9s '/2) + Ztgé. thzt.“t“t'] K +
[222; yss.(vs+gs/2)(VS.+gsu/2)
+ . y 2 z .][J(J+1) - K2] +
{Egt' 2t2t' t t

JK KJ K

Ho JJ3(J+1)3 + H0 K2J2(J+1)2 + H0 K“J(J+1) + H0 K5

Table IV

Constants involved
in energy terms

14

Classical Interpretation of Energy Terms

Classical interpretation of
energy term

 

s
Aws
x ., x
as 2tzt.
Y u at Y
88 8 sltlt.
Ae’ Be
2
Aext {t
nt' nt,s
J JR K
De ’ De ' De
A B
a , a
s s
K
nt , nt

Y er
“5 “t“t'

Y I: Y
38 Ltl
J JK KJ K

harmonic oscillator energy

fourth-order correction to harmonic
oscillator energy having same
quantum dependence

first-anharmonic corrections

second-anharmonic corrections.

rigid rotor energies of molecule in
equilibrium configuration

Coriolis term - first-order term
representing vibration-rotation
interaction

third-order vibration-rotation
correction to Coriolis term

centrifugal distortion corrections
to rigid rotor energies

corrections to A and Be in excited'
vibrational stat s

third-order vibration-rotation
interaction terms

corrections to D J, DJK, and DeK in
excited vibratiofial sgates

fourth-order corrections to “3A

fourth-order corrections to as8

fourth-order centrifugal distortion
corrections

15

II. Frequency Expression

 

In our work we are solely concerned with transi-
tions in absorption which take the molecule from a level
within the rotational fine structure superimposed upon the
ground vibrational state (all v3 = 0) to a level within the
rotational fine structure superimposed upon an upper vibra-
tional state (one or more vs # 0). To obtain the frequency
expression representing a general transition of this sort,
one subtracts the energy expression representing the ground
state from that representing the upper state. It is
desireable to have an expression general enough to represent
all possible transitions between the rotational fine struc-
ture levels superimposed upon the ground and upper vibra-
tional states. In the ground vibrational state v3 = 0 for

all "s"; 2 = 0 for all values of "t” since the ground state

t
is non-degenerate; J and K represent the ground state
quantum numbers associated with the total angular momentum
and its component along the symmetry axis respectively. In
the upper vibrational state one or more of the v3 are non-

zero; the 2 corresponding to those vB which are non-zero

t
for degenerate modes are themselves non—zero; J + AJ and
K + AK represent the upper state values of the J and K
quantum numbers.

Table V lists the generalized frequency expres-
sion representing a general vibration-rotation transition

from the ground state to any upper vibrational state,

assuming negligible inversion probability and the absence of

16

Table V Amat-Nielsen Generalized Frequency Expression

AK
(vn'vn+l"°"Vt'A£t’vt+l'A2t+l"") AJK(J)

(w + Aw )v +
S S

s s
2ggg:xss'[(vs+gs/2)(Vs'+gsv/2)‘gsgsu/4] +
2 IX A2 A2 . +

EEEv ltzt' t t

X§§§:§§=Y33.3nl(vs+93/2)(vs.+gs./2)(vsn+gSn/2)-gsqsugsu/8] +

23E§§E:Y'£t2t'(v3+gS/2)AztA£t' +

Ao[(K+AK)2-K2] +

Bo[(J+AJ)(J+1+AJ)-J(J+l) — (K+AK)2+K2] +
[-2AeztctzA2t+2t{nt+zsnts(vs+gs/2)}A2t][K+AK] +
-DOJ[(J+AJ)2(J+1+AJ)2-J2(J+1)2] +
-DgK[(K+AK)2(J+AJ)(J+l+AJ)-K2J(J+1)] +
-DOK[(K+AK)“-K“] +

['ZSGSAV3+23§§:YS:I (vsvsl+vsgSl/2+Vslgs/2)
+2t2t'72 2 'AltAzt.][(K+AK)2] +
t‘t' t t

B

[‘2303 vs+zsis'ysg'(vsvs'+vsgs'/2+Vs'gs/2)
sss'

B _ 2
+{t2t.y£ 2 'AltAzt.][(J+AJ)(J+l+AJ) (K+AK) ] +
tét' t t

{tntJAzt[(K+AK)(J+AJ)(J+1+AJ)] +

{tntKA2t[(K+AK)3] +
'J 2 2

2333 vs[(J+AJ) (J+1+AJ) 1 +

{sengs[(K+AK)2(J+AJ)(J+1+AJ)] +

K
{333 vs[(K+AK)“] +

17

HOJ[(J+AJ)3(J+1+AJ)3-J3(J+l)3] +
HgK[(K+AK)2(J+AJ)2(J+1+AJ)2-K2J2(J+1)2] +
H§J[(K+AK)“(J+AJ)(J+1+AJ)-K“J(J+1)] +

HOK[(K+AK)5-K6]

18

any accidental resonances. With the proper selection rules
this expression should represent vibration-rotation spectra,
Raman spectra, electric field-induced spectra, microwave

spectra, etc. Note the following substitutions:

A
A0 = Ae - 23 as (gs/2) + X§§;.Ys§-(9sgs'/4>r
_ __ B B
Bo _ Be {a as (gs/2) + 235;.Yss'(gsgs'/4)'
D: = D: - {s Bsm(gs/2), m = J, JR, K,

which were made by way of grouping together all the terms
which have exactly the same quantum dependences.

A diagram, taken from Ref. (1), of the observed
frequencies of the various methyl halide fundamentals is
shown in Fig. 2. The band 2v5 is also shown as a dashed
line because this band is in Fermi resonance with v1. The
unperturbed position of v1 would be between the indicated
positions of 2v5 and v1.

The atomic motions involved in the normal vibra-
tions of a CH3X molecule are indicated in Fig. 3. This
diagram is also taken from Ref. (1).

Schematically, one may consider that superimposed
upon the vibrational energy levels are rotational energy
levels - rigid rotor levels corrected by higher order
effects. Splendid diagrams are given on p. 28 of Ref. (1)
for non-degenerate states and on p. 404 of Ref. (1) for
degenerate states split by the Coriolis interaction.

A vibration-rotation band may be represented

schematically as the set of allowed transitions from the

l9

ooom ocmm ooum coma

 

m

9 a)

 

 

 

 

 

 

 

 

 

 

 

_———-k

 

mmfiocmsvmum sowumunfl> Housmespssh meadow

 

 

 

 

 

 

EU
com a:
-
m>
Hmmo
ummmo
Hommo
ammo

 

Harps: m .mss

 

Fig. 3 CH3X Normal Modes

9»

 

V5

20

 

 

V2

21

rotational levels within the ground vibrational state up to
rotational levels within the upper vibrational state. A
parallel band consists of transitions from a non-degenerate
ground state to a non-degenerate upper state; a perpendicu-
lar band consists of transitions from a non-degenerate
ground state to a degenerate upper state.

Infrared vibration-rotation spectra are described
by the electric dipole selection rules on Avs, Amt, AJ, and
AK. For a pure harmonic oscillator the allowed electric
dipole selection rule on the vibrational quantum number is
Avs = *l. The presence of electrical or mechanical an-
harmonicity permits, in general, Avs = *l, *2, *3, ..., but
with greatly reduced intensity relative to the fundamentals.
Of course, for transitions in absorption only positive Av8
exist. The selection rule on Azt for a particular band can
be obtained from Amat's Rule (2). Discussion of this will
be deferred until Chapter III. We shall merely note here

that for a parallel band A2 = 0, and for a perpendicular

t
band A2 f 0 in general. The dipole selection rules on AJ
and AK are:

Parallel band

AK = 0

AJ = 0, *1 (AJ # 0 when J = 0);

Perpendicular band

AK = *1

AJ = 0' t1.

CHAPTER III

SINGLE-BAND FREQUENCY EXPRESSION

Let us consider the analysis of a single band of
a methyl halide molecule. Analyzing a band consists of
determining the best set of estimators of the coefficients
involved in the frequency expression appropriate to that
band. These coefficients are molecular constants or linear
combinations of molecular constants. This can be done in
a relatively crude manner with graphs [see Ref. (§)], or in
a much more sophisticated and precise manner by means of a
least squares computer fit.

In general, the frequency expression listed in
Table V cannot be used directly in the analysis of a band.
A primary requirement in either a graphical or least squares
analysis is that all the terms in the frequency expression
be linearly independent of one another. This means that if
X1, X2, ..., Xn represent the quantum dependences of the
various terms, it must not be possible to represent any xi
in the form

X. = aX + bX

l l 2 i-l i+l
Consider the specific case of the v“ band of a

+ ... + ex + fx + ... + mxn.

methyl halide molecule. This represents a degenerate
carbon-hydrogen stretching mode. The methyl halide molecule
has six normal modes of vibration, of which three, v1, v2,
and v3, are non-degenerate, and three, vs, v5, and v5, are

degenerate. In the case of v“, Av“ = l and Avi = 0 for

i a l, 2, 3, 5, and 6. The AK and AJ selection rules are
22

23

those appropriate to a perpendicular band, viz.,
AK = *1,
AJ = 0, *l.
The selection rule on A24 is given by Amat's Rule
as described below. In general, Amat's Rule places restric-
tions on the allowed values of A24 such that

AK - {t Ant = *3p, p = o, 1, 2, ....

There is a restriction on the possible values of it in the

state vt,
I it I = vt, vt * 2, ...,

or, since Avt = vt and Alt = it for a transition originating

in the ground vibrational state,

I A”: = AV AV * 2’ .000

t I t’ t
Since Avg = l is the only non-zero Avs for v”, one has the
two conditions on Ann:
1. AK - AlA = *3p, p = 0, 1, 2, ...
2. I AM, | = 1, 3, ....
Since AK 8 *1, the only solution possible is
*l - A2. = 0 or
A2. = AK.

When A£A = AK is substituted into the frequency
expression obtained by specializing Table V to the v. band,
several terms are found to be linearly .dependent. In
particular, the Coriolis term splits into a constant term

plus a term with the same quantum dependence as A0, viz.,

['ZAeCuz + n. + 2nuul[A£u(K + AK)] =

24

[~2Ae;.,z + n. + 2n44][AK(K + AK)] =

z + 1/2 n. + nun][(AK)2]

z

{-AeCu
+ I-Ae§u + 1/2 nu + nun][(K + AK)2 - K2].
In the same manner the an term, with quantum dependence
[AEH(K + AK)3], is linearly dependent upon several other
terms and is treated in a similar manner, viz.,

nuKlA2u(K + AK)3] = n.K[AK(K + AK)3] =

nuKIAK(K3 + 3K2AK + 3KAK2 + AK3)] =

1/4 nsK[(K + AK)l+ - K“] same quantum dep. as -DoK
+ 3/2 nAK[(K + AK)2] same quantum dep. as -auA
- 1/2 nAK[(K + AK)2 - K2] same quantum dep. as A0
- 1/4 nuK same quantum dep. as v0.

The final frequency expression, suitable for a
least squares computer fit of v. (with k = l), is given in
Table VI. Note, in particular, that AC cannot be obtained
alone from such a single-band analysis. The sum (AO - Aecqz
+ 1/2ns + nun ‘ l/anK) is obtained as the coefficient of
[(K + AK)2 - K2]. The third-order n terms are probably
quite small, however the Coriolis coefficient, Aecuz, is
certainly not negligible compared to A0. Unless qu can be
estimated accurately by some other means, such as calculating
it theoretically, the value of A0 cannot be accurately deter-
mined. A similar thing happens in the case of several other
terms for which the estimator of the coefficient obtained
from a least squares fit represents a sum of several indi-

vidual molecular parameters.

It should be noted that the expression in Table VI

25

Table VI Frequency Expression Suitable for Single-Band
Fit Of Va or 2V”.

AKAJK(J) - {Bo[(J+AJ)(J+l+AJ)-J(J+l) - (K+AK)2+K2]

-DOJ[(J+AJ)2(J+1+AJ)2-J2(J+1)2]
-DgK[(K+AK)2(J+AJ)(J+l+AJ)-K2J(J+l)]} =
“0(Vu) or

Vo(2Vu i.) or as appropriate - [k(AK)2(AecAz-1/4nuK)] +
“0(2Vu ll)
[A0 - kAecgz - 1/2knuK][(K+AK)2-K2] +

K - 1/4knAK][(K+AK)~-K~] +

[-D
{-a:A + 3/2knuK][(Avu)(K+AK)2] +
[-a.Bl[(Av.){(J+AJ)(J+1+AJ) - (K+AK)2}] +
[n.J][(kAK)(K+AK)(J+AJ)(J+1+AJ)] +
[8.J][(Av.)(J+AJ)2(J+1+AJ)2] +
[BgK][(Avu)(K+AK)2(J+AJ)(J+1+AJ)] +
[3.K1[(Av.)(K+AK)“1 +
[HOJ][(J+AJ)3(J+1+AJ)3-J3(J+l)3] +
[HgK][(K+AK)2(J+AJ)2(J+1+AJ)2-K2J2(J+l)2] +
[HgJ][(K+AK)“(J+AJ)(J+1+AJ)-K“J(J+1)] +

[HOK1[(K+AK)6-K61

 

Set k = 1 for the vs band, k = -2 for the 2v“ band.

 

26

is only one of several equivalent frequency expressions
which could be written, even after the linear dependences
have been removed. It happens to be the one most convenient
for our purposes.
Consider now a single-band fit of the 2v” gang of

a methyl halide. In this case the selection rule on A2.
is obtained from Amat's Rule in the following manner. For
2v“, AVA = 2 and Avi = 0 for i = l, 2, 3, 5, and 6. The
two conditions on A2“ are

1. AK - A2” = *3p, p = O, 1, 2, ...,

2. |Az.| = o, 2, 4,

Two solutions are possible, a perpendicular component (AI

= *1):
*1 - A2. = *3 (p = 1) or
-A2q = 12,

and a parallel component (AK = 0):

0 - A24 = 0 (p = 0) or

A2. = 0.
Both of these can be represented as a general selection rule
for the 2v“ band:

A2. = -2AK.

In the same manner as for v4, the Coriolis term
splits into a constant term plus a term with the same
quantum dependence as A0, viz.,

['ZAeCuz + n. + 3n..][AzA(K + AK)] =
{-ZAean + n. + 3ngn][—2AK(K + AK)] =

[2AeCuz - nu - 3nuu][(AK)2]

27

+ [ZAeCuz - n. - 3n..1[<K + AK)2 - K21.

Similarly for nAK:

nuKlAlu(K + AK)3] = -2n.KIAK(K + AK)3] =

- 1/2 an[(K + AK)“ — K“] same quantum dep. as -DOK
- 3 nuK[(K + AK)2] same quantum dep. as -2aAA
+ an[(K + AK)2 - K2] same quantum dep. as A0

+ 1/2 nuK same quantum dep. as v0.

The frequency expression appropriate for a single-
band fit of 2v“ (with k = -2) is listed in Table VI. The
quantity Ao cannot be obtained alone, but rather only the

linear combination (A0 + 2Aecqz - nu - 3n“. + flu“)-

CHAPTER IV

SIMULTANEOUS FREQUENCY EXPRESSION

It was pointed out in Chapter III that for any

single band, the fact that Al is proportional to AK means

t
that the Coriolis term is necessarily linearly dependent
upon the A0 term, so that one can obtain from a least
squares fit only a numerical value for a linear combination
of the two coefficients. The key to obtaining accurate
values of the individual molecular constants, however, is
the fact that the constant of proportionality for overtone
bands is different from that for fundamentals, viz., ARA

= AK for v4 and A1“ = -2AK for 2v“. If the transition
frequencies of VA and 2V. are fit simultaneously to an
expression general enough to represent both bands, the fact
that Air takes on different values for the two bands intro-
duces an extra variable into the quantum dependence of the
Coriolis term. In other words, the term [(K + AK)2 - K2] is
now linearly independent of [(Azu)(K + AK)]. A least
squares simultaneous fit of the data of the two bands will
yield individual values of A0 and the Coriolis coefficient.
Similarly, nuK is also linearly independent of the other
terms in this case.

Table VII gives a frequency expression suitazle
for a simultaneous least squares fit of v. and 2v“. It is
assumed that values of AVA and A2. will be input for eacn
transition, along with AK, AJ, K, J, the frequency of tde

transition, and the weight assigned to it. Again, it Should

28

29

Table VII Frequency Expression for Simultaneous Fit of
v“ and 2v“.

AKAJK(J) - {Bo[(J+AJ)(J+l+AJ)-J(J+1)-(K+AK)2+K2]

-DOJ[(J+AJ)2(J+1+AJ)2-J2(J+l)2]

-DgK[(K+AK)2(J+AJ)(J+1+AJ)-KZJ(J+1)]} =

vo(v4) or‘
“0(2Vu i) or as appropriate +
“0(2vull)

[Ao][(K+AK)2-K2] +
{-2Aec.z+n.][(Az.)(K+AK)] +
[-DOK][(K+AK)“-K“] +
{-auA][(AVn)(K+AK)2] +
[n.K1[(A2A><K+AK)31 +
[BAK1[(Av.>(K+AK)“1 +
[HOK][(K+AK)5-K5] +
[-agB][(Avu){(J+AJ)(J+1+AJ)-(K+AK)2}] +
[nAJ][(A2.)(K+AK)(J+AJ)(J+1+AJ)] +
[BAJ][(AVu)(J+AJ)2(J+l+AJ)2] +
[sflK][(Av,)(K+AK)2(J+AJ)(J+1+AJ)] +
[HOJ][(J+AJ)3(J+1+AJ)3-J3(J+l)3] +
[HgK][(K+AK)2(J+AJ)2(J+1+AJ)2-K2J2(J+1)2] +
[HEJ][(K+AK)“(J+AJ)(J+l+AJ)-K“J(J+l)] +
[nun][(Avu+l)(A2g)(K+AK)] +

[Yuf][(AVA)(AVA+2)(K+AK)2] +

[ A
qufin

[YAE][(AVu)(AVu+2){(J+AJ)(J+1+AJ)'(K+AK)2}] +

1((A2A)2(K+AK)2] +

[Ynfgkl[(Alu)2{(J+AJ)(J+1+AJ)-(K+AK)2}]

30

be noted that this is not the only possible form of a valid
expression. It happens to be the one most convenient for
us, however, and it is the one which was used in the analyses.
It may be noted that it would be also possirle to
obtain a value of AC and the Coriolis coefficient by analy—
zing vs and 2v“ individually and then combining the results.
From a least squares fit of v. to the formula of Table VI
(with k = 1) one obtains a numerical value for the quantity

2 + ...), and from a fit of 2v. to the formula

(Ao - AeCu
of Table V1 (with k = -2), a numerical value of (A0 + 2Aecuz
+ ...). Then if the n terms are neglected, one can solve
the two equations for values of A0 and Aean.
However, the method of simultaneously analyzing
VA and 2v“ is definitely superior to that of analyzing the
two bands individually and combining the results. First,
confidence intervals (statistical limits of accuracy; ex-
plained in detail later) are obtained for the individual
quantities Ab and (Aecuz - l/ZnA) from a simultaneous fit.
From individual fits of vs and 2v“, confidence intervals
are obtained for the quantities (Ao - Angz + ...) and
(A0 + 2Aecuz + ...) respectively. It is not at all apparent
how one goes about determining from these the confidence
intervals on the individual quantities. Secondly, the
simultaneous analysis method is superior because of the
mathematical nature of the least squares fitting process.

A least squares fit will obtain the best possible fit of the

given data to the given equation. The ”best" fit is defined

31

as that fit (set of estimators of the coefficients) for which
the weighted sum of the squares of the deviations, (vobs -

vcalc)’ is a minimum. This does not necessarily ensure the
"physically best" fit, however. If the data is less than
perfect, the fit may yield a biased set of estimators. The
advantage of the simultaneous fit is that the computer is
forced to select the one set of estimators of the coeffi-
cients which best represents both bands. Single-band
analyses yield a set of estimators of the appropriate
coefficients for each band. In the case of one band being
perturbed, those estimators which are directly comparable
between the bands, e. g., agB, may be considerably different.
Such a discrepancy is obvious only for those coefficients
which are the same for both bands, but some or all of the
rest are likely to be adversely affected since the entire
set of estimators is adjusted in obtaining the "best" fit.
This is obviously a bad state of affairs since the molecular
parameters are constants of the molecule and not merely of
the band.

If both bands are relatively unperturbed a simul-
taneous fit is the best procedure because individual values
of A0 and (Aecgz - l/ZnA) are obtained, the single set of
estimators of the coefficients is obtained from about twice
the amount of data involved in a single—band fit, and
confidence intervals are obtained for the individual mole-
cular constants. The values of the molecular constants

thus obtained should be closer to the "true" values than

32

those of either hand alone. If one band is considerably
perturbed it may be desireable to determine the values of
some of the coefficients from a single-band fit of the good
band. These coefficients would then be held constant in the
simultaneous fit. A simultaneous fit of the good band plus
the ”unperturbed" parts of the poorer one should yield the
best estimate of A0 and the other parameters available froa
the data. If, say, vs were the good band and iv. the per-
turbed band, single-band fits would yield a very good value
of (Ao - Aecuz + ...) and probably a quite poor value of

(A0 + ZAGCAZ + ...). A value of A0 determined from these
would be rather untrustworthy. In a simultaneous fit, how-
ever, the data of the good band, which is predominant both
in quantity and statistical weight, is likely to "hold in
line" the unperturbed data of the poorer band and force it
to fit reasonably well. The value of Ao obtained from such
a fit, although somewhat uncertain in precision, is probably
the best that can be obtained from the given data.

All too often it happens that one of the bands is
quite badly perturbed. One must treat each case on its own
merits. If enough unperturbed lines of the poorer band can
be identified, a simultaneous fit can probably be made. The
results will be less precise than one would wish, but will
be of some value. If the band is too badly perturbed the
results will be so untrustworthy as to be nearly worthless.

If one is lucky a substitute band may be availaile

to replace the badly perturbed band. This was the case for

33

methyl fluoride, as described in Chapter IX. The v. band
appeared to be quite badly perturbed, while the 2vA band did
not seem too bad. Fortunately, the data from V3 + v. of
methyl fluoride was available from a recent thesis by W. E.
Blass (Z). The v3 + v. and 2v. bands could be fit simul-
taneously to obtain A0, Aecuz + ..., and the other molecular
constants. For future reference, the frequency expression
appropriate to this fit is listed in Table VIII. It is
generally not too difficult to obtain a substitute for a
perturbed vs. Any band of the type vn + v., where Vn rep-
resents a non-degenerate transition, will do nearly as well.
It is likely to be very difficult, however, to obtain a
substitute for a perturbed 2vg. A band of the type ”n + 2v“
would do quite well. However, such bands seem to be so weak
that it is a very difficult matter to obtain an acceptable
high-resolution spectrum.

There exist a few other methods by which values
of Ao have been or can be determined. Two important methods
are described in some detail in Appendix I. The first is
the zeta-sum method. This is the method by which nearly all
previous values of Ao have been estimated for the methyl
halides. The principles behind this method and its appli—
cation are discussed in Appendix Ia. A comparison of our
method with the zeta-sum method is presented below, since we
feel that our values represent a considerable improvement
over those determined from zeta-sums.

A second method which shows great promise is

34

Table VIII Frequency Expression for Simultaneous Fit of
V3 + V“ and 2v“.

AKAJK(J) - {Bo[(J+AJ)(J+1+AJ)—J(J+1)-(K+AK)2+K2]

-DoJ[(J+AJ)2(J+1+AJ)2-J2(J+l)21

-DgK[(K+AK)2(J+AJ)(J+l+AJ) -K2J(J+1)]} =

' v°(v3 + vs) or
“0(2V“ l) or as appropriate +
__vo(2v.||)

[Ao][(K+AK)2-K2] +

 

 

{-2Ae642+nul[(Alu)(K+AK)] +
{-DOK][(K+AK)”-K”] +

[nuK][(Alu)(K+AK)3] +

[HOK][(K+AK)5-K6] +
[n.J][(Az.)(K+AK)(J+AJ)(J+1+AJ)] +
[HOJ][(J+AJ)3(J+1+AJ)3-J3(J+l)3] +
[HgK][(K+AK)2(J+AJ)2(J+1+AJ)2-K2J2(J+l)2] +
[HEJ][(K+AK)”(J+AJ)(J+l+AJ)-K“J(J+l)] +
{(-agA) from 2v.) or

[ X[(Avu)(K+AK)2] +

{(-a3A-aAA) from v3+vuL

x[(Av.){(J+AJ)(J+1+AJ)-(K+Ax)2}] +

 

[[(-auB) from Zvu] or

B

{(-631'a98) from V3+VhL

[(BuJ) from 2vu] or
‘ J J x[(AVI.)(J+AJ)2(J+1+AJ)2] +
[(33 +Bu ) from v3+v4]

"[(BiK) from Zvu] or

LI<8JK JK

]X[(Avn)(K+AK)2(J+AJ)(J+1+AJ)] +
3 +8“ ) from.v3+vgl

F-[(BA,IK) from ZVQ] or

 

K K ]x[(Av.)(K+AK)“]-
L[(s3 +3. ) from v3+vu]

35

Raman spectroscopy. As shown in Appendix Ib, it is possible
to determine Ao directly from a full Raman spectrum of a
symmetric top molecule. For the Raman v4 band the permitted
selection rules are AK = *l, *2, AJ = 0, *l, *2. The A2.
selection rules are found to be A2. = AK for transitions
with AK = *l, and A2“ = -l/2 AK for transitions with AK = *2.
Because of this fact, the Coriolis term is linearly inde-
pendent of the Ao term, and the coefficients can be obtained
individually in the same manner as for the simultaneous fit.
' In fact, the frequency formula of Table VII should apply

to such a Raman analysis with the exception that only one
vibrational constant (v0) is obtained. With the advent of
the laser as a source, Raman spectroscopy has received new
life and has the potential of eventually surpassing infrared
spectroscopy in many areas.

A third method of determining A0 in certain very
special cases is that applied by Maki and Hexter (g). They~
obtained an estimate of A0 for CH31 from a study of the
Coriolis resonance between the K = 4, -2 levels of v3 + v5
and the K = 3, +2 levels of v5. These bands were already
known to be in Fermi resonance. This method is obviously
of limited applicability and is probably of limited accu-
racy.

