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MEASUREMENT OF HYPERFINE STRUCTURE IN DEUTERATED ACETYLENES VIA MOLECULAR-BEAM MICROWAVE SPECTROSCOPY

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300 N. Zeeb Road
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8306000
Tack, Leslie Martin
MEASUREMENT OF HYPERFINE STRUCTURE IN DEUTERATED
ACETYLENES VIA MOLECULAR-BEAM MICROWAVE SPECTROSCOPY
The University o f Arizona
University
Microfilms
International
PH.D.
1982
300 N. Zeeb Road, Ann Arbor, M I 48106
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MEASUREMENT OF HYPERFINE STRUCTURE IN
DEUTERATED ACETYLENES VIA MOLECULAR-BEAM
MICROWAVE SPECTROSCOPY
by
L e s lie M artin Tack
A D is s e rta tio n Submitted to the Faculty o f the
DEPARTMENT OF CHEMISTRY
In P a rtia l F u lfillm e n t o f the Requirements
For the Degree o f
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
19
8 2
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THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Final Examination Committee, we certify that we have read
.,
...
, , Leslie Martin Tack
the dissertation prepared by ___________________________________________________
entitled Measurement of Hyperfine Structure in Deuterated Acetylenes
via Molecular-Beam Microwave Spectroscopy
and recommend that it be accepted as fulfilling the dissertation requirement
.,
_ Doctor of Philosophy
for the Degree of ______________________
.
Date
Date
D
_
-
Date
Date
/z /e /s z L
Date
Final approval and acceptance of this dissertation is contingent upon the
candidate's submission of the final copy of the dissertation to the Graduate
College.
I hereby certify that I have read this dissertation prepared under my
direction and recommend that it be accepted as fulfilling the dissertation
requirement.
Dissertation DirectcBx
Date
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STATEMENT BY AUTHOR
This d is s e rta tio n has been submitted in p a rtia l f u lf illm e n t o f
requirements fo r an advanced degree a t The U n iv e rs ity o f Arizona and is
deposited in the U n iv e rs ity L ib ra ry to be made a v a ila b le to borrowers
under ru le s o f the L ib ra ry .
B rie f quotations from th is d is s e rta tio n are allow able w itho ut
special perm ission, provided th a t accurate acknowledgment o f source is
made. Requests fo r permission fo r extended quotation from or reproduction
o f th is manuscript in whole or in p a rt may be granted by the head o f the
major department o r the Dean o f the Graduate College when in his judgment
the proposed use o f the m a terial is in the in te re s ts o f scholarship. In
a ll o th e r instances, however, permission must be obtained from the author.
SIGNED:
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ACKNOWLEDGMENT
I would f i r s t lik e to thank my research d ir e c to r , Professor
S.G. K ukolich, fo r his valuable assistance and guidance in the completion
o f the p ro je c ts contained in th is work.
I have met few s c ie n tis ts w ith
his professional competence and I w ill always appreciate the o p p o rtu n ity
I had under his d ire c tio n a t Arizona.
In a d d itio n , I would lik e to acknowledge the re s t o f my d is s e rta ­
tio n committee:
Professor Mike B a rfie ld fo r his valuable discussions on
th e o re tic a l m a tters; Professors Robert Feltham and Robert Bates fo r th e ir
ca re fu l e d itin g o f th is d is s e rta tio n ; Professors B i ll Hetherington and
Peter Bernath fo r complementing my d is s e rta tio n committee on short
no tice .
Many graduate students and research groups have helped me through
th e ir generous loans o f equipment and personal tim e.
They are too
numerous to l i s t here and I hope th is w ill not understate my a p pre ciation
and regard f o r each in d iv id u a l.
I would f i n a l l y lik e to thank Jeannette Gerl fo r typ in g th is
d is s e rta tio n in the sh o rt amount o f time a llo te d her.
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TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS.
. . • ............................................................................. vi
LIST OF TABLES......................................................................................................v ii
ABSTRACT..................................................................................................................v i i i
1. INTRODUCTION.................................................................................................. 1
2. THEORY OF HYPERFINE STRUCTURE FOR ROTATING MOLECULES....................4
In tro d u c tio n ......................................................................................... 4
In te ra c tio n o f a Nucleus w ith Molecular F ie ld s .......................5
E le c tro s ta tic In te ra c tio n ...................................................... 5
Magnetic In te ra c tio n s .............................................................. 8
M a trix Elements and Coupling Schemes.......................................... 10
Symmetry C o n s id e ra tio n s ..................................................................12
3. EXPERIMENTAL DETAILS...................................................................................14
In tro d u c tio n ......................................................................................... 14
Stark-Modulated Microwave Spectrometer.......................................14
Molecular-Beam Maser Spectometer.................................................. 16
Measurements on In d iv id u a l Molecules.......................................... 19
Chloroacetylene-D......................................................................19
Propyne-D3 .................................................................................. 21
Cyanoacetylene-D ......................................................................25
4.
ANALYSIS OF DATA.......................................................................................... 26
In tro d u c tio n ......................................................................................... 26
Data Analysis fo r In d iv id u a l Molecules.......................................27
C h lo ro a ce tyle n e -D ......................................................................27
Cyanoacetylene-D..........................................................................27
Propyne-D3 ......................................................................................30
5.
DUSCUSSION OF RESULTS................................................................................... 31
In tro d u c tio n ..........................................................................................31
Indivudual Molecules..........................................................................31
Chi oroacetyl ene-D..........................................................................31
C yanoacetylene-D ..........................................................................34
P ropyne-D o......................................................................................36
Deuterium Quadrupole Coupling in Molecules...............................37
iv
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V
TABLE OF CONTENTS— Continued
Page
APPENDIX:
PROGRAM FOR CALCULATION OF HYPERFINE STRUCTURE.....................44
REFERENCES................................................................................................................... 53
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LIST OF ILLUSTRATIONS
Figure
Page
1. E le c tro s ta tic in te ra c tio n s w ith nucleus in atoms
and molecules.......................................................................................... 5
2. Block diagram o f Stark-modulated spectrometer ................................. 15
3. Block diagram o f beam-maser spectrom eter.............................................17
4.
Hyperfine components o f J = 1 -* 0 tr a n s itio n , Cl CCD..................... 22
5. D e riva tive maser spectra o f the hyperfine components
o f the J = 1 -* 0 tr a n s itio n s , CD3 CCH....................................... 23
6.
Hyperfine components o f the J = 1 -»■ 0 tr a n s itio n , NCCCD . . . .
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24
LIST OF TABLES
Table
Page
1.
Measured and ca lcu la te d frequencies f o r components
o f J = 1 -*■ 0 tra n s itio n o f Cl CCD...............................................28
2.
M olecular constants fo r Cl CCD............................................................... 28
3.
Measured and ca lcu la te d frequencies f o r components
o f J = 1 -* 0 tr a n s itio n o f NCCCD...............................................29
4.
Molecular constants fo r NCCCD............................................................... 29
5.
Measured frequencies fo r J = 1
6.
M olecular coupling parameters and ro ta tio n a l constant fo r
CD3CCH................................................................................................. 30
7.
Comparison o f gas and liq u id phase hyperfine components
in NCCCD..............................................................................................36
8.
Chemical s h if t tensor elements o f
9.
Comparison o f ca lcu la te d and experimental deuterium
quadrupole coupling strengths fo r some selected
m o le c u le s ......................................................................................... 40
10.
Quadrupole coupling strengths along the C-D bond ....................... 40
11.
Quadrupole coupling constants and bond distances
fo r the C-D moiety in several deuterated
acetylenes......................................................................................... 41
0 tra n s itio n in CDgCCH. . . 30
14
N in NCCCD ........................... 36
v ii
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ABSTRACT
This work describes the measurement o f hyperfine s tru c tu re in a
series o f deuterated acetylenes via molecular-beam microwave spectroscopy.
Measurements o f s p in -ro ta tio n constants were used to c a lc u la te the
paramagnetic c o n trib u tio n to the chemical s h ie ld in g o f the concerned
nucleus.
Where p o ssible, comparisons w ith NMR measurements were made.
Measurements o f the deuterium quadrupole coupling determined in th is work
are compared w ith previous measurements on the same or s im ila r systems.
A review o f the th e o re tic a l work done in th is area is presented as w ell as
a discussion o f trends observed from high p re cisio n measurements o f
deuterium quadrupole coupling.
A computer program th a t ca lcu la te s hyper­
fin e s tru c tu re fo r up to fo u r coupled nuclei o f a r b itr a r y spin is
presented.
v iii
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CHAPTER 1
INTRODUCTION
Beam - MASER(microwave a m p lific a tio n by stim u la ted emission o f
ra d ia tio n ) instrum ents have yie ld e d some o f the sharpest and most stable
resonances to be found in spectroscopy.
U ltim a te ly , a MASER device w ill
become the universal time standard w ith s t a b ilit y o f the order one p a rt
in 1 0 14.
For the m olecular spectroscopist the MASER instrum ent is an
extremely accurate measuring device fo r both strong and weak in te ra c tio n s
in molecules. R esolution requirements are d ic ta te d by what in te ra c tio n s
are being in v e s tig a te d .
R otational energies can be ro u tin e ly measured by
conventional microwave o r even in fra re d techniques.
The lit e r a t u r e is
f u l l o f measurements o f ro ta tio n a l energies made w ith good Stark-modulated
4
microwave spectrometers where the re s o lu tio n was about one p a rt in 10 or
5
even 10 . The observation o f hyperfine s tru c tu re , which w ill perturb the
ro ta tio n a l tr a n s itio n s , may re q u ire higher re s o lu tio n fo r f u l l e lu c id a tio n
o f the e ffe c t.
This is p a r tic u la r ly tru e fo r hyperfine s tru c tu re in
deuterated species.
Resolution o f the order on p a rt in 10^ is u su ally
needed; fo rtu n a te ly , the beam-maser instrum ent meets th is requirement.
Hyperfine s tru c tu re in ro ta tio n a l tra n s itio n s arise s when one
considers the f i n i t e extension o f the nucleus.
There are e le c tr ic and
magnetic in te ra c tio n s th a t s p l i t the ro ta tio n a l lin e ; e le c tr ic in te ra c ­
tio n s dominate in diamagnetic molecules.
Analysis o f the data taken on
1
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2
the beam-maser spectrometer w ill y ie ld in te ra c tio n strengths fo r e le c tr ic
quadrupole, s p in -ro ta tio n and sp in -sp in in te ra c tio n s .
The deuterium quad­
rupole in te ra c tio n is o f special in te re s t because the e le c tr ic f ie ld
gra d ie n t a t the deuterium nucleus arises p rim a rily from external charges
in the molecule; thu s, there is a probe fo r estim ating the charge d i s t r i ­
bution in the molecule.
E le c tro n ic wave fu n ction s can be tested fo r
power to preduct the quadrupole coupling
th e ir
a t the deuterium s ite .
The s p in -ro ta tio n and sp in -sp in in te ra c tio n s are magnetic e ffe c ts .
The former re s u lts from the in te ra c tio n o f the magnetic moment o f the
nucleus w ith the magnetic f ie ld produced by the ro ta tio n o f the molecule.
Ramsey^ was f i r s t to p o in t out the in tim a te re la tio n s h ip between the spinro ta tio n constant and the paramagnetic c o n trib u tio n to the chemical
s h ie ld in g (which is represented by a second-rank te n s o r).
O rd in a rily ,
from a purely th e o re tic a l p o in t o f view, th is would be a d i f f i c u l t c o n tr i­
bution to c a lc u la te ; however, s p in -ro ta tio n constants can be f a i r l y
a ccu rate ly measured (to w ith in 10 %) from the data taken from a beam
experiment.
Where c a lc u la tio n s are a v a ila b le f o r determ ining the diamag­
n e tic c o n trib tu io n (a much more tra c ta b le problem since i t only depends on
knowledge o f ground s ta te wave fu n c tio n s ) experimental and calculated
re s u lts can be combined to obtain the to ta l chemical s h ie ld in g .
data are a v a ila b le (where the to ta l chemical s h ie ld in g
I f NMR
is measured)there
is a consistency check to use.
S pin-spin in te ra c tio n s are u su ally ca lcu la ted from the molecular
geometry and nuclear moments to aid in the data analysis process.
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3
Each o f these in te ra c tio n s is discussed in more d e ta il in
Chapter I I .
A computer program was w ritte n to c a lc u la te hyperfine s tru c ­
tu re f o r up to fo u r coupled nuclei o f a r b itr a r y spin.
program is provided in the Appendix.
A l i s t i n g o f th is
Chapter I I I gives a complete d e scrip ­
tio n o f the microwave instrum e ntatio n used to make the measurements.
Analysis o f the data along w ith the derived spectrscopic constants is
presented in Chapter IV; wherever possible these re s u lts are compared
w ith previous works (u s u a lly in v o lv in g less p re c is io n ) in the same or
s im ila r systems.
