Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients of neon gas

Abstract

Second virial coefficients for the density dependence of a number of electric properties are calculated for neon gas. Employing an accurate CCSD(T) potential for the Ne-2 van der Waals dimer and interaction-induced electric dipole polarizabilitiesand hyperpolarizabilities obtained from CCSD response theory, we evaluated the dielectric, refractivity, Kerr and ESHG second virial coefficients using both a semiclassical and a quantum statistical approach. The results cover a wide range of temperatures and are expected to be more reliable than the available experimental and empirical data. Quantum effects are found to be important only for temperatures below 100 K. The frequency-dependence of the refractivity virial coefficient is found to be small, but not negligible. For frequencies in the visible region it accounts for a few percent of the final results. For the ESHG virial coefficient of neon, frequency dependence is found to be very important, accounting for 20 - 25% of the second virial coefficient at the typical frequencies employed in experiments.

Full text

Ab initio calculation of the refractivity and hyperpolarizability
second virial coefficients of neon gas
CHRISTOF HA
¨TTIG
1
, JAVIER LO
´PEZ CACHEIRO
2
, BERTA FERNA
´NDEZ
2
, and ANTONIO RIZZO
3
*
1
Forschungszentrum Karlsruhe, Institute of Nanotechnology,
PO Box 3640, D-76021 Karlsruhe, Germany
2
Department of Physical Chemistry, Faculty of Chemistry,
University of Santiago de Compostela,
E-15782 Santiago de Compostela, Spain
3
Istituto per i Processi Chimico-Fisici del Consiglio Nazionale delle Ricerche,
Area della Ricerca, via G. Moruzzi 1, loc. S. Cataldo, I-56124 Pisa, Italy
(Received 23 October 2002; accepted 1 February 2003)
Second virial coefficients for the density dependence of a number of electric properties are
calculated for neon gas. Employing an accurate CCSD(T) potential for the Ne2van der Waals
dimer and interaction-induced electric dipole polarizabilities and hyperpolarizabilities
obtained from CCSD response theory, we evaluated the dielectric, refractivity, Kerr and
ESHG second virial coefficients using both a semiclassical and a quantum statistical approach.
The results cover a wide range of temperatures and are expected to be more reliable than the
available experimental and empirical data. Quantum effects are found to be important only
for temperatures below 100 K. The frequency-dependence of the refractivity virial coefficient is
found to be small, but not negligible. For frequencies in the visible region it accounts for a few
percent of the final results. For the ESHG virial coefficient of neon, frequency dependence is
found to be very important, accounting for 20–25% of the second virial coefficient at the
typical frequencies employed in experiments.
1. Introduction
In previous work [1–6] we carried out accurate
theoretical studies of the second virial coefficients for a
number of electric and magnetic properties of helium
and argon gases. These coefficients describe the second-
order contribution to the pressure dependence of
properties caused by pair interactions in the gas phase.
In a semiclassical approximation [7] second virial
coefficients for properties are in general obtained as
integrals over the intermolecular (or interatomic)
coordinates of the interaction-induced contribution to
the properties weighted by a Boltzmann factor, which is
calculated from the interaction potential. Within the
semiclassical approximation the calculation of second
virial coefficients is thus in principle a simple task once
accurate data or reliable models are available for the
potential and the interaction-induced properties. The
latter are, however, difficult to obtain from experimental
data [8, 9]. Additionally, the ab initio calculation of
interaction-induced properties is a non-trivial problem,
particularly if frequency-dependent or nonlinear
properties are involved, as for the refractivity and
the hyperpolarizability virial coefficients. To obtain
accurate results for such properties, highly correlated
wavefunctions and large one-particle basis sets must be
employed, making the calculation of the properties
computationally expensive.
The semiclassical approximation becomes exact in
the limit of high temperatures and is often a good
approximation to a quantum statistical treatment at and
even below room temperature. However, for very low
temperatures and for light molecular (or atomic)
systems, quantum effects contribute significantly to the
virial coefficients and must be accounted for to obtain
accurate results. A fully quantum statistical approach
requires the calculation of the expectation values of the
interaction-induced properties for the rovibrational
bound and—even more important—continuum states
of the dimer [4]. Depending on the vibrational spectrum
of the van der Waals complex and the occurrence of
quasi-bound states or resonances in the continuum, the
evaluation of the required Slater sums can become a
challenging task [4].
In previous publications [1–4] we studied, with
helium and argon, two very different prototype systems.
*Author for correspondence. e-mail: rizzo@ipcf.cnr.it: Web
http://www.icqem.pi.cnr.it/rizzo/ar.html
MOLECULAR PHYSICS, 10 July 2003, VOL. 101, NO. 13, 1983–1995
Molecular Physics ISSN 0026–8976 print/ISSN 1362–3028 online #2003 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/0026897031000109374

