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MEASUREMENT OF HYPERFINE STRUCTURE INDEUTERATED ACETYLENES VIA MOLECULAR-
BEAM MICROWAVE SPECTROSCOPY.
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University Micr6films
International 300 N. Zeeb Road Ann Arbor, MI48106
8306000
Tack, Leslie Martin
MEASUREMENT OF HYPERFINE STRUCfURE IN DEUTERA TED ACETYLENES VIA MOLECULAR-BEAM MICROWAVE SPECTROSCOPY
The University of Arizona
University Microfilms
International 300N. Zeeb Road, AnnArbor,MI48106
PH.D. 1982
MEASUREMENT OF HYPERFINE STRUCTURE IN
DEUTERATED ACETYLENES VIA MOLECULAR-BEAM
MICROWAVE SPECTROSCOPY
by
Leslie Martin Tack
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF CHEMISTRY
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
198 2
THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Final Examination Committee, we certify that we have read
by Leslie Martin Tack the dissertation prepared
entitled Measurement of Hyperfine Structure in Deuterated Acetylenes
via Molecular-Beam Microwave Spectroscopy
and recommend that it be accepted as fulfilling the dissertation requirement
D f Doctor of Philosophy for the egree 0
( 2.--J>-8'2--Date
1'2---~-~ '--Date
/) - >J--f' 2-Date
, J2.-Y-!C Date
1::Z/fI/e2 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.
Date
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
Comparison of gas and liquid phase hyperfine components in NCCCD ................. .
Chemical shift tensor elements of 14N in NCCCD .. .
Comparison of calculated and experimental deuterium quadrupole coupling strengths for some selected molecules ................ .
Quadrupole coupling strengths along the C-D bond
Page
28
28
29
29
30
30
36
36
. 40
40
11. Quadrupole coupling constants and bond distances for the C-D moiety in several deuterated acetyl enes. . . . . . . . . . . . . . . . . . . . . . . 41
vii
ABSTRACT
This work describes the measurement of hyperfine structure in a
series of deuterated acetylenes via molecular-beam microwave spectroscopy.
Measurements of spin-rotation constants were used to calculate the
paramagnetic contribution to the chemical shielding of the concerned
nucleus. Where possible, comparisons with NMR measurements were made.
Measurements of the deuterium quadrupole coupling determined in this work
are compared with previous measurements on the same or similar systems.
A review of the theoretical work done in this area is presented as well as
a discussion of trends observed from high precision measurements of
deuterium quadrupole coupling. A computer program that calculates hyper
fine structure for up to four coupled nuclei of arbitrary spin is
presented.
viii
CHAPTER 1
INTRODUCTION
Beam - MASER(microwave amplification by stimulated emission of
radiation) instruments have yielded some of the sharpest and most stable
resonances to be found in spectroscopy. Ultimately, a MASER device will
become the universal time standard with stability of the order one part
in 1014 .
For the molecular spectroscopist the MASER instrument is an
extremely accurate measuring device for both strong and weak interactions
in molecules. Resolution requirements are dictated by what interactions
are being investigated. Rotational energies can be routinely measured by
conventional microwave or even infrared techniques. The literature is
full of measurements of rotational energies made with good Stark-modulated
microwave spectrometers where the resolution was about one part in 104 or
even 105. The observation of hyperfine structure, which will perturb the
rotational transitions, may require higher resolution for full elucidation
of the effect. This is particularly true for hyperfine structure in
deuterated species. Resolution of the order on part in 107 is usually
needed; fortunately, the beam-maser instrument meets this requirement.
Hyperffne structure in rotational transitions arises when one
considers the finite extension of the nucleus. There are electric and
magnetic interactions that split the rotational line; electric interac
tions dominate in diamagnetic molecules. Analysis of the data taken on
2
the beam-maser spectrometer will yield interaction strengths for electric
quadrupole, spin-rotation and spin-spin interactions. The deuterium quad
rupole interaction is of special interest because the electric field
gradient at the deuterium nucleus arises primarily from external charges
in the molecule; thus, there is a probe for estimating the charge distri
bution in the molecule. Electronic wave functions can be tested for their
power to preduct the quadrupole coupling at the deuterium site.
The spin-rotation and spin-spin interactions are magnetic effects.
The former results from the interaction of the magnetic moment of the
nucleus with the magnetic field produced by the rotation of the molecule. 1 Ramsey was first to point out the intimate relationship between the spin-
rotation constant and the paramagnetic contribution to the chemical
shielding (which is represented by a second-rank tensor). Ordinarily,
from a purely theoretical point of view, this would be a difficult contri
bution to calculate; however, spin-rotation constants can be fairly
accurately measured (to within 10%) from the data taken from a beam
experiment. Where calculations are available for determining the diamag
netic contribtuion (a much more tractable problem since it only depends on
knowledge of ground state wave functions) experimental and calculated
results can be combined to obtain the total chemical shielding. If NMR
data are available (where the total chemical shielding is measured) there
is a consistency check to use.
Spin-spin interactions are usually calculated from the molecular
geometry and nuclear moments to aid in the data analysis process.
3
Each of these interactions is discussed in more detail in
Chapter II. A computer program was written to calculate hyperfine struc
ture for up to four coupled nuclei of arbitrary spin. A listing of this
program is provided in the Appendix. Chapter III gives a complete descrip
tion of the microwave instrumentation used to make the measurements.
Analysis of the data along with the derived spectrscopic constants is
presented in Chapter IV; wherever possible these results are compared
with previous works- (usually involving less precision) in the same or
similar systems. In Chapter V is presented a discussion of the interpre
tation of the measurements. Every attempt will be made to maximize the
information that can be obtained from each measurement. A review of the
theoretical work done in conjunction with these measurements will be
presented; problems in comparing theoretical and experimental results will
be discussed.
CHAPTER 2
THEORY OF HYPERFINE STRUCTURE FOR ROTATING MOLECULES
A. Introduction
The Hamiltonian used to analyze the data taken in this work is
as follows:
(1)
where H t is the rotational energy of the molecule which includes effects ro . due to centrifugal distortion and rotation-vibration interactions. These
effects are discussed in standard texts. 2,3
For the systems studied in this work, the rotational energy will
be much greater than the hyperfine interactions (approximately a factor
of 103). The hyperfine structure can then be viewed as a perturbation
of the rotational energy levels and the analysis will always include a
IIline center frequencyll which corresponds to the line we would see in the
absence of hyperfine sturcture.
The hyperfine Hamiltonian will include electric quadrupole, spin-
rotation, and spin-spin interactions. Each interaction will be discussed
and a scheme for calculating hyperfine structure for all interactions
will be presented.
