ALBERT

Notes: A coupled electron pair approximation is derived and illustrative calculations are presented. The present approximation, which we refer to as the unitary coupled electron pair approximation (UCEPA), provides multiconfigurational (MC) reference capability and is as computationally tractable as multireference configuration interaction (MRCI). The method is capable of yielding size-consistent energies if the MC reference function is of the complete active space (CAS) variety. The coefficient matrix of the resultant set of simultaneous linear equations is evaluated using internal/external orbital space partitioning within a unitary group approach (UGA) treatment of the state space. We demonstrate the accuracy of the method on several small benchmark molecules for which full CI results are known, and on nontrivial studies on the singlet–triplet splitting in methylene and the electron affinity of the oxygen atom.

Notes: The recently formulated multiconfiguration-based unitary coupled electron pair approximation (UCEPA) is compared with multireference configuration interaction (MR-CISD) calculations, including all single and double excitations, for the molecules in this study. The electronic states of the molecules in this study are not only of experimental interest, but represent a challenge to any formalism to accurately predict the energy separations of the low-lying electronic states. The equilibrium geometries and fundamental vibrational frequencies of the three lowest electronic states (i.e., 1A1, 3A‘, and 1A‘) of aminonitrene H2N2, and phosphinonitrene, H2PN, have been determined using a split-valence basis with polarization functions on the heavy atoms and a small complete active space self-consistent-field (CASSCF) description of the active space. Both MR-CISD and UCEPA calculations have been performed at the equilibrium structures using larger basis sets to accurately determine the relative energetics of the electronic states. The equilibrium geometries and vibrational frequencies of the two lowest electronic states (i.e., 1A' and 3A‘) of phosphinocarbene, H2PCH, have been determined using a larger than double zeta basis set, augmented with polarization and diffuse functions, and a CASSCF description of the active space. Both MR-CISD and UCEPA calculations were performed on the equilibrium structures and predict that the singlet lies between 10.4 and 11.8 kcal/mol lower in energy than the triplet. The use of a generalized valence bond (GVB) reference function within UCEPA is introduced and is shown to be a useful approximation.

Notes: We present a general procedure for studying intersystem crossing effects in bimolecular chemical reactions, along with an application of this to the O+H2 reaction. In this procedure, we use previously derived singlet and triplet potential energy surfaces that were based on high quality multireference configuration interaction (MRCI) nonrelativistic electronic structure calculations, and the coupling surface is obtained from lower level complete active space self-consistent field (CASSCF) calculations using the effective nuclear charge one-electron Breit–Pauli expression for the spin-orbit interaction. We find that the resulting spin-orbit splittings match the known values for O(3P), O(1D), and OH(2Π) sufficiently accurately to be useful for dynamics calculations. Also, the electronic basis can be truncated to seven states (1 3A′, 1 3A″, and 1 1A′) without seriously distorting these asymptotic splittings. We show that the seven states may be exactly decoupled into a set of four, which contains the singlet, and a set of three states from the triplets. We find that the spin-orbit matrix elements vary smoothly with geometry, so that a relatively simple function can be used to interpolate matrix elements for all geometries. The cross sections for reaction are calculated using a trajectory surface-hopping (TSH) approach in conjunction with a "diabatic" representation based on the nonrelativistic potentials and the CASSCF spin-orbit coupling matrix. An application of this approach is presented to the O+H2 reaction, using the 1 1A′ state of Dobbyn and Knowles, and 1 3A′ and 1 3A″ states of Walch and Kuppermann [slightly modified so that they are asymptotically degenerate in the product (H+OH) region]. The states show a singlet–triplet (S–T) crossing that is generally on the product side of the barrier on the triplet surfaces. The TSH results indicate that only a few percent of the trajectories undergo intersystem crossing (either from singlet to triplet, or vise versa) at the S–T crossing, so the effect of these transitions on measurable properties of the reaction dynamics is small. However, those trajectories that undergo triplet to singlet transition have much higher product rotational excitation than those that react on the triplet alone. We find that a much larger fraction of trajectories (20%–40%) undergo hopping between the two triplet states, and this leads to an averaging of the dynamical results for the two states. © 2000 American Institute of Physics.

