J. Fernández Rico

Molecular integrals with Slater basis. I: General approach

A method for the calculation of many‐center integrals involving Slater‐type orbitals (STOs) is reported. The method is based on the separation of the short‐ and long‐range terms of the potentials generated by charge distributions. The calculation of three‐center nuclear attraction integrals and many‐center electron repulsion integrals, when the short‐range contributions are negligible, is formulated in terms of multipolar moments of charge distributions. Recurrence relations for obtaining the multipolar moments, which enable us to reduce their calculations to the evaluation of some basic integrals, are reported.

Analysis of the molecular density

The minimal deformation criterion, previously proposed for the partition of the molecular density into atomic contributions, is updated and extended. For any Gaussian basis set, these atomic contributions are expanded in a series of real spherical harmonics by radial factors. The terms with l=0 determine the spherical parts of the atomic clouds and the remaining ones, their deformations. This detailed description is complemented with a simplified representation of the molecular density in terms of atomic charges and multipoles. Moreover, these descriptions give a simple way to calculate the electrostatic potential of the molecule as well as the electrostatic interaction between molecules.

Molecular integrals with Slater basis. II. Fast computational algorithms

A new algorithm for the calculation of molecular integrals involving STOs is reported. The algorithm enables us to obtain every two‐center one‐electron integral and the long‐range many‐center one‐ and two‐electron integrals. The efficient implementation of the algorithm is discussed and its performance is thoroughly tested. The analysis on the stability of the relations employed in the calculation of multipolar moments is included. Futhermore, the computer time required to carry out each step (construction of basic matrices, calculation of multipolar moments, and calculation of two‐electron integrals) has also been analyzed. The range of validity of this approach is shown in several molecular integrals.

Analysis of the molecular density: STO densities

A partition of the molecular density for Slater basis sets (STO), which parallels one previously developed for Gaussian basis sets (GTO), is reported. The atomic fragments are expanded in spherical harmonics times radial factors. Each fragment contains all the one-center charge distributions centered in the atom plus the part of every two-center distribution assigned to the atom by the partition criterion. The performance of the procedure is analyzed, concluding that the analysis gives highly accurate representations of the molecular density at a very low cost. Moreover, the results of the analysis are illustrated with the study of the densities in CO and H 2 O and the comparison of the atomic densities obtained from STO and GTO molecular calculations.

Recurrence relations for the expansion of Slater‐type orbitals about displaced centers

We show here that the coefficients for the expansion of any Slater‐type function over a displaced center can be obtained by three simple (and numerically stable) recurrence relations starting at the expansion coefficients of the 1s or 2s orbitals. We present also compact (and very fast to compute) formulas for these initial quantities.

Molecular integrals with Slater basis. V. Recurrence algorithm for the exchange integrals

The numerical and analytical procedures, used for calculating exchange integrals with Slater functions, fail to give high accuracy for large quantum numbers and some values of the parameters. We propose here an algorithmic approach based on recurrence relations which start from some simple functions. An extensive study in which exponents, interatomic distances, and quantum numbers were varied proves that the new procedure is fully reliable. Finally, the cost of the procedure is analyzed, including the effect of the number of terms needed to attain a given accuracy.

Molecular integrals with Slater basis. III. Three‐center nuclear attraction integrals

We discuss the calculation of the general three‐center nuclear attraction integrals with Slater type orbitals using a one‐center expansion method which exploits the algorithms previously developed by us for long‐range integrals. From the analysis of the numerical stability, accuracy, and computational cost we conclude that the reported procedure is suitable for the calculation of these integrals with any desired precision. Moreover, we have found that most of these integrals can be obtained with high accuracy at a very low computational cost, but the procedure could be too expensive for a few of them.

Molecular integrals with Slater basis. IV: Ellipsoidal coordinate methods for three-center nuclear attraction integrals

We develop and check an updated version of the ellipsoidal coordinate methods for the calculation of three‐center nuclear attraction integrals with Slater‐type orbitals (STOs). A first set of recurrence relations allows us to express these integrals in terms of some basic potentials. The basic potentials are classified in short‐ and long‐range types, and then calculated in specific ways from auxiliary functions obtained by using a new set of both stable and fast recurrence relations. Numerical tests show that the procedure enables to reach very high accuracy in the integrals with a low computational cost.

An economical technique for forcing convergence in conventional SCF methods

We present an economical technique for ensuring convergence of the open‐ and closed‐shell SCF methods. In this technique, the number of operations required is proportional to the square of the number of basis functions and all the employed quantities are present in any conventional SCF procedure. We test their efficacy with several numerical calculations.

Density and binding forces in diatomics

In a recently reported method, the molecular density is partitioned in minimally deformed atomic contributions, which are expanded in spherical harmonics times radial factors. Here we use this representation to express the electrostatic potential of the molecule, the force on its nuclei, and the conformational variations of energy in terms of some simple integrals of the atomic radial factors. As a first application, we analyze the relationship between the density and the binding forces (and the bonding energy) in the diatomic molecules of the first row atoms. Two types of forces act on each nucleus: the self-pulling exerted by its own cloud and the external force due to the remaining atoms. The self-pulling comes only from the dipole type term of the atomic density. The external force comes from the other clouds and nuclei and is dominated by the effective charges which depend on the outermost region of the charge term. Analyzing the progressive deformations of the atoms when they approach each other, the forces associated with these deformations and their contributions to the energy, one has a detailed description of the chemical bond which is complementary, and in many aspects more appealing, than the conventional ones.