Slater-type Atomic Orbitals Research Articles

Angular momentum in molecular quantum mechanical integral evaluation

Solid-harmonic derivatives of quantum-mechanical integrals over Gaussian transforms of scalar, or radial, atomic basis functions create angular momentum about each center. Generalized Gaunt coefficients limit the amount of cross differentiation for multi-center integrals to ensure that cross differentiation does not affect the total angular momentum. The generalized Gaunt coefficients satisfy a number of other selection rules, which are exploited in a new computer code for computing forces in analytic density-functional theory based on robust and variational fitting of the Kohn–Sham potential. Two-center exponents are defined for four or more solid-harmonic differentiations of matrix elements. Those differentiations can either build up angular momentum about the centers or give forces on molecular potential-energy surfaces, thus generalized Gaunt coefficients of order greater than the number of centers are considered. These 4- j generalized Gaunt coefficients and two-center exponents are used to compute the first derivatives of all integrals involving all the Gaussian exponents on a triplet of centers at once. First all angular factors are contracted with the corresponding part of the linear-combination-of-atomic-orbitals density matrix. This intermediate quantity is then reused for the nuclear attraction integral and the integrals corresponding to each basis function in the analytic fit of the Kohn–Sham potential in the muffin-tin-like, but analytic, Slater–Roothaan method that allows molecules to dissociate into atoms having any desired energy, including the experimental electronic energy. The energy is stationary in all respects and all forces precisely agree with a previous code in tests on small molecules. During geometry optimization of an icosahedral C 720 fullerene computing these angular factors and transforming them via the 4- j generalized Gaunt coefficient takes more than sixty percent of the total computer time. These same angular factors could be used in identical fashion with Gaussian transforms of Slater-type and numerical radial atomic orbitals.

Nonstationary Theory of Perturbations for Open-Shell Atoms in the Hartree–Fock Approximation

For a multielectron open-shell system exposed to an external time-dependent perturbation, the Hartree–Fock nonstationary equations are obtained in terms of density operators. Using them as a basis, equations of nonstationary “coupled” perturbation theory are suggested in orbital representation within the framework of the two-operator variant of the Roothaan method for an open shell. The perturbation theory corrections to the orbitals have been found in the form of expansions in unperturbed orbitals which are assumed to be calculated in the LCAO approximation in the basis of Slater-type atomic orbitals. The dynamic polarizability of open-shell atoms of substances from Li to F and Sc has been calculated as an even-power series of the frequency of incident radiation.

Fast and accurate evaluation of three‐center, two‐electron Coulomb, hybrid, and three‐center nuclear attraction integrals over Slater‐type orbitals using the SD transformation

AbstractIn this work, Slater‐type atomic orbitals (STOs) are expanded over B functions to exploit the compact Fourier transform of these functions. The analytic representations of three‐center nuclear attraction integrals and multicenter bielectronic integrals over B functions involve semiinfinite highly oscillatory integrals, which can be transformed into infinite series. These integrals must be evaluated precisely and quickly. This work describes an efficient method for the evaluation of these integrals based on the nonlinear SD transformation, recently developed by Safouhi. The SD method was applied to multicenter integrals over B functions, and it is shown that this method is highly efficient compared with alternatives previously used. The SD approach should lead to a definitive suite of ab initio Slater software. The convergence properties were analyzed and they showed that the approximations obtained using the SD approach converge to the exact values of the semiinfinite integrals without any constraint. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Linear Approximation of Basis Set Orbital Exponents for Isoelectronic Series of Atoms

The orbital exponents of Slater type atomic orbitals (AOs) in isoelectronic series of atoms may be approximated by the linear dependence on the nuclear charge using a technique developed for optimization of AO basis sets in Hartree–Fock–Roothaan calculations. This approach yields the analytical Hartree–Fock wave functions for any ion in the isoelectronic atomic series without optimization of orbital exponents. The approximated linear equations for atomic orbital basis sets of B, C, O, and F in the ground state are presented as an example.

