Despite the remarkable thermochemical accuracy of Kohn–Sham density‐functional theories with gradient corrections for exchange‐correlation [see, for example, A. D. Becke, J. Chem. Phys. 96, 2155 (1992)], we believe that further improvements are unlikely unless exact‐exchange information is considered. Arguments to support this view are presented, and a semiempirical exchange‐correlation functional containing local‐spin‐density, gradient, and exact‐exchange terms is tested on 56 atomization energies, 42 ionization potentials, 8 proton affinities, and 10 total atomic energies of first‐ and second‐row systems. This functional performs significantly better than previous functionals with gradient corrections only, and fits experimental atomization energies with an impressively small average absolute deviation of 2.4 kcal/mol.

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Density‐functional thermochemistry. III. The role of exact exchange

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

Previous attempts to combine Hartree–Fock theory with local density‐functional theory have been unsuccessful in applications to molecular bonding. We derive a new coupling of these two theories that maintains their simplicity and computational efficiency, and yet greatly improves their predictive power. Very encouraging results of tests on atomization energies, ionization potentials, and proton affinities are reported, and the potential for future development is discussed.

A New Mixing of Hartree-Fock and Local Density-Functional Theories

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

6.2.2. Definition of Effective Properties 3064 6.3. Response Properties to Magnetic Fields 3066 6.3.1. Nuclear Shielding 3066 6.3.2. Indirect Spin−Spin Coupling 3067 6.3.3. EPR Parameters 3068 6.4. Properties of Chiral Systems 3069 6.4.1. Electronic Circular Dichroism (ECD) 3069 6.4.2. Optical Rotation (OR) 3069 6.4.3. VCD and VROA 3070 7. Continuum and Discrete Models 3071 7.1. Continuum Methods within MD and MC Simulations 3072

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Quantum mechanical continuum solvation models.

A fifth-order perturbation comparison of electron correlation theories

The Gaussian‐2 theoretical procedure (G2 theory), based on a b i n i t i o molecular orbital theory, for calculation of molecular energies (atomization energies, ionization potentials,electron affinities, and proton affinities) of compounds containing first‐ (Li–F) and second‐row atoms (Na–Cl) is presented. This new theoretical procedure adds three features to G1 theory [J. Chem. Phys. 9 0, 5622 (1989)] including a correction for nonadditivity of diffuse‐s p and 2d f basis set extensions, a basis set extension containing a third d function on nonhydrogen and a second p function on hydrogen atoms, and a modification of the higher level correction. G2 theory is a significant improvement over G1 theory because it eliminates a number of deficiencies present in G1 theory. Of particular importance is the improvement in atomization energies of ionic molecules such as LiF and hydrides such as C2H6, NH3, N2H4, H2O2, and CH3SH. The average absolute deviation from experiment of atomization energies of 39 first‐row compounds is reduced from 1.42 to 0.92 kcal/mol. In addition, G2 theory gives improved performance for hypervalent species and electron affinities of second‐row species (the average deviation from experiment of electron affinities of second‐row species is reduced from 1.94 to 1.08 kcal/mol). Finally, G2 atomization energies for another 43 molecules, not previously studied with G1 theory, many of which have uncertain experimental data, are presented and differences with experiment are assessed.

Gaussian-2 theory for molecular energies of first- and second-row compounds

The direct (recomputation of two‐electron integrals) implementation of the gauge‐including atomic orbital (GIAO) and the CSGT (continuous set of gauge transformations) methods for calculating nuclear magnetic shielding tensors at both the Hartree‐Fock and density functional levels of theory are presented. Isotropic 13C, 15N, and 17O magnetic shielding constants for several molecules, including taxol (C47H51NO14 using 1032 basis functions) are reported. Shielding tensor components determined using the GIAO and CSGT methods are found to converge to the same value at sufficiently large basis sets; however, GIAO shielding tensor components for atoms other than carbon are found to converge faster with respect to basis set size than those determined using the CSGT method for both Hartree‐Fock and DFT. For molecules where electron correlation effects are significant, shielding constants determined using (gradient‐corrected) pure DFT or hybrid methods (including a mixture of Hartree‐Fock exchange and DFT exchange...

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A comparison of models for calculating nuclear magnetic resonance shielding tensors

The low‐lying dns2→dn+1s1 excitation energies of the first row transition metal atoms Sc–Cu are calculated using fourth‐order M≂ller–Plesset perturbation theory (MP4) as well as quadratic configuration interaction (QCI) techniques with large spd and spdf  basis sets. The MP4 method performs well for Sc–Mn but fails dramatically for Fe–Cu. In contrast, the QCI technique performs uniformly for all excitation energies with a mean deviation from experiment of only 0.14 eV after including relativistic corrections. f functions contribute 0.1–0.4 eV to the excitation energies for these systems. The highly correlated d10 state of the Ni atom is also considered in detail. The QCI technique obtains the d9s1→d10 splitting of the Ni atom with an error of only 0.13 eV. The results show that single‐configuration Hartree–Fock based methods can be successful in calculating excitation energies of transition metal atoms.

Highly correlated systems. Ionization energies of first row transition metals Sc–Zn

The Gaussian‐1 theoretical procedure is extended and tested on compounds containing second‐row atoms (Na–Cl). This is a composite procedure based on ab initio molecular orbital theory, utilizing large basis sets (including diffuse‐sp, double‐d, and f‐polarization functions) and treating electron correlation by Mo/ller–Plesset perturbation theory and by quadratic configuration interaction. Total atomization energies for a set of 24 species agree with accurate experimental data to an accuracy of better than 3 kcal/mol in most cases, SO2 being the notable exception. Similar agreement is achieved for ionization energies, electron affinities, and proton affinities. The method is used to assess experimental data for a number of other compounds having less accurate atomization energies.

Gary W. Trucks

Papers

A fifth-order perturbation comparison of electron correlation theories

Gaussian-2 theory for molecular energies of first- and second-row compounds

A comparison of models for calculating nuclear magnetic resonance shielding tensors

Highly correlated systems. Ionization energies of first row transition metals Sc–Zn

Gaussian‐1 theory of molecular energies for second‐row compounds