Optimized Cluster Expansions for Classical Fluids. I. General Theory andVariational Formulation of the Mean Spherical Model and Hard SpherePercusYevick EquationsHans C. Andersen and David Chandler Citation: J. Chem. Phys. 57, 1918 (1972); doi: 10.1063/1.1678512 View online: http://dx.doi.org/10.1063/1.1678512 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v57/i5 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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1918 GOLDING, NEWMAN, RAE, AND TENNANT

be good particularly since off-diagonal hyperfine matrix elements have been neglected. The hyperfine lines in the calculated spectrum are simply drawn in about the mid points, noting that second order perturbation requires a small increase in the hyperfine splittings equal to (2A 2/ g{3H) (= 3.7 G) with increasing fields.

* Present address: Chemistry Division, D.S.I.R. Wellington, New Zealand.

1 L. V. Bershov, A. S. Marfunin, and R. M. Mineyeva, Geo­chemistry (USSR) 1966, 464.

2 V. M. Vinokurov, M. M. Zaripov, and V. G. Stepanov, Fiz. Tverd. Tela 6,1130 (1964) [Sov. Phys. Solid State 6,870 (1964)].

3 F. Holuj, Can. J. Phys. 46, 287 (1968). 4 V. M. Vinokurov, Geochemistry (USSR) 1966, 996.

5 A. Manoogian, Can. J. Phys. 46, 129 (1968). 6 A. Manoogian, Can. J. Phys. 46, 1029 (1968). 7 A. D. Rae, J. Chern. Phys. 50, 2672 (1969). 8 R. M. Golding, Applied Wave Mechanics (Van Nostrand,

London, 1969). 9 B. Bleaney and K. W. H. Stevens, Rept. Progr. Phys. 16, 108

(1953) . 10 B. Bleaney and R. S. Rubins, Proc. Phys. Soc. (London)

77,103 (1961). 11 B. Bleaney and D. J. E. Ingram, Proc. Roy. Soc. (London)

205, 336 (1951). 12 L. M. Matarrese, J. Chern. Phys. 34, 336 (1961). laB. E. Warren, Z. Kryst. 72, 42 (1930). 14 J. Zussman, Acta Cryst. 12, 309 (1959). 15 R. H. Newman and W. C. Tennant, New Zealand Depart­

ment of Scientific and Industrial Research, Report No. C.D. 2109, 1969.

THE JOURNAL OF CHEMICAL PHYSICS VOLUME 57. NUMBER 5 1 SEPTEMBER 1972

Optimized Cluster Expansions for Classical Fluids. I. General Theory and Variational Formulation of the Mean Spherical Model and Hard Sphere

Percus-Y evick Equations*

HANS C. ANDERSEN

Department of Chemistry, Stanford University, Stanford, California 94305

AND

DAVID CHANDLER

School of Chemical Sciences, University of Illinois, Urbana, Illinois 61801

(Received 24 April 1972)

Computationally convenient theoretical methods for calculating the thermodynamic properties and pair correlation functions of a classical mUlticomponent fluid are presented. The Mayer cluster series for the Helmholtz free energy and pair correlation function are transformed using topological reduc­tion to more compact forms involving a renormalized potential. Then the convergence of the two series is improved by an optimal choice of the renormalized potential. The result is two rapidly convergent series which are useful both for ionic solutions and for simple liquids with short range intermolecular forces. When these series are truncated, very accurate and convenient approximations are obtained for both types of fluids. Another set of results is a variational formulation of the mean spherical model and hard sphere Percus-Yevick equations for the pair correlation functions of multicomponent fluids. The varia­tional formulations greatly facilitate the process of solving the equations numerically. Each of these results can be extended to models for molecular fluids with only a moderate increase in computational effort.

I. INTRODUCTION

The Mayer cluster theoryl-s is the starting point for most of our theoretical understanding of the equilib­rium properties of fluids. In its original form it is directly applicable only to dilute gases of molecules with short range intermolecular forces, but it has been generalized in a variety of ways and has been greatly strengthened by powerful mathematical tools, such as topological reduction and functional differentiation.sb .d

In this paper we discuss one of these generalizations, which arises when the intermolecular potential is separated into two parts, a "reference" part and a perturbation, and the perturbation poter.tial is then eliminated in favor of a "renormalized" potential by a summation over chains (see below for definitions of these terms).4-8 More specifically we will be concerned

with applying topological reduction and functional differentiation techniques to the derivation of com­putationally convenient approximations for the thermo­dynamic and structural properties of various types of classical fluids of spherically symmetric particles.

This is the first paper in a series which will apply these methods. The second paper9 generalizes the methods to models for molecular fluids. In the third paperlO numerical results for simple liquids and ionic solutions will be presented and compared with the results of Monte Carlo and molecular dynamics simulations. Subsequent papers will be devoted to theoretical computations for molecular fluids and inhomogeneous ionic solutions.

The major emphasis in this work is on the develop­ment and use of approximations which we have de­scribed above as "computationally convenient." An

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CLUSTER EXPANSIONS FOR CLASSICAL FLUIDS. I 1919

approximation is "computationally convenient" if it requires only those numerical procedures which can easily and quickly be performed accurately with readily available computational facilities. For example, a one-dimensional numerical integration over an infinite interval is such a procedure, but a three- or six-dimen­sional integration is much more difficult and costly (especially if it must be performed many times). An iterative method for finding the minimum value of a function of several variables using the N ewton­Raphson techniquell can be very rapidly convergent, but an iterative solution of a nonlinear integral equa­tion can be difficult and costly. Monte Carlo and molecular dynamics calculations are also not "con­venient," in the sense used here. One other factor affects the "computational convenience" of an ap­proximation. Many statistical mechanical approxima­tions require a knowledge of the properties of a hard sphere fluid. Because of Monte Carlo and molecular dynamics studies, the free energy and pair correlation function are now accurately known for a hard sphere fluid and have been presented in very convenient analytic form.12 The three- and four-particle correlation functions are known much less accurately. Even the tabulation of the three-particle function requires a three dimensional array of numbers for each value of the density. Thus, the less information about the hard sphere fluid that an approximation requires, the more convenient that approximation is.

A major motivation for obtaining accurate, com­putationally convenient approximations for atomic fluids is that these are the approximations which are most likely to be useful in describing the properties of fluids of nonspherical molecules. Molecular fluids are intrinsically much more complicated than atomic fluids, and if a theoretical method can just barely cope with an atomic fluid then there is little reason to hope that it will lead to meaningful results for molecular liquids.

