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STRUCTURE AND DYNAMICS OF VAN DER WAALS MOLECULES FROM MICROWAVE SPECTROSCOPY

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300 N 2EEBRD , ANN ARBOR Ml 18106
8203501
KEENAN, MICHAEL ROBERT
STRUCTURE AND DYNAMICS OF VAN DER WAALS MOLECULES FROM
MICROWAVE SPECTROSCOPY
University of Illinois at Urbana-Chatnpaign
University
Microfilms
I n t e r n £ t t J O n a I 300 N Zeeb Road. Ann Arbor, MI 48106
PH.D. 1981
STRUCTURE AND DYNAMICS OF
VAN DER WAALS MOLECULES
FROM MICROWAVE SPECTROSCOPY
BY
MICHAEL ROBERT KEENAN
B.S., State University of New York, 1976
THESIS
Submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Chemistry
in the Graduate College of the
University of Illinois at Urbana-Champaign, 1981
Urbana, Illinois
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
THE GRADUATE COLLEGE
JUNE 1981
W E HEREBY RECOMMEND T H A T T H E THESIS BY
MICHAEL ROBERT KEENAN
ENTITLED
STRUCTURE AND DYNAMICS OF
VAN DER WAALS MOLECULES FROM MICROWAVE SPECTROSCOPY
BE ACCEPTED IN PARTIAL F U L F I L L M E N T OF T H E REOUIREMENTS FOR
T H E DEGREE O F
DOCTOR OF PHILOSOPHY
MJL2_Q^.
Director of Thesis Research
Head of DepaY
Department
Committee on Final Examination!
2J^t^
'/>vrX^^
t Required for doctor's degree but not for master's
Chairman
ACKNOWLEDGEMENTS
I greatly appreciate having had the opportunity to work with Dr.
W.H. Flygare.
He was a tremendous source of ideas and insight and his
enthusiasm was inspiring.
Without him, the work described here would
have been impossible.
I would like to thank Terry Balle for building the microwave
spectrometer, and Ed Campbell and Bill Buxton for keeping it in top
working condition.
In addition, assistance from all of the members of
the research group is appreciated.
The support of the University of Illinois through teaching and
research assistantships, the E.I. Dupont Co. for a fellowship and the
National Science Foundation for general support of the research is
gratefully acknowledged.
Most of all, I wish to thank my wife, Ginny, for her support and
encouragement through the years and for her help in preparing this
thesis.
iv
TABLE OF CONTENTS
CHAPTER
I.
Page
INTRODUCTION
A.
B.
II.
D.
E.
F.
10
11
21
35
52
61
Introduction
Hindered Rotor Model
Results and Discussion
70
71
72
XENON NUCLEAR QUADRUPOLE COUPLING IN XENON HYDROGEN
CHLORIDE
A.
B.
C.
V.
Introduction
General Analytical Methods
Krypton Hydrogen Chloride and Xenon
Hydrogen Chloride
Argon Hydrogen Bromide and Krypton
Hydrogen Bromide
Argon Deuterium Fluoride
Summary of Results and Conclusion
A HINDERED ROTOR MODEL FOR ESTIMATING ANGULAR EXPECTATION
VALUES IN THE RARE GAS-HYDROGEN HALIDES
A.
B.
C.
IV.
1
4
STRUCTURE AND DYNAMICS OF RARE GAS-HYDROGEN HALIDE
VAN DER WAALS MOLECULES
A.
B.
C.
III.
Background and Review
Experimental Method
Introduction
Spectra and Results
Analysis of the Xenon Nuclear Quadrupole
Coupling Constant
82
83
89
ARGON CYANOGEN CHLORIDE
A.
B.
C.
D.
E.
Introduction
Spectra and Results
Molecular Structure
Intramolecular Force Field
Nuclear Quadrupole Coupling
97
99
103
115
125
v
TABLE OF CONTENTS
CHAPTER
APPENDIX I .
Page
THE ROTATIONAL CONSTANTS FOR AN
ATOM-LINEAR MOLECULE COMPLEX
128
APPENDIX I I . RELATION OF MONOMER TENSOR PROPERTIES
TO ATOM-LINEAR MOLECULE COMPLEXES
135
REFERENCES
138
VITA
143
1
CHAPTER I
INTRODUCTION
A. Background and Review
Intermolecular forces have been under intensive invesitgation for
several decades.
These forces play a major role in the understanding
of the structure and properties of condensed phases as well as the
deviations of gases from ideal behavior.
On the molecular scale,
intermolecular forces can be described in terms of the intermolecular
potential energy surface.
Numerous theoretical and experimental
techniques have been used m
attempts to obtain potential surfaces
2-4
and many reviews are available.
Theoretical understanding of the
long range electrostatic, induction and dispersion
exchange forces is well advanced.
and short range
However, the detailed shape of the
potential near the potential minimum is currently difficult to obtain
from first principles.
From an experimental standpoint, nearly all
phenomena of chemical interest bear relation to the potential energy
surface.
Bulk phase phenomena such as virial and transport properties
are intimately related to the intermolecular potential but are complicated in that averaging over a large number of interactions may have
to be performed.
Individual intermolecular interactions are probed
directly in various molecular beam scattering experiments (elastic,
3 6
7
inelastic and reactive) ' and rotational line shape analyses.
All of
2
the above methods suffer, however, from not being equally sensitive to
all parts of the potential.
Indeed, the same experiment may probe
different parts of the potential as an experimental condition such as
temperature
or molecular beam energy is changed.
Perhaps the best
potential surface information results from spectroscopic studies, in
the gas phase, of bound states of molecular complexes or van der Waals
molecules.
Van der Waals molecules are weakly bound molecular complexes which
may be characterized as having binding energies on the order of kT at
room temperature, where k is the Boltzmann constant and T is the temperature.
From simple equilibrium considerations, production of van der
Waals molecules is favored by high pressures and/or low temperature.
Infrared spectra were originally observed in highly compressed rare gaso
hydrogen h a l i d e mixtures and were l a t e r i n t e r p r e t e d i n terms of van der
9
Waals molecules.
More r e c e n t l y , spectroscopy has been performed on
10-12
supersonic gas expansions
which takes advantage of the low i n t e r n a l
temperature generated in these
jets.
Spectroscopy of van der Waals
molecules has included a l l frequency ranges and the l i t e r a t u r e has been
13-15
r e c e n t l y reviewed.
Studies of the r o t a t i o n a l s p e c t r a of van der Waals molecules present
the g r e a t e s t i n t e r e s t h e r e .
Klemperer and coworkers f i r s t found micro16
wave and radiofrequency s p e c t r a for ArHCl
and have since s t u d i e d the
v i b r a t i o n a l ground s t a t e of s e v e r a l complexes by employing the molecular
10,17
beam e l e c t r i c resonance t e c h n i q u e .
Low r e s o l u t i o n microwave
s p e c t r a including v i b r a t i o n a l s a t e l l i t e s have been observed f o r hydrogen
3
bonded species in relatively high pressure s t a t i c gas mixtures.
With
the recent development, in our laboratory, of a method for obtaining
12
high resolution rotational spectra of weakly bound molecular complexes,
the body of information concerning van der Waals molecules has considerably increased.
Rotational spectra yield a wealth of information about van der
Waals molecules and their potential surfaces in the region of the potent i a l minimum.
The extraction of this information for several molecular
systems i s the topic of this d i s s e r t a t i o n .
A rotational spectrum r e s u l t s
from the absorption or emission of radiation at frequencies corresponding
to changes in the s t a t e of molecular rotational angular momentum.
Since
angular momentum i s proportional to the molecular moment of i n e r t i a ,
which in turn i s a function of the atomic coordinates, analysis of the
rotational spectrum serves to locate the region of the potential energy
minimum.
Hyperfine interactions, which appear as perturbations in the
pure rotational spectrum, give further structural and orientational
information directly related to the anisotropy of the potential surface.
Since van der Waals molecules are not rigid r o t o r s , centrifugal d i s t o r tions of spectroscopic properties yield information about the shape of
the potential near i t s minimum and if coupled with assumed forms for the
potential, properties such as the binding energy can be estimated.
Finally, certain spectroscopic features such as rare gas nuclear quadrupole coupling result solely from formation of the weak van der Waals
bond.
Analyses of these features lend a direct insight into the basic
nature of the van der Waals binding interaction.
4
B. Experimental Method
The spectroscopic method employed in the studies presented here
involves pulsed Fourier transform microwave spectroscopy carried out in
a Fabry-Perot cavity with a synchronized pulsed supersonic nozzle as the
molecular source.
The operational theory and details of the apparatus
12,19
have been given before.
spectrometer.
Figure 1-1 shoxre a block diagram of the
Briefly, microwave power at frequency v is supplied to
the Fabry-Perot cavity by the master oscillator (MO) which is phase
locked to a frequency standard.
The local oscillator (LO), which is
used in the superheterodyne detection, is phase locked to the MO at v-30
MHz.
A gas mixture used to generate the molecular complexes is pulsed
into the cavity and, after a suitable delay, is polarized by a microwave
pulse formed by opening the PIN diode 1 switch for 3-7 us.
The micro-
wave pulse dies away with the cavity relaxation time after which PIN
diode 2 is opened and the coherent molecular emission at the resonant
frequency, v^, is detected in the superheterodyne receiver.
The result-
ing time domain signal is digitized and subsequent signals averaged.
An
equal number of signals are taken with the gas absent to subtract out any
coherent noise.
The final time domain record is Fourier transformed to
yield the frequency spectrum.
In practice, 256 point time domain records
are obtained by digitizing the molecular emission signals at rates of 1-2
MHz.
After 256 zeros are added, this record is Fourier transformed to
give a 256 point spectrum having 3;9-1.9 kHz/point resolution.
Figure
1-2 shows the time domain record of the 1=5/2, F=5/2 to 1=5/2, F=7/2
LO
v -30 MHz
Frequency
Stabilizer
<Xf
xer
30 MHz
Monitor
Detector
a
Pin
Diode
MO
Frequency
Standard
v
Pulsed
no
Nozzle
ffie
Pin
Tuner .JUL Diode
H3--Q
Circulator
Mixer
<S)
Block diagram of the microwave spectrometer.
A±30MHz
I [Diffusion
'—' Pump
Mixer
•
Frequency
Stabilizer
Figure I-l.
Detector
iMixer
$
A
t
T
Display
Display
6
hyperfine component of the O Q Q - 1,, transition in ArClCN and Figure 1-3
shows the corresponding frequency spectrum.
The spectral lines show a
characteristic doubling which is a Doppler effect involving the natures
of the gas expansion and the standing wave electric field in the cavity.
A detailed theory explaining this effect is available in the literature.
2
As far as spectroscopic information is concerned, the true molecular
resonance frequency lies at the midpoint of the Doppler pairs.
The
spectral lines have half widths at half height of about 5 kHz and the
frequency uncertainty is estimated to be about 2 kHz.
The key experimental feature that permits the study of van der Waals
molecules is the use of a pulsed supersonic nozzle expansion as the
molecular source. The adiabatic expansion rapidly cools the internal
degrees of freedom and the large stagnation pressures available give the
conditions of low temperature and high pressure needed to form significant quantities of molecular complexes.
In addition, since the molecules
are observed in the free molecular flow region of the beam, collisional
line broadening effects are minimal.
Dimer formation proceeds by 3 body
collisions which are proportional to p^D where p is the stagnation
pressure and D is the nozzle diameter.
Use of a pulsed nozzle is advan-
tageous since it permits a greater p and D than a continuous nozzle for
given pumping speeds.
The nozzle used in the present studies is a t h m
plate flat orifice, approximately 0.5-1 mm in diameter, bolted onto a
solenoid valve.
For complexes involving rare gases, the gas mixtures
consisted of 1-4% of the dipolar specie seeded in the rare gas and were
held at room temperature and at 0.5-2.5 atra pressure.
The estimated
TIME
(/JLS)
Figure 1-2. Time domain record of the 1=5/2, F=5/2 -+ I'=5/2, F*=7/2 hyperfine
component of the OQQ -*- 1 ^ transition in ArClCN.
7403.18
7403.10
7403.02
MHz
Figure 1-3. The frequency spectrum corresponding to the time domain in Figure 1-2.
The resolution is 3.9 kHz per point.
<»
9
21
rotational temperature of the expanding gas is 1-10 K.
This low tem-
perature greatly facilitates the observation of the rotational spectra
for low lying rotational levels.
In general, the signal strength is
proportional to the number density difference, Ap(J,K)/p0, between the
J and K levels.
Table I-l compares Ap/pQ for several transitions in
KrHCl at temperatures of 0.5 K and 300 K.
Table I-l.
For J=0-M=1 and J=3"M=4
Comparison of the number density differences
between rotational levels of KrHCl at temperatures of 0.5 K and 300 K.
0.5 K
300 K
Ap(0,l)/pQ
2.3xl0~2
7.4xl0"8
Ap(3,4)/pQ
1.2xl0_1
2.1xl0~6
Ap(9,10)/p
5.9xl0"5
1.4xl0-5
there are large gains in Ap/p
over a 300 K gas.
J=9, however, there is a fall off in gain.
For J transitions over
10
CHAPTER II
STRUCTURE AND DYNAMICS OF RARE GAS-HYDROGEN HALIDE
VAN DER WAALS MOLECULES
A.
Introduction
The simplest, strongly anisotropic, van der Waals interaction is
represented by the class of van der Waals molecules that involve the
binding between a rare gas atom (X) and a hydrogen halide (HY). As
such, the X-HY complexes, in general, and ArHCl, in particular, have
been intensively investigated.
Gas phase spectroscopic studies of X-HY
8
have ranged from the infrared
radio frequencies.
'
'
22
and far infrared
to the microwave and
X-HY has several characteristics which
permit a great deal of information to be gained about it from relatively
little data.
X can be treated, for the most part, as a structureless
particle probing HY in the region of the potential minimum.
X-HY can
also be treated, in certain situations, as a pseudodiatomic molecule,
since the mass of H is small compared to the mass of the halogen.
only
Finally,
3 modes of vibration are added to the atom-molecule system upon
complexation.
These are easily visualized in terms of bending and stretching
of the van der Waals bond.
Due to this simplicity, there is relatively
little ambiguity in interpreting the spectroscopic findings in terms of
the structure and potential of X-HY".
The studies of several X-HY complexes presented here serve a twofold
11
purpose.
The primary aim is to increase the body of knowledge concerning
van der Waals molecules.
In particular, the X-HY interaction is studied
over a series of complexes to see how the interaction may correlate with
the properties of the X and HY monomers.
The second goal is to investigate
spectroscopic features peculiar to particular complexes such as the centrifugal distortion of the Br nuclear quadrupole coupling constant in the
HBr complexes and the HF nuclear spin-spin interaction in ArHF.
These
interactions serve to give more insight into the intermolecular potential
and to test some assumptions made in the spectral and structural analyses.
B.
General Analytical Methods
Due to the small mass of H as compared to the rare gas and halogen
atoms, the pure rotational energies of X-HY can be described by the
symmetric top Hamiltonian for all orientations of HY with respect to X.
The extreme cooling caused by the nozzle expansion places essentially all
molecules in the ground vibrational and K=0 rotational states, thus, the
rotational Hamiltonian, H R , is given by
%
33
= B 0 J 2 - DjJ4
,
(II-l)
where ~BQ = (B0+C0)/2 is the rotational constant, Dj is the centrifugal
distortion constant and J is the rotational angular momentum.
In addition,
nuclei with spin angular momenta, I, may couple with the rotational angular momentum, J, to form a total angular momentum, F.
This latter
coupling appears as a perturbation on the pure rotational state and
results in the observed rotational spectrum hyperfine structure.
Three
12
types of coupling have been observed in the X-HY complexes, and the
hyperfine Hamiltonian is given by
H
HF=HQ+HSR+HSS'
(II
"2>
H Q describes the nuclear quadrupole coupling for nuclei have 1 ^ 1 and
involves the interaction between a nuclear quadrupole moment and the
electric field gradient at that nucleus.
Hg^ is the spin-rotation inter-
action which couples a nuclear magnetic moment to the rotational molecular
magnetic moment and H g s , the nuclear spin-spm coupling, is the classical
interaction between two magnetic dipoles.
For all cases considered here,
the rare gas nuclei do not couple and a first order perturbation treatment
for the resulting two spin system gives
H = i x1 [(2h'&2
Q
i=l
a
+
hij->i)
-4*2)
2I1(2IjL-l)(2J-l)(2J+3)
(H-3)
H
SR " c l<&.-#
H«
SS =
~ (2J-l)(2J+3) &K.'V<h'$
+
c
2<fc'#
+ 3(
l2'V(lvJ)
- 2(l!^ 2 )^]
where x 1 , c and d a are the appropriate coupling constants.
Matrix
elements of the coupling operators in Eq. (II-3) are readily computed in
the coupled basis given by
Aj
-,
'tl = F
ol
(H-4)
J, + I, = F
13
where I
refers to the nucleus that is the most strongly coupled. The
matrix elements are composed from
35—37
<JW2*V£2| " i W ' V
= J(J+1)6
(j^F^FM^IJI^F'Mp)
= I±(I1+l)«PiP,6Fpl
(jI^^FMpl^.JlJI^^F'Mp)
FlFi6FF'
= [F 1 (F 1 +1)-J(J+1)-I 1 (I 1 + 1)J /26 F
F,5FF,
(JI 1 F 1 I 2 FM F |I 2 .J|JI 1 F'I 2 F'M F ) = (-l) 1 + I l + I 2 + J + 2 F i + F
\,
x[(2F1+l)(2F[+l)I2(I2+l)(2I2+l)J(J+l)(2J+l)]
1 F;
Z 1
Fl n
2
A «*•
i 2 i 2 FJ ( J J 1^
<JI1F1I2FMF|l1.I2|jI1FiI2F'MF>
=
1+I +I +J+F +F'+F
( - Dn ^0 l ^ n ^ l
x [(2F 1 +1) ( 2 F ' + 1 ) I 2 ( I 2 + 1 ) (212+1)1^^(^+1) (21^+1)] *2
\
Fi l] (h F[ i)
J
i2 i 2
F] (I X I L
6
FF"
jj
where 6 . is the Kronecker delta and the quantities in braces are 6-j
symbols.
The Hamiltonian matrices in Eqs. (II-2) and (II-3) are then
readily constructed by matrix multiplication and addition and diagonalized
14
to yield the energy levels.
In the special (and most common) case that
only a single nucleus couples with J,
energy levels are given by
H = IL, + H„ F is diagonal and the
33
-E(J,I,F)=B o J(J+l)-D J J 2 (J+l) 2 - x a
(3/4)G(G+l)-1(1+1)J(J+1)
21(21-1)(2J-1)(2J+3)
2
+ —a. gxlO -3 + cG/2
where F = I + J , G = 2 I - J =
(II-6)
F(F+1)-I(I+1)-J(J+1), and g is a second
order quadrupole energy correction taken from Appendix II in Townes and
Schawlow.
33
To obtain the spectroscopic constants for X-HY, the
experimental spectral frequencies were fit to B , D T and the appropriate
coupling constants by a nonlinear least squares procedure involving
diagonalization of H m
spin case.
the two spin case or Eq. (II-6) in the single
In all cases, the Hamiltonian in Eq. (II-2) was sufficient
to fit the data within the experimental uncertainty.
The molecular constants of X-HY derived from the spectral data
reflect vibrationally averaged rather than equilibrium properties of the
complex.
Considering the weak nature of the van der Waals bond, the
vibrational amplitudes are expected to be large.
As a starting point
for determining the structure of X-HY, we assume that the complex is held
rigidly fixed in its vibrationally averaged configuration and that all
properties of free HY are preserved unchanged upon complexation.
particular, the H-Y distance is fixed at the free HY bond length.
In
Under
15
these assumptions, the structure of X-HY can be described in terms of
two parameters, a length and an angle.
Figure
II-l
dinate system used in the X-HY structure analysis.
shows the coor-
The structural para-
meters have been chosen, for convenience, to be the rare gas-halogen
distance, R , and the angle, 9, between HY and R .
The angle, a, between
HY and the a-inertial axis is not a free parameter but is fixed once R
and 8 are known.
An alternative parameterization in which the length
is taken between X and the HY center of mass is discussed in Appendix I.
It suffers, however, from the fact that given a fixed atomic arrangement,
the structural parameters vary with isotopic substitution.
In any case,
the two parameterizations are easily interconverted by simple geometry.
The rotational constant of X-HY is very insensitive to the location
of H, and thus 9, since H is much lighter than X or Y (this will be
discussed at length in the next section).
Fortunately, the hyperfine
constants can be used to derive angular information.
The measured
nuclear quadrupole and nuclear spin-spin coupling constants, x a and d„ ,
c
*
EL *
respectively, are simply those components of the respective tensors
along the a-axis of the complex.
Under the constancy of HY properties
assumption discussed above, the coupling constant tensors for X-HY can
be obtained by rotating the free HY tensors to the principal axes of the
complex according to (see Appendix II )
P ^ ( c o s ^ a -1)P 0 ,
(II-7)
where P represents either X„ or d 'and P. is the corresponding property
of free HY.
The angle a is shown in Figure (II-l).
In practice, a and
®
a-axis
R
Figure II-l. Coordinates used to describe the structure of rare gas (X)-hydrogen
halide (HY) van der Waals molecules.
a*
17
the structural angle, 9, differ by only a couple tenths of a degree and
can often be used interchangeably.
Once a has been determined by using
Eq. (II-7), the evaluation of RQ is straightforward.
An iterative
technique is used whereby the rotational constant B = (B+C)/2 is calcu38
lated
as a function of R and 9 until B 0 and a are reproduced. It is
noted that the sign of cosa
cannot be determined from Eq. (II-7).
The
question of whether a is acute or obtuse can be settled, however, by
resort to isotopic substitution in the HY subunit since, as a first
approximation, the structure is invariant to isotopic substitution.
In
all X-HY complexes, a has been found to be acute.
The internal dynamics of van der Waals molecules which undergo
large amplitude vibrations are not totally understood.
Information
about the intermolcular potential may be obtained, however, by considering the centrifugal distortion constant, D,, and the structures of X-HY
determined above, along with simple assumptions about the form of the
potential.
In these comlexes, D, arises from stretching of the van der
Waals bond, taken to be along the X-Y axis, as well as small effects due
to the change in the bond angle, 9, with rotation.
Neglecting the effects
due to changes in 9, the van der Waals stretching motion can be treated
by considering X-HY to be a pseudodiatomic molecule.
In this model,
the radial interaction between X and HY is described by an effective
radial potential averaged over the bending mode.
Dj is related to the
van der Waals stretching frequency, v g , and force constant, k , by the
usual
diatomic expression
39
18
4B_
(H-8)
where
(H-9)
and the reduced mass, u , is given by
(11-10)
X
HY
By assuming a functional form for the angle averaged effective radial
potential, k
and the molecular structure can be used to estimate the
potential parameters.
The choice, made here, is to describe the radial
interaction in terms of the Lennard-Jones 6/12 potential given by
V(R) = e
Re\ 1 2 _2{Re
(11-11)
where e is the well depth and R e is the equilibrium mternuclear
separation.
Other choices could be made for the form of V(R). However,
it has been found that the final results do not depend very much on the
specific form chosen.
Expanding V(R) in a Taylor series about R=Rg gives
36
(R-RJ 2 - 252
2
R'
R.
e
"e
V(R) = -e + ^
(R-P)3 + .
