TY - JOUR

T1 - On derivatives of the energy with respect to total electron number and orbital occupation numbers. A critique of Janak's theorem *

AU - Baerends, E. J.

PY - 2020/3/3

Y1 - 2020/3/3

N2 - The relation between the derivative of the energy with respect to occupation number and the orbital energy, ∂E/∂ni = ϵi, was first introduced by Slater for approximate total energy expressions such as Hartree–Fock and exchange-only LDA, and his derivation holds also for hybrid functionals. 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, ∂E/∂N. How to handle the lack of a defined derivative ∂E/∂N at the integer point, is demonstrated using the Lagrange multiplier technique to enforce constraints. The well-known straight-line behaviour of the energy as derived from statistical physical considerations [J.P. Perdew, R.G. Parr, M. Levy and J.L. 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.

AB - The relation between the derivative of the energy with respect to occupation number and the orbital energy, ∂E/∂ni = ϵi, was first introduced by Slater for approximate total energy expressions such as Hartree–Fock and exchange-only LDA, and his derivation holds also for hybrid functionals. 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, ∂E/∂N. How to handle the lack of a defined derivative ∂E/∂N at the integer point, is demonstrated using the Lagrange multiplier technique to enforce constraints. The well-known straight-line behaviour of the energy as derived from statistical physical considerations [J.P. Perdew, R.G. Parr, M. Levy and J.L. 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.

KW - DFT

KW - fractional electrons

KW - Janak's theorem

KW - occupation numbers

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U2 - 10.1080/00268976.2019.1612955

DO - 10.1080/00268976.2019.1612955

M3 - Article

AN - SCOPUS:85077433930

VL - 118

SP - 1

EP - 20

JO - Molecular Physics

JF - Molecular Physics

SN - 0026-8976

IS - 5

M1 - e1612955

ER -