For helium gas, a full quantum statistical treatment is
essential for the calculation of second virial coefficients
for low temperatures, because of its low atomic mass
and the low boiling point. Moszynski et al. [10, 11] have
shown that for helium the perturbative treatment of
quantum effects diverges at low temperatures and thus
cannot be applied. However, since the He2dimer is only
very weakly bound, its rovibrational spectrum does not
include narrow quasi-bound states and the Slater sums
required in the quantum statistical approach can easily
be calculated with good accuracy. Conversely, the Ar2
dimer has a much deeper potential and its rovibrational
spectrum contains numerous quasi-bound states [4].
Some of them give rise to very narrow resonances, which
results in a difficult integration procedure. This makes
an accurate evaluation of the Slater sums rather
cumbersome. Due to its larger atomic weight and the
higher boiling point, the quantum effects on the second
virial coefficients of argon gas are very small—negligible
compared to errors caused by the remaining basis set
and correlation errors in the ab initio calculation of the
interaction-induced properties.
Here we present the results of an investigation of
second virial coefficients for the electric properties of
neon gas as an example of an intermediate case, where
quantum effects are expected to be small but non-
negligible. Results are reported for the semiclassical
approximation and for a full quantum statistical
approach. Besides the comparison with experiment, an
important aspect of the present study is to obtain some
insight into the accuracy of the semiclassical approx-
imation for different molecular weights and depths of
the van der Waals potential.
Only few experimental data for second virial coeffi-
cients of neon gas are present in the literature. The gas
virial coefficient was determined experimentally by
Gibbons [12] and by Dymond and Smith [13] for
temperatures ranging from 44 to 550 K. Many authors
have used these results to check the quality of the
intermolecular potentials. Employing an ab initio poten-
tial curve Eggenberger et al. [14] evaluated, among other
thermodynamic properties, the gas second virial coeffi-
cient for the same temperature range. The authors found
quantum effects to be smaller than or comparable in size
to the experimental error (1cm
3mol1) for tempera-
tures down to about 60 K. However, the deviations were
found to increase with decreasing temperature when
compared to the available experimental data.
The interaction-induced polarizabilities of Ne2have
been studied by many authors, mainly theoretically
using empirical or ab initio data. The results of these
studies are far from being satisfactory. Frommhold and
Proffitt [15] studied the collision-induced Raman
spectra of neon and found the dipole–induced dipole
(DID) model for the polarizability anisotropy to be
inconsistent with the experimental results. Interaction-
induced polarizability anisotropies from quantum
chemical calculations gave even worse agreement [16].
The authors claimed their calculations to be too
inaccurate.
Dacre evaluated the neon pair polarizability using the
configuration interaction (CI) method. He found that the
dielectric second virial coefficient calculated from these
results (0:15 cm6mol2) did not agree well with the
available experimental data [17, 18]. A comparison with
an empirical model for the pair polarizability anisotropy,
derived by Meinander et al. [19] from moment analysis in
depolarized light scattering experiments, gave for neon
an agreement not as good as that found for other noble
gases. Ceccherini et al. [20] obtained a model for the
interaction-induced polarizability that allowed the repro-
duction of the available measurements of the interaction-
induced binary depolarized Raman spectra. When
compared with the ab initio results [17], deviations
attributed to the inaccuracy of the ab initio calculations
and the experimental data were found.
Maroulis quite recently reported the results of a
study of static interaction (hyper)polarizabilities of
several systems, including the neon dimer. The author
used a finite many-body perturbative approach and
coupled cluster (CC) theory [21]. Very recently his
studies also extended to mixed dimers (i.e. NeAr) [22].
In [21] the author employs a 7s5p4d1fbasis set and
shows that the effect of triple excitations, as shown by
the comparison between the results of a CC singles
and doubles (CCSD) approach and those obtained
using a CCSD wavefunction with a perturbative
treatment of triple excitations (CCSD(T)), is non-
negligible—on the order of a few percent, although the
size of the basis set is not sufficient to allow for
definitive conclusions.
Experimentally, the dielectric second virial coefficient
of neon has been determined by Orcutt and Cole [23]
at 323 K. The value is (0:30 0:10) cm6mol2. The
authors pointed out that for all the six gases they
studied the coefficients are less than those predicted at
that time by theories of fluctuations and pair polariz-
abilities. Vidal and Lallemand [24] determined at
298.15 K a value of (0:24 0:04) cm6mol2whereas
Huot and Bose [25] measured values of (0:07 0:02),
(0:10 0:30) and (0:12 0:06) cm6mol2at 77.4,
242.95 and 323.15 K, respectively. When trying to
reproduce the experimental behaviour theoretically, all
attempts failed. In the most recent reference in the
literature, Bulanin et al. [26] calculated the coefficients
for a wide range of temperatures using HFD interac-
tion potentials and the CI results taken from the work
of Dacre [17, 18]. With these data it was not possible
1984 C. Ha
¨ttig et al.

to reproduce the experimental temperature dependence
of the coefficient. Also, theoretical predictions based on
the DID model or an empirical model developed by
Meinander [27], which accounts also for short-range
effects, are in disagreement with measurements carried
out, e.g. by Burns et al. [28] and by Achtermann et al.
[29], for the second refractivity virial coefficient of
neon.
In the next section we present the results of accurate
coupled cluster calculations for the potential and for the
frequency-dependent interaction-induced polarizability
and hyperpolarizability of the Ne2van der Waals dimer.
In section 3 these results are employed to evaluate
refractivity, Kerr and electric field-induced second
harmonic generation (ESHG) second virial coefficients,
using both the semiclassical approximation and a
quantum statistical approach. In the last section we
give some concluding remarks.
2. Ab initio calculation of the potential and the
interaction-induced (hyper)polarizability
For the present investigation we use an approach
similar to that adopted in our previous work on argon
and helium [1, 2]. Both the interaction potential and the
interaction-induced properties are calculated using size-
consistent coupled-cluster methods and the counterpoise
correction proposed by Boys and Bernardi [30] to
account partially for basis set superposition errors.
The latter correction can be formulated for a general
size-extensive property Pin the case of a homonuclear
dimer as
PðRÞ¼PAAðSAAjRÞ2PAðSAAjRÞ;ð1Þ
where PAAðSAAjRÞis the property of the dimer
evaluated in the dimer basis set SAAat the interatomic
distance Rand PAðSAAjRÞis the property of the
monomer evaluated in the dimer basis set for the same
geometry.
The potential of the Ne2dimer has been investigated
previously by several authors using a large variety of
methods and basis sets. A number of very accurate
potential curves have been published in the literature
[31–35]. Most of them are, however, only available close
to the equilibrium distance, located at re310 pm,
which provides insufficient information for the calcula-
tion of virial coefficients. Therefore, we computed the
Ne2potential using the CCSD(T) model and the d-aug-
cc-pV5Z basis set [36–40] for 33 interatomic distances
ranging from 3 to 30 bohr (158.75 to 1587.5 pm). These
calculations were carried out using the coupled cluster
program developed by Koch et al. [41–43]. The same
potential has been used recently by Halkier et al. [44] in
a study of the nuclear quadrupole coupling constant of
21Ne and it was shown to give an accurate description of
the van der Waals interaction with re¼310:5 pm and
De¼127:9Eh(or 40.40 K) close to the highly accurate
results by Gdanitz of 310:07 pm and 131:5Eh, respec-
tively [35]. A HFD-B-like [45] analytic function was then
employed to fit the ab initio calculated points
VðRÞ¼ 
kB
V*ðRa0=rmÞ;ð2Þ
V*ðrÞ¼A*exp ð*rþ*r2ÞFðrÞX
2
n¼0
c6þ2n
r6þ2n;ð3Þ
FðrÞ¼
exp ðD=r1Þ2