The Hartree–Fock Approximation Calculation of Electric Polarizability of Atoms with an Open Shell

Within the framework of the restricted Hartree–Fock method, equations of stationary “coupled” perturbation theory have been obtained for atomic-molecular systems with an open shell in orbital representation. Corrections to the Hartree–Fock orbitals are sought in the form of expansions in unperturbed orbitals which are assumed calculated in the LCAO approximation. The resulting inhomogeneous algebraic system for the expansion coefficients admits an exact solution. The static polarizability of atoms with an open shell from Li to Sc has been calculated with the use of an optimized basis set of Slater-type atomic orbitals.

Hartree-Fock-Roothaan calculations of dipolar polarizabilities for closed-shell atoms

Methods are suggested for optimization of Slater type atomic orbitals in polarizability calculations of closed-shell atoms using “bound” perturbation theory (algebraic version of the Hartree-Fock method). The size and composition of the basis set of atomic orbitals providing the Hartree-Fock limit for perturbation parameters are considered. Dipolar polarizabilities are calculated for He, Be, Ne, and Mg atoms and their isoelectronic series.

Analytic evaluation of three-electron atomic integrals with Slater wave functions

The formulas of Fromm and Hill and of Remiddi, for the three-electron correlated atomic integral using Slater-type atomic orbitals, are shown to be equivalent in their common region of applicability. The demonstration required the derivation of an identity connecting dilogarithm functions which is either new or not widely known. It is then shown how to modify the more general of these formulas (that of Fromm and Hill) to eliminate the necessity of branch tracking in the complex plane and thereby to achieve a straightforwardly computable formalism. By obtaining a formal cancellation of the individual-term singularities in the Fromm-Hill formula, problems of numerical instability are also avoided. The work is validated with a computer program and sample calculations.

STOP: A slater-type orbital package for molecular electronic structure determination

A general ab initio package using Slater-type atomic orbitals is presented. This package, called STOP, uses the one-center two-range expansion method to evaluate the multicenter electronic integrals. Thoroughly optimized numerical techniques, in particular, convergence accelerators and suitable Gauss quadratures, are used in the algorithms which provide accurate numerical values for all these integrals. STOP thus provides wavefunctions for general molecular structures at the self-consistent field level for the first time over a Slater-type orbital basis. © 1996 John Wiley & Sons, Inc.

Correlation effects and charge fluctuations in .pi. systems: a combined PPP and localized many-particle analysis

A local correlation approach is employed to elucidate the many-particle character of the π electronic system in hydrocarbons, heterocompounds, and a collection of other π networks. The self-consistent-field (SCF) wave function |ψ SCF >, which is the prerequisite to derive the correlated ground state |ψ 0 >, is calculated in the Pariser-Parr-Pople (PPP) scheme. The matrix elements occurring in the SCF precursor as well as the many-particle part are calculated ab initio in a minimal basis of Slater-type atomic orbitals (AOs). In the determination of |ψ 0 >, singly and doubly excited configurations are considered. The former ones optimize the one-particle density beyond the results from the SCF calculation

Recursive calculation of closed-form scattering factors for atomic orbitals

Recurrence relations are derived for calculating closed-form expressions for the Fourier–Bessel transform of electronic charge densities represented by Gaussian- and Slater-type atomic orbitals. The derivation is based on hypergeometric and Legendre function representations of the transform and the use of their recurrence properties. The scattering-factor calculations are discussed in detail. A table of closed-form expressions for Gaussian-type orbitals, not hitherto available, has been calculated using the new method and is available on request.

Valence‐bond calculations on ZNO and HGO using integrals computed through the semiempirical AM1 method

AbstractValence‐bond calculations have been carried out on ZnO and HgO using a basis set of Slatertype atomic orbitals and the one‐ and two‐electron integrals as computed in the semiempirical AM1 molecular orbital method. The zero differential overlap approximation has been used to calculate integrals between atomic orbital Slater determinants using the rules for matrix elements between determinants formed by orthogonal orbitals. Diabatic and adiabatic curves have been analyzed for the two systems, and results compared with molecular orbital AM1 results. © 1992 John Wiley & Sons, Inc.