4
5
B. Interaction of a Nucleus with Molecular Fields
1. Electrostatic Interaction.
The general electrostatic interaction between a charged nucleus of
finite size and the electrons and nuclei in the rest of the molecule is
given by:
He1 . = f f Pe(r:)pn~~)dTedTn (2)
Te Tn ~
where P e(re) is the charge density of the electrons and nuclei in the ... volume element dT at the position r relative to the center of the rele-e e -vant nucleus, P n(rn) is the nuclear charge density of the nucleus con-
.> cerned in the volume element dTn at the position rn relative to the cen-
ter of the nucleus, and r is the magnitude of the radius vector joining
dT and dTn as shown in Fig.1. p (~) is negative for electrons and posi-e e e tive for positive charges.
If electronic charges more distant than the radius R of the
nucleus are considered, l/r may be expressed, using the cosine law, as
follows: r 1=_1 +_n_p
r r 2 1 e re
where Po is the Legendre polynomial of cose ,so that, ]V en
Pl = cose en
1 2 P = - (3cosa -1) 2 2 en
(3)
(4a)
(4b)
The first term in the expansion corresponds to an electric monopole, the
second to the electric dipole, and the third to the electric quadrupole
moment.
.",.--......... ... , '\ ,,,.,,.,'
I
dr" i P" R
\ \ , ,
I I \ I \ I \ , , / " / ' ......... _ ---"filii'
Fig. 1. Electrostatic interactions with nucleus in atoms and
molecules.
6
7
The monopol e term wi 11 1 ead to no hyperfi ne structure and is there
fore of no interest. The dipole term can be shown to vanish from parity
considerations; thus, the only important term is the quadrupole moment.
Higher moments are generally too small to be considered. 4
Ramsey has shown the electrostatic quadrupole interaction can be
written as:
2 _ e qJQ ~">2 3~-> 22
HQ - 21(21-1)J(2J-1) [3(I·J) +"2 I·J -I J ] (5 )
where Q is the arbitrary nuclear constant and is called the magnitude of
the electric quadrupole moment and is given by:
r ~ 2 2 eQ = J P (r) (3z - r ) dT (6)
n n m =1 n n n I
The integral is evaluated for m1=1 as indicated. eqJ is defined as the
electric field gradient coupling constant and is given by:
r ~ 2 1 eqJ = ~P \ r. ) (3cos e - 1)3 'dT e e =J ez r e
r > mJ e e
(7)
-.I-
where e is the angle between r and the z axis. As written above there ez e is an implicit J dependence on qJ' The quadrupole interaction can be
rewritten with explicit J dependence by utilizing the following relation
ship:
(8)
8
where Ve is the potential from all charges external to the nucleus and Zo
is along the symmetry axis of the molecule. Substituting (8) into (5)
gives: 2 -eQ a V 2
_ . a Zo -l ~ 3 ..£. .,.1. 2 2 HQ - 2I(2I-l)(2J+3)(2J-l) {3(I'J) + 2 I·J - I J } (9)
The above expression is only applicable to linear molecules. Thaddeus,
Krisher, and Loubser5 give the quadrupole interaction in asymmetric rotors
Fig. 4. Hyperfine components of the J = 1 -+ 0 transition. eleen. (a) Recorder tracing of Fl = 3/2 components. Frequency relative to 10 358 017 kHz. {b) Fl = 5/2 components. Frequency relative to 10 377 921 kHz. (c) Fl = 1/2 component. Frequency relative to 10 393 892 kHz.
r- I I -- I I -40 -20 0 20 40
Fig. 5. Derivative maser spectra of the hyperfine components of the J = 0 to J = 1 transition, CD3CCH.
kHz N W
24
(a)
+44 +33 +22 +11 O' -u -22 -33 -44 -55
'-2 1'1 i81
-~-- V-----t- (b)
-+48 +40 +32 +24 +16 -+8 0 -a -16 -24 -32 -40 -48
(c)
-9 -6 -3 0 +3 -t6 +9
Fig. 6. Hyperfine components of the J = 1 ~ 0 transition. NCCCD. (a) F = 2 components. Frequencies relative to 8 4431386.0 kHz. (b) F, = 1 components. Frequencies relative to 8 442 092.0 kHz. (t) F, = 0 component. Frequency relative to 8 445 318.0 kHZ.
25
3. Cyanoacety1ene-D
Cyanoacety1ene is a molecule of strong astrophysical interest;
rotational lines of its various isotopes have been observed in inter
stellar clouds since 1971. 21 The first microwave measurements were made
in 1950 by Westenberg and Wi1son. 22 Tyler and Sheridan23 looked at the
microwave spectrum of an excited vibrational state and determined the
structure. DeZafra 24 accurately measured the rotational constant and
hyperfine structure for the ground vibrational state on a beam-maser
spectrometer. A thorough review of previous measurements is available in
the 1iterature. 25 We have accurately measured the rotational constant and
hyperfine structure of the deuterated species via a molecular-beam micro
wave,spectrometer, fully resolving the hyperfine structure due to 14N and
D.
A 20 cm cylindrical TM010 mode brass cavity was used. The cavity
was machined until the resonant frequency was slightly higher than the
desired frequency. The cavity could then be tuned down in frequency by
inserting a quarter-inch nylon rod into the cavity. The beam was gener
ated by use of a single hole (0.15 mm) supersonic nozzle. A pressure of
85 torr was maintained behind the nozzle by cooling the sample in an ice
water bath.
Cyanoacety1ene-D was prepared by the method outlined by Mallinson
and Fayt. 26
Derivative spectra of all hyperfine components are shown in Fig. 5.
CHAPTER 4
ANALYSIS OF DATA
Introduction
A computer program was written to calculate hyperfine structure of
a rotational transition for up to four nuclei of arbitrary spin. The
general coupling scheme is as follows:
11 + J = Fl
12 + Fl + F2
I + F = F 3 3
(24)
Expressions for the matrix elements in this coupled scheme were taken
from the papers of Cook and DeLucia6 and Thaddeus, Krisher, and Loubser. 5
This program was checked against a previous coupled-scheme program and
against the uncoupled basis set calculation (graciously provided by Dr.
Sid Yeung). The case of hyperfine structure in the presence of three
equivalent spins has been previously discussed in detail.
