Spherically symmetrical properties of two-center overlap integrals over arbitrary atomic orbitals and translation coefficients for Slater-type orbitals

The orthogonality relations are derived for the rotation coefficients of two-center overlap integrals over arbitrary atomic orbitals (AAOs) and expansion coefficients for translation of Slater-type orbitals (STOs). Using these formulas, a very interesting theorem regarding the angular dependence is established. If we add the products of all the overlap integrals or all the translation coefficients with the same n and l values, but different m values, the result is independent of orientation. The final results are of a simple structure and are, therefore, especially useful for machine computations of multielectron multicenter molecular integrals by expanding one- and two-center electron charge density over STOs in terms of STOs about a new center.     

New translation method for STOs and its application to calculation of two-center two-electron integrals

The new translation method for Slater-type orbitals (STOs) previously tested in the case of the overlap integral is extended to the calculation of two-center two-electron molecular integrals. The method is based on the exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. Fair uniform convergence and stability under wide changes in molecular parameters are obtained for all studied two-center hybrid, Coulomb, and exchange repulsion integrals. Ten-digit accuracy in the final numerical results is achieved through multiple precision arithmetic calculation of common angular coefficients and Gaussian numerical integration of some of the analytical formulas resulting for the radial integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 91–100, 2000     

Application of modified Neumann expansion for analytical evaluation of two-center,two-electron integrals over slater-type orbitals

A modified form of the Neumann expansion in terms of products of orthogonal polynomials for the inverse interelectronic distance r¹₁₂ is proposed. This expansion has been applied in order to derive a unified analytical formula for two-center and two-electron integrals over Slater-type orbitals. The results are equivalent to those given recently by Yasui and Saika, but the expansion itself can be used for building up a realistic algorithm for evaluation of three- and four-electron integrals determined by using correlated variational wave functions.     

Implementation of atomic basis set composed of 1s Gaussian and 1s Slater-type orbitals to carry out quantum mechanics molecular calculations

A mixed atomic basis set formed with ls Slater-type orbitals and 1s floating spherical Gaussian orbitals is implemented. Evaluation of multicenter integrals is carried out using a method based on expansion of binary products of atomic basis functions in terms of a complete basis set, and a systematic analysis is performed. The proposed algorithm is very stable and furnishes fairly good results for total energy and geometry. An LCAO-SCF test calculation is carried out on LiH. The trends observed show that there are some combinations of mixed orbitals that are appropriate to describe the system. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 604–609, 1999     

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.     

Three-center expansion of electron repulsion integrals with linear combination of atomic electron distributions

A new systematic way of constructing auxiliary basis functions for approximating the evaluation of electron repulsion integrals is proposed and applied to SCF and MCSCF wavefunction calculations. In the approximation, the one-electron density is expanded in terms of a linear combination of atomic electron distributions (LCAD), and the four-center two-electron repulsion integrals are reduced to the three- and two-center quantities. This results in a high-accuracy approximation as well as a large reduction in disk storage and input/output requirement, proportional to N³ rather than N⁴, N being the number of basis functions. Numerical results indicate that the error from the present approximation decreases as the size of molecular basis functions increases and that the LCAD version of MCSCF calculations requires only a fractional amount of the CPU time required in the conventional procedure without loss of accuracy.     

Efficiency of the algorithms for the calculation of Slater molecular integrals in polyatomic molecules

The performances of the algorithms employed in a previously reported program for the calculation of integrals with Slater-type orbitals are examined. The integrals are classified in types and the efficiency (in terms of the ratio accuracy/cost) of the algorithm selected for each type is analyzed. These algorithms yield all the one- and two-center integrals (both one- and two-electron) with an accuracy of at least 12 decimal places and an average computational time of very few microseconds per integral. The algorithms for three- and four-center electron repulsion integrals, based on the discrete Gauss transform, have a computational cost that depends on the local symmetry of the molecule and the accuracy of the integrals, standard efficiency being in the range of eight decimal places in hundreds of microseconds.     