;for rD;
1;for rD:
8
<
:
ð4Þ
The parameters ,*,*,rm,c6,c8and c10 were
determined by a weighted least-squares fit; the multi-
plicative constant A*and the parameter Dwere taken
from [45]. The original points are listed in table 1 for
some selected internuclear distances, whereas the values
of the parameters are given in table 2.
As in [2], to give a measure of the quality of the
potential energy curve we computed the second gas
virial coefficient BðTÞ, which in the classical approach is
given by the expression
BðTÞ¼2pNAZ1
0
exp VðRÞ=kBT1

R2dR;ð5Þ
whose validity is limited to temperatures high enough
for quantum effects to be negligible. kBis the Boltzmann
constant. The results obtained by integrating the
Table 1. Ne
2
potential calculated at the
CCSD(T)/d-aug-cc-pV5Z level.
R/bohr VðRÞ=Eh
3.0 91163.5497
3.7 15519.8232
4.7 827.3348
5.15 64.0232
5.3 29.5414
5.5 95.9638
5.8 127.0377
6.0 125.4761
6.5 97.3739
7.0 67.2280
8.0 30.2824
10.0 7.3516
15.0 0.5917
20.0 0.1023
30.0 0.0087
Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients 1985

analytic function given above between 0.1 and 500 bohr
are given and compared to available experimental data
in table 3 and in figure 1. The agreement between our
estimates and experiment is fair for all temperatures,
which suggests that quantum effects are probably small
for this property down to 40 K. The sensitivity of the
semiclassical integral in equation (5) to the features of
the potential and the important role that regions of very
low interatomic distances maintain in the determination
of BðTÞhave been discussed in [2].
Since the virial coefficients are—at least for noble
gases—only weakly frequency dependent [1, 3], the
dispersion of the interaction-induced properties was
approximated to second order in the frequency as
ð!;!Þ¼þ!2Sð4ÞþOð!4Þ;ð6Þ
jjðOÞ¼jj þ!2
Lðjj AÞþOð!4
iÞ;ð7Þ
Kð!;0;0;!Þ¼jj þ!2DKð2ÞþOð!4Þ;ð8Þ
with Oemployed here and below as a shorthand
notation for the sequence !0;!1;!
2;!
3and
!2
L¼!2
0þ!2
1þ!2
2þ!2
3. This reduces the computa-
tional efforts considerably and it leads to virial
coefficients which are also correct through Oð!2Þ.In
addition to the static interaction polarizability—both
the isotropic part ave and the anisotropy ani—and
the hyperpolarizability jj, the interaction-induced
Cauchy moments Sð4Þand the dispersion coefficients
for the isotropic and the dc-Kerr hyperpolarizability
ðjj AÞ¼Djjð2Þand DKð2Þare thus needed. At
the CCSD response level these properties and dispersion
coefficients are available analytically (i.e. without
employing fits) using the program described in [46–49],
which is available in the DALTON 1.2 package [43].
These calculations were carried out for 32 interatomic
distances between 3 and 20 bohr and employed the
d-aug-cc-pVQZ basis set of Dunning and co-workers
[36–40], augmented with a ð3s3p2d1f1gÞset of diffuse
midbond functions [1, 3, 50, 51]. The resulting basis set
is denoted in the following as d-aug-cc-pVQZ-33211. An
investigation carried out at R¼5:0 au (see table 4)
indicates that with the d-aug-cc-pVQZ-33211 basis the
interaction-induced polarizabilities and hyperpolariz-
abilities are obtained within a few percent of the
CCSD basis set limit. To interpolate the interaction
properties between the calculated points, analytic
functions were fitted to the ab initio results. A selection
Table 3. The gas second virial coefficient B(T) (in
cm
3
mol
1
). Semiclassical results and comparison with
experimentally available data.
T/K This work Exp. Ref.
44 46.36 46.1 [12]
50 36.02 35.4 [12]
60 24.33 24.9 [12]
70 16.58 17.1 [12]
80 11.10 12.8 [13]
100 3.89 6.0 [13]
125 1.45 0.4 [13]
150 4.78 3.2 [13]
200 8.60 7.7 [13]
250 10.65 10.0 [13]
300 11.88 11.3 [13]
350 12.66 12.1 [13]
400 13.17 12.7 [13]
450 13.52 13.1 [13]
500 13.76 13.4 [13]
550 13.92 13.7 [13]
Table 2. Parameters of the weighted least-squares fits of the
CCSD(T)/d-aug-cc-pV5Z potential for Ne
2
to a HFD-B-
like function, equations (2)–(4). The values of the
parameters are given for Rin bohr and V(R) in hartree;
the conversion from hartree to K (i.e. the Boltzmann
constant in atomic units) was taken as 315 773:0KE
1
h:
Parameter Value
A895717.95
12.041788
0.2289419
c62.6178124
c81.5141754
c10 0.5777010
rm2.768605 A
˚
D1.36
=kB36.895522 K
Figure 1. Temperature dependence of the gas second virial
coefficient. Comparison is made with previously available
results.
1986 C. Ha
¨ttig et al.

