Correlation Energy Functionals Research Articles

A new Wigner‐like correlation‐energy functional from coordinate scaling requirements

To arrive at the very best approximations to the correlation energy in density functional theory, it is worthwhile to investigate the properties of the exact correlation-energy functional and to design functionals that obey as many constraints as possible. Examples of such requirements include uniform and nonuniform coordinate scaling. An attempt to obey the uniform scaling constraints, the only ones known at the time of designing, led in 1990 to the development of the Wilson–Levy (WL) correlation-energy functional, Wigner-like with gradients. In this article, we present a new Wigner-like functional that obeys most coordinate scaling requirements and gives results competitive with ones obtained from other correlation-energy functionals. Numerical results obtained by means of this new functional are presented. Ideas for further development of this kind of functional are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 523–532, 1998

Correlation potentials for the He atom and the hydrogen molecule: A comparison between the correlation factor approach and DFT correlation energy functionals

Two points about correlation potentials have been dealt with in this article. The first one is related to the shape of some of the most representative correlation potentials applied to the ground state of the He atom. It is shown here that both LDA and two-body density correlation potentials compare well with that obtained through the quantum chemistry definition of correlation energy. This is an interesting result because, in previous works, it had been shown that none of the correlation potentials compared well with the Kohn–Sham one. The gradient-corrected correlation potentials exhibit a very different behavior to that of both exact potentials (quantum chemistry and Kohn–Sham ones). The other question posed here refers to how a reference to the two-body density must modify DFT functionals for the correlation energy, when a multideterminant wave function is needed. This question has been addressed by analyzing the variation of correlation potentials as the bond length of the H2 molecule increases. The results show that an external reference to the two-body density qualitatively improves DFT correlation potentials and also that only those functionals explicitly depending on two-body density can give the quantitative correct trends. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 143–156, 1998

New functionals for correlation energy deduced in the framework of the correlation factor approach

The general characteristics of two-body density functionals (TBDF) are explored and two new correlation energy functionals are derived using the correlation factor approach. The optimization of the parameters entering the above functionals requires exact and accurate atomic correlation energies (ACE). We revised the ACE values in the literature and obtained a new set of “exact” ACE for atoms with 2 ≤ Z ≤ 10. Unfortunately, there exist some inaccuracies in the ACE values of the second-row atoms, which make unsuitable the inclusion of them in the optimization. The ACE calculated for the first period with the above functionals, using the optimized sets of parameters, are in excellent agreement with the exact ones, while the corresponding values calculated for the second-row atoms are between the precision margins estimated by us for the exact values. © 1997 John Wiley & Sons, Inc.

Density functional theory for open-shell systems using a local-scaling transformation scheme. I. Single-density energy functional

A rigorous approach of density functional theory (DFT) for open-shell multifermionic systems is devised, using a local-scaling transformation (LST) scheme involving a single scalar function f(r). Within the orbit θN induced by a model wave function (MWF) _Ψ, the total energy of space or spin degenerate or nondegenerate states is expressed as an exact functional of the single-particle density ρ(r). In the first step, it is shown how the reduced density functions and matrices (RD Fs and Ms) of any order (s=1,…,N−1) of an (open or closed) N-fermion system can be expressed as functionals of the one-fermion charge density. The spatial components (which depend on the spin configuration of the fermions) of the RD Fs and Ms of orders 1 and 2, and the resulting charge and spin distribution and correlation densities, are functionals of the one-fermion charge density. We form the manifolds of the charge and spin distribution and correlation energy functionals, from which the theory can be extended to degenerate states of a spinless Hamiltonian. For multielectronic systems, the spin densities and spin–pair correlations as well as the spin–orbit and spin–spin interactions are determined by the function ρ(r). In the second step, it is shown how the expectation values of s-particle operators, in particular those of spin-including mono- and biparticle operators, are functionals of the monoparticle density ρ(r). We give general expressions for the expectation values of spin-free, spin–field, spin–orbit and spin–spin interaction operators in degenerate states. We show how to express the energy functional of the spin manifolds of a system described by a Schrödinger Hamiltonian. The use of an LST, which preserves spin symmetry, to build this functional must fulfill certain conditions in order to maintain the space symmetry of the system. We investigate the dependence of the Weizsaecker and extended Thomas–Fermi kinetic energy terms and of the Coulomb attraction, repulsion, and exchange potential energy terms on the choice of the MWF, i.e., on the induced orbit and on the spin configuration of the system. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 241–256, 1997

Density-functional theory for excited states.

