The Exact Second Order Corrections and Accurate Quasiparticle Energy Calculations in Density Functional Theory

Abstract

We develop a second order correction to commonly used density functional approximations (DFA) to eliminate the systematic delocalization error. The method, based on the previously developed global scaling correction (GSC), is an exact quadratic correction to the DFA for the fractional charge behavior and uses the analytical second derivatives of the total energy with respect to fractional occupation numbers of the canonical molecular orbitals. For small and medium-size molecules, this correction leads to ground-state orbital energies that are highly accurate approximation to the corresponding quasiparticle energies. It provides excellent predictions of ionization potentials, electron affinities, photoemission spectrum and photoexcitation energies beyond previous approximate second order approaches, thus showing potential for broad applications in computational spectroscopy.

Koopmans spectral functionals in periodic-boundary conditions

“…In the present implementation, we use a Taylor expansion of Eq. 3 retaining only the terms up to second order in f i 42,58,59,81,82 ; although not strictly necessary, this approximation allows to simplify the expression for the KI corrections and potentials, and at the same time it does not affect the Hartree contribution in Eq. 3, which is the dominant one and exactly quadratic in the occupations, while the residual difference in the xc contribution has a minor effect on the final results (see section 4.1).…”

“…In particular, the exact condition of the piece-wise linearity (PWL) of the total energy as a function of the total number of electron 28 , or equivalently of the occupation of the HOMO, is extended to the entire manifold, leading to a generalized PWL of the energy as a function of each orbital occupation 21,23 . In KS-DFT the deviation from PWL has been suggested [29][30][31][32][33] as a definition of electronic self-interaction errors (SIEs) 34 , and in recently developed functionals, such as DFT-corrected [35][36][37][38][39][40][41][42] , range-separated [43][44][45] or dielectric-dependent hybrid functionals [46][47][48][49] , it has been recognized as a critical feature to address. The generalized PWL of KC functionals leads to beyond-DFT orbital-density dependent functionals, with enough flexibility to correctly describe both ground states and charged excitations 27,[50][51][52][53][54] .…”