Crystallographic point and space group symmetries in the one-and two-body density of crystals and generalized Patterson functions: Fourier path integral Monte Carlo case studies of novel high-temperature/high-pressure ⁴He quantum crystals

Abstract

We present a complete formal analysis of crystallographic point and space group symmetries in the local one- and two-body density ρ(x) and ρ₂(x₁,x₂) of crystals and from there develop the mathematical theory of generalized Patterson functions. The two-body density ρ₂(x₁,x₂) and ρ(x₁) ρ(x₂) may be written as symmetrized lattice Fourier series over all reciprocal lattice vectors K with Fourier transforms that are functions of the relative vector r = x₁ – x₂. In the new formalism derived here the usual Patterson function turns out to be the lowest-lying K = 0 lattice Fourier transform in the Fourier series representation of the uncorrelated product ρ(x₁)ρ(x₂) of one-body densities. All other K ≠ 0 Fourier transforms in the Fourier series of ρ(x₁)ρ(x₂) may be regarded as generalized Patterson functions. In complete analogy to these generalized Patterson functions the Fourier transform functions in the Fourier series of the fully correlated two-body density ρ₂(x₁,x₂) may be regarded as K = 0 and K ≠ 0 fully correlated generalized Patterson functions. It is shown that the former generalized Patterson functions represent the uncorrelated long-range limits for large interparticle distances of the latter, fully correlated, generalized Patterson functions. Both types of generalized Patterson functions may be cast in the form of symmetry-adapted trigonometric series with expansion functions that depend on the relative distance r = |r| and on the polar angle of r. The symmetrized trigonometric series representations of the generalized Patterson functions are given here explicitly in specific applications of the new formalism to crystallographic space group P6₃/mmc. In exact path integral Monte Carlo case studies of novel high-temperature/high-pressure ⁴He quantum crystals in the hexagonal close-packed structure we present and discuss exact numerical results for the one-body density ρ(x) and for expansion functions, for both types of generalized Patterson functions, in the symmetrized trigonometric series representations of the generalized Patterson functions. The numerical results for the two types of generalized Patterson functions are compared to each other.