In the following section the zeta-sum method is
compared with our method. The methods and variants listed
below are ordered, in our opinion, from the least accurate

to the most accurate presently available.

36

1. Application of the zeta-sum rule to Q-branch analyses
of v2, v5, and v5 using Q-branch maxima (most previous
values of Ao seem to have been determined in tLiS
manner).

2. Application of the zeta-sum rule to Q-oranch analyses
of v2, v5, and v5 using the leading edges of the Q—
branches.

3. Application of the zeta-sum rule to Q-branch analyses
of v2, v5, and v5 using subband origins.

4. Application of the zeta-sum rule after full single-hand
analyses of the rotational fine structure of V“, v5,
and v5.

5. Solution for AC from numerical values of (Ao - Aecuz
+ ...) and (A0 + 2Aec.z + ...) obtained from Q-branch
analysis of v. and 2v“.

6. Solution for AC from numerical values of (A0 - AeChz

z
+ 000) and (A0 + ZAeCh

+ ...) obtained from single—
band fits of resolved rotational fine structure of v.
and 2v“.

7. Determination of AC from simultaneous fit of resolved
rotational fine structure of v“ and 2v“.

A Q-branch analysis means a fit of tne observed
frequencies of the wide unresolved Q-branches to the
formula [see Ref. (5)]

vo(QK) = [vo + A'(1-2C) - B'] i 2[A'(1-C) - B']K

+ [(A' - B') - (A' - 3")]K2.

This formula, as taken from Ref. (2), is written in the

37

older notation of A” and 8' representing the lower state
constants and A' and B' the upper state constants. However,
an equivalent expression can be easily obtained by speciali-
zing Table V to the desired band with AK = *1, AJ = O, and

J = constant.

In the past, most Q—branch analyses seem to have
been done using the maxima of the Q-branches. Since the
Q-branches are wide (ml cm-l) and sometimes irregularly
shaped, determination of the positions of the maxima is a
process of somewhat limited accuracy. Any perturbations
present may give anomalous intensity distributions in the
Q-branches or shift the maxima. Furthermore, even in the
ideal case, the maxima of the Q-branches occur at different
values of J for different Q-branches, whereas the formula
was set up for constant J.

A slightly better procedure would be to use the
sharp leading edges of the Q-branches. While the leading
edges still represent varying J-values, they are often
easier to measure and should suffer less from intensity
anomalies.

A Q-branch analysis should be done in the manner
described by Brown and Edwards (2). In this more refined
method, the true subband origins (for J a 0) are found by
graphing or fitting the RRK(J), RPK(J), PRK(J), and/or
PPK(J) lines subband by subband. Q-branch fits using these
subband origins should yield the best results available

from this sort of procedure.

38

If one has the rotational fine structure resolved
in the v2, v5, and V6 bands, however, it is rather pointless
to make a Q-branch analysis. If one has access to a good
sized computer it is much more fruitful to make single-band
frequency analyses of each band. Then the zeta-sum rule
can be applied to the results of these fits. Applied in
this manner, the zeta-sum rule should yield reasonably good
values of A0.

If large computer facilities are not available,
Q-branch fits of the subband origins for v. and 2v“ will
yield numerical values of (A0 - Aechz + ...) and (A0 + ZAecAz
+ ...) respectively. These can be solved for A0 and Aecuz.

The methods of obtaining A0 and Aecuz from com-
bined single-band fits and from simultaneous fits of v. and
2v“ have been discussed in detail in the first part of this
chapter. The advantages of using the simultaneous analysis

method have also been discussed in detail.

CHAPTER V

LEAST SQUARES ANALYSIS OF TEL LATA

The first part of this chapter contains a descrip-
tion of the mathematics involved in the least squares
method. This is taken mainly from Hildebrand's "Introduc-
tion to Numerical Analysis" (19).

Assume we have available a set of numerical values,
f(xi) 5 fi’ taken at various discrete values of variable x,
xi, over a particular region. Suppose we have reason to

believe that a function, y(x), of a chosen general series

 

form should closely approximate the "true" function, f(xi),
over this region. In general, y(x) will have the form
y(x) = Xk=3 ak ¢k(x).

where the ¢k(x) are (n+1) known, appropriately chosen
functions, linearly independent of one another, and the ak
are (n+1) constants which are to be determined. We wish
to obtain the set of constants, ak, which gives the best
possible agreement (according to a chosen criterion) between
y(x) and the set f(xi) over the given region.

Suppose we define the "deviation" or "residual"

at any point, xi, as
n

2:0

f(xi) "' Y(Xi) E f(xi) - z ak ¢K(xi).
The least squares criterion for the "best possible fit" is
that the weighted sum of the squares of the deviations

should be a minimum, viz.,

Zia-1: W(Xi) [f(xi) - zK=3 ak ¢K(xi)]2 = minimum,

where N is the number of sets of data.

39

40

This imposes the conditions
a N __ n 2-
.... { {i=1 w(xi)[f(xi) Xk=0 ak ¢k(xi)] } - 0.
= O, l, ..., n, or
{Nw<x)¢<x)tf(x)-2 na ¢<x>1=o
i=1 i r i i k=0 k k i '

These are the normal equations, (n+1) simultaneous linear

 

equations in the (n+1) unknown quantities a0, a1, ..., an.
The formation of the set of normal equations is
illustrated below for a very simple example. Suppose the
equation
y = A + Bx + C2
is thought to adequately represent a physical process for

which N sets of data, fi' have been taken, each with weight

W.

1, at points (xi, 21). It should be noted that variables

x and 2 can be quite general. For example, 2 might represent

x2, sin x, etc., or might represent a function of a different

2'
variable, such as 2'3, tan 2', e , etc.

In terms of the previous notation, y1 = y(xi,zi),

fi = f(xi,zi), ¢1 = 1, ¢2 = x, ¢3 = z. The set of normal

equations (three equations in three unknowns) is

N N N _ N
1° Azi=lwi + Bzi=lwixi + C2i=lwizi ‘ zi=1wifi'
N N 2 N _ N
2' A2i=lwlxi + Bzi=lwixi + C2i=lwixizi ’ Xi=lwixifi'
N N N 2 _ N
3' A2i=lwizi + Bli=lwixizi + Czi=1wizi ‘ Xi=lwizif"

Since xi, 2 fi' and wi are all known (observed) quantities,

it
N

the sums ii=lwixizi' etc. are known constant quantities.

The set of three equations in three unknowns, A, 2, and C,

can be solved for these quantities.

Suppose now that the frequency expression of

41

Table VII is to be fit by least squares. Identifying terms
with those in the previous definition, y = 2k=8 ak ¢k' one
has the set of terms listed in Table IX. In principle, the
normal equations are formed in the same manner as for the
simple case just illustrated. In practice, a computer is
necessary to handle the sheer mass of calculations.

In solving these normal equations for large num-
bers of coefficients a computer is even more necessary. In
addition, the direct method of substituting equations into
one another becomes so complicated and inefficient that the
more general and more powerful methods of numerical matrix
inversion must be used.

The set of normal equations,

n N N
2k=0 aklzi=1w(xi)¢r(xi)¢k(xi)1 = [Ziglw(xi)¢r(xi)f(xi)lr

with r = O, l, ..., n, can be represented as a matrix

equation
ss=2
where
M = ”M M 7

.— 00 M01 "'
M

On
10 M11 "' m1n

 

 

LMnO Mn1 "' M’nn_

rk = 11-? w<xi)¢r<xi)¢k(xi>, r,k = o, 1, ..., n,

= ‘.N 1

I2 3

 

 

Nr = Xi w(xi)¢r(xi)f(xi), r = 0, l, ..., n,

42

Table IX Normal Equation Terms

Coefficient Quantum Dependence

a0 = v0 ¢o = 1.0

a1 = A0 ¢1 = [(K+AK)2-K2]

a2 = l-ZAeCuz+nu] ¢2 = [(Azu)(K+AK)l

a3 = l-DOK] ¢3 = [(K+AK)“-K“]

a» = l-auA] ¢u = [<Av.)(K+AK>21

as = nuK ¢5 = [(Afiu)(K+AK)3]

as = BuK ¢6 = [(Av.)<K+AK)“1

a7 = HoK ¢7 = [(K+AK)6-K5]

a3 = [-a.B] ¢8 = [(Av.){(J+AJ)(J+1+AJ)
-(K+AK)2}1

a9 = n43 ¢9 = [(Alu)(J+AJ)(J+1+AJ)(K+AK)]

alo = er ¢1o = [(Avu)(J+AJ)2(J+l+AJ)2]

all = eflK ¢11 = [(Avu)(K+AK)2(J+AJ)(J+1+AJ)]

a12 = HOJ ¢12 = [(J+AJ)3(J+1+AJ)3-J3(J+1)3]

313 = HgK ¢13 = [(K+AK)2(J+AJ)2(J;I;AJ)2
-K J (J+1)2]

a1. = HEJ ,1, = [(K+Ax)u(J+AJ)(ggigfigll)]

f1 = AKAJKw) - {B°[(J+AJ)(J+l+AJ)-J(J+l)-(K+AK)2+K2]

-DOJ[(J+AJ)2(J+1+AJ)2-J2(J+l)2]

-DgK[(K+AK)2(J+AJ)(J+1+AJ)-K2J(J+l)]}

43

m
ore o

 

 

n)

n

The solution, obtained by inverting fl, is

5=Efi-
This gives the set of coefficients, ak, as a column vector,
A.

Many kinds of matrix inversion techniques are
available. Most computer installations will have library
routines. The various methods will not be discussed here.
Some may be more efficient than others in a given case, but
most will be suitable as long as they are mathematically
accurate.

The remainder of this chapter is concerned with
the application of least squares fitting methods to the data
for the v“ and Zvu bands of a methyl halide molecule. For
a simultaneous least squares fit of v. and 2v“ each set of
data, representing one transition, will consist of Av“ (= 1

for v“, 2 for 2v“), AK, AJ, K, J, v s (the observed fre-

ob
quency of the line), and w (the weight assigned to the line).
There will be as many sets of data as lines in the fit. We
shall assume that the frequencies of the transitions have
been measured and are known.

The least squares fits were performed on the

Michigan State University Control Data 3600 computer using

the program FALSTAF (Erequency Ahalysis by Eeast §quares To

44

All-powerful Eormula). We obtained the basic least squares
program from Dale Pimple of Sandia Corporation. It was
written by M. A. Efroyemson of Esso Research and Engineering
Company and is described in "Mathematical Methods for Digital
Computers“ (ll). Input and output sections were rewritten
to accept the data in a more convenient format and print out
the quantities of interest to us. The program was also
double-precisioned to handle large numbers with more accu-
racy. The FALSTAF program is listed in Appendix II.

The results of a typical least squares fit are

composed of three main parts: (1) the estimators of the
coefficients, (2) the predicted frequencies, and (3) the
various statistical quantities. As discussed previously,
the estimators of the coefficients represent the best values
(in a least squares sense) of the molecular constants, or
combinations of them, which are available from the data.
The calculated or predicted frequencies are obtained from
the coefficients just found. Other lines, not already in-
cluded in the data, can be calculated, then found in the
spectrum, and finally added to the data in a new fit.
Generally, after each fit a ”predicted spectrum” is calcu-
lated, in other words, the AK and AJ quantities are allowed
to take on their permitted values and all transition fre-
quencies are calculated on the basis of the previously deter-
mined coefficients up to specified maximum values of K and J.

From our standpoint, the most important statis-

tical quantities are the standard deviation of the fit and

45

the simultaneous confidence intervals on the coefficients.
If there are N observations (line frequencies in the data
list) and p coefficients to be determined, the weighted
standard deviation of the fit is
- _ N 2 1n
5 - [N/(N P)Xi=1 wi(Avi) ] .

where Avi = (v )i - (vcalc)i' The main value of this

obs
statistical quantity is that it serves as a criterion of
“goodness" of the fit and as a comparison with other fits.
If Mii-l is the diagonal element, corresponding
to the coefficient ai, of the inverse of the normal equation
matrix, then the standard error associated with ai is

w(ai) = sZMii'l.

The simultaneous confidence interval corresponding
to ai is defined as [tSw(ai)], where S = pFa(p, N-p), and
the Fa(p, N-p) are tabulated values of the F-distribution
(12, 13). The value of the coefficient and its simultaneous
confidence interval are then written [ai t S¢(ai)]. The
interpretation of the simultaneous confidence interval is
that if the data in the fit is considered to be one sampling
of a normally-distributed infinite population of similar
sets of data, then the probability that all of the quantities

[a1 1 S¢(ai)] will include the "true" value of ai is (1 - a).

CHAPTER VI

METHODS OF OBTAINING AND TREATING SPECTRA

The spectra of the methyl halides and related mole-
cules were obtained on the Michigan State University high—
resolution infrared vacuum recording spectrometer. This
spectrometer, which employs an f/5 Pfund-Littrow monochro-
mator of focal length approximately one meter, has been des-
cribed in detail elsewhere (12, l§!.£§)‘

The source generally used for the 2V4 spectra near

6000 cm“1

was a commercial 300w zirconium lamp. These lamps
could not be used in the v“ region near 3000 cm"1 because
the glass envelOpe cut off about half of the energy. A new
zirconium arc source was built, having a water-cooled brass
housing and a sapphire window. It used the electrodes taken
from old 300w zirconium arcs. All v“ spectra and some of
the 2v” spectra were obtained using this source.

The foreoptics leading into the spectrometer con-
sisted of an all-mirror system. The infrared energy from
the source (along with the visible radiation) was sent
through an 80 cm long multiple traverse cell containing the
gas under study. The number of traversals could be varied
so as to optimize conditions among pressure, path length,
and output energy. The spectrometer contained two Bausch
and Lomb “certified precision" eschelette gratings, a 600
line/mm 212mmx158mm grating blazed at 1.6u, and a 300 line/
mm 254mle28mm grating blazed at 50. These were mounted

back-to-back on a worm gear-driven turntable and could be

46

47

switched with relative ease. The spectrometer was normally
used in "single-pass" configuration, in which the light
struck the grating only once. It could be adjusted, however,
to be used in "double-pass" configuration in which the light
diffracted from the grating was intercepted and sent back
to the grating for a second diffraction. This greatly in-
creased the resolution, but reduced the available energy,
especially for the calibration "fringes”. Thus, double-
passing was useful only in certain regions of the spectrum.
The infrared output energy was measured with
Eastman-Kodak lead sulphide detectors, P-type for v“ and
N-type for 2v“. These were cooled to -l96°C by means of a
liquid nitrogen bath. The infrared beam was chopped at 90
cycles/sec. The lead sulphide detector output was amplified
by a Tektronix Type 122 preamplifier followed by a phase-
sensitive amplifier. The amplifier output was recorded as
one trace on a Leeds and Northrop Model G two—pen recorder.
The second pen of the recorder traced the ”frin—
ges." These were Edser-Butler bands of visible light in the
higher orders of the grating, detected with an RCA lPZl
or selected 931A photomultiplier. The fringe trace (lower
trace in Fig. 4) served as a frequency ”ruler" to calibrate
the spectrum since the fringes are equally spaced in fre-
quency. Our calibration methods have already been described
in "Wavelength Standards in the Infrared" (11) and will be
described in more detail below.

The pens were first aligned parallel to the ver-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

48

 

 

 

 

 

 

V‘I-(N

 

 

 

 

 

 

 

 

 

 

 

 

quHMEwm 0m owfiscaa

 

 

 

 

 

 

<f’IflAk/Ifi,

 

 

49

tical grid lines on the chart paper. At the beginning and
end of each chart it was necessary to run a "dynamic pen
separation." This consisted of traces obtained by imposing
the same fringe signal on both pens simultaneously. As
shown in Fig. 4, this gave two identical traces slightly
displaced from each other. The displacement or "pen separa-
tion" resulted from the necessity of displacing the recorder
pens slightly to allow them to cross without colliding.

The displacement was a few millimeters, but could depend on
how the pens were adjusted for a particular run. The pen
separation was obtained by measuring the distance by which
the tOp trace was displaced relative to the lower trace.

It was important to obtain a dynamic pen separation (taken
while the chart was moving) since the pen separation was
generally different for a moving chart than for a stationary
one.

Aside from the pen separation sections, a typical
infrared trace consisted of a set of calibration lines, the
spectrum to be analyzed, and a second set of calibration
lines. An idealized trace is shown in Fig. 5. It is
assumed that throughout the entire chart the grating has
moved as smoothly as possible and that the fringe pen has
traced out a continuous series of fringes.

Each set of calibration lines consisted of a band,
or portion of a band, of a simple molecule. The frequencies
of the lines in these bands have been measured independently

to high accuracy [see Refs. (lg, 12, 32)]. These established

50

o m .
m # econ aoflumunaamo om om m m m m m

 

o0 HO «0 mo «0 mo H ¢ pawn coflumubwamo

 

 

 

 

 

 

 

ssuuomdm omuflammoH m .mfls

51

a set of known frequencies at each end of the chart.

The charts on which the high-resolution spectra
were recorded could be as long as 40 yards. This encompassed
a few hundred cm-l. The relative scale was generally 2 - 4
inches/cm.1 along the chart. The spectra were photographed
in 15 inch sections using a Nikkon F camera with special
minimum distortion copying lens. The camera was mounted
very rigidly on a specially constructed copying stand. The
paper was carefully aligned parallel to the plane of the
film.

The developed film was measured on a Hydel semi-
automatic digitized film reader connected to an IBM 526
printing summary card punch. A schematic diagram of the
Hydel optical system is shown in Fig. 6. An image of the
film, held flat by vacuum on the projection stage, was pro-
jected onto a large (24"x24”) ground-glass viewing screen.

A fixed image of a reticle was simultaneously projected onto
the center of the viewing screen. The image of the film was
moved about relative to the reticle image by means of hori-
zontal and vertical traverse controls. These traverse con-
trols were connected to the projection stage with precisely
machined and calibrated screws. The operator aligned the
reticle image with a desired point on the film image and
punched a readout button. This caused the (x, y) coordinates
of that point, relative to an internal coordinate system, to
be punched out on a computer card.

The Hydel measurement of a spectrum proceded as

52

 

 

 

 

 

 

 

 

   

Houufifi

-..?

Hamcmncoo Houcmawmu
mesa : fluomnoum

---‘----

maowumn I1 Edam

mama "
Enos

7 \ . .

n

-

g

.‘

.

.

.

-

-

summon | | | | “3.."qu
mcwsmfiy l. \ b \\ ....

g mowuso amp? m .3...“

53

follows: Each frame on the film covered about 15" of chart.
The film was measured one frame at a time. For each frame,
two cards were always present. The first was a ”parameter
card" which contained the frame number, the fringe number
of the leftmost fringe in the frame (or the fringe which
was to be the first one measured), and a number identifying
the operator. The second card in each frame was a "rotation
card.” This consisted of six measurements along the image
of a vertical grid line on the chart. This was used to
correct for any rotation of the chart image relative to the
Hydel axis system. Normally the first and last frames on
each film strip were of the dynamic pen separations. For
these frames, in addition to the parameter card and rotation
card, measurements were made of the fringe traces, alterna-
ting between the upper and lower traces (see Fig. 4). These
were called "pen separation cards." All the other frames
contained portions of the infrared spectrum and fringe
traces. For these frames, in addition to the parameter and
rotation cards, the fringes were first measured in sequence,
starting with the fringe specified on the parameter card.
As many cards as necessary were used for these. Next the
infrared lines in the frame were measured, with no require-
ments on which ones were measured or in what order they were
measured. Again, as many cards as necessary were used.

The raw data was converted into useable form by
means of the computer program SCAN, listed in Appendix III.

A typical, though short, data deck is included. The numbers

54

at the far right are the codes: 1 - parameter card, 2 -
rotation card, 3 - pen separation card, 4 — calibration card
(explained later), 5 - fringe card, and 6 - infrared card.
The data deck, as fed into the first run of SCAJ, consisted
of the heading or identification cards, two pen separation
sections (codes 1, 2, 3, 3, 3, ..., l, 2, 3, 3, 3, ...), and
as many spectrum sections (codes 1, 2, 5, 5, 5, ..., 6, 6,
6, ..., l, 2, 5, 5, 5, ..., 6, 6, 6, ...) as there were
frames containing sections of spectra.

The SCAN program first corrected each of the first
two pen separation frames for rotation and obtained the pen
separations by subtracting the corrected x-coordinates of
the lower fringes from the corresponding ones of the upper
trace. These were averaged for each frame, and finally a
grand average pen separation was obtained for the chart.

Then, for each successive frame the program cor-
rected for rotation, counted the fringes starting from the
one specified in the parameter card, corrected each infra-
red line for pen separation, and obtained the "fringe num-
ber" of each infrared line. The fringe number represented
the number of the nearest fringe on the left plus the frac-
tional distance from that fringe to the next fringe. These
fringe numbers of the infrared lines constituted the main
output of the first SCAN run.

With the fringe numbers and frequencies known for
the calibration lines, the chart could be calibrated. This

was done by making a linear least squares fit of the cali-

55

bration lines to the formula v = A + Bf, where v represents
the frequency and f the fringe number. A modification of
FALSTAF known as FITTUM was used to make the fit. This
yielded values for the coefficients A and B. Since the
fringes were equally spaced in frequency, the frequencies
of all the infrared lines in the spectrum could be easily
calculated.

The values of A and B were punched onto a "cali-
bration card" (code 4). This card was included in the data
deck in a second SCAN run. In this case, both the fringe
numbers and the frequencies of the lines in the spectrum

were output.

CHAPTER VI I

ANALYSIS OF c141 32R

Spectra of v“ of CH3Br, with band origin near

3050 cm"1

cm-1 to 3170 cm-1, were run on the 300 line/mm grating with

and with useable region extending from about 2970

the spectrometer in single-pass configuration. Two charts
of vs, 1065-Br and ll65-Br, were analyzed. They were cali-
brated with HCl (1-0) (lg) on the low-frequency side and

HCN (0,0,1) (12) on the high-frequency side. The standard

deviations of the calibration fits were both 0.004 cm-l.

The details of the experimental conditions under which the
spectra were run are given in Table X.

Spectra of the 2v“ band were run on the 600 line/
mm grating with the spectrometer in both single and double-

pass configurations. The parallel component of 2v. has its

band origin near 6045 cm-1 and extends from about 6020 cm-1

to 6070 cm-1. The perpendicular component has band origin

near 6095 cm-1. Its useable portion extends from 6080 cm-

1

1

to 6220 cm- . On the low frequency side of PQ2(J) it is
overlapped by the much stronger lines of the parallel com-
ponent. Three charts of Zvu, 0565-Br, 0665-Br, and l465-Br,
were analyzed. The calibration bands for all of these were
N20 (3,0,1) (22) on the low-frequency side and HCJ (0,0,2)
(12) on the high—frequency side. In addition, a number of
lines of the HCl (2—0) band (lg) were run on the low-fre-
quency side of the l465-Br chart. The standard deviations

of the calibration fits for all the charts were about 0.005

56

57

x~.o.ovzom

sob 1H.c.m.osz

e ¢.o as on zumbm cum MN .m.m cos loamoaom umummva saw
cum MN Am.o.ovzom

a s.m as OH zumbm 3 com .d.@ cos Aa.o.mvo~z umumsso .>~
OHM MN Am.o.ovzum

e m.a as OH zumbm 3 com .d.o oos Aa.o.meowz umumsmo saw
. sob AH.o.ovzom

e v.m as m mumbm sum as .d.m oom nonavaom “mummaa :>
sob Aa.o.o.zom

s v.m as m mumbm can an .d.m com .onavaom Hmsmmoa 3»

 

  

 

 

 

  

SE mmswa
onwumum

  

momma—

coaumunwamo

.mmmum uouomumo 00H90m_ Ho .Q.m unabo_ baub—

 

Anmmmuv mGOfluHUGOU asunmaaummxm x manna

58

cm-l. Details of the experimental conditions under which

the spectra were run are given in Table X.

In the CH3Br spectra it was a relatively simple
matter to assign a large number of transitions, i. e.,
identify the set of selection rules and quantum numbers
(AK, AJ, K, and J) which characterize each transition. The
"missing lines" in the subband series, complemented by the
subband intensity alternation, made the initial assignment
of many lines unambiguous. These two effects are discussed
below.

Qne very striking feature of the methyl bromide
spectra, and of the spectra of any methyl halide-type
molecule, is the fact that every third Q-branch is noticably
more intense than its neighbors. Theoretically the ratio
is 2:1. The RRK(J), etc. lines of these subbands are also
twice as intense as the corresponding lines of the neighbor-
ing subbands. This intensity alternation is due to the 63v
symmetry of the molecule [see King, p. 301 (11)] and makes
the subbands for which K a 0, 3, 6, 9, ... twice as intense
as their neighbors. The intensity alternation is quite
clear in the methyl bromide rapid scan spectra, Fig. 7. In
practice, the RQ0(J) Q-branch is not necessarily the most
intense, so that making assignments on this basis could lead
to a misassignment by 3 in K. Such a mistake might easily
be made in the case of vs in Fig. 7.

The initial assignments of many lines were made

with certainty on the basis of the "missing lines" in the

59

Fig. 7 CHaBr Survey Spectra - v“ and 20“

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'03 ‘00 I. a,
5‘
'0.
‘09
n
°12
3000 aI-l 3050 3100 3150
A A A A
p ‘00 3‘. l
2'0 H R06
R
' Q9
4
0050 c. 6100
, L 5150 6200
‘ n

 

60

subband series. These are due to the requirement that K be
less than or equal to J, in other words, the z-component of
the angular momentum cannot be greater than the total angular
momentum. For example, in the KAK = +3 subband the PR3(3)
line is the first line of the RR3(J) series which exists.
There is a distinct gap where the lines RR3(0 - 2) would
have been. Even in the rapid scan spectra in Fig. 7 the
gaps in the RR3(J), RR6(J), and RR9(J) subbands are quite
obvious. In Fig. 8 the R03(J) Q-branch and the first
several RR3(J) lines from the high-resolution spectrum of
the 20A band are shown.