The program input data consisted of the spin of each nucleus and
its associated hyperfine interaction strengths. The effective rotational
constant would be thrown in as a parameter so that we could fit to the
actual experimentnl frequencies. Good trial interaction strengths were
obtained from previous measurements on the same or similar systems. The
actual experimental frequencies were also read in to calculate the
26
27
standard deviation for the fit, defined as foll ows: \Ii - ri
) 1 /2 °fit = 0: obs. calc. (25)
i N-l
where N is the number of lines being fit. The interaction strengths and
rotational constant were varied to fit the experimental frequencies. The
error in each parameter was estimated by varying each one in turn until the
standard deviation changed significantly.
Data Analysis for Individual Molecules
Chloroacetylene-D
The hyperfine Hamiltonian consisted of the deuterium quadrupole
and chlorine quadrupole and spin-rotation interaction strengths. Second
order quadrupole energies were calculated for chlorine from the tables
given by Townes and Schawlow2 and added to the first order energies.
Table 1 lists the measured and calculated frequencies for each component.
The hyperfine constants are given in Table 2 and where possible compared
with those obtained from previous work.
Cyanoacetylene-D
The hyperfine Hamiltonian consisted of the nitrogen quadrupole,
nitrogen spin-rotation, and deuterium quadrupole interactions. The
calculated and observed frequency for each component is given in Table 3
as well as the standard deviation for the fit. Hyperfine interaction
strengths where possible are given in Table 4 and compared to previous
works.
Table 1. Measured and calculated frequencies for components of the J = 1 + 0 transition of C1CCD. Frequencies are given in kHz. The Standard deviation for the fit was Q~ 1 kHz.
Component (F 1 ; F) Measured
(1/2;1/2,3/2) 10 393 892.0
(3/2;1/2) 10 358 050.6
(3/2;3/2) 10 357 975.3
(3/2;5/2) 10 358 017.1
(5/2;3/2) 10 377 912.9
(5/2;5/2) 10 377 975.3
(5/2;7/2) 10 377 931.6
Table 2. Molecular Constants for C1CeD
eqQ(D)
eqQ(Cl)
This Work
208.5 + 1.5 kHz
-79 739.5 + 1 kHz
1.3 + .1 kHz
Calculated
10 393 892.1
10 358 050.6
10 357 975.2
10 358 017.1
10 377 912.9
10 377 975.5
10 377 931.6
Previous Work (Ref.)
225 + kHz
-79 670. + 100 kHz
-79 660 kHz
C (Cl )
B 5 186 973.9 + .1 kHz 5187 010. kHz
aRef . 16
bRef . 15
28
29
Table 3. Measured and calculated frequencies for components of the J=l to J=O transition of CNCCD. Frequencies are given in kHz.
aMeasured from data taken on NCCCH (Ref. 23). b Ref. 23.
Calculated
8442051.51
8442084.11
8442130.58
8443343.39
8443366.93
8443413.39
8445318.03
Previous Work
-4318.8 + 1.2 kHz a
1.05 + 0.26 kHza
4221581.67 + 44 kHzb
30
Propyne-03
The hyperfine Hamiltonian consisted of the deuterium quadrupole,
spin-rotation, and mutual spin-spin interactions. The spin-spin inter
action was calculated from the known nuclear moments and structure and was
not treated as a variable parameter. The measured and calculated frequen
cies for the observed components are given in Table 5. Hyperfine inter
action strengths are given in Table 6 and compared with previous work.
Table 5. Measured frequencies for J = 1 ~ 0 transitions in OC3CCH. Frequencies in kHz. F value is the total angular momentum in the J = 1 state.
F
1
2
3
Measured
14 711 522.6
14 711 530.7
14 711 565.9
Calculated
1411522.6
14711530.8
14711565.9
Table 6. Molecular coupling parameters and rotational constant for C03CCH.
Parameter
eqaaQ
eqzzQ
Co
0(0-0)
B
OJ
-55.0 + 0.5 kHz
174.0 + 6.0 kHz
0.06 + 0.08 kHz
-0.25a kHz
7 355 767.0 + 1.0 kHz
2.8b kHz
a, calculated for nuclear moments and structure, b from previous measure.
CHAPTER 5
DISCUSSION OF RESULTS
A. Introduction
Attention will now be turned to the interpretation of the measure
ments made in this work. The results for each individual molecule will
be discussed followed by a general discussion on trends in measurements
of deuterium quadrupole coupling strengths. A review of the theoretical
work done in this area will be presented. Accurate spin-rotation con-14 35. stants were measured for Nand Cl ln cyanoacetylene-D and
chloroacetylene-D, respectively. A discussion of chemical shielding in
these isotopes will be presented and comparisons with NMR measurements
will be made. B. Individual Molecules
Chloroacetylene-D
By examination of Table II it is noted that the values of eqQ(D)
reported by this work and that of Weiss and Flygare16 agree within the
earlier limits of experimental error (i.e., ~ 18 kHz); however, the
measurement from this work has a much smaller uncertainty (~ 1.5 kHz). The
values of eqQ(Cl) do not agree within the limits of experimental error.
Since the frequency measurements in this work are more accurate by two
orders of magnitude and there is second order quadrupole effects are
included for chlorine in our analysis, there is more confidence in the
present results.
31
32
The analysis also yielded an accurate measurement for the spin
rotation interaction strength for 35Cl . The spin-rotation interaction
strength for the kth nucleus is defined as:
C(K) = -I <J 2> M /J(J+l) (26) 9 9 gg
M are the spin-rotation tensor diagonal elements. For a linear molecule gg l. <Jg
2> is simply J(J + 1) and c(k) = -M = -M Calculation of these xx yy
tensor elements from spin-rotation measurements enables the estimation of
the paramagnetic and diamagnetic contributions to the total chemical
shielding. The diagonal elements of the shielding tensor for nucleus k
along the gth principal axis are given by:
o(K) = o(d) + (p) gg gg 0 gg (27)
O~g and O~g are, respectively, the diamagnetic and paramagnetic contribu
tions to the shielding tensor elements and are calculated from the
following expressions: d e2 -3 2 2
o = (-2 ) < OII r . K (r. K - (r. K) ) 10> gg amc 1 1 1 9
p Ogg
e2 hCMgg -3 2 2 = (--2){ 2 egG - LZ, r i K [r £ K - ( r i K) g]}
amc )IN K gg
(28a)
(28b)
)IN is the nuclear magneton, gK is the nuclear 9 value, r£k is the vector
from nucleus k to the other £ nuclei, Ggg is the rotational constant along
th th . . 1 . e 9 prlnclpa aX1S. d 0gg cannot be found directly from experimental data; one must rely
on molecular orbital calculations. In contrast, the paramagnetic contri
bution can be determined from measured spin-rotation constants and know
ledge of the molecular geometry. This is very useful since calculation
of the paramagnetic term would require evaluation of matrix elements
33
coupling the ground state to excited states with nonzero orbital angular
momentum. The intimate relationship between the spin-rotation interaction
and chemical shielding was first pointed out by RamSey.l
Since chloroacetylene is a linear molecule, obtaining the compo
nents of O~g is especially simple. Evaluating the right-hand side of
equation 26b and using our measured value of C(35Cl ) = 1.3 kHz, we obtain
op = op = a = -420 ppm The average value of oP is one-third the trace xx gg .
of the shielding tensor, -280 ppm.
