Basis set convergence of explicitly correlated double-hybrid density functional theory calculations

The basis set convergence of explicitly correlated double-hybrid density functional theory (DFT) is investigated using the B2GP-PLYP functional. As reference values, we use basis set limit B2GP-PLYP-F12 reaction energies extrapolated from the aug(')-cc-pV(Q+d)Z and aug(')-cc-pV(5+d)Z basis sets. Explicitly correlated double-hybrid DFT calculations converge significantly faster to the basis set limit than conventional calculations done with basis sets saturated up to the same angular momentum (typically, one "gains" one angular momentum in the explicitly correlated calculations). In explicitly correlated F12 calculations the VnZ-F12 basis sets converge faster than the orbital A(')VnZ basis sets. Furthermore, basis set convergence of the MP2-F12 component is apparently faster than that of the underlying Kohn-Sham calculation. Therefore, the most cost-effective approach consists of combining the MP2-F12 correlation energy from a comparatively small basis set such as VDZ-F12 with a DFT energy from a larger basis set such as aug(')-cc-pV(T+d)Z.     

Semiempirical hybrid functional with improved performance in an extensive chemical assessment

It is demonstrated that there is still scope for improvement in the quality of conventional, semiempirical hybrid exchange-correlation functionals in density-functional theory. A new functional, denoted B97-3, is determined from a fit to eight chemical properties (316 data points). For a series of 25 chemical assessments (850 data points) including 17 assessments and 10 chemical properties absent from the fitting data, B97-3 provides the lowest or joint-lowest mean absolute error on 15 occasions, compared to 6, 5, and 4 occasions for B3LYP, PBE0, and B97-2, respectively [A. D. Becke, J. Chem. Phys. 98, 5648 (1993); M. Ernzerhof and G. E. Scuseria, J. Chem. Phys. 110, 5029 (1999); C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 (1999); P. J. Wilson, T. J. Bradley, and D. J. Tozer, J. Chem. Phys. 115, 9233 (2001)]. Mean absolute errors from B97-3 are, on average, 21%, 18%, and 12% smaller than from these three functionals. The most notable improvements are obtained for classical reaction barriers, where the error reductions are 60%, 54%, and 27%.     

The self-interaction error and the description of non-dynamic electron correlation in density functional theory

The self-interaction error (SIE) plays a central role in density functional theory (DFT) when carried out with approximate exchange-correlation functionals. Its origin, properties, and consequences for the development of standard DFT to a method that can correctly describe multi-reference electron systems by treating dynamic and non-dynamic electron correlation on an equal footing, is discussed. In this connection, the seminal work of Colle and Salvetti on wave function-based correlation functionals that do no longer suffer from a SIE is essential. It is described how the Colle–Salvetti correlation functional is an anchor point for the derivation of a functional multi-reference DFT method.     

B2-PPW91: a promising double-hybrid density functional for the electric response properties

A new double-hybrid density functional, termed B2-PPW91, is presented which includes the Becke88 (B88) exchange in conjunction with Perdew-Wang91 (PW91) gradient-corrected correlation functional. The fitting parameters are obtained by minimization of mean absolute error of the static dipole polarizability of 4d transition metal monohalides against the CCSD(T)∕aug-cc-pVTZ∕SDD results. The performance of proposed functional has been assessed for estimation of other response properties, such as dipole moment and excitation energy, for the same species. We then proceed to explore the validity of B2-PPW91 method for calculation of the dipole polarizability of some 5d transition metal monofluorides. In all cases, the improvement compared to common density functional methods and even previously reported double-hybrid functionals such as B2-PLYP and mPW2-PLYP has been observed. This indicates that the utility of double-hybrid density functional methods can be further extended to study linear and non-linear optical properties of transition metal containing molecules.     

A short-range correlation energy density functional with multi-determinantal reference

We introduce a short-range correlation density functional defined with respect to a multi-determinantal reference which is meant to be used in a multi-determinantal extension of the Kohn–Sham scheme of density functional theory based on a long-range/short-range decomposition of the Coulomb electron–electron interaction. We construct the local density approximation for this functional and discuss its performance on the He atom.     

Scaled density functional theory correlation functionals

We show that a simple one-parameter scaling of the dynamical correlation energy estimated by the density functional theory (DFT) correlation functionals helps increase the overall accuracy for several local and nonlocal functionals. The approach taken here has been described as the "scaled dynamical correlation" (SDC) method [Ramachandran, J. Phys. Chem. A 2006, 110, 396], and its justification is the same as that of the scaled external correlation (SEC) method of Brown and Truhlar. We examine five local and five nonlocal (hybrid) DFT functionals, the latter group including three functionals developed specifically for kinetics by the Truhlar group. The optimum scale factors are obtained by use of a set of 98 data values consisting of molecules, ions, and transition states. The optimum scale factors, found with a linear regression relationship, are found to differ from unity with a high degree of correlation in nearly every case, indicating that the deviation of calculated results from the experimental values are systematic and proportional to the dynamic correlation energy. As a consequence, the SDC scaling of dynamical correlation decreases the mean errors (signed and unsigned) by significant amounts in an overwhelming majority of cases. These results indicate that there are gains to be realized from further parametrization of several popular exchange-correlation functionals.     

