Essential read: Perdew's "Density functional theory and the band gap problem" http://onlinelibrary.wiley.com/doi/10.1002/qua.560280846/abstract

The long-standing underestimation of the band gaps of semiconductors and insulators, by DFT calculations, has been totally resolved [AIP Advances, 4, 127104 (2014), http://dx.doi.org/10.1063/1.4903408].  Both LDA and GGA potentials lead not only to accurate descriptions of the electronic and related properties of materials, including band gaps, but also to predictions that are confirmed by experiment. Details are provided in the above article that is accessible free of charges.

A fifty (50) year misunderstanding of DFT was a the source of the recalcitrant underestimation; the underestimation is due to the fact that results from calculations that do not search for and verifiably attain the absolute minima of the occupied energies (i.e., the ground state), do not generally possess the full, physical content of DFT. Consequently, the band gaps from these calculations are generally in error. This misunderstanding was not realized for so long due to the fact that the community believed that the results one gets with a self-consistent calculation using a single basis set are those of the ground state of the material. They are not.  THEY ARE STATIONARY SOLUTIONS AMONG A POTENTIALLY INFINITE NUMBER OF SUCH SOLUTIONS.

Specifically, it was believed, to judge by the thousands of calculations in the literature, that a given basis set, judiciously selected, can lead to the ground state energy following self-consistency iterations; IT GENERALLY DOES NOT. It leads to one of the n’ in the second Hohenberg-Kohn theorem (See the inequality below). It is only by beginning calculations with a relatively small basis set that one augments with one orbital (at a time) that one searches for the basis set that leads absolute minima of the occupied energies, i.e., the ground state; the corresponding charge density (n) is that of the ground state, as per the uniqueness theorem of DFT.  Please note that many, successively augmented basis sets lead to the same ground state [i.e., n], with vastly different unoccupied energies! Let the system under study be subject to v(r) and possess the ground state charge density n, then the second Hohenberg-Kohn theorem follows.

Second Hohenberg-Kohn theorem:

Energy functional with n’ and v(r) greater than Energy functional with n and v(r).

It is only one of the stationary solutions that yield the lowest possible occupied energies that describe the material: It is the one, among the very many leading to the absolute minima of the occupied energies, with the smallest basis set. This basis set is the optimal basis set. Larger basis sets that contain the optimal basis set also lead to the same occupied energies [and the same n]; however, the larger they are, the more they lower some unoccupied energies, including some of the lowest ones, by virtue of the Rayleigh theorem for eigenvalues:  There is the explanation of the many different results for the band gaps of semiconductors and insulators, even from single basis set calculations that employ the same potential and computational formalism or method – but different basis sets!

If one is not familiar with the Rayleigh theorem and the related artifact consisting of the unphysical (totally not due to the Hamiltonian) lowering of some unoccupied energies when large basis sets, containing the optimal one, are utilized, the second Hohenberg theorem states that these lowered, unoccupied energies do not belong to the spectrum of the Hamiltonian of the system under study. The true spectrum of the Hamiltonian is a unique functional of the ground state charge density n, as per the first Hohenberg-Kohn theorem. Hence, lowered, unoccupied energies different from their corresponding ones obtained with the optimal basis set, do not belong to the true spectrum of the Hamiltonian.

Yes, the optimal basis set and larger basis sets containing it lead to the same ground state energy (and charge density n)! The destruction of the physical content of some unoccupied energies, lowered below their corresponding values obtained with the optimal basis, was the problem. Conventional wisdom added to the persistence of the gap problem by leading to the choice of as large as possible basis sets to ensure completeness. In DFT, completeness is practically defined by the attainment of the absolute minima of the occupied energies and not by any other considerations.

Let us note that the above definitive resolution of the band gap problem does not invoke self-interaction correction or derivative discontinuity. These categories may be needed to correct the fact that results (occupied or unoccupied) from single basis set calculations do not generally possess the full, physical content of DFT. This resolution does not employ any Ad Hoc potentials, many of which have been introduced, at least in part, to solve the band gap problem. Please refer to the AIP Article at the top: It discusses the fact that many Ad Hoc potentials are not totally DFT potentials – given the requirement of being obtained as the functional derivative of the exchange-correlation energy! This situation actually complicate matters as far as the physical content of results obtained with Ad Hoc potentials is concerned.

