Correct form of the electron wavefunction in periodic solids

TL;DR

This paper corrects the form of the electron wavefunction in periodic solids, resolving inconsistencies with boundary conditions and simplifying electronic structure calculations.

Contribution

It introduces a mathematically consistent form of the wavefunction that depends only on the band index, un(r), aligning with Bloch theorem and boundary conditions.

Findings

Simplifies electronic structure calculations

Unifies band and bond pictures of electrons

Resolves mathematical inconsistencies in wavefunction form

Abstract

The Bloch wavefunction leads either to mathematically impossible consequences or suggests that the ground state energy is a function of size and shape when the geometry of large crystals is considered in detail. It is incompatible with the assumption underlying the Born-von Karman periodic boundary condition. The source of the difficulty is the incorrect dependence of the Bloch wavefunction on the wavenumber index k. The mathematically impossible consequences can be overcome if the periodic part of the electron wavefunction is represented as un(r), which is dependent only on the band index, n, and is independent of the wavenumber index k. This correct form of the wavefunction is consistent with the Bloch theorem and with all other properties of Bloch wavefunctions. The correct form is also consistent with the Born-von Karman periodic boundary condition. The correct form of the electronic wavefunction in a periodic solid has profound consequences. It simplifies the calculation of electronic structure as only one wavefunction per band, un(r), needs to be evaluated. It brings about a conceptual unification between the band picture favored by physicists and the bond picture favored by chemists. The correct form of the electron wavefunction will simplify the understanding of many phenomena involving valence electrons.