Toward a density functional description of liquid pH2

A finite-temperature density functional approach to describe the properties of parahydrogen in the liquid-vapor coexistence region is presented. The first proposed functional is zero-range, where the density-gradient term is adjusted so as to reproduce the surface tension of the liquid-vapor interface at low temperature. The second functional is finite-range and, while it is fitted to reproduce bulk pH(2) properties only, it is shown to yield surface properties in good agreement with experiments. These functionals are used to study the surface thickness of the liquid-vapor interface, the wetting transition of parahydrogen on a planar Rb model surface, and homogeneous cavitation in bulk liquid pH(2).     

Water cluster anions studied by the long-range corrected density functional theory

Long-range corrected density functional theory (LC-DFT) is applied to a series of small water cluster anions(n= 2-6) to compute their vertical detachment energies (VDEs). The LC scheme is shown to eliminate an unphysical overestimation of the electron-water attraction in the hybrid functional by properly accounting for the long-range exchange repulsions. It is shown that a correct correlation energy behavior for a rapidly varying density is also important for describing a spatially extent, excess electron. The one-parameter progressive (OP) correlation functional, which satisfies this condition, leads to a remarkable improvement in the calculated VDE over the conventional one. The LC-BOP method produces highly accurate VDEs with a mean absolute deviation of 13.8 meV from the reference CCSD(T) results, reducing the error of B3LYP by more than 15 times. LC-BOP is found to be more accurate than MP2 which yields an excess electron underbound by 43.6 meV. The effect of basis sets on the calculated VDE is also examined. The aug-cc-pVDZ basis set with an extra diffuse function is found to be more accurate and reliable than the extended Pople-type basis sets used in the previous works. The extrapolation of the calculated VDE of different electron binding motifs is compared with the VDEs of experimentally observed three isomers (Verlet, J. R. R.; Bragg,A. E.; Kammrath, A.; Cheshnovsky, O.; Neumark, D. M. Science 2005, 307, 93).     

Scaling dynamical correlation energy from density functional theory correlation functionals

The scaling of dynamical correlation energy in molecules obtained by the correlation functionals of density functional theory (DFT) is examined. The approach taken is very similar to the scaled external correlation method of Brown and Truhlar but is based on the observation that DFT correlation functionals, especially the LYP, appear to represent the dynamical portion of the correlation energy in molecules. We examine whether higher accuracy in atomization energies can be gained by scaling without significant deterioration of the structural and spectroscopic properties of the molecules using four DFT functionals (BLYP, OLYP, B3LYP, and O3LYP) on 19 molecules including the six molecule AE6 database, the latter being representative of a much larger, 109 molecule training set. We show that, with molecule specific scale factors, nearly perfect agreement with experiment can be achieved in atomization energies without increasing the average errors in other molecular properties relative to the DFT calculation. We further show that it is possible to find optimal scale factors which reduce the mean unsigned error per bond to levels comparable to those of some multilevel multicoefficient methods.     

Improving the orbital-free density functional theory description of covalent materials

The essential challenge in orbital-free density functional theory (OF-DFT) is to construct accurate kinetic energy density functionals (KEDFs) with general applicability (i.e., transferability). During the last decade, several linear-response (LR)-based KEDFs have been proposed. Among them, the Wang-Govind-Carter (WGC) KEDF, containing a density-dependent response kernel, is one of the most accurate that still affords a linear scaling algorithm. For nearly-free-electron-like metals such as Al and its alloys, OF-DFT employing the WGC KEDF produces bulk properties in good agreement with orbital-based Kohn-Sham (KS) DFT predictions. However, when OF-DFT, using the WGC KEDF combined with a recently proposed bulk-derived local pseudopotential (BLPS), was applied to semiconducting and metallic phases of Si, problems arose with convergence of the self-consistent density and energy, leading to poor results. Here we provide evidence that the convergence problem is very likely caused by the use of a truncated Taylor series expansion of the WGC response kernel. Moreover, we show that a defect in the ansatz for the first-order reduced density matrix underlying the LR KEDFs limits the accuracy of these KEDFs. By optimizing the two free parameters involved in the WGC KEDF, the two-body Fermi wave vector mixing parameter gamma and the reference density rho* used in the Taylor expansion, OF-DFT calculations with the BLPS can achieve semiquantitative results for nine phases of bulk silicon. These new parameters are recommended whenever the WGC KEDF is used to study nonmetallic systems.     

