LDA+U methods with different definitions of the occupation number operator [16] are available for both the collinear and non-collinear calculations by the following keyword 'scf.Hubbard.U':
scf.Hubbard.U on # On|Off, default=off
It is noted that the LDA+U methods can be applied to not only LDA but also GGA.
The occupation number operator is specified by the following keyword
'scf.Hubbard.Occupation':
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 which is the most primitive
conserved quantity in a Hubbard model.
For the details of the operators onsite, full, and dual, see
Ref. [16].
The effective U-value in eV on each orbital of species defined by
<Definition.of.Atomic.Species
Ni Ni6.0S-s2p2d2 Ni_CA13S
O O5.0-s2p2d1 O_CA13
Definition.of.Atomic.Species>
is specified by
<Hubbard.U.values # eV
Ni 1s 0.0 2s 0.0 1p 0.0 2p 0.0 1d 4.0 2d 0.0
O 1s 0.0 2s 0.0 1p 0.0 2p 0.0 1d 0.0
Hubbard.U.values>
The beginning of the description must be '
Hubbard.U.values',
and the last of the description must be 'Hubbard.U.values
'.
For all the basis orbitals, you have to give an effective U-value in eV
in the above format. The '1s' and '2s' mean the first and second s-orbital,
and the number behind '1s' is the effective U-value for the first s-orbital.
The same rule is applied to p- and d-orbitals.
As an example of the LDA+U calculation, the density of states for
a nickel monoxide bulk is shown for cases with an effective U-value
of 0 and 4 (eV) for d-orbitals of Ni in Fig. 27, where the input
file is 'Crys-NiO.dat' in the directory 'work'.
We see that the gap increases due to the introduction of a Hubbard
term on the d-orbitals.
The occupation number for each orbital is output to the file '*.out'
in the same form as that of decomposed Mulliken populations which
starts from the title 'Occupation Number in LDA+U' as follows:
***********************************************************
***********************************************************
Occupation Number in LDA+U and Constraint DFT
Eigenvalues and eigenvectors for a matrix consisting
of occupation numbers on each site
***********************************************************
***********************************************************
1 Ni
spin= 0
Sum = 8.591857905308
1 2 3 4 5 6 7 8
Individual -0.0024 0.0026 0.0026 0.0038 0.0051 0.0051 0.0888 0.0950
s 0 0.1671 0.0005 -0.0006 0.0040 0.0000 0.0005 -0.0124 0.0000
s 1 -0.9856 -0.0030 0.0039 -0.0227 -0.0000 -0.0072 0.0066 0.0000
px 0 0.0010 0.0004 0.0011 -0.0131 0.0004 0.0001 -0.0261 -0.0291
py 0 0.0010 0.0006 -0.0008 -0.0130 0.0000 0.0009 -0.0271 -0.0000
pz 0 0.0010 -0.0012 -0.0001 -0.0131 -0.0004 0.0001 -0.0261 0.0291
px 1 0.0067 0.0023 0.0066 -0.0792 -0.0161 0.0123 0.5594 0.7062
py 1 0.0068 0.0041 -0.0053 -0.0801 -0.0000 -0.0162 0.5797 0.0002
pz 1 0.0067 -0.0070 -0.0005 -0.0792 0.0161 0.0123 0.5594 -0.7063
d3z^2-r^2 0 0.0002 -0.0781 -0.0105 0.0002 0.0023 0.0014 0.0002 0.0108
dx^2-y^2 0 0.0004 -0.0105 0.0781 0.0004 -0.0013 0.0024 0.0003 -0.0062
dxy 0 0.0004 -0.0009 -0.0002 0.0246 -0.0421 -0.0251 0.0794 -0.0050
dxz 0 -0.0001 0.0008 -0.0010 0.0269 0.0000 0.0478 0.0795 0.0000
dyz 0 0.0004 0.0004 0.0008 0.0246 0.0420 -0.0251 0.0794 0.0050
d3z^2-r^2 1 -0.0023 0.9875 0.1327 -0.0033 -0.0262 -0.0159 -0.0001 -0.0069
dx^2-y^2 1 -0.0040 0.1326 -0.9875 -0.0056 0.0151 -0.0275 -0.0002 0.0040
dxy 1 0.0091 0.0233 0.0052 -0.5578 0.7055 0.4249 -0.0749 0.0157
dxz 1 0.0189 -0.0180 0.0233 -0.5964 -0.0003 -0.7958 -0.0748 -0.0000
dyz 1 0.0091 -0.0110 -0.0212 -0.5578 -0.7052 0.4255 -0.0749 -0.0157
9 10 11 12 13 14 15 16
Individual 0.0952 0.2456 0.9902 0.9974 0.9975 1.0060 1.0060 1.0137
s 0 0.0002 0.9859 -0.0036 -0.0001 0.0000 -0.0000 0.0000 -0.0000
.....
...
The eigenvalues of the occupation number matrix of each atomic site
correspond to the occupation number to each local state given by
the eigenvector.
The LDA+U functional possesses multiple minima in the degree of freedom
of the orbital occupation, leading to that the SCF calculation tends
to be trapped to some local minimum. To find the ground state with
an orbital polarization, a way of enhancing explicitly the orbital
polarization is available by the following switch :
For collinear cases
<Atoms.SpeciesAndCoordinates # Unit=AU
1 Ni 0.0 0.0 0.0 10.0 6.0 on
2 Ni 3.94955 3.94955 0.0 6.0 10.0 on
3 O 3.94955 0.0 0.0 3.0 3.0 on
4 O 3.94955 3.94955 3.94955 3.0 3.0 on
Atoms.SpeciesAndCoordinates>
For non-collinear cases
<Atoms.SpeciesAndCoordinates # Unit=AU
1 Ni 0.0 0.0 0.0 10.0 6.0 40.0 10.0 0 on
2 Ni 3.94955 3.94955 0.0 6.0 10.0 40.0 10.0 0 on
3 O 3.94955 0.0 0.0 3.0 3.0 10.0 40.0 0 on
4 O 3.94955 3.94955 3.94955 3.0 3.0 10.0 40.0 0 on
Atoms.SpeciesAndCoordinates>
The specification of each column can be found in the section 'Non-collinear DFT'.
Since the enhancement treatment for the orbital polarization is
performed on each atom, you have to set the switch for all the atoms
in the specification of atomic coordinates as given above.
The enhancement for the atoms switched on is applied during
the first few self-consistent (SC) steps, then no more enhancement
are required during the subsequent SC steps. It is also emphasized that
the enhancement does not always give the ground state, and that it can work
badly in some case. See Ref. [16] for the details.
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