Journal of Structural Chemistry, Vol. 41, No. 5, 2000
RADIAL DISTRIBUTION FUNCTION
OF A HARD-SPHERE FLUID
UDC 539.196.536.758
Yu. T. Pavlyukhin
A hard-sphere fluid (8788 particles) is modeled by the Monte Carlo method for 41 occupation coefficients
in the range of η = 0.10-0.50 (step 0.01). The radial distribution functions were determined at 512 points
in an interval of up to five hard sphere radii. In this interval, the number of analyzed particle pairs was
from 1.8 ⋅109 to 9.0 ⋅109 (η = 0.10-0.50). The two-variable function g(r,η) was analytically expressed
using least-squares analysis; standard deviation from the Monte Carlo data was of the order of 0.001.
An equation of state is suggested for a hard-sphere fluid (standard deviation 0.002). A direct comparison
shows that at high densities the accuracy of the expressions is one order of magnitude higher than that
of the best relations reported in the literature.
INTRODUCTION
In analysis of the structure and properties of the liquid phases of real substances, liquids with temperatures close
to crystallization points are conveniently called melts to distinguish them from higher-temperature liquids proper [1]. As
shown by experiment, metallic, ionic, and dielectric melts with appropriate densities have similar structures [2]. Their
universal behavior is dictated by a sharp increase in the particle interaction potential due to particle attraction. The attraction
forces produce an insignificant effect on the structure and properties of a liquid. Model systems of particles with a spherically
symmetric pair interaction potential and analogous short-range behavior are called simple fluids. In this case, a melt and
a liquid are equivalent. A hard-sphere (HS) fluid is the simplest model system.
The constant interest in HS systems arises from two major factors.
1. This is a universal model of the behavior of a wide range of liquids at high temperatures and the simplest model
with a liquid–crystal phase transition. Since the advent of numerical methods in statistical physics, numerous Monte-Carlo
(MC) and molecular dynamic (MD) simulations have been carried out to model HS systems, and ample experimental data
have been accumulated in this field. All innovations in the equilibrium statistical mechanics of fluids, for example, density
functional theory of nonhomogeneous classical fluids are primarily tested on HS fluid [3-5]. This system is simple and
convenient from the viewpoint of theory and is physically meaningful.
2. HS is a basis to construct perturbation theory for soft sphere (SS) and Lennard-Jones (LJ) potentials, which are
more realistic (theory of simple fluids suggested by Weeks, Chandler, and Andersen (WCA) [6]) [6-8]. Elaboration of this
theory [9] permits ab initio calculations of all equilibrium properties of liquids to an accuracy of the order of 1% for SS
and slightly more than 1% for LJ to densities preceding a transition to the solid state. The accuracy of such calculations
is comparable to that of numerical simulations reported in the literature. For RDF, the calculations are less accurate (5-10%
error for the first peak at high densities).
Thus the properties of an HS system (equation of state, free energy) and RDF serve as a basis for constructing
the theory of simple fluids. For real substances similar in properties to simple fluids, it now seems possible to recover
parameters from the equilibrium properties (density, temperature dependence of vapor pressure, compressibility, thermal
expansion, etc.) of fluids in a chosen particle interaction potential and thus to describe their properties numerically over a
wide range of pressures and temperatures. It is therefore important not only to develop theoretical approaches in simple
Institute of Solid State Chemistry and Mechanochemistry, Siberian Branch, Russian Academy of Sciences,
Novosibirsk. Translated from Zhurnal Strukturnoi Khimii, Vol. 41, No. 5, pp. 988-1004, September-October, 2000. Original
article submitted May 7, 1999.
0022-4766/00/4105-0809$25.00 ©2001 Plenum Publishing Corporation
809
fluid theory but also to analyze the accuracy of the assumptions used and the results obtained. This primarily refers to the
accuracy of the initial data for HS systems.
The purpose of this work is an MC simulation of an HS system in a density range of the liquid state and
determination of RDF and equilibrium properties of the system with a maximal possible accuracy. Our aim was analytical
expression of the results, which is the most exact and convenient form of expression for further use in WCA theory since
further calculations demand labor-consuming efforts. It should be noted that basic results for this system were obtained in
the 1960-1970ies on assemblies of 100-500 particles. These works were mainly aimed at investigating the physical properties
of HS systems, whereas computations for further use of numerical data as basic data in simple fluid theory were of secondary
importance. A complete list of references on numerical computations is found in [2, 10-12].
1. MC SIMULATION OF A HARD-SPHERE SYSTEM
A diameter of a hard sphere is set to be unity. For the HS problem, the packing coefficient η is used along with
the density of particles ρ:
η=
π
ρ.
6
(1)
A transition to a crystal occurs when η ≈ 0.48; accidental close packing of hard spheres (Bernal number) takes place for
η = 0.637; finally, η = 0.7405 corresponds to close packing in an fcc crystal [2, 10-12].
The starting HS fluid was prepared from an fcc crystal (η = 0.40) containing 13×13×13 cubic unit cells (8788
particles), which was melted. Successive compression or expansion of the computation region was used to prepare and set
to equilibrium 41 fluid samples with occupation coefficients η = 0.10-0.50 with a step of 0.01. The simulation used the
periodic boundary conditions. MC simulation with successive item-by-item examination of particles was effected by using
an arbitrary shift of each particle with Gaussian distribution. The mean square amplitude of the shift was determined as
follows. The mean distance DL between particles in a liquid was estimated as
 0.637
DL = 

 η 
1/3
.
(2)
For the chosen range of variation of η, the distance was varied from 1.85 to 1.08. The mean square displacement at each
Cartesian coordinate ∆S was chosen to be
∆S =
(DL − 1)
.
3
8√
(3)
The ∆S value varied from 0.062 to 0.006. During the successive exhaustion procedure, each atom was arbitrarily displaced
(NS times per cycle) with a mean square value (3). It was chosen that
2
 2DL − 1 
NS = 2
.
 DL − 1 
(4)
The NS value is 20-386 (η = 0.10-0.50). It was assumed that after the cycle the system ‘‘forgot’’ the initial state; then
RDF was constructed. A total of 4096 RDF were constructed for each density. RDF was calculated in the range of distances
1.0-5.0; the number of points was 512 (distance quantization step ∆ = 2−7 sphere diameter). We thought that for the chosen
cell size (from 35.8 to 21.0 sphere diameter) we managed to avoid particle correlation to themselves due to the periodic
boundary conditions. In the range of distances in question, the overall RDF function had a statistical sample (the number
of particle pairs analyzed) Np from 1.8⋅109 to 9⋅109 events. Given the above RDF computation conditions, the uncertainty
of RDF determination was estimated at (512/Np)0.5 ≅ 0.0005-0.0002.
The realized precision of simulation poses a number of specific problems. Thus for the experimental determination
of N∆ — the number of particle pairs lying in the range of mutual distances [R, R+∆], one needs to choose an optimal
criterion 0 < ξ < 1 according to
810
4π
ρ[(R + ∆)3 − R3]g(R + ξ∆) = N∆.
3
(5)
We proceeded from the relation
∆
N∆ = 4πρ∫ g(R + x)(R + x)2dx =
0
4πρg′(R + ξ∆)
4π
ρ[(R + ∆)3 − R3] g(R + ξ∆) +
3
∆2  1 2

(6R + 8R∆ + 3∆2) − ξ(6R2 + 6R∆ + 2∆2) + ...,
6  2

(6)
assuming the term in square brackets at the RDF derivative to be zero. This provides an accuracy of RDF in (5) of the
order of ∆2; in our case, it equals 2−14, which is approximately 4 times lower than the statistical accuracy of RDF simulation
and is quite acceptable. The typical choice ξ = 0.5 leads to an unacceptable error of the order of ∆.
RDF is used to determine important thermodynamic quantities. Thus the equation of state for HS (U(r) is the pair
interaction potential) [10] is
ZHS ≡
P
4η dU(r)
=1− ∫
g(r)r3dr = 1 + 4ηg(1).
ρT
T dr
(7)
The relation for reduced compressibility [10] is
 ∂ρ   d

χHS ≡ T  =  η ZHS
∂
d
η
P
 T 

−1
= 1 + 24η∫[g(r) − 1]r2dr.
(8)
The g(1) value was estimated by least-squares fitting of the first 16 points (1.0 ≤ r ≤ 1.125) of RDF by cubic polynomial
and its extrapolation to the value at r = 1.0. A phenomenological equation of state for HS is well known [13], which shows
a ‘‘surprising accuracy’’ [10] of agreement with the results of simulation:
ZCS =
1 + η + η2 − η3
(1 − η)3
.
(9)
Indeed, relation (9) gives a surprising accuracy of results, and the standard deviation for 0.10 ≤ η ≤ 0.50 is 0.012 when the
value of (9) changes from 1.52 to 13.0. For this range of occupation coefficients, (9) was refined and we obtained a close
relation using least-squares fitting:
ZHS =
1 + η + η2 − 0.63523η3 − 0.10205η4 − 3.3666η5 + 4.2848η6
(1 − η)3
.
(10)
Expansion of (10) into a power series of η yields correct virial coefficients up to η3 inclusive (the coefficient at
η in the numerator was fixed). The standard deviation for (10) is 0.002, which is 6 times smaller than that for (9), and
is therefore used below. Relation (10) was derived for relatively high densities. For η < 0.1, it is desirable to use the virial
expansion, whose coefficients are defined up to the term η7 inclusive [2, 10, 11].
Instead of (10) and (8), below we use
3
Z=
χHS − 1 1
ZHS − 1
and χ =
+ .
3
4η
24η
(11)
For convenient representation of the results, the least-squares procedure was used to approximate an array of initial
data containing 41×512 RDF values using a specially selected two-variable function
gHS(r, η) = 1 + γ(η)[gV(r) + ∆gV(r)] + (1 − γ(η))[gN(r) + gM(r) + gL(r)].
(12)
In (12), the first term in brackets contains the known terms of RDF expansion in terms of density gV(r) and a correction
to them ∆gV(r) (low densities); the second term is the RDF selected for high densities. The behavior of RDF is defined
by the gN(r) function for short distances, by gL(r) for long distances, and by gM(r) for intermediate distances. The γ(η)
811
function provides a smooth transition from low to high density limit. It was obtained that the function equals
  η 
γ(η) = exp −

  0.381
15

.