(11-12)
The harmonic force constant is related to the quadratic term of this
potential by
19
k s = 36e/Re2
\
(H-13)
and, through second order, the rotation-vibration interaction constant,
4
uby 39,40
<xe, Ais given
a e = 36(B e 2 /v s ) .
(11-14)
In the ground vibrational state, a e relates the vibrationally averaged
rotational constant,BQ, to the equilibrium rotational constant, B e ,
according to
B
o = Be " K
= B e " 18(B e 2 /v s ).
(H-15)
Taking
B0
2
'
Eq. (11-15) is solved by the quadratic formula to obtain B
(H-16)
and the
equilibrium separation is calculated from
Rg=|
8*2y B e \ *
S_]
#
(II_17)
Using this value for R e and the value of k s derived from Dj, Eq. (11-13)
Is solved for the Lennard-Jones estimate of the effective radial well
depth, e.
The treatment of the van der Waals bending motion poses more severe
problems.
In contrast to the stretching motion which Is simply described
in terras of the stretching between two heavy particles, the bending modes
involve complicated, large amplitude motion by a light atom in two dimen-
20
sions.
It is still useful, however, to consider the bending in terms of
a harmonic oscillator model for at least
qualitative and comparative
purposes.
By analogy with ArHCl
'
and from evidence to be presented later,
it appears that the equilibrium geometry of X-HY is linear.
The measured
bond angle results from averaging over the large amplitude bending vibrations.
The two dimensional isotropic harmonic oscillator is the model
appropriate to the bending motion of a linear system.
The bending poten-
tial is given by
V(9) = f k ^ 2
(11-18)
where the bending force constant, k., is related to the bending frequency,
v D , and bending reduced mass, u^, by
(11-19)
The energy levels of the two dimensional oscillator are given by
(E) = <T>+<V) = hvb(n+l)
(11-20)
where T and V are the kinetic and potential energies, respectively, and n
is the vibrational quantum number.
From the virial theorem
<T) = (V)
thus, from Eqs. (11-18) - (11-21) and for n=0
(11-21)
21
In other words, an estimate of the bending potential can be gained simply
from the structure of X-HY and a value of u^.
,.
Mfc, is taken to be
The bending reduced mass,
42
where r is the bond length of HY.
The choice of bending angle and re-
duced mass is not unique and the choice made here is not the one conventionally made.
However, with this choice
lVf~ + — I" *" = 1
-
'
(11-24)
MY;
where I is the moment of inertia of free HY and the bending problem becomes
couched m
the more physically meaningful picture of a hindered internal
rotation.
C. Krypton Hydrogen Chloride and Xenon Hydrogen Chloride
KrHCl was the first molecular complex studied using the microwave
spectrometer described in Section I-B.
As such, it served to demonstrate
the high resolution and sensitivity achievable with this spectroscopic
technique.
The J=3->J=4 and J=4-*J=5 rotational transitions in the ground
vibrational state were readily observed and recorded for the 12 isotopes
consisting of all combinations of
82
84
86
35
37
Kr,
Kr,
Kr, H, D,
CI and
CI.
The measured KrHCl transition frequencies are listed in Table II-l.
The spectrometer's high sensitivity is established by noting that the
22
Table II-l.
Observed transition frequencies for KrHCl and those calculated
with Eq. (II-6) and the constants in Table II-3.
Isotope
J,F
J',F'
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
82
35 ,
KrH CI
3,9/2
3,3/2,5/2
3,7/2,9/2
3,7/2
3,5/2
4,5/2,7/2
4,9/2,11/2
4,9/2
4,7/2
4,9/2
4,5/2,7/2
4,9/2,11/2
4,7/2
4,5/2
5,7/2,9/2
5,11/2,13/2
5,9/2
5,7/2
9665.464
9671.933
9672.779
9675.345
9679.246
12089.008
12089.514
12091.586
12096.326
9665.464
9671.930
9672.780
9675.349
9679.245
12089.011
12089.516
12091.580
12096.327
0
3
-1
-4
1
-3
-2
6
-1
84
3,9/2
3,9/2
3,3/2,5/2
3,7/2,9/2
3,7/2
3,5/2
3,7/2
4,11/2
4,5/2,7/2
4,9/2,11/2
4,9/2
4,7/2
4,9/2
4,7/2
4,5/2,7/2
4,9/2,11/2
4,7/2
4,5/2
4,5/2
5,11/2
5,7/2,9/2
5,11/2,13/2
5,9/2
5,7/2
9596.022
9598.592
9602.486
9603.333
9605.902
9609.801
9613.208
11995.399
12002.213
12002.723
12004.774
12009.524
9596.022
9598.591
9602.485
9603.334
9605.902
9609.795
9613.214
11995.403
12002.212
12002.717
12004.780
12009.523
0
1
1
-1
0
6
-6
-4
1
6
-6
1
3,7/2,9/2
3,7/2
3,5/2
4,11/2
4,5/2,7/2
4,9/2,11/2
4,9/2
4,7/2
4,9/2
4,5/2,7/2
4,9/2,11/2
4,7/2
4,5/2
5,11/2
5,7/2,9/2
5,11/2,13/2
5,9/2
5,7/2
9529.757
9536.223
9537.064
9539.637
9543.534
11912.592
11919.403
11919.903
11921.969
11926.713
9529.756
9536.220
9537.069
9539.638
9543.531
11912.591
11919.401
11919.906
11921.969
11926.713
1
3
-5
-1
3
1
2
-3
0
0
3,3/2,5/2
3,7/2,9/2
4,5/2,7/2
4,9/2,11/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2 5,11/2,13/2
9315.690
9316.364
11643.709
11644.112
9315.690
9316.364
11643.710
11644.111
0
0
-1
1
KrH
35
C1
86
35
KrII
Kru Cl
UL
82
KrH
37
C1
3 9/2
'
3)3/2j5/2
23
Table I I - l , Continued
Isotope
J,F
J',F»
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
8
3,9/2
3,3/2,5/2
3,7/2,9/2
3,7/2
3,5/2
4,11/2
4,5/2,7/2
4,9/2,11/2
4,9/2
4,7/2
4,9/2
4,5/2,7/2
4,9/2,11/2
4,7/2
4,5/2
5,11/2
5,7/2,9/2
5,11/2,13/2
5,9/2
5,7/2
9241.135
9246.243
9246.915
9248.938
9252.017
11551.541
11556.929
11557.322
11558.952
11562.696
9241.137
9246.242
9246.913
9248.941
9252.017
11551.544
11556.923
11557.322
11558.951
11562.699
-2
1
2
-3
0
-3
6
0
1
-3
KrH 3 7 Cl
3,3/2,5/2
3,7/2,9/2
3,7/2
4,5/2,7/2
4,9/2,11/2
4,5/2,7/2
4,9/2,11/2
4,7/2
5,7/2,9/2
5,11/2,13/2
9179.979
9180.650
9182.681
11474.113
11474.514
9179.979
9180.650
9182.681
11474.113
11474.513
0
0
0
0
1
™. 35 r-i
KrD CI
3,3/2,5/2
3,7/2,9/2
3,7/2
4,5/2,7/2
4,9/2,11/2
4,9/2
4,5/2,7/2
4,9/2,11/2
4,7/2
5,7/2,9/2
5,11/2,13/2
5,9/2
9537.198
9538.382
9541.973
11920.916
11921.622
11924.502
9537.197
9538.384
9541.972
11920.915
11921.621
11924.503
1
-2
1
1
1
-1
84
35
KrD CI
3,9/2
3,3/2,5/2
3,7/2,9/2
3,7/2
3,5/2
4,11/2
4,5/2,7/2
4,9/2,11/2
4,9/2
4,7/2
4,9/2
4,5/2,7/2
4,9/2,11/2
4,7/2
4,5/2
5,11/2
5,7/2,9/2
5,11/2,13/2
5,9/2
5,7/2
9458.627
9467.644
9468.825
9472.413
9477.832
11824.482
11833.983
11834.688
11837.569
11844.185
9458.626
9467.641
9468.826
9472.411
9477.838
11824.484
11833.983
11834.688
11837.568
11844.182
1
3
-1
2
-6
-2
0
0
1
3
V.H 3 7 P-.
KrH CI
86
82
24
Table I I - l , Continued
Isotope
J,F
J',F»
Observed
(MHz)
86
35
KrD CI
3,3/2,5/2
3,7/2,9/2
3,7/2
4,11/2
4,5/2,7/2
4,9/2,11/2
4,9/2
4,7/2
4,5/2,7/2
4,9/2,11/2
4,7/2
5,11/2
5,7/2,9/2
5,11/2,13/2
5,9/2
5,7/2
9401.276
9402.457
9406.046
11741.517
11751.038
11751.740
11754.616
11761.239
9401.272
9402.459
9406.048
11741.522
11751.031
11751.737
11754.620
11761.241
4
-2
-2
-5
7
3
-4
-2
82
37
KrD CI
3,3/2,5/2
3,7/2,9/2
4,5/2,7/2
4,9/2,11/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2
5,11/2,13/2
9191.338
9192.276
11488.553
11489.106
9191.339
9192.275
11488.551
11489.108
-1
1
2
-2
84 v 37_.
KrD CI
3,9/2
3,3/2,5/2
3,7/2,9/2
3,7/2
3,5/2
4,11/2
4,5/2,7/2
4,9/2,11/2
4,9/2
4,7/2
4,9/2
4,5/2,7/2
4,9/2,11/2
4,7/2
4,5/2
5,11/2
5,7/2,9/2
5,11/2,13/2
5,9/2
5,7/2
9114.707
9121.819
9122.756
9125.591
9129.867
11394.174
11401.668
11402.228
11404.497
11409.730
9114.705
9121.821
9122.757
9125.586
9129.872
11394.173
11401.672
11402.228
11404.501
11409.723
2
-2
-1
5
-5
1
-4
0
-4
7
86
37 ,
KrD CI
3,3/2,5/2
3,7/2,9/2
4,5/2,7/2
4,9/2,11/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2
5,11/2,13/2
9055.487
9056.421
11318.772
11319.333
9055.486
9056.422
11318.774
11319.331
1
-1
-2
2
Calculated
(MHz)
Residue
(kHz)
25
AF=-1 hyperfine components were observed in the J=3->-J=4 transition in
KrH
AF—1
CI.
KrH
CI is present at 43.0% isotopic abundance and the
components comprise only 0.07% of the total transition intensity.
on
In addition, the spectra of
easily assigned.
o7
KrH
CI (2.8% isotopic abundance) were
Table II-l also exhibits the high spectral resolution
inherent in this technique.
within a couple of kHz.
All transition frequencies are good to
Subsequent to the KrHCl study, the resolution
was further increased by a fuller understanding of the instrumental
effects on the line shape and a slightly more sophisticated data handling
procedure.
132
35
XeH CI was ini28
tially assigned using the molecular beam electric resonance technique.
The rotational spectrum of the single isotope
The microwave spectrum of
132
35
XeH CI was re-examined in this lab to obtain
a more precise value of its centrifugal distortion constant which was
useful in determining the
131
Xe nuclear quadrupole coupling constant in
131
XeHCl (see Chapter IV). In addition, the microwave spectra of
129
35
129
37
129
35
132
35
XeH
CI,
XeH
CI,
XeD
CI and
XeD
CI were assigned and the
measured frequencies are listed in Table II-2. Using the rotational
constants, centrifugal distortion constants and CI nuclear quadrupole
coupling constants derived from the spectra, a more complete structure was
determined for XeHCl and greater information about the intermolecular
potential was obtained by referring to the results of isotopic substitution.
The analyses of the spectra, molecular structures and potentials of
KrHCl and XeHCl proceeded in an identical fashion for both complexes.
26
Table II-2.
Observed transition frequencies for XeHCl and those calculated
with Eq. (II-6) and the constants in Table II-3.
Isotope
129 v „35
XeH Cl
129 v TT37 ,
XeH Cl
129 v „35 ,
XeD Cl
132
XeH 3 5 Cl
132
J
35
XeD JJ Cl
J,F
J',F*
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
0.6
3,3/2,5/2
3,7/2,9/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2
5,11/2,13/2
6,9/2,11/2
6,13/2,15/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2
5,11/2,13/2
6,9/2,11/2
6,13/2,15/2
7,11/2,13/2
7,15/2,17/2
7951.4371
7952.4462
9939.1278
9939.7248
11926.1776
11926.5781
13912.6138
13912.8994
7951.4365
7952.4464
9939.1273
9939.7246
11926.1798
11926.5776
13912.6137
13912.8988
3,3/2,5/2
3,7/2,9/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2
5,11/2,13/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2
5,11/2,13/2
6,9/2,11/2
6,13/2,15/2
7621.9843
7622.7796
9527.2647
9527.7326
11431.9749
11432.2877
7621.9861
7622.7807
9527.2615
9527.7315
11431.9756
11432.2886
3,3/2,5/2
3,7/2,9/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2
5,11/2,13/2
6,9/2,11/2
6,13/2,15/2
4,5/2,7/2
4,9/2,11/2
5,7/2,9/2
5,11/2,13/2
6,9/2,11/2
6,13/2,15/2
7,11/2,13/2
7,15/2,17/2
7833.4321
7834.7333
9791.8363
9792.6065
11749.5909
11750.1065
13706.7637
13707.1307
7833.4320
7834.7338
9791.8355
9792.6060
11749.5927
11750.1059
13706.7631
13707.1309
4,5/2,7/2
4,9/2,11/2
6,9/2,11/2
6,13,2,15/2
7,11/2,13/2
7,15/2,17/2
5,7/2,9/2
5,11/2,13/2
7,11/2,13/2
7,15/2,17/2
8,13/2,15/2
8,17/2,19/2
9890.3078
9890.9044
13844.2962
13844.5786
15820.3100
15820.5230
9890.3096
9890.9054
13844.2926
13844.5770
15820.3107
15820.5251
-1.8
-1.0
6,9/2,11/2
6,13/2,15/2
7,11/2,13/2
7,15/2,17/2
8,13/2,15/2
8,17/2,19/2
7,11/2,13/2
7,15/2,17/2
8,13/2,15/2
8,17/2,19/2
9,15/2,17/2
9,19/2,21/2
13638.1782
13638.5446
15584.9436
15585.2190
17531.0282
17531.2446
13638.1784
13638.5447
15584.9431
15585.2191
17531.0286
17531.2444
-0.2
-0.1
-0.2
0.5
0.2
-2.2
0.5
0.1
0.6
-1.8
-1.1
3.2
1.1
-0.7
-0.9
0.1
-0.5
0.8
0.5
-1.7
0.6
0.6
-0.2
3.6
1.6
-0.7
-2.1
0.5
-0.1
-0.4
0.2
27
Table II-3.
Isotope
Spectroscopic Constants for KrHCl and XeHCl.
B (MHz)
Dj(kHz)
X (MHz)
82
KrH 3 5 Cl
1209.3075(6)
7.431(15)
-29.268(9)
84
KrH 3 5 Cl
1200.6255(7)
7.389(16)
-29.250(7)
86
KrH 3 5 ci
1192.3386(5)
7.273(12)
-29.253(6)
82
KrH 3 7 Cl
1164.7454(2)
6.935(5)
-23.235(33)
84
KrH 3 7 Cl
1156.0596(5)
6.795(12)
-23.106(6)
86
KrH 3 7 Cl
1147.7735(2)
6.691(4)
-23.139(8)
82
KrD 3 5 Cl
1192.4708(4)
6.612(8)
-40.842(14)
84
KrD 3 5 Cl
1183.7735(5)
6.530(12)
-40.805(6)
86
KrD 3 5 Cl
1175.4763(11)
6.489(23)
-40.849(14)
82„ „37„,
KrD Cl
1149.1999(8)
6.126(19)
-32.237(122)
84T, „37„,
KrD Cl
1140.5069(8)
6.026(17)
-32.211(9)
1132.2113(8)
5.909(18)
-32.232(119)
XeH 3 5 Cl
994.14484(12)
3.8130(15)
-34.761(42)
129
37
^yXeHJ/Cl
952.92241(38)
3.4927(63)
-27.359(94)
129 v „35„,
XeD Cl
979.40964(11)
3.4520(14)
-44.816(38)
86
KrD 3 7 Cl
129
132
XeH 3 5 Cl
989.26113(34)
3.7762(31)
-34.670(20)
132
XeD 3 5 Cl
974.50773(6)
3.4221(4)
-44.629(54)
U n c e r t a i n t i e s represent the standard deviation of the least squares f i t .
28
Good fits of the observed data were obtained by least squares fitting of
the spectra to the single nuclear spin energy levels given in Eq. (II-6).
Although deuterium nuclear quadrupole coupling gave rise to some line
broadening in the deuterated species, its effects on the energy levels
were ignored in this analysis as were the Cl spin-rotation interactions.
The spectroscopic constants for all isotopes of KrHCl and XeHCl are listed
in Table II-3.
The frequencies calculated with Eq. (II-6) and the con-
stants in Table II-3 are shown in Tables II-l and II-2 for KrHCl and
XeHCl, respectively, and, in general, the fits are very good.
The structures of KrHCl and XeHCl were derived from the measured
rotational and Cl nuclear quadrupole coupling constants using the method
outlined in Section II-A and known properties of HC1.
The appropriate
properties of free HC1 are listed in Table II-4.
Table II-4.
Isotope
H 3 5 C1
H
35C1
37C1
D^'Cl
D
a
Rotational constants, bond lengths and Cl
nuclear quadrupole coupling constants for
4 isotopic species of HC1
B 0 (MHz) a
r
312989.297
312519.121
161656.238
161183.122
1.28387
1.28386
1.28124
1.28123
0
(A)b
See Ref. 43.
Calculated from B .
c
See Ref. 44; X Q ( 37 C1) = x Q ( 35 Cl)/1.2688.
X Q (MHz) C
-67.61893
-53.294
-67.39338
-53.115
29
The structures of KrHCl and XeHCl are shown in Table II-5. All angles
are acute as will be shown below.
The structures of KrHCl and XeHCl show little change upon isotopic
subsitution at the rare gas and halogen positions.
A dramatic deuterium
isotope effect is observed, however, and merits some comment.
The
deuterated complexes show a much smaller bond angle, 6, and an increased
bond length, R , as compared to the hydrogen species.
Due to a fractional
mass change of near unity, there is a large difference in the manner that
the angle is averaged over the van der Waals bending vibration with the
result that H and D sample significantly different portions of the
potential surface.
Qualitatively, the lighter H is, on the average,
farther away from the heavy atom axis than D and, consequently, there is
less repulsion between the heavy atoms and they are allowed to approach
more closely.
The deuterium isotope effect in the structures of KrHCl and XeHCl
can be used, to some advantage, in testing the assumption that these
complexes are linear at equilibrium.
For a linear two dimensional iso-
tropic harmonic oscillator, Eq. (11-22) predicts that (9 2^ should scale
as li^"'2 for a given force constant.
Since u^ approximately doubles in
going from the H to D isotopes, the deuterated complexes should have
angles given by
e(XDCl) *fe (j^eOEHCl).
(II-25)
Using Eq. (11-25), 9 is predicted to be 31.7° for KrDCl and 29.0° for
XeDCl.
These values compare well with the measured values of 30.5° and
30
Table II-5. Structures of KrHCl and XeHCl where 9 is determined from x a -
Isotope
R
0(
A
)
9(Deg.)
82
KrH 35 Cl
4.1107
37.74
84
KrH 35 Cl
4.1106
37.75
86 35,,,
KrH Cl
4.1105
37.75
82
KrH 37 Cl
4.1101
37.63
84
KrH 37 Cl
4.1100
37.73
86
KrH 37 Cl
4.1099
37.70
82
KrD 3 5 C l
4.1253
30.52
84
KrD 35 Cl
4.1251
30.54
86
KrD 35 ci
4.1250
30.51
82
KrD 37 Cl
4.1248
30.50
84
KrD 37 Cl
4.1246
30.52
86
KrD 37 Cl
4.1245
30.50
W5C1
4.2753
34.50
129
XeH 3 7 Cl
4.2747
34.54
129
XeD 3 5 ci
4.2875
27.91
132
XeH 3 5 Cl
4.2752
34.56
132
XeD 3 5 Cl
4.2873
28.03
12
31
27.9 , respectively, considering the level of approximation.
It appears
that the most likely equilibrium configuration is linear and that the
measured angle results solely from vibrational averaging.
While the structures of KrHCl and XeHCl have been determined above
by resort to an analysis of the Cl nuclear quadrupole coupling, it is
possible, in principle, to determine the structures from rotational constants alone if the structure is unchanged upon isotopic substitution.
The problem encountered here is that the rotational constants are fairly
insensitive to the location of the hydrogen.
Since a large amount of
isotopic data was available for KrHCl, an attempt was made to derive the
structure of KrHCl from its rotational constants alone.
In this analysis,
it is assumed that the six hydrogen isotopes have identical structures as
do the six deuterium isotopes.
differ from each other, however.
The hydrogen and deuterium isotopes may
It is also assumed that the H-Cl dis-
tance is rigidly fixed at the bond length of free HC1.
To estimate the
45
molecular geometry, a grid search method
is employed whereby the rota-
tional constants are calculated for various assumed structures and the
minimum in f is sought, where
f =
i ( I oi- ¥ ci ) 2
1=1
'
(II
"26)
and where Bo0 and Bc are the observed and calculated rotational constants,
respectively.
Table II-6 lists typical results from this calculation
where the minimum of f and the corresponding bond length is given as a
function of 9.
The absolute minima were found at Ro=4.100A and 9=63.7
o
o
for KrHCl and at R =4.116A and 9=45.7 for KrDCl. While there are a range
32
Table II-6. Structure of KrHCl as
from rotational constants
alone.
6
0
°
\c-0.&
^CB^-B^CMHz )
iW-nw
rv.r ui
o
z
p
B
< c-r
ol
B
•) osaz )
ci
0
4.1163
.472
4.1326
.484
10
4.1159
.449
4.1317
.440
20
4.1146
.384
4.1293
.322
30
4.1125
.288
4.1252
.171
40
4.1096
.181
4.1195
.051
50
4.1059
.086
4.1123
.045
60
4.1015
.031
4.1037
.241
70
4.0965
.042
4.0939
.730
80
4.0909
.144
4.0830
1.581
90
4.0849
.355
4.0713
2.837
100
4.0787
.678
4.0593
4.496
110
4.0725
1.106
4.0472
6.507
120
4.0664
1.617
4.0355
8.767
130
4.0608
2.173
4.0247
11.13
140
4.0559
2.728
4.0151
13.42
150
4.0518
3.231
4.0072
15.44
160
4.0487
3.632
4.0013
17.03
170
4.0468
3.891
3.9977
18.05
180
4.0462
3.980
3.9965
18.40
a
r
= 1.28387A\
H-Cl
VCI
=
1,28124i
'
p
33
of angles that will adequately fit the rotational constants, it is clear
that H(D) is located between the heavy atoms.
It is also noted that the
large deuterium isotope effect is reproduced.
The relatively large
values of 9 are expected since an increasing angle will compensate for
the expected shortening of the Kr-Cl bond as substitution with heavier
isotopes occurs.
The fact that 9 is acute in XeHCl can also be established by referring to the results of Cl isotopic substitution.
Table II-7 shows the
129
Xe-Cl distances, R Q , calculated using both acute and obtuse choices
35
37
for 9. Since the fractional mass change in going from
Cl to
Cl is
very small, the expected isotope effect is small and since R
over the van der Waals stretch, R
the Cl mass increases.
is averaged
is expected to decrease slightly as
Clearly, as demonstrated in Table II-7, only the
acute choice for 9 fulfills both these conditions.
