Interpolation across a muffin-tin interstitial using localized linear combinations of spherical waves

TL;DR

This paper introduces a localized interpolation method for 3D functions between spheres using spherical waves, applicable to charge densities in condensed matter, with explicit formulas and symmetry considerations.

Contribution

It presents a novel localized interpolation technique using unitary spherical waves with explicit formulas, applicable to non-periodic structures and complex densities.

Findings

Effective interpolation of charge densities in condensed matter.

Explicit formulas for localized basis functions and their potentials.

Application to materials like Si, ZnSe, and CuBr.

Abstract

A method for 3D interpolation between hard spheres is described. The function to be interpolated could be the charge density between atoms in condensed matter. Its electrostatic potential is found analytically, and so are various integrals. Periodicity is not required. The interpolation functions are localized structure-adapted linear combinations of spherical waves, socalled unitary spherical waves (USWs), centered at the spheres where they have cubic-harmonic character. Input to the interpolation are the coefficients in the cubic-harmonic expansion of the target function at and slightly outside the spheres; specifically, the values and 3 first radial derivatives. To fit this, we use USWs with 4 negative energies. Each interpolation function is actually a linear combination of these 4 sets of USWs with the following properties: (1) It is centered at a specific sphere where it has a specific cubic-harmonic character and radial derivative. (2) Its value and first 3 radial derivatives vanish at all other spheres and for all other cubic-harmonics. It is therefore highly localized. Explicit expressions are given for these value-and-derivative (v&d) functions and their Coulomb potentials in terms of the USWs and their structure matrix, including those with zero energy, as well as for integrals over the interstitial of the v&d functions and of their products. Use of point- and space-group symmetries can significantly reduce matrix sizes and the number of v&d functions. For open structures, additional constraints are needed to pinpoint the interpolated function deep in the interstitial. This requires the USW set at one extra energy. As examples, we consider a constant density and the valence-electron densities in zinc-blende structured Si, ZnSe, and CuBr.