(13)
The least-squares analysis of the unknown functions in (12) used the following additional constraints.
1. Following [8], RDF (12) must satisfy the equation of state (7) and reduced compressibility (8) for the equation
of state (10). For the functional expression (12), these conditions must be satisfied by both expressions in brackets in (12).
2. The experimental RDF was defined on a limited interval (R ≤ 5). To achieve the maximal accuracy of estimation
for RDF behavior at long distances, the statistical weight of the points in the range 2.5-5.0 was increased 24 fold. This
further improves the behavior of the Fourier components g(k) at small k.
3. The Fourier component of the direct correlation function for small k is inversely proportional to compressibility,
which at large densities is of the order of 0.01. Therefore when these quantities are calculated, the relative error of
approximation (12) increases 100 fold. To avoid false oscillations of this quantity at small k, we additionally minimized
its deviation from the theoretical estimate. We chose the interval k = 0-3.0 (100 points more with a statistical weight of
24). The theoretical estimate was chosen according to WCA theory (the procedure is described in Sect. 2) but this was
done for Eq. (10) but not (8).
4. The unknown functions in (12) were represented as the sum of basis set functions whose Fourier images may
be calculated analytically. The basis set was chosen to be (µ > 0):
Φ(µ, α, β, n; z) = exp [−µz] cos [αz + β]zn.
(14)
If we additionally define the parameters
p=√
 
