# Calculation of the effective Coulomb interaction.

### This page gives hints on how to calculate the effective Coulomb interaction 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.

LDA+U as well as DFT+DMFT requires as input values the effective Coulomb interaction. Two ways to compute them are available in ABINIT.

Firstly, the constrained Random Phase Approximation [Aryasetiawan2004] ucrpa allows one to take into account the screening of the Coulomb interaction between correlated electrons, by non-interacting electrons. For non-entangled bands (ucrpa= 1), the bands excluded from the polarisability can be specified either by a band index (ucrpa_bands) or an energy window (ucrpa_window) [Amadon2014].

For entangled bands (ucrpa= 2}), the scheme used in ABINIT [Shih2012],[Sakuma2013],[Amadon2014] uses a band and k-point dependent weight to define the polarisability, using Wannier orbitals as correlated orbitals.

This method is well adapted to compute the effective interaction for the same orbitals used in DFT+DMFT. To use the same orbitals as in DFT+U, the Wannier functions can be ajusted such that the bare interaction is close to the bare interaction of atomic orbitals as used in DFT+U (see tutorial).

Secondly, a linear response method [Cococcioni2005] is implemented. The implementation is not yet in production. The implementation in ABINIT takes into account the truncated atomic orbitals from PAW and therefore differs from the original work [Cococcioni2005] treating full atomic orbitals. In particular, considerably higher effective values for U are found.
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### 2. Related lesson(s) of the tutorial.

- The lesson on the calculation of effective interactions U and J by the cRPA method shows how to determine the U value with the constrained Random Phase Approximation [Aryasetiawan2004] using projected Wannier orbitals. Prerequisite : DFT+U.
- The lesson on the determination of U for DFT+U shows how to determine the U value with the linear response method [Cococcioni2005], to be used in the DFT+U approach. Prerequisite : DFT+U.
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### 3. Related input variables.

Compulsory input variables:

... ucrpa [calculation of the screened interaction U with the Constrained RPA method]

Basic input variables:

... ucrpa_bands [For the calculation of U with the Constrained RPA method, gives correlated BANDS]
... ucrpa_window [For the calculation of U with the Constrained RPA method, gives energy WINDOW]
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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/v7/Input: t23.in t24.in t25.in t78.in t79.in
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### 5. References.

[Amadon2014] B. Amadon, T. Applencourt and F. Bruneval, "Screened Coulomb interaction calculations: cRPA implementation and applications to dynamical screening and self-consistency in uranium dioxide and cerium", Phys. Rev. B 89, 125110 (2014).
DOI: 10.1103/PhysRevB.89.125110.
[Aryasetiawan2004] F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Biermann and A. I. Liechtenstein, "Frequency-dependent local interactions and low-energy effective models from electronic structure calculations", Phys. Rev. B 70, 195104– (2004).
DOI: 10.1103/PhysRevB.70.195104.
[Cococcioni2005] M. Cococcioni and S. de Gironcoli, "Linear response approach to the calculation of the effective interaction parameters in the LDA+U method", Phys. Rev. B 71, 035105 (2005).
DOI: 10.1103/PhysRevB.71.035105.
[Sakuma2013] R. Sakuma and F. Aryasetiawan, "First-principles calculations of dynamical screened interactions for the transition metal oxides MO (M=Mn, Fe, Co, Ni)", Phys. Rev. B 87, 165118 (2013).
DOI: 10.1103/PhysRevB.87.165118.
[Shih2012] B.-C. Shih, Y. Zhang, W. Zhang and P. Zhang, "Screened Coulomb interaction of localized electrons in solids from first principles", Phys. Rev. B 85, 045132 (2012).
DOI: 10.1103/PhysRevB.85.045132.

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