JOURNAL OF CHEMICAL PHYSICS
VOLUME 117, NUMBER 4
22 JULY 2002
Quantum hydrodynamic model for the enhanced moments of inertia
of molecules in helium nanodroplets: Application to SF6
Kevin K. Lehmanna) and Carlo Callegarib)
Department of Chemistry, Princeton University, Princeton, New Jersey 08544
共Received 28 August 2001; accepted 29 April 2002兲
The increase in moment of inertia, ⌬I, of SF6 in helium nanodroplets is calculated using the
quantum hydrodynamic approach 关Callegari et al., Phys. Rev. Lett. 83, 5058 共1999兲; 84, 1848
共2000兲兴, which we extend here to an explicit three-dimensional treatment. Three plausible helium
densities are reconstructed by interpolation of previously published ‘‘density cuts’’ in terms of an
expansion into cubic harmonics 共several interpolation strategies are presented兲. This allows us to
predict a value of ⌬I that ranges from as low as 30 u•Å2 to as high as 318 u•Å2 . The lower limit
reproduces the prediction of Kwon et al. 关J. Chem. Phys. 113, 6469 共2000兲兴, who use the same
hydrodynamic model and an unpublished density based upon a Path Integral Monte Carlo
calculation. These values can be compared with the experimentally measured ⌬I 共310⫾10 u•Å2 )
for large (N⭓103 He atoms兲, and with Fixed Node, Diffusion Monte Carlo calculations by Lee,
Farrelly, and Whaley 关Phys. Rev. Lett. 83, 3812 共1999兲兴, which found ⌬I⫽290–305 u•Å2 for N
⫽8 –20 helium atoms. The present results show that the value of ⌬I obtained from the
hydrodynamic model is quite sensitive to physically reasonable variations in the helium density;
therefore one has to be careful as to which density to use. Because the model is based upon the
assumption that the helium is in the ground ‘‘quasienergy’’ state of the helium-molecule
time-dependent potential, we propose that calculations should be done using densities calculated at
0 K rather than at finite temperature. We have extended our original algorithm to also handle
irregular boundaries. We find that in the present case the calculated value of ⌬I only changes by a
few percent. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1486443兴
there is consensus that the additional moment of inertia, ⌬I,
arises from the kinetic energy of helium motion that is correlated with the rotation of the solute. In the case of
SF6 – HeN 共Ref. 7兲 a fixed frame, fixed node, diffusion Monte
Carlo 共DMC兲 calculation on small clusters with N⫽8 –20
has recovered rotational level spacings (⬀B eff) in excellent
agreement with those observed for SF6 in a 共much larger兲 He
nanodroplet. Such calculations give limited dynamical information, however, and thus leave open the question of how to
physically characterize the helium motion. Two quantitatively predictive dynamical models have been put forth that
invoke very distinct types of helium motion.8 –10 In one,8,9 a
proposed ‘‘nonsuperfluid fraction’’ of helium is calculated
using Path Integral Monte Carlo 共PIMC兲 methods, and is
assumed to rotate rigidly with the molecule, provided that
the molecule–He potential is sufficiently anisotropic relative
to the induced rotational energy. To date, this ‘‘two-fluid
model’’ has been applied to SF6 and OCS.8,9 For SF6 it gives
a result in excellent agreement with experiment: the predicted increase in the effective moment of inertia, ⌬I two-fluid
⫽327 u•Å2 almost exactly matches the experimental value
⌬I exp⫽310⫾10 u•Å2 共Ref. 4兲 derived from the rovibrational spectrum.11
The second approach, introduced by our group,10 is
based upon a hydrodynamic treatment for the helium flow.
Its key assumptions are: 共1兲 a continuum fluid; 共2兲 constant
solvation density in the frame rotating with the molecule
共adiabatic following兲; 共3兲 ideal fluid flow 共aviscous and irro-
I. INTRODUCTION
There is considerable current interest in the spectroscopy
of atoms and molecules solvated in liquid helium. In particular, 4 He nanodroplets1,2 are currently the best system to
study, on a microscopic scale, some unique physical properties: He is the sole substance known to remain a liquid as
T→0 K, and is also the most easily accessible superfluid.
These properties are also relevant to chemistry, as they make
4
He droplets an almost ideal ‘‘matrix’’ for the production and
characterization of novel species.3 Notably, even large and
highly anisotropic solute molecules exhibit rotationally resolved spectra, though with rotational constants, B eff , often
considerably smaller than for the same molecule in the gas
phase. Physically this means that the effective moments of
inertia for rotation of molecules in liquid helium, I eff , are
several times larger than those of the isolated 共gas phase兲
molecules.1,4 – 6 The number of molecules and aggregates being formed and detected in He droplets is steadily increasing,
as is their complexity. A theory capable of quantitative predictions for I eff will be valuable inasmuch as the rotational
constants of an unknown spectral band can be used to identify the chemical carrier of the transition strength, as is often
done in gas phase spectroscopy.
Although the exact dynamics still has to be elucidated,
a兲
Electronic mail: lehmann@princeton.edu
Current address: Condensed Matter Physics, M/C 114-36, Caltech, Pasadena, CA 91125. Electronic mail: cal@caltech.edu
b兲
0021-9606/2002/117(4)/1595/9/$19.00
1595
© 2002 American Institute of Physics
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
1596
J. Chem. Phys., Vol. 117, No. 4, 22 July 2002
K. K. Lehmann and C. Callegari
tational兲. The ‘‘classical look’’ of condition 共1兲 might give
the impression that this approach is no longer valid when the
variation of helium density is large on an atomic length
scale, which is exactly our situation. In reality the hydrodynamic equation used, Eq. 共13兲, can be derived from the variational optimization of a many-body helium wave function of
the form,
冋
⌿⫽ ␺ 共 R 兲 exp ⫺
im
ប
兺i ␾ 共 r i 兲
册
,
共1兲
where ␺ (R) is the highly correlated ground state wave function, m the atomic mass of 4 He, and 兺 i ␾ (r i ) a phase function written as a sum of one-body terms. Condition 共2兲 follows from the assumption that the rotational frequency of the
molecule is much less than the excitation frequencies of the
helium that are coupled to the rotation.
The hydrodynamic approach has been applied to a number of linear molecules; its estimates of the additional
moment of inertia, ⌬I h , were found to be in good agreement
with ⌬I exp for the heavier molecules, including OCS.
