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Toward an All-Around Semilocal Potential for Electronic Exchange
Abstract
We test local and semilocal approximations of the exchange potential for a variety of systems including atoms, molecules, and atomic chains. In particular, we focus on a recent universal extension of the Becke−Johnson exchange potential [Räsänen, E.; Pittalis, S.; Proetto, C. R. J. Chem. Phys. 2010, 132, 044112]. It is shown that when this potential is used together with the Becke−Roussel approximation to the Slater potential [Becke, A. D.; Roussel, M. R. Phys. Rev. A 1989, 39, 3761−3767], a good overall agreement is obtained with experimental and numerically exact results for several systems, and with a moderate computational cost. Thus, this approximation is a very promising candidate in the quest for a simple and all-around semilocal potential.
... The RPP potential, which differs from the original BJ potential by just the extra term À ∇n j j 2 =ð4nÞ in Eq. (13), approaches zero asymptotically, is exact for any single-orbital system (in contrast with BJ) as well as for the homogeneous electron gas. It has been shown to be as accurate as the BJ potential for the ionization potential, electron affinity, and polarizability of finite systems 131 . We mention that, as originally proposed 128 , D also contains an extra term depending on the paramagnetic density-current j, however this term is not relevant here since it is zero in the present context (non-magnetic solids). ...
We conducted a large-scale density-functional theory study on the influence of the exchange-correlation functional in the calculation of electronic band gaps of solids. First, we use the large materials data set that we have recently proposed to benchmark 21 different functionals, with a particular focus on approximations of the meta-generalized-gradient family. Combining these data with the results for 12 functionals in our previous work, we can analyze in detail the characteristics of each approximation and identify its strong and/or weak points. Beside confirming that mBJ, HLE16 and HSE06 are the most accurate functionals for band gap calculations, we reveal several other interesting functionals, chief among which are the local Slater potential approximation, the GGA AK13LDA, and the meta-GGAs HLE17 and TASK. We also compare the computational efficiency of these different approximations. Relying on these data, we investigate the potential for improvement of a promising subset of functionals by varying their internal parameters. The identified optimal parameters yield a family of functionals fitted for the calculation of band gaps. Finally, we demonstrate how to train machine learning models for accurate band gap prediction, using as input structural and composition data, as well as approximate band gaps obtained from density-functional theory.
... This is due to that, (i) the expression given by BJ takes an asymptotic value far outside a finite system that depends on the eigenvalue of highest occupied KS orbital ε I , where i = I (HOMO), and, (ii) BJ have stated that the potential may be manually shifted with the system-dependent constant χ = lim |r|→∞ v BJ x (r) to achieve an asymptotic value of zero. Various modifications of the BJ potential have been shown to give band structures closer to higher-order theory [16,17], polarizabilities [15], as well as atomic and molecular properties [18,19]. In contrast, the main feature of the AK13 functional is a potential that reproduces the same asymptotic behavior, but from a consistent energy-potential pair that avoids any theoretical or practical issue with model potentials [20]. ...
The recent non-empirical semi-local exchange functional of Armiento and K\"ummel, the AK13 [PRL 111, 036402 (2013)] incorporates a number of features reproduced by higher-order theory. The AK13 potential behaves analogously with the discontinuous jump associated with the derivative discontinuity at integer particle numbers. Recent works have established that AK13 gives a qualitatively improved orbital description compared to other semi-local methods, and reproduces a band structure closer to higher-order theory. However, its energies and energetics are inaccurate. The present work further investigates the deficiency in energetics. In addition to AK13 results, we find that applying the local-density approximation (LDA) non-self-consistently on the converged AK13 density gives very reasonable energetics with equilibrium lattice constants and bulk moduli well described across 14 systems. We also confirm that the attractive orbital features of AK13 are retained even after full structural relaxation. Hence, the deficient energetics cannot be a result of the AK13 orbitals having adversely affected the quality of the electron density compared to that of usual semi-local functionals; an improved orbital description and good energetics are not in opposition. We also prove that the non-self-consistent scheme is equivalent to using a single external-potential dependent functional in an otherwise consistent KS-DFT scheme. Furthermore, our results also demonstrate that, while an internally consistent KS functional is presently missing, non-self-consistent LDA on AK13 orbitals works as a practical non-empirical computational scheme to predict geometries, bulk moduli, while retaining the band structure features of AK13 at the computational cost of semi-local DFT.
... In this case, the flexibility of the real-space method, allowing for the calculation of many different properties of a wide variety of systems, is again an advantage. Octopus has therefore been used to benchmark the performance of xc functionals whose potential has a correct asymptotic behavior [162] when calculating ionization potentials and static polarizabilities of atoms, molecules, and hydrogen chains. ...
Real-space grids are a powerful alternative for the simulation of electronic
systems. One of the main advantages of the approach is the flexibility and
simplicity of working directly in real space where the different fields are
discretized on a grid, combined with competitive numerical performance and
great potential for parallelization. These properties constitute a great
advantage at the time of implementing and testing new physical models. Based on
our experience with the Octopus code, in this article we discuss how the
real-space approach has allowed for the recent development of new ideas for the
simulation of electronic systems. Among these applications are approaches to
calculate response properties, modeling of photoemission, optimal control of
quantum systems, simulation of plasmonic systems, and the exact solution of the
Schr\"odinger equation for low-dimensionality systems.
