Statistical Mechanical Models with Effective Potentials: Deﬁnitions, Applications, and Thermodynamic Consequences Frank H

JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 1 1 JULY 2002

Statistical mechanical models with effective potentials: Deﬁnitions, applications, and thermodynamic consequences Frank H. Stillinger Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544 Hajime Sakai Department of Chemistry, Princeton University, Princeton, New Jersey 08544 Salvatore Torquato Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544 ͑Received 14 February 2002; accepted 2 April 2002͒ Realistic interactions that operate in condensed matter systems can exhibit complicated many-particle characteristics. However, it is often useful to seek a more economical description using at most singlet and pair effective interactions that are density dependent, to take advantage of the theoretical and computational simpliﬁcations that result. This paper analyzes the statistical mechanical formalism required to describe thermal equilibrium in that kind of approach. Two distinct interpretations are available for the role of density dependence. Either one can be treated with internal consistency, but generally they lead to differing thermodynamic predictions. One regards the density dependence of effective interactions as merely a passive index for the state at which the optimal choice of those effective interactions was determined ͑Case I͒. The other treats the density as an active variable on the same footing as particle coordinates ͑Case II͒. Virial pressure and isothermal compressibility expressions in terms of particle distribution functions are displayed for both cases. Under special circumstances it is possible for the two interpretations to yield the same pressure isotherms; the conditions producing this coincidental concordancy have been explored by means of density expansions. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1480863͔

I. INTRODUCTION AND HISTORICAL BACKGROUND We have assumed for simplicity here that only a single spe- cies is present ͑the multicomponent generalization is Regardless of whether it is close to, or far from, a state ͒ straightforward , and ri comprises all nuclear positions of the of equilibrium, the behavior of any material sample depends ⌽ ith particle. Also, the zero of energy for N has been chosen fundamentally on the structure of its constituent particles and to correspond to widely separated particles, all in their on their mutual interactions. For the majority of substances lowest-energy conformation. In the event that structureless that command research attention in the physical and biologi- particles were under consideration, u1 would vanish identical sciences, those structures and interactions tend to be very cally except near confining container walls. The requirement complex. In theory, a full characterization would require that Eq. ͑1.1͒ be satisfied identically for all N, with the same solving the Schro¨dinger equation for all electrons, in an ar- functions ui , uniquely determines these component func- bitrary configuration of the system’s nuclei, at least for the tions. ground electronic state. In practice this is a difficult or im- The scientific literature reveals many theoretical propos- possible task. Both conceptual understanding and numerical als to replace the exact many-body potential ͑1.1͒ with a studies motivate the search for a more economical descrip- mathematically simpler, and therefore more tractable, form tion. ⌽ N* while attempting to retain physical essentials. The most Quite generally, the many-particle interaction potential popular approach approximates ⌽ with a linear combina- ⌽ ¨ N N that emerges from solving the Schrodinger equation for tion of ͑at most͒ ‘‘effective’’ one-body and two-body inter- electrons can be resolved into separate one-body, two-body, actions, ..., N-body contributions: ⌽ Х⌽ ϭ ͑ ͒ϩ ͑ ͒ ͑ ͒ N N* ͚ u1* ri ͚ u2* ri ,rj . 1.2 i iϽ j ⌽ ͑ ͒ϭ ͑ ͒ϩ ͑ ͒ N r1¯rN ͚ u1 ri ͚ u2 ri ,rj i iϽ j Of course some criterion must be advanced for the selection

of the ui*’s, including speciﬁcation of the function search ϩ ͒ϩ ϩ ͒ ͚ u3͑ri ,rj ,rk uN͑r1 rN . space. The results generally will depend on temperature T, Ͻ Ͻ ¯ ¯ i j k system volume V, and either N ͑closed system͒ or chemical ͑1.1͒ potential ␮ ͑open system͒, presuming that the system is close

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to a state of thermal equilibrium. So far as the present inves- seeks a continuous variation in the pair potential, as the num- tigation is concerned, we shall assume that isothermal con- ber density ␳ of the entire system changes, so that the short- ditions apply, and thus shall concentrate on the effects of V range order expressed by the pair correlation function g(2) and N. The implications of temperature dependence for ef- remains invariant.18 The density range over which this in- fective interactions will be reserved for later consideration. variance can be maintained offers some insight into the sta- Early proposals for effective pair potentials assumed that tistical geometry of particle packing. the full expansion ͑1.1͒ converged rapidly with respect to Yet another source of density-dependent effective pair order, and that terms beyond three-body contributions could potentials has emerged from an adaptation19,20 of the so- be neglected. This assumption was viewed as well justified called ‘‘reverse Monte Carlo’’ procedure.21,22 This approach for insulating atomic and molecular systems ͑e.g., noble starts with a given pair distribution function at fixed number gases, water, saturated hydrocarbons͒. One of the first pro- density, often determined experimentally for a real sub- posals to invoke density-dependent pair potentials in this stance, and then applies an iterative statistical algorithm to context was put forward by Sinanogˇlu;1–3 it utilized invari- extract a pair potential that would produce the given distri- ance of the thermodynamic energy as its basic criterion. bution function at the same temperature and density. Conse- Rushbrooke and Silbert4 adopted a contrasting strategy, quently, the final objective is the same as that of the Rush- which was also followed by Rowlinson and collaborators,5–7 brooke and Silbert work mentioned above.4 The presence of that the effective pair interaction should reproduce the pair many-body interactions in the initial system can be expected correlation function of the system with the full many-body to induce temperature and density effects on the outcome of potential. Neither of these approaches assumed that it was this procedure. 23–26 necessary to consider a u1* differing from u1 . A substantial group of published works has dealt In the event that the particles of the system bear perma- with consistency issues that appear to arise in statistical me- nent electrostatic charges, local polarization effects become chanics in the presence of density-dependent effective inter- important, and confer an intrinsically many-body character actions. Specifically, this involves reconciling the Ornstein– on the true interaction ⌽. For molten salts8 the obvious and Zernike expression for isothermal compressibility27 ͓i.e., the traditional way to simplify induced polarization involves structure function S(k) for k→0͔ with the Clausius virial renormalizing the bare Coulomb interactions by the inverse theorem for pressure, extended to the case of density- of Dϱ , the density-dependent high-frequency dielectric con- dependent pair interactions. This issue also forms a major stant. For some purposes it may be desirable to include Born subject of the present paper. cavity energies for each ion.9 Two opposing viewpoints are possible when using nomi- The case of highly polar, but electrostatically neutral, nally density-dependent effective interactions. On the one molecules presents an analogous situation. The modeling of hand, if interest only requires modeling a given substance water and aqueous solutions provides an obvious and signifi- over a narrow density range, it may suffice to determine an cant application area. The presence of large multipole mo- effective interaction only at the midpoint of that range, and ments on the bare molecules induces excess electrostatic mo- then subsequently to treat the resulting effective interaction ments, whose influence affects the choice of an effective pair as though it were density independent. Within this interpre- potential, and requires inclusion of effective self-energy tation the compressibility formula and the Clausius virial terms.10 theorem in its conventional form are automatically consis- Molten metals11–13 and semiconductors14,15 present an- tent. On the other hand, density dependence of effective in- other broad class of substances for which effective interac- teractions can be regarded as an intrinsic property of a tions offer a useful modeling tool. For these cases, valence model, under experimental control by varying the system and conduction band electrons dominate the underlying volume ͑closed system͒, or by varying the chemical potential physics, and their thermal excitations ͑to excited electronic ͑open system͒. This second viewpoint requires modification states͒ generally confer additional temperature dependence of the Clausius virial relation,8,23–27 and thus requires a rein- upon the effective interactions, beyond what would be terpretation of the Ornstein–Zernike compressibility for- present for insulating substances. mula. Both viewpoints can be independently developed as Assuming that nuclear motions and local order in a sub- thermodynamically consistent formalisms, but they generally stance of interest obey classical statistical mechanics, the yield distinct predictions. It is not always possible a priori to corresponding canonical partition function can be interpreted decide which approach is preferable; the choice depends on as an inner product. This tactic generates a variational crite- the application. rion for selection of a ‘‘best’’ effective pair potential, regard- The principal objective of the present paper is to exam- less of how rapidly expression ͑1.1͒ converges.16 Such a ine formal aspects of these contrasting viewpoints, and to variational approach in principle accounts for particle corre- show that each is internally consistent. The following Sec. II lations of all orders by optimally approximating the full- reviews some general statistical mechanical formulas needed potential multidimensional Boltzmann factor. Furthermore it for our analysis. Section III considers the well-known mean- can serve as the basis for developing density expansions of field approximation, perhaps the simplest example of an ef- effective interactions.17 fective interaction model. The aforementioned iso-g(2) pro- In addition to the examples just mentioned, another cess is revisited in Sec. IV, emphasizing how the two source of density-dependent pair interactions arises in a interpretations ͑viewpoints͒ lead to different thermodynamic purely theoretical context. This is the ‘‘iso-g(2)’’ process that predictions. Section V demonstrates the existence of a math-

