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A new determination of the ground state interatomic potential for He2
Ronald A. Aziz a
; Frederick R. W. McCourt b
; Clement C. K. Wong b
a
Department of Physics, University of Waterloo, Waterloo, Canada b
Guelph-Waterloo Centre for Graduate
Work in Chemistry, University of Waterloo, Waterloo, Canada
Online Publication Date: 20 August 1987
To cite this Article Aziz, Ronald A., McCourt, Frederick R. W. and Wong, Clement C. K.(1987)'A new determination of the ground state
interatomic potential for He2
',Molecular Physics,61:6,1487 — 1511
To link to this Article: DOI: 10.1080/00268978700101941
URL: http://dx.doi.org/10.1080/00268978700101941
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MOLECULARPHYSICS,1987, VOL. 61, No. 6, 1487-1511
A new determination of the ground state interatomic
potential for Hez
by RONALD A. AZIZ
Department of Physics, University of Waterloo,
Waterloo, Ontario, N2L 3G1, Canada
FREDERICK R. W. McCOURT and CLEMENT C. K. WONG
Guelph-Waterloo Centre for Graduate Work in Chemistry,
University of Waterloo, Waterloo, Ontario, N21 3G1, Canada
(Received 10 February 1987; accepted 31 March 1987)
A simple accurate potential of the HFD-B form, which appears to be the
best characterization of the He-He interaction constructed to date, is presented.
It has been fitted to low temperature second virial coefficient data and recent
accurate room temperature viscosity data, while at the same time pinning the
repulsive wall to the value calculated by Ceperley and Partridge at 1Bohr. It
possesses a well depth of 10.948K, considerably deeper than many of the recent
empirical or ab initio potentials. It reproduces, within experimental error, such
dilute gas properties as second virial coefficients, viscosities and thermal conductivities
over a wide temperature range. It also predicts, within experimental
error, such microscopic properties as differential cross sections, high energy
integral cross sections and backward glory oscillations in the integral cross
sections. Finally, it accounts for nuclear magnetic relaxation in 3He and supports
a weakly bound state in the 4He interaction.
1. Introduction
The problem of constructing accurate potential energy functions is an old one
which, even for relatively simple interactions such as those between rare gas atoms,
has not yet been fully resolved. For the calculation of beam scattering cross sections,
second virial coefficients and transport properties of the rare gases to within a few
percent, certain model potentials appear to be adequate [11. Included in this set of
model potentials are the Hartree-Fock plus damped dispersion (HFD) [21, the
Tang-Toennies (TT) [3] and the exchange coulomb (XC) [41 models. For more
accurate calculations, however, it is still necessary to appeal to multiproperty fits
[5]. Such an approach was employed for the determination of an accurate He-He
potential by Aziz et al. [6] in 1979 using the HFD form in conjunction with what
were then considered to be the best available experimental data. Their HFDHE2
potential was obtained by fitting accurate second virial coefficients at intermediate
temperatures [7] (98K to 423 K) as well as reliable high-temperature transport
coefficient data [8, 9]. It has nearly the correct Hartree-Fock short range repulsion
[101 and long-range attraction [11] and supports a single (weakly) bound state
[12]. Moreover, it represents well the isotopic differences in the viscosity [13] at
temperatures below 100 K (where the transport coefficient data are less reliable) and
it predicts differential cross sections reasonably well [141. Finally, Kalos et al. [15],
using a Green function Monte Carlo (GFMC) method to calculate the properties of
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1488 R.A. Aziz et al.
ground-state 4He, concluded that of the accurate He-He potentials available [6, 14,
16, 17] in 1981, the HFDHE2 potential gave the best agreement with experiment.
Since 1981, additional He-He potentials have been presented. Among them are
those of Ng et al. [4], Douketis et al. [18], Tang and Toennies [19], and Feltgen et
al. [20]. In particular, the HFIMD potential of [20] was obtained by inversion of
the backward glory oscillations appearing in the integral cross sections and associated
with the identical particle scattering of 4He and 3He. Feltgen et al. employed a
physically realistic (but mathematically complicated) two parameter model. In addition
to the scattering data (which determined the potential only in the region 1.8 A
to 2.2A), the full HFIMD potential also included all available ab initio data. It
supports a bound state, as does the HFDHE2 potential [12, 20], and coincides with
the ab initio potentials of Burton [21] and Liu and MacLean [22]. For separations
greater than 3A, it agrees with the HFDHE2 potential but for separations less than
2/~, it is considerably softer. An earlier repulsive potential wall obtained by
Foreman et al. [23], also derived from (high-energy) integral cross-section data, lies
between the HFDHE2 and HFIMD potentials, but is closer to the latter than to the
former.
There is a growing body of evidence to indicate that the majority of the modern
potentials are too repulsive. Firstly, Ceperley and Partridge [24] have employed
quantum Monte Carlo methods to determine the exact Born-Oppenheimer interaction
energy of two helium atoms with internuclear separations between 0.5 A and
1.8A. Their results indicate that the HFDHE2 potential is too repulsive below
1.8 A. Secondly, Stebbings and coworkers [25] measured absolute differential cross
sections for small-angle elastic scattering in He-He collisions at keV energies and
found that their data were consistent with a potential less repulsive than the
HFDHE2 potential. Kalos and Whitlock [26] have continued to find nonetheless
that, for their GFMC calculations, which depend critically upon the potential well,
the HFDHE2 potential remains superior to the HFIMD potential.