A Kohn-Sham formalism for the treatment of excited states within the density-functional theory (DFT) is presented. DFT exchange and correlation energy functionals for excited states are defined. Explicit expressions for these functionals are derived by generalizing a recent DFT perturbation theory. A computational scheme for the treatment of excited states within the DFT is suggested. Differences of Kohn-Sham eigenvalues are shown to be well-defined approximations for excitation energies. Correction terms to these approximations are presented. Perturbation theory expansions for band gaps are discussed. \textcopyright{} 1996 The American Physical Society.

Scaling and virial theorems in current-density-functional theory.

Starting from a constrained-search formulation of current-density-functional theory, we obtain rigorous scaling and virial relations for the kinetic, exchange, and correlation energy functionals of electronic systems in strong magnetic fields.

Density functional theory and molecular clusters

AbstractDensity functional theory (DFT) methods, including nonlocal density gradient terms in the exchange and correlation energy functionals, were applied to various types of molecular clusters: H‐bonded, ionic, electrostatic, and London. Reliable results on the structure and stabilization energy were obtained for the first two types of cluster as long as Becke3LYP and Becke3P86 functionals and basis sets of at least DZ + P quality were used. DFT methods with currently available functionals failed completely, however, for London‐type clusters, for which no minimum was found on the potential energy surfaces. DFT interaction energy exhibits the same basis set extension dependence as the Hartree‐Fock (HF) interaction energy. Therefore, the Boys‐Bernardi function counterpoise procedure should be employed for elimination of the DFT basis set superposition error. © 1995 John Wiley & Sons, Inc.

An analysis of the integrand occurring in correlation energy functionals

An analysis of the integrand occurring in correlation energy functionals

An analysis of the integrand occurring in correlation energy functionals

AbstractAn analysis of the integrand occurring in current density functional calculations is presented, concentrating attention on correlation energy functionals and the atomic regions, i.e., the regions of space surrounding atomic centers. The analysis follows the structure of a previously proposed numerical integration scheme for three‐dimensional integrals occurring in electronic structure calculations. The scheme is based on the choice of density‐based weight functions that naturally partition the space into «atomic» volumes (in which the integration is performed in terms of spherical coordinates) and «diatomic» volumes (in which the integration is performed in terms of confocal elliptical coordinates). From this analysis, a simplified procedure for the atomic «internal» integrations is developed, whereas preliminary results are discussed for the atomic and diatomic «external» integrations. The numerical tests are performed on the C60 molecule in the symmetrical configuration. © John Wiley & Sons, Inc.

Applicability to atoms of a large set of correlation energy functinals

AbstractAn extensive study of the application of a large set of correlation energy functionals to atomic systems is presented. Hartree–Fock densities and, sometimes, generalized‐valence‐bond wave functions are used. The functionals have been grouped into four sets: (I) local density and local spin density, (II) local spin density with self‐interactioon correction, (III) local spin density with gradient correction, and (IV) methods found on the correlation factor approach. In this bench mark, we report the results of calculations made for the ground states of the first and second rows, their anions and cations, and the two‐ and four‐isoelectronic series. © 1994 John Wiley & Sons, Inc.

A microscopic model of dangling bonds at interfaces of ordered and disordered semiconductors†

Abstract A microscopic model of an isolated silicon dangling bond is developed. A self-consistent mean-field approximation is applied to study the electronic structure of the dangling bond. Atomic relaxation is taken into account phenomenologically. Thermal and optical transition energies, correlation energy, equilibrium displacements and wave function structure are calculated.

Development of non-local Wigner-like correlation-energy density functionals through coordinate scaling requirements and evaluation of contraint obedience in the construction of density functionals

Uniform coordinate scaling requirements for a correlation-energy density functional in density functional theory (DFT) led to the development of the Wilson—Levy (WL) functional. The numerical results of the functional, which is non-local and Wignerlike, are compared to results of recent and traditional functionals. Values calculated from the correlation-energy functionals of both the original Wigner and the VWN are also tabulated. Additional functionals were constructed to address the question as to whether constraint-obedience is a valuable tool for correlation-energy functional construction. Comparison of functionals with varying degrees of constraint obedience, including functionals that obey few or no contraints, is made for both local and non-local formulas. Parameterization of the WL functional is explained and the functional derivative of both the close-shell and open-shell forms of the WL is presented.

Electric dipole polarizabilities for helium-like ions from correlated wavefunctions. A density-functional approach

Static electric dipole polarizabilities for helium-like ions H −, He, Li +, Be 2+, B 3+, and Ne 8+ are calculated from their respective Hylleraas wavefunctions. However, instead of using the wavefunctions directly, we employ the corresponding densities in a density-based approach proposed recently (by Harbola) for calculating polarizabilities. The only approximation made is that of the local density for the exchange and correlation energy functionals. The use of density makes our calculations much simpler than those based on the wavefunctions. They lead however to essentially the same results as the wavefunctional calculations in cases where the results are known.

Weighted-density exchange and local-density Coulomb correlation energy functionals for finite systems: Application to atoms.

A nonlocal exchange energy functional developed within the weighted spin-density approximation (WSDA) and a local Coulomb correlation energy functional are combined within the density-functional scheme for the calculation of the electronic structure of many-electron systems. Both functionals include alternative forms of the pair-correlation functions governing the size and shape of the Fermi and Coulomb holes. Self-consistent calculations have been performed for all atoms from He to Rn within the exchange-only WSDA scheme. These have been compared to the results of similar calculations using Becke's exchange functional. Besides, the results of calculations for all atoms from He to Ar, including both exchange and correlation, are compared with those of the self-interaction-corrected method. In addition to obtaining accurate total, exchange, and correlation energies, we perform a successful calculation of the s\ensuremath{\rightarrow}d interconfigurational energies (ICE's) for transition-metal atoms. The average error of the calculated ICE values for 3d atoms is only 0.2 eV.

Electron correlation contributions to one-electron properties from functionals of the Hartree–Fock electron density

Equations for the electron correlation contributions to one-electron properties calculated with correlation energy functionals of the Hartree–Fock density are derived. For any functional that is of an explicitly algebraic form, these contributions are limited to the occupied-virtual block of the one-electron density matrix. This deficiency results in very poor values of one-electron properties, as demonstrated by numerical examples. Implications of these observations to the future of the density functional theory (DFT)-based electronic structure methods are discussed.

A numerical integration scheme for the evaluation of correlation energy functionals

AbstractA numerical integration scheme is presented for three‐dimensional integrals occurring in electronic structure calculations, concentrating attention on the evaluation of the correlation energy through a density‐functional expression. The scheme is based on the choice of density‐based weight functions that naturally partition the space into “atomic” volumes (in which the integration is performed in terms of spherical coordinates) and “diatomic” volumes (in which the integration is performed in terms of confocal elliptical coordinates). Such a choice is justified on the basis of the analytical behavior of the integrand. The attainable accuracy and the required computational effort within the proposed scheme are discussed in detail in a test application on the C60 molecule in the symmetrical configuration. Finally, a comparison with previously proposed schemes is presented. © 1993 John Wiley & Sons, Inc.

Examination of several exchange-correlation energy functionals by accurate self-consistent atomic calculations

Several local and nonlocal exchange-correlation functionals are tested through accurate Kohn–Sham self-consistent calculations for six nobel gas atoms and three isoelectronic series (with two, four, and ten electrons and atomic numbers up to 20). The calculated exchange, correlation, and total energies have been compared with those from accurate Hartree–Fock calculations and accurate estimates of correlation energies available recently. Improvements over local-density approximation are found for all the tested nonlocal exchange-correlation functionals. In particular, nonlocal correlation energy functionals are shown to perform significantly better than the local approximations when both are used in conjunction with the nonlocal Becke’s exchange energy functional. However, in most cases, all the functionals fail to predict the correct trends of exchange and correlation energies as the atomic charge increases.

Spectroscopic constants of diatomic molecules computed correcting Hartree-Fock or general-valence-bond potential-energy curves with correlation-energy functionals.

The Kohn-Sham energy with exact exchange [using the exact Hartree-Fock (HF) exchange but an approximate correlation-energy functional] may be computed very accurately by adding the correlation obtained from the HF density to the total HF energy. Three density functionals are used: local spin density (LSD), LSD with self-interaction correction, and LSD with generalized gradient correction. This scheme has been extended (Lie-Clementi, Colle-Salvetti, and Moscardo--San-Fabian) to be used with general-valence-bond (GVB) energies and wave functions, so that the extra correlation included in the GVB energy is not counted again. The effect of all these approximate correlations on HF or GVB spectroscopic constants (${\mathit{R}}_{\mathit{e}}$,${\mathrm{\ensuremath{\omega}}}_{\mathit{e}}$, and ${\mathit{D}}_{\mathit{e}}$) is studied. Approximate relations showing how correlation affects them are derived, and may be summarized as follows: (1) the effect on ${\mathit{R}}_{\mathit{e}}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathit{e}}$ depends only on the correlation derivative at ${\mathit{R}}_{\mathit{e}}$, and (2) the effect on ${\mathit{D}}_{\mathit{e}}$ depends mainly on the correlation difference between quasidissociated and equilibrium geometries. A consequence is that all the correlation corrections tested here give larger ${\mathrm{\ensuremath{\omega}}}_{\mathit{e}}$ and ${\mathit{D}}_{\mathit{e}}$ and shorter ${\mathit{R}}_{\mathit{e}}$ than the uncorrected HF or GVB values. This trend is correct for ${\mathit{D}}_{\mathit{e}}$ for both HF and GVB. For ${\mathit{R}}_{\mathit{e}}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathit{e}}$, it is correct in most cases for GVB, but it often fails for the HF cases. A comparison is made with Kohn-Sham calculations with both exchange and correlation approximated. As a final conclusion, it is found that, within the present scheme, a qualitatively correct HF or GVB potential-energy curve, together with a correlation-energy approximation with correct dissociation behavior, is crucial for obtaining good estimates of spectroscopic constants.

Results of the correlation energy functionals versus the wavefunction type

The results of applying several correlation energy functionals (Gombas, Vosko-Wilk-Nussair, Perdew, Becke, Colle-Salvetti, Moscardó-San-Fabián and Lee-Yang-Parr) to different wave-function types (RHF, GVB and CI) were analyzed. The test was carried out on atomic systems (He, Li, Be, F and Ne) and on H 2 and LiH molecules at several internuclear distances. The potential energy curve of the hydrogen molecule was calculated; its accuracy was analyzed by checking the results for (i) some molecular properties such as R e and D e and (ii) the Born-Oppenheimer vibrational quanta, as obtained from the Kolos and Wolnievicz potential energy curve.

Calculation of the total atomic binding energy with recently proposed kinetic-, exchange-, and correlation-energy functionals

The functionals adopted in the present work are those of DePristo and Kress [Phys. Rev. A35, 438 (1987)] for the kinetic energy, of Cedillo et al. [Phys. Rev. A38, 1697 (1988)] for the exchange energy, and of Wilson and Levy [Phys. Rev. B41, 12930 (1990)] for the correlation energy. In addition to the kinetic-, exchange-, and correlation-energy functionals, the total-energy functional used in the present calculations contains also the functional describing the interaction of the electrons with the atomic nucleus, and the functional describing the classical or direct part of the interaction among the electrons. Using these functionals, a two-parameter and a three-parameter variational density-functional calculation of the total atomic binding energy of the Ne atom is carried out. The electron (number) density of the Ne atom is modeled by using hydrogenlike one-electron radial wavefunctions (with the 2s radial function orthogonalized to the 1s radial function) containing two/three variational parameters. It is found that the calculated correlation energies are in reasonable agreement with the correlation energy value obtained from the Wilson and Levy expression with a Hartree-Fock electron density.