B. K. Novosadov

Besselian: calculation of multicenter matrix elements

Abstract The article is an authors review and includes also some new developments in calculation of multicenter matrix elements for investigating molecular structures using exponentially decreasing atomic orbitals. The Bessel atomic orbitals (B-functions) as basis functions having the simplest representation in momentum space among the spherical exponential type atomic orbitals (ETO) are considered. The important integral representation for the B-functions product allows one to reduce four-center molecular electron repulsion matrix elements to integrals over the unit three-dimensional cube. The operator representation of multicenter integrals results in calculation of the latters with scalar B-functions, so the computing of the matrix elements of the spherical AOs with high angular momentum quantum numbers reduces to differentiation of the integrals of the simplest ETOs. For this purpose a method is suggested to replace the differentiation procedure by Fourier transforms in polyspherical coordinates in Rn (n=6,9). The results of numerical calculations confirm the efficiency of the Besselian algorithms suggested allowing them to compete with the Gaussian ideology.

CALCULATION OF 4-CENTER COULOMB REPULSION MATRIX ELEMENTS IN A BASIS OF EXPONENTIAL TYPE SPHERICAL AO USING 9-DIMENSIONAL POLYSPHERICAL HARMONICS

A method for calculating 4-center Coulomb repulsion integrals in a basis of exponential type AO with regular sectorial harmonics as angular terms is proposed. The initial integrals are represented as a partial differentiation operator with respect to the Cartesian coordinates of the centers of AO, acting on the scalar function which is a 4-center integral of s functions. Differentiation is performed by calculating the Fourier transform of this scalar function in 9-dimensional Euclidean space with the help of the sectorial harmonic argument summing theorem. Thus compact representation of quantum-chemical multicenter integrals is obtained in a basis of exponential type functions with arbitrary angular parts.

Method of Calculating Two-Center Two-Particle Matrix Elements from a Screened Coulomb Potential in an Exponential AO Basis

An analytical method is suggested for calculating matrix elements from an exponentially screened two-particle Coulomb potential in a basis set of exponential functions specified on different centers in space. The two-particle potentials of this type may be used to approximate the short-acting parts of atom–atomic interactions in molecules in quantum-chemical structural calculations.

Theory and calculations of second-order anharmonic corrections to the radial distribution function of polyatomic molecules

A theory of the thermal averages of normal coordinates was suggested for polyatomic molecules. Second-order approximation equations were obtained on the basis of iterations of the Bloch integral equation. These equations can be used to calculate anharmonic corrections to the radial distribution function and the parameters that determine the intensity of fast electron scattering by molecules in gas-phase electron diffraction.

Calculation of Second- and Higher-Order Force Constants Including Electron Correlation Effects and Hartree-Fock Force Constant Scaling

A formalism suitable for practical implementation is suggested for computation of quadratic, cubic, and quartic force constants using the configuration interaction (CI) method. Expressions are compared which involve the Hartree-Fock (HF) harmonic and anharmonic force constants calculated by the HF and CI methods. Also, physical assumptions are formulated to perform scaling of the diagonal harmonic and, which is more important, of anharmonic HF force constants with a common scaling factor. This approach is consistent with the data of ab initio quantum-chemical calculations. A table is presented which compares the results of valence force constant calculations for a series of simple organic molecules.

A NEW METHOD OF CALCULATING MULTICENTER MATRIX ELEMENTS IN MO LCAO THEORY USING AN EXPONENTIAL TYPE BASIS

Using integral representation of the product of reduced Bessel functions (RBF) specified on different centers and a new generalized integral identity for RBF one can prove that the 4-center integral of Coulomb repulsion in an exponential type AO basis may be expressed as a three-dimensional integral over the volume of a cube with an edge 1. A new method of calculating the multicenter matrix elements of quantum chemistry in an exponential AO basis is suggested based on this representation. Numerical calculations of a number of multicenter integrals using this algorithm illustrate the efficiency of the method.