The prominence of density functional theory in the field of electronic structure computation stems from its ability to usefully balance accuracy and computational effort. At the base of this ability is a functional of the electron density: the exchange-correlation energy. This functional satisfies known exact conditions that guide the derivation of approximations. The strongly constrained and appropriately normed (SCAN) approximation stands out as a successful, modern, example. In this Letter, we demonstrate how the SU(2) gauge invariance of the exchange-correlation functional in spin current density functional theory allows us to add an explicit dependence on spin currents in the SCAN functional (here called JSCAN)—and similar meta-generalized-gradient functional approximations—solely invoking first principles. In passing, a spin-current dependent generalization of the electron localization function (here called JELF) is also derived. The extended forms are implemented in a developer’s version of the crystal23 program. Applications on molecules and materials confirm the practical relevance of the extensions.
Spin-current density functional theory (SCDFT) is a formally exact framework designed to handle the treatment of interacting many-electron systems including spin-orbit coupling (SOC) at the level of the Pauli equation. In practice, robust and accurate calculations of the electronic structure of these systems call for functional approximations that depend not only on the densities and currents but also on spinors explicitly. Here we extend the generalized Kohn-Sham (GKS) approach of [Seidl et al., Generalized Kohn-Sham schemes and the band-gap problem, Phys. Rev. B 53, 3764 (1996)] to SCDFT. This framework entails the prominent cases of hybrid forms and meta-generalized-gradient-approximations. We clarify that the exchange-correlation potentials conjugate to the currents need to be computed within the GKS approach only when the spin currents are included in the functional form explicitly. We analyze the consequence of this fact for various approximations and numerical procedures for the evaluation of SOC effects. The practical power of the extended approach is demonstrated by calculating the spin-orbit induced/enhanced band splittings of inversion-asymmetric single-layer MoSe2 and inversion-symmetric bulk α−MoTe2. Key to these results is the capacity to account for SOC self-consistently while employing energy functionals and effective potentials that depend (implicitly or explicitly) on spin currents.
Several semilocal exchange potentials usually employed in the framework of density-functional theory are tested and compared with their exact counterpart, the exchange optimized effective potential (OEP), as applied to the jellium-slab model of a metal-vacuum interface. Driven by their explicit dependence on the ground-state density, its gradient, and its kinetic-energy density, the three analyzed semilocal exchange potentials approach their respective asymptotic limits faster than in the case of the OEP, all of them having an asymptotic scaling of the form −αe2/z+V∞, with α<1. Here, we provide the leading analytic asymptotics of the three model potentials under study, and we find that none of them exhibits the exact OEP slab asymptotics −e2/z. While the so-called Becke-Roussel potential's leading asymptote is close to its exact OEP counterpart, the other two model potentials under study approach a material-dependent positive constant value far into the vacuum, resulting in considerably overestimated ionization potentials.
The knowledge of electronic properties of matter is the key to the understanding of its properties and to propose useful applications. To model hybrid organic/inorganic systems with the plane-wave approach, large supercells with many atoms are usually necessary to minimize artificial interactions between periodic images. For such systems, accurate approximations to the exchange-correlation functional of density functional theory, such as hybrid functionals, become computationally expensive, and cheaper approaches need to be considered. Here, we apply the local modified Becke-Johnson exchange-correlation potential to free molecules and surfaces and study its accuracy for calculated ionization potentials. This quantity being important to understand the band alignment of composite heterogeneous systems, we demonstrate the application of the potential to the electronic structure calculation of an exemplary composite semiconductor/molecule system, namely, a F6-TCNNQ molecule adsorbed on a hydrogenated Si(111) surface.
This chapter reviews past and ongoing efforts in using high-throughput ab-initio calculations in combination with machine learning models for materials design. The primary focus is on bulk materials, i.e., materials with fixed, ordered, crystal structures, although the methods naturally extend into more complicated configurations. Efficient and robust computational methods, computational power, and reliable methods for automated database-driven high-throughput computation are combined to produce high-quality data sets. This data can be used to train machine learning models for predicting the stability of bulk materials and their properties. The underlying computational methods and the tools for automated calculations are discussed in some detail. Various machine learning models and, in particular, descriptors for general use in materials design are also covered.
One-dimensional (1D) systems are useful laboratories aiming further improvement of electronic structure calculations. In order to simulate electron-electron interactions, two types of expressions are commonly considered: soft-Coulomb and exponential. For both cases, in the context of density-functional theory (DFT), 1D systems can be employed to gain insight into the ingredients accurate exchange-correlation (XC) density functionals must incorporate. A question of major interest is the treatment of strongly interacting situations, one of the main modern challenges for DFT. In this manuscript, we propose a generalization of preexisting XC potentials which can be applied to investigate the transition from weak to strong interactions. Specifically, we employ the intriguing behavior of electrons confined in one dimension: the spin-charge separation, for which spin and charge are decoupled to form two independent quasiparticles, spinons, and chargons. By means of Friedel oscillations, our results indicate it is possible to reproduce the weak-strong interaction transition by using a simple strategy we name, from previous works, spin-charge separation correction (SCSC). In addition, SCSC also yields good results in reproducing the constancy of the highest occupied Kohn-Sham eigenvalues upon fractional electron charges.
The Becke–Roussel (BR) potential [Phys. Rev. A 1989, 39, 3761] was proposed as an approximation to the Slater potential, which is the Coulomb potential generated by the exact exchange hole. In the present work, a detailed comparison between the Slater and BR potentials in solids is presented. It is shown that the two potentials usually lead to very similar results for the electronic structure; however, in a few cases, e.g., Si, Ge, or strongly correlated systems like NiO, the fundamental band gap or magnetic properties can differ markedly. Such differences should not be neglected when the computationally expensive Slater potential is replaced by the cheap semilocal BR potential in approximations to the exact-exchange Kohn–Sham potential, such as the one proposed by Becke and Johnson [J. Chem. Phys. 2006, 124, 221101].
