mirror of https://github.com/abinit/abinit.git
89 lines
4.6 KiB
Markdown
89 lines
4.6 KiB
Markdown
---
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description: How to compute the electronic self-energy (due to electron-electron interaction)
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authors: MG
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---
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<!--- This is the source file for this topics. Can be edited. -->
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This page gives hints on how to compute the electronic self-energy (due to electron-electron interaction) with the ABINIT package.
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## Introduction
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In principle, the exact self-energy can be obtained by solving self-consistently
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the set of coupled integro-differential equations proposed by
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Hedin [[cite:Hedin1965]]. The fundamental building blocks of Hedin's equations
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are, besides $\Sigma(1,2)$, the Green's function of the interacting many-body system,
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$G(1,2)$, the Green's function of an appropriate non-interacting system, $\Go(1,2)$,
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and the irreducible polarizability, $\tchi(1,2)$, which, through the inverse
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dielectric matrix $\ee^{-1}(1,2)$, re-normalizes the static Coulomb potential,
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resulting in the dynamical screened interaction $W(1,2)$. Finally, the vertex
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function $\Gamma(1,2,3)$ describes the interactions between virtual holes and
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electrons.
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A typical self-energy calculation combines a quasi-particle band structure
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with a screened interaction and possibly a vertex correction to the QP
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corrections ([[optdriver]]=4).
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In the frequency domain, the GW self-energy $\Sigma(\omega)$ can be evaluated in ABINIT
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with two different, more effective, techniques:
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* integration with a plasmon-pole model (PPM)
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* integration with contour deformation (CD).
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In the former case, the frequency dependence of $\ee^{-1}(\omega)$, is modeled with a
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simple analytic form, and the frequency convolution is carried out analytically.
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In the latter approach, the integral is evaluated numerically extending the
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functions in the complex plane in order have a smoother integrand.
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Four different plasmon pole models (PPMs) are available in ABINIT. The choice
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of the particular PPM to be used is controlled by the variable [[ppmodel]].
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The first two options ([[ppmodel]] = 1, 2) refer to approximations employed in
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the pioneering implementations of the GW formalism: the plasmon-pole models of
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Godby-Needs [[cite:Godby1989]] (GN) and Hybertsen and Louie [[cite:Hybertsen1986]] (HL).
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The contour deformation technique is activated by setting the input variable
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[[gwcalctyp]] to 2. The integration along the imaginary axis requires the
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calculation of $\ee^{-1}(\omega)$, for purely imaginary frequencies.
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The frequency mesh for the quadrature is governed by the input variable [[nfreqim]], and can be very
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coarse since the integrands is very smooth in this region.
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The evaluation of the residue of the poles requires the calculation of $\ee^{-1}(\omega)$
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on a fine mesh along the real axis. This regular mesh, sampling the interval
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[0, +∞], is defined by the two input variables [[nfreqre]] and [[freqremax]].
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The CD approach requires many evaluations of $\ee^{-1}(\omega)$ and can therefore be
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computationally highly demanding. On the other hand, it is the preferred
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approach for calculating the QP correction of low-lying states. Moreover, it
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is the only technique available in ABINIT to compute the imaginary part of
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$\Sigma(\omega)$ and the spectral function $A(\omega)$.
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It is possible to disable the full computation, and actually do an Hartree-Fock,
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screened exchange, COHSEX or hybrid functional calculation.
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The calculation is done in a precomputed basis set, that can be Kohn-Sham
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(e.g. PBE) or generalized Kohn-Sham (e.g. HSE06).
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As vertex corrections, the bootstrap kernel and others can be included in the
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self-consistent W.
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The Faleev method ([[cite:Faleev2004]]), is implemented.
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Convergence over the number of unoccupied band is much improved with respect
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to usual implementations of GW, thanks to the "extrapolar" method.
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The frequency meshes, used e.g. for integration along the real and imaginary
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axes are described in [[topic:FrequencyMeshMBPT]].
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## Related Input Variables
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{{ related_variables }}
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## Selected Input Files
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{{ selected_input_files }}
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## Tutorials
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* The first tutorial on GW ([[tutorial:gw1|GW1]]) deals with the computation of the quasi-particle band gap of Silicon (semiconductor), in the GW approximation (much better than the Kohn-Sham LDA band structure), with a plasmon-pole model.
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* The second tutorial on GW ([[tutorial:gw1|GW2]]) deals with the computation of the quasi-particle band structure of Aluminum, in the GW approximation (so, much better than the Kohn-Sham LDA band structure) without using the plasmon-pole model.
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* [[tutorial:paral_mbt|The tutorial on Parallelism of Many-Body Perturbation calculations (GW)]] allows to speed up the calculation of accurate electronic structures (quasi-particle band structure, including many-body effects).
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