Unified analytical treatment of one-electron two-center integrals with noninteger n Slater-type orbitals

A unified treatment of one-electron two-center integrals over noninteger n Slater-type orbitals is described. Using an appropriate prolate spheroidal coordinate system with the two atomic centers as foci, all the molecular integrals are expressed by a single analytical formula which can be readily and compactly programmed. The analysis of the numerical performance of the computational algorithm is also presented. Received: 1 April 1999 / Accepted: 2 July 1999 / Published online: 2 November 1999     

A Hartree-Fock-Roothaan analogon using the principle of variance minimization

The variance-minimizing Roothaan-like equation derived in a preceding paper [1] gives rise to a double iteration procedure. The procedure is tested by application on some simple atomic systems, using Slater-type basis functions. The integrals needed for atomic systems and Slater-type basis functions are solved.     

Fast and stable algorithm for the analytical computation of two-center Coulomb and overlap integrals over Slater-type orbitals

Proceeding from analytical expressions for two-center kernel functions that we derived recently, we present new analytical formulas for the two-center Coulomb and overlap integrals over Slater-type orbitals. These formulas are of an exceptionally simple analytical structure and high numerical efficiency. An especially important point is that for the most frequently needed ranges of discrete quantum numbers, the formulas are completely stable in the cases of nearly equal scaling parameters or vanishing interatomic distances, except for one particular case of the Coulomb integral. No special asymptotic formulas are needed any more to compute the two-center integrals over Slater-type orbitals in these case. Furthermore, a largely recursive formulation makes the integral evaluation very economical and fast. In particular, we assess the numerical performance of a new kind of angular momentum recurrences that we have proposed in a previous article [W. Hierse and P.M. Oppeneer, J. Chem. Phys. 99 , 1278 (1993)]. © 1994 John Wiley & Sons, Inc.     

Applications of fourier transforms in molecular orbital theory. Calculation of optical properties and tables of two-center one-electron integrals

Fourier transform methods initiated by Geller and Harris are applied to the calculation of optical properties of molecules. Tables of one-electron two-center integrals needed for the accurate computation of molecular absorption and optical activity are calculated by the Fourier transform method. A general theorem is derived which allows the angular part of the integrals to be treated by means of projection operators. The radial parts of the integrals are treated by the methods of Harris. The results are obtained in a simple closed form which avoids the usual transformation to local coordinates. The two-center integrals evaluated include matrix elements of the momentum operator, the dipole moment operator, the tensor operator , the quadrupole moment operator, and the angular momentum operator. These are evaluated between 1s, 2s, and 2p Slater-type atomic orbitals located on different atoms. The results are expressed as functions of the Slater exponents and of the relative coordinates of the two atoms.     

Density-functional expansion methods: generalization of the auxiliary basis

The formulation of density-functional expansion methods is extended to treat the second and higher-order terms involving the response density and spin densities with an arbitrary single-center auxiliary basis. The two-center atomic orbital products are represented by the auxiliary functions centered about those two atoms, and the mapping coefficients are determined from a local constrained variational procedure. This two-center variational procedure allows the mapping coefficients to be pretabulated and splined as a function of internuclear separation for efficient look up. The splines of mapping coefficients have a range no longer than that of the overlap integrals, and the auxiliary density appears as a single point-multipole expansion to all nonoverlapping atoms, thus allowing for the trivial implementation of a linear-scaling algorithm. The method is tested using Gaussian multipole expansions, and the effect of angular and radial completeness is explored. Several auxiliary basis sets are parametrized and compared to an auxiliary basis analogous to that used in the self-consistent-charge density-functional tight-binding model, and the method is demonstrated to greatly improve the representation of the density response with respect to a reference expansion model that does not use an auxiliary basis.     

New method for the direct calculation of electron density in many-electron systems. I. Application to closed-shell atoms

Analytical properties of hydrogen-like atomic orbitals (HAO ) that are used in the MOLCAO approach to the quantum theory of molecules have been studied. Addition and expansion theorems for HAO have been proved, both in coordinate and momentum representations. A close relation has been established between HAO and the reduced Bessel functions of half-integer indices. New methods are suggested to calculate integrals for atomic and molecular form factors, and multicenter integrals, for the HAO basis in the MO LCAO theory.     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

The scaling approach to multicenter molecular integrals with Slater-type orbitals

A scaling approach to multicenter molecular integrals with Slater-type orbitals (STOs) is presented. The result is significant in that it shows (1) the existence of a simple relationship between multicenter integrals and (2) an implied computational savings. Operation count estimates indicate that the significant savings would occur for a system having large numbers of STOs on each atom.     