The question now arises, what should be done with this result, or,
is there anything to compare it to? If NMR data were available, the
measurement of the paramagnetic part of the shielding could be combined
with a measured value of the total shielding. Note that the corresponding
components of each cannot be subtracted unless the total shielding aniso
tropy was actually measured in the NMR experiment. Sometimes calculations
are available that give the anisotropy of the diamagnetic part of the
nuclear shielding. This data can be combined with the measured paramagne
tic shielding and comparison with NMR measurements can be made. A good
example of this consistency check is the beam-maser work on CH3l3CN.
There are problems with obtaining nuclear shielding constants for
chlorine (as well as the higher halogens). First, you must remember that
the shielding constant of a nucleus in a particular compound cannot be
directly measured in an NMR experiment; only the "chemical shift", which
is the different chemical environment is obtained. Chemical shifts of
chlorine, bromine and iodine have been mostly measured relative to the
corresponding halide ion in aqueous solution. Since the ion shifts are
themselves dependent on the nature of the counter-ion, salt concentration
34
and temperature, they are not ideal references. In addition, 35Cl or
37Cl NMR signals have considerable line widths (several hundred ppm) due
to quadrupolar relaxation, the chemical shift covering a range of about
1000 ppm. Thus, due to experimental difficulties only shielding data for
a few covalent chlorine compounds have been reported. The error bars
associated with these measurements are usually quite large, e.g., a(35Cl ) =
-340 ~ 100 ppm in chloroform. No chemical shift data for chloroacetylene
has been reported. The possibility of ever being able to sort out the
diamagnetic and paramagnetic contributions to the chemical shielding in 35 Cl compounds is not encouraging. The chemical shift in the gas phase
would have to be measured relative to a common reference such as aqueous
NaCl solution. This is a non-trivial problem.
Cyanoacetylene-D
It is noted from Table IV that the deuterium quadrupole coupling
strength along the C-D bond is accurately measured at 203.5 ~ 1.5 kHz.
Included are the hyperfine constants measured on CNCCH by Robert DeZafra24
with the molecular-beam apparatus at Columbia University. The hyperfine 14 constants on N are not expected to be very different for the two iso-
topic species and they are found to be the same within experimental error.
Although DeZafra was able to obtain high resolution in his measurements
(FWHM = 3 kHz) he was unable to resolve the splitting due to the spin
rotation interaction on hydrogen. For the deuterated species the spin
rotation interaction only causes a relative shift of the hyperfine lines
since no new quantum numbers are introduced when it is included in the
Hamiltonian. Its contribution os small with a correspondingly high
35
uncertainty. The spin-rotation interaction strength for hydrogen would
be expected to be about a factor of seven greater.
Vold,27 et al., have measured the 13C, D and 14N nuclear magnetic
relaxation of cyanoacetylene-D in toluene solution. Analysis of these
relaxation times gives liquid phase values for 14N and 0 quadrupole
coupling constants. Chemical shift anisotropies for 13C were also
measured. Table VII compares gas phase values with the liquid NMR results.
There is a noticeable difference (4%) between eqQ(14N) in the liquid and
the solid phase. The solvent undoubtedly plays a role in how it affects
cyanoacetylene-cyanoacetylene hydrogen bonding. The difference is less
than that found in solid vs. gas phase measurements (6-14%) of quadrupole
coupling constants.
Analysis of the data also yielded a value for the spin-rotation
interaction strength on 14N: C(14N) = 1.1 + 0.2 kHz. Proceeding in a
similar way for chloroacetylene, cr P (14N) = -593 ppm.
The diamagnetic contribution to the chemical shift can be com
puted from ab initio calculations or more conveniently by Flygare's atom
dipole method. The former was chosen for its simplicity and the good
agreement it provides with ab initio calculations. The diamagnetic
and paramagnetic tensor elements for 14N in cyanoacetylene are given in
Table VIII. The diamagnetic and paramagnetic tensor elements can be
combined to obtain the components of the total shielding tensor. We find
that crAV = 18 ppm. Unfortunately, chemical shift data for cyanoacetylene
is not yet available (owing to referencing problems); however, the value
obtained is in reasonable agreement with that of 25 ppm given for CH3CH.
Note that this value is relative to the bare 14N nucleus.
36
Table 7. Comparison of gas and liquid phase hyperfine constants in CNCCD.
Parameter
eqQ( 14N)
eqQ(D)
C(N)
aThis work. b Ref. 27.
Gas Phasea
-4318.0 + 0.1 kHz
203.5 + 1.5 kHz
1. 1 + 0.2 kHz
Liquid Phaseb
-4 140 + 50 kHz
200 + 2 kHz
Table 8. Chemical shift tensor elements of 14N in NCCCD (all values in ppm).
Tensor element
xx
yy
zz
Average
Propyne-D3
d CJ
446
446
348
413
-593
-593
o
-395
CJ
-147
-147
348
18
By examination of Table 6 it is noted that the deuterium quadru
pole coupling along the a-rotational axis is accurately determined at
-55.0 ~ 0.5 kHz; however, the coupling strength along the C-D bond direc
tion (which is what we are really interested in has a larger uncertainty
(~ 6 kHz). This is so because when the error is propagated into eqQ along
the C-D bond direction the majority of the error comes from the uncertainty
in the molecular structure. This is an inherent problem with CD 3X type
molecules. With the assumption that the C-D bond is cylindrically
symmetric the value along the C-D bond is given as:
eq Q = egQ(C-D) (3cos 2e-l) aa z
where e is the C-C-H angle which has an uncertainty of ~ 0.5°. The
37
(29)
value of eqQ along the C-D bond of 174 kHz is in excellent agreement with
the previous value although the uncertainty is reduced by a factor of
three.
