A new density functional (DF) of the generalized gradient approximation (GGA) type for general chemistry applications termed B97-D is proposed. It is based on Becke's power-series ansatz from 1997 and is explicitly parameterized by including damped atom-pairwise dispersion corrections of the form C(6) x R(-6). A general computational scheme for the parameters used in this correction has been established and parameters for elements up to xenon and a scaling factor for the dispersion part for several common density functionals (BLYP, PBE, TPSS, B3LYP) are reported. The new functional is tested in comparison with other GGAs and the B3LYP hybrid functional on standard thermochemical benchmark sets, for 40 noncovalently bound complexes, including large stacked aromatic molecules and group II element clusters, and for the computation of molecular geometries. Further cross-validation tests were performed for organometallic reactions and other difficult problems for standard functionals. In summary, it is found that B97-D belongs to one of the most accurate general purpose GGAs, reaching, for example for the G97/2 set of heat of formations, a mean absolute deviation of only 3.8 kcal mol(-1). The performance for noncovalently bound systems including many pure van der Waals complexes is exceptionally good, reaching on the average CCSD(T) accuracy. The basic strategy in the development to restrict the density functional description to shorter electron correlation lengths scales and to describe situations with medium to large interatomic distances by damped C(6) x R(-6) terms seems to be very successful, as demonstrated for some notoriously difficult reactions. As an example, for the isomerization of larger branched to linear alkanes, B97-D is the only DF available that yields the right sign for the energy difference. From a practical point of view, the new functional seems to be quite robust and it is thus suggested as an efficient and accurate quantum chemical method for large systems where dispersion forces are of general importance.

Semiempirical GGA-type density functional constructed with a long-range dispersion correction.

We present two new hybrid meta exchange- correlation functionals, called M06 and M06-2X. The M06 functional is parametrized including both transition metals and nonmetals, whereas the M06-2X functional is a high-nonlocality functional with double the amount of nonlocal exchange (2X), and it is parametrized only for nonmetals.The functionals, along with the previously published M06-L local functional and the M06-HF full-Hartree–Fock functionals, constitute the M06 suite of complementary functionals. We assess these four functionals by comparing their performance to that of 12 other functionals and Hartree–Fock theory for 403 energetic data in 29 diverse databases, including ten databases for thermochemistry, four databases for kinetics, eight databases for noncovalent interactions, three databases for transition metal bonding, one database for metal atom excitation energies, and three databases for molecular excitation energies. We also illustrate the performance of these 17 methods for three databases containing 40 bond lengths and for databases containing 38 vibrational frequencies and 15 vibrational zero point energies. We recommend the M06-2X functional for applications involving main-group thermochemistry, kinetics, noncovalent interactions, and electronic excitation energies to valence and Rydberg states. We recommend the M06 functional for application in organometallic and inorganometallic chemistry and for noncovalent interactions.

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The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals

Multiwfn is a multifunctional program for wavefunction analysis. Its main functions are: (1) Calculating and visualizing real space function, such as electrostatic potential and electron localization function at point, in a line, in a plane or in a spatial scope. (2) Population analysis. (3) Bond order analysis. (4) Orbital composition analysis. (5) Plot density-of-states and spectrum. (6) Topology analysis for electron density. Some other useful utilities involved in quantum chemistry studies are also provided. The built-in graph module enables the results of wavefunction analysis to be plotted directly or exported to high-quality graphic file. The program interface is very user-friendly and suitable for both research and teaching purpose. The code of Multiwfn is substantially optimized and parallelized. Its efficiency is demonstrated to be significantly higher than related programs with the same functions. Five practical examples involving a wide variety of systems and analysis methods are given to illustrate the usefulness of Multiwfn. The program is free of charge and open-source. Its precompiled file and source codes are available from http://multiwfn.codeplex.com.

Multiwfn: a multifunctional wavefunction analyzer.

We present an analysis of the performances of a parameter free density functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amount of exact exchange. The results obtained for structural, thermodynamic, kinetic and spectroscopic (magnetic, infrared and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chemistry and condensed matter physics.

Toward reliable density functional methods without adjustable parameters: The PBE0 model

Although density functional theory is widely used in the computational chemistry community, the most popular density functional, B3LYP, has some serious shortcomings: (i) it is better for main-group chemistry than for transition metals; (ii) it systematically underestimates reaction barrier heights; (iii) it is inaccurate for interactions dominated by medium-range correlation energy, such as van der Waals attraction, aromatic−aromatic stacking, and alkane isomerization energies. We have developed a variety of databases for testing and designing new density functionals. We used these data to design new density functionals, called M06-class (and, earlier, M05-class) functionals, for which we enforced some fundamental exact constraints such as the uniform-electron-gas limit and the absence of self-correlation energy. Our M06-class functionals depend on spin-up and spin-down electron densities (i.e., spin densities), spin density gradients, spin kinetic energy densities, and, for nonlocal (also called hybrid)...

