Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. III. Orthogonal bonded functions

By using the group symmetrical localized molecular orbitals (SLMOs) as configuration-generating orbitals (CGOs) of many-electron wave functions, the symmetry adaptation of many-electron spaces is greatly simplified, and novel orthogonal bonded functions (OBFs), as complete space- and spin-adapted antisymmetrized products, are introduced. The corresponding programs for the solutions of OBFs are developed. © 1993 John Wiley & Sons, Inc.     

Unitary group approach to the many-electron problem. III. Matrix elements of spin-dependent Hamiltonians

This is the final paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians. In this paper we derive the reduced matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables a direct evaluation of the matrix elements of spin-dependent Hamiltonians in the spin-orbit basis. An alternative (indirect) method, which employs the use of U(2n) ↓ U(n) × U(2) subduction coefficients, is also discussed.     

PCILON. perturbative configuration interaction using localized orbitals and numerical integration. I. Numerical integration techniques for the calculation of Hamiltonian matrix elements between localized orbitals

Modern techniques for multidimensional numerical integration, Korobov's and Sobol's formulas namely, are used for the direct computation of matrix elements between the localized molecular orbitals needed for a configuration interaction calculation by a perturbation method. A minimal orbital basis of Slater functions is used for formaldehyde and ethylene taken as example. The resulting precision is satisfactory.     

The Hylleraas-CI method in molecular calculations. III. Implementation and numerical verification of a three-electron many-center theory

The general theory of three-electron Hylleraas-Configuration-Interaction method using linear correlation factors of the form r_ij has been implemented for molecular systems using cartesian Gaussians as basis sets. A brief review of the theory and the form of the three-electron integrals is presented. Additionally, a table of numerical values of some selected three-electron integrals is given. Results from test calculations on H₃ using the full form of the theory are presented for some simple basis sets. A discussion of the computational problems that need to be overcome before this approach is competitive with traditional methods is included.     

Unitary group approach to the many-electron problem. I. Matrix element evaluation and shift operators

This is the first paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians directly in the spin-orbit basis. In this paper we present a new and complete derivation of the matrix elements of the U(n) generators in the electronic Gel'fand basis. The approach employed differs from previous treatments in that the matrix elements of nonelementary generators are obtained directly. A general matrix element formula is derived which explicitly demonstrates the segment level formalism obtained previously by Shavitt using different methods. A simple relationship between the matrix elements of raising and lowering generators is determined which indicates that in CI calculations, only the matrix elements of raising generators need be calculated. Some results on the matrix elements of products of two generators are also presented.     

Transferable integrals in a deformation density approach to crystal orbital calculations. IV. Evaluation of angular integrals by a vector-pairing method

A general method for performing angular integrations is presented. The method depends on the fact that the integral must be invariant under rotations of the coordinate system, and it also makes use of combinatorial analysis. In most cases the method presented is computationally much faster than alternative methods of angular integration using Condon–Shortley coefficients. Applications to charge density analysis and Fourier transforms are discussed, and a general formula for the action of angular momentum projection operators on functions of the Cartesian coordinates is derived. A general angular integration formula for an m-dimensional space is also given.     

Exponential transformation of molecular orbitals. II. General formulation for UHF calculations and application to diatomics and molecular fragments

The exponential transformation of the molecular orbitals, that has been previously used to achieve a process with a convergence of quadratic quality in SCF closed-shell calculations [J. Chem. Phys. 72 , 1452 (1980)] has been extended to UHF determinantal wave functions built from different orbitals for different spins. Explicit formulas are given for the first and second derivatives of the energy to be varied. The method is illustrated by UHF calculations for systems described as standard singlets (Li₂ and F₂) or triplets (NH) at the RHF approximation level, as well as for CH, CH₂, CH₃ molecular fragments in their valence states.     

Configuration interaction matrix elements. I. Algebraic approach to the relationship between unitary group generators and permutations

Matrix elements of unitary group generators between spin-adapted antisymmetric states are shown to be proportional to spin matrix elements of so-called “line-up” permutations. The proportionality factor is given explicitly as a simple function of the orbital occupation numbers. If one bases the theory on ordered orbital products, the line-up permutations are given a priori. The final formulas have a very simple structure; this is a direct consequence of the fact that the spin functions have been taken to be geminally antisymmetric.     

Configuration interaction matrix elements. II. Graphical approach to the relationship between unitary group generators and permutations

The explicit expressions for the matrix elements of unitary group generators between geminally antisymmetric spin-adapted N-electron configurations in terms of the orbital occupancies and spin factors, given as spin function matrix elements of appropriate orbital permutations, are derived using the many-body time-independent diagrammatic techniques. It is also shown how this approach can be conveniently combined with graphical methods of spin algebras to obtain explicit expressions for the spin factors, once a definite coupling scheme is chosen. This method yields explicit expressions for the orbital permutations defining the spin factors. However, if desired, the explicit determination of line-up permutations can be avoided in this approach, since they are implicitly contained in the orbital diagrams. It also clearly indicates why the geminally antisymmetric spin functions have to be used when a simple formalism is desired.     