129
Table II-7. The bond lengths of
XeHCl for both acute and
obtuse choices for 6.
9 is acute
Isotope
R0(A)
129
4.2753
XeH 3 5 Cl
8 is obtuse
R 0 ( 35 C1)-R 0 ( 37 C1)A
R Q (R)
4.2162
0.0006
129 X e H 37 c l
4.2747
R o ( 35 Cl)-R 0 ( 37 Cl) (A)
-0.0025
4.2187
The harmonic bending and stretching force constants and frequencies
for KrHCl and XeHCl are shown in Table II-8.
Also shown in Table II-8
are the Lennard-Jones estimates of the effective radial potential well
OClCnCTiOOOOICOOOHOCOCTiCTiCTi
oo(~.t^r-»oooooo>crvooofricocotnin
H r l r l H H H N r H N N W N N N N N
tncooN<f<rsr<rn^coo>NN>j>jN
COrO<r«a--<rv>-HiHHHrHrHr^f^f^HH
vovovovovovDeooooooooooofOcocomm
O O O O O O O O O O O O C M C M C M C M C M
r^-i~-i^.ooococricoooo\oHa\oo>srHrHtHCMHrHinmvOvDUOvOHOi-ll^vO
CMCMCMCMCMCMcMCMCMCMCMCNCOcOcOCOcO
vOvf<rHl^rOCMvD-*cMvOi-<OCMCO<rcO
^j-^vrr^^fLnocriOocriOoovo^ocMoo
r~.r~r~rNp^r^i^vor^r~>£ii^.vovOvo<tco
•<r-*-<r-d-<r<rcocococococoininm<rvt
CTiOOHcMCMOOOH^mr-sOOCMCPiO
vor-.r^oococoorHHinminr».r^oooH
COCOCOCOCOCOHrHrHHHHCOCOCOHH
vOvOvOvOvOvOCMCMCNCMCMCNvOvOvOCMCM
HHrHHHHcoCOCOcOCOfOHHHCOCO
voinminvovOiHi-HOcMcMcoooHmm
inmmininmi^.r^r^f->.r^r-»oicrichOO
H H r H i - H H i H H H r H H H r H H H H C M C M
O O O O O O O O O O O O O O O O O
ooooooooooooooooo
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mcNicMoooor^r^vococMHOcor^cMP^r~CMCMCMiHHr-lrOCOCOCOCOCOrOcOCO-<r-tf
cocofocococococococococorococococo
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O N f f l i f l - T - J M N v O t n - 1 H W N v O M ( 0 CO
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CMcMCMCMCMCMCMeMcNCMCMCMCNCMcMCMcM
H H H H H r - H r - l H H H i - l r H O O U O U
o o o o o u u o a o o
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mininr^i^r^ininmr^rvr^cococococo
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SSCS3QQ
S S B H i S K P Q i a f l Q O i J t J I ) <y
U U U ^ u p j j U ^ U U
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^ i l ^ ^ W W W W W W W ! ^ Wov CM en o> CM
C M « a - v O C M ^ r ^ O C M < - v O CMO-vOCMCOCMCNrO
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35
depths and equilibrium bond lengths.
The striking observation, made
in examining the results of Table II-8, is that the potential parameters
exhibit a large difference between the H and D isotopes of a given
complex.
The true intermolecular, on the other hand, should be iso-
topically invariant within the Born-Oppenheimer approximation.
It
must be kept In mind that the potential parameters derived above do not
describe the true intermolecular interaction but only an averaged
effective potential.
They will be expected to vary with D substitution
due to strong coupling of the radial and angular potentials and the
greatly different angle averaging experienced by the two species. The
deuterated complexes, which undergo smaller amplitude bending vibrations,
are, on the average, sampling nearer to the potential minimum than H,
giving the appearance of a stronger bond.
In particular, the estimates
of the potential well depths made here cannot be expected to give more
than an order of magnitude estimate of the interaction energy.
They
still find use, however, in comparing the interaction energies of
similiar complexes.
D.
Argon Hydrogen Bromide and Krypton Hydrogen Bromide
The studies of KrHCl and XeHCl presented in the previous section,
along with the prior investigation of ArHCl, permit the rare gas-hydrogen halide Interaction to be examined as a function of the rare gas
atom.
To explore the interaction as a function of the halogen, studies
of the ArHBr and KrHBr van der Waals molecules were undertaken.
The
36
rotational spectra of four isotopic species of ArHBr and eight isotopic
species of KrHBr were assigned.
In broad outline, the information
obtained about ArHBr and KrHBr paralleled that derived from the spectra
of KrHCl and XeHCl.
However, additional small effects such as a centri-
fugal distortion of the Br nuclear quadrupole coupling constant and the
Br spin-rotation interaction were first observed here.
Their effects
demonstrate, once again, that small perturbations in the rotational
spectra can be studied using the Fabry-Perot Fourier transform spectrometer.
Finally, an analysis of the centrifugal distortion of the Br
nuclear quadrupole coupling constant yielded additional information about
the intermolecular potential.
In particular, it is a measure of the
coupling between the radial and angular motions.
Microwave transitions of ArHBr and KrHBr were measured m
quency band covering 8.0 to 12,4 GHz.
the fre-
The observed transition frequencies
for ArHBr and KrHBr are listed in Tables II-9 and 11-10, respectively.
Due to the high J transitions measured here (J=3+J=4 and J=4-*-J=5 for
ArHBr and J=5-»J=6 up to J=8^J=9 for KrHBr), no resolvable hyperfine
structure attributable to deuterium nuclear quadrupole coupling was
observed.
However, these high J transitions provided a unique opportunity
to study the centrifugal distortion of the Br nuclear quadrupole coupling
constant, an effect which was found to increase with J(J+1).
Poor signal-
to-noise resulting from isotopic dilution caused the second order and
centrifugal distortion effects of the Br nuclear quadrupole coupling to
be incompletely resolved for
82
86
KrHBr and
KrHBr.
In these cases, center
frequencies of the second order doublets are reported.
The estimated
Table II-9. Observed and Calculated Transition Frequencies for ArHBr.
Isotope
J,F
-*•
J',F'
Observed
(MHz)
Calculated3
(MHz)
Residue
(kHz)
Calculated'5
(MHz)
Residue
(kHz)
3,5/2
3,7/2
3,9/2
3,7/2
3,5/2
3,3/2
3,9/2
4,7/2
4,9/2
4,11/2
4,9/2
4,5/2
4,7/2
4,11/2
4,5/2
4,7/2
4,11/2
4,9/2
4,7/2
4,5/2
4,9/2
5,7/2
5,9/2
5,13/2
5,11/2
5,7/2
5,9/2
5,11/2
8810.4083
8833.8364
8848.8732
8848.8983
8853.8595
8853.8693
8892.2805
11019.0770
11047.4871
11059.5804
11059.6210
11062.5235
11062.5513
11103.0270
8810.4116
8833.8383
8848.8704
8848.8965
8853.8558
8853.8691
8892.2837
11019.0791
11047.4903
11059.5778
11059.6175
11062.5234
11062.5485
11103.0308
-3.3
-1.9
2.8
1.8
3.7
0.2
-3.2
-2.1
-3.2
2.6
3.5
0.1
2.8
-3.8
8810.4095
8833.8362
8848.8738
8848.8991
8853.8572
8853.8690
8892.2807
11019.0761
11047.4874
11059.5816
11059.6206
11062.5237
11062.5504
11103.0275
-1.2
0.2
-0.6
-0.8
2.3
0.3
-0.2
0.9
-0.3
-1.2
0.4
-0.2
0.9
-0.5
3,5/2
3,7
3,9/2
3,7/2
3,5/2
3,3/2
3,9/2
4,7/2
4,9/2
4,11/2
4,9/2
4,5/2
4,7/2
4,11/2
4,5/2
4,7/2
4,11/2
4,9/2
4,7/2
4,5/2
4,9/2
5,7/2
5,9/2
5,13/2
5,11/2
5,7/2
5,9/2
5,11/2
8744.2856
8763.8321
8776.4224
8776.4452
8780.5969
8780.6036
8812.7075
10935.0812
10958.8068
10968.9300
10968.9656
10971.3934
10971.4186
11005.2501
8744.2856
8763.8344
8776.4203
8776.4432
8780.5968
8780.6017
8812.7098
10935.0827
10958.8084
10968.9300
10968.9626
10971.3939
10971.4172
11005.2520
-1.5
-2.3
2.1
2.0
0.1
1.9
-2.3
-1.5
-1.6
0.0
3.0
-0.5
1.4
-1.9
8744.2843
8763.8331
8776.4224
8776.4448
8780.5976
8780.6016
8812.7079
10935.0808
10958.8066
10968.9313
10968.9645
10971.3941
10971.4184
11005.2500
0.2
-1.0
0.0
0.4
-0.7
2.0
-0.4
0.4
0.2
-1.3
1.1
-0.7
0.2
0.1
Table II-9. Continued
Isotope
ArD79Br
ArD81Br
a
J,F
-y
J',F'
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
3,5/2
3,7/2
3,7/2,9/2
3,5/2
3,3/2
3,9/2
4,7/2
4,9/2
4,11/2
4,9/2
4,7/2
4,5/2
4,11/2
4,5/2
4,7/2
4,9/2,11/2
4,7/2
4,5/2
4,9/2
5,7/2
5,9/2
5,13/2
5,11/2
5,9/2
5,7/2
5,11/2
8634.4250
8672.0002
8695.8376
8703.6936
8703.7919
8765.0150
10804.8226
10850.2344
10869.4041
10869.4298
10874.0712
10874.0890
10938.6084
8634.4277
8672.0015
8695.8339
8703.6918
8703.7915
8765.0169
10804.8238
10850.2377
10869.4032
10869.4242
10874.0713
10874.0879
10938.6084
-2.7
-1.3
3.7
1.8
0.4
-1.9
-1.2
-3.3
3,5/2
3,7/2
3,7/2,9/2
3,5/2
3,3/2
3,9/2
4,7/2
4,9/2
4,11/2
4,9/2
4,7/2,9/2
4,11/2
4,5/2
4,7/2
4,9/2,11/2
4,7/2
4,5/2
4,9/2
5,7/2
5,9/2
5,13/2
5,11/2
5,9/2,11/2
5,11/2
8573.5899
8604.9064
8624.8826
8631.4705
8631.5348
8682.6943
10726.5936
10764.4880
10780.5517
10780.5727
10784.4712
10838.3841
8573.5927
8604.9072
8624.8791
8631.4684
8631.5352
8682.6960
10726.5960
10764.4900
10780.5498
10780.5690
10784.4681
10838.3884
Calculated15
(MHz)
Residue
(kHz)
5.6
-0.1
1.1
-3.2
8634.4263
8672.0001
8695.8370
8703.6933
8703.7918
8765.0147
10804.8212
10850.2352
10869.4065
10869.4269
10874.0729
10874.0882
10938.6055
-1.3
0.1
0.6
0.3
0.1
0.3
1.4
-0.8
-2.4
2.9
-1.7
0.8
-0.3
-2.8
-0.8
3.5
2.1
-0.4
-1.7
-2.4
-2.0
1.9
3.7
3.1
-4.3
8573.5914
8604.9060
8624.8823
8631.4701
8631.5353
8682.6934
10726.5929
10764.4874
10780.5529
10780.5719
10784.4708
10838.3854
-1.5
0.4
0.3
0.4
-0.5
0.9
0.7
0.6
-1.2
0.8
0.4
-1.3
0.9
Calculated from Eq. (II-6) without the spin-rotation interaction and the data in Table 11-11.
^Calculated with spin-rotation interaction.
U3
00
39
Table 11-10.
Observed transition frequencies for KrHBr and those
calculated with Eq. (II-6) and the constants in Table 11-12.
Isotope
82
84
86
KrH 7 9 Br
KrH ? 9 Br
KrH 7 9 Br
J,F
->•
J',F'
Observed
(MHz)
Calculated
(MHz)
5,11/2,13/2
5,7/2,9/2
6,13/2,15/2
6,9/2,11/2
7,15/2,17/2
7,11/2,13/2
6,13/2,15/2
6,9/2,11/2
7,15/2,17/2
7,11/2,13/2
8,17/2,19/2
8,13/2,15/2
8304.6126
8307.1662
9688.0939
9689.9217
11071.0618
11072.4406
8304.6127
8307.1666
9688.0931
9689.9218
11071.0629
11072.4398
5,13/2
5,11/2
5,9/2
5,7/2
6,15/2
6,13/2
6,9/2
6,11/2
7,17/2
7,15/2
7,11/2
7,13/2
8,19/2
8,17/2
8,13/2
8,15/2
6,15/2
6,13/2
6,11/2
6,9/2
7,17/2
7,15/2
7,11/2
7,13/2
8,19/2
8,17/2
8,13/2
8,15/2
9,21/2
9,19/2
9,15/2
9,17/2
8207.6450
8207.6570
8210.2011
8210.2126
9574.9845
9575.0017
9576.8177
9576.8259
10941.8204
10941.8467
10943.2014
10943.2204
12308.1250
12308.1578
12309.1972
12309.2278
8207.6449
8207.6571
8210.2021
8210.2085
9574.9834
9575.0029
9576.8173
9576.8270
10941.8223
10941.8477
10943.2021
10943.2219
12308.1232
12308.1537
12309.2004
12309.2274
5,11/2,13/2
5,7/2,9/2
6,13/2,15/2
6,9/2,11/2
7,15/2,17/2
7,11/2,13/2
8,17/2,19/2
8,13/2,15/2
6,13/2,15/2
6,9/2,11/2
7,15/2,17/2
7,11/2,13/2
8,17/2,19/2
8,13/2,15/2
9,19/2,21/2
9,15/2,17/2
8115.1330
8117.6909
9467.0846
9468.9074
10818.5304
10819.9159
12169.4657
12170.5371
8115.1349
8117.6902
9467.0801
9468.9097
10818.5350
10819.9126
12169.4634
12170.5392
Residue
(kHz)
-0.1
-0.4
0.8
-0.1
-1.1
0.8
0.1
-0.1
-1.0
4.1
1.1
-1.2
0.4
-1.1
-1.9
-1.0
-0.7
-1.5
1.8
4.1
-3.2
0.4
-1.9
0.7
4.5
-2.3
-4.6
3.3
2.3
-2.1
40
Table 11-10.
Isotope
82
84
86
KrH 8 1 Br
KrH 8 1 Br
KrH 8 1 Br
Continued
J',F'
Observed
(MHz)
Calculated
(MHz)
Res idue
(kHz)
5,11/2,13/2
5,7/2,9/2
6,13/2,15/2
6,9/2,11/2
8,15/2,17/2
8,11/2,13/2
6,13/2,15/2
6,9/2,11/2
7,15/2,17/2
7,11/2,13/2
9,17/2,19/2
9,13/2,15/2
8201.5542
8203.6992
9567.8264
9569.3653
10933.6016
10934.7620
8201.5547
8203.7018
9567.8246
9569.3622
10933.6039
10934.7617
-0.5
-2.6
5,13/2
5,11/2
5,7/2,9/2
6,15/2
6,13/2
6,9/2
6,11/2
7,17/2
7,15/2
7,11/2
7,13/2
8,19/2
8,17/2
8,13/2
8,15/2
6,15/2
6,13/2
6,9/2,11/2
7,17/2
7,15/2
7,11/2
7,13/2
8,19/2
8,17/2
8,13/2
8,15/2
9,21/2
9,19/2
9,15/2
9,17/2
8104.5871
8104.5966
8106.7341
9454.7158
9454.7330
9456.2522
9456.2618
10804.3706
10804.3894
10805.5270
10805.5432
12153.5098
12153.5327
12154.4136
12154.4356
8104.5847
8104.5949
8106.7312
9454.7172
9454.7328
9456.2541
9456.2628
10804.3720
10804.3920
10805.5287
10805.5447
12153.5088
12153.5327
12154.4119
12154.4334
2.4
1.7
2.9
5,11/2,13/2
5,7/2,9/2
6,13/2,15/2
6,9/2,11/2
7,15/2,17/2
7,11/2,13/2
8,17/2,19/2
8,13/2,15/2
6,13/2,15/2
6,9/2,11/2
7,15/2,17/2
7,11/2,13/2
8,17/2,19/2
8,13/2,15/2
9,19/2,21/2
9,15/2,17/2
8012.0718
8014.2148
9346.8145
9348.3458
10681.0831
10682.2376
12014.8492
12015.7498
8012.0736
8014.2146
9346.8123
9348.3455
10681.0836
10682.2381
12014.8487
12015.7504
J,F
-»•
1.8
3.1
-2.3
0.3
-1.4
0.2
-1.9
-1.0
-1.4
-2.6
-1.7
-1.5
1.0
0.0
1.7
2.2
-1.8
0.2
2.2
0.3
-0.5
-0.5
0.5
-0.6
41
Table 11-10.
Continued
Isotope
J,F
+
J',F'
Observed
(MHz)
Calculated
(MHz)
5,11/2
5,13/2
5,9/2
5,7/2
6,13/2,15/2
6,11/2
6,9/2
7,17/2
7,15/2
7,11/2,13/2
8,19/2
8,17/2
8,13/2
8,15/2
6,13/2
6,15/2
6,11/2
6,9/2
7,15/2,17/2
7,13/2
7,11/2
8,19/2
8,17/2
8,13/2,15/2
9,21/2
9,19/2
9,15/2
9,17/2
8107.1076
8107.0990
8110.6176
8110.6571
9457.9654
9460.5050
9460.4878
10808.3710
10808.3830
10810.2820
12158.3028
12158.3151
12159.7912
12159.8039
8107.1055
8107.1000
8110.6153
8110.6575
9457.9676
9460.5055
9460.4902
10808.3708
10808.3813
10810.2813
12158.3013
12158.3170
12159.7930
12159.8019
5,11/2,13/2
5,9/2
5,7/2
6,13/2,15/2
6,11/2
6,9/2
7,17/2
7,15/2
7,11/2,3/2
8,19/2
8,17/2
8,13/2
8,15/2
6,13/2,15/2
6,11/2
6,9/2
7,15/2,17/2
7,13/2
7,11/2
8,19/2
8,17/2
8,13/2,15/2
9,21/2
9,19/2
9,15/2
9,17/2
8005.9785
8008.9313
8008.9566
9339.9358
9342.0544
9342.0643
10673.4468
10673.4627
10675.0560
12006.5084
12006.5230
12007.7588
12007.7711
8005.9811
8008.9330
8008.9592
9339.9343
9342.0542
9342.0608
10673.4477
10673.4599
10675.0527
12006.5086
12006.5251
12007.7600
12007.7716
Residue
(kHz)
2.1
-1.0
2.3
-0.4
-2.2
-0.5
-2.4
0.2
1.7
0.7
1.5
-1.9
-1.8
2.0
-2.6
-1.7
-2.6
1.5
0.2
3.5
-0.9
2.8
3.3
-0.2
-2.1
-1.2
-0.5
42
no
frequency uncertainty is 5 kHz for the
Qfl
Kr and
Kr species.
The observed data were fit to the ground vibrational state, K=0
symmetric top energy levels given by Eq. (II-6).
In the first analysis,
the Br spin-rotation constant was assumed to be zero.
It was necessary,
however, to make the Br nuclear quadrupole coupling constant, xa>
a
function of J and Eq. (II-6) was modified by replacing X with X ao +D v J(J+l),
where X ao is Che coupling constant extrapolated to the J=0 state and Dv
represents the centrifugal distortion of X a .
The need for D Y became
readily apparent in trying to assign the spectra.
Figure II-2 compares
the observed AF=+1 hyperfine structure for the J=4->J=5 transition in
79
ArH Br with that calculated with and without D... As can be seen,
there is a large difference in the calculated splittings, especially for
the F=ll/2->F=13/2 and F=9/2-*F=ll/2 pair.
is also reversed for both pairs.
The ordering of the transitions
In addition to the effects on the AF=+1
transitions, the AF=0 frequencies are shifted by up to 120 kHz.
The
J(J+1) dependence of the energy levels upon Dy as well as its interpretation m
terms of vibrational bend-stretch interaction constant will be
explained below.
The spectroscopic constants, as determined from the least squares
fit of the data to Eq. (II-6) without the Br spin-rotation interaction,
are listed in Table 11-11, Column I, for ArHBr and Table 11-12 for KrHBr.
The transition frequencies calculated with these constants and Eq. (II-6)
are listed in Tables II-9 and 11-10, respectively.
The quoted uncertain-
ties represent the standard deviations from the least squares fit.
Upon
comparing the measured and calculated frequencies for ArHBr in Table
11-11, Column I, a clear trend emerges for all isotopes; all AF=0 lines
PORTION OF OBSERVED J = 4 - ^ J = 5 SPECTRUM IN ArHBr
11/2-13/2
9/2-11/2
5/2-7/2 7/2-9/2
j
,
/-
Calculated
With Dv
V
/ •
\
/
v
\
/
\
A
I
11059.57
Figure II-2.
/
A
Calculated
Without Dv
W
11059.63
/
/
V
i
1062.52
11062.56
The effect of the centrifugal distortion of the Br nuclear quadrupole coupling
constant on the AF = + 1 hyperfine structure of the J=4-KJ=5 transition in ArHBr.
.eu>
44
Table 11-11.
Spectroscopic constants for ArHBr.
Column I is the fit
without and Column II is the fit with the Br spin-rotation
interaction.
ArH 79 Br
I
B0(MHz)
Dj(kHz)
X (MHz)
D„(kHz)
II
1106.66928(48)
12.397(11)
173.204(17)
18.25(74)
1106.66951(17)
12.397(4)
173.199(6)
18.42(26)
1.09(13)
ctkHz)
ArD
1087.50854(48)
8.494(11)
276.227(17)
14.58(74)
Table 11-12.
Isotope
8
84
W 9 7B9 r
KrH Br
86KrH79Br
82
KrH 8 1 Br
KrH 8 1 Br
86
KrH 8 1 Br
84
KrD 7 9 Br
84KrD81Br
84
a
I
II
1097.58095(31)
12.226(7)
144.796(11)
15.10(48)
79
Br
1097.58109(15)
12.226(3)
144.793(5)
15.21(23)
0.67(12)
ArD8 1 Br
II
I
BQ(MHz)
Dj(kHz)
Xa (MHz)
Dy(kHz)
c(kHz)
ArH8 1 Br
:L087.50889(27)
8.497(6)
276.223(9)
14.73(41)
0.94(20)
I
1078.59281(51)
8.350(11)
230.884(20)
12.00(87)
II
1078.59315(22)
8.352(5)
230.878(8)
12.26(37)
1.02(16)
Spectroscopic constants for KrHBr.
B 0 (MHz)
692.31400(14)
684.22952(15)
676.51549(36)
683.71030(41)
675.62442(14)
667.91018(13)
675.84862(14)
667.40217(20)
Dj(kHz)
2.6579(14)
2.5973(11)
2.5364(30)
2.6003(40)
2.5266(11)
2.4634(10)
2.1900(10)
2.1384(15)
Xa(MHz)
228.18(9)
228.26(14)
228.41(36)
191.26(30)
190.78(14)
190.78(14)
318.18(14)
265.85(19)
D x (kHz) a
(7.57)
7.57(29)
(7.57)
(5.88)
5.88(28)
(5.88)
5.05(30)
4.76(40)
Values of D Y in parentheses were fixed in the least squares fit.
45
are shifted down in frequency relative to the AF=+1 components.
This
trend is characteristic of the spin-rotation, thus, a second fit of the
ArHBr data was made in which the Br spin-rotation constant was included
in Eq. (II-6).
