The Hypernetted Chain Equations for Periodic Systems

Martin Panholzer

doi:10.1007/s10909-017-1771-5

Abstract

Starting from the general inhomogeneous Fermi hypernetted chain equations, the equations for periodic systems are derived by simple Fourier transform. It is shown how the symmetry reduces the size of the involved quantities. First results for a one-dimensional (1D) model system are presented. The results allow a reliable estimation of the numerical demand even for realistic 3D systems, such as solids. It is shown that treatment of this systems is feasible with moderate computational resources.

Highlights

Jastrow correlated trial wave functions, i.e., a wave function which to a Slater determinant includes two-particle correlations, are widely used in quantum Monte Carlo (MC) techniques [1,2]
The Fermi hypernetted chain method (FHNC) is an effective scheme to calculate a large class of cluster diagrams
The optimal correlation function is obtained by minimization of the energy expectation value

Summary

Introduction

Jastrow correlated trial wave functions, i.e., a wave function which to a Slater determinant includes two-particle correlations, are widely used in quantum Monte Carlo (MC) techniques [1,2]. Less well known are the analytic methods to calculate expectation values with this wave function [3,4,5,6]. An additional advantage of the used formulation of the FHNC method is that the functional form of the method allows a parameter-free optimization of the two-particle correlation function. In order to keep the numerical demand low, one has to utilize the symmetries of the system. A different discretization and coordinates are proposed by Krotscheck [14] It allows a further reduction in numerical demand by reducing the resolution of the center of mass coordinate, it sacrifices the simplicity of the relevant equations and reduces the resolution and the accuracy of the result. The presented method extends this approach by explicitly including a nonconstant density

The Inhomogeneous HNC Equations

Definition of the Fourier Transform

Results for a 1D Model System

Conclusions