Deuterium Quadrupole Coupling in Molecules
It is useful at this point to make a connection with the theore
tical work done in this area. High precision measurements of deuterium
quadrupole coupling have come out of this work; to predict the interaction
strengths obtained here would require equal precision in accounting for
the electronic distributions in these molecules. The problem is tractable
from a purely theoretical end for several reasons: (1) the deuteron contri
butes almost insignificantly to the electric field gradient; (2) there
are no complications due to Sternheimer anti-shielding which is prominent
in heavy atoms.
The electric field gradient at the deuteron is the sum of an
electronic and nuclear contribution: 2 2
3z - rO * eq(D) = +IZn Dn5 'n - e<y I
rOn (30)
Z is the charge of nucleus n in units of e, the index n is the sum over n
the n nuclei, i is the index for the sum over the electrons in the mole-
cule. It should be remembered that what is measured in the lab is the
38
product eqQ and not eq. The measured quadrupole coupling can be converted
to the electric field gradient within the accuracy of the measurement
and the known value of Q for the deuteron. Reid and Vaida 29 have calcu--27 2 lated Q for the deuteron to be 2.860 x 10 cm based on an electric
field gradient computed from an 87 term electronic wave function. Their
analysis included effects for vibrational averaging.
The first term in eq. 28 is easily calculated from the molecular
geometry. Imagine at this point a molecule consisting of only bare nuclei;
now fill in the electron density in the molecule. This is the difficult
part of the problem.
There have been two approaches in the estimation of deuterium quad-30 rupole coupling in molecules. Snyder has computed electric field grad-
ients at molecular deuterons from~ initio wave functions in a double
zeta basis. Barfield,31 et al., have done the same calculations employing
semi-empirical molecular orbital theory (i.e., SCF-MO in the INDO approx
imation). Both methods of calculation give consistently high values for
the deuterium quadrupole coupling. This state of affairs is probably due
to an underestimation of the electronic contribution. The ab initio
approach seems to work better when dealing with a C-D moiety. Table 9
compares the results between the calculated (by ab initio and semi
empirical methods) with experimental measurements.
There are several qualitative and quantitative observations to be
made concerning the calculations and measurements: (1) eqQ ii positive for
a deuterium bonded to a first row atom. This is due to incomplete
shielding by bonding electrons of the nucleus to which the deuteron is
bonded. (2) As nuclei further away are considered from the deuteron and
39
its associated electronic distribution the electronic and nuclear
contributions effectively cancel one another. (3) The calculated quadru
pole couplings are consistenly higher than the measured values (about 20%).
Snyder30 has responded to the third observation by inclusion of
p basis functions on the deuteron and d basis functions on the carbon atom
bonded to the deuteron. This has resulted in the lowering of calculated
values to within 8% of experimental measurements.
Observations (1) and (2) are well documented in Table 9 where
quadrupole coupling strengths along the C-D bond are listed for eight
molecules. The first four molecules involve quadrupole coupling on an
acetylenic deuterium; the final four are CD3X type molecules. All
measurements except for those on FCCD are high precision.
After examination of Table 10 two trends are immediately obvious:
(1) the deuterium quadrupole coupling is not very sensitive to sUbstitution
within a group where the deuterium is bonded to carbon of given hybridi
zation (sp or sp3) (2) eqQ(sp) > eqQ( sp3) by about 20%.
Let us consider the first trend. We have available in this work
for the first time comparison of quadrupole coupling constants on acety
lenic deuteriums where the measurements have been precise enough to show
real differences. It is interesting to ponder these fi~ite differences
with the available structural data. Table 11 gives the quadrupole coupling
for several acetylenic deuterons along with the bond distance of the C-D
moiety. Propyne-D l and cyanoacetylene-D have virtually the same bond
distance for the C-D fragment yet the dueterium quadrupole coupling is
significantly different (about 10%). Although qualitative statements must
40
Table 9. Comparison of calculated and experimental deuterium quadrupole coupling strengths for some selected molecules. All values are in kHz. DZ = double zeta basis, SE = semi-empirical, DZ + P = double zeta basis plus polarization wave functions.
Molecule OZ OZ + P SE expo
CH30 226 208 268.7 192a
DCN 248 213 314.9 194b
HOO 378 339 311.7 307.9c
NH20 308 310 304.8 291 d
a Ref. 32. b Ref. 33. c Ref. 34. d Ref. 35.
Table 10. Quadrupole coupling strengths along the c-o bond direction. All values are in kHz.
Molecule eqQ Ref.
Cl - C :: C - 0 208.5 + 1.5 This work -CN - C:: C - 0 203.5 + 1.5 This work F - C:: C - 0 212 + 10 16 -CH3 - C:: C - 0 228 + 2 36 -C03CN 168 + 4 37 -COll 166 + 5 38 -C03Br 175 + 3 39 -C03C :: C ., H 174 + 6 36 -
41
Table 11. Quadrupole coupling constants and bond distances for the C-D moiety in several deuterated acetylenes.
a b
c
Molecule
CH3C :: C - D
Cl - C :: C - D
NC - C :: C - D
Ref. 40.
Ref. 41-
Ref. 25.
eqQ(kHz)
228 + 2
208 + 2
204 + 2
C-D(A)
1.058 + 0.002a
1.0550 + 0.00005b
1.057 + O.OOlc
be taken with caution we might say the difference can be explained by more
effective shielding on the carbon nucleus bonded to deuterium in
cyanoacetylene. Flygare16 has pointed out that for acetylenes a 10%
change in the nuclear shielding of the carbon atom directly bonded to
deuterium will have the same effect as a 100% change for the next nearest
carbon atom. Since the bond distance of the C-D moiety is the same for
both molecules the significant difference in the deuterium quadruple
coupling points to the chemical shielding on the adjacent carbon atom.
Similar reasoning can be applied to explain the difference in quadrupole
coupling in C1CCD and NCCCD. The C-D bond distance is slightly smaller
for C1CCD leading to a higher nuclear contribution (Dositive) +0 the
quadrupole coupling. If the nuclear shielding on the adjacent carbons
were the same C1CCD would be expected to have a larger quadrupole coupling.
Note that these arguments are extremely qualitative and may have little
basis. The electronic contribution from chlorine has been completely
42
ignored. As a second row atom, computing its contribution to the
electric field gradient at the deuterium atom is a non-trivial problem.
Barfield3l has established distance criteria for the inclusion of
various contributions to the deuterium quadrupole coupling in propyne-Dl .
He evaluated two and three center integrals associated with atomic centers
in the molecule. The three center terms were found to make a contribution
of -53 kHz to the quadrupole coupling. The point of this is that signifi
cant differences in quadrupole coupling may be directly related to these
terms and therefore transcend qualitative arguments.
