Extensivity of Energy and Electronic and Vibrational Structure Methods for Crystals

A pedagogical proof is presented for the extensivity of energies of metallic and nonmetallic crystals that proceeds by elucidating the asymptotic distance dependence of the effective chemical interactions: kinetic, Coulomb, exchange, and correlation. On this basis, a guideline for the size-consistent design of electronic and vibrational methods is proposed. This guideline underscores the significance of the distinct use of the intermediate and standard normalization of wave functions for extensive and intensive quantities, includes the extensive and intensive diagram theorems as the unambiguous criteria for determining size consistency of a method for extensive and intensive quantities, and introduces the extensive-intensive consistency theorem, which stipulates the precise balance between the determinant spaces reached by extensive and intensive operators. Electronic and vibrational methods for crystals are reviewed that are inspired by these formal analyses or developed in accordance with the guideline.