In Chapter V is presented a discussion o f the in te rp re ­
ta tio n o f the measurements.
Every attem pt w ill be made to maximize the
in fo rm a tion th a t can be obtained from each measurement.
A review o f the
th e o re tic a l work done in conjunction w ith these measurements w ill be
presented; problems in comparing th e o re tic a l and experimental re s u lts w ill
be discussed.
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CHAPTER 2
THEORY OF HYPERFINE STRUCTURE FOR ROTATING MOLECULES
A.
In tro d u c tio n
The Hamiltonian used to analyze the data taken in th is work is
as fo llo w s :
H ■ Hr o t. + Hh f.
(,)
where H . is the ro ta tio n a l energy o f the molecule which includes e ffe c ts
r o t.
due to c e n trifu g a l d is to r tio n and ro ta tio n -v ib r a tio n in te ra c tio n s .
These
2 3
e ffe c ts are discussed in standard te x ts . 5
For the systems studied in th is work, the ro ta tio n a l energy w ill
be much g re a te r than the hyperfine in te ra c tio n s (approxim ately a fa c to r
3
o f 10 ). The hyperfine s tru c tu re can then be viewed as a p e rtu rb a tio n
o f the ro ta tio n a l energy le v e ls and the analysis w ill always include a
" lin e center frequency" which corresponds to the lin e we would see in the
absence o f hyperfine s tu rc tu re .
The hyperfine Ham iltonian w ill include e le c t r ic quadrupole, spinro ta tio n , and sp in -sp in in te ra c tio n s .
Each in te ra c tio n w ill be discussed
and a scheme fo r c a lc u la tin g hyperfine s tru c tu re fo r a ll in te ra c tio n s
w ill be presented.
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5
B.
1.
In te ra c tio n o f a Nucleus w ith Molecular Fields
E le c tro s ta tic In te ra c tio n .
The general e le c tr o s ta tic in te ra c tio n between a charged nucleus o f
f i n i t e size and the e le ctron s and nuclei in the re s t o f the molecule is
given by:
H
'el
e
n
where Pe( r ) is the charge density of the ele ctron s and nuclei in the
volume element dx
e
a t the p o s itio n r
e
r e la tiv e to the center o f the re le -
vant nucleus, D ( r ) is the nuclear charge density o f the nucleus conn n
cerned in the volume element dx
n
a t the p o s itio n r „ r e la tiv e to the cenn
t e r o f the nucleus, and r is the magnitude o f the radius vecto r jo in in g
dxg and dxn as shown in Fig. 1. PeCr"e) is negative fo r ele ctron s and posi­
tiv e fo r p o s itiv e charges.
I f e le c tro n ic charges more d is ta n t than the radius R o f the
nucleus are considered, 1 / r may be expressed, using the cosine law, as
fo llo w s :
+ - V ,
e
♦ \ p 2 + e
where P is the Legendre polynomial o f cose
&
The f i r s t
second
(3)
6n
, so th a t,
P1 = coseen
(4a)
P2 = 1 (3cos2een- l )
(4b)
term
in the expansioncorresponds to ane le c tr ic monopole, the
to the e le c tr ic d ip o le , and the th ir d to the e le c tr ic
quadrupole
moment.
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6
p.
Fig. 1.
E le c tro s ta tic in te ra c tio n s w ith nucleus in atoms and
molecules.
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7
The monopole term w ill lead to no hyperfine s tru c tu re and is th e re ­
fo re o f no in te re s t.
The d ip o le term can be shown to vanish from p a rity
co n sid e ra tio n s; thus, the only im portant term is the quadrupole moment.
Higher moments are g e n e ra lly too small to be considered.
4
Ramsey has shown the e le c tro s ta tic quadrupole in te ra c tio n can be
w ritte n as:
2
n
Sq-iW
- A J»p
HQ = 21(21-1)J ( 2J-1)
q -A
po
+ 2 I,J - I J ]
where Q is the a r b itr a r y nuclear constant and is c a lle d the magnitude o f
the e le c tr ic quadrupole moment and is given by:
'p />
, t
( r x)
(3z 2 - r 2 )d
Knv n '
n
n
n
m j-i
eQ
(6 )
The in te g ra l is evaluated fo r nij =I as in d ic a te d ,
eqj is defined as the
e le c tr ic f i e l d gradie nt coupling constant and is given by:
eqj -
p C r)
r >R
e
(3cos2e
mj ‘ J
- 1l - i j dt
(7)
re
where e
is the angle between r and the z a x is .
ez
e
is an im p lic it J dependence on q j.
As w ritte n above there
The quadrupole in te ra c tio n can be
re w ritte n w ith e x p lic it J dependence by u t iliz in g the fo llo w in g r e la tio n ­
ship :
n = ^ L 1 SVI
qJ
2J+3 e 3 2
azo
-
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(Q \
8
0
where V is the p o te n tia l from a ll charges external to the nucleus and Zg
is along the symmetry axis o f the molecule.
g ive s:
S u b s titu tin g ( 8 ) in to (5)
?
-eQ ^
HQ = 21 (2 1 -1 )(2J+3) ( 2J-1)
+
I
I-J - I 2 J2}
(9)
The above expression is only a p p lic a b le to lin e a r molecules. Thaddeus,
5
K ris h e r, and Loubser give the quadrupole in te ra c tio n in asymmetric ro to rs
fo r the Kth coupling nucleus as:
'q = 2 l j r f l ) J ( 2 J - l ) x^ K " ) 2 +
l\
(10)
l\
where, fo r an asymmetric ro to r:
q j = 21 <Jg2 > qgg/o (J +1)
(11)
q
is the e le c tr ic f i e l d g radie nt tensor along the g
p rin c ip a l in e ryy
2
t i a l axis and <J^ > are the average values o f the square o f the compoi , L.
nents o f J along the g
2.
p rin c ip a l a x is .
Magnetic In te ra c tio n s
There are two types o f magnetic in te ra c tio n s .
The f i r s t corres­
ponds to the in te ra c tio n o f the magnetic moment o f the nucleus o f in te re s t
w ith the magnetic f ie ld produced by the ro ta tio n o f the molecule and is
re fe rre d to as the s p in -ro ta tio n in te ra c tio n .
The second corresponds to
the magnetic in te ra c tio n o f the two moments and w ill be re fe rre d to as the
sp in -s p in in te ra c tio n .
The s p in -ro ta tio n in te ra c tio n fo r nucleus K is given by Thaddeus,
5
K ris h e r, and Loubser as:
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9
Hsr = C (K )tK- j '
(12a)
where C(K) = I <J 2> N__/J(J+1)
g
9
yg
(12b)
Ngg are the diagonal elements o f the s p in -ro ta tio n ten sor in the p r in ­
c ip a l axis system.
When the re are two n u cle i present w ith non-zero spin the re may be
a measurable in te ra c tio n between the moments.
C la s s ic a lly , the d ip o le -
d ip o le in te ra c tio n is given as:
«-A-
U i*y 9
P1 p 2
= ^
r
- j /- A *
\ /-A *
—
3 ( y , *r ) (y • r )
r
b
2
(13)
where r is the ve cto r connecting the two d ip o le s . Thaddeus, K ris h e r,
5
and Loubser w rite the s p in -s p in in te ra c tio n between n u clei K and L as
the c o n tra c tio n o f two symmetric dyadics:
Hs s =
S:R
(14)
where
s = I (V
l
+
(15a)
R = r KL ( " 3 r LKr LK + r LK ^
^15b^
average r over symmetric wave fu n c tio n we o b ta in :
•=R ” * j T w - n - ( f
( " > )tr ■ J (J + 1 > 11
d 6a)
where d . = Z \ <J 2> R /(J + l)(2 J + 3 )
(16b)
cj
yy
S u b s titu tin g (17) in to (14) the tensor c o n tra c tio n can be w ritte n in
terms o f s c a la r product operators
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10
2
HjS = ^
r
f
{ l 2 ( I l- '', ) ( I K 'J ) + ( I K-J ) ( I L 'J ) >-<I L - V J 1
‘ 17>
For symmetric tops d j w ill s im p lify to :
-^2
i j
s
i
l
F
3k2
3
j
!
,
-
j
p
y
H
1
- -|sin^f>
-
7
—
}
<1 8 >
where <j> is the angle between the vector connecting the two nuclei and the
symmetry axis o f the molecule.
Thus, the sp in -s p in in te ra c tio n strength
can be ca lcu la te d from knowledge o f the nuclear moments and the m olecular
s tru c tu re .
C.
M a trix Elements and Coupling Schemes
As can be seen from examination o f the hyperfine H am iltonian, the
Ham iltonian m a trix is determined by evaluation o f m a trix elements fo r the
-i-
_ l __c
s c a la r product operators I K*J and I K*JL in an appropriate coupling scheme.
The molecules in th is work represent the determ ination o f hyper­
fin e s tru c tu re fo r several general cases.
Chloroacetylene-D and
cyanoacetylene-D represents the case o f two nuclei w ith great by unequal
couplings.
Two papers have applied the method o f irre d u c ib le tensor
operators fo r the most general case o f N coupling n u c le i: Thaddeus,
c
K ris h e r, and Loubser in 1964; Cook and Delucia in 1971. The basic
d iffe re n c e s between the two papers are th a t Cook and DeLucia included
terms in the Hamiltonian m a trix o f f diagonal in J ; however, they om itted
the sp in -sp in in te ra c tio n in the hyperfine H am iltonian.
A program fo r c a lc u la tin g hyperfine s tru c tu re up to fo u r coupling
nuclei o f a r b itr a r y spin was w ritte n based on these two papers.
The
coupling scheme fo r the general case o f N coupling nuclei is as fo llo w s :
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Basis fu n ctio n s in th is coupling scheme are as fo llo w s :
I i *•• ^n*Fi] **• FN - l9F >
( 20)
^
U p-i'
The m a trix elements fo r the scala r product operators I L-J and I ^ * I K are
diagonal in the to ta l angular momentum quantum number F.
_L JL
fo r I^*J are given by:
<F' i
•••
M atrix elements
FL-1 >
fi
= (-1 ) r [J ( J + l) (2J+1 ) I L( (IL+1 ) ( 2 I l +1 ) ] 1 / 2
i FL
i
I.L F.'L- I f L - 1
i
>11
1
F ^ I l ) i= l
T /o (l F-l - li F-l I-l
(2F!+1) (2F.+1) l/2 {
1
1
(F. F . ^ 1 )
L-2
where r = ( L - l) + J
(F ! _ 1 + I . + F.) + ( F ^
M a trix elements fo r
+ I L + FL)
(21a)
(21b)
fo r K<L are given by:
<F1 ••• FL - 1 I! l ’ I k I PK
FL-1>
= ( - 1 ) S[ I k ( I k+ 1 )(2 Ik+ 1 )(Il ) ( I l + 1 )(2 Il +1) ] 1 / 2
T
F1 F
^
K K K -lf
IF
) L
T
L
F1
L - l.
(22)
f k *k 1
;
v
fl- i l i
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where S - (L-K) ♦ ( F|W ♦
D.
+ Fy
^ ( F j.,
♦ I , ♦ F ,) ♦ ( F ^ , +I L+FL)
Symmetry Considerations
When dealing w ith molecules where there are equivale nt coupling
n u c le i, such as propyne-D^, i t is advantageous to make use o f the
symmetry r e s tr ic tio n s imposed on the spin fu n ction s to reduce the energy
m a trix .
This problem has been discussed in the lit e r a t u r e over the la s t
7 8 9
t h ir t y y e a rs .’ ’
The program used to analyze a ll molecules in th is work
did not impose i n i t i a l symmetry r e s tr ic tio n s on the spin fu n c tio n s .
This
re s u lts in increased CPU time (due to the sp in -sp in in te ra c tio n ) but main­
ta in s the program's useful g e n e ra lity .
In any event, the symmetry argu­
ments th a t are invoked to reduce the energy m a trix w ill be o u tlin e d .
More
d e ta ile d explanations are avaiable in the lit e r a t u r e . ^ 0
The deuteron has a spin o f one, re q u irin g the to ta l wave fu n c tio n
to be symmetric w ith exchange o f the two deuterons.
Propane-D^ has
symmetry and the to ta l wave fu n ctio n s must transform as the irre d u c ib le
representations o f the p o in t group.
The to ta l wave fu n c tio n is w ritte n
as an in v e rs io n , r o ta tio n a l, and spin fu n c tio n :
^ =
( ^
( 23)
) w i l l transform as the irre d u c ib le representation o f
K=0, (i^r ^ v ) w ill be e ith e r A-j o r A2 to give a product A-j.
For
The symmetry
o f the spin fu n c tio n is determined by the to ta l s p in , I j = 1 ^
+ I Q2 + ^
The to ta l deuterium spin and i t s associated symmetry are tabulated below
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
13
I-j.
Symmetry
3
A-j
2
E
1
A1 + E
0
a2
1 ^.
= 2 is symmetry forbidden because the cross product does not contain
A-|.