α2
µ2 + 
and tan (θ) =
α
,
µ
(15)
then for (14) we have
∞
∫Φ(µ, α, β, n; r−1)rdr =
1
Γ(n + 1) sin [(n + 1)θ + β] Γ(n + 2) sin [(n + 2)θ + β]
+
.
pn+1
pn+2
(16)
The Fourier components are calculated in a similar way.
Previously, we defined the first three terms of RDF expansion in terms of density:
gV (r) = ρg1V (r) + ρ2g2V (r) + ρ3g3V (r).
(17)
The first term is trivial [10] and differs from zero for 0 ≤ r ≤ 2:
rg1V (r) =
1 4
4π
3
(r − r 2 +
r ).
16
3
4
(18)
For the second term, an exact analytical expression was derived in [14] (for r > 3, this function is identically zero). The
expression of [14], however, is cumbersome and does not suit our purposes. Therefore its representation was found by the
least-squares method:
4
rg2V (r) = ∑ CsΦ(µs, αs, βs, 0; z)z3 + Ds z s+3,
(19)
s=1
where the function on the right side depends on the variable z = R − 3. The parameters of Eq. (19) are listed in Table 1.
This approximation for the interval 0.9 ≤ r ≤ 3 has a standard deviation of 0.001 from the exact value. For the third function
in (17), only numerical data are available which are tabulated in [15]. Accordingly, for r > 4, the function g3V (r) in (17)
is identically zero. When one uses the data of [15], an analogous representation is of the form (z = r − 4)
812
8
rg3V (r) = ∑ CsΦ(µs, αs, βs, 0; z)z3 + Ds z s−1.
(20)
s=5
For the interval 0.9 ≤ r ≤ 4, this approximation has a standard deviation of 0.004 from the values given in [15]. The
parameters of (20) are listed in Table 1. Below we use the following properties of (18)-(20):
gV (1) = 1.3090ρ + 1.2586ρ2 + 1.0148ρ3,
(21)
∫ gV (r)r 2dr = 0.7418ρ − 1.2047ρ2 + 1.6124ρ3 ≡ ∆χV .
(22)
∞
1
A correction to RDF expansions in terms of density ∆gV (r) (here and below z = r − 1) equals
13
r∆gV (r) = ∑ CsΦ(µs, αs, βs, ns; z);
Cs = C0s + C2sρ2 + C4sρ4;
s=1
βs = β0s + β2sη2.
αs = α0s + α2sη2;
µs = µ0s + µ2sη2;
(23)
The coefficients of (23) are given in Table 2. Two quantities — C9 and C13 — are determined from conditions (11) for
the equation of state (10), taking into consideration (21) and (22):
TABLE 1. Parameters of Relations (19) and (20)
S
CS
µS
αS
βS
DS
1
2
3
4
5
6
7
8
2.303
0.606
0.204
1.180
–0.1184
–0.001241
–0.003183
0.002772
–1.055
–3.88
–3.91
–2.88
–0.266
0.400
–0.405
0.237
2.227
13.70
20.50
7.3840
4.549
11.30
16.0
8.69
2.2675
2.82
–2.84
2.43
0.91
1.74
–1.14
0.01
0
0
0
0
0.6621
1.8354
1.1015
0.1872
TABLE 2. Parameters of (23)
S
nS
C0S
C2S
C4S
µ0S
µ2S
α0S
α2S
β0S
β2S
1
2
3
4
5
6
7
8
9
10
11
12
13
0
1
2
3
4
5
6
1
0
1
2
3
1
–6.581
–42.97
1.32
3.38
–5.34
–30.4
–50.0
51.1
—
0.277
–0.024
–0.013
—
11.09
45.3
3.67
–9.83
–87.5
90.9
135.6
–95.28
—
10.0
–3.65
0.334
—
–4.41
–20.3
–4.12
13.1
80.1
–180.1
–264.6
–6.16
—
–7.85
3.23
–0.315
—
1.96
2.344
3.18
3.99
3.17
6.05
7.09
6.01
1.144
1.144
1.144
1.144
2.08
–6.58
0.369
–1.13
0.263
2.41
8.75
8.70
91.6
0.553
0.553
0.553
0.533
–0.624
5.72
3.11
11.6
19.3
3.43
26.3
32.5
3.52
0.0
0.0
0.0
0.0
0.0
6.88
1.68
2.75
–6.58
10.3
–12.8
–10.8
–18.4
0.0
0.0
0.0
0.0
0.0
1.32
0.028
–0.641
–1.79
1.15
0.803
0.088
–1.45
0.0
0.0
0.0
0.0
0.0
0.24
1.42
5.48
9.98
–11.2
11.1
12.4
2.58
0.0
0.0
0.0
0.0
0.0
813
gV (1) + ρ4∆gV (1) = Z;
(24)
∞
∆χV + ρ4 ∫ ∆gV (r)r 2dr = χ.
(25)
1
The other functions in (12) are of the form
rgL(r) = C exp (−µz) cos (αz + β);
C = −1.13 η + 15.58 η2 − 17.41 η3;
α = 6.541 − 3.698 η + 9.138 η2;
µ = 2.433 − 3.69 η;
(26)
β = −5.6 + 18.7 η − 17.59 η2;
12
rgM(r) = ∑ CsΦ(µs, αs, βs, ns; z);
Cs = C0s + C1sη + C2sη2;
s=1
µs = µ0s + µ1sη;
αs = α0s + α1sη.
(27)
The parameters of the function (27) are found in Table 3. The values βs = 0 (s = 1, ..., 9) and βs = −π/2 (s = 10, 11, 12).
rgN (r) = exp (−µz)[A cos (αz − 0.849 ) + Bz + (31.1 − 15.3 η) µ2η4z2];
µ = 7.145 + 13.0 η2;
α = 7.63 + 11.29 η2,
(28)
where the coefficients A and B are determined analogously to (24) and (25):
gN (1) + gM (1) + gL(1) = Z.
(29)
∫ [gN(r) + gM(r) + gL(r)]r 2dr = χ.
(30)
∞
1
Using these expressions makes it possible to approximate the two-variable function gHS(r, η) with a standard
deviation of 0.001 for η = 0.10-0.50. In view of the additional conditions, for the interval r = 1.0-2.5 this deviation is 4
times larger than that for r = 2.5-5.0 (0.0016 and 0.0004, respectively). For smaller η, the behavior of (12) is determined
by the terms of the virial expansion; therefore (12) gives an expression for RDF over the whole range of existence of an
equilibrium HS fluid which is accurate to 0.001.
RDF for HS (12) is exactly zero for r < 1. However, the applications (for example, calculations using WCA theory
of simple fluids) demand the function
TABLE 3. Parameters of (27)
814
S
nS
C0S
C1S
C2S
µ0S
µ1S
α0S
α1S
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
0
5
10
15.0
–4.67
–74.2
–406.0
254.0
12816.0
10852.0
–3827.0
