Chemical Physics 159 ( 1992) 89-97 North-Holland

The radial distribution function for two-dimensional Lennard-Jones fluids: computer simulation results

F. Cuadros and A. Mulero Departamento de Fisica. Universidad de Extremadura, Ctra. Portugal s/n. 06071 Badajoz, Spain

Received 3 June I99 1

Computer simulations were performed to obtain the radial distribution functions for the two-dimensional Weeks-Chandler- Andersen reference system and the full Lennard-Jones system over a wide range of densities and temperatures. The results for the shape (first and second maxima) are compared with other computer simulation and theoretical results. The high-temperature approximation proposed by Weeks-Chandler-Andern was also tested and found to be quantitatively valid only for the highest fluid temperatures and densities, in a small section of the temperature-density plane. The main difference from previous tests of

this approximation is that the radial distribution function for the reference system is obtained directly instead of from a hard-disk system. All these results are necessary for a later and complete analysis of the validity of the Weeks-Chandler-Andersen theory and its consequences in the study of the adsorption of rare gases onto flat surfaces.

1. Introduction

the fluid’s structure may conveniently be de- scribed, for spherical molecules, through the radial distribution function (RDF), g(r), which is propor- tional to the local number density of particles at a given distance, r, from a fixed reference particle [ 11. It is very important to know its shape because this contains the principal structural information, and the location and height of the first maximum because the calculation of pressure depends considerably on these two factors.

For three-dimensional (3D) systems, the RDF can be obtained from theoretical approximations [ 2,3 1, scattering experiments [ 41 and computer simulation [ 1,561. In the two-dimensional (2D) case, only the first [ 7,8] and the last [ 9- 121 ways are feasible.

Fehder [lo] has analysed the shape and location of two principal maxima of the RDF for 2D Len- nardJones (2DLJ) fluids through molecular dy- namics simulations. In particular, he finds three per- sistent “subsidiary features” which appear as small “shoulders” in the overall structure. These.features are located at the base of the first maximum, and at the base and large radius side of the second maxi- mum. They are most prominent at intermediate den-

sities, and their exact locatio; depends on both tem- perature and density. Some experimental measurements have shown these same features [ 13,141 but others have not [ 15,161. Moreover they are not seen in the Monte Carlo results of Tsien and Valleau [ 111. Both facts contribute to the contro- versy [lo].

There are also RDF values for a 2DLJ fluid ob- tained from the Percus-Yevick (PY ) [ 7 ] and hyper- netted-chain (HNC) theories [ 8 1, for more than one hundred thermodynamic states. Both theories give RDFs in agreement with earlier computer simula- tion results [ 10, 1 1 1. The principal difference is that the PY theory gives a better position for the first maximum and then better values of the thermody- namic properties, whereas the HNC theory yields a better height for that maximum.

2. WCA perturbation theory

Various perturbation theories for describing the fluid state have been developed over the last decades. The most successful is the one proposed by Weeks, Chandler and Andersen (WCA) [ 5,6,17-201. In this theory the full intermolecular potential, is separated

0301-0104/92/$05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.

90 F. Cuadros. A. Mulero /Radial distribution function for 2D Lennard-Jones fluids

into a reference part, U,(r), containing all the short- range repulsive forces, and a perturbation part, U,(r), containing all the long-range attractive forces. The separation of the potential is made by introducing a perturbation parameter, A, as follows:

UA(r) = UJ(r)+IU,(r) ,

where r is the interparticle distance.

(1)

Of course, from eq. ( 1 ), if 1=0 the potential is purely repulsive, while for A= 1, the full potential is obtained. As 1 varies continuously from 0 to 1, the attractive and the repulsive parts of the potential play different roles in determining the thermodynamics and structure of fluids.

When the Lennard-Jones potential [ 2 I] is used, the reference repulsive potential is given by

U,(r)=&(r)+l, r<rm=2’/6,

=o, r>r,,

while the attractive part is

(2)

U,(r)=-1, r<r,,

=Uu(r) , c-r,. (3)

In these equations, r, is the minimum of the po- tential, i.e. the distance at which the force changes from repulsive to attractive, and the W potential is

U,(r)=4[(l/r)‘2-(1/r)6]. (4)

In these and the following equations all the prop- erties are expressed in reduced LJ units.