This paper contains two sets of results for atomic fluids. The first is a set of cluster expansions for the free energy and pair correlation functions of a multi­component fluid of particles interacting with each other by a hard sphere potential plus a perturbation potential. For certain fluids these series converge very rapidly even for very long ranged perturbations such as the Coulomb potential. As a result when these series are truncated after the first few terms, they give com­putationally convenient approximations for a variety of fluids. For example, both ionic solutions and simple liquids are described very accurately by the same approximations.l° The second set of results is a varia­tional formulation of the mean spherical model (MSM)13,14 and hard sphere Percus-Yevick (PY)15,16 integral equations for multicomponent fluids. The variational formulation leads to a great simplification in the solving of these integral equations. Both of these two sets of results can be generalized to apply to

molecular fluids with only a moderate increase in the amount of computation needed to implement them.

Many of the theoretical methods used in this paper are not new. In particular the procedure of separating the potential into two parts and the technique of topological reduction have been used many times.3-8 However, they have usually been used to obtain elegant formal results or limiting laws or series expansions which are ordered in terms of an ordering parameter. Results of this type are often not useful for numerical computation of thermodynamic and structural proper­ties of fluids for experimentally interesting conditions. In particular, if the ordering parameter for an infinite series is not small enough, then a large number of terms must be evaluated, and many of these terms involve multidimensional integrals and hard sphere fluid correlation functions for three or more particles.

The work presented herein was motivated by recent studies of the mode expansion theoryl7-2o for classical fluids, which is formally different from the Mayer cluster theory but which also involves the use of re­normalized potentials. For the optimal choice of renormalized potential, the mode expansion provides very accurate thermodynamic properties for model ionic solutions18 ,19 and the Lennard-Jones fluid20 and pair correlation functions for the latter. The results in the present paper were obtained by applying the idea of an optimized renormalized potential to the Mayer cluster theory.

In the following section the usual Mayer diagram series for the Helmholtz free energy and pair correlation function of a multicomponent atomic fluid will be reviewed. The series will then be transformed, using topological reduction, to a form more useful as a starting point for the subsequent sections. In Sec. III these new series are transformed again to a more compact form involving a renormalized potential. Section IV discusses the method for choosing the renormalized potential in an optimal way and discusses the relationship between this "optimized cluster expansion" and the optimized random phase approxima­tion of the mode expansion theory and the MSM theory. Section V discusses the convergence of cluster series for ionic solutions and the reasons why the optimized cluster theory is rapidly convergent for these solutions. In Sec. VI the application of the optimized cluster theory to simple liquids is discussed. In Sec. VII the variational formulations of the MSM and hard sphere PY equations are presented. Section VIII contains some concluding remarks.

II. DIAGRAM SERIES FOR THE FREE ENERGY AND PAIR CORRELATION FUNCTION

In this section we review the diagrammatic expres­sions for the Helmholtz free energy and pair correlation function of a homogeneous multicomponent fluid. Throughout we shall rely heavily on the work of

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1920 H. C. ANDERSEN AND D. CHANDLER

Morita and Hiroike,3d which introduces several enor­mous simplifications of the combinatorial problems arising from the use of diagrams. In particular, their Lemmas 4 and 6 make it a straightforward matter to sum classes of diagrams by introducing renormalized potentials. Also, using their approach the combinatorial problems involved in dealing with multicomponent fluids are no more complicated than those of single­component fluids. A convenient discussion of their method is included in Ref. 3(b).

The fluid of interest is an m-component mixture of spherical particles in a volume Vat a temperature T. We shall use Greek subscripts to denote the species. There are Na particles of type a, and the average number density of particles of type a is

Pa=Na/V.

The particles interact with each other through a distance-dependent potential composed of a hard sphere part and a perturbation. Thus the total potential energy of interaction between a particle of type a and a particle of type f3 is

=Ua/3(r), (2.1)

where dafj is the distance of closest approach of an a particle and a f3 particle.21 The potential energy for the collection of particles is the sum of the pair interactions. If we define the hard sphere part of the potential, Udafj(r), as

(2.2) then we can write

Wafj(r) = Udafj(r)+uafj(r), O::::;r< 00. (2.3)

We shall use the subscript d to denote the properties of a hypothetical multicomponent reference system whose intermolecular potentials are just the hard sphere potentials Udafj(r). Equation (2.3) states that the total potential of two particles is the sum of the reference potential Ua and u, which we shall call the perturbation part of the potential. It is clear that Eqs. (2.1)-(2.3) do not define the perturbation uniquely for r<dafj. For this physically impossible separation of two particles, the perturbation can be chosen to have any finite functional form. This flexibility in the choice of per­turbation in the "unphysical region" will be exploited below in obtaining optimized renormalized potentials.

We now define several functions which are needed for cluster diagrams. The Mayer cluster function is

ja(3(rl, r2)= exp[ -Wa(3([ rl-r2 [)/kBT]-1. (2.4)

Here, kB is Boltzmann's constant, and T is the absolute temperature. The cluster function for the reference system is

jdafj(rl, r2)= exp[ -Uda/3([ rl-r2 [)jkBT]-1. (2.5)

Also we define cJ>afj(rl, r2) as

cJ>afj(rl, r2) = -Uafj([ rl-r2 [)/kBT. (2.6)

Each of these three functions may be regarded as an mXm matrix of functions of two position variables.

Diagrams for multicomponent fluids are most con­veniently defined by the method of Morita and Hiroike.3d

The same definitions are used by Stell and Lebowitz.3b

A diagram is a collection of circles (or vertices) and bonds connecting the vertices. There are two types of circles, white and black. Each white circle has a species label and a position associated with it, but the black vertices are unlabeled. Each circle is associated with an m dimensional function of a position variable. (In this paper that function is always the set of m number densities which are in fact independent of position.) Each type of bond has associated with it an mXm matrix of functions of two positions. (The quantities f, fd, and q, are typical examples of such matrices.) The value of a diagram is defined in terms of these functions and an integration over the positions and a summation over the species labels which can be assigned to each black circle. Also the value of a diagram includes a combinatorial factor which is determined by the topological structure of the diagram. The reader is referred to Refs. 3b and 3d for a detailed presentation of the definitions.

The excess Helmholtz free energy of the fluid, relative to that of an ideal gas at the same temperature, volume, and species densities, is denoted by £lA. If we define

(2.7)

we have the following diagrammatic expression for (1:

(1= V-I (Sum of all more than singly connected (irreducible) diagrams with no white circles, two or more black P circles, and at most one f bond connecting any two circles). (2.8)

The pair distribution functions, gafj(rl, r2), are given by

PaPfj[ga/l(rl, r2)-1]

=PaP/lha/l(rl, r2)

= {Sum of all connected diagrams with two white P circles (one of species a at rl and one of species f3 at r2), any number of black p circles, at most one j bond connecting any two circles, and no articulation circles}. (2.9)

An articulation circle is a black circle whose removal leaves a disconnected diagram with one or more of the disconnected parts containing no white circle.

