Ab initio electronic structure platform

Developed by Andrey Kutepov

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 a main challenge. The ab initio electronic structure platform will offer several first-principle many body diagrammatic approaches (LDA, HF, scGW, LQSGW) with different levels of approximations to Σ to study electronic structure, ground and excited state properties of weakly to moderately correlated materials in their (periodic) crystalline state - based on their known atomic structure. When combined with DMFT or G-RISB the properties of strongly correlated materials can be computed.

Our ab initio electronic structure platform is implemented in a fully relativistic way (based on Dirac equations), which is unique for both scGW and LQSGW. It is based on the stand-alone software package FlapwMBPT.

Capabilities:

•  The ab initio electronic structure platform is implemented in the FLAPW 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 the solutions of radial equations and in the interstitial regions as linear combinations of plane waves.
• All included methods are designed to evaluate electronic spectra and total energies. Exception: the total energy is not well defined in the LQSGW method.

Our ab initio electronic structure platform will include:

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 V_xc, 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" V_xc:  Σ=V_xc.

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 (Fock) 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 with a static dielectric screening.

(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. 

We will offer GW in a one shot quasiparticle (QP) mode (G₀W₀), a fully self-consistent mode (scGW) enabling the computation of total energies, and a linearized variant of the QP self-consistent mode (LQSGW).

• 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 experiment.
•  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's function, which replaces the need to solve the Dyson equation. The success of QSGW relies on the fact that the QP approximation roughly 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 functions. 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 Greens 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.

A principal complication of GW as compared to Hartree-Fock is the appearance of a dynamic part in the self-energy as a result of screening effects included in the simplest one-loop approximation for the polarizability. 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. 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. Both scGW and LQSGW can handle up to 20-25 atoms per unit cell.

Related publications

Linearized self-consistent quasiparticle GW method: Application to semiconductors and simple metals.
Andrey L. Kutepov, Viktor S. Oudovenko, and Gabriel Kotliar,
Comput. Phys. Comm., 219, p. 407 (2017)

Electronic structure of Pu and Am metals by self-consistent relativistic GW method.
A. Kutepov, K. Haule, S.Y. Savrasov, and G. Kotliar,
Phys. Rev. B 85, 155129 (2012)