An efficient second-order MC SCF method for long configuration expansions

A new second-order optimisation procedure for general MC SCF wavefunctions is described. The method shows greatly improved convergence as compared to previous methods. Using a determinant-based direct CI procedure which avoids the construction of a formula tape, very long complete active space (CAS SCF) wavefunctions can be handled. Energy averages of several states can also be optimised. Sample calculations for CH₂, FeO, and the vinoxy radical CH₂CHO with up to 178916 configurations are presented.     

Three-dimensional effects on the linear adiabatic molecular wavefunctions in the H + H2 system

The adiabatic molecular wavefunctions in the H + H₂ system are obtained in one dimension by solving the double-well potential problem. In three dimensions, the corresponding linear adiabatic molecular wavefunctions are obtained. A comparison between these wavefunctions clearly suggests that the probability of reaction is smaller in three dimensions.     

Population analyses of valence-bond wavefunctions and BeH2

A method for calculating occupation numbers and making a population analysis is proposed. The method is designed particularly for use with wavefunctions written in terms of non-orthogonal bases. An application to the valence-bond wavefunction of BeH₂ is given and the results are compared with other population analysis schemes which have been proposed.     

Molecular symmetry. IV. The coupled perturbed Hartree–Fock method

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the coupled perturbed Hartree–Fock (CPHF) formalism, a key step in the analytical computation of energy first derivatives for configuration interaction (CI) wavefunctions, and energy second derivatives for Hartree–Fock (HF) wavefunctions. One possible computational strategy is to construct Fock-like matrices for each nuclear coordinate in which the one- and two-electron integrals of the usual Fock matrix are replaced by the integral first derivatives. “Skeleton” matrices are constructed from the unique blocks of electron-repulsion integral derivatives. The correct matrices are generated by applying a symmetrization operator. The analysis is valid for many wavefunctions, including closed- or open-shell spin-restricted and spin-unrestricted HF wavefunctions. To illustrate the method, we compare the computer time required for setting up the coupled perturbed HF equations for eclipsed ethane using D_3h symmetry point group and various subgroups of D_3h. Computational times are roughly inversely proportional to the order of the point group.     

Density functional studies of single molecule magnets

A method for the calculation of the second-order anisotropy parameters of single molecular magnets from the single particle orbitals is reviewed. We combine this method with density functional calculations to predict the magnetic anisotropy parameters of several single molecule magnets: Mn₁₂-acetate, Mn₁₀, Co₄, Fe₄, Cr₁ and V₁₅. Comparison with available experimental data shows that it is possible to predict these values quite accurately from density functional wavefunctions.     

Optical properties and paramagnetic susceptibility of europium gallium garnet

A single crystal of europium gallium garnet has been grown by the flux method and the experimental variation with temperature of its paramagnetic susceptibility has been studied up to 600 K. The crystal field parameters have been calculated for D₂ symmetry. From the derived wavefunctions, the magnetic susceptibility versus temperature was calculated. The experimental and simulated susceptibilities are in good agreement.     

Electron momentum spectroscopy of the valence orbitals of H2O and D2O: Quantitative comparisons using Hartree—Fock limit and correlated wavefunctions

The large discrepancies found earlier between experimental measurements and calculations based on near Hartree—Fock wavefunctions for the valence orbital electron momentum distributions of H₂O are reinvestigated. New and improved electron momentum spectroscopy measurements for the valence orbitals of H₂O and D₂O, together with existing experimental data, have been placed on a common intensity scale using the binding energy spectra. Investigation of possible vibrational effects by means of new measurements of the momentum distributions of D₂O indicates no detectable differences with the H₂O results, within experimental error. A quantitative comparison of these experimental results with both the shapes and magnitudes of momentum distributions calculated in the PWIA and THFA approximations using new, very precise Hartree—Fock (single-configuration) wavefunctions is made. These wavefunctions, which include considerable polarization and which are effectively converged at the HF limit for total energy, dipole moment and momentum distribution permit establishment of basis set independence. The significant discrepancies between theory and experiment which still remain for the momentum distributions of the 1b₁, 3a₁ and 2a₁ orbitals at the THFA level are largely removed by CI calculations of the full ion—neutral overlap amplitude. These CI wavefunctions for the final ion and neutral ground states, generated from the accurate HF limit basis sets, recover up to 88% of the correlation energy. The present work clearly shows the need for adequate consideration of electron correlation effects in describing the low-momentum parts of the 1b₁, 3a₁ and 2a₁ electron distributions, a region which is of crucial importance in problems related to chemical bonding and reactivity. The high level of quantitative agreement obtained between experiment and calculations using sufficiently sophisticated wavefunctions provides support for the essential validity of the plane wave impulse approximation as used in the interpretation of EMS experiments on small molecules.     