To convert these diagram series into forms more useful for obtaining renormalized potentials, we expand jafJ in powers of cf>afj:

fa/l(rl, r2)=jdafj(rl, r2)+[1+faafj(rl, r2)] co

xL: (n!)-I[cf>afj(rl, r2)]n. (2.10) n-l

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CLUSTER EXPANSIONS FOR CLASSICAL FLUIDS. I 1921

Thus each of the 1 bonds in these series can be regarded as the sum of an h bond, any number of powers of cp bonds, and any number of powers of cp bonds times an Id bond. Making this substitution we find:

<l= V-I I Sum of all more than singly connected diagrams with no white circles, two or more black P circles, at most one Id bond, and any number of cp bonds connecting any two circles}, (2.11)

PaP/l[gall(fl, f2)-1]

= I Sum of all connected diagrams with two white circles (one of species a at fl and one of species (3 at f2), any number of black P circles, and the same restrictions on the numbers of bonds connecting any two circles as in (2.11)}. (2.12)

It is convenient at this point to eliminate the reference system Id bonds in favor of reference system hd bonds, where hd denotes the functions gall(fl, f2) -1 for the hard sphere fluid. This procedure is a straightforward example of topological reduction and is facilitated by the use of Lemma 6 of Ref. 3d. The details will be omitted. The results are:

<l=(td+ V-I {Sum of all more than singly con­nected diagrams with no white circles, two or more black P circles, at most one hd bond and any number of cp bonds connecting any two circles, at least one cp bond, and no reference articulation pair of circles}. (2.13)

A reference articulation pair of circles is a pair of circles whose removal leaves a disconnected diagram at least one of whose disconnected parts contains only reference system bonds and at least one black circle. We also find:

PaPIlI~a/l(rl' f2)-1]

=PaPllha/l(fl, f2)

=PaP/lhda/l(fl, f2)+{Sum of all connected diagrams with two white P circles (one of species a at fl and one of species (3 at f2), any number of black P circles, at most one hd bond and any number of cp bonds con­necting any two circles, at least one cp bond, no articulation circles, and no reference articulation pair of circles}. (2.14)

Note that <ld and gda/l(fl, f2) are the free energy and pair correlation functions of the hard sphere reference fluid.

Equations (2.13) and (2.14) provide the starting point for further topological reductions in Sec. III.

m. SUMMATION OF PERTURBATION CHAINS AND RINGS

In the Mayer ionic cluster theory,1.2 the renormalized potential is obtained by summing chains of cp bonds.

+ o-----e- - --0

+ o----iII • ...---t ..... ---0

+ 0- - - -t.t----il ..... - - ~ • .---4 .... - --0 FIG. 1. Diagrammatic definition of the renormalized potential

which is defined as the sum of generalized chains, Eq. (3.1). In each diagram, the white circle at the left is of species ll/ and is at rl, and the white circle at the right is of species (3 and is at r2. Each circle denotes a p circle, each solid line denotes a cf> bond, and each dashed line denotes an hd bond.

The renormalized potential can be defined in a variety of other ways. We shall use a definition suggested by work on the 'Y expansion for classical fluidsf-s In this section we review the chain summation procedure and reformulate the cluster series in terms of this re­normalized potential.

We define the renormalized potential22 ea/l(rl, r2) as

PaP/leall(fl, r2)

= I Sum of all chains of one or more cp bonds connecting two white P circles (one of species a at fl and one of species (3 at f2) together with the results of all possible ways of inserting hd bonds at the end or in the middle of the chain}. (3.1)

These diagrams are depicted in Fig. 1. (The symmetry number of each of these diagrams is unity.) We can write this series as

00

PaP/leall(r) = PaP/l L ea/l(n) (r), (3.2)

where PaP/lea/l(n) is the value of the sum of all such chains with n perturbation bonds. We shall call these "generalized chains." In (3.2) we have used the fact that e and e(n) depend only on the difference between their two vector arguments. It should be noted that e(n) should contain no diagrams with two hd bonds attached end to end since in the series (2.13) and (2.14) no diagram contains a reference articulation pair of circles. Hence there are a maximum of n+ 1 hd bonds in e(n), and e(n) contains 2n+1 diagrams.

Generalized chains and diagrams involving them can be expressed more compactly in terms of reference system hypervertices used in the 'Y expansion.4-8 A hypervertex is a hybrid between a circle and a bond in diagram theory. The hypervertex useful for the present case is, like a bond, a function of two position co­ordinates and two species indices:

Fda/l(fl, f2)=P~a~(fl-f2)+PaP/lhdall(fl, f2)' (3.3)

[This particular quantity actually depends only on the

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1922 H. C. ANDERSEN AND D. CHANDLER

PaP{3C aJ2) (r1h) = ~

FIG. 2. Diagrammatic representation of generalized chains using the hypervertices of Refs. 4-8. The large hatched circles with two small circles on the circumference are Fd hypervertices, defined in Eq. (3.3). See Fig. 1 for the meaning of other symbols and the labeling of the white circles.

difference between its vector arguments. The function 8(r) is a Dirac delta function and 8a (3 is the Kronecker delta function.] This hypervertex is represented graphically as a large circle with two small circles, black or white, on its circumference, and as we shall use them each small circle is either white or has only one cp bond coming from it. In evaluating a diagram, integra­tion is to be performed over each small black circle and for purposes of finding combinatorial factors the hypervertex is the same as a single circle. Then ea,s(n) (rl' r2) can be expressed as one diagram, a chain of n+ 1 F d hypervertices connected by n cp bonds. See Fig. 2.

The summation in (3.2) can easily be performed using Fourier transforms. If we let a caret denote the Fourier transform of a function, then, for example,

¢a,s(k) = J dr exp( -ik· r)CPa,s(r). (3.4)

It is easily shown that

Pap,sea,s(n)(r) = (2'71)-3Jdk exp(ik·r)

X I [Fd(k)cP(k) ]nFd(k) }a,s. (3.5)

In (3.5) the multiplication processes are matrix multi­plication. If we define

P(k) =Fd(k)cP(k),

then the Fourier transform of Pap,sea(3(n) (r) is

I [p(k)]nFd(k) }a,s,

and substitution into (3.2) gives

Pap,sea,s(r) = (211')-3J dk exp(ik·r)

(3.6)

(3.7)

X IP(k)[l-P(k)]-IFd(k) } all, (3.8)

where the inverse in this equation is a matrix inverse, and 1 denotes the mX m identity matrix. This result is a slight generalization of one obtained by Hauge8 in studies of the l' expansion.

We shall need another quantity which is related to the renormalized potential. Let

PaPIl"iJlall(rl, r2)

= I Sum of diagrams with two white P circles (one of species ex at rl and the other of species {3 at r2) which are connected by two or more generalized chains and which may

or may not have an hd bond connecting the white circles},

= [l+hdall(rl, r2)]{ Sum of diagrams with two white P circles (one of species ex at rl and the other of species {3 at r2) which are connected by two or more generalized chains}. (3.9)

Using Lemma 4 of Ref. 3d, we find

'It a,s(rl, r2) = gda,s(rl, r2){ exp[ea,s(rl, r2)]

-l-ea,s(rl, r2)}. (3.10)

The renormalized potential e and this new function 'It which contains two or more powers of the renormalized potential, will now be used to rewrite the diagrammatic series of interest in a more compact form.