of the original points are reported in table 5, whereas the
fitting functions and the parameters are given in table 6.
In figures 2 to 4 the present results for the interaction
properties are compared with the latest available
empirical functions. For the interaction-induced iso-
tropic polarizability we compare the present results with
those obtained by Frommhold and Proffitt using the
data obtained from the polarized Raman spectrum [15].
The parameters for the function (see equation (16) in
[15]) were taken as l
t¼135 a3
o,rt¼0:626 aoand
A6¼224 a9
o, corresponding to the set that gave the
best agreement with respect to the spectral results. The
empirical function shows a steeper decrease than the ab
initio calculated curve in the region of short Rvalues,
whereas the agreement for long interatomic distances is
good (figure 2). We also compare to the results of
Frommhold and Proffitt for the anisotropic contribu-
tion, obtained from the depolarized Raman spectrum
[15]. The parameters for the function (equation (11) in
[15] in this case) were taken as l¼720 a3
o,ro¼0:509 ao
Table 4. Basis set convergence of the CCSD results for the
interaction-induced polarizabilities ave,ani and
hyperpolarizabilities jj,?. The calculations have
been carried out at R= 5.00 bohr. The results are given in
atomic units (au).
Basis Set ave ani jj ?
d-aug-cc-pVDZ-33211 0.0358 0.2473 15.8325 5.2775
d-aug-cc-pVTZ-33211 0.0292 0.2415 9.4955 3.1652
d-aug-cc-pVQZ-33211 0.0281 0.2389 9.3566 3.1189
d-aug-cc-pV5Z-33211 0.0277 0.2379 9.4075 3.1358
Table 5. Interaction-induced polarizabilities ave and ani, Cauchy moments Save ð4Þand Sani ð4Þ, isotropic
hyperpolarizability jj and hyperpolarizability dispersion coefficients Djjð2Þand DKð2Þas a function of the internuclear
distance R calculated at the CCSD/d-aug-cc-pVQZ-33211 level. All values given in au.
Rave Saveð4Þani Sani ð4Þjj Djjð2ÞDKð2Þ
3.00 0.31554 0.42553 0.28010 1.01960 14.5927 166.764 345.166
3.50 0.22391 0.51976 0.21375 0.26831 15.6518 21.0307 39.6770
4.00 0.12894 0.38192 0.24078 0.21884 17.9822 49.7398 99.2923
4.50 0.06440 0.23332 0.25306 0.29131 14.1495 46.2748 93.2433
5.00 0.02814 0.12446 0.23892 0.34408 9.35690 34.5483 70.0892
5.50 0.01044 0.05849 0.20944 0.35229 5.41368 22.5601 46.0781
6.00 0.00282 0.02360 0.17594 0.32803 2.75407 13.1938 27.1574
7.00 0.00081 0.00047 0.11856 0.24598 0.38196 3.14428 6.66690
8.00 0.00073 0.00225 0.08050 0.17319 0.14906 0.06507 0.30548
10.00 0.00023 0.00094 0.04115 0.08925 0.12556 0.42641 0.79582
12.00 0.00009 0.00035 0.02377 0.05143 0.05314 0.19417 0.36394
14.00 0.00004 0.00015 0.01495 0.03228 0.02251 0.08331 0.15744
16.00 0.00001 0.00005 0.01000 0.02156 0.00853 0.03331 0.06390
18.00 0.00001 0.00002 0.00701 0.01511 0.00299 0.01214 0.02360
20.00 0.00000 0.00001 0.00511 0.01101 0.00108 0.00428 0.00852
Table 6. Parameters of the weighted least-squares fits to the
interaction-induced properties. The fit functions have
been chosen as follows: PðRÞ¼exp ððRsÞ2=Þþ
P3
n¼0A6þ2nR62nfor ave and Saveð4Þ;P(R) =
exp ððbR3þRsÞ=ÞþA3R3þP2
n¼0A6þ2nR62n
for ani and Sanið4Þ, and PðRÞ¼Df½1ð1þbRþ
cR2Þexp ðaðRrÞÞ21gþexp ð1exp ½aðRerÞÞ
A6R6for jj;Djj ð2Þand DKð2Þ:
ave Saveð4Þ
s0.783255 1.87398
5.34016 5.13093
A6297.685 1244.76
A86905.68 36669.9
A10 59806.0 376798
A12 151689 1139134
ani Sanið4Þ
b8.00900 10313.1712 103
s3.80920 5.31723
0.986526 1.50426
A340.9336 88.2682
A6105.888 631.809
A86125.93 3698.18
A10 28987.1 0.
jj Djjð2ÞDKð2Þ
D33.4726 136.798 131.356
b0.950276 0.534505 0.639462
c0.313117 0.144986 0.111701
a1.69236 1.53458 1.75300
r3.47856 3.79049 5.00979
e1.12857 0.973356 0.913738
A655417.2 236713 375150
Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients 1987

and ¼0:3992 1024 cm3(2.6939 au). The agreement
with the present results is worse than for the isotropic
component, especially in the region of short Rdistances
(dashed curve in figure 3). For the anisotropy in figure 3
we compare also with the empirical function obtained by
Meinander et al. from the depolarized collision-induced
light scattering spectrum [19] (dotted curve, equation
(3 a) in [19], parameters: B¼50 
A3,0¼0:3989 
A3
(2.6919 au), A¼0:664 
A9,r0¼0:294 222 
A) and with
the results of Ceccherini et al. [20] (dashed-dotted curve,
equation (12) in [20], parameters: r1¼0:755 au,
r0¼0:8426 au, G¼101 au, C6¼6:8 au). The latter
curve gives the best overall agreement with our ab initio
values.
The results of the recent study by Maroulis [21] are
also included in figures 2 to 4. Although the interaction-
induced properties were computed employing rather
sophisticated methods as CCSD and CCSD(T), the
quality of the results is probably still highly influenced
by the limited basis set used in [21]. We also note that
the effect of linked triple excitations for second hyper-
polarizabilities is much larger in the orbital-relaxed
finite field coupled cluster method used in these
calculations than in the CC response approach adopted
in the present work [52]. Maroulis mentions numerical
instabilities in the finite field method, and these are more
than evident in figure 4, where the average parallel
interaction-induced hyperpolarizability as computed in
[21] for Rbetween 7 and 10 au shows no intention of
adapting to the expected long range R6pattern.
3. Calculation of the virial coefficients
3.1. Semiclassical approximation
Within the semiclassical approximation, the refractiv-
ity second virial coefficient is given by the expression [53]
BRð!; TÞ¼N2
A
60
4pZ1
0
aveð!; RÞ
exp VðRÞ=kBTR2dR:ð9Þ
The dielectric second virial coefficient BðTÞis defined as
the static (!¼0) limit of BRð!; TÞ. Inserting for
aveð!; RÞthe expansion in Cauchy moments,
equation (6), one obtains [1]:
BRð!; TÞ¼BðTÞþ!2Bð2Þ
RðTÞþ...;ð10Þ
Figure 3. Interaction-induced static anisotropic polarizabil-
ity in terms of Rand the analytical function fitted to it (see
text). The empirical curves obtained by Frommhold and
Proffitt [15], Meinander et al. [19] and Ceccherini et al.
[20] are also plotted, together with the CCSD(T) results of
Maroulis [21].
Figure 2. Interaction-induced static isotropic polarizability
in terms of Rand the analytical function fitted to it (see
text). The empirical curve obtained by Frommhold and
Proffitt [15] is also plotted, together with the CCSD(T)
results of Maroulis, see [21].
Figure 4. Interaction-induced static isotropic hyperpolariz-
ability in terms of Rand the analytical function fitted to it
(see text). Also enclosed are the CCSD(T) results of
Maroulis [21].
1988 C. Ha
¨ttig et al.