Conjusteau et al.13 have experimentally established that for
lighter molecules, specifically HCN and DCN, adiabatic following breaks down, and that the sign of the observed deviations is consistent with the approximations of the hydrodynamic theory in these cases. Lee, Farrelly, and Whaley7 had
previously demonstrated numerically that the adiabatic following approximation breaks down for SF6 if the free molecule rotational constant is increased by a factor of 10 over
its known value, making it similar in size to that of HCN and
DCN.
The hydrodynamic model has been used by Kwon et al.9
to calculate ⌬I h for SF6 ; the reported value, ⌬I h⫽31 u•Å2 ,
is taken by Kwon et al. as evidence of the failure of the
hydrodynamic approach since it is only about 10% of ⌬I exp .
This result would indeed suggest that the hydrodynamic
model has failed for this molecular system, however no discussion was given as to the robustness of this result with
respect to changes in the density used. In the present work,
we present the results of calculations which use several different density functions, based upon both thermally averaged
and zero-temperature helium densities around SF6 . We demonstrate a large sensitivity of the calculated ⌬I h to reasonable variations of the density, and we present reasons why
the zero-temperature density should be used in hydrodynamic calculations of the helium contribution to the effective
moment of inertia.
Because of the different symmetry of the system 共O h)
compared to the linear molecules we treated in Ref. 10
(C ⬁ v), it is no longer possible to remove one degree of
freedom by separation of variables. As a result, both the helium density and the velocity potential have to be calculated
in three dimensions; this requires several changes to our
original approach; in addition we extend our original algorithm so that irregular boundaries could be used. The computational procedures used in the present case are described
in Sec. II B for the He density, and in Sec. II C for the velocity potential; the results are presented in Sec. III and discussed in Sec. IV.
II. CALCULATIONS
A. Notation
We use spherical coordinates (r, ␪ , ␸ )⬟(r,⍀); explicit
dependence on coordinates may be omitted when obvious.
The SF6 共solute兲 molecule is oriented with the S–F bonds
lying along the Cartesian axes which instantaneously coincide for the laboratory-fixed and molecule-fixed frames of
reference; ␳ (r,⍀) is the three-dimensional helium density
around the solute molecule; ␾ (r,⍀) is the velocity potential
共see Sec. II C兲; v(r,⍀) the Eulerian velocity of the fluid in
the laboratory frame. O h is the point group of the SF6 molecule, which also determines the symmetry of the helium
density; C 2 , C 3 , and C 4 are the associated symmetry axes;
␳ 2 (r), ␳ 3 (r), ␳ 4 (r) are radial cuts of the density along those
axes, e.g., along ⍀ 2 ⫽( ␲ /4,0), ⍀ 3 ⫽ 关 arccos(1/冑3), ␲ /4兴 ,
⍀ 4 ⫽( ␲ /2,0), respectively; ␳ 0 (r) is the spherically averaged
density at each r, defined as (1/4␲ ) 兰 ␳ (r,⍀)d⍀. T L (L
⫽0, 4, 6, 8, 10, . . . 兲 are cubic harmonics,16 i.e., the linear
combinations of the spherical harmonics, Y LM , that transform as A 1g in the O h point group. In this work we will only
use harmonics up to L⫽8, for which the explicit expressions
are
T 0 ⫽1,
共2兲
T 4⫽
冑
7␲
Y ⫹
3 40
T 6⫽
冑
冑7 ␲
␲
Y 60⫺
共 Y 64⫹Y 6,⫺4 兲 ,
2
2
T 8⫽
冑33␲
4
⫹
1
4
Y 80⫹
冑
冑
1
2
5␲
共 Y 44⫹Y 4,⫺4 兲 ,
6
冑
共3兲
共4兲
7␲
共 Y 84⫹Y 8,⫺4 兲
6
65␲
共 Y 88⫹Y 8,⫺8 兲 .
6
共5兲
Several papers present a procedure to calculate these and
higher harmonics. We cite among these the original work of
von der Lage and Bethe,16 where the term ‘‘Kubic Harmonics’’ is first introduced, that of Fox and Ozier,17 and the
works of Fehlner and Vosko, who give powerful quadrature
formulas for integration of functions with octahedral
symmetry.18,19 We use here the popular normalization
兰 (T L ) 2 d⍀⫽4 ␲ , i.e., the T L have unit rms value on the
sphere.
B. Reconstructing the helium density
For the cylindrically symmetric molecules dealt with in
our earlier hydrodynamics work10 it was convenient to use
helium density functional 共DF兲 theory14 to calculate the helium density around a static solute molecule. For systems
with cylindrical or higher symmetry, DF is computationally
many orders of magnitude less expensive than Quantum
Monte Carlo 共QMC兲 methods. Speed comes at the price of a
more approximate treatment, compared to QMC, but the DF
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem. Phys., Vol. 117, No. 4, 22 July 2002
Moments of inertia in helium nanodroplets
treatment has been shown to accurately reproduce the static
properties of pure and doped He clusters,20 at least when the
dopant–helium interaction is not too large.
Currently, DF has not been implemented in three dimensions; because of this, and of the fact that we want to compare our results to those of Kwon et al.,9 we chose to use
published densities that have been calculated by QMC methods. Two of those exist: one calculated by Barnett and
Whaley15 and based upon a DMC calculation with a second
order estimator 共hereafter, density I兲; the other calculated by
Kwon et al.9 by path integral quantum Monte Carlo 共hereafter, density II兲. We will describe our procedure as applied to
density I; further details that only pertain to density II are
postponed to a later section.
We were unable to obtain the three dimensional densities
from the authors of the respective papers. We then decided to
focus on density I, for which a more extensive set of data
samples has been published. We extract the radial isotropic
density ␳ 0 (r), and the radial cuts ␳ 2 (r), ␳ 3 (r), ␳ 4 (r) by
digitizing the plots in Figs. 7 and 8 of Ref. 15, for a cluster of
69 He atoms, the largest for which density cuts were reported.
The standard approach to interpolate a sparse set of data
is to write the interpolating function as a series expansion in
terms of complete set of functions that are a basis for the
given symmetry, and to use the available data to determine
the coefficients of the lowest terms of the expansion. More
explicitly we write
␳ 共 r,⍀ 兲 ⫽ ␳ 0 共 r 兲 ⫹a 4 共 r 兲 T 4 共 ⍀ 兲 ⫹a 6 共 r 兲 T 6 共 ⍀ 兲
⫹a 8 共 r 兲 T 8 共 ⍀ 兲 ⫹•••.
共6兲
Formally, it would be straightforward to obtain the expansion coefficients, from a L (r)⫽(4 ␲ ) ⫺1 兰 ␳ (r,⍀)T L d⍀, if
enough sampling directions were available that the integral
could be reliably calculated. Instead we assume that the density contains only the terms explicitly given in Eq. 共6兲. We
then evaluate Eq. 共6兲 along ⍀ 2 , ⍀ 3 , ⍀ 4 , thus obtaining a
system of linear equations which, when inverted, gives the
equations for a 4 (r),a 6 (r), and a 8 (r),
a 4共 r 兲 ⫽
1
143冑21
关 189␳ 0 共 r 兲 ⫺128␳ 2 共 r 兲 ⫺189␳ 3 共 r 兲
⫹128␳ 4 共 r 兲兴 ,
a 6 共 r 兲 ⫽⫺
冑
1
55
2
关 5 ␳ 0 共 r 兲 ⫹32␳ 2 共 r 兲 ⫺27␳ 3 共 r 兲
13
⫺10␳ 4 共 r 兲兴 ,
a 8 共 r 兲 ⫽⫺
冑
8
65
共7兲
共8兲
3
关 35␳ 0 共 r 兲 ⫺16␳ 2 共 r 兲 ⫺9 ␳ 3 共 r 兲
187
⫺10␳ 4 共 r 兲兴 .
共9兲
Figure 1 shows the expansion coefficients thus calculated. We refer to this scheme as ‘‘direct expansion.’’ It is
apparent from this figure that the density in the first solvation
shell 共the most important for determination of the moment of
1597
FIG. 1. The expansion coefficients, ␳ 0 (r),a 4 (r),a 6 (r), and a 8 (r) as a function of the He–SF6 radial distance.
inertia兲 is not fully converged with the present truncation of
the density expansion. One of the consequences is that there
are regions where the density calculated by Eq. 共6兲 becomes
negative; this is not a serious problem, however, as it only
occurs in regions that can be considered as occupied by the
SF6 molecule, where the density can be safely set to the
threshold value ␳ min required by the hydrodynamic equations
共see below兲. It is a more serious problem that even in regions
where it is positive, such a density may be a poor representation of the true density. We later implemented a much better approach than the direct expansion of the helium density
by Eq. 共6兲. Nevertheless the results obtained by this rather
crude method are presented for completeness since they help
demonstrate the sensitivity of the results to changes in the
assumed density function.
This lack of convergence arises from the large variation
of the density 共at fixed r) in the first solvation shell, in particular near the SF6 repulsive wall. The same problem has
plagued the series expansion of the intermolecular potential
between a spherical atom and a nonspherical molecule; expansions of such potentials into a series of orthogonal functions are notorious for their terrible convergence properties at
short distances, due to the fact that near the repulsive wall, at
fixed r, the potential at different ⍀ can vary by several orders of magnitude. It has been recognized that a superior
approach is an indirect expansion, where one first expresses
the radial cuts of the potential 共at fixed ⍀) as an analytic
function of r and of a set of parameters, and subsequently
expand each of these parameters into a series of spherical
harmonics. With a judicious choice, the parameters are
slowly varying with angle, and therefore much better convergence can be achieved. The method has been successfully
applied to numerous potentials, e.g., to He–CH4 共Ref. 21兲
and He–SF6 共Ref. 12兲 共incidentally, the latter potential is the
one that was used to calculate the helium densities we utilize
here兲.
A functional form f (r) that was found to closely reproduced the radial density cuts is what we call a sum of ‘‘modified half Gaussians,’’ explicitly 共see also Fig. 2兲,
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
1598
J. Chem. Phys., Vol. 117, No. 4, 22 July 2002
K. K. Lehmann and C. Callegari
FIG. 2. Radial density cuts, ␳ 0 (r), ␳ 2 (r), ␳ 3 (r), and
␳ 4 (r) as a function of the He–SF6 radial distance, and
their fits to ‘‘modified half Gaussians’’ 共see text兲. Also
shown is the physical interpretation of the fitting parameters.
2
w i⫽
再
冉 冏 冏冊
兺
i⫽1
A i exp ⫺
w⫺
i
共 r⬍c i 兲
f 共 r 兲⫽
w⫹
i
bi
r⫺c i
wi
共 r⭓c i 兲 ;
b i⫽
再
⫹c 1,6T 6 (⍀)⫹c 1,8T 8 (⍀) 其 兴. Note that for convenience in
tabulating the expansion coefficients, this form is slightly
different than that used for the density, Eq. 共6兲.
Finally we write
共10兲
,
b⫺
i
共 r⬍c i 兲
b⫹
i
共 r⭓c i 兲 .
2
␳ 共 r,⍀ 兲 ⫽
Note that the exponential functions are Gaussians only for
b i ⫽2 共hence the term ‘‘modified’’兲, and that f (r) is continuous and has a continuous first derivative as long as b ⫾
i ⬎1.
We fitted ␳ 0 (r), ␳ 2 (r), ␳ 3 (r), ␳ 4 (r) with the functional
form of Eq. 共10兲 by least squares minimization, obtaining the
parameters reported in Table I. The density cuts and the best
fitting curves are shown in Fig. 2; in all cases we found that
the fitting function would reproduce the original data within
the limit of their noise. Therefore we find it convenient, and
justified, to use the best fitting analytic function as a smooth
interpolation of the original data.
An expansion into cubic harmonics was then calculated
by use of Eqs. 共6兲–共9兲, with ␳ replaced by the parameter to
be expanded 关for example, c 1 (⍀)⫽c 1,0兵 1⫹c 1,4T 4 (⍀)
兺
i⫽1
冋冏
A i 共 ⍀ 兲 exp ⫺
r⫺c i 共 ⍀ 兲
w i共 ⍀ 兲
冏 册
b i (⍀)
.
共11兲
We will refer to this scheme as ‘‘indirect expansion.’’