Exchange and correlation energy in density functional theory

Several different versions of density functional theory (DFT) that satisfy Hohenberg–Kohn theorems are characterized by different definitions of a reference or model state determined by an N‐electron ground state. A common formalism is developed in which exact Kohn–Sham equations are derived for standard Kohn–Sham theory, for reference‐state density functional theory, and for unrestricted Hartree–Fock (UHF) theory considered as an exactly soluble model Hohenberg–Kohn theory. A natural definition of exchange and correlation energy functionals is shown to be valid for all such theories. An easily computed necessary condition for the locality of exchange and correlation potentials is derived. While it is shown that in the UHF model of DFT the optimized effective potential (OEP) exchange satisfies this condition by construction, the derivation shows that this condition is not, in general, sufficient to define an exact local exchange potential. It serves as a test to eliminate proposed local potentials that are not exact for ground states. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 521–525, 2000     

Polarizability and second hyperpolarizability evaluation of long molecules by the density functional theory with long-range correction

Polarizabilities and second hyperpolarizabilities of polyacetylene and a hydrogen chain are evaluated by density functional theory (DFT) using a hybrid generalized gradient approximation functional with correct long-range electron-electron interactions. The well known catastrophic overestimate of the hyperpolarizabilities for molecular systems of enhanced length is corrected by the two-electron repulsion operator decomposition technique, integrating the distance-dependent nonlocal exchange effects for long-range interaction, while neither the asymptotically corrected exchange functional for long-range interaction nor ordinary hybrid methods seem to be capable of overcoming the serious drawback of the DFT in polarizability/hyperpolarizability evaluation.     

Steric effect: a quantitative description from density functional theory

The concepts of steric energy, steric potential, and steric charge are introduced within the density functional theory framework. The steric energy, representing a hypothetical state with all electrons packed into the lowest orbital and other effects entirely excluded, is a measure of the intrinsic space occupied by an electronic system. It is exclusive, repulsive, and extensive, and it vanishes for homogeneous electron gas. When Bader's zero-flux boundary condition is adopted, atoms in molecules are found to achieve balanced steric repulsion among one another with vanished steric energy density interfaces. A few molecular systems involving conformation changes and chemical reactions have been investigated to examine the relative contribution of the steric and other effects, providing insights for a few controversial topics from a different perspective.     

Application of semiempirical long-range dispersion corrections to periodic systems in density functional theory

Ewald summation is used to apply semiempirical long-range dispersion corrections (Grimme, J Comput Chem 2006, 27, 1787; 2004, 25, 1463) to periodic systems in density functional theory. Using the parameters determined before for molecules and the Perdew-Burke-Ernzerhof functional, structure parameters and binding energies for solid methane, graphite, and vanadium pentoxide are determined in close agreement with observed values. For methane, a lattice constant a of 580 pm and a sublimation energy of 11 kJ mol(-1) are calculated. For the layered solids graphite and vanadia, the interlayer distances are 320 pm and 450 pm, respectively, whereas the graphite interlayer energy is -5.5 kJ mol(-1) per carbon atom and layer. Only when adding the semiempirical dispersion corrections, realistic values are obtained for the energies of adsorption of C(4) alkenes in microporous silica (-66 to -73 kJ mol(-1)) and the adsorption and chemisorption (alkoxide formation) of isobutene on acidic sites in the micropores of zeolite ferrierite (-78 to -94 kJ mol(-1)). As expected, errors due to missing self-interaction correction as in the energy for the proton transfer from the acidic site to the alkene forming a carbenium ion are not affected by the dispersion term. The adsorption and reaction energies are compared with the results from M?ller-Plesset second-order perturbation theory with basis set extrapolation.     

Application of double-hybrid density functionals to charge transfer in N-substituted pentacenequinones

A set of N-heteroquinones, deriving from oligoacenes, have been recently proposed as n-type organic semiconductors with high electron mobilities in thin-film transistors. Generally speaking, this class of compounds self-assembles in neighboring π-stacks linked by weak hydrogen bonds. We aim at theoretically characterizing here the sequential charge transport (hopping) process expected to take place across these arrays of molecules. To do so, we need to accurately address the preferred packing of these materials simultaneously to single-molecule properties related to charge-transfer events, carefully employing dispersion-corrected density functional theory methods to accurately extract the key molecular parameters governing this phenomenon at the nanoscale. This study confirms the great deal of interest around these compounds, since controlled functionalization of model molecules (i.e., pentacene) allows to efficiently tune the corresponding charge mobilities, and the capacity of modern quantum-chemical methods to predict it after rationalizing the underlying structure-property relationships.