In the few years since the HFDHE2 potential was constructed, new virial coefficient
data for 3He [27] and 4He [28-31] have been measured at various standards
laboratories, new ab initio calculations of the dispersion coefficients have appeared
[32] in the literature, and new measurements of the transport properties have been
made [33-37]. It is now evident that the HFDHE2 potential as well as the newer
ones are inadequate [24, 38, 39] to the task of predicting all these data. The thrust
for producing a new potential is many-fold, the least of which is the determination
of an accurate characterization of the helium interaction. These are:
(1) The International Practical Temperature Scale below 18 K [27-31] is to be
redefined in terms of an ideal gas thermometer using helium gas. This
requires an accurate knowledge of the virial coefficients, in particular, the
second virial coefficient. To this end, many standards laboratories have been
measuring them. Rather than expressing the experimental values in terms of
an empirical correlation function, it would be preferable [38] to use a potential
as the correlating 'instrument '.
(2) The National Bureau of Standards in Washington 140] is redetermining the
universal gas constant R from measurements of the sound velocity in helium
and argon. To assign an error to their determination, they require a knowledge
of the viscosity and thermal conductivity of these gases at the triple
point of water (273-16K). Values derived from an accurate potential would
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Interatomic potential for He 2 1489
be more physically based than those expressed in terms of an empirical
correlating function.
(3) In many-body studies, the original HFDHE2 potential has served as an
excellent 'effective' pair potential, in the sense that pair-wise treatment of the
atoms in the condensed phase (neglecting many-body effects) predicts
condensed-phase properties reasonably well [15, 26]. Should a newlydetermined
and more precise potential behave similarly, it might then be
possible to draw important conclusions regarding the relative magnitudes of
the various short-range and long-range many-body forces [41].
(4) The more recent accurate second virial coefficient data seem to require for
their prediction a well depth greater than that of any potential so far
obtained by purely ab initio methods [21, 22, 42]. This may in fact indicate
that a reassessment of ab initio methods is in order.
This paper presents a new empirical potential of the HFD-B form which either
reproduces or is fully consistent with both short-range and long-range ab initio
calculations, as well as with second virial coefficient measurements of 3He and 4He
over an extended temperature range, and with new more precise transport property
measurements. In addition, it accounts well for differential [14, 43] and integral [20,
44, 45] scattering cross section measurements.
2. The HFD--B(HE) potential
The potential form chosen to represent the He-He interaction is the so-called
HFD-B form of Aziz and Chen [46] which has certain advantages over the HFD-C
form. It is, for example, considered to be a more 'realistic' form in that it reproduces
the spectroscopic spacings of the rare gas dimers better than does the HFD-C
form [47]. Moreover, it does not 'turn over' and become negative at very small
separations.
The form of the HFD-B potential is
V(r) = ~V*(x),
where
with
where
(1)
2
V*(x) = A* exp (--ct*x + fl*x2) -- F(x)~, C2j+6/X2j+6, (2)
j=O
I )2]F(x)= exp - -1 , x<D,
1, x >~D,
(3)
/x
= - . (4)
r m
For the dispersion coefficients, we have assumed the ab initio values of Thakkar and
Koide et al. [32]. The potential has therefore four adjustable parameters, namely, fl,
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1490 R.A. Aziz et al.
Table 1.
Parameters for HFD--B(HE) potential
A* 1.8443101(5)
at* 10.43329537
ca 1.36745214
ca 0.42123807
Clo 0.17473318
Cr/a.u. 1.461
Ca/a.u. 14.11
Clo/a.u. 183.5
fl* --2"27965105
fl - 0"259660
D 1"4826
k/K 10"948
rJA 2"963
a/A 2"6369
Not all figures displayed are significant. Displayed digits are given specifically for the
avoidance of round-offerrors. (5) means 105.
D, e, and rm, determined by fitting the potential to the low temperature (1.47 K to
20-3 K) aHe second virial coefficient data of Matacotta et al. [27], the low temperature
(2-60K to 27.1 K) 4He second virial data of Berry [28], and the accurate
viscosity data of Vogel [33, 37] (298 K to 641 K), while at the same time pinning the
repulsive wall to the value calculated by Ceperley and Partridge [24] at 1.0 Bohr.
We decided to use the accurate viscosity data of Vogel [33, 37] rather than the
virial coefficient at 25~ to establish the location of the lower repulsive region of the
potential wall because there appears to be no agreement as to what value the virial
coefficient should have 1-7,48-51]. The viscosity data have stated error bars ranging
from 0.1 per cent at 300K to 0.3 per cent at 600K. Previous work on neon [52],
and on argon and krypton [47] indicate that these error limits are realistic in that
potentials consistent with such data are capable of predicting other properties sensitive
to the same region of the potential. The parameters of the present potential are
given in table 1.
3. Method of calculation
3.1. Second virial coefficients
Because of the high degree of accuracy required for the present study, full
quantum mechanical virial coefficients have been calculated using established procedures
1-54--56]. The quantum expressions contain three terms, the perfect gas
contribution, the contribution from bound states, and the dynamical contribution
associated with the scattering phase shifts. Spin statistics must also be taken into
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Interatomicpotentialfor He2 1491
I.|'
Q,!
r,o
<
o.
-Ih$.
Figure 1.
i
t
i
%%%
-I'
-I.$
0 2 4 6 8 I0
E/cm-I
Low energy behaviour of qo for 4He-4He scattering (dotted line) and aHe-aHe
scattering (solidline),showingapproach to n and O,respectively,as Ek~ O.
account so that the second virial coefficientfor a boson gas is given by [54]
21 + 1 (mk.T'~3/2
N \--~--~] B(T)
= + 89 (I+1),~.~ (2/+ 1) exp -kBT] 1
+ I ~ (21+ l)[exp (- Et~:")
fold k.TJ-1]}
fo{ }_-8 (I+ 1) ~ (2/+ 1)q,+I2(2/+l)q, exp(-x) dx,
I even 1 odd
(5a)
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1492 R.A. Aziz etal.
while that for a fermion gas is given by
2l+l(mksT)3/2N--~ ~(1)....
=--89 .... ~sTj--1]
+ (I + 1),~,oaa(21+ 1) exp -- kBTJ- 1
8 I 2 (21+l)t h+(I+l)~(21+l)t h exp(-x) dx, (5b)
g I. l even 1 odd
where x is the reduced energy
Ek
x = (6)
kBT"
Equation (5 a), with nuclear spin quantum number I = 0, applies to 4He and equation
(5 b), with I = 1/2, applies to 3He.
