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

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 new implementation of the conductor-like screening solvation model (COSMO) in the GAUSSIAN94 package is presented. It allows Hartree−Fock (HF), density functional (DF) and post-HF energy, and HF and DF gradient calculations:  the cavities are modeled on the molecular shape, using recently optimized parameters, and both electrostatic and nonelectrostatic contributions to energies and gradients are considered. The calculated solvation energies for 19 neutral molecules in water are found in very good agreement with experimental data; the solvent-induced geometry relaxation is studied for some closed and open shell molecules, at HF and DF levels. The computational times are very satisfying:  the self-consistent energy evaluation needs a time 15−30% longer than the corresponding procedure in vacuo, whereas the calculation of energy gradients is only 25% longer than in vacuo for medium size molecules.

Quantum Calculation of Molecular Energies and Energy Gradients in Solution by a Conductor Solvent Model

This paper presents an evaluation of the performance of time-dependent density-functional response theory (TD-DFRT) for the calculation of high-lying bound electronic excitation energies of molecules. TD-DFRT excitation energies are reported for a large number of states for each of four molecules: N2, CO, CH2O, and C2H4. In contrast to the good results obtained for low-lying states within the time-dependent local density approximation (TDLDA), there is a marked deterioration of the results for high-lying bound states. This is manifested as a collapse of the states above the TDLDA ionization threshold, which is at ??HOMOLDA (the negative of the highest occupied molecular orbital energy in the LDA). The ??HOMOLDA is much lower than the true ionization potential because the LDA exchange-correlation potential has the wrong asymptotic behavior. For this reason, the excitation energies were also calculated using the asymptotically correct potential of van Leeuwen and Baerends (LB94) in the self-consistent field step. This was found to correct the collapse of the high-lying states that was observed with the LDA. Nevertheless, further improvement of the functional is desirable. For low-lying states the asymptotic behavior of the exchange-correlation potential is not critical and the LDA potential does remarkably well. We propose criteria delineating for which states the TDLDA can be expected to be used without serious impact from the incorrect asymptotic behavior of the LDA potential

Molecular excitation energies to high-lying bound states from time-dependent density-functional response theory: Characterization and correction of the time-dependent local density approximation ionization threshold

A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initioMO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initioSCF–MO calculations.

Optimization of equilibrium geometries and transition structures

It is shown that large scale MCSCF linear response (MCLR) calculations can be carried out efficiently using an iterative algorithm where the linear transformations are carried out directly, i.e., without explicitly constructing the MCLR matrices. Calculations are presented on H2O of frequency dependent polarizabilities with configuration spaces containing up to 128 283 determinants.

Linear response calculations for large scale multiconfiguration self‐consistent field wave functions

We describe an efficient implementation of the quadratic response function for a multiconfiguration self‐consistent field reference wave function. The quadratic response function determines the hyperpolarizability and its residues determine the two‐photon transition matrix elements and the transition matrix elements between excited states. We report sample calculations for the hyperpolarizability of Ne and for the two‐photon transition matrix elements of Ne and H2.

Quadratic response functions for a multiconfigurational self‐consistent field wave function

Molecular magnetizabilities have been calculated at the Hartree–Fock level for a series of diamagnetic molecules: H2O, NH3, CH4, PH3, H2S, CO2, CSO, CS2, and C3H4 Gauge invariance is imposed by the use of London atomic orbitals The results are compared to those obtained with the IGLO (individual gauge for localized orbitals) method and are found to converge faster to the basis set limit Magnetizabilities obtained from basis sets of different quality never differ by more than 4% for the London method, compared to 20% for IGLO The Hartree–Fock limit may be approached using London basis sets of moderate size, in contrast to calculations of molecular polarizabilities which require large basis sets to be reliable Comparison with experiment shows that the Hartree–Fock method overestimates experimental susceptibilities by 5%–10%

Hartree-Fock limit magnetizabilities from London orbitals

An atomic integral-direct implementation of molecular linear-response properties and excited-state one-electron properties is presented for the coupled cluster models CCS, CC2, and CCSD. Sample calculations are presented for the polarizability of N2 and for excited-state one-electron properties and transition-properties of furan.

Integral-direct coupled cluster calculations of frequency-dependent polarizabilities, transition probabilities and excited-state properties

A Mo/ller–Plesset energy functional (Lagrangian) which is variational in all variables (the Lagrange multipliers, the orbital rotation parameters, and the orbital energies) is constructed. The variational property ensures that the responses of the orbitals and orbital energies to order n in geometrical perturbations determine the energy derivatives to order 2n+1. The Lagrange multipliers satisfy the somewhat stronger 2n+2 rule. The multipliers, orbital rotations, and orbital energy responses are determined from coupled perturbed Hartree–Fock‐type equations using an exponential parametrization of the orbitals. This ensures that the orbital rotations and energy responses are treated in the same way and calculated from a single set of linear equations. Explicit expressions for energy derivatives up to third order are derived for the second‐order Mo/ller–Plesset energy.

Poul Jo

Papers

Linear response calculations for large scale multiconfiguration self‐consistent field wave functions

Quadratic response functions for a multiconfigurational self‐consistent field wave function

Hartree-Fock limit magnetizabilities from London orbitals

Integral-direct coupled cluster calculations of frequency-dependent polarizabilities, transition probabilities and excited-state properties

Mo/ller–Plesset energy derivatives