The last point to be addressed is the effect of the hybridization
on the carbon atom to which the deuterium is bonded. Millett and Dailey42
measured deuterium quadrupole coupling strengths in the liquid phase on a
series of hydrocarbons and observed from the measurements that eqQ(sp» 2 3 eqQ(sp » eqQ(sp). This conclusion must be taken with caution on experi-
mental and theoretical grounds. When making comparisons of deuterium quad
rupole coupling in different systems one must note the error bar associated
with the reported measurement. Many of the molecules they were comparing
had uncertainties of the order + 15 kHz. Absolute trends can truly be
established when measurements have been shown to differ within the limits
of experimental error. Furthermore, in support of their conclusion they
pointed to the work of Fung,43 et al., who studied the dependence of the
deuterium quadrupole coupling on carbon hybridization via a simple MO
description. They treated the C-D moiety as an isolated fragment insensi
tive to substituent effects. This is an oversimplification and their
43
basis set was minimal. Clearly, the later treatments of snyder30 and
Barfield31 demonstrate the importance of not treating the c-o moiety as
an isolated fragment.
APPENDIX
PROGRAM FOR CALCULATION OF HYPERFINE STRUCTURE
The program KUPLD calculates how a rotational level splits due to
electric quadrupole, spin-rotation and spin-spin interactions. Hyperfine
structure can be calculated for the most general case of four coupling
nuclei of arbitrary spin. The order in which the nuclei are coupled is
immaterial as a full matrix diagonalization is always a part of the calcu
lations. As a check the order of the coupling can be reversed even during
the same run since the program will accept more than one set of data.
The algorithm is simple enough to follow. The philosophy in which
the program was written was such that a person could calculate hyperfine
structure in rotational transitions even if their background in the area
was limited. One merely inputs the rotational level, spin of each nucleus
and their associated interaction strengths.
Note that the program can not differentiate between equivalent and
non-equivalent spins; no attempt was made to simplify the energy matrix
for special cases (e.g., CD3X type molecules).
44
PROGRAM KUPLDlINPUT,OUTPUT,TAPES-INPUT,TAPE6-0UTPUTI C THIS PROGRAM CALCULATES THE SPLITTING OF ROTATIONAL C LEVELS DUE TO QUAORUPOLE,SPIN-ROTATION,AND SPIN-SPIN C INTERACTIONS FOR THE CASE Of fOUR NUCLEI OF ARBITRARY SPIN. C THE BASIS STATES ARE WRITTEN INTHE COUPLED SCHEME AND ARE C OF THE FOR" :J,11,Fl,I2,FZ,I3,F3,1~,F).
REAL Hl3,31,Ul3,31,DJl61,CJl41,QI41,El31 REAL J,ll,IZ,13,14,J2,ISS,IJ COMMON/QSRSS/J,11,IZ,13,I~ COMMON/SOE/Q,BROT
C THE INPUT DATA ARE AS FOLLOWS:J IS THE ROTATIONAL LEVEL, CII IS THE SPIN OF NUCLEUS ONElTHE SAME FOR IZ,I3,I~I.
C THE INTERACTION STRENGTHS ARE NOW READ IN. THE C OJ'S ARE THE SPIN-SPIN CONSTANTS BETWEEN THE fOU~ NUCLEI. C OJll' IS THE SPI~-SPIN INTERACTION STRENGTH BETWEEN C NUCLEI ONE AND TWO,OJIZI IS THE SSIS BETWEEN NUCLEI ONE C AND THREE,OJI31 IS THE SSIS BETWEEN NUCLEI ONE C AND fOUR,DJI~1 IS THE SS15 BETWEEN NUCLEI TWO AND THREE C OJISI IS THE SSIS BETWEEN NUCLEI TWO AND FOUR,OJlbl IS THE C SSIS BETWEEN NUCLEI THREE AND FOUR. THE CJ'S ARE C THE SPIN-ROTATION INE\TERACTION STRENGTHS. CJlll IS C THE SPIN-ROTATION INTERACTION STRENGTH FOR NUCLEUS ONE C lANALOGOUS FOR CJIZI ••• CJ(~I. THE Q'S ARE THE QUADRUPOLE C INTERACTION STRENGTHS FOR NUCLEI ONE THPOUGH FOUR.
930 FORHAT(IHO,'THE QUADRUPOLE COUPLING CONSTANTS AREt) WRITElb,9101 lQlII,I-I'~) DO 31 NI- 1,3 DO 31 NZ- 1,3 H(NI,NZI- O.
31 CONTINUE C THE HAHILTONIAN MATRIX IS INDEXED:
Nl-l HZ-I
C IN THIS PART OF THE PROGRAM THE SPIN-SPIN
45
C INTERACTION IS CALCULATED DO 10 FI-ABSCJ-ll"AB5CJ+Ill DO 10 F2-ABSCFI-12"ABSIF1+IZ' DO 10 F3-ABSCF2-13"ABSCF2+13' DO 10 F-ABSCF3-I4"ABSCF3+14' DO 20 FIP-ABSCJ-Il"ABSCJ+Il' DO 20 F2P-ABSCFIP-I2',ABSCFIP+IZ' DO 20 F3P-ABSIF2P-13"ABSCFZP+13' DO 20 FP-ABSCF3P-141,ABSCF3P+141 IFIF.NE.FPI GO TO 30
C THE NEXT STATE~ENT ACTS AS SWITCH TO BYPASS CALCULATION C OF THE SPIN-SPIN INTERACTION IF ALL INTERACTION STRENGTHS C ARE SET EQUAL TO ZERO. THIS IS USE FULL AS CALCULATION C OF THE SPIN-SPIN PART USES UP QUITE A B[T OF ~XECUT[ON TIME.
IFCOJCll.EQ.0.AND.DJIZI.EQ.O.ANO.OJI31.EQ.0.ANO 1.DJI51.EQ.0.AND.DJlbl.EQ.01 GO TO 15
KOUNT-l DO 100 L-l,3 DO 100 lL-L+l,4 DO 200 FlPP-ABSIJ-[11,ABSIJ+I11 DO 200 F2PP-ABSIF1PP-IZI,ABSIFIPP+IZI DO ~OO F3PP-ABSIFZPP-I31,ABSIFZPP+[31 DO 200 FPP-ABSIF3PP-I41,ABSIF3PP+I41 IFIF.EQ.FPPI THEN HINl,NZI-HIN1,NZ'+ DJIKOUNTI.l1.5.IIJIFl,FZ,F3,F,FlPP,F2PP,
If3PP,FPP,LI.IJIFIPP,FZPP,F3PP.FPP,FIP,FZP,F3P,FP,LLI 2+IJIFl,FZ,F3,F,FlPP,FZPP,F3PP,FPP,lll·IJIFlPP,FZPP, 3F3PP,FPP,FIP,FZP,F3P,FP,LI,-IISSIF1,FZ,F3, 4F,FIPP,F2PP,F3PP,FPP,L,LLI·JZeF1PP,FZPP,F3PP,FPP, 5FIP,F2P,F3P,FPI I I
END IF 200 CONTINUE
KOUNT- KOUNT + 1 100 CONTINUE
15 DO 300 l-1,4 HINl,N21-HINl,NZI + CJILI.IJIFI,FZ,F3,F,FIP,F2P,F3P,FP,ll
61 CONTINUE CALL SORTCE,3) DO 63 1-1,3 WRITEr6,6ZIElII
63 CONHNUE 6Z fORMATIIX,F15.51
GO TO 11 END SUBROUTINE OUllX,MI DII'IENSION Xl3,31 DO ZO K-l,f1,10 WRITEI6,ZI
Z FORMATllHOI 00 20 J-l,M Ll-K
1
20 10
1
2 3 It
lU-K+9 IF ILU.GT.,..) lU-M WRITEr6,11 rXlJ,ll,l-Ll,LU) ~ORMATlIHO,10G13.71 IFIlU.EQ.,...AND.J.EC.MI GO TO 10 CONTINUE RETURN END SUBROUTINE JACOB I H, U, M, H, IFU, FIN) H IF THE ARRAY TO BE DIAGONALIZED. U IS THE UNITARY MATRIX USED FOR FORMATION OF THE EIGENVECTORS. M IS THE ALLOTTEe ORDER OF THE MATRICES H, U IN CALLING PROGRAM N IS THE ORDER OF H, lU) USED. N CANNOT BE GREATER THAN M IFU MUST BE SET EQUAL TO ONE IF EIGENVALUES AND EIGENVELTORS ARE TO BE COMPUTED. IFU MUST BE SET EQUAL TO TWO IF ONLY EIGENVALUES ARE TO BE COMPUTED. FIN IS THE INDICATOR FOR SHUT-OFF, THE FINAL LARGEST OFF DIAGONAL ELEMENT. THE SUBROUTINE OPERATES ONLY ON ThE ELEMENTS OF H THAT ARE TO THE RIGHT OF THE MAIN DIAGONAL. CAll ERRSET lZOe,0,Z5,1,01 CALL ERRSET l207,256,25,1,0) CALL TRAPSlO,20,201 DIMENSION HIM,MI, UlM,MI
~REPARATORY OPERATION GO TO rl,~), IFU DO 3 I - 1, N DO 2 J - 11 N utI,") - 0.0 Un,It - 1.0 IF eN .EQ. II GO TO 100 NSl - N - 1 SUM - 0.0 DO 6 I - 1, NSI IAl - I + 1 00 5 J - U1, N
5 SUM - SUM+Hrl,J).HlI,J) 6 CONTINUE
IF ISUM .LT. FIN.FINI GO TO 100 OFFMAX - SQRTeSUf1+SUM) HN - N
47
c C SCANNING FOR URGE OFF DIAGONAL ELEMENT C
C
7 OFFKAX • OFFI1AX/HN 8 MEKO • 1
00 17 I· 1, NS1 IA1 • I + 1 DO 17 J • I Al, N IF (ABSIHII,J» .LT. OFFMAX) GO TO 17 I1EI10 • 2
C TRANSFORMATION C
HII • HIld) HJJ • HIJ,J) HIJ • HII,J) TANG • SIGNIZ.O, IHII-HJJII'HIJ
1 II ABSIHII-HJJ) + SORTIIHII-HJJI"Z + HIJ"Z'~.OI) COSIN • 1.0/S0RTll.0 + TANG'*ZI SINE • TANG'COSIN 151 • I - 1 IF (lSI .EO. 0) GO TQ 10 DO 9 K· 1, lSI HKI • HIK,I) HIK,I) • HKI'COSIN + HIK,J)'SINE HIK,JI·. -HKI'SINE + HIK,JI'COSIN
lCZ •• J+l.'·IJ+l."·SlXJCJ,Il,Fl,Il,J,l.1 END IF ELSE IFCM.EQ.Z' THEN IFIFZ.NE.FZP.OR.F3.NE.f3P.OR.F.NE.FP) THEN I~-O. ELSE lxZ-Z.*FlP+IZ+FZ+J+Il+l. IJ-C-11**CIXZ'*SQRTIIZ*IZ.*lZ+1.,
1.CIZ+l.I.J·IZ.·J+1.1.IJ+1.I.IZ.·FlP+1.I.IZ.·Fl+1.11 Z.SIXJIF1,IZ,FZ,IZ,FIP,1.).SIXJIJ,Fl,Il,FlP,J,1.1 E~D If ELSE (F(M.EQ.3) THEN IFCF3.NE.F3P.OR.F.NE.FPI THEN 13-0. ELSE (X3 a Z.·FZP+Fl+FlP+13+F3+IZ+J+Il IJ-I-l'.·IIX31.SQRTC13·IZ.*13+1.I*I13+1.1.
lCZ.*FZP+l.I*Cz.*rZ+l.I*IZ.*FlP+l.'*CZ.*Fl+l.l. Z J*IZ.*J+l.)*IJ+l.11 *SlXJI~Z,I3,F3,13,F2P,1.1* 3SIXJCF1,FZ,IZ,FZP,FIP,1.I*SIXJIJ,Fl,Il,FlP,J,1.1
END IF ELSE (FCM.EQ.4' THE~ IFCF.NE.FPI THEN IJ-O. ELSE IX4-Z.*f3P+14+F+FZ+FZP+I3+Fl+FlP+IZ+Il+J+1. !J·C-11**IIx41*SQ~TII4*CZ.*I4+1.1*II4+1.1.