First we consider the series for a, Eq. (2.13). Each of the diagrams in the series contains at least one cP bond. Only two diagrams contain only one cP bond. They are represented in Fig. 3. We denote their sum as VaHTA.

VaHTA = t I: Pap,sJ drldr2gda,s(rl, r2)CPlla(r2, rl) all

= tv I: PaPIlJ drgdall(r )CP,sa(r). (3.11) a(j

The quantity aHTA is the "high temperature approxima­tion"20,23,24 correction which when added to ad gives a correct to within first order in T-I.

Next we consider the diagrams in the series for a which are generalized rings of perturbation bonds. A generalized ring is a ring of cP bonds connected by Fd hypervertices. Examples are shown in Fig. 4. We define

"" V aRING = I: <R (n), (3.12)

n=2

where <R(n) is the generalized ring with n cP bonds. It is easy to demonstrate that

men) = V(2n )-1 I: Pap,sJ dreall(n-1l (r )cp,sa(r), (3.13) a{3

since a generalized ring is like a generalized chain whose ends have been connected by a cP bond. The factor of (2n )-1 accounts for the fact that the symmetry number of a ring with n bonds is 2n while the symmetry number of a chain with n-l bonds is unity. When the r integra­tion in (3.13) is converted into a k integration and use is made of (3.7) and (3.6), we find a very simple result,

where Tr denotes the trace of a matrix. Substitution

Va HTA = e---e + ...=-=.:. FIG. 3. The two diagrams

in a which have only one <t> bond. See Fig. 1 for the meaning of the circles and bonds.

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CLUSTER EXPANSIONS FOR CLASSICAL FLUIDS. I 1923

into (3.12) then gives

aRINO= -[1/2(21IY]Jdk{TrP(k)+ In det[l- P(k)] I,

(3.15)

where det denotes the determinant of a matrix. Here we have used the fact that for any diagonalizable matrix A, Tr InA = In detA, and the assumption that all the eigenvalues of I-P(k) are positive.25

Using topological reduction, the diagrams other than rings and aHTA diagrams in the series for a can be re­expressed in terms of e and it bonds rather than rf> bonds; in other words the potential is replaced by a renormalized potential. The result is

a = (td+aHTA +aRING+ B2+ V-I {Sum of all more than singly connected diagrams with no white circles, three or more black P circles, hd, e, and it bonds, no more than one bond of each type connecting any pair of circles, no it bond con­necting two circles which are also connected by another bond, no black circles with only a e bond and an hd bond or only two e bonds attached to it, no reference articulation pairs of circles, and at least two it bonds or two e bonds or one it and one e bond I. (3.16)

In Eq. (3.16),

B2 = t L Pap~Jdrhda~(rH[ea~(r)] a~

co

+t L PaP~fdrgda~(r) L (n!)-I[ea~(r)]n. (3.17) a~ n-3

An analogous topological reduction can be carried out for the series for ga~(rI, r2). The result is

Pap~[ga/l(rI, r2)-1]

=: PaP/l[gda/l(rl, r2) -1 +ea~(rl, r2)

+hda/l(rl, r2)ea/l(rI, r2)+ita~(rl, r2)]

+ {sum of all connected diagrams with two white P circles (one of species a at rl and one of species f3 at r2), at least one black P

circle, hd, e, and it bonds, no articulation circles and no reference articulation pairs of circles, the same restrictions on the numbers of bonds connecting any pair of circles or attached to anyone black circle as in (3.16), and atleast one e or it bond}. (3.18)

Equations (3.16) and (3.18) are the principal results of this section. They are closely related to those derived by Stell and Lebowitz6 for Coulombic fluids. They can also be regarded as generalizations of the cluster expansions derived by Friedman3a for the case in which the perturbation is Coulombic for all r.

In the next section we discuss the optimal choice of perturbation potential for rapid convergence of these series.

Va RING ~ + ~

+ t1 + •••

FIG. 4. Diagrammatic representation of generalized rings, containing Fd hypervertices and cf> bonds.

IV. OPTIMAL CHOICE OF PERTURBATION

Equations (3.16) and (3.18) for the free energy and pair correlation functions of a fluid are formally exact infinite series. Both series possess a rather strange property; namely, each term (except the first two) in each series depends on the value of the perturbation potential ua~(r) for r<da~, i.e., inside the hard cores, despite the fact that both sums must obviously be independent of the value of the perturbation for such physically impossible in terparticle separations. Although the complete sums are not dependent upon the choice of perturbation inside the cores, the rates of convergence of the series are crucially dependent upon the choice. In this section we discuss a way of choosing the per­turbation so as to make the convergence very rapid for some interesting fluids.

There are several conceivable choices. We propose choosing ua~(r) for r<da~ to guarantee that

( 4.1)

This requires a different choice of perturbation inside the core for each value of the temperature and number densities. To justify this choice we must show two things: first, that such a choice is possible, and, secondly, that such a choice does indeed improve the convergence.

This choice is possible because it can be shown from the diagrammatic definitions of ea~(r) and aRING that

5VaRING/5rf>a~(r, r')=tPaP/lea~(r, r'). (4.2)

(The proof of this equation requires Lemma 2 of Ref. 3d.) If we now regard aRING as a functional of the perturbation Ua~(r), which is restricted to being dependent only on the magnitude of its argument and to being a symmetric function of its species indices, then (4.2) leads to

ea/l(r) = - (kBT /21rr2paP/l) [5aRING/5ua/l (r )]. (4.3)

The perturbation Ua~(r) for r~da/l is to be held equal to its physically correct value. We can now regard aRING as a functional of the perturbation inside the core. It can easily be shown using (3.15) that aRING is positive definite. Thus it has some minimum value which is nonnegative. We choose the perturbation inside the core to be the one which causes aRINO to attain its minimum value, and barring mathematical pathologies, this minimum corresponds to a stationarity point.

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1924 H. C. ANDERSEN AND D. CHANDLER

Then for this choice (4.1) follows immediately from (4.3); that is, the statement that

OaRING/Ouar(r) =0, r<dar, (4.1')

is equivalent to Eq. (4.1). The demonstration that the choice of perturbation

defined by (4.1) actually does enhance the convergence of Eqs. (3.16) and (3.18) is postponed to Secs. V and VI, where it will be discussed for the problems of ionic solutions and simple liquids. In the remainder of this section we show the relationship between Eq. (4.1) and the optimized random phase approximation of the mode expansion theory and the mean spherical model (MSM). We also discuss the physical meaning of the optimization condition.