We perform benchmark calculations for the ionization potential and electronic affinity of atoms and small molecules using several semilocal exchange-correlation functionals of density-functional theory with improved asymptotic behavior. We are particularly interested in a new generalized-gradient approximation for exchange [Armiento and Kümmel, Phys. Rev. Lett. 2013, 111, 036402] that provides an energy functional whose functional derivative yields a potential with better decay behavior. We find that it yields energies that are worse than traditional energy functionals and potentials that are less accurate than functionals that model directly the exchange-correlation potential. However, we find that this functional offers a excellent balance between the quality of the energy and of the potential and is therefore a good compromise for applications that require at the same time reasonable energies and good potentials.
It is known that the separation of electrons into spinons and chargons, the spin-charge separation, plays a decisive role when describing one-dimensional (1D) strongly correlated systems [Phy. Rev. B 2012, 86 075132]. In this paper, within the density-functional theory (DFT) formalism, we extend the investigation by considering a model for the third electron fractionalization: the separation into spinons, chargons and orbitons, the last associated with the electronic orbital degree of freedom. Specifically, we deal with two exact constraints of exchange correlation (XC) density functionals: (i) the constancy of the highest occupied (HO) Kohn-Sham (KS) eigenvalues upon fractional electron numbers and (ii) their discontinuities at integers. By means of 1D discrete Hubbard chains and 1D H-2 molecules in the continuum, we find that spin-charge separation yields almost constant HO KS eigenvalues, whereas the spin-orbital counterpart can be decisive when describing derivative discontinuities of XC potentials at strong correlations.
The exact-exchange (EXX) potential, which is obtained by solving the
optimized-effective potential (OEP) equation, is compared to various
approximate semilocal exchange potentials for a set of selected solids (C, Si,
BN, MgO, Cu$_{2}$O, and NiO). This is done in the framework of the linearized
augmented plane-wave method, which allows for a very accurate all-electron
solution of electronic structure problems in solids. In order to assess the
ability of the semilocal potentials to approximate the EXX-OEP, we considered
the EXX total energy, electronic structure, electric-field gradient, and
magnetic moment. An attempt to parameterize a semilocal exchange potential is
also reported.
We derive an exchange energy functional of generalized gradient form with a corresponding potential that changes discontinuously at integer particle numbers. The functional is semilocal, yet incorporates key features that are connected to the derivative discontinuity of Kohn-Sham density-functional theory. We validate our construction for several paradigm systems and explain how it addresses central well-known deficiencies of antecedent semilocal methods, i.e., the description of charge transfer, properly localized orbitals, and band gaps. We find, e.g., an improved shell structure for atoms, eigenvalues that more closely correspond to ionization energies, and an improved description of band structure where localized states are lowered in energy.
We propose a general method for obtaining accurate valence and Rydberg excitation energies from standard density-functional approximations in adiabatic linear-response time-dependent density-functional theory. The method consists in modeling the sum of Hartree (Coulomb) and exchange-correlation potentials, vHXC(r), by the Hartree-exchange-correlation potential of the corresponding partially ionized system in which a fraction of electron charge (δ=0.15 to 0.30, depending on the functional) is removed from the highest occupied Kohn-Sham orbital level. The model potential is less repulsive and closer to exact in valence and near asymptotic regions, so it yields more accurate Kohn-Sham orbitals and orbital eigenvalues. By applying this scheme to conventional local, semilocal, and hybrid density-functional approximations, we improve their accuracy for Rydberg excitations by almost an order of magnitude without sacrificing the already good performance for valence transitions.
The density of states and the energy loss near-edge structure of the oxygen K edge in NiO are calculated using different models for the exchange-correlation functional. The results are compared to each other and to experimentally acquired energy loss spectra. It is found that only when using the modified Becke-Johnson potential are the calculated spectra in good agreement with the experimental data. This can be achieved at much less computational costs than with more sophisticated calculation methods but with as good results.
The modified Becke-Johnson exchange potential [ F. Tran and P. Blaha Phys. Rev. Lett. 102 226401 (2009)] (TB-mBJ) is tested on various types of solids which are difficult to describe theoretically: nonmagnetic semiconducting transition-metal oxides and sulfides, metals (Fe, Co, Ni, and Cu), and (anti)ferromagnetic insulators (e.g., YBa2Cu3O6). The results for the band gap and other quantities such as the magnetic moment or electric field gradient are analyzed in detail, in particular to have a better understanding of the mechanism which leads to improved (or sometimes worse) results with the TB-mBJ potential compared to the standard local density and generalized gradient approximations.
Direct approximation of exchange-correlation potentials is a promising approach to accurate prediction of molecular response properties. However, little is known about ways of obtaining total energies from model potentials other than by using the Levy-Perdew virial relation. We introduce and explore several alternative formulas which arise as line integrals of potentials taken along density scaling and aufbau-filling paths, and which are not limited to the exchange term. The relaxed-orbital variant of the aufbau-path energy expression is shown to be closely related to the Slater-Janak theorem. Although the Levy-Perdew relation generally yields reasonable energies for all model exchange potentials, the relaxed-orbital aufbau path gives better results for those potentials that predict accurate highest-occupied orbital eigenvalues, such as the potential of Räsänen, Pittalis, and Proetto [J. Chem. Phys. 132, 044112 (2010)]. The ideas presented in this work may guide the development of new types of density-functional approximations for exchange and correlation.
The common way to obtain energies from Kohn-Sham exchange potentials is by using the Levy-Perdew virial relation. For potentials that are not functional derivatives (i.e., nearly all model exchange potentials in existence), this approach leads to energy expressions that lack translational and rotational invariance. We propose a method for constructing potential-based energy functionals that are free from these artifacts. It relies on the same line-integration technique that gives rise to the Levy-Perdew relation, but uses density scaling instead of coordinate scaling. The method is applicable to any exchange or correlation potential that depends on the density explicitly, and correctly recovers the parent energy functional from a functional derivative. To illustrate our approach we develop a properly invariant generalized gradient approximation for exchange starting from the model potential of van Leeuwen and Baerends.
We propose an approach to approximate the exchange and correlation (XC) term in density functional theory. The XC potential is considered as an electrostatic potential, generated by a fictitious XC density, which is in turn a functional of the electronic density. We apply the approach to develop a correction scheme that fixes the asymptotic behavior of any approximated XC potential for finite systems. Additionally, the correction procedure gives the value of the XC derivative discontinuity; therefore, it can directly predict the fundamental gap as a ground-state property.
Standard density functional approximations greatly overestimate the static polarizability of long-chain polymers, but Hartree-Fock or exact exchange calculations do not. We show that simple self-interaction corrected approximations afford a viable alternative for accurate polarizability calculations within density functional theory.
We present a generalization of the electron localization function (ELF) that can be used to analyze time-dependent processes. The time-dependent ELF allows the time-resolved observation of the formation, the modulation, and the breaking of chemical bonds, and can thus provide a visual understanding of complex reactions involving the dynamics of excited electrons. We illustrate the usefulness of the time-dependent ELF by two examples: the π-π* transition induced by a laser field, and the destruction of bonds and formation of lone pairs in a scattering process.
Semilocal density functionals such as the local-spin-density and generalized-gradient approximations are known to overestimate the polarizabilities and especially the hyperpolarizabilities of long-chain molecules, the latter by as much as a factor of 10 or more in model hydrogen chains. These quantities are much better predicted by exact-exchange methods such as Hartree-Fock or optimized effective potential. We show here that the semilocal functionals, after full or scaled-down Perdew-Zunger self-interaction correction (SIC), are about as good as the exact-exchange methods for these quantities. As is the case for the exact-exchange methods, SIC is fully nonlocal and exact for all one-electron densities, and (more relevantly to the electrical response) tends to maintain an integer number of electrons on each H2 chain unit to a greater extent than the semilocal functionals do. In this study, the SIC energy is minimized directly, without an optimized effective potential.
The shape corrections applied to the local-density and generalized gradient approximations to the Kohn-Sham potentials for molecular response calculations of density functional theory were investigated. The calculations of polarizability and excitation energy of molecules were presented. The shape corrections were shown to concern primarily the asymptotic part of the potential and affect Rydberg type transitions. The shape corrected potentials were found to produce high quality polarizability, hyperpolarizability and reliable Cauchy moments for molecules.
DFT schemes based on conventional and less conventional exchange-correlation (XC) functionals have been employed to determine the polarizability and second hyperpolarizability of π-conjugated polyacetylene chains. These functionals fail in one or more of several ways: (i) the correlation correction to α is either much too small or in the wrong direction, leading to an overestimate; (ii) γ is significantly overestimated; (iii) the chain length dependence is excessively large, particularly for γ and for the more alternant system; and (iv) the bond length alternation effects on γ are either underestimated or qualitatively incorrect. The poor results with the asymptotically correct van Leeuwen–Baerends XC potential show that the overestimations are not related to the asymptotic behavior of the potential. These failures are described in terms of the separate effects of the exchange and the correlation parts of the XC functionals. They are related to the short-sightedness of the XC potentials which are relatively insensitive to the polarization charge induced by the external electric field at the chain ends. © 1998 American Institute of Physics.
An approximate Kohn-Sham (KS) exchange potential vxsigmaCEDA is developed, based on the common energy denominator approximation (CEDA) for the static orbital Green's function, which preserves the essential structure of the density response function. vxsigmaCEDA is an explicit functional of the occupied KS orbitals, which has the Slater vSsigma and response vrespsigmaCEDA potentials as its components. The latter exhibits the characteristic step structure with ``diagonal'' contributions from the orbital densities |psiisigma|2, as well as ``off-diagonal'' ones from the occupied-occupied orbital products psiisigmapsij(!=1)sigma*. Comparison of the results of atomic and molecular ground-state CEDA calculations with those of the Krieger-Li-Iafrate (KLI), exact exchange (EXX), and Hartree-Fock (HF) methods show, that both KLI and CEDA potentials can be considered as very good analytical ``closure approximations'' to the exact KS exchange potential. The total CEDA and KLI energies nearly coincide with the EXX ones and the corresponding orbital energies εisigma are rather close to each other for the light atoms and small molecules considered. The CEDA, KLI, EXX-Visigma values provide the qualitatively correct order of ionizations and they give an estimate of VIPs comparable to that of the HF Koopmans' theorem. However, the additional off-diagonal orbital structure of vxsigmaCEDA appears to be essential for the calculated response properties of molecular chains. KLI already considerably improves the calculated (hyper)polarizabilities of the prototype hydrogen chains Hn over local density approximation (LDA) and standard generalized gradient approximations (GGAs), while the CEDA results are definitely an improvement over the KLI ones. The reasons of this success are the specific orbital structures of the CEDA and KLI response potentials, which produce in an external field an ultranonlocal field-counteracting exchange potential.
The capability of density-functional theory to deal with the ground state of strongly correlated low-dimensional systems, such as semiconductor quantum dots, depends on the accuracy of functionals developed for the exchange and correlation energies. Here we extend a successful approximation for the correlation energy of the three-dimensional inhomogeneous electron gas, originally introduced by Becke J. Chem. Phys. 88, 1053 1988, to the two-dimensional case. The approach is based on nonempirical modeling of the correlation-hole functions satisfying a set of exact properties. Furthermore, the electron current and spin are explicitly taken into account. As a result, good performance is obtained in comparison with numerically exact data for quantum dots with varying external magnetic field, and for the homogeneous two-dimensional electron gas, respectively.
We report on the background, current status, and current lines of development of the octopus project. This program materializes the main equations of density-functional theory in the ground state, and of time-dependent density-functional theory for dynamical effects. The focus is nowadays placed on the optical (i.e. electronic) linear response properties of nanostructures and biomolecules, and on the non-linear response to high-intensity fields of finite systems, with particular attention to the coupled ionic-electronic motion (i.e. photo-chemical processes). In addition, we are currently extending the code to the treatment of periodic systems (both to one-dimensional chains, two-dimensional slabs, or fully periodic solids), magnetic properties (ground state properties and excitations), and to the field of quantum-mechanical transport or “molecular electronics.” In this communication, we concentrate on the development of the methodology: we review the essential numerical schemes used in the code, and report on the most recent implementations, with special attention to the introduction of adaptive coordinates, to the extension of our real-space technique to tackle periodic systems, and on large-scale parallelization. More information on the code, as well as the code itself, can be found at http://www.tddft.org/programs/octopus/.
The accurate prediction of electronic response properties of extended molecular systems has been a challenge for conventional, explicit density functionals. We demonstrate that a self-interaction correction (SIC) implemented rigorously within Kohn-Sham theory via the optimized effective potential (OEP) yields polarizabilities close to the ones from highly accurate wave-function-based calculations and exceeding the quality of exact-change OEP. The orbital structure obtained with the OEP-SIC functional and approximations to it are discussed.
In this work we analyze the properties of the exchange-correlation potential in the Kohn-Sham form of density-functional theory, which leads to requirements for approximate potentials. Fulfilment of these requirements is checked for existing gradient-corrected potentials. In order to examine the behavior of approximate potentials over all space we compare these potentials with exact Kohn-Sham potentials calculated from correlated densities using a newly developed iterative procedure. The main failures in the existing gradient-corrected potentials arise in the asymptotic region of the atom where these potentials decay too fast and at the atomic nucleus where the potentials exhibit a Coulomb-like singular behavior. We show that the main errors can be corrected by a simple potential in terms of the density and its gradients leading to considerably improved one-electron energies compared to the local-density approximation. For Be and Ne it is shown that the electron density is improved in the outer region.
Ab initio calculations of the static longitudinal polarizability of molecular hydrogen model chains have been performed at different levels of approximation to investigate the effects of including electron correlation. Unlike uncoupled and coupled Hartree-Fock calculations for which a split-valence atomic basis set already provides suitable longitudinal polarizability estimates, the techniques of the Mo/ller-Plesset partitioning leading to successive electron corrections, namely, MP2, MP3, and MP4, and the coupled-cluster ansatz including all double excitations, all single and double excitations, and all single and double excitations as well as a perturbational estimate of the connected triple excitations require at least additional polarization functions and a triple-zeta-type basis set in order to give suitable polarizability values. It has also been shown that including electron correlation decreases the longitudinal polarizability values and that the electron correlation effects are overemphasized when using a too small basis set. Within the Mo/ller-Plesset treatment of electron correlation, the relative importance of the different orders and the different classes of substitutions used in the intermediate states has been investigated. The double substitutions present the largest electron correlation correction to the coupled Hartree-Fock longitudinal polarizabilities per unit cell. If the atomic basis set is sufficiently extended, the third-order contribution is dominant.
The Kohn-Sham exchange potential of finite systems is shown to approach different asymptotic limits on nodal surfaces of the energetically highest-occupied orbital than in other regions. This leads to barrier-well structures in the near asymptotic region, which have a strong influence on virtual orbitals and thus on excitation energies. Common approximations for the exchange potential do not exhibit these features. These asymptotic structures, however, can be correctly described by effective exact-exchange methods. Conditions for the presence of an asymptotic barrier well in the full exchange-correlation potential are discussed.
This work assesses the Heyd-Scuseria-Ernzerhof (HSE) screened Coulomb hybrid density functional for the prediction of lattice constants and band gaps using a set of 40 simple and binary semiconductors. An extensive analysis of both basis set and relativistic effects is given. Results are compared with established pure density functionals. For lattice constants, HSE outperforms local spin-density approximation (LSDA) with a mean absolute error (MAE) of 0.037 A for HSE vs 0.047 A for LSDA. For this specific test set, all pure functionals tested produce MAEs for band gaps of 1.0-1.3 eV, consistent with the very well-known fact that pure functionals severely underestimate this property. On the other hand, HSE yields a MAE smaller than 0.3 eV. Importantly, HSE correctly predicts semiconducting behavior in systems where pure functionals erroneously predict a metal, such as, for instance, Ge. The short-range nature of the exchange integrals involved in HSE calculations makes their computation notably faster than regular hybrid functionals. The current results, paired with earlier work, suggest that HSE is a fast and accurate alternative to established density functionals, especially for solid state calculations.
The authors propose a novel approach to the problem of polarizabilities and dissociation in electric fields from the static limit of the Vignale-Kohn (VK) functional. The response to the purely scalar part of the VK response potential is considered. This potential has ground-state properties that notably improve over the full VK response density and over usual (semi-)local functionals. The correct qualitative behavior of our potentials means that it is expected to work well for polarizabilities in cases such as the H(2) chain, and it will also correctly dissociate open-shell fragments in a field.
Density Functionals for Non-relativistic Coulomb Systems in the New Century.- Orbital-Dependent Functionals for the Exchange-Correlation Energy: A Third Generation of Density Functionals.- Relativistic Density Functional Theory.- Time-Dependent Density Functional Theory.- Density Functional Theories and Self-energy Approaches.- A Tutorial on Density Functional Theory.
The Fermi hole curvature C(r,s) is defined as the Laplacian of the parallel-spin pair distribution, evaluated at zero separation r’=r for a pair of Fermions in a many-Fermion system. It has been used by a number of authors to discuss electron localization, properties of the exchange and correlation hole, and exchange and correlation energies of inhomogeneous electron gases. Here, the discussion of this quantity is extended in two directions. First, for the special case of a single-determinant many-electron state, it is shown that a previously derived macroscopic expression for C can be generalized in a simple fashion to apply to current-carrying states. Second, it is shown that a recently given interpretation of C(r,s), in terms of relative kinetic energy of pairs, is valid for a general many-Fermion state and is not limited to the single-determinant case investigated previously.
The long-standing problem of the large overestimation of polymer polarizabilities in density-functional theory is reexamined and largely solved using an exact exchange method. We have built an accurate optimized effective potential as the sum of a fixed potential and a linear combination of basis sets based on our direct optimization method. This effective potential properly develops a linear counteracting depolarization field, and it significantly improves recent results from approximate optimized potentials. The controversial case of hydrogen chains is now correctly described and the failure of the local density approach is attributed to the large self-interaction error in systems with a non-integer number of electrons.
An exchange potential functional is constructed from semi-local quantities and is shown to reproduce hydrogen chain polarizabilities with the same accuracy as exact exchange methods. We discuss the exchange potential features that are essential for accurate polarizability calculations, i.e., derivative discontinuities and the potential step structure. The possibility of a future generalization of the methods into a complete semi-local exchange-correlation functional is discussed.
Previous models for exchange (Becke and Roussel, Phys. Rev. A: 39, 3761 (1989)) and for correlation (Becke, J. Chem. Phys. 88, 1053 (1988)) are, in a simple and natural way, generalized to include explicit dependence on current density J. First-principles incorporation of J into exchange-correlation density functionals, as proposed here, is crucial for further progress in the study of magnetic effects in density-functional theory. Key words: density-functional theory, exchange-correlation functionals, current density.
The concept of the electron localization function (ELF) is extended to two-dimensional (2D) electron systems. We show that the topological properties of the ELF in two dimensions are considerably simpler than in molecules studied previously. We compute the ELF and demonstrate its usefulness for various physical 2D systems focusing on semiconductor quantum dots that effectively correspond to a confined 2D electron gas. The ELF visualizes the shell structure of harmonic quantum dots and provides insight into electron bonding in quantum-dot molecules. In external magnetic fields, the ELF is found to be a useful measure of vorticity when analyzing the properties of quantum-Hall droplets. We show that the current-dependent term in the ELF expression is important in magnetic fields.
We introduce in this work a new approach to the identification of localized electronic groups in atomic and molecular systems. Our approach is based on local behavior of the Hartree–Fock parallel‐spin pair probability and is completely independent of unitary orbital transformations. We derive a simple ‘‘electron localization function’’ (ELF) which easily reveals atomic shell structure and core, binding, and lone electron pairs in simple molecular systems as well.
We develop a coordinate‐space model for dynamical correlations in an inhomogeneous electron gas. The model treats opposite‐spin and same‐spin pairs separately, and it also accounts properly for correlation contributions to the kinetic energy. Furthermore, it gives identically zero correlation energy in the case of one‐electron systems. Applications to the uniform electron gas and to the atoms H through Ar are reported.
Predicting the polarizabilities of extended conjugated molecules with semilocal functionals has been a long-standing problem in density functional theory. These difficulties are due to the absence of a term in the typical semilocal Kohn−Sham exchange potentials that has been named “ultranonlocal”. Such a term should develop in extended systems when an external electric field is applied, and it should counteract the field. We calculate the polarizabilities of polyacetylene molecules using the recently developed extended Becke−Johnson functional. Our results show that this functional predicts the polarizabilities with much better accuracy than typical semilocal functionals. Thus, the field-counteracting term in this functional, which is semilocal in the Kohn−Sham orbitals, can realistically describe real molecules. We discuss approaches of constructing an energy functional that corresponds to this potential functional, for example, via the Levy−Perdew virial relation.
In these notes I have given a personally flavored exposé of static density-functional theory (DFT). I have started from standard many-body physics at a very elementary level and then gradually introduced the basic concepts of DFT. Successively more advanced topics are added and at the end I even discuss a few not yet published theories. The discussion represents many of the personal views of the author and there is no attempt at being comprehensive. I fully realize that I am often 'unfair' in treating the achievements of other researchers. Many topics of standard DFT are deliberately left out like, e.g., time-dependence, excitations, and magnetic or relativistic effects. These notes represent a compilation of a series of lectures given at at the EXC!TING Summer School DFT beyond the ground state at Riksgränsen, Sweden in June of 2003.
We approximate the exchange-correlation energy of density functional theory as a controlled extrapolation from the slowly varying limit. While generalized gradient approximations (GGA's) require only the local density and its first gradient as input, our meta-GGA also requires the orbital kinetic energy density. Its exchange energy component recovers the fourth-order gradient expansion, while its correlation energy is free of self-interaction error. Molecular atomization energies and metal surface energies are significantly improved over GGA, while lattice constants are little changed.
A formula is derived for the approximate calculation of the correlation energy, starting from knowledge of the one-electron and two-electron density matrices, calculated by the Hartree-Fock method.
The results that we have obtained in test calculations seem to be accurate enough (average error 2.5%, highest error 8%)
The material world of everyday experience, as studied by chemistry and condensed-matter physics, is built up from electrons
and a few (or at most a few hundred) kinds of nuclei . The basic interaction is electrostatic or Coulombic: An electron at
position r is attracted to a nucleus of charge Z at R by the potential energy −Z/|r − R|, a pair of electrons at r and r′ repel one another by the potential energy 1/|r − r′|, and two nuclei at R and R′ repel one another as Z′Z/|R − R′|. The electrons must be described by quantum mechanics, while the more massive nuclei can sometimes be regarded as classical
particles. All of the electrons in the lighter elements, and the chemically important valence electrons in most elements,
move at speeds much less than the speed of light, and so are non-relativistic.
We present a computer package aimed at the simulation of the electron–ion dynamics of finite systems, both in one and three dimensions, under the influence of time-dependent electromagnetic fields. The electronic degrees of freedom are treated quantum mechanically within the time-dependent Kohn–Sham formalism, while the ions are handled classically. All quantities are expanded in a regular mesh in real space, and the simulations are performed in real time. Although not optimized for that purpose, the program is also able to obtain static properties like ground-state geometries, or static polarizabilities. The method employed proved quite reliable and general, and has been successfully used to calculate linear and non-linear absorption spectra, harmonic spectra, laser induced fragmentation, etc. of a variety of systems, from small clusters to medium sized quantum dots.
We have performed self-consistent calculations for first and second row atoms using a variant of density-functional theory, the optimized effective potential method, with an approximation due to Krieger, Li and Iafrate and a correlation-energy functional developed by Colle and Salvetti. The mean absolute deviation of first-row atomic ground-state energies from the exact non-relativistic values is 4.7 mEh in our scheme, as compared to 4.5 mEh in a recent configuration-interaction calculation. The proposed scheme is significantly more accurate than the conventional Kohn-Sham method while the numerical effort involved is about the same as for an ordinary Hartree-Fock calculation.
We present a computer package designed to generate and test norm-conserving pseudo-potentials within Density Functional Theory. The generated pseudo-potentials can be either non-relativistic, scalar relativistic or fully relativistic and can explicitly include semi-core states. A wide range of exchange–correlation functionals is included.
DOI:https://doi.org/10.1103/PhysRev.90.317
A self-consistent set of equations is derived for an atomic central potential such that the energy given by the orbitals for the potential is minimized. It is shown that this effective potential behaves like -e2r for large r values. The equations have been solved for carbon, neon, and aluminum, and the resulting total energies exceed the Hartree-Fock total energies by less than 0.005%. The theory leads to an effective, local, central exchange potential analogous to the Xα potential.
This review provides a perspective on the use of orbital-dependent functionals, which is currently considered one of the most promising avenues in modern density-functional theory. The focus here is on four major themes: the motivation for orbital-dependent functionals in terms of limitations of semilocal functionals; the optimized effective potential as a rigorous approach to incorporating orbital-dependent functionals within the Kohn-Sham framework; the rationale behind and advantages and limitations of four popular classes of orbital-dependent functionals; and the use of orbital-dependent functionals for predicting excited-state properties. For each of these issues, both formal and practical aspects are assessed.
Accurate treatment of the electronic correlation in inhomogeneous electronic systems, combined with the ability to capture the correlation energy of the homogeneous electron gas, allows to reach high predictive power in the application of density-functional theory. For two-dimensional systems we can achieve this goal by generalizing our previous approximation [Phys. Rev. B 79, 085316 (2009)] to a parameter-free form, which reproduces the correlation energy of the homogeneous gas while preserving the ability to deal with inhomogeneous systems. The resulting functional is shown to be very accurate for finite systems with an arbitrary number of electrons with respect to numerically exact reference data.
We extend the Becke-Johnson approximation [J. Chem. Phys. 124, 221101 (2006)] of the exchange potential to two dimensions. We prove and demonstrate that a direct extension of the underlying formalism may lead to divergent behavior of the potential. We derive a cure to the approach by enforcing the gauge invariance and correct asymptotic behavior of the exchange potential. The procedure leads to an approximation which is shown, in various quasi-two-dimensional test systems, to be very accurate in comparison with the exact exchange potential, and thus a considerable improvement over the commonly applied local-density approximation. Comment: submitted to Phys. Rev. B on July 9th, 2009
The Becke-Johnson exchange potential [A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, 221101 (2006)] has been successfully used in electronic structure calculations within density-functional theory. However, in its original form, the potential may dramatically fail in systems with non-Coulombic external potentials, or in the presence of external magnetic or electric fields. Here, we provide a system-independent correction to the Becke-Johnson approximation by (i) enforcing its gauge-invariance and (ii) making it exact for any single-electron system. The resulting approximation is then better designed to deal with current-carrying states and recovers the correct asymptotic behavior for systems with any number of electrons. Tests of the resulting corrected exchange potential show very good results for a hydrogen chain in an electric field and for a four-electron harmonium in a magnetic field.
A modified version of the exchange potential proposed by Becke and Johnson [J. Chem. Phys. 124, 221101 (2006)10.1063/1.2213970] is tested on solids for the calculation of band gaps. The agreement with experiment is very good for all types of solids we considered (e.g., wide band gap insulators, sp semiconductors, and strongly correlated 3d transition-metal oxides) and is of the same order as the agreement obtained with the hybrid functionals or the GW methods. This semilocal exchange potential, which recovers the local-density approximation (LDA) for a constant electron density, mimics very well the behavior of orbital-dependent potentials and leads to calculations which are barely more expensive than LDA calculations. Therefore, it can be applied to very large systems in an efficient way.
A model exchange-correlation potential constructed with Kohn-Sham orbitals should be a functional derivative of some density functional. Several necessary conditions for a functional derivative are discussed including: (i) minimization of the total-energy expression by the ground-state solution of the Kohn-Sham equations, (ii) path independence of the van Leeuwen-Baerends line integral, and (iii) net zero force and zero torque on the density. A number of existing model potentials are checked for these properties and it is found that most of the potentials tested are not functional derivatives. Physical properties obtained from potentials that have no parent functionals are ambiguous and, therefore, should be interpreted with caution.