The frequencies of all the spectral lines were
obtained in the manner described in Chapter VI. The
spectra were photographed, measured on the Hydel film reader,
and the raw data run through the SCAN program. This gave
fringe numbers for all the lines. The spectra were cali-
brated as described. The raw data from the Hydel was run
through SCAN again, this time with the calibration formula
included. This yielded an output listing the fringe number
and frequency of each line in the spectrum.

The resolution limits were about 0.04 cm-1 in the

1 in the 2v“ spectra. By our defini-

Vg spectra ond 0.06 cm-
tion, this was the minimum separation of two "useable" lines.
It generally ran slightly less than the true Rayleigh cri-
terion distance. Lines which had neighbors so close were

strongly downweighted in the fits.

After a considerable number of lines had been

61

 

62

assigned by inspection, further assignments were made using

ground state combination differences. These were calcu-

lated on the computer using the formula (1, 22)
G.S.C.D.(AJ1'AJ2’K'J) =

AKAJ1K(J-AJ1) - AKAJ2K(J-AJ2) =
_ 2 JK _ _ J 2 _ 2

(Bo K D0 )(A2 A1) + ( Do )(Az Al ),

where J is the quantum number of the common upper state and
A1 : (J-AJi)(J+l-AJi), i = l or 2.

Very accurate values of Bo = 0.3185537 cm'l, DOJ = 3.33x10'7

1 K = 4.27x10-6 cm‘l

cm_ , and D: , available from microwave
work (13), were used to calculate a table of ground state
combination differences.

Using the table of calculated ground state combina-
tion differences it was possible to assign RPK(J) lines when
the RRK(J) lines of the same subband series had already been
identified. Likewise, PRK(J) lines could be obtained when
PPK(J) lines had been assigned. In some cases it was
possible to use the leading edge of the Q-branches in order

to obtain the first few lines in an R

RK(J) or PPK(J) series
when the assignment of these lines was not immediately
obvious. The ground state combination differences were also
very useful for checking some series assigned by inspection.
If the RQK(J) line predicted from the observed first member
of the RRK(J) series fell on the leading edge of the Q-
branch and the rest formed a smooth progression through the

Q-branch, the assignment of the RRF(J) lines was checked

quite accurately.

63

An interesting but annoying feature of the CH3Br
spectra was a small rotational isotOpe effect. This arose
from the presence of two isotopic species of methyl bromide,
CH379Br and CHgalBr, present in almost equal abundance. The
observed isotope effect manifested itself as a gradual
broadening and eventual splitting of the lines in each sub-
band as J increased. The lines in a subband started out
initially sharp, became appreciably broadened around J = 12,
and were actually split into two components around J = 25.
This splitting is due to the difference in the values of Bo
for the two isotOpic species. There was no noticeable dif-
ference of vibrational band origins for the two species, due
apparently to the fact that the halogen atom participates
very little in the v“ vibrational mode. This could be an-
ticipated since the band origins of CHgI, CHaBr, and CE3C1
are not much different.

The annoying feature of this isotOpe effect was
that only the low-J lines of any subband were really sharp.
The most useful lines, those of J z 12 — 25, were broadened
and therefore much more difficult to measure accurately.

At high enough J that the two components of each doublet
were resolved, the intensities had fallen so low that the
individual lines were quite poor. Since there was no pos-
sibility of analyzing the spectra of the individual iso-

(av)Br. The

topic species, the analysis was made for CH3
frequencies of the centers of the broadened lines and the

average frequencies of the resolved doublets were used.

64

Also the microwave values of 30' DOJ, and 03“ for each

species were averaged to obtain the values for the average

molecule CH3(aV)

Br (values given previously).

Since all the lines in the spectra were not
equally good, weights had to be assigned to each transition
frequency. The weighting system used was based on our esti-
mate of how well each line could be measured relative to the
best lines in the spectrum. The very best lines were con-
sidered uncertain by an amount equal to the standard devia-
tion of the calibration fit, at a minimum. Other less per-
fect lines were in error by this amount plus any uncertainty
in determining the exact center of the line. If a line
happened to be broad, or irregular, or blended with another
line, there was considerable difficulty in measuring its
exact center.

Because of the manner in which the weights enter
the least squares fit, e. g., X. w.x.z. the weights were

i i i 1’
assigned according to the scheme

"1 a (Avi)2.
where Avi represents the total uncertainty in the measured
frequency of the line. The best lines were assigned weight
1.00. These were considered to have an inherent uncertainty
equal to the standard deviation of the calibration fit. A
line with estimated uncertainty twice that value received a
weight of 0.25.
As mentioned before, two charts of the v“ band and

three charts of the 2v4 band were measured. In many cases

65

the weights assigned to the same line from different charts
were different. The weighted average frequency was obtained
from the formula

“av = [21=1 wivil/[Zi=g "1]-
The average weight of the average frequency was

wav = [21=1 "ll/n‘

As a rule, the course of procedure in analyzing
the Spectra of vs and 2va was to first analyze the individual
bands as well as possible, and then combine the data of both
bands into a simultaneous fit. This step was bypassed in
the case of methyl bromide. The data of both bands were
good enough that an excellent simultaneous fit was obtained
immediately. The data of both bands were fit to the fre-
quency expression of Table VII. On the basis of the coef-
ficients determined from this fit, predicted spectra were
calculated for Va and 2v“. From these, new lines could be
found and perturbed series could be identified. The only
badly perturbed subband was KAK = +7 of 2v“. The frequencies
of the lines in this subband were simply left out of the fit.
Small, localized perturbations occured in a few subbands.
These were handled acceptably by simply leaving out several
lines on either side of the perturbed region.

The process of fitting, predicting new lines, and
refitting with those lines included in the fit was continued
until no more lines could be identified in the spectra. Then
a final fit was made to a frequency expression involving

only those terms whose coefficients were "significant" in

66

the next to last fit. Significance was defined as the value
of the coefficient being larger than its 95% simultaneous
confidence interval.

The results of the final simultaneous analysis of

vs and ZVA of CH3(av)

Br are given in Table XI. The esti-
mators of the coefficients obtained from the least squares

fit are listed, along with their 95% simultaneous confidence

J

intervals. Microwave values of Bo’ Do

, and DgK were input
as known quantities. The terms corresponding to these quan-
tities were subtracted from each transition frequency before
it was entered into the fit. The available data were appa-
rently not sufficient to allow the program to determine
values for the last five quantities listed in Table VII,
hence, these were taken to be zero. In any case, they were
expected to be extremely small. Values of the coefficients
BuJ, BiK, naJ, HOJ, HgK, HgJ were obtained in the next to
last fit, but were not significant according to the above
criterion. The terms corresponding to these coefficients
were left out of the final fit, hence, these coefficients
were also assumed to be zero. The standard deviation of the
final simultaneous fit was 0.006 cm-l. A list of the final
assignments, observed and calculated frequencies, deviations,
and assigned weights for CH3Br is given in Appendix IV.

The two bands were also fit individually in order
to compare the results with those from the simultaneous fit.

The results of the single-band fits for v4 and Zvu are shown

in Tables XII and XIII respectively. These fits were made

67

Table XI Coefficients of Simultaneous Fit of CH33r v“

 

and 2Vuo
Coefficient Value (cm-1) 95% s. c. i.
00(0.) 3056.35254097 0.00400511
vo(2v. 1) 6095.37929332 0.00876727
00(2v.||) 6046.12775517 0.00755994
A0 5.12908907 0.00097086
Aecaz - 1/2 n. 0.30493035 0.00070940
00K 0.00003663 0.00003277
«HA 0.02849233 0.00038622
flux -0.00008277 0.00003732
an -0.00018444 0.00000621
auK -0.00001623 0.00000931
HOK 0.00000011 0.00000009

All other coefficients were insignificant and were set = 0.

 

Bo * 0.3185537
DOJ * 0.000000333
02K * 0.0000042?

 

* Microwave values of 12CH3(aV)Br taken from Ref. (13).

 

fl, ——v—

Standard deviation of fit = 0.006 cm-l.

 

68

Table XII Coefficients of Single-Band Fit of CH33r v“.

 

Coefficient Value (cm-l) 95% s. c. i.
v0 3056.04303491 0.00305341
2

AC - ABC“ + l/an

+ nu. - 1/2m,K 4.82460782 0.00039907
a.A - 3/2n.K 0.02846680 0.00028663
00K - 1/4m,K 0.00009631 0.00001946
«.3 -0.00021128 0.00001523

All other coefficients were insignificant and were set = 0.

 

 

Bo * 0.3185537

DOJ * 0.000000333

02K * 0.00000427
(av)

* Microwave values of 12CH3 Br taken from Ref. (21).

 

Standard deviation of fit a 0.004 cm‘l.

 

69

Table XIII Coefficients of Single-Band Fit of CflaBr 2v“.

1

 

Coefficient value (cm- ) 95% s. c. 1.
v0 6095.98964858 0.01222403
n 2
A0 + éAeCu ' nu
- 3n., + “HK 5.74130602 0.01015969
2a.A + 3n.K 0.05817770 0.00638908
a.B -0.00017740 0.00003036

All other coefficients were insignificant and were set = C.

 

 

Bo * 0.3185537

DOJ * 0.000000333

03K * 0.00000427
(av)

* Microwave values of 12CH

...—

Br taken from Ref. (33).

Standard deviation of fit = 0.006 cm-l.

 

70

to the expression in Table VI with the appropriate value of
k included.

A value of A0 = 5.1291 . 0.0009 cm‘1 was obtained
from the simultaneous analysis of vu and 2vn of CH3(aV)Br.
For comparison, the combined results of the two single-band
fits yielded A0 = 5.131 t 0.010 cm-l. The simultaneous con-
fidence intervals indicate that this result is less accurate
than that obtained from the simultaneous fit by a factor of
more than ten.

Previous values of A0 for methyl bromide, obtained
by application of the zeta-sum rule (Appendix Ia) to
analyses using only the unresolved Q-branches of v8, v5,

and v5, are given by Herzberg (1) as 5.08 cm.1 and Burke

(22) as 5.126 cm-l. No estimates of accuracy are listed.
The value of the Coriolis term coefficient,
obtained from the simultaneous analysis, was [Aetuz - l/Znn]
a 0.3409 1 0.0007 cm-l. Under the approximations n4 2 0
and Ae . A0, then c.” = 0.0594.
The band origins are represented by the pure
vibrational terms
Vo(Vg) a (wu+Awu) + l/2x1. + l/2x2. + l/2x3. + 3x.“

+ xh5 + x36 + X + y-terms,

lulu
Vo(2Vq l) ' 2(wg+Amq) + x1“ + X2“ + X3“ + SKA“ + 2Xn5

+ 2Xg6 + 4x£ + y-terms,

«in
VO(2VQII) 3 2(Uh+Awu) + X1“ + X2“ + X3“ + Bxkk + 2Xg5
+ 2ng + y—terms.

Numerical values of these band origin constants were ob-

71

tained from the simultaneous least squares fit. When the
second-anharmonic y-terms were neglected, values of two of

the anharmonic terms could be extracted: qu = -20.977 cm'1

and xiulu = 12.313 cm-l.
The results of the methyl bromide analysis demon-
strate very clearly the usefulness of the method of simul-
taneous analysis in determining A0. It was very fortunate
that methyl bromide was the first molecule to be analyzed
in this manner. Subsequent work showed that either vs or
2v4, or both, were badly perturbed for all of the other
methyl halides. This made the analyses much more difficult
and the results much less trustworthy. None of the simul-

taneous analyses attempted for the other methyl halides

was nearly as satisfactory as that for methyl bromide.

CHAPTER VI I I

ANALYSIS OF €831

Spectra of v“ of CH31, with band origin near 3060
cm.1 and with useable region extending from about 3000 cm-1
to 3180 cm-1, were run on the 300 line/mm grating with the
spectrometer in single-pass configuration. Two charts of 0“,
0365-1 and 0465-1, were analyzed. They were calibrated with
HCl (l-0) (18) on the low-frequency side and HCN (0,0,1) (12)
on the high-frequency side. The standard deviations of the
calibration fits for both charts were 0.004 cm-l. The de-
tails of the experimental conditions under which the Spectra
were run are given in Table XIV.

Spectra of the 2vn band were run on the 600 line/
mm grating with the spectrometer in both single and double-
pass configurations. The parallel component of 20. has its

1 1

band origin near 6052 cm- and extends from about 6030 cm-

to 6080 cm-1. The perpendicular component has its band

origin near 6102 cm-1. Its useable region extends from

1 to 6220 cm-1. On the low frequency side,

about 6080 cm"
from about P02(J) and lower, it is overlapped by the much
stronger lines of the parallel component. Four charts of
Zvn, 0165-1, 0265-1, 0166-1, and 0266-1, were analyzed.) The
calibration bands for all of these were N20 (3,0,1) (32) on
the low-frequency side and HCN (0,0,2) (12) on the high-
frequency side. The standard deviations of the calibration
fits of all four bands were about 0.006 cm-l. Details of

the experimental conditions under which the spectra were

72

73

xon A~.o.ovzum

e m as om zumna out AN .d.n ooo AH.o.meoNz Huoomo Jam
son A~.o.ovzom

s m as om zumba one nu .d.m ooo AH.o.mvoNz Huooao saw
one AN Am.o.ovzom

s o.a as ea zumna 3 oom .d.o ooo Aa.o.mvo~z Humome Jam
out AN A~.o.ovzom

s o.a as ea zumnd 3 oom .d.n ooo Aa.o.meo~z Humoao 5>N
son AH.o.ovzom

e e.o as o samba oun AN .d.n oom Aouaeaom Humeeo :5
son Aa.o.ovzom

a e.o as o dumbd out AN .d.n oom nouavaom Humomo 5»

 

  

 

 

 

ES mmsfia mucus
mswusnm coflumuawaso— uumno_ vaun—

  

nu d0H H mo ,
sung .mmmum Houowumc wOHSOm Ho .m.m

AHmmov mcowuwpaoo Hmucmfiwuomxm >Hx wands

 

74

run are given in Table XIV.

Survey spectra of the Va and 20. bands of CH3I
are shown in Fig. 9. Even in these greatly compressed
spectra the resolved rotational fine structure is evident.
Perturbations are obvious in both bands. Note the anomalous
intensity of the R06(J) Q-branch of “hr and the split RQ5(J)
Q-branch in both bands. Such obvious perturbations serve
as a warning to proceed with caution in analyzing the bands.

In the high-resolution spectra of CH3I a great
many lines could be assigned by inspection. The presence
of perturbations made assignments in some subbands uncertain,
however. The RQl(J) through RQ7(J) Q-branches of Zvu were
obviously split. When the RRK(J) series could be identified
for these subbands, sharp discontinuities were found corres-
ponding to the split in the Q-branch.

The frequencies of all the lines in each of the
six charts were obtained as described in Chapter VI. The
line frequencies were weighted according to how well the
frequencies seemed to be determined. The frequencies from
all the charts of each band were combined into a weighted

1 in the

average. The resolution limits were about 0.04 cm-
vs band and 0.06 cm-1 in the 2v“ band.

Since both bands showed considerable evidence of
being perturbed, the most productive course of action seemed
to be to first analyze the bands individually. They were
then combined only after the best possible individual fits

had indicated which lines were the least perturbed.

75

Fig. 9 CH31 Survey Spectra - vn and 20A

 

 

 

 

 

 

 

 

 

 

 

’°3 “no “ l"3
’0‘
“0, 3°:
3000 an'1 3050 3:00 3150
’03 1no 2v. 1 .03
2" II
'0.

‘09

6050 en'1 6100 6150 6200

 

76

Microwave values of BO 2 0.2502167 cm"1 DoJ =

I
-1 K = 3.29Xl0‘5 cm"1 (22) were used to

2.09)<10"7 cm , and D:
calculate a set of ground state combination differences, as
described in Chapter VII. Many new lines were assigned with
the aid of this table.

The lines of each band which had been assigned with
reasonable certainty were fit to the single-band frequency
expression of Table VI with the appropriate k included.

OJ, and nix, for

Note that the terms corresponding to 80' D
which microwave values were available, were subtracted from
each transition frequency before it was included in the fit.
On the basis of coefficients obtained from these fits, pre-
dicted spectra were calculated. From these new lines were
assigned, included in the data for new fits, and the whole
process repeated until no further lines were found.

In practice, the process was considerably more
difficult than that indicated above, because of perturba-
tions. Subbands which were badly perturbed were left out of
the fit from the start, and were considered only after pre-
dicted spectra were available for comparison. Otherwise,
the usual procedure in handling localized perturbations was
to tentatively discard subbands or portions of subbands (not
individual lines) which did not seem to fit with the
majority of the other lines. Of course, there must be a
predominant population of unperturbed lines which fit well.

This procedure worked well for V0 of CH3I. Al-

though the KAK s -6, -5, +6, +7, and +8 subbands all seemed

77

to be somewhat perturbed and were left out of the final fit,
there remained 364 apparently unperturbed lines of the other
subbands. These were fit to the expresbion in Table VI.

The results are shown in Table XV. The standard deviation
of the fit was 0.005 cm-l.

The perpendicular component of 2V0 proved to be
impossible to treat in this manner. The entire P-side of
the band, P02(J) and lower, was overlapped by the parallel
component. No perpendicular band transitions could be
assigned with any degree of certainty in this region. A
few of the PP9(J) series should have been available, but
none could be identified since 01 + v“ also ran through this
region. In addition to the loss of nearly half of the band,
the subbands KAK = +1 through +7 were all perturbed to
varying degrees. R05(J) was very badly split and no RR5(J)
lines could be identified. In the subbands on either side
of R05(J) the Splitting could be observed to be moving
through the Q-branches. In those in which the RRK(J)
series could be traced over a considerable distance, sharp
discontinuities appeared in the line separations. Fig. 10
shows the RQ3(J) region of 20a in which the split Q-branch

and the corresponding shift in the R

R3(J) lines is clearly
shown.

A discussion of the perturbation in the 2V“ band
is given by Mme. Joffrin-Graffouillere and M. Nguyen Van
Thanh (35). They interpret the perturbation as a Fermi

resonance between the CH3I 2v“ perpendicular component and

78

Table XV Coefficients of Single-Band Fit of CH3I vn.

 

Coefficient Value 95% s. c. i.
v0 3069.75141674 0.00469999
2

A0 - AeCn + l/an

+ 044 - 1/2m.K 4.83319667 0.00137344
GuA - 3/2m.K 0.03056726 0.00117964
00K - 1/4n.K 0.00008539 0.00008007
6.3 -0.00012605 0.00001117
84K 0.00012004 0.00008937
HOK -0.00000247 0.00000134

All other coefficients were insignificant andwere set = 0.

 

so * 0.2502167
DOJ * 0.00000021
03K * 0.0000033

 

* Microwave values for 12CH31 taken from Ref. (32).

 

Standard deviation of fit = 0.010 cm-l.

 

79

 

80

the band v2 + V“ + 2V5.

On the basis of single-band fits to lines of sub-
bands not obviously perturbed, it was not possible to decide
which series, if any, were really unperturbed. There were
not enough lines to give any sort of useful values of the
coefficients.

Fortunately, a simultaneous fit of V4 and 2v“
proved to be feasible. From the perpendicular component of
204, 97 lines of the types RPO(J), RRO(J), RR8(J), and
RR9(J), together with 56 low-J lines of the 20. parallel com-
ponent, were found to fit very well in a simultaneous fit
with the 364 unperturbed lines of v4. The fit was made to
the frequency expression in Table VII. These series of 20.
were assumed to be relatively unperturbed because they fit
so well with the lines of v4. When the other series were
included they did not fit at all. The standard deviation
of the final simultaneous fit was 0.007 cm‘l.

The results of the simultaneous analysis of v. and
Zva of CH3I are listed in Table XVI. As for CHgBr, the data
were insufficient to determine the last five terms of Table

VII. These were assumed to be zero. values of the coef-

J J JR KJ
' Ho ’ Ho ’ Ho

next to last fit, but proved not significant. The final fit

ficients BAJ, BJK’ flu were determined in the

was made with these terms removed.

1 was obtained

A value of A0 = 5.134 . 0.003 cm-
from the simultaneous fit. The 95% simultaneous confidence

interval on this quantity has been listed. Because of the

81

 

Table XVI Coefficients of Simultaneous Fit of CH3I v“
and 28A.

Coefficient Value 95% s. c.
vo(v.) 3060.05691147 0.00493212
vo(2v. 1) 6101.88830686 0.01778452
vo(2v.||) 6052.03783417 0.01064939
Ao 5.13425925 0.00332773
Aer.” - 1/2 n. 0.30151603 0.00340549
DoK 0.00008500 0.00008000
a.A 0.03108683 0.00088588
nAK 0.00028459 0.00011305
8.3 -0.00012219 0.00000328
8.x 0.00008169 0.00003738
HOK -0.00000215 0.00000083

All other coefficients were insignificant and were set = C.

 

so * 0.2502167
DOJ * 0.00000021
ng * 0.0000033

 

* Microwave values for 12CH3I taken from Ref. (23).

 

Standard deviation of fit = 0.007 cm-l.

 

f

82

lack of good data for 2V4: this value of A0 is statistically
less accurate than that obtained for methyl bromide. It
may also be less trustworthy. Not only are there fewer
lines and fewer subbands of 2v. represented in the fit, but
there are probably also some slightly perturbed lines in-
cluded in the fit.

Previous values of A0 for CH3I, obtained by appli-
cation of the zeta-sum rule to Q-branch analyses of the three

degenerate fundamentals, are given by Herzberg (1) as 5.077

cm-l, by Burke (12) as 5.104 cm-l, and by Jones and Thompson
(26) as 5.119 cm-l. Maki and Hexter (8) obtained a value of
1

5.158 * 0.02 cm- from a study of the Coriolis resonance
between V3 + v5 and vs.

Under the approximations n. a 0 and A8 = A0, one
obtains from our results caz = 0.059. Jones and Thompson
(36) obtained exactly this same value.

In the same manner as described for CH33r, one

finds values for the two first-anharmonic consrants x“. 2

= 12.46 cm.-1

-1
-26.57 cm and x2428 .

CHAPTER IX

ANALYSIS OF CH3F

For methyl fluoride, AC was obtained from a simul-
taneous fit of V3 + V4 and 2v“ after vs proved to be too
badly perturbed. Details of how 0. and 2V“ were recorded
and analyzed are given below. Details of the analysis of
V3 + v. are given in a thesis by W. E. Blass (Z) and subse-
quent paper by Blass and Edwards (22).

Spectra of CH3F V4: with band origin near 3006

1 l

-1 to 3150 cm- , were

cm and extending from about 2940 cm-
run on the 300 line/mm grating with the spectrometer in
single-pass configuration. Two charts of vs, 0465-F and
0565-F, were measured and analyzed. They were calibrated
with HCl (l-O) (12) and HCN (0,0,1) (12). The standard
deviations of the calibration fits were 0.004 cm-l. Details
of the experimental conditions under which the spectra were
run are given in Table XVII.

Spectra of the Zvu band were run on the 600 line/
mm grating with the spectrometer in both single and double-
pass configurations. The perpendicular component had band

origin near 6000 cm-1. Its useable region extended from

1 to 6130 cm-1. No parallel component could

about 5940 cm-
be identified. Two charts of 2v», 0365-F and 0166-F, were
measured and analyzed. These were calibrated with HCl (2-0)
(12) and HCN (0,0,2) (12). The standard deviations of the
calibration fits were both 0.005 cm—l. Details of the

experimental conditions under which the spectra were run

83

84

non A~.o.e.zom

a m as mm zumom one AN .d.n eoo Aoumvaom muooao eem
cum MN .m.o.ovzom

e o.a as NH zumne 3 oom .n.o ooo loumeaom sumomo ea~
son AH.o.o.zom

e e.o es m.H dumba one as .d.m oom .onaeaom mumomo 5»
son AH.o.oezom

e e.o as N mumna one AN .d.n eon Aoaaeaom s-moeo .5

 

 

 

 

   

.m mo, .96

. mpamn
spam .mmmum .Houomu0p_ mousom_ no .m.m

coauswbwaso uusso_ wasn—

  

Asmmoo ndoauaodoo assessauomxm HH>x manna

85

are given in Table XVII.

The V3 + V4 band of CH3F was analyzed by W. L.
Blass as part of a thesis at Michigan State University. A
detailed description of this band, ground state combination
differences, single-band analyses, perturbations, etc. are
given in his thesis, along with a listing of the line assign-
ments and frequencies in this band. Many of these points
are also included in the subsequent paper by Blass and
Edwards (22).

Survey spectra of VA, 2Vu, and V3 + Va are shown
in Fig. 11. Even in these compressed spectra the resolved
rotational fine structure is evident. Figure 12 shows the
R06(J) region of the 2V4 band of CH3F. This is a splendid
example of a highly-resolved Q-branch. The individual tran-
sitions which make up the Q-branch are resolved and easily
measurable. This region is typical of our high-resolution
records. The resolution limits were =0.04 cm-1 for Vs and
=0.06 cm.1 for 2Vuo

The frequencies of the lines in the V4 and 2V.
bands were obtained as described in Chapter VI. The line
frequencies were weighted on the basis of how well the fre-
quencies seemed to be determined. The frequencies from both
charts of each band were combined into a weighted average
frequency.

It was our expectation that Va and 2Vu of C33?
would be fit simultaneously to obtain A0 and the other

molecular parameters. The bands were fit individually to

"oo .-

| MIN“,

”WW “"1“

”I 'i

WW1": WWW (WWW w

4‘”

 

.. A .1
WWMWWW M

 

 

87

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m; A s
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. I -..11 w . t.
r A .
.
.
.

.
_
_

. H A _

4H
w
A
1 nl l4. ...Iuv. I .11l. .lIl'lluollllllr .Illrlllltlu \ ll: lllnll Il+lfFilll1Tlil41|.1 ILIJII «II l-)vu 1.le Inl+lllllll .I‘Itll ...I .J.‘
O . o .
.

_.
.
A
.

A
. :3 2......

.W A . ....

1

|

f
...

I

 

 

 

 

 

 

 

 

 

 

 

 

 

L 7
h»;
.1!

h.--
5
c-
I)

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L.
rt...
as
3

 

A
~o.—_1p——b— —o—— A

 

 

 

 

 

‘ 4...- (.mL—y

+ 4—;HP—L—o—f4—a——
1
~.-«—*'--+ ...-
0.,
v
n
4..—

 

 

 

III I. 1.}

 

 

 

1.

