The inverse problem for simple classical liquids: a density functional approach

A recently introduced algorithm for solving the inverse problem for simple classical fluids (i.e. the deduction of the interatomic interaction from structural data), which is based on the fundamental-measure free-energy density functional for hard spheres, is analysed in comparison with other methods. In a benchmark test for the Lennard-Jones system near the triple point, it is comparable with about ten simulations in the iterative predictor–corrector scheme proposed some years ago by Levesque, Weis, and Reatto. The method is used to extract the effective pair potential of Kr from very accurate experimental neutron scattering structure factor data. The inverse problem, i.e. the deduction of the interatomic interaction from structural data obtained from scattering experiments, has been the object of much attention [1–15] in the physics of liquids. The determination of the interatomic interaction in condensed matter is of fundamental importance. Although many-body forces are always present in condensed systems, even in monatomic systems, an effective state-dependent two-body interaction (a pair potential, φ(r)) is still an important and useful quantity. The insensitivity in a dense fluid of the pair radial distribution function g(r) to the exact shape of the pair potential φ(r) plays a major role in the solution of the direct problem, i.e. φ(r) → g(r). As a result, the solution of the inverse problem, i.e. g(r)→ φ(r), requires a highly accurate and non-perturbative theory for the fluid structure. A non-perturbative theory should be equally applicable to quite disparate potentials (e.g., the hard-sphere and Coulomb ones), and the quest for such a theory has led to many developments. The simulation of model fluids provides the testing ground for theoretical methods and has played a key role in addressing both the ‘direct’ and ‘inverse’ problems. The first non-perturbative accurate theory of fluid structure, the modified hypernettedchain (MHNC) theory, was based on the ansatz of universality of the bridge functions [16]. Using the bridge functions for hard spheres, it proved accurate for the ‘direct’ problem, and motivated several other integral equation approximations [17, 18]. It also led to the first successful results for the solution of the inverse problem [7]. Yet, these ‘inverse’ results were not accurate enough in certain density–temperature regions of the fluid, and for certain types of liquids. The predictor–corrector method of Levesque, Weis, and Reatto (LWR) [11, 12], which is based on the MHNC scheme and on simulations, overcomes these drawbacks (at least in the one-component case): it can be applied to any liquid and gives reliable results even near the triple point. Applications to realistic systems (e.g. liquid Ga [13]) demonstrated the power of this approach. One should bear in mind, however, that in the iterative predictor–corrector algorithm each ‘corrector’ step is represented by a full computer simulation for a fluid with a given ‘predictor’ pair potential, and about ten 0953-8984/97/070089+10$19.50 c © 1997 IOP Publishing Ltd L89 L90 Letter to the Editor such iteration steps are required near the triple point. The initial guess is provided by the MHNC equation with the hard-sphere bridge functions. The convergence of the iterative predictor–corrector procedure is due to the general high accuracy of the approximation of universality of the bridge functions. The generalization of the predictor–corrector formalism to the binary case is straightforward, but its realization [14] is by no means trivial and even fails in some cases: for small concentrations of the minority component the statistical errors in the computer simulation (‘corrector’) steps accumulate and the simulation does not lead to satisfactory results. A recent development in density functional theory is the fundamental-measure functional (FMF) for the free energy of hard spheres and hard convex bodies [19–24]. The FMF, which is based on geometrical rather than van-der-Waals-like considerations, brings together the Percus–Yevick [25] and scaled-particle [26] theories. It is the first free-energy functional for hard spheres with adequate properties of crossover between different effective dimensionalities of the fluid, which result from spatial confinement of the fluid by external potentials. Due to its accuracy, the FMF enabled the extension of the approximation of universality of the bridge functions to that of universality of the bridge functional [22]. This provides an accurate non-perturbative theory for the static structure of fluids [22, 15] which also enables one to overcome [15] the problems encountered by the predictor– corrector algorithm for the inverse problem for mixtures. As we recently demonstrated [15], the interaction potentials extracted from the simulation pair correlation data are accurate to such an extent that this method can become a more efficient alternative to the use of simulations in the inversion problem. In this letter we highlight the special features of this method [22, 15] in comparison with other possible schemes for solving the inverse problem. We further test the density functional method by comparison with the LWR-simulation predictor–corrector results [11, 12] for the Lennard-Jones (LJ 12–6) system near the triple point. We find that the potential obtained by our method is comparable to that obtained by about ten simulation predictor– corrector steps. We apply this method to very accurate scattering experiments on Kr [27], and extract the effective pair potentials from the experimental structure factor data. The exact diagrammatic MHNC equation has the following form: βφ(r)+ b(r) = g(r)− 1− c(r)− ln g(r) (1) where c(r) is the usual Ornstein–Zernike direct correlation function given by h(r)− c(r) = ρ0 ∫ dr′ c(|r − r′|)h(r ′) (2) where h(r) = g(r)−1, ρ0 is the bulk number density, and where hereafter we denote 1/kBT as β. In terms of the Fourier transforms h̃(k) and c̃(k), and the structure factor S(k), it reads S(k) = 1+ ρ0h̃(k) = 1 1− ρ0c̃(k) . (3) When the bridge function is ignored, b(r) = 0, we have the hypernetted-chain (HNC) approximation φHNC(r) = 1 β (g(r)− 1− c(r)− ln g(r)). (4) Given a pair correlation function g(r), the right-hand side of this equation is fully specified (with c(r) given by the Ornstein–Zernike relation (2)), so we can regard it as a functional of g(r): φHNC(r) = 8HNC[g(r)]. (5) Letter to the Editor L91 We thus have the following exact relation: φ(r) = 8HNC[g(r)]− 1 β b(r) (6) and the strategy for the solution of the inverse problem depends on how the information about the bridge function, b(r), is given. Like in the diagrammatic analysis (see the discussion and the list of references in [16]), we consider the following two possibilities, (a) and (b). (a) The bridge function is given as a functional of the pair potential: