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

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 operator '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_CA11S O O5.0-s2p2d1 O_CA11 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

*********************************************************** *********************************************************** 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 0 on 2 Ni 3.94955 3.94955 0.0 6.0 10.0 40.0 10.0 0 0 on 3 O 3.94955 0.0 0.0 3.0 3.0 10.0 40.0 0 0 on 4 O 3.94955 3.94955 3.94955 3.0 3.0 10.0 40.0 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.