Databases for transition element bonding: metal-metal bond energies and bond lengths and their use to test hybrid, hybrid meta, and meta density functionals and generalized gradient approximations

We propose a data set of bond lengths for 8 selected transition metal dimers (Ag(2), Cr(2), Cu(2), CuAg, Mo(2), Ni(2), V(2), and Zr(2)) and another data set containing their atomization energies and the atomization energy of ZrV, and we use these for testing density functional theory. The molecules chosen for the test sets were selected on the basis of the expected reliability of the data and their ability to constitute a diverse and representative set of transition element bond types while the data sets are kept small enough to allow for efficient testing of a large number of computational methods against a very reliable subset of experimental data. In this paper we test 42 different functionals: 2 local spin density approximation (LSDA) functionals, 12 generalized gradient approximation (GGA) methods, 13 hybrid GGAs, 7 meta GGA methods, and 8 hybrid meta GGAs. We find that GGA density functionals are more accurate for the atomization energies of pure transition metal systems than are their meta, hybrid, or hybrid meta analogues. We find that the errors for atomization energies and bond lengths are not as large if we limit ourselves to dimers with small amounts of multireference character. We also demonstrate the effects of increasing the fraction of Hartree-Fock exchange in multireference systems by computing the potential energy curve for Cr(2) and Mo(2) with several functionals. We also find that BLYP is the most accurate functional for bond energies and is reasonably accurate for bond lengths. The methods that work well for transition metal bonds are found to be quite different from those that work well for organic and other main group chemistry.     

Optimized effective potentials yielding Hartree-Fock energies and densities

It is commonly believed that the exchange-only optimized effective potential (OEP) method must yield total energies that are above corresponding ground-state Hartree-Fock (HF) energies except for one- and two-electron systems. We present a simple procedure for constructing local (multiplicative) exchange potentials that reproduce exactly the HF energy and density in any finite basis set for any number of electrons. For any finite basis set, no matter how large, there exist infinitely many such OEPs, which questions their suitability for practical applications.     

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.     

On the test of different atomic exchange functionals

The yet unknown exchange-correlation energy functional and the lack of a systematic way to find such a functional has led to propose different expressions for exchange, on the one hand, and correlation on the other, by several authors. A common factor in the design of exchange functionals is to reproduce different kinds of energies rather than the important atomic shell structure. We present a strategy for testing the behavior of exchange functionals based on the reproduction of atomic shell structure. A variety of well-known exchange functionals are tested on He (closed shell), Li (open shell), and Be (closed subshell). Important differences are found among them. Finally, we propose an exchange energy functional that includes a Becke-88-type expression and, in addition, a nongradient correction term that leads to improved shell structure behavior. © 1995 John Wiley & Sons, Inc.     

The limitations of Slater's element-dependent exchange functional from analytic density-functional theory

Our recent formulation of the analytic and variational Slater-Roothaan (SR) method, which uses Gaussian basis sets to variationally express the molecular orbitals, electron density, and the one-body effective potential of density-functional theory, is reviewed. Variational fitting can be extended to the resolution of identity method, where variationality then refers to the error in each two-electron integral and not to the total energy. However, a Taylor-series analysis shows that all analytic ab initio energies calculated with variational fits to two-electron integrals are stationary. It is proposed that the appropriate fitting functions be charge neutral and that all ab initio energies be evaluated using two-center fits of the two-electron integrals. The SR method has its root in Slater's Xalpha method and permits an arbitrary scaling of the Slater-Gàspàr-Kohn-Sham exchange-correlation potential around each atom in the system. The scaling factors are Slater's exchange parameters alpha. Of several ways of choosing these parameters, two most obvious are the Hartree-Fock (HF) alpha(HF) values and the exact atomic alpha(EA) values. The former are obtained by equating the self-consistent Xalpha energy and the HF energies, while the latter set reproduces exact atomic energies. In this work, we examine the performance of the SR method for predicting atomization energies, bond distances, and ionization potentials using the two sets of alpha parameters. The atomization energies are calculated for the extended G2 set of 148 molecules for different basis-set combinations. The mean error (ME) and mean absolute error (MAE) in atomization energies are about 25 and 33 kcal/mol, respectively, for the exact atomic alpha(EA) values. The HF values of exchange parameters alpha(HF) give somewhat better performance for the atomization energies with ME and MAE being about 15 and 26 kcal/mol, respectively. While both sets give performance better than the local-density approximation or the HF theory, the errors in atomization energy are larger than the target chemical accuracy. To further improve the performance of the SR method for atomization energies, a new set of alpha values is determined by minimizing the MAE in atomization energies of 148 molecules. This new set gives atomization energies half as large (MAE approximately 14.5 kcal/mol) and that are slightly better than those obtained by one of the most widely used generalized-gradient approximations. Further improvements in atomization energies require going beyond Slater's functional form for exchange employed in this work to allow exchange-correlation interactions between electrons of different spins. The MAE in ionization potentials of 49 atoms and molecules is about 0.5 eV and that in bond distances of 27 molecules is about 0.02 A. The overall good performance of the computationally efficient SR method using any reasonable set of alpha values makes it a promising method for study of large systems.     