Somewhat oversimplifying answer: Hybrids introduce some amount of exact exchange, which is something that goes into the exact many-body wavefunction. This in turn gives some amount of added interpretability of the Kohn-Sham eigenvalues, since they become some amount more many-body like (if you are lucky, hybrids can do quite badly for magnetic metals).

Fleshing the above out a little: Adding exact exchange decreases an error in the long-range potential of the (semi-)local functionals (this is usually referred to as the self-interaction error), which messes with the ionization potential and electron affinity. Fixing this improves the HOMO-LUMO gap, and its solid-state counterpart the band gap.

Now: Read the Perdew review recommended earlier!

I don't know if anything makes codes particularly good at 2D when it comes to hybrids. Being able to cut periodic boundary conditions can be useful, I suppose, and then you have candidates like FHI-AIMS or GPAW. But I don't know whether they are ultimately the best or fastest for the problem. VASP has speeded up hybrid calculations quite a bit in its last version (as seen here: https://www.nsc.liu.se/~pla/blog/2015/09/21/vasp541/), that may be useful knowledge to you.

Dear Ali,

You have to be slightly careful when talking about the DFT band-gap, because it depends what you're talking about. If you compute the energy of the N+1, N and N-1 electron systems (the Delta SCF method) you get a different band-gap than if you just compute the difference between the lowest unoccupied and highest occupied Kohn-Sham states. You can't actually do the Delta SCF method in a plane-wave basis (because you can't have a charged unit cell in a periodic simulation, its energy would be infinite), which adds to the confusion in the literature and means most people just report the difference in the Kohn-Sham eigenvalues.

Physically, the discrepancy arises because the true, exact Kohn-Sham exchange-correlation functional depends on the total electron number and changes as the total electron count passes through each integer; this is generally referred to as the "derivative discontinuity".

The band-gap calculation is also complicated by the self-interaction error which arises in the occupied states in standard DFT, and in the unoccupied states in Hartree-Fock. If you consider the "true" band-structure, then semi-local DFT has a spurious self-interaction in the occupied states, which over-delocalises them and forces them up in energy, thus reducing the band-gap; Hartree-Fock has the same problem but for the *unoccupied* states, so they are over-delocalised and forced up in energy, which increases the band-gap.

Empirically it's found that if you mix some semi-local functional (e.g. LDA or PBE) and some Fock operator, then the self-interaction error in the top occupied band and lowest unoccupied band can be made approximately the same, and the band-gap can be made to be close to experiment or higher-level theoretical prediction. Of course this is just averaging out the self-interaction by introducing it for all states approximately equally, so every state now over-delocalises (though not as much) and, whilst the band-gap is good, other properties are unreliable.

A cheap alternative to correct the self-interaction in DFT for localised atomic-like states (e.g. for d- and f-block elements) is to introduce a modest Hubbard U term and do a DFT+U calculation. The Hubbard U introduces a potential which favours localising the states to which it is applied, thus re-localising them (compared to standard Kohn-Sham DFT) and lowering their energy. Interestingly, the Hubbard U term actually has a derivative discontinuity. Determining the appropriate U can be tricky, but once you've done that it is a simple calculation to perform and computationally straightforward.

Further to the references you've already had, if you'd like to read more about these issues I recommend you look into the problems of the "derivative discontinuity", and the related issue of "Koopman's Theorem" and "Janak's Theorem".

If you want to really try to solve this for your calculations, there are some self-interaction correction (SIC) methods, but they have other problems (different Hamiltonian for each state, loss of orthonormality etc); you might like to look instead at either Screened Exchange, or the method of "Optimised Effective Potential" (OEP) and variants, which try to determine from first principles what a good, self-interaction-free Kohn-Sham potential is for your system, e.g.

"Optimized effective potential using the Hylleraas variational method", by

T. W. Hollins, S. J. Clark, K. Refson, and N. I. Gidopoulos

Phys. Rev. B 85, 235126, DOI: 10.1103/PhysRevB.85.235126

Hope that helps,

Phil Hasnip

Mr. Askari,

You need basic knowledge on DFT. Please pay attention to:

1. Computational Molecular Science, P. Schreiner, W. Allen, M. Orozco, P. Willett (Eds.), Vols 1 - 6 (pp. 1 - 3041), Wiley, Chichester, 2014.

2. Encyclopedia of Computational Chemistry, PvR Schleyer, N. Allinger, J. Gasteiger, P. Kollman, H. Schaefer III, P. Schreiner (Eds.), Vols. 1 - 5 (pp. 1 - 3375), Wiley, Chichester, 1998.