Derivative discontinuities in density functional theory

Abstract

Fifty years after the original formulation of density functional theory (DFT), subtle consequences of the mathematical mappings underlying its formalism continue to merit new views. In this article, we discuss the origin, the importance, and the challenges associated with finding the derivative discontinuity of the exchange-correlation (XC) energy of DFT at integer–electron numbers. We show how even the energy of a quantum electron gas with finite volume and number of electrons displays such derivative discontinuities, but continuous density functional approximations to the XC functional miss them entirely. We discuss some of the practical problems that arise due to this lack of derivative discontinuities in standard functionals, and explain new ways to recover them.

Keywords:

• density functional theory
• fractional number of electrons
• exchange-correlation energy
• derivative discontinuity
• molecular dissociation

Acknowledgements

We acknowledge support from the National Science Foundation CAREER program [grant number CHE-1149968]. Adam Wasserson also acknowledges support from the Alfred P. Sloan Foundation and the Camille Dreyfus Teacher-Scholar Awards Programs.

Notes

1. We will refer to Ref. [26] as PPLB.