Universal density functional approach to the calculation of correlation energies of atoms

A new local density functional approach for the calculation of correlation energies of many-electron atomic systems is proposed by using the exact results for the correlation energy of a two-electron system bound by a harmonic oscillator external potential. This is motivated by the fact that the correlation energy is a universal functional of the electron density, and the form of this functional is independent of the external potential. The calculated numerical results for the correlation energies show very good agreement with the standard values reported in the literature. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 461–465, 1997     

Structure of the optimized effective Kohn—Sham exchange potential and its gradient approximations

An analysis of the structure of the optimized effective Kohn-Sham exchange potential v_x and its gradient approximations is presented. The potential is decomposed into the Slater potential v_s and the response of v_s to density variations, v_resp. The latter exhibits peaks that reflect the atomic shell structure. Kohn—Sham exchange potentials derived from current gradient approaches for the exchange energy are shown to be quite reasonable for the Slater potential, but they fail to approximate the response part, which leads to poor overall potentials. Improved potentials are constructed by a direct fit of v_x with a gradient-dependent Padé approximant form. The potentials obtained possess proper asymptotic and scaling properties and reproduce the shell structure of the exact v_x. © 1996 John Wiley & Sons, Inc.     

Efficient exact exchange approximations in density-functional theory

Two approaches to approximate the Slater potential component of local exact exchange of density-functional theory are investigated. The first approach employs density fitting of the electrostatic potential integrals over two occupied orbitals and the other approach approximates the "exact" Slater potential with the potential derived from the Becke-Roussel [Phys. Rev. A. 39, 3761 (1989)] model of the exchange hole. In both cases significant time savings can be achieved for larger systems compared to the calculation of the numerical Slater potential. It is then analyzed how well the orbitals obtained from the various total exchange potentials reproduce Hartree-Fock energies and molecular properties. A large range of atoms and small molecules has been utilized, including the three DNA bases adenine, thymine, and cytosine.     

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.     

Energy landscapes of nucleophilic substitution reactions: a comparison of density functional theory and coupled cluster methods

We have carried out a detailed evaluation of the performance of all classes of density functional theory (DFT) for describing the potential energy surface (PES) of a wide range of nucleophilic substitution (SN2) reactions involving, amongst others, nucleophilic attack at carbon, nitrogen, silicon, and sulfur. In particular, we investigate the ability of the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA as well as hybrid DFT to reproduce high-level coupled cluster (CCSD(T)) benchmarks that are close to the basis set limit. The most accurate GGA, meta-GGA, and hybrid functionals yield mean absolute deviations of about 2 kcal/mol relative to the coupled cluster data, for reactant complexation, central barriers, overall barriers as well as reaction energies. For the three nonlocal DFT classes, the best functionals are found to be OPBE (GGA), OLAP3 (meta-GGA), and mPBE0KCIS (hybrid DFT). The popular B3LYP functional is not bad but performs significantly worse than the best GGA functionals. Furthermore, we have compared the geometries from several density functionals with the reference CCSD(T) data. The same GGA functionals that perform best for the energies (OPBE, OLYP), also perform best for the geometries with average absolute deviations in bond lengths of 0.06 A and 0.6 degrees, even better than the best meta-GGA and hybrid functionals. In view of the reduced computational effort of GGAs with respect to meta-GGAs and hybrid functionals, let alone coupled cluster, we recommend the use of accurate GGAs such as OPBE or OLYP for the study of SN2 reactions.     