where the second-order coefficient Bð2Þ
RðTÞis calculated
analogous to equation (9), but with aveð!; RÞreplaced
by Saveð4;RÞ. The second Kerr virial coefficient has
contributions related to the interaction-induced polariz-
ability anisotropy and to the dc-Kerr hyperpolarizability
Kð!; RÞ, respectively
BKð!; TÞ¼ N2
A
16201
5kBTX
Kð!; TÞþX
Kð!; TÞ
¼B
Kð!; TÞþB
Kð!; TÞ;ð11Þ
with X
Kð!; TÞand X
Kð!; TÞdefined as
X
Kð!; TÞ¼4pZ1
0
anið!; RÞani ð0;RÞ
exp VðRÞ=kBTR2dR;ð12Þ
X
Kð!; TÞ¼4pZ1
0
Kð!; RÞ
exp VðRÞ=kBTR2dR:ð13Þ
As in [1], X
Kð!; TÞand X
Kð!; TÞare expanded through
second order in !using the relationships equations (6)
and (8), but without employing Kleinmann symmetry, in
contrast to [1]. Finally, in the semiclassical approxima-
tion the second virial coefficient for the isotropic
hyperpolarizability jjðOÞis obtained as [1, 54]
BðO;TÞ¼ NA
8p0
4pZ1
0
jjðO;RÞ
exp VðRÞ=kBTðÞR2dR:ð14Þ
The frequency dependence of BðO;TÞis again approxi-
mated using the expansion equation (7) through second
order:
BðO;TÞBð0Þ
ðTÞþ!2
LBð2Þ
ðTÞ;ð15Þ
where Bð0Þ
ðTÞand Bð2Þ
ðTÞare calculated similar to
equation (14), but again with jjðO;RÞreplaced by
the static interaction-induced hyperpolarizability and
by jjð0ÞA, respectively.
To compute the virial coefficients we employed the fits
for the Ne2potential and for the interaction-induced
properties reported in tables 2 and 6, respectively. The
required integrals were evaluated by means of numerical
quadrature.
3.2. Quantum statistical calculations
The quantum statistical expression for the second
dielectric virial coefficient was derived in detail by
Moszynski and co-workers [10, 11, 55]. In a quantum
statistical treatment the average over the semiclassical
expression for the pair density exp ðVðRÞ=kBTÞis
essentially replaced by a derivative of the trace of the
quantal density operator with respect to external fields,
where the derivative taken is the same as that taken from
the (quasi-)energy to obtain the respective underlying
property of the dimer. Thus the dielectric second virial
coefficient is obtained as
BðTÞ¼2pkBT
3Z1
0
@2WðR;FÞ
@F2

F¼0
R2dRð16Þ
¼2p
3Z1
0
WðR;0ÞaveðRÞR2dR;ð17Þ
where Fis the strength of an external homogenous
electric field and WðR;FÞis the Slater sum [11, 56]:
WðR;FÞ¼l3
BhRjexp ðHðFÞÞjRi
þl3
Bð1Þ2I
2Iþ1hRjexp ðHðFÞÞjRi:ð18Þ
In the last equation we have introduced the field-
dependent Hamiltonian HðFÞ¼ðhh2=2Þr2þVðRÞ
ð1=2ÞzzðRÞF2þOðF3Þ, the thermal wavelength
lB¼ð2phh2=kBTÞ1=2and the nuclear spin quantum
number I. The reduced mass is denoted by .
Inserting the complete set of eigenfunctions of Hð0Þ,
corresponding to the bound and continuum rovibra-
tional states of the (unperturbed) dimer, one obtains the
expression [11]:
BðTÞ
¼2pl3
B
3X
1
J¼0
ð2Jþ1Þ1þð1ÞJþ2I
2Iþ1

X
n
exp EnJ
kBT

hnJ ðRÞjaveðRÞjnJ ðRÞi
þZ1
0
exp 
hh2k2
2kBT

hkJ ðRÞjaveðRÞjkJ ðRÞi 
hh2kdk;
ð19Þ
where Jis the rotational quantum number and nJ (kJ )
are the radial parts of the bound (continuum) eigen-
functions of the field-independent Hamiltonian. See
[4, 11] for details.
A similar approach can also be used for the other
second virial coefficients. For some properties, however,
additional assumptions have to be made; e.g. if frequency
dependence has to be considered [57]. If products of
interaction-induced properties contribute, as for example
for the Kerr second virial coefficients, the exact quantum
statistical expressions become more involved due to the
appearance of an extra integration which accounts for
the non-vanishing commutator between the Hamiltonian
Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients 1989

for the intermolecular motion Hð0Þand the interaction-
induced properties PðRÞ[57, 58]. In such cases it is
convenient to introduce the additional assumption that
these commutators are negligible. This approximation
becomes exact in the classical limit, i.e. for small thermal
wavelengths, and it provides an upper bound for BKðTÞ
[58]. Under these conditions the (approximate) quantum
statistical expression for second virial coefficients is
obtained by replacing the pair density exp ðVðRÞ=kBTÞ
by the (field-independent) Slater sum WðR;0Þin the
semiclassical expressions given in section 3.1. For
example, the (approximate) expression for the Kerr
second virial coefficients is
BKð!; TÞ¼ N2
A
16201
5kBTX½anið!; RÞani ð0;RÞ;T
þX½Kð!; RÞ;T;ð20Þ
with
X½PðRÞ;T
¼4pl3
BX
1
J¼0
ð2Jþ1Þ1þð1ÞJþ2I
2Iþ1

