Boundary conditions for augmented plane-wave methods

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.

Topological Characterization of the Electron Density Laplacian in Crystals. The Case of the Group IV Elements

“…Using Gauss theorem, it can be expressed as a flux of the density gradient across the muffin surfaces: ∫ cell L ( r ) d r = − ∑ α ∮ S normalα ( ∇ ρ α − ∇ ρ i ) · d S where α runs over the atoms in the cell, ρ i and ρ α are the density function forms in the interstitial, and the α muffin (eq ), respectively. This result has two important consequences: the scriptG and scriptK forms of the kinetic energy are not equivalent, the one entering the total energy expression being scriptG , and the integral of L ( r ) within the topological (∇ρ) basins is not zero. Note that, although the sum in eq can be easily computed to correct the integral of L over the cell, it is not possible to do the same to the atomic expectation values of L , except in the cases where no muffin crosses the interatomic surface.…”