The results of this fit are shown in Column II of Table
11-11 and the frequencies calculated with these constants and Eq. (II-6)
are listed in Table II-9. The measured spin-rotation constants are
generally about 1 kHz, implying a shift in the J=4>J=5, AF=0 lines of
about 5 kHz.
Since this is approximately the uncertainty in the fre-
quency measurements, c is imprecisely determined.
However, its value
is close to that which would be predicted on the basis of the molecular
structures as shown below.
The KrHBr data were not fit with a spin-
rotation constant since the value of the constant is expected to be
smaller and its effects on the AF=+1 transitions measured in KrHBr much
less pronounced.
In this case, the spin-rotation interaction was not
expected to add meaningfully to the fit.
The vibrationally averaged structures for ArHBr and KrHBr were
determined, in the usual way, from the rotational constants, Br nuclear
quadrupole coupling constants and known properties of HBr.
The appro-
priate free molecular properties of HBr are listed in Table 11-13, and
the structures of ArHBr and KrHBr are shown in Table 11-14.
The struc-
tures of ArHBr and KrHBr listed in Table 11-14 indicate that 6 is acute.
This was determined by examining the effects of Br isotopic substitution.
Table 11-15 compares the structures of ArHBr and KrHBr computed assuming
a (and thus
0) is acute in one case and obtuse in the other.
Clearly,
only for the acute choice does the isotope effect have the proper magnitude
46
Table 11-13.
Isotope
H
81Br
H
7 9 BB rr
D
«i
D 81 Br
a
Spectroscopic and structural properties of HBr.
,A b
B (MHz)*
250358.510
250280.582
127357.639
127279.757
1.42426
1.42426
1.42144
1.42144
X0(MHz)
c Br (kHz) v
532,.30590
444,,68066
530,,6315
443.,2799
290.796
313.370
145.82
157.26
See Ref. 43.
Calculated from B .
See Ref. 46 for HBr constants and Ref. 47 for DBr constants.
Table 11-14.
Structures of ArHBr and KrHBr.'
Isotope
a(deg)
O(deg)
R0(A)
ArH 79 Br
ArH 81 Br
ArD 79 Br
ArD ai Br
82
KrH 7 9 Br
84
KrH 7 9 Br
86KrH79Br
42.12
42.11
34.43
34.42
38.11
38.10
38.09
38.05
38.10
38.10
31.11
31.10
42.08
42.07
34.38
34.38
38.04
38.03
38.02
37.98
38.02
38.02
31.00
31.00
4.1464
4.1463
4.1820
4.1819
4.2573
4.2572
4.2571
4.2572
4.2571
4.2570
4.2809
4.2808
82
KrH°J-Br
86
KrH 8 jBr
84
KrD 7 9 Br
KrD 8 1 Br
84
1
rR_Br=l.42426 A, rD_Br=l.42144 A.
'See Figure II-l.
47
Table 11-15.
Comparison of the Br Isotope Effect in the Structures of
ArHBr and KrHBr when a is assumed acute and obtuse.
Acute Angle
Isotope
RQ(A)
ArH 79 Br
4.1464
Obtuse Angle
R0(79Br)-R0(81Br)(A)
R0(X)
4.1198
-0 .0006
0.0001
ArH 8L Br
4.1463
ArD 79 Br
4.1820
82
4.1204
4.1238
-0 .0003
0.0001
Q1
ArD
Br
KrH 7 9 Br
4.1819
4.1251
4.2573
4.2291
-0 .0005
0.0001
82
KrH 8 1 Br
4.2572
84
KrH 7 9 Br
4.2572
4.2296
4.2290
-0 .0005
0.0001
84
KrH 8 1 Br
4.2571
86
KrH 7 9 Br
4.2571
4.2295
4.2288
-0 .0006
0.0001
86
KrH 8 1 Br
4.2570
84
KrD 7 9 Br
4.2809
4.2294
4.2204
-0 .0013
0 .0001
84
KrD
81
Br
4.2808
R0(79Br)-Ro(8lBr)(A)
4.2217
48
and sign (see Section II-B).
The ArHBr and KrHBr structures reported in Table 11-14 are appropriate to the ground vibrational and J=0 rotational states of the respective complexes.
Since the bond angles are determined from x a and x a is
a function of J, there are small but measurable changes in 9 as J increases.
The Br nuclear quadrupole coupling constant for any state J is
given by
*aJ
= X
ao
+ D
J(J+1)
X
'
(11-27)
For the J=5 level of ArH 79 Br, xa=173.752 MHz as compared to x a =17 3.199
MHz for J=0.
0.03°.
This difference translates into a decrease in 8 of about
While this structural change is small, it has a dramatic effect
on the transition
frequencies and must be accounted for m
any potential
surface derived for these systems.
It is also of interest to compare the measured spin-rotation constants of HBr with those predicted on the basis of the structures derived
above.
It is again assumed that the electronic properties of HBr are
unchanged upon complex formation.
The spin-rotation constant is, in
33
general, proportional to the inverse moment of inertia of a molecule,
thus, the free HBr spin-rotation constant, c Q , must be reduced by
Io/lD=b/B0, where I D and b are the moment of inertia and rotational constant of free HBr and I 0 is the average of the moments of inertia perpendicular to the a-axis of the complex.
Also, since only the average of
the two perpendicular components of the spin-rotation tensor are measured,
the average of the projections of c 0 onto the b- and c-axes of the complex
49
must be taken.
c =
Following the method of Appendix II,
-^-{l+cos 2 a>
.
(11-28)
Table 11-16 lists the measured and predicted values of c for ArHBr.
Table 11-16.
In
Measured values of the spin-rotation constant
of ArHBr and those calculated with Eq. (11-28),
Isotope
c(measured)(kHz)
ArH 79 Br
ArH°J;Br
ArD 79 Br
ArD 81 Br
1.09
0.67
0.94
1.02
(13)
(12)
(20)
(16)
c(calculated)(kHz)
1.00
1.06
1.05
1.12
Q-l
all cases,
save ArH
Br, the measured and predicted values are the same
within the standard deviations from the least squares fit.
It must be
emphasized, however, that due to the small size of c, the measured values
should be considered more uncertain than the standard deviations indicate.
The harmonic, bending and stretching, force constants and frequencies
for ArHBr and KrHBr as well as the Lennard-Jones estimates of their
effective radial potentials are given in Table 11-17.
These were calcu-
lated according to the prescriptions of Section II-A and the related
discussion involving KrHCl and XeHCl applies equally well here.
The new effect observed in this study which had not previously been
seen in the rare gas-HCl complexes is the centrifugal distortion, D^, of
the halogen nuclear quadrupole coupling constant, X a .
D\ and the J(J+1)
Table 11-17.
Isotope
Harmonic Force Constants and Frequencies and Lennard-Jones parameters for ArHBr and KrHBr.
ys(amu)
o ,
ks(mdyn-A-J-)
i
vs/c(cm~-L)
o^
ub(amu A'1)
<
o
k^mdyn-A)
_i
v,/ (cm -1)
—l
e(cm x )
o
R e (A)
79
Br
26.64174
0.0076
22.1
2.027210
0.001136
30.8
89
4.0816
ArH 81 Br
26.86010
0.0076
21.9
2.027483
0.001136
30.8
89
4.0817
26.75261
0.0106
26.0
4.001671
0.001291
23.4
126
4.1280
26.96738
0.0106
25.9
4.002752
0.001291
23.4
127
4.1282
KrH 79 Br
40.45381
0.0132
23.6
2.030124
0.001698
37.7
165
4.2191
VKrHU 7 9 Br
R
40.93517
0.0132
23.4
2.030190
0.001700
37.7
165
4.2192
ArH
79
ArD Br
ArD 81 Br
82
8
86
KrH 79 Br
41.40519
0.0133
23.3
2.030253
0.001702
37.7
165
4.2193
82
KrH 81 Br
40.95941
0.0132
23.4
2.030405
0.001708
37.8
164
4.2192
84
KrH 81 Br
41.45295
0.0133
23.3
2.030472
0.001700
37.7
165
4.2194
86
KrH 81 Br
41.93501
0.0133
23.2
2.030535
0.001700
37.7
166
4.2195
84
KrD 79 Br
41.19752
0.0152
25.0
4.013155
0.001948
28.7
192
4.2458
84
KrD 81 Br
41.70903
0.0152
24.9
4.014269
0.001948
28.7
192
4.2459
51
dependence of the energy levels on D
can be explained in terms of
coupling between the radial and angular potentials.
Consider a simple
potential consisting of harmonic terms in the radial and angular coordinates, as assumed before, along with a single bend-stretch interaction
term,
v(R,e)=|ks(R-Re)2 + ike 2 + lk I e 2 (R-R e ),
(11-29)
where the measured bending force constant is given by
2
kb - |g|- = k + kx(R-Re) .
(11-30)
Consider, a l s o , t h e change i n x a with R.
From Eq. ( I I - 7 ) , assuming 9^a,
&* £ - 3 x 0 c o s 9 s i n 9 ^ | .
(11-31)
Using t h e chain r u l e and r e a l i z i n g
h2
/
0
a
and
/~2~j
4ir^u b k b
dk b
IT
dR
that
=
k
l
,
(II-32)
l
we have i n terms of v a r i a t i o n s
|-tibe5kIcosesine6R.
<5X =
(11-33)
4h
6R is easily related to J using the simple diatomic theory of centrifugal
distortion by
38
R D
o J
6R = ^ - J ( J + l )
^Bo
(11-34)
thus,
5x
=—p ° °
8TTB0
J
i
e5cos9sin9J(J+l)
(11-35)
52
and
3u, X R Djk
D O o j i e5cos6slne>
(11-36)
8h z B Q
*
To test Eq. (11-36), a rough calculation of the expected value of DY for
A
79
ArH
79
Br and KrH
Br can be made.
To a first approximation, k-r = dkb/dR
can be estimated from the changes in k b and R Q upon deuterium substitution.
D
is predicted to be 13.0 kHz for ArH
Br and 6.4 kHz for KrH 79 Br.
A
These values compare very favorably with the measured values of 18.4 and
7.6 kHz, respectively, considering the crudeness of the calculation.
It
is also noted that upon °^-Br substitution, Dv is reduced by the rctio of
X (H 9 Br)/x(H 81 Br) as is predicted by this model.
The absence of signi-
ficant effects in KrHCl and XeHCl is consistent with the roughly ten-toone ratio in magnitude of the quadrupole coupling constants in free HBr
and HC1, respectively.
In conclusion, D.. appears to be a sensitive
measure of the coupling between the angular and radial potentials and
should be of great help in determining the true potential surfaces for
these
molecules.
E.
Argon
Deuterium Fluoride
In the previous two sections, it has been shown that to determine
the structures of complexes containing HC1 or HBr, extensive use must be
made of the halogen nuclear quadrupole coupling constant.
For complexes
containing HF, however, this information is unavailable since H
a quadrupolar nucleus.
19
F lacks
In this section.the assignment of the rotational
spectrum of ArDF is reported.
The structure of ArDF is obtained from
53
the rotational constant and analyses of the DF nuclear spin-spin and D
nuclear quadrupole coupling constants.
These constants are found to
yield structural information consistent with one another and equivalent
to that obtained from the halogen coupling constants in the corresponding
HC1 and HBr complexes.
In addition, the rotational spectrum of ArHF has
been re-examined and the structure of ArHF determined with help of the
HF spin-spin constant.
The observed transition frequencies for ArDF and ArHF are listed in
Table 11-18.
Figure II-3 shows the time domain record for the J=C->-J=l
transition in ArDF and Figure II-4 is the corresponding frequency spectrum.
The hyperfine structures of the J=0+J=1 transitions indicate the
presence of D nuclear quadrupole coupling in ArDF and the H(D)F nuclear
spin-spin interaction m
both complexes.
The spin-spin interaction can
be described by the classical interaction between two magnetic dipoles
• .u
_3(m-&)Qt2-$.
H s s = R 3 Ul ^2
Jl
where R is the vector from nucleus 1 to nucleus 2.
(H-37)
The nuclear magnetic
moments, U J , are related to the nuclear g-values and angular momenta,1^,
by
JJ± = "N»l^i
(II
'
where u^ is the nuclear magneton.
d = y N 2 g 1 g 2 /R 3
,
~38)
Defining the spin-spin constant, d, as
•
(11-39)
54
Table 11-18.
Isotope
Observed and calculated transition frequencies for ArDF
and ArHF.
J,F, ,F ->• J',Fj_,F'
Label
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
a
6079.3690
6079.3700
-1.0
b
6079.4525
6079.4526
-0.1
c
6079.4628
6079.4611
1.7
d
6079.5110
6079.5112
-0.2
e
6079.5283
6079.5287
-0.4
12157.4101
12157.4095
0.6
12157.5037
12157.5043
-0.6
6131.1112
6131.1111
0.1
6131.1355
6131.1361
-0.6
6131.1420
6131.1415
0.5
12260.5707
12260.5707
0.0
for
Figure
II-4
ArDF
0,1,3/2
1,0,1/2
0,1,1/2
1,0,1/2
0,1,3/2
1,2,3/2
0,1,1/2
1,2,3/2
0,1,3/2
1,2,5/2
0,1,3/2
1,1,3/2
0,1,1/2
1,1,3/2
0,1,3/2
1,1,1/2
0,1,1/2
1,1,1/2
1,1,3/2
2,1,3/2
1,1,1/2
2,1,1/2
1,2,3/2
2,3,5/2
1,2,5/2
2,3,7/2
0,1/2,1
1,1/2,1
0,1/2,0
1,1/2,1
0,1/2,1
1,3/2,1
0,1/2,0
1,3/2,1
0,1/2,1
1,3/2,2
1,1/2,1
2,3/2,2
1,3/2,1
2,5/2,2
1,3/2,2
2,5/2,3
}
}
}
}
}
ArHF
}
}
240
TIME (^s)
Figure I I - 3 .
Time domain record of the J=0-KJ=1 transition in ArDF. The signal was
digitized at 1 us per point and averaged over 10 pulses.
Ul
Ul
6079.350
6079.525 MHz
Figure II-4. Frequency spectrum corresponding to the time domain in Figure II-3.
1.95 kHz per point and +/- label the Doppler pairs.
The resolution is
Ul
57
Ramsey
34
has shown that
(II-40)
H
+
ss - (2J-1K2J+3) K V ^ V ^
3<l2'J)(Il-J> " 2 ^ 1 ' ^ ) J 2 ] •
The total Hamiltonian used to fit the data for ArDF is given by Eqs.
(II-l)-(II-3).
For ArHF, the same Hamiltonian is used with the exception
that no quadrupole term is included.
The data were fit using a nonlinear
least squares routine involving diagonalization of the Hamiltonian matrix
as described in Section II-A. A preliminary fit was made by assuming
that the spin-rotation constants, c^, were zero. The structures of ArDF
and ArHF were then roughly determined and the spin-rotation constants
calculated from Eq. (11-28).
For the final fit, the c L were held fixed
at these calculated values. The derived spectroscopic constants for
ArDF and ArHF are listed in Table 11-19.
Table 11-19.
B0(MHz)
Dj(kHz)
Xa(kHz)
da(kHz)
c„(kHz) b
c£(kHz) b
Spectroscopic constants for ArDF and ArHF .
ArDF
ArHF
3039.8529(4)
59.53(6)
194.7(20)
12.4(11)
-0.05
1.25
3065.7099(4)
70.90(6)
49.0(19)
-0.3
1.2
Uncertainties represent one standard deviation from the
least squares fit.
b'Held fixed in the least squares fit,
58
As with the halogen nuclear quadrupole coupling constants in the HC1
and HBr complexes, the measured spin-spin and D nuclear quadrupole coupling
constants in ArDF are simply those components, along the a-inertial axis,
of the respective second rank tensors.
Making the usual assumption that
all properties of DF are preserved on complex formation, the D nuclear
quadrupole coupling constant, X a , is related to that in free DF, X 0 , by
X a = i<3cos2a-l> x 0 .
(11-41)
Similarly, the measured spin-spin constant, d , is given by
d a = Y<3COS 2 CX-1> d0.
(11-42)
The appropriate properties of free DF and HF are listed in Table 11-20.
Table 11-20.
Spectroscopic constants for HF and DF.
HF
»
X 0 (kHz) d
d(kHz) d
c H (kHz) d
c F (kHz) d
DF
616365.6(4)a
0.925595
143.375(25)
-71.128(24)
307.637(20)
aSee Ref. 48.
c
Calculated from
b
d
See Ref. 49.
See Ref. 43.
325585.0(3)b
0.923243
354.238(78)
22.170(35)
-5.755(19)
158.356(45)
V
Since both X a and d a are measured in ArDF, two values for a are
obtained, a=33.2(2)° from X a and ct=32.7(22)° from da.
As shown
59
by Eq. (11-39), the free DF value of the spin-spin constant can be calculated from the known properties of DF and depends on (l/r-^), where r is
the DF internuclear separation. X 0 , on the other hand, depends on the
electric field gradient, q, at the deuterium nucleus.
Since the two values
of a are the same within experimental uncertainty and it is unlikely that
two such diverse properties as (l/r^) and q would change by the same
amount on complex formation, it appears that the assumption made above
holds for these quantities within the resolution of this experiment.
Having determined a from either X a or d a , the structures for ArDF
were derived in the usual manner (see Section II-A). The structures of
ArDF and ArHF are shown in Table 11-21.
Table 11-21.
6(deg) b
R0(A)b
9(deg) c
R0(A)C
a
It is noted that unlike in the
Structures of ArDF and ArHFa.
ArDF
ArHF
32.2(21)
3.5352(15)
32.7(2)
3.5349(2)
41.1(5)
3.5445(2)
See Figure II-l; r D F = 0.923243A, r H F = 0.925595A.
Spin-spin interaction used to determine the angle.
C
D quadrupole coupling used to determine the angle.
HC1 and HBr complexes, the Ar-F distance, R Q , decreases upon D substitution.
A possible explanation for this phenomenon is that the
lighter HF molecule is undergoing much larger amplitude vibration in
60
the radial mode than is the corresponding HCl or HBr complex.
The motion
is consequently more complicated and the arguments made for HCl and HBr
no longer hold.
In general, the heavier of two isotopic species will
exhibit the shorter bond.
With the greatest fractional mass change upon
D substitution occuring in HF, it is likely that the mass effects more
than compensate for the repulsion effects seen in the HCl and HBr species
and result in the shortening of the bond.
The harmonic force constants and frequencies and Lennard-Jones potential parameters for ArHF and ArDF are listed in Table 11-22.
Table 11-22.
These require
Harmonic force constants and frequencies and
Lennard-Jones potential parameters for ArDF
and ArHF.
ks(mdyn/A)
v s / c (cm" )
ArDF
0.0170
45.8
ArHF
0.0142
42.5
vjb/,c(cm_1)
Rg(A)
e(cm -1 )
0.00392
64.1
3.463
143.0
0.00301
78.1
3.465
119.2
kb(mdyn-A)
no comments other than those already made with respect to KrHCl and
XeHCl in Section II-B.
In summary, the studies of the spectra and structures of ArDF and
ArHF presented here show that the nuclear spin-spin interaction, when
observable, can be used to gain structural information about van der
Waals molecules in complexes which lack quadrupolar nuclei.
There is
also no evidence that either the bond length of free DF or the electric
61
field gradient at the D nucleus is significantly perturbed on complex
formation.
F.
Summary of Results and Conclusion
The studies of rare gas-hydrogen halide van der Waals molecules
presented here, along with prior investigations of X-HY, allow an opportunity to examine the X-HY interaction as a function of both the rare gas
and halogen atoms.
Table 11-23 summarizes the structures, force constants
and Lennard-Jones potential parameters for the major isotopes of all
studied complexes involving Ar, Kr or Xe and HF, HCl or HBr.
These
constants were calculated using the methods of Section II-A.
The struc-
tural constants display expected trends.
The rare gas-halogen distance,
R 0 , increases with increasing size of the respective atoms.
For a given
rare gas atom, the structural angles, 9, remain relatively constant in
going along the series from HF to HBr.
This latter effect results from
a decreased anisotropy in the potential, along that series, as measured
by kb.
The trend is easily explained by realizing that as the halogen
becomes bigger, the corresponding hydrogen halide looks more spherical.
One question concerning the structure of van der Waals molecules has
been whether the van der Waals bond length in a dimer can be explained in
terms of the van der Waals radii of the constituent molecules.
rare gas hydrogen halides, it appears to be possible.
In the
If it is assumed that
RQ is the sum of the van der Waals radius of the rare gas atom, R ^ and of
the hydrogen halide, R^y,
and
R x * s taken to be 5
1.9, 2.0
and
2.2 A for
62
Table 11-23.
Summary of the structures, force constants and LennardJones potential parameters for the rare gas-hydrogen halides
as calculated using the method of Section II-A.
*o
9
k
s
k
b
R
e
e
HY
radiusa
(A)
(deg.)
o
(mdyn/A)
ArHF
3.5445
41.1
0.0142
0.00301
3.465
119
1.64
ArDF
3.5349
32.7
0.0170
0.00392
3.463
143
1.63
KrHF
3.6467
38.9
0.0182
0.00378
3.585
163
1.65
KrDF
3.6399
30.2
0.0210
0.00533
3.584
189
1.64
XeHFC
3.8152
35.6
0.0210
0.00537
3.763
208
1.62
XeDF
3.8111
29.4
0.0237
0.00594
3.763
235
1.61
ArHCld
4.0065
41.3
0.0117
0.00151
3.942
128
2.11
ArDCld
4.0249
33.4
0.0134
0.00180
3.966
148
2.12
KrHCl
4.1106
37.8
0.0155
0.00216
4.064
179
2.11
KrDCl
4.1251
30.5
0.0171
0.00259
4.081
199
2.13
XeHCl
4.2753
34.5
0.0190
0.00311
4.237
239
2.08
XeDCl
4.2875
27.9
0.0205
0.00370
4.251
259
2.09
ArHBr
4.1464
42.1
0.0076
0.00114
4.082
89
2.25
ArDBr
4.1820
34.4
0.0106
0.00129
4.128
126
2.28
KrHBr
4.2572
38.0
0.00132
0.00170
4.219
165
2.26
KrDBr
4.2809
31.0
0.0152
0.00195
4.246
192
2.28
0
0
(mdyn•A)
(A)
(cm"1)
(X)
Calculated assuming RQ is the sum of the X and HY van der Waals radii
where the radii for X are taken from Ref. 50.
b
See Ref. 25.
c
See Ref. 26.
d
See Ref. 16.
63
Ar, Kr and Xe, respectively, Rjjy can be computed.
results of this calculation.
Table 11-23 shows the
It shows that remarkably constant values
are obtained for the van der Waals radii of each of HF, HCl and HBr.
In
particular, R ^ equals 1.63(1), 2.11(2) and 2.27(2) A for HF, HCl and HBr,
respectively.
Using tabulated values for the van der Waals radii of the
51
halogens,
is obtained.
o
a constant value for the contribution of H to Ryy* 0.30(2) A,
The magnitude of this contribution is approximately that
which is expected.
Consider KrHCl, where the angle, 8, between HCl and
the Kr-Cl mternuclear axis is 37.8 and the van der Waals radius of H
51
°
is taken to be
1.2 A. If Kr is brought up along the Kr-Cl internuclear
axis until it touches H, the Kr-H distance is given by the sum of their
o
o
van der Waals radii, 3.2 A. Since the H—Cl distance is known to be 1.28 A,
the Kr-Cl distance can be computed from simple trigonometry and is found
o
to be 4.11 A which is exactly the measured value. While this exact
agreement is certainly fortuitous, it shows that a consistent picture of
the structures of X-HY can be obtained by considering the van der Waals
radii of the monomers.