–20.5
3.47
–35.6
288.0
–66.0
23.6
380.0
2021.0
–1132.0
–51435.0
–56645.0
20622.0
102.0
–16.8
197.0
–1535.0
77.0
–32.9
–520.0
–2708.0
1782.0
58992.0
57560.0
–28119.0
–125.0
19.4
–280.0
2052.0
5.65
8.30
6.55
7.85
7.33
12.7
12.7
7.05
7.92
1.89
6.67
6.73
–0.999
–7.31
–0.241
–0.549
–0.505
–6.03
–4.89
7.83
–6.05
1.094
1.27
2.38
1.50
30.1
25.2
20.9
9.83
6.60
6.22
40.0
12.5
5.71
46.6
18.1
0.341
6.11
2.54
–0.58
4.61
3.08
2.55
–1.41
0.242
3.540
–0.0527
0.618
 UHS(r) 
γHS(r, η) = exp 
gHS(r, η).
 T 
(31)
For the region r < 1, (31) contains an uncertainty. One can show, however [6-8], that this function and the first
three derivatives are continuous at a point r = 1. Therefore to calculate (31) we use (12) continued into the region r < 1
[8]:
gHS(r, η) = d0 + (r − 1)d1 + (r − 1)2d2 .
(32)
Another technique is continuous extension of the RDF logarithm [7]:
ln[gHS(r, η)] = d0 + (r − 1)d1 + (r − 1)2d2 .
(33)
For RDF (12), the most convenient realization of (32) or (33) is numerical. For this purpose, one can calculate
16 values of function (12) or its logarithm in the range r = 1.0-1.125 and approximate them by a square polynomial, that
is, determine d0, d1, and d2 for (32) or (33). The uncertainty in choosing between (32) and (33) is unimportant since (31)
is used in perturbation theory and the differences between (32) and (33) are higher-order infinitesimal corrections. However,
expression (33) seems to be preferable because this is a less sharp dependence in the region r < 1.0.
2. RDF APPROXIMATION BASED ON SOLUTION OF THE PERCUS–JEVICK EQUATION
To our knowledge, (12) is the first analytical expression suggested for RDF of an HS system based on MC data
and possessing a high accuracy. The most complete numerical data are presented in [16]. Here RDF values are tabulated
for eight density values at 60 points (1 < R < 2.3) obtained by the MC method for a system of 108 particles. Therefore for
applications we used an approach based on a modification of the exact solution of the Percus–Jevick equation for hard
spheres [6-8]. This solution was first obtained in [17, 18]. Derivation of an expression for the direct correlation function
cPY(r) demands the least effort. The calculations, however, require the pair distribution function, which is related to the
Fourier image of the direct correlation function by a simple Fourier transform according to the Ornstein–Zernike relation
[10]
hPY(k) ≡
cPY(k)
4π
sin (kr)[gPY(r) − 1]rdr =
.
k∫
1 − ρcPY(k)
(34)
Although the direct correlation function is simply a cubic polynomial, direct use of (34) in particular computations in
inconvenient. The analytical expressions for dHS < r < 5dHS [19] and the numerical method of calculating gPY(r) [20] are
found in the literature. The solution techniques were reviewed in [21]. We have realized a simple and convenient scheme
of calculating gPY(r), not limited by an interval or form of data representation. The scheme is stable, and computations do
not take much effort. We proceeded from the set of equations proposed in [22] to determine the direct correlation function
and the correlation function hPY(r) = gPY(r) − 1 in the Percus–Jevick approximation (all variables below are dimensionless;
x = r/dHS):
1
xcPY(x) = −q′(x) + 12η∫ q(t − x)q′(t)dt,
(35)
x
1
xhPY(x) = −q′(x) + 12η∫ (x − t)hPY(|x − t|)q(t)dt.
(36)
x
Here in (35) and (36), q(t) is a square polynomial [22], which is
q(x) = 0, x ≥ 1,
(37)
π
ρ d3HS ,
6
(38)
η=
815
where η is the packing coefficient. Let us introduce designations:
q(x) = d2x2 + d1x + d0;
q(1) = 0;
(39)
3η
,
2(1 − η)2
1 + 2η
.
2(1 − η)2
(40)
d1 =
d2 =
When relations (39), (40) are valid, solutions of Eqs. (35) and (36) may be obtained independently. Let us seek a solution
of (36). For this, we introduce notation
xhPY(x) = xgPY(x) − x ≡ Θ(x) − x;
Θ(x) = 0, x ≤ 1.
(41)
For x > 1, (36) takes the form
1
Θ(x) − c1 x − c2 = ∫Θ(x − t)Q(t)dt,
(42)
0
where
1
Q(x) = 12ηq(x);
1
c1 = 1 − ∫Q(t)dt;
c2 = 1 − ∫Q(t)tdt.
0
0
(43)
Equation (42) is differentiated three times. After simple transformations, we obtain the expressions (reduced to the form
not containing the derivatives of the function Θ(x) of distinct orders in the same relation):
1
Θ′(x) − c1 + α2(c1x + c2) = β2Θ(x − 1) + ∫Θ(x − t)[Q′(t) − α2Q(t)],
(44)
0
Θ′′(x) + α2[c1 − α2(c1x + c2)] + α1(c1x + c2) = Θ′(x − 1)β2 +
1
Θ(x − 1)(β1 − α2 β2) + ∫Θ(x − t)[Q′′(t) + Q(t)(α22 − α1) − α2Q′(t)],
(45)
0
Θ′′′(x) + α2Θ′′(x) + α1Θ′(x) + α0Θ(x) = β2Θ′′(x − 1) + β1Θ′(x − 1) + β0Θ(x − 1).
(46)
Here Q′′′(x) ≡ 0, and the following notation is introduced:
α2 = −12ηq(0); α1 = −12ηq′(0); α0 = −12ηq′′(0);
β2 = −12ηq(1); β1 = −12ηq′(1); β0 = −12ηq′′(1).
(47)
Equations (42), (44), and (45) define the boundary conditions for solving Eq. (46). In Eqs. (42), (44)-(46), the left side
