GW Approximation Research Articles

Electronic properties and polaronic dynamics of semi-Dirac system within Ladder approximation

We investigate the electronic properties of the semi-Dirac system and its polaronic dynamics when coupled with a Fermi bath with quadratic dispersion. The electronic anisotropic transport properties and the semiclassical dynamics of the semi-Dirac system are studied, including the density-of-states, conductivity, transport relaxation rate, specific heat, electrical current density, and free energy. The attractive polaron formed as the semi-Dirac impurity dressed with the particle–hole excitations in a two-dimensional system are studied both analytically and numerically. The pair propagator, self-energy, spectral function are being detailly calculated and discussed. The method of medium T-matrix approximation (nonself-consistent), which equivalent to the partially dressed interaction vertex by summing over all ladder diagrams, is applied, and compared with some other methods (for many-body problem), like the leading-order 1/N expansion (GW approximation), Hartree–Fock theory, and the Nozieres–Schmitt–Rink theory. Since we focus on the weak-coupling region, the mean-field approximation is also applicable. The polaron properties is related to the anisotropic effective masses of the semi-Dirac system. That’s in contrast to the polarons formed in the surface of normal Dirac systems which has an isotropic dispersion, since the anisotropic dispersion of the semi-Dirac systems results in anisotropic effective mass and anisotropic charge carrier transport. Besides, the symmetry between electron and hole is also broken since the effective masses of the electron and hole are different, which are affected by the polaronic effect when the semi-Dirac material is deposited on a polar substrate, like hBN. The self-localization, short-range potential, and experimental methods as well as the possible formation of the Bose polaron on the surface of semi-Dirac system are also discussed in the end. Our results are useful also for the investigation of two/three-dimensional bosonic polaron as well as the polarons in other solid state systems, like the magnetic matter or the topological systems.

Atomic structures and orbital energies of 61,489 crystal-forming organic molecules

Data science and machine learning in materials science require large datasets of technologically relevant molecules or materials. Currently, publicly available molecular datasets with realistic molecular geometries and spectral properties are rare. We here supply a diverse benchmark spectroscopy dataset of 61,489 molecules extracted from organic crystals in the Cambridge Structural Database (CSD), denoted OE62. Molecular equilibrium geometries are reported at the Perdew-Burke-Ernzerhof (PBE) level of density functional theory (DFT) including van der Waals corrections for all 62 k molecules. For these geometries, OE62 supplies total energies and orbital eigenvalues at the PBE and the PBE hybrid (PBE0) functional level of DFT for all 62 k molecules in vacuum as well as at the PBE0 level for a subset of 30,876 molecules in (implicit) water. For 5,239 molecules in vacuum, the dataset provides quasiparticle energies computed with many-body perturbation theory in the G0W0 approximation with a PBE0 starting point (denoted GW5000 in analogy to the GW100 benchmark set (M. van Setten et al. J. Chem. Theory Comput. 12, 5076 (2016))).

Determination of the ionization energy and the electron affinity of organic molecular crystals from first-principles: dependence on the molecular orientation at the surface

I demonstrate that the ionization energy (IE) and the electron affinity (EA) of organic molecular crystals can be predicted from first-principles. Here, I describe the induced electronic polarization and the electrostatic effects upon crystalline IE and EA. I also demonstrate that the electronic polarization mainly originates from the screened Coulomb interaction inside the crystalline bulk phase, and that the electrostatic contribution to IE and EA crucially depends on the orientation of the molecule at the surface. The former is well described by the GW approximation, while the latter is reasonably estimated by the difference in frontier orbital energy between the gas phase and the surface at the level of a generalized gradient approximation to the density functional theory. The present methodology enables us to demonstrate the impact of the electrostatic effect upon the energy level of the injected charge at a multi-monolayer surface of an organic semiconductor thin film.