As an added bonus in the above analysis, a con-
sistent set of van der Waals radii for the hydrogen halides have been
determined.
The force constants and potential parameters listed in Table 11-23
show the large variations upon D substitution that were discussed at length
in Section II-B.
These variations indicate the limits of present analysis
since, within the Born-Oppenheimer approximation, the intermolecular potential
should be invariant to isotopic substitution.
To further the analysis,
a clue is taken from the investigation of the centrifugal distortion of
64
the Br nuclear quadrupole coupling constant (see Section II-C), and bendstretch interaction constants are added to the simple harmonic potential.
The more complete potential is given by
V(R,8)=2k sl (R-R e ) 2 +2k bl 6 2 +k b2 e 2 (R-R e )+k b3 e 2 (R-R e ) 2 .
(11-43)
The measured stretching and bending force constants are then approximated
by
k
= k
^
= khl+
s
si
+ 2k
b3(^
(II
" 44)
and
21^(R-R e ) + 2k b 3 (R-R e ) 2 .
(11-45)
It is also noted that RQ is a function of 9 that is approximated here as
the first two terms of a Taylor series expansion about 6=0,
R 0 = R'0 + c<B>2
.
(11-46)
R'0 is the rare gas-halogen distance averaged over the stretching vibration
but fixed at 6=0, and ks-j_ and k,, are the isotopically invariant stretching
and bending force constants for X-HY in its equilibrium configuration.
Using the measured stretching force constants and bond angles for the
hydrogen and deuterium isotopes of X-HY, Eqs. (11-44) and (11-46) are
readily solved to give k , and RQ.
These, in turn, are used in the
development following Eq. (11-11) to obtain isotopically invariant estimates
of the Lennard-Jones well depth, e, and internuclear separation, R e .
With
R e known, Eq. (11-45) can be solved, using the data from isotopic substitution, to give kbx-
The isotopically invariant potential parameters
65
for the X-HY complexes are given in Table 11-24.
Table 11-24.
ksl(mdyn/A)
ArHF
KrHF*
XeHFb
ArHClc
KrHCl
XeHCl
ArHBr
KrHBr
a
The bonding trends exhi-
Isotopically invariant force constants and
potential parameters for the rare gas-hydrogen
halides. The attractive energy, E, calculated
from the multipole potential is also given.
k ^ (mdyn'A)
0.0218
0.0252
0.0295
0.0166
0.0202
0.0233
0.0166
0.0192
0.0116
0.0194
0.0133
0.0015
0.0022
0.0032
0.0014
0.0021
R£(A)
e (era- )
e/eArHF
3.454
3.577
3.758
4.005
4.112
4.277
4.211
4.297
182
226
291
186
239
298
206
247
1.0
1.2
1.6
1.0
1.3
1.6
1.1
1.4
E(ciri
101
123
143
96
119
142
98
121
) E /EArHF
1.0
1.2
1.4
1.0
1.2
1.4
1.0
1.2
See Ref. 25.
b
See Ref. 26.
c
See Ref. 15.
bited in Table 11-24 follow, for the most part, simple chemical intuition.
It is expected that the binding energy should increase with the polarizability of X for a given HY.
the polarizability.
The binding energy does not scale, however, as
It is also surprising, perhaps, that the binding
energy remains relatively constant as the halogen is changed.
To put the expected trends in binding energy on a somewhat more
quantitative basis, consider the intermolecular potential as expanded in
a multipole series.
Buckingham has given a multipole expansion for the
attractive potential between a linear and spherical molecule.
While
66
this potential does not contain repulsive terms and may not be convergent
at the small distances considered here, it is thought that the attractive
energies should serve as a rough guide to the bonding interaction.
The
leading term in the multipole expansion is given (for the linear configuration by
where a is a polarizability, p is a dipole moment and U is an ionization
potential.
Table 11-24 lists the multipole energies calculated with
Eq. (11-47) and reduced binding and multipole energies where all energies
have been normalized to the values of ArHF.
It is seen from Table
11-24
that the multipole expansion does an excellent job of explaining the
relative binding energies observed in the rare gas-hydrogen halides.
The
absolute values of the binding energies cannot be determined from the
multipole expansion.
Comparison is possible, however, with detailed
potential surfaces derived from spectroscopic data for ArHCl
52
and KrHCl
41
52
, ArHBr
-1
. The binding energies of 183, 186, and 224 cm
for ArHCl,
ArHBr and KrHCl, respectively agree reasonably well with the values of
186, 206, and 239 cm"
listed in Table 11-24.
This good agreement indicates
that the simple potential in Eq. (11-43) is doing a reasonably good job at
reproducing the potential near the potential minimum.
As shown above, a multipole expansion of the potential can be used
to account for the trends in the binding energy observed in the rare gashydrogen halides.
Evidence is presented here that the same potential can
also be used to account for the anisotropy of the binding interaction as
67
measured by the angle 9. The full expression, given by Buckingham , for
the long range part of the potential between a linear and spherical
molecule is
(C±+Cd)
v( R cm5 ecm') - V(R
-' e ^
R*m
,
/
- 1
(3cos e
-1)
" "'cm
- -; I
6c, ^ + l£d
Rcm(
HY
^
^ m
a
u
HY HY
a «_U I
ll-
L +c 'i- -^
lC± + Cd 3«HY
(11-48)
.
4A
3
cose„„
ll ' 3~ cos 9 cm+ cj&d a^y
cm
A
where
u u
3 U„Ui:
y HY
-± = <*xu
a
2
UX+UHY
XaHY
(11-49)
Q is the molecular quadrupole moment of HY, A., and Aj_ are components of
the tensor describing the dipole-quadrupole polarizability of HY.
R c m is
the length from X to the center of the mass of HY and 6 c m is the angle
HY makes with R,cm.
The other quantities have been previously defined.
R c m and e c m are easily derived in terms of the structures listed in Table
11-23 by using simple trigonometry.
In the limit of harmonic oscillation,
the vibrationally averaged angle is given by (see Eq. (11-22))
< 9 cm 2 > = H / t t i r V b ) *
(11-50)
where the reduced mass, u b , is approximately given by the moment of
inertia of free HY and the force constant, k b , is related to the intermolecular potential by
1
S=
32V
3 fl
2
9
cm
(11-51)
cm = 0
D
68
Using Eqs. (11-48), (11-50) and (11-51), the vibrationally averaged
angles, <^9
)= ( G 2 ) > can be calculated.
Table 11-25 summarizes the
structures of X-HY and includes the angles calculated from the multipole
expansion.
The agreement between the observed and calculated angles is
very good considering the approximations inherent in this model, namely,
that the multipole expansion is valid at the small distances considered
here, that the bending motion is harmonic, that the repulsive potential
is isotropic, and that
<9cm> = ( O 1 " 2
"
coa-1«co8necm>1/n).
(11-52)
In addition, the average angle is relatively insensitive to k b since it
varies only as the inverse fourth root of the force constant.
However,
all observed trends in the observed average angle are reproduced by the
multipole potential.
The calculated angles increase along the series
X-HF to X-HBr and decrease along the series Ar-HY to Xe-HY as is found
experimentally.
The large difference in the average angle between the
H and D isotopes of a given complex is also reproduced but is due mainly
to the factor of 2 difference in the bending reduced mass.
The present
results indicate that the anisotropy in the X-HY potential surface is
dominated by long range forces and that the multipole expansion may be
useful m
modeling the attractive part of the X-HY potential.
In conclusion, it is believed that the foregoing analysis, while
still fairly crude, is giving a more realistic and consistent picture
of the rare gas-hydrogen halide van' der Waals interaction than has
generally been the case thus far.
69
Table 11-25.
Comparison of the observed angles in the rare gas-hydrogen
halides with those calculated from the multipole expansion
in Eq. (11-48).
O
o
0
Complex
Rcm(A)
ArHF
3.5096
41.6
38.3
a
ArDF
3.4607
33.5
31.9
a
KrHF
3.6105
39.3
36.4
25
KrDF
3.5637
31.0
30.3
25
XeHF
3.7772
35.6
35.0
26
XeDF
3.7339
29.5
29.2
26
NeDCl
3.799
47
42.4
32
ArHCl
3.9795
41.7
38.5
16
ArDCl
3.9668
34.0
32.5
16
KrHCl
4.0822
38.1
36.5
a
KrDCl
4.0652
31.0
30.7
a
XeHCl
4.2457
34.8
34.9
a
XeDCl
4.2260
28.3
29.4
a
ArHBr
4.1331
42.2
43.4
a
ArDBr
4.1529
34.7
36.9
a
KrHBr
4.2431
38.2
41.0
a
KrDBr
4.2506
31.2
34.7
a
T h i s Chapter.
<8 c m >(observed)
<8>(calculated)
Reference
70
CHAPTER III
A HINDERED ROTOR MODEL FOR ESTIMATING ANGULAR EXPECTATION
VALUES IN THE RARE GAS-HYDROGEN HALIDES
A.
Introduction
The internal electric fields and electric field gradients in van
der Waals molecules which arise from monomer charge distributions have
found great utility in explaining polarization effects in these complexes.
53-55
In the rare gas (X)-hydrogen halide (HY) complexes, they
have been used to explain the rare gas nuclear quadrupole coupling
16 31
as well as the X-HY dipole moments. '
arising from HY m
'
'
The fields and field gradients
X-HY are generally expressed by a multipole series
involving expectation values of the Legendre polynomials, (P^(cos8)) ,
where 6 is the angle between HY and the vector R connecting the HY center
of mass and X.
(P2(cos8)) can be determined from the halogen nuclear
quadrupole or the hydrogen-halogen nuclear spin-spin coupling and
(P,(cos9)> can be derived from the X-HY dipole moment, u, if it is available.
Unfortunately, it is not possible to measure u using the current
spectrometer owing to the difficulty of placing Stark electrodes into
the Fabry-Perot cavity and (P2(cos9)) is the only angular expectation
value available in this case.
There are currently no ways to directly
obtain (P,j/(cos8)) for l>2 from the microwave results.
Defining e^as the angle derived from <Pa(cos9)) , it is not, in gen-
71
eral, true that 6 = 6
for m ^ n. In fact, in the limit that HY undergoes
free internal rotation, 9^ decreases monotonically from 90
30.56 for 1= 4.
for A = 1 to
In this chapter, a simple method is presented to
estimate (p2,(cos9)> with Z i 2 if (P9i.(cos9)) is known.
The (P-,(cos9)>
i
computed with this method are compared with experimental values derived
from spectroscopic data and with calculations based on detailed potential
surfaces where available.
Finally, a multipole expansion of the potential
between X and HY is used to calculate (P2(cos8)) and some insight into the
nature of the true X-HY potential surface is gained.
B.
Hindered Rotor Model
To estimate the angular expectation values, (P^(cos9)), the bending
motion of the linear X-HY complex is treated as a hindered internal rotation of the HY subunit.
Fixing the HY center of mass to X distance at
its vibrationally averaged value,R, HY experiences a potential V(6) and
the Hamiltonian for the hindered rotor is given by
H = BJ 2 + V(9)
,
(III-D
where B is the ground state rotational constant and J is the angular
momentum of HY.
V(9) is cylmdrically symmetric about R and is conveni-
ently expressed in terms of Legendre polynomials as
00
V (8) = g 1 V £ [1-P£ (cos8)]
(III-2)
In the limit that all V. go to zero, HY becomes a free rotor and its motion
72
is governed by the rigid rotor eigenfunctions
|JM> = ^ ( 9 , 4 0 ,
where Y
(III-3)
(9,<|>) is a Spherical Harmonic.
Since only the ground state
(J = M = 0) expectation values are desired here and V(9) does not couple
states with different M, the hindered rotor eigenfunctions are expanded
in the basis
|J) = |JO) = Y JO (8,4) = ( 2 J±I) 1/2 P J (cos9).
(III-4)
In the basis of Eq. (III-4), the Kamiltonian matrix elements are given by
H J K = [BJ(J+1) + ^
v j 6 J K - JJ1V^<j|PJl(cose)|K), (III-5)
where the matrix elements of the Legendre polynomials are given by37
P
and 6
JK
=
<J|P£(cos6)|K> = [(2J+1) (2K+1)]
1/2
(
J
Q
2
* M
(III-6)
is the Kronecker delta. In Eq. 6, L Q Q is a 3-j symbol. The
expectation values,(Pjj, (cos8)^, are then obtained by constructing and diagonalizing H and transforming P
to the same basis. At most, 15 basis
functions were required to attain convergence for the potentials considered here.
C. Results and Discussion
To a first approximation, the potential experienced by HY is given
by the leading term in Eq. (III-2)
73
V(9) = Vj-jVP^cose)] = V1(l-cos8)
where V^ is half the barrier height.
,
(III-7)
The one adjustable parameter in V(8)
can be fixed by requiring V(9) to reproduce the measured value of
(P2(cos6)) within the hindered rotor approximation.
all (P^(cos9)} are readily calculated as above.
With V^ determined,
Since (P^(cos9)>depends
only on the ratio V^/B, the calculation need only be done once with
(PJ2/(cos9)) tabulated as a function of V^/B.
(P£(cos9)) can also be fit
to a polynomial in (^(cose)) given by
(III-8)
<P£(cos6)>
= az + b^<P2(cos6)> + c£(P2(cos6)>2 + dJl<P2(cose)>3.
The coefficients for Eq.(III-8) with 1= 1, 3 and 4 are listed in Table III-l.
Table III-l.
Coefficients for the equation
<P2,(cos6)> = az + b£<P2(cos9)> + ^ ^ ( c o s g ^ 2 + dJl(P2(cos6)>3.
1
H
H
H
1
3
4
0.2697
0.0022
-0.0100
1.8959
0.0482
0.1321
-2.3510
0.8511
-0.4722
Use of Eq. (III-8) reproduces the (P£(cos9)>
d
*
1.3032
0.1112
1.3246
to within 0.5% of their
tabulated values over the range of angles appropriate to the X-HY complexes.
Figure III-l shows 6^ as a function of V,/B.
Clearly, this model gives the
correct limiting behavior and it is seen that the algebraic approximation
(i.e., 9 m = 92 for all m) becomes increasingly bad as the most weakly bound
0
10
20
30
40
V,/B
Figure II-l.
6 £ as a function of the hindered rotor barrier height.
50
75
complexes are approached.
In the course of this calculation, the height of the barrier to
internal rotation is obtained.
X-HY complexes.
Table III-2 gives the barrier height for the
These barriers are probably too high, in absolute terms,
because a P-, (cos 8) potential is relatively broad thus requiring a greater
well depth to achieve the correct expectation values.
however, all expected trends are observed.
In relative terms,
For a given HY, the barrier
height scales roughly as the polarizability of X and for a given X, the
anistropy decreases from HF to HBr.
This last trend is expected since HY
looks more spherical as the halogen becomes larger relative to H.
The hindered rotor model can be tested by comparing <P1(cos6)> as
computed above with those derived from dipole moment measurements.
a simple electrostatic model, °
/» ,
^
» ^
(l
(P^cose)) = ~ - - ^
.
S?k\
[1 " — — )
_1 R +
f
2
"HY\
Using
.
,
[ ^ - + - 2 ^ ) <P2(cos9)>
,
where uand u Q are the X-HY and HY dipole moments, respectively, Q is the
HY quadrupole moment,a is a static polarizability and ct^y is the polarizability of HY along R given by (see Appendix II)
ai
a
HY
=
3(a|| ~a0.V < V c o s 6 ^
Table II-2 compares the 9
+a
HY -
for several complexes.
(111-10)
In general, the hindered
rotor method is much better than the algebraic method for 6 2 > 35°.
Below
35°, the algebraic method is comparable to slightly better than the hindered rotor.
This agrees with Figure III-l where the 9^ are seen to
greatly diverge at^ 35°.
While the algebraic method seems to work in some
76
Table III-2.
Comparison of measured values of 6, with those estimated
from the hindered rotor model and detailed potentials
where available.
2xVL
9°
(cm~^-)
NeDCl
ArHCl
ArDCl
KrHCl
KrDCl
XeHCl
XeDCl
ArHF
ArDF
KrHF
KrDF
XeHF
XeDF
ArHBr
ArDBr
KrHBr
KrDBr
a
30
112
149
173
221
261
326
222
322
292
450
456
550
84
108
136
170
See Ref. 41.
See Ref. 52.
c
See Chapter II.
b
9°
Measured
6°
6j
Hindered
Rotor
Experimental
Potential
47
65
56
41.7
34.0
38.1
31.0
34.8
28.3
41.6
33.5
39.3
31.0
35.7
29.5
42.2
34.7
38.2
31.2
47.9
34.5
40.4
29.1
34.6
46.3
36.0
41.2
32.5
37.0
29.3
46.2
35.4
42.9
32.5
38.1
30.8
47.2
36.8
41.4
32.7
—
48.4
34.0
—
—
37.8
28.2
50.8
34.0
—
—
47.5-50,.5a
37.0-37 ,3a
43.0-44,.3b
32.9-33,.0B
Reference
32
16
16
27,c
27,c
28, c
c
24,c
26,c
*
49.8-53, 5
' h
37.6-38.,0b
25
25
26
26
31,c
31,c
c
c
77
cases in estimating (P^(cos6)) , it will do worse in estimating (Po(cos9))
and (P^(cos9)).
For X-HY, the algebraic method will actually give the
wrong sign for (P3(cos9)) in 1/3 of the complexes and for (P^(cos9)) in
all but one complex.
Table III-2 also compares (p^(cos6)) as calculated above with those
obtained from detailed potential surfaces derived from spectroscopic
data.
41 52
'
Both methods give comparable results and, in particular,
neither method predicts the relatively small values of 9-^ for 9 2 < 35°.
It is interesting that this simple hindered rotor model appears to give
angular expectation values as well as a much more sophisticated calculation.
This is especially surprising in the light that experimental
values of 9j_ were used in deriving these surfaces.
Finally, the hindered rotor method can be used to calculate the
angular expectation values for any barrier given as a sum of Legendre
polynomials.
Assuming that the anisotropy in the X-HY potential surface
arises primarily from the attractive part of the potential, the barrier
5 57 58
can be described by the multipole expansion ' '
„
V(6) = - Zi c(n,i,ind)
n,Jl
+
where again w e assume R is fixed.
P (cos9)
c(n,S,,dis) _
_
„n
,
(III-ll)
Expressions for the induction and
dispersion coefficients, c(n,a,type), are listed in Table III-3.
Q
c(9,£,ind)/R
accounts for about 5% of the attractive energy in X-HY
and since the dispersion terms are difficult to calculate and drop off
faster than the induction terms, c(9,£,dis) was
not included in V ( 8 ) .
9o calculated with the barrier in Eq. (III-ll) is compared, in Table
78
Table III-3.
Explicit expressions for the induction and disperson
coefficients for the multipole potential in Eq.(III-ll).a
on
c ( 6 , 0 , i n d ) = y 2 a „x
HY =
c ( 6 , 0 , d i s ) = 42- =Ux+U
- = . 2ax a„„
„2„a
c(6,2,ind) =
= M
x
is
c(7,l,ind) = i | y Q a x
( a I I ~°±)
c ( 6 , 2 , d i s ) = c d " 3a ^
HY
c(7,3,ind) = ^ Q a
d
Cj
m
c(7,l,dis) = j c d ( A | | + 2 ^ ) / ^
c(8,0,ind) = | y 2 q x + |
Q2ax
26 ,n. x j+. 2u^q
o„2„
c ( 8 , 2 , m d ) = ^funa
7
c(8,4,ind) = ^ W
7
^
_ 4 _ ,...
4
4
4
c ( 7 , 3 , d i s ) = j c d (Ai i - -j Aj_)/aHY
^+ i12
2„
£ . nQ/a
X
~
+ £ Q2a
7
X
c ( 9 , l , m d ) = 12yQqx + 36
7
^ ^
na
c(8,0,dis) =-|c d (3d[p+7d| 2) ) b
4
c(8,2,dis) = y cd(13dfp+15d_^;)D
c(8,4,dis) =lic d (d| ( p-d| 2 >) b
on
c ( 9 , 3 , i n d ) = ^ i * a x + 6uQqx + 4 fiax
W xx +
c(9,5,ind) = ±
±^u$a
+ i 20
^ pax
3
7
u , Q, £2 and $ a r e the 2 - , 4 - , 8 - , and 16-pole moments of HY, a and q are
2 - and 4 - pole p o l a r i z a b i l l t i e s , U i s an i o n i z a t i o n p o t e n t i a l and Aii and
Aj_ are components of the t e n s o r d e s c r i b i n g the d i s t r i b u t i o n of p o l a r i z a b i l i t y in HY.
b
See Ref. 58.
79
III-4 , with the measured value
Table III-4.
The agreement
8~ for several complexes computed using the
multipole potential in Eq. (III-ll) and the
hindered rotor model.
4
°°2
ArHF
KrHF
XeHF
NeDCl
ArHCl
KrHCl
XeHCl
for several complexes.
Measured
Multipole
Potential
41.6
39.3
35.7
47.0
41.7
38.1
34.8
40.5
37.5
35.1
45.9
40.7
37.8
35.4
Difference
1.1
1.8
0.6
1.1
1.0
0.3
-0.6
Reference
a
a
26
32
16
a
a
See Chapter II.
is remarkable and better than that using the harmonic oscillator model
proposed in Section II-E.
It is noted that 92 from the multipole
potential is generally smaller than the measured value and, while they
are certainly the same within the uncertainty of the model and properties
of HY, anisotropy in the repulsive part of the potential would be expected
to have the opposite sign and, in effect, make the potential look more
isotropic.
The multipole barrier for ArHCl is shown in Figure
given by (all values are in cm
)
III-2
and is
80
e
*>
o
60
-
40
-
20
0
-180
-140
-100
-60
-20
20
60
100
140
180
THETA
Figure III-2.
The multipole potential barrier for ArHCl from Eq. (111-12).
oo
o
81
V(9) = 58.6 - 25.3 P1(cos9) - 13.6 P2(cos9) - 14.3 P3(cos9)
-4.0 P4(cos9) - 1.4 P 5 (cos9).
(111-12)
While the experimental X-HY potential surfaces
'
'
'
are generally
parameterized in terms of Pi(cos9) and P 2 (cos9), it is clear from Eq.
(111-12) that the P-(cos8) term plays a large role in the anisotropy of
the surface and must be accounted for.
It is also evident that the
octupole and hexadecapole moments as well as the higher order polarizabillties of HY are needed to get good agreement with experiment.
Unfor-
tunately, these are unknown for most molecules and are difficult to
estimate.
It is hoped, however, that the long range potential in Eq.
(III-ll) can be matched to an ab initio or empirical repulsive potential
to reproduce the microwave data as well as scattering and line broadening
results.
82
CHAPTER IV
XENON NUCLEAR QUADRUPOLE COUPLING IN XENON HYDROGEN CHLORIDE
A.
Introduction
The first determination of a nuclear quadrupole coupling constant
for a rare gas atom in a van der Waals molecule was reported in a recent
paper on KrHF.
That study was extended by investigating the
nuclear quadrupole coupling in XeHCl.
Xe
The nuclear quadrupole inter-
action arises through a coupling of a nuclear quadrupole moment with the
62
electric field gradient at that nucleus.
