# The band gap discontinuity in DFT???

by new_986
Tags: band, discontinuity
 P: 35 Hi everybody.... I've read about the band gap problem of Density functional theory, there is a discontinuity of the band gap when an electron add to the Kohn-Sham system I have two questions could anybody answer me? please 1- why this discontinuity happens anyway? 2- why we need to add an electron to the Kohn-Sham system to calculate the band gap? with my regards Nawzad A.
 Sci Advisor P: 3,119 ad 1: As far as I know, the problem is due to not representing well the exchange with common functionals. ad 2: With DFT you calculate the energy of the ground state. So you have to compare a system where the Valence band is full and the conduction band is empty with one where the valence band contains just one electron.
P: 35
 Quote by DrDu ad 1: As far as I know, the problem is due to not representing well the exchange with common functionals. ad 2: With DFT you calculate the energy of the ground state. So you have to compare a system where the Valence band is full and the conduction band is empty with one where the valence band contains just one electron.
Dear DrDu...
The Kohn-sham represent the wave function determined by the Slater which known as Slater determinant which is also used in Hartree-Fock method to overcome the symmetry of the electrons am I right?
If this is true, Is this means the Slater determinant does not represent the exchange well? and how please?

PF Patron
P: 220

## The band gap discontinuity in DFT???

Up to early 2000's it was common to use this approach (add electron, Subtract electron) to compute the band gap using DFT. However, This is not the practice anymore as there are more RAM memory available. The current practice is to calculate high quality electronic density of states using dens K-points grid , fine Fourier Transform grid, and accurate convergence criteria for the electronic iterations. The band gap can be readily obtained then.
P: 3,119
 Quote by new_986 Dear DrDu... The Kohn-sham represent the wave function determined by the Slater which known as Slater determinant which is also used in Hartree-Fock method to overcome the symmetry of the electrons am I right? If this is true, Is this means the Slater determinant does not represent the exchange well? and how please?
No, the Kohn-Sham Orbitals are never used to form a Slater determinant. They are little more than a computational device to find a good approximation mainly for the kinetic energy.
P: 3,119
 Quote by Useful nucleus To add to the answer of the 2nd Q: Up to early 2000's it was common to use this approach (add electron, Subtract electron) to compute the band gap using DFT. However, This is not the practice anymore as there are more RAM memory available. The current practice is to calculate high quality electronic density of states using dens K-points grid , fine Fourier Transform grid, and accurate convergence criteria for the electronic iterations. The band gap can be readily obtained then.
Interesting. But how do you calculate the density of states? Time dependent DFT?
PF Patron
P: 220
 Interesting. But how do you calculate the density of states? Time dependent DFT?
Still within the framework of DFT , one can obtain the density of states.
Time dependent DFT generally can improve the width of the band gap specially for semiconductors with narrow gap.
 Sci Advisor P: 3,119 I still don't understand how. Do you have a reference?
 PF Patron P: 220 The following reference explains the so called Tetrahedron method. "Singular integrals over the Brillouin zone: the analytic-quadratic method for the density ofstates" M H Boon, M S Methfessel and F. M. Mueller 1986 J. Phys. C: Solid State Phys. 19 5337
P: 35
 Quote by DrDu No, the Kohn-Sham Orbitals are never used to form a Slater determinant. They are little more than a computational device to find a good approximation mainly for the kinetic energy.
"The Eigen states can be expressed in the form of Slater determinant" this is what I saw in the following reference:
" J. Kohanoff and N. Gidopoulos , Density functional theory: basics, new trends and applications (January 31, 2002).

I thing this means the same what I said, If I'm wrong please tell me.
thanks

 Sci Advisor P: 3,119 You may arange the KS orbitals in a Slater determinant if you want. But this Slater determinant is only a wavefunction for the hypothetical wavefunction of non-interacting particles which feel the KS potential. It is not a representation of the true wavefunction of the interacting system. Especially it cannot be used to calculate expressions for exchange or correlation (it should be clear that independent particles aren't correlated). However, to calculate the one body expectation values (especially of the kinetic energy), you don't need to form a determinant of the orbitals.
P: 3,119
 Quote by Useful nucleus The following reference explains the so called Tetrahedron method. "Singular integrals over the Brillouin zone: the analytic-quadratic method for the density ofstates" M H Boon, M S Methfessel and F. M. Mueller 1986 J. Phys. C: Solid State Phys. 19 5337
I had a look at the paper, but methinks it does not adress how to calculate the density of states in DFT. You may determine some gap in the spectrum of the KS orbitals but there is no reason why this gap should coincide with the gap for e.g. natural orbitals as Koopmans theorem does not hold for KS orbitals.
 P: 1 Hartree Fock method has the exact exchange energy for a single slater determinant. DFT has an approximation for the exchange in the exchange-correlation functional. Most hybrid DFT methods like BL3YP add some exchange from the HF method. Also, band gaps are often not well represented by DFT because of the fictitious Kohn-Sham orbitals. One can still often gain insight from the KS orbitals though. Also a major reason for the disparity is that the exchange correlation functional has a cusp when plotting ionization energy and electron affinity vs. the number of electrons. This cusp is often not built into the functionals.
 Sci Advisor P: 3,119 I think that nowadays the band gap is often calculated solving the Hedin or GW equations. These use often as a starting point the Kohn Sham orbitals. Nevertheless this theory has not much to do with density functional theory per se.
 P: 35 I've done a correction of band gap using GW approximation but I still have no idea about the physical reason of the discontinuity of the KS band gap