Short-range exchange and correlation energy density functionals: beyond the local-density approximation

We propose approximations which go beyond the local-density approximation for the short-range exchange and correlation density functionals appearing in a multideterminantal extension of the Kohn-Sham scheme. A first approximation consists of defining locally the range of the interaction in the correlation functional. Another approximation, more conventional, is based on a gradient expansion of the short-range exchange-correlation functional. Finally, we also test a short-range generalized-gradient approximation by extending the Perdew-Burke-Ernzerhof exchange-correlation functional to short-range interactions.     

Long-range and short-range Coulomb correlation effects as simulated by Hartree–Fock,local density approximation,and generalized gradient approximation exchange functionals

 Exchange functionals used in density functional theory (DFT) are generally considered to simulate long-range electron correlation effects. It is shown that these effects can be traced back to the self-interaction error (SIE) of approximate exchange functionals. An analysis of the SIE with the help of the exchange hole reveals that both short-range (dynamic) and long-range (nondynamic) electron correlation effects are simulated by DFT exchange where the local density approximation (LDA) accounts for stronger effects than the generalized gradient expansion (GGA). This is a result of the fact that the GGA exchange hole describes the exact exchange hole close to the reference electron more accurately than the LDA hole does. The LDA hole is more diffuse, thus leading to an underestimation of exchange and stronger SIE effects, where the magnitude of the SIE energy is primarily due to the contribution of the core orbitals. The GGA exchange hole is more compact, which leads to an exaggeration of exchange in the bond and the nonbonding region and negative SIE contributions. Partitioning of the SIE into intra-/interelectronic and individual orbital contributions makes it possible to test the performance of a given exchange functional in different regions of the molecule. It is shown that Hartree–Fock exchange always covers some long-range effects via interelectronic exchange while self-interaction-corrected DFT is lacking these effects. Received: 25 May 2002 / Accepted: 7 October 2002 / Published online: 21 January 2003 Correspondence to: E. Kraka e-mail: kraka@theoc.gn.se Acknowledgements. This work was supported financially by the Swedish Natural Science Research Council (NFR). Calculations were done on the supercomputers of Nationellt Superdatorcentrum (NSC), Link?ping, Sweden. The authors thank the NSC for a generous allotment of computer time.     

On the conjoint gradient correction to the Hartree—Fock kinetic and exchange energy density functionals

The kinetic and the exchange energy functionals are expressed in the form T[ρ] = C_TF∫ drρ^5/3(r)f_t(s) and K[ρ] = C_D∫ drρ^4/3(r)f_K(s), where C_TF = (3/10)(3π²)^2/3 and C_D = −(3/4)(3/π)^4/3 are the Thomas-Fermi and the Dirac coefficients, respectively, and s = |∇ρ(r)|/C_sρ^4/3(r), with C_s = 2(3π²)^1/3. These expressions are used to perform a comparison of f_T(s) and f_K(s) in terms of their generalized gradient expansion approximations. It is shown that f_κ(s) and is congruent to f_T(s) in the range characteristic of the interior regions of atoms and many solids and that the second-order gradient expansion of the kinetic energy provides a rather reasonable approximation to the generalized gradient expansion approximation of both the kinetic and the exchange energy functionals. © 1996 John Wiley & Sons, Inc.     

Description of core excitations by time-dependent density functional theory with local density approximation, generalized gradient approximation, meta-generalized gradient approximation, and hybrid functionals

Time-dependent density functional theory (TDDFT) is employed to investigate exchange-correlation-functional dependence of the vertical core-excitation energies of several molecules including H, C, N, O, and F atoms. For the local density approximation (LDA), generalized gradient approximation (GGA), and meta-GGA, the calculated X1s-->pi* excitation energies (X = C, N, O, and F) are severely underestimated by more than 13 eV. On the other hand, time-dependent Hartree-Fock (TDHF) overestimates the excitation energies by more than 6 eV. The hybrid functionals perform better than pure TDDFT because HF exchange remedies the underestimation of pure TDDFT. Among these hybrid functionals, the Becke-Half-and-Half-Lee-Yang-Parr (BHHLYP) functional including 50% HF exchange provides the smallest error for core excitations. We have also discovered the systematic trend that the deviations of TDHF and TDDFT with the LDA, GGA, and meta-GGA functionals show a strong atom-dependence. Namely, their deviations become larger for heavier atoms, while the hybrid functionals are significantly less atom-dependent.     

Link between the kinetic‐ and exchange‐energy functionals in the generalized gradient approximation

An approximate kinetic‐energy functional of the generalized gradient approximation form was derived following the “conjointness conjecture” of Lee, Lee, and Parr. The functional shares the analytical form of its gradient dependency with the exchange‐energy functionals of Becke and Perdew, Burke, and Ernzerhof. The two free parameters of this functional were determined using the exact values of the kinetic energy of He and Xe atoms. A set of 12 closed‐shell atoms was used to test the accuracy of the proposed functional and more than 30 others taken from the literature. It is shown that the conjointness conjecture leads to a very good class of kinetic‐energy functionals. Moreover, the functional developed in this work is shown to be one of the most accurate despite its simple analytical form. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Asymptotic properties of the exchange energy density and the exchange potential of finite systems: relevance for generalized gradient approximations

It is shown that generalized gradient approximations (GGAs) for exchange only, due to their very limited form, quite generally can not simultaneously reproduce both the asymptotic forms of the exchange energy density and the exchange potential of finite systems. Furthermore, mechanisms making GGAs formally approach at least one of these asymptotic forms do not improve the corresponding quantity in the relevant part of the asymptotic regime of atoms. By constructing a GGA which leads to superior atomic exchange energies compared to all GGAs heretofore but does not reproduce the asymptotic form of the exact exchange energy density it is demonstrated that this property is not important for obtaining extremely accurate atomic exchange energies. We conclude that GGAs by their very concept are not suited to reproduce these asymptotic properties of finite systems. As a byproduct of our discussion we present a particularly simple and direct proof of the well known asymptotic structure of the exchange potential of finite spherical systems.     

Effective homogeneity of the exchange-correlation and non-interacting kinetic energy functionals under density scaling

Correlated electron densities, experimental ionisation potentials, and experimental electron affinities are used to investigate the homogeneity of the exchange-correlation and non-interacting kinetic energy functionals of Kohn-Sham density functional theory under density scaling. Results are presented for atoms and small molecules, paying attention to the influence of the integer discontinuity and the choice of the electron affinity. For the exchange-correlation functional, effective homogeneities are highly system-dependent on either side of the integer discontinuity. By contrast, the average homogeneity-associated with the potential that averages over the discontinuity-is generally close to 4/3 when the discontinuity is computed using positive affinities for systems that do bind an excess electron and negative affinities for those that do not. The proximity to 4/3 becomes increasingly pronounced with increasing atomic number. Evaluating the discontinuity using a zero affinity in systems that do not bind an excess electron instead leads to effective homogeneities on the electron abundant side that are close to 4/3. For the non-interacting kinetic energy functional, the effective homogeneities are less system-dependent and the effect of the integer discontinuity is less pronounced. Average values are uniformly below 5/3. The study provides information that may aid the development of improved exchange-correlation and non-interacting kinetic energy functionals.     

An application of correlation energy density functionals to atoms and molecules

Four density functionals — including that recently introduced by Perdew ((1986) Phys Rev B33: 8822)—are tested for first-row atoms, hydrides and dimers. Calculated contributions of the correlation energy to the ionization potentials and electron affinities of atoms and to the dissociation energies of molecules are compared with empirical values which were reevaluated for this purpose. An improvement over Hartree-Fock is found in all cases if the self-interaction or the gradient correction are included in the density functional, although there is a rather large variation in the accuracy of the predictions.     

First-principles local density approximation (generalized gradient approximation) +U study of catalytic CenOm clusters: U value differs from bulk

Ceria possesses strong catalytic properties for CONO(x) removal and H(2) production. Clusters often show more intriguing functionalities than their bulk counterparts. Here, the geometric and electronic structures of Ce(n)O(m) (n=1-4,m=2n-1,2n) clusters are studied for the first time using the projected augmented wave method in density functional theory with detailed assessment of the exchange-correlation functional and the Hubbard parameter U. We note that the U value strongly affects the electronic structures of the oxygen-deficient Ce(n)O(2n-1) clusters, though less so on the stoichiometric Ce(n)O(2n). Furthermore, the local density approximation (LDA)+U method is more accurate than the generalized gradient approximation+U in describing the localization of the 4f electrons of the Ce(n)O(m) clusters. The calculated vibration frequency of the CeO molecule with the LDA+U (U=4 eV) is 818.4 cm(-1), in close agreement with experimental values of 820-825 cm(-1) for the low lying states. Different optimal U values were noted for the ceria cluster (4 eV) and its bulk (6 eV), due to quantum-size and geometric effects. The largely reduced formation energy of an oxygen vacancy indicates that the catalytic effect of the Ce(n)O(m) clusters are far greater than bulk CeO(2).     

A density-functional study of the possibility of noncollinear magnetism in small Mn clusters using SIESTA and the generalized gradient approximation to exchange and correlation

We investigated the possibility of noncollinear magnetism in small Mn(n) clusters (n=2-6) using the density-functional method SIESTA with the generalized gradient approximation (GGA) to exchange and correlation. The lowest-energy states identified were collinear, with the atomic spin magnetic moments pointing in the same direction, for Mn(2) and Mn(3), and noncollinear for Mn(4), Mn(5) and, most decidedly, Mn(6). These SIESTA/GGA results, which are compared with those of an earlier SIESTA study that used the local spin density approximation, are qualitatively in keeping with the result obtained by VASP/GGA calculations.     

A generalized gradient approximation for exchange derived from the model potential of van Leeuwen and Baerends

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.     

Comparative performance of exchange and correlation density functionals in determining intermolecular interaction potentials of the methane dimer

We have calculated the interaction potentials of the methane dimer for the minimum-energy D(3d) conformation using the density functional theory (DFT) with 90 density functionals chosen from the combinations of nine exchange and 10 correlation functionals. Several hybrid functionals are also considered. While the performance of an exchange functional is related to the large reduced density gradient of the exchange enhancement factor, the correlation energy is determined by the low-density behavior of a correlation enhancement factor. Our calculations demonstrate that the correlation counterpart plays an equally important role as the exchange functional in determining the van der Waals interactions of the methane dimer. These observations can be utilized to better understand the seemingly unsystematic DFT interaction potentials for weakly bound systems.     

Performance of recently developed kinetic energy density functionals for the calculation of hydrogen binding strengths and hydrogen-bonded structures

 The accuracy of predicted hydrogen binding energies and equilibrium structures for a benchmark set of molecules is compared for some recently developed density functionals, Becke's three parameter hybrid method with the Lee, Yang, and Parr (LYP) correlation functional (B3LYP), Becke's half and half functional combined with the LYP correlation functional (BHLYP), Perdew, Burke and Ernzerhof functional (PBE), Van Voorhis, Scuseria exchange correlation functional (VSXC), the hybrid Perdew, Burke and Ernzerhof functional (PBE1PBE), and meta-generalized gradient approximation (meta-GGA). Overall, the hybrid functionals which contain a portion of Hartree–Fock exchange (B3LYP, BHLYP, and PBE1PBE) yield the most accurate results. The kinetic-energy-density-dependent functionals, VSXC and meta-GGA, are significantly less accurate. Received: 10 December 1999 / Accepted: 5 March 2000 / Published online: 21 June 2000     

Padé approximation for the polynomial representation of the correlation energy density functional

Polynomials in the one-third power of the density were recently proposed to represent approximately the correlation energy density functional [S. Liu and R.G. Parr, Phys. Rev. A 53 (1996) 2211]. Studied in the present work is a Padé-approximant in that same variable which overcomes weaknesses in the polynomial form. Numerical results for atoms and molecules show that Padé forms fairly reproduce the experimental values, and are comparable in accuracy with other commonly used local functionals for the correlation energy.     

Generalized gradient approximation adjusted to transition metals properties: Key roles of exchange and local spin density

Perdew-Burke-Ernzerhof (PBE) and PBE adapted for solids (PBEsol) are exchange-correlation (xc) functionals widely used in density functional theory simulations. Their differences are the exchange, μ, and correlation, β, coefficients, causing PBEsol to lose the Local Spin Density (LSD) response. Here, the μ/β two-dimensional (2D) accuracy landscape is analyzed between PBE and PBEsol xc functional limits for 27 transition metal (TM) bulks, as well as for 81 TM surfaces. Several properties are analyzed, including the shortest interatomic distances, cohesive energies, and bulk moduli for TM bulks, and surface relaxation degree, surface energies, and work functions for TM surfaces. The exploration, comparing the accuracy degree with respect experimental values, reveals that the found xc minimum, called VV, being a PBE variant, represents an improvement of 5% in mean absolute percentage error terms, whereas this improvement reaches ~11% for VVsol, a xc resulting from the restoration of LSD response in PBEsol, and so regarded as its variant.     

Is combining meta‐GGA correlation functionals with the OPTX exchange functional useful?

The recent generalized gradient approximation (GGA) density functional OCS1 of Handy and Cohen is implemented in the deMon code and tested on a carefully selected set of problems. OCS1 is found to be accurate for molecular atomization energies, transition metal–ligand bonds, and systems with intramolecular hydrogen bonds. However, OCS1 encounters problems for systems with intermolecular hydrogen bonds. It also tends to elongate bond lengths systematically, and sometimes significantly. The OPTX exchange is combined with three meta‐GGA correlation functionals, Lap3, τ1, and τ2, the latter reported for the first time. The new meta‐GGA scheme OPTX exchange plus τ2 correlation called Oτ2 yields improved molecular geometries, NMR shielding constants, and an improved barrier height for the H+H₂ reaction. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Exact theorems concerning noninteracting kinetic energy density functional in D dimensions and their implications for gradient expansions

The focal point of the present work is the single-particle kinetic energy density tensor in D dimensions. This quantity enters both differential and various integral forms of the virial theorems, which are again set up in D dimensions. Major new results lie in (i) demonstrating that, by one-dimensional quadrature, it is possible to construct the Pauli potential directly from the kinetic energy tensor, without the need for functional differentiation and (ii) generating the gradient expansion for the kinetic energy tensor, in D dimensions. © 1995 John Wiley & Sons, Inc.