In a free ground state rare
gas atom, the electronic distribution about the nucleus is spherically
symmetric, thus, the field gradient and coupling constant are identically zero.
The measurement of a nonzero nuclear quadrupole coupling
131
constant for
Xe in XeHCl indicates that the free Xe electron distri-
bution is to some extent perturbed in the complex.
field gradient is interpreted m
The observed electric
terms of Sternheimer antishieldmg
and formation of the weak Xe-HCl van der Waals bond.
Subsequent to the study presented here, slight modifications to
the spectrometer and experimental technique permitted the rotational
spectrum of
131
XeHCl to be examined in lower rotational states than had
been previously possible.
The lower J rotational transitions have a
more highly resolved hyperfine structure which allowed better values of
the Xe nuclear quadrupole coupling constants to be obtained and a more
83
complete analysis of the effect to be made.
study are presented elsewhere.
The results of this latter
The major conclusions concerning the
nature of the Xe quadrupole coupling remain unchanged, however.
The main
purpose of the present chapter is to complete the reporting of the experimental results and to outline the origin of the rare gas nuclear quadrupole coupling in van der Waals molecules.
As an aside, it is indicated
how useful information may be obtained from poorly resolved spectra by
using the effects of isotopic substitution as a predictive tool.
B. Spectra and Results
The spectra of
131
XeHCl and
131
XeDCl are greatly complicated by
the presence of two quadrupolar nuclei in these species (deuterium
coupling is ignored).
There are more than 100 hyperfine components in
the spectra observed here.
This large number of components greatly
reduces the intensity of any particular component, thus lowering the
signal-to-noise ratio.
In addition, severe overlapping of the spectral
lines and the Doppler doubling phenomenon lower the effective resolution
and make the unambiguous assignment of the spectra difficult.
The
hyperfine Hamiltonian is given in Eq. (II-3) and is the sum of quadrupole
Hamiltomans for Cl and Xe.
The Hamiltonian matrix elements were calcu-
lated using the method described in the discussion following Eq. (II-3).
For
129
131
XeH
XeHCl, the Cl coupling constant, x . » was fixed at its value in
Cl (-34.76 MHz) and the hyperfine Hamiltonian matrix was con-
structed and diagonalized as a function of the Xe coupling constant, x v ,
AS
84
until plots of the predicted spectra suitably matched the observed
transition frequencies.
value of x x
After the spectral assignment and an approximate
were obtained in this manner, the data were fit by least
squares involving diagonalization of the Hamiltonian matrix to give
Xv
= -4.9(2)MHz.
are listed in Table
The observed transition frequencies for
IV-1 .
131
35
Ji
XeH Cl
Figures IV-1 and IV-2 show the predicted
spectra as a sum of Lorentzians as well as the measured frequencies.
A second evaluation of x X e can be obtained by considering the centrifugal distortion in XeHCl.
In a pseudodiatomic model of the van der
Waals stretching vibration, the centrifugal distortion constant, Dj, is
proportional to the inverse square of the stretching reduced mass (see
Eqs. (II-8), (11-10) and (11-16)). Using the precisely known values of
Dj in
129
XeH 3 5 Cl (3.8130(15)kHz) and
132
XeH 35 Cl(3.7762(31)kHz), this
scaling law is found to hold within the experimental uncertainly.
the scaling law and the value
be 3.7882(45)kHz.
of Dj in
With
129
35
XeH Cl, Dj is calculated to
It is now noted that as x v
changes, the line positions
move with respect to the hypothetical center frequency, v Q .
Thus, by
using the best determined line from each R branch (J=4,F1=5/2, 7/2, F=8/2,
10/2-*J=5, F 1 =7/2, 9/2, F=10/2, 12/2 and J=6, F-,=9/2, 11/2, F=12/2, 14/2+
J=7, F^ll/2, 13/2, F=14/2, 16/2), the center frequencies, \>0, can be
calculated as a function of Xx e
obtained from
and
the
rotational constant, B 0 , and Dj
33
v Q = 2B0(J+1) - 4Dj(J+l) 3 .
(IV-1)
Figure IV-3 shows a plot of D, as a function of x x • Using this graph
85
Table IV-1.
Observed
Isotope
J
131
4 -*• 5
XeH 3 5 Cl
J'
L31
XeH 3 5 Cl and
131
XeD 3 5 Cl.
Fx,F
F1',F'
7/2,8/2
9/2,10/2
7,2,10/2
5/2,8/2
9/2,12/2
7/2,10/2
9/2,8/2
11/2,10/2
9906.855
11/2,10/2
9/2,10/2
13/2,12/2
11/2,12/2 }
9906.888
11/2,14/2
13/2,16/2
9906.934
9/2,12/2
11/2,14/2
9906.973
11/2,8/2
9/2,10/2
13/2,10/2
}
11/2,12/2
13866.704
11/2,14/2
9/2,12/2
13/2,16/2
11/2,14/2 }
13866.731
13/2,12/2
15/2,14/2
13866.956
15/2,14/2
17/2,16/2
13866.967
15/2,12/2
13/2,10/2
13/2,14/2
15/2,16/2
17/2,14/2
15/2,12/2
15/2,16/2
17/2,18/2
15/2,18/2
17/2,20/2
13867.004
13/2,16/2
17/2,18/2
13867.016
7/2,10/2
5/2,8/2
9/2,12/2
7/2,10/2
}
9758.986
6 +7
11/2,14/2
9/2,12/2
13/2,16/2
11/2,14/2 }
13660.713
7 + 8
13/2,16/2
11/2,14/2
15/2,18/2
13/2,16/2 }
15610.676
6 -> 7
XeD JJ Cl
Frequencies for
4 -»-5
Frequency (MHz)
9906.271
}
9906.387
13866.981
J=4-vJ =5
F F—-F 1 , F'
7/2, 10/2^9/2, !2/2
5/2, 8/2~*7/2, 10/2
J
11/2, 14/2—13/2, 16/2
11/2, 10/2—13/2,12/2
9/2, 10/2—N/2, 12/21
J
9/2,12/2—11/2,14/2
7/2, 8/2—9/2,10/2
9 / 2 , 8/2—11/2,10/2
9906.2
9906.9 MHz
'0
Figure IV-1. The J=4+J=5 transition in 131,XeHCl predicted with the constants in Table IV-2.
The vertical lines are the measured frequencies.
03
J = 6—J = 7
F' F'
F
r rF
r
11/2, 14/2—13/2, 16/2
9/2, 12/2— H/2, 14/2
15/2, 16/2—17/2,
15/2, 12/2—17/2,
13/2, 10/2—15/2,
13/2, (4/2—15/2,
18/2
14/2
12/2
16/2
Jj
11/2, 8/2—13/2,10/2
9/2, 10/2—11/2,12/2
15/2, 18/2—17/2, 2 0 / 2
15/2, 14/2—17/2,16/2/
13/2,16/2—15/2,18/2
13/2, 12/2—15/2,14/2J
13866.7
Figure IV-2.
13867.0 MHz
The J=6-hJ=7 transltxon in 131,
^""-XeHCl predicted with the constants in Table IV-2.
The vertical lines represent the measured frequencies.
oo
3.90
X
Q 3 3.7882
3.70
Xa(
Xe)MHz
Figure IV-3. Graph of Dj vs assumed values for Xa(
131,
Xe)
00
oo
89
and the calculated value of Dj for
131
35
XeH Cl, x X e is estimated to be
-5.0(3) MHz, which is consistent with the value obtained from the spectra.
131
35
XeD Cl, onlyone hyperfine component was obtained in
In the case of
each R branch and the above method was the only way to obtain x y • The
value derived for x_
in
131
35
XeD Cl is -6.0(5) MHz.
The spectroscopic
AS
constants of
IV-2.
131
XeH 3 5 Cl and
The values of x v
131
XeD 3 5 Cl obtained here are listed in Table
derived in the subsequent study
55
of the Xe
nuclear quadrupole coupling are -4.641(50) MHz and -5.89(20) MHz for
131
XeH 3 5 Cl and
131
Table IV-2.
X e D 3 5 C l , respectively.
Spectroscopic constants for
XeD 3 5 Cl.
131
35
XeH Cl and
131
B0(MHz)
131
XeH 3 5 Cl
131 X e D 35 c l
a
C.
D J (kHz) 3
X c l (MHz) a X
(MHz)
990.86264(40)
3.7844
-34.76
-4.9(2)
976.11581(9)
3.4321
-44.82
-6.0(5)
Assumed value.
Analysis of the Xenon Nuclear Quadrupole Coupling Constant.
The existence of a nonzero nuclear quadrupole coupling constant for
Xe in
XeHCl indicates the presence of a nonzero electric field gradient
at the Xe nucleus. The electric field gradient,qaa,along the a-inertial
axis is related to the measured coupling constant, x> °y
90
eQ
where h
is Planck's constant, e is the proton charge and Q is the
nuclear quadrupole moment.
For x = -4.9(2) MHz and Q = -0.12x10
the electric field gradient at the
q aa = -5.6(2)xl014SC-cm~3
where SC is a statcoulomb.
will be considered here.
131
Xe
cm ,
131
Xe nucleus is found to be
,
(IV-3)
Four contributions to this field gradient
Two contributions result from the electrostatic
potential experienced at the Xe nucleus due to the neighboring HCl
charge distribution.
The effects of overlap of the Xe and HCl electronic
distributions and possible charge transfer will also be considered.
The coordinate system used to describe the interaction between an
atom and a cylindrically symmetric molecule is shown in Figure IV-4.
The electrostatic potential energy experienced by an elctron at the
atomic position (r,8) due to the neighboring molecule can be expressed
in terms of a multipole series
as
-eV = -ERerP,(cos9) - q^er P2(cos9) - ... ,
(IV-4)
where P (cosG) is a Legendre polynomial and E R and q ™ are the electric
field and field gradient, along R, measured at the atomic nucleus and
arising solely from the molecular charge distribution.
E R and q ^ can
be expressed in terms of the permanent electric moments of the linear
molecule as
2u
3Q_
4Q
En = -o Pi (cos9) + — J f P„(cos8) + __ P.(cos6) + ...
R
i
I
^ 3
Rj
R4
(IV-5)
91
R
Figure IV-4.
Coordinates used to describe the electrostatic
interactions between an atom and a linear molecule,
92
and
q
RR
6„
" ~f p i ( c o s
=
e)
12Qm
2 on
1 p (cos6)
—r
?
" ~T p 3 ( c o s e> " • • '
R
where u, Q
R
5
<IV_6>
R
and P, are the permanent electric dipole, quadrupole and
octupole moments, respectively.
E„ and q„„ can be calculated for XeHCl
by using the known structure of XeHCl (see Table II-5) and properties of
HCl.
They are found to be
E_ = 4.59xl04SC-cm~2
R
(IV-7)
and
q„„ = 13.87xl02SC-cm~3
RR
.
(IV-8)
The first term on the right hand side of Eq. (IV-4) gives rise to
dipole polarization of Xe in XeHCl.
When the polarizable Xe atom is
placed in an electric field, its initial spherical symmetry is distorted
by the perturbation.
The interaction energy arises in second order of
perturbation theory.
To estimate the effect of dipole polarization on q,
the mixing of the 5p and 6s states of Xe is considered.
accounts for about 20% of the Xe polarizability.
This mixing
Using the perturbation
Hamiltonian
Ex = - E R er cos 6,
(IV-9)
the wave function for the p electron correct to first order is given by
i>
|
= ( l - b 2 ) ^ <j>5p + hj>6s
•
(IV-10)
93
where 6
5p
and A. are zero order Xe atomic wave functions, and assuming
&
6s
b«l,
=
_ ^6e\eT
cos9
l<l>5p>= ER<uz>
_- EEr_5^„
5
(IV-11)
hv
'5P6 . 5p 6s
where v is the frequency of the corresponding transition in Xe.
(u
)
can be estimated from the oscillator strength, f, of this transition in
X e 6 6 by
(u) = (3fhe2/8Tr2mv)'2
where m is the electron mass.
,
(IV-12)
The excitation can now be related to the
field gradient at the Xe nucleus.
The field gradient operator is given by
q - -2(^3") (3cos 2 e 1 -i),
(iv-13)
where the sum runs over all charges, e., outside of the Xe nucleus.
Realizing that s electrons and closed shells do not contribute to q,
determination of the field gradient due to polarization, q , reduces to
the evaluation of
q p = b2(<(>5p[^3(3cos29-l)|t(,5p).
Integrating over angles for a p
i
SP - "*2<5»-<
3>„
i \ r 3/
av
where b
(IV-14)
electron gives
.'
(IV 15)
"
represents the fractional p hole character in the 5p electron
94
distribution.
Using analytical Hartee-Fock atomic orbitals for Xe
<l/r3>
is found to be 112xl0 24 cm -3 .
av
67
,
Thus, q„ = -7.0xl012SC'cnT3
P
and accounts for about 1% of the observed field gradient.
The second term on the right of Eq. (IV-4) causes quadrupole polarization of Xe and results in the Sternheimer antishielding effect.
In this effect, the external field gradient, q ^ , arising from HCl, induces a quadrupole moment in the electronic distribution of Xe which in
turns induces a field gradient, qin<j> at the Xe nucleus.
The Sternheimer
shielding factor, You, is defined in terms of the external and induced
field gradients by
Y» = - lind
q
(IV-16)
RR
and arises in first order of perturbation.
the Xe nucleus, q
Thus, the field gradient at
, is the sum of the external and induced field grad-
ients or
= 1 R R C 1 ^-) •
%s
< IV " 17 >
The difficulty in this analysis is the determination of the shielding
factor.
Too for Xe was calculated, in this lab, by using a variational
procedure to solve for the Xe atomic wave function perturbed by
H
l
=
iRR
er2 p
2(cos9>-
(IV-18)
The details of the calculation have been given before
relativistic and electron reorganization effects.
'
and neglect
Once the perturbed
wave function is known,qincl is computed with the aid of Eq. (IV-13)
95
and Ym obtained from its definition in Eq. (IV-16).
Y*. for Xe is found
to be -138 and the field gradient becomes
q sg = -5.4xl0 14 SC-cm"3 .
(IV-19)
This effect is quite large and accounts for the entire measured field
gradient in
131
XeHCl within the experimental uncertainties.
subsequent studies of the
Xe coupling in XeHCl
In the
and XeHF,
it was
found that all field gradients could be explained solely in terms of
Sternheimer shielding if YM = -153.
This experimentally derived value
compares well with the calculated value of Yoo= -138 shown above and a
value of Yoo = -177 calculated from relativistic Hartree-Fock-Slater
-i,
theory.
7 0
For completeness, two additional contributions to the field gradient
are considered here.
In XeHCl, it might be expected that the electron
distributions of Xe and HCl will begin to overlap.
This will necessitate
electron rearrangements to maintain conformance to the Pauli principle.
It appears unlikely that this reorthogonali2ation
will have much effect
on the field gradient at the Xe nucleus since only the shape of the outer
electron distribution should be severely affected and q depends on
(l/r3) which Is only large near the nucleus. Thus,it is assumed that
q ro &0.
Finally, if charge is transferred from Xe to HCl in a Lewis acid-
Lewis base type interaction, a field gradient, q ct , will arise from the
partial depletion of a p electron.
for polarization:
The treatment is similar to that
96
%t - - c 2 4 K p > a v >
where c
<iv-2o>
represents the fraction of an electron transferred.
In the
model considered here,
q
aa » %s
+
q
+
P
q
ro
+
q
ct
•
< IV " 21 >
Using the previous results, q ct is found to be -1.8xl0^3 SC-cm
c 2 = 0.0004.
or
In other words, on the order of 0.04% of an electron
appears to be transferred from Xe to HCl.
In conclusion, the electric field gradient at the Xe nucleus in
XeHCl is found to arise primarily through the Sternheimer shielding
effect, and that charge transfer plays a minor, if any, role in this
complex.
97
CHAPTER V
ARGON CYANOGEN CHLORIDE
A.
Introduction
The assignment of the rotational spectrum of the weak complex formed
between an argon atom and the linear cyanogen chloride molecule is reported here. ArClCN is a planar asymmetric rotor and its rotational
constants, centrifugal distortion constants and diagonal elements of the
Cl and N nuclear quadrupole coupling tensors are determined.
This study
represents the first time that complete centrifugal distortion and normal
mode analyses have been performed for the weak van der Waals bond between
a rare gas atom and a molecule.
These data allow a reasonably quantita-
tive determination of the bending and stretching frequencies and average
molecular structure in the part of the molecule involving the van der
1L.
Waals bond.
Furthermore, the measurement of the
N and
35
Cl nuclear
quadrupole coupling in ArClCN allows the first comparison between the
average structure as described above with that obtained by projecting
the free C1CN molecule's nuclear quadrupole coupling constants onto the
principal inertial axes of the ArClCN van der Waals molecule.
The agree-
ment is quite good thereby justifying the use of this latter method of
determining structures of weakly bound rare gas-molecule complexes.
The van der Waals binding Interaction between an atom and a linear
molecule has been previously investigated in inertially linear complexes
98
such as the rare gas-hydrogen halides
asymmetric ArOCS
is T-shaped.
72
and ArC0 2
71
(see Chapter II), as well as the
complexes.
Like ArOCS and ArC02, ArClCN
However, ArClCN provides certain features which permit a
more intimate look at the van der Waals interaction.
Only the structure
was determined for ArOCS, but for ArC0 2 , an analysis of the centrifugal
distortion was performed.
Upon complexation, two weak modes of vibration
are introduced into the atom-molecule system.
These correspond to a
stretch and an in-plane bend of the van der Waals bond.
C 2 „ symmetry, only two of the Kivelson-Wilson T'S
Since ArC0 2 has
depend on the weak
modes and the bending and stretching force constants are directly calculated from them.
ArClCN, on the other hand,has C s symmetry and all
four of the linearly independent T'S depend on the two weak modes.
This
allows the force constants to be fitted and the coupling between the
modes to be obtained from a bend-stretch interaction constant.
methods of vibrational analysis
can also be used to obtain further
information about the complex once the force field is known.
tional frequencies, structure of the average configuration
estimate of the inertial defect
Standard
are derived for ArClCN.
76
The vibraand an
The fact that
the estimated inertial defect differs by only 2% from the experimental
value indicates that a normal mode analysis retains a large measure of
validity for these weakly bound complexes and, in addition, confirms the
78
planarity of ArClCN.
Finally, the spectrum of ArClCN shows the effects of Cl and N nuclear quadrupole coupling.
In an inertially linear complex, nuclear
quadrupole coupling is used to determine the vibrationally averaged
99
structure of the complex.
This procedure depends on the assumption that
the measured coupling constant simply represents the averaging of a
monomer coupling constant over the van der Waals modes.
While there is
evidence that this assumption is reasonable in the rare gas-hydrogen
53
halides, a study of the N coupling in NCCN"-HF
54
and (HCN)2
that it certainly is not valid in hydrogen bonded species.
shows
In planar
ArClCN, however, the component of the nuclear quadrupole coupling tensor
perpendicular to the plane does not depend on an average over the van
der Waals modes.
This allows a quantitative measure of how much the
coupling constants of free C1CN are perturbed by complexation with Ar.
It is found that there is definitely a small change.
However, it is
reassuring that the angle between C1CN and the b-inertial axis as determined from the rotational constants alone and from N and Cl quadrupole
coupling agree quite well.
B. Spectra and Results
The Hamiltonian for ArClCN is the sum of rotational and nuclear
quadrupole parts
H = HR + HQ .
(V-l)
79
The rotational Hamiltonian, HR, is appropriate to the semi-rigid rotor
and involves the A, B and C rotational constants as well as the quartic
centrifugal distortion constants.
The quadrupole Hamiltonian, HQ, includes
both Cl and N nuclear quadrupole coupling and is given by
H 0 = Q(C1):V(C1) + Q(N):V(N) ,
(V-2)
where Q(i) and V(i) are the nuclear quadrupole moment and electric field
gradient tensors, respectively, for nucleus i. Matrix elements of H are
calculated in the symmetric rotor basis and with the angular momentum
coupling scheme
fcl + fo - I
(V 3)
"
I + J =F
1
<\l
<\l
Following Flygare and Gwmn 80, matrix elements of the 2 quadrupole
Hamiltonian in the basis of Eq. (V-3) are given by
(IC1INIJFMF|HQ|IC1INI'J'F'M'F) ,
^
.
u
t
f
. 1/2/ J 2 J'\-l Cl I' 2)
x[(2I+l)(2I'+l)]
(_J0 j ]
j j, j FJ
I I' 2 I
1
^
W
(V-4)
fl l'2
2
(-1) (^l5ci%) e Q c l (q J J f ) c l + (-l) (!ININIClJe QN(qjji)N
I
C1 2 IC1^
"^1° W
I
/ XN
2
%
^N ° h
where e is the proton charge, QJ is the nuclear quadrupole moment of
nucleus i, and (qTT,) is the electric field gradient at nucleus 1.
JJ l
The quantities in parentheses and braces are 3-j and 6-j symbols, respectively, and 6., is the Kronecker delta. Using the method of Benz,
81
et.al., ^ J J ' \ is readily calculated in the symmetric rotor basis
101
eq J J t = <JKj|Mr [J'K'J) = <JKJ| V 2Q | J'K' J> ,
3z
(V-5)
3 V
—j
^-s the electric field gradient in the direction defined by
8z
the angular momentum and V„_ is the corresponding spherical component
where
of the field gradient tensor.
V'
2q
The electric field gradient components,
,are constant in a body-fixed axis system and are related to Vor. by
<£U
V2o =
q=ZV2qD^(-Y-3Hx)
,
37
(V-6)
(2)
where D M , M
is the matrix representing the 5-dimensional
irreducible
representation of the 3-dimensional rotation group and (aBY) are the
Euler angles locating the body-fixed axes in the space-fixed frame.
Since V 2 Q is real,
2
v2o=v2o=
.
ZD^WX-DVJ
q=-2
M
(v-7)
4
In addition, the symmetric rotor wave functions are given by
,(J) J2J+1^
*&'Wq
D(J) (aeY)
MK
'
(v 8)
"
thus ,
<JKJ|V20|J'K'J) = [(2J+l)(2J'+l)] ^(-l) J+K (_j I ^
_\ ^ V 2 _ q . (V-9)
Defining the spherical components of nuclear quadrupole coupling tensor
as
X ^ = e2QV'q ,
"
(V-10)
102
Eq. (V-4) becomes
I'+I
(^I^JKFMJIHQIICX^I'J'K'F'M'P)
I' 2 ]
N
= 6FF,6MpM, i z | i
x[(2l+l)(2I'+l)(2J+l)(2J , +l)] ^ , j ' F ]
(l
+1 +F+K
_K _2q K-)
(V
*n)
("I I'2
c - i ^ l ^ c i V (x2.q)cl + (-D1 }jb&±.{ (X2.q)N
I
2 X
C1
Cl\
°y
-^I
X
Nz %
1-%°%
For a near prolate planar asymmetric rotor, the axes
x, y and z corres-
pond to the principal inertial axes b, c and a, respectively, and the
spherical components of X are related to the usual Cartesian components by
x
20
= x
X2+1
=
aa
i ^ X a b
*2±2 - &
2
(V"12>
(X bb -X cc ) •
It is seen from Eq. (V-ll) that H Q is diagonal in F and M F <
Thus,to
calculate the exact quadrupole hyperfine energies it is simply necessary
to set up the matrix of Eq. (V-ll) in blocks of F, add the rotational
energies calculated in the same basis and diagonalize the resulting
matrix.