The mode expansion17- 20 provides infinite series for the free energy of a classical fluid of particles interacting through a hard core potential plus a perturbation. By functional differentiation with respect to the perturba­tion, a series for g is found. These series, like the cluster series, depend upon the functional form chosen for the perturbation inside the hard cores. The one-mode approximation for the free energy, called the random phase approximation (RPA), is precisely equal to the terms in (3.16) up to and including the ring sum. When this is functionally differentiated to obtain g(r) the result is

gall(RPA)(r) = gdall(r) + eall (r ). (4.4)

This is precisely equivalent to keeping only the first three terms on the right side of the cluster series Eq. (3.18). In the optimized random phase approximation (ORPA), the perturbation is chosen so that

(4.5) Since

The physical meaning of the optimization condition is not entirely clear despite the empirical fact that it does improve the convergence (see Secs. V and VI). Within the context of the RPA, optimization is ob­viously related to the excluded volume effect produced by the hard core forces and must be performed in order to make g(RPA) given by (4.4), into a physically reasonable approximation. Nevertheless the same renormalized potential should also be used even when going beyond the RPA. The optimization condition can be partially understood as a way of taking into account the "screening" of the perturbation by the short ranged repulsive forces. In the Mayer ionic cluster theory, chains of cp bonds are summed (without including the diagrams that have hd bonds). The result is a screened Debye-Hiickel potential. The meaning of the screening is that cooperative behavior among large numbers of particles can reduce the effect of the perturbation. These chains of cp bonds take no account of the fact that short ranged hard sphere interactions can also produce some screening. The inclusion of the Fd hypervertices in the definition of the renormalized potential does account for some of this effect, but not enough of it.26 The optimization procedure apparently is needed to do the rest. Another aspect of the meaning of optimization is that for some choices of the per­turbation inside the hard cores, the ring and chain sums do not exist. This happens when the matrix 1-P (k) is singular for some values of k. These divergences are artifacts of the renormalized potential technique and represent spurious singularities that have no relation­ship to the properties of real fluids. 27 The optimization procedure eliminates these divergences.

v. CONVERGENCE OF CLUSTER SERIES FOR r<dall, (4.6) IONIC SOLUTIONS

the optimal perturbation for the ORPA is exactly the same as the perturbation which makes the renormalized potential of the cluster series equal to zero inside the cores. This establishes the connection between the ORPA and the optimized cluster theory.

The connection between the optimized cluster theory and the MSM is easily established because in Ref. 19 the relationship between the MSM and ORPA was given. The latter, together with Eq. (4.4), implies

ga/l(MSM) (r) = gda/l(PY) (r )+ea/l(PY) (r), (4.7)

where e(PY) denotes the generalized chain sum evalu­ated using the Percus-Yevick approximation for hd (rather than the exact hard sphere functions) and using the perturbation that makes

eall(PY) (r) = 0, r<dall. (4.8)

This shows that the MSM approximation for the pair correlation function is very similar to the first few terms of .the cluster series (3.18) when the latter is optimized according to (4.1).

Cluster series similar to (3.16) and (3.18) have been applied many times to the study of ionic solutions. In this section we discuss the factors which affect the convergence of these series and show how the optimum choice of perturbation, defined by (4.1), improves the convergence. For this choice, the series converge quickly enough to suggest some very simple and ac­curate approximations for the properties of ionic solu­tions in the range of concentration between 0 and 2M.

In order to make the discussion specific, we shall assume that the fluid under consideration is an aqueous symmetric primitive model electrolyte solution. In this model the ions are represented as hard spheres with charges embedded at their centers and the Coulombic interaction between ions is reduced by a factor of E-l, where E is the dielectric constant of the solvent. The positive and negative ions have the same diameter and the same magnitude of charge. Most of our considera­tions are more generally applicable, however.

Three factors are important for determining the convergence of the cluster series: the magnitudes of the

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CLUSTER EXPANSIONS FOR CLASSICAL FLUIDS. I 1925

bond functions, the ranges of the bond function, and the values of the densities. At 1M concentration the e and q, bonds are reasonably short ranged due to shielding of the Coulomb potential. Let us denote this range by 1. (If the perturbation is chosen to be Coulombic at all distances, then 1 is approximately ct, the Debye shielding length.) The average range of the hd functions is approximately d, the mean hard sphere diameter for the reference fluid. The magnitude of the hd bond is of order unity. Let us denote the average magnitude of the e and q, bonds by e and q" respectively. Then the magnitude of a typical diagram with n black circles in the series for a is

(5.1)

where a is the number of e bonds in the diagram, b is the number of q, bonds, p is the total number density of the reference fluid, s is an integer between 0 and n-l which is determined by the connectivity of the diagram, and A is a numerical factor of order unity which depends on the particular diagram and which is slightly density dependent. The order of magnitude of the diagrams in the series for g can be estimated in a similar way.

The usual way to describe primitive model elec­trolytes is to choose the perturbation potential to be Coulombic for all values of r, including the range of physically inaccessible values inside the hard cores. The generalized chain sum can then be performed analytically for this particular model.6 The renormalized potential is

e",jS(rl, r2)= - (z",zjSe2/kB TE I rl- r2 \) exp(-K I rl- r2 \),

(5.2)

K=[(4'/1'e2/EkT) L p"Z",2]1/2. (5.3) '"

Here, z'" is the charge in electronic units on the ions of species a, and e is the magnitude of the electronic charge. With this choice of perturbation, however, the series do not converge quickly enough to be useful. For example, the reference system, high temperature, ring diagram, and B2 contributions can be evaluated, since they require only a knowledge of the properties of a dilute hard sphere fluid and the evaluation of one dimensional numerical integrals. This approximation is very close to the "Debye-Htickellimiting law + B2"

approximation, or "DHLL+B2" approximation.3s The only differences are that the latter has jd rather than hd functions in the B2 integrals, includes only the second virial coefficient contribution to the reference system free energy, and the hd functions are not included in the chains. These differences are small at the concentrations of interest and the hd bonds in the chains exactly cancel one another for this particular model. The DHLL+B2

approximation is exact in the limit of low concentra­tion, but its accuracy deteriorates as the concentration is increased. For example, Rasaiah and Friedman28

have concluded that, for 1-1 salts at 1M concentration,

this approximation is not accurate enough for compari­son with experimental data.

We now consider why this series does not converge well at these higher concentrations. For a 1-1 sym­metric electrolyte, with an ionic diameter of 4 A, pd3 is approximately 0.077 at 1M. At 25°C the Debye shielding length K-1 is about 3.0 A, and so p13 is ap­proximately 0.034. The maximum value of q,,,,~(r) is its value at r=d",,,,, which is about 0.135. Hence q,<0.135. Thus when the perturbation is chosen to be Coulombic for all values of r, the parameters pd3, pP, and q" appearing in (5.1), are all small. All the diagrams in the free energy series (3.16) have n~3 and a+b~2, and subsequent terms have more and more powers of the small parameters. Thus the important reason why the series do not converge quickly enough at 1M must be that the parameter e is too large. For r~d",~, e",jS(r) is small, but for r<d",{3, e",{3(r) is unbounded and diverges like l/r as r approaches zero. A reasonable estimate of the average value of e is the following dimensionless number:

For the ionic solution under consideration, at a con­centration of 1M, this estimate for e gives 3.1. Thus the behavior of the renormalized potential inside the hard core, i.e., for physically inaccessible separations, makes the usual ionic cluster theory inaccurate at concentrations of about 1M.29 This difficulty can ob­viously be remedied by using the optimization condi­tion, Eq. (4.1), in choosing the renormalized potential.

