Boundary conditions for augmented plane wave methods

TL;DR

This paper justifies the mathematical foundation of the augmented plane wave method, clarifying the domain of the Hamiltonian and the form of kinetic energy, and advises against using discontinuous basis functions.

Contribution

It provides a rigorous mathematical justification for the augmented plane wave method and clarifies the implications for basis function continuity.

Findings

Mathematically justified the use of the Rayleigh-Ritz principle with discontinuous derivatives

Extended the domain of the Hamiltonian to include functions with discontinuous derivatives

Argued that discontinuous basis functions should be avoided

Abstract

The augmented plane wave method uses the Rayleigh-Ritz principle for basis functions that are continuous but with discontinuous derivatives and the kinetic energy is written as a pair of gradients rather than as a Laplacian. It is shown here that this procedure is fully justified from the mathematical point of view. The domain of the self-adjoint Hamiltonian, which does not contain functions with discontinuous derivatives, is extended to its form domain, which contains them, and this modifies the form of the kinetic energy. Moreover, it is argued that discontinuous basis functions should be avoided.