 

 

\. |-I #11 . I‘l1( 'Illlil

 

 

 

 

 

 

 

 

 

 

 

 

 

..Jl -
L
T
_ _...._++--_
|
I
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l
—+--lo—.—IL—L—-y-<——o —4r -0 —9 1 4—4) ...—2-1.
i
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L4.—
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-> .. _ ..
l

. ...... ......t.6..hfim8eAse... a... ...: ..

|
’ f3

88

the expression listed in Table VI (with the appropriate
value of k included). As usual, in these and subsequent
fits, the terms involving Bo = 0.8517935 cm-l, DoJ = 2.015

x10"6 cm-l, and DgK - 1.4652xlO'"S cm-1

(21) were subtracted
from each line frequency before it was entered into the fit.
In the high-resolution spectra of 2V., two series,
KAK = -4 and -5 were obviously perturbed, having badly split
Q-branches. Single-band fits of 2Vq soon indicated, in
addition, that the KAK a +6 through +9 series did not fit
well with the rest of the band. Eventually 162 lines of
the 2Vq perpendicular component were established as being
apparently the least perturbed, although several obvious
biases still existed. A fairly good single-band fit (stan-
dard deviation 0.0l3 cm-l) was obtained for these lines.
The results are given in Table XVIII.

The Va band was obviously badly perturbed. The
strong perturbation on the R—side, which Pickworth and
Thompson (22) remarked upon in 1954, was quite obvious in
our high-resolution spectra. While the normal Q-branches
were wide and partly resolved, tailing off to the high
frequency side, the KAK = + 4 Q-branch was spread out over
a considerable distance and the KAK a +5 and higher
Q-branches appeared abnormally narrow. Lines from these
subbands did not fit at all with the rest of the band.

Even with these series eliminated, the rest of the band
fit very poorly. It was difficult to decide which lines,

if any could be reasonably called "unperturbed." However,

89

Table XVIII Coefficients of Single-Band Fit of CH3F Zvu.

 

Coefficient ,Value (cm-1) 95% s. c. i.
Vo 6001.37242480 0.01230728
2

A0 + 2AeCu - nu

- 3n.. - n.K 5.97016254 0.00304190
26.A - 3n.K -o.02275383 0.00188018
noK + 1/2n.K -0.00079118 0.00006587
6.3 -0.00112620 0.00004187
6.x 0.00003197 0.00002629

All other coefficients were insignificant and were set = 0.

 

‘fi_

so * 70.8517935
soJ * 0.000002015
03K * 0.000014652

 

* Microwave values for 12CH3F taken from Ref. (31).

 

Standard deviation of fit = 0.013 cm'l.

 

v

90

it was finally found that 92 lines of this band gave a
fairly good fit to the theoretical formula (standard devia-
tion 0.020 cm-l) and apparently a fairly well determined
set of coefficients. The results are listed in Table XIX.

In Spite of the fact that the V. and 2V. bands
fit fairly well individually, a simultaneous fit of the
"unperturbed" lines of both bands gave such a poorly deter-
mined set of coefficients and reproduced the data so poorly
as to be nearly worthless. No reasonable estimate of A0
and Aer.z could be obtained from this fit.

The situation was considerably improved by making
use of the "unperturbed" lines of the V3 + V. band, origi-
nally analyzed by Blass. The frequency expression to which
the data were fit is given in Table VIII.

The results of the simultaneous fit of 196 lines
of V3 + V. and 162 lines of 2V. are given in Table XX. In
view of the perturbed nature of both bands, the fit seemed

1

fairly good (standard deviation 0.015 cm- ). A value of

A0 = 5.104 t 0.002 cm"1 was obtained. For comparison,

previous values of A0 for CH3F, obtained from applications
of the zeta-sum rule to Q-branch analyses of the three
degenerate fundamentals, are listed by Herzberg (1) as
5.100 cm-l, Pickworth and Thompson (22) as 5.11 cm'l, and

Andersen, Bak, and Brodersen (22) as 5.095 cm-l. Smith and

1 obtained through private communi-

Mills (39) used 5.081 cm-
cation with Andersen, in their calculations.

Under the approximations n. 2 0 and Ae 2 A0 one

91

Table XIX Coefficients of Single-Band Fit of CH3F V..

 

Coefficient Value (cm-1) 95% s. c. i.
vo 3005.28088945 0.01638834
2

A0 - AeCu + 1/2n4

+ ... - 1/2n.K 4.64091916 0.00327381
a.A - 3/2n.K 0.01009577 0.00452974
6.3 -0.00144265 0.00012461
s.K 0.00023992 0.00023220

All other coefficients were insignificant and were set = 0.

 

1 v — i

so * 0.8517935
soJ 2 0.000002015
03K * 0.000014652

 

* Microwave values for 12CH3F taken from Ref. (21).

 

Standard deviation of fit = 0.017 cm-l.

 

w

Table XX Coefficients of Simultaneous Fit of CH3F V3 + V.
and 2"]...

Coefficient

92

1)

Value (cm-

95% s. c.

 

V°(V3+v.) 4057.65311000 0.01446841
VO(2v. l) 6000.50578000 0.01522262
Ao 5.10427102 0.00201388
Ae;.z - 1/2n. 0.43346464 0.00145493
DoK -0.00012201 0.00007953
n.K 0.00135226 0.00018699
03A+a.A 0.02386673 0.00150811
a.A -o.00942605 0.00111255
(63A) (0.033)

a3B+a.B 0.01083222 0.00012198
a.B -0.00112829 0.00005408
(agB) (0.0119)

83K+B.K 0.00002868 0.00002956
8.K 0.00002992 0.00003391
(83K) (-0.000001)

g3J+s.J -0.00000014 0.00000026
B.J 0.00000073 0.00000009
(BaJ) (-0.0000008)

3€K+afK -0.00000393 0.00000218
six 0.00004555 0.00000209
(05K) (-0.00005)

All other coefficients were insignificant and were set = 0.

 

Standard deviation of fit = 0.015 cm-l.

 

93

obtains c.2 2 0.085 from our results. Andersen, Bak, and
Brodersen (22) obtained (.2 2 0.093.

Our analyses must, however, be considered as, at
best, a qualified success. There exists one glaring discre-

1

pancy. Our value of a.A = -0.009 cm- , obtained essentially

from the data of 2V. alone, does not correspond to the value

c.A - +0,008 cm"1

listed by Andersen, Bak, and Brodersen,
and also by Pickworth and Thompson. Their results were
obtained from the V. band, and, in fact, are just the result

which was obtained when we fit the V. band alone. This

value of 0.A = +0.008 cm"l appears to be correct, and that
obtained from 2V. wrong, for the reasons outlined below.

From V3 + V. we obtained 03A+d.A a +0.024 cm-l.

Our value of a.A yielded G3A = +0.033 cm’l. Smith and Mills
(22) obtained dgA = +0.011 cm-1 from an analysis of V3, and
also 2V3. This seems to be correct, since the appearance
of the V3 band permits no other conclusion than 03A 2 033.
There seems to be no difficulty with 633 and 0.3; our results
were in good agreement with previous results (22, 22, 22).
Furthermore, when a.A = +0.008 cm-1 was used, the result
03A - +0.012 cm-1 was obtained from our results, in excellent
agreement with Smith and Mills.

Hence, none of our values of the coefficients are
to be trusted implicitly. There does exist some corrobora-
ting evidence, however, for at least the values of A0 and

Aec.z. Single-band fits of the V3 + V. and 2V. bands

yielded the values of the coefficients (Ao - Aer.z + ...) =

94

1

4.6717 . 0.0015 cm- and (Ao + 2Ae;.z + ...) = 5.970 2 0.003

cm- respectively. When the above quantities were calcu-
lated using the values of A0 and Aer.z from the simultaneous
fit of these two bands, the numerical results were identical
with those above within the confidence intervals. Since the
individual A0 and Aec.z values were obtained from the data
of both bands simultaneously, this is strong evidence that

the values of these two parameters, at least, are reasonably

correct.

CHAPTER X

ANALYSIS 01" CH 3CN

Methyl cyanide, CH3CN, is not one of the methyl

halides, but is a C axially symmetric molecule. The CN

3v
radical takes the place of the halogen atom, lying along the
symmetry axis above the apex of the CH3 radical. The fact
that CH3CN has six atoms rather than five means that there

is one more non-degenerate mode and one more degenerate

mode of vibration. Because of relabeling, the V5 mode of
CH3CN is the one in which the atomic motions are essentially
the same as in V. of the methyl halides. Indeed, this band
occurs at nearly the same frequency as the V. methyl halide
bands.

Only one Spectrum of V5 of CH3CN was analyzed,
0266-CN. It was run on the 300 line/mm grating with the
spectrometer in single-pass configuration. The band origin
was near 3009 cm-1. The spectrum was calibrated with HCl
(1-0) (12) and HCN (0,0,1) (12). The standard deviation
of the calibration fit was 0.004 cm-l. The experimental
conditions under which the spectra were run are given in
Table XXI.

Like V5, only one spectrum of 2V5 was run. The
reason for this was the extremely poor quality of the 2V5
spectrum. For some reason, 2V5 was extremely weak, much
weaker relative to V5 than any of the methyl halide 2V. bands

relative to their V. fundamentals. In order to obtain suf-

ficient absorption it was necessary to increase the pressure
95

96

 

E m 88 on

EC 58m

Zlmnm

mlmflm

 

a» ema 20 mo,
base .mmeum

nouomumbfl

zonooao mam

zouoomo m5

 

 

  

Mon A~.o.ovzum
one AN .d.n ooo .H.o.m.o~z
son Aa.o.o.zom
one MN .d.m com Aouavaum
ES mmsfla spawn

mounom_ Ho .m.u mcwusum dofiusunwaso

 

 

_ unmno— Masai

Azommo. nnosusosoo Headmasumdxm Hxx manna

97

so much that the spectrum was nearly ruined by pressure
broadening. The single 2V5 chart, 0166-CN. was run on the
600 line/mm grating with the spectrometer in single-pass
configuration. The band was calibrated with N20 (3,0,1)

(22) and HCN (0,0,2) (12). The standard deviation of the
calibration fit was 0.006 cm-1. Details of the experimental

conditions under which the Spectra were run are given in

Table XXI.

Survey spectra of the V5 and 2V5 bands of cnch
are shown in Fig. 13. Under high resolution the V5 band
was quite good, with the resolution limit about 0.04 cm-1.
Because of the extreme pressure broadening of lines in 2V5

the effective resolution in this spectrum is probably no
1

better than 20.2 cm- .
Assignment of lines in the two bands presented no
difficulty, although there were few identifiable lines
in the 2V5 spectrum. Apparently both bands were nearly
unperturbed. The P-side of V5 was badly overlapped, so
that most of the lines in this band were RRK(J) types. All
of the 2V5 lines were RRK(J) types.
The V5 band was first analyzed alone. The 125
lines assigned in V5 gave a quite good fit (standard devia-

1

tion 0.009 cm- ). The results are given in Table XXII.

Microwave values of so = 0.30684219 cm-1, ooJ

cm-1, and DgK 1

s 1.27x10'-8
- 5.901xlo-6 cm‘ (31) were input and held
constant. Of course, the few lines of the 2V5 band could

not be meaningfully fit alone.

98

Fig. 13 CH3CN Survey Spectra - V5 and 2V5

P 3
Q3 00 V 5

 

 

 

 

 

 

 

 

 

 

 

3000 c.‘1 3050 3100
A I 1

R00 2V5 l

295 ll

 

 

 

 

 

 

 

 

 

 

 

5950 cm' 6000 6050 6100

99

Table XXII Coefficients of Single-Band Fit of CH3CN V5.

1

 

Coefficient Value (cm- ) 95% s. c. i.
vo 3008.69693446 0.01493334
2

A0 ' AeCS + 1/2n5

+ .55 -1/2n5K 4.96070950 0.00667738
65A - 3/2n5K 0.03225829 0.00217414
653 0.00005397 0.00003211
85K -0.00002691 0.00001136

All other coefficients were insignificant and were set = 0.

 

fir

so * 0.30684219
DOJ * 0.0000000127
03K * 0.000005901

 

* Microwave values for 12CH3CN taken from Ref. (21).

 

‘—

Standard deviation of fit - 0.009 cm-1.

 

100

The results of the simultaneous fit of 125 lines
of V5 and 20 lines of 2V5 are given in Table XXIII. They

are quite poor, as would be expected under the circumstances,

although the standard deviation of the fit was 0.008 cm-1

There was simply not enough data from 2V5 to permit an

accurate calculation of A0. The value of A0 obtained from
-1

the simultaneous fit was 5.03 t 0.06 cm

101

Table XXIII Coefficients of Simultaneous Fit of CH3CN V5

 

and 2V5.
Coefficient Value (cm-1) 95% s. c. i.
VO(V5) 3009.11122113 0.01541944
vo(2V5 1) 6005.95747459 0.23498695
Ao 5.02644866 0.06439253
Aecsz + 1/2 .5 0.32828684 0.02075757
DoK 0.02382226 0.00162153
65A -0.l4282842 0.02742596
nsK 0.02476328 0.00146676
653 0.00007866 0.00001360
85K 0.00655623 0.00027944
HoK -0.00004544 0.00000185

All other coefficients were insignificant and were set = 0.

 

so * 0.30684219
soJ * 0.0000000127
03K 0.000005901

 

* Microwave values for 12CH3CN taken from Ref. (21).

 

Standard deviation of fit = 0.008 cm-l.

 

CHAPTER XI

ANALYSIS OF CH3C1

Spectra of V. of CH3Cl, with band center near 3044
cm-1 were run on the 300 line/mm grating with the Spectro-
meter in single-pass configuration. Two charts of V., 0465-
C1 and 0565-Cl, were measured and analyzed. They were cali-
brated with HCl (1-0) (12) and HCN (0,0,1) (12). The stan-
dard deviations of both calibration fits were 0.004 cn-1.
The experimental conditions under which the Spectra were run
are given in Table XXIV.

Spectra of the 2V. band were run on the 600 line/
mm grating with the spectrometer in both single and double-
pass configurations. The parallel component had its hand

origin near 6015 cm-1; the perpendicular component had its

band origin near 6065 cm-1. Like the corresponding bands

of CfiaBr and CH3I, its P-side was lost due to overlap of the
parallel component. Two charts were measured, 0365-Cl and
0166-Cl. They were calibrated with N20 (3,0,1) (22) and

HCN (0,0,2) (19) . The standard deviations of the calibration

fits were 0.006 cm-1. Details of the experimental conditions

under which the Spectra were run are given in Table XXIV.
Survey spectra of the V. and 2V. bands of CH3C1
are shown in Fig. 14. The resolved rotational fine struc-
ture is evident even in these rapid scan spectra.
Also evident, however, in even these highly com-
pressed spectra is the very pronounced perturbation in the

2V. band. Under high-resolution the effects of the per-

102

103

sob .~.o.o.zom

e e as om zumbe one he .d.m ooe .H.o.m.o~z Houoofio .em
can as A~.o.o.zom

e e.a as ea zumbe 3 com .d.o ooo AH.o.m.o~z Honmome .em
sob AH.o.ovzom

e e.o as m Alene one be .d.m oom .osaeaom Houmomo .5
sob AH.o.ovzom

e e.o as o eumba one AN .d.m oom .ouaeaom Honmoeo .5

 

 

    

ES embed
mswumum

spawn L .
newusunwaso Dunno—babb—

  

 

uouoeueb_ moHSOm—lwo .m.m

Aaummov mGOflufibGOU Heucmfiwummxm >Hxx dance

104

Fig. 14 CH3C1 Survey Spectra - V. and 2V.

 

 

 

 

 

'03 "‘ ‘03
'0‘ '00 '0‘
“09
“on
30:10 cm‘1 3350 3100 3150
2v. II P03 R00 his .1.
R06
1 R93
11
1 1 1
‘ a 1 i.
i 1 1 1 1 1 1
WWW 11.11 ’ *
111 WWWW 1
-1 m
6000 on 5050 6100 6150

 

105

turbation are even more pronounced. The R02(J) Q-branch is
Split and spread over a wide region. The neighboring Q-
branches are also badly split and pushed about. It was

quite impossible to obtain any sort of reasonable unperturbed
fit of this band, either alone or in a simultaneous fit with
V.. Alamichel, Bersellini, and Joffrin—Graffouillere (21)
have interpreted the perturbation as a Fermi resonance
between 2V. and V. + V5 + V5 + V3.

The V. band of CH3C1 was considerably better,
though by no means perfect. Most of the center of the band
appeared to be somewhat perturbed, but a quite good single-
band fit was obtained from the PP3(J) series and the RR4(J)

through RR12(J) series. Microwave values of Bo = 0.443402

cm‘l, soJ = 6.04Xl0‘7 cm-1, and ng = 6.60210‘6 cm"1 (23)
for CH3(av)Cl were taken as known quantities. The results

of the single-band fit of V. are given in Table XXV. The
standard deviation of the fit was 0.010 cm-l.
The 2V. band was simply too badly perturbed to
use in a simultaneous fit, hence Ao could not be determined
from these bands. No substitute band of the type Vn + 2V.
was available, nor is one likely to soon become available.
Previous values of A0, determined by means of the
zeta-sum rule method, are given by Herzberg (1) as 5.097
cm-1 and Burke (22) as 5.069 cm'l. This represents one case

where the zeta-sum rule is still the only method for deter-

mining Ao because of perturbations in the 2V. Spectrum.

106

Table XXV Coefficients of Single-Band Fit of CH3C1 V..

 

Coefficient Value (cm-1) 95% s. c. i.
vo 3043.34494889 0.06030692
2

A0 ‘ ABC“ + 1/2ng

+ ... - 1/2...K 4.37359326 0.00566803
2.1 - 3/2m.K -0.07489814 0.00595875
soK - 1/4m.K 0.00140707 0.00016057
2.3 0.00101681 0.00009425
HoK 0.00000150 0.00000025

All other coefficients were insignificant and were set = 0.

 

so 2 0.443402
DOJ * 0.000000604
03K * 0.00000660

 

* Microwave values for 12CH3C1 taken from Ref. (22).

 

Standard deviation of fit a 0.011 cm'l.

 

CHAPTER XII

ANALYSIS or CH3D

This section on CH3D is included only for the sake
of completeness, since both the V. and 2V. bands were too
badly perturbed to treat in the usual manner.

Two charts of CH3D V., 0465-D and 0565-D, and two
charts of 2V., 0365-D and 0166-D, were measured. The cali-
bration bands were the same as those for CH3F (Chapter IX)
and the standard deviations of the calibration fits were the
same. The experimental conditions under which the spectra
of CH3D were run are given in Table XXVI. Survey spectra of
V. and 2V. are shown in Fig. 15.

The spectrum of 2V. of CH3D under high resolution
was quite beautiful. All the individual lines, even in the
perpendicular band Q-branches, are resolved. Unfortunatly,
the appearence of this band belies its true nature. The
assignments of most of the lines were quite obvious and unam-
biguous. They Simply did not fit the theoretical formula,
however. In the end, no satisfactory fit was obtained for
this band. The results of such fits have not even been in-
cluded here. For future reference, the values of B0 =

3.88047 cm‘l, soJ = 5.277x10'5 cm'l, and 03K

= 1.238x10‘“
cm-1 were given us by Bruce D. Olson (22). These seemed to
be quite good, since a table of ground state combination
differences calculated from these constants permitted accu-
rate identification of a large number of lines in the 2V.

spectrum.

107

108

E m.m

E v.m

E «.9

   

 

 

xOQ

es mm zumbe use us .d.n
one MN

as we znmbe 3 com .d.o
Non

EE N mlmnm one MN .m.m
Mon

as m mlmnm one MN .m.m

a mo ._ %
.mmmum Houomumb mOHSOm no .m.m

 

.m.e.o.zom

 

BE embed
mcwumum

 

 

 

 

ooo .onmvaom onooao .em
.m.o.ovzom
oeo .onmvaom oumomo sew
AH.o.o.zom
oom .onaeaom oumomo .5
.H.o.ovzom
oom .onaeaom oumoeo .5
mbsmn
coaumunflamo uumno wanna
Anmmuv mcowuflbaou chcmEHummxm H>xx manna

109

 

!
e3. 88 ace. 7.6 on:

was: 5...). mug-(o 33.. .3 use

A. sen

 

 

4 q u 4
90.: Oman 0°°n All". Oman

- ... o ......f... i

e?

x
3:3 3...: 3.0.33-0 3:: .2 ....

rpm was :9 I muuommm >m>usm ammo ma .mflm

110

The V. band was not even attempted. The pertur-
bations in this band have been the subject of intensive
investigations by Olson and co-workers (22). The band seems
to be too badly perturbed to have any hope of making a
reasonable "unperturbed" fit.

Olson (22) has suggested to us that, in his
opinion, only V3 + V. of the possible alternative bands
might be useable. The lack of a good fit to 2V., however,
makes a determination of A0 for this molecule unfeasible

by our method.

CHAPTER XIII

STRUCTURAL CONSIDERATIONS

Once Ao had been determined for methyl bromide,
it seemed that the calculation of the structure of this
molecule would be a simple and productive project. This
quickly proved to be a much more difficult undertaking than
it appeared at first glance.

The structure of methyl bromide is specified by

three structural parameters (see Fig. 16) CH' and

. rCBr' r
B or d. The two moment of inertia equations, corresponding
to A0 and Bo, for a single isotOpic molecular species are
not sufficient to completely determine these three unknown
parameters. Thus, measurements of A0 and Bo for a second
isotopic species, for example Br or C substituted, are
needed. For the non-deuterated species, Ao can be assumed
to be the same, since substitution of Br or C Should make
little difference in the positions of the hydrogen atoms.
Very accurate values of B0 are available from microwave work
for the various isotopic species.

Let us consider the problem of determining a

"ground state" or ”r " structure directly from measured

0
values of A0 and ED for two isotOpic species of CH3Br.
Assume that the values of A0 and Bo are available for the
isotopic species 12CH379Br and 12CH3elBr (Ao can be assumed
to be the same for both species). Thus the structural para-

meters r , and B are to be determined from the moment

B

CBr’ rCH
of inertia equations corresponding to IC

111

(for 12CH37QBI),

112

I B"

o (for 12CH3elBr), and I°A(for both) - three equations in

three unknowns. A major assumption implicit here is that
the ground state structure is identical for both species.
The principal axes system is chosen as shown in
Fig. 16, with the origin at the center of mass, the z-axis
along the symmetry axis, and the x-axis chosen (arbitrarily)
such that one of the hydrogen atoms lies in the x-z plane.
For convenience, ”s" represents the distance from the bromine
atom to the center of mass of the particular molecular
species under consideration. The equations will be written
in terms of rCBr' rCH’ and B. The set of equations can be
made simpler in appearence if they are left written in terms
of the above three parameters plus 3 and s' for the two
species, and the s and s' quantities are related to the
structural parameters through the two center of mass equa-
tions. The moment of inertia and center of mass equations

which are solved for the ro-structure are given below, with

B 30
I

I0 ,

M- V,

MBr' and 3 being unique to 12CH379Br and I0 <r

and 3' being unique to 12CH331Br:

A

1. Io = BMHIrCHsin(u/2 - 8)]2

2' IoB g ”3:32 + MC(rCBr ‘ 3’2 + MH[(rCBr ‘ 3)

+ rCHcos(n/2 - 8)]2 + 2MH{[(rCBr - s) + rCHcos(n/2 — 8)]2

+ [rCHsin(n/2 - s)cos30°]2}

3. MBrs - MC(rCBr - s) - 3MH[(rCBr - s) + rcncos(w/2 - 8)]

Be: 012 _u2 _u
4. Io MBr s + Mc(rCBr s ) + MHHrCEr s )

113

Fig. 16 CH3Br Structural Parameters

A
Io
a A
I
I
I
a: o
c
'0
------ I“
I, > o
1'
1’
1’
2:/'
B
I
o rCBr
B
Q C
a

114

+ rCHcos(w/2 - 5)]2 + 2MH{[(rCBr - s') + rCHcos(w/2 - 8)]2

+ [rCHsin(n/2 - B)cosBO°]2}

5. MBr's' - MC(rCBr - s') - 3MH[(rCBr - s') + rCHcos(n/2 - 3)]

The reduction of these equations to formulas for
rCBr' rCH' and B is quite complicated but straightforeward.
The final values were calculated on the computer using a
program which substituted the known quantities into the
above equations and performed the complicated calculations
quickly and accurately.

Calculations of the ro-structure were carried out
for the four pairs of isotopic species (12CH379Br and
12CH381Br), (13CH379Br and 13CH3813r), (12CH379Br and
13CH379Br), and (12CH381Br and 13CH3813r). The results are
shown in Table XXVII.

The results are obviously very inconsistant.
Costain (22) discusses this problem. He points out that
the molecular parameter Bo (and similarly A0) is propor-
tional to the average over the zero-point vibrations of the

reciprocal of the "true" moment of inertia, viz.,

B a h 1
o 8n2c ‘ X m r 2 '
i i i

where the ri are the "true" instantaneous distances of the

 

 

atoms from the b-axis. The effective ground state moment

B

of inertia, Io , is defined through

h 1

B 8 ,
2 B
81 0 IO

0

 

115

 

.me .oHH .NH .moa mmo.a Hmmm.a umsmmmoms .umsommoNH
,mv .oaa .NH .moa mmo.a Hmma.a ummkmmoms .ummkmmows
.mo .moa .«m .moa aoH.H vamm.a umsmmmomH .umakmmoMs
.me .moa .HH .oaa « HHH.H < mmmm.a nmsmmmows .ummkmmomH
Amlolmva Announnvu mun umou cobwuoo ma ousuosuum

sodas Eoum nmfiommm UHQOHUOmH

nHmvm—fidhflm Hflgfiogflm OH HH>NN CHQMB

116

hence IoB is actually given by

-1
I°B=< 1 2>.
Xi miri

In setting up the moment of inertia equations in the manner

 

described above, one implicitly defines the effective atomic

distances, (ro)1, such that

-l
B: 2_ 1
I0 21 mi(ro)i ‘ 2 °
2. m.r.
111

When isotOpic substitutions are made, the zero-

 

point vibrations change, hence the averages over the zero-
point vibrations also change. This makes the ro distances
slightly different for each species.