X
n
exp EnJ
kBT

hnJ ðRÞjPðRÞjnJ ðRÞi
þZ1
0
exp 
hh2k2
2kBT

hkJ ðRÞjPðRÞjkJ ðRÞi 
hh2kdk:
ð21Þ
The evaluation of the quantum statistical expressions
for the virial coefficients requires two numerical
integrations, one over the interatomic distance Rand
the other over the wavenumber k, plus an (in principle)
infinite summation over the angular momentum J. The
summation over n(the bound states supported by the
potential of the dimer) ranged in our case from 0 to 2 for
the lowest (J=0) rotational quantum number, with
D0¼15:40 cm1,D1¼2:50 cm1and D2< 0.03 cm
1
.
Experimentally [59, 60] three stable vibrational levels
were detected, with the highest lying very close to the
dissociation limit: D0¼16:3cm
1,D1¼2:6cm
1and
D20:01 cm1. Cybulski and Toczylowski [33] found
two vibrational levels in their ab initio study exploiting a
CCSD(T) wavefunction and using, among others, an
aug-cc-pV5Z basis set supplemented with a set of
3s3p2d2f1gmidbond functions. The recent study by
Gdanitz [35], improving slightly on the potential curve
of [33], determines again three vibrational levels for
J=0 in the neon dimer, with D0¼16:406 cm1,
D1¼2:789 cm1and the third and highest level placed
only 0.004 cm1below the dissociation limit. For the
integration over Rwe used an even-spaced grid ranging
from 0.5 to 100 bohr with a step width of 0.0005 bohr for
J<250, and a different grid, going from 0.5 to 200 bohr
with a step width of 0.0002 bohr for J250. These
grids were used both for the solution of the vibrational
Schr€odinger equation and for the evaluation of the
matrix elements hðRÞjPðRÞjðRÞi. The atomic mass of
Ne was taken as 19.992 4356 amu [61]. For the integra-
tion over the wavenumber kan even-spaced grid cannot
be applied—at least not for all values of J—since for
certain values of Jvery narrow resonance peaks appear,
which require more elaborate integration schemes. We
employed the algorithm described in [4], which accounts
explicitly for the quasi-bound states and which allows
for an accurate and efficient integration of the matrix
elements over k. For J<250, this integration was
carried out for kfrom 0.5 to 200 au, whereas for J250
the interval went from 1.0 to 300 au. Finally, the
summation over Jwas carried out up to J¼500.
These parameters were chosen such that convergence
within about 0.01% in the summations was reached.
The fact that indeed convergence within this accuracy is
reached in the numerical integrations is corroborated by
the comparison between quantum statistical and semi-
classical results for the higher temperatures, where the
latter become exact. The two approaches should give
practically identical results, which is what we find within
the specified accuracy.
The total errors in the virial coefficients are thus
dominated by the errors caused by the limited number
of available data points for the potential and the
interaction-induced properties and, in particular, by
the approximations made in the quantum-chemical
calculations, i.e. the incompleteness of the one-electron
basis set and the residual electron correlation error.
All calculations for the second virial coefficients were
carried out with a modified version of the KORG
program [62] originally written by I. Cacelli, and which
has been extended for the calculation of second virial
coefficients. For the present task we have improved the
code such that the fits to the ab initio calculated points,
reported in section 2, can be used for the calculation of
second virial coefficients without employing intermedi-
ate spline interpolations.
Conversion factors between au, SI and esu units for
the properties presented and discussed in the next
section can be found in [3].
4. Results and discussion
4.1. Dielectric and refractivity second virial coefficients
The results for the dielectric and refractivity second
viral coefficients of neon are given in table 7. Both
quantum statistical and semiclassical results are
reported and compared to the experimental values
1990 C. Ha
¨ttig et al.

available. In figure 5 the data are plotted against the
temperature. As anticipated in the introduction, due to
the mass of the Ne atom being in between that of helium
and argon, the contribution of the quantum effects to
the virial coefficient is in between those obtained for
helium and argon [4]. Only at temperatures below 100 K
do the quantum effects start to be significant (>5%) for
neon. At the experimentally studied temperatures, these
effects are negligible and the semiclassical values can be
expected to give accurate estimates of the coefficients.
Frequency dispersion is also rather weak, being con-
tained between 5% (at low temperatures) and 3% (at
high temperatures) for a wavelength of 632.8 nm. The
effect of frequency is only slightly more accentuated in
the quantum statistical calculations at low temperatures.
When comparing with the available experimental data,
we can see that, as happened with the helium and argon
results [4], the spread of the experimental results is very
large and most values have large error bars. This makes
any discussion on the supposed agreement or disagree-
ment between theory and experiment quite useless. The
experimental datum at 77.4 K carries a probably too
optimistic error bar, given the level of accuracy of our
study. Therefore, and as concluded also in previous
papers devoted to the study of helium and argon, we
advocate new experimental determinations which might
shed some light on the comparison with the theoretical
data. Recent advances in the field are in this respect
particularly promising [8].
Table 7. The dielectric and refractivity second virial coefficients (cm
6
mol
2
).
Semiclassical Quantum statistical Experimental
T/K BðTÞBð2Þ
RðTÞBðTÞBð2Þ
RðTÞBðTÞ;BRðTÞ
40.0 0.0659 0.6718 0.0551 0.5810
77.4 0.0525 0.4997 0.0492 0.4715 0.07 0.02 [25]
a
100.0 0.0546 0.4868 0.0524 0.4683
140.0 0.0614 0.4948 0.0600 0.4841
180.0 0.0690 0.5151 0.0681 0.5080
220.0 0.0766 0.5386 0.0759 0.5337
242.95 0.0808 0.5525 0.0803 0.5485 0.10 0.03 [25]
a
273.15 0.0863 0.5706 0.0859 0.5676
298.15 0.0906 0.5853 0.0903 0.5830 0.24 0.04 [24]
a
0.14 0.14 [28]
b
0.11 0.22 [28]
c
0.06 0.09 [63]
d
303.15 0.0915 0.5882 0.0912 0.5861 0.11 0.02 [29]
e
323.0 0.0949 0.5997 0.0947 0.5980 0.30 0.10 [64]
a
323.15 0.0949 0.5998 0.0947 0.5981 0.12 0.06 [25]
a
0.22 0.02 [65]
a
370.0 0.1025 0.6259 0.1025 0.6252
410.0 0.1088 0.6473 0.1088 0.6473
a
B;
b
BRðl¼632:8 nm) at T=298.2 K;
c
BRðl¼457:9 nm) at T=298.2 K;
d
BRðl¼632:8 nm) at T=299 K;
e
BRðl¼632:8 nm); cf. computed 0:157 cm6mol
2
.
Figure 5. Temperature dependence of the dielectric second
virial coefficient BðTÞ—semiclassical (SC) and quantum
statistical (QS) contributions—and of the refractivity
second virial coefficient BRð!; TÞcomputed at
l¼632:8 nm. The available experimental data for BðTÞ
and BRð!; TÞare also shown.
Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients 1991