contains the function of x; the right side, the function of (x − 1) or (x − t). Equation (46) is solved sequentially in the
intervals n ≤ x ≤ (n + 1). For this, we introduce the functions reduced to these intervals:
Θn(y) ≡ Θ(x − n), Θ0(y) ≡ 0, 0 ≤ y ≤ 1.
(48)
For n = 1, from Eq. (46) we have an ordinary differential equation
d3
d2
dx
dx2
Θ (x) + α2
3 1
Θ1(x) + α1
d
Θ (x) + α0Θ1(x) = 0.
dx 1
(49)
According to the general rules, we have
Θ1(x) = ∑ C1λ exp (λx),
λ
816
(50)
where three eigenvalues of λ are determined from a cubic equation
λ3 + α2λ2 + α1λ + α0λ = 0.
(51)
For all η < 1, one root in (51) is real and larger than zero, and the other two roots are conjugate complex. The presence
of a positive root in Eq. (51) indicates that the corresponding exponential in the solution for r → ∞ increases without bound.
In the solution, this is certainly compensated by a decreased coefficient at this exponential. The solution in general tends
to zero. In view of this uncertainty, the standard procedures for solving Eqs. (44)-(46) must be used with care. Therefore
below we give a method of solving which is free from these disadvantages.
In (50), C1λ are unknown coefficients determined from the initial conditions (42), (44), and (45) according to
d2
dx2
Θ1(0) − c1 − c2 = 0,
(52)
d
Θ (0) − c1 + α2(c1 + c2) = 0,
dx 1
(53)
Θ1(0) + α2[c1 − α2(c1 + c2) + α1(c1 + c2) = 0.
(54)
The boundary conditions (52)-(54) are equivalent to the condition that the function yHS(x) = gHS(x) − c(x) is continuous
together with the two derivatives at the point x = 1. Then the solution Θn(x) in the interval n ≤ x ≤ (n + 1) will be (the index
λ in the notation C1λ is omitted)
Θn(x) = ∑[Pn(x) + Cn] exp (λx).
(55)
λ
Here Pn(x) is a homogeneous polynomial of (n−1)st degree (in (55) it is assumed that P1(x) ≡ 0), and Cn is an unknown
constant. Note that the introduction of functions (48) leads to the fact that, by definition, for the range n ≤ x ≤ (n + 1) in
(55) all the functions are reduced to the variable 0 ≤ x ≤ 1. Therefore the uncertainty in the behavior of the solution
corresponding to the first positive eigenvalue in (51) at long distances does not appear and the suggested scheme of solution
is stable for x → ∞. If the solution Θn−1(x) in the range (n − 1) ≤ x ≤ n is known (i.e., the coefficients of the polynomial
Pn−1(x) are known), then the coefficients of the homogeneous polynomial Pn(x) are determined from the identity (for each λ)
d3
d2
dx
2
P (x) + (3λ + α2)
3 n
dx
Pn(x) + (3λ2 + 2λα2 + α1)
d
P (x) =
dx n
d2
d
P (x) + (β1 + 2λβ2) Pn−1(x) + (λ2β2 + λβ1 + β0)(Pn−1(x) + Cn−1).
2 n−1
dx
dx
(56)
For the homogeneous polynomial Pn(x), this equation permits one to determine all of its coefficients. Three Cn constants
for the three eigenvalues of λ are found from the boundary conditions at a point x = n using Eqs. (42)-(45) analogously to
(52)-(54):
1
Θn(0) − c1n − c2 = ∫Θn−1(1 − t)Q(t)dt,
(57)
0
1
d
Θ (0) − c1 + α2(c1n + c2) = β2Θn−1(0) + ∫Θn−1(1 − t)[Q′(t) − α2Q(t)]dt,
dx n
(58)
0
d2
dx 2
Θn(0) + α2[c1 − α2(c1n + c2)] + α1(c1n + c2) =
817
1
d
Θ (0)β2 + Θn−1(0)(β1 − α2 β2) + ∫Θn−1(1 − t)[Q′′(t) + Q(t)(α22 − α1) − α2Q′(t)]dt.
dx n−1
(59)
0
The direct correlation function is determined from (35) according to
cPY(x) = B0 + B1x + B3x3,
(60)
where, as is known [10],
B0 = −2d2 + 8(1 − η)d1d2 + 4 ηd2(2d2 + 3d1) = −
B1 = 6 ηd12 − 8(1 − η)d1d2 = η
B3 = −2ηd22 = − η
(1 + 2η)2
(1 − η)4
,
3(2 + η)2
,
2(1 − η)4
(1 + 2η)2
.
2(1 − η)4
(61)
(62)
(63)
The Percus–Jevick approximation for hard spheres is reduced to the requirements
cPY(x) = 0, x > 1;
(64)
gPY(x) = 0, x ≤ 1,
(65)
i.e., these functions are discrete for x = 1. However, the function required for the calculations,
yPY(x) = gPY(x) − cPY(x) = exp [βUHS(x)]gPY(x),
(66)
together with the three derivatives is continuous for x = 1. Note that condition (65) is exact, whereas (64) is approximate.
Therefore the solution of the Percus–Jevick equation needs a refinement. It was suggested phenomenologically [8] that the
following function be added to (66):
∆y(x) =
A
exp [−µ(x − 1)] cos [µ(x − 1)], x > 1,
x
(67)
where A and µ are the parameters to be determined. The functions (66) and (67) were expanded [8] into a Taylor series
for x ≥ 1, and the expansion was continued into the region of x < 1. The same procedure was suggested [7] for the logarithm
of function (67). These expansions are necessitated by the fact that addition (67) is a fast oscillating function, and in the
case of a continuous extension to the region x < 1 at large densities the total RDF is not a monotonically increasing function.
Therefore ordinary continuous extension of RDF to the region x < 1 is not satisfactory. We used the approach of [7] —
introduction of an addition
∆y(x) =


