# Solid foundation for obtaining the band gap from density functional theory

Looking around a lot of solid calculations, the band gap is usually estimated by DFT. But I have some doubts of that within my knowledge, so I'm asking the question.

The usual process to generate density is the Kohn-Sham approach where a density of fictitious system is set to the same of real ground state system. The KS orbitals are unphysical quantities just for construction of density.

However, many people often says that using better functionals give better band gaps (the orbital energy differences). I would understand that better functionals yield more accurate ground-state potential energy surfaces, because there's the variational principle for densities formulated by the Hohenberg-Kohn theorem. But I'm not sure that DFT is made to explain excitations of the system.

So, I'm wondering that

(1) there is any theoretical foundation that fictitious system should give good descriptions of band gaps. (or quasi-particle behaviors)

(2) or physicists are just trying to get better estimations for band gaps keeping reasonable ground-state quantities.

It is true that density functional theory (DFT) describes a fictitious system and the Kohn-Sham orbitals are only approximately similar to the true (quasiparticle) states. However, energies of two states are given accurately by DFT: the ionization potential (I), which is the Kohn-Sham eigenvalue of the highest occupied state, and the electron affinity (A), which is the eigenvalue of the lowest unoccupied state plus a derivative discontinuity. The difference between these quantities is exactly equal to the band gap. You can find this expression as Eq. (23) in Ref. :

$$I(M) - A(M) = \epsilon_{M+1}(M) - \epsilon_{M}(M) + C,$$

where $$\epsilon_N(M)$$ is the $$N$$th eigenvalue of an $$M$$-electron system and where the derivative discontinuity of the exchange-correlation potential with respect to the number of particles is given by Eq. (9) of the same paper:

$$C = \left. \frac{\delta E_{xc}}{\delta n(\mathbf r)} \right|_{M+\delta} - \left. \frac{\delta E_{xc}}{\delta n(\mathbf r)} \right|_{M-\delta}.$$

That is, by adding an infinitesimal number of electrons to an $$M$$-electron system in its ground state, the derivative of the xc-energy jumps by a finite amount. This jump is generally not taken into account by xc-functionals used in practice, which leads to the underestimation of band gaps. To answer your questions:

1) There is no theoretical foundation to suggest that a fictitious system would somehow inherently be better at estimating band gaps than computing the real system. We use DFT because it's cheap and reasonably accurate for many applications. And, as described above, DFT is in principle capable of yielding the exact band gap, but the numerical approximations used in practice are not quite there yet.

2) It depends on what you mean by "better". You could have empirical functionals that are accurate for predicting the band gaps of some materials but perform poorly for a wider class of systems and/or other properties. It is typically the case that including a portion of exact exchange improves the band gap, but this approach is empirical. It is not theoretically fully justified and doesn't solve the derivative continuity issue, which is inherent to DFT. It just happens to work well in practice, especially for functionals like B3LYP, where there parameters are empirically tuned by fitting to a set of molecules.

See Refs. -, which are the classical works describing this issue.

 - John P. Perdew and Mel Levy, "Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities", Phys. Rev. Lett. 51, 1884 (1983).

 - L. J. Sham and M. Schlüter, "Density-Functional Theory of the Energy Gap", Phys. Rev. Lett. 51, 1888 (1983).

 - John P. Perdew et al., "Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy", Phys. Rev. Lett. 49, 1691 (1982).

In band theories, electrons are independent in the crystal. Unit cells have an average density, but variations would be as random as in the free electron gas. This is a poor approximation for narrow bands, like the $$3d$$ bands of the transition metal compounds or the $$4f$$ bands of the rare earths.

Then on-site Coulomb interactions are strong. Many such compounds are insulators, although there are partially filled bands in the theory. Nickel oxide is the classic example: experimentally a clear insulator, a metal in the local-density approximation. (In that particular case, calculations with spin-ordering produce a small gap in LSDA, but it does not work for rare-earth compounds.)

There are ways to incorporate such on-site correlations, LSDA+U calculations, where $$U$$ is the energy required for moving an electron from one site to another one. This can also be calculated ab initio with atomic Hartree-Fock calculation, or it can be taken from experiment.