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ematical constraint on effective interactions that, if imple- ⌽* ͉␳ process. Subsequent usage of N (r1¯rN d) for theoretical mented, would enforce equality between the thermodynamic modeling or computer simulation would not necessarily be predictions that emerge from the two alternative viewpoints. ␳ restricted to number density d , but could in principle entail The final Sec. VI presents conclusions and discusses some ␳Þ␳ any other number density d . One must also bear in related matters. We have also included an Appendix devoted mind that a series of similar optimizations could also be to rigid sphere models with density-dependent effective col- accomplished, each belonging to a discrete set or a continu- lision diameters. ␳ ous range of d’s; but for any one of these the corresponding ␳ d serves only as an index to the optimization, and implies II. GENERAL RELATIONS no density commitment for subsequent application. For that reason, we can refer to ␳ as merely a ‘‘passive’’ variable. The canonical ensemble offers the most convenient sta- Case II. This alternative treats ␳ as a true independent tistical mechanical setting in which to develop our analysis. variable for the effective interactions, on the same math- In the interest of simplicity, it will be supposed in the fol- ematical footing as r r . Consequently, for this case we lowing that N identical structureless particles reside in vol- 1¯ N write ⌽*(r r ,␳). ‘‘Experimental’’ control over this ␳ ume V, at inverse temperature ␤ϭ1/k T, and that they are N 1¯ N B can be exercised by varying either N,orV, or both. By subject to periodic boundary conditions. Given an effective contrast with the preceding Case I, ␳ now plays the role of an potential ⌽* , the corresponding canonical partition function N ‘‘active’’ variable. has the following form:28 Suppose for the moment that the passive-␳ Case I inter- ͑␤ ͒ϭ͑␭3N ͒Ϫ1 ͑␤ ͒ ZN ,V T N! Y N ,V , pretation is applicable, with just one-particle and two- ͑2.1͒ particle effective interactions, as suggested earlier in Eq. ͑␤ ͒ϭ ͵ ͵ ͓Ϫ␤⌽ ͑ ͔͒ ͑1.2͒, and which we now rewrite as follows: Y N ,V dr1 drN exp N* r1 rN , V ¯ V ¯ where ␭ is the mean thermal deBroglie wavelength. Con- ⌽*͑ ͉␳ ͒ϭ *͉͑␳ ͒ϩ͚ *͑ ͉␳ ͒ ͑ ͒ T N r1¯rN d Nu1 d u2 rij d . 2.4 nection to thermodynamics occurs through the Helmholtz iϽ j free energy F, which is obtained from ZN , In writing this expression we have recognized that all u1*’s Ϫ␤ ␤ ͒ϭ ␤ ͒ ͑ ͒ F͑N, V ln ZN͑ ,V . 2.2 are identical and position independent, and we have followed the conventional assumption that the u*’s are spherically In particular, this connection can serve as the starting point 2 symmetric. In this circumstance, the usual Clausius virial for developing various thermodynamic functions in density relation gives the pressure p, with no contribution from the power series, i.e., in cluster expansions.29 singlet interactions u* . Whether it is inferred from a time- As a result of the free translation permitted by the peri- 1 averaged virial quantity, or by applying a volume scaling to odic boundary conditions, the particle density will be strictly the canonical partition function,31 the resulting expression constant throughout the system. This will be true for all for the pressure is the following: phases that might be present, either singly or in coexistence, and whether those phases are fluid or crystalline. Particle ␤ ϭ␳Ϫ͑␤␳2 ͒ ͵ ͓ ͑ ͉␳ ͒ ͔ configurational order is revealed by the correlation functions p /6 dr12r12 du2* r12 d /dr12 g(n) of orders nϾ1, whose definition in the effective- ϫ ͑2͒ ␳ ␤͒ ͑ ͒ interaction canonical ensemble is substantially identical in g ͑r12 , , . 2.5 form to that for conventional ͑density independent͒ 27 interactions:30 Furthermore, the Ornstein–Zernike relation that connects the isothermal compressibility VnN! ͑n͒͑ ͒ϭ ͵ ͵ ͒ ͑ ͒ ץ ץg r1 rN n Ϫ ͒ drnϩ1 drN ␬ ϭϪ͑ ¯ N ͑N n !Y N V ¯ V T ln V/ p N,␤ 2.6 ϫ ͓Ϫ␤⌽*͑ ͔͒ ͑ ͒ to local density fluctuations via g(2) is also valid for Case I: exp N r1¯rN . 2.3 Periodic boundary conditions ensure that all g(n) possess ␳␬ ␤ϭ ϩ␳ ͵ ͓ ͑2͒͑ ␳ ␤͒Ϫ ͔ ͑ ͒ translational invariance; rotational invariance can only be ex- T / 1 dr12 g r12 , , 1 . 2.7 pected for homogeneous fluid phases in the large system limit. It must be stressed that the pair correlation function g(2) to As pointed out in Sec. I, two distinct meanings, or inter- be inserted in this relation must be the infinite-system limit ͑ ͒ pretations, can be attached to the density dependence of ef- function for which the large-r12 asymptote is indeed 1 , and ⌽ fective interactions N* . To keep these cases separate, we the upper integration limit is then allowed to pass to infinity. shall use correspondingly distinct notations. The logical consistency between the two expressions ͑2.5͒ Case I. This involves prior determination of an optimal and ͑2.7͒ that involve the pressure is just the same as for effective interaction, using input data for the system of inter- conventional statistical mechanical models with true density- ␳ est, at number density d . We denote that effective interac- independent pair potentials ͑e.g., the Lennard-Jones 12-6 po- ⌽* ͉␳ ␳ ͒ tion by N (r1¯rN d). The vertical bar isolates d from the tential model . configurational variables to emphasize that this quantity The active-␳ Case II provides an illuminating contrast. serves only as an index or identifier for the optimization The analog of Eq. ͑2.4͒ for this alternative is

pair interactions appearing in Eq. ͑3.1͒ can be replaced by ⌽ ͑ ␳͒ϭ ͑␳͒ϩ ͑ ␳͒ ͑ ͒ N* r1 rN , Nu1* ͚ u2* rij , . 2.8 density-dependent singlet effective interactions, ¯ iϽ j N The explicit density dependences of the effective singlet and ͑l͒͑ ͒Х ͑ ͒ ͚ u2 rij ͚ u1* i , pair interactions appearing here generate additional terms iϽ j iϭ1 8,23,24,26 contributing to the pressure in its virial-theorem form ͑3.3͒ u*͑i͒ϭϪA␳. ␳͒ 1 ͑ ץ ␳͒ ␤␳3͑ du1* u2* r12 , ␤pϭ␳ϩ␤␳2 ϩ ͵ dr ͫ -␳ This approximation can be generated as the leading perturץ d␳ 2 12 bation term in a suitably crafted systematic expansion, the ␳͒ 34,35 ͑ ץ r u* r12 , so-called ␥ expansion. Ϫ 12 2 ͬ ͑2͒͑ ␳ ␤͒ ͑ ͒ g r12 , , . 2.9 ץ ␳ 3 r12 In view of the fact that u1* is independent of position within the system, an alternative ͑but equivalent͒ formulation The existence of the new terms reflects the additional iso- of the mean field approximation is to replace Eq. ͑3.3͒ by the thermal reversible work that must be performed, while following: changing volume V, in order to modify the interactions among any fixed set of particles in the interior of the system. ͑l͒͑ ͒Х ͑l͒ ͑ ͒ ͑ ␳͒ ͚ u2 rij ͚ u2 * i, j , Unless the new Case II active contributions to the iϽ j iϽ j pressure accidentally sum to zero over the density range of ͑ ͒ ͑ ͒ 3.4 interest see Sec. V below , the isothermal compressibility ͑ ͒ ␬ ͑ ␳͒ u l *͑i, j͒ϭ͓u*͑i͒ϩu*͑ j͔͒/͑NϪ1͒ϭϪ2A␳/͑NϪ1͒. T will be modified from its Case I passive counterpart. 2 1 1 However it is important to realize that correlation functions In other words, the collection of long-range attractive inter- (n) ͑ ͒ g , Eq. 2.3 , are identical for the two classes of effective actions becomes replaced by position-independent, very ⌽* ͉␳ ϭ␳ ⌽* ␳ interactions N (r1¯rN d ) and N (r1¯rN , ), pro- weak pair interactions. For either formulation, the effective vided that the configuration-variable dependences of these interaction for the entire N-body system is two alternatives are the same. In other words, short- and long-range particle order in the system under fixed N, V, ␤ ⌽ ϭ ͑s͒͑ ͒Ϫ ␳ ͑ ͒ N* ͚ u2 rij NA . 3.5 conditions cannot depend on whether the effective interac- iϽ j tions would, or would not, change if V were to be changed. The implication is that the Ornstein–Zernike relation ͑2.7͒ The standard interpretation of the mean field approxima- ␬ tion conforms to Case II of the preceding Sec. II. That is, ␳ generally will not yield the correct T for Case II. becomes a true ‘‘active’’ variable in the N-body potential function, which in the notational convention introduced earlier would be written ⌽ (r r ,␳). Consequently the pres- III. MEAN FIELD APPROXIMATION N 1¯ N sure should be evaluated from the extended form of the virial One of the simplest, and most familiar, examples of ef- expression, Eq. ͑2.9͒, with the familiar result equal to that for (s) fective interactions emerges from the mean field approxima- the short-range-interaction system (p ) supplemented by tion. Within the domain of classical statistical mechanics for the mean field correction continuum systems, this approximation usually appears in ͑ ͒ p͑␳,␤͒ϭp s ͑␳,␤͒ϪA␳2. ͑3.6͒ connection with a strategy to separate the full N-body potential into short-range ͑s͒ and long-range ͑l͒ components. Such Under the assumption that the short-range interactions are a separation implicitly underlies the venerable van der Waals those for rigid spheres, this is just the result produced by equation of state,32 as well as the more modern revision cre- Longuet-Higgins and Widom in their extension of the van ated by Longuet-Higgins and Widom.33 der Waals equation.33 Suppose that the initial N-body potential consisted only The passive-␳ Case I interpretation ͑that is unnatural for of pair terms, and let the separated form be written the mean field approximation͒ would establish the effective interactions, whether singlet or pair, at a chosen distin- ⌽ ͑ ͒ϭ͚ ͓ ͑s͒͑ ͒ϩ ͑l͒͑ ͔͒ ͑ ͒ guished density ␳ , and then insist on maintaining exactly N r1¯rN u2 rij u2 rij . 3.1 d iϽ j the same effective interactions at any subsequently used sys- (l) tem number density ␳. In the notational convention defined Here it will be supposed that u2 comprises of primarily attractive interactions, and is bounded and integrable, earlier, one would write ϱ ␲ ͵ 2 ͑l͒͑ ͒ ϭϪ ͑ ͒ ⌽*͑r r ͉␳ ͒ϭ͚ u͑s͒͑r ͒ϪNA␳ . ͑3.7͒ 4 r u2 r dr 2A. 3.2 N 1¯ N d 2 ij d 0 iϽ j The basic idea of the mean field approximation is to postu- The last term on the right-hand side of this expression (l) late that u2 is pointwise sufficiently weak so as to have merely amounts to a constant downward shift of all configu- negligible effect on structure, but sufficiently long ranged so rational energies, and can have no influence on the pressure, that it encompasses a macroscopic domain of neighbor par- which is just that of the short-range-interaction system, p(s). ͑ ␬(s) ticles whose number therefore scales with the overall den- The corresponding isothermal compressibility T would be sity ␳͒. If that is the case, then the entire set of long-range given by the Ornstein–Zernike formula, Eq. ͑2.7͒, using the

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͒Ͻ ͑ ͒ pair correlation function for the short-range-interaction sys- G͑kmin 0, 4.3 tem. Note that the relation between the Case I and Case II ͑ ͒ ␳ isothermal compressibilities is as follows: then restriction 4.2 would be violated at kmin if were to exceed the upper limit 1/␬ ϭ1/␬͑s͒Ϫ2A␳2. ͑3.8͒ T T ␳ ϭϪ ͒ ͑ ͒ * 1/G͑kmin . 4.4 In spite of the fact that the pair correlation function is the same regardless of which of Case I or Case II is invoked, it An important group of applications leading to an upper ␳ ͑ ͒ ϭ is directly relevant for evaluating isothermal compressibility density limit * of the form 4.4 have kmin 0. In particular only for the former; Eq. ͑3.8͒ must be used for the latter. this occurs for the rigid sphere model where the invariant g(2) is that for low or vanishing density.18 This implies that the passive-␳ Case I isothermal compressibility continuously IV. ISO-g„2… PROCESSES approaches zero as ␳ approaches ␳* from below, i.e., the iso- The introduction and examination of ‘‘iso-g(2) g(2) system becomes arbitrarily ‘‘stiff’’ at ␳*. Evidently this processes’’ 18,36 has been motivated by the need to under- indicates that at ␳* the many body system has run out of stand more deeply and in technical detail the connections particle arrangements that would be required to maintain the between interparticle interactions and the equilibrium statis- invariant pair correlation pattern under further density in- tical correlations that they induce. As the name suggests, crease. ϭ these processes seek to identify changes in pair potentials In those instances where kmin 0, we can expand G(k) that manage exactly to preserve pair correlation functions as around the origin of k space as follows: invariants as the system number density varies isothermally. ͑ ͒ϭ ϩ 2ϩ ͑ 4͒ Put another way, the objective is to have density change and G k G0 G2k O k , pair potential change exert precisely canceling effects on the ͑4.5͒ (2) Ͻ Ͼ pair correlation function g (r). No singlet effective inter- G0 0, G2 0. actions u* are involved. Detailed studies have thus far been 1 This can then be related to the small-k behavior of the Fou- carried out for the rigid rod, disk, and sphere systems,18 as rier transform of the direct correlation function, C(k),37 well as the square-well system in three dimensions.36 One straightforward example of an iso-g(2) process would be to G͑k͒ Ϫ1 C͑k͒ϭ ϭ ϩO͑k4͒. ͑4.6͒ maintain the exact zero-density form of the pair correlation 1ϩ␳G͑k͒ ␦␳ϩ͑G /G2͒k2 function ͑i.e., the Boltzmann factor for the initial pair poten- 2 0 tial͒ as the density increases from zero, and to inquire Here we have set the density deficit below ␳* equal to ␦␳, whether this condition can only be applied up to some maxi- ␳ϭ␳ Ϫ␦␳ ͑ ͒ mum density ␳*Ͼ0. In the following, we will assume for * . 4.7 convenience that the pair correlation function to be held con- When ␦␳ is small, only the leading term on the right-hand stant has emerged from a system with pair interactions of side of Eq. ͑4.6͒ needs to be considered, and this term pro- finite range. duces a large and narrow peak at the origin. An inverse Fou- (2) The pair potentials created by an iso-g process are not rier transform applied to that leading term then yields the effective interactions in the usual sense of having emerged r-space form of the direct correlation function from an approximation method. Nevertheless, they are density dependent, and the resulting many-body systems can be c͑r ,␳͒Х͑2␲͒Ϫ3 ͵ dk exp͑Ϫik r ͒ interpreted according to either the passive-␳ Case I or the 12 • 12 active-␳ Case II protocol. The former involves a sequence of ϫ͕Ϫ ͓␦␳ϩ͑ 2͒ 2͔͖ ͉␳ 1/ G2 /G0 k model systems with pair interactions u2*(rij d), each cre- (2) ␳ 2 ated specifically to reproduce the target g at a given d , ϭϪ͑G /4␲G r ͒exp͓Ϫ͉G ͉͑␦␳/G ͒1/2r ͔. ␳ 0 2 12 0 2 12 but only at that d . The latter envisions a single model sys- ␳ ͑4.8͒ tem with pair interactions u2*(rij , ) that continuously change form as the system volume varies continuously. This Yukawa function continuously loses its exponential The structure factor is defined by damping upon approach to the upper density limit ␳*. The 4 ͑ ͒ S͑k͒ϭ1ϩ␳G͑k͒, O(k ) terms neglected in Eq. 4.8 can be expected to add ͑ ͒ short-range corrections to this long-range Yukawa result. 4.1 38 ͑ ͒ The Percus–Yevick ͑PY͒ and the hypernetted chain G͑k͒ϭ␳ ͵ dr exp͑ik r ͓͒g 2 ͑r ͒Ϫ1͔. 12 • 12 12 ͑HNC͒39 approximations supply connections between the di- In an iso-g(2) setting, S(k) becomes strictly a linear function rect correlation function and the pair interactions that are ␳ → ␳␬ ␤ present to produce it. Both approximations agree that in the of , as does its k 0 limit, the Case I quantity T / , Eq. ͑2.7͒. Under thermal equilibrium conditions, the structure asymptotic large-distance regime, factor obeys the restriction ͑ ␳͒ϳϪ␤ ͑ ͒ ͑ ͒ c r12 , u2* r12 . 4.9 ͑ ͒у ͑ ͒ S k 0. 4.2 ͑ ͒ Equation 4.8 above then implies that u2* will adopt a repel- Consequently, if G(k) has an absolute minimum that is nega- ling Yukawa form at large separations, which in the ␳→␳* tive at some kmin , limit becomes Coulombic,

*͑ ͉␳͒ *͑ ␳͒ϳ͑ 2 ␲␤ ͒ *͑␳͒ϭ ␳ϩ ␳2ϩ ␳3ϩ ͑ ͒ u2 r12 ,u2 r12 , G0/4 G2r12 u1 u1,1 u1,2 u1,3 ¯ , 5.3 ϫexp͓Ϫ͉G ͉͑␦␳/G ͒1/2r ͔. 0 2 12 *͑ ␳͒ϭ ͑ ͒ϩ ͑ ͒␳ϩ ͑ ͒␳2ϩ u2 r12 , u2,0 r12 u2,1 r12 u2,2 r12 ¯ . ͑4.10͒ ͑5.4͒ With either the Case I or Case II interpretation, this long- 0 range effective pair potential has a dominating influence on Notice that the first of these should have no constant (␳ ) the system pressure. For the passive-␳ Case I, insertion of term. The leading term of the second simply represents the form ͑4.10͒ into the conventional virial expression ͑2.5͒ gen- ‘‘bare’’ pair interaction of two particles in isolation. erates a pressure contribution that has the character of a Under the conventional circumstances, where ‘‘true’’ in- simple-pole divergence: teractions are involved that are short ranged and density independent, the pair correlation function can also be devel- 2 ␤p͑␳,␤͒ϳ͑␳*͒ /2␦␳ ͑Case I͒. ͑4.11͒ oped in a density power series using the cluster expansion 39,40 This simple pole contribution is also present in the active-␳ technique. The resulting expansion can be represented in Case II interpretation, of course, but it is itself dominated by the following manner: -␳ conץ/*uץ a double-pole contribution that arises from the 2 ͑2͒͑ ␳͒ϭ ͓Ϫ␤ ͑ ͔͕͒ ϩ ͑2,1͒͑ ͒␳ tribution appearing in the extended virial relation ͑2.9͒: g r12 , exp u2 r12 1 C r12 ͑ ͒ 3 2 ϩ 2,2 ͑ ͒␳2ϩ ͖ ͑ ͒ ␤p͑␳,␤͒ϳ͑␳*͒ /2͑␦␳͒ ͑Case II͒. ͑4.12͒ C r12 ¯ . 5.5 This contrasting pair of pressure results vividly illustrates the (2,j) ␳ Here the C (r12) would be independent, and u2 would different thermodynamic implications that can emerge from ͑ ͒ be identified with u2,0 in Eq. 5.4 above. But when the alternative interpretations. We remind the reader that Eq. ␳-dependent effective interactions are under consideration, ͑ ͒ 4.11 expresses the pressure behavior of a sequence of sys- and the cluster sums are themselves constructed using those ␳ ␳ tems indexed by and each examined at its indexing , effective interactions, a strict power series for g(2) would ͑ ͒ ␳ whereas Eq. 4.12 refers to a single system for which is a require expanding each C(2,j), and then collecting all terms fundamental variable of the potential energy function. In in Eq. ͑5.5͒ with a common order in ␳. Once this is done, and ␳ view of this distinction, note that the vanishing at * of the all power series are inserted in Eq. ͑5.1͒ above, each ␳ order sequence of isothermal compressibilities for Case I cannot generates its own constraint to enforce Case I–Case II con- ͑ ͒ correctly be inferred from asymptote 4.11 by differentiation cordancy. All constraints of orders up to ␳n extracted from ͓ with respect to density which would erroneously suggest Eq. ͑5.1͒, when enforced, assure that Case I and Case II ␦␳ 2͔ ͑ ͒ proportionality to ( ) ; in fact it follows from Eq. 4.1 pressures will agree through order ␳nϩ2. ␦␳ ␳ above that the proportionality is linear in at *: The leading-order constraint ͓O(␳0)͔ from Eq. ͑5.1͒ is ␳␬ ␤ϭ͉ ͉␦␳ ͒ ͑ ͒ trivial: T / G0 ͑Case I . 4.13 ϭ ͑ ͒ V. CONCORDANCY CONDITION u1,1 0, 5.6

Although Case I and Case II interpretations generally and simply shows that the series ͑5.3͒ begins at quadratic will produce distinct pressure predictions, special circum- order. The next order ͓O(␳)͔ leads straightforwardly to the stances exist under which those distinctions will vanish. condition: Identifying those circumstances proceeds from the requirement that the conventional and the extended forms of the 1 virial expression for pressure, Eqs. ͑2.5͒ and ͑2.9͒, respec- u ϭ ͵ dr u ͑r ͒exp͓Ϫ␤u ͑r ͔͒. ͑5.7͒ 1,2 4 12 2,1 12 2,0 12 tively, produce the same result. This is equivalent to demand- ing that the Case II interactions satisfy the condition This last relation can be viewed as a specification of the ␳͒ ͑ ץ ␳͒ ␳͑ du1* u2* r12 , constant u1,2 once that u2,1(r12) has been selected. Alterna- ϭ ͵ dr g͑2͒͑r ,␳͒. ͑5.1͒ ␳ 12 tively, if interest lies in the subset of effective potentials thatץ d␳ 2 12 have only short-range pair terms ͑and no singlet effective As pointed out in Sec. III above, the singlet effective inter- interactions͒, then Eq. ͑5.7͒ reduces to actions u1* could formally be absorbed into the definition of u2* . Thus the concordancy condition for the two interpreta- ϭ ͵ ͑ ͒ ͓Ϫ␤ ͑ ͔͒ ͑ ͒ tions would reduce to the following: 0 dr12 u2,1 r12 exp u2,0 r12 , 5.8 u*͑r ,␳͒ץ ϭ ͵ 2 12 ͑2͒͑ ␳͒ ͑ ͒ -␳ g r12 , . 5.2 which requires that u2,1 be orthogonal to the bare-pairץ dr12 0 potential Boltzmann factor. For the purposes of this section, however, Eq. ͑5.1͒ offers the Proceeding to the next order ͓O(␳2)͔, it is necessary to more convenient starting point. account for contributions to g(2) that ͑a͒ arise from C(2,1) ϫ ͑ ͒ The singlet and pair effective interactions can both be (r12), and b arise from the already-constrained effective developed into density power series pair function u2,1(r12). One finds

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1 corporation of the effects of real many-body interactions can u ϭ ͵ dr exp͓Ϫ␤u ͑r ͔͒ͭ ͓2u ͑r ͒ 1,3 6 12 2,0 12 2,2 12 often be well approximated by adroit choice of ‘‘effective’’ singlet and pair potentials, which generally can be expected Ϫ␤͑ ͑ ͒͒2͔ϩ ͑ ͒ ͵ ͑ ͒ ͑ ͒ͮ to display both temperature and density dependence. u2,1 r12 u2,1 r12 dr3 f r13 f r32 , The present study has been concerned with the conse- ͑5.9͒ quences of density dependence in effective potentials. Two distinct viewpoints, or interpretations, have been identified. where f is the Mayer cluster bond for the bare pair interaction The first ͑Case I͒ involves determination of effective interac- ͑ ͒ϭ ͓Ϫ␤ ͑ ͔͒Ϫ ͑ ͒ ␳ f rij exp u2,0 rij 1. 