The first order quadrupole energies can be obtained in the
scheme above by considering only the AJ=0 and AK>0,±2 matrix elements.
The data for ArClCN were fit by an iterative procedure consisting
of the following steps:
1. Estimates were made of the rotational parameters and quadrupole
coupling constants, and both the first order and exact energy levels
were calculated as shown above.
2. The differences between the calculated first order and exact
energies were used to adjust the measured frequencies which allowed a
least squares fit based on the first order expression to be performed.
The results of this fit gave the diagonal elements of the coupling constant tensors and center frequencies of the transitions.
3. The line centers were fit to the Watson parameters A", B", C",
T
l>
T
2 ' T aaaa' T bbbb
and T
cccc
us
inS
82
tne
program written by Kirchhoff.
82
To was calculated by invoking the planarity conditions
and was
held fixed in the fit as were the off-diagonal elements of the quadrupole
coupling tensors which were estimated from the molecular structure (see
Section V-E).
Since the higher order quadrupole corrections are fairly
insensitive to small changes in rotational and quadrupole parameters,
only 2 iterations were required to attain self-consistency.
Table V-l
shows the observed transition frequencies as well as those calculated
from the fit of the hyperfine structure in 2 above.
The fit is very
good with the standard deviation of the residuals being about 2 kHz.
The line centers are listed in Table V-2 and Table V-3 shows the spectroscopic constants for ArClCN.
C.
Molecular Structure
The vibrationally averaged structure of ArClCN can be derived from
104
Table V-1.
JK
Observed and c a l c u l a t e d frequencies for ArClCN. The c a l c u l a t e d
frequencies are obtained from Eq. (V-ll) and data i n Tables
V-2 and V-3.
- 1 C J ' K 1 I K «hi
[,F
-*•
I',F'
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
Ar 35 ClCN
3
13" 4 04
5/2,11/2
3/2 ,9/2
3/2
3/2
5/2
5/2
^O"1!!
5
05~ 5 14
1 -2
^ 1 ^12
7/2
5/2
9/2
7/2
5/2,13/2
3/2 ,11/2
3/2 ,9/2
3/2 7/2
5/2 11/2
5/2 ,9/2
7293.5468
7294.2585
7301.1910
7306.5914
7306.7150
7307.2577
7293.5489
7294.2583
7301.1900
7306.5923
7306.7137
7307.2571
1/2
5/2
3/2
3/2
5/2
1/2
3/2,3/2
5/2 5/2
1/2
5/2
3/2
3/2
5/2
1/2
1/2
5/2
3/2
5/2
7/2
3/2
3/2, 1/2
5/2 3/2
7382.3654
7382.9091
7383.5704
7402.3848
7403.1015
7403.3984
7418.8524
7418.9009
7382.3647
7382.9086
7383.5696
7402.3842
7403.0979
7403.3977
7418.8546
7418.9057
5/2 15/2
3/2 13/2
5/2,13/2
5/2 11/2
5/2
3/2
5/2
5/2
15/2
13/2
13/2
11/2
7662.4832
7663.6175
7682.1778
7683.1972
7662.4810
7663.6160
7682.1815
7683.1972
5/2 5/2
5/2 5/2
3/2,3/2
1/2 1/2
5/2 5/2
3/2 3/2
3/2 5/2
5/2, 7/2
1/2 3/2
3/2 5/2
5/2 7/2
3/2 5/2
5/2 7/2
1/2 3/2
5/2 3/2
3/2,1/2
5/2 3/2
3/2
5/2
3/2
3/2
5/2
5/2
3/2
5/2
3/2
3/2
3/2
1/2
5/2
1/2
5/2
1/2
5/2
5/2
7/2
5/2
1/2
5/2
3/2
5/2
7/2
3/2
7/2
7/2
5/2
9/2
5/2
5/2
3/2
1/2
9876.2879
9876.4974
9876.6020
9883.6233
9883.9012
9884.2403
9885.5922
9886.1392
9886.3490
9895.7508
9896.0942
9896.4123
9896.5883
9896.8947
9901.0766
9911.4650
9911.5038
9876.2900
9876.4989
9876.6048
9883.6246
9883.9028
9884.2406
9885.5899
9886.1394
9886.3496
9895.7498
9896.0904
9896.4111
9896.5860
9896.8938
9901.0746
9911.4672
9911.5049
1/2
5/2
3/2
3/2
5/2
1/2
-2.1
0.2
1.0
-0.9
1.3
0.6
0.7
0.5
0.8
0.6
3.6
0.7
-2.2
-4.8
2.2
1.5
-3.7
0.0
-2.1
-1.5
-2.8
-1.3
-1.6
-0.3
2.3
-0.2
-0.6
1.0
3.8
1.2
2.3
0.9
2.0
-2.2
-1.1
105
Table V - 1 .
"V+f
Continued
JK K
' :I +I
I,F
I',F'
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
Ar 35 ClCN
4
14" 5 05
2
02""3i3
5
14" 5 23
4
13
-4
22
1/2,9/2
5/2,13/2
3/2,11/2
3/2,5/2
3/2,9/2
5/2,7/2
3/2,7/2
5/2,11/2
5/2,9/2
1/2,11/2
5/2,15/2
3/2,13/2
3/2,7/2
3/2,11/2
5/2,9/2
3/2,9/2
5/2,13/2
5/2,11/2
10478, 4498
10478, 5578
10479, 1345
10485, 3092
10485, 4204
10485. 8510
10488, 7376
10488, 8226
10489. 3186
10478 .4505
10478 .5569
10479 .1357
10485 .3081
10485 .4197
10485 .8518
10488 .7350
10488 .8252
10489 .3186
-0.7
0.9
-1.2
3/2,5/2
5/2,7/2
3/2,3/2
3/2,5/2
5/2,7/2
3/2,7/2
5/2,9/2
5/2,3/2
5/2,5/2
5/2,3/2
3/2,7/2
5/2,9/2
1/2,5/2
5/2,3/2
5/2,5/2
1/2,3/2
5/2,1/2
1/2,3/2
5/2,1/2
5/2,7/2
5/2,9/2
3/2,5/2
5/2,5/2
3/2,7/2
5/2,7/2
5/2,9/2
5/2,5/2
3/2,7/2
3/2,3/2
3/2,9/2
5/2,11/2
1/2,7/2
5/2,3/2
1/2,5/2
5/2,3/2
5/2,3/2
1/2,5/2
5/2,1/2
12213.2438
12213.7278
12213.8580
12217.2176
12217.6096
12221.5084
12222.2299
12223.3418
12223.5600
12223.6625
12229.7545
12230.5968
12230.8318
12231.6507
12231.9218
12240.0669
12240.0831
12240.3551
12240.5969
12213.2444
12213.7274
12213.8583
12217.2191
12217.6110
12221.5124
12222.2341
12223.3394
12223.5573
12223.6610
12229.7544
12230.5973
12230.8359
12231.6480
12231.9185
12240.0692
12240.0811
12240.3534
12240.5949
-0.6
0.4
-0.3
-1.5
-1.4
-4.0
-4.2
2.4
2.7
1.5
3/2,13/2
5/2,15/2
1/2,11/2
3/2,13/2
5/2,15/2
1/2,11/2
12403.1276
12403.7029
12403.7870
12403.1270
12403.6995
12403.7910
3/2,11/2
5/2,13/2
1/2,9/2
5/2,5/2
3/2,11/2
5/2,13/2
1/2,9/2
5/2,5/2
12858.8818
12859.6706
12859.8196
12863.9040
12858.8820
12859.6683
12859.8187
12863.9070
-0.8
2.6
-2.6
0.0
1.7
2.0
-0.2
2.3
0.9
-3.0
106
Table V-1. Continued
I,F
-1K+1
K
I',F'
-1 K +1
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
Ar 35 ClCN
12' 3 21
5/2,9/2
3/2,9/2
5/2,11/2
1/2,7/2
5/2,9/2
3/2,9/2
5/2,11/2
1/2,7/2
13312.5822
13330.9137
13331.9352
13332.1766
13312.5822
13330.9144
13331.9335
13332.1776
0.0
-0.7
1.7
-1.0
5
15-606
1/2,11/2
5/2,15/2
3/2,13/2
1/2,13/2
5/2,17/2
3/2,15/2
13638.5364
13638.6056
13639.0566
13638.5342
13638.6065
13639.0579
2.2
-0.9
-1.3
2
H~220
3/2,7/2
5/2,9/2
1/2,5/2
3/2,7/2
5/2,9/2
1/2,5/2
13750.1401
13751.4827
13751.9306
13750.1394
13751.4835
13751.9306
0.7
-0.8
0.0
3
03" 4 14
5/2,7/2
5/2,9/2
3/2,5/2
5/2,5/2
3/2,7/2
3/2,9/2
5/2,11/2
1/2,7/2
1/2,5/2
5/2,9/2
5/2,11/2
3/2,7/2
5/2,7/2
3/2,9/2
3/2,11/2
5/2,13/2
1/2,9/2
1/2,7/2
14412. 2356
14412. 8662
14413. 0025
14418. 2724
14418. 7803
14426. 8087
14427. 6076
14427. 7782
14433. 4958
14412 ,2363
14412 .8660
14413 .0008
14418 .2725
14418 ,7816
14426 ,8062
14427 .6083
14427 ,7791
14433 ,4964
-0.7
0.2
1.7
-0.1
-1.3
2.5
-0.7
-0.9
-0.6
2 -2
12 Z 21
1/2,5/2
3/2,7/2
5/2,5/2
5/2,7/2
3/2,5/2
1/2,5/2
3/2,7/2
5/2,5/2
5/2,7/2
3/2,5/2
14716.2840
14716.3485
14716.5308
14717.1908
14717.2134
14716.2839
14716.3482
14716.5288
14717.1917
14717.2149
0.1
0.3
2.0
-0.9
-1.5
3 -3
13 J 22
1/2,7/2
5/2,11/2
3/2,9/2
1/2,7/2
5/2,11/2
3/2,9/2
15217.1295
15217.2348
15217.6790
15217.1288
15217.2355
15217.6790
0.7
-0.7
0.0
4
1/2,9/2
5/2,13/2
3/2,11/2
1/2,9/2
5/2,13/2
3/2,11/2
15892.3797
15892.4990
15893.1368
15892.3775
15892.4997
15893.1383
2.2
-0.7
-1.5
3
14
-4
23
107
Table V - 1 .
Continued
v+r'^+i I , F
I',F'
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
Ar 35 ClCN
4
04- 5 15
3/2,11/2
5/2,13/2
1/2,9/2
3/2,13/2
5/2,15/2
1/2,11/2
16521.9899
16522.7221
16522.8438
16521.9914
16522.7200
16522.8444
-1.5
2.1
-0.6
5
15" 5 24
1/2,11/2
5/2,15/2
3/2,13/2
1/2,11/2
5/2,15/2
3/2,13/2
16744.2662
16744.3898
16745.1402
16744.2684
16744.3885
16745.1393
-2.2
1.3
0.9
Ar 37 ClCN
OOO" 1 !!
5/2,5/2
5/2,5/2
5/2,3/2
7/2,5/2
7269.866
7270.577
1
X
-2
01 n.2
5/2,3/2
7/2,5/2
3/2,1/2
7/2,3/2
9/2,5/2
5/2,1/2
9724.302
9725.131
9725.438
02" 3 i3
7/2,3/2
9/2,5/2
5/2,1/2
9/2,3/2
11/2,5/2
7/2,1/2
2
12021.547
12022.384
12022.626
Table V-2.
J
Line centers for ArClCN .
-*•.]•', „,
-lK+l K_iK+i
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
Ar35ClCN
!3-404
7298.2691(8)
7298.2706
-1.5
^O^ll
7398.9402(7)
7398.9404
-0.2
7670.3592(10)
7670.3597
-0.5
or2i2
9891.5025(5)
9891.5036
-1.1
4
14" 5 05
10482.5052(7)
10482.5047
0.5
2
02-313
12225.1817(5)
12225.1781
3.6
5
14" 5 23
12399.7199(12)
12399.7213
-1.4
4
13~ 4 22
12854.3276(10)
12854.3251
2.5
3
12- 3 21
13325.2184(10)
13325.2192
-0.8
5
15" 6 06
13641.7369(12)
13641.7361
0.8
2
H-220
13743.0464(12)
13743.0455
0.9
3
03~ 4 14
14422.2449(7)
14422.2454
-0.5
210-2
12 21
14716.5635(9)
14716.5696
-6.1
3
5
05" 5 14
1
3
13" 3 22
15220.1227(17)
15220.1202
2.5
4
14~ 4 23
15896.8037(12)
15896.7983
5.4
4
04~ 5 15
16517.7069(12)
16517.7086
-1.5
5
15~ 5 24
16749.5614(12)
16749.5644
-3.0
109
Table V-2.
K
Continued
-l K +l"' K-1K+1
Observed
(MHz)
Calculated
(MHz)
Residue
(kHz)
Ar 37 ClCN
^O"1!!
7267.264
'oi-'u
9721.081
2
a
02~ 3 i3
12018.070
The uncertainties represent one standard deviation from the least
squares fit of the hyperfine structure.
Table V-3.
Spectroscopic constants for Ar
Constant
Value (MHz) a
A"
6152.5411(21)
B"
1577.0362(8)
C"
1246.7514(6)
T
l
-0.5760(2)
T
2
-0.09728(8)
x3b
2.059(3)
T
aaaa
-0.2348(17)
T
bbbb
-0.05597(10)
T
cccc
-0.02160(7)
Xaa
Cl
bb
Cl
Xcc
Cl b
x
ab
N
x
x
*aa
N
Xbb
N
XCc
Nb
x
x
a
C1CN.
37.9468(23)
-79.5239(20)
41.5771(43)
21.9
1.6403(22)
-3.4571(20)
1.8168(42)
0.95
ab
Values in parentheses represent one standard deviation from the least
squares fit.
Assumed value held fixed in fit.
Ill
its rotational constants by treating the CICN subunit as a rigid linear
rod having properties identical to those of free CICN. The appropriate
properties of CICN are shown in Table V-4. In the limit that CICN remains
Table V-4. Molecular constants of CICN.
Constant
Value
B0(MHz)a
5970.820(10)
X C1 (MHz) b
o
-83.27519(40)
x*0mz)h
r
-3.62277(90)
ci-c& C
1.627(1)
rc_N&c
1.166(1)
O—l c
kg-, _c(mdyn-A )
5.284(14)
°-l c
kc_N(mdyn-A )
17.982(80)
k
0.3502(5)
ci-c-N(mdyn^)C
See Ref. 83.
See Ref. 84.
c
See Ref. 85.
b
linear, the structure of ArClCN is necessarily planar and can be described by 2 parameters. These are conveniently taken to be the distance
R
from the CICN center of mass to Ar and the angle 9 cm between Rcm and
the CICN axis (see Figure V-1).
In this coordinate system, the non-zero
moments of inertia are given by (s ee Appendix I where z is along Rcm and
y is perpendicular to the molecular plane)
i a-axis
Figure V-1.
Coordinates used to describe the structure of ArClCN.
X
x
xx
=
yy
= I
h
+
^ l C N 0 0 3 9 cm
D + IC1CN
(V-13)
2
X
X
where
zz
=
^lCN 3 1 1 1 9cm
xz = x zx = "^lCN^^cm^^cm
I Q ^ C N *"S
t ie m o m e n t
*
°f inertia of free CICN and I_ is a pseudo-
diatomic moment of inertia given by
h
_ mArmClCN 2
mA + 0 1 * 0 1 1
Ar CICN
(V_14>
•
The 2 structural parameters can be obtained from any pair of A, B and C
by solving the moment equations.
The structures determined for the 3
possible choices of rotational constants are listed in Table V-5.
Table V-5.
The structures of ArClCN as derived from the
effective moments of inertia.
Pair of l°a
useda
1° and 1°
aa
bb
1° and 1°
aa
cc
1° and 1°
bb
oc
9.
R^tf)
9 cm (deg)
3.629
81.5
3.645
81.5
none b
3.645
O
0
The effective moments of inertia are obtained by I =(h/8ir A)
etc. where A, B and C are the T free rotational constants in
Table V-7.
No angle is consistent with the observed I.. .
114
It is apparent from Table V-5 that the calculated structures vary
considerably with the choice of rotational constants.
Indeed, no rigid
planar structure is consistent with the measured B and C.
This is a
consequence of the nonrigidity of the van der Waals complex as indicated
by its large inertial defect,
AI = I c c - I a a ^ b b = 2 - 7 5
The zero superscripts on the 1°
amu
^2
(v"15>
•
indicate that these are effective moments
of inertia which are derived directly from the measured rotational cono
stants.
The structures obtained from the I a a are the usual r
79
structures.
The problems posed by the inertial defect in calculating the structure
can be largely overcome if the harmonic force field is known.
It then
becomes possible to compute the moments of inertia, I^a,and the structure
of the average configuration.
Discussion of this point will be taken up
in Section V-D.
Consideration of Eq. (V-13) shows that the 2 structures having the
same R
and with angles related by 8 c m + 9^m = IT will yield the same
rotational constants.
This ambiguity in the structure determination,
namely, whether 6 c m is acute or obtuse can be settled by reference to
the results of isotopic substitution in the linear molecule. In this
37
"^S
case,
Cl was substituted for J J C1 in CICN. Assuming the geometry of
the complex remains constant upon isotopic substitution, the rotational
37
constants of Ar CICN were predicted on the basis of the 2 possible
structures (T-shaped when 6 c m is acute or L-shaped when 9 c m is obtuse).
These predicted rotational constants are compared with the measured ones
in Table V-6.
Clearly, ArClCN is T-shaped with the C-Ar line nearly
Table V-6. Measured rotational constants of Ar 37CICN and
those predicted on the basis of the two structures
consistent with the Ar3->C1CN rotational constants.
Measured
A(MHz)
B(MHz)
C(MHz)
6040.4
1554.0
1226.9
perpendicular to the CICN axis.
T-Shaped
L-Shaped
6046.
1552.
1226.
6006.
1562.
1231.
Additional structural parameters such
as the C-Ar distance and the Ar-C-N angle are easily derived from R c m
and 9 c m by simple trigonometry.
D.
Intramolecular Force Field
The effects of centrifugal distortion in the rotational spectrum of
ArClCN can be used to obtain information about its intramolecular force
field.
T
abab
Assuming planarity, four linearly independent T'S ( T a a a a , ^bbbb'
and
T
aabb^
are
related to the force field by
c.3
J
n
ag
F
=
i
3N-6 3N-6
(i)
74
,
,.s
la^ i where R., is one of the internal coordinates in Figure V-2,
3^
i
is the inverse of the force constant matrix and N=4 is the number of
atoms.
t a a a a and T b b b b are taken directly from the fit of the line
Figure V-2. Internal coordinates of ArClCN. a, and cu are the
m-plane and out-of-plane angles, respectively.
117
centers whereas T a b a b and T a a b b must be derived from any two of T^, T
and
TCCCC.
82
2
In analogy with the inertial defect, ArClCN exhibits a
82
large Tplanarity defect
(AT = -1.54(10)kHz).
This causes the calcu-
lated T a b a f a and T a a b b to vary with which pair of
to compute them.
T;J.}
T2 and
TCCCC
is used
Table V-7 shows this variability in T a b a b and T a a b b
and reflects the range of uncertainty expected on the basis of model
error alone.
It is to be noted that the model error greatly exceeds the
measurement error.
Strictly speaking, Eq. (V-16) is valid only when the laa
refer to the equilibrium configuration of the molecule.
and Jag
These are
seldom known and,in practice, vibrationally averaged moments of inertia
and structures are used.
In the procedures which follow, 2 sets of
calculations were performed.
The first set employed the effective
moments of inertia, I^a, and the r Q structure.
ft
The second set used the
average moments, I^a, and the structure of the average configuration which
were derived in the course of the first set of calculations as shown
below.
Only the results of this second set of calculations are reported
here.
The J^g in Eq. (V-16) depend solely on the masses and geometry of
74
ArClCN. They can be calculated using the method of Kivelson and Wilson
and expressions for the J^g are given in Table V-8. Alternatively, the
(i)
86
Jog can be obtained from the matrix expression
-1
a^6
Jn.»2fi
B I I X
•top
% fy % ^
ffc
(V-17)
where J a o i s a 3N-6 dimensional column v e c t o r c o n t a i n i n g the J a g , G
is
Table V-7. Derived Molecular Constants of ArClCN.
Tn and T 2
T 1 and T C C C C
x 2 and T C C C C
A(MHz)b
6152.369
6152.381
6152.384
B(MHz)b
1576.893
1576.905
1576.907
C(MHz)b
1246.701
1246.683
1246.679
T
aaaa( kHz )
-234.8(17)
-234.8(17)
-234.8(17)
T
bbbb( kHz )
-55.97(10)
-55.97(10)
-55.97(10)
T
aabb(kHz)
53.7(14)
25.3(18)
25.3(18)
^abab^Hz)
-310.3(11)
-286.5(15)
-281.7(17)
AI(amu>A )
2.739
2.747
2.748
50(13)
63.93(9)
62.52(16)
40(2)
35.17(1)
34.60(1)
12(7)
17.55(2)
17.06(4)
F ^ (mdyn-A L)
F
(mdyn*A)
F (mdyn)
25
X
a
Numbers in parentheses represent one standard deviation from the least
squares fit.
This represents the T-free rotational constant .
119
Table V-8.
The non-zero derivatives J
in terms of the internal
coordinates shown in Figure V-2.
R-^, R 2 and RQ are the
C-Cl, C-Ar, and C-N lengths, respectively, cc^ and o, are
the in-plane and out-of-plane Cl-C-N angles and a 2 is the
Ar-C-N angle,
m- is the mass of Cl, m 2 of Ar and nig of N.
The X£ and z^ are the principal axis coordinates of atom i
and 9. is the counterclockwise angle between the positive
x-axis and R J .
i
Internal
Coordinate
1
I,
R,
2m,z,sm9.,
2m-,x-LCOs9..
2
R2
2m 2 z 2 sin9 2
2m 2 x 2 cos9 2
- — = - I Izz,z cos9 2 +I xx x 2 sin6 1
3
R„
3
2m 0 z.sin6o
3 3
3
2nux 0 cos9o
3 J
3
-
(l)
xx
(i)
zz
j(i)
xz
1 fl
Iyy
2m 3
4
5
2
-2m1R;.z cos9 1
2m..R,x,sin9,
2m2R2Z2cos92
^n^RjX^inG;,
z cos8-+I
x-sineJ
r
I z„cos9»+I x„sin9 "I
I yy L zz 3
3 xx 3
3J
2m,
Ri1 r
•,
2m 1 R
=—- I z z ; z.. sin9..-I x x x.. cos9-j|
yy
2m9R_ _
—£_£ I
Iyy L
ZpSinOg-I
»
Xgcose.
J
120
42
the inverse of the well known G matrix
it
and B is the matrix which trans%
forms the Cartesian coordinates,Xi9to the internal coordinates,Rj,by
R = B X .
(V-18)
$ is the 3N dimensional column vector whose components are the atomic
coordinates in the principal inertial axis system and I a is the 3N x 3N
87
dimensional auxilliary matrix introduced by Meal and Polo.
It is
formed by placing N identical 3x3 blocks along the diagonal.
These
blocks have the form
/0 0 0\
i""
/0
0 0 l U N f l
\0 - 1 0 /
\l
0 -l\
/0
1 0>
0
0 , ^ Z = (-1 0 0 ].