I-j.= 0 is allowed but u n in te re s tin g because i t does not y ie ld any
h yp e rfin e s p lit t in g .
Basis fu n ctio n s are w ritte n as |I^.,J,F> and the coupling scheme
is I T + J = F.
10
Kukolich and Cogley have worked out the m a trix elements
fo r three e q u iva le n t deuterons in CDgBr w ith the above coupled scheme.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 3
EXPERIMENTAL DETAILS
A.
In tro d u c tio n
I n i t i a l measurements were made on a Stark modulated microwave
spectrometer patterned a fte r the instrum ent developed by Hughes and
W ils o n .^
Hyperfine s tru c tu re due to c h lo rin e and nitrogen was e a s ily
resolved but only p a rtia l re s o lu tio n could ever be achieved due to deuter­
ium quadrupole coupling.
modulated spectrometer.
or
more
Typical re s o lu tio n was 80-100 kHz on the Stark
Resolution could be increased by a fa c to r o f 10
by going to the beam experiment where the s p lit t in g due to
deuterium could be f u l l y resolved.
Each instrum ent w ill be described as
w ell as the s p e c ific experimental con dition s fo r a ll measurements.
B.
Stark Modulated Microwave Spectrometer
A block diagram o f the Stark modulated spectrometer is shown in
Fig. 2.
The absorption c e ll consisted o f a 4 meter section o f C-band wave­
guide h e rm e tica lly sealed by the compression o f 0 -rin g s against Mylar
windows.
Thus, tra n s itio n pieces could be connected to the absorption
c e ll w ith o u t breaking the vacuum.
The system was evacuated by use o f a
liq u id nitrogen tra p in series w ith an o il d iffu s io n and mechanical pump.
The microwave ra d ia tio n was generated by re fle x k ly s tro n tubes.
Frequency s ta b iliz a tio n was achieved through a phase-1ock loo p, using a
sta b le 10 MHz quartz c ry s ta l as reference.
The te n th harmonic o f the
14
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Valve
Valve
Stark Cell
Detector
Klystron
Oscillator
Synchronizer
Crystal
Oscillator
Fig. 2.
Stark
Modulator
Lock-In
Amplifier
Frequency
Counter
Clmrt
Recorder
Block diagram o f Stark-modulated microwave spectrometer.
cn
16
the c ry s ta l was displayed on a frequency counter.
A sweep generator
would slow ly change the c ry s ta l frequency, thereby causing the k ly s tro n
frequency to slo w ly change also .
One o f the main features' o f the instrum ent is the Stark modulato r .
12
I t ap p lie s a 0-1000 V square wave to the Stark e le c tro d e , which
was supported by grooves cut in the c e ll and in su la te d from i t by s p l i t
Teflon tu b in g .
modulator.
References fo r lo c k -in d e te ctio n also came from the Stark
The output from the c ry s ta l d e tecto r is f ilt e r e d and a m p lifie d ,
follow ed by lo c k -in d e te ctio n a t 1 0 'kHz w ith a time constant o f 3 seconds.
The output o f the lo c k -in a m p lifie r is displayed on a lin e a r c h a rt re co r­
der along w ith frequency markers derived from the counter.
Sample pres­
sure would be reduced to obtain the best re s o lu tio n w hile m aintaining
adequate signal to noise.
C.
The pressure was t y p ic a lly 1-3 microns.
Molecular-Beam Maser Spectrometer
The beam maser spectrometer was patterned a 'fte r the one developed
by Gordon, Zeiger and Townes.
13 14
’
In a m olecular beam microwave e x p e ri­
ment the e le c tr ic ra d ia tio n is perpendicular to
the v e lo c ity o f the molecules; thu s, the f i r s t order Doppler e ffe c t
vanishes.
Pressure broadening e ffe c ts are n e g lig ib le as the beam chamber
pressure is maintained a t 10" ' 7 t o r r .
These e ffe c ts are in h e re n tly present
in a Stark c e ll experiment y ie ld in g a lin e w idth o f 50 kHz o r more.
Resolution o f th is magnitude is u su a lly about a fa c to r o f 10 too low to
resolve deuterium hyperfine s tru c tu re .
A block diagram o f the beam-maser spectrometer is shown in Fig. 3.
The beam source is a s ta in le s s stee l supersonic nozzle, a 3 inch c y lin d e r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
vacuum
beam
e n velo p e
focuser
source
covitv
MULTIPLIER
CHAIN
x IOOO
PHA SE
LOCKED
KLYSTRON
a DOUBLER
P H A SE
MODULATOR
SUPER­
HETERODYNE
MICROWAVE
RECEIVER
W ITH 3 0 MHz
I.F
T
TU N EA B LE
CRYSTA L
OSCILLATOR
FREQUENCY
C O UNTER
8 M ARKER
Fig. 3.
A M P L IFIE R ,
PHASE
S E N S IT IV E
DETECTO R
a 2 0 0 Hz
OSCILLATOR
I
CHART
RECORDER
Block diagram o f the molecular beam maser
spectrometer system.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
w ith a 0.15 mm hole a t i t s head.
T y p ic a lly , a pressure o f 100 T orr was
maintained behind the nozzle, causing the ro ta tio n a l temperature to be
very low (20-40°K).
This is a favorable c o n d itio n because hyperfine
s tru c tu re was measured f o r the f i r s t ro ta tio n a l tr a n s itio n throughout th is
work.
A fte r e x itin g the nozzle, the molecules pass through a section o f
q u a rte r-in c h copper tubing in thermal contact w ith a liq u id nitrogen tra p .
The beam is co llim a te d by a 6.3 mm hole in the b a ffle separating the
source and beam regions.
The beam enters the focusser which consists o f
4 s ta in le s s steel rods, 1/4" in diameter and 6 " long.
A lte rn a te rods are
held a t p o te n tia ls o f up to 20 kV. to produce a la rg e , ra d ia l e le c tr ic
f ie ld g ra d ie n t.
Molecules w ith p o s itiv e Stark c o e ffc ie n ts , u su a lly upper
s ta te s , w ill be drawn to the axis o f the focusser.
t iv e Stark c o e ffic ie n ts are d e flecte d away.
Molecules w ith nega­
Thus, use o f the quadrupole
focusser allow s s p a tia l f i l t e r i n g o f states which g re a tly increases the
s e n s itiv it y o f the instrum ent.
The beam then enters the microwave c a v ity which is a c y lin d r ic a l
section o f copper o r brass tubing and 19 cm long.
TMqiq m°de c a v itie s were used.
For a ll experiments
For th is mode the resonant frequency o f
the c a v ity depends on i t s in s id e diameter and not on i t s length.
Sections
o f copper o r brass tubing could be accu rate ly machined to achieve the
required resonant frequency.
siz e .
Fine tuning o f the c a v ity depended upon it s
Small c a v itie s could be temperature tuned by heating the c a v ity
e le c t r ic a lly .
This method is not p ra c tic a l fo r larg e c a v itie s which were
fin e tuned by in s e rtin g a q u a rte r-in c h brass rod ( fo r higher frequencies)
or nylon rod ( fo r lower frequencies) in to the c a v ity .
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
S tim u la tin g ra d ia tio n was in je c te d in to the c a v ity via a 10 or
20 dB d ire c tio n a l coupler.
microwave re c e iv e r.
Emission was detected w ith a superheterodyne
The s tim u la tin g and lo ca l o s c illa to r s were phase-
locked to a harmonic o f the same sta b le 10 MHz c r y s ta l; however, the
interm ediate reference frequencies are s u ita b ly chosen so th a t the
s tim u la tin g and lo ca l o s c illa to r s are always 29.4 MHz a p a rt.
The s tim u la tin g power induces resonant emission.
The microwave
ra d ia tio n coupled out o f the c a v ity is mixed w ith the lo ca l o s c illa to r
in a balanced mixer to produce a beat frequency o f 29.4 MHz. To increase
s e n s itiv it y the c ry s ta l o s c illa t o r is frequency modulated, so th a t a fte r
f i l t e r i n g and a m p lific a tio n , the IF signal is demodulated and brought
through a lo c k -in d e te cto r referenced by the chopper a t 225 Hz w ith a
time constant o f 15 seconds.
The tenth harmonic o f the sta b le 10 MHz c ry s ta l is displayed on a
frequency counter, referenced to the 60 kHz transm ission o f WWVB, Boulder,
CO.
The output o f the lo c k -in de te cto r is displayed on a lin e a r ch a rt
recorder w ith automatic frequency markers derived from the counter.
D.
1.
Measurements on In d iv id u a l Molecules
Chloroacetylene-D
Chloroacetylene ro ta tio n a l tra n s itio n s were f i r s t measured by
Westenberg, G oldstein and W ils o n ^ who obtained the ro ta tio n a l constant,
c h lo rin e quadrupole coupling constant and d ip o le moment.
Weiss and
F ly g a re ^ have pre vio u sly measured the deuterium quadrupole coupling
stre ngth in a series o f deuterated acetylenes using a Stark modulated
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
microwave spectrometer.
In the case o f chloroacetylene-D the s p lit t in g
due to deuterium was on ly p a r t ia lly resolved and lin e shape routine s
were used to e x tra c t the coupling stre n g th .
We have made measurements
on th is isotope using a beam maser spectrometer, enabling us to go beyond
the Doppler lim ite d lin e w id h t (50 kHz) and f u l l y resolve the s p lit t in g due
to deuterium by o b ta in in g lin e w id th s o f 3kHz.
A 20 cm c y lin d r ic a l TMq-jq c a v ity was used.
The resonant frequency
o f the c a v ity could be adjusted by in s e rtin g a q u a rte r-in c h brass rod ( fo r
higher frequencies) or nylon rod ( fo r lower frequencies) in to the c a v ity .
The beam was generated by use o f a sin g le hole (0.15 m )
supersonic nozzle.
A presssure o f 100 t o r r was maintained behind the nozzle by cooling the
gas in a c h lo ro fo rm /^ slu sh (-6 3 °C ).
modulated a t 200 Hz.
The s tim u la tin g signal was frequency
D e riva tiv e spectra were recorded w ith lo c k -in detec­
tio n a t 200 Hz w ith a time constant o f 10 seconds.
Chloroacetylene was prepared by the method o f Bashford, Emelius
and B is c o e .^
E ighty per cent deuteration was obtained by exchange w ith
a lk a lin e D2 O as confirmed by the IR spectrum.
An i n i t i a l low re s o lu tio n measurement was performed using the
Stark modulated spectrometer to check the frequencies reported fo r the
J=l-*0 tr a n s itio n .
cie s.
No resonances were observed a t o r near these frequen­
Using the values f o r the ro ta tio n a l constant and c h lo rin e quadru­
pole coupling stre ngth reported by Westenberg we calcu la ted the expected
frequencies fo r the hyperfine components o f the f i r s t ro ta tio n a l tr a n s i­
tio n .
The ca lcu la te d frequencies were 2 MHz above the frequencies reported
e a r lie r .
Resonances were observed very close to these ca lcu la ted
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
frequencies.
I t should be mentioned th a t the spacings o f the measured
resonances were close to those reported by Weiss and Flygare.
D e riv a tiv e recorder traces o f a ll hyperfine components are shown
in Fig. 4.
The components are labeled f o r the J=1 le v e l since to f i r s t
order a l l hyperfine m a trix elements vanish f o r J=0.
2.
Propyne-D3
I n i t i a l measurements on propyne were made by Gordy
18
, et a l.
Previous work in v o lv in g deuterium quadrupole e ffe c ts was also reported
e a r lie r ,
19 20
*
however, the spectra obtained were not w ell resolved.
We
have remeasured the hyperfine components o f th is tr a n s itio n w ith a molecular-beam spectrom eter.
Three hyperfine components were com pletely
resolved by o b ta in in g an order o f magnitude increase in re s o lu tio n over
the previous work.
Propyne-D^ spectra were recorded using a s in g le TMq^ q mode copper
c a v ity 10 cm long.
The resonance o f the c a v ity was coarse adjusted by
in s e rtin g a piece o f brass shim in the lengthwise s lo t o f the c a v ity and
the in s id e diameter was changed by tu rn in g fo u r screws which squeeze the
»
c a v ity .
Fine tuning was achieved by e le c t r ic a lly heating the c a v ity to
the optimum temperature.
Propyne-D^
was prepared by re a ctin g dim enthylsulfate-D g w ith
sodium a c e ty lid e in DMSO.
tio n on a vacuum lin e .
The product was p u rifie d by fra c tio n a l d i s t i l l a ­
A pressure o f 100 t o r r was maintained behind the
nozzle source by coo lin g the gas in a pyridine/N g slush (-42° C).
A recorder tra c in g o f the strongest components o f the J=l->0
tr a n s itio n f o r propyne-Dg is shown in Fig. 5.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
F-7/2
kHz
+35
+28
+21
+14
0
+7
-14
7
-21
-28
-35
-42
F=5/2
F-3/2
F=l/2
kHz
-36
-27
0
•9
-18
+18
+27
+36
+45
kHz
■8
Fig. 4.