Molecular integrals over the gauge-including atomic orbitals. II. The Breit-Pauli interaction

Each accompanying coordinate expansion (ACE) formula is derived for each of the orbit-orbit interaction, the spin-orbit coupling, the spin-spin coupling, and the contact interaction integrals over the gauge-including atomic orbitals (GIAOs) by the use of the solid harmonic gradient (SHG) operator. Each ACE formula is the general formula derived at the first time for each of the above molecular integrals over GIAOs. These molecular integrals are arising in the Breit-Pauli two-electron interaction for a relativistic calculation. We may conclude that we can derive a certain ACE formula for any kind of molecular integral over solid harmonic Gaussian-type orbitals by using the SHG operator. The present ACE formulas will be useful, for example, for a calculation of a molecule in a uniform magnetic field, for a relativistic calculation, and so on, with the GIAO as a basis function.     

Poisson equation for molecular exchange,hybrid and coulomb electron repulsion integrals

The various multicenter exchange, hybrid and Coulomb electron repulsion integrals that occur in molecular quantum mechanics are shown to satisfy a Poisson equation in which an overlap integral plays the role of a source distribution function. Two-, three-and four-center exchange integrals arise from four-center source functions; two- and three-center hybrid integrals arise from three-center distributions; and one- and two-center Coulomb integrals have two-center sources.     

A new parametrizable model of molecular electronic structure

A new electronic structure model is developed in which the ground state energy of a molecular system is given by a Hartree-Fock-like expression with parametrized one- and two-electron integrals over an extended (minimal + polarization) set of orthogonalized atom-centered basis functions, the variational equations being solved formally within the minimal basis but the effect of polarization functions being included in the spirit of second-order perturbation theory. It is designed to yield good dipole polarizabilities and improved intermolecular potentials with dispersion terms. The molecular integrals include up to three-center one-electron and two-center two-electron terms, all in simple analytical forms. A method to extract the effective one-electron Hamiltonian of nonlocal-exchange Kohn-Sham theory from the coupled-cluster one-electron density matrix is designed and used to get its matrix representation in a molecule-intrinsic minimal basis as an input to the parametrization procedure--making a direct link to the correlated wavefunction theory. The model has been trained for 15 elements (H, Li-F, Na-Cl, 720 parameters) on a set of 5581 molecules (including ions, transition states, and weakly bound complexes) whose first- and second-order properties were computed by the coupled-cluster theory as a reference, and a good agreement is seen. The model looks promising for the study of large molecular systems, it is believed to be an important step forward from the traditional semiempirical models towards higher accuracy at nearly as low a computational cost.     

Translation of STO charge distributions

Barnett and Coulson's zeta-function method (M. P. Barnett and C. A. Coulson, Philos. Trans. R. Soc., Lond. A 1951, 243, 221) is one of the main sources of algorithms for the solution of multicenter integrals with Slater-type orbitals. This method is extended here from single functions to two-center charge distributions, which are expanded at a third center in terms of spherical harmonics times analytical radial factors. For s-s distributions, the radial factors are given by a series of factors corresponding to the translation of s-type orbitals. For distributions with higher quantum numbers, they are obtained from those of the s-s distributions by recurrence. After analyzing the convergence of the series, a computational algorithm is proposed and its practical efficiency is tested in three-center (AB/CC) repulsion integrals. In cases of large basis sets, the procedure yields about 12 correct significant figures with a computational cost of a few microseconds per integral.     

Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals

Multicenter integrals over noninteger n Slater type orbitals with integer and noninteger values of indices u of screened Coulomb type potentials, f(u)(eta,r)=r(u-1)e(-etar), and their first and second derivatives with respect to Cartesian coordinates of the nuclei of a molecule are described. Using complete orthonormal sets of Psi(alpha) exponential type orbitals and rotation transformation of two-center overlap integrals, these integrals are expressed through the noncentral potential functions depending on the molecular auxiliary functions A(k) and B(k). The series expansion formulas derived for molecular integrals of screened Coulomb potentials and their derivatives are especially useful for the computation of multicenter electronic attraction, electric field, and electric field gradient integrals. The convergence of series is tested for arbitrary values of parameters of potentials and orbitals.