The perfect gas contribution and the bound state contributions are important
only at low temperatures. For 4He, our potential supports only a single low-lying
bound state at an energy of 1.684mK, while for 3He, it supports no bound states.
The existence or nonexistence of a bound state is reflected in the behaviour of the
phase shifts associated with the partial waves for the corresponding orbital angular
momentum quantum number via Levinson's theorem 1-57-1,which states that the
phase shift for a partial wave associated with a bound states changes by it for each
bound state in the well. This is clearly illustrated in figure 1 where the phase shift ~/o
is plotted as a function of collision energy Ek at low energies. For 4He, the r/0 phase
shift approaches ~ as Ek approaches zero, while for 3He, it goes to zero as Ek
approaches zero, in accordance with Levinson's theorem. The approach of ~/0 to rr
for 4He is further illustrated in the inset showing its behaviour near Ek = 0.01 cm - 1.
The relative importance of the individual contributions to B(T)at low temperatures
is illustrated in table 2.
Two independent sets of virial coefficient values are presented. In one of the
calculations, the phase shifts were first obtained at a very fine mesh of reduced wave
numbers q, specifically, 477 q-values between q = 0.08 and 27-00 for #He (between
0.07 and 26.99 for 3He) for the calculation of low-temperature virial coefficients (up
to 27 K) and 685 reduced q-values between 0.08 and 80.00 for high-temperature
calculations (up to 623 K). Further details of the partitioning of the energy intervals
is given in table 3. Thermal averaging was accomplished using a Simpson's rule
numerical integration. In the other calculation, the phase shifts were obtained at a
slightly coarser energy mesh, using 268 Ek values between 0-01 cm-1 (Keysers) and
8000 cm-1, with the thermal averaging carried out using a Simpson's composite rule
over energies between 0.01 cm-1 and 10.0cm-1 combined with a cubic spline integration
over those energies above 10.0 cm- 1. Further details regarding these meshes
may also be found in table 3. The two sets almost converge. For example, for 3He at
1.47 K, the two sets of calculations differ by only 0.26cc/mol in 174cc/mol, and are
practically identical at 20 K. Differences at room temperature are in the order of
0-04 cc/mol in 12 cc/mol.
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Interatomic potential for He2 1493
Table 2. Various contributions to the second virial coefficient of helium-4 on the basis of
the HFD-B(HE) potential
T (K) B (ideal) B (binding) B (phase shift) B (total)
3"00 --13"61 --0.122 --106-22 --119"95
4.00 --8"84 --0-060 --76"08 --84-98
5"00 --6"33 --0"034 --57"93 --64.29
8"00 --3'13 --0'011 --30"30 --33"43
10"00 --2-24 --0"006 --20"93 --23"17
15.00 --1'22 --0.002 --8"33 --9"55
20-00 --0"79 --0.001 --2.01 --2'80
27.00 --0"50 --0.001 2"85 2"35
30"00 --0"43 --0.000 4-22 3'79
35"00 --0"34 --0"000 5"94 5"60
40"00 --0"28 --0-000 7"21 6-93
50"00 --0"20 --0"000 8"91 8"71
60"00 --0"15 --0"000 9-98 9.82
70"00 --0"12 --0"000 10"68 10"56
80"00 --0"10 --0-000 11.16 11"06
90"00 -- 0"08 -- 0.000 11"49 11"41
100"00 --0-07 --0"000 11'74 11-67
125"00 --0"05 --0.000 12"07 12"02
150-00 --0"04 --0.000 12.20 12"16
175"00 --0"03 --0"000 12"23 12-20
200.00 --0"03 --0"000 12"20 12"17
250"00 --0"02 --0-000 12.04 12"02
300-00 --0"01 --0.000 11"85 11"84
Table 3.
From To Step size From To Step size
q-Integration mesh for virial calculations
0.08 3.00 0-020 27-00 61.0 0.200
3.00 6.00 0.025 61.0 80.0 0.050
6.00 27.00 0.100
Ek-Integration mesh for virial calculations
0.01 0-10 0.0025 30-0 100.0 5.0
0.10 1.00 0.050 100.0 200.0 10.0
1.00 6.00 0.10 200.0 1000.0 20.0
6.00 20.0 0.50 1000.0 2000.0 50.0
20.0 30.0 1.0 2000.0 8000.0 100.0
Ek-Integration mesh for TI calculation in aHe
0"01 0.10 0.0025 20"0 30.0 1.0
0.10 1.0 0.050 30.0 100.0 5-0
1-0 6-0 0"100 100.0 200.0 10"0
6.0 20.0 0.500 200.0 1000-0 20.0
3.2. Transport properties
Quantum temperature-dependent reduced cross sections have been calculated
using an adaptation of the classical reduced integral program of O'Hara and Smith
[59]. In this approach, a Chebyshev approximation is found for the quantal cross
sections which are calculated using standard formulas [60] which include appropri-
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1494 R.A. Aziz et al.
ate spin and statistical effects 1-61]. The required phase shifts were obtained using
the quantal phase shift routine of LeRoy 1,62] which employs a numerical integration
and a Gaussian quadrature of the JWKB correction to the phase shift. The
Clenshaw-Curtis quadrature 1,58]was used to perform the energy integration in the
required collision integrals 1,61].O'Hara and Smith argue that this approach is very
efficient and reliable because Clenshaw-Curtis integration is almost as accurate as
gaussian quadratures for the same number of abscissae and, in addition, the abscissae
for higher order integrations overlap those of lower order integrations. An
important feature of the Clenshaw-Curtis quadrature is its error estimate. This
reliable estimate is based on the same function evaluations needed in the quadrature
formula. Therefore, only the computation necessary to achieve a specified accuracy
is done. Third Chapman-Cowling approximation expressions 1,63] have been used
to calculate the viscosity and thermal conductivity coefficients for comparison with
experimental values.
