Koopmans' Theorem - Counterpart in Density Functional Theory

Counterpart in Density Functional Theory

Kohn-Sham (KS) density functional theory (KS-DFT) admits its own version of Koopmans' theorem (sometimes called the DFT-Koopmans' theorem) very similar in spirit to that of Hartree Fock theory. The theorem equates the first (vertical) ionization energy of a system of electrons to the negative of the corresponding KS HOMO energy . More generally, this relation is true even when the KS systems describes a zero-temperature ensemble with non-integer number of electrons for integer and . When considering electrons the infinitesimal excess charge enters the KS LUMO of the N electron system but then the exact KS potential jumps by a constant known as the "derivative discontinuity". It can be shown that the vertical electron affinity is equal exactly to the negative of the sum of the LUMO energy and the derivative discontinuity.

Unlike the approximate status of Koopmans' theorem in Hartree Fock theory (because of the neglect of orbital relaxation), in the exact KS mapping the theorem is exact, including the effect of orbital relaxation. A sketchy proof of this exact relation goes in three stages. First, for any finite system determines the asymptotic form of the density, which decays as . Next, as a corollary (since the physically interacting system has the same density as the KS system), both must have the same ionization energy. Finally, since the KS potential is zero at infinity, the ionization energy of the KS system is, by definition, the negative of its HOMO energy and thus finally:, QED.

While these are exact statements in the formalism of DFT, the use of approximate exchange-correlation potentials makes the calculated energies approximate and often the orbital energies are very different from the corresponding ionization energies (even by several eVs!).

A tuning procedure is able to "impose" Koopmans' theorem on DFT approximations thereby improving many of its related predictions in actual applications.

In approximate DFTs one can estimate to high degree of accuracy the deviance from Koopmans' theorem using the concept of energy curvature.