Kohn-Sham Gap Research Articles

Can the Kohn-Sham gap be larger than the fundamental gap?

It is often stated that the exact Kohn-Sham HOMO-LUMO gap underestimates the fundamental gap. Here, using the Hubbard dimer as a model Hamiltonian, we show numerically that the exact Kohn-Sham gap can be larger than the fundamental gap.

Many-Body Calculations of Excitons in Two-Dimensional GaN

We present an ab initio study on quasiparticle (QP) excitations and excitonic effects in two-dimensional (2D) GaN based on density-functional theory and many-body perturbation theory. We calculate the QP band structure using GW approximation, which generates an indirect band gap of 4.83 eV (K→Γ) for 2D GaN, opening up 1.24 eV with respect to its bulk counterpart. It is shown that the success of plasmon-pole approximation in treating the 2D material benefits considerably from error cancellation. On the other hand, much better gaps, comparable to GW ones, could be obtained by correcting the Kohn–Sham gap with a derivative discontinuity of the exchange–correlation functional at much lower computational cost. To evaluate excitonic effects, we solve the Bethe–Salpeter equation (BSE) starting from Kohn–Sham eigenvalues with a scissors operator to open the single-particle gap. This approach yields an exciton binding energy of 1.23 eV in 2D GaN, which is in good agreement with the highly demanding GW-BSE results. The enhanced excitonic effects due to reduced dimensionality are discussed by comparing the optical spectra from BSE calculations with that by random-phase approximation (RPA) for both the monolayer and bulk GaN in wurtzite phase. Additionally, we find that the spin–orbit splitting of excitonic peaks is noticeable in 2D GaN but buried in the bulk crystal.

Shell DFT-1/2 method towards engineering accuracy for semiconductors: GGA versus LDA

The Kohn-Sham gaps of density functional theory (DFT) obtained in terms of local density approximation (LDA) or generalized gradient approximation (GGA) cannot be directly linked to the fundamental gaps of semiconductors, but in engineering there is a strong demand to match them through certain rectification methods. Shell DFT-1/2 (shDFT-1/2), as a variant of DFT-1/2, is a potential candidate to yield much improved band gaps for covalent semiconductors, but its accuracy depends on the LDA/GGA ground state, including optimized lattice parameters, basic Kohn-Sham gap before self-energy correction and the amount of self-energy correction that is specific to the exchange-correlation (XC) functional. In this work, the LDA/GGA as well as shDFT-1/2 results of six technically important covalent semiconductors Si, Ge, GaN, GaP, GaAs and GaSb were tested, with an additional ionic insulator LiF for comparison. The impact of XC flavor (LDA, PBEsol, PBE and RPBE), either directly on the gap value, or indirectly through the optimized lattice constant, is examined comprehensively. Moreover, the impact of XC flavor on LDA/GGA and shDFT-1/2 gaps was investigated under the condition of fixed experimental lattice constants. In-depth analysis reveals the rule of reaching the best accuracy in calculating the electronic band structures of typical covalent semiconductors. Relevant parameters like lattice constant, self-consistency in shDFT-1/2 runs, as well as the exchange enhancement factor of GGA, are discussed in details.

Exact exchange-correlation potentials for calculating the fundamental gap with a fixed number of electrons

Capturing the discontinuous shift by $\Delta$ in the exact exchange-correlation (xc) potential is the standard proposal for calculating the fundamental gap, $E_\mathrm{g}$, from the Kohn-Sham (KS) gap, $\varepsilon_\mathrm{g}$, within KS density functional theory (DFT), as $E_\mathrm{g} = \varepsilon_\mathrm{g} + \Delta$, yet this discontinuity is absent from existing approximations. The '$N$-centered' formulation of ensemble DFT artificially maintains a total electron number, $N$, in order to yield $E_\mathrm{g}$ not through a discontinuous shift in the xc potential but via the ensemble-weight derivative of the xc energy. Within the $N$-centered approach we calculate exact xc potentials for a one-dimensional finite system and show analytically that $\Delta$ can in fact be interpreted as a discontinuous shift in the exact $N$-centered ensemble xc potential, thereby extending to charged excitations an exact property of uncharged excitations. We show that applying the Levy-Zahariev 'shift-in-potential' procedure in this context relocates the discontinuous shift to the unimportant periphery of the system, so that the exact xc potential in effect is free of discontinuities and thus the inability of a local functional to capture discontinuous behavior is inconsequential.