0
0/
\0
(V-19)
0 0,
If J is the matrix whose columns are the Jag then
%~%I $
and Eq. (V-16) becomes
W ' f i
x'x !., ^ 8 *
(V 21)
"
4TT •LaccLBB-LYY SS
The force field of ArClCN was obtained by assuming the force constants
in CICN remain unchanged upon complex formation and using Eq. (V-16) to
fit the 4 t's to the (F~ ) . of the van der Waals modes.
It was found
that 3 inverse force constants were required to obtain a good fit of the
centrifugal distortion constants.
These correspond to the stretching of
the Ar-C distance, F~T, the bending of the Ar-C-N angle, F j 5 , and a term
121
-1
coupling the two modes, F2^.
from F
Table
by matrix inversion.
V-7
The force constant matrix, F, was obtained
The inverse force constants are shown in
for the three sets of T ' S .
The force constants listed in
Table V-9 were obtained by taking the weighted averages of the corresponding (F
) . . in Table V-7 before the matrix inversion.
While there
is still a fair amount of uncertainty in F due to the model errors, it
is reassuring that the interaction constant is negative as would be ex<- A 4 2
pected.
The normal coordinate analysis is the starting point from which
molecular information can be obtained from the force field.
This analy-
sis involves solving the secular equation
nij r\j
?\j
*Xi
f\j
A.^ = X x = 4ir^c2o)2 where oi^ is a vibrational frequency m
cm - 1 , c is
the speed of light in vacuum and L transforms the normal coordinates,
Q x , to internal coordinates by
<v 23)
1 1 1 '
"
G is the kinetic energy matrix in internal coordinates given by
«- « _ 1 £
where M~
(v 24)
-
is the inverse of the 3N x 3N dimensional diagonal matrix having
the atomic masses along the diagonal.
The vibrational frequencies for the
van der Waals modes are listed in Table V-9.
Information from the normal coordinate analysis can be used to
122
Table V-9. Vibrational force constants and frequencies for the van der
Waals modes of ArClCN and properties of ArClCN calculated
from the force field. The structure of the average configuration of ArClCN is also given.
Constant
Value
°-l
k s = F22(mdyn-A )
kB = F55(mdyn-A)
k
I ~ F25^mdyn^
0.0185
0.0333
-0.0092
vs/c(cm-1)
36.0
vB/c(cm"1)
26.3
Taaaa(kHz)
-234.6
Tbbbb(kHz)
-55.86
Taabb(kHz)
24.99
^abab(kHz)
"284-3
AI(amu-A2)
2.81
I* (amu-A2)
82.626
aa
Ibb(amu-A2)
*
323.67
02
Icc(amu-A )
406.03
W*>
3 649
9cm(deg)
82.40
Rc_ Ar (£)
'
3.622
0C1-C-Ar(des)
87
a(deg)a
10.25
a
a is the angle between CICN and the b inertial axis.
'09
a 87
compute the Coriolis coupling constants ?ij.
These, in turn, can be
used to obtain the moments of inertia of the average configuration and
an estimate of the inertial defect.
?
=%($
< v - 25 >
•
where <^ transforms the normal coordinates to the mass weighted Cartesian
coordinates, qx, according to
a = M ^ X = %Q.
(V-26)
-1
^-1 -1
From Eqs. (18), (23) and (26) and the fact that G = (L )L , it is
seen that
<»? = M~ ** B G L .
(V-27)
ft
76
The moments of inertia of the average configuration, Iact, are given by
x
ft
aa = C
3N_6
" T Z d s e s a thar)
* s=l
,
(V-28)
where the sum runs over all vibrational modes having degeneracy d and
e g (har) depends only on the harmonic force constants.
.aap
a
d n e«
(har)\ -= -
.
Aaa_ 4 y
6h
8-rr (osc L
t^s
a
\^_
2
c
s
, (V-29)
.
where
A aa o
»2
t_^
6Qg
N
=
x __£ ^ ^
i-l
2
f
(v _ 30)
The structure of the average configuration is derived from the L ^ as
shown in Section V-C and is listed in Table V-9. The average structure
computed in this fashion has the advantage that the inertial defect is
124
largely corrected for.
This permits a single structure to reproduce the
ft
three I o a . In addition, the structure of the average configuration,
which depends on (Rx)» has a well defined physical meaning.
This con-
trasts with the structures derived from the effective moments,Iaa,which
are defined only in terms of their method of computation.
Finally, the inertial defect, AL. can be calculated and compared with
the measured value in Eq. (V-15).
AI is the sum of vibrational, centri-
fugal and electronic parts,
AI = A I v i b + A I c e n t + A I e l e c .
(V-31)
The vibrational part accounts for 96% of AI in ArClCN and is the part
taken into consideration m
the computation of the 1 ^ above.
Following
77 78
Oka and Morino,
'
AI ., f° r a planar molecule in its ground vibra-
tional state and having the ArClCN geometry is given by
6
AI
vib
=
~2
2TT C
8=1
,
_
2.
° S'#S (U)s-0J
3_
t
(V-32)
)
where u>t refers to the out-of-plane bending mode.
The much smaller
centrifugal contribution is given by
Alcent " ^abab ( £ T
+
if
+
|r)
(V
"33)
and the electronic contribution is assumed to be zero. Using Eqs. (V-31),
(V-32) and (V-33), AI is calculated to be 2.81 amu-A
as compared to the
measured value of 2.75 amu-A . This good agreement indicates that useful
information about the intramolecular potential can be obtained from an
125
analysis of the centrifugal distortion and that a normal mode analysis
can be used to advantage in weakly bound complexes.
The agreement also
helps to confirm the planarity of ArCLCN since it fits the planar model
11
so well.
E.
7 8
Nuclear Quadrupole Coupling
In the limit that the electric field gradients at the Cl and N
nuclei are not perturbed upon complexation, the observed nuclear quadrupole coupling constants are obtained by rotating the free CICN coupling
constant tensors X
to the principal axes of the complex and averaging
over the ground vibrational state.
As shown in Appendix II,
— cos a- i
0
0
- i
0
- y cosasincT
0
) ,
(V-34)
3
2
1 - T cos a
where xlj is the coupling constant of the atom l in free CICN with l = Cl
or N and a is the angle between CICN and the b-inertial axis of the complex.
From Eq. (V-34), it is seen that the out-of-plane components do not
depend on a vibrational average but are simply given by
X1 = - - x j
cc
2 °
•
(V-35)
In ArClCN, -2x C 1 = -83.1542(86) and -2x^c = -3.6336(84) as compared to
X C 1 = -83.2752(4) and x? = -3.6228(9).
r*
O
*
While -2xJL and x? a *e the same
1*1—
o
Cl
Cl
within 2 standard deviations, -2x c c and x o clearly differ slightly.
indicates that the field gradient along the c-axis is perturbed to the
This
126
extent of about 0.15^.
Previous work on complexes containing a rare gas atom has not provided any evidence that the field gradients in the binding partner are
perturbed on complex formation.
It is clear, however, from studies of
the rare gas nuclear quadrupole coupling in
131
55
83
25
XeHCl
and
KrHF
and
53
N coupling in hydrogen bonded complexes such as NCCN---HF
54
and (HCN)„
that dipolar species severely perturb the field gradients in their partners.
Unfortunately, it is difficult to separate the perturbation of
the field gradient from the effects of vibrational averaging in the cases
Cl
above. The measurement of x
in ArClCN represents the first quantitative
cc
measure of how much the field gradient changes on forming a weak complex
Cl
Cl
with a rare gas atom. x a a and x b b cannot be used in the same fashion,
however, since both undergo vibrational averaging.
While /a) is known
from the rotational constants, the relation
(cos 2 a) = cos 2 (a)
does not, in general, hold.
Cl
X
(V-36)
It is conceivable that the small change in
Cl
Cl
results from large offsetting changes in x a a and x b b .
However, the
Cl
magnitude of the change in x
c
is about that to be expected from polar-
ization of CICN by a dipole moment induced in Ar.
In addition, while
Eq. (V-36) does not hold exactly, it is found to be approximately correct
as shown in the next paragraph.
The main use of nuclear quadrupole coupling in van der Waals molecules has been to aid in the determination of the vibrationally averaged
structures of inertially linear molecules.
This procedure depends on the
validity of Eqs. (V-34) and (V-36).
Since a can be determined from the
rotational constants of ArClCN, it is interesting to compare it with a
as obtained from the coupling constants.
PI
Using Eqs. (V-34) and (V-35) ,
TVT
a is found to be 9.98°(1) from x b b and 10.1(1) from x b b • This compares
o
with a = 10.25(4) from the rotational constants. As the most pronounced
effects of changes in x 0 on the structure will be felt at the small
pa
angles considered here,
it appears that no serious problems should be
encountered in using quadrupole coupling data in the structural determination of a rare gas containing complex.
128
APPENDIX I
THE ROTATIONAL CONSTANTS FOR AN ATOM-LINEAR MOLECULE COMPLEX
The moments of inertia, I, for a molecule in an arbitrary axis
system, (r,s,t), are given in terms of the atomic Cartesian coordinates
by 3 8
I rr =2> x (s x 2 + tx2) ,
i
^s -
(AI-1)
-ZWi
i
and cyclic permutations, where the sums run over all atoms having mass
mx.
The purpose of this appendix is to express the moments of inertia
in terms of the structural parameters of a molecular complex instead of
the Cartesian coordinates.
The appropriate parameters for an atom-linear
molecule complex are the distance, R, from the linear molecule center of
mass to the atom and the angle, 9, between the linear molecule and R.
Figure AI-1 shows an atom-linear molecule complex as located in two
coordinate systems, (u,v,w) and (x,y,z).
Both coordinate systems have
their origin at the linear molecule center of mass.
The w and z axes
coincide with one another and are perpendicular to the plane of the
complex.
In the (u,v,w) system, the linear molecule lies along v and in
the (x,y,z) system, R lies along x.
The moments of inertia of Lhe complex are easily derived in the
(u,v,w) system by using Eq. (AI-1) and realizing that
Y
complex
cm
V
near
cm
\6
R
X
U
Figure AI-1.
Coordinate systems used to describe the structure of an atom-linear molecule
complex. The complex lies in the uv and xy planes and the w and z axes are
perpendicular to this plane.
to
VO
130
1m
H mivi
=
h
<AI-2>
*
where the sum runs over all atoms in the linear molecule and I. is the
moment of inertia of the linear molecule. Using primed I's to denote
the moments of inertia with the linear molecule center of mass as origin,
the nonzero moments of inertia are
I' = I 0 + m„R cos'9 ,
uu
*•
a
x•'
vv
== m m R p 2s ei.l» n 2
9
a ^
^
>
(AI-3)
*ww = h + m a R 2
and
I'
uv = -nuR
a sin9cos9 ,
where m a is the masa of the atom.
These moments of inertia can be re-
lated to those in an axis system parallel to (u,v,w) but having its origin
at the center of the mass of the complex by using the Parallel Axis
Theorem.89
Using unprimed I's to denote these moments of inertia, I u u
becomes
2
I,,,, = I„ + m-R2cos29 - (ma+mj) maRcos9
ma + m^
where m^ is the mass of the linear molecule.
>
(AI-4)
Defining a pseudodiatomic
rotational constant, I., by
=
mam0
2
R
h , 1^ »
Eq. (AI-4) becomes
(AI 5)
"
131
X
uu
= Z
Z
+
I d cos2 9 .
(AI-6)
In a similar fashion, the other moments of inertia are obtained:
V
" I d sin2e
I ™ = I £ + Id
(AI-7)
I u v = - I.cos8sin9 .
To obtain the principal axis moments of inertia, the inertia tensor,
whose components are given by Eqs. (AI-6) and (AI-7) can be diagonalized.
The rotational constants are then obtained from A = (h/8ir I aa ) , etc.
In practice, it is found that one principal axis nearly coincides
with $. To put the moment of inertia tensor into a more diagonal form,
it can be transformed from the (u,v,w) axes to the (x,y,z) system by a
rotation of TT/2-6 about w. Using the usual two dimensional coordinate
transformation and realizing that cos (IT/2-8) = s m 8
and that sin(Tr/2-9)
= cos9
I(x,y,z) ={l(u,v,w) jg,
(AI-8)
where
sin8 -cos9 0\
C =1 cose sine 0
* * 0
0
1
.
(AI-9)
Using Eqs. (AI-6) - (AI-9), the moments of inertia in an axis system
parallel to (x,y,z) whose origin is at the center of mass of the complex
are found to be
x
xx
=
I*sin29
x
yy " Td
+
I
£cos
e
(AI-10)
x
T
zz -
d
+ Z
l
I x y = Iyj^ = - l£COs9sin9 .
The moment of inertia tensor in Eq. (AI-10) can be diagonalized by a
rotation of coordinates through an angle Y about the z-axis
38
where Y
is given by
tan 2Y = 2I x y /(I x x -I y y )
(AI-11)
and the principal moments of inertia are
I a a = I x x cos 2 Y+ 2IxycosysinY + I yy sin 2 Y
2
Ibb = I
U xx sin Y - 21
XX""
I
CC
=
1
ZZ
cosYsmY + I yy cos 2 Y
-^y^-W-u,
. o.yy
(AI-12)
•
In the case that IJj<<I(3 in Eqs. (AI-10), the principal moments of
inertia in Eqs. (AI-12), can be expanded in powers of (^/I^).
l(x,y,z)
is nearly diagonal, thus, Y is small and tan 2Y can be approximated by
tan 2Y % 2Y = 2I x y / (I xx -I yy ) •
(AI-13)
Using Eqs. (AI-10) and (AI-12), the binomial expansion gives (through
second order in I^/I^)
Yft (I„7ld)cos9sin9 - (I!!//Id)2(cos29-sin20).
(AI-14)
Likewise, sinY^Y and cosY%l-Y/2, hence Eqs. (AI-12) become
X
aa = W 1 " 7 * )
+
21^7(1-^/2) + hyy2
I b b = IxyY2 - 2IxyY(l-Y/2) + I y y ( l - Y 2 )
Ice
=
(AI-15)
•'•zz'
S u b s t i t u t i n g Eqs. (AI-10) and (AI-14) i n t o Eqs. (AI-15) and expanding
through second o r d e r m
(I^/I^),
x
aa = I d [ ( l £ / I d ) s m 2 e
x
bb
=
Id[L
Ice - ^
+
-
(I£/Id)2cos29sin29]
(I)i/Id)cos29 + (l£/ld)2cos26sin2e]
(AI-16)
+ xe •
The rotational constants measured for the rare gas hydrogen halides
are the averages of those about the b and c inertial axes,
B 0 = (B+C)/2 = (h/8TT2) (1/I bb + l/lcc)/2
(AI-17)
Applying the binomial expansion to Eqs. (AI-16),
1/I bb £ [l-(I£/Id)cos29 + (Iji/Id)2(cosA9-cos2esin29)]/Id
1/I CC % [l-(I£/Id) + (I^/Id) 2 ]/^
,
(AI-18)
and the rotational constant is given by
B Q = Bd[l-(Bd/2b)(l+cos26) + (Bd/b)2(l-cos20sin26+cos49)/2] ,
(AI-19)
2
where B d = (h/8tr Id) and b is the rotational constant of the linear
134
molecule.
For ArHCl, use of Eq. (AI-19) predicts B 0 to within spectro-
scopic accuracy.
Eqs. (AI-14) and (AI-15) can be used to derive the structures for
complexes that fulfill the requirement that l£ < < ^j.
In the rare gas-
hydrogen halides, the angles, a, obtained from analyses of the hyperfine
structure are related to 9 and Y by
9 = a + Y % a +
(B~ 0 /b)cosasina ,
where it has been assumed that B, £ B Q and 9%a.
(AI-20)
B d is then obtained as
the solution to Eq. (AI-19) and R from
R = (h/8TT2uBd)J5
•
(AI-21)
The structures derived in this manner compare well with those obtained
using the exact numerical technique of Chapter II.
135
APPENDIX II
RELATION OF MONOMER TENSOR PROPERTIES TO ATOM-LINEAR MOLECULE COMPLEXES
The spectroscopic properties of atom-linear molecular complexes
that relate to tensorial properties of the linear molecule are simply
those tensor components referred to the principal inertial axes of the
complex.
Under the assumption that all monomer properties remain un-
changed upon complex formation, the tensors, P 0 , describing linear
molecule properties, are diagonal in an axis system where the molecule
lies along one of the axes (taken to be the x-axis) and have the form
Px 0 0 \
0 Py 0
P0 -
%
*00P2/
-
/ Pn 0 o\
0 Pj. 0
\o
.
(AII-1)
o ?J
Due to the cylindrical symmetry of the linear molecule, one principal
plane can be chosen arbitrarily.
It is chosen here to coincide with the
plane defined by the atom-molecule system and z is taken to be perpendicular to this plane.
P Q is transformed to the principal axes of the
complex by a rotation of coordinates through a (see Figure II-l) about z:
cosa sina 0\ /P|| 0 o\/cosa -sina 0\
F. *I-sina cosa 0
0 B 0
sina cosa 0 ,
l
*
o
o
i/\o
Cl i ) \ 0
0
1/
(AII-2)
where P_ is the tensor in the principal inertial axis system of the complex.
Performing the indicated matrix multiplications,
~
f
B| cos2a+Pxsin2a
(Pn -P,)sinacosa
\
-*- 0
(P||-F, )sinacosa
P,|Sin2a+PJ_cos2a
0
0 \
0
. (AII-3)
P:JLI
136
Eq. (AII-3) is general but three specific applications are shown below.
1.
The nuclear quadrupole and nuclear spin-spin coupling are des34
cribed by traceless tensors.
1
In these cases, ?]_
=
- "T P,i
1
=
~ "T p
where P Q is the free linear molecule coupling constant and Eq. (AII-3)
becomes
/P0(3cos2a-l)/2
P r = -3P 0 sinacosa/2
*
\
0
-3P 0 sinacosa/2
P Q (l-3cos 2 a/2)
0
0
\
0
.(AII-4)
- P 0 /2 /
For the rare gas-hydrogen halides, x+a and the nuclear quadrupole coupling constant, Xa>
or
nuclear spin-spin constant, d a , for the complex is
given by
P a = P0(3cos2a-l)/2
2.
, P = x, d .
(AII-5)
The spin-rotation tensor, c 0 , for the linear molecule has c,. = 0
and c, = c Q where c Q is the spin-rotation constant.
The spin-rotation
tensor for the complex, c, is then found from Eq. (AII-3) to be
c0sin a
c = c0sinacosa
^ \
0
c0sinacosa
c Q cos 2 a
0
0
) .
(AII-6)
In the rare gas-hydrogen halides, the average of the two components of
c perpendicular to the a-axis is measured, thus,
c « cQ(l + cos2a)/2.
3.
(AII-7)
The polarizability, a, of the linear molecule with respect to
the principal axes of the complex can be obtained by using the full form
of Eq. (AII-3).
The polarizability along the a-axis is useful in inter-
137
preting the dipole moments of the rare gas-hydrogen halides and is
given by
a
a
= a
| i c o s oti+ o s i n a = (a||-ai ) c o s z a + o i .
(AII-8)
a a is easily given in terms of the average polarizability, a" = (a,,+2a|)/3,
and the anisotropy of the polarizability Aa = (a||-ai ) as
a a = - A a ( | cos2a - h
+ a
.
(All-9)
138
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VITA
Michael Robert Keenan was born on September 6, 1953, in Rochester,
New York where he attended Cardinal Mooney High School.
After obtaining
a Diploma in June, 1971, he studied medical laboratory technology at
Monroe Community College in Rochester and was awarded an A.A.S. degree,
With Distinction, in May, 1974. He enrolled in the State University
College at Brockport, New York, majoring m
B.S. degree, Summa Cum Laude, in May, 1976.
chemistry, and received a
In August, 1976, he entered
graduate school at the University of Illinois.
He served both as a
teaching assistant and a research assistant and held a DuPont Industrial
Fellowship.
His Ph.D. degree in physical chemistry will be obtained in
October, 1981.
He is a member of the American Physical Society, the
American Association for the Advancement of Science and Phi Kappa Phi
and is co-author of the following publications:
T.J. Balle, E.J. Campbell, M.R. Keenan, and W.H. Flygare, A New Method
for Observing the Rotational Spectra of Weak Molecular Complexes: KrHCl,
J. Chem. Phys. 71, 2723 (1979), (Communication).
T.J. Balle, E.J. Campbell, M.R. Keenan, and W.H. Flygare, A New Method
for Observing the Rotational Spectra of Weak Molecular Complexes: KrHCl,
J. Chem. Phys. 72, 922 (1980).
E.J. Campbell, M.R. Keenan, L.W. Buxton, T.J. Balle, P.D. Soper, A.C. Legon,
and W.H. Flygare, 8 ^Kr Nuclear Quadrupole Coupling in KrHF: Evidence for
Charge Transfer, Chem. Phys. Lett. 70, 420 (1980).
M.R. Keenan, E.J. Campbell, T.J. Balle, L.W. Buxton, T.K. Minton, P.D.
Soper, and W.H. Flygare, Rotational Spectra and Molecular Structures of
ArHBr and KrHBr, J. Chem. Phys. 72, 3070 (1980).
A.C. Legon, P.D. Soper, M.R. Keenan, T.K. Minton, T.J. Balle, and W.H.
Flygare, The Rotational Spectra of Weakly Bound Dimers of Carbon Monoxide
and the Hydrogen Halides HX (X=F, Cl, and Br), J. Chem. Phys. 73, 583
(1980).
144
M.R. Keenan, L.W. Buxton, E.J. Campbell, T.J. Balle, and W.H. Flygare,
!^lxe Nuclear Quadrupole Coupling and the Rotational Spectra of XeHCl,
J. Chem. Phys. 73, 3523 (1981).
M.R. Keenan, T.K. Minton, A.C. Legon, T.J. Balle, and W.H. Flygare,
Microwave Spectrum and Molecular Structure of the Carbon Monoxide-Hydrogen
Bromide Molecular Complex, Proc. Natl. Acad. Sci. USA 77^, 5583 (1980).
E.J. Campbell, L.W. Buxton, T.J. Balle, M.R. Keenan, and W.H. Flygare,
The Gas Dynamics of a Pulsed Supersonic Nozzle Molecular Source as
Observed with a Fabry-Perot Cavity Microwave Spectrometer, J. Chem. Phys.
74, 829 (1981).
L.T«. Buxton, E.J. Campbell, M.R. Keenan, T.J. Balle, and W.H. Flygare,
The Rotational Spectrum, Nuclear Spin-Spin Coupling, Nuclear Quadrupole
Coupling, and Molecular Structure of KrHF, Chem. Phys. 54., 173 (1981).
M.R. Keenan, L.W. Buxton, E.J. Campbell, A.C. Legon, and W.H. Flygare,
Molecular Structure of ArDF: An Analysis of the Bending Mode in the
Rare Gas-Hydrogen Halides, J. Chem. Phys. 74, 2133 (1981).
83
E.J. Campbell, L.W. Buxton, M.R. Keenan, and W.H. Flygare,
Kr and
132-Xe Nuclear Quadrupole Coupling and Quadrupole Shielding in KrHCl
and XeDCl, Phys. Rev. A (to be published).
M.R. Keenan, D.B. Wozniak, and W.H. Flygare, Rotational Spectrum,
Structure, and Intramolecular Force Field of the ArClCN van der Waals
Complex, J. Chem. Phys. (to be published).
M.R. Keenan and W.H. Flygare, A Hindered Rotor Method for Estimating
Angular Expectation Values m the Rare Gas-Hydrogen Halides, Chem. Phys.
Lett, (to be published).
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