If we choose the perturbation not to be purely Coulombic but to be Coulombic only outside the core and fix its value inside the core according to the optimization condition (4.1), the convergence is improved in two ways. First, it makes e",{3(r) finite for all r and zero for a significant range of interparticle distances. This effectively makes the parameter e much smaller. Secondly, it greatly decreases the values of those diagrams which have a pair of circles connected by both an hd bond and a e bond. The latter bond is now nonzero only for r~d",{3 but the former is of order pd3 rather than of order unity for these values of r. Thus, instead of (5.1), the order of magnitude of a typical diagram in (3.16) is now

Ap(pd3),,-.+t-l (p13). eaq,b, (5.4)

where t is the number of pairs of circles connected by both a e bond and an hd bond. This choice of perturba­tion has other, unfavorable, effects which should also be realized. The renormalized potential, e, and q, are now larger outside the cores and of longer range than for the usual choice of perturbation. These effects are more than counterbalanced, however, by the two favorable effects described above.

Thus the convergence of the free energy series, Eq. (3.16), is enhanced when the perturbation inside the core is chosen to make the renormalized potential zero

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1926 H. C. ANDERSEN AND D. CHANDLER

inside the core. If the renormalized potential is weak enough and short ranged enough, and if the hard sphere density is low enough, the sum of diagrams in Eq. (3.16) can be neglected because each diagram has an order of magnitude given by (5.4) with n;:::3, t;:::O, and a+b;::: 2. This leads to the following approximation:

a~ad+aHTA+aRING+B2. (5.5)

This is a computationally convenient approximation since the first term on the right can be evaluated from our knowledge of hard sphere fluids and the next three terms can be calculated by performing one-dimensional numerical integrations, once the optimal perturbation potential is determined. We shall call this the ORPA+ B2 approximation, because the first three terms on the right are equivalent to the optimized random phase approximation. A detailed discussion of this approxima­tion and its application to model ionic solutions will be the subject of a future paper.lO Preliminary numerical calculations show that Eq. (5.5) is very accurate for 1-1 and 2-2 aqueous electrolytes for concentrations between ° and 2M.30

To obtain an analogous approximation for the pair correlation function we neglect all the diagrams in the sum in Eq. (3.18). We find

gafj(r1, r2)~gdafj(r1' r2)+eafj(rt, r2)

+hdafj(r1, r2)eafj(rt, r2)+'Itafj(r1, r2), (5.6)

or, more simply,

ga/J(r1, r2)~gdafj(r1, r2) expeafj(r1, r2). (5.7)

For lack of a better name, we shall call this the exponen­tial approximation. The neglected terms are all finite and make a contribution to gafj of order

(5.8)

where n is the number of black circles and A, s, t, a, and b are defined above. Each neglected diagram has at least one black circle (n;::: 1) and either n+t;::: 2 and a+b;::: 1 or n+t= 1 and b= 1. Hence if the density and renormalized potential are small enough and if the latter is short ranged enough, Eq. (5.7) will be a good approximation.

Comparison of the exponential approximation, Eq. (5.7), with Monte Carlo data for 1-1 electrolytes indicates that it is a useful approximation. A detailed test of its accuracy will be given in Paper III of this series.!O

Equations (5.5) and (5.7) are the principal results of this section. They represent approximate expressions for the free energy and pair correlation function of a fluid and are obtained by neglecting an infinite class of diagrams in the appropriate formally exact series. For best results, the perturbation which is used to evaluate these expressions must be the optimized perturbation which is chosen according to the criterion that the renormalized potential should be zero inside the hard

cores. The neglect of these diagrams is justifiable when the renormalized potential is small and short ranged enough and/or the density is small enough. These approximations will be useful for theoretical discussions of ionic solutions.

VI. CONVERGENCE OF CLUSTER SERIES FOR SIMPLE LIQUIDS

In this section we discuss the usefulness of the optimization condition (4.1) for obtaining rapidly convergent cluster series for simple liquids. We will find that the optimized series converge rapidly enough to make Eqs. (5.5) and (5.7) accurate for simple fluids with short range forces as well as for ionic solu­tions. A remarkable property of these approximations applied to simple liquids is that each is asymptotically correct in each of the following four limits: the limit of low density, the limit of high density, the limit of high tem­perature, and the limit 'Y~O, where ,,-1 is the range of the perturbation potential in the'Y expansion.4- s

In Ref. 20, the ORPA was applied to the fluid whose reference system is the hard sphere fluid and whose perturbation potential is the atrractive part of the Lennard-Jones potential. Numerical calculations of the optimal perturbation inside the core were performed; in effect e(r) was evaluated. A striking result of these calculations is that the optimal e (r) is small at high densities even for low temperatures. For example, even though I q,(r) I > 1 at the triple point temperature, le(r)1 SO.05 at the triple point. The physical meaning of this is clear from (4.4), namely that at high densities (even for low temperatures) the perturbation can have only a very small effect on the structure of a fluid. The hard core forces dominate the structure and the effect of the perturbation, when suitably renormalized, is small. The mathematical effect of this is to make the e function small at high densities for all temperatures. From (3.10) it is seen that 'It is also small. Moreover, e and ~, which as in Sec. V are the average magnitude of the strengths of e and 'It, are decreasing functions of density at high density.

For a one-component fluid with a short ranged potential, the range of the renormalized potential is of order d except perhaps in the critical region. Hence the magnitude of a typical diagram in the infinite series for a (3.16) is

(6.1)

where n, a, b, and A are defined as in Sec. V, and, for each diagram in the sum, n;:::3 and a+b;:::2. As dis­cussed above, for simple liquids e is small at high densities for all temperatures. It is also small at high temperatures for all densities since it is of order T-1. It is finite at low densities, where it is almost equal to q,. Thus it is reasonable to expect the series for a to conv~rge quickly for low density and for high density and for high temperature. Also since each of the diagrams is at least second order in the strength of the

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CLUSTER EXPANSIONS FOR CLASSICAL FLUIDS. I 1927

renormalized potential and second order in pd3, it is a reasonable approximation to neglect the infinite series in (3.16) completely, giving, once again, the ORPA + B2 approximation, Eq. (5.5). The analogous result for g is obtained by neglecting all the diagrams in (3.18), which have at least one black circle. The result is the exponential approximation, Eq. (5.7). The neglected terms are all finite and makes a contribution to g,,(3 of order

(6.2)

where n~ 1 and either n+t~ 2 and a+b~ 1 or n+t= 1 and b= 1. The above discussion shows that Eqs. (5.5) and (5.7) when applied to simple liquids with short ranged forces are correct in the limit of high density or of low density or of high temperature provided the optimization condition (4.1) is satisfied. In addition, these results are correct in the limit ')'-70 in the ')' expansion. Hence it is not surprising that they are quite accurate approximations over much of the temperature­density plane.