On the use of spatial symmetry in ab initio calculations. Transformation of the two-electron integrals from atomic orbital to localized molecular orbital basis

Anm ⁵-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.     

An analysis of the through-bond interaction using the localized molecular orbitals with ab initio calculations—I : Lone-pair orbital interactions in cis- and trans-hydrazines

Ab initio SCF MO calculations using STO-3G basis set were performed on the cis- and trans- hydrazines. The cannonical MOs obtained by these calculations were then transformed into the localized MOs. With the use of the localized MOs thus obtained, the variation in the lone-pair orbital energies of the molecules were pursued in the light of the through-space and/or the through-bond interactions between the specified localized MOs. As a result of this analysis, it was found that ; (a) the effect of the inner shell orbitais, l s electrons of N atoms, is not negligibly small, (b) the effect of the through-bond interaction is not so larger than the through-space interaction, and (c) the large contribution of the through-space interaction is caused from the indirect as well as direct interactions between two lone-pairs.     

Influence of electron correlation and degeneracy on the Fukui matrix and extension of frontier molecular orbital theory to correlated quantum chemical methods

The Fukui function is considered as the diagonal element of the Fukui matrix in position space, where the Fukui matrix is the derivative of the one particle density matrix (1DM) with respect to the number of electrons. Diagonalization of the Fukui matrix, expressed in an orthogonal orbital basis, explains why regions in space with negative Fukui functions exist. Using a test set of molecules, electron correlation is found to have a remarkable effect on the eigenvalues of the Fukui matrix. The Fukui matrices at the independent electron model level are mathematically proven to always have an eigenvalue equal to exactly unity while the rest of the eigenvalues possibly differ from zero but sum to zero. The loss of idempotency of the 1DM at correlated levels of theory causes the loss of these properties. The influence of electron correlation is examined in detail and the frontier molecular orbital concept is extended to correlated levels of theory by defining it as the eigenvector of the Fukui matrix with the largest eigenvalue. The effect of degeneracy on the Fukui matrix is examined in detail, revealing that this is another way by which the unity eigenvalue and perfect pairing of eigenvalues can disappear.     

The use of localized molecular orbitals and the polarization propagator to identify transmission mechanisms in nuclear spin-spin couplings

An extension of the IPPP (inner projections of the polarization propagator) method to theoretically analyze transmission mechanisms of indirect nuclear spin-spin couplings is presented. The localization technique used is modified so that all the canonical molecular orbitals of a compound may be localized to represent chemical bonds, lone pairs, and the corresponding antibonding molecular orbitals. These localized molecular orbitals, together with the polarization propagator, are used to obtain an intuitive picture of how a coupling is generated as a sum of terms, each one consisting of two particle-hole single excitations. This picture can be used to identify underlying transmission mechanisms and quantitatively evaluate their importance toward the total coupling. The paramagnetic spin-orbit and the spin-dipole interactions are studied in detail.     

Correlation problems in atomic and molecular systems III. Rederivation of the coupled-pair many-electron theory using the traditional quantum chemical methodst

The equations of the coupled-pair many-electron theory (CPMET ) for the closed shell systems are rederived both in the spin-orbital and orbital forms without the use of second quantization, Wick's theorem or the technique of Feynman-like diagrams. Only the Slater rules are used for the calculation of necessary matrix elements. A comparison with earlier papers shows clearly the usefulness and conceptual simplicity of the mathematical methods of quantum field theory both in the derivation of the CPMET , in spin-orbital form, and in the process of excluding spin variables.     

A matrix partitioning approach to the calculation of intermolecular potentials. General theory and some examples

A general expression for the interaction energy of two molecules is obtained by using a matrix partitioning method. The wavefunction of the whole system is expanded in terms of antisymmetrized products of free-molecule functions and using a matrix perturbation scheme it is possible to describe the interaction energy in terms of free-molecule quantities (like frequency-dependent polarizabilities and density or spin density matrices) that, in principle, can be evaluated at any level of approximation. By way of example, the interaction potentials are calculated for (N₂)₂ and Ne₂ using wavefunctions of HF and TDHF form. The results are in substantial accord with those available in the literature. Application of these potentials to the calculation of macroscopic properties, however, leads to considerable errors. From the analysis of our results it appears that the dispersion energy is underestimated, probably on account of the neglect of intrasystem correlation energy in the TDHF approximation. The use of more sophisticated methods of evaluation of dynamic polarizabilities will not involve any extension of the approach presented in this work.