A multireference second-order perturbation theory using a restricted active space self-consistent field wave function as reference (RASPT2/RASSCF) is described. This model is particularly effective for cases where a chemical system requires a balanced orbital active space that is too large to be addressed by the complete active space self-consistent field model with or without second-order perturbation theory (CASPT2 or CASSCF, respectively). Rather than permitting all possible electronic configurations of the electrons in the active space to appear in the reference wave function, certain orbitals are sequestered into two subspaces that permit a maximum number of occupations or holes, respectively, in any given configuration, thereby reducing the total number of possible configurations. Subsequent second-order perturbation theory captures additional dynamical correlation effects. Applications of the theory to the electronic structure of complexes involved in the activation of molecular oxygen by mono- and binuclear copper complexes are presented. In the mononuclear case, RASPT2 and CASPT2 provide very similar results. In the binuclear cases, however, only RASPT2 proves quantitatively useful, owing to the very large size of the necessary active space.

Multiconfigurational quantum chemical methods (CASSCF/CASPT2) have been used to study the chemical bond in the actinide diatoms Ac₂, Th₂, Pa₂, and U₂. Scalar relativistic effects and spin−orbit coupling have been included in the calculations. In the Ac₂ and Th₂ diatoms the atomic 6d, 7s, and 7p orbitals are the significant contributors to the bond, while for the two heavier diatoms, the 5f orbitals become increasingly important. Ac₂ is characterized by a double bond with a ³âˆ‘_g^-(0_g⁺) ground state, a bond distance of 3.64. Ã…, and a bond energy of 1.19 eV. Th₂ has quadruple bond character with a ³D_g(1_g) ground state. The bond distance is 2.76 Ã… and the bond energy (D₀) 3.28 eV. Pa₂ is characterized by a quintuple bond with a ³âˆ‘_g^-(0_g⁺) ground state. The bond distance is 2.37 Ã… and the bond energy 4.00 eV. The uranium diatom has also a quintuple bond with a ⁷O_g (8_g) ground state, a bond distance of 2.43 Ã…, and a bond energy of 1.15 eV. It is concluded that the strongest bound actinide diatom is Pa₂, characterized by a well-developed quintuple bond.

The two-photon spectrum of the 2¹A_g â†� 1¹A_g transition in trans-stilbene has been calculated at the complete active space self-consistent field (CASSCF) level of theory. Energies were obtained at the complete active space second-order perturbation (CASPT2) level of theory, while the geometries of both the initial and final states were optimized at the CASSCF level. The energy and the geometry optimizations were performed using an active space of 14 electrons in 14 active Ï€ orbitals. The vibrational frequencies of both states and the two-photon transition (TPT) cross-section were calculated with a smaller active space where the two lowest Ï€ orbitals were kept inactive. A newly implemented algorithm, in the quantum chemical package Molcas was used to determine the two-photon transition intensity. This method requires only the linear response of the CASSCF wavefunction. Furthermore, the vibronic structure of this TPT was studied. The Franck-Condon factors were obtained by calculating the overlap between the vibrational states involved, which were determined from the force fields of both the initial and final states, at the CASSCF level of theory. The results are in agreement with experiment.

The results of a theoretical study of the ground state, 1¹A_g, and of the lowest ¹B_u states oftrans-stilbene are presented. The vertical and adiabatic excitation energies of the lowest ¹B_ustates have been computed using multiconfigurational SCF theory, followed by second-order perturbation theory. It is shown that the two lowest excited states are separated by a small energy gap in the Franck−Condon region. They are the 1¹B_u, characterized by the HOMO→LUMO single excitation substantially localized on the ethylenic moiety, and the 2¹B_u, formed by a combination of one electron excitations localized mainly on the benzene rings. The most intense transition is found to be the lowest in energy when the interaction between different states is included at the level of second-order perturbation theory. The vibronic structure of emission and absorption spectra of the two lowest ¹B_u states have been determined within the Franck−Condon approximation. The spectrum calculated for the 1¹B_ustate agrees with the experimental spectrum, while the low intensity band computed for the 2¹B_u state has no experimental counterpart. It is concluded that this band is buried in the strong 1¹B_u absorption and therefore not observed.

This article presents a numerical quadrature intended primarily for evaluating integrals in quantum chemistry programs based on molecular orbital theory, in particular density functional methods. Typically, many integrals must be computed. They are divided up into different classes, on the basis of the required accuracy and spatial extent. Ideally, each batch should be integrated using the minimal set of integration points that at the same time guarantees the required precision. Currently used quadrature schemes are far from optimal in this sense, and we are now developing new algorithms. They are designed to be flexible, such that given the range of functions to be integrated, and the required precision, the integration is performed as economically as possible with error bounds within specification. A standard approach is to partition space into a set of regions, where each region is integrated using a spherically polar grid. This article presents a radial quadrature which allows error control, uniform error distribution and uniform error reduction with increased number of radial grid points. A relative error less than 10^−14 for all s-type Gaussian integrands with an exponent range of 14 orders of magnitude is achieved with about 200 grid points. Higher angular lquantum numbers, lower precision or narrower exponent ranges require fewer points. The quadrature also allows controlled pruning of the angular grid in the vicinity of the nuclei.

The structure and vibrational frequencies of the UO₂ molecule have been determined using multiconfigurational wave functions (CASSCF/CASPT2), together with a newly developed method to treat spin−orbit coupling. The molecule has been found to have a (5fφ)(7s), ³Î¦_u, Ω = 2 ground state with a U−O bond distance of 1.77 Ã…. The computed antisymmetric stretching σ_u frequency is 923 cm^-1 with a 16/18 isotope ratio of 1.0525 which compares with the experimental values of 915 cm^-1 and 1.0526, respectively. Calculations of the first adiabatic ionization energy gave the value 6.17 eV, which is 0.7 eV larger than the currently accepted experimental result. Reasons for this difference are suggested.