The accuracy of semi-local functionals

Contributed by David Bowler
Reposted from Atomistic Computer Simulations with permission

The great strength of density functional theory (DFT) in its purest form is that the energy only depends on the charge density, whether its magnitude alone (LDA), or with its gradient (GGA) or beyond. This simplicity is, of course, also the source of some of its largest errors: poor band gap predictions, excessive delocalisation of charge density, poor description of charge transfer, and problems with band widths. Many of these issues can be solved by the addition of a judicious amount of exact exchange, giving hybrid functionals. But exchange is very expensive to calculate, and requires the use of orbitals in the place of a charge density. And there is the associated question of what fraction of exchange to mix (there are many hybrid functionals available, each with a different fraction of exchange) and even whether the exchange interactions should be screened (and if so, by how much…). An interesting recent review and extended opinion piece from one of the key practitioners in the field[1] gives more details on some of these problems, and is well worth a read.

There have been recent attempts to construct semi-local (GGA) functionals which have at least some of the features of exact exchange, in the hope of improving the accuracy. The first of these[2] constructed the potential as a functional of the charge density which matched the OEP (optimised effective potential - an approach to solving the exact exchange problem) very well for atoms. It has been modified[3] to create a potential that can be used in solids, and which gives good band gaps (known as the mBJ or TB-mBJ potential). However, this is only a potential, and because of its form, it is not possible to create a corresponding energy functional. This limits the applicability to post-hoc corrections of the band structure. (The potential can also diverge where there are large areas of vacuum, such as surfaces.)

A similar attempt has been made to create consistent potential and energy functionals which also reproduce key features of the exact exchange[4,5]. One of these, the derivative discontinuity (DD), describes a change in the gradient of the energy with respect to electron number at integer fillings; this behaviour is missing from most semi-local functionals. The AK13 functional[4] restores this by considering the asymptotic behaviour of the functional, giving reasonable accuracy for band gaps, without the need for fitting (the mBJ functional has parameters that are fit to improve the accuracy).

A very recent paper, which presents a comparison of results and potentials from these functionals[6], shows, as ever, that there is no single functional that gives good results in all situations (though in principle the exact density functional should do this). As DFT practitioners we must choose whether we want to use highly fitted functionals which give excellent accuracy within certain limits, or to use functionals which respect certain important limits and behaviours for the electron gas. There is no universal function, just as there is no universal basis set, and part of the practice of DFT simulations is to quantify carefully the limitations and effects of the approaches we choose.

[[1] Rev. Mod. Phys. 87, 897 (2015) DOI:10.1103/RevModPhys.87.897

[[2] J. Chem. Phys. 124, 221101 (2006) DOI:10.1063/1.2213970

[[3] Phys. Rev. Lett. 102, 226401 (2009) DOI:10.1103/PhysRevLett.102.226401

[[4] Phys. Rev. Lett. 111, 036402 (2013) DOI:10.1103/PhysRevLett.111.036502

[[5] Phys. Rev. B 91, 035107 (2015) DOI:10.1103/PhysRevB.91.035107

[[6] Phys. Rev. B 91, 165121 (2015) DOI:10.1103/PhysRevB.91.165121