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FlapwMBPT: Full-potential Linearized Augmented Plain Wave method and Many Body Perturbation Theory

main developer: Andrey Kutepov
hosted by: Brookhaven National Laboratory

FlapwMBPT (Full-potential Linearized Augmented Plain Wave method and Many Body Perturbation Theory) is a computer code implementing many body diagrammatic approaches to study electronic structure and ground state properties of materials in their (periodic) crystalline state.

The electronic structure is a basic building block for the theoretical calculation of material properties.  It can be understood as an eigenvalue problem in which the determination of the electron self-energy matrix Σ is the main challenge. FlapwMBPT offers several first-principle and many body diagrammatic approaches with different levels of approximations to Σ. Currently available approximations include the Local Density Approximation (LDA) which we use to generate an initial approximation for the Green function in many body approaches, but it can be used on its own as well. Also available are a self-consistent Hartree-Fock approximation and a self-consistent GW approximation (scGW). The latter can be combined with a linearized quasi-particle approximation to form our version of the self-consistent quasiparticle GW method (LQSGW). Both scGW and LQSGW are implemented in a fully relativistic way (based on Dirac equations).

An essential new feature of the code consists in the recently implemented self-consistent diagrammatic approaches which go beyond GW approximation. This feature (so-called scGWΓ, with Γ being the vertex function) can be considered as a new ab initio, diagrammatic tool to study materials with weak to modest strength of correlations.

Capabilities:

  • FlapwMBPT is implemented in the LAPW basis set with local orbital extensions. FLAPW abbreviates the full–potential linear augmented plane waves method in its non-relativistic or (scalar–)fully relativistic form, depending on the material under study. In the FLAPW method the space is subdivided into non–overlapping muffin–tin (MT) spheres and so-called interstitial regions. Inside the MT spheres the Bloch states are represented as linear combinations of solutions of radial equations and in the interstitial regions as linear combinations of plane waves.
  • All included methods are designed to evaluate electronic spectra, susceptibilities, and total energies. Exceptions:  The total energy is not well defined in LQSGW. Certain diagrammatic limitations (approximations) exist in the evaluation of susceptibilities with scGWГ.

FlapwMBPT modules in more detail:

LDA (density functional theory in its local density approximation)
LDA is the simplest theoretical method to study electronic structure properties for weakly correlated materials and is useful when one needs to get quick answers, especially for complex materials with many atoms per unit cell. Typically, one can study crystals up to approximately 30-40 atoms per unit cell. LDA is also used as an initial approximation for more advanced methods (such as GW, DMFT, G-RISB).

It is based on the Kohn-Sham density functional theory (DFT) in which the free energy is expressed as a functional of the electron density and then extremized to obtain the total free energy of the material.

All correlation effects that go beyond a Hartree Coulomb potential in DFT are caught by the so-called exchange correlation potential Vxc, which is not exactly known. In DFT's local density approximation (LDA) this term is approximated using the free energy of the electron gas at a given density -  and, ultimately, Σ, the self-energy in the electronic structure eigenvalue problem, is replaced by this local "LDA exchange correlation potential" Vxc:  Σ=Vxc.

HF (Hartree-Fock)
The Hartree-Fock module solves self-consistently the Hartree-Fock equations. A principal difference as compared to LDA is the presence of a non-local (but static) self-energy. LDA often results in a metallic electronic structure even if experiments find an insulator. Therefore, HF is often used when one wants to know if the inclusion of the non-local exchange term leads to a band gap opening.

The Hartree-Fock module can handle up to 20-25 atoms per unit cell and is also used within a so-called Hybrid Functional approach, a mix of LDA and Hartree-Fock.

(self-consistent) GW:
Common GW approximations solve (self-consistently) the simplest set of Hedin’s equations, which are based on a first order expansion of the self-energy Σ in the screened Coulomb interaction W and neglect vertex corrections: Σ=GW. The Green's function G can be defined in various ways, leading to different variants of the GW method. The full Green's function is obtained using the Dyson equation. 

FlapwMBPT  contains a one shot quasiparticle (QP) mode (G0W0), a fully self-consistent mode (scGW) enabling the computation of total energies, a linearized variant of the QP self-consistent mode (LQSGW), and our new self-consistent diagrammatic approach scGWΓ.

In Hybertsen and Louie's "one shot" G0W0 method one uses the LDA Kohn-Sham ("quasiparticle") Green's function G0, which is successful for sufficiently weak correlations. W0 is obtained in the random phase approximation (RPA) from the bare Coulomb potential screened by the polarization P. However, in comparison to self-consistent approaches, a major disadvantage of this approach is its strong dependence on the starting point. Electronic spectra obtained in LDA should already be sufficiently accurate in order to ensure that "one shot" G0W0 is able to provide results close to experiments.

Hedin's original self-consistent scheme uses the full Green's function. However, the "fully self-consistent" GW (scGW) approach without vertex corrections exhibits certain intrinsic theoretical problems and corresponding calculations overestimate band gaps in semiconductors and insulators, and band widths in metals. In addition, it is computationally expensive.

The quasiparticle self-consistent GW (QSGW) is essentially equivalent to the fully scGW method but uses a special QP construction for the Green function, which replaces the need to solve the Dyson equation. The success of QSGW relies on the fact that the QP approximation considerably cancels out the error associated with the absence of higher order diagrams in the self energy Σ and the polarizability P. QSGW is computationally more expensive than G0W0 but it doesn’t depend on a starting point and often shows improvements over G0W0 for materials where LDA doesn’t provide a good starting point.

Our "linearized" version of QSGW (LQSGW) takes the eigenvalues and wave functions within LDA or HF as starting points. Then it calculates the polarizability, screened Coulomb interaction, electron self-energy, and single particle Green's function. It achieves good scaling by using the space-imaginary time method (going back and forth between real space and momentum space, performing only multiplications and Fourier transforms). Afterwards a conversion from an interacting Green’s function into a free G0 for the purpose of computing the polarization function and self-energy is carried out by linearizing the frequency dependent electron-self energy near zero frequency.

Note that, in contrast to scGW, the total energy is ill defined in LQSGW. LQSGW is often better than scGW in the calculated one-electron spectra if one deals with simple metals or narrow band semiconductors. For large gap insulators and at least some of the actinides, scGW provides better accuracy (as compared to LQSGW) for the one-electron spectra. scGW (or LQSGW) often provides better accuracy of calculated electronic structure as compared to LDA or Hartree-Fock with the price of being a lot more time consuming. Both scGW and LQSGW can handle up to 20-25 atoms per unit cell.

One way to improve the accuracy of the scGW method is to include skeleton diagrams of higher order (vertex corrections) in the self energy and the polarizability. scGWГ then solves (self-consistently) the Hedin’s equations with a non-trivial three-point vertex. Ultimately in this approach approximations are introduced purely diagrammatically. Furthermore there is no quasi-particle approximation involved. Instead the Green function is renewed on every iteration from Dyson’s equation. All diagrams take into account the full frequency-dependence of the screened interaction, which also is updated on every iteration. scGWГ can handle up to 5-6 atoms per unit cell.

related publications

Ground state properties of 3d metals from self-consistent GW approach.
A.L. Kutepov,
arXiv:1707.01995, accepted manuscript in J. Phys.: Condens. Matter

Self-consistent solution of Hedin's equations: Semiconductors and insulators.
Andrey L. Kutepov,
Phys. Rev. B 95, 195120 (2017)

Electronic structure of Na, K, Si, and LiF from self-consistent solution of Hedin's equations including vertex corrections.
Andrey L. Kutepov,
Phys. Rev. B 94, 155101 (2016)