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 -