Density functionals with broad applicability in chemistry.

A recently suggested procedure for the systematic optimization of gradient-corrected exchange-correlation functionals [A. D. Becke, J. Chem. Phys. 107, 8554 (1997)] has been applied to the extended G2 test set [L. A. Curtiss et al., J. Chem. Phys. 106, 1063 (1997)], which consists of the standard heats of formation of 148 molecules. The limit of reproduction of the experimental data in this test set is found to be 1.78 kcal/mol mean absolute error, with a maximum of 8.89 kcal/mol error for the ozone molecule. This compares rather well with previous results for G2 theory itself (1.58 and 8.2 kcal/mol, respectively). We show that fair stability can be obtained by our optimization procedure.

Optimized density functionals from the extended G2 test set

A recent concept for the simulation of delocalized exact exchange [A.D. Becke, J. Chem. Phys. 112 (2000) 4020] involves a variable t σ that depends only on local values of the non-interacting kinetic energy density and of the charge density. Here, we show that this variable is highly indicative of details of atomic and molecular electronic structure, and reflects common chemical concepts such as atomic shells, molecular bonds and lone-pair regions, in a clear and intuitive manner. Explanations for this behavior are given in terms of localized orbitals, as well as on kinematic grounds. On the example of a few simple chemical reactions, it is demonstrated that bond formation and bond breaking are reflected in this variable, allowing a simple “sectioning” of the potential energy surface in terms of species involved.

Chemical content of the kinetic energy density

We examine and compare two previously introduced functions of the one-particle density matrix that are suitable to represent its off-diagonal structure in a condensed form and that have illustrative connections to the nature of the chemical bond. One of them, the Localized-Orbital Locator (LOL) [J. Molec. Struct. (THEOCHEM) 527, 51 (2000)], is based only on the noninteracting kinetic-energy density τ and the charge density ρ at a point, and gives an intuitive measure of the relative speed of electrons in its vicinity. Alternatively, LOL focuses on regions that are dominated by single localized orbitals. The other one, the Parity Function P [J. Chem. Phys. 105, 11134 (1996)], is a section through the Wigner phase-space function at zero momentum, and contains information about the phase of the interference of atomiclike orbital contributions from bound centers. In this paper, we discuss the way in which these functions condense information in the density matrix, and illustrate on a variety of examples of unusual chemical bonds how they can help to understand the nature of “covalence.”

Two functions of the density matrix and their relation to the chemical bond

In a recent paper [H. L. Schmider and A. D. Becke, J. Chem. Phys. 108, 9624 (1998)], we applied a systematic method for the determination of exchange-correlation functionals within the generalized gradient approximation (GGA) to the extended G2 test set of standard heats of formation of Curtiss et al. [J. Chem. Phys. 106, 1063 (1997)]. In the present work, we apply a similar methodology that goes beyond the GGA by taking second-order gradients and the (noninteracting) kinetic-energy density into account. The resulting improvement in the reproduction of thermochemical data brings us very close to the quality of G2 theory itself. Our lowest mean absolute error for standard heats of formation, 1.60 kcal/mol, is only marginally greater than the G2 value (1.58 kcal/mol). The corresponding largest deviation is 9.97 kcal/mol, as compared to 8.2 kcal/mol for G2 theory.

Density functionals from the extended G2 test set: Second-order gradient corrections

The structure of the one-particle reduced density matrix when expressed in a Cartesian Gaussian basis set is investigated. A set of exact linear dependency conditions between products of basis functions, which result from the angular behaviour of the basis functions, is discovered. Some of these exact linear dependencies hold simultaneously in both position and momentum spaces making it possible to alter the one matrix while keeping both the position and momentum densities fixed. The magnitude of this space is easily predicted for the Pople and Dunning-Hay basis sets commonly used in quantum chemical calculations, and we give simple rules for their enumeration. It is further shown that alteration of the one-matrix component in this space alters the eigenvalue structure of the one-matrix and therefore has consequences for N-representability. Using the one-matrix corresponding to a wavefunction as a starting point, the eigenvalue change is always in the same direction, that is small eigenvalues get more negative while large ones become more positive. For independent particle model wavefunctions, which are already extreme in their eigenvalues, no change is possible without breaking the N-representability conditions.

Hartmut L. Schmider

Papers

Optimized density functionals from the extended G2 test set

Chemical content of the kinetic energy density

Two functions of the density matrix and their relation to the chemical bond

Density functionals from the extended G2 test set: Second-order gradient corrections

Linearly dependent subspaces and the eigenvalue spectrum of the one-particle reduced density matrix