DFT+U+V

This tutorial aims at explaining how to perform DFT+U+V calculations in Octopus. This correspond to adding not only an on-site Hubbard interaction U, but also an intersite interaction V. As a prototypical example, we will consider bulk silicon.

The DFT+U+V method, as well as its performances and implementation details are discussed in Ref.1

Input

The input file we will use is the following one:

CalculationMode = gs
PeriodicDimensions = 3

Spacing = 0.5

%LatticeVectors
  0.0 | 0.5 | 0.5
  0.5 | 0.0 | 0.5
  0.5 | 0.5 | 0.0
%

a = 10.18
%LatticeParameters
 a | a | a
%

%ReducedCoordinates
 "Si" | 0.0 | 0.0 | 0.0
 "Si" | 1/4 | 1/4 | 1/4
%

nk = 2
%KPointsGrid
  nk |  nk |  nk
 0.5 | 0.5 | 0.5
 0.5 | 0.0 | 0.0
 0.0 | 0.5 | 0.0
 0.0 | 0.0 | 0.5
%

ExtraStates = 4

%KPointsPath
 10 |  10 |  15
 0.5 | 0.0 | 0.0  # L point
 0.0 | 0.0 | 0.0  # Gamma point
 0.0 | 0.5 | 0.5  # X point
 1.0 | 1.0 | 1.0  # Another Gamma point
%

DFTULevel = dft_u_acbn0
AOLoewdin = yes
UseAllAtomicOrbitals = yes
SkipSOrbitals = no

ACBN0IntersiteInteraction = yes
ACBN0IntersiteCutoff = 7

ExperimentalFeatures = yes

Most of the variable are the same as in the introduction tutorial on solids [Getting started with periodic systems](../Getting started with periodic systems), and the DFT+U related variables are explained in the tutorial DFT+U. For convenience, we are doing here a ground-state calculation with a k-point grid and a k-point path. This allows to get the bandstructure as a direct output of a ground-state calculation. However, the unoccupied states might not be converged with this approach. To perform a proper band-structure calculation, see the tutorial [Getting started with periodic systems](../Getting started with periodic systems).

Compared to a more conventional DFT+U calculation, we note few differences here:

Output

After running Octopus using the above input, we can look at the output. In the static/info file, we find the direct and indirect bandgap


Direct gap at ik=    2 of  0.0304 H
Indirect gap between ik=    8 and ik=    2 of  0.0177 H

  Direct gap at ik=    6 of  0.1346 H
  Indirect gap between ik=    6 and ik=   24 of  0.1148 H

As a comparison, without adding the +U+V, one finds at the LDA level a much smaller bandgap


Direct gap at ik=   14 of  0.1346 H
Indirect gap between ik=    2 and ik=   11 of  0.1148 H

  Direct gap at ik=    1 of  0.1041 H
  Indirect gap between ik=    3 and ik=    9 of  0.0810 H

The effect of bandgap opening can be understood more precisely from the bandstructure. As one can see, the conduction bands have been rigidly shifted toward higher energies, as expected from more advance calculations such as hybrid functionals or GW calculations.

Band structure of bulk silicon from DFT+U+V. The zero of energy has been shifted to the maximum of the occupied bands.
Band structure of bulk silicon from DFT+U+V. The zero of energy has been shifted to the maximum of the occupied bands.

References


  1. N. Tancogne-Dejean, and A. Rubio, Parameter-free hybridlike functional based on an extended Hubbard model: DFT+U+V, Physical Review B 102 155117 (2020); ↩︎