5.10 tions at a set of one or more fixed densities d , and then ␳ Only the two quantities u and u (r ) make their first treating each such determination as though d were simply a 1,3 2,2 12 ␳ appearance in Eq. ͑5.9͒, and are thus constrained for the first nonvarying parameter; the number density in a theoretical time. or simulational application of interest is then permitted to ␳ It is clear that this systematic process could be continued vary from d while treating the interaction functions as in- to higher orders, though the outcomes rise rapidly in com- variant quantities. The second ͑Case II͒ treats the density plexity with increasing order. At O(␳n), one has first appear- dependence of effective interactions as a legitimate variable ances of u1,n and u2,nϪ1 . If in fact the various density ex- in the Hamiltonian of the many-particle system, on a math- pansions ͑5.3͒–͑5.5͒ are convergent, the resulting ematical footing equivalent to that of particle configurational constrained effective interactions u1* and u2* as represented coordinates. These separate viewpoints generally produce ͑ ͒ ͑ ͒ by density series 5.3 and 5.4 should produce concordant distinct thermodynamic functions. In particular, the virial results for Case I and Case II pressures, at least up to the pressure appears in two different forms, Eqs. ͑2.5͒ and ͑2.9͒ density of a phase transition. It should also be pointed out ␳ for Case I and for Case II, respectively. But in spite of this that if the effective pair potential u2*(r12 , ) were a given quantity, then Eq. ͑5.1͒ could be used to determine that sin- distinction, both interpretations enjoy internal consistency. ␳ Which of Case I or Case II is to be the preferred interpreta- glet function u1*( ) which would produce Case I–Case II pressure agreement, even across phase transitions. tion in any given application cannot be decided a priori, but We close this Sec. V with a brief examination of the must rest upon physical details of the specific materials and result obtained by applying the concordancy constraint to the phenomena under consideration. iso-g(2) process discussed in Sec. IV. In this circumstance, Section V explores the possibility that imposition of suit- the usual compressibility expression, Eq. ͑2.7͒ applies to able constraints on the space of effective interactions, at least both Case I and Case II by construction. It can be written in at the singlet plus pair level, might cause the virial pressures the form for the Case I and Case II interpretations to become equal functions of the density. This special circumstance indeed pץ ␤ ϭ͓ ϩ␳ ͑ ͔͒Ϫ1 ͑ ͒ seems to be realizable, and in the event that only short-range ␳ͪ 1 G 0 , 5.11ץ ͩ ␤ interactions are present, the constraints can be stepwise implemented in increasing integer orders in density. The situ- where G(0) is the excess pair correlation function spatial ͓ ͑ ͔͒ ation is less straightforward if the particles comprised in the integral Eq. 4.1 . In view of the fact that G(0) is density ͑ (2) ͑ ͒ system bear electrostatic charges e.g., fused salts, electro- independent for iso-g processes, the preceding Eq. 5.11 ͒ can immediately be integrated to yield lytic solutions , because density expansions are inappropri- ate. This aspect of the subject deserves an in-depth examination in a future study. ␤pϭ͓G͑0͔͒Ϫ1 ln͓1ϩ␳G͑0͔͒. ͑5.12͒ Although the present work has concentrated on density In other words, the constrained version of the iso-g(2) pro- as a variable, we recognize that parallel issues exist concern- cess possesses an elementary explicit pressure equation of ing temperature dependence that could arise from any proce- state over its full density range. Finally, it should be noted dure used to assign effective interactions. The analogs of our that in the limit of vanishing G(0), the expression shown in Case I and Case II interpretations would hinge on whether ͑ ͒ Eq. 5.12 reduces identically to the ideal gas equation of the temperature in the effective interactions were to be state. ␳ treated simply as an indexing parameter Td similar to d above ͑from which the operative temperature could depart͒, or whether T were to be regarded as a ‘‘true’’ variable of the VI. CONCLUSIONS AND DISCUSSION effective interactions, controllable through the total energy in The desire to construct and analyze realistic models for a microcanonical ensemble. It is clear that the respective technologically important substances is hindered by the com- predictions of the heat capacities generally would differ, al- plexity of the constituent particles and their interactions. For though suitably constraining the space of effective interac- reasons of practicality, it is desirable to limit the mathemati- tions should once again produce concordancy. This is an- cal representation of interactions to low order in particle other feature of the general subject whose full analysis must number, for example to singlet and pair functions only. In- await a later examination.

