Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory

Abstract

The conventional analysis of Perdew and Levy, and Sham and Schlüter shows that the functional derivative discontinuity of the exchange-correlation density functional plays a critical role in the correct prediction of bandgaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for bandgap prediction with all common types of exchange-correlation functionals have been derived without invoking the concept of exchange-correlation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital (HOMO) in describing electron removal. (2) Every term in the total energy functional contributes to the energy gap because of the discontinuity of the derivative of the density (or density matrix) with respect to the number of electrons, ((∂ρ s (r , r))/∂N ) v s , at integers. (3) Consistent with the Perdew-Levy-Sham-Schlüter conclusion that the exact Kohn-Sham energy gap differs from the fundamental bandgap by a finite correction due to the functional derivative discontinuity of the exchange-correlation energy, we show that the exchange-correlation functional cannot be an explicit and differentiable functional of the electron density, either local or nonlocal. The last result is further strengthened when we consider Mott insulators. There, the exact exchange-correlation functional needs to have an explicitly discontinuous (nondifferentiable) dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spin-up and spin-down number of electrons.

The van der Waals interactions in rare-gas dimers: the role of interparticle interactions

“…Accordingly, our findings are also in support of the key feature of the LC hybrid functionals for systems with Coulomb interactions, which have recently been found to provide supreme performance for a very wide range of applications [57,58], especially for problems related to the asymptote of the XC potential [59][60][61][62][63][64][65][66], self-interaction errors [67,68], fundamental gaps [69][70][71][72][73][74][75][76][77][78][79][80][81][82], and charge-transfer excitations [83][84][85][86][87][88][89]. Besides, empirical atom-atom dispersion potentials [51,55,56,[90][91][92] or MP2 correlation energy [43,53,[93][94][95] can be added to the KS-DFT energy in order to improve the description of noncovalent interactions (e.g., vdW interactions).…”

Kohn-Sham band gaps and potentials of solids from the optimised effective potential method within the random phase approximation

“…45,60,61,69 This is a consequence of the fact that the energy functional, unlike in the, e.g., LDA case, is not an explicit and differentiable functional of the electron density, as discussed recently by Yang and coworkers. 70 The close relation between the quasi particle correction and the derivative discontinuity leads to similar expressions for them. To obtain the quasi-particle energy, the quasi-particle equation…”

Self-consistent double-hybrid density-functional theory using the optimized-effective-potential method

“…(10) and using the fact that the HOMO spin orbital dominates in this limit over all occupied spin orbitals [50], or by considering the derivative of the HF exchange energy with respect to the electron number (at fixed potential, i.e. at fixed orbitals) and using the chain rule with either the one-particle density or the oneparticle density matrix [50,51]. Similarly, Eq.…”

“…For the self-consistent OEP DH approximations, these derivatives can be expressed in terms of frontier spin orbital energies, like in exact KS DFT, [51,52,54,55] …”