-4
0
44
46
+12
H yperfine components o f the J = 1 -* 0 tr a n s itio n .
(a) Recorder tra c in g o f F-j = 3/2 components.
C1CCD.
Frequency
r e la tiv e to 10 358 017 kHz. (b) F-| = 5/2 components.
Frequency r e la tiv e to 10 377 921 kHz. (c) F] = 1/2 component.
Frequency re la tiv e to 10 393 892 kHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
N
X
o
II
i
o
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
components of
Fig. 5.
o
CM
Derivative maser spectra
tra n s itio n , CD,CCH.
of
-o
the hyperfine
o
CM
the J = 0 to
•"D
24
■F-l
‘
F-3
>2
+44 +33 +22 +11 O' -11 -22 -33 -44 -55
F-2
Cb)
+48 +40 +32 +24 +15 46
-9-6-3
Fig. 6 .
0 -8 -16 -24 -32 -40 -48
0 +3 46 46
Hyperfine components o f the J = 1
0 tr a n s itio n . NCCCD.
(a) Fn = 2 components. Frequencies r e la tiv e to
Frequencies re la tiv e
8 443 386.0 kHz. (b) F, = 1 components.
to 8 442 092.0 kHz. (c) F-, = 0 component. Frequency
r e la tiv e to 8 445 318.0 kHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
3.
Cyanoacetylene-D
Cyanoacetylene is a molecule o f strong astrophysical in te r e s t;
ro ta tio n a l lin e s o f i t s various isotopes have been observed in in te r s t e lla r clouds since 1971.
21
The f i r s t microwave measurements were made
in 1950 by Westenberg and Wilson.
22
T y le r and Sheridan
23
looked a t the
microwave spectrum o f an excite d v ib ra tio n a l s ta te and determined the
s tru c tu re .
DeZafra
24
accurately measured the ro ta tio n a l constant and
hyperfine s tru c tu re fo r the ground v ib ra tio n a l s ta te on a beam-maser
spectrom eter.
the lit e r a t u r e .
A thorough review o f previous measurements is a v a ila b le in
25
We have accu rate ly measured the ro ta tio n a l constant and
hyperfine s tru c tu re o f the deuterated species via a molecular-beam m icrowave,spectrometer, f u l l y re so lvin g the hyperfine s tru c tu re due to
14
N and
D.
A 20 cm c y lin d r ic a l
mode brass c a v ity was used.
The c a v ity
was machined u n til the resonant frequency was s lig h t ly higher than the
desired frequency.
The c a v ity could then be tuned down in frequency by
in s e rtin g a q u a rte r-in ch nylon rod in to the c a v ity .
The beam was gener­
ated by use o f a sin g le hole (0.15 mm) supersonic nozzle.
A pressure o f
85 t o r r was maintained behind the nozzle by cooling the sample in an ice
water bath.
Cyanoacetylene-D was prepared by the method o u tlin e d by M allinson
and Fayt.^®
D e riva tive spectra o f a ll hyperfine components are shown in Fig. 5.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 4
ANALYSIS OF DATA
In tro d u c tio n
A computer program was w ritte n to c a lc u la te hyperfine s tru c tu re o f
a ro ta tio n a l tra n s itio n fo r up to fo u r nuclei o f a r b itr a r y spin.
The
general coupling scheme is as fo llo w s :
i , + j = F,
I 2 - F, + F2
I_
LO + F„ + F
13 + F3 = F
(24)
Expressions fo r the m a trix elements in th is coupled scheme were taken
from the papers o f Cook and DeLucia
6
and Thaddeus, K ris h e r, and Loubser.
5
This program was checked against a previous coupled-scheme program and
against the uncoupled basis set c a lc u la tio n (g ra c io u s ly provided by Dr.
Sid Young).
The case o f hyperfine s tru c tu re in the presence o f three
equivale nt spins has been pre vio u sly discussed in d e ta il.
The program in p u t data consisted o f the spin o f each nucleus and
i t s associated hyperfine in te ra c tio n strengths.
The e ffe c tiv e ro ta tio n a l
constant would be thrown in as a parameter so th a t we could f i t to the
actual experimental frequencies.
Good t r i a l in te ra c tio n strengths were
obtained from previous measurements on the same o r s im ila r systems.
actual experimental frequencies were also read in to c a lc u la te the
26
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The
standard d e via tio n fo r the f i t , defined as fo llo w s :
_ /v
6f i t
U
v1
obs.
- r* ,
•, n
c a lc . \ l / 2
(Pi
>
where N is the number o f lin e s being f i t .
/nr.x
<25>
The in te ra c tio n strengths and
ro ta tio n a l constant were varied to f i t the experimental frequencies.
The
e rro r in each parameter was estimated by varying each one in tu rn u n til the
standard d e via tio n changed s ig n ific a n tly .
Data Analysis f o r In d iv id u a l Molecules
Chloroacetylene-D
The hyperfine Hamiltonian consisted o f the deuterium quadrupole
and c h lo rin e quadrupole and s p in -ro ta tio n in te ra c tio n strengths.
Second
order quadrupole energies were calculated fo r c h lo rin e from the tables
2
given by Townes and Schawlow and added to the f i r s t order energies.
Table 1 l i s t s the measured and calcu la ted frequencies fo r each component.
The hyperfine constants are given in Table 2 and where possible compared
w ith those obtained from previous work.
Cyanoacetylene-D
The hyperfine Hamiltonian consisted o f the nitrogen quadrupole,
nitrogen s p in -ro ta tio n , and deuterium quadrupole in te ra c tio n s .
The
ca lcu la te d and observed frequency fo r each component is given in Table 3
as w ell as the standard d e via tio n f o r the f i t .
Hyperfine in te ra c tio n
strengths where possible are given in Table 4 and compared to previous
works.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 1.
Measured and ca lcu la te d frequencies fo r components o f the
J = 1 -»- 0 tr a n s itio n o f C1CCD.
Frequencies are given in kHz.
The
Standard d e v ia tio n fo r the f i t was Q*1 kHz.
Component (F^;F)
Measured
Calculated
( l / 2 ; l/ 2 , 3 / 2 )
10 393 892.0
10 393 892.1
(3 /2 ; 1 / 2 )
10 358 050.6
10 358 050.6
(3 /2 ;3 /2 )
10 357 975.3
10 357 975.2
(3 /2 ; 5/2)
10 358 017.1
10 358 017.1
(5 /2 ;3 /2 )
10 377 912.9
10 377 912.9
(5 /2 ; 5/2)
10 377 975.3
10 377 975.5
(5 / 2; 7/ 2)
10 377 931.6
10 377 931.6
Table 2.
Molecular Constants fo r Cl CCD
This Work
eqQ(D)
208.5 + 1.5 kHz
eqQ(Cl)
-79 739.5 + 1 kHz
Previous Work (R ef.)
225 + kHz
-79 670. + 100 kHz
-79 660 kHz
C(C1)
B
1.3 + .1 kHz
5 186 973.9 + .1 kHz
—
5187 010. kHz
aRef. 16
bRef. 15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
29
Table 3.
Measured and ca lcu la te d frequencies fo r components o f the J=1
to J=0 tr a n s itio n o f CNCCD.
Component (F^;F)
Frequencies are given in kHz.
Measured
Calculated
(l ; i )
8442051.56
8442051.51
(1 ;2 )
8442084.10
8442084.11
(1 SO)
8442130.42
8442130.58
(2 ;1 )
8443343.19
8443343.39
(2 ;3 )
8443366.70
8443366.93
(2 ;2 )
8443413.26
8443413.39
(0 ; 1 )
8445318.05
8445318.03
Table 4.
Molecular Constants fo r CNCCD
This Work
Previous Work
------
eqQ(D)
203.5 + 1.5 kHz
eqQ(N)
-4318.0 + 1 kHz
-4318.8 + 1.2 kHza
C(N)
1.1 + 0.2 kHz
1.05 + 0.26 kHza
C(D)
-.1 + 0.4 kHz
B
4221580.1 + 0.1 kHz
------
4221581.67 + 44 kHzb
aMeasured from data taken on NCCCH (Ref. 23).
bRef. 23.
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30
Propyne-D3
The h yp erfin e Hamiltonian consisted o f the deuterium quadrupole,
s p in -ro ta tio n , and mutual sp in -sp in in te ra c tio n s .
The sp in -sp in in t e r ­
a ctio n was ca lcu la te d from the known nuclear moments and s tru c tu re and was
not tre a te d as a v a ria b le parameter.
The measured and ca lcu la te d frequen­
cies fo r the observed components are given in Table 5.
Hyperfine in t e r ­
a ctio n strengths are given in Table 6 and compared w ith previous work.
Table 5.
Measured frequencies fo r J = 1
Frequencies in kHz.
0 tra n s itio n s in DC^CCH.
F value is the to ta l angular momentum in the
J = 1 s ta te .
F
Measured
Calculated
1
14 711 522.6
1411522.6
2
14 711 530.7
14711530.8
3
14 711 565.9
14711565.9
Table 6 .
Molecular coupling parameters and ro ta tio n a l constant fo r
cd3 cch.
Parameter
eqaaQ
eqzzQ
CD
D(D-D)
B
DJ
-55.0 + 0.5 kHz
174.0 + 6.0 kHz
0.06 + 0.08 kHz
-0 .2 5 3 kHz
7 355 767.0 + 1 .0 kHz
2 .8 b kHz
a, ca lcu la te d fo r nuclear moments and s tru c tu re , b from previous measure.
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CHAPTER 5
DISCUSSION OF RESULTS
A.
In tro d u c tio n
A tte n tio n w ill now be turned to the in te rp re ta tio n o f the measure­
ments made in th is work.
The re s u lts fo r each in d iv id u a l molecule w ill
be discussed follow ed by a general discussion on trends in measurements
o f deuterium quadrupole coupling strengths.
work done in th is area w ill be presented.
stants were measured fo r
14
N and
chloroacetylene-D , re s p e c tiv e ly .
35
A review o f the th e o re tic a l
Accurate s p in -ro ta tio n con-
Cl in cyanoacetylene-D and
A discussion o f chemical sh ie ld in g in
these isotopes w ill be presented and comparisons w ith NMR measurements
w ill be made.
B.
In d iv id u a l Molecules
Chloroacetylene-D
By examination o f Table I I i t is noted th a t the values o f eqQ(D)
reported by th is work and th a t o f Weiss and Flygare
16
agree w ith in the
e a r lie r lim its o f experimental e rro r ( i . e . , + 18 kHz); however, the
measurement from th is work has a much sm aller u n c e rta in ty ( + 1 .5 kHz).
values o f eqQ(Cl) do not agree w ith in the lim it s o f experimental e rro r.
Since the frequency measurements in th is work are more accurate by two
orders o f magnitude and there is second order quadrupole e ffe c ts are
included fo r c h lo rin e in our a n a ly s is , there is more confidence in the
present re s u lts .
31
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The
32
The analysis also yielded an accurate measurement f o r the spinr o ta tio n in te ra c tio n strength f o r
35
Cl.
The s p in -ro ta tio n in te ra c tio n
strength f o r the k^*1 nucleus is defined as:
C(K) =
- I <J 2> M/J(d+1)
g
y
yy
(26)
Mgg are the s p in -ro ta tio n tensor diagonal elements.
T <J 2> is simply J(J + 1) and
L
g
= -M
xx
= -M
•
yy
For a lin e a r molecule
Calculation o f these
tensor elements from s p in -ro ta tio n measurements enables the estim ation o f
the paramagnetic and diamagnetic c o n trib u tio n s to the to t a l chemical
s h ie ld in g .
The diagonal
along the g
th
elements o f the s h ie ld ing tensor f o r nucleus k
p rin c ip a l axis are given by:
(K) _
gg ~
(d) , _(p)
gg
gg
(97)
' ’
d
n
and aP are, re s p e c tiv e ly , the diamagnetic and paramagnetic co n trib u yy
yy
tio n s to the shie ld ing tensor elements and are calculated from the
o
fo llo w in g expressions:
°gg '
> <0I ^ K
2
( r iK -<r iK ) g) l ° >
hcM
=
0
^
9
<28a)
9
[ ^ - ( r iK ) g ] >
<28b>
ypg is the nuclear magneton, g^ is the nuclear g value, r ^ is the vector
from nucleus k to the other i n u c le i, Ggg is the ro ta tio n a l constant along
th
the g
p rin c ip a l axis.
CTgg cannot be found d i r e c t l y from experimental data; one must re ly
on molecular o r b ita l c a lc u la tio n s .
In c o n tra s t, the paramagnetic c o n t r i ­
bution can be determined from measured s p in -ro ta tio n constants and know­
ledge o f the molecular geometry.
This is very useful since c a lc u la tio n
o f the paramagnetic term would require evaluation o f m atrix elements
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33
coupling the ground state to excited states w ith nonzero o r b it a l angular
momentum.