Notes: A new quasidegenerate perturbation theory is developed that describes the interactions of electronic states of interest with energetically low-lying excited states variationally and with more high-lying excited states perturbatively. The states of interest, the low-lying excited states and the more high-lying excited states, define primary, secondary, and external subspaces, respectively. The task of determination of the lowest solutions of the full configuration interaction (CI) problem is shown to be equivalent to the task of searching iteratively for an optimal primary subspace within the model space spanned by the initial unperturbed primary and secondary states. It is also shown that the present approach, which we refer to as the self-consistent quasidegenerate perturbation theory (SC-QDPT), theoretically satisfies the following criteria: (1) it avoids instabilities due to intruder states; (2) it ensures the additivity of the energy for noninteracting subsystems; (3) the projection of the correlated wave functions on the model space coincides with the optimal primary subspace; and (4) the energies of the primary states will be restricted below by the full CI limit. Furthermore, by use of an exponential ansatz the model space effective Hamiltonian takes into account finite-order primary-external perturbations exactly. Some of these conclusions are corroborated by the results of application of the lowest-order approximation, SC-QDPT(SD), of the method on the beryllium atom and on the reaction Be+H2, using a simple computer realization. © 1998 American Institute of Physics.

Notes: A unitary wave operator exp(G) is used to relate a multiconfigurational reference function Φ to the full, potentially exact, electronic eigenfunction Ψ=exp(G)Φ. If the reference function Φ is of a generalized complete-active-space (CAS) form, then the energy, computed as 〈Φ||exp(−G)H exp(G)||Φ〉 is size extensive; here H is the full N-electron Hamiltonian. The Hausdorff expansion of exp(−G)H exp(G) is truncated at second order as part of our development. The parameters which appear in the cluster operator G are determined by making this second-order energy stationary. Applications to the widely studied H2O (at the double zeta basis level) and lowest and first excited 1A1 states of BeH2 are performed in order to test this method on problems where "exact'' results are known.

Notes: Equations for the determination of the cluster coefficients in a full coupled cluster theory involving single, double, and triple cluster operators with respect to an independent particle reference, expressible as a single determinant of spin-orbitals, are derived. The resulting wave operator is full, or untruncated, consistent with the choice of cluster operator truncation and the requirements of the connected cluster theorem. A time-independent diagrammatic approach, based on second quantization and the Wick theorem, is employed. Final equations are presented that avoid the construction of rank three intermediary tensors. The model is seen to be computationally viable, size-extensive, high-level description of electron correlation in small polyatomic molecules.

Notes: A variational formulation finite element method is developed for calculation of vibrational wave functions in a domain spanned by close-coupled, or Jacobi, coordinates R and γ. C1 tensor-product basis functions, which allow straightforward separation of kinetic and overlap integrals into products of one-dimensional integrals, are used. Furthermore, representation of the potential energy surface in terms of the same tensor-product basis functions used to represent the wave functions allows the potential energy integrals to also be written as a sum of products of one-dimensional integrals. Factorization of the integrals, together with expression of one-dimensional integrals in analytic or rapidly convergent power series form, reduces the computational effort of calculation of all matrix elements to a small, and arguably insignificant, level. It is shown that the theoretical error in eigenvalue, i.e., O(h6) for bicubic Hermite functions, is achieved for a number of rare gas van der Waals triatomics for which surfaces have been previously published. We also present illustrative calculations on NeHCl and 2A′ and 2A″ NeHCl+, which have not been previously studied, for surfaces calculated at the CCSD(T)/cc-pVTZ level. © 2001 American Institute of Physics.

Notes: Abstract A canonical quasidegenerate Rayleigh-Schrödinger perturbation theory, correct through fourth order in the energy, is explored for a block-diagonal unperturbed Hamiltonian. The theory is developed completely within a Lie Algebra in Hilbert space. Explicit equations forn-particle transition elements in terms of solutions of simultaneous linear equations are presented. A two-dimensional anisotropic anharmonic oscillator is used to provide numerical results. The perturbation theory is shown to be stable under small separation of model and complement spaces. An iterative variant of the fourth order perturbation theory is introduced; the iterative variant is related to the non-iterative one in much the same way as nondegenerate coupled cluster theories are related to nondegenerate perturbation theory. The quasidegenerate coupled cluster theory appears to be stable in the presence of multiple intruder states.