We present applications of the recently introduced ``Generalized SIC-Slater''
scheme which provides a simple Self-Interaction Correction approximation in the
framework of the Optimized Effective Potential. We focus on the computation of
static polarizabilities which are known to constitute stringent tests for
Density Functional Theory. We apply the new method to model H chains, but also
to more realistic systems such as C4 (organic) chains, and less symmetrical
systems such as a Na5 (metallic) cluster. Comparison is made with other SIC
schemes, especially with the standard SIC-Slater one.
We present a new coordinate-space model of spherically averaged exchange-hole functions in inhomogeneous systems that depends on local values of the density and its gradient and Laplacian, and also the kinetic energy density. Our model is completely nonempirical, incorporates the uniform-density electron gas and hydrogenic atom limits, and yields the proper 1/r asymptotic exchange potential in finite systems. Comparisons of model exchange energies, holes, and potentials with exact Hartree-Fock results in selected atoms are very encouraging.
An accurate spin-polarized exchange-only Kohn-Sham (KS) potential is constructed from a consideration of the optimized-effective-potential (OEP) method. A detailed analysis of the OEP integral equation for the exchange-only case results in a set of conditions which are manifestly satisfied by the exact OEP; these conditions are employed to construct an approximate OEP, Vxσ, and therefore an approximate KS exchange-only potential as a functional of KS orbitals. Further, it is shown that this Vxσ can be derived analytically based on a simple approximation of the Green’s functions in the OEP integral equation. The constructed potential, although approximate, contains many of the key analytic features of the exact KS potential: it reduces to the exact KS result in the homogeneous-electron-gas limit, approaches -1/r as r→∞, yields highest occupied-orbital energy eigenvalues ɛmσ that satisfy Koopmans’s theorem, and exhibits an integer discontinuity when considered as a function of fractional occupancy of the highest-energy occupied single-particle state of a given spin projection σ. In addition ɛmσ nearly exactly satisfies Janak’s theorem. The approximate OEP is a simple but remarkably accurate representation of the exact, numerically derived exchange-only OEP.
We present accurate optimized-potential-model (OPM) results for spherical spin-polarized atoms emphasizing the precise construction of the OPM exchange potential from the numerical solution of the OPM integral equation, especially for large r. The results are used to discuss the quality of the local spin-density approximation (LSDA) and a generalized-gradient expansion (GGA) [A. D. Becke, Phys. Rev. A 38, 3098 (1988)] for describing these atoms. It is shown that the LSDA can produce substantial errors (beyond what is known from unpolarized atoms) for quantities which are directly related to the spin polarization of these systems. In particular, the LSDA overestimates the magnetization density in the interior of Cu by a factor of 2. While the GGA improves integral quantities like total ground-state and exchange energies, remarkably it is less successful for energy differences like Ex↑-Ex↓. Most important, however, it is not able to reduce the LSDA's errors for local quantities like the difference between spin-up and spin-down exchange potentials and magnetization densities significantly nor does it reverse the LSDA's incorrect ordering of the two highest occupied majority-spin eigenvalues of Cr and Cu.
We propose a simple analytic representation of the correlation energy εc for a uniform electron gas, as a function of density parameter rs and relative spin polarization zeta. Within the random-phase approximation (RPA), this representation allows for the r-3/4s behavior as rs-->∞. Close agreement with numerical RPA values for εc(rs,0), εc(rs,1), and the spin stiffness alphac(rs)=∂2εc(rs, zeta=0)/deltazeta2, and recovery of the correct rslnrs term for rs-->0, indicate the appropriateness of the chosen analytic form. Beyond RPA, different parameters for the same analytic form are found by fitting to the Green's-function Monte Carlo data of Ceperley and Alder [Phys. Rev. Lett. 45, 566 (1980)], taking into account data uncertainties that have been ignored in earlier fits by Vosko, Wilk, and Nusair (VWN) [Can. J. Phys. 58, 1200 (1980)] or by Perdew and Zunger (PZ) [Phys. Rev. B 23, 5048 (1981)]. While we confirm the practical accuracy of the VWN and PZ representations, we eliminate some minor problems with these forms. We study the zeta-dependent coefficients in the high- and low-density expansions, and the rs-dependent spin susceptibility. We also present a conjecture for the exact low-density limit. The correlation potential musigmac(rs,zeta) is evaluated for use in self-consistent density-functional calculations.
Generalized gradient approximations (GGA{close_quote}s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. {copyright} {ital 1996 The American Physical Society.}
The electron density, its gradient, and the Kohn-Sham orbital kinetic energy density are the local ingredients of a meta-generalized gradient approximation (meta-GGA). We construct a meta-GGA density functional for the exchange-correlation energy that satisfies exact constraints without empirical parameters. The exchange and correlation terms respect two paradigms: one- or two-electron densities and slowly varying densities, and so describe both molecules and solids with high accuracy, as shown by extensive numerical tests. This functional completes the third rung of "Jacob's ladder" of approximations, above the local spin density and GGA rungs.
The electrical response of molecular chains is dramatically overestimated
by local and semilocal density functionals. We show that Kohn-Sham density-functional
theory yields accurate linear and nonlinear polarizabilities when the exact
exchange energy is employed together with the corresponding exact Kohn-Sham
potential vx(r). We further show that approximations
to vx(r) that are very accurate
for the ground-state energy can nevertheless fail badly for the response because
of potential barriers that have little effect on the ground-state energy but
strongly affect the electron mobility.
The optimized effective potential (OEP) for exchange was introduced some time ago by Sharp and Horton and by Talman and Shadwick. The integral equation for the OEP is difficult to solve, however, and a variety of approximations have therefore been proposed. These are explicitly orbital dependent and require the same two-electron integrals as Hartree-Fock theory. We have found a remarkably simple approximate effective potential that closely resembles the Talman-Shadwick potential in atoms. It depends only on total densities and requires no two-electron integrals.