GW self-energy approximations

The many-body perturbation theory as a high-level theory encapsulates in principle all the physics of a many-particle system. The complex formalism makes an understanding of the physical principles difficult. Therefore, a reduction of the amount of information contained in the self-energy by means of approximations is a necessary step in order to be able to gain an idea of the underlying fundamental interactions. In the following, an overview is presented of the frequently applied approximations to the self-energy, and ways of how to improve accuracy or to massively speed up calculations.

Hartree-Fock self-energy

By setting Σ = 0, one receives the known Hartree approximation, while the Hartree-Fock approximation is reproduced by replacing the dynamically screened interaction W(1,2) by the static electron-electron interaction v(1,2) :

Σ_x =iv(1,2)G(1,2) (2.74) with Σ_x as Hartree-Fock self-energy. From (2.73) it can be seen that the GW approxi-mation is nothing else but a dynamically screened version of the HF theory, allowing a many-electron system to respond, and thus relax upon an external perturbation potential (beyond Koopmans theorem). The Hartree-Fock self-energy causes massive overestimation of the electronic band gap of materials due to too strong exchange effects. Thus, finding a more reasonable approximation is mandatory to reach a better level of agreement with experimental data.

COHSEX self-energy

The Coulomb-hole screened exchange (COHSEX) approach is a very delicate approximation to the self-energy Σ due to reduction of computational complexity to a large degree. It is static (no frequency sampling) and summation over empty bands is eliminated. The COHSEX self-energy is composed of quantum and classical terms. The first one is the screened exchange term :

Σ^SEX(1,2) = −G(1,2)W(1,2, ω= 0) =−X

i

φ_i(1)φ_i(2)W(1,2, ω= 0) (2.75) This is identical to (2.74), exceptv is replaced by W, which decreases HF-exchange effects by taking into account polarization (exchange damping). The screened exchange term ac-counts for the Pauli principle, and thus the fermionic nature of electrons. The sum in (2.75) for the Green’s function represented by KS-wave functions φ_i runs only over the occupied

bands, which is the massive benefit of this approximation. However, unoccupied bands are only eliminated in the Green’s function and have to be taken into account in the calculation of polarization function explicitly.

The second part of COHSEX is the Coulomb hole term : Σ^COH(1,2) = 1

2δ_1,2W_p(1,2, ω= 0) (2.76) with W_p =W −V as a local and static polarization. The Coulomb hole term is a classical term, representing a shift in energy due to instantaneous polarization, once an electron is added or removed. Approximations (2.75) and (2.76) can be used to either solve Eqs.(2.65-2.69) for the self-energy fully iteratively or in a non-self consistent manner within the COH-SEX approximation.

An interesting aspect of COHSEX is that it is capable of describing band structures of different structural phases of electronically correlated materials properly, for instance, in the case of VO₂, where LDA or GGA’s are not able to capture the true nature of the band structure of low temperature anti-ferromagnetic monoclinic phase of VO₂, which incorrectly predict a metallic band structure. By contrast, it’s been shown by Gatti [29] that the full self-consistent COHSEX scheme captures the correct band structure of both high and low temperature phases of VO₂through successive update of the KS-wave functions. Therefore, COHSEX is capable of restoring the true character of the band structure and is further a good starting point for more accurate calculations targeting dynamical correlations in solids.

single-shot GW

Based on the fundamental set of equations (2.65- 2.69) in the GW approximation, one solves the Hedin-Pentagon for a dynamical self-energy in a one-shot manner. Strinati, Mattausch and Hanke [30], Hybertsen and Louie [31, 32] and Godby, Schlutier and Sham [33, 34], used the best possible initial guess forGandW from mean field theories, such as LDA, or semi-local functionals, and performed only one cycle of the self-consistent Hedin-Equations. This approach is known as one-shot GW orG₀W₀, which is frequently used for the calculation of band structure of various materials with remarkable success. After performing a single-shot of Hedin-Pentagon, the QP energies for band structures of semi-conductors and insulators are calculated within the GW-approximation as first-order corrections to the Kohn-Sham energies. Through linearization of the self-energy around the KS-energies, one obtains a

perturbative expression for QP energies :

^QP_n =^KS_n +Z hψn|(Σ(^KS)−V_xc^KS)|ψni (2.77) withZ as the renormalization factor ranging from 0 to 1, describing the correlation grade in materials. Values close to 1 indicate an electronically less correlated system, meaning that a simple QP description of the many-electron system is justified to properly account for charged excitations. Furthermore, as obvious from (2.77) the quality of the results heavily depends on the starting point. Usually, as mentioned above,G₀W₀ is started from the local or semi-local DFT reference orbitals (LDA,GGA), leading to a considerable improvement of band gaps compared to DFT-hybrid functionals, for instance. Hence, the G₀W₀ approach is way superior to all DFT-hybrid functionals in terms of accuracy. However, it is computa-tionally costly due to explicit frequency sampling of the dielectric matrix (ω), inversion of a potentially large (ω) matrix, and summation over empty bands at each frequency point.

Nevertheless, the computational load is affordable on today’s computers.

The dependency of G₀W₀ results on the choice of starting wave functions is an issue;

however it is considerably reduced by iterating the Hedin-Pentagon multiple times, instead of running only one iteration. This is discussed in the following.

multi-shot GW’s

To further increase the accuracy, a partial self-consistent scheme is employed, as systemat-ically applied for the first time by Kresse et al. [35] on a number of semi-conductors and insulators with promising results. Partial self-consistency of Hedin-Eqs. means performing a full self-consistent cycle in G but keeping the dynamical screened interaction W at the mean-field level. This scheme is known as GW₀, and proved to be a promising approach for accurate prediction of the band gaps of a wide range of solids [35], probably due to fortuitous systematic error cancellations [37].

Further iteration both in G and W leads to the fully self-consistent scheme which is from computational point of view the most time consuming GW variant. Within the fully self-consistent GW scheme band widths and gaps are typically overestimated in comparison to experimental references. The overestimation is due to underscreening of W(ω) caused by spectral weight transfer from the QP peak to the satellite part of the spectral function, as illustrated in Fig. 2.3. This is a direct consequence of inclusion of Z factor in the Green’s function (G=Z_i/(ω−_i−Γ)), constructing the polarization function (P =−iG G) which is then wrongly attenuated by a factor of Z² upon each iteration. The weight transfer is enhanced upon self-consistency resulting in too strong W(ω), and finally overestimation of

Figure 2.3: Typical features of a diagonal spectral function A_ii in the GW theory are shown, namely, a QP peak and a satellite at lower energies. The spectral weight Z under the QP peak determines the validity of the QP approximation. For Z far from 1 the QP approximation should be abandoned. In the case of non-interacting electrons the spectral function has no broadening and is characterized by a delta peak. Figure adopted from Ref. [25].

band widths and gaps. The extreme case is HF where there is no screening inW(ω) causing massive overestimation of the gaps. One way to solve the underscreening of W(ω) is to include vertex corrections Γ in Σ =G W Γ and in the screeningP. However, until now there is no convenient way of how to treat vertex corrections in both Σ andP simultaneously and properly. However, an approximation using test charges was introduced by Kresse [36].

Plasmon Pole Approximation

The dynamic character of W imposes a cumbersome computational burden. In order to reduce the computational load related to frequency dependency of the dielectric function (ω), one proceeds with the single pole approximation, provided the dielectric function is not too structured. This allows to practically skip the computationally most demanding part of the GW calculations, as otherwise for each frequency point of the dielectric function (ω), an inversion of a quite large matrix and a summation over a large number of empty bands have to be performed. However, in the Plasmon Pole approximation (PPA), inversion and summation are carried out only at two frequencies, namely, at zero and plasma frequency according to the following fit scheme :

⁻¹

GG⁰(q, ω) =δ_GG⁰ + Ω²_GG0

ω²(q)−(eω_GG⁰ −iη)² (2.78)

where Ω(q) and eω(GG⁰) are the two fit parameters.

The small parameter η in the denominator ensures the correct time-ordering.

The justification for PPA is that the general behavior of the dielectric function ⁻¹(ω) can exhibit a single pole character which can simply be approximated by a model dielectric function at zero and an imaginary frequency according to the Godby-Needs [37] or Hy-bertsen and Louie [32] PP schemes. Furthermore, Eq. (2.78) allows to calculate ⁻¹(ω) everywhere in the complex plane analytically.

The critical point of PPA is that if results depend on the imaginary frequency, PPA is no longer adequate, and an explicit frequency sampling of the dielectric function is indeed unavoidable. This is the consequence of many poles or poles lower than the electronic band gap appearing in ⁻¹(ω), leading to a breakdown of the PP approximation.