In order to solve the moment of inertia equations
one must assume that the ro quantities are not changed by
isotOpic substitution, since, otherwise, new unknown para-
meters enter into the problem. The ro-structures for
different isotopic species obtained in this manner are
usually inconsistant. Our results have borne out this
conclusion.

Costain points out that these large variations
arise essentially from the attempt to force the ro para-
meters to exactly reproduce the IO values from which they
were obtained. There is no reason why an artificial struc-
ture of this sort, which is known to be not consistant among
different isotopic species, should exactly reproduce the
effective moments of inertia. The criterion that correct-

ness is defined by the closeness with which the calculated

117

parameters reproduce the original data should be reconsidered
in this case.

The problem of determining a meaningful structure
from the ground state constants is discussed by Kraitchman
(g2) and Costain (fig). Both stress the point that the assum-
ption of identical structures for different isotopic species
is, at best, valid only for the equilibrium structure.

The equilibrium structure is generally the ideal
of structural investigations. In the first place, the equi-
librium structure is theoretically the same for all the iso-
tOpic species of a molecule. Also, it should be directly
and simply comparable among different molecules of the same
type. Secondly, although the equilibrium structure is not
a true physical structure (the molecule is never more than
instantaneously in the equilibrium state), all sorts of
other structures, ground state, upper state, average, etc.
can be determined from the equilibrium structure. Also,
structures directly comparable to electron diffraction, etc.
structures can be calculated from the equilibrium structure.

The equilibrium structure would be calculated by
means of the moment of inertia and center of mass equations
listed previously, with the difference that now IeA, IeB,
and 183' would be used. This means that theoretically the
equations are exactly true then - the quantities rCBrIe'

rCH'e' and file are actually the same for both species. Un-

fortunately, the equilibrium constants Ae and B8 are related

A

s and

to the ground state constants A0 and Bo through the a

118

asB constants which must be determined experimentally for
each of the six vibrational modes of the molecule. The
relations are
asA(gs/2),

B = B + Z 6 asB(gs/2).
Until the six values of asA and asB, or at least the value
of the above sums, have been determined spectrosc0pically,
the quantities Ae and Be cannot be determined.
Kraitchman (2E) and Costain (g2) discuss the cal-

culation of a much more consistant structure known as the
"substitution structure." Kraitchman develops the formulas
for determining the distance of a substituted atom in an iso-
topic molecule from the center of mass of the original
"normal“ molecule. These are developed as an aid to obtain-
ing equilibrium structures when the equilibrium moments of
inertia are known. In this context, the basic assumption,
that the structure is identical for both species, is entirely
valid.

For axially symmetric molecules two cases have to
be considered. If the substituted atom lies on the symmetry
axis (e. g., substitution of 81Br for 79Br in CHaBr) then
the distance from the substituted atom to the center of mass
of the original molecule (CH37gBr) is given by

Izl = [u'1(IB' - IB)1”9,

B and 13'

where I represent the moments of inertia about a
principal axis perpendicular to the symmetry axis for the

”normal" and isotopically substituted molecule reSpectively,

119

and u - MAm/(M + Am), M being the mass of the "normal“ mole-
cule and (M,+ Am) being the mass of the isotOpically substi-
tuted molecule.

For the substitution of an off-axis atom (e. g.,
substitution of D for one H in CH3Br) Kraitchman derives
expressions for the distance of that atom from the center
of mass of the original molecule. Since deuterium substi-
tutions were never used in our calculations, these expres-
sions are not reproduced here.

Costain suggests that these same expressions, while
exact only for the equilibrium structure, are nevertheless
very useful in calculating a “substitution" structure. Al-
though this substitution structure is still quite artificial,
it should be a great deal more consistant than the ro-struc-
tures.

Briefly, a complete substitution structure requires
measurement of the moments of inertia for enough isotopically
substituted species that each independent atom in the mole-
cule has been substituted once. For example, to obtain a
complete substitution structure for CHaBr one should measure
A0 and so for 12CH379Br, lzcnaalsr, 13CH379Br, and 13CH3813r,

and A0, B , and Co for at least one of the singly-deuterated

0
species, and preferably all of them.

One point must be considered, however, before pro-
ceeding with the calculation of a substitution structure of

CH3Br. The basic assumption underlying all of this is,

again, that all of the isotopic species have identical sub—

120

stitution or rs-structures. Since we are now dealing with
ground state effective moments of inertia this assumption
does not necessarily hold. Most certainly it does 22£_hold
in the case of substitution of deuterium for hydrogen. The
bond lengths are known to generally shrink appreciably when
deuterium is substituted for hydrogen (2g). Thus, determi-
nation of the positions of the H atoms using D substitution
seemed unlikely to yield satisfactory results.

The final procedure which we settled upon was as
follows. Fortunately for us, Schwendeman and Kelley (21)

had already determined the r substitution bond lengths

CBrls
from microwave measurements of the B0 values for the four

non-deuterated species of CH33r. Using each isotopic mole-
cule in turn as the basis molecule, they calculated rCBrls
for each species as follows: Referring to Fig. 17, assume
that 12CH379Br is taken as the basis molecule. A substitu-
tion of 81Br for 79Br permits a calculation of distance 2
(the distance from the substituted atom to the center of
mass of the original molecule),

Izl = tu‘1(IOB' - IOB>11fl,

3' refer to 12CH379Br and 12CfigslBr respec-

where IoB and I0
tively, and u = MAm/(M + Am), where M represents the mass
of the former species and (M + Am) the mass of the latter.
Again, with 12CH379Br as the basis molecule, a substitution
of 13C for 12C will give distance 2',

Iz'l = [u"1(IoB” - IOB)11”.

Bu

where IoB refers to 12CHg79Br and I0 refers to 13CH379Br;

121

Fig. 17 CH3Br Substitution Parameters

Br ¢<;>1.z

 

C
H H
H

Assume CH379Br is the basis molecule and CH381Br is the
substituted molecule. Then

 

22 = My} AM AIOB

MAM
where
AM = M' - M,
M = total mass of CH379Br molecule,
M' = total mass of CH331Br molecule,
AIOB = IOB' - IOB.

122

u' is the appropriate redefinition of u above. The sum of

the two distances gives r for 12CH37gBr.

CBrIs

We used Schwendeman and Kelley's values of rCBrls

as known quantities, and from our value of A0 and the micro-
wave values of B0 for each species as given by Schwendeman

and Kelley (21), calculated the r and 8 parameters for

CH
each isotOpic species. This did not yield a true substi-
tution structure (as defined by Costain), but gave some sort

of effective values for r and B. For want of a better

CH
name, however, they will be referred to here as substitution
values. Our results of these calculations are listed in

Table XXVIII, along with Schwendeman and Kelley's values of

rCBrls' It is clear that these results are much better than
the ro-structural parameters, but there is still considerable
inconsistancy.

Table XXIX shows results obtained in just the
same manner except that the appropriate center of mass equa-
tion was used instead of the IoB equation. These results
are remarkably consistant among all the species. For this
reason we feel that these constitute our best results for
the structure of methyl bromide.

Table XXX gives the single set of values which we
consider to be the best average structure of methyl bromide.
This consists of an average of Schwendeman and Kelley"s
and 8.

values of r and an average of our values of r

CBrls CH
For comparison, results obtained by Miller, Aamodt, Dous-

manis, Townes, and Kraitchman (fig) who combined values of

123

.aMMv umdamm was cmewcnmsnom scum mwsHs> m_umuu

.1.

 

.NN .moa .vm .aoa soa.a ammm.~ umsmmeomH
.em .moa .mm .moa moa.a mmmm.a ummsmmoms
.NH .mOH .me .moa moH.H ammm.a umsmmmoas
.HH owes .me .moa a mcH.H « mamm.a ummsmmoms
I I I.I mu Hmo U0>Humo ma wusuosuum
Am 0 my: Am 0 Hmvm H a u scans scum newcomm camouOmH
moH one .soH .m_HmUu Eosm pm>uumo assumesusm Honsuosuum mu HHH>xx manna

124

.AMMJ.»0HH0M use sssmusmssow Bonn nosas>

 

n H
_ mun «
.em .oaa .sm ones emo.a mmmm.a umssmmomH
.wm oOHH .bm abcd mmooH mammoH HmmhmmUmH
.mm ooaa .mm ohoa mmo.H mmmm.a Hmsmmmoma
.mm .oaa .mm .soH emo.a mamm.a ummsmmows
I I . .r.I .. ,mu. umu om>auso as ousuosuum
Am 0 my: Am 0 Hmvm H a H scans scum mmaommm OHQOHOmH
sowussvm an
no Housmo can . 0H .m_umou scum om>auma m:

d

mumumssusm Asusuosuum H xHxx manna

125

.xHxx manna Scum mumumEmumm HousuosHUm mo emsuo>¢ +

.mva amaaom was swamcsmsnom scum mmsHs> m_umuu «

 

.8 22 .mm .2: a 2:4 a 284 adv .Hm um £33m
+.mm .oaa +.om .aoa em emo.a .« mmmm.a monum>m numb .xuos mssa
Amlolmva Amlolumvm mun umuu musuosuuu on no mousom

momuobd Umwm I mumumfimusm amusuosuum mu xxx manna

126

B0 for the four non-deuterated isotopic species of CH38r
with values of B0 - Co for two doubly-deuterated species.
The agreement between their result and ours is excellent.
Our conclusion is that, while one can calculate
various scrts of artificial structures which are more or
less consistant among themselves, determination of a really
satisfying and meaningful structure must probably wait until

equilibrium molecular constants are available.

CHAPTER XIV

CONCLUSION

A method has been develOped here for determining
accurate values of A0 for axially symmetric molecules. It
consists essentially of simultaneously analyzing a degenerate
fundamental band, Vt' and its first overtone, th. In such
an analysis the A0 and Coriolis terms in the frequency ex-
pression representing those bands are linearly independent
of each other. A least squares fit of the data of both
bands to a frequency expression general enough to represent
both simultaneously yielded individual values of these para-
meters. The frequency expression used in the analyses was
the appropriate specialization of the Amat-Nielsen generali-
zed frequency expression.

The data which was fit to this expression consis-
ted of frequencies of the individual transitions in the re—
solved rotational fine structure - lines of the types PPK(J),

P R R

RK(J), and in cases where the Q-branches
were resolved, PQK(J) and RQK(J). In some cases, lines of
the overtone parallel component were also included. The
least squares fits were performed on a large computer capa-
ble of inverting the large normal equation matrices.
Excellent values of A0, Aecsz + ..., and the other

molecular constants were obtained for CH3Br. None of the
other methyl halides or methyl halide-types yielded compar-

able results, due to perturbations in one or more of the

bands.
127

128

Descriptions of our method of obtaining A0 and
its application to CH3Br and CH31 have been recently pub-
lished in the Journal of Molecular Spectroscopy (32, 2g, 21).

Accurate values of A0 for the methyl halides are
essential for the calculation of molecular structures. As
noted in Chapter XIII, our value of A0 for CH3Br could be
combined with microwave values of ED for four isotopic
species of CflgBr to obtain a consistant, though artificial,
"substitution structure."

Calculation of the equilibrium structures of these
molecules is our main goal. It will be a difficult goal to
realize. Accurate values of Ae and Be are needed, and these
can be obtained only after values of “BA and 083 for each of
the six normal modes of the molecules have been measured
experimentally.

The equilibrium structure is a very worthwhile
goal, however. There are many advantages to having such a
structure. The equilibrium structures should be directly
comparable among isotoPic species of a molecule and among
different molecules of the same type. Also, many of tne
other interesting structures (ground state, upper state,
average, r. m. 3., etc.) can be calculated from the equili-
brium structure.

Accurate equilibrium structures for the methyl
halides should also provide a firm base for determining
accurate and consistant sets of force constants and poten-

tial functions for these molecules. In this way it should

129

be possible to eventually arrive at a detailed understanding
of the interactions involved in one of the few many-body
problems in nature in which a theory is available of an

accuracy equalling that of the experimental data.

10.

11.

12.

LIST OF REFERENCES

G. Herzberg, "Infrared and Raman Spectra of Poly-
atomic Molecules," Van Nostrand, Princeton, New
Jersey, 1945.

M. Goldsmith, G. Amat, and H. H. Nielsen, J. Chem.
Phys. 231 1178 (1956);

G. Amat, M. Goldsmith, and 1
Phys. 21, 838 (1957);

I4:
I:

H. Nielsen, J. Chem.
G. Amat and H. H. Nielsen, J. Chem. Phys. 27, 845
(1957); "'

G. Amat and H. H. Nielsen, J. Chem. Phys. 29, 665
(1958); '—-

G. Amat and H. H. Nielsen, J. Chem. Phys. 36. 1859
(1962); '—-

M. Grenier-Besson, G. Amat, and 5. H. Nielsen, J.
Chem. Phys. 26, 3454 (1962);

S. Maes, J. M01. Spectry. 2, 204 (1962);
Stewart K. Kurtz, thesis, Ohio State University, 1960.

B. T. Darling and D. M. Dennison, Phys. Rev. 57, 128
(1940). .—

G. Amat, Compt. rend. 250, 1439 (1960).

H. Allen and P. Cross, "Molecular Vib—Rotors,“ Wiley,
New York, 1963.

I. M. Mills, Mol. Phys. 8, 363 (1964).
W. E. Blass, thesis, Michigan State University, 1962.
A. G. Maki and R. Hexter, to be published.

R. G. Brown and T. H. Edwards, J. Chem. Phys. 38, 384
(1958).

F. B. Hildebrand, WIntroduction to Numerical Analysis,Ct

McGraw-Hill, New York, 1956.

M. A. Efroymsen, "Mathematical Methods for Digital
Computers,” Ralston and Wilf (Eds.), p 191, New
York, 1960.

Henry Scheffe, "The Analysis of '.'ariance,”c p. 78 ff,
Wiley, New York, 1959.

130

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

131

J. W. Boyd, thesis, Michigan State University, 1962.
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 Spectrosc0py and
Molecular Structure, Columbus, Ohio.

D. B. Keck, forthcoming thesis, Michigan State Uni-
versity, 1967.

K. N. Rao, C. J. Humphreys, and D. H. Rank, "Wave-
length Standards in the Infrared," Academic Press,
New York, 1966.

D. H. Rank, D. P. Eastman, B. S. Rao, and T. A.
Wiggins, J. Opt. Soc. Am. 53, l (1962).

D. H. Rank, G. Skorinko, D. P. Eastman, and T. A.
Wiggins, J. M01. Spectry. 2, 518 (1960).

D. H. Rank, D. P. Eastman, B. S. Rao, and T. A.
Wiggins, J. Opt. Soc. Am. 51, 929 (1961).

Gerald W. King, "Spectrosc0py and Molecular Struc-
ture," Holt, Reinhart, and Winston, New York, 1964.

W. E. Blass and T. H. Edwards, to be published.

W. O. J. Thomas, J. T. Cox, and W. Gordy, J. Chem.
Phys. 22, 1718 (1954).

Richard J. Burke, thesis, University of Maryland, 1954.

C. Joffrin—Grafouillere and Nguyen Van Thanh, to be
published.

E. W. Jones and H. W. Thompson, Proc. Roy. Soc.,
London 288A, 50 (1965).

D. A. Steiner and W. Gordy, J. Mol. Spectry. 21, 291
(1966). "_

J. Pickworth and H. W. Thompson, Proc. Roy. Soc.,
London A222, 443 (1954).

F. A. Andersen, B¢rge Bak, and S. Brodersen, J. Chem.
Phys. 22, 989 (1956).

W. L. Smith and I. M. Mills, J. Mol. Spectry. 11, 11
(1963).

C. Alamichel, A. Bersellini, and C. Joffrin-Grafouil-
1ere, to be published.

32.
33.

34.
35.

36.

37.

38.

39.

4o.

41.
42.

43.

132

B. W. Olson, private communication.

B. W. Olson, 1966 Symposium on Molecular Spectroscopy
and Structure, Columbus, Ohio.

C. C. Costain, J. Chem. Phys. 22, 864 (1958).
J. Kraitchman, Am. J. Phys. 21, 17 (1958).

C. H. Townes and A. L. Schawlow, "Microwave Spectro-
SCOpy," p. 54, McGraw-Hill, New York, 1955.

R. H. Schwendeman and J. D. Kelley, J. Chem. Phys. 42,
1132 (1965) . ‘—

S. L. Miller, L. C. Aamodt, G. Dousmanis, C. E.
Townes, and J. Kraitchman, J. Chem. Phys. 20, 1112
(1952) . "‘

T. L. Barnett and T. H. Edwards, J. Mol. Spectry. 29,
347 (1966).

T. L. Barnett and T. H. Edwards, J. Mol. Spectry. 32,
352 (1966).

T. L. Barnett and T. H. Edwards, to be published.

D. R. J. Boyd and H. C. Longuet-Higgins, Proc. Roy.
Soc. A213, 55 (1952).

E. H. Richardson, S. Brodersen, L. Krause, and E. L.
Welch, J. Mol. Spectry. 8, 406 (1962).

APPENDIX I

ALTERNATE METHODS OF OBTAINING A0

a. Use of the Zeta-Sum Rule

Almost all previous determinations of A0 for the
methyl halides have been based on the zeta-sum rule. The
theory of the zeta-sum rule has been developed by D. R. J.
Boyd and H. C. Longuet-Higgins (43). For a C3v type mole-
cule one has

It ctz = (# atoms on symmetry axis) - 2 + B/ZA,
which becomes for a CH3X type molecule
it ctz = B/2A.

The application of the zeta-sum rule in the deter-
mination of Ao proceeds as follows: From single-band
analyses of vs, v5, and v5 (Q-branch analyses are sufficient
at a minimum) one determines the numerical values, repre-
sented by C“, C5, and C5, of the coefficients

[A - Aecuz + see] 8 Cu

O
z -
[A0 - AeCS + 6.0] - C5
2
[A0 - Aecs + see] 3 C6.
Then

2 _
3A0 ‘ Ae(§k + ‘52 + :62) + too = C“ + C5 + C6:
or

3A0 - Ae(BO/2AO) + one = C“ + C5 + C6.

Then, to the approximations Ae a A0 and the n-terms (repre-
sented + ...) being negligible, one finds

Ao . (1/3)(C. + C5 + C5) + Bo/6'

The zeta-sum rule is exactly true only to the

133

134

approximation of purely harmonic vibrations. This arises
from the :t2 constants being defined in terms of purely

harmonic quantities, the £133“ (2),

a a B Y _ B Y
z;sas'a' 21 (Lisa Iis'a' ILis"cI" zisa )
a

The 1130 are coefficients involved in the small-vibration
expansion of the potential energy of the molecule. As such,
they are inherently harmonic, since all usual small-vibration

expansions assume harmonic vibrations.

b. Analysis of Raman Spectra

Ao can be determined for the methyl halides
directly from a fully resolved Raman spectrum of the molecule
in the gas phase. Let us consider the Raman spectrum of the
v. band. If available, the infrared v. band might be fit
along with the Raman spectrum for statistically better deter-
mination of the coefficients.

As in the case of the simultaneous analysis of v.
and 2vs, the vital factor in the determination of AC from
Raman spectra is the selection rule on A2.. The selection
rules on Al. for Raman spectra are given by Mills (6) or
may be derived from Amat's Rule (4) for the v. Raman band:

AK - A2. - t3p, p = 0,1,2,...
|AI.| = l, 3, ...
Then, for AK = *1,
*1 - A2. - 0, p = 0
A2. = *1 or

A“, = AK:

and for AK = *2,
*2 - An. = *3,
-A2. = *1 or
A2. = -1/2AK.

If lines

135

of both the AK = *1 and AK = *2 types

are available from the Raman spectrum, the A0 and Coriolis

terms are linearly independent and the two coefficients

can be determined individually, just as in the case of the

simultaneous fit of v. and 2v“. Such an analysis has al-

ready been carried out by Richardson, Brodersen, Krause,

and Welch (42) for CH3D.

APPENDIX II

LISTING OF FALSTAF PROGRAM

FALSTAF was the least squares computer program
which was used to analyze the spectra considered in this
thesis. Using this program the observed individual tran-
sition frequencies were fit to the apprOpriate speciali-
zations of the Amat-Nielsen generalized frequency expres-
sion. The program listed here is that used for the final

simultaneous analysis of v. and 2v. of methyl bromide.

136

CIOITC)O(7C)OC1

L1

('1 C“)

(‘2

O

LISTING OF
DE CHSBR N04 AND 2Nu4.

FALSTAF PROGRAM FOR LtAST SQUARES SIMULTANEOUS ANALYSIS

PROGRAM FALSTAF

DIMENSION DATA(?D)nVECTORI21,21):AVEIQOIISIGMAIZOI.COEN(20).

137

15!GMCO(2O)DINDEXIZO)DFVALI15'SP4I9CONFINT(20)1IHEAD‘20)9KDEL(1000))

?IJDEL(1000).KAY(1000):JAYIIOOO).FREQUESIIDOO)IHHTI1000).DEV(1000I.

SINCRVIB(1DOOIINGT(1000)

COMMON NOIN.

INDEX:

COEN. BZRO: UZROJK, DZROJ

TYPE DOUBLE VECTOR.SlGMA,SIGY.SIGMCO.COEN.AVE
IFWT = 1.THEN ALL NHTS : 1,0

IFSTEP = 1. DO NOT PRINT EACH STtP

IFRAN = 1 DO NOT PRINT RAH SUNS AND SQUARES
lFAvE = 1 DO NOT PRINT AVERAGES

IFRESP = 1 DO NOT PRINT RESIDUAL SUNS SQUARES
IFCOEN = 1 Do NOT PRINT PARTIAL COEFFICIENTS
IFPRED = 1 Do NOT CALC PREDICTED VALUES
IFCNST = 1 DO NOT HAVE CONST TtRM IA EQUATION

100

107

102

101

103
104

120

125
124
125

READING AND INPUT
CALL FAULIIOI

TOL 3 .00000001 S EFIN = .00000001 $ EFOUT =
IFSTEP=0 $ IFRAN=0 $ IFAVEsfl $ IFRESD=0 $ IFCOtN=0 S IFPREon
IFCNST=1 $ NOTIMES=0 $ VAR=0,0 $ K=0 S FLEVEL=0.0 S NOENT=0

NOMIN=0 $ NOMAX=0

PRINT 107

FORMAT (141)

NZILCH = 0

NOIN = 0

REWIND 50

DO 103 NU“ = 1. 100

READ 101a (IHEADIM), M=1.12)
FORMAT (12A6)

IF (IHEAD(2) - 6HENDHED) 103,104.105
pRINT 101: IIHEADIMII M31112)
CONTINUE

READ 105p NOVAR

FORMAT (I?)

NVP1 = NOVAR + I

DO 120 I = 1INV91

DO 120 J = lnNVPI

VECTOR (IDJ) 3 0.0

SUMNT = 0.0

DO 125 IPROB = 1,4,?

IPROB = 1. 2. 3. 4 IMPLIES CONFIDENCE LEVELS OF

DO 124 IDEGF : 1, 5

IDEGF = 1. 2o 3. 4. 5 IMPLIES DEGREES OF FREEDOM

40,‘60. 120. INFINITE

READ 123' (FVAL(INOVOIDEGFIIPROR),INOV=1113)
INOV = 1. 2. .... 10. 11. 12. 13 IMPLIES NUMBER OF VARIABLES (P)

1: 2: 000! 10: 12: 15a 20
FORMAT I15F4,?)

