Choice of DFT+$U$ scheme; simplified or general

The general forms of DFT+$U$ methods with different definitions of the occupation number operator, the functional form, and the choice of the double counting term are available in OpenMX for both collinear and noncollinear calculations [21,22] by the following two keywords:

  scf.Hubbard.U                 on  # on|off, default=off
  scf.DFTU.Type                 2   # 1:Simplified(Dudarev)|2:General, default=1

scf.DFTU.Type=1 corresponds to the so-called 'simplified rotationally invariant form' by Dudarev et al. [25] as implemented in the previous versions of OpenMX [20], where only $U$ (Hubbard $U$) plays a role. In more general DFT+$U$ schemes, not only $U$ but also $J$ (Hund's coupling $J$) are used as input parameters. Note that the keyword 'scf.SpinPolarization' should be always switched on, meaning that 'on' or 'nc' has to be specified, whenever the DFT+U methods are used.

The occupation number operator [20] is specified by the following keyword:

  scf.Hubbard.Occupation        dual  # onsite|full|dual, default=dual

Among three occupation number operators, only the dual operator satisfies a sum rule that the trace of occupation number matrix gives the total number of electrons. For the details of the operators onsite, full, and dual, see Ref. [20].

The $U$ and $J$ values in eV on each orbital of species defined by

  <Definition.of.Atomic.Species
    Ni Ni6.0S-s2p2d2f1  Ni_CA13S
    O  O5.0-s2p2d1      O_CA13
  Definition.of.Atomic.Species>

are specified by

  <Hubbard.U.values   # eV
    Ni 1s 0.0 2s 0.0 1p 0.0 2p 0.0 1d 5.0 2d 0.0 1f 0.0
    O  1s 0.0 2s 0.0 1p 0.0 2p 0.0 1d 0.0
  Hubbard.U.values>

and

  <Hund.J.values      # eV
    Ni 1s 0.0 2s 0.0 1p 0.0 2p 0.0 1d 0.9 2d 0.0 1f 0.0
    O  1s 0.0 2s 0.0 1p 0.0 2p 0.0 1d 0.0
  Hund.J.values>

The beginning of the description must be <Hubbard.U.values (<Hund.J.values), and the last of the description must be Hubbard.U.values> (Hund.J.values>) for $U$ ($J$). For all the basis orbitals, you have to give $U$ and $J$ values in eV as in the above format. The '1s' and '2s' mean the first and second $s$-orbital, and the number behind '1s' is the $U$ (or $J$) value for the first $s$-orbital. The same rule is applied to $p$- and $d$-orbitals. When scf.DFTU.Type=1, only $U$ values are read and thus users do not need to specify $J$ values.