Accurate optical spectra of solids from pure time-dependent density functional theory

We present accurate optical spectra of semiconductors and insulators within a pure Kohn-Sham time-dependent density-functional approach. In particular, we show that the onset of the absorption is well reproduced when comparing to experiment. No empirical information nor a theory beyond Kohn-Sham density-functional theory, such as $GW$, is invoked to correct the Kohn-Sham gap. Our approach relies on the link between the exchange-correlation kernel of time-dependent density functional theory and the derivative discontinuity of ground-state density-functional theory. We show explicitly how to relate these two quantities. We illustrate the accuracy and simplicity of our approach by applying it to various semiconductors (Si, GaP, GaAs) and wide-gap insulators (C, LiF, Ar).

The role of edge magnetism on the Kohn-Sham gap and fundamental energy gap of graphene quantum dots with zigzag edges

Energy gap engineering in graphene nanostructures is one of the most important topics towards development of graphene-based electronics. In this work, based on the density functional theory, the role of the edge magnetism on the size dependence of Kohn-Sham gap and fundamental energy gap for C6nnH6n (n=2−16) hexagonal graphene quantum dots (GQDs) with zigzag edges is studied. We found a transition from a nonmagnetic to an antiferromagnetic state at a certain critical diameter (∼ 3 nm), characterized by the opening of a Kohn-Sham gap as a consequence of the exchange interaction between localized edge states. Furthermore, the fundamental gap is obtained from the difference between the calculated vertical ionization and electron affinity energies. Such approximation includes relaxation in the exchange correlation potential when the electron is added to the system, which might be useful for GQDs transport properties interpretation. We found a scaling rule for the fundamental gap dependence on quantum dot size, providing a practical way to predict this property for large GQDs with zigzag edges, which currently in most demanding approaches, such as GW, is unfeasible.

Discovering two-dimensional topological insulators from high-throughput computations

We have performed a computational screening of topological two-dimensional (2D) materials from the Computational 2D Materials Database (C2DB) employing density functional theory. A full \textit{ab initio} scheme for calculating hybrid Wannier functions directly from the Kohn-Sham orbitals has been implemented and the method was used to extract $\mathbb{Z}_2$ indices, Chern numbers and Mirror Chern numbers of 3331 2D systems including both experimentally known and hypothetical 2D materials. We have found a total of 46 quantum spin Hall insulators, 7 quantum anomalous Hall insulators and 9 crystalline topological insulators that are all predicted to be dynamically stable. Roughly one third of these were known prior to the screening. The most interesting of the novel topological insulators are investigated in more detail. We show that the calculated topological indices of the quantum anomalous Hall insulators are highly sensitive to the approximation used for the exchange-correlation functional and reliable predictions of the topological properties of these materials thus require methods beyond density functional theory. We also performed $GW$ calculations, which yield a gap of 0.65 eV for the quantum spin Hall insulator PdSe$_2$ in the MoS$_2$ crystal structure. This is significantly higher than any known 2D topological insulator and three times larger than the Kohn-Sham gap.

Exact partition potential for model systems of interacting electrons in 1-D

We find the numerically exact partition potential for 1-D systems of interacting electrons designed to model diatomic molecules. At integer fragment occupations, the kinetic contribution to the partition potential develops sharp features in the internuclear region that nearly cancel corresponding features of exchange-correlation. They occur at locations that coincide with those of well-known features of the underlying molecular Kohn-Sham potential. For non-integer fragment occupations, we demonstrate that the fragment Kohn-Sham gaps determine the kinetic part of the partition potential. Our results highlight the importance of non-additive noninteracting kinetic and exchange-correlation energy approximations in density-embedding methods at large internuclear separations and the importance of non-additive noninteracting kinetic energy approximations at all separations.