µ2
A
exp −µ(x − 1) − (x − 1)2 , x ≤ 1.
x
2


(68)
The conditions for choosing the parameters A and µ were proposed in [8]. To improve agreement of RDF with the results
of numerical experiments, it was suggested that RDF be constructed for
ηW = η −
η2
;
16
d3W d3HS 6
=
=
ηW
πρ
η
(69)
but not for parameter (38). A new radius of hard spheres dW [8] is determined in accordance with (69) and (38) (for the
given density). This procedure improves agreement between the periods of RDF oscillations in numerical experiments and
computations [8]. We have performed a special test to check the optimality of the suggested numerical coefficient in (69)
by comparing the standard deviation of the calculated RDF with the results of our MC simulation. It appeared that relation
(69) is close to optimal. If in the functions we explicitly define the dependences on dW, dHS, and r, then, given (67) and
818
(68) in the approximation of [8], the desired functions will equal (Verlet–Weis formula)
yVW(dHS, r) = gPY(dW, r) + ∆y(x) ≡ gVW(dHS, r)
yVW(dHS, r) = gPY(dW, r) + ∆y(x)
yVW(dHS, r) = −cPY(dW, r) + ∆y(x)
gVW(dHS, r) = 0
d ≤ r ≤ ∞;
dW ≤ r < d;
0 ≤ r < dW;
0 ≤ r < d.
(70)
Here, as above, x = r/dHS. We emphasize that yVW(dHS, r) and gVW(dHS, r) are the only functions defined according to
(70), whereas cVW(dHS, r) is not defined. This function may be obtained from the Ornstein–Zernike relation, and its form
was analyzed in [23]. For the functions introduced in this way in (70) and defined as in [23], the first equality in (66) is
not valid, i.e., for cVW(dHS, r), condition (64) is naturally not valid.
Second, it was assumed [8] that for r = dHS the value of RDF satisfied the equation of state for hard spheres
Z(η) ≡
P
= 1 + 4ηgVW (r = dHS),
ρT
(71)
which is specified additionally and independently. Recall that the Percus–Jevick equation is approximate; therefore the
equations of state determined from the RDF value (for r = dHS) and compressibility do not coincide. Therefore addition
(67) serves the purpose of not only improving agreement of RDF with simulation but also eliminating the above discrepancy.
In [8], the equation of state (9) from [13] was used. The A parameter is determined from (67), (70), and (71):
A=
 d

Z(η) − 1
Z(η) − 1
− gVW (dW, r = d) =
− gPY , ηW .
4η
4η
d
 W

(72)
Instead of (9), one can use another known equation of state obtained from the Pade approximant of the virial expansion
[24] (σ = 4η):
Z ≡
1 + 0.063507σ + 0.017329 σ2
P
=1+σ
.
ρT
1 − 0.561493 σ + 0.081313 σ2
(73)
The coefficients of dependence (73) were estimated in [24] by using the values of six virial coefficients. Another three
versions of expression (73) were given after the seventh virial coefficient was calculated in [25]. We compared the methods
of choosing Z(η) in (72) according to (9), (10), and (73). The accuracy does not change since these dependences correct
the behavior of RDF only within the narrow region of r ≈ 1. The µ parameter is found from the equation of compressibility
of a system of hard spheres (8):

