Interpolation Across a Muffin-Tin Interstitial Using Localized Linear Combinations of Spherical Waves

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, the so-called unitary spherical waves (USWs), ${\ensuremath{\psi}}_{RL}\left(\ensuremath{\varepsilon},\mathbf{r}\right),$ centered at the spheres $R,$ where they have cubic-harmonic character $L.$ Input to the interpolation are the coefficients in the cubic-harmonic expansions of the target function at and slightly outside the spheres; specifically, the values and the three first radial derivatives labeled by $d=0$ (value) and 1--3 (derivatives). To fit this, we use USWs with four negative energies, $\ensuremath{\varepsilon}={\ensuremath{\epsilon}}_{1},{\ensuremath{\epsilon}}_{2},{\ensuremath{\epsilon}}_{3}$, and ${\ensuremath{\epsilon}}_{4}.$ Each interpolation function, ${\ensuremath{\varrho}}_{dRL}\left(\mathbf{r}\right),$ is actually a linear combination of these four 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 the first three radial derivatives vanish at all other spheres and for all other cubic-harmonic characters, and is therefore highly localized, essentially inside its Voronoi cell. Value-and-derivative (v) functions were originally introduced and used by Methfessel [Phys. Rev. B 38, 1537 (1988)], but only for the first radial derivative. Explicit expressions are given for the v functions and their Coulomb potentials in terms of the USWs at the four energies, plus ${\ensuremath{\epsilon}}_{0}\ensuremath{\equiv}0$ for the potentials. The coefficients, as well as integrals over the interstitial such as the electrostatic energy, are given entirely in terms of the structure matrix, ${S}_{RL,{R}^{\ensuremath{'}}{L}^{\ensuremath{'}}}\left({\ensuremath{\epsilon}}_{n}\right)$, describing the slopes of the USWs at the five energies and their expansions in Hankel functions. For open structures, additional constraints are installed to pinpoint the interpolated function deep in the interstitial. The strong localization of the v functions makes the method uniquely suited for complicated structures. Use of point- and space-group symmetries can significantly reduce matrix sizes and the number of v functions. As simple examples, we consider a constant density and the valence-electron densities in zinc-blende structured Si, ZnSe, and CuBr.