Local vertex corrections from exchange-correlation kernels with a discontinuity

The fundamental gap of an interacting many-electron system is given by the sum of the single-particle Kohn-Sham gap and the derivative discontinuity. The latter can be generated by advanced approximations to the exchange-correlation (XC) energy and is the key quantity to capture strong correlation with density functional theory (DFT). In this work we derive an expression for the derivative discontinuity in terms of the XC kernel of time-dependent density functional theory and demonstrate the crucial role of a discontinuity in the XC kernel itself. By relating approximate XC kernels to approximate local vertex corrections we then generate beyond-GW self-energies that include a discontinuity in the local vertex function. The quantitative importance of this result is illustrated with a numerical study of the local exchange vertex on model systems.

Modelling realistic TiO2 nanospheres: A benchmark study of SCC-DFTB against hybrid DFT.

TiO2 nanoparticles (NPs) are nowadays considered fundamental building blocks for many technological applications. Morphology is found to play a key role with spherical NPs presenting higher binding properties and chemical activity. From the experimental point of view, the characterization of these nano-objects is extremely complex, opening a large room for computational investigations. In this work, TiO2 spherical NPs of different sizes (from 300 to 4000 atoms) have been studied with a two-scale computational approach. Global optimization to obtain stable and equilibrated nanospheres was performed with a self-consistent charge density functional tight-binding (SCC-DFTB) simulated annealing process, causing a considerable atomic rearrangement within the nanospheres. Those SCC-DFTB relaxed structures have been then optimized at the DFT(B3LYP) level of theory. We present a systematic and comparative SCC-DFTB vs DFT(B3LYP) study of the structural properties, with particular emphasis on the surface-to-bulk sites ratio, coordination distribution of surface sites, and surface energy. From the electronic point of view, we compare HOMO-LUMO and Kohn-Sham gaps, total and projected density of states. Overall, the comparisons between DFTB and hybrid density functional theory show that DFTB provides a rather accurate geometrical and electronic description of these nanospheres of realistic size (up to a diameter of 4.4 nm) at an extremely reduced computational cost. This opens for new challenges in simulations of very large systems and more extended molecular dynamics.

Convergence behavior of the random phase approximation renormalized correlation energy

Based on the random phase approximation (RPA), RPA renormalization [J. E. Bates and F. Furche, J. Chem. Phys. 139, 171103 (2013)] is a robust many-body perturbation theory that works for molecules and materials because it does not diverge as the Kohn-Sham gap approaches zero. Additionally, RPA renormalization enables the simultaneous calculation of RPA and beyond-RPA correlation energies since the total correlation energy is the sum of a series of independent contributions. The first-order approximation (RPAr1) yields the dominant beyond-RPA contribution to the correlation energy for a given exchange-correlation kernel, but systematically underestimates the total beyond-RPA correction. For both the homogeneous electron gas model and real systems, we demonstrate numerically that RPA renormalization beyond first order converges monotonically to the infinite-order beyond-RPA correlation energy for several model exchange-correlation kernels and that the rate of convergence is principally determined by the choice of the kernel and spin polarization of the ground state. The monotonic convergence is rationalized from an analysis of the RPA renormalized correlation energy corrections, assuming the exchange-correlation kernel and response functions satisfy some reasonable conditions. For spin-unpolarized atoms, molecules, and bulk solids, we find that RPA renormalization is typically converged to 1 meV error or less by fourth order regardless of the band gap or dimensionality. Most spin-polarized systems converge at a slightly slower rate, with errors on the order of 10 meV at fourth order and typically requiring up to sixth order to reach 1 meV error or less. Slowest to converge, however, open-shell atoms present the most challenging case and require many higher orders to converge.

Fundamental gaps with approximate density functionals: the derivative discontinuity revealed from ensemble considerations.

The fundamental gap is a central quantity in the electronic structure of matter. Unfortunately, the fundamental gap is not generally equal to the Kohn-Sham gap of density functional theory (DFT), even in principle. The two gaps differ precisely by the derivative discontinuity, namely, an abrupt change in slope of the exchange-correlation energy as a function of electron number, expected across an integer-electron point. Popular approximate functionals are thought to be devoid of a derivative discontinuity, strongly compromising their performance for prediction of spectroscopic properties. Here we show that, in fact, all exchange-correlation functionals possess a derivative discontinuity, which arises naturally from the application of ensemble considerations within DFT, without any empiricism. This derivative discontinuity can be expressed in closed form using only quantities obtained in the course of a standard DFT calculation of the neutral system. For small, finite systems, addition of this derivative discontinuity indeed results in a greatly improved prediction for the fundamental gap, even when based on the most simple approximate exchange-correlation density functional--the local density approximation (LDA). For solids, the same scheme is exact in principle, but when applied to LDA it results in a vanishing derivative discontinuity correction. This failure is shown to be directly related to the failure of LDA in predicting fundamental gaps from total energy differences in extended systems.

