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 $l$ 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 $8~\text{eV}$ and a J of $0.9~\text{eV}$ for $l=2$ 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 $(2l+1)\times(2l+1)$ density matrix (one entry for each $m,m'$ combination) is written out to the file n_mmp_mat_out. The format of the matrices are $7\times7$ 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, $l$ 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 $l$ 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.