## Usage of the LDA+U approach

It is possible to perform Fleur calculations incorporating Hubbard U parameters for certain
orbitals at certain atoms. The implementation of the approach complies with the publication
*Shick et al., Implementation of the LDA+U method using the full-potential linearized augmented plane-wave basis, Phys. Rev. B 60, 10763 (1999)*.
This implies that the U parameter actually doesn't act on certain Kohn-Sham orbitals but on those
parts of the LAPW basis functions featuring the angular momentum quantum number for which the
U parameter is specified.

For each atom species adding a U parameter together with a J parameter for a certain quantum number in the
MT sphere is straight forward. For this an `ldaU`

tag has to be inserted below
the `energyParameters`

, `electronConfig`

, and `nocoParams`

tags and above
the `lo`

tags. The `ldaU`

tag includes specifications for the U parameter in `ldaU/@U`

, and
specifications for the J parameter in `ldaU/@J`

. Both parameters have to be provided in eV. Additionally
it has to be specified in `ldaU/@l_amf`

whether the U is treated in terms of the around-mean-field
limit (`T`

) or in terms of the atomic limit (`F`

). For each atom species up to 4 different U parameters
can be specified, each within a separate `ldaU`

tag. An example for the specification of a U parameter of
and a J of for is shown below.

```
<ldaU l="2" U="8.0" J="0.9" l_amf="F"/>
```

In DFT+U calculations the additional Hamiltonian contributions due to the U parameter depend on the density matrix for the considered angular momentum quantum number in the respective MT sphere. Similar to the density this matrix also has to be determined self-consistently.

In the Fleur input file the mixing for the density matrix is specified in the separate `calculationSetup/ldaU`

tag. One can choose
to either perform a linear mixing by setting `calculationSetup/ldaU/@l_linMix`

to `T`

or to perform the mixing consistently
with the density by setting it to `F`

. For linear mixing the mixing parameter is defined in `calculationSetup/ldaU/@mixParam`

.
An example for the setup of a `calculationSetup/ldaU`

tag is provided below.

```
<ldaU l_linMix="F" mixParam="0.05" spinf="1.00"/>
```

In every SCF iteration for each spin and U parameter in each MT sphere the density matrix (one entry for
each combination) is written out to the file `n_mmp_mat_out`

. The format of the matrices are blocks of space
separated complex numbers where each line of the matrix stretches out to two lines in the file. The real and imaginary part of the
entries are also space separated. If a file `n_mmp_mat`

is provided in the working directory the density matrix provided in
that file is taken as an input and the file is directly renamed into `n_mmp_mat_old`

so that it is not read in again in the
next iteration. Advanced users may define density matrix starting points in this way. The density matrix is also written to
the `out.xml`

file within `densityMatrixFor`

sections. The attributes of these tags uniquely identify a density matrix by
providing spin, atom type, index of the U parameter, quantum number, U parameter, and J parameter.

Note: If a U parameter and a local orbital (LO) are defined for the same atom species and quantum number, Fleur will write
out `LO and LDA+U for same l not implemented`

. Nevertheless you can use Fleur in this way. What is not implemented is the effect of
the U parameter on the extra radial function used for the LO. This is not problematic if the LO is used to represent a
semicore state and the U parameter is supposed to have an effect on higher lying valence or conduction band states. In fact, in such a situation this
code behavior is explicitly wanted. On the other hand if the LO and the U are made for the same states in mind the calculation will
not work as intended.

The user should also be aware that implementations of the LDA+U approach in FLAPW typically only affect the MT spheres. This implies that depending on the localization of the states, adequate U parameters may slightly depend on the MT sphere radii. In general U parameters used for calculations with different DFT implementations are not necessarily quantitatively comparable. For localized states these differences in different methods have limited implications on results. If, however, a U parameter is applied to correct the eigenenergies of very delocalized states, the application of the U only in the MT speres may lead to strongly deviating results in comparison to other approaches that may apply the U in regions of other sizes or on the basis of Wannier functions.