Evaluation of one-centre electron interaction integrals over slater type atomic orbitals

Simplified expressions for one-centre electron interaction integrals in general and Slater-Condon parameters as well as repulsion integrals of the type (aa|aa) in particular have been obtained over Slater-type atomic orbitals. Results calculated using these expressions are in agreement with those due to other authors.

Convergence of the separable exchange approximation in electron-molecule scattering using Slater-type basis sets

The convergence of a finite L2 expansion of the exchange interaction is studied in the single-centre close-coupling formulation of electron-molecule scattering at the static exchange level. It satisfies no variational principle; some confusion about the convergence process may have resulted in the past from the emphasis given to a diagonal representation of the exchange operator. A number of expansion sets are tested, which include short-range multicentre molecular orbitals derived from Slater-type atomic basis sets and long-range one-centre numerical orbitals. An expansion set optimising the rate of convergence is derived. Illustrative results are presented for e-H2 and e-N2 scattering and compared with previous calculations in order to assess the accuracy of the method and its range of applicability.

Analytic evaluation of two-center molecular integrals

By using the Fourier-transform technique, the explicit expressions for the one-electronndashtwo-center overlap integrals of Slater-type atomic orbitals up to 3d are derived. The final expressions are analytic, simple, and independent of local coordinates. Furthermore, they do not contain the non-closed-form of exponential integrals which were presented in expressions given in earlier work. It is shown that the two-electron--two-center Coulomb integrals, as well as the hybrid integrals, can simply be expressed in terms of these integrals. The numerical instability arising from the situation in which the exponents of the two orbitals are almost equal is discussed, and a solution for this problem based on a Taylor-series expansion of the integral is suggested.

Unified analytical evaluation of two-center, two-electron integrals over Slater-type orbitals

A unified analytical expression is given for all types of two-center electron-repulsion integrals over Slater-type atomic orbitals. By means of the Neumann expansion of the inverse interelectronic distance (1/r12) in prolate spherical coordinates, Coulomb, hybrid, and exchange integrals are expressed in terms of the function W1‖ m ‖ introduced by Kotani. The present formulation allows a preliminary estimation of the number of terms in the expansion to be retained for a desired accuracy, resulting in a significant saving of computation time and working storage. The method is implemented on a computer for various type atomic orbitals from 1s to 4f with an accuracy of 12 decimal places.

A quantum-chemical model of calculating the bond energy of crystals

A model has been obtained from a strict expression of crystal energy by introducing S and valence approximations. The Mulliken approximation has been used for double-centered exchange and three-center integrals. The calculations have been carried out for Slater type atomic orbitals, varying the screening parameters. The bond energy has been represented as a sum of the energy for forming partial differential electronic densities, EA, the interatomic interaction energy EIA, and the correlation energy of atomic states in the crystal, E S . The computed bond energies for four oxides of alkaline-earth metals and six oxides of transition (iron group) metals are in good agreement with experimental data.

Matrix elements of the Coulomb Green function between Slater orbitals

Analytic expressions are given for integrals of the Coulomb Green function with Slater type atomic orbitals. The results involve hypergeometric functions.

Extremely compact formulas for molecular two-center one-electron integrals and Coulomb integrals over Slater-type atomic orbitals

Extremely compact analytical formulas are derived for molecular two-center one-electron integrals and Coulomb integrals, which occur in linear-combination-of-atomic-orbitals calculations with Slater-type orbitals (STO's). The derivation is based on the connection of the STO's with the reduced Bessel functions (RBF's) and makes use of the convolution theorems of RBF's. The final results are of a surprisingly simple structure and are, therefore, especially useful for practical applications.

Generalized X-ray scattering factors. An approximate method for the two-centre case with Slater-type orbitals

A general method is described for calculating the Fourier transform of a product of two Slater-type atomic orbitals located on different atomic centres. The method is approximate., but can be carried to any desired degree of accuracy.

Analytical evaluation of multicenter integrals for Slater-type orbitals. III. Two-center exchange integrals

The general two-center exchange integrals with an arbitrary combination of quantum numbers of Slater-type atomic orbitals are evaluated analytically. The final results are expressed through the new auxiliary functions for which have been obtained recurrence relations and analytical formulas.