Pyrene-stabilized acenes as intermolecular singlet fission candidates: importance of exciton wave-function convergence

Singlet fission (SF) is a photophysical process considered as a possible scheme to bypass the Shockley–Queisser limit by generating two triplet-state excitons from one high-energy photon. Polyacene crystals, such as tetracene and pentacene, have shown outstanding SF performance both theoretically and experimentally. However, their instability prevents them from being utilized in SF-based photovoltaic devices. In search of practical SF chromophores, we use many-body perturbation theory within the GW approximation and Bethe–Salpeter equation to study the excitonic properties of a family of pyrene-stabilized acenes. We propose a criterion to define the convergence of exciton wave-functions with respect to the fine k-point grid used in the BerkeleyGW code. An open-source Python code is presented to perform exciton wave-function convergence checks and streamline the double Bader analysis of exciton character. We find that the singlet excitons in pyrene-stabilized acenes have a higher degree of charge transfer character than in the corresponding acenes. The pyrene-fused tetracene and pentacene derivatives exhibit comparable excitation energies to their corresponding acenes, making them potential SF candidates. The pyrene-stabilized anthracene derivative is considered as a possible candidate for triplet–triplet annihilation because it yields a lower SF driving force than anthracene.

Formation of normal surface plasmon modes in small sodium nanoparticles.

Formation of surface plasmon modes in sodium nanoclusters containing 20-300 atoms was studied using the G0W0 approximation. It is shown that in the small Na nanoparticles up to 2 nm in size, the loss function Im[ε-1] is dominated by a single peak corresponding to localized surface plasmon resonance (LSPR). For particles of 2 nm and more, a resonance corresponding to surface plasmon polariton (SPP) oscillations begins to form, as well as a resonance corresponding to volume plasmon (VP) excitations. Considering the above, the linear size of a particle in the range of 0.7-3.7 nm can be estimated as the lower limit for metal nanodevices operating with SPPs. On the example of spherical nanoparticles consisting of a silicon core coated with sodium atoms, it is shown that the LSPR mode is selectively suppressed while the SPP mode is not. Such composite structures can be considered as an example of nanoplasmonic devices with selectively tuned characteristics.

Электронная структура и нелинейная диэлектрическая восприимчивость γ-фазы оксида теллура

The structural electronic and nonlinear optical properies of the γ-TeO2 crystal is studied by ab inito quantum mechanical calculations. The incluence of 5d electrons localization effect is taken into account in calculations by using Habbard-like correction term (LDA+U approximation). The obtained within such approach structural parameters are obtained in a good agreement with experimental data. The electronic bund structure is studied within quasiparticle G0W0 approximation, which is known as the most accurate method. It was found that the γ-TeO2 crystal is wide gap semiconductor with non-direct band gap. The analysis of the chemical bond is performed using Maximaly Localized Wannier functions. It is established that the valence electrons of ozigen atoms are belong to sp3-hybridization states and valence of tellurium atoms are equal to four.

Ab Initio Many-Body Perturbation Theory Calculations of the Electronic and Optical Properties of Cyclometalated Ir(III) Complexes

Cyclometalated Ir(III) compounds are the preferred choice as organic emitters in organic light-emitting diodes. In practice, the presence of the transition metal surrounded by carefully designed ligands allows fine-tuning of the emission frequency as well as good efficiency of the device. To support the development of new compounds, experimental measurements are generally compared with absorption and emission spectra obtained from ab initio calculations. The standard approach for these calculations is time-dependent density functional theory (TDDFT) with a hybrid exchange-correlation functional like B3LYP. Because of the size of these compounds, the application of more complex quantum chemistry approaches can be challenging. In this work, we used many-body perturbation theory approaches, in particular the GW approximation with the Bethe-Salpeter equation (BSE) implemented in Gaussian basis sets, to calculate the quasiparticle properties and the absorption spectra of six cyclometalated Ir(III) complexes, going beyond TDDFT. In the presented results, we compared standard TDDFT simulations with BSE calculations performed on top of perturbative G0W0 and accounting for eigenvalue self-consistency. Moreover, in order to investigate in detail the effect of the DFT starting point, we concentrated on Ir(ppy)3 and performed GW-BSE simulations starting from different DFT exchange-correlation potentials.

Improving the efficiency of G0W0 calculations with approximate spectral decompositions of dielectric matrices.

Recently, it was shown that the calculation of quasiparticle energies using the G0W0 approximation can be performed without computing explicitly any virtual electronic states, by expanding the Green function and screened Coulomb interaction in terms of the eigenstates of the static dielectric matrix. Avoiding the evaluation of virtual electronic states leads to improved efficiency and ease of convergence of G0W0 calculations. Here, we propose a further improvement of the efficiency of these calculations, based on an approximation of density-density response functions of molecules and solids. The approximation relies on the calculation of a subset of eigenvectors of the dielectric matrix using the kinetic operator instead of the full Hamiltonian, and it does not lead to any substantial loss of accuracy for the quasiparticle energies. The computational savings introduced by this approximation depend on the system, and they become more substantial as the number of electrons increases.

Questaal: A package of electronic structure methods based on the linear muffin-tin orbital technique

This paper summarises the theory and functionality behind Questaal, an open-source suite of codes for calculating the electronic structure and related properties of materials from first principles. The formalism of the linearised muffin-tin orbital (LMTO) method is revisited in detail and developed further by the introduction of short-ranged tight-binding basis functions for full-potential calculations. The LMTO method is presented in both Green’s function and wave function formulations for bulk and layered systems. The suite’s full-potential LMTO code uses a sophisticated basis and augmentation method that allows an efficient and precise solution to the band problem at different levels of theory, most importantly density functional theory, LDA+U, quasi-particle self-consistent GW and combinations of these with dynamical mean field theory. This paper details the technical and theoretical bases of these methods, their implementation in Questaal, and provides an overview of the code’s design and capabilities. Program summaryProgram Title: QuestaalProgram Files doi:http://dx.doi.org/10.17632/35jxxtzpdn.1Code Ocean Capsule:https://doi.org/10.24433/CO.3778701.v1Licensing provisions: GNU General Public License, version 3Programming language: Fortran, C, Python, ShellNature of problem: Highly accurate ab initio calculation of the electronic structure of periodic solids and of the resulting physical, spectroscopic and magnetic properties for diverse material classes with different strengths and kinds of electronic correlation.Solution method: The many electron problem is considered at different levels of theory: density functional theory, many body perturbation theory in the GW approximation with different degrees of self consistency (notably quasiparticle self-consistent GW) and dynamical mean field theory. The solution to the single-particle band problem is achieved in the framework of an extension to the linear muffin-tin orbital (LMTO) technique including a highly precise and efficient full-potential implementation. An advanced fully-relativistic, non-collinear implementation based on the atomic sphere approximation is used for calculating transport and magnetic properties.

Relaxation of electrons in quantum-confined states in Pb/Si(111) thin films from master equation with first-principles-derived rates

Atomically thin films of Pb on Si(111) provide an experimentally tunable system comprising a highly structured electronic density of states. The lifetime of excited electrons in these states is limited by both electron–electron (e–e) and electron–phonon (e–ph) scattering. We employ the description by a master equation for the electronic occupation numbers to analyze the relative importance of both scattering mechanisms. The electronic and phononic band structures, as well as the matrix elements for electron–phonon coupling within deformation potential theory were obtained from density functional calculations, thus taking into account quantum confinement effects. For the relaxation dynamics, the contribution of impact ionization processes to the lifetime is estimated from the imaginary part of the electronic self-energy calculated in the GW approximation. By numerically solving rate equations for the occupations of the Pb-derived electronic states coupled to a phononic heat bath, we are able to follow the distribution of the electronic excitation energy to the various modes of Pb lattice vibrations. While e–e scattering is the dominant relaxation mechanism, we demonstrate that the e–ph scattering is highly phonon-mode-specific, with a large contribution from surface phonons. At electron energies of about 0.3 eV above the Fermi surface, a ‘phonon bottleneck’ characteristic of relaxation in nanostructures with well-separated electronic states is observed. The time scales extracted from the simulations are compared to data from pump-probe experiments using time-resolved two-photon photoemission.

Electronic structure and core electron fingerprints of caesium-based multi-alkali antimonides for ultra-bright electron sources

The development of novel photocathode materials for ultra-bright electron sources demands efficient and cost-effective strategies that provide insight and understanding of the intrinsic material properties given the constraints of growth and operational conditions. To address this question, we propose a viable way to establish correlations between calculated and measured data on core electronic states of Cs-K-Sb materials. To do so, we combine first-principles calculations based on all-electron density-functional theory on the three alkali antimonides Cs3Sb, Cs2KSb, and CsK2Sb with x-ray photoemission spectroscopy (XPS) on Cs-K-Sb photocathode samples. Within the GW approximation of many-body perturbation theory, we obtain quantitative predictions of the band gaps of these materials, which range from 0.57 eV in Cs2KSb to 1.62 eV in CsK2Sb and manifest direct or indirect character depending on the relative potassium content. Our theoretical electronic-structure analysis also reveals that the core states of these systems have binding energies that depend only on the atomic species and their crystallographic sites, with largest shifts of the order of 2 eV and 0.5 eV associated to K 2p and Sb 3d states, respectively. This information can be correlated to the maxima in the XPS survey spectra, where such peaks are clearly visible. In this way, core-level shifts can be used as fingerprints to identify specific compositions of Cs-K-Sb materials and their relation with the measured values of quantum efficiency. Our results represent the first step towards establishing a robust connection between the experimental preparation and characterization of photocathodes, the ab initio prediction of their electronic structure, and the modeling of emission and beam formation processes.

Evening out the spin and charge parity to increase {T}_{c} in {{rm{Sr}}}_{2}{{rm{RuO}}}_{4}

Unconventional superconductivity in {{rm{Sr}}}_{2}{{rm{RuO}}}_{4} has been intensively studied for decades. However, the nature of pairing continues to be widely debated. Here we develop a detailed ab initio theory, coupling quasiparticle self-consistent GW approximation with dynamical mean field theory (DMFT), including both local and non-local correlations to address the subtle interplay among spin, charge and orbital degrees of freedom. We report that the superconducting instability has multiple triplet and singlet components. In the unstrained case the triplet eigenvalues are larger than the singlets. Under uniaxial strain, the triplet eigenvalues drop and the singlet components increase. This is concomitant with our observation of spin and charge fluctuations shifting closer to wave-vectors favoring singlet pairing. We identify a complex mechanism where charge fluctuations and spin fluctuations co-operate in the even-parity channel under strain leading to increment in critical temperature (Tc), thus proposing a novel mechanism for pushing the frontier of critical temperature (Tc) in unconventional ‘triplet’ superconductors.

Quasiparticle and Optical Properties of Hydrogen Titanate and Its Defective Systems: An Investigation by Density Functional Theory with Hubbard Correction, Many‐Body Perturbation Theory, and Bethe–Salpeter Equation

Hydrogen‐rich titanium oxides have recently attracted attention as active photocatalysts with excellent photoabsorption. Herein, the optical properties of hydrogen titanate (H2Ti3O7) are calculated both for the perfect crystal and in the presence of a high concentration of oxygen, hydroxyl, and hydrogen vacancies (VO, VOH, and VH, respectively). Spin‐polarized density functional theory calculations with the Hubbard correction (DFT+U) predict VH as the dominant defect, as a consequence of the low formation energy and the insignificant accompanied structural rearrangement. By the GW approximation, it is found that VO and VOH create localized deep defect states below the conduction band (by 2.11 and 3.30 eV, respectively), whereas VH leads to a shift of the Fermi level moved inside the valence band. All the vacancies result in excitonic transitions in the infrared and visible regions in the absorption spectra, as calculated by solving the Bethe–Salpeter equation (BSE). Overall, comparison between single‐particle theories and BSE results shows that excitonic effects are very important in this material. Delocalized excitons are observed in the case of VO and VH, which can be important to suppress electron–hole recombination, thus contributing to the enhancement of the photocatalytic activity of the material.

Stochastic Resolution of Identity for Real-Time Second-Order Green's Function: Ionization Potential and Quasi-Particle Spectrum.

We develop a stochastic resolution of identity approach to the real-time second-order Green's function (real-time sRI-GF2) theory, extending our recent work for imaginary-time Matsubara Green's function [ Takeshita et al. J. Chem. Phys. 2019 , 151 , 044114 ]. The approach provides a framework to obtain the quasi-particle spectra across a wide range of frequencies and predicts ionization potentials and electron affinities. To assess the accuracy of the real-time sRI-GF2, we study a series of molecules and compare our results to experiments as well as to a many-body perturbation approach based on the GW approximation, where we find that the real-time sRI-GF2 is as accurate as self-consistent GW. The stochastic formulation reduces the formal computatinal scaling from O(Ne5) down to O(Ne3) where Ne is the number of electrons. This is illustrated for a chain of hydrogen dimers, where we observe a slightly lower than cubic scaling for systems containing up to Ne ≈ 1000 electrons.

Practical GW scheme for electronic structure of 3d-transition-metal monoxide anions: ScO-, TiO-, CuO-, and ZnO.

The GW approximation to many-body perturbation theory is a reliable tool for describing charged electronic excitations, and it has been successfully applied to a wide range of extended systems for several decades using a plane-wave basis. However, the GW approximation has been used to test limited spectral properties of a limited set of finite systems (e.g., frontier orbital energies of closed-shell sp molecules) only for about a decade using a local-orbital basis. Here, we calculate the quasiparticle spectra of closed- and open-shell molecular anions with partially and completely filled 3d shells (shallow and deep 3d states, respectively), ScO-, TiO-, CuO-, and ZnO-, using various levels of GW theory, and compare them to experiments to evaluate the performance of the GW approximation on the electronic structure of small molecules containing 3d transition metals. We find that the G-only eigenvalue self-consistent GW scheme with W fixed to the PBE level (GnW0@PBE), which gives the best compromise between accuracy and efficiency for solids, also gives good results for both localized (d) and delocalized (sp) states of 3d-transition-metal oxide molecules. The success of GnW0@PBE in predicting electronic excitations in these systems reasonably well is likely due to the fortuitous cancellation effect between the overscreening of the Coulomb interaction by PBE and the underscreening by the neglect of vertex corrections. Together with the absence of the self-consistent field convergence error (e.g., spin contamination in open-shell systems) and the GW multisolution issue, the GnW0@PBE scheme gives the possibility to predict the electronic structure of complex real systems (e.g., molecule-solid and sp-d hybrid systems) accurately and efficiently.

Optical Properties of Layered Hybrid Organic-Inorganic Halide Perovskites: A Tight-Binding GW-BSE Study.

We present a many-body calculation of the band structure and optical spectrum of the layered hybrid organic-inorganic halide perovskites in the Ruddlesden-Popper phase with the general formula A2'An-1MnX3n+1, where n controls the thickness of the primarily inorganic perovskite layers. We calculate the mean-field band structure with spin-orbit coupling, quasi-particle corrections within the GW approximation, and optical spectra using the Bethe-Salpeter equation. The model is parametrized by first-principles calculations and classical electrostatic screening, enabling an accurate but cost-effective study of large unit cells and corresponding n-dependent properties. A transition of the electronic and optical properties from quasi-two-dimensional behavior to three-dimensional behavior is shown for increasing n, and the nonhydrogenic character of the excitonic Rydberg series is analyzed. For methylammonium lead iodide perovskites with butylammonium spacers, our n-dependent 1s and 2s exciton energy levels are in good agreement with those from recently reported experiments, and the 1s exciton binding energy is calculated to be 302 meV for n = 1, 97 meV for n = 5, and 37 meV for n = ∞ (bulk MAPbI3). A calculation for an exfoliated n = 1 bilayer predicts a very large 1s exciton binding energy of 444 meV.

Real-Space Based Benchmark of G0W0 Calculations on GW100: Effects of Semicore Orbitals and Orbital Reordering.

Using an implementation based on real-space wave functions, we perform G0W0 calculations of the HOMO and LUMO energies of molecules and atoms in the GW100 set. Our main conclusions are as follows: (1) Different implementations of the G0W0 approximation show much better agreements for HOMO (highest occupied molecular orbital) energies than for LUMO (lowest unoccupied molecular orbital) energies. The mean absolute differences between the results calculated with different pseudopotential codes range from 100 to 200 meV. For delocalized LUMOs, all-electron codes that use local orbital basis tend to predict much higher energies than those calculated with plane-wave basis or real-space methods. (2) The effects of semicore electrons in pseudopotentials can explain some of the large discrepancies between results calculated with different GW implementations. For molecules or atoms that include I, Xe, and Ga, pseudopotential-based calculations that exclude semicore electrons produce results that agree better with all-electron calculations. For polar molecules such as NaCl and BrK, however, it is necessary to include semicore electrons for alkaline metal elements to get correct LUMO energies. (3) More than 20 molecules show rearrangement of the order between LUMO (or HOMO) with other orbitals due to GW corrections. Such orbitals that switch order with LUMO are unbound and delocalized in space. The predicted LUMO GW levels (or electron affinity) can be corrected by 2.0 eV if we consider the rearrangement of orbitals in GW calculations. In all, our work clarifies some of the discrepancies between different GW codes and sets a benchmark for real-space implementations of the G0W0 approximation.

SternheimerGW: A program for calculating [formula omitted] quasiparticle band structures and spectral functions without unoccupied states

The SternheimerGW software uses time-dependent density-functional perturbation theory to evaluate GW quasiparticle band structures and spectral functions for solids. Both the Green’s function G and the screened Coulomb interaction W are obtained by solving linear Sternheimer equations, thus overcoming the need for a summation over unoccupied states. The code targets the calculation of accurate spectral properties by convoluting G and W using a full frequency integration. The linear response approach allows users to evaluate the spectral function at arbitrary electron wavevectors, which is particularly useful for indirect band gap semiconductors and for simulations of angle-resolved photoelectron spectra. The software is parallelized efficiently, integrates with version 6.3 of Quantum Espresso, and is continuously monitored for stability using a test farm. Program summaryProgram Title:SternheimerGWProgram Files doi:http://dx.doi.org/10.17632/7ywzjng9t5.1Licensing provisions: GNU General Public License v3.0Programming language: Fortran 2003Nature of problem: The lack of the exchange–correlation discontinuity in density-functional theory (DFT) leads to a systematic underestimation of the band gap between conduction and valence states. Many-body perturbation theory in the GW approximation provides an effective solution to this problem, as well as other limitations faced by DFT in the description of electronic excitations. However, the GW method comes with its own set of limitations: (i) The perturbation of the system is typically expressed in terms of the unoccupied states, and achieving numerical convergence with respect to these states is often cumbersome. Since the underlying DFT codes rely only on the occupied states, their default behavior is often ill-suited to provide a sufficient amount of empty states. Combined with the large number of parameters that need to be converged, this limits the use of GW codes by non-expert users and automatic scripts. (ii) Currently, GW codes require that a homogeneous k-point mesh is used in the calculation. Hence, features close to the band edges are often only accessible via prohibitively expensive dense k-point meshes or interpolation techniques. (iii) To evaluate the frequency convolution of the Green’s function G and the screened Coulomb interaction W, many current GW calculations rely on approximations such as the plasmon-pole approximation, the analytic continuation, or the contour deformation. The relative merits and accuracy of the various approximations are not fully understood.Solution method: In SternheimerGW, we address (i) by replacing the summation over unoccupied states with a linear response equation. The solution depends on the occupied states only, and it employs linear response algorithms already provided by the Quantum ESPRESSO suite to compute phonons and related properties. As an additional benefit, transforming the problem in a linear response equation removes the restriction to homogeneous k-point meshes (ii), so that the GW self-energy for any arbitrary point can be evaluated. Finally (iii), we provide the possibility to perform a full frequency convolution along the real frequency axis. This feature can serve as a benchmark for approximate integrations using models or analytic continuation.

Potassium titanyl phosphate (KTP) quasiparticle energies and optical response

The KTiOPO4 (KTP) band structure and dielectric function are calculated on various levels of theory starting from density-functional calculations. Within the independent-particle approximation an electronic transport gap of 2.97 eV is obtained that widens to about 5.23 eV when quasiparticle effects are included using the GW approximation. The optical response is shown to be strongly anisotropic due to (i) the slight asymmetry of the TiO6 octahedra in the (001) plane and (ii) their anisotropic distribution along the [001] and [100] directions. In addition, excitonic effects are very important: The solution of the Bethe–Salpeter equation indicates exciton binding energies of the order of 1.5 eV. Calculations that include both quasiparticle and excitonic effects are in good agreement with the measured reflectivity.

Quasiparticle electronic structure of 1T’-MoS2 within GW approximation

Two-dimensional transition metal dichalcogenides, such as MoS2, exhibit several polymorphs, namely semiconducting 1H, metallic 1T, and semi-metallic 1T’. Recent experiment [Xinmao Yin et al., Nat. Commun. 8, 486 (2017)] showed an inverted gap of 0.5 eV and a fundamental gap of 0.1 eV in the absorption spectrum of the semi-metallic 1T’-MoS2. We carry out first-principles calculations on the electronic band structure of 1T’-MoS2. Since the transition across the fundamental gap occurs at a non-high-symmetry k-point, the choice of k-point sampling is crucial. Our converging result regarding k-point sampling shows that two bands touch at Fermi energy. It indicates the absence of fundamental gap. We report that spin-orbit interaction induces an opening of this fundamental gap of about 0.06 eV, which is smaller than the gap observed in experiment. To see the effects of electron-electron interaction on this fundamental gap, we calculate the quasiparticle electronic band structure within the GW approximation.