END IF END IF RETURN END fUNCTION ISSCF1,FZ,F3,F,FIP,FZP,F3P,FP,L,LLI COMMON/QSRSS/J,Il,IZ,13,I4 REAL J,Il,IZ,I3,I4,ISS (FIL.EQ.l.AND.LL.EQ.ZI THEN IFCFZ.NE.FZP.OR.F3.NE.F3P.OR.f.NE.FPI THEN ISS-O. ELSE IXS- J+ll+(Z+FlP+fl+FZ+l. ISS-l-l' •• lIXS'.SQRTIIZ •• IZ+l.'.IIZ+l.I*IZ*
3SIXJ,Fl,fZ,IZ,FZP,FIP,I.'·SiXJ,Il,Fl,J,FlP,11,1.1 END IF ELse If,L.fQ.l.AND.LL.EO.4' THEN tf C F .NE .FP I THEN ISS-O. ELSE IX7-Z.·FIP+2 •• F3+F2P+F2+I4+I3+IZ+Il+J+1.+F ISS-'-I'··'IX7'.SORT"2.·F3P+1.'.'Z.·F3+1.'.
fND IF ELse IF'L.EQ.2.AND.LL.EQ.31 THEN IF'f.NE.fP.OR.F3.NE.F3P.OR.Fl.NE.FlPI THEN ISS-O. ELSE !Xe-F3+FZP+FZ+F1+I3+IZ+1. ISS-'-II •• 'IXBI·SQRT'CZ.·I3+1.I.CI3+1.I·I3
1·'Z •• I2+1.1·IIZ+l.I.IZ.CZ •• fZP+l.J·IZ •• FZ+l.ll. 2SIXJCFZ,I3,F3,I3,FZP,1.J*SIXJ,12,F2,F1,FZP,IZ,1.1 ~ND IF fLSE IFCL.EQ.Z.AND.LL.EQ.41 THEN IFIF.NE.FP.OR.FI.NE.FlPI THEN ISS-O. ELSE IK9-F+Z.·F3+Z.·FZP+F1+14+13+IZ ISS-C-ll •• ,IX9J·SQRTIIZ.·F3P+1.1.IZ.·F3+1.1.
END IF ELSE IFCL.EQ.3.AND.LL.EQ.41 THEN IF,F.NE.FP.OR.FZ.NE.FZP.OR.Fl.NE.FlPI THEN IS'5-0. ELSE IXI0-F3+F3P+FZ+F+I4+I3+1. ISS-,-II •• IIXIOJ.SQRT,CZ.·F3P+l.I.IZ •• F3+1.1.
C FUNCTION VQ CALCULATES THE QUADRUPOLE INTERACTION C FOR THE K TH COUPLED NUCLEUS' UP TO FOUR NUCLEI C CAN BE COUPLEDJ
FUNCTION VOlfl.F2,r3,r,FlP,F2P,F3P,FP,MI COMHON/QSRSSI J,ll,I2,I3,I4 REAL J,ll,12,13,I4 IF , H • E Q • 11 THE N ,f'f.ne.fp.or.f3.ne.f3p.or.fZ.ne.fZp.or.fl.ne.flp.
END IF ELSE IFCM.EQ.3J THEN IFIF3.NE.F3P.OR.F.NE.FP.or.i3.lt.1J THEN VQ-O. elSE IX13-Z.$FZP+13+F3+F1+IZ+J+Il+FlP VQ-C-1I •• IIX131·SQRTIIZ.·FZP+1.1.12 •• FZ+1.1.
liZ. Hl P +1. I. I Z. $f 1 + 1.1 I. S CRT II 12. *13+ 1. I. I Z •• I 3 + Z. I. 2IZ.·13+3.1'/118 •• 131*12.·I3-1.)11 3·SQRTII12.·J+l.I*12.·J+Z.I.12.·J+3.11/118 •• J).12 •• J-1.))) 4·SIXJIF2,13,F3,I3,FZP,2.I.SIXJIF1,FZ,I2,F2P,F1P,Z.I. 5SIXJIJ,F1,Il,FlP,J,2.1
END IF ELSE IFIM.EC.41 THEN IFCF.NE.FP.or.i4 •• t.11 THEN VQ-O. £LSE IXS-2.*f3P+14+F+F2+I3+Fl+IZ+FZP+J+Il+F1P VQ-I-ll··IIX81·SCRTII12 •• I4+1.1*IZ.·14+2.1*li •• I4+3.)1
1/1 18 •• I 4 1 • I 2 •• 14-1 • ) 1 1 *S QR TC 1 12 •• J + 2. I. 12 •• J+ 1. I. I 2 •• J + 3 • ) ) Z/IIS •• J).'2.*J-l.II)·SIXJIF3,14,F,14,F3P,2.) 3·SIXJIFZ,F3,I3,F3P,FZP,2.I.SIXJIF1,F2,IZ,F2P,FIP,2.)* 4SIXJIJ,Fl,Il,FIP,J,2.1 5. S QR TC 12 •• F3 P + 1. ) * I Z •• F 3 +1 • ) *1 Z •• F 2P + 1. )* I 2 •• F 2 + 1. ). 6IZ.·FIP+l.J*12.*F1+1.1)
END IF rNO IF RETURN END FUNCTION SIXJIJ1,J2,J3,Ll,L2,L31 REAL J1,JZ,J3,L1,L2,L3 IIMIN-INTIAMAX1IJI+JZ+J3,JI+L2+L3,Ll+JZ+L3,Ll+L2+J31+0.~I IIMAX-INTIAMIN1IJ1+J2+L1+L2,J2+J3+LZ+L3,J1+J3+Ll+L3)+O.5) W-O.O IFI"001IZMIN,2J.EC.11 GO TO 10 SlGN--1.0 GO TO 20
1*DELTAIL1,LZ,J3).W RETURN END FUNCTION DELTAIA,B,C' IFIA+B-C.GE.-0.1.AND.-A+B+C.GE.-0.l.AND.A-B+C.GE.-0.l)
1GO TO 10 DELTA-O.O RETURN
10 DELTA-SQRTIFACIA+B-C)*FACIA-B+C'*FACI-A+B+C)/FACIA+B+C+1') RETURN END FUNCTION FAC ()() N-INTIX+0.5) HIN) ZO,30,10
20 WRITEI6,100)X 100 FOR~ATIIHO,'FACTORIAL',G16.B,'IS REQUIRED. UNITY IS ASSUM
leD' , FAC-1.0 RETURN
30 FAC-l.O RETURN
10 FAC-l.O DO 1 J-l,N FAC-FAC*J
1 CONTINUE R ETUR N END FUNCTION JZIF1,FZ,F3,F,FIP,FZP,F3P,FP) REAL JZ IFIF1.NE.F1P.OR.FZ.NE.F2P.OR.F3.NE.F3P.OR.F.NE.FP) THEN 8Z-0. ELS E JZ·Z. END IF RETURN eND SUBROUTINE SORTfA,B' INTEGER B DI~ENSION AlB' lFIB.EQ.l' RETURN L -B-1 00 10 l-l,L t(. B-1 00 ZO J-l,K IFIAIJ)-AIJ+1"ZO,ZO,30