# The DFT+U approximation

### This page gives hints on how to perform a DFT+U calculation with the ABINIT package.

##### Copyright (C) 2016-2017 ABINIT group (BAmadon)

Mentioned in help_features#5.

### Table of content:

- 1. Introduction.
- 2. Related lesson(s) of the tutorial.
- 3. Related input variables.
- 4. Selected input files.
- 5. References.

### 1. Introduction.

This feature is available only in PAW. The DFT+U framework is described in [Anisimov1991] and [Liechtenstein1995]. In ABINIT, the DFT+U approximation is implemented inside the PAW atomic spheres only. Two choices of double counting are provided: the Full Localized limit and the Around Mean Field approximation. Our implementation is described in [Amadon2008a]. It follows the main lines of [Bengone2000]. See also [Czyzyk1994]. Forces and stress are implemented. For details on keywords (lpawu,upawu,jpawu,usedmatpu,dmatpuopt,dmatudiag) see keyword usepawu in input variables.

In both the output and log files, we can find:

- The DFT+U contribution of energy which is contained inside the PAW Spherical terms in the output file.

- The Decomposition of the LDA+U energy is given (Interaction energy, Double counting term, and sum of the two) in the log file.

- The orbital density matrix (n_{m,m'}^{\sigma}), also called occupation matrix (corresponding to Eq.(9) of [Bengone2000] and Eq.(1) of [Liechtenstein1995], see also [Amadon2008a] and variable dmatpuopt) is also given for each atom in the basis of real spherical harmonics. It is given at each SCF step in the log file: one can thus check the convergency of the calculation.

Consistency between total energy and forces in DFT+U have been checked.

The implementation of DFT+U in ABINIT allows also to impose a starting density matrix in order to compare the energy of various electronic configuration (see keywords usedmatpu and dmatpawu).

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### 2. Related lesson(s) of the tutorial.

- The lesson on DFT+U shows how to perform a DFT+U calculation using ABINIT, and will lead to compute the projected DOS of NiO. Prerequisite : PAW1.
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### 3. Related input variables.

Compulsory input variables:

... jpawu [value of J for PAW+U]
... lpawu [value of angular momentum L for PAW+U]
... upawu [value of U for PAW+U]
... usepawu [USE PAW+U (spherical part)]

Useful input variables:

... dmatpawu [initial Density MATrix for PAW+U]
... usedmatpu [USE of an initial Density MATrix in Paw+U]

Relevant internal variables:

... %natpawu [Number of AToms on which PAW+U is applied]

Input variables for experts:

... atvshift [ATomic potential (V) energy SHIFTs]
... dmatpuopt [Density MATrix for PAW+U OPTion]
... dmatudiag [Density MATrix for paw+U, DIAGonalization]
... f4of2_sla [F4 Over F2 ratio of Slater integrals]
... f6of2_sla [F6 Over F2 ratio of Slater integrals]
... macro_uj [MACRO variable that activates the determination of the U and J parameter (for the PAW+U calculations)]
... natvshift [Number of ATomic potential (V) energy SHIFTs (per atom)]
... normpawu [NORMalize atomic PAW+U projector]
... pawujat [PAW+macro_UJ, ATom number]
... pawujrad [PAW+macro_UJ, sphere RADius]
... pawujv [PAW+macro_UJ, potential shift (V)]
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### 4. Selected input files.

WARNING : as of ABINITv8.6.x, the list of input files provided in the specific section of the topics Web pages is still to be reviewed/tuned. In some cases, it will be adequate, and in other cases, it might be incomplete, or perhaps even useless.
The user can find some related example input files in the ABINIT package in the directory /tests, or on the Web:

tests/v5/Input: t08.in t14.in t15.in t16.in t19.in t20.in t33.in t37.in t38.in t39.in t40.in

tests/v7/Input: t21.in t22.in t44.in
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### 5. References.

[Amadon2008a] B. Amadon, F. Jollet and M. Torrent, "γ and β cerium: LDA+U calculations of ground-state parameters", Phys. Rev. B 77, 155104 (2008).
DOI: 10.1103/PhysRevB.77.155104.
[Anisimov1991] V. I. Anisimov and O. Gunnarsson, "Density-functional calculation of effective Coulomb interactions in metals", Phys. Rev. B 43, 7570–7574 (1991).
DOI: 10.1103/PhysRevB.43.7570.
[Bengone2000] M. Alouani, P. Blöchl, O. Bengone and J. Hugel, "Implementation of the projector augmented-wave LDA+U method: Application to the electronic structure of NiO", Phys. Rev. B 62, 16392 (1994).
[Czyzyk1994] M. T. Czyzyk and G. A. Sawatzky, "Local-density functional and on-site correlations: The electronic structure of La2CuO4 and LaCuO3", Phys. Rev. B 49, 14211 (1994).
[Liechtenstein1995] A. I. Liechtenstein, V. I. Anisimov and J. Zaanen, "Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators", Phys. Rev. B 52, R5467–R5470 (1995).
DOI: 10.1103/PhysRevB.52.R5467.

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