Note that for an arbitrary function f, f (x)⫽ f (x), therefore c 1,0 in the example above 共and likewise any other parameter兲 is not the entry in row ␳ 0 of Table I. Rather the
isotropic part of each parameter has to be calculated selfconsistently, by constraining the spherical average of the
density to match ␳ 0 (r). We call these ‘‘spherically averaged
parameters,’’ and report them in row L⫽0 of Table I. In the
same table, we also report the other cubic harmonic expansion coefficients for each parameter.
TABLE I. Best fit parameters for radial cuts of density I to Eq. 共10兲 are given in rows ␳ 0 through ␳ 4 . The row labeled L⫽0 contains the ‘‘spherically averaged
parameters’’ for which Eq. 共12兲 best fits the ␳ 0 curve; the last three rows contain the dimensionless coefficients of the higher cubic harmonics for the expansion
of each parameter in the ‘‘indirect expansion’’ scheme 共see text兲.
␳0
␳2
␳3
␳4
L⫽0
L⫽4
L⫽6
L⫽8
a
A1
(Å⫺3)
c1
共Å兲
0.0715
0.093
0.205
0.0506
4.518
4.493
4.170
4.801
0.0786
⫺0.569
0.242
0.216
4.721
0.046
⫺0.010
⫺0.026
w⫺
1
共Å兲
w⫹
1
共Å兲
b⫺
1
b⫹
1
A2
(Å⫺3)
c2
共Å兲
w⫺
2
共Å兲
Parameters for fits to radial density and density cuts along C n axes
0.619
0.674
2.499
1.493
0.0268
7.557
1.190
0.489
0.685
2.374
1.642
0.0178
8.029
1.779
0.418
0.592
2.400
1.649
0.0444
7.561
1.129
0.559
0.856
2.857
1.692
0.0298
7.64a
1.275
Cubic harmonic expansion coefficients for each line shape parameter
0.690
0.564
2.530
1.230
0.0268
7.664
1.268
0.134
0.045
0.052
⫺0.090
⫺0.103
⫺0.006
⫺0.046
⫺0.023
⫺0.002
0.013
0.016
0.211
⫺0.014
⫺0.113
⫺0.158
0.141
⫺0.002
0.190
0.027
0.010
0.086
w⫹
2
共Å兲
b⫺
2
b⫹
2
1.295
1.730
0.927
1.578
2.403
2.782
2.692
2.157
1.387
1.867
1.535
1.576
1.220
0.045
⫺0.121
0.116
2.675
⫺0.047
⫺0.022
⫺0.019
1.360
⫺0.079
⫺0.049
0.136
Fixed.
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem. Phys., Vol. 117, No. 4, 22 July 2002
Moments of inertia in helium nanodroplets
1599
TABLE II. Fitting parameters for density II. See Table I for details.
␳0
␳2
␳3
␳4
L⫽0
L⫽4
L⫽6
L⫽8
A1
(Å⫺3)
c1
共Å兲
0.0863
0.0919
0.165
0.0677
4.384
4.382
4.112
4.741
0.0944
⫺0.266
0.130
0.054
4.430
0.037
⫺0.006
⫺0.002
w⫺
1
共Å兲
w⫹
1
共Å兲
b⫺
1
b⫹
1
冕␳
c2
共Å兲
w⫺
2
共Å兲
Parameters for fits to radial density and density cuts along C n axes
0.500
0.635
2.682
1.480
0.0266
7.350
1.075
0.455
0.586
2.648
1.474
0.0269
7.329
1.080
0.396
0.501
2.458
1.489
0.0313
7.346
0.949
0.551
0.659
2.972
1.553
0.0205
7.635
1.436
Cubic harmonic expansion coefficients for each line shape parameter
0.473
0.586
2.758
1.443
0.0266
7.376
1.105
0.086
0.066
0.054
0.001
⫺0.097
0.009
0.104
⫺0.011
⫺0.019
⫺0.006
0.007
0.016
0.003
⫺0.001
⫺0.006
⫺0.001
⫺0.013
0.022
⫺0.009
0.003
0.021
⫹
兲 d⍀
共 r,⍀,A 1,0 ,c 1,0 , . . . ,b 2,0
w⫹
2
共Å兲
b⫺
2
b⫹
2
1.282
1.282
1.070
1.347
2.054
2.088
2.497
1.969
1.442
1.442
1.295
1.527
1.260
0.054
⫺0.028
⫺0.006
2.094
⫺0.067
0.033
0.017
1.428
0.039
⫺0.015
0.000
of He atoms in a cluster is increased the density of complete
shells increases, but not significantly. Figure 12 of Ref. 9,
which reports the isotropic density for a cluster of 100 He
atoms, then gives us an upper limit for the density of a 64 He
atoms cluster calculated at 0.3125 K. As a higher temperature will have, if any, the effect of delocalizing the He density, i.e., of smoothing the first solvation shell, we think that
the above proves that our replacement for the spherically
averaged density is acceptable. A posteriori the agreement
between the value of ⌬I h we obtain with this density and the
value calculated by Kwon et al. also justifies our approximations.
In summary, density II represents a cluster of 64 He
atoms and is constructed using the radial density cuts from a
PIMC calculation at 0.3125 K from Ref. 9, and the spherically averaged density from a PIMC calculation at 0.625 K
from Ref. 8. The fitting parameters, and their expansion into
cubic harmonics, that reproduce density II are reported in
Table II.
Finally we also fitted the superfluid fraction of density II,
reconstructed from Figs. 2 and 3 of Ref. 8. The fitting parameters, and their expansion into cubic harmonics, that reproduce this ‘‘superfluid density’’ are reported in Table III.
The self-consistent calculation was performed by fitting
␳ 0 (r), as recreated from its analytic expression, to the function,
共 4 ␲ 兲 ⫺1
A2
(Å⫺3)
共12兲
⫹
as free
by least-square minimization, with A 1,0 ,c 1,0 , . . . ,b 2,0
parameters of the fit. The integral in Eq. 共12兲 was numerically evaluated at each iteration by 6-point quadrature, with
the formula provided in Ref. 18.
While, as we will see, density I produces satisfactory
results which are consistent with ⌬I exp , it does not agree
with the much lower value found by Kwon et al.9 Upon visual inspection, it is evident that the PIMC density used by
Kwon et al., calculated for 64 He atoms, is considerably
more isotropic than the DMC density we use 共compare Fig. 3
of Ref. 9 with Fig. 8 of Ref. 15兲. The question then arises:
How sensitive to density anisotropy is the prediction of the
hydrodynamic treatment? Unfortunately Kwon et al. only
report the radial density cuts ␳ 2 (r), ␳ 3 (r), and ␳ 4 (r), but not
the spherically averaged density, ␳ 0 (r), which is an essential
ingredient of our approach. Nevertheless, the importance of
this test grants, in our opinion, the extra approximations
we’ll make to bypass this problem. Lacking an isotropic density consistent with the above PIMC density cuts, we replace
it with that obtained from the data 共for 64 He atoms兲 in Figs.
2共a兲 and 2共b兲 of Ref. 8. The latter density is calculated at
0.625 K, as opposed to 0.3125 K, but we do not expect the
difference to be substantial. Our opinion is based on the following facts: Figure 7 of Ref. 15 tells us that as the number
C. Hydrodynamic calculations
1. Hydrodynamic equations
Because the helium motion is assumed to be irrotational
(“⫻v⫽0), the Eulerian velocity of the fluid in the laboratory frame, v, can be written as the gradient of a scalar func-
TABLE III. Fitting parameters for the superfluid fraction of density II. See Table I for details.
␳0
␳2
␳3
␳4
L⫽0
L⫽4
L⫽6
L⫽8
A1
(Å⫺3)
c1
共Å兲
0.0439
0.0408
0.0681
0.0306
4.401
4.321
4.143
4.698
0.0478
⫺0.164
0.090
⫺0.034
4.419
0.035
⫺0.002
⫺0.004
w⫺
1
共Å兲
w⫹
1
共Å兲
b⫺
1
b⫹
1
A2
(Å⫺3)
c2
共Å兲
w⫺
2
共Å兲
Parameters for fits to radial density and density cuts along C n axes
0.512
0.652
2.719
1.462
0.0162
7.339
1.083
0.372
0.651
1.964
1.628
0.0179
7.224
1.035
0.432
0.461
3.163
1.441
0.0204
7.264
0.866
0.488
0.720
2.758
1.645
0.0136
7.466
1.427
Cubic harmonic expansion coefficients for each line shape parameter
0.468
0.613
2.773
1.480
0.0162
7.367
1.125
0.070
0.093
0.015
0.010
⫺0.126
0.010
0.135
0.035
⫺0.050
0.093
⫺0.020
0.014
0.003
⫺0.007
⫺0.056
0.008
⫺0.054
0.039
0.036
⫺0.005
⫺0.010
w⫹
2
共Å兲
b⫺
2
1.278
1.277
1.111
1.403
2.029
1.857
2.197
1.779
1.450
1.459
1.403
1.622
1.261
0.054
⫺0.018
0.004
2.046
⫺0.029
0.026
⫺0.033
1.439
0.029
0.001
0.020
b⫹
2
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
1600
J. Chem. Phys., Vol. 117, No. 4, 22 July 2002
K. K. Lehmann and C. Callegari
tion, v⫽⫺“ ␾ , where ␾ is termed the velocity potential. In
order to calculate ␾ we need to specify the motion of the
SF6 , which we will assume to be a classical rotation with
angular velocity ␻ around the z axis. Because of the octahedral symmetry of the helium solvation density around SF6 ,
the net rotational kinetic energy is independent of the axis
about which rotation takes place. Imposing continuity gives
the following equation:
“• 共 ␳ “ ␾ 兲 ⫽
冉 冊
⳵␳
⳵t
⫽⫺ 共 “ ␳ 兲 • 共 ␻⫻r兲 ,
共13兲
LF
where the second equality comes from the assumption that
the helium density remains time-independent in the
molecule-fixed frame. LF means that the partial derivative is
taken with respect to a point fixed in the laboratory reference
frame; the corresponding partial with respect to a point fixed
in the molecular frame is zero by construction. The boundary
conditions on ␾ are that in the direction normal to the
boundary the fluid velocity matches the velocity of the
boundary, i.e., ⳵ ␾ / ⳵ n⫽“ ␾ •n̂⫽⫺( ␻⫻r)•n̂, with n̂ the unit
vector normal to the boundary surface and pointing out of
the fluid.
The solution to Eq. 共13兲 is linear in ␻ and thus is calculated for unit ␻ . Equating the helium kinetic energy with a
rotational kinetic energy defines ⌬I h :
冕␳ ␾
冋 冕 ␾ 冉 ⳵␳⳵ 冊 冕 ␳ ␾
1
1
⌬I h ␻ 2 ⫽ m He
2
2
1
⫽ m He ⫺
2
兩 “ 兩 2 dV
t
dV⫹
共14兲
册
共 “ ␾ 兲 •dS .
共15兲
Equation 14 holds for any velocity potential; Eq. 共15兲 has
been derived from Eq. 共14兲 using vector identities and Eq.
共13兲. As such, the two estimates for ⌬I h need only be equal
for a ␾ that is solution to Eq. 共13兲. It can also be shown that
the net orbital angular momentum produced by the helium
flow is, for the solution of Eq. 共13兲, JHe⫽⌬I h ␻. Note that,
consistent with common convention, dS in the second integral of Eq. 共15兲 is defined to point out of the fluid region.
With our choice of coordinates 共SF6 centered at r⫽0
and oriented with its S–F bonds lying on the Cartesian axes兲
␾ must be invariant to reflection in the xy plane, therefore its
␪ domain can be restricted to the interval 关 0,␲ /2兴 and the
condition ⳵ ␾ / ⳵ ␪ ⫽0 imposed in the plane ␪ ⫽ ␲ /2 共the xy
plane兲. Reflection in each of the four symmetry planes containing the z axis is equivalent to inverting the direction of
rotation, i.e., to a change in the sign of ␾ . Therefore the ␸
domain can be restricted to the interval 关 0,␲ /4兴 and ␾ ⫽0
imposed in the planes ␸ ⫽0 and ␸ ⫽ ␲ /4.
In most of our calculations, the inner and outer boundaries in the radial direction are spherical surfaces at r⫽r min
and r⫽r max 共this condition can be relaxed, see Sec. II C 3兲.
On both boundaries the condition ⳵ ␾ / ⳵ n⫽0 was imposed,
which implies that no helium flux out of the domain of the
solution was allowed. Since either ␾ or ⳵ ␾ / ⳵ n are zero on
all the boundary surfaces, the surface integral in Eq. 共15兲 is
zero.
2. Numeric solution
The inhomogeneous partial differential equation, Eq.
共13兲 was solved numerically on a uniformly-spaced grid of
points. The radial grid consists of up to 201 points; r min is
always 3.5 Å, for r max we tried the values 10 and 15 Å. Grids
with up to 161 points were used for each angular coordinate,
in the restricted domains specified above.
Equation 共13兲 was cast into the more convenient form,
ⵜ 2 ␾ ⫹(“ ln ␳)•(“␾⫹␻⫻r)⫽0, converted to a 7-point
finite-difference equation in the chosen grid of points, and
solved by Gauss-Seidel iteration with successive
over-relaxation.22 This equation becomes singular when ␳
⫽0, therefore we clip the density to a threshold value, ␳ min ,
i.e., all grid points where ␳ ⬍ ␳ min are assigned the value ␳
⫽ ␳ min . Solutions were iterated until the rms change in ␾ on
the grid points was less than a fixed fraction, ⑀ (⫽10⫺6 in
this work兲 of the rms value of ␾ on the grid points. In the
Gauss–Seidel method, the change in the value of each grid
point is proportional to the error in the finite difference equation evaluated at that point, so this convergence criteria is
equivalent to one based on the residuals in this equation. In a
test calculation with the convergence criteria tightened to ⑀
⫽10⫺10, the norm of ␾ was approximately 0.01% lower and
⌬I h approximately 0.04% higher than when ⑀ ⫽10⫺6 was
used. After an initial transient, the convergence estimator, the
norm of ␾ , and the predicted values of ⌬I h all appear to
converge exponentially, approximately with the same rate,
with the number of Gauss–Seidel iteration steps. For the
largest grid used, with ⬇5.1 million points, a typical run
required about 1100 iteration cycles 共several hours of computation time on a personal computer兲 with an overrelaxation parameter of 1.9 ( ␻ in the notation used in Ref.
22兲. We tested ␳ min values in the range 10⫺7 –10⫺5 Å , and
found that the estimate for ⌬I h changed by ⬃1% 共with
higher values of ␳ min giving lower values of ⌬I h ). Refining
the grid from 101⫻81⫻81 to 201⫻161⫻161 points
changed the estimate for ⌬I h by ⱗ0.5%.
3. Nonspherical boundaries
The use of a rigid, spherical inner boundary and a lower
bound on the density are mathematical artifices introduced to
accelerate convergence with respect to grid size and GaussSeidel iteration. It is natural to ask whether these rather ad
hoc choices significantly affect the final value of the kinetic
energy. The velocity of the fluid, ⫺“ ␾ , need not approach
zero in the region where the density goes to zero; rather, it
will vary considerably upon small changes of the boundary
conditions. While the effect of the choice of boundary condition will propagate into a region of higher density, by conservation of flux it should damp as the density gets larger. It
is clear on physical and mathematical grounds that in the
limit that ␳ min→0 and “ ␾ bounded, the kinetic energy of
the solution will be independent of the exact boundary conditions. However, since the equation for ␾ depends only
logarithmically on the density 关through the term “(ln ␳)兴,
convergence may be legitimately expected to be slow. At the
urging of the referee, we have modified our code to include a
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem. Phys., Vol. 117, No. 4, 22 July 2002
FIG. 3. The first order estimator procedure of Fox 共Ref. 23兲. In a normal
5-point finite-difference scheme, ␾ 0 ⫽g( ␾ 1 , ␾ 2 , ␾ 3 , ␾ 4 ), where g is the
finite-difference operator for the equation being solved. If one or more
points lie outside of the boundary 共here, left of the thick line兲, their values
have to be estimated at the beginning of each Gauss–Seidel iteration by use
of the boundary conditions and the ␾ values from the previous iteration. In
the example shown, this is done by solving for ␾ 1 by using the set of
equations: ( ⳵ ␾ / ⳵ n) B⫽( ␾ 1 ⫺ ␾ C)/k and ␾ C⫽ ␾ 0 ⫹( ␾ 2 ⫺ ␾ 0 )h, with k⫽C1
and h⫽0C/02. The procedure is easily generalized to three dimensions. The
normal to the surface is along ⫺“(ln ␳), which is estimated by numerical
derivatives of the function ln␳, which is tabulated on each grid point.
curved inner boundary surface, defined as the loci of
␳ (r,⍀)⫽ ␳ min , and we did not clip the density to ␳ min at
large r.
To overcome the problem that the inner boundary no
longer falls exactly on grid points, we use the first order
estimator discussed by Fox23 共see Fig. 3兲, generalized from
two to three dimensions. In general, programming this procedure involves treatment of many special cases, but in the
present problem is simplified by the fact that at the inner
boundary, “ ␳ remains nearly parallel to r̂, and thus the
boundary always moves by less than one radial grid point for
a unit change in the ␪ or ␸ grid. Calculations were performed for the ‘‘indirect expansion’’ of density I, using ␳ min
values of 10⫺5 and 10⫺7 Å⫺3 . Also, r min was reduced to a
value of 2.5 Å so that the surface defined by ␳ ⫽ ␳ min is
contained inside the grid for all ⍀. The ⌬I h thus calculated
differed by less than 1% from those calculated with a spherical inner boundary and with the density clipped to ␳ min in the
outer region. We consider this to be a convincing demonstration that the kinetic energy is not significantly affected 共to be
precise, no more than by the choice of grid size兲 by the use
of a computationally easier boundary condition in a region
where the density is very small.
Moments of inertia in helium nanodroplets
1601
FIG. 4. Radial kinetic energy density, calculated at unit ␻ , for DMC and
PIMC He densities. Plotted values includes the r 2 Jacobian factor.
We repeated the calculation using the indirect expansion
form of the DMC density 关Eq. 共11兲 and Table I兴. With r max
⫽10 Å, the resulting values for ⌬I h were 281.1–281.7 u•Å2 .
At this radius, however, the He density is still significant.
The calculation was repeated with r max⫽15 Å, at which
point all the calculated densities are on the order of
10⫺5 Å⫺3 or less. This gave values of 316.4 –317.5 u•Å2
for ⌬I h . Figure 4 shows the radial kinetic energy density
共including the r 2 weighting兲. About 49% of the kinetic energy is carried by helium in the first solvation shell, while the
rest by helium in the second solvation shell.
Calculations were also performed using the indirect expansion form of the PIMC density 关Eq. 共11兲 and Table II兴.
This PIMC density gave ⌬I h ⫽29.47– 29.56 u•Å2 , far below any value from the DMC densities. About 85% of the
kinetic energy is carried by helium in the first solvation shell.
Our results can be compared with a value of 31 u•Å2 for ⌬I h
reported by Kwon et al., computed with the total PIMC helium density. We take the close agreement between our results and those of Kwon et al. 共when referred to the same
densities兲 as support for the adequacy of our parameter-based
interpolation to reproduce the full three dimensional helium
density distribution.
Kwon and Whaley8 also reported the ‘‘superfluid’’ densities along the same symmetry axes, as calculated using
PIMC and their proposed superfluid estimator. Repeating the
hydrodynamic calculation using the interpolating form of
these densities gave ⌬I h ⫽ 14.58 –14.64 u•Å2 . This can be
compared to the ‘‘superfluid’’ contribution to ⌬I h of 22 u•Å2
reported by Kwon et al.9
III. NUMERICAL RESULTS
The first calculations were performed using the direct
expansion into cubic harmonics of the DMC density. The
two integral estimates for ⌬I h are found to be 171 and 168
u•Å2 , respectively. Convergence with respect to grid size,
density cut off and convergence criteria appear to be within a
few percent. Given the radial range of the data available for
cubic harmonic expansion, the calculations were performed
for r min⫽3.5 Å and r max⫽10 Å. The calculated ⌬I h are
⬇55% of the value of ⌬I exp⫽310⫾10 u•Å2 extracted from
the observed effective rotational constant of SF6 in 4 He
nanodroplets.4
IV. DISCUSSION
Our results show that the calculated value of ⌬I h for SF6
in helium is extremely sensitive to the details of the helium
density, particularly the angular anisotropy. Values of ⌬I h
calculated using DMC densities span the range from 55% to
102% of ⌬I exp . A calculation using more recently reported
PIMC densities gave a much lower ⌬I h of only 10% of
⌬I exp . This can be contrasted with the results reported in
Ref. 10, where for heavier rotors, the theory appeared to
systematically overestimate the size of ⌬I h . Similarly, the
hydrodynamic approach was found to overestimate the in-
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
1602
J. Chem. Phys., Vol. 117, No. 4, 22 July 2002
creased moment of inertia when applied to the model problem of a planar rotor interacting with a rigid ring of He
atoms.30
The values of ⌬I h calculated with the PIMC and DMC
densities differ by an order of magnitude. This was quite
surprising to us, based upon the qualitative similarity of the
two densities. It is easily seen that, if the density is scaled by
a uniform factor, ␾ will not change and ⌬I h will scale by the
same factor. Qualitatively, one can compare the peak values
for the PIMC and DMC densities. For the raw densities the
peak lies along the C 3 and is about 20% higher for the PIMC
density. For the angular average of the densities, the PIMC
one still has a higher peak value, by about 25%. In all cases
large differences are observed in the first solvation shell and
much smaller ones in the second solvation shell. Both of
these facts would, in themselves, suggest that the kinetic
energy should be higher for the PIMC density. The PIMC
based density does have a significantly reduced angular anisotropy compared to the DMC density. For example, the
maximum value for 兩 d␳ /d␸ 兩 共which is the source term in the
hydrodynamic equation兲 is 0.46 Å⫺3 for the DMC density
and 0.28 Å⫺3 for the PIMC density. This would suggest that
the kinetic energy of the PIMC density should be less than
that of the DMC density, as is calculated. While these differences are significant, they are far below the resulting difference in kinetic energy. A principal result of this work is the
demonstration of this high sensitivity, which was far higher
than we had expected at the outset.
Given the sensitivity of the results reported above to the
density 共and thus to the angular anisotropy of the potential,
which is poorly characterized experimentally兲 we do not believe it is possible at this time to reach any definitive conclusions about the accuracy of the hydrodynamic model for
this particular molecule. Even given a fixed potential, there
appears to be a substantial variation in the density calculated
by the DMC and PIMC methods. While DMC calculations
have ‘‘trial function bias,’’ the differences are much larger
than the remaining bias as estimated by Barnett and
Whaley.15 The PIMC calculations suffer possible bias based
upon the use of the ‘‘primitive action’’ for the He–SF6 interaction potential.24 No results have yet been published that
would allow the reader to assess the convergence of the published PIMC densities with respect to this bias, nor with
respect to length of simulation run. It has been recently
reported25 that for the case of a Na⫹ cation in helium, a
similar PIMC calculation26 gave an even qualitatively incorrect radial helium density, which was blamed on metastable
trapping of the PIMC simulations.
If we take the reported DMC and PIMC results as accurate, we can ask which should be used in the hydrodynamic
calculations. Since experiments are done at a finite temperature of 0.38 K, one might expect that PIMC calculations at
0.3125 K would be superior to the ground state 共i.e., zero
temperature兲 DMC results. However, as discussed above, the
hydrodynamic model can be derived from the variational
minimization of a trial wave function 关Eq. 共1兲兴 that consists
of the helium ground state wavefunction multiplied by a
phase function written as a sum of one body terms. This
‘‘Feynman-type’’ of approximation is widely used in the
K. K. Lehmann and C. Callegari
theory of liquid helium, and should provide a rigorous bound
on the zero temperature effective moment of inertia.27,28 We
plan to present a detailed derivation of this statement in a
later publication. The relevance to the present discussion is
that the quantum theory we have for the use of the hydrodynamic approach is based upon excitations of the ground state
of the system. As such, this approach is only theoretically
consistent if we us as input the ground state helium density
for the system, i.e., that which is directly estimated by DMC
calculations. Of course, a PIMC density can be used as well
if it is calculated at a temperature low enough that the density
is sufficiently close to that of the ground state. Based upon
the liquid drop model estimates of the excitation energies of
liquid helium,29 this would be expected to be the case for
calculations done at 0.3125 K, but if it is not, then the hydrodynamic calculations should be done with the ground
state helium density. Of course, such a zero temperature
theory cannot predict the temperature dependence of the effective rotational constants, but experiments on SF6 have
shown that there is a negligible temperature dependence of
this quantity over the range T⫽0.15–0.5 K.
The fixed node, Diffusion Monte Carlo calculations of
Lee et al.7 were in excellent agreement with the experimental rotational excitation energies. From the effective rotational constants reported in that work, ⌬I⫽290–305 u•Å2
can be calculated for clusters with N⫽8 –20 helium atoms.
This suggest that, barring cancellation of errors, the nodal
structure of the wave function assumed in that work 共which
had the nodal properties of the rigid rotor wave function for
the SF6 ) should be a reasonable description of the true many
body wave function for this system. It is useful, therefore, to
examine how consistent the hydrodynamic model is with the
functional form assumed in that work. The many body wave
function of Eq. 共1兲 gives the amplitude in the frame rotating
with the SF6 . Quantization of the SF6 rotational motion in
the adiabatic approximation results in a full wave function
that is the product of the helium wave function, Eq. 共1兲,
times the rigid rotor wave function for the SF6 . Since the
helium ground state wave function is nodeless, this wave
function will have, in the limit that the one-body phase terms
m ␾ /បⰆ1, the same exact nodes as those assumed in the
DMC calculation of Lee et al. The maximum value for ␾ for
our solution is ⬃5 Å2 ␻ , and for the J⫽1 level ␻
⬇4 ␲ B eff , where B eff is the effective moment of inertia of
SF6 in helium, 1.04 GHz. This gives a maximum hydrodynamic phase of ⬇0.02 over the entire solution space, and
thus the above inequality is everywhere satisfied. We would
like to point out that for the model problem of a planar rotor
coupled to a ring of helium atoms 共which can be solved
exactly兲, the errors in the equivalent fixed node approximation was of this same size and yet the DMC estimate of the
rotational excitation energy had errors of at most a few
percent.30 We can conclude, in contrast with unsubstantiated
claims made by Kwon et al.,9 that the fixed nodes assumed
in the DMC calculation of Lee et al. are entirely consistent
with the results of the present quantum hydrodynamic treatment.
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem. Phys., Vol. 117, No. 4, 22 July 2002
ACKNOWLEDGMENTS
The authors wish to acknowledge Professor Giacinto
Scoles for many helpful discussions. This work was supported by the National Science Foundation and the Air Force
Office of Scientific Research.
J. P. Toennies and A. F. Vilesov, Annu. Rev. Phys. Chem. 49, 1 共1998兲.
A special issue on helium nanodroplets was published by J. Chem. Phys.
115共22兲 共2001兲.
3
K. K. Lehmann and G. Scoles, Science 279, 2065 共1998兲.
4
M. Hartmann, R. E. Miller, J. P. Toennies, and A. F. Vilesov, Phys. Rev.
Lett. 95, 1566 共1995兲.
5
C. Callegari, R. Schmied, K. K. Lehmann, and G. Scoles, in Ref. 2, p.
10090.
6
C. Callegari, A. Conjusteau, I. Reinhard, K. K. Lehmann, and G. Scoles, J.
Chem. Phys. 113, 10535 共2000兲.
7
E. Lee, D. Farrelly, and K. B. Whaley, Phys. Rev. Lett. 83, 3812 共1999兲.
8
Y. Kwon and K. B. Whaley, Phys. Rev. Lett. 83, 4108 共1999兲.
9
Y. Kwon, P. Huang, M. V. Patel, D. Blume, and K. B. Whaley, J. Chem.
Phys. 113, 6469 共2000兲.
10
C. Callegari, A. Conjusteau, I. Reinhard, K. K. Lehmann, G. Scoles, and
F. Dalfovo, Phys. Rev. Lett. 83, 5058 共1999兲; 84, 1848 共2000兲.
11
We should point out that this level of agreement is far better than what
should be expected given the considerable uncertainty in the anisotropy of
the He–SF6 interaction potential used 共the two parameters that determine
the anisotropy of the He–SF6 potential of Ref. 12 have reported values of
⫺0.6⫾0.3 and 0.14⫾0.14). To be conservative, one should therefore only
compare theoretical values 共which have so far all been obtained with the
same potential兲, although we take the agreement between models that use
1
2
Moments of inertia in helium nanodroplets
1603
rather different assumptions as an indication that the potential of Ref. 12
might be significantly more accurate than the error estimates would suggest.
12
R. T Pack, E. Piper, G. A. Pfeffer, and J. P. Toennies, J. Chem. Phys. 80,
4940 共1983兲.
13
A. Conjusteau, C. Callegari, I. Reinhard, K. K. Lehmann, and G. Scoles,
J. Chem. Phys. 113, 4840 共2000兲.
14
M. Casas, F. Dalfovo, A. Lastri L Serra, and S. Stringari, Z. Phys. D: At.,
Mol. Clusters 35, 67 共1995兲.
15
R. N. Barnett and K. B. Whaley, J. Chem. Phys. 99, 9730 共1993兲.
16
F. C. von der Lage and H. A. Bethe, Phys. Rev. 71, 612 共1947兲.
17
K. Fox and I. Ozier, J. Chem. Phys. 52, 5044 共1970兲.
18
W. R. Fehlner, S. B. Nickerson, and S. H. Vosko, Solid State Commun. 19,
83 共1976兲.
19
W. R. Fehlner and S. H. Vosko, Can. J. Phys. 54, 2159 共1976兲.
20
F. Dalfovo, Z. Phys. D: At., Mol. Clusters 29, 61 共1994兲.
21
U. Buck, K. H. Kohl, A. Kohlhase, M. Faubel, and V. Staemmler, Mol.
Phys. 55, 1255 共1985兲.
22
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. Vetterling, Numerical
Recipes 共Cambridge University Press, Cambridge, 1986兲.
23
L. Fox, Numerical Solution of Ordinary and Partial Differential Equations 共Pergamon Press, New York, 1962兲, Chap. 21.
24
D. M. Ceperley, Rev. Mod. Phys. 67, 279 共1995兲.
25
M. Buzzacchi, D. E. Galli, and L. Reatto, Phys. Rev. B 64, 094512
共2001兲.
26
A. Nakayama and K. Yamashita, J. Chem. Phys. 112, 10966 共2000兲.
27
A. J. Leggett, Physica Fennica 8, 125 共1973兲.
28
K. Kim and W. F. Saam, Phys. Rev. B 48, 13735 共1993兲.
29
D. M. Brink and S. Stringari, Z. Phys. D: At., Mol. Clusters 15, 257
共1990兲.
30
K. K. Lehmann, J. Chem. Phys. 114, 4643 共2001兲.
Downloaded 24 Jun 2003 to 128.112.71.198. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp