Kohn-Sham Orbitals Research Articles

Daubechies wavelets for linear scaling density functional theory.

We demonstrate that Daubechies wavelets can be used to construct a minimal set of optimized localized adaptively contracted basis functions in which the Kohn-Sham orbitals can be represented with an arbitrarily high, controllable precision. Ground state energies and the forces acting on the ions can be calculated in this basis with the same accuracy as if they were calculated directly in a Daubechies wavelets basis, provided that the amplitude of these adaptively contracted basis functions is sufficiently small on the surface of the localization region, which is guaranteed by the optimization procedure described in this work. This approach reduces the computational costs of density functional theory calculations, and can be combined with sparse matrix algebra to obtain linear scaling with respect to the number of electrons in the system. Calculations on systems of 10,000 atoms or more thus become feasible in a systematic basis set with moderate computational resources. Further computational savings can be achieved by exploiting the similarity of the adaptively contracted basis functions for closely related environments, e.g., in geometry optimizations or combined calculations of neutral and charged systems.

Hierarchy of model Kohn-Sham potentials for orbital-dependent functionals: a practical alternative to the optimized effective potential method.

We describe a method for constructing a hierarchy of model potentials approximating the functional derivative of a given orbital-dependent exchange-correlation functional with respect to electron density. Each model is derived by assuming a particular relationship between the self-consistent solutions of Kohn-Sham (KS) and generalized Kohn-Sham (GKS) equations for the same functional. In the KS scheme, the functional is differentiated with respect to density, in the GKS scheme--with respect to orbitals. The lowest-level approximation is the orbital-averaged effective potential (OAEP) built with the GKS orbitals. The second-level approximation, termed the orbital-consistent effective potential (OCEP), is based on the assumption that the KS and GKS orbitals are the same. It has the form of the OAEP plus a correction term. The highest-level approximation is the density-consistent effective potential (DCEP), derived under the assumption that the KS and GKS electron densities are equal. The analytic expression for a DCEP is the OCEP formula augmented with kinetic-energy-density-dependent terms. In the case of exact-exchange functional, the OAEP is the Slater potential, the OCEP is roughly equivalent to the localized Hartree-Fock approximation and related models, and the DCEP is practically indistinguishable from the true optimized effective potential for exact exchange. All three levels of the proposed hierarchy require solutions of the GKS equations as input and have the same affordable computational cost.

Comparison of Dyson, Hartree-Fock, Kohn-Sham, natural, and natural-bond orbitals: electron momentum spectroscopy of CH4

The Dyson, Hartree-Fock, Kohn-Sham, natural, and natural-bond orbitals of methane (CH4) were compared with the experimental electron momentum distributions. The Dyson, Hartree-Fock, Kohn-Sham theories can present a direct concise interpretation and well reproduce the experimental results, but the natural-bond orbital cannot.

Natural orbital functional theory and pairing correlation effects in electron momentum density

Occupation numbers of natural orbitals capture the physics of strong electron correlations in momentum space. A natural orbital density functional theory based on the antisymmetrized geminal product provides these occupation numbers and the corresponding electron momentum density. A practical implementation of this theory approximates the natural orbitals by the Kohn–Sham orbitals and uses a mean-field approach to estimate pairing amplitudes leading to corrections for the independent particle model. The method is applied to weakly doped La2CuO4.

Density functional tight binding

This paper reviews the basic principles of the density-functional tight-binding (DFTB) method, which is based on density-functional theory as formulated by Hohenberg, Kohn and Sham (KS-DFT). DFTB consists of a series of models that are derived from a Taylor series expansion of the KS-DFT total energy. In the lowest order (DFTB1), densities and potentials are written as superpositions of atomic densities and potentials. The Kohn-Sham orbitals are then expanded to a set of localized atom-centred functions, which are obtained for spherical symmetric spin-unpolarized neutral atoms self-consistently. The whole Hamilton and overlap matrices contain one- and two-centre contributions only. Therefore, they can be calculated and tabulated in advance as functions of the distance between atomic pairs. The second contributions to DFTB1, the DFT double counting terms, are summarized together with nuclear repulsion energy terms and can be rewritten as the sum of pairwise repulsive terms. The second-order (DFTB2) and third-order (DFTB3) terms in the energy expansion correspond to a self-consistent representation, where the deviation of the ground-state density from the reference density is represented by charge monopoles only. This leads to a computationally efficient representation in terms of atomic charges (Mulliken), chemical hardness (Hubbard) parameters and scaled Coulomb laws. Therefore, no additional adjustable parameters enter the DFTB2 and DFTB3 formalism. The handling of parameters, the efficiency, the performance and extensions of DFTB are briefly discussed.

High resolution (e, 2e) spectroscopy of dimethyl ether

We report high-resolution (e, 2e) spectra of dimethyl ether (H3COCH3) in combination with high-level theoretical calculations. The observed binding energy spectra and momentum distributions were compared with Hartree–Fock, density functional theory (DFT), and symmetry-adapted-cluster configuration-interaction (SAC-CI) calculations. In general, the theoretical momentum distributions based on Dyson orbital (DO) and Kohn–Sham orbital (KSO) agree well with the experimental momentum distributions except the discrepancies in the low momentum regions of 2b1 and 1a2 orbitals. The present high resolution experimental data provide a benchmark for the future rigorous theories.

Kohn-Sham orbitals and potentials from quantum Monte Carlo molecular densities

In this work we show the possibility to extract Kohn-Sham orbitals, orbital energies, and exchange correlation potentials from accurate Quantum Monte Carlo (QMC) densities for atoms (He, Be, Ne) and molecules (H2, Be2, H2O, and C2H4). The Variational Monte Carlo (VMC) densities based on accurate Jastrow Antisymmetrised Geminal Power wave functions are calculated through different estimators. Using these reference densities, we extract the Kohn-Sham quantities with the method developed by Zhao, Morrison, and Parr (ZMP) [Phys. Rev. A 50, 2138 (1994)]. We compare these extracted quantities with those obtained form CISD densities and with other data reported in the literature, finding a good agreement between VMC and other high-level quantum chemistry methods. Our results demonstrate the applicability of the ZMP procedure to QMC molecular densities, that can be used for the testing and development of improved functionals and for the implementation of embedding schemes based on QMC and Density Functional Theory.

Symmetry-adapted perturbation theory with Kohn-Sham orbitals using non-empirically tuned, long-range-corrected density functionals.

The performance of second-order symmetry-adapted perturbation theory (SAPT) calculations using Kohn-Sham (KS) orbitals is evaluated against benchmark results for intermolecular interactions. Unlike previous studies of this "SAPT(KS)" methodology, the present study uses non-empirically tuned long-range corrected (LRC) functionals for the monomers. The proper v xc (r)→0 asymptotic limit is achieved by tuning the range separation parameter in order to satisfy the condition that the highest occupied KS energy level equals minus the molecule's ionization energy, for each monomer unit. Tests for He2, Ne2, and the S22 and S66 data sets reveal that this condition is important for accurate prediction of the non-dispersion components of the energy, although errors in SAPT(KS) dispersion energies remain unacceptably large. In conjunction with an empirical dispersion potential, however, the SAPT(KS) method affords good results for S22 and S66, and also accurately predicts the whole potential energy curve for the sandwich isomer of the benzene dimer. Tuned LRC functionals represent an attractive alternative to other asymptotic corrections that have been employed in density-functional-based SAPT calculations, and we recommend the use of tuned LRC functionals in both coupled-perturbed SAPT(DFT) calculations and dispersion-corrected SAPT(KS) calculations.

Scandium carbide/cyanide alloyed cluster inside fullerene cage: synthesis and structural studies of Sc3(μ3-C2)(μ3-CN)@Ih-C80.

A new Sc3(C2)(CN)@Ih-C80 metallofullerene encaging a scandium carbide/cyanide alloyed cluster was prepared and investigated. Sc3(C2)(CN)@Ih-C80 was synthesized by the arc-discharging method and isolated by HPLC. Its experimental (13)C NMR spectrum with two signals clearly confirms an icosahedral C80 cage, and theoretically calculated (13)C NMR peaks agree well with the experimental results. Further, theoretical calculations disclosed that the endohedral μ3-C2 and μ3-CN moieties are situated on each side of the triangular shaped Sc3 unit to form a scandium carbide/cyanide alloyed cluster. Kohn-Sham molecular orbitals reveals the electronic structure of (Sc(3+))3(C2)(2-)(CN)(-)@C80(6-), in which two anions, μ3-C2(2-) and μ3-(CN)(-), construct and stabilize this special molecule together. The FTIR and Raman spectra of Sc3(C2)(CN)@Ih-C80 were analyzed by comparison of the experimental and calculated results to further confirm its structure and to uncover cluster-based vibrational modes.

Structure and electronic absorption spectra of isotruxene dyes for dye-sensitized solar cells: Investigation by the DFT, TDDFT, and QTAIM methods

The electronic structure of two photosensitizing isotruxene dyes used in Gratzel photoelectro-chemical cells has been investigated by the quantum-chemical methods of the density functional theory (DFT) and Bader’s topological analysis of the electron density. It has been shown that intramolecular interactions in the dye molecules stabilize the planar relative arrangement of isotruxene and cyano-acrylic structural fragments, which provides π-conjugation over the whole molecular backbone. Based on the time-dependent density functional theory (TDDFT) calculations and analysis of the Kohn-Sham orbitals, it has been found that the first vertical electronic transition in the spectra of the isotruxene dyes is a combination of a poorly resolved lπ*-type excitation and a well-resolved π-π* excitation. The π-π* excitation admixture is associated with the asymmetric structure of the isotruxene fragment and contributes significantly to the intensity of the first electronic transition.

Electron dynamics in charged wet TiO2 anatase (001) surface functionalised by ruthenium ions

The surface of charged wet TiO2 anatase (001) functionalised by ruthenium ion at ambient temperatures is studied by computational modelling. Response of this model to photoexcitations at ambient temperatures is explored with the Redfield density matrix equation of motion on the basis of Kohn–Sham orbitals. The parameters of the Redfield equation are on-the-fly non-adiabatic couplings for electronic degrees of freedom obtained along the ab initio molecular dynamics nuclear trajectories. The main results in this study are the following: (1) optical properties of the doped models such as light absorption intensity and transition energies can be tuned by modifying total charge; (2) electron and hole relaxation rates depend on the initial excitation; and (3) in the doped model, excitations of lower energy provide quicker relaxation. Results of computational modelling would benefit understanding of the mechanism of electron transfer processes on the surface of ruthenium-doped TiO2.

DFTB Parameters for the Periodic Table: Part 1, Electronic Structure.

A parametrization scheme for the electronic part of the density-functional based tight-binding (DFTB) method that covers the periodic table is presented. A semiautomatic parametrization scheme has been developed that uses Kohn-Sham energies and band structure curvatures of real and fictitious homoatomic crystal structures as reference data. A confinement potential is used to tighten the Kohn-Sham orbitals, which includes two free parameters that are used to optimize the performance of the method. The method is tested on more than 100 systems and shows excellent overall performance.

Efficient construction of exchange and correlation potentials by inverting the Kohn–Sham equations

Given a set of canonical Kohn-Sham orbitals, orbital energies, and an external potential for a many-electron system, one can invert the Kohn-Sham equations in a single step to obtain the corresponding exchange-correlation potential, vXC(r). For orbitals and orbital energies that are solutions of the Kohn-Sham equations with a multiplicative vXC(r) this procedure recovers vXC(r) (in the basis set limit), but for eigenfunctions of a non-multiplicative one-electron operator it produces an orbital-averaged potential. In particular, substitution of Hartree-Fock orbitals and eigenvalues into the Kohn-Sham inversion formula is a fast way to compute the Slater potential. In the same way, we efficiently construct orbital-averaged exchange and correlation potentials for hybrid and kinetic-energy-density-dependent functionals. We also show how the Kohn-Sham inversion approach can be used to compute functional derivatives of explicit density functionals and to approximate functional derivatives of orbital-dependent functionals.

Geometric, electronic, and optical properties of a boron-doped aluminum cluster of [formula omitted]: A density functional theory study

We present the physicochemical properties for the lowest-energy isomer of a boron-doped aluminum cluster of B2Al21-. The isomer was obtained by basin-hopping minimization based on the density functional theory, starting from a face-sharing bi-icosahedral structure in which two boron atoms were endohedrally doped to each icosahedron. The lowest-energy isomer is a triangular form in which an aluminum cage encapsulates two boron atoms endohedrally. The electronic structure was analyzed by projecting Kohn–Sham orbitals onto the spherical harmonics; occupied and unoccupied frontier orbitals are dominantly G- and H-symmetries, respectively. Optical absorption is mainly assigned to G to H transitions.

An improved treatment of empirical dispersion and a many-body energy decomposition scheme for the explicit polarization plus symmetry-adapted perturbation theory (XSAPT) method

We recently introduced a low-cost quantum chemistry method for computing intermolecular interactions, combining a monomer-based self-consistent field calculation (the "explicit polarization" method, XPol) with pairwise-additive symmetry adapted perturbation theory (SAPT). The method uses Kohn-Sham (KS) orbitals in the SAPT formalism but replaces the SAPT dispersion and exchange-dispersion terms with empirical potentials ("+D"), and we called this method XPol+SAPT(KS)+D. Here, we report a second-generation version of this approach, XPol+SAPT(KS)+D2 or XSAPT(KS)+D2 for short, in which we have modified the form of the empirical atom-atom dispersion potentials. Accurate binding energies are obtained for benchmark databases of dimer binding energies, and potential energy curves are captured accurately for a variety of challenging systems. We suggest that using different asymptotic corrections for different monomers is necessary to get good binding energies in general, especially for hydrogen-bonded complexes. As compared to our original "+D" formulation, the second-generation "+D2" method accurately reproduces not only total binding energies but also the various components of the interaction energy, and on this basis we introduce an energy decomposition scheme that extends traditional SAPT energy decomposition to systems containing more than two monomers. For (H2O)6, the many-body contribution to the interaction energy agrees well with that obtained from traditional Kitaura-Morokuma energy decomposition analysis in a large basis set.

Functional derivatives of meta-generalized gradient approximation (meta-GGA) type exchange-correlation density functionals

Meta-generalized gradient approximation (meta-GGA) exchange-correlation density functionals depend on the Kohn-Sham (KS) orbitals through the kinetic energy density. The KS orbitals in turn depend functionally on the electron density. However, the functional dependence of the KS orbitals is indirect, i.e., not given by an explicit expression, and the computation of analytic functional derivatives of meta-GGA functionals with respect to the density imposes a challenge. The practical solution used in many computer implementations of meta-GGA density functionals for ground-state calculations is abstracted and generalized to a class of density functionals that is broader than meta-GGAs and to any order of functional differentiation. Importantly, the TDDFT working equations for meta-GGA density functionals are presented here for the first time, together with the technical details of their computer implementation. The analysis presented here also uncovers the implicit assumptions in the practical solution to computing functional derivatives of meta-GGA density functionals. The connection between the approximation that is invoked in taking functional derivatives of density functionals, the non-uniqueness with respect to the KS orbitals, and the non-locality of the resultant potential is also discussed.

Isoorbital indicators for current density functional theory

Exchange-correlation density functionals at the meta-generalised gradient approximation level typically use the Kohn–Sham orbitals as an ingredient to create the isoorbital indicator , where is the von Weizsäcker kinetic energy density and is the kinetic energy density of the Kohn–Sham orbitals. When a magnetic field is included in the treatment, the Kohn–Sham orbitals vary with the vector-potential gauge, as does and implicitly the exchange-correlation energy. However, it is a known result in current density functional theory that the exact exchange-correlation energy is gauge independent. Three different gauge-independent replacements for have been proposed by other authors. Only one of these is, however, known to stay between 0 and 1. We modify one of the others, the Maximoff–Scuseria isoorbital indicator, so that it also stays between 0 and 1. We investigate the ability of the isoorbital indicators evaluated in a system with density and paramagnetic current density to reproduce the value of in a system having no paramagnetic current, but the same density . We provide the first example known to us where the Tao–Perdew isoorbital indicator goes outside of the interval from 0 to 1.

Electronic structure and photoelectron spectra of nickel(II) acetylacetonate

Density functional theory and photoelectron spectroscopy are used to study the electronic structure of nickel acetylacetonate Ni(acac)2. Based on the calculated energies and composition of the molecular orbitals and the pattern of localization of the electron density, the nature of the gas-phase photoelectron spectrum bands of Ni(acac)2 is examined, and a new interpretation of it is proposed. The energies of the Kohn-Sham orbitals are assigned to the experimental vertical ionization energies in the approximation of the extended Koopmans theorem, which enables, with consideration given to the dependence of the Koopmans defect on the nature of the electronic level, to obtain a good agreement between the calculated orbital energies and experimental data on ionization energies.

Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals

The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, ``nature's glue'. (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.

Regularization of the Coulomb singularity in exact exchange by Wigner-Seitz truncated interactions: Towards chemical accuracy in nontrivial systems

Hybrid density functionals show great promise for chemically accurate first-principles calculations, but their high computational cost limits their application in nontrivial studies, such as exploration of reaction pathways of adsorbents on periodic surfaces. One factor responsible for their increased cost is the dense Brillouin-zone sampling necessary to accurately resolve an integrable singularity in the exact exchange energy. We analyze this singularity within an intuitive formalism based on Wannier-function localization and analytically prove Wigner-Seitz truncation to be the ideal method for regularizing the Coulomb potential in the exchange kernel. We show that this method is limited only by Brillouin-zone discretization errors in the Kohn-Sham orbitals, and hence converges the exchange energy exponentially with the number of $k$ points used to sample the Brillouin zone for all but zero-temperature metallic systems. To facilitate the implementation of this method, we develop a general construction for the plane-wave Coulomb kernel truncated on the Wigner-Seitz cell in one, two, or three lattice directions. We compare several regularization methods for the exchange kernel in a variety of real systems including low-symmetry crystals and low-dimensional materials. We find that our Wigner-Seitz truncation systematically yields the best $k$-point convergence for the exchange energy of all these systems and delivers an accuracy to hybrid functionals comparable to semilocal and screened-exchange functionals at identical $k$-point sets.