Structure of small molecular hydrogen clusters

The ground state energies and structural properties of small (H₂) _N ,N=2?7, are calculated using the variational Monte Carlo method. These wavefunctions include both short- and long-range correlation effects that are important in the binding of van der Waals clusters. We have investigated these clusters using shadow wavefunctions and found that the coupling to shadow variables raises the energy in all cases, implying that the ground states of these small clusters are properly described as quantum liquids rather than solid structures.     

Direct minimization of the energy functional in LCAO-MO density matrix formalism

A general theory is presented for the optimization of the coefficients of orbitals and configuration interaction expansion in the case of multiconfiguration wavefunctions containing all single excitations. The orbital coefficients are optimized by suitable orthogonal transformations of the atomic basis; the Cl coefficients are determined solving the usual secular problem. The energy minimization is performed directly by a gradient approach. The method works both for ground and excited states and no convergence difficulties are met. Computational examples are given for H₂O and H₂S molecules.     

Modification of dipole-length and dipole-velocity matrix elements: Application to 1sσg-2pσu transitions of H2+

A procedure, based on approximate variational improvement of the wavefunctions, is developed for modifying electric dipole matrix elements calculated in the length and velocity forms. Its relationship to the recently proposed method of Roginsky et al. is investigated and test calculations are carried out for 1sσ_g-2pσ_u, transitions of H₂⁺. The results indicate that it may be an effective procedure in molecular calculations.     

Direct minimization of the energy functional in the LCAO-MO density matrix formalism

A previously proposed method of energy minimization is developed for MC SCF wavefunctions formed by all-pair excitations for a closed-shell system. The orbital coefficients are optimized by a gradient approach using a suitable orthogonal transformation of the atomic basis, while optimum CI coefficients are determined solving the usual secular problem for the lowest eigenvalue, after each optimization of the orbitals. Applications to LiH and NH₃ molecules show that the method is numerically well stable, and is capable of accounting for a large part of the correlation energy giving results which compare well with those of the conventional CI method.     

Energy shift in (dtμ)e due to the finite size of the muonic molecular ion (dtμ)+

The energy shift due to the presence of the extended (dtμ)₁₁ pseudonucleus (in its first excited state with one unit of angular momentum) in the quasihydrogenlike system (dtμ)₁₁ e _1s is estimated using perturbation theory up to second order with two choices of zeroth order electron wavefunctions. The energy shift is found to be 0.50 meV using pure Coulomb electron wavefunctions and 0.58 meV using electron wavefunctions calculated with a potential modified to take partial account of the finite size of (dtμ)₁₁. In both cases, the perturbation Hamiltonian is expanded in multipoles, retaining terms up to and including octupole terms.     

Limits on the localized interpretation of molecular orbital wavefunctions

A new measure of orbital localizability is proposed. The actual improvement in classical electrostatic interpretations of electronic energy contributions on localization of CH₄, NH₃, H₂O, HF and Ne LCAO wavefunctions is computed to be small. Substantial valence LMO spatial overlapping remains.     

Variational method for calculating molecular multipole polarizabilities using atom centered extended trial functions

An improvement of the variational method for calculating the electronic multipole polarizabilities is proposed. This modification allows the computation of the polarizabilities at any point and the results are compatible with the relations existing for a change of origin. It is applied to H₂, HF, CO and N₂ by using SCF wavefunctions developed on a limited basis. The computed polarizabilities are systematically too large but this discrepancy is attributed to the fact that the ground state is too far from the exact wavefunction.E.R.A. au C.N.R.S. No. 22.     