CONTINUE

CONTINUE

.00000001 $ NOPROB=0

NPR?
NPR?
MPRZ
MPRZ
MPRZ
NPR?
MPR2
MPR2

138

READ 140. BZRO. DZROJK. DZROJ
140 FORMAT (F15.8)
READ 141. NORFTS
141 FORMAT (12)
D“ 160 L. = 10 "DDU
L1 = 3*L - 2 S L2 = 3*L
READ 14S. (KDELIN).JDEL(N).KAY(N).JAYIN).FREOOOS(N).HHT(N).
INCRVIBIN). N = L1. L2)
145 FORMAT (3(A1.A1.12.1X.12.1X.F8.3.1X.F4.2.1X.11.2X))
DO 159 N 2 L1. L2
IF (NHTIN)) 109. 110.109
110 NZILCH = NZILCH + 1
109 CONTINUE
IF (KOELIN) - IHF) 146,161,146
146 SUMNT = SUMNT + NHTIN)
NODATA : N
159 CONTINUE
160 CONTINUE
161 AVENT = SUMwT/NODATA
DO 510 N = 1. NnDATA
NHTIN) = NHTIN) / AVENT
IF (KOELIN) . 1H0) 162.165.164
162 KDELIN) = -1 $ GO TO 165
163 KDELIN) = 0 $ GO TO 165
164 KDELIN) = +1 I GO TO 165
165 IF (JOELIN) - 1P0) 166.167.168
166 JDELIN) -1 $ 00 To 169
167 JDELIN) 0 $ GO TO 169
168 JDELIN) +1 N GO TO 169
169 CONTINUE

'J

UtLTAl = KAY(N) + KOELIN)
DELTA2 = KAYIN)

DELTA3 = JAYIN) * 1 + JOELIN)
DELTA4 : JAY(N) . JDELIN)
DELTAS = JAYIN) * 1

DELTAb : JAYIN)
IF (INCRVIB(N) . 1) 210. 190.200
FORMS DATAIL) FOR NU'4

190 DATA(1) 3 1.0 T DATAIZI = 0.0 $ DATA(3) = 0.0
LDEL = KDtL(N)
GO TO 210

200 IF (KDEL(N)) 201.202.201
FORMS DATAIL) FOR 2NU-4 PERPENDICULAR

201 DATA(2) = 1.0 T DATA(1I = 0.0 S DATAIS) = 0.0
30 TO 203
FORMS DATAIL) FOR 2NU-4 pARALLEL

202 DATAI3) 3 1.0 T DATAIl) = 0.0 $ DATAIZ) = 0.0

203 LDEL = “2*KOELIN)
210 AA 3 INCRVIB(N)6(DELTA4ODELTA3 - DELTAlttZ)

AB = INCRVIB(NI¢DELTA1*04

AC : LDEL'DELTAl*DELTA40DELTA3

AD 3 INCRVIB(N)6DELTA4662«DELTA3662

AE : INCRVIBIN)oDELTAlstZ‘DELTA4tDELTAS

AF : DELTA4PDELTA3~DELT46*DELTAS . DELTAltt2+DtLTA20i2

(I

530
550
560
540
510

565
566

567
570
580
590
600
610
620
630
640
645

650
651
652
653
655
670

139

AG = DELT41**2*DELTA4*DELTA3 - DtLTA2*t2*DELTAthELTAS

AH : DELTA4¢t2iDELTA3¢i2 - DELTAbttZtDELT45it2

AI = INCRVIBIN)6DELT41**2

AJ = LDEL*DELTA1**3

AK : DELTAlit4 - DELTA20fi4

AL 3 DELTA3ifi3tDELTA4663 - DELTAOii3tDELTA6tt3

AM a DELTA3tt2tDELTA4662OOELTAltt2 ' DELTA5tt2tDELT46t*2tDELT42**2
AN 3 DELTASiDELTA4tDELTAlit4 - DtLTASiDELTAétDtLTAztt4

A0 8 DELTAitto w DELTA2646

AP = (INCQVIB(N) * 1)*LDEL*DELT41

A0 a (INCRVIBINIttz + 2aINCRvIBINIItDELTA1..2

AR 3 LDEL'*2ODELTA1¢02

AS a (INCRVIBIN)i*2 + ZOINCRVIB(N)I‘IEELTA4tDELTA3 - DELTAltt2)

AT = LDEL**2*(DELTA4*DELTAS 9 DELTAlt*2I

OATAI4) = DELTA1**2 - DELTA2¢¢2
DATAIS) = -2.*LDEL*DELTA1
OATAI6) : AK

0ATAI7) = AI

DATAIBI = AJ

DATAI9) = AA

OATAIIO) 8 AB

DATAI11) = A0

SUBTOFF = BZRO.AF - DZROJK.AG - DZROJtAH

DATAINOVAR) = FREOORSIN) - SuBTOFF

RUN = N

WRITE TAPE 50. RUN,(DATA(L). L=1.NOVAR).SUBTOFI,IDBAND
MAIN PROGNAM

DO 540 I 3 19 NUVAR

VECTOR(1.VOVAR+1) = VECTORII,NOVAR*1) + DATAIIIRNHTIN)
DO 540 J = I. NOVAR U

VECTORII.J) a VECTORII.J) 4 DATAII)*DATA(J)*HHT(N)
VECTORINV91.NVPII = VECTORINVP1.NVP1) + NHTIN)

RENIND 50

NOSTAT = NODATA - NzILCH

NOVMI = NOVAR ~ 1
NOVPL = NOVAR 6 1
DMAXM : 0.0

PRINT 90. NOPROP,NODATA.NOVAR.VECTURIDOVPLpNOVPL)
IF (IFRAN) 900. 580. 650

PRINT 15

pRINT 20. (IIVFCTORIIINOVPL).I=IIN0VMII

PRINT 25. VECTOP(NOVAR,NOVPL)

PRINT 30

pRINT 35: ((IDJOVECT0R(IIJ)OJ:1)NOVMI)DI=1IN0VMI)
PRINT 40: IIIVECTOR(IINOVAR),I=11NOVMI)

pRINT 45p VECTOR(N0VARIN0VARI

GO TO 650

CALCULATION OF RESIDUAL SUNS 0F SQUARES AND CROSS PRODUCTS
IF(IFCNST) 900.651.735

IF(VECTORINOVPL.NOVPLII 652:652.655

pRINT 654
GO TO 910
OD 660 I = 1. NOVAR
DO OOU J 3 I: NOVAR

MPR

NPR
MPR

NPR

MPH
MPR
NPR
MPR

660 VECTOR
/ VECTOR (NOVPL.
DO 690
AVEII)
IF (IFAVE)
PRINT 50

PRINT 20.

PRINT 25,

II (IFRESD)
PRINT 55
PRINT 35.
PPINT 40:
PRINT 45:

680
690
700
710
720
730
735
740
750
760
770
780
781
782
790
791
792
798
796

794
796
797
795
810
500
.20
840
841
550
830
060
870
874
675
880
885
.90
900
1000
1001
1002
1003
1005
1010
1015
1016
1017
1021
1020
1030
1035

NOSTEP
ASSIGN

DEFR

DO 800

IF(VECTOR(I.I))

PRIN

PRIN

T

T

(I.

I

-
' .

J)

VECTOR(I.NOVPL)

1

= VECTOR

900.

(Ind)

NOVPLII
. NOVAR

710:

735

140

(VtCTOR(IINOVPL)

/ VFCTOR(NOVPL.NOVPL)

(I.AVP(I).I=1.N0VMII
AVE(NOVAR)

((I.J.VECTOR(I.J).J=1.N0VMI).I=1,NUVNI)

900:

740.

780

(I.VECTOR(I.NOVAR),I=1.N0VNII

VECTOR(NOVAR.NOVAR)

1

1520

I

795,
00 T0 910
FORMAT
1LFM TERMINATED I

(31H ERROR RESIDUAL SQUAR: VARIABLE

795.

SIGMAII) =
60 TO 800
FORMAT

SIGMAII) '
VECTORII

DO 830

191

I.) I) 8 3 II
VECTURIIIJ)
VECTOR(J8I)
IF (IFCCEV)
PRINT 60
NOVN?

DO 885

1P1

I

To NUMBER
= VECTORINOVPL.NOVPLI

1.NOVAR

I

1.0

792,794,810

" 100

(1H010H VARIABLE 15.13H IS CONSTANT )

.I
I
I +
J :

DSORT(VECTOR(I.I))

)
1

1P1.

1.0

1.NOVNI

NOVAR
VECTORII.J)

VFCTOR(I.JI

900.

= NOVNI -

I

I +

PRINT 65.
PRINT 40,
CONTINUE
= NOSTEP + 1

NOSTEP

1

1
(I.
(I

IF (VECTOR(
NSTPMl
PRINT 1004.
DO TO 1381

SIGY
DEFR

PPIN

T

60 T0

VMIN
VNAX

SIGNA(NOVAR)

870.

1

. NOVM?

1000

/(

SIGNA(II* SIGNA(J))

J.VECTOR(I.J).J=IP1.NOVNI)

.VECTOR(I.NOVAR).I=1.NOVMI)

NOVAR.NOVAR)) 1002.1002.1010
= NOSTEP -

NSTPNl

=DIIF'T‘1'0
IF (DFFR I
1019

1681
0,0
0.0

NOIN = 0

1

1017.1017.
,NOSTEP

1020

* DSORT(VECTOR(NOVAR,NOVAR)/ DEFR)

I4051H

a VECTOR (J.NOVPL) NPR?

NPR?
NPR?

NPR2

NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?

IS NEGATIVE.PROBNPR2

NPR?

NPR?
NPR?
NPR?

MPR?
NPR?
MPR?
NPR?
MPR2
NPR?
NPR?

NPR?
NPR?
NPR?

NPR?
NPR?
NPR?

NPR?
NPR?
NPR?

MPR?
NPR?
NPR?

1040
1041
1042
1045
1060
1000
1090
1100
11?0
1130
1140
1150

904
1155
1170
1180
1190
1160
1110
1210
1220
1050
1230

906
1235
1240
1260
1245
1305
1310
1511
1512
1516
1314
1515
1520
1330
1340
1645
1346
1350
1360
1561
1670
1390
1691
1592
1694
1400
14?0
1430
1450
1460

141

00 1050 1 = 1.00v01

IF (VECTOR (1.1)) 1042.1050.1060

PRINT 1044. I. NOSTEP

GO TO 1581

IF<VECTOR(I.I) - TOL) 1050,1080.1080

VAR = VECTOR(I.NOVAR) o VECTOR<NOVAR.I) / VtCTORII.I)
IF(VAP)1100.1050.1110

NOIN = NOIN + 1

INDEX(NOIV) = I
COEN(NOIN) = VECTOR(I.NOVAR) t SIGMA(NOVAR) / SIGMA (I)
SIGMCO(NOIN) = (SIGY / SIGMA(I)) t DSCRT(VECTOH(I.1))
IF (VMIN) 1160.1170.904

PRINT 906

GO TO 910

VMIN = VAR

«\IONIN : I

00 TO 1050

IF(VAR - VMIN)1050.1050.1170

IF (VAR - VMAX)1050.1050.1210

VNAX : VAQ

NOMAX = I

CONTINUE

IF (NOIN) 903.1?40.1245

PRINT 907

00 TO 910

PRINT 65. SIGY

GO TO 1550

CNST = 0.0

IF (IFSTEP) 900.1510.1320

IF (NOENT) 1311.1311.1513

PRINT 91. NOSTEP,K

GO TO 1314

PRINT 92. NOSTEP.K

PRINT70, FLFVEL.SIGY.(INDEX(J):COEN(J)oSIGNCO(J).J=1.NOIN)

GO TO NUMdER. (1320,1580)

FLEVEL = VMIN * DFFR / VECTOR (NOVAR.KOVAR)
IF(EFOUT + FLEVFL) 1550. 1350. 1040

K : NOMIV

NOENT = 0

GO TO 1091

FLEVEL = VNAX * DEFR / (VECTOR(NOVAR,NOVAR)~ VMAX)
IF (EFIN - FLEVPL) 1$70.1361,1380

IF (EFIN) 1380,1380.1670

K : NOMAX

NOENT = K

IF(K) 1592.1392.1400

PRINT 1395. NOSTEP

00 TO 910

00 1410 I = 1.NOVAR

IF (l'K) 1430.141011430

DO 1440 J a 1, NOVAR

IF (J-K) 1460.144001460

VECTOR(I.J) = VFCTOR(I.J) - (VECTORIInK) 0 VECTOR (K,J)
'(K9K7)

/ VECTOR

MPRZT
MPRZE
MPRZE
MPRPI
MPRPI
MPRQE
MPR?L
MPR2L
MPRZL
MPRQL

NPRZE

MPRZC
NPR2L
NPRPT
NPRZL
MPR?E
MPR?I
NPRZT
MPRZL
MPRPI
NPRZC

NPRQT

MPRZI

NPRQE

NPR91
NPR2L
NPRZI
NPRQL
NPRZI
NPR?I
NPR2(
MPRZE
NPRZI

MPR2E

NPRZI
NPRZI
NPR21
NPRZI
NPRQI
NPR?(
NPRZI

1440
1410
1470
1490
1:00
1480
1510
1530
1540
1520
1550
1560
1380
1381
1570
1D71
1580
1581
1582
1586

7005

P086

1&30

1651
1b60

1061
1670
1680
1690
1695
1700
1710
1720
1725

1745

142

CONTINuE

CONTINUE

DO 1480 l = 1. NOVAR

IF (I-K) 1500.1400.1500

VECTOR (I.K) = - VECTOR (I.K) / VECTOR (K.K)
CONTINUE

DC 1520 J = 1. NOVAR

IF (J'K) 1540.1520.1540

VECTOR(K.J) = VECTOR (KaJ) / VECTOR (KnK)
CONTINUE

VFCT0R(K.K) = 1.0 / VECTOR(K.K)

GO TO 1000

PRINT 75. NOSTEP

IF (TESTER) 900, 1580,1570

ASSIGN 1980 T0 NUMBER

GO TO 1510 _

PRINT 1586. (L. VECTOR(L.L).L=1.NOVNI)

I‘( IFPREU) 900.15821910

CONTINUt

FORMAT (24H0 DIAGONAL ELEMENTS //?0H VAR.N0. VALUE/l
1(1H I 7. F16.6I)

OUTPUT SECTION

PRINT 2085

FORMAT (* OBSERVED VS CALCULATED RESULTS* //)
PRINT 2086

FORMAT (3X,3HRUN116X18HOBS TREQ,DX,9HCALC FREQ:4X:5HRESID06Xo

XOHNEIGHT)

DNAXM = 0.0 $ SRESZNT = 0.0

00 1750 N = 1. MUDATA

READ TAP: 50. RUN.(DATA(L). L=1.NOVAF)oSUBTOFF.ICBAND
YPRED = CVST * RUBTOFF

00 1630 I = 1, MOIN

K = INOEX(I)

YPRED = YPRFD + COEM(I)*DAT4(K)

DEVIN) = DATA(NOVAR) ~(YPRED - SURTOFF)
NGTU‘J) ‘3 NHT(N)

AHDEV = D&V(N)*o2*WGT(N)

IF (ABDEV - DMAXN) 1660.1651,1651

DNAXM = ABOEV

SPESZHT = SRESZNT + DEVIN)**2*NHT(NI
NHT(N) = NHT(N)¢AVENT

IF (NOTINES) 7999.166111725
IF (KDEL(V)) 1670.1680.1690

KDEL(N) = 1HP 1 GO TO 1695

KDEL(N) = 1H0 0 Go TO 1695

KDELIN) = lHR $ GO TO 1695

IF(JOFL(N>) 1700.1710.1720

JOEL(N) = 1HP 0 Go TO 1725

JDEL(N) = 1H0 0 GO TO 1725

JUEL(N) = 1HR 0 -GO TO 1725

PRINT 1745. RUN,KDEL(N).JDEL(N>.KAY(N).JAY(N>.IRECOBS(N).YPRED.

10EV(NI.NHT(N).APDEV

FORMAT (1X.F5.0.6X;A1.Al,I201H9,12:3XoF—99405X0FgoquXpF7g4p4X:

1F5.?.5X.1UX.F15.8)

NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?
NPR?

NPR?

NPR?
NPR?

143

1750 CONTINUE
RENIVD so
PRINT 1760. DMAXM
1760 FORNAT(///9H DNAXN = , F15.8,///)
‘ STANDARD DEVIATIONS FROM RESIDUALS
S2 : (NOSTATtSHFSZWT)/((NOSTAT-NOVAR)*VECTOR<NUVPL,NOVPL))
1770 PRINT 1775. $2
1775 FORMAT (//24H SUN 50. 0F RESIDUALS = . F14,a)
1780 STDDEV = SURTF(82)
1785 PRINT 1790. STDDEV
1790 FORMAT (39H STD. DEV. CALCULATED FROM RESIDUALS = . F10.8)
(3 CONFIDENC: INTERVALS
C HERE WE WISH A 95 PERCENT PROBABILITY
IPROB = 1 $ CONFLEV = 95.
DFGFR = NDDATA - VOVAR
1+ (DEGFR - 55.) 2183.2183.2184
2186 IDEGF = 1 % GO TO 2192
2184 IF (DEGFR - 50.) 2185.2185,2186
2135 IDEGF = 2 $ 80 TO 2192
2136 IF (DEGFR - 90.) 2137.2187,2188
2187 IDEGF = 3 5 GO TO 219?
?135 IF (DEGFR - 120.) 2189. 2189, 2190
= 4

P

2189 IDESF $ GO TO 2192
2190 IDEGF - 5
2192 IF (NOVAR‘IUI 2001.2001.2002
2001 INOV = NOVAR S GO TO 2007
2002 IF (NOVAR ' 15) 2003.2003.2004
9335 INOV = 11 $ GO TO 2007
2004 IF (NOVAR ’ 17) 2008.2003.2006
2003 INOV = 12 T GO TO 2007
2006 INOV = 13
2007 CIRATIO : SORTF (NOVARtFVALIINOV.IDEGF.IPROH))
00 2010 I 3 1. NOIN
9010 CONFINTII) 3 CIRATIO*SIGWC0(I)
PRINT 2011. CONFLEV
2311 FORMATI1H4.*COEFF. STD ERRORS. CONF INT USING CORR STD DEVi. //
120R CONFIDENCE LEVEL = F5.2//45H FINAL COEFFICIENTS WITH CONFIDENC
2E INTERVALS .// 20X. 11HCOEFFICIENT, 14X.18HSTD ERROR 0F COEFF.
37X, 19HCOVFIDENCE INTERVAL //I
2020 PRINT 2015. INDEXII).COEN(I).SIGNC0(II.CONFINT(I)
2016 FORMAT (12H COEFF OF X(12.1H). 5X.EIB.8.7X.E18.8.7X.E18.8)
C REFIT SECTION
REwIND 51
NOTIMES 3 NOTIMES * 1 S NODEL 3 0
SUMwT = 0.0
IF (NOTINES ' NORFTS) 7001. 7001. 7900
7001 00 7021 N=I.NODATA
READ TAPE 50. RUN.(UATAILI.L=1.N0VAR).SUBTOFF.IDBAND
ABDEV = DEVINItt2*NGT(N)
IF (DMAXM ' ABDEV) 7002.7010,7020
7002 PRINT 7005. N
7005 FORMAT I/ItDIUNT FORM MAX OI RESIDUALS CORRECTLY. SET NHTIN) = 0 F
109 PT. NO.'. 133

144

NHT(\1) = 0.0
GO TO 7020
7010 MODEL = NUDEL + 1
pRINT 701D, NOTI“ES, N
7015 FORMAT (141.21H REFIT OF DATA NUMBER . 12/ 15H DATA POINT NUMBER
1 3 I4: 26* HAS REEN RFMOVED FROM FIT)
NHT(N) : 090
7020 WRITE TAPE 51!KDEL(M)IJDEL(N),KAY(N)IVAY(N))FRtOOBS<N)9NHT(N)oRUN1
1(DATA(L).L=1.NOVAR),SUUT0FF.INCRVIB(N).IDBAND
SUMNT = SUVWT + NHT(N)
7021 CONTINUE
RENIND 50 S RFRIND 51
AVEWT = SUMNT/NDDATA
DO 7022 I 3 1.NVP1
DO 7022 J = loNVP1
7022 VFCT0R(I:J) = 0.0
DO 7045 N = 1, NDDATA
READ TAP: 511KPEL(N)1JDEL(k)[KAY(N)vaY(N)'FRtOOBS(N)DwHT(N)pRUNp
1(FATA(L):L=10NOVAR),SUBTOFF:INCRVIB(N).IDBAND
wHT(N) = NHT(N) / AVENT
WRITE TAPE 50o RU~0(DATA(L): L=11N0VAR)13U8TOFT:IDBAND
DD 7050 I = lgNnVAR
VECTOR (I: NOVAR+1) = VECTOR (I, NOVAR+1) + DATA(1)0NHT(N)
DO 7050 J = 1, MOVAR ‘
VECTOR (IOJ) = VECTDR(I:J) * DATA(I)*LATA(J)tNHT(N)
7050 CONTIAUF
VECTOR(NV91.NV91) = VECTOR<NVP1,NVP1) + HHT(N)
7U45 CONTINUE
RFNIND 50 S RFNIND 51
GO TO 565
7900 CONTINUE
7999 CONTINUE
10 FORMAT (F14.8)

15 FORMAT (1H 49H SUM OF VARIABLES//)MPR2I
20 FORMAT (1H 11H SUM x< 12.3H) = 015.8.8M SUM xt 12.3M) 3015.8.
15H SUM xc 12,3H) =015.8.8H SUM x<12.3H) =DlS.8 >
25 FORMAT (17H SUM Y =015.8 >
30 FORMAT(1H0 70H RAH SUM OF SQUARES AMPR2|
1M0 CROSS PRODUCTS/I )
35 FORMAT (1H 7H X(12,7H) vs x(12.3H) : 015.8.
1 6H X(12.7H) VS X(12.3H) = C15.8.
2 6H x<I2.7H) vs X(I2.3H) = 015.8 )
40 FORMAT (1H 7H XT12.12H) vs v =015.a.
1 OH X(12112H) VS Y 301598!
2 6H xc12.12M) vs v =015.a )
45 FORMAT (1H 21H y vs Y =015.8)
50 FORMAT (1H063H AVERAGE VALUE OFMPRZI

- VARIABLES/l )

55 FORMAT(1H077H RESIDUAL SUMS 0F SQUAMPRZi

-RES AND CROSS PRODUCTS//)

60 FURMAT(1H069H PARTIAL CORRELATIMPR2:

-ON COEFFICIENTS//)
65 FORMAT (ZDHO STANDARD ERROR OF Y = F14.8 )
70 FORMAT (11H F LEVEL F14.8/29H STANDARD ERROR 0F Y z F14.8/

7b

90 FORMAT

91 FORMAT
92 FORMAT
96 FORMAT

654
905
906
907
1004
1019
1344
1395
910

9999

3175

{

23R 0F COE‘

FORMA

IQATA
2EEDOM

13H
25:3H

FORMA
FORMA
FORMA
FORMA
FORMA
FORMA
FORMA
FORMA
CDNTI

T

//

(10H COMPLETED
(22H4STEPNISE
15 //18H

56H
I16H

NO OF

F12.2 //)

(9HDSTEP No.15 /19h
(94OSTEP No.15 /20H
(5H RUN

F60013H

F10.5/(14H

F10.5))
ZERO NUMBER OF DATA.
IN CONTROL CARD.
VMIN PLUS,
ERROR.NOIN MINUS.
(1H037HY SQUARE NON-POSITIVE.TERMINATE STEP I
(1H029H NO MORE DEGREES FREEDOM STEP 1 b )

(1H010H SQUARE X915.17H NEGATIVE.
K=O.

T I
T

T
T (
T

T
T
T I
NUE

31H

26H

12H

READ 9999a

FORMA
IF
STOP
END

SIMULTANEOUS FIT
IDENTICAL
pUNCHFD
UATA(1)
DATA(2)
DATAIS)
DATAIA)
DATAIS)

FIT 1
OETAI
CUEFF
COFFF
COEFF
COEFF
CDEFF
COEFF
COEFF
CDEFF
COEFF
CDEFF
COEFF
COEFF
CCEFF
COEFF
CDEFF
COEFF
COEFF
CDEFF
COEFF
CCEFF
COEFF
COEFF
CDEFF
CUEFF
COEFF

IN

T (

S
N A
0F
OF
OF
OF
0F
OF
0F
0F
OF
0F
OF
OF
OF
0F
0F
OF
OF
0F
OF
OF
OF
OF
0F
OF
OF

A8)

(KONTIN

AA
AB
AC
AD
AE
AF
AG
AH
AI
AJ
AK
AL
AM
AN
AD
AP
A0
AR
AS
AT

(42H ERROR
(29H ERROR:

STEP I6:

KONTIN

BHCONTINUE ) 3175:

T0 THAT
DECK 0F
= NUZRO
NUZRO
NUZRO
AZRO

8
ALPHA84
BFTAK4
ETAJ4
BFTAJA
BETAJK4
M7RO
DZROJK
DZROJ
ALPHAA4
ETAK4
DZROK
HZROJ
H7ROJK
H7ROKJ
H7ROK
ETA44
GAMMA-A(44)
GAMMA-A(L4L4)
GAMMA-8(44)
GAMMA-8(L4L4)

THIS PROGRAM

VARIABLES

F10.5,5H

OF CHSBR NU4 AND
REPORTED
RESULTS.
FOR NU-4
FOR ?NU'4 PERPENDICULAR
FOR 2NU'4 PARALLEL

145

VARIABLE COEFFICIENT
X-IS:F15.8:F15.8))

V5.20H STEPS OF REDRESSIUN)
REGRESSION IIIZF PROBLEM N0

VARIABLE REMOVED 18)

VARIABLE ENTERING IO)

F10.5.5H F10.5,3H
F10.D:6H F10.S,3H

F1005,
F10.5:5H

SO LONG.)

PRCBLEM TERMINATED)
SOLONG)

SOLONG )
5)
SOLONG 15,6H STEPS)
7H SOLONG)

100. 3175

2NU4.

IN FAPERS( OBJECT HERE IS TO

AE‘ZETA + GARBAGE

STD ERRMPRZI

MPRZI
MPRZI

110 //13H NO OF MPRZI
110 //60H NEIGHTED DEGREES OF FRMPRZ‘

MPR2|
MPH?!
MPRZI

F10.MPR2!

MPRZI
MPRZI
MPR2I
MPRZI
MPR21
MPR2I
MPRZI
MPRZI
MPR2|

12

DATA(5)
DATAI7)
DATAIB)
UATA(9)

DATAI10) =
DATAIll) 3

AND FROM

SURTOFF =
ENUHEO

AK
AI
AJ
AA

A8
A0

NOVAR

BZROtAF

146

DZROJ‘AH

THF MIRROMAVE QUANTITIES NE FORMED
DZROJKtAG -

4.173.62?.922.692.532.422.632.272~212-162'092'011'93
4,086.322,842-612045203420252’182'122'082.001'921.84
4'903.152,762.552.372.252.172.102o041~991'921|841'75
3.923,072.68?.492.292.172.092.021.961.9llo8511751'66
3.843.002.602.372.212.102.011.941.881.831.751.671-57
7.565,394.514.023.703.473.303.173.072.982.842n702-55
7.315.184.313.833.513.293.122.992o892-502'662'522'67
7,084.984.133.653.343.122.952.822.722.832.502.652.20
6,354.793.953.453.172.962.702.662.562.472.642.192.05
6,.34,613.783.323.022.802.642.512.412.322.182.041-88
0.3185537
0.00000427
0.000000333

0
PP
up
PP
pp
PP
pp
pp
pp
pp
PP
pp

OD
up
on
PP
pp
PP
pp
pp
RR

4. 6
4. 9
4:12
4,16
4.19
4.23
32 4
3. 7
3:10
3.15
3.17

3. 4
3. 8
3:11
1. 6
1: 9
1:13
1:19
1:25
0. 1

6020.451
3018.540
5016.657
3014.129
6012.248
3009.777
6030.854
3028.955
3027.054
3025.164
6022.644

6045.077
6040.566
6058.641
6086.742
6084.845
6082.352
6078.573
6074.882
6100.689

:2)
-.
\4

4b.

HO“ II II II

CD

,0
o
d‘

0.07
0.05
0.00
0.05
1.00
0.14
1.00
0.07

0.07
0.07
0.67
0.00
0.01
0.05
0.00
0.00
0.00

‘2290.

DZROJK.

