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

A gauge origin independent formalism for the calculation of molecular magnetic properties is presented. Origin independence is obtained by using London’s gauge invariant atomic orbitals, expanding the second quantization Hamiltonian in the external magnetic field and nuclear magnetic moments, and using the resulting expansion terms as perturbation operators in response function calculations. To ensure orthonormality of the molecular orbitals, a field‐dependent symmetrical orthonormalization is employed. In this way the gauge dependence of the London orbitals is transferred to the Hamiltonian. The resulting perturbation operators may be used to calculate magnetic properties from any approximate ab initio wave function.

An electronic Hamiltonian for origin independent calculations of magnetic properties

The problem of finding orbitals, which make the total energy stationary, when the ground state or reference state is of the multiconfigurational form, is analyzed in this paper. Particular attention is given to the quadratically convergent methods. Orbital variations are generated by unitary transformations and an expansion of the total energy through second order in the parameters used to define these transformations provides a characterization of the stationary point, which is reached at convergence, as well as estimates of second order properties. Effective one‐particle potentials are analyzed and it is emphasized that only certain blocks of the Fock matrix are determined by the generalized Brillouin theorem. In an analysis of the convergence properties of the various methods it is shown that the Hartree–Fock procedure can be brought to converge for stable states of a given symmetry if the blocks of the Fock matrix, which are not determined by the Brillouin theorem, are chosen in an appropriate fashion...

Optimization of orbitals for multiconfigurational reference states

Coupled cluster singles and doubles linear response (CCLR) calculations have been carried out for excitation energies and dipole transition strengths for the lowest excitations in LiH, CH+, and C4 and the results compared with the results from a CI‐like approach to equation of motion coupled cluster (EOMCC). The transition strengths are similar in the two approaches for single molecule calculations on small systems. However, the CCLR approach gives size‐intensive dipole transition strengths, while the EOMCC formalism does not. Thus, EOMCC calculations can give unphysically dipole transition strengths, e.g., in EOMCC calculations on a sequence of noninteracting LiH systems we obtained a negative dipole strength for the lowest totally symmetric dipole allowed transition for 19 or more noninteracting LiH systems. The CCLR approach is shown to be a very attractive ‘‘black box’’ approach for the calculation of transition moments.

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Calculation of size‐intensive transition moments from the coupled cluster singles and doubles linear response function

Nuclear shielding calculations are presented for multiconfigurational self‐consistent field wave functions using London atomic orbitals (gauge invariant atomic orbitals). Calculations of nuclear shieldings for eight molecules (H2O, H2S, CH4, N2, CO, HF, F2, and SO2) are presented and compared to corresponding individual gauges for localized orbitals (IGLO) results. The London results show better basis set convergence than IGLO, especially for heavier atoms. It is shown that the choice of active space is crucial for determination of accurate nuclear shielding constants.

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Multiconfigurational self-consistent field calculations of nuclear shieldings using London atomic orbitals

High‐order Mo/ller–Plesset perturbation calculations have been carried out for several small molecules and compared to full configuration interaction (FCI) results. The convergence of the Mo/ller–Plesset series is found to depend crucially on the one‐electron basis sets. Addition of diffuse basis functions leads in some cases to divergent behavior of the Mo/ller–Plesset series, even for highly single reference dominated systems as Ne and HF. The results thus questions the usefulness of higher‐order perturbation calculations as a vehicle for obtaining arbitrary accuracy of quantum chemical calculations and raises the fundamental theoretical question: When does Mo/ller–Plesset perturbation theory converge for many‐electron systems in extended basis sets?

Poul Jo

Papers

An electronic Hamiltonian for origin independent calculations of magnetic properties

Optimization of orbitals for multiconfigurational reference states

Calculation of size‐intensive transition moments from the coupled cluster singles and doubles linear response function

Multiconfigurational self-consistent field calculations of nuclear shieldings using London atomic orbitals

Surprising cases of divergent behavior in Mo/ller–Plesset perturbation theory