3.3. Nuclear magnetic relaxation
The 3He nucleus possesses a nuclear spin I = 89whereas the 4He nucleus has
nuclear spin ! = 0, and consequently 3He gas has a nuclear magnetization. As a
consequence of the nonzero nuclear spin, two colliding 3He atoms may form either
a singlet or a triplet nuclear spin state. In the singlet state, with IT ----0, the interaction
is simply V(r) but, in the triplet state, with Ix = 1, the atoms also interact
weakly via the magnetic dipole-dipole interaction and, in principle, also via a
spin-rotation interaction (rotation of the short-lived 3He2 collision pair) so that the
interaction potential is
V(r, I) = V(r) + Vmag(r, 11"), (7)
where Vmagis orders of magnitude smaller than the central potential V(r) which we
have been discussing earlier.
Traditional nuclear spin relaxation experiments were first carried out on 3He gas
by Chapman and Richards [64]. They determined that the relaxation mechanism
was that of dipolar relaxation 1-65].As expected, the longitudinal relaxation time T1
was found to be very long. A kinetic theory description of the dipolar relaxation in
3He was given in 1967 by Chen and Snider [66] and a detailed numerical calculation
of 7"1,using the McLaughlin-Schaefer (MS) 1-10] and Beck 1-16] potentials
was first made by Shizgal 1-67]. He used a distorted-wave Born treatment with the
dipole-dipole interaction serving as the perturbation.
Recent measurements of T~ in 3He at low temperatures were made by Chapman
1,68]. He also used Shizgars code and compared his data with values of Tt calculated
from the MS 1,10], Beck 1,,16]and Bruch-McGee [17] potentials. He obtained
good agreement between measured and calculated values of T1. We have performed
essentially the same calculations as Shizgal and Chapman for the Beck and MS
potentials as well as for the HFDHE2 potential of Aziz et al. I-6] and the present
HFD-B potential.
Our distorted-wave Born approximation (DWBA) calculations were based upon
formulae given by Shizgal 1,67], properly allowing for the fact that relaxation only
occurs via those collisions which correspond to the formation of a triplet nuclear
spin state of the 3He2 complex (in which the dipolar relaxation mechanism is
operative). In calculating the accurate wave functions required at each collision
energy, we have used a modification of the Le Roy phase shift-time delay code 1,62],
starting our integration from the inner turning point and employing 8000 steps in
the outward integration. The actual step size depends upon the energy at which the
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Interatomic potential for He2
Table 4.
1495
R (au) from scattering [23] Ab initio [24] HFD-B(HE) HFDHE2 [6]
Strongly repulsive region of helium potential (in K)
1.00 308 500 291900 291 300 543800
1.25 182900 174600 175500 299900
1.50 108500 105000 104800 165300
1.75 64330 61400 61990 91 120
2.00 38 150 35 800 36 340 50090
2.25 22620 20950 21090 27 360
2.50 13410 11920 12100 14820
2.75 7955 6 852 7 962
2.8346 6 666 5 540 5633 6439
3.00 4 717 3800 3 819 4 240
calculation is made. We have employed 228 collision energies spanning the range
0-01 cm-i to 1000cm-1 for the energy averages. As in one set of virial coefficient
calculations, we have combined a composite Simpson's rule integration from
0.01cm -1 to 10.0cm -1 and a cubic spline integration from 10.0cm -1 to
1000 cm- 1. Further details on the mesh can be found in table 3.
4. Results and discussion
4.1. Repulsive wall
The repulsive wall of our potential was pinned at an internuclear separation of
one Bohr to the value obtained by Ceperley and Partridge using quantum Monte
Carlo techniques to determine the Born-Oppenheimer interaction energies at
separations of 1.0 to 3.0 Bohr. In table 4, we present a comparison between values
for the repulsive wall of the new HFD--B(HE) and the earlier HFDHE2 potentials
with the interaction energies calculated by Ceperley and Partridge [24] at the same
separations. Also shown are the results inferred from high energy integral crosssection
measurements of Foreman et al. [23]. As might be expected from our fitting
procedure, our values are in excellent agreement with those of reference [24] for the
whole range of separations. Our potential lies somewhat lower than the scattering
potential especially at larger separations but is certainly in better agreement with it
than is the earlier potential of Aziz et al. [6]. The repulsive walls of the HFD-B(HE)
and HFDHE2 potentials are shown in figure 2 along with that obtained by
Foreman et al. [23] from high energy integral cross section measurements.
4.2. Second virial coefficients
Probably the most reliable low temperature virial coefficients for +He are those
of Berry [28] which, in the range 2.6 to 27-1 K, have error bars ranging from 1 to
0.2 cc/mol. Smoothed fits to these data have been provided by Plumb [30] and by
Steur [69] (the latter is a surface fit to the data of Berry [28], Kemp et al. [70] and
Kell et al. [49]). Other data considered include those of Kemp et al, [70] (27-1 to
172K); those of Kell et al. [49] (values at 298.15 and 623.15K), Holste et al. 1980
[50] (100 to 300K) and NBS values (273 to 423K) as cited by Guildner and
Edsinger [51].
Recent data for 3He have been obtained by Matacotta et al. [27] for the range
1.47 to 20.3 K with error bars ranging from +__0-5 cc/mol at 1.5 K to _ 0.2 cc/mol at
DownloadedBy:[CDLJournalsAccount]At:12:0113January2009
1496 R.A. Aziz et al.
x_
105~
10 4
103
102
10
f i
\\ He-H~
X
.0 1.6 2.2 2.8
Figure 2. Behaviour of the repulsive walls for the HFD-B(HE) (solid line), HFDHE2
(dashed line) potentials together with the repulsive wall obtained in [23] from inversion
of high-energy integral cross section data (dotted line).