Preliminary numerical calculations indicate that Eqs. (5.5) and (5.7) are very accurate for the Lennard­Jones fluid. 31 Equation (5.5), with the B2 term omitted, is just the ORPA expression for the free energy. It was found to be very accurate for the single fluid phase except for the very low density, low temperature part of the phase diagram.2o The B2 term makes (5.5) correct in the limit of low density for all temperatures, without affecting the accuracy of the high density results. Also a comparison of (5.7) with the ORPA result (4.4) shows that (5.7) in general predicts a larger effect of the perturbation on g(r), especially in the first peak. Comparison of the ORPA with molecular dynamics calculations indicates that the ORPA does in fact tend to underestimate this effect. Hence (5.7) is a significant improvement.

Equations (5.5) and (5.7) are the principal results of this section, as they were for Sec. V. They are ac­curate, computationally convenient approximations for the properties of simple fluids as well as ionic solutions. Detailed numerical tests of these approximations for simple liquids are the subject of Paper III of this series. to

VII. VARIATIONAL FORMULATION OF THE MEAN SPHERICAL MODEL AND HARD SPHERE PERCU8-YEVICK EQUATIONS

The mean spherical modeP3,14 (MSM) and hard sphere Percus-YevickI5 ,16 (PY) integral equations have provided very useful approximations for the properties of some simple fluids, such as hard spheres and primitive model electrolytes. One reason for their usefulness is that these equations are exactly solvable for some interesting special cases, namely the PY equation for a one component fluid of hard spheres32 ,33 and for a hard sphere mixture with additive diameters16 and the MSM

for a symmetric primitive model electrolyte,l4 The other reason is that they give a reasonably accurate repre­sentation of the thermodynamic properties of some fluids.

When integral equations cannot be solved analy­tically, iterative numerical solution methods are often used. These methods often lead to difficulties. Attention must be paid to the effects of grid size, truncation error, and round-off error in the evaluation of numerical integrals, and the convergence of the iterated solutions to a final answer can be painfully slow. In this section we derive a variational formulation of the MSM and hard sphere PY equations. This formulation greatly facilitates the solution of these equations. There are two basic motivations for obtaining the variational princi­ples. First, there are some interesting simple fluids, such as a mixture of hard spheres with nonadditive diameters,34 for which it would be illuminating to have the solution of the PY equation. Secondly, molecular analogues of these equations, together with their variational formulations, can be constructed,9 leading to investigations of molecular fluids using the same equations that have been so useful for simple liquids.

The MSM and hard sphere PY approximations are most easily stated in terms of the direct correlation function. The latter, denoted c"/l(rl, T2), is defined by the Ornstein-Zernike (OZ) equation

g,,(3(r1, rz)-l=c,,/l(rl, r2)+ L p.Jdr3C"y(r1, T3) y

For a fluid of particles interacting through a hard sphere potential plus a perturbation, the MSM approximation is defined by

ga/l(r1, r2)=0, 1 r1- T21 <da/l, (7.2)

Ca/l(r1, r2)=l/>a(3(rl, r2), 1 Tl- r21 ~da(3. (7.3)

Equation (7.2) is exact, but Eq. (7.3) is an approxima­tion. For hard spheres, l/>a(3(r) is zero for r>da/l, and the hard sphere PY equation is defined by (7.2) and

(7.4)

Thus the hard sphere PY approximation can be re­garded as the MSM approximation applied to hard spheres. In each case the unknown functions are g,,(3(r) for r~d,,(3 and c"fJ(r) for r~da(3.

The OZ equation can be given a graphical interpreta­tion. If we imagine solving this equation by iteration we have the following result:

p"p/lg"/l(r1, T2)

= P"P(3+ {Sum of all diagrams with two white p circles (one of species a at Tl and one of species fj at r2) which are connected by a chain of one or more C bonds}. (7.5)

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1928 H. C. ANDERSEN AND D. CHANDLER

o o + 0---0

+ 0--_._--0

+ 0-- -.-- _____ --0

+ • • •

FIG. 5. Diagrammatic representation of the pair correlation function in terms of the direct correlation function, Eq. (7.6). In each diagram the white circle at the left is of species Ci and is at rl, and the white circle at the right is of species fJ and is at r2. Each circle is a p circle and the bonds with unequally sized dashes are c bonds. The two white circles on the top line are a single diagram, the value of which is PaPp, which is the first term on the right in Eq. (7.5).

This equation is given diagrammatically in Fig. 5. To formulate the variational principle it is convenient to define the following functional:

V5' = {The diagram in which two black p circles are connected by one c bond plus the sum of all diagrams which are rings con taining black p circles and c bonds} . (7.6)

These diagrams are shown in Fig. 6. The significance of this functional depends on the fact that when it is functionally differentiated with respect to c,,{1(rl, r2) it yields a result which is simply related to the chain diagrams in (7.5). The exact statement is

This follows from Lemma 2 of Ref. 3d. This result is exact, and (7.6) and (7.7) may be regarded as an alternative formulation of the OZ equation.

To apply the MSM and hard sphere PY approxima­tions, we use Eq. (7.2) and the fact that for a homogene­ous fluid cC/{1(rl, r2) and gC/{1(rl, r2) are functions of / rl-r2/' Then we find that the MSM and hard sphere PY approximations are expressed by the following set of equations:

(7.8a) where

5'=! 1: p"p{lJdrca{l(r) ,,(l

-H271')-3fdk{Trq(k)+ In det[l-q(k)J}, (7.8b)

(7.8c) and

(7.8d)

Here the ring sum has been evaluated using the same method that led to (3.15). In other words, the MSM solution for c"fj(r) inside the hard cores is the set of functions which makes the functional 5' stationary with

respect to changes of c inside the cores. Equations (7.8) are the variational formulation of the MSM and hard sphere PY equations.

In practice the way to use the variational principle is to assume a simple functional form for c,,{l(r) for r<dC/fl' containing a finite number of adjustable con­stants, aI, "', an, and to vary the constants to make 5' stationary:

i= 1, "', n. (7.9)

These equations for the constants can be solved itera­tively. Once c(r) is known inside the cores, g(r) can be obtained using the Fourier transform of the OZ equation. A similar numerical problem arises in finding the optimal perturbation for 'the random phase ap­proximation in the mode expansion theory.2o Experi­ence with the latter problem indicates that the iterative Newton-Raphson methodll for solving (7.9) converges very quickly and that accurate solutions of the varia­tional problem can be obtained using a linear variational function with four basis functions in the one component case.