Improved density functionals for water

The accuracy of existing density functional methods for describing the noncovalent interaction energies in small water clusters is investigated by testing 25 density functionals against a data set of 28 water dimers and 8 water trimers whose structures are taken from the literature and from simulations. The most accurate functionals are found to be PW6B95 with a mean unsigned error of 0.13 kcal/mol and MPWB1K and B98 with mean unsigned errors of 0.15 kcal/mol; the best functional with no Hartree-Fock exchange is mPWLYP, which is a GGA with a mean unsigned error of 0.28 kcal/mol. In comparison, the most popular GGA functionals, PBE and BLYP, have mean unsigned errors of 0.52 and 1.03 kcal/mol, respectively. Since GGAs are very cost efficient for both condensed-phase simulations and electronic structure calculations on large systems, we optimized four new GGAs for water. The best of these, PBE1W and MPWLYP1W, have mean unsigned errors of 0.12 and 0.17 kcal/mol, respectively. These new functionals are well suited for use in condensed-phase simulations of water and ice.     

Design of an exchange functional with correct asymptotics

Knowledge of asymptotic conditions on exchange allows for a better design of exchange energy expressions in density functional theory. By working inside an exchange-only framework, the fulfillment of such conditions by some of the most widely used exchange functionals is discussed. In turn, we propose a model expression which partially meets the energetics and asymptotics of both the exchange energy density and potential. Improvement upon the local spin density approximation without the use of generalized gradient corrections is also presented. Hartree-Fock orbitals are employed to build electron densities. © 1997 John Wiley & Sons, Inc.     

On the accuracy of frozen density embedding calculations with hybrid and orbital-dependent functionals for non-bonded interaction energies

We analyze the accuracy of the frozen density embedding (FDE) method, with hybrid and orbital-dependent exchange-correlation functionals, for the calculation of the total interaction energies of weakly interacting systems. Our investigation is motivated by the fact that these approaches require, in addition to the non-additive kinetic energy approximation, also approximate non-additive exact-exchange energies. Despite this further approximation, we find that the hybrid/orbital-dependent FDE approaches can reproduce the total energies with the same accuracy (about 1 mHa) as the one of conventional semi-local functionals. In many cases, thanks to error cancellation effects, hybrid/orbital-dependent approaches yield even the smallest error. A detailed energy-decomposition investigation is presented. Finally, the Becke-exchange functional is found to reproduce accurately the non-additive exact-exchange energies also for non-equilibrium geometries. These performances are rationalized in terms of a reduced-gradient decomposition of the non-additive exchange energy.     

Generalized gradient approximation model exchange holes for range-separated hybrids

We propose a general model for the spherically averaged exchange hole corresponding to a generalized gradient approximation (GGA) exchange functional. Parameters are reported for several common GGAs. Our model is based upon that of Ernzerhof and Perdew [J. Chem. Phys. 109, 3313 (1998)]. It improves upon the former by precisely reproducing the energy of the parent GGA, and by enabling fully analytic evaluation of range-separated hybrid density functionals. Analytic results and preliminary thermochemical tests indicate that our model also improves upon the simple, local-density-based exchange hole model of Iikura et al. [J. Chem. Phys. 115, 3540 (2001)].     