The intim ate re la tio n s h ip between the s p in -ro ta tio n in te ra c tio n
and chemical sh ie ld ing was f i r s t pointed out by RamseyJ
Since chloroacetylene is a lin e a r molecule, obtaining the compo­
nents o f oj? is e s p e cia lly simple. Evaluating the right-hand side o f
yy
35
equation 26b and using our measured value o f C( Cl) = 1.3 kHz, we obtain
°vv
= °nn
= ° ~ "420 PPmXX
yy
The average value o f a P is o n e -th ird the trace
o f the shie ld in g tensor, -280 ppm.
The question now a ris e s , what should be done w ith th is r e s u lt , o r,
is there anything to compare i t to?
r f NMR data were a v a ila b le , the
measurement o f the paramagnetic p a rt o f the shie ld in g could be combined
w ith a measured value o f the to t a l s h ie ld in g .
Note th a t the corresponding
components o f each cannot be subtracted unless the to ta l sh ie ld ing aniso­
tro py was a c tu a lly measured in the NMR experiment.
Sometimes c a lc u la tio n s
are a v a ila b le th a t give the anisotropy o f the diamagnetic p a rt o f the
nuclear s h ie ld in g .
This data can be combined w ith the measured paramagne­
t i c s h ie ld ing and comparison w ith NMR measurements can be made.
A good
13
example o f th is consistency check is the beam-maser work on CHg CN.
There are problems w ith obtaining nuclear s h ie ld ing constants f o r
c h lo rin e (as well as the higher halogens).
F i r s t , you must remember th a t
the shie ld in g constant o f a nucleus in a p a r tic u la r compound cannot be
d i r e c t l y measured in an NMR experiment; only the "chemical s h i f t " , which
is the d i f f e r e n t chemical environment is obtained.
Chemical s h i f t s o f
c h lo rin e , bromine and iodine have been mostly measured r e la t iv e to the
corresponding halide ion in aqueous s o lu tio n .
Since the ion s h i f t s are
themselves dependent on the nature o f the cou n te r-io n , s a l t concentration
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34
and temperature, they are not ideal references.
37
In a d d itio n ,
35
Cl or
Cl NMR signals have considerable lin e widths (several hundred ppm) due
to quadrupolar re la x a tio n , the chemical s h i f t covering a range o f about
1000 ppm.
Thus, due to experimental d i f f i c u l t i e s only s h ie ld in g data f o r
a few covalent c h lo rin e compounds have been reported.
The e r ro r bars
associated w ith these measurements are usu ally q u ite la rg e , e . g ., a (
-340 + 100 ppm in chloroform.
has been reported.
35
C l) =
No chemical s h i f t data f o r chloroacetylene
The p o s s i b i l i t y o f ever being able to s o rt out the
diamagnetic and paramagnetic c o n trib u tio n s to the chemical sh ie ld ing in
35
Cl compounds is not encouraging.
The chemical s h i f t in the gas phase
would have to be measured r e la t iv e to a common reference such as aqueous
NaCl s o lu tio n .
This is a n o n - t r iv ia l problem.
Cyanoacetylene-D
I t is noted from Table IV th a t the deuterium quadrupole coupling
strength along the C-D bond is accurately measured a t 203.5 + 1 . 5 kHz.
Included are the hyperfine constants measured on CNCCH by Robert DeZafra
with the molecular-beam apparatus a t Columbia U n ive rs ity .
constants on
14
24
The hyperfine
N are not expected to be very d i f f e r e n t f o r the two is o ­
to p ic species and they are found to be the same w ith in experimental e rro r.
Although DeZafra was able to obtain high re s o lu tio n in his measurements
(FWHM = 3 kHz) he was unable to resolve the s p l i t t i n g due to the spinro ta tio n in te ra c tio n on hydrogen.
For the deuterated species the spin-
r o ta tio n in te ra c tio n only causes a r e la t iv e s h i f t o f the hyperfine lin e s
since no new quantum numbers are introduced when i t is included in the
Hamiltonian.
I t s c o n trib u tio n os small w ith a correspondingly high
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f
35
u n c e rta in ty.
The s p in -ro ta tio n in te ra c tio n strength f o r hydrogen would
be expected to be about a fa c to r o f seven greater.
Void,
27
e t a l . , have measured the
13
C, D and
re la x a tio n o f cyanoacetylene-D in toluene s o lu tio n .
re la x a tio n times gives l i q u i d phase values f o r
coupling constants.
measured.
14
14
N nuclear magnetic
Analysis o f these
N and D quadrupole
Chemical s h i f t anisotropies f o r
13
C were also
Table VII compares gas phase values w ith the liq u id NMR re s u lts .
There is a noticeable d iffe re n c e (4%) between eqQ(^N) in the li q u i d and
the s o lid phase.
The solvent undoubtedly plays a ro le in how i t a ffe c ts
cyanoacetylene-cyanoacetylene hydrogen bonding.
The d iffe re n ce is less
than th a t found in s o lid vs. gas phase measurements (6-14%) o f quadrupole
coupling constants.
Analysis o f the data also yielded
in te ra c tio n strength on ^ N : C (^N ) = 1.1
a value f o r the s p in -ro ta tio n
+ 0.2 kHz.
Proceeding in a
s im ila r way f o r chloroacetylene, a*3 ( ^ N ) = -593 ppm.
The diamagnetic c o n trib u tio n to the chemical s h i f t can be com­
puted from ab i n i t i o c a lc u la tio n s or more conveniently by Flygare's atom
dipole method.
The former was chosen f o r i t s s im p lic it y and the good
agreement i t
provides
w ith ab i n i t i o c a lc u la tio n s .
and paramagnetic tensor elements f o r
Table V I I I .
14
The diamagnetic
N in cyanoacetylene are given in
The diamagnetic and paramagnetic tensor elements can be
combined to obtain the components o f the to t a l s h ie ld ing tensor.
th a t
= 18 ppm.
We fin d
U nfortunately, chemical s h i f t data f o r cyanoacetylene
is not y e t a v a ila b le (owing to referencing problems); however, the value
obtained is in reasonable agreement w ith th a t o f 25 ppm given f o r CH^CH.
Note th a t t h is value is r e la t iv e to the bare
14
N nucleus.
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36
Table 7.
Comparison o f gas and l i q u i d phase hyperfine constants in CNCCD.
Parameter
Gas Phase
g
Liquid Phaseb
eqQ(14N)
-4318.0 + 0.1 kHz
-4 140 + 50 kHz
eqQ(D)
203.5 + 1 . 5 kHz
200 + 2 kHz
C(N)
1.1 + 0.2 kHz
—
aThis work.
bRef. 27.
Table 8.
Chemical s h i f t tensor elements o f 14N in NCCCD ( a ll values in ppr
Tensor element
a^
ap
xx
446
-593
-147
yy
446
-593
-147
zz
348
0
348
413
-395
18
Average
a
Propyne-Dg
By examination o f Table 6 i t is noted th a t the deuterium quadru­
pole coupling along the a -ro ta tio n a l axis is accurately determined at
-55.0 + 0 . 5 kHz; however, the coupling strength along the C-D bond d ire c ­
tio n (which is what we are r e a lly inte re ste d in has a la rg e r un certainty
(+ 6 kHz).
This is so because when the e rro r is propagated in to eqQ along
the C-D bond d ire c tio n the m a jo rity o f the e rro r comes from the un certa inty
in the molecular s tru c tu re .
This is an inherent problem w ith CDgX type
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37
molecules.
With the assumption th a t the C-D bond is c y l i n d r i c a l l y
symmetric the value along the C-D bond is given as:
eq
dd
Q = eq^ C~D) (3cos2e - l)
£
(29)
where 0 is the C-C-H angle which has an u n certa inty o f + 0.5°.
The
value o f eqQ along the C-D bond o f 174 kHz is in e x c e lle n t agreement with
the previous value although the uncertainty is reduced by a fa c to r of
three.
Deuterium Quadrupole Coupling in Molecules
I t is useful a t th is p o in t to make a connection with the theore­
t i c a l work done in th is area.
High precision measurements o f deuterium
quadrupole coupling have come out o f th is work; to p re d ic t the in te ra c tio n
strengths obtained here would require equal precision in accounting fo r
the e le c tro n ic d is tr ib u tio n s in these molecules.
The problem is tra c ta b le
from a purely th e o re tic a l end f o r several reasons: (1) the deuteron c o n t r i ­
butes almost i n s i g n i f i c a n t ly to the e le c t r ic f i e l d gra d ie n t; (2) there
are no complications due to Sternheimer a n ti-s h ie ld in g which is prominent
in heavy atoms.
The e l e c t r i c f i e l d gradient at the deuteron is the sum o f an
e le c tro n ic and nuclear c o n trib u tio n :
o_2
eq(D) = +J.Z
Zn is
the
cule.
°V ^
r Dn
the charge o f nucleus n
n n u c le i, i is the index f o r
2
-2
- e<v*l
-
2
DV rDi
l*>
(30)
in un its o f e, the index n
is the sum over
r Di
the sum over the electrons in the mole­
I t should be remembered th a t what is measured in the lab is the
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38
product eqQ and not eq.
The measured quadrupole coupling can be converted
to the e l e c t r i c f i e l d gradient w ith in the accuracy o f the measurement
and the known value o f Q f o r the deuteron.
late d Q f o r the deuteron to be 2.860 x 10
-27
Reid and Vaida
29
have calcu-
2
cm based on an e le c t r ic
f i e l d gradient computed from an 87 term e le c tro n ic wave fu n c tio n .
Their
analysis included e ffe c ts f o r v ib ra tio n a l averaging.
The f i r s t term in eq. 28 is e a s ily calculated from the molecular
geometry.
now f i l l
Imagine a t th is p o in t a molecule co n sistin g o f only bare n u c le i;
in the ele ctron density in the molecule.
This is the d i f f i c u l t
pa rt o f the problem.
There have been two approaches in the estim ation o f deuterium quadrupole coupling in molecules.
Snyder
30
has computed e le c t r ic f i e l d grad­
ie n ts a t molecular deuterons from_ab i n i t i o wave functions in a double
zeta basis.
B a r fie ld ,
31
e t a l . , have done the same c a lc u la tio n s employing
semi-empirical molecular o r b it a l theory ( i . e . , SCF-MO in the INDO approx­
im ation).
Both methods o f c a lc u la tio n give c o n s is te n tly high values fo r
the deuterium quadrupole coupling.
This state o f a f f a i r s is probably due
to an underestimation o f the e le c tro n ic c o n trib u tio n .
The ab i n i t i o
approach seems to work b e tte r when dealing w ith a C-D moiety.
Table 9
compares the re s u lts between the calculated (by ab i n i t i o and semiem pirical methods) w ith experimental measurements.
There are several q u a lit a t iv e and q u a n tita tiv e observations to be
made concerning the c a lc u la tio n s and measurements: (1) eqQ is p o s itiv e f o r
a deuterium bonded to a f i r s t row atom.
This is due to incomplete
s h ie ld ing by bonding electrons o f the nucleus to which the deuteron is
bonded. (2)
As nuclei f u r th e r away are considered from the deuteron and
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39
i t s associated e le c tro n ic d i s t r ib u t i o n the e le c tro n ic and nuclear
c o n trib u tio n s e f f e c t i v e l y cancel one another. (3) The calculated quadru­
pole couplings are con sistenly higher than the measured values (about 20%).
Snyder
30
has responded to the t h i r d observation by inc lu s io n o f
p basis fun ction s on the deuteron and d basis functions on the carbon atom
bonded to the deuteron.
This has resulted in the lowering o f calculated
values to w ith in 8% o f experimental measurements.
Observations (1) and (2) are well documented in Table 9 where,
quadrupole coupling strengths along the C-D bond are l i s t e d f o r eig ht
molecules.
The f i r s t fo u r molecules involve quadrupole coupling on an
a ce tyle n ic deuterium; the f in a l fo u r are CDgX type molecules.
A ll
measurements except f o r those on FCCD are high p re c isio n .
A fte r examination o f Table 10 two trends are immediately obvious:
(1) the deuterium quadrupole coupling is not very s e n s itiv e to s u b s titu tio n
w ith in a group where the deuterium is bonded to carbon o f given h y b rid i3
3
zation (sp or sp ) (2) eqQ(sp) > eqQ(sp ) by about 20%.
Let us consider the f i r s t trend.
We have a v a ila b le in t h is work
f o r the f i r s t time comparison o f quadrupole coupling constants on acetyle n ic deuteriums where the measurements have been precise enough to show
real d iffe re n c e s .
I t is in te re s tin g to ponder these f i n i t e differences
w ith the a v a ila b le s tru c tu ra l data.
Table 11 gives the quadrupole coupling
f o r several a ce tylenic deuterons along w ith the bond distance o f the C-D
moiety.
Propyne-D-| and cyanoacetylene-D have v i r t u a l l y the same bond
distance f o r the C-D fragment y e t the dueterium quadrupole coupling is
s i g n if ic a n t l y d i f f e r e n t (about 10%).
