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

Static and frequency-dependent polarizabilities and first hyperpolarizabilities have been calculated for HF and Ne using full configuration interaction (FCI) and a hierarchy of coupled cluster models: coupled cluster singles (CCS), an approximate coupled cluster singles and doubles model (CC2), coupled cluster singles and doubles (CCSD), an approximate coupled cluster singles, doubles, and triples model (CC3), and coupled cluster singles, doubles, and triples (CCSDT). A previous study of BH concerning FCI benchmarking has been extended to include CC3 and static CCSDT values. Systematic improvements of the polarizabilities and the hyperpolarizabilities are found going from CCS to CCSD and from CCSD to CC3 or CCSDT. Little or no improvement of the polarizabilities and no improvement of the hyperpolarizabilities are seen when going from CCS to CC2. The CCSD results represent a significant improvement over CCS and CC2 but are again surpassed by the CC3 results which agree very well with the FCI values. The re...

Polarizabilities and first hyperpolarizabilities of HF, Ne, and BH from full configuration interaction and coupled cluster calculations

We outline a method for the calculation of nonlinear properties such as dynamic hyperpolarizabilities for self‐consistent‐field (SCF) wave functions. In this method, two‐electron integrals are only addressed in the evaluation of Fock matrices and Fock matrices with one‐index transformed integrals. These matrices are determined directly in terms of integrals evaluated in the atomic orbital basis, avoiding expensive integral transformations between atomic and molecular orbital bases as well as storing and retrieving the two‐electron integrals. The method is double direct—direct in the sense of constructing Fock matrices from atomic integrals, and direct in the sense of solving the response equations iteratively using direct linear transformations of a generating matrix times trial vectors. Applications can be performed on species of the same size as in direct SCF. The cost of evaluating a single nonlinear molecular property is comparable to that of optimizing the wave function. Additional properties can be obtained at little extra cost by solving all response equations simultaneously. As a demonstration, we calculate the static and dynamic hyperpolarizabilities of para‐nitroaniline.

Direct atomic orbital based self‐consistent‐field calculations of nonlinear molecular properties. Application to the frequency dependent hyperpolarizability of para‐nitroaniline

We have developed a multiconfigurational electron propagator (MCEP) technique for the theoretical determination of ionization potentials for general open shell and highly correlated atomic and molecular systems. In order to do this, we have used and extended the generalized spin‐symmetry adapted operators of Pickup and Mukhopadhyay. To properly account for correlation effects we have additionally included ionization and electron affinity operators analogous to the ‖Γ〉〈0‖ state transfer operators necessary in multiconfigurational linear response. MCEP ionization potentials and ionization process probabilities have been evaluated for both O2 and N2 and used to carry out detailed examination and interpretation of the respective PES and ESCA spectra. The MCEP results are extremely encouraging for both principal and shake‐up I.P.′s. For example, using 〈5s4p1d〉 contracted Gaussian basis sets the principal valence ionization potentials to bound ionic states are calculated within ±0.3 eV of experiment for both N2...

Multiconfigurational electron propagator (MCEP) ionization potentials for general open shell systems

Multiconfigurational self‐consistent field (MCSCF) response is used to study the electric field dependence of magnetizabilities and nuclear shielding constants for N2, C2H2, HCN, and H2O. London perturbation‐dependent atomic orbitals are used to ensure gauge origin independence. The computed magnetizabilities and shielding derivatives show a strong electron correlation dependence. The N2 results confirm the conclusions of previous ab initio studies. For the other molecules, this is the first study of the above magnetic properties beyond the SCF approximation.

Electric field dependence of magnetic properties: Multiconfigurational self‐consistent field calculations of hypermagnetizabilities and nuclear shielding polarizabilities of N2, C2H2, HCN, and H2O

Based on an analysis of the first residues of coupled cluster response functions we devise variational functionals from which the transition moments for n-photon excitations can be calculated as nth derivatives. Combining these functionals with variational perturbation theory, we obtain a new approach for the derivation of multiphoton transition moments which allows us to utilize the full strengths of variational perturbation theory without the roundabout way via residues of response functions. Coupled cluster multiphoton transition moments derived by this approach are formally equivalent to the one identified from the first residues of the ground state response functions. The introduction of the variational functionals makes the mathematical structure of the transition moments more transparent and provides an interpretation of intermediates in terms of responses of excited state vectors and Lagrangian multipliers. 2n+1 and 2n+2 rules are formulated for the transition moments and build the basis for a str...

Poul Jo

Papers

Polarizabilities and first hyperpolarizabilities of HF, Ne, and BH from full configuration interaction and coupled cluster calculations

Direct atomic orbital based self‐consistent‐field calculations of nonlinear molecular properties. Application to the frequency dependent hyperpolarizability of para‐nitroaniline

Multiconfigurational electron propagator (MCEP) ionization potentials for general open shell systems

Electric field dependence of magnetic properties: Multiconfigurational self‐consistent field calculations of hypermagnetizabilities and nuclear shielding polarizabilities of N2, C2H2, HCN, and H2O

Multiphoton transition moments and absorption cross sections in coupled cluster response theory employing variational transition moment functionals