Bridge Functions Calculated From A Second-Order Approximation For Hard Spheres And A Lennard-Jones Fluid

Results for the bridge function of hard sphere and Lennard-Jones uids are reported. Our method uses the theory of inhomogeneous uids to obtain a higher-order approximation. The agreement of our results with simulations is good.


Introduction
The standard method of calculating the pair and direct correlation functions, g(R) and c(R), respectively, of a simple uid with central forces combines the Ornstein-Zernike (OZ) relation, where r i is the position of molecule i, R ij = jr i ?r j j, and = N=V (N is the number of molecules in the system, V is the volume), and h(R) = g(R) ?1, with (R) = h(R) ?c(R) = ln y(R) ?B(R); (2) where y(R) = exp u(R)]g(R) (3) and B(R) are called the cavity and bridge functions, respectively.In equation (3), = 1=kT , k is Boltzmann's constant, T is the temperature and 52 D.Henderson, S.Soko lowski u(R) is the pair potential, i.e., the energy of interaction between a pair of molecules.
So far everything has been exact.Equation ( 1) is a de nition of the direct correlation function and equation ( 2) is a de nition of the bridge function.Various theories are obtained by approximating the bridge function.For example, the Percus-Yevick (PY) approximation is B(R) = ln y(R) ?y(R) + 1 (4) and the hypernetted chain (HNC) approximation is B(R) = 0: (5) Other approximations are possible.
The PY and HNC approximations tend to fail at high densities and low temperatures.In this note, we report a method for calculating the bridge function and compare our results for hard spheres, where u(R) = 1; R < d 0; R > d; (6) and for a Lennard-Jones uid, where with computer simulations or accurate empirical expressions.

Generalized Ornstein-Zernike relation
The OZ relation may be generalized to an inhomogeneous uid, where the density, (r), is not constant.This generalized OZ equation is h(r 1 ; r 2 ) = c(r 1 ; r 2 ) + Z (r 3 )h(r 1 ; r 3 )c(r 2 ; r 3 )dr 3 : (8) Generally, equation ( 8) is used when the the inhomogeneity is due to a surface or a large molecule.However, following Attard 1] it can be applied equally well to the case where the inhomogeneity is due to one of the uid molecules.Equation (8) can be solved using any of the common approximations, for example, the PY approximation.In most cases, applying a given approximation to equation (8) gives better results than applying it to equation (1).For example, the fourth virial coe cient is given correctly by PY + equation (8) but is not given correctly by PY + equation (1).We refer to the results of equations ( 1) and ( 8) with the PY approximation as the PY and PY2 results, respectively.We use a similar notation with other approximations.
To have a set of equations which can be solved, we also need a relation between (r) and the pair correlation functions.Here we use the Lovett-Mou-Bu -Wertheim (LMBW) equation 2] r (r 1 ) = ?(r 1 )rv(r 1 ) + (r 1 ) Z c(r 1 ; r 2 )r (r 2 )dr 2 ; (9) Bridge Functions calculation : : : 53 where v(r 1 ) is the potential due to the particle which is regarded as the source of the inhomogeneity.In our, case v(r 1 ) is just the pair potential.Note that the rst term on the RHS of equation ( 9) can be combined with the LHS to give an equation for ry(r).
Attard 1] has solved this system of equations for a hard sphere uid where the source of the inhomogeneity is a hard sphere of varying size.His emphasis was on the triplet and pair correlation functions.In the extreme case where the source sphere is in nitely large, one has the case considered by Plischke and Henderson 3].In the calculations reported here our emphasis is on the calculation of the bridge function B(R) for the case where the source particle has the same size as the uid particles.Our methodology and numerical procedure is that of Attard 1].
The procedure is to solve equation (8), using the PY approximation, and equation ( 9) to obtain y(R) and thus, g(R).We then use equation ( 1) to obtain c(R) and then B(R).To solve equations ( 8) and (9) we use the numerical algorithm of Attard.The function y(r 1 ; r 2 ) is expanded in a series of Legandre polynomials (we use 75).Our step size is R = 0:04d with R max = 7:5d.Given an k-th iterate for (r) and y(r 1 ; r 2 ), we compute h(r 1 ; r 2 ) and c(r 1 ; r 2 ) and the coe cients in the expansion of h and c in terms of the Legandre polynomials.The (k + 1)th iterate for (r) and the Legandre transform of y(r 1 ; r 2 ) are obtained from equation ( 9) and the Legandre transform of equation ( 8).The (k+1)th iterate for y(r 1 ; r 2 ) is then obtained by the inverse Legandre transformation.The iteration is continued until convergence is achieved.Detailed formulae are given in Attard's paper.Also we have obtained some results using the HNC approximation.We include only a few HNC2 results here.
The computations were performed in parallel on an 18 processor Silicon Graphics Power Challenge computer located at UAM.

Results
Values for g(R), y(R), and B(R) for hard spheres at a fairly high density are given in gures 1-3.The agreement of the PY2 and HNC2 results with the Grundke-Henderson 4] (GH) and Malijevsk y-Lab ik 5] (ML) ts of simulation data is good.Both the PY and HNC results are less satisfactory.We report results for these three functions for a Lennard-Jones uid in gures 4-6.The PY2 results are in better agreement with the computer simulation results 6-7] than are the PY results.The convergence of our procedure for calculating the PY2 results become less satisfactory at low temperatures and higher densities.For the moment, we cannot draw any conclusions about the usefulness of this method for such interesting states, except that they will be di cult to obtain.

Conclusions
The inhomogeneous OZ equation provides an accurate method for calculating the bridge function, at least for hard spheres and a Lennard-Jones uid whose density is not too high and whose temperature is not too low.The calculations are somewhat time consuming and have large memory requirements.None-the-less, it seems a useful method for the calculation of the bridge function of a uid.
We wish to express gratitude to CONACYT of M exico (Grant No. 4186-E9405 and el Fondo para C atedras Patrimoniales de Excelencia) for nancial support of this project.We thank Dr. Karl Johnson for providing us with his unpublished data for the Lennard-Jones radial distribution function.

Figure 1 .
Figure 1.Hard sphere radial distribution function, g(R), as a function of R for d 3 =0.7.The distance scale is in units of the hard sphere diameter.The solid and broken curves give the PY2 and HNC2 results, respectively.The points give the results of the GH t of computer simulations.

Figure 2 .
Figure 2. Hard sphere cavity function, y(R), as a function of R for d 3 =0.7.The distance scale is in units of the hard sphere diameter.The solid and broken curves give the results of the inhomogeneous PY2 and HNC2 results, respectively.The PY results for y(R) are rather poor inside the core.For example, y(0)=18.65 for the PY approximation.The points give the GH t.

Figure 3 .
Figure 3. Hard sphere bridge function, B(R), as a function of R for d 3 =0.7.The distance scale is in units of the hard sphere diameter.The solid, long dashed and short dashed curves give the results of the PY2, HNC2, and the GH t, respectively.The points give the ML t.

Figure 4 .
Figure 4. Lennard-Jones radial distribution function, g(R), as a function of R for T =1.5 and =0.6.The solid and dashed curves give the PY2 and PY results, respectively.The circles give the simulation results of Johnson.

Figure 5 .
Figure 5. Lennard-Jones cavity function, y(R), as a function of R for T =1.5 and =0.6.The curves have the same meaning as in gure 4. The circles give the simulation results of Llano-Restrepo and Chapman.

Figure 6 .
Figure 6.Lennard-Jones bridge function, B(R), as a function of R for T =1.5 and =0.6.The curves have the same meaning as in gure 4 and the circles have the same meaning as in gure 5.