Although q u a lit a t iv e statements must
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40
Table 9.
Comparison o f c alcu la ted and experimental deuterium quadrupole
coupling strengths f o r some selected molecules.
A ll values are in kHz.
DZ = double zeta basis, SE = sem i-em p irical, DZ + P = double zeta basis
plus p o la riz a tio n wave fu n ctio n s .
Molecule
DZ
CH3D
226
DCN
DZ + P
SE
exp.
208
268.7
192a
248
213
314.9
194b
HDO
378
339
311.7
307.9C
nh2d
308
310
304.8
291d
aRef. 32.
bRef. 33.
CRef. 34.
dRef. 35.
Quadrupole coupling strengths along the C-D bond d ire c tio n .
Table 10.
A ll values are in kHz.
Molecul e
eqQ
Ref.
Cl - C = C - D
208.,5 + 1.5
This work
CN - C = C - D
203.,5 + 1.5
212 + 10
228 + 2
This work
F - C= C - D
ch3
- C: e c - d
16
36
CD3CN
168 + 4
37
CD3C1
166 + 5
38
CD3Br
175 + 3
39
174 + 6
36
cd3c
= C- H
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41
Table 11.
Quadrupole coupling constants and bond distances f o r the C-D
moiety in several deuterated acetylenes.
eqQ(kHz)
Molecule
C-D(A)
228 + 2
1.058 + 0 . 002a
Cl - C = C - D
208 + 2
1.0550 + 0.00005b
NC - C = C - D
204 + 2
1.057 + 0.001C
ch3 c
= C- d
aRef. 40.
bRef. 41.
CRef. 25.
be taken w ith caution we might say the d iffe re n c e can be explained by more
e ff e c tiv e shie ld in g on the carbon nucleus bonded to deuterium in
1g
cyanoacetylene. Flygare
has pointed out th a t f o r acetylenes a 10%
change in the nuclear sh ie ld in g o f the carbon atom d i r e c t l y bonded to
deuterium w i l l have the same e ff e c t as a 100% change f o r the next nearest
carbon atom.
Since the bond distance o f the C-D moiety is the same fo r
both molecules the s ig n if ic a n t d iffe re n c e in the deuterium quadruple
coupling points to the chemical shie ld ing on the adjacent
carbon atom.
S im ila r reasoning can be applied to explain the d iffe re n c e in quadrupole
coupling in Cl CCD and NCCCD.
The C-D bond distance is s l i g h t l y smaller
f o r C1CCD leading to a higher nuclear c o n trib u tio n ( p o s itiv e ) +o the
quadrupole coupling.
I f the nuclear sh ie ld ing on the adjacent carbons
were the same Cl CCD would be expected to have a la rg e r quadrupole coupling.
Note th a t these arguments are extremely q u a lit a t iv e and may have l i t t l e
basis.
The e le c tro n ic
c o n trib u tio n from c h lo rin e has been completely
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42
ignored.
As a second row atom, computing i t s c o n trib u tio n to the
e l e c t r i c f i e l d gradient a t the deuterium atom is a n o n - t r iv ia l problem.
B a rfie ld
31
has established distance c r i t e r i a f o r the in clu sio n o f
various c o n trib u tio n s to the deuterium quadrupole coupling in p r o p y n e - D ^ .
He evaluated two and three center in te g ra ls associated w ith atomic centers
in the molecule.
The three center terms were found to make a c o n trib u tio n
o f -53 kHz to the quadrupole coupling.
The p o in t o f th is is th a t s i g n i f i ­
cant d iffe ren ces in quadrupole coupling may be d i r e c t l y re la ted to these
terms and the re fo re transcend q u a lita tiv e arguments.
The la s t po int to be addressed is the e ff e c t o f the h y b rid iz a tio n
on the carbon atom to which the deuterium is bonded.
Mi 11e t t and Dailey
42
measured deuterium quadrupole coupling strengths in the liq u id phase on a
series o f hydrocarbons and observed from the measurements th a t eqQ(sp)>
2
3
eqQ(sp )> eqQ(sp ). This conclusion must be taken w ith caution on exp e ri­
mental and th e o re tic a l grounds.
When making comparisons o f deuterium quad­
rupole coupling in d i f f e r e n t systems one must note the e r ro r bar associated
w ith the reported measurement.
Many o f the molecules they were comparing
had u n c e rta in tie s o f the order + 15 kHz.
Absolute trends can t r u l y be
established when measurements have been shown to d i f f e r w ith in the l i m i t s
o f experimental e rro r.
Furthermore, in support o f t h e i r conclusion they
pointed to the work o f Fung,
43
et a l . , who studied the dependence o f the
deuterium quadrupole coupling on carbon h y b rid iz a tio n via a simple MO
d e s c rip tio n .
They treated the C-D moiety as an is o la te d fragment inse n s i­
t i v e to s u b stitu e n t e ffe c ts .
This is an o v e rs im p lific a tio n and t h e i r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
43
basis set was minimal.
B a rfie ld
31
C le a rly, the l a t e r treatments o f Snyder
30
and
demonstrate the importance o f not tre a tin g the C-D moiety as
an is o la te d fragment.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX
PROGRAM FOR CALCULATION OF HYPERFINE STRUCTURE
The program KUPLD calcu la tes how a ro ta tio n a l level s p l i t s due to
e l e c t r i c quadrupole, s p in -ro ta tio n and spin-spin in te ra c tio n s .
Hyperfine
s tru c tu re can be calcu la ted f o r the most general case o f fou r coupling
nuclei o f a r b it r a r y spin.
The order in which the nuclei are coupled is
immaterial as a f u l l m atrix dia gon alization is always a p a rt o f the calcu­
la tio n s .
As a check the order o f the coupling can be reversed even during
the same run since the program w i l l accept more than one set o f data.
The algorithm is simple enough to fo llo w .
The philosophy in which
the program was w r itte n was such th a t a person could ca lc u la te hyperfine
s tru c tu re in ro ta tio n a l tra n s itio n s even i f t h e i r background in the area
was lim ite d .
One merely inputs the ro ta tio n a l le v e l, spin o f each nucleus
and t h e i r associated in te ra c tio n strengths.
Note th a t the program can not d i f f e r e n t i a t e between equivalent and
non-equivalent spins; no attempt was made to s im p lify the energy matrix
f o r special cases ( e . g . , CDgX type molecules).
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
PROGRAM K U P L D ( I N P U T # O U T P U T » T A P E 5 « I N P U T » T A P E 6 - 0 U T P U T )
T H I S PROGRAM CAL C UL A T E S THE S P L I T T I N G OP ROT A T I ONAL
L E V E L S DUE TO O U A D R U P O L E » S P I N - R O T A T I O N # A N D S P I N - S P I N
I N T E R A C T I O N S FOR THE CASE OF FOUR N U C L E I OF ARBI T RARY S P I N .
THE B A S I S ST AT ES ARE WRI T T E N I N T H E COUPLED SCHEME AND ARE
OF THE FORM ! J » 1 1 » F I > I 2 » F 2 , 1 3 » F 3 , U # F > .
REAL H ( 3 f 3 ) » U ( 3 f 3 ) » 0 J ( 6 ) » C J ( 4 ) » Q ( 4 ) » E ( 3 )
REAL J » I 1 » I 2 » I 3 » I 4 « J 2 » I S S » I J
C0HM0N/0SRSS/J*I1»I2»I3>I4
C OMMO N / S OE / Q » B R O T
C THE I N P U T DATA ARE AS F OL L OWS t J I S THE ROT AT I ONAL LEVEL *
C I 1 I S THE S P I N OF NUCLEUS ONE ( THE SAME FOR I 2 » I 3 » I A > .
11 R E A D ( 5 > 6 0 0 ) J » I l » I 2 * I 3 » I < r
600 F0RMAT(5F10.0)
C
C
C
C
C
W R I T E ! 6* 7 0 0 ) J » 1 1 # I 2 » 1 3 , IA
700 FORMAT<1X» • J - ' t F5 . 1 * ' 11* • » F 5 . 1 » • I 2 - ' » F 5 . 1 >
l*I3 » 'fF 5 .1 » 'I< i» *,F 5 .1 )
I F ( J . E O . O ) CALL PMDSTOP
THE I N T E R A C T I O N STRENGTHS ARE NOW READ I N . THE
D J ' S ARE THE S P I N - S P I N CONSTANTS BETWEEN THE FOUR N U C L E I .
DJ ( 1 ) I S THE S P I N - S P I N I N T E R A C T I O N STRENGTH BETWEEN
N U C L E I ONE AND T W 0 . D J I 2 ) I S THE S S I S BETWEEN N U C L E I ONE
AND THREE » D J ( 3 ) I S THE S S I S BETWEEN
N U C L E I ONE
AND F OUR* D J ( A ) I S THE S S I S BETWEEN N U C L E I TWO AND THREE
D J ( 5 > I S THE S S I S BETWEEN N UCL E I TWO AND F 0 U R # D J < 6 > I S THE
S S I S BETWEEN N U C L E I THREE AND F OUR. THE C J ' S ARE
THE S P I N - R O T AT I ON I N E V T E R A C T I O N ST RENGT HS. C J ( 1 ) I S
THE S P I N - R Q T A T I D N I N T E R A C T I O N STRENGTH FOR NUCLEUS ONE
( ANALOGOUS FOR C J ( 2 ) . . . C J ( A ) . THE O ' S ARE THE OUADRUPOLE
I N T E R A C T I O N STRENGTHS FOR N U C L E I ONE THPOUGH F OUR.
R E AD( 5» 6 6 ) BROT
66 FORMAT(F1 0. 0)
W R I T E ( 6 » 7 6 ) BROT
7 6 F OR MAT ( 1 X » ’ THE ROT AT I ONAL CONSTANT I S ' » F 1 5 . 5 )
R E A D ( i t BOO > ( D J ( I ) # I * l > 6 )
READ ( 5 » 8 0 0 ) ( CJ ( I ) » I ■ 1 . A )
R E AD ( 5» 8 0 0 ) ( O ( I ) . I - l . A )
BOO F 0 R M A T 1 6 F 1 0 . 0 )
WRITE(6»900)
9 0 0 F ORMAT ( 1 H 0 » ' T H E S P I N - S P I N CONSTANTS A R E ' )
WRITE(6»910) ( D J ( I ) » I " l » 6 )
910 F0RMAT(1X»F10.3)
WRITE( 6»920)
9 2 0 F ORMAT ( 1 H 0 » * THE S P I N - R O T A T I O N CONSTANTS A R E M
WRITE(6»910) ( C J ( I ) » I » 1 . A )
WRITE(6>930)
9 3 0 F O R M A T ! 1 H 0 » « T H E OUADRUPOLE COUP L I NG CONSTANTS A R E * )
WRI T E ( 6» 9 1 0 ) ( Q ( I ) » I * 1 * 4 )
DO 3 1 N l - 1 # 3
DO 3 1 N 2 " 1» 3
H( N1 »N 2)■ 0.
3 1 C O NT I NUE
C THE H A M I L T O N I A N M AT R I X I S I N D E X E D :
Nl *1
N2-1
C I N T H I S PART OF THE PROGRAM THE S P I N - S P I N
C
C
C
C
C
C
C
C
C
C
C
C
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C I N T E R A C T I O N I S CAL CU L A T E D
DO 1 0 F l - A B S U - I D . A B S d + I l )
DO 1 0 F 2 - A B S ( F 1 - I 2 ) . A B S ( F 1 + 1 2 )
DO 1 0 F 3 - A B S ( F 2 - I 3 ) » A B S ( F 2 + I 3 )
DO 1 0 F - A B S < F 3 - U ) , A B S t F 3 d 4 )
DO 2 0 F 1 P - A B S ( J - I 1 ) » A B S ( J + I 1 )
DO 2 0 F 2 P - A B S ! F 1 P - I 2 ) , A B S < F 1 P * I 2 )
00 20 F 3 P - A B S ( F 2 P - I 3 ) » A B S ( F 2 P * I 3 )
DO 2 0 F P - A B S ( F 3 P - I A ) » A B S ( F 3 P + I A )
I F ( F . N E . F P ) GO TO 3 0
C THE NEXT ST AT EMENT ACTS AS S WI T CH TO BYPASS C A L C UL A T I ON
C OF THE S P I N - S P I N I N T E R A C T I O N I F ALL I N T E R A C T I O N STRENGTHS
C ARE SET EQUAL TO Z E R O. T H I S I S USEFULL AS CA L C U L A T I O N
C OF THE S P I N - S P I N PART USES UP Q U I T E A B I T OF E X E C U T I O N T I M E .