The first problem of all perturbation theories is that the properties of the reference system are unknown. Fortunately, it can be related to systems of hard spheres [ 19,22,23], disks [24,25,26] or rods [27] (in three, two and one dimensions respectively) through a scaling procedure. The second is to calcu- late the perturbation effects on the thermodynamic properties and RDF. This problem is solved through a high-temperature expansion (HTE), or l-expan- sion, up to the second term [ 11.

For 2DLJ systems, we described a test of the scal- ing procedure in a previous paper [ 261. In the pres- ent work, we check the validity of the simplest form of the HTE quantitatively.

3. High-temperature approximation

When the second-term approximation of the HTE is used together with the WCA theory, the thermo- dynamic properties of a fluid can be calculated if one knows the RDF for each value of density, tempera- ture, interparticle distance and 1, gA( r) (see the equations for the 3D case in refs. [ 5 ] or [ 12 ] ) . From a theoretical point of view, therefore, an approxi- mate method of calculating this function is needed. The simplest is that proposed by WCA [ 17-l 91, called the high-temperature approximation (HTA).

The physical argument of the HTA is that where the density and temperature of a fluid are high, its structure is mainly determined by the strong repul- sive forces. Then WCA propose that the RDF of a partially coupled or full LJ system (gA (r) for A# 0) can be approximated to the RDF of the reference sys- tem (g,(r) forJ_=O). That is,

gA(r) =gO(r) . (5)

The application of this last equation in the calcula- tions of thermodynamic properties is equivalent to cutting off the HTE at the first term [ 1,121, thus needing to know go(r) only. In particular, in the WCA theory, thego( unknown theoretically, is related to the RDF of a hard-sphere system, ghS( r) (in the 3D case), which can be obtained from computer simu- lation or from integral equations (PY commonly). Moreover, the potential energy per particle and the pressure calculated using the HTA must be linearly dependent on the perturbation parameter 1 [ 201.

For intermediate and low densities, where eq. ( 5 ) is not applicable, WCA proposes an additional hy- pothesis by studying the spectrum (Fourier trans- form) of the distribution function:

6(k) =p s [g(r) - 1 ] eeikrddr, (6)

where k is a wavevector. The hypothesis is that for all liquid densities, the approximation

ii,(k) =&o(k) (7)

is accurate for wavevectors ka x (in reduced LJ units) [ 17,181. Physically, eq. (7) means that the re- sponse of fluid to a disturbance of wavelength greater than 2x/k is dominated by the repulsives forces.

The validity of the HTA and its consequences in

F. Cuadros, A. Mulero /Radial distribution function for 2D Lmnard-Jones fluids 91

the WCA theory have already been tested for 3D [ 5,6,20,22 ],2D [ 12,241 and one-dimensional ( 1D) systems [27]. It is important to keep in mind that this approximation will be applicable for higher den- sities in the lower dimensional cases [ 271. In the 3D and ID cases the conclusions are similar: although eq. (5) is not always quantitatively valid, the WCA separation of the potential leads to good values of the thermodynamic properties [ 5,271. The HTA is then reasonable in the calculation of these properties if the temperature and density are sufficiently high.

The first test of validity of the HTA for 2DLJ sys- tems was made by Steele [ 241 who explored its ap- plicability to a theory of monolayer physical adsorp- tion onto flat surfaces, and showed that the temperatures and densities of interest are generally too low for eq. (5) to be successful, this being the principal difficulty. In particular, he finds that the WCA-HTA potential energy agrees with that ob- tained from computer simulation only for reduced densitiesp greater than 0.6 (deviations less than 101) at intermediate temperatures (from 0.7 to 1, in re- duced LJ units).

Subsequently, the extension of Steele’s theory up to periodic surfaces [ 28 ] requires the knowledge of the values of fi (k) and to use eq. ( 7 ) . Monson, Steele and Henderson [28] compared the E(k) obtained from Monte Carlo simulations with the &i,(k) ob- tained from the RDF of HD systems (for r=O.95). They found that eq. (7) seems correct (fork> n) for 2D systems too.