Equations (7.8) are the principal results of this section. They provide a convenient way of solving the MSM and hard sphere PY equations for simple fluids, and they can be generalized to apply to certain models for molecular liquids.9 •

VIII. CONCLUDING REMARKS

The purpose of this paper has been to use cluster theory and topological reduction to obtain rapidly convergent formal expansions for the thermodynamics and structural properties of fluids, convenient ap­proximations based on truncating these formal ex­pansions, and variational formulations of the mean spherical model and hard sphere Percus-Yevick equations. Some applications of these results will be discussed in future papers, and their generalizations to molecular fluids will also be presented.

In this work we have restricted our attention to a class of artificial and unrealistic models, namely those for which the particles interact with each other through hard sphere forces plus an additional perturbation. More realistic models describe the repulsions with continuous repulsive potential energy functions. How-

~ + ' \ + .:----

+ • • •

FIG. 6. Diagrammatic definition of the functional which is defined in Eq. (7.6) for an alternative formulation of the Ornstein­Zernike equation. The circles and bonds have the same meaning as in Fig. 5.

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CLUSTER EXPANSIONS FOR CLASSICAL FLUIDS. I 1929

ever, once the free energy and pair correlation functions are known for a fluid with a "hard sphere plus per­turbation" potential, the corresponding properties for a "soft sphere plus perturbation" potential can easily be obtained.35 ,20

* Work supported by the National Science Foundation and the donors of the Petroleum Research Fund as administered by the American Chemical Society.

1 J. Mayer and M. Mayer, Statistical Mechanics (Wiley, New York, 1940).

2 J. E. Mayer, J. Chem. Phys. 18, 1426 (1950). a Some useful references on cluster theory are: (a) H. L.

Friedman, Ionic Solution The01'y (Interscience, New York, 1962). (b) G. Stell, in The Equilibrium The01'y of Classical Fluids, edited by H. L. Frisch and J. L. Lebowitz (Benjamin, New York, 1964). (c) G. E. Uhlenbeck and G. W. Ford, in Studies in StaUstical Mechanics, edited by J. de Boer and G. E. Uhlenbeck Vol. I, (North-Holland, Amsterdam, 1962). (d) T. Morita and K. Hiroike, Progr. Theoret. Phys. 25, 537 (1961).

4 J. L. Lebowitz, G. Stell, and S. Baer, J. Math. Phys. 6, 1282 ( 1965).

6 G. Stell, J. L. Lebowitz, S. Baer, and W. Theumann, J. Math. Phys. 7,1532 (1966).

6 G. Stell and J. L. Lebowitz, J. Chem. Phys. 49, 3706 (1968). 7 P. C. Hemmer, J. Math. Phys. 5, 75 (1964). 8 E. H. Hauge, J. Chem. Phys. 44, 2249 (1966). 9 D. Chandler and H. C. Andersen, J. Chem. Phys. 57, 1930

(1972) . 10 H. C. Andersen, D. Chandler, and J. D. Weeks, J. Chem.

Phys (to be published). 11 D. R. Hartree, Numerical Analysis (Oxford, London, 1955). 12 L. Verlet and J.-J. Weis, Phys. Rev. A 5, 939 (1972). 13 J. L. Lebowitz and J. Percus, Phys. Rev. 144, 251 (1966). 14 E. Waisman and J. L. Lebowitz, J. Chem. Phys. 52, 4307

(1970); 56, 3086, 3093 (1972). 16 J. K. Percus and G. J. Yevick, Phys. Rev. 110, 1 (1958). 16 J. L. Lebowitz, Phys. Rev. 133, A895 (1964). 17 H. C. Andersen and D. Chandler, J. Chem. Phys. 53, 547

(1970). 18 D. Chandler and H. C. Andersen, J. Chem. Phys. 54, 26

(1971) . . 19 H. C. Andersen and D. Chandler, J. Chem. Phys. 55, 1497 (1971).

20 H. C. Andersen, D. Chandler, and J. D. Weeks, J. Chem. Phys. 56, 3812 (1972).

21 We do not assume that da~ is a sum of the radii of the two hard spheres, i.e., that da~= (daa+d~~) 12.

22 Strictly speaking, the renormalized potential is actually -kBTe, but for simplicity we shall call e the renormalized potential.

23 R. Zwanzig, J. Chem. Phys. 22, 1420 (1954). 24 J. D. Weeks, D. Chandler, and H. C. Andersen, J. Chem.

Phys. 54, 5237 (1971); D. Chandler and J. D. Weeks, Phys. Rev. Letters 25, 149 (1970).

26 The fact that P(k) [and hence also 1-P(k) ] is a diagonaliz­able matrix follows from the definition (3.6) when use is made of the facts that both ~(k) and Fa(k) are real symmetric matrices and that the latter has positive-definite eigenvalues for all k. It is necessary to assume that all the eigenvalues of 1-P(k) are positive for all k in order for the chain summations to exist. At very high temperatures, these eigenvalues are all unity for all k, since P(k) is of order liT. If, as the temperature is lowered, an eigenvalue for some value of k became negative, then presum­ably there would be at least one value of k which had a zero eigenvalue. In this case the matrix inverse in (3.8) would not exist. This is related to the so-called" RPA catastrophe" [see J. C. Wheeler and D. Chandler, J. Chem. Phys. 55,1645 (1971)]. The optimization procedure discussed in Sec. IV insures that this type of catastrophe is avoided.

26 In fact, the use of simple ct>-bond chains without Fd hyper­vertices but with the optimization condition does take into account much of the excluded volume effect and is closely related to the ideas of the MSM.

27 These singularities are identical with those arising in the RPA. See J. C. Wheeler and D. Chandler, Ref. 25.

28 J. C. Rasaiah and H. L. Friedman, J. Chern. Phys. 48, 2742 (1968) .

29 One might suggest that another reason for the lack of rapid convergence is that the number of terms with given values of n, s, a, and b grows very rapidly as these values are increased. It will be clear from the following discussion that this is not the important reason.

30 ORPA+B2 results for the osmotic pressure and internal energy of 1-1 symmetric aqueous electrolyte solutions agree with Monte Carlo results essentially perfectly for all salt concentra­tions between 0 and 2M. For 2-2 electrolytes in the same range of concentration the configurational energies of the ORPA+B2

and Monte Carlo methods differ by at most 7%. 31 To apply these equations to the Lennard-Jones fluid it is

necessary to take into account the softness of the repulsive part of the Lennard-Jones potential. This can be done using the methods described in Refs. 35 and 20 .

32 E. Thiele, J. Chern. Phys. 39, 474 (1963). 33M. S. Wertheim, Phys. Rev. Letters 10, 321 (1963). .4 J. L. Lebowitz and D. Zomick, J. Chern. Phys. 54, 3335

(1971). 36 H. C. Andersen, J. D. Weeks, and D. Chandler, Phys. Rev.

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