IF(DJd).EQ.O.AND.DJ<2).EQ.O.AND.DJ(3).£Q.O.AND
l . D J ( 5 ) . E Q . O . A N D . D J < 6 ) . E Q . O > GO TO 15
KOUNT * 1
DO 1 0 0 L - 1 . 3
DO 1 0 0 L L - L d . A
DO 2 0 0 F 1 P P - A B S ( J - I 1 ) » A B S ( J + I 1 )
DO 2 0 0 F 2 P P « A B S ( F 1 P P - I 2 ) , A B S ( F 1 P P + I 2 )
DO 2 0 0 F 3 P P - A B S ( F 2 P P - I 3 ) , A B S ( F 2 P P * 1 3 )
DO 2 0 0 F P P - A B S ( F 3 P P - I A ) , A B $ < F 3 P P * I < . )
I F ( F . E Q . F P P ) THEN
H ( N 1 * N 2 ) - H ( N 1 » N 2 > * D J ( KOUNT) ♦ ( 1 . 5 * ( I J ( F 1 , F 2 . F 3 # F , F 1 P P , F 2PP,
1F3PP,FPP,LJ*IJ(F1PP,F2PP,F3PP,FPP,F1P,F2P»F3P»FP,LL>
2 * I J I F 1 . F 2 . F 3 . F . F 1 P P . F 2 P P . F 3 P P . F P P . L L ) * I J ( F 1 P P » F2PP »
3F3PP»FPP»F1P»F2P»F3P»FP»L))-(ISS(F1»F2»F3»
AF,F1PP,F2PP,F3PP,FPP,L»LL)*J2<F1PP,F2PP,F3PP»FPP,
5F 1 P .F 2 P .F 3 P .F P )))
END I F
2 0 0 C ON T I NU E
KOUNT - KOUNT ♦ 1
1 0 0 C ONT I NUE
15
DO 3 0 0 L - l . A
H C N 1 , N 2 ) - H < N 1 , N 2 ) ♦ C J ( L > * I J ( F 1 , F2 , F 3 . F , F 1 P. F2 P . F 3 P , F P . L )
1+0(L)*V0(F1,F2,F3,F,F1P,F2P,F3P,FP,1>
3 0 0 CONT I N UE
30
N2-N2 *1
2 0 C ON T I NU E
N l-N ld
N2-1
1 0 CO N T I N U E
WRITE( 6 * 4 0 )
AO F ORMAT d H O . * UNDI AGNOl I ZED H A M I L T O N I A N • J
M» 3
CALL O U T J H . M )
W-0.
DD 5 0 1 - 1 . 3
W- W ♦ A B S ( H d . I ) )
5 0 CONTI NUE
F I N - ( W / 3 . ) * 1 .OE-12
CALL J A C O B I H . U , 3 . 3 . 1 . F I N )
WRITE( 6 , A 1 )
A1 F O R M A T ! 1 H 0 . ' D I A G N O L I ZED H A M I L T O N I A N ' )
CALL O U T ( H . M )
WRITE(6.A2)
A2 F O R M A T d H O . ' T R A N S F O R M A T I O N M A T R I X ' )
CALL O U T C U . N )
WRITE(6.60)
60 FORMATdHO."EIGENVALUES” )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
00 61 1 - 1 * 3
E( I ) - H ( 1 * 1 )
E(I) -E d )
♦ 2»*BR0T
6 1 CO N T I N U E
CALL S O R T ( E * 3 )
OQ 6 3 1 - 1 * 3
WRITE( 6 * 6 2 ) E ( I )
6 3 CO N T I N U E
6 2 F OR MA T ( 1 X * F 1 5 . 5 )
GO TO 1 1
END
SUBROUTI NE O U T ( X * M )
DI MENSI ON * ( 3 * 3 )
DO 2 0 K - l * M * 1 0
WRI TE( 6 * 2)
2 F OR MA T ( 1 H 0 )
DO 2 0 J - 1 * M
LL-K
LU-K+9
I F ( L U . G T . M ) L U- M
WRITE( 6 * 1 ) ( X ( J * L ) * L - L L » L U )
1 F0RMAT(1H0*10G13.7)
t F ( L U . E Q . M . A N D . J . E O . M ) GO TO 10
2 0 C ON T I NU E
1 0 RETURN
END
S U B RO U T I NE
JACOB ( H , U* M* N* I F U , F I N )
H I F THE ARRAY TO BE 0 1 A G O N A L I Z E 0 .
U I S THE U N I T A R Y MA T R I X USED FOR F ORMAT I ON OF THE E I G E N V E C T OR S •
M I S THE AL L OT T ED ORDER OF THE MAT RI C E S H , U I N C A L L I N G PROGRAM
N I S THE ORDER OF H, ( U ) USE D.
N CANNOT BE GREATER THAN M
I F U MUST BE SET EQUAL TO ONE I F E I GE N V A L U E S AND E I GE NVECT ORS ARE
TO BE COMPUT ED.
I F U MUST BE SET EQUAL TO TWO I F ONLY E I GE NV AL UE S ARE TO BE
COMPUT ED.
F I N I S THE I N D I C A T O R FOR S H U T - O F F , THE F I N A L LARGEST OFF DI AGONAL
E L E ME N T .
THE SUBROUT I NE OPERATES ONLY ON THE ELEMENTS OF H THAT ARE TO THE
R I G H T OF THE MA I N D I A G O N A L .
CALL ERRSET ( 2 0 8 * 0 * 2 5 * 1 * 0 )
CALL ERRSET ( 2 0 7 , 2 5 6 , 2 5 , 1 , 0 )
CALL T R A P S ( 0 » 2 0 , 2 0 )
DI MENSI ON
H ( M , M )* U(M*M)
1
2
3
*
5
6
PREPARATORY OPE R AT I ON
GO TO ( 1 * 4 ) ,
IFU
DO 3
I - 1* N
DO 2
J ■ 1* N
U d * J ) - 0.0
U ( I * 1> - 1 . 0
I F (N . E Q . 1)
GO TO 1 0 0
NS1 ■ N - 1
SUM > 0 . 0
DO 6
I - 1 * NS1
IA1 - I ♦ 1
DO 5
J - IA1*
N
SUM - S U M + H ( I * J ) * H ( I * J )
CONTI NUE
I F ( SUM . L T . F I N P F I N )
GO TO 1 0 0
OFFMAX ■ S O R T ( S U M + S U M )
HN - N
R eproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
non
48
S C A N N I N G FDR
r> n o
7
6
LARGE
OFF
DF FMAX ■ OF F MA X / H N
NEMO - 1
DO 1 7
I - 1# NS1
IA1 • I ♦ 1
DO 1 7
J ■ I Alt N
I F ( ABS( H ( I , J ) ) . L T .
NEMO • 2
DI AGONAL
OFFMAX)
ELEMENT
GO TO 1 7
T RANS F ORMAT I ON
HII - H U , I )
HJJ - H I J , J )
HIJ ■ H( I , J )
TANG ■
SIGNI2.0, (HII-HJJ)>*HIJ
1
/ ( ABS(HII-HJJ) ♦ SORT*(HII-HJJ )**2 ♦ H IJ * * 2 * * . 0 ) >
COSIN ■ l . O / S O R T f l . O ♦ TANG** 2)
SINE • TANG*C0SIN
IS1 - I - 1
IF (IS 1 .EO. 0)
GO TO 1 0
DO 9
K ■ 1, IS1
HKI ■ H ( K , 1)
H ( K » I ) « H K I * C OS I N ♦ H ( K , J ) * S I N E
H ( K» J ) ■ —H K I * S I N E ♦ H ( K , J ) AC OS I N
9 C ONT I N UE
10 H U , I )
■H II*C 0 S IN **2 ♦ HJJ*SINE**2
♦ H I J * S I NE * C OS I N * 2 . C
H( J»J ) - H II*S IN E **2
♦ HJJ*C0SIN»*2 - HIJ*SINE*C05IN*2.0
JS1 ■ J - 1
I F I1A1 . G T . JS1)
GO TO 12
DO 1 1
K - I A 1 , JS1
HIK « H I l i K I
H ( I , K ) • HIK»COSIN ♦ H ( K , J ) * S I N E
H ( K , J ) • —H I K * S I N E ♦ H ( K , J ) * COSI N
11 CONTI NUE
12 H ( I , A ) ■ 0 . 0
I F ( J . E O . N)
GO TO 6
J A1 ■ J ♦ 1
DO 1 3
K > JA1, N
HIK ■ H ( I , K)
H ( I , KI
■HIK*COSIN ♦ H ( J , K ) * S I N E
H( J, K ) • -HIK*SINE + H(J,K)*COSIN
1 3 CONTI NUE
3 5 GO TO ( 1 5 , 1 7 ) ,
IFU
1 5 DO 1 6
K - 1, N
UK I - U ( K , I )
U ( K , I ) ■ UKI+COSIN ♦ U ( K , J ) *SI NE
U(K,J) ■ -UKI*SINE ♦ U(K,J)*CDSIN
1 6 CO N T I N U E
1 7 C O NT I NUE
GO TO ( 1 6 , 6 ) ,
MEMO
1 8 I F ( OFFMAX . G T . F I N )
GO TO 7
1 0 0 RETURN
END
C F U N C T I O N I J CAL CUL AT ES M A T R I X ELEMENTS FOR THE
C SCALAR PRODUCT OPERATOR I * J I N THE COUPLED SCHEME
FUNCTI ON I J ( F 1 , F 2 , F 3 , F , F 1 P , F 2 P , F 3 P , F P , M )
COMMON / OS R S S / J , I I , 1 2 , 1 3 , I A
REAL J , I 1 , I 2 , I 3 » I 4 » I J
I F ( M . E Q . l ) THEN
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
49
IFtFI.NE.F1P.OR.F2.NE.F2P.OR.F3.NE.
1 F 3 P . 0 R . F . N E . F P ) THEN
1J-0.
EL SE
IX 1-J 4I14F1
lJ -< -l» **U X lJ *S Q R T (Il*(Il4 l)*< 2 .*li4 l.)*J *
l<2.*j4l.)*(j4l.n*SlXJ(J»Il,Fl,Il,J,l.)
END I F
E L S E I F ( N . E 0 . 2 ) THEN
I F I F 2 . N E . F 2 P . 0 R . F 3 . N E . F 3 P . 0 R . F . N E . F P ) THEN
I J-0.
ELSE
IX2»2.*F1P+I2*F2+J*I1*1.
I J - t - 1 >* * ( 1 X 2 ) * S O R T ! 1 2 * ( 2 . * 1 2 « 1 . )
l*(I2 4 l.)*J *(2 .*J 4 l.)*(j4 l.)*(2 .*F lP 4 l.)*(2 .*F l4 l.))
2 * S I X J t F l » I 2 . F2#1 2 . F I P , 1 . > * S I X J I J , F 1 , I I . F I P , J , l . )
END I F
E L SE I F ( M . E 0 . 3 ) THEN
I F ( F 3 . N E . F 3 P . 0 R . F . N E . F P ) THEN
13*0.
EL SE
1 X 3 * 2 .+F2P4F1*F1P*I3*F3*I2 +J+I1
IJ*(-l)**(IX 3)+SQ R T(13*(2.*I3+l.>*tI3+l.l*
1(2.*F2P+1.)*(2.*F2+1.)*(2.*F1P*1.)*(2.*F1»1.)*
2 J * ( 2 . * J 4 l . ) * ( J + l . ) » *SIXJ(F2»I3»F3»I3#F2P»1.)*
3SIXJ(F1,F2»I2,F2P,F1P,1.)*SIXJ(J»F1,I1»F1P,J,1.)
END I F
ELSE I F ( H . E Q . A ) THEN
I F ( F . N E . F P ) THEN
1J*0.
EL SE
m - 2 . * F 3 P * I < i + F +F 2 4 F 2 P * I 3 * F l * F l P * I 2 +I l * J * l .
I J * ( - 1 ) * ♦ ( J X A ) * S OR T ( I 4 * ( 2 . * I < i 4 l . ) * ( 1 * 4 1 . ) *
l(2.*F3P 4l.)*(2.*F34l.)*(2.*F2P 4l.)*(2.4F24l.)
2 *(2 .*F lP 4 l.)*< 2 .*F l4 l.)*j*(2 .*j4 l.)*U 4 l.))4
3SIXJ(F3,I4,F,I*.F3P,1.)»SIXJ(F2,F3,I3»F3P#F2P,1.)
< t * S I X J « F l , F 2 , I 2 » F 2 P , F 1 P , 1 . ) * S I X J ( 4 , F I t I l » F 1 P»J, 1 . )
END I F
END I F
RETURN
END
FUNCTI ON I S S ( F 1 » F 2 » F 3 » F » F 1 P » F 2 P » F 3 P » F P » L » L L )
C0MM0N/QSRSS/J»I1»I2»I3»IA
REAL J » ! 1 » I 2 » 1 3 . 1 A * I S S
I F I L . E 0 . 1 . A N D . L L . E 0 . 2 ) THEN
I F ( F 2 . N E . F 2 P . 0 R . F 3 . N E . F 3 P . 0 R . F . N E . F P ) THEN
ISS-O.