Weeks and Broughton [ 121 also tested the HTA and its influence on the feasibility of applying the van der Waals picture of melting in 2DLJ systems. They compare the g(r) and the go(r) obtained from mo- lecular dynamics (MD) computer simulations for high densities (p>O.75). For T close to 1 the agree- ment seems satisfactory, whereas for probably the lowest liquid temperature, TcO.45, they find notice- able differences, which is physically reasonable.

A wide check of both applications requires a better study of the HTA in 2DLJ systems. In particular, one needs to know the temperature-density range in which it is quantitatively correct, and that in which it is only qualitatively correct, by which we mean un- derstanding the influence of the HTA on the calcula- tion of thermodynamic properties.

4. Molecular dynamics simulation

We have run MD computer simulations of two dif- ferent 2D systems at several thermodynamic states (fig. 1). The first system was the WCA reference sys- tem (il= 0 in eq. (2) ) and the second the full LJ sys- tem (2 = 1 in eq. (2) ). For each system and state the simulation was carried out as described in ref. [ 261. The computational system is isochoric and adiabatic, and hence isoenergetic (isolated). The usual peri- odic boundary conditions are used [ 1 ] and the po- tential is cut off at a reduced interparticle distance of 2.5. The total time-step number is 5000 for each run. A particle neighbour table [ 291, which is renewed each 20 time-steps, is used to compute the interpar- title forces.

The total number of particles was 256 in all runs. Bruin et al. [30] have shown that higher numbers (specifically 780, 3 120 and 10864) do not have any significant effect on the results.

5. Results

The values of temperature and density of the sim-

SIMULATED STATES

21 A a d x x**

02 0.3 04 07 08 C t Cnticol point 0 Triple point

Fig. I. Thermodynamic states selected in our simulation. Sym- bols mean the different values of percentage deviations (Ag in eq. (9) ) with respect to the HTA: + to Ag> 50%, A to SO%aAga IO%, x to IO%>Agg> 1% and * to A& 1%. As a ref- erence, the critical and triple points of a 2D fluid are also shown ( refs. [ 34,35 ] ) All values are in reduced LJ units.

92 F. Cuadros. A. Mulero / Radial distribulion function for 2D Lmnard-Jonesjluids

ulated systems are shown in fig. 1 in reduced IJ units. These states correspond to the 2D fluid phase and some of these have been studied elsewhere [ 7- 12, 24,26,3 1,321 being of interest in the study of the ad- sorption of fluids (rare gases) on solids (graphite).

IneachruntheRDFfor,&Oand1=1,g(r)and g,(r), was obtained each Arz0.02 units of the re- duced interparticle distance using

g2(r)= _.?e.L 2xr Arp ’ (8)

where nL( r) is the time-averaged number of particles situated at a distance rf Ar from a given particle, when the interparticle interaction potential is given by eq. ( 1). The statistical error was estimated to be of the order of 1W. Some examples of these calcu- lated functions are shown in figs. 2-7. The positions and heights of the two principal maxima are listed in tables 1 and 2.

In each of the two systems considered, the heights ofboth maxima, g(r,), g(r2), g,,(r,) andg,(r,), in- crease when the temperature decreases or the density increases. Nevertheless, the second maximum is less sensitive to both parameters. The positions, calcu- lated with an error of f 0.02, agree in behaviour with that reported by Fehder [ lo] for the full LJ system. The position of the first maximum, r,, is not sensitive to either temperature or density, except at high tem-

RDF T = 0.625 , RO = 0.5

Fig. 2. RDF for the 2D full LJ fluid and for the WCA reference system at T~O.625 andp=0.5.

RDF T=0.7 ; RO = 0.6

2.6

2.6

2.4

2.2

2.0

1.8 i

T1.6 -

-$4-

12-

1.0 -

0.6 -

0.6 1

I I, I, 5 I I I , I I I 0.9 1.1 1.3 1.5 1.7 19 2.1 23 25

Fig. 3. RDF for the 2D full LJ fluid and for the WCA reference system at T=0.7 andp0.6.

RDF T=1.5 ; RO = 0.6

2.4 -I

2.2 -

2.0 -

1.8 -

1.6 -

1.4 -

';‘- .-&1.2 -

1.0 -

0.6 7

0.6 -I

a

.

02 .

0.0 l , I , I , I , , I , I , I , 0.9 1 1 1.3 15 1.7 1.9 21 23 25

l LJ system r 0 WA reference system

Fig. 4. RDF for the 2D full LJ fluid and for the WCA reference system at T= 1.5 andp0.6.

peratures or densities, where it shifts to lower values. The position of the second maximum, r2, also shifts to lower values when the density or temperature increases.

In general, our results for the full LJ system agree with previously published results [ 7,8, lo- 12 1. A few of them are listed in table 3. The most important dif- ferences are found in comparing with the Tsien-Val- leau values [ 111 at low density. We think that the

.

F. Cuadros. A. Mulero /Radial dimibution function for 2D Lmnard-Jones fluids 93

RDF T = 2.0 ; RO = 0.3

Fig. 5. RDF for the 2D full LJ fluid and for the WCA reference system at T=2.0andp=0.3.

RDF T = 0.5 , RO = 0.7

3.4 t*

32 *

30

2.0 - D

Tl.8 - : -ml .6 - “%

14-

1.2- q

to- *

08-

06: o 0.4 - *

Fig. 6. RDF for the 2D full LJ fluid and for the WCA reference system at T=O.S andpz0.1.

principal cause for this is the choice of the cut-off dis- tance, 3 in the Tsien-Valleau simulation. Theoretical results [ 7,8] show a small discrepancy too. In agree- ment with Glandt and Fitts [8], the HNC theory seems to give better values for the first maximum. We find also that the position of both maxima, and the value of the second, are good in both the PY and HNC theories.

As one can see in figs. 2-4, the existence of the first

RDF T = 1.0 : RO = 0.8

O2 P

00 I I , , I 1 1 , , 1 , / ,

09 ,l 13 15 17 19 21 23 2.5

Fig. 7. RDF for the 2D full LJ fluid and for the WCA reference system at T= 1.0 andp0.8.

“feature” reported by Fehder is evident at all tem- peratures for intermediate densities. Its position is practically the same in both systems, 1.4 to 1.6, shift- ing to smaller values when the density increases. We find that it is practically temperature independent, while Fehder notices a slight shift to smaller values with increasing temperature [ lo]. On the other hand, the next two features of Fehder are not evident. Only the third, located at distances between 2.3 and 2.5, seems to appear at intermediate densities (figs. 2-4 ) . Moreover an additional feature appears at the lowest density reported here, p= 0.3, located at distances be- tween 1.9 and 2.0 (fig. 5).

Finally, the test of the quantitative validity of the HTA is shown in figs. 2-7 and table 4. The shape of g,,(r) is more strongly density dependent than that of g(r), especially the height of the first maximum. The positions of the maxima are practically coincident, with a small shift forgo(r) to smaller values or greater values for the first and second maxima, respectively. The shift of the first maximum implies that, for high densities (p> 0.7), go(r) is greater than g(r) at very small interparticle distances, r-z r,,, (figs. 6 and 7). The height of the second maximum and the overall structure of both RDFs are very similar at high tem- perature and/or density (figs. 6 and 7). Neverthe- less, the principal difference is to be found in the height of the first maximum. The percentage devia-

94

Table I

F. Cuadros, A. Mukro /Radial distribution function for 2D Lennard-Jones fluids

Position and height of the two maxima in the RDF of the 2D full LJ system.

T P rl g(rl) r2 g(r2)