4.2. Second Kerr and ESHG virial coefficients
The results obtained for the second Kerr virial
coefficient BKð!; TÞare given in table 8, where the two
contributions B
Kð!; TÞand B
Kð!; TÞare further split
into their static and dispersion parts. Figure 6 shows
the behaviour of BKð!; TÞfor neon for a wavelength of
632.8 nm.
For neon, as for helium, the deviations from linearity
found in the experimental work by Buckingham and
Dunmur dated 1968 were negligible [66]. A linear fit to
the Kerr constant was found to be adequate for densities
of more than 103mol cm3. To our knowledge, no
other experimental studies of the density dependence of
the Kerr constant of neon appear in the literature. Our
study of [3] shows that in the case of helium the two
contributions B
Kð!; TÞand B
Kð!; TÞessentially tend to
cancel each other at temperatures approaching standard
conditions, with their sum BKð!; TÞchanging sign
approximately around room temperature. This beha-
viour is perfectly in line with the experimental evidence
of [66]. The effect does not appear in the case of neon,
where, as figure 6 clearly shows, BKð!; TÞ—at least
for l¼632:8 nm—decreases smoothly as Tincreases,
remaining positive at least up to 410 K. Here the
B
Kð!; TÞcontribution decreases steadily as the tem-
perature increases, whereas B
Kð!; TÞremains practically
unchanged and from 30 times to twice smaller than the
other contribution. The effect of pair interactions in
neon is thus small but still stronger at room temperature
than predicted and observed for helium.
Quantum effects are significant only below 50–60 K,
where they decrease the value of BKð!; TÞby more
than 5%. Above 180 K the difference between quantum
statistical and semiclassical results in table 8 is less
than 1%. This is also the limit of accuracy of the
numerical integration employed for these calculations.
Frequency dispersion modifies the static values of
B
Kð!; TÞby 2% and those of B
Kð!; TÞby 10%,
rather independent of the temperature or the approach
(either quantum statistical or semiclassical). Due to the
predominance of the polarizability contribution over
the whole investigated temperature range (see table 8),
the overall effect on BKð!; TÞremains contained.
The results for the second ESHG virial coefficient of
neon are collected in table 9, where both the static
Table 8. The Kerr second virial coefficients ðV2m8mol21035Þ. We define B
Kð!; TÞ¼Bð0Þ;
Kþ!2Bð2Þ;
Kand analogously
B
Kð!; TÞ¼Bð0Þ;
Kþ!2Bð2Þ;
K, with !assumed to be given in au.
Semiclassical Quantum statistical
T/K Bð0Þ;
KðTÞBð2Þ;
KðTÞBð0Þ;
KðTÞBð2Þ;
KðTÞBð0Þ;
KðTÞBð2Þ;
KðTÞBð0Þ;
KðTÞBð2Þ;
KðTÞ
40.00 29.27 56.54 0.94 11.08 26.87 51.93 0.88 9.66
77.40 11.77 22.76 0.65 7.84 11.37 21.97 0.64 7.39
100.00 8.68 16.75 0.62 7.36 8.47 16.34 0.61 7.07
140.00 5.97 11.47 0.61 7.07 5.88 11.29 0.60 6.90
180.00 4.58 8.75 0.61 7.02 4.53 8.65 0.61 6.90
220.00 3.73 7.10 0.63 7.05 3.70 7.03 0.63 6.97
242.95 3.37 6.41 0.64 7.09 3.35 6.35 0.64 7.03
273.15 3.00 5.68 0.65 7.15 2.98 5.64 0.65 7.10
274.00 2.99 5.66 0.65 7.15 2.97 5.62 0.65 7.10
298.15 2.75 5.20 0.66 7.21 2.74 5.16 0.66 7.17
303.15 2.70 5.11 0.67 7.22 2.69 5.08 0.66 7.18
323.15 2.54 4.79 0.67 7.26 2.53 4.76 0.67 7.23
340.00 2.42 4.56 0.68 7.30 2.41 4.53 0.68 7.28
370.00 2.23 4.19 0.69 7.37 2.22 4.16 0.69 7.35
410.00 2.01 3.78 0.71 7.47 2.01 3.76 0.71 7.45
Figure 6. The Kerr second virial coefficient BKð!; TÞas a
function of the temperature computed at l¼632:8nm
(V2m8mol21035).
1992 C. Ha
¨ttig et al.

