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

Publisher Website
Google Scholar
Abstract
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, '.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 '.r/ plays a major role in the solution of the direct problem, i.e. '.r/! g.r/ .A s a result, the solution of the inverse problem, i.e. g.r/!'.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 hypernetted- chain (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