1.0

0.7

0.625

2.0 0.30 1.12 1.69

0.40 1.12 1.82

0.50 1.10 1.97

0.60 I .08 2.13

0.70 I .08 2.44

0.75 1.06 2.61

0.80 1.06 2.80

1.5 0.30 1.12 1.81

0.40 1.12 1.97

0.50 1.10 2.10

0.60 1.10 2.29

0.70 1.10 2.56

0.75 I .08 2.73

0.80 I .08 3.01

0.30 1.12 2.25

0.40 1.12 2.24

0.50 1.12 2.37

0.60 1.12 2.55

0.70 1.10 2.80

0.75 1.10 3.01

0.80 1 .OS 3.26

0.30

0.40

0.50

0.60

0.70

0.75

2.71

2.82

2.8 1

2.81

3.07

3.32

0.30

0.40

0.50

0.60 0.70

0.75

1.12 1.12

1.12

1.12

1.10

1.10

1.12

1.14

1.12

1.12

1.12

1.10

1.12

1.12

1.12

1.12

1.12

1.12

3.14

2.90

3.02

2.98

3.12

3.38

0.55 0.60

0.70

0.75

3.24 2.22 1.39

3.34 2.26 1.47

3.50 2.20 1.56

0.50 0.70

0.75

0.45 0.75

3.42 2.24 1.54

3.63 2.22 1.59

3.72 2.22 1.61

2.24 1.06

2.26 1.09

2.20 1.14

2.22 1.25

2.16 1.33

2.10 1.42

2.24 1.10

2.22 1.19

2.24 1.29

2.16 1.41

2.10 1.48

2.24 1.09 2.22 1.14

2.22 1.24

2.22 1.39

2.16 I .49

2.14 1.55

2.24 I .22

2.22 1.23

2.24 1.30

2.24 1.45

2.16 1.54

2.28 1.29

2.22 1.32

2.24 I .48

2.18 1.56

tions with respect to the HTA, calculated at this point (r, ) from the expression

Ag= 100 g(r,)-_go(r,)

g(rr ) (9)

are listed in table 4. For low temperatures and low, or even intermedi-

ate, densities (fig. 2) the structure of g(r) is very different from that of gO( r) , so that deviations greater than 50% are found in r, (symbols + in fig. 1) . Devia- tions greater than 10% (symbols A in fig. 1) are found for low densities and high temperatures (fig. 5 ), intermediate densities and temperatures (fig. 3 ), and even high densities when the temperature is low

Table 2

F. Cuadros, A. Mulero /Radial distribution function for 2D Lmnard-Jones fluids 95

Position and height of the two maxima in the RDF of the 2D WCA reference system.

T P rl go(r)) r2 g0(rd

2.0

1.5

1.0

0.7

0.625

0.30 0.40

0.50

0.60

0.70

0.75

0.80

0.30

0.40

0.50

0.60

0.70

0.75

0.80

0.30

0.40

0.50

0.60

0.70

0.75

0.80

0.30

0.40

0.50

0.60

0.70

0.75

0.30

0.40

0.50

0.60

0.70

0.75

0.60

0.70

0.75

0.70

0.75

0.75

1.12

I.10

I.10

1.10

I .06

I .06

I .06

1.12

1.10

1.10

1.10

I .08

1.06

I .06

1.12

1.12

1.12

1.10

1.08

1.08

I .08

1.16

1.12

1.10

1.10

1.10

I .08

1.12

1.12

1.12

1.10

1.10

1.10

1.12

1.10

1.10

1.10

1.10

1.10

1.36

1.54

I .75

2.04

2.37

2.58

2.78

I .40

1.58

1.80

2.10

2.52 2.71

2.98

1.42 1.62 1.89

2.23

2.68

2.95

3.29

1.44

1.69

1.95

2.36

2.87

3.18

I .44

1.67

2.01

2.37

2.91

3.24

2.45

3.01

3.38

3.04

3.41

3.48

2.36

2.30

2.28

2.18

2.14

2.10

2.36

2.28

2.24

2.16

2.10

2.44

2.40

2.38

2.24

2.16

2.14

2.58

2.32

2.44

2.16

2.38

2.38

2.24

2.16

2.40

2.24

2.16

2.22

2.18

2.18

I .02

I .06

1.12

1.22

1.30

1.38

I .05

1.13

I .26

1.37

1.44

1.02 1.06 1.15

1.29

I .42

I.51

1.06 1.18

1.38

1.45

1.09 1.20

1.37

1.48

1.19

1.40

1.50

1.41

1.51

1.54

(fig. 6 ) . For these states, therefore, the attractive part of the potential plays a role in imposing order on the fluid. Finally, deviations greater than 1% (symbols x in fig. 1) are found even at high densities or tem- peratures (fig. 4). In conclusion, only for the highest densities (~~0.75 and 0.8) and temperatures

(T> 1 ), symbols + in fig. 1, is the HTA quantita- tively correct (fig. 7), with deviations in r, less than the likely error in the calculated RDFs. The structure of the system is thus primarily determined by the short-range repulsive forces.

