Theoretical unification of hybrid-DFT and $\text{DFT}\hspace{0.17em}+\hspace{0.17em}U$ methods for the treatment of localized orbitals

Hybrid functionals serve as a powerful practical tool in different fields of computational physics and quantum chemistry. On the other hand, their applicability for the case of correlated $d$ and $f$ orbitals is still questionable and needs more considerations. In this article we formulate the on-site occupation dependent exchange correlation energy and effective potential of hybrid functionals for localized states and connect them to the on-site correction term of the $\text{DFT}\hspace{0.17em}+\hspace{0.17em}U$ method. The resultant formula indicates that the screening of the onsite electron repulsion is governed by the ratio of the exact exchange in hybrid functionals. Our derivation provides a theoretical justification for adding a $\text{DFT}\hspace{0.17em}+\hspace{0.17em}U$-like on-site potential in hybrid-DFT calculations to resolve issues caused by overscreening of localized states. The resulting scheme, hybrid $\text{DFT}\hspace{0.17em}+\hspace{0.17em}{\mathrm{V}}_{\mathrm{w}}$, is tested for chromium impurity in wurtzite AlN and vanadium impurity in 4H-SiC, which are paradigm examples of systems with different degrees of localization between host and impurity orbitals.

Figure 1

Schematic diagram of the defect orbitals of the positively charged and the neutral Cr${}_{\mathrm{Al}}$ point defect in w-AlN.

Figure 2

Single particle charge densities and their change due to the correction of the functional. The charge density of the gap states of the neutral Cr${}_{\mathrm{Al}}$ point defect in w-AlN (see Fig. 1) are shown in the upper part of the figure, i.e., (a) the highest occupied defect orbital $a^{\prime }$, (b) the occupied lower lying split $e$ defect orbital, and (c) the lowest unoccupied split $e$ defect orbital as calculated with mHSE method and plotted with the isosurface value is 0.05. The following (d), (e), and (f) figures show the change of the electron density of the corresponding defect orbitals due to the correction V${}_{\mathrm{w}}$ of the mHSE hybrid functional with $w=-1.6$ (see text for more explanation). The isosurface values are 0.005, 0.01, and 0.0005 Å${}^{-3}$, respectively. The red (dark gray) and blue (light gray) lobes indicate increased and decreased localization, respectively.

Figure 3

Total and partial density of the states of the host w-AlN and the Cr impurity in the positively charged Cr${}_{\mathrm{Al}}$ point defect, respectively. The red filled curves show the total DOS of the host, while blue and green filled curves show the $d$ and $sp$ partial DOS of the Cr. These later curves were scaled up to be visible. (a) and (b) The results of the calculations with mHSE and mHSE + V${}_{\mathrm{w}}$ exchange correlation functional (see text for more explanation).

Figure 4

Total and partial density of the states of the host w-AlN and the Cr impurity in the neutral Cr${}_{\mathrm{Al}}$ point defect, respectively. The red filled curves show the total DOS of the host, while blue and green filled curves show the $d$ and $sp$ partial DOS of the Cr. These later curves were scaled up to be visible. (a) and (b) The results of the calculations with the mHSE and mHSE + V${}_{\mathrm{w}}$ exchange correlation functional (see text for more explanation).

Figure 5

Variation of Kohn-Sham (KS) eigenvalues and the charge corrected total energy difference (see text for more explanation) with respect to the strength of the correction parameter $w$ of the mHSE + V${}_{\mathrm{w}}$ method in the case Cr${}_{\mathrm{Al}}$ in w-AlN. The variance of the highest occupied KS orbital in the neutral charge states and the unoccupied states in the positively charged state are shown as obtained on the fix geometry of the neutral state. The total energy difference is calculated from the total energies of the two charge states with applied charge correction. The valence band edge is chosen to possess the zero value on the energy scale.

Figure 6

Change of the total charge density and the spin density upon the correction of the hybrid functional mHSE. (a) The summed charge density of the occupied KS orbitals in the band gap and (b) the spin density of the Cr${}_{\mathrm{Al}}$ in w-AlN are shown with the isosurface value of 0.05. (c) The change of the total charge density while (d) is the change of the spin density as a response to the additional potential V${}_{\mathrm{w}}$ with $w=-1.6$ eV. In both cases the red (dark gray) and blue (light gray) lobes represent increase and decrease of the density, respectively. The isosurface values in (c) and (d) were chosen to be 0.002 and 0.01 Å${}^{-3}$, respectively.

Figure 7

Schematic diagram of the defect orbitals of the neutral and positively charged V${}_{\mathrm{Si}}$ point defect in 4H-SiC.

Figure 8

Total and partial density of the states of the host 4H-SiC and the V impurity in the neutral V${}_{\mathrm{Si}}$ point defect, respectively. The red filled curves show the total DOS of the host, while blue and green filled curves show the $d$ and $sp$ partial DOS of the vanadium. These later curves were scaled up to be visible. (a) and (b) The results of the calculations with HSE06 and HSE06 + V${}_{\mathrm{w}}$ exchange correlation functional (see text for more explanation).

Figure 9

Variation of Kohn-Sham (KS) eigenvalues and the charge corrected total energy difference with respect to the strength of the correction parameter $w$ of HSE06 + V${}_{\mathrm{w}}$ method in the case V${}_{\mathrm{Si}}$ in 4H-SiC (see text for more explanation). The variance of the highest occupied KS orbital in the neutral charge states and the lowest unoccupied state in the positively charged state are shown. The total energy difference is calculated from the total energies of the two charge states with applied charge correction. The valence band edge is chosen to possess the zero value on the energy scale.