Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. I. Three-electron integrals

A method to evaluate the nonrelativistic electron-repulsion, nuclear attraction and kinetic energy three-electron integrals over Slater orbitals appearing in Hylleraas-CI (Hy-CI) electron structure calculations on atoms is shown. It consists on the direct integration over the interelectronic coordinate r _ij and the sucessive integration over the coordinates of one of the electrons. All the integrals are expressed as linear combinations of basic two-electron integrals. These last are solved in terms of auxiliary two-electron integrals which are easy to compute and have high accuracy. The use of auxiliary three-electron ones is avoided, with great saving of storage memory. Therefore this method can be used for Hy-CI calculations on atoms with number of electrons N ≥ 5. It has been possible to calculate the kinetic energy also in terms of basic two-electron integrals by using the Hamiltonian in Hylleraas coordinates, for this purpose some mathematical aspects like derivatives of the spherical harmonics with respect to the polar angles and recursion relations are treated and some new relations are given.     

Evaluation of the Boys Function using Analytical Relations

The analytical relations for Boys function F _m (x) are presented. These relations are useful in the fast and more accurate calculations of multicenter molecular integrals over Gaussian type orbitals (GTOs). The formulas obtained are numerically stable for all values of m and x.     

Evaluation of integrals over STOs on different centers and the complementary convergence characteristics of ellipsoidal‐coordinate and zeta‐function expansions

One‐electron integrals over three centers and two‐electron integrals over two centers, involving Slater‐type orbitals (STOs), can be evaluated using either an infinite expansion for 1/r₁₂ within an ellipsoidal‐coordinate system or by employing a one‐center expansion in spherical‐harmonic and zeta‐function products. It is shown that the convergence characteristics of both methods are complimentary and that they must both be used if STOs are to be used as basis functions in ab initio calculations. To date, reports dealing with STO integration strategies have dealt exclusively with one method or the other. While the ellipsoidal method is faster, it does not always converge to a satisfactory degree of precision. The zeta‐function method, however, offers reliability at the expense of speed. Both procedures are described and the results of some sample calculation presented. Possible applications for the procedures are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 1–13, 1999