Two-electron valence indices from the Kohn-Sham orbitals

The recent Hartree-Fock (HF) difference approach to the chemical valence indices (ionic and covalent), formulated in the framework of the pair-density matrix, is implemented within the Kohn-Sham (KS) density functional theory (DFT). The valence numbers are quadratic in terms of displacements of the molecular spin-resolved charge-and-bond-order (CBO) matrix elements, relative to values in the separated atoms limit (SAL). It is shown that the global valence represents a generalized “distance” quantity measuring a degree of similarity between the two CBO matrices: the molecular and SAL. Numerical values for typical molecules exhibiting single and multiple bonds demonstrate that the KS orbitals give rise to these new bond valences in good agreement with both chemical and HF predictions. This KS bond multiplicity analysis is applied to the chemisorption system including the allyl radical and a model surface cluster of molybdenum oxide. It is concluded that the quadratic valence analysis represents a valuable procedure for extracting useful chemical information from standard DFT calculations. © 1997 John Wiley & Sons, Inc.     

The calculation of electron localization and delocalization indices at the Hartree–Fock, density functional and post-Hartree–Fock levels of theory

 Localization, λ(A), and delocalization indices, δ(A,B), as defined in the atoms in molecules theory, are a convenient tool for the analysis of molecular electronic structure from an electron-pair perspective. These indices can be calculated at any level of theory, provided that first- and second-order electron densities are available. In particular, calculations at the Hartree–Fock (HF) and configuration interaction (CI) levels have been previously reported for many molecules. However, λ(A) and δ(A,B) cannot be calculated exactly in the framework of Kohn–Sham (KS) density functional theory (DFT), where the electron-pair density is not defined. As a practical workaround, one can derive a HF-like electron-pair density from the KS orbitals and calculate approximate localization and delocalization indices at the DFT level. Recently, several calculations using this approach have been reported. Here we present HF, CI and approximate DFT calculations of λ(A) and δ(A,B) values for a number of molecules. Furthermore, we also perform approximate CI calculations using the HF formalism to obtain the electron-pair density. In general, the approximate DFT and CI results are closer to the HF results than to the CI ones. Indeed, the approximate calculations take into account Coulomb electron correlation effects on the first-order electron density but not on the electron-pair density. In summary, approximate DFT and CI localization and delocalization indices are easy to calculate and can be useful in the analysis of molecular electronic structure; however, one should take into account that this approximation increases systematically the delocalization between covalently bonded atoms, with respect to the exact CI results. Received: 13 February 2002 / Accepted: 24 April 2002 / Published online: 18 June 2002     

Density functional LCAO calculations for solids: Comparison among Hartree–Fock,DFT local density approximation,and DFT generalized gradient approximation structural properties

In this article, the results of a recently implemented DFT a posteriori and Kohn-Sham (KS ) linear combination of atomic orbital computational scheme for solids are presented. The equilibrium lattice parameters, bulk moduli, and lattice energies are calculated for eight crystallized systems. Local density approximation (LDA ) and generalized gradient approximation (GCA ) functionals and potentials are used. The maps of the Hartree-Fock (HF ) and Ks electronic densities and band structures are depicted. The KS results confirm the trend of the a posteriori scheme. Very good agreement between calculated and experimental lattice energies has been found for GGA potentials. © 1995 John Wiley & Sons, Inc.     

Exact density functionals for two-electron systems in an external magnetic field

In principle, the extension of density functional theory (DFT) to Coulombic systems in a nonvanishing magnetic field is via current DFT (CDFT). Though CDFT is long established formally, relatively little is known with respect to any generally applicable, reliable approximate E(XC) and A(XC) functionals analogous with the workhorse approximate functionals (local density approximation and generalized gradient approximation) of ordinary DFT. Progress can be aided by having benchmark studies on a solvable correlated system. At zero field, the best-known finite system for such purposes is Hooke's atom. Recently we extended the exact ground state solutions for this two-electron system to certain combinations of nonzero external magnetic fields and confinement strengths. From those exact solutions, as well as high-accuracy numerical results for other field and confinement combinations, we construct the correlated electron density and paramagnetic current density, the exact Kohn-Sham orbitals, and the exact DFT and CDFT exchange-correlation energies and potentials. We compare with results from several widely used approximate functionals, all of which exhibit major qualitative failures, whether in CDFT or in naive application of ordinary DFT. We also illustrate how the CDFT vorticity variable nu is a computationally difficult quantity which may not be appropriate in practice to describe the external B field effects on E(XC) and A(XC).     

Tight-binding density functional theory: an approximate Kohn-Sham DFT scheme

The DFTB method is an approximate KS-DFT scheme with an LCAO representation of the KS orbitals, which can be derived within a variational treatment of an approximate KS energy functional. But it may also be related to cellular Wigner-Seitz methods and to the Harris functional. It is an approximate method, but it avoids any empirical parametrization by calculating the Hamiltonian and overlap matrices out of DFT-derived local orbitals (atomic orbitals, AO's). The method includes ab initio concepts in relating the Kohn-Sham orbitals of the atomic configuration to a minimal basis of the localized atomic valence orbitals of the atoms. Consistent with this approximation, the Hamiltonian matrix elements can strictly be restricted to a two-center representation. Taking advantage of the compensation of the so-called "double counting terms" and the nuclear repulsion energy in the DFT total energy expression, the energy may be approximated as a sum of the occupied KS single-particle energies and a repulsive energy, which can be obtained from DFT calculations in properly chosen reference systems. This relates the method to common standard "tight-binding" (TB) schemes, as they are well-known in solid-state physics. This approach defines the density-functional tight-binding (DFTB) method in its original (non-self-consistent) version.     

Electronic Structure of Rh(2)(&mgr;-CO)(CO)(2)(H(2)PCH(2)PH(2))(2). An Example of a Non-A-Frame Structure

Calculations based on density functional theory (DFT) and Hartree-Fock configuration interaction (HF-CI) methodology have been carried out to investigate the rhodium-rhodium coupling in Rh(2)(CO)(2)(dppm)(2), 1 (dppm = Ph(2)PCH(2)PPh(2)) and in Rh(2)(&mgr;-CO)(CO)(2)(dppm)(2), 2. DFT geometries, obtained with the Dgauss program, are in good agreement with those determined from X-ray, but HF geometries, calculated using the same basis sets, yield bond distances systematically too long. Calculations indicate that the rhodium atoms in 1 are linked by a single bond. The insertion of a semibridging carbonyl between the two metal atoms leads to a shortening of the rhodium-rhodium distance and also to a noticeable weakening of the metal-metal interaction. Both effects, and also the stabilization of the HOMO of 2, are related to an observed change from square planar to tetrahedral of the ligand environment of the Rh atom proximal to the inserted CO. Both MO analysis and bond characterization from the topology of the charge density confirm the existence of a bonding interaction between the semibridging carbonyl and the distal rhodium atom. The electronic structures of the dicationic complex [Rh(2)(CO)(3)(dppm)(2)](2+) and of the A-frame-like, isoelectronic system Rh(2)Br(2)(&mgr;-CO) (dppm)(2) are also discussed. The electron deformation density is derived from 2 by means of several methodological approaches, namely, HF, HF-CI, DFT, and DFT + gradient corrections. The HF deformation density obtained in the plane containing the metals and the three CO ligands is discussed, as well as the "correlation density" obtained from the difference maps DFT - HF and CI - HF.     

Application of Exact Analytic Total Energy Functional for Hooke’s Atom to He, Li+ and Be++: An Examination of the Universality of the Energy Functional in DFT

We employ the recently generated energy density functional for Hooke’s atom [Ludeña EV et al (2004) Intern J Quantum Chem 99:297], to which we introduce a simplification for the kinetic energy term, to evaluate the total energy of the helium atom and of the two-electron ions Li⁺ and Be⁺⁺. Using accurate representations for the one-particle densities of these systems we show that the energy density functional for Hooke’s atom leads, in these cases, to energy values that are below the exact ones. We discuss the implication of this finding with respect to the existence of a universal energy functional in DFT.     

Calculation of longitudinal polarizability and second hyperpolarizability of polyacetylene with the coupled perturbed Hartree-Fock/Kohn-Sham scheme: where it is shown how finite oligomer chains tend to the infinite periodic polymer

The longitudinal polarizability, α(xx), and second hyperpolarizability, γ(xxxx), of polyacetylene are evaluated by using the coupled perturbed Hartree-Fock/Kohn-Sham (HF/KS) scheme as implemented in the periodic CRYSTAL code and a split valence type basis set. Four different density functionals, namely local density approximation (LDA) (pure local), Perdew-Becke-Ernzerhof (PBE) (gradient corrected), PBE0, and B3LYP (hybrid), and the Hartree-Fock Hamiltonian are compared. It is shown that very tight computational conditions must be used to obtain well converged results, especially for γ(xxxx), that is, very sensitive to the number of k(->) points in reciprocal space when the band gap is small (as for LDA and PBE), and to the extension of summations of the exact exchange series (HF and hybrids). The band gap in LDA is only 0.01 eV: at least 300 k(->) points are required to obtain well converged total energy and equilibrium geometry, and 1200 for well converged optical properties. Also, the exchange series convergence is related to the band gap. The PBE0 band gap is as small as 1.4 eV and the exchange summation must extend to about 130 A? from the origin cell. Total energy, band gap, equilibrium geometry, polarizability, and second hyperpolarizability of oligomers -(C(2)H(2))(m)-, with m up to 50 (202 atoms), and of the polymer have been compared. It turns out that oligomers of that length provide an extremely poor representation of the infinite chain polarizability and hyperpolarizability when the gap is smaller than 0.2 eV (that is, for LDA and PBE). Huge differences are observed on α(xx) and γ(xxxx) of the polymer when different functionals are used, that is in connection to the well-known density functional theory (DFT) overshoot, reported in the literature about short oligomers: for the infinite model the ratio between LDA (or PBE) and HF becomes even more dramatic (about 500 for α(xx) and 10(10) for γ(xxxx)). On the basis of previous systematic comparisons of results obtained with various approaches including DFT, HF, Moller-Plesset (MP2) and coupled cluster for finite chains, we can argue that, for the infinite chain, the present HF results are the most reliable.     

eQE: An open‐source density functional embedding theory code for the condensed phase

In this work, we present the main features and algorithmic details of a novel implementation of the frozen density embedding (FDE) formulation of subsystem density functional theory (DFT) that is specifically designed to enable ab initio molecular dynamics (AIMD) simulations of large‐scale condensed‐phase systems containing 1000s of atoms. This code (available at http://eqe.rutgers.edu ) has been given the moniker of embedded Quantum ESPRESSO (eQE) as it is a generalization of the open‐source Quantum ESPRESSO (QE) suite of programs. The strengths of eQE reside in a hierarchical parallelization scheme that allows for an efficient and fully self‐consistent treatment of the electronic structure (via the addition of an additional DIIS extrapolation layer) while simultaneously exploiting the inherent symmetries and periodicities in the system (via sampling of subsystem‐specific first Brillouin zones and utilization of subsystem‐specific basis sets). While bulk liquids and molecular crystals are two classes of systems that exemplify the utility of the FDE approach (as these systems can be partitioned into weakly interacting subunits), we show that eQE has significantly extended this regime of applicability by outperforming standard semilocal Kohn–Sham DFT (KS‐DFT) for large‐scale heterogeneous catalysts with quite different layer‐specific electronic structure and intrinsic periodicities. eQE features very favorable strong parallel scaling for a model system of bulk liquid water composed of 256 water molecules, which allows for a significant decrease in the overall time to solution when compared to KS‐DFT. We show that eQE achieves speedups greater than one order of magnitude ( ) when performing AIMD simulations of such large‐scale condensed‐phase systems as: (1) molecular liquids via bulk liquid water represented by 1024 independent water molecules (3072 atoms with a 25.3× speedup over KS‐DFT), (2) polypeptide/biomolecule solvation via (gly )₆ solvated in (H₂O)₃₉₅ (1230 atoms with a 38.6× speedup over KS‐DFT), and (3) molecular crystals via a 3 × 3 × 3 periodic supercell of pentacene (1940 atoms with a 12.0× speedup over KS‐DFT). These results represent a significant improvement over the current state‐of‐the‐art and now enable subsystem DFT‐based AIMD simulations of realistically sized condensed‐phase systems of interest throughout chemistry, physics, and materials science.     

Orbital-dependent correlation energy in density-functional theory based on a second-order perturbation approach: success and failure

We have developed a second-order perturbation theory (PT) energy functional within density-functional theory (DFT). Based on PT with the Kohn-Sham (KS) determinant as a reference, this new ab initio exchange-correlation functional includes an exact exchange (EXX) energy in the first order and a correlation energy including all single and double excitations from the KS reference in the second order. The explicit dependence of the exchange and correlation energy on the KS orbitals in the functional fits well into our direct minimization approach for the optimized effective potential, which is a very efficient method to perform fully self-consistent calculations for any orbital-dependent functionals. To investigate the quality of the correlation functional, we have applied the method to selected atoms and molecules. For two-electron atoms and small molecules described with small basis sets, this new method provides excellent results, improving both second-order Moller-Plesset expression and any conventional DFT results significantly. For larger systems, however, it performs poorly, converging to very low unphysical total energies. The failure of PT based energy functionals is analyzed, and its origin is traced back to near degeneracy problems due to the orbital- and eigenvalue-dependent algebraic structure of the correlation functional. The failure emerges in the self-consistent approach but not in perturbative post-EXX calculations, emphasizing the crucial importance of self-consistency in testing new orbital-dependent energy functionals.     

Structural Characteristics of Graphane‐Type C and BN Nanostructures by Periodic Local MP2 Approach

The structural characteristics of fully‐hydrogenated carbon and boron nitride mono‐ and multilayer slabs, together with nanotubes derived from the slabs, are investigated mainly by means of periodic local second‐order Møller–Plesset perturbation (LMP2) calculations and the results are compared with Hartree–Fock (HF), density functional theory (DFT), and dispersion function‐augmented DFT (DFT‐D) obtained ones. The investigated systems are structurally analogous to (111) and (110) slabs of diamond, where the hydrogenated (111) slab of diamond corresponds to the experimentally known graphane. Multilayering of monolayers and nanotubes is energetically favorable at the LMP2 level for both C and BN, while HF and DFT are not able to reproduce this behavior for CH systems. The work highlights the importance of utilizing methods capable of properly describing weak interactions in the investigation of dispersively‐bound systems such as the multilayered graphanes and the corresponding nanotubes.     

Intermolecular potentials based on symmetry-adapted perturbation theory with dispersion energies from time-dependent density-functional calculations

Recently, three of us have proposed a method [Phys. Rev. Lett. 91, 33201 (2003)] for an accurate calculation of the dispersion energy utilizing frequency-dependent density susceptibilities of monomers obtained from time-dependent density-functional theory (DFT). In the present paper, we report numerical calculations for the helium, neon, water, and carbon dioxide dimers and show that for a wide range of intermonomer separations, including the van der Waals and short-range repulsion regions, the method provides dispersion energies with accuracies comparable to those that can be achieved using the current most sophisticated wave-function methods. If the dispersion energy is combined with (i) the electrostatic and first-order exchange interaction energies as defined in symmetry-adapted perturbation theory (SAPT) but computed using monomer Kohn-Sham (KS) determinants, and (ii) the induction energy computed using the coupled KS static response theory, (iii) the exchange-induction and exchange-dispersion energies computed using KS orbitals and orbital energies, the resulting method, denoted by SAPT(DFT), produces very accurate total interaction potentials. For the helium dimer, the only system with nearly exact benchmark values, SAPT(DFT) reproduces the interaction energy to within about 2% at the minimum and to a similar accuracy for all other distances ranging from the strongly repulsive to the asymptotic region. For the remaining systems investigated by us, the quality of the SAPT(DFT) interaction energies is so high that these energies may actually be more accurate than the best available results obtained with wave-function techniques. At the same time, SAPT(DFT) is much more computationally efficient than any method previously used for calculating the dispersion and other interaction energy components at this level of accuracy.     

Kohn-Sham Density Matrix and the Kernel Energy MethodSCI北大核心CSCD

The kernel energy method(KEM) has been shown to provide fast and accurate molecular energy calculations for molecules at their equilibrium geometries.KEM breaks a molecule into smaller subsets,called kernels,for the purposes of calculation.The results from the kernels are summed according to an expression characteristic of KEM to obtain the full molecule energy.A generalization of the kernel expansion to density matrices provides the full molecule density matrix and orbitals.In this study,the kernel expansion for the density matrix is examined in the context of density functional theory(DFT) Kohn-Sham(KS) calculations.A kernel expansion for the one-body density matrix analogous to the kernel expansion for energy is defined,and is then converted into a normalizedprojector by using the Clinton algorithm.Such normalized projectors are factorizable into linear combination of atomic orbitals(LCAO) matrices that deliver full-molecule Kohn-Sham molecular orbitals in the atomic orbital basis.Both straightforward KEM energies and energies from a normalized,idempotent density matrix obtained from a density matrix kernel expansion to which the Clinton algorithm has been applied are compared to reference energies obtained from calculations on the full system without any kernel expansion.Calculations were performed both for a simple proof-of-concept system consisting of three atoms in a linear configuration and for a water cluster consisting of twelve water molecules.In the case of the proof-of-concept system,calculations were performed using the STO-3 G and6-31 G(d,p) bases over a range of atomic separations,some very far from equilibrium.The water cluster was calculated in the 6-31 G(d,p) basis at an equilibrium geometry.The normalized projector density energies are more accurate than the straightforward KEM energy results in nearly all cases.In the case of the water cluster,the energy of the normalized projector is approximately four times more accurate than the straightforward KEM energy result.The KS density matrices of this study are applicable to quantum crystallography.     

Comparing ab initio density-functional and wave function theories: the impact of correlation on the electronic density and the role of the correlation potential

The framework of ab initio density-functional theory (DFT) has been introduced as a way to provide a seamless connection between the Kohn-Sham (KS) formulation of DFT and wave-function based ab initio approaches [R. J. Bartlett, I. Grabowski, S. Hirata, and S. Ivanov, J. Chem. Phys. 122, 034104 (2005)]. Recently, an analysis of the impact of dynamical correlation effects on the density of the neon atom was presented [K. Jankowski, K. Nowakowski, I. Grabowski, and J. Wasilewski, J. Chem. Phys. 130, 164102 (2009)], contrasting the behaviour for a variety of standard density functionals with that of ab initio approaches based on second-order M?ller-Plesset (MP2) and coupled cluster theories at the singles-doubles (CCSD) and singles-doubles perturbative triples [CCSD(T)] levels. In the present work, we consider ab initio density functionals based on second-order many-body perturbation theory and coupled cluster perturbation theory in a similar manner, for a range of small atomic and molecular systems. For comparison, we also consider results obtained from MP2, CCSD, and CCSD(T) calculations. In addition to this density based analysis, we determine the KS correlation potentials corresponding to these densities and compare them with those obtained for a range of ab initio density functionals via the optimized effective potential method. The correlation energies, densities, and potentials calculated using ab initio DFT display a similar systematic behaviour to those derived from electronic densities calculated using ab initio wave function theories. In contrast, typical explicit density functionals for the correlation energy, such as VWN5 and LYP, do not show behaviour consistent with this picture of dynamical correlation, although they may provide some degree of correction for already erroneous explicitly density-dependent exchange-only functionals. The results presented here using orbital dependent ab initio density functionals show that they provide a treatment of exchange and correlation contributions within the KS framework that is more consistent with traditional ab initio wave function based methods.     

Density functional response theory calculations of three-photon absorption

Three-photon absorption probabilities delta(3PA) have been calculated through application of a recently derived method for cubic response functions within density functional theory (DFT). Calculations are compared with Hartree-Fock (HF) and with a coupled cluster hierarchy of models in a benchmarking procedure. Except for cases having intermediate states near resonance, density functional theory is demonstrated to be in sufficient agreement with the highly correlated methods in order to qualify for predictions of delta(3PA). For the larger systems addressed, a set of acceptor A and donor D substituted pi-conjugated systems formed by trans-stilbene and dithienothiophene (DTT), we find noticeable differences in the magnitude of delta(3PA) between HF and DFT, although similar trends are followed. It is shown that the dipolar structures, TS-AD and DTT-AD, have substantially larger delta(3PA) than other types of modifications which is in accordance with observations for two-photon absorption. This is the first application of density functional theory to three-photon absorption beyond the use of few-state models.     

Higher order alchemical derivatives from coupled perturbed self-consistent field theory

We present an analytical approach to treat higher order derivatives of Hartree-Fock (HF) and Kohn-Sham (KS) density functional theory energy in the Born-Oppenheimer approximation with respect to the nuclear charge distribution (so-called alchemical derivatives). Modified coupled perturbed self-consistent field theory is used to calculate molecular systems response to the applied perturbation. Working equations for the second and the third derivatives of HF/KS energy are derived. Similarly, analytical forms of the first and second derivatives of orbital energies are reported. The second derivative of Kohn-Sham energy and up to the third derivative of Hartree-Fock energy with respect to the nuclear charge distribution were calculated. Some issues of practical calculations, in particular the dependence of the basis set and Becke weighting functions on the perturbation, are considered. For selected series of isoelectronic molecules values of available alchemical derivatives were computed and Taylor series expansion was used to predict energies of the "surrounding" molecules. Predicted values of energies are in unexpectedly good agreement with the ones computed using HF/KS methods. Presented method allows one to predict orbital energies with the error less than 1% or even smaller for valence orbitals.     

Parallel implementation of γ‐point pseudopotential plane‐wave DFT with exact exchange

Semi‐local functionals commonly used in density functional theory (DFT) studies of solids usually fail to reproduce localized states such as trapped holes, polarons, excitons, and solitons. This failure is ascribed to self‐interaction which creates a Coulomb barrier to localization. Pragmatic approaches in which the exchange correlation functionals are augmented with small amount of exact exchange (hybrid‐DFT, e.g., B3LYP and PBE0) have shown to promise in rectifying this type of failure, as well as producing more accurate band gaps and reaction barriers. The evaluation of exact exchange is challenging for large, solid state systems with periodic boundary conditions, especially when plane‐wave basis sets are used. We have developed parallel algorithms for implementing exact exchange into pseudopotential plane‐wave DFT program and we have implemented them in the NWChem program package. The technique developed can readily be employed in Γ‐point plane‐wave DFT programs. Furthermore, atomic forces and stresses are straightforward to implement, making it applicable to both confined and extended systems, as well as to Car‐Parrinello ab initio molecular dynamic simulations. This method has been applied to several systems for which conventional DFT methods do not work well, including calculations for band gaps in oxides and the electronic structure of a charge trapped state in the Fe(II) containing mica, annite. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Density approximation to the average Hartree-Fock exchange potential for atoms

A non-local density-based approximation to the average Hartree-Fock (HF) exchange potential is developed. The new potential is formulated within the spin-dependent version of the weighted density approximation, and is based on a novel form of the (exchangeonly) pair-correlation function for electrons in finite systems. The results for total energies and one-electron orbital energies of atoms are reasonably accurate in comparison with those obtained using the exact average HF potential, or the exact orbital-dependent HF potential.     

Corrected small basis set Hartree‐Fock method for large systems

A quantum chemical method based on a Hartree‐Fock calculation with a small Gaussian AO basis set is presented. Its main area of application is the computation of structures, vibrational frequencies, and noncovalent interaction energies in huge molecular systems. The method is suggested as a partial replacement of semiempirical approaches or density functional theory (DFT) in particular when self‐interaction errors are acute. In order to get accurate results three physically plausible atom pair‐wise correction terms are applied for London dispersion interactions (D3 scheme), basis set superposition error (gCP scheme), and short‐ranged basis set incompleteness effects. In total nine global empirical parameters are used. This so‐called Hartee‐Fock‐3c (HF‐3c) method is tested for geometries of small organic molecules, interaction energies and geometries of noncovalently bound complexes, for supramolecular systems, and protein structures. In the majority of realistic test cases good results approaching large basis set DFT quality are obtained at a tiny fraction of computational cost. © 2013 Wiley Periodicals, Inc.