20.3 K. Revised values corrected for the third virial coefficient have been determined
by McConville [71] and smoothed by us by a least squares procedure. The high
temperature data have been determined by the Burnett method. In addition,
Gammon [7] presented values from 98 to 423 K derived from acoustic measure-
ments.
In table 5, we present various sets of data, their corresponding error bars and the
deviations of the predictions based on the present potential. It predicts, in addition
to the data of Berry [28] and Matacotta et al. [27] to which it was fitted, the data of
Kell et al. [49], Kemp et al. [70], Gammon [7] and Holste et al. [50] to within
experimental error. The predicted sets of data encompass the extensive temperature
range from 1.47 to 623 K ! The value of the second virial coefficient of Waxman and
Davis [48] at 298-15 K appears to be too high by about 0-05 cc/mol, or about four
times their estimated error. The values determined at this temperature by Kell et al.
(11.83 + 0.03cc/mol), Gammon (11.86 __+0-05co/tool) and Holste et al.
(11.81 + 0-1 cc/mol) are all lower; the predictions based on our potential agree with
all of these measurements, given their less optimistic error bounds. Agreement with
the recent measurements of Cameron and Seidel [72] is less satisfactory. Deviation
plots for 4He and aHe are given in figures 3 and 4, respectively. The HFDHE2
potential clearly does not fare as well as does the new HFD-B(HE) potential,
especially at low temperatures.
4.3. Transport data
4.3.1. Viscosity
In addition to reproducing the single value of the viscosity at 300 K which was
used in its determination, the present HFD-B(HE) potential also predicts to within
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Interatomic potential for He 2 1497
4
2
0
-2
T
o 1.0
E
ro
0.5I i
x
0
133
I
,._, -0. 5
O
rj
m
0.4
0.2
i , i i i l ~ l l i ! i
~He: Berry Data (Smoothed)
Error Bars • | to +-0.2 cc real-I\
k
\
. . . . . :
i i , llO I i i lib m , , ,
2 6 14 22 2B
! i , ! i !
4He: Kemp et al Data
Error Bars • Icc mol-I
i
3O
~ b
I I I I I I I
20 50 lOO 140 lBO
0
-0. 2
I | i i i ! , ! , 1
4He: Gammon Data
Error Bars • mol "I
1 I i i i i I i , | i I i
lOO 150 200 250 300 350 400 450
T/K
Figure 3. Deviation plots for 4He second virial coefficient. Deviations from the data of
Berry [28] (upper plot), Kemp et al. [70] (middle plot) and Gammon [7] (lower plot).
Potentials identified as in figure 2.
Figure 4.
T
,-~ 4
0
E
0
O
C3_ 2
X
OJ
m
i
o 0
o
CI3
-2
, , i ! , , 1 i
3He: Motocotto et o l Data
Error Bars • 5 to • 2cc mol -I
I
|
I
\
\
\
I I [ I I I I I I I
16 204 8 12
T/K
Deviation plots for aHe second virial coefficient. Deviations from the data of
Matacotta et al. 1-27].Potentials identified as in figure 2.
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1498 R.A. Aziz et al.
0
,.0
"0
0
0
.o
"0
J..
am
am
am
mm
~4
. . . . . . . . . . ~ o ~ ~ ~ ~
I I I I I I I I I I I I I I I I I I I
~ ~ I ~ 0 ~ 1 ~ ~ I I ~
TTT' ' ' ' TTT ' I I ' T ' ' ' I ' ' T' ' '
6 6 6 6 6 6 o 6 6 6 6 6 o o o 6 6 o 6 o 6 o 6 6 6 o 6 6
I I I I I I I I I I I I I I I I I
~ ~ 1 ~ ~ 1 ~ ~ ~
i i i I I I i i i I I I i I I I I i I I I
i i i I I I i i i I I I i I I I I i I I I
m m m ~0
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lnteratomic potential for He 2 1499
6 6 6 6 6 6 6 o o 6 6 6 o o o 6 o o o 6 o o 6 o o6 6 o 6 o 6 o 6 6 6 o
t I I I I I I I I I I I I I I I I I I I I I I I
~ 0 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 O0 ~ 0 0 0 0 0 0 0 0
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
I I I I I I I I I I I I I I I I I I
m ~ ~ o ~ ~ ~ ~ ~ ~o ~ . . . . . . . . .
o o o o o o o o o 6 o o 6 6 6 6 6 6 o 6 o 6 6 6 66 6 6 6 6 6 6 6 6 6 6
0= ~
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1500 R.A. Aziz et al.
~
0
E ""
0 0 0 0 0 0 0
I I I I I I I I
o 6 o 6 6 6 6 6
I I I I I I I I
~'~
. . . . . . . . . . . .~- o ~
I I I I I I I I I I I I I I
I I I I I I 11 I I I
6 6 6 d 6 o d 6 6 d d o d 6 6 d
I I I I I I I I I I I I I
I I I I I I I I I I I I I I
6 6 o 6 6 6 6 6 6 6 6 6 6 6 o 6
I TTTT ~ ~ll~,I
,~rq
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Interatomic potential for He 2 1501
I I I I I I I I I I I I ~11 I I
I I I I I I
66666 66666666666 6 ~
I I I I I I I I I I I I ~ I I I I
1 I I I 1 I I I I I I
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 o ~ 6
I I I I I
~r~
o r " o-~
.~oo
,.el "0
~8
o~
8~
+-- .~-+
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1502 R.A. Aziz et al.
r-i
X
Ol
X
OJ
I
0
(J
v
CD
C3
9
6
:3
0
-3
i ! i ! 1
He-He; Vogel Data
Error Bars zO.I% to •
280
=,
I I I I
400 520 640
4
3
2
1
0
-1
i ! ! !
He-He: Clarke-Smith Data
Error Bars • lZ
_ _ ,. 9 ,
, I I I I
100 200 300 400
i ! i i i !