Bð0Þ
ðTÞand the dispersion Bð2Þ
ðTÞcontributions are
included. Since in experiment it is customary to measure
the pressure dependence of a gas with respect to a
reference gas, it is often convenient to report a relative
second virial hyperpolarizability coefficient defined as
bðO;TÞ¼BðO;TÞ=ðOÞ(Ois in this case !;0;0;!).
For the case of the electric field-induced second-
harmonic generation we give in table 9 the coefficient
bðl¼514:5 nm, T), estimated both in the semiclassical
and in the quantum statistical approximations, and
obtained by assuming as reference ESHG parallel com-
ponent of the hyperpolarizability the experimental
value of 122:20:4 au taken from studies of Shelton
and Donley [67–69] and neglecting the error bars.
In figure 7, bðl¼515:4 nm, T) is plotted together
with the curves of the coefficients obtained at
two more wavelengths: 488 and 1024 nm (the refer-
ence values jj;ESHGð488:0nmÞ¼123:50:5au and
jj;ESHGð1024 nmÞ¼109:90:5 au were also taken
from [67–69]).
Quantum effects reduce the value of bð!; TÞin
absolute value by up to 10% around 40 K, and the effect
drops to 1–2% at temperatures around 100 K. Around
room temperature the difference between quantum
statistical and semiclassical results drops to less than
0.5%, which is again comparable with the errors
connected with the rather elaborate and delicate
procedure which has to be adopted in the quantum
calculations (see section 3.2). The frequency dispersion
is far from being negligible. For a wavelength of
514.5 nm it increases the static value (in absolute
value) by 20–25% in the whole range of temperatures
of table 9, rather independent of the type of approach.
The values of bð!; TÞcomputed for neon are of the
same order of magnitude as those predicted for helium
in [2], and thus they are an order of magnitude larger
than predicted for argon [2].
Table 9. The hyperpolarizability (ESHG) second virial coefficients Bðl;TÞ¼Bð0Þ
ðl;TÞþ6!2Bð2Þ
ðl;TÞðC4m7J3mol11068Þ
and relative coefficients bðl;TÞ¼Bðl;TÞ=jj;ESHGðcm3mol1Þ. To compute bðl¼514:5nm;TÞwe have assumed
jj;ESHG ¼122:2au:a
T/K Semiclassical Quantum statistical
Bð0Þ
ðTÞBð2Þ
ðTÞbðl¼514:5nm;TÞBð0Þ
ðTÞBð2Þ
ðTÞbðl¼514:5nm;TÞ
40.00 11.15 57.68 1.820 10.44 50.51 1.682
77.40 7.78 40.48 1.270 7.59 38.16 1.232
100.00 7.37 38.05 1.202 7.24 36.49 1.176
140.00 7.22 36.66 1.174 7.14 35.70 1.158
180.00 7.32 36.54 1.186 7.26 35.86 1.175
220.00 7.50 36.87 1.212 7.45 36.35 1.202
242.95 7.61 37.15 1.229 7.57 36.69 1.220
273.15 7.77 37.56 1.252 7.73 37.17 1.245
298.15 7.90 37.93 1.271 7.87 37.58 1.265
303.15 7.93 38.00 1.275 7.90 37.66 1.269
323.00 8.03 38.30 1.291 8.00 37.98 1.285
340.00 8.12 38.55 1.304 8.09 38.26 1.298
370.00 8.28 39.00 1.327 8.25 38.74 1.322
410.00 8.47 39.59 1.357 8.45 39.36 1.352
a[67]. The value is 122.2 0:4 au.
Figure 7. The relative hyperpolarizability (ESHG) second
virial coefficient bð!; TÞ¼Bð!; TÞ=jj;ESHGð!Þas a
function of the temperature computed at l¼488 nm,
l¼514:5 nm and l¼1024 nm (1068 C4m7J3mol1).
The reference values were taken from experiment:
jj;ESHGðl¼488 nmÞ¼123:5au [68]; jj;ESHGðl¼514:5nmÞ
¼122:2au [67] and jj;ESHG ðl¼1024nmÞ¼ 109:9au [69].
Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients 1993

5. Summary and conclusions
In the present work we extend our previous ab initio
studies of the second virial coefficients for linear and
nonlinear properties of helium and argon gases to neon,
considered as an intermediate case as far as the binding
energy of the van der Waals dimer is concerned. As
shown in [4] the importance of quantum effects on the
interatomic (or in general intermolecular) motion for
virial coefficients depends critically on this property.
Also, the evaluation of the quantum statistical expres-
sions for the virial coefficients becomes cumbersome
with increasing binding energy and the presence of an
increasing number of bound and quasi-bound rovibra-
tional states.
For the calculation of the dimer properties an
approach similar to that used in [1, 2] for the Ar2
dimer has been employed. An accurate CCSD(T)
potential and CCSD interaction-induced electric dipole
polarizabilities and hyperpolarizabilities have been
computed, and the frequency dependence of the proper-
ties has been accounted for through second order using
analytic dispersion coefficients. When comparing the
interaction-induced isotropic and anisotropic polariz-
abilities to the previous empirical curves, we observe
that the agreement is acceptable for the asymptotic
long-range behaviour. It becomes rather poor at short
range, where all semi-empirical models considered here
fail to some extent.
The dielectric, refractivity, Kerr and hyperpolariz-
ability second virial coefficients have been evaluated
using both a semiclassical and a quantum statistical
approach. It is found that for neon, quantum effects are
negligible (<1–5%) for temperatures larger than 100 K.
Only few experimental data are available for virial
coefficients of electric properties of neon gas. As for
argon, also for neon the experimental data on the
dielectric and refractivity second virial coefficient carry
at times very large uncertainties. As a consequence, and
in spite of the large distribution of the experimental
points, comparison between the present highly accurate
ab initio results and experiment can only be defined as
fair. A renewed effort from the experimentalists would be
welcome.
The Kerr second virial coefficient of neon is dominated
(contrary to what has been observed in [4] for helium)
in the whole temperature range by the polarizability
contribution B
Kð!; TÞ, which is approximately 30–35
times larger than the hyperpolarizability contribution at
small temperatures (where quantum effects account for
10% of the coefficient) and twice to three times larger at
temperatures around 400 K. The deviations from linear-
ity in the dependence of the Kerr constant of neon on the
density, not detected in the experiment of Buckingham
and Dunmur [66], are thus predicted to be small but still
non-negligible. The frequency dispersion of the Kerr
second virial coefficient is quite contained.
The ESHG second virial coefficient of neon assumes
values comparable to those computed for helium in [3],
and it displays a remarkable frequency dispersion.
Quantum effects are again negligible above 100 K.
This paper is dedicated to Bjoern Roos, who is
responsible for the meeting of two of the authors in
a memorable first edition of the European School in
Quantum Chemistry just a few years ago. We look
forward to hearing of his past, present and future
contributions in the field of science for many years to
come. We thank MichalJaszun
´ski for kindly providing
detailed information on his study of the potential of
Ne2. This work has been supported by the European
Research and Training Network ‘Molecular Properties
and Molecular Materials’ (MOLPROP), Contract No.
HPRN-CT-2000-00013, and by the Spanish Comisio
´n
Interministerial de Ciencia y Tecnologı
´a (PB98-0609-
C04-01 Project and FP99-44446531C FPI Grant).
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Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients 1995