Modeling charged defects inside density functional theory band gaps

Density functional theory (DFT) has emerged as an important tool to probe microscopic behavior in materials. The fundamental band gap defines the energy scale for charge transition energy levels of point defects in ionic and covalent materials. The eigenvalue gap between occupied and unoccupied states in conventional DFT, the Kohn–Sham gap, is often half or less of the experimental band gap, seemingly precluding quantitative studies of charged defects. Applying explicit and rigorous control of charge boundary conditions in supercells, we find that calculations of defect energy levels derived from total energy differences give accurate predictions of charge transition energy levels in Si and GaAs, unhampered by a band gap problem. The GaAs system provides a good theoretical laboratory for investigating band gap effects in defect level calculations: depending on the functional and pseudopotential, the Kohn–Sham gap can be as large as 1.1eV or as small as 0.1eV. We find that the effective defect band gap, the computed range in defect levels, is mostly insensitive to the Kohn–Sham gap, demonstrating it is often possible to use conventional DFT for quantitative studies of defect chemistry governing interesting materials behavior in semiconductors and oxides despite a band gap problem.

Kohn-Sham band gaps and potentials of solids from the optimised effective potential method within the random phase approximation

We present an implementation of the optimised effective potential (OEP) scheme for the exact-exchange (EXX) and random phase approximation (RPA) energy functionals and apply these methods to a range of bulk materials. We calculate the Kohn-Sham (KS) potentials and the corresponding band gaps and compare them to the potentials obtained by standard local density approximation (LDA) calculations. The KS gaps increase upon going from the LDA to the OEP in the RPA and finally to the OEP for EXX. This can be explained by the different depth of the potentials in the bonding and interstitial regions. To obtain the true quasi-particle gaps the derivative discontinuities or G0W0 corrections need to be added to the RPA-OEP KS gaps. The predicted G0W0@RPA-OEP quasi-particle gaps are about 5% too large compared to the experimental values. However, compared to G0W0 calculations based on local or semi-local functionals, where the errors vary between different materials, we obtain a rather consistent description among all the materials.

Ab Initio Electronic Gaps of Ge Nanodots: The Role of Self-Energy Effects

Nanostructuring of a material leads to enormous effects on its excited state properties. This study, through the application of different state-of-the-art ab initio theoretical tools, investigates the effect of size on the electronic gap of germanium nanocrystals highlighting similarities and differences with respect to equivalent silicon nanostructures. We performed both GW and ΔSCF calculations for the determination of their electronic structure. While it is known that ΔSCF corrections to the Kohn–Sham gap vanish for extended systems, the two approaches were expected to be equivalent in the limit of small clusters. However, it has been recently found that for hydrogenated Si clusters the ΔSCF gaps are systematically smaller than the GW ones, while the opposite is true for Ag clusters. In this work we find that the GW gaps are larger than the ΔSCF ones for all the Ge dots, with the exception of the smallest one. Such crossing between the ΔSCF and the GW gap values was not expected and has never been obse...

The Kohn–Sham gap, the fundamental gap and the optical gap: the physical meaning of occupied and virtual Kohn–Sham orbital energies

A number of consequences of the presence of the exchange-correlation hole potential in the Kohn-Sham potential are elucidated. One consequence is that the HOMO-LUMO orbital energy difference in the KS-DFT model (the KS gap) is not "underestimated" or even "wrong", but that it is physically expected to be an approximation to the excitation energy if electrons and holes are close, and numerically proves to be so rather accurately. It is physically not an approximation to the difference between ionization energy and electron affinity I-A (fundamental gap or chemical hardness) and also numerically differs considerably from this quantity. The KS virtual orbitals do not possess the notorious diffuseness of the Hartree-Fock virtual orbitals, they often describe excited states much more closely as simple orbital transitions. The Hartree-Fock model does yield an approximation to I-A as the HOMO-LUMO orbital energy difference (in Koopmans' frozen orbital approximation), if the anion is bound, which is often not the case. We stress the spurious nature of HF LUMOs if the orbital energy is positive. One may prefer Hartree-Fock, or mix Hartree-Fock and (approximate) KS operators to obtain a HOMO-LUMO gap as a Koopmans' approximation to I-A (in cases where A exists). That is a different one-electron model, which exists in its own right. But it is not an "improvement" of the KS model, it necessarily deteriorates the (approximate) excitation energy property of the KS gap in molecules, and deteriorates the good shape of the KS virtual orbitals.