d
χ =  ηZ
η
d


−1
= 1 + 24η ∫ [gVW (d, r) − 1]r 2dr.
(74)
The solution of the Percus–Jevick equation for the occupation factor ηW defines compressibility according to [10]:
∞
χW = 1 + 24ηW ∫ [gPY(ηW, r) − 1]rdr =
0
(1 − ηW)4
(1 + 2ηW)2
.
(75)
Then an addition associated with correction (68) of RDF is isolated in (74), and the equation for µ takes the form
δ
χ = χW − 24ηW ∫ gW (dW, x)x2dx + 12η ⋅
1
d
A
, δ=
.
µ
dW
(76)
A good approximate solution of (76) is
µ=
24 A
η gW(1)
(77)
[8]. This is because the difference between χ and χW is of the order of 1/200 [8].
819
The proposed method is convenient and indispensable in RDF calculations for long distances. In addition to stability
it offers an advantage of numerical calculation of only unknown coefficients in (55). At fixed density, this may be done
only once to obtain an analytical expression for RDF.
3. DISCUSSION OF RESULTS
Figure 1 compares RDF of a hard-sphere fluid for several values of occupation factor obtained by simulation and
by calculation using formula (12). The second RDF peak has a distinct ‘‘shoulder’’ at large densities and is split for a
system of accidentally close-packed spheres (η = 0.637) [26, 27]. Formula (12) is in good agreement with simulation data;
in particular, the behavior of RDF in the region of the second peak is well approximated. The standard deviation of the
calculation by (12) and WCA calculation by (70) from the results of Monte Carlo simulation is 0.001 and 0.015, respectively
(η = 0.10-0.50). Figure 2 shows the dependence of this value on the occupation factor. In the region of a dense fluid,
WCA theory does not adequately describe the results of the numerical simulation. Figures 3 and 4 demonstrate the
dependences of the deviation of the calculated RDF from the results of simulation on the distance. It is noteworthy that
the WCA calculation shows an explicit dependence of the error on the range of values n < r < n + 1 (n is an integer). This
may be associated with the form of Eq. (46), which is a lag differential equation. Formula (12) permits easy calculations
using the Fourier components of RDF. Figure 5 shows the structural factors of HS for some occupation factors
∞
S(k) = 1 + ρh(k) = 1 + 4πρ∫ | g(r) − 1 |
0
sin (kr)
rdr.
k
Fig. 1. RDF of HS fluid. Solid curves — MC simulation data,
dots — calculation by formula (12). Curves 1-6 are shifted upward
by 0-5 units, respectively.
820
(78)
Fig. 2. Standard deviation of RDF obtained by
Monte-Carlo simulation from the RDF calculated in
terms of WCA theory (1) by formula (70) and by
formula (12) suggested in this work (2).
Fig. 3. Difference between RDF obtained by Monte-Carlo simulation and WCA
calculations by formula (70). Curves 1-8 are shifted upward by 0.0, 0.03, ...,
0.21, respectively.
Figure 6 gives the direct correlation function
∞
1
h(k)
c(r) = 2 ∫
sin (kr)kdk.
+
ρh(k)
1
2π r 0
(79)
821
Fig. 4. Difference between RDF obtained by Monte-Carlo simulation and
calculations by formula (12) suggested in this work. Curves 1-8 are shifted
upward by 0.0, 0.03, ..., 0.21, respectively.
Fig. 5. Structural factor of HS fluid. Solid curves — WCA calculation by formula (70),
dots — calculation by formula (12). Curves 1-6 are shifted upward by 0-5 units, respectively.
822
Fig. 6. Direct correlation functions of HS. Solid curves — WCA
calculation by formula (70), dots — calculation by formula (12). For
r < 1, all functions are multiplied by the coefficient 0.1. Curves 1-6
are shifted upward by 0-5 units, respectively.
This function was analyzed in the framework of WCA theory [23]. In the region r > 1, the calculations using (12) and (70)
differ widely.
In conclusion we note that based on the above analysis one can recommend using relation (12) for RDF of a
hard-sphere fluid in exact WCA calculations [9] (with an error of the order of 1% and less) over the whole region of its
existence (η ≤ 0.50).
This work was supported by RFFR grant No. 00-03-32523.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
A. R. Ubelode, Melted State of Substances [in Russian], Metallurgiya, Moscow (1982).
B. Older and W. Hoover, Physics of Simple Liquids. Statistical Theory [Russian translation], G. Temperly, J.
Rowlinson, and J. Rashbook (eds.), Mir, Moscow (1971), pp. 81-135.
T. V. Ramakrishnan and M. Yussouf, Phys. Rev. B, 19, No. 3, 2775-2794 (1979).
R. Evans, Adv. Phys., 28, No. 2, 143-200 (1979).
M. Baus, J. Stat. Phys., 48, Nos. 5/6, 1129-1146 (1987).
J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem. Phys., 30, No. 12, 5237-5247 (1971).
H. C. Andersen, J. D. Weeks, and D. Chandler, Phys. Rev. A, 4, No. 4, 1597-1607 (1971).
L. Verlet and J.-J. Weis, ibid., 5, No. 2, 939-952 (1972).
H. S. Kang, S. C. Lee, and T. Ree, J. Chem. Phys., 82, No. 1, 415-423 (1985).
R. Balescu, Equilibrium and Nonequilibrium Statistical Methods, Wiley, New York (1975).
C. A. Croxton, Liquid State Physics — A Statistical Mechanical Introduction, Cambridge University Press, London
(1974).
823
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
824
D. Leveck, J.-J. Weis, and J.-P. Ansen, in: Monte Carlo Methods in Statistical Physics, K. Binder (ed.), Springer,
Berlin (1979).
N. F. Carnahan and K. E. Stirling, J. Chem. Phys., 51, No. 2, 635-636 (1969).
B. R. A. Nijboer and L. Van Hove, Phys. Rev., 85, No. 5, 777-783 (1952).
F. E. Ree, R. N. Keeler, and S. L. McCarthy, J. Chem. Phys., 44, No. 9, 3407-3425 (1966).
J. A. Barker and D. Henderson, Mol. Phys., 21, No. 1, 187-191 (1971).
M. S. Wertheim, Phys. Rev. Lett., 10, No. 8, 321-323 (1963).
E. J. Thiele, J. Chem. Phys., 39, No. 2, 474-479 (1963).
W. R. Smith and D. Henderson, Mol. Phys., 19, No. 3, 411-415 (1970).
J. W. Perram, ibid., 30, No. 5, 1505-1509 (1975).
J. A. Barker and D. Henderson, Rev. Mod. Phys., 48, No. 4, 587-673 (1976).
R. J. Baxter, Aust. J. Phys., 21, 563-569 (1968).
D. Henderson and E. W. Grundke, J. Chem. Phys., 63, No. 2, 601-607 (1975).
F. H. Ree and W. G. Hoover, ibid., 40, No. 4, 939-950 (1964).
F. H. Ree and W. G. Hoover, ibid., 46, No. 11, 4181-4197 (1967).
J. L. Finney, Nature, 266, 309-314 (1977).
Yu. I. Naberukhin, Structural Models of Fluid [in Russian], Novosibirsk University Press, Novosibirsk (1981).
A. R. Denton and N. W. Ashcroft, Phys. Rev. A, 39, No. 9, 4701-4708 (1989).