ACKNOWLEDGMENTS the virial integral is concentrated at pair contact. Conse- quently we can make the following formal replacement in- ͑ ͒ One of the authors S.T. gratefully acknowledges sup- volving the Dirac delta function: port by the Engineering Research Program of the Office of u*͑r ,␳͒␤ץ Basic Energy Sciences at the Department of Energy, and by ͩ 2 12 ͪ ͑2͒͑ ␳͒ , ␳ g r12ץ -the Petroleum Research Fund as administered by the Ameri can Chemical Society. da͑␳͒ →g͑2͓͒a͑␳͔͒ͩ ͪ ␦͓r Ϫa͑␳͔͒, ͑A8͒ d␳ 12 APPENDIX: EFFECTIVE INTGERACTIONS FOR RIGID which is analogous to the replacement that converts the usual SPHERES virial expression Eq. ͑2.5͒ to an expression involving just the ͑ ͒ 43 The purpose of this Appendix is to record a few results contact pair correlation function Eq. A3 above. Equation ͑ ͒ ͑ ͒ for the specific model of rigid spheres with density- A8 converts Eq. 2.9 to the following: ␳ dependent collision diameters a( ). The conventional rigid d ln a3͑␳͒ sphere model, with a fixed collision diameter, provides the ␤pϭ␳ͭ 1ϩ͑2␲/3͒␳a3͑␳͒g͑2͓͒a͑␳͒,␳͔ͫ1ϩ ͬͮ d ln ␳ natural backdrop. The pressure for that fixed-a model can be expressed in the following form: ϵ␳F͓␳a3͑␳͔͒. ͑A9͒ ␤ ϭ␳ ͑␳ 3͒ ͑ ͒ p f a , A1 From this expression it follows that the Case II compressibil- where f possesses a convergent power series that generates ity is given by the virial series for the pressure 2 ␳␬ ␤ϭͭ ͓␳ 3͑␳͔͒ ͑ ͒ϭ ϩ ϩ ϩ ͑ ͒ T / F a f x 1 f 1x f 2x ¯ . A2 This function is related to the contact value of the rigid d ln ␳a3͑␳͒ Ϫ1 sphere pair correlation function as follows:41 ϩ␳a3͑␳͒FЈ͓␳a3͑␳͔͒ ͮ . ͑A10͒ d ln ␳ ␳ 3͒ϭ ϩ ␲ ͒␳ 3 ͑2͒ ϭ ␳ 3͒ ͑ ͒ f ͑ a 1 ͑2 /3 a g ͑r12 a, a . A3 The isothermal compressibility can simply be expressed in 1 O. Sinanogˇlu, Chem. Phys. Lett. 1, 340 ͑1967͒. 2 terms of f, O. Sinanogˇlu, Adv. Chem. Phys. 12, 283 ͑1967͒. 3 T. Halciogˇlu and O. Sinanogˇlu, J. Chem. Phys. 49,996͑1968͒. ␳␬ ␤ϭ ␳ 3͒ϩ␳ 3 ␳ 3͒ Ϫ1 ͑ ͒ 4 G. S. Rushbrooke and M. Silbert, Mol. Phys. 12,505͑1967͒. T / ͓ f ͑ a a f Ј͑ a ͔ . A4 5 J. S. Rowlinson, Mol. Phys. 12, 513 ͑1967͒. The first-order freezing transition from isotropic fluid to 6 G. Casanova, R. J. Dulla, D. A. Jonah, J. S. Rowlinson, and G. Saville, face-centered-cubic crystal occurs over the density range:42 Mol. Phys. 18, 589 ͑1970͒. 7 R. J. Dulla, J. S. Rowlinson, and W. R. Smith, Mol. Phys. 21,299͑1971͒. 0.943Ͻ␳a3Ͻ1.014, ͑A5͒ 8 F. H. Stillinger, in Molten Salt Chemistry, edited by M. Blander ͑Interscience-Wiley, New York, 1964͒, p. 1. Note that a factor ␤ is missing and over that coexistence range f must be inversely propor- from the second term on the right-hand side of Eq. ͑47͒,p.36. tional to ␳ so that the pressure is constant. 9 Reference 8, p. 21. 10 H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma, J. Phys. Chem. 91, The Case I interpretation simply utilizes expressions 6269 ͑1987͒. ͑A1͒–͑A5͒ with insertion of the density-dependent collision 11 W. A. Harrison, Pseudopotentials in the Theory of Metals ͑W. A. Ben- diameter a(␳). If this kind of effective-diameter rigid sphere jamin, New York, 1966͒. 12 model is to be realistic in its description of the freezing tran- P. Egelstaff, An Introduction to the Liquid State ͑Academic, New York, ␳ 1967͒. sition, a( ) would have to be a constant across the coexist- 13 D. Stroud and N. W. Ashcroft, Phys. Rev. B 5, 371 ͑1972͒. ence range in order for the pressure to remain constant. But 14 A. Arnold, N. Mauser, and J. Hafner, J. Phys.: Condens. Matter 1,965 within either homogeneous phase interval of density, this ef- ͑1989͒. 15 W. Jank and J. Hafner, Phys. Rev. B 41, 1497 ͑1990͒. fective diameter could have nontrivial density variation. The 16 ͑ ͒ ␳ F. H. Stillinger, J. Phys. Chem. 74, 3677 1970 . only requirements that must be obeyed by a( ) are that it 17 F. H. Stillinger, J. Chem. Phys. 57, 1780 ͑1972͒. have a convergent density expansion, 18 F. H. Stillinger, S. Torquato, J. M. Eroles, and T. M. Truskett, J. Phys. Chem. 105, 6592 ͑2001͒. ͑␳͒ϭ ϩ ␳ϩ ␳2ϩ ͑ ͒ 19 a a0 a1 a2 ¯ , A6 A. P. Lyubartsev and A. Laaksonen, Phys. Rev. E 52, 3730 ͑1995͒. 20 A. P. Lyubartsev and A. Laaksonen, Comput. Phys. Commun. 121–122, to permit a pressure virial series to exist, and that the obvious 57 ͑1999͒. close-packing condition be satisfied, 21 R. L. McGreevy and L. Pusztai, Mol. Simul. 1, 359 ͑1988͒. 22 D. A. Keen and R. L. McGreevy, Nature ͑London͒ 344, 423 ͑1990͒. 3 1/2 ␳͓a͑␳͔͒ р2 . ͑A7͒ 23 P. Ascarelli and R. J. Harrison, Phys. Rev. Lett. 22,385͑1969͒. 24 W. Jones, J. Phys. C 6, 2833 ͑1973͒. Analysis of the Case II interpretation for the effective- 25 N. G. Almarza, E. Lomba, G. Ruiz, and C. F. Tejero, Phys. Rev. Lett. 86, diameter rigid sphere model begins with the extended form 2038 ͑2001͒. of the pressure virial expression, Eq. ͑2.9͒. The new contri- 26 J. Hafner, From Hamiltonians to Phase Diagrams ͑Springer-Verlag, New York, 1987͒, p. 342. bution to be considered involves the density derivative of the 27 ͑ ͒ ␳ L. S. Ornstein and F. Zernike, Phys. Z. 27, 761 1926 . effective pair interaction u2*(r12 , ). Because the singular 28 T. L. Hill, Statistical Mechanics ͑McGraw-Hill, New York, 1956͒,p.91, rigid sphere interaction is involved, the entire contribution to Eq. ͑16.67͒.

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29 J. E. Mayer and M. G. Mayer, Statistical Mechanics ͑Wiley, New York, 37 J. P. Hansen and I. R. McDonald, Theory of Simple Liquids ͑Academic, 1940͒,Chap.13. New York, 1976͒, p. 101. 30 Reference 27, Chap. 6. 38 J. K. Percus and G. J. Yevick, Phys. Rev. 110,1͑1958͒. 31 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of 39 J. M. J. Van Leeuwen, J. Groeneveld, and J. De Boer, Physica ͑Amster- Gases and Liquids ͑Wiley, New York, 1954͒, p. 134. dam͒ 25, 792 ͑1959͒. 32 Reference 27, Chap. 12. 40 S. A. Rice and P. Gray, The Statistical Mechanics of Simple Liquids ͑In- 33 H. C. Longuet-Higgins and B. Widom, Mol. Phys. 8,549͑1964͒. ͒ 34 P. C. Hemmer, J. Math. Phys. 5,75͑1964͒. terscience, New York, 1965 , Chap. 2. 41 35 J. L. Lebowitz, G. Stell, and S. Baer, J. Math. Phys. 6, 1282 ͑1965͒. Reference 28, p. 216. 36 H. Sakai, S. Torquato, and F. H. Stillinger, J. Chem. Phys. 117, 297 42 W. G. Hoover and F. H. Ree, J. Chem. Phys. 49, 3609 ͑1968͒. ͑2002͒, following paper. 43 Reference 28, p. 214.

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