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# Category:Many-body perturbation theory

**Many-body perturbation theory** includes screening and renormalization effects beyond the density-functional theory (DFT). It is based on the Green's-function formalism and can be derived and visualized in terms of a diagrammatic expansion of, e.g., the electron interacting with other electrons. Instead of describing the electrons by means of Kohn-Sham (KS) orbitals, the renormalized (or dressed) propagators yield quasiparticle orbitals.

## Theory

### Random-phase approximation (RPA)

GW and RPA are post-DFT methods used to solve the many-body problem approximatively.

RPA stands for the random-phase approximation and is often used as a synonym for the adiabatic connection fluctuation-dissipation theorem (ACFDT). RPA/ACFDT provides access to the correlation energy of a system and can be understood in terms of Feynman diagrams as an infinite sum of all bubble diagrams, where excitonic effects (interactions between electrons and holes) are neglected. The RPA/ACFDT is used as a post-processing tool to determine a more accurate ground-state energy.

### Constrained random-phase approximation

The constrained random-phase approximation (CRPA) is a method that allows calculating the effective interaction parameter , and for model Hamiltonians. The main idea is to neglect the screening effects of specific target states in the screened Coulomb interaction of the method. Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher-level theory, such as dynamical-mean-field theory (DMFT).

### GW method

The GW approximation goes hand in hand with the RPA since the very same diagrammatic contributions are taken into account in the screened Coulomb interaction of a system often denoted as W. However, in contrast to the RPA/ACFDT, the GW method provides access to the spectral properties of the system by means of determining the energies of the quasi-particles of a system using a screened exchange-like contribution to the self-energy. The GW approximation is currently one of the most accurate many-body methods to calculate band-gaps.

### Bethe-Salpeter equations (BSE)

VASP offers a powerful module for solving time-dependent DFT (TD-DFT) and time-dependent Hartree-Fock equations (TDHF) (the Casida equation) or the Bethe-Salpeter (BSE) equation^{[1]}^{[2]}. These approaches are used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional, or *GW* approximations. VASP also offers the TDHF and BSE calculations beyond the Tamm-Dancoff approximation (TDA)^{[3]}.

### Second-order Møller-Plesset perturbation theory (MP2)

There are three implementations available:

**MP2**^{[4]}: this implementation is recommended for very small unit cells, very few k-points and very low plane-wave cuttofs. The system size scaling of this algorithm is N⁵.**LTMP2**^{[5]}: for all larger systems this Laplace transformed MP2 (LTMP) implementation is recommended. Larger cutoffs and denser k-point meshes can be used. It possesses a lower system size scaling (N⁴) and a more efficient k-point sampling.**stochastic LTMP2**^{[6]}: even faster calculations at the price of statistical noise can be achieved with the stochastic MP2 algorithm. It is an optimal choice for very large systems where only relative errors per valence electron are relevant. Keeping the absolute error fixed, the algorithm exhibits a cubic scaling with the system size, N³, whereas for a fixed relative error, a linear scaling, N¹, can be achieved. Note that there is no k-point sampling and no spin polarization implemented for this algorithm.

## How to

Practical guides to different diagrammatic approximations are found on following pages:

- ACFDT: ACFDT/RPA calculations.
- GW: Practical guide to GW calculations.
- BSE: BSE calculations.
- Using the GW routines for the determination of frequency dependent dielectric matrix: GW and dielectric matrix.
- MP2 method: MP2 ground state calculation - Tutorial.

## References

- ↑ S. Albrecht, L. Reining, R. Del Sole, and G. Onida, Phys. Rev. Lett.
**80**, 4510-4513 (1998). - ↑ M. Rohlfing and S. G. Louie, Phys. Rev. Lett.
**81**, 2312-2315 (1998). - ↑ T. Sander, E. Maggio, and G. Kresse, Phys. Rev. B
**92**, 045209 (2015). - ↑ M. Marsman, A. Grüneis, J. Paier, and G. Kresse, J. Chem. Phys. 130, 184103 (2009).
- ↑ T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 146, 104101 (2017).
- ↑ T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 148, 064103 (2018).