Calculation of coulombic lattice potentials: II. Spherical harmonic expansion of the Green function

Abstract

The modified Green function appropriate for calculation of coulombic lattice potentials is developed in a spherical harmonic expansion. This is derived from the corresponding Ewald sum in Cartesian coordinates, by applying Gegenbauer's addition theorem for modified spherical Bessel functions to the screened Coulomb potentials resulting from Laplace transformation with respect to the scalar convergence parameter, and Bauer's expansion to the plane waves. It is useful where the charge-density distribution about each nucleus is represented by a spherical harmonic expansion. Radial coefficients of the spherical harmonics are attenuated exponentially, and orthogonality reduces determination of electrostatic lattice potentials to one-dimensional quadratures. This use of the Green function is contrasted with conventional approaches based on point-multipole representations, in which important information on the diffuseness of electronic charge density around the nuclei may be lost in the calculation of multipole coefficients. Possible applications of this result in electronic structure calculations are briefly discussed.

Mean potential of Bethe in the classical problem of calculating bulk electrostatic potentials in crystals

“…without any unwanted surface contribution [20,23,33] by definition. Actually, the universal character of b ( ) U r accounts for this result.…”

“…Paper from the effects of the ionic overlap [11,68], modifies the three-dimensional point-charge lattice potentials [20,69].…”

“…Based on the Poisson summation formula, the Ewald approach [15] is powerful and reliable for practical calculations [16][17][18][19]. As a result, this method is also used for testing other approaches [20,21] and even for improving the convergence upon purely direct summation [22,23]. The physical reason responsible for the high rate of convergence within the Ewald approach is the smearing of point charges there, as discussed in the general case by Bertaut [24].…”

Multiple charge spreading as a generalization of the Bertaut approach to lattice summation of Coulomb series in crystals

“…where the prime on the summation sign implies that the contribution of h = 0 is actually omitted, as follows from formula (7) derived later on. The structure factor F (h), by definition, is determined as [22,28]…”

Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

“…In particular, the pressure due to a positive point source and a distributed sink of equal magnitude can be summed over the points of an infinite lattice to give a Green function satisfying periodic boundary conditions. In a rectangle, this can be represented either as a rapidly converging infinite series of logarithmic functions (Marshall 1999) or as an Ewald series of exponential integrals (Marshall 2002), and in a parallelepiped, as a series of modified Bessel functions or an Ewald series of error functions (Marshall 2000). Homogeneous Dirichlet or Neumann boundary conditions on the faces of two-or three-dimensional rectangular regions can be satisfied by additive combination of these periodic Green functions.…”