DZROJ.
NORFTS

F‘HFJP*HFJF‘HPJP*H

TURJNFURDNFCRJN

PP
PP
PP
PP
PP
pp
PP
PP
PP
PP
PP

QP
OP
OP
PP
PP
PP
PP
PP
PP

CHSBR

CHSBR

CHSBR

d. 7
4.10
4.13
4,17
4.20
4.24
31 5
3. 8
3.11
3.15
3.18

3. 6
3. 9
3.12
1. 7
1.11
1.14
1.20
1.26
0. 2

3019.809
3017.913
3016.016
3015.497
3011.634
3009.087
3030.217
3028.622
3026.428
5023.901
3022.017

6041.813
6039.912
6038.022
6086.118
6083.588
6081./05
6077.976
6074.183
6100.059

0.02
0.07
0.07
0.05
0.01
0.00
0.14
0.40
0.14
0010
0.07

0.07
0.67
0.03
0.00
0.00
0.05
0.10
0.00
0.00

HFJP‘HTAF#F*HF*P‘H

TONJMFURJNDOR3M

PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP

up
0P
op
PP
PP
PP
PP
PP
RP

4. 8
4:11
4.15
4.18
4:21
3: 3
3o 6
3. 9
3.12
3.16
5:19

3: 7
3.10
6:13
10 8
1:12
1015
1:21
1027
09 3

3019.179
3017.288
3014.755
3012.866
3010.999
3031.493
3029.587
3027.688
3025.791
3023.272
3021.390

6041.184
6039.278
6037.380
6085.473
6082.965
6081.091
6077.353
6073.543
6099.445

0.12
0.05
0.02
0.02
0.01
0.12
1.00
0.07
1.00
0.07
0.02

0.17
0.4

0.67
0.00
0.01
0.01
0.10
0.00
0.01

HW‘P‘HIJF*HOJF*HFJ

RJNFURJNTURJNTV

APPENDIX III

LISTING OF SCAN PROGRAM

SCAN was used to translate the raw data from the
Hydel film reader into fringe numbers and ultimately into
frequencies for the transitions comprising the spectra.

Included is a typical, though short, data deck.

147

148

LISTING OF SCAN PROGRAM

PROGRAM SCAN
DIMENSION XMI6). YMIb). DELXI6). xROTIb). FRNGX(50). PSEPI50).
1NFGSEPISD). KHE01I3).‘HE02(3)0IPSEPI3).IHEAO(10)
COMMON THETA. DEL!
READ 4. KhEO1I1).KHFDZII).KHE01(2).KHE02(2)
4 FORMAT (A8.88.10X.A8.A8)
1 REAO 3. (IHEADII). 131.10)
PRINT 3. IIHEADII). 131.10)
IF (IHEADIIG) ' OHEND HEAD) 1.2.1
2 IH = 1
NCAL : 1
1P 2 0
NFRM = 0
PENSEP = 0.
101 READ 1009 (”MII)0 YM(I)O I 3 1:6)! ICODEO NFIN
IF IICOOE ” D ) 20091011205
205 IF (ICOOE ' 7) 210.210.200
200 PRINT 206
206 FORMAT(///. IMPROPER 000E NUMBER IN THIS FRAME - ENTIRE FRAME N0 0
1000 till)
60 TO 101
210 CONTINUF
GO TO (1000.2000.3000.4000.5000.6000.102). ICODE
102 IF (NFIN - 1H0) 103. 110. 103
110 GO TO 1
103 STOP-
‘000 NIR = 1
[FRAME I XMIl)
IFRMG = Y~(1)
IOP = XMI?)
NOFR a 0
IF (NFRM g 2) 101.7000.101
7000 FRNn : IFRNG
NFRM a NFRM o 1
CALL LSTAN<XM.YM)
GO TO (200302004). IH
9004 PRINT 9701
7003 PRINT 9702
PRINT 3. (IHEADII). 131.9)
PRINT 9200.IFRAME.KHE01(IH).KHE02(IR).IOP.IFRNG.THETA.(DELX(I).
1131.6)
00 T0 101
3000 IH 31
DO 3007 J 3 10622
IF IX”(J)) 101.101.3003
3003 IF (X“(J*1)) 101.10123002
3002 IP : IP 0 1
PSEPIIPIRROTATETXMIJ‘1).Y%(J*1).THETA) ' ROTATEIXM(J).YM(J)'THETA)
JCNT I (J‘1)/2
3007 IPSFPIJCNT) : PSE°(IP)

4000

5000
“002
“001

6000
6100

6600

66‘0
6200

.002

6003

6004
£006

6011
6020
‘001

‘015

6010

6005

7000

7001

149

PRINT 9100. (IPSEPII).I=1.3)

GO TO 101

NCAL : ?

A = Y“(1) * XM(2)*0.00001

B = YN(?I * XM(3)*0.00001 + YM(3)*0.0000000001
PRINT 9400; A: R

GO TO 101

00 5001 J = 1.6

IF rxv(J)1 $002,101.5002

NOFR : MOFR . 1

FRNGX¢NOFR) : ROTATF(XM(J).YMIJ).THETA)
NFGSE”(MOFR) x FHNGXIVDFR) - FRNGXCNDFR-i)
GO TO 101

GO TO (6100.62001. MIR

PRINT 9500. 1FRAMF.(NFGSEP(I).I=1.NOFR)
PRINT 970?

MIR 3 2

GO TO (6600.66501. NCAL

PRINT 9A00. IFRAHE

GO T0 6200

PRINT 9650. IFRAME. A. 8

DO 6001 J3116

IF IX“IJ)I 6002.101.6002

XIR : PFNSEP . ROTATEIXMIJ).YM(J).THETA)
DO 6003 I=1.NOFR

IF IXIR - FRNGXIII) 6004.6005.6003
CONTINUF

00 TO 6010

IF (I-lI 6015.6015.6006

FRAnT = (XlR-FRNGXII'1))/IFRNGX(I)-FRNGX(I91))
FRK = I v 2

FRNO a FRAG + FPK + FRACT

INTFRP 8 3H

GO TO (9000.90501. NCAL

CONTINUE

CONTINUE

00 T0 101

FRACT : (XIR - FRNGY(1))/(FRNGX(2) - FRVGXI1))
FRNO I FRAG + FRACT

INTFRP : SHFXL

GO TO 6011

FRACT = (X19 - FRNGX(VOFR))lTFRNGXIMOFR) - FRNGX(NOFR-1))
FNOFR = NCFR

FRNO : rRAG + FRACT + FNOFR - 1.

INTFRP : aHEXR

GO TO 6011

FRK = I - 1

FRNn - FRAG . FRK

GO TO 6011

IH a ?

PSEPSUM = 0

PIP = IP

DO 7001 J ‘ 101°

pSEpSUM = PSEPSUM + PSERIJ)

jI

 

fi.‘ .‘AAO' .
l

150

PFNSEP : PSEPSUM/RIP

PRINT 9700. PENSEP

GO TO 101

PRINT 9651.

GO TO 6020

FRED : A 6 PtPRNU

PRINT 9652. F900.

GO TO 6020

3 FORMAT c1nA81
100 FORMAT 0612;500:1XI011101’

9200 FORMATIIOH FRAME MO.I$.5X2A8.1OX7HOP. N0.15./5X19HFIRST FRINGE IS
1N0.15/5x1TMROTATION ANGLE ISF9.6.8H RADIANS/9X17HIROTATIDN FITS T0
96¢F5.1.1M.I.9H MICRONSII)

9300 FORMAT (38H OBSFRVEO PEV SEPARATIONS (IN MICRONS).6(15.1H.I)

9400 FORMAT(SZH THE FDLLONING CALIBRATION CONSTANTS HAVE BEEN INPUT/5X
138A =F10.4.5X3HB =F9.8//52M THE FRINGE POSITIONS AND FREQUENCIES w
2ILL 8F OUTPUT/1H1)

9500 FORMAT(?SH FRINGE SEPARATIONS IN FRAMEI5.10H IMICRONSTIISX1OI7II

9600 FORMAT ¢4eH0FRINGE ROSITIDNS 0F LINES MEASURED IN FRAME N0.. ISII

9650 F0RMAT<SSH FRINGE POSITIONS AND FREQUENCIES MEASURED IN FRAME N0.
115/5X30HCALIBRATION CONSTANTS USED AREIIOXSHA =F10.4.5X3HB =F9.6//
25X15MFRINGE POSITIONlDXlBHORSERVED FREQUENCY/1

9000 FRNO. IMTERD

Q0‘50

FREQ: INTFRP

9651 FORMAT (F16030 EX: A3)
9652 FORMAT (F16.3. F27.3. 5X. A3) _
9700 FORMAT (32HOPEN SEPARATION FOR THIS CHART =F5.1. 8H MICRONS)
0701 FORMAT (181)
9702 FORMAT (1RD)
END
FUNCTION ROTATE (ALPHA. BETA. GAMMA)
RDTATE s (1. - GAMMA*GAMMA/2.)¢ALPHA - GAMMA-BETA
END
SUBROUTINE LSTAN (X.Y)
DIMENSION XI6). Y(6I. XCALC<6I. OELXI6>
COMMON THETA. UFLX
SUMY : n
SUM! a n
SUMYY = 0
SUMYX = 0
DO 20 I = 1.6
SUMY - SUNY + YIII
SUMX a SUVX 0 XII)
SUMYY = SUMYY + YIIT*Y(I)
20 SUMYX = SUMYX + YIII*X(I)
THETA = ISUMYX - SUMY*(SUWX/6.))/ISUMYY - SUMY*(SUMY/6.))
DO 30 I 3 1:6
30 DELXII) = (SUMX-THETAPSJMYII6. * TMETA.Y<I> - X(I)
END

LINE POSITIONS LINE FREQUENCIES
FREQUENCIES FOR 0365-F
5537.42404 0.5359293000

8 000 7

036

W

Wmflxmn Lila-Earn.

END
s-F

1975621326 1976425947 1977630197
198213112? 1302622877 1462231147
1810131116 1810922932 1972131199

1977634538 1977637419 1978341477
1482222831 1640631159 1661322867
1994222961 2149431281 2170822915

2313131217

51 000
1591821707
1186630907
154883110“
1913131199

17 515
1912320089
1242923369
2162893369
3106423367
1366627875
1713628441
1948630055
2989129351
2633928774
299212934?

18 526
2T20620060
1242623204
2467923204
3405423204
4048723204
1175329635
1491129836
1949929149
2373230459
2559931168
2917678241
3311129240
3906931333
4494129383
4510629081

19 545
2335120039
1078893415
2005923415
2943623169
3876123169
1083330128
1481528548
1809129060
2189227698
2609627476
2858829401
3213329159
3495928845
3918331619
4942328640

2335622880
7
0590127155
1210029931
4572722889
1938322981
7
1913924962
0403623362
7394623362
3261323362
1408827950
1752428201
1987330103
7392328283
2665328355
8009129061
7
2120024583
4401323204
9337123204
3262623204
4211623204
1220423565
4534929836
9064228049
2371630459
2635228297
2953229137
3392627673
3962028742
4221630089
4563829385
7
2335625254
1234623415
2167123415
3109623169
4031623169
11715291425
1503728835
1863127460
9253327698
2668628043
2924628224
3248328745
3534827975
3935630753
4426130446

2474631505

1589431687
1308931068
1673330815
2036131022

1914629135
1552923362
2478823362

1479027875
1815330226
2041130317
2351128604
2738329425
3070129157

2120829112
1555823204
2497423204
3416123204
4367923204
1285829351
1668629369
2094928369
2371630459
2740329708
3033629713
3444330450
3997828376
4254129057
4584130312

2336330296
1377123415
2315123415
3256623169
4188323169
1191129659
1588629259
1952327884
2331629095
2680628134
2969630219
3288629047
6583928276
4020427943
4542827647

151

2497223024

1588336190
1332822845
1697422795
2059923000

1915633405
1699023362
2638123362

1542128014
1840730527
2063829999
2436329844
2845629137
3123629668

2120932495
1707323204
2653123204
3571623204
4509623204
1300629070
1761328859
2155728371
2422931363
2767929571
3075828403
3544129529
4028528701
4304929057

2336134004
1531923415
2467823415
3402823169
4351823169
1241730643
1616328810
2006127669
2381628013
2734228566
3052628469
3340629637
3656129001
4134328673

1588537611
1426730960
1791431094

1916737401
1848723362
2796223362

1604728739
1880130162
2119199579
2510728758
2881129669
3197329810

2121336324
1858123204
2802223204
3735123204

1372929070
1830529013
2277628900
2467630941
2803129493
3149129097
3588628475
4086433600
4323529702

2336638024
1688123415
2625823415
3559323169
4513123169
1377329002
1675129807
2033327579
2454928295
2787728304
3145627668
3427631145
3667029336
4165627737

1587341437
1449922936
1816622817

1917141681
2008123362
2947623362

1676229233
1914430055
2246329446
2567328680
2926528498
3240629792

2121441439
2006623204
2952423204
3897623204

1417329638
1888630294
2305629037
2520330056
2865828404
3221629293
3828830231
4174129211
4373630749

2336841497
1845223415
2779123169
3717823169

1451630186
1748028609
2114529590
2497629371
2831828991
3194829273
3463729470
3720629208
4200327471

000000000O‘U'IUIUIUINHOOOIOO‘O‘O‘O‘OO‘U‘U‘UIUINHO‘OOO‘OOWWU’INHGMUMHM

0365'
0365'
0365‘
0365'
0365'
0365'
0365'
0365‘
0365'
0365'
0365‘
0365‘
0365'
0365‘
0365‘
0365'
0365'
0365‘
0365'
0365'
0365‘
0365'
0365'
0365‘
0365'
0365'
0365'
0365'
0365‘
0365'
0365‘
0365'
0365'
0365'
0365‘
0365‘
0365'
0365‘
0365‘
0365'
0365‘
0365‘
0365'

I.

Email—"‘7.- -

VI

APPENDIX IV
LISTING OF OUTPUT FROM SIMUL‘Z‘ANEOUS

FIT OF CHgBr v“ AND 2v”

The data listed here were part of the output of

the final simultaneous least squares computer fit of CH3Br

us and 2v“, and correspond to the values of the molecular

constants listed in Table XI. Listed here are:

1. the assignment of each transition [AKAJK(J)],

2.

3.

the observed frequency of each transition,

the calculated frequency for each transition, obtained
from the values of the molecular constants listed in
Table XI,

the deviations (“ob ),

the weight assigned to each transition.

- v
s calc

152

153

LISTING 0P FINAL DATA FROM 81MULTANEOUS ANALYSIS OF CHSBR Nu4 AND 2NU4
dY MEANS OF FALSTAF PROGRAM LISTtD IN APPENDIX II.

ASSIGN. OBS. FREQ. CALC. FREQ. RESID. NHT.
PP 5. 5 3011.8740 3011.8717 0.0023 0.00
PP 5, 6 3011,2400 3011,2371 0.0029 0.00
PP 5. 7 3010.5910 3010.6031 -0,0121 0.00
PP 5. 8 3009,9950 3009,9695 0.0255 0.00
PP 5, 9 3009,3420 3009,3364 0.0056 0.00
PP 5.10 3008.7210 3008.7039 0.0171 0.00
PP 5.11 3008,0960 3008,0719 0.0241 0.01
PP 4. 4 3021.7130 3021.7120 0.0010 0.05
PP 4. 5 3021.0810 3021.0768 0.0042 0.30
PP 4, 6 3020.4510 3020.4421 0.0089 0.07
PP 4. 7 3019.8090 3019.8079 0.0011 0.02
PP 4. 8 3019.1/90 3019,1741 0.0049 0.12
PP 4. 9 3018.5400 3018.5409 -0.0009 0.14
PP 4,10 3017,9130 3017,9081 0.0049 0.07
PP 4.11 3017.2880 3017,2758 0.0122 0.05
PP 4.12 3016.6570 3016.6441 0.0129 0.05
PP 4.13 3016.0160 3016.0128 0.0032 0.07
PP 4.15 3014.7550 3014.7520 0.0030 0.02
PP 4,16 3014,1290 3014,1223 0.0067 0.07
PP 4.17 3013.4970 3013.4933 0.0037 0.05
PP 4.18 3012.8660 3012.8648 0.0012 0.02
PP 4,19 3012.2480 3012,2368 0.0112 0.05
PP 4.20 3011.6340 3011,6095 0.0245 0.01
PP 4.21 3010.9990 3010.9827 0.0163 0.01
PP 4.23 3009,7770 3009,7310 0.0460 0.00
PP 4.24 3009.0870 3009.1060 -0,0190 0.00
PP 3. 3 3031.4930 3031.4979 -0,0049 0.12
PP 3, 4 3030.8540 3030,8622 -0.0082 0.05
PP 3. 5 3030.2170 3030.2268 -0.0098 0.14
PP 3, 6 3029,5870 3029,5920 -0.0050 1.00
PP 3. 7 3028,9550 3028,9576 -0,0026 1.00
PP 3. 8 3028.3220 3028.3236 -0,0016 0.40
PP 3. 9 3027,6880 3027,6902 -0.0022 0.07
PP 3.10 3027.0540 3027.0572 -0,0032 0.14
PP 3.11 3026.4280 3026.4247 0.0033 0.14
PP 3.12 3025,7910 3025,7927 -0.0017 1.00
PP 3.13 3025.1640 3025.1612 0.0028 1.00
PP 3.15 3023,9010 3023,8997 0.0013 0.10
PP 3.16 3023.2/20 3023,2698 0.0022 0.07
PP 3.17 3022.6440 3022.6404 0.0036 0.07
PP 3.18 3022.0170 3022.0115 0.0055 0.07
PP 3,19 3021.3900 3021,3832 0.0068 0.02
PP 3.20 3020,7640 3020,7555 0.0085 0.05
PP 3.21 3020.1370 3020.1283 0.0087 0.05
PP 3,22 3019,5120 3019,5017 0.0103 0.05
PP 3.23 3018.8910 3018.8757 0.0153 0.01
PP 3.24 3018.2580 3018.2503 0.0077 0.01
PP 3.25 3017.6400 3017.6255 0.0145 0.01

 

ASSIGN.

PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP

3.26
3.27
3.23
3.30
3031
3332
3.33
3.34
3.35
3.36
3.38

154

035. FREQ.

3017.0070
3016.3690
3015.7650
3014.5160
3013.9030
3013.2840
3012,6270
3012.0920
3011.4150
3010.8000
3009.5620
3008.9800
3007,7400
3041.2230
3040.5930
3039.9590
3039.3190
3038.6660
3038.0480
3037.4130
3036.7840
3036.1410
3035.5130
3034.8860
3034.2950
3033.0000
3032.3560
3031.7340
3031.1110
3030.4/80
3029.8570
3029.2290
3026.7220
3026.0090
3025.4810
3024.8960
3023.6130
3022.9870
3022.3640
3050.9070
3050.2570
3049.6420
3048.9890
3098.3320
3047.7230
3047.0830
3046.4940
3045.8190
3045.1860
3044.5940
3043.9280
3043.2910

CALC. FREQ.

3017,0013
3016,3777
3015,7548
3014.5107
3013,8897
3013,2693
3012,6496
3012.0306
3011.4122
3010,7945
3009.5613
3008,9457
3007,7167
3041.2292
3040.5929
3039,9571
3039,3217
3038,6867
3038,0521
3037.4180
3036.7844
3036.1512
3035,5185
3034.8862
3034.2545
3032,9925
3032.3623
3031.7325
3031.1034
3030,4747
3029,8466
3029.2191
3026.7145
3026,0898
3025,4658
3024,8423
3023,5972
3022.9/56
3022,3547
3050,9052
3050.2685
3049,6321
3048,9962
3048,3607
3047,7256
3047.0909
3046.4567
3045,8228
3045.1895
3044,5566
3043.9241
3043.2922

RESID.

0,0057
0.0113
0.0102
0.0053
0.0133
0,0147
0,0074
0.0214
0.0028
0.0055
0.0207
0,0343
0.0233
“0.0062
0.0001
0.0019
'0.0027
-0.0007
“0.0041
“0.0050
“0.0004
'0.0102
-0.0055
“0.0002
0.0005
0.0075
.030063
0.0015
0.0076
0.0033
0.0104
0.0099
0.0075
-0.0808
0.0152
0.0037
0.0158
0.0114
0.0093
0.0018
'0.0115
0,0099
'0.0072
'0.0087
-0.0026
“0.0079
-0.0027
-0,0038
“0.0035
-0.0026
0.0039
-0.0012

HHT.

0.02
0.01
0.01
0.02
0.00
0.01
0.00
0.02
0.04
0.02
0.00
0.00
0.00
0.05
0.02
0.05
0.05
0.02
0.07
0.12
0.02
0.01
0.05
0.10
0.10
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.01
0.01
0.00
0.01
0.01
0.00
0.01
0.03
0.05
0.07
0.07
0.07
0.10
0.04
0.04
0.12

 

ASSIGN.

PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP
PP

1.15
1.16
1.17
1.18
1.19
1.20
1.26
1.27
1.29
1.30
1.31
1.32
0. 6
0. 7
0. 8
0. 9
0.10
0.11
0.12
0.13
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.26
0.27
0.29
0.30
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
1.15
1.16
1.17
1.18
1.19
1020
1.21
1.22
1.23
1.24

155

088. FREQ.

3042.0330
3041.4070
3040.7/40
3040.1440
3039.5160
3038.8880
3035.1360
3034.5080
3033.2620
3032.6400
3032.0240
3031.4050
3056.7050
3056.0690
3095.4340
3054.8030
3054.1/10
3053.5340
3052.9070
3052.2/60
3051.0160
3050.3820
3049.7490
3049.1260
3048.5000
3047.8650
3047.2600
3046.6900
3045.9890
3045.3/00
3044.1180
3043.4910
3042.2920
3041.6280
3041.0150
3040.3890
3039.7600
3039.1440
3038.5270
3067.9060
3037.2880
3056.6/30
3099.8/80
3059.2290
3058.5840
3057.9350
3057:3060
3056.6440
3055.9930
3055.3390
3054.6900
3054.0260

CALC. FREQ.

3042,0296
3041.3991
3040.7691
3040,1396
3039.5106
3038.8822
3035.1230
3034.4984
3033.2511
3032.6284
3032.0062
3031.3847
3056.7076
3056.0720
3055,4385
3054.8045
3054.1710
3053,5379
3052.9053
3052.2731
3051.0101
3050.3793
3049.7490
3049,1192
3048,4899
3047.8611
3047.2329
3046.6051
3045.9779
3045.3512
3044.0995
3043,4145
3042.2263
3041.6030
3040,9804
3040.3583
3039,7369

"3039.1161

3038,4959
3037.8763
3037,2574
3036.6392
3059,9322
3059.3012
3058.6707
3058,0406
3057.4110
3056.7819
3056.1533
3055.5252
3054,8976
3054.2/05

RESID.

0.0034
0,0079
0.0049
0.0044
0,0054
0.0058
0.0130
0.0096
0.0109
0.0116
0.0178
0.0203
'0.0026
'0.0038
.0.0045
“0.0015
0.0000
'0.0039
0.0017
0.0029
0.0059
0.0027
“0.0000
0.0068
0.0101
0.0039
0,0271
0,0449
0.0111
0.0188
0.0185
0.0165
0,0257
0.0250
0,0346
0.0307
0.0231
0,0279
0.0311
0.0297
0.0306
0,0338
-0.0542
.030722
.030867
-0.1056
'0.1050
-0.1379
“0.1603
.0.1862
-0.2076
“0.2445

NHT.

0,05
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0,01
0.01
0.00
0.00
0.02
0.07
0,30
0.30
0,30
0,40
0.25
0.05
0.07
0.07
0.01
0,04
0.04
0,02
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0,00
0.00
0.00
0.00
0.00
0,00
0.00
0.00
0.00
0.00
0.00
0,00
0.00
0.00

0.00~

0.00
0,00
0.00

 

ASSIGN.

RR
RP
RP
RR
RR
RP
RR
RR
RR
RR

RR‘

RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR

1.25
1.26
1.27
1.28

156

088. FREQ.

3053.3/00
3052,7620
3052.0/90
3051.4420
3050.7920
3050.5510
3080.2200
3080.8600
3081.4990
3082.1370
3082.7/60
3083.4200
3084,0560
3084.7010
3085.3430
3085.9890
3086.6260
3087.8970
3088.5500
3089.2060
3089.8310
3090.4750
3091.1260
3094.9920
3095.6200
3096.9090
3097.5480
3098.1940
3098.8950
3099,4330
3100.1600
3100.7860
3102.0630
3102.7000
3089,6570
3090.3020
3090.9420
3091.5350
3092,2210
3092.8950
3093.4990
3094.1680
3094,7820
3095,4220
3096.7020
3097.3420
3098.0020
3098,6250
3099,2690
3099,9130
3100,5990
3101,2000

CALC. FREQ.

3053.6440
3053.0180
3052.3926
3051,7677
3051.1434
3050.5197
3080.2250
3080,8633
3081.5019
3082.1408
3082.7800
3083.4194
3084,0591
3084.6991
3085.3393
3085,9797
3086.6203
3087,9023
3088.5436
3089,8268
3090,4686
3091.1107
3094.9660
3095,6090
3096.8953
3097,5386
3098.1820
3098.8255
3099.4690
3100.1126
3100.7562
3102.0437
3102.6874
3089.6638
3090,3023
3090,9410
3091.5800
3092.2193
3092.8588
3093.4985
3094,1385
3094,7787
3095.4191
3096,7005
3097.3414
3097.9826
3098,6239
3099,2654
3099.9070
3100,5488
3101.1908

RESID.

-0.2740
-0.2860
-0.3136
-0.3257
'0.3514
0.0313
'0.0050
-0,0033
“0.0029
“0.0038
'0.0040
0.0006
-0.0031
0.0019
0,0037
0.0093
0.0057
-0,0053
0,0064
0.0209
0.0042
0.0064
0.0153
0.0260
0.0110
0.0137
0.0094
0.0120
0.0195
0.0140
0.0174
0.0296
0.0193
0.0126
'0.0068
-0.0003
0.0010
0.0050
0.0017
“0.0038
0.0005
“0.0005
0.0033
0.0029
0.0015
0.0006
0,0194
0,0011
0.0036
0.0060
0.0102
0.0092

NHT.

0.00
0.00
0.00
0.00
0.00
0.00
0.10
0.10
0.30
0.25
0,14
0.30
0,14
0,14
0.14
0,07
0.07
0.02
0.02
0.01
0,05
0.01
0.01
0,00
0.00
0.01
0,01
0.01
0.02
0,01
0.01
0.00
0.01
0.00
0.25
1,00
1.00
1.00
1.00
1.00
1.00
0,40
0.40
0,40
1,00
1.00
0.00
0,07
0.07
0.02
0,02
0.00

J

W“‘ Will-“K
. 1

III I 11.1]

ASSIGN.

RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RP
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR

3022
3.23
3.24
3.25
3.27
3023
3.29
3030

157

OBS. FREQ.

3101.8420
3102.4830
3103.1280
3103.7/30

3105.0990.

3105.7050
3106.3490
3106.9910
3107.6340
3083.9430
3083.2880
3082.6430
3082.0250
3081.3980
3080.1260
3079.4650
3099.0380
3099.6740
3100.3160
3100.9540
3101.5960
3102.2350
3102.8/50
3103.5150
3104.1560
3105.4410
3106.0350
3107.3570
3108.0030
3108.6500
3109.2930
3109.9320
3110.5670
3111.2270
311108380
3112.5030
3113.7910
3114.4330
3115,0730
3115.7190
3116,3690
3117.0090
3103.3330
3108.9/20
3109.6160
3110.2920
3110.8950
3111.5340
3112.1’50
3112.8200
3114.1000
3114.7380

CALC, PREQ,

3101.8328
3102.4750
3103.1173
3103,7597
3105,0448
3105,6875
3106.3302
3106,9730
3107,6159
3083.9308
3083,2954
3082.6604
3082.0257
3081,3914
3080.1240
3079.4909
3099,0376
3099.6762
3100.3151
3100.9541
3101.5934
3102.2329
3102.3/26
3103.5125
3104.1526
3105,4334
3106.0/41
3107.3558
3107.9969
3108.6382
3109.2/95
3109.9210
3110,5626
3111,2044
3111.8462
3112.4881
3113.7721
3114.4142
3115,0564
3115.6986
3116,3409
3116.9832
3108,3439
3108.9826
3109.6214
3110.2605
3110,8997
3111,5392
3112.1/88
3112.8186
3114,0987
3114.7390

RESID.

0.0092
0.0080
0.0107
0.0133
0.0142
0.0175
0.0188
0.0180
0.0181
0.0122
-0,0074
“0.0174
.0.0007
0.0066
0.0020
'0.0059
0,0004
'0.0022
0.0009
-0.0001
0.0026
0.0021
0,0024
0.0025
0.0034
0.0076
0.0109
0.0012
0.0061
0.0135
0.0110
0,0044
0.0226
0.0115
0,0149
0.0189
0.0180
0,0100
0.0204
0.0201
0,0250
'0.0109
'0.0106
-0,0054
-0,0085
'0.0047
“0.0052
'0.0038
0.0014
0.0013
”0.0010

NHT,

0.01
0.01
0.02
0,02
0,01
0.01
0.02
0.02
0.02
0.00
0.00
0.01
0,05
0,01
0.00
0,00
0,14
0.10
0.07
0,30
0,07
0.14
0.12
0.07
0.12
0.05
0.02
0,02
0.02
0,05
0.01
0.01
0,00
0.00
0.01
0.01
0.02
0.01
0.01
0.01
0.02
0,02
0,07
1.00
1.00
1.00
0.40
0,40
0,14
0.12
0,05
0,01

 

ASSIGN.

RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR

5.17
5.18
5.19
5:20
5021
5022
5023
5.25
6. 6
6. 7
6. 8
6. 9
6010
6011
6012
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6021
6022
6:23
6.24
6.25
6.27
6.28
6.29
6030
6.31
6032
6033
6.34
6.35
6036
6.37
6.38
6.40
6.41
7. 7
7. 8
7. 9
7010
7012
7.13
7014
7015
7016

158

OBS. FREQ.

3116.0190
3116.6600
3117,2900
3117.9340
3118.5/60
3119.2110
3119.8480
3120.4/80
3121.1160
3117.5850
3118,2220
3118.8610
3119.5010
3120.1420
3120.7180
3121.4160
3122,0550
3122.6970
3123.3360
3123.9/30
3124.6150
3125.2560
3125.9000
3126.5410
3127.1860
3127.8200
3128.4640
3129.1010
3129,7450
3131.0300
3131.6/00
3132.3110
3132.9580
3133.5960
3134.2270
3134.8130
3135.5230
3136.1570
3136.7960
3137.4410
3138.0800
3139.3630
3140.0050
3126.7420
3127.3/70
3128.0190
3128.6580
3129.9380
3130.5/50
3131.2200
3131.8600
3132.4990

CALC, 1REQ,

3116.0200
3116,6607
3117.3016
3117,9425
3118,5835
3119.2247
3119.8659
3120,5072
3121.1486
3117.5801
3118,2187
3118,8575
3119.4965
3120,1357
3120.7/50
3121,4145
3122.0542
3122.6940
3123,3339
3123,9739
3124.6141
3125.2544
3125,8948
3126.5353
3127.1758
3127,8165
3128,4572
3129.0980
3129.7388
3131.0206
3131.6616
3132.3026
3132.9436
3133,5846
3134.2256
3134.8666
3135,5076
3136.1485
3136.7895
3137.4303
3138.0712
3139.3527
3139,9933
3126.7436
3127.3822
3128.0209
3128,6597
3129.9379
3130.5772
3131.2166
3131,8562
3132.4958

RESID,

'0.0010
'0.0007
I'0.0116
-0,0005
'0.0075
-0.0137
-0,0179
-0.0292
“0.0326
0.0049
0.0033
0.0035
0.0045
0.0063
0.0030
0.0015
0.0008
0.0030
0.0021
.0.0009
0.0009
0.0016
0.0052
0.0057
0.0102
0.0035
0.0068
0,0030
0.0062
0,0094
0.0004
0.0084
0.0144
0.0114
0.0014
0.0064
0,0154
0.0085
0.0065
0,0107
0.0088
0.0103
0,0117
“0.0016
'0.0052
-0,0019
'0.0017
0.0001
'0.0022
0.0034
0.0038
0.0032

HHT,

0.01
0.01
0,00
0,00
0,01
0.01
0.00
0,00
0,00
1.00
0.40
0.40
0,40
0,40
0,40
0.40
0.40
1.00
0,40
0.25
0,14
0.14
0,07
0.05
0,02
0.02
0.02
0,02
0.01
0.01
0,01
0.01
0,00
0,00
0.01
0.00
0,00
0.01
0.01
0.01
0.00
0.01
0,01
0.07
0.03
0,30
0.10
0,14
0.12
0.04
0.07
0,04

 

ASSIGN.

RR 7,17
RR 7.18
RR 7.19
RR 7.20
RR 7.21
RR 7.22
RR 7,26
RR 7.24
RR 7.25
RR 7.26
RR 7.27
RR 8. 8
RR 8. 9
RR 8010
RR 8.11
RR 8.12
RR 8.13
RR 8.14
RR 8.15
RR 8.16
RR 8.17
RR 8.18
RR 8.19
RR 8.20
RR 8021
RR 9. 9
RR 9010
RR 9011
RR 9.13
RR 9.14
RR 9015
RR 9.16
RR 9,17
RR 9.18
RR 9,20
RR 9022
RR 9.23
RR 9.24
RR 9026
RR 9.27
RR 9.28
RR 9.29
RR 9.30
RR11011
RR11.13
RRI1.14
RR11015
RR11.16
RR11017
RR12012
RR12.13

159

OBS. FREQ.

3133.1360
3133.7820
3134.4210
3135.0600
3135.7070
3136.3460
3136.9890
3137.6290
3138.2780
3138.8970
3139.5570
3135.8280
3136.4690
3137.1070
3137.7470
3138.3900
3139.0260
3139.6650
3140.3080
3140.9450
3141.5840
3142.2270
3142.8630
3143.5010
3144.1420
3144.8390
3145.4/70
3146.1170
3147.3950
3148.0320
3148.6710
3149.3090
3149.9490
3150.5940
3151.2310
3151.8/20
3153.1520
3153.7880
3154.4270
3155.7030
3156.3470
3156.9840
3157.6190
3158.3580
3162.6410
3163.9090
3164.5440
3165.1960
3165.8280
3166.4680
3171.3920
3172.0380

CALC, FREQ,

6166,1656
6166,7754
6134.4154
3135.0554
6165.6955
6166,6656
3136,9758
3137,6161
3138,2563
6168,8967
3139,5370
3135,8321
6166.4705
6167,1091
3137,7477
6168,6866
6169,0255
6169.6645
6140.6067
6140.9429
6141.5822
6142,2216
3142.8611
6146.5006
6144.1402
6144,8464
6145.4816
6146,1199
6147,6969
6148.0656
6148,6746
6149,6161
3149,9519
3150.5909
3151.2299
3151,8689
6156.1470
3153.7361
6154.4256
6155.7065
6156,6426
3156,9817
6157,6208
3158,2598
6162.6276
3163.9027
6164,5405
3165,1783
3165,8162
3166,4541
3171,3975
6172,0648

RESID.

0.0004
0.0066
0.0056
0.0046
0.0115
0,0104
0.0132
0.0129
0,0217
0.0003
0.0200
’O.0041
'0.0015
.0.0021
.0.0007
0,0064
0.0005
0.0005
0,0046
0.0021
0.0018
0.0054
0.0019
0.0004
0.0018
'0.0044
“0.0046
'0.0029
-0,0019
'0.0036
“0.0033
-0,0041
“0.0029
0.0031
0.0011
0.0031
0.0050
0.0019
0,0017
'0.0005
0.0044
0.0023
'0.0018
0.0982
0,0167
0,0066
0,0065
0,0177
0.0118
0.0139
-0,0055
0,0032

HHT,

0.01
0.02
0.02
0,02
0.02
0,01
0.01
0.01
0.00
0.00
0.00
0.12
0.12
0.07
0,07
0.07
0,07
0.12
0,07
0.04
0.01
0,01
0,01
0.01
0,00
0.05
0.10
0.10
0.25
0.25
0,14
0,12
0.12
0.12
0,12
0.12
0,05
0,05
0.05
0,05
0.01
0,01
0.01
0.00
0.01
0,01
0.01
0.00
0,01
0.00
0.02
0,02

 

‘fn" - ' Infi 1 .-

",1"

ASSIGN.

RR12.14
RR12615
RR12.16
RR12317
RR12618
RR12.19
OP 3. 4
GP 3. 6
0R 3. 7
GP 30 8
GR 3. 9
OR 3.10
QR 3911
OR 3.12

160

035. FREQ.

3172.6/40
3173.3110
3173.9470
3174.5900
3175.2260
3175.8630
6043.0170
6041.8130
604141840
6040.5660
6039.9120
6039.2/80
6038.6410
6038.0220
6037.3800
6037.7400
6036.7470
6036.1210
6035.4950
6042.7490
6042.1020
6041.4820
6040.8350
6040.2040
6039.5660
6030,9390
6038.3140
6037.6150
6037.0470
6048.1670
6048.8200
6049.4900
6050.0980
6050.7280
6051.3800
6052.0220
6052.6600
6053.3040
6053.9410
6054.5980
6088.6610
6088.0160
6087.3800
608647450
6086.1180
6085.4130
6084.8450
6083.5880
6082.9650
6082.3920
6081.7050
6081.0910

CALC, FREQ.

3172.6722
3173,3095
3173.9469
3174.5843

317502217

3175.8592
6043.0655
6041,7982
6041.1658
6040,5341
6039,9032
6039,2731
6038,6439
6038,0154
6037.3878
6036,7610
6036,7610
6036.1351
6035.5100
6042,7202
6042,0870
6041,4544
6040,8227
6040.1918
6039.5617
6038,9324
6038,3039
6037,6762
6037,0494
6048,1645
6048,8051
6049,4464
6050,0884
6050.7311
6051.3744
6052.0184
6052,6631
6053,3084
6053,9543
6054,6008
6088,6590
6088,0242
6087,3902
6086,7569
6086,1244
6085,4927
6084,8618
6083,6025
6082,9741
6082,3465
6081,7198
6081.0939

RESID.

0.0018
0,0015
0.0001
0.0057
0,0043
0.0035
0.0115
0.0182
0,0319
0,0088
0,0049
.0 .0029
0.0066
“0.0073
0,9790
'0.0140
'0.0141
'0.0150
0.0288
0.0150
0,0276
0.0123
0,0122
0,0043
0.0066
0.0101
-0.0012
.0.002‘
0.0025
0,0149
0.0116
0,0096
“0.0031
0.0056
0.0036
'0.0031
“0.0099
'0.0133
'0.0020
0.0020
-0,0062
'030102
'0.0119
-0,0064
'0.0197
'0.0168
-0,0145
'0.0091
0.0055
-0,0148
-0,0029

HHT.

0.02
0.01
0.02
0.01
0,00
0.00
0.07
0.07
0.17
0.07
0,67
0.40
0,67
0.03
0.67
0,00
0.17
0.00
0.00
0907
0.07
0.07
0.07
0.07
0.07
0,07
0,03
0,07
0.07
0.03
0,01
0.03
0.07
0.07
0.17
0,17
0.67
0.67
0.67
0,67
0.01
0.05
0.01
0.00
0,00
0.00
0.01
0,00
0.01
0.05
0.05
0.01

 

ASSIGN.

PP
PP
PP
PP
PP
PP
RP
RP
RP
RP
RP
RP
RP
RP
99
99
99
RP
RP
RP
RP
RP
R9
R9
RP
RP
RP
RP
RP
RP
RP
89
RR
RR
RR
RR
89
RR
88
RR
RR
RR
an
an
80
RR
89
RR
RR
RR
RR
RR

1.19
1620
1921
1.25
1.26

C)
.
‘OODV(%U1&(NKJH

0.13
0.14
0.15
0616
0.18
0.19
0620
0621
0.22
0.23
0.24
0.25

161

085. FREQ.

6078.5730
6077.9730
6077.3530
6074.8820
6074.1830
6073.5430
6100.6890
6100.0590
6099.4450
6098.8090
6098.1780
6097.5640
6096.9310
6096.3040
6095.6560
6095.0250
6094.3950
6093.7660
6093.1400
6092.5170
6091.8860
6091.2520
6090.0000
6089.3850
6088.7590
6088.1290
6087.5190
6086.9030
6086.2810
6085.6740
6085.0590
6084.4370
6102.5/50
6103.2530
6103.9030
6104.5470
6105.1870
6105.8290
6106.4770
6107.1170
6107,7590
6108.4030
6109.0490
6109.6980
6110.3520
6110.9960
6111.6540
6112.9460
6113.5970
6114.2430
6114.8950
6116,8560

CALC, FREQ,

6078,5991
6077,9776
6077,3570
6074,8838
6074,2679
6073,6528
6100,7148
6100.0784
6099,4428
6098,8079
6098.1738
6097,5404
6096.9078
6096,2160
6095,6450
6095,0147
6094,3853
6093,7567
6093.1289
6092.5019
6091,8758
6091.2505
6090.0025
6089,3798
6088.7580
6088.1671
6087.5170
6086,8979
6086,2796
6085.6623
6085.0459
6084,4304
6102.6283
6103.2675
6103.9075
6104.5481
6105,1895
6105,8315
6106.4741
6107.1175
6107.7615
6108.4061
6109,0514
6109,6973
6110.3438
6110.9909
6111.6387
6112.9359
6113,5854
6114,2355
6114,8861
6116,8413

RESID.

“0.0261
-0,0046
'0.0040
-0,0018
-0,0849
'0.1098
-0,0258
.D.0194
0.0022
0.0011
0.0042
0,0236
0.0232
0.0280
0.0110
0.0103
0.0097
0,0093
0.0111
0,0151
0.0102
0.0015
.0.0025
0.0052
0.0010
.0.0081
0.0020
0.0051
0,0014
0.0117
0.0131
0.0060
'0.0533
-0,0145
-0,0045
.0.0011
'0.0025
-0,0025
0.0029
'0.°005
.0.0025
'0.0031
-0,0024
0.0007
0,0082
0.0051
0.0153
0.0101
0,0116
0.0075
0.0089
0,0147

NHT.

0.00
0.10
0.10
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.01
0.00
0,00
0.00
0.01
0.05
0,05
0.05
0.05
0.10
0.10
0.05
0.10
0,01
0.05
0.05
0.01
0.01
0.05
0.05
0.01
0.01
0.00
0.00
0.01
0.01
0.05
0.05
0.25
0.10
0.05
0.10
0.10
0.10
0.10
0.10
0.00
0.01
0,00
0.00
0.00
0.01

ASSIGN,

RR
RR
RR
RR
PR
PR
PR
PR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RP
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR

0.24
0.25
0.26
0.27

1.12

162

095, FREQ.

6117,5100
6118,1630
6118,8100
6119.4/60
6102,7830
6103,4290
6104,0990
6104,7630
6113,2980
6113,9400
6114,5050
6115,2250
6115,8230
6116,5020
6117,1460
6117,7920
611941150
6119,7480
6120,3850
6121,0210
6121,6540
6122,2980
6127,3180
6126,6900
6126,0310
6124,7820
6124,1450
6124,4900
6125,1630
6125,7110
6126,4050
6127,0450
6127,6890
6128,3640
6128,9810
6129,6250
6130,2120
6130,9110
6161,5600
6132,2030
6132,8540
6134,1440
6134,7810
6165,4410
6156,0850
6135,5720
6136,2070
6166,8440
6137,4920
6138,1340
6138,7110
6139,4110

CALC, FREQ,

6117,4941
6118,1475
6118,8013
6119,4557
6102,8047
6103,4551
6104,1061
6104,7576
6113,2974
6113,9366
6114,5764
6115,2170
6115,8582
6116,5001
6117,1426
6117,7858
6119,0740
6119,7191
6120,3648
6121,0110
6121,6579
6122,3054
6127,3028
6126,6721
6126,0423
6124,7849

6124,1573 °

6124,4899
6125,1297
6125,7701
6126,4112
6127,0530
6127,6953
6128,3383
6128,9819
6129,6262
6130,2710
6130,9164
6131,5624
6132,2090
6132,8562
6134,1521
6134,8010
6135,4503
6136,1002
6135,5656
6136,2059
6136,8469
6137,4884
6138,1306
6138,7734
6139,4168

RESID.

0,0159
0,0155
0,0087
0.0203
'0.0217
'0.0261
'0.0071
0.0054
0.0006
0.0034
0.0086
0,0080
-0.0052
0.0019
0,0034
0.0062
0.0410
0,0289
0,0202
0.0100
-0,0039
'0.0074
0,0152
0.0179
'0.0113
.000029
'0.0123
0.0001
0.0033
0.0009
-0,0062
'0.0080
'0.°063
-0,0043
'0.0009
“0.0012
0.0010
20.0054
'0.0024
'0.0060
.0.0022
“0.0081
'010200
-0,0093
'0.°152
0.0064
0.0011
'0.0029
0.0036
0,0034
'0.0029
'0.0058

“HT.

0.01
0,01
0,01
0,00
0,05
0,01
0,00
0.01
0.10
0,05
0,00
0.00
0,01
0,01
0,10
0.10
0,00
0,00
0,00
0,10
0,10
0,10
0.05
0,01
0,01
0,00
0,01
0,10
0,10
0,05
0,05
0,10
0,05
0,10
0,05
0,10
0,05
0,05
0,10
0,05
0,01
0,01
0,01
0,01
0,01
1,00
1,00
1,00
1,00
1,00
1.00
1,00

ASSIGN.

RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR
RR

3:10
3011
3.12
3.13
3.14
3.16
3.17
3.13
3.19
3020
3021
3.22
3023
3.24
3.25
3026
3.27
3028
3.29
3.30
3.32

3.33.

6010

163

085. FREQ.

6140.0570
6140.6930
6141.3490
6141.9990
6142.6410
6143.9250
6144.5850
6145.2650
6145.8140
6146.5610
6147.1170
6147.8200
6148.4690
6149,1340
6149.7700
6190.4100
6151,0960
6151.7600
6152.3950
6153,0600
6154.3800
6155.0240
6150.3840
6151.0260
6151.6640
6152.3120
6152.9750
6154.9090
6155.5470
6156.1880
6156.8260
6157.3550
6157.9960
6158.6360
6159.2790
6159.9260
6160.5610
6161.2110
6161.8530
6162.4990
6163.1460
6164,4280
6165.0840
6165.7270
6166.3730
6167.0200
6168.9670
6168.0820
6160.7190
6169,3600
6169,9960
6170.6650

CALC, FRED,

614000608
614007054
614103505
6141,9963
6142.6426
6143.9368
6144,5847
614502331
6145,8821
6146.5316
6147.1816
6147.8321
614804831
614901345
6149,7864
6150.4386
6151.0917
6151.7450
615203987
6153.0529
615403625
615500180
6150:3757
615100201
6151,6649
6152.3104
615209563
615418974
6155,5455
615601940
6156.8431
6157,3565
6157.9977
6158.6395
6159,2615
6159,9246
616005682
6161,2123
6161.6563
616205019
616301476
616414404
616500676
6165.7352
6166.3834
6167.0320
6163,9807
616800656
616807072
6169,3493
6169.9920
6170.6351

RESID.

I'0.0036
.0.012‘
'0.0015
0.0027
'0.0016
.0.0118
0.0003
0.0019
'0.0081
.0.0006
’0.0046
.0.0121
'0.0141
'0.0005
'0.008‘
.0.0288
0,0043
0,0150
'0.0037
0.0071
0.0175
0.0060
0.0033
0.0059
0.0191
0.0016
0.0187
0.0116
0.0015
”0.0060
'0.0171
-0,0015
'0.0017
.0.0035
'0.0026
0,0012
'0.0072
-0,0013
.0.0038
'0.0029
-0,0016
'0.0124
“0.0036
'0.0082
'0.0104
-0,0120
.0.0137
0,0164
0.0118
0,0107
0,0040
“0.0001

HHT,

1.00
0.25
1.00
1,00
1,00
0.01
0,05
0.05
0,05
0.05
0,05
0.05
0.05
0.01
0.01
0.01
0,01
0.01
0,01
0.01
0.01
0.01
0.01
0.01
0.01
0,01
0.01
0.01
0.01
0.01
0.01
0,25
0,25
0.25
0,05
0.10
0.10
0,10
0.10
0.05
0,01
0.05
0,05
0.01
0.01
0,01
0.00
0.01
1.00
0,10
0,25
1.00

ASSIGN.

RR
00
RR
an
RR
R:
an
RR
an
an
an
an
an
an
an
RR
an
RR
RR
an
an
an
RR
RR
an
RR
an
an
an
RR
RR
RR
an
RR
an
RR
an
RR
RR
an
RR
an
99
PP
PP
PP
99
PP
99
90
99
99

6011
6.12
6.13
6.14
6.16
6.17
6.18
6.19
6.20
6.21
6022
6.23
6.24
6925
6.26
6.29
8. 8
8. 9
8.10
8011
8012
8.13
8.14
8.16
8.17
8.18
8.19
5020
8.21
8022

8023

9. 9
9010
9011
9.12
9.13
9.15
9.16
9.17
9.18
9.19
9020
9. 9
9:10
9011
9.14
9.15
8. 8
8: 9
8.10
8.11
8.12

164

0850 FREQ:

6171,2820
6171,9230
6172,5630
6173,2240
6174,4930
6175,1270
6175,8050
6176,4450
6177,0950
6177,7370
6178,3940
6179,0420
6179,6890
6180.3220
6180,9920
6182,9360
6189,0700
6189,7100
6190.3530
6191.0020
6191,6400
6192,2790
6192,9380
6194,2320
6194,8760
6195.5080
6196.1630
6196.8090
6197,4490
6198.1020
6198.7270
6199,4110
6200.0520
6200.6930
6201,3410
6201,9810
6203,2680
6203,9110
6204,5600
6205,1990
6205,8510
6206,4990
5994.9450
5994,3140
5993.6520
5991.7920
5991.1400
6007.0770
6006,5160
6005.8850
6005.2980
6004.6420

CALC, FREQ,

6171,2789
6171,9231
6172,5679
6173,2132
6174,5053
6175,1520
6175,7993
6176,4470
6177,0952
6177,7438
6178,3929
6179,0424
6179,6923
6180,3427
6180,9935
6182,9482
6189,0935
6189,7655
6190,3781
6191,0212
6191,6648
6192,3088
6192,9534
6194,2438
6194,8897
6195,5361
6196,1829
6196,8601
6197,4778
6198.1258
6198,7743
6199,4043
6200,0466
6200,6896
6201,3325
6201,9762
6203,2649
6203,9099
6204,5554
6205,2013
6205,8476
6206,4943
5994,3129
5993,6847
5993,0575
5991,1818
5990,5585
6006,7114
6006,0820
6005,4536
6004,8261
6004,1995

RESID.

0.0031
-0,0001
'0.0049

0.0108
'0.0123
-0,0250

0,0057
-0,0020
'0.0002
'0.0068

0.0011
“0.0004
'0.0033
.030207
-0,0015
.0.0122
'0.0235
“0.0255
'0.0251
-0,0192
.0'0248
.0'0298

-0,0154>

'0.0116
'0.0137
'0.0281
“0.0199
'0.0211
~0.0288
'0.0236
-0,0473
0,0067
0.0054
0.0037
0.0085
0.0048
0.0031
0.0011
0.0046
'0,0023
0,0034
0.0047
0.6321
0,6293
0,5945
0,6102
0,5015
0.3656
0.4340
0,4314
0,4719
0.4425

HHT,

1,00
1.00
0.10
0,25
0.25
0,10
0,05
0,05
0,05
0.05
0,05
0,05
0,05
0.01
0,01
0.01
0.10
0,05
0,05
0,05
0,05
0,01
0.01
0,01
0,01
0.01
0.01
0.01
0.01
0.01
0.00
0,25
0.10
0.10
0,05
0,05
0.05
0,05
0.05
0.05
0,01
0.01
0.00
0,00
0.00
0,00
0,00
0.00
0,00
0,00
0.00
0.00

"'00'0090011001100110“