Variational Monte Carlo derivatives

When a mathematical model and the variation which affects it6 are both simple in form, it is sometimes possible to derive analytically any desired information relating to the behavior of the system. When this is the case the Monte Carlo method may offer little or no advantage. However, it many problems it is impractical to obtain the desired results entirely by analytical methods. It is in this situation that the Monte Carlo method becomes a most valuable tool. 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

Ceperley and Bernu [64] introduced a method that addresses these problems. It is a generalization of the standard variational method applied to the basis set exp(-f ) where is a basis of trial functions 1 s a < m. One performs a single-diffusion Monte Carlo calculation with a guiding function that allows the diffusion to access all desired states, generating a trajectory R(t), where t is imaginary time. With this trajectory one determines matrix elements between basis functions = ( a( i) I /3(fi + t)) and their time derivatives. Using... 

The second fbnn of this themm, (4.69). was obtained by Widom, B. J. chem. Phys. 39,2808 (1963), and Jackson, J. L. and Klein, L. S. Phys. Fluids 7,228 (1964). A special case, testricted to a system of hard spheres, and in a form appropriate for Monte Carlo simulation, was first put forward by Byckling, E. nysica 27, 1030 (1961). More extensive computer calculations for homogeneous fluids have been made by Adams, D. J. Mol. Phys. 28, 1241 (1974) Romano, S. and Singer, K. Mol. Phys. 37,1765 (1979) and Powles, J. G. Mol. Phys. 41, 715 (1980). The first proof of the constancy of A of (4.69) in an inhomogeneous system is in Widom, B. J. slat. Phys. 19, 563 (1978). The first form of the theorem, (4.68), was obtained more recently by de Oliveira, M. J., personal communication (1979) Robledo, A. and Varea, C. J. star. Phys., 26,513 (1981) have derived the second form by a variational principle from the grand potential of a non-uniform fluid as a functional of the density. Snider, N. S. J. chem. Phys. 55, 1481 (1971). 

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