Mapping Kohn-Sham eigenenergies onto vertical ionization energies and electron affinities

Each Kohn-Sham eigenenergy can be mapped onto a difference between the interacting ground-state energies of successive integer-electron systems via a coupling-constant integration. Occupied Kohn-Sham energies can be mapped onto vertical ionization energies and virtual Kohn-Sham energies can be mapped onto vertical electron affinities. This mapping is unique for nondegenerate Kohn-Sham energies, but degenerate Kohn-Sham levels can be mapped onto multiple ionization energies or electron affinities. Exact expressions for the first ionization and electron affinity energies lead to exact formal expressions for the difference between the Kohn-Sham gap and the fundamental gap.

Efficient Band Gap Prediction for Solids

An efficient method for the prediction of fundamental band gaps in solids using density functional theory (DFT) is proposed. Generalizing the Delta self-consistent-field (ΔSCF) method to infinite solids, the Δ-sol method is based on total-energy differences and derived from dielectric screening properties of electrons. Using local and semilocal exchange-correlation functionals (local density and generalized gradient approximations), we demonstrate a 70% reduction of mean absolute errors compared to Kohn-Sham gaps on over 100compounds with experimental gaps of 0.5-4eV, at computational costs similar to typical DFT calculations.

Kohn-Sham potential with discontinuity for band gap materials

We model a Kohn-Sham potential with a discontinuity at integer particle numbers derived from the GLLB approximation of Gritsenko et al. We evaluate the Kohn-Sham gap and the discontinuity to obtain the quasiparticle gap. This allows us to compare the Kohn-Sham gaps to those obtained by accurate many-body perturbation theory based optimized potential methods. In addition, the resulting quasiparticle band gap is compared to experimental gaps. In the GLLB model potential, the exchange-correlation hole is modeled using a GGA energy density and the response of the hole to density variations is evaluated by using the common-denominator approximation and homogeneous electron gas based assumptions. In our modification, we have chosen the PBEsol potential as the GGA to model the exchange hole, and add a consistent correlation potential. The method is implemented in the GPAW code, which allows efficient parallelization to study large systems. A fair agreement for Kohn-Sham and the quasiparticle band gaps with semiconductors and other band gap materials is obtained with a potential which is as fast as GGA to calculate.

Spin gaps and spin-flip energies in density-functional theory

Energy gaps are crucial aspects of the electronic structure of finite and extended systems. Whereas much is known about how to define and calculate charge gaps in density-functional theory (DFT), and about the relation between these gaps and derivative discontinuities of the exchange-correlation functional, much less is know about spin gaps. In this paper we give density-functional definitions of spin-conserving gaps, spin-flip gaps and the spin stiffness in terms of many-body energies and in terms of single-particle (Kohn-Sham) energies. Our definitions are as analogous as possible to those commonly made in the charge case, but important differences between spin and charge gaps emerge already on the single-particle level because unlike the fundamental charge gap spin gaps involve excited-state energies. Kohn-Sham and many-body spin gaps are predicted to differ, and the difference is related to derivative discontinuities that are similar to, but distinct from, those usually considered in the case of charge gaps. Both ensemble DFT and time-dependent DFT (TDDFT) can be used to calculate these spin discontinuities from a suitable functional. We illustrate our findings by evaluating our definitions for the Lithium atom, for which we calculate spin gaps and spin discontinuities by making use of near-exact Kohn-Sham eigenvalues and, independently, from the single-pole approximation to TDDFT. The many-body corrections to the Kohn-Sham spin gaps are found to be negative, i.e., single particle calculations tend to overestimate spin gaps while they underestimate charge gaps.