Examination of charge alternation in CH4 and CH3F from ab initio LCAO SCF MO wavefunctions and their localized bond orbitals

The question of whether or not fluorine substitution produces charge alternation is examined for CH₄ and CH₃F. Two sets of ab initio LCAO SCF MO wavefunctions (one a 3 G STO based one, the other a double zeta based one) are analyzed via charge density, localized CH bond moment, and population analysis calculations. Although both sets of wavefunctions show a slightly more negative H region in CH₃F relative to CH₄, in qualitative agreement with earlier work by Pople et al., the differences are small, and their sources are not clear. For example, in the 3 G calculations the CH localized orbital is the essential source of the increased density in CH₃F, while for the double zeta calculations the increased density is due to the tail of an F lone-pair orbital trans to the CH bond. Consideration of details of these studies as well as those from large STO based SCF MO wavefunctions by Arrighini et al., suggests that one will need very accurate wavefunctions to resolve the problem unambiguously.The Radiation Laboratory is operated under contract with the U.S. Atomic Energy Commission. This is AEC document no. COO-38-847.     

On the transition functional method

The transition operator method of calculating ΔE_SCF ionization and excitation energies using a single calculation, is generalized to embrace any energy change and any variational treatment using approximate wavefunctions. The transition functional obtained is justified using perturbational and variational arguments. The applications of the method are discussed, along with a concrete example, namely a semi-empirical VB treatment of Li₃ and Li₃⁺.     

Broken orbital symmetry and the description of valence hole states in the tetrahedral [CrO4]2− anion

The localization of ligand-based valence holes in the tetrahedral complex ion [CrO₄]^2? in a crystalline environment is studied by SCF calculations on the hole states, with progressively less restrictions on the spatial symmetry of the molecular orbitals. The final wavefunctions are obtained by constructing, from the symmetry broken SCF solutions, wavefunctions that exhibit again the proper transformation properties under the operations of T _d . The crystal environment of the [CrO₄]^2? anion is represented by a point charge model. In contrast with the situation for core hole states, the projection afterwards into T _d symmetry is important. The final ionization energies, which are obtained from projected C _3v adapted SCF solutions, are reduced considerably (?3 eV) with respect to the T _d ΔSCF results, but the ordering of the states has not changed essentially. The calculated ionization energies compare favourably with results of XPS experiments on Na₂CrO₄. The evaluation of the energies of projected symmetry broken SCF solutions requires the calculation of hamiltonian matrix elements between determinantal wavefunctions built from mutually non-orthogonal orbital sets. An efficient method for the calculation of such matrix elements is presented.     

Effects of anharmonicity on non-radiative transitions in polyatomic molecules

The effect of anharmonicity on the non-radiative transition in large molecules is examined within the Morse potential surface model. The vibrational wavefunctions are assumed to be the product of the harmonic and Morse oscillator wavefunctions. The method of factorization introduced by Gelbart et al. is used for the evaluation of a density weighted Franck-Condon factor. As an example, we choose the intersystem crossing ³ B _1u→¹ A _1g in benzene. The numerical calculation shows that the anharmonicity causes an increase by a numerical factor ～ 10³ in the non-radiative transition rate. The electronic energy distribution over the vibrational modes in the final state is determined and compared with that obtained using the harmonic potential surface model.     

A comparison of variational and non-variational internally contracted multiconfiguration-reference configuration interaction calculations

Summary The internally contracted multiconfiguration-reference configuration interaction (CMRCI) method and several non-variational variants of this method (averaged coupled pair approximation (ACPF), quasidegenerate variational perturbation theory (QD-VPT), linearized coupled pair many electron theory (LCPMET)) have been employed to compute potential energy functions and other properties for a number of diatomic molecules (F₂, O₂, N₂, CN, CO) using large basis sets and full valence CASSCF reference wavefunctions. In most cases the variational CMRCI wavefunctions yield more accurate spectroscopic constants than any of the employed non-variational methods. Several basis sets are compared for the N₂ molecule. It is found that atomic natural orbital (ANO) contractions led to significant errors in the computedr _e , _e , andD _e values.