ELSE
1X5* J 4 I 1 4 I 2 4 F 1 P 4 F 1 4 F 2 4 1 ,
IS S -(-ll**(IX 5 )*S 0 R T ((2 .*I2 4 l.)*(I2 4 l. )*I2*
1(2.*I141.)+(I141.)*I1*(2.*F1P41.)4(2.*F141.))
2 *S IX J (F l,I2 # F 2 ,I2 ,F lP ,l.)*S IX J (Il,F l,J ,F lP ,n .l.)
END I F
EL SE I F U . E 0 . 1 . A N D . L L . E 0 . 3 ) THEN
I F ( F 3 . N E . F 3 P . 0 R . f . N E . F P ) THEN
ISS-O.
ELSE
IX6-2.*F242.*F1P4I34I24I14F34J
T S S - C —1 * * * ( I X 6 ) * S Q R T ( < 2 . * F 2 P 4 l . ) * t 2 . * F 2 4 l . ) ♦
H2.*F1P41. I* I2 .* F l4 l.)* I3 * ( 2 .* I3 * l.)* ( I3 4 l.» *
2 C 2 . * I 1 4 1 .) ^ ( I1 4 1 .» * U ) * S IX J « F 2 » I3 # F 3 # I3 » F 2 P # 1 . )*
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3 S I X J ( F 1 , F 2 , I 2 , F 2 P , F 1 P , 1 . > * S I X J <I 1 , F I , J , F 1 P # I I , 1 . )
END I F
EL SE I F C L . E 0 . 1 . A N D . L L . E Q . 4 ) THEN
I F ( F . N E . F P ) THEN
ISS-O.
EL SE
IX7-2.*F1P*2.*F3«F2P*F2*I4*I3*I2*I1*J*1.*F
I S S - ( - l ) * * ( I X 7 ) * S Q R T < ( 2 . * F 3 P + 1 . ) * ( 2 . * F 3 * 1 . )♦
1(2.*F2P*1.1*(2.*F2*1.)*(2»*F1P*1.|*(2.*F1*1.)'*
2 (2 .*I4 « 1 .)*(I4 + 1 .)*I4 *(2 .*I1 « 1 .)*(I1 *1 .)*I1 )
3*SIXJ(F3,I4,F,I4,F3P,l.)*SIXJ(F2,F3,l3»F3P,F2P,l.)*
4SIXJCF1,F2#I2»F2P»F1P,1.)*SIXJ<I1,F1,J,F1P,I1,1.)
END I F
EL SE I F ( L . E Q . 2 . A N D . L L . E 0 . 3 > THEN
I F ( F . N E . F P . 0 R . F 3 . N E . F 3 P . 0 R . F 1 . N E . F 1 P ) THEN
ISS-O.
ELSE
IX8-F3*F2P«F2+F1+I3«I2*1.
ISS-(-l)**(IX6)*SO R T((2.*I3*l.)*(2 3«l.)*I3
1 *(2 .*12+1.) * ( I 2 *1 .)*I2*(2.*F 2P »1.)*(2.*F 2 *1 .))*
2SIXJIF2,I3»F3,I3»F2P,1.J*SIXJ(I2,F2»F1,F2P#I2,1.)
END I F
EL SE I F ( L . E Q . 2 . A N D . L L . E 0 . 4 ) THEN
I F 1 F . N E . F P . O R . F 1 . N E . F 1 P ) THEN
ISS-O.
ELSE
IX9-F+2.*F3+2.*F2P+F1+I4+13+I2
I S S - ( - 1 ) * * ( 1X9) * S 0 R T U 2 . * F 3 P +1 . ) * ( 2 . * F 3 + 1 . ) *
1 ( 2 . * F 2 P +1. ) * ( 2 . * F 2 +1. ) * ( 2 . * I 4 + 1 . ) * ( I 4 + 1 . ) * I 4 *
2 ( 2 . * 1 2 + 1 . ) * ( 12 + 1 . ) * I 2 ) * S I X J ( F 3 , I 4 , F , I 4 , F 3 P , 1 . > *
3SIXJ<F2»F3>I3,F3P,F2P,1.)*SIXJ(I2,F2,F1,F2P,I2,1.)
END I F
ELSE I F ( L . E Q . 3 . A N D . L L . E 0 . 4 ) THEN
I F I F . N E . F P . 0 R . F 2 . N E . F 2 P . O R . F I . N E . F 1 P ) THEN
ISS-O.
ELSE
IX10-F3+F3P+F2+F+I4+I3+1.
IS S -(-ll**(IX 10 )*S 0 R T ((2.*F 3 P + 1 .) * <2.*F3+1.I*
1 (2 .*14*1.1*114*1.)*14*(2.*13*1.)*(13*1. )*I3>*
2SIXJ(I3#F3»F2»F3P,I3»1.)*SIXJ(F3»I4»F»I4»F3P>1.)
END I F
END I F
RETURN
END
C F U N C T I O N VO CAL CULATES THE OUADRUPOLE I N T E R A C T I O N
C FOR THE K TH COUPLED NU C L E U S l UP TO FOUR N U C L E I
C CAN BE COUPL ED)
FUNCTI ON V 0 ( F 1 . F 2 * F 3 » F » F 1 P » F 2 P » F 3 P » F P » M)
COMMON / QS R S S / J , I I , I 2 , 1 3 # 1 4
REAL J * I I , 1 2 , 1 3 , 1 4
I F ( H . E O . l ) THEN
if(r.n e.fp .o r.f3.n e.f3p .o r.r2.n e.f2p .o r.fl.n e.flp .
l o r . i l . l t . l )
then
VQ-O.
else
l« ll-J + ll+ fl
v q -(-l)**(lx ll)*s q rt< ( (2 .*j*l.)*(2 .*J *2 .)*(2 .*J *3 . )
1 )/(B .*J *(2 .*J -1 .)))*s q rt(l(2 .* i1 *1.)* ( 2 .* i1*2.)*(2.*<1*3.)>
2 /( B .* I l * ( 2 . * i l - l . ) ) ) * s i x j ( f l , i l , j , 2 . , j > i l )
end i f
EL SE I F ( M . E Q . 2 ) THEN
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
I F ( F 2 . N E . F 2 P . 0 R . F 3 . N E . F 3 P . 0 R . F . N E . F P . o r . i 2 . l t . 1 ) THEN
VQ-O.
ELSE
1X12- 2.*F1P+I2+F2+J+I1
V0»(-1>**(IX12)*SQRT(<2.*F1P+1.)*(2.*F1+1.I)
1 *S O R T(((2.*I2+l.)*(2.*I2+2.)*(2.*12+3.))/((£.*12)
2 * ( 2 . * 1 2 - 1 . I ) > * S O R T < C ( 2 . * J * 1 . ) * ( 2 . * J * 2 . ) * < 2 . * J + 3 . 1)
3/(<6.*J>*(2.*J-1.1>)*SIXJ(F1,I2,F2#I2»F1P,2.)
**SIXJ(J»F1»I1,F1P,J,2.)
END I F
ELSE I F ( H . E Q . 3 I THEN
I F I F 3 . N E . F 3 P . D R . F . N E . F P . o r . i 3 . l t . i l THEN
VO-O.
ELSE
IX13-2.*F2P+I3+F3+F1*I2+J+I1+F1P
V Q - ( - l > * * ( 1 X1 3 1 ♦SORT( ( 2 . * F 2 P + 1 . ) * < 2 . * F 2 + 1 . I *
1 ( 2 . * F 1 P * 1 . I* ( 2 . * F 1 + 1 . ) )*SORT( ( ( 2 . * 1 3 * 1 . ) * ( 2 . * I 3 * 2 . I *
2 (2 .*1 3 + 3 .))/((B .*I3 )*(2 .*I3 -1 .)I)
3*SQ R T(((2,*J+1.)*(2.*J+2.)*(2.*J+3.I)/((8.*J)*(2.*J-1.)1)
A*SIXJ(F2»I3»F3#I3»F2P»2.)*5IXJ(F1»F2»I2»F2P»F1P»2.I*
5SIXJ(J*F1>I1#F1P,J,2.>
END I F
EL SE I F ( H . E O . A ) THEN
I F ( F . N E . F P . o r . i * . 1 1 . 1 1 THEN
VQ-O.
EL SE
IX8«2.*F3P+IA+F+F2+I3+F1+I2+F2P+J+I1+F1P
V 0 « { - 1 ) * * ( I X 8 ) * S 0 R T ( ( ( 2 . * I <i + l . I * < 2 . * I * + 2 . ) * ( £ . * 1 4 + 3 . I I
1 / ( ( B . * I 4 ) * ( 2 » * I 4 - 1 . ) I l * S O R T ( ( ( 2 . * J +2 . ) * ( 2 . * J + 1 . ) * ( 2 . * J +3. ) I
2/((8.*J)*(2.*J-l.l))*SlXJIF3»IA»F»I<i#F3P»2.)
3*SIXJ(F2.F3#I3fF3P»F2P,2.)*SIXJ(Fl,F2,I2,F2P»FlP»2.l*
ASI X J( J . F 1 , I 1 , F 1 P , J , 2 . I
5*SQRT((2.*F3P+1.)*(2.*F3+1.)*(2.*F2P+1.)*(2.*F2+1.I*
6(2.*F1P+1.)*(2.*F1+1.)I
END I F
END I F
RETURN
END
FUNCTI ON S I X J ( J 1 * J 2 » J 3 » L l » L 2 * L 3 )
REAL J 1» J 2 » J 3 r L I >L 2» L 3
IZMIN-INT(AMAX1(J1+J2+J3»J1+L2+L3*L1+J2+L3»L1+L2+J3)+0.5I
I Z M A X - I N T ( A M I N K J1 + J 2 + L 1 + L 2 * J 2 + J 3 + L 2 + L 3 . J 1 ♦ J3 + L1 + L 3 1 + 0 . 5 )
U-0.0
I F ( H O D ( I Z H I N » 2 ) . E O * 1 ) GO TO 10
SIGN— 1 . 0
GO TO 2 0
10 S I G N - 1 . 0
2 0 DO 1 J - I Z H I N , I Z M A X
Z-J
SIGN-( - 1 . I*SIGN
A-Z-J1-J2-J3
B-Z-fl1-L2-L 3
C-Z-LI-J2-L3
D-Z-L1-L2-J3
E- J1 + J2+L1+L2-Z
F-J2+J3+L2+L3-Z
G-J1+J3+LI+L3-Z
IF(A.LT.-O.l.OR.B.LT.-O.l.OR.C.LT.-O.l.OR.D.LT.-O.l
1 . O R . E . L T . - O . l . O R . F . L T . - O . l . O R . G . L T . - O . l ) GO TO 1
W«W+SIGN*FAC<Z+1.0)/(FAC(A)*FACCB>*FAC(C)*FAC<D)*FAC<E>*
I F AC ( F I * F AC ( G 11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
CO N T I N U E
S I X J - D E L T A t J l » J 2 , J 3 ) * D E L T A ( J l » L 2 » L 3 ) * D E L T A( L l » J 2 , L3)
1ADELTA(L1»L2»J3>*W
RETURN
END
FUNCTI ON D E L T A ! A , B , C )
IFIA^B -C .G E.-0.1.AND .-A*B *C.G E.-0.1 . AND.A-B+C.GE.-O.l)
I S O TO 1 0
DELTA-0.0
RETURN
1 0 OELTA- SORTCFAC( A 4 B - C ) * F A C < A - B + C • ♦ F A C ! - A » B + C ) / F A C ! A * B + C * l ) )
RETURN
END
FUNCTI ON F AC( X)
N - I N T ( X*0.5>
I F IN ) 20# 3 0 * 1 0
20 WRITE(b#100)X
1 0 0 F O R M A T C l H O . ' F A C T O R I A L ' . G l b . B j ' I S REOUI RED.
U N I T Y I S ASSUM
1 E D* )
FAC-1.0
RETURN
30 FAC-1.0
RETURN
10 F A C -1 .0
DO 1 J ■ 1 # N
F A C - F AC♦ J
1 CONT I NUE
RETURN
ENO
FUNCTI ON J 2 ( F I # F 2 , F 3 # F » F I P , F 2 P » F 3 P , F P )
REAL J 2
I F ( F 1 . N E . F 1 P . 0 R . F 2 . N E . F 2 P . 0 R . F 3 . N E . F 3 P . 0 R . F . N E . F P ) THEN
32-0.
ELSE
J2-2.
ENO I F
RETURN
END
S UB ROUT I NE S O R T « A , B >
I NTEGER B
DI MENSI ON A ( B )
I F ( B . E O . l ) RETURN
L-B-l
DO 1 0 I - 1 , L
K-B-I
DO 2 0 J - 1 , K
IFIAIJ)-A(J +1H20,20,30
30 T EN P-A IJ+l)
A!
20
10
J+l) ■ A ! J )
A ( J >- T E MP
CONT I NUE
CONTI NUE
RETURN
END
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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~
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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