96

Table 3

F. Cuadros. A. Mulero /Radial distributionjitnction for 2D Lmnard-Jones fluids

Position and height of the two maxima in the RDF of the 2D full LJ system calculated from the PY and HNC theories [ 7,8] and other computer simulations [ IO- 121.

T P rl g(rl) h g(h) Ref.

I .44 0.50 1.12 2.13 1101 0.63 1.12 2.29 2.23 [lOI

0.95 0.25 1.12 2.3 2.30 1.25 1111 0.32 1.10 2.48 2.24 1.12 171

1.10 0.50 0.95 0.54 1.02 0.58 1.07 0.63

0.95 0.67

0.98 0.70 0.95 0.74 I.0 0.84

0.7 0.25 1.12 3.4 2.27 0.54 1.10 2.9 2.25 0.60 I .09 3.04 2.19

0.68 0.70 1.12 3.11 2.20 0.7 0.74 1.10 3.2 2.17

0.75 1.08 3.50 2.14

0.625 0.25 0.54

0.55 0.54 0.67 0.74

1.12 1.12

1.12 1.10 1.10

3.6 2.22 1.3 3.0 2.25 1.3

3.2 2.22 1.4 3.71 2.14 1.52 3.6 2.20 1.6

1.12 2.31 1.10 2.5 1.12 2.54 1.12 2.55 1.10 2.81 1.10 2.70 1.10 3.17 1.07 2.91 1.09 2.82 1.10 3.1 1.08 3.56

2.25 1.35 2.23 2.20 2.16 1.29 2.18 1.28 2.14 1.38 2.17 I .36 2.20 2.17 1.45 2.15 1.61

1.3 1.3 1.36

1.5 1.58

101 111 101 101 71 81 71 51 101 111 121

[III [III [81 1101 [Ill 181

1111 1111

1111 [71 [Ill

Table 4 Percentage deviations with respect to the HTA at the first maximum, Ag in eq. (9).

P T

0.45 0.50 0.55 0.625 0.7 1.0 1.5 2.0

0.30 - 118% 88% 59% 29% 24% 0.40 - 74% 67% 38% 25% 18% 0.50 - 50% 44% 25% 17% 13% 0.60 - 32% 26% 19% 14% 9% 4.4% 0.70 - 12.5% 1 I% 7.2% 7% 4.5% 1.6% 3% 0.75 7% 6.5% 3.6% 4.3% 4.4% 2% 0.7% 1.2% 0.80 - -0.9% 1% 0.7%

6. Conclusions

The calculated RDFs are in agreement with earlier computer simulation results. Their overall structure and particular dependence of the position and height

of two maxima on temperature and density confirm the observations of Fehder [ lo]. We also confirmthe existence of at least two “subsidiary features”, in agreement with Fehder but in disagreement with the results of Tsien and Valleau [ 111. These additional

F. Cuadros. A. Mulero /Radial distribution function for 2D Lennard-JonesJluids 97

structures may reflect an alternative configuration for local ordering within the fluid. Their physical expla- nation, importance and implications have already been commented on by Fehder [ lo]. In particular, he remarks that the existence of such substructures is very important because it could have serious impli- cations for the statistical mechanics of dense fluids.

Direct comparison ofg( r) and g,-,(r) permits us to determine the temperature-density range where the HTA is quantitatively correct according to the per- centage deviations required. At this point, it is im- portant to recall that both functions have been ob- tained through computer simulation, while, in both the WCA theory and the Steele analysis of the HTA, theg( r) from simulations is compared with theg,( r) obtained from the approximation g, ( r ) = g,,, ( r) . We find that the deviations are less than 1% only for high temperatures and densities, representing only a small area in the density-temperature plane (symbols * in fig. 1).

Since the HTA is not always quantitatively correct, as in the 3D case, several questions must be revised. First, its qualitative validity, that is, its influence on the calculations of thermodynamic properties (pres- sure and potential energy principally). Second, whether the linearity of these properties is main- tained as was found to be the case in 3D [ 201. And third, whether the van der Waals picture of melting is valid at high temperature and density [ 5,12,33]. To answer these questions correctly we have begun more extensive calculations and analysis, similar to those performed by Rull et al. [ 5 ] and Valderrama et al. [20].

Acknowledgement

We gratefully acknowledge the help of Dr. Marvin Bishop with the computer programs.

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