Improved constraint satisfaction in a simple generalized gradient approximation exchange functional

Though there is fevered effort on orbital-dependent approximate exchange-correlation functionals, generalized gradient approximations, especially the Perdew-Burke-Ernzerhof (PBE) form, remain the overwhelming choice in calculations. A simple generalized gradient approximation (GGA) exchange functional [A. Vela, V. Medel, and S. B. Trickey, J. Chem. Phys. 130, 244103 (2009)] was developed that improves substantially over PBE in energetics (on a typical test set) while being almost as simple in form. The improvement came from constraining the exchange enhancement factor to be below the Lieb-Oxford bound for all but one value of the exchange dimensionless gradient, s, and to go to the uniform electron gas limit at both s = 0 and s → ∞. Here we discuss the issue of asymptotic constraints for GGAs and show that imposition of the large s constraint, lim(s→∞)s(1/2)F(xc)(n,s)<∞, where F(xc)(n, s) is the enhancement factor and n is the electron density, upon the Vela-Medel-Trickey (VMT) exchange functional yields modest further improvement. The resulting exchange functional, denoted VT{8,4}, is only slightly more complicated than VMT and easy to program. Additional improvement is obtained by combining VT{8,4} or VMT exchange with the Lee-Yang-Parr correlation functional. Extensive computational results on several datasets are provided as verification of the overall performance gains of both versions.     

Complex model optical potential for electron scattering from oxygen

We report calculations of the total (elastic plus inelastic) cross-sections for the e-O system over a wide energy range 10–5000 eV. A local complex optical potential is calculated for the system using the atomic wavefunctions at the Hartree-Fock level. The real part of potential is composed of the attractive static, correlation polarisation and exchange potential. The imaginary component of the complex potential is a function of target charge density, incident electron energy and mean excitation energy. The resulting complex potential is treated in variable phase approach to yield complex phase shifts and total cross-sections. The total and elastic cross-sections are compared with the other available results. The agreement is excellent for total cross-sections with other theoretical work at energies greater than 100 eV. We have excellent agreement for elastic cross-sections with the experimental and theoretical results at all the energies. The elastic differential cross-sections are presented at 50 eV and 1000 eV. We fit the absorption cross-section values to Bethe asymptotic formula in high energy range (≥ 5500 eV). Ionisation cross-sections above 1000 eV are also deduced from the theory.     

Asymptotic exchange energy of heteronuclear dimers

A simple expression for the asymptotic exchange energy of heteronuclear dimers is derived from the surface integral method. A five-dimensional hypersurface, consisting of all spherical surfaces centered at the nucleus of the atom with higher ionization energy, more appropriate for the case where the two atoms have different ionization energies, is used in the surface integral. All integrals are carried out analytically. Compared with the exchange energy of Smirnov and Chibisov, which is also obtained from the surface integral method with a hypersurface consisting of all infinite planes perpendicular to the internuclear axis, the present result is much simpler. The exchange energies of alkali hydrides are computed as an illustration. It is shown that the present method and the method of Smirnov and Chibisov are complementary to each other.     

Evidence of conformational exchange averaging in the thermal rotational spectrum of ethyl cyanoformate

The Stark modulated low resolution microwave spectrum of ethyl cyanoformate between 21.5 and 24.0 GHz at 210, 300, and 358 K, which shows the J + 1 <-- J = 8 <-- 7 bands of three species, is compared to simulations based on electronic structure calculations at the MP2/6-311++G theory level. Calculations at this theory level reproduce the relative energies of the syn-anti and syn-gauche conformers, obtained in a previous study, and indicate that the barrier to conformer exchange is approximately 360 cm(-1) higher in energy than the syn-anti minimum. Simulated spectra of the eigenstates of the calculated O-ethyl torsional potential function reproduce the relative intensities and shapes of the lower and higher frequency bands which correspond to transitions of the syn-anti and syn-gauche conformers, respectively, but fail to reproduce the intense center band in the experimental spectra. A model incorporating exchange averaging reproduces the intensity of the center band and its temperature dependence. These simulations indicate that a large fraction of the thermal population at all three temperatures undergoes conformational exchange with an average energy specific rate constant, , of approximately 25 GHz. This model can explain anomalies present in rotational spectra of many other compounds composed of mixtures of conformers.