DFT is a method that describes a system as a single-body Hamiltonian of one electron moving in the mean-field by approximating the Schrodinger equation, which describes entangled electrons in the many-electrons system. It also explains the affection of other electrons using Vxc potential (exchange-correlation potential) in the mean-field. Furthermore, the approximation method used in KS-DFT is named after Vxc.
With LDA and GGA, the most general semi-local functionals Vxc of each point is determined based on the electron density of the point and the next point, so an error occurs at the correlation energy and exchange energy, which needs to be corrected. 
There are several methods to correct the correlation and exchange energy errors. The most common ones are DFT+U, van der Waals correction, and hybrid functional. We will introduce these subjects for three weeks.
In this weekly tip, we will find out the DFT+U method and the affection of the U parameter to the electronic structure.
1. DFT+U to correct correlation energy
DFT+U is a correction method for the correlation energy caused by the mean-field approximation.
When electrons fill in an orbital, the energy differs based on whether the other electron exists in the orbital. This energy difference caused by the electron’s correlation is called correlation energy, and the U parameter of DFT+U is the explicit correction term for the energy, which differs because of the electrons’ repulsion force.
DFT+U affects largely and mainly to the localized d and f orbitals. Therefore, it is suitable to correct transition metal oxides, which usually have the d and f orbitals as valence orbitals.
By calculating the energy corrected correlation energy using the DFT+U Hamiltonian (Hubbard Hamiltonian), which adds the U parameter to the Kohn–Sham Hamiltonian, DFT+U can be used at a more precise calculation for the underestimated band gap, formation energy, magnetism, etc.
When applying the U parameter for the DFT+U calculation, we need to caution the value of the U parameter that has different values based on the orbital, element, materials, and bonding configuration. In general, we need to determine the proper U parameter based on the empirical parameter, so it is better to refer to other papers and conduct a test. 
2. Applying DFT+U in the MatSQ
The DFT+U correction keywords are added at the Correction Section of Scripting Option: Template. Then, the DFT+U Calculation and U parameter keywords are also added under the &SYSTEM.
The U parameter keyword must be set for each element. As seen in the figure below, the order of the elements is located in the ATOMIC_SPECIES section in the input script.
For the PbTiO3, which was used as an example in this weekly tip, Ti has partially occupied the 3d orbital as a valence orbital, thus applying the U parameter . As seen in the following figure, the Ti was placed in the third order; therefore, a proper U parameter value should be placed next to the (3) in the U parameter input window.
3. The affection of DFT+U to the electronic structure of PbTiO3
The band structure of PbTiO3, which is obtained using the PBE functional, is as follows.
According to the band structure, PbTiO3 has an indirect band gap between the high symmetry points X and Z. The valence band maximum (VBM) was located at the X point; and the conduction band minimum (CBM), at the Z point.
The computational band gap of PbTiO3, which was obtained using the PBE functional, is smaller than the experimental band gap of 3.45 eV . However, we can reduce the error by correcting the correlation energy through the DFT+U calculation
The most proper U parameter can be determined by checking the band gap of various U parameter values.
According to the results, when the U parameter is set too large, the band gap is diminished. Therefore, it can consider 5.0 eV as the most proper U parameter. On the other hand, the calculated band gap was obtained as 2.65 eV at the U parameter that was set as 5.0 eV. The corrected +U band gap of 2.65 eV is better than the PBE band gap of 1.91 eV, but it still smaller than the experimental band gap of 3.45 eV.
Because of the DFT+U correction, this error affects only the localized Ti-3d orbital in the PBE functional. It can be improved by using the hybrid functional, but it also consumes more time than the DFT+U calculation.
In this aspect, the best advantage of the +U method is that it can correct errors moderately with a similar calculation time compared to the GGA method. However, for a complicated and big model, it might cost the calculation more because of bad convergence.
The band structures obtained at the DFT calculation (GGA) and DFT+U calculation (U=5.0 eV) has some difference.
When applying the DFT+U correction, not only the band gap has increased but also the electronic structure has changed. Therefore, we can obtain a more precise and new electronic structure with the correlation energy corrected by applying DFT+U.
This weekly tip introduced the DFT+U method, which is the first method among the three exchange-correlation energy correction methods, and explained how we can apply it in the MatSQ, showing the changed electronic structure when applied.
The DFT+U correction largely affects the d and f orbitals, which have a huge correlation effect, so it is particularly correct and well for the transition metal oxide to have the d and f orbitals as valence orbitals. With this method, the underestimated band gap can be corrected as well. Furthermore, it can apply to the formation energy calculation, magnetism calculation, etc. However, using the GGA functional, the calculation result might not be perfectly corrected. For this, please check another correcting method in the weekly tip series.
Get the proper band gap information through the DFT+U calculation in Materials Square!
 Anisimov, V. I., Zaanen, J., & Andersen, O. K. (1991). Band theory and Mott insulators: Hubbard U instead of Stoner I. Physical Review B, 44(3), 943.
 Cococcioni, M., & De Gironcoli, S. (2005). Linear response approach to the calculation of the effective interaction parameters in the LDA+ U method. Physical Review B, 71(3), 035105.
 Gou, G. Y., Bennett, J. W., Takenaka, H., & Rappe, A. M. (2011). Post density functional theoretical studies of highly polar semiconductive Pb (Ti 1− x Ni x) O 3− x solid solutions: effects of cation arrangement on band gap. Physical Review B, 83(20), 205115.
 Peng, C. H., Chang, J. F., & Desu, S. B. (1991). Optical properties of PZT, PLZT, and PNZT thin films. MRS Online Proceedings Library Archive, 243.
Sign up for free and get $10.
This post is licensed under a Creative Commons Attribution-Nonprofit-ShareAlike 4.0 International License.