On derivatives of the energy with respect to total electron number and orbital occupation numbers. A critique of Janak's theorem

TL;DR

This paper critiques Janak's theorem in density functional theory, arguing that the relation between energy derivatives and orbital energies is invalid for exact theories due to the nonexistence of systems with noninteger electron numbers.

Contribution

It demonstrates the invalidity of extending Slater's relation to exact Kohn-Sham DFT and clarifies the proper handling of energy derivatives at integer electron numbers.

Findings

Janak's theorem does not hold for exact DFT.

Energy as a function of electron number is not differentiable at integers.

Straight-line energy behavior is a nonphysical idealization.

Abstract

The relation between the derivative of the energy with respect to occupation number and the orbital energy, $\partial E/\partial n_i = \epsilon_i$, was first introduced by Slater for approximate total energy expressions such as Hartree-Fock and exchange-only LDA, and his derivation holds for hybrid functionals as well. We argue that Janak's extension of this relation to (exact) Kohn-Sham density functional theory is not valid. The reason is the nonexistence of systems with noninteger electron number, and therefore of the derivative of the total energy with respect to electron number, $\partial E/\partial N$. How to handle the lack of a defined derivative $\partial E/\partial N$ at the integer point, is demonstrated using the Lagrange multiplier technique to enforce constraints. The well-known straight-line behavior of the energy as derived from statistical physical considerations [J.P. Perdew, R. G. Parr, M. Levy and J.J. Balduz, Phys. Rev. Lett. 49, 1691 (1982)] for the average energy of a molecule in a macroscopic sample ("dilute gas") as a function of average electron number is not a property of a single molecule at $T=0$. One may choose to represent the energy of a molecule in the nonphysical domain of noninteger densities by a straight-line functional, but the arbitrariness of this choice precludes the drawing of physical conclusions from it.