He-He: guevara et al Data
Error Bare •
2
0 ~ ~ '
--~ I l i I l I I I
000 1:300 1600 1900 2200
T/K
Figure 5. Deviationplotsfor viscosity:Percentagedeviation from data ofVogel[33] (upper
plot), Clarke and Smith [73] (middleplot) and Guevara et al. [74] (lowerplot).
___0-2 per cent the other highly accurate viscosity measurements of Vogel [33, 37],
as well as the intermediate temperature (77 K-300 K) viscosity data of Clarke and
Smith [73] (77 to 300 K) to within _+1 per cent and the high temperature (1100 K-
1800K) viscosity data of Guevara et al. [74] to within ___0-6 per cent. Above
1800 K, where temperature assignment is probably difficult, the deviations between
calculated and experimental results would appear to suggest a potential which is too
repulsive, even though the present potential is consistent with the interaction energies
calculated by Ceperley and Partridge for the full range of separations (1.0-
3.0 Bohr) of their calculations. In this context, it is perhaps interesting to note that
the Los Alamos viscosity measurements for neon, argon, and krypton also demonstrate
the same behaviour above 1800K [75, 74, 76] with respect to other recent
potentials. It does not agree particularly well with the older low temperature viscosity
data but, as these data are considered to be unreliable (cf. [6]), we have
ignored them. Deviation plots for both the HFDHE2 and HFD--B(HE) potentials
are given in figure 5. As can be seen, calculations based upon the current HFDB(HE)
potential give better agreement with the experimental data at all temperatures
below 1800K than do calculations based upon the HFDHE2 potential.
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Interatomic potential for He2
Table 6. Rms deviations for selected transport properties of helium-4
1503
Temperature Error Maximum
range bars Rms ___per cent
Data (K) (per cent) deviation deviation
Viscosity //~P
Vogel 1-33] 298-623 _0.1 to -t-0.3 0.340(0-12) -0.04 to -0.17
Vogel [37]'~ 298-641 +0.1 to +0-3 0.146(0.05) -0.02 to -fill
Clarke and 77-374 _ 1 0.825(0.57) -0.20 to - 1.0
Smith [73]
Clarke and 120-360 ___1 0.934(0.63) - 0.25 to - 1"2
Smith [73]
(smoothed)
Guevara et al. 1100-1800 +0.6 2-32(0.38) -0.53 to 0-59
[74]
Guevara et al. 1100-2150 ___0.6 11.4(1.50) -0.53 to 3.7
[74]
Kestin et al. 300.65 +0.3 0.218(0.11) -0.11
[36]
Thermal conductivity/mW (mK)-1
Assael et al. 308"15 +0.2 0.21(0.13) 0.13
[35]
Kestin et al. 300.65 -t-0.3 0.11(0.07) 0"07
[36]
Acton and 4-20 ___1"1 0.11(0.73) - 1"2 to 0"34
Kellner [34]
Acton and 4-20 _ 1-1 0.10(0.68) -1.0 to 0"37
Kellner [34]
(smoothed)
Haarman [77] 328-468 +0-3 0.64(0.33) 0.21 to 0"42
Jody et al. [8] 400-2500 -t-2.0 to +4.7 10.8(2-25) -0"05 to 4.8
i Data obtained using edge-correction C.
Values in parentheses refer to rms percentage deviation.
4.3.2.~Thermal conductivity
Although earlier viscosity data below 77 K may not be reliable, recent low
temperature thermal conductivity data, which depend upon the same f~t2.2)* collision
integral as does the viscosity [63-1, obtained by Acton and Kellner [34] have
stated accuracies of + 1-1 per cent. There is, however, appreciable scatter in their
measurements and as a result, we have smoothed the data using a least squares
procedure. The thermal conductivities are predicted by our potential to within
experimental error (the raw data to __+1.1 per cent and the smoothed data to ___1-0
per cent). Other thermal conductivity data considered are those of Assael et al. [35-1,
Kestin et al. [36], Haarman [77-1, and Jody et al. [8]. The data of Haarman,
obtained using a transient hot-wire technique, are not quite predicted to within
experimental error but the more recent data of Kestin et al. [36] and Assael et al.
[35-1, who used the same technique but with an improved apparatus, are predicted
to within their stated error bounds (+ 0.3 and + 0-2 per cent respectively). Details of
the rms and rms percentage deviations for the transport properties are given in
table 6.
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1504 R.A. Aziz et al.
C
0
-el
.,o
0 1000
~r--~
@
.f-I
@ 100@ C
0 D
>~
LO L I0
,-4 0
L
n 1
~- .1
-r
rn
4He-'He HFO-B
E=7.99x10-lsd[57.87 K]
20 40 60 80
Center-og-mass scatterin 9 angle
Figure 6. Differential cross section for 4He-4He scattering at E k = 57.87K as calculated
from the HFD-B(HE) potential (solid line) and from the ESMSV fitted potential
(points)givenin 1-14,43].
4.3. Differential cross sections
Differential collision cross sections were measured for crossed beams of 4He-4He
by Farrar and Lee [43] and for aHe-4He by Burgmans et al. [14]. To assess the
ability of a potential to predict differential collision cross section (DCCS) data, the
best approach would be to predict values in the laboratory frame-of-reference,
averaging over the appropriate experimental conditions. Since sufficient information
to allow such a calculation to be performed was not available to us, we have taken a
different approach [78]. The ESMSV(4He--4He) [43, 14] and ESMMSV(aHe-4He)
[14] potentials which were fitted to the original data have been employed to generate
pseudodata in the centre-of-mass frame of reference. The pseudodata generated
in this way have then been treated as standard data with which predictions based on
other potentials may be compared. The use of such pseudodata for testing the
present potential construct depends upon the accuracy with which the experimental
fit potential describes the beam data. Any averaging over the experimental conditions
would be expected to mask the differences between the two calculated values
to some extent. The results for the HFD-B potential are presented in figures 6-8
with the dotted line representing the pseudodata. The agreement is seen to be
excellent, especially for low relative energies for both the 4He-4He and 3He-4He
cases.
4.4. Integral collision cross sections
Integral collision cross sections were measured for crossed beams of 4He--4He
and 3He-3He by both the Bonn [44] and Goettingen [20] groups as well as for
3He--4He crossed beams by the latter group. This property is treated in a way
analogous to that used for calculating the DCCS. The results are presented in
figures 9, 10. In the case of the 4He--4He scattering we show the pseudodata
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Interatomic potential for He 2 1505
, | ,
E
o 4He- "He HFO-B
.#
o lo00 E=I. 01 xl0-~aJ [731. 5:3 K]
(]l 'm
(#
(/) .#
o) 100
o) E
o 3
L
L 10
,..4 0
L
9 L
L.J
@
~- .1
q-
4--t
[]
0 20 40 60 80
Center-o?-moss scotter in9 on 91 e
Figure 7. Differential cross section for 4He-4He scattering at Ek = 731-53 K as calculated
from the HFD-B(HE) potential (solid line) and from the ESMSV fitted potential
(points) given in [43].
obtained using the potential obtained from inversion of the pure 4He scattering data
(solid circles), together with the results obtained using the HFD-B(HE) potential
reported here and the results obtained using the recommended HFIMD potential
from [20]. For the aHe-aHe scattering, we show only the results of calculations with
the present HFD-B(HE) potential and the HFIMD potential of [20]. In the first
case, we see that the solid curve representing the HFD-B(HE) potential and the
E
0
o IOOO
@ ~
o
o ~" 10
o
.1
t4-
n
aHe- 4He HFO-B
E=7. 99xlO-ISJ [57. 87 K]
i i i I
0 20 40 60 80
Center-o?-moss scatterin 9 on91e
Figure 8. Differential cross section for 4He-aHe scattering at E k = 57.87 K as calculated
from the HFD-B(HE) potential (solid line) and from the ESMMSV potential (points)
of [14].
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1506 R.A. Aziz et al.
I000
r"
0
4He- +He
3ID
~o I00
oo
9
C
I0 '
loo looo 10000
Velocity / m s -j
Figure 9. Integral cross section as a function of relative speed for +He-+He as calculated
from the HFD-B(HE) potential (solid line), the HFIMD potential (dashed line).
Included also are pseudodata (dots) calculated from the reference potential obtained
by Feltgen et al. 1"20]from inversion of their +He--+He scattering data: However, they
cannot easily be distinguished on the scale of this figure as they are masked by the
solid line.
pseudodata overlap, while the HFIMD results lie considerably lower at low velocities.
In the second case, we see that the results of the HFD-B(HE) calculations lie
above the HFIMD results. For both +He--4He and 3He-3He scattering, the HFD-
I000
c-
O
3
ID
t/)
tt~
o
t..)
Figure 10.
c~
3H~- 3H~
I00
10 ~
loo looo 10000
Velocity / m s-'
Integral cross section as a function of relative speed for 3He-3He scattering as
calculated from the HFD-B(HE) potential (solid line) and the HFIMD potential
(dashed line). No pseudodata are shown in this case, since we were unable to obtain an
appropriate reference potential from [20].
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lnteratomic potential for He2 1507
180 , , ,
60 y
co 140
o 120
E
~ 100 ,'/~
p-
80
60
40 J I I
0 5 10 15 20
Temperature / Kelvin
Figure 11. Behavour of the longitudinal relaxation time T1 as a function of temperature as
calculated using the MS, HFDHE2 and HFD-B(HE) potentials. The calculations for
the HFDHE2 and HFD-B(HE) potentials are almost superimposed. Also shown in
this figure are the experimental data of Chapman [68]. --, HFD-B(HE); ,
HFDHE2;- - -, MS.
B(HE) calculations are closer to the experimental results than are the HFIMD
calculations. It is possible that this difference is merely due to an incorrect c6 value
used in the HFIMD potential.
4.5. Nuclear magnetic relaxation in 3He
The results of our calculations of pT1 (p is the helium gas density in gm cm -a)
for the MS, HFDHE2 and HFD-B(HE) potentials are displayed in figure 11.
Unfortunately, Chapman's [68] raw data were never published, so that we have had
to extract experimental values of pT~ from his figure 6. Moreover, we have assigned
(optimistically) error bars of __+2-5gm cm-as to the values so extracted. All three
potentials give results which are in fairly good agreement with the experimental
values, the results for the MS potential faring somewhat better than the
(indistinguishable) results obtained from the HFDHE2 and HFD-B(HE) potentials.
Because of this, the N.M.R. data do not provide us with as strong a check on the
potential as we would like. Shizgal [79] has also pointed out that the value T~,mi.
which occurs at T~ninis quite sensitive to the values of the potential parameters tr
and e. In view of the fact that all other properties for He are relatively insensitive to
the precise value of e, accurate measurements of T~ in the vicinity of the minimum
would provide useful bounds on the correct value of e. Finally, since both the
spherical He-He interaction is now so accurately known and experimental values of
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1508 R. A. Aziz et al.
Table 7. Calculated integral cross sections (/A2)for selected potentials.
4He-4He aHe-aHe
Primary beam
velocity Feltgen--4t HFD--B HFDHE2 HFIMD HFD-B HFDHE2 HFIMD
/m sec- 1 Pseudodata (Calculated-pseudodata) Calculated Differences~
80 172-56 -1.47 -7.71 -11-13 50.07 -2.18 -3.21
85 134.69 - 1.22 -6.37 -9.22 48.93 -2.04 -2"97
90 104.98 -1.01 -5"26 -7.64 48.85 -1.96 -2.83
95 81.79 -0.83 -4.34 -6.31 49.57 - 1.91 -2.74
100 63.91 -0-68 -3-57 -5.21 50.88 -1.90 -2.70
120 29-46 -0-30 -- 1.66 --2-37 57-98 - 1-98 -2-76
140 31.65 -0.14 - 1.03 - 1.26 61.64 - 1.98 -2"76
160 51-23 -0.13 -1"13 -1-16 59.04 -1.81 -2.54
180 73.20 -0.20 -1-49 -1.49 52.08 -1.53 -2-17
200 87.38 -0.25 - 1"67 - 1.74 43.67 - 1.23 - 1.77
240 86"90 -0.25 - 1.47 - 1"72 29.63 -0"76 - 1"09
280 67"59 -0.20 - 1-09 - 1.45 22"56 -0.44 -0.60
320 48.29 -0-15 -0-79 - 1.16 21.83 -0.26 -0.26
360 36-21 -0-ti -0"62 --0-87 25"40 -0-17 -0-06
400 31.26 -0.08 -0-50 -0.57 30-61 -0.16 +0"03
440 31"04 --0.06 --0"37 --0"26 35-21 --0"15 +0-07
480 33"77 --0-03 --0"24 +0"01 38"18 --0"14 +0"06
520 38"03 --0.01 --0"16 +0-17 39"43 --0"13 +0-03
~fPotential fitted to 4He-4He data by Feltgen et al. 1-20].
Only the differences between the present HFD-B(HE) potential and the HFDHE2 and
HF1MD potentials are given.
T1 are available for 3He at low temperatures, it would be worthwhile attempting an
accurate ab initio calculation of the nuclear spin-dependent anisotropic components
of the 3He-3He interaction.
5. Summaryand conclusions
A new ground state He-He potential has been proposed in order to provide
more accurate calculations of the equilibrium and nonequilibrium properties of
isotopic gaseous helium over an extended temperature range (1.4K to 2200K).
Recent ab initio values of the short-range (1 a.u. to 3 a.u.) repulsion energies have
been taken into account in the determination of the new potential by pinning the
repulsive wall at 1a.u. to the value obtained there by Ceperly and Partridge [24]. In
addition, the present potential has also been fitted to the shear viscosity [37] at
298.15K, the second virial coefficient of 3He [27] at 1.47K, and the second virial
coefficient of 4He [28] at 2.60 K. It has then been tested against a large number of
additional experimental data. Good quantitative agreement has been found with all
reliable measurements of second virial coefficients [7, 27-29, 49, 69-71], shear viscosity
[33, 36, 37, 73, 74] and thermal conductivity coefficients [34-36], both for
aHe and 4He. Additionally, good quantitative agreement has been obtained with
differential [14, 43] and integral [20] scattering data and with low temperature
nuclear spin relaxation measurements [68] in 3He.
Determination of a new potential for ground state He has been undertaken
because the previous HFDHE2 potential of Aziz et al. [6], although the best yet
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Interatomic potential for He 2 1509
available, has been found to be slightly deficient for the calculation of highly precise
low temperature 3He and 4He second virial coefficients and the transport properties
at high temperatures. For example, the HFDHE2 potential predicts a B(T) value for
4He at 2.6 K which differs by 3 cm3mol-1 from the most accurate available experimental
value [28], lying outside the error bars (__+1cm3tool-1) and a B(T) value for
3He at 1.47 K which differs by 4 cm3mol- 1 from the most accurate available experimental
value [27], also lying outside the experimental error bars (_.+0.5 cm3mol-1).
Although it is difficult to interpret any comparison between the individual
parameter values determining the two (slightly different) potential forms, a comparison
of the depth and location of the potential well and the value at which the
potentials become zero is significant. The present HFD-B(HE) potential has its
minimum at e/k = 10.95K, rm = 2-963A while the HFDHE2 potential has its
minimum at e/k = 10-80K, rm= 2-967 A, so that the new potential is slightly deeper
with the bowl shifted slightly inwards with respect to the HFDHE2 potential. The
corresponding a values are tr(HFD-B(HE))= 2.637A and ~r(HFDHE2)= 2"639A.
These values are consistent with the improved agreement with the low temperature
virial coefficient measurements, since the low temperature virial coefficient is largely
sensitive to the shape and volume of the potential well. The HFDHE2 potential was
also in strong disagreement with the ab initio repulsion energies obtained by Ceperley
and Partridge [24], being of the order of 80 per cent too high over much of the
domain of the calculations (1.0 a.u. to 3.0 a.u.). It is the overestimate of the repulsive
wall that accounts for the incorrect prediction of the high temperature transport
coefficients by the HFDHE2 potential.
This research was supported in part by grants in aid of research (R.A.A. and
F.R.W.M.) from the National Science and Engineering Research Council of Canada.
The authors would like to thank Dr. Alec Janzen, who assisted with the integral
cross section calculations, M. F. Slaman, who helped in coding some of the preliminary
computer programs, Professor R. J. Le Roy for guidance in the use of his phase
shift-time delay code, and Prof. W.-K. Liu for helpful discussions regarding our
DWBA calculations of the 3He nuclear spin relaxation times. They would also like
to acknowledge many helpful discussions with Dr. G. T. McConville, who also
kindly supplied virial data prior to publication, and Drs. P. P. M. Steur, M. Moldover,
J. C. Holste, K. R. Hall and E. Whalley for help in the assessment of the virial
data.
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[25] STEBBINGS,R. F. (private communication).
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Metrologia (to be published).
[28] BERRY,K. H., 1987(private communication).
[29] GUGAN,D., 1982, Temperature: Its Measurement and Control in Science and Industry
(American Institute of Physics), p. 33.
[30] PLUMB,H. H., 1982, Temperature: Its Measurement and Control in Science and Industry
(American Institute of Physics), p. 77.
[31] KEMP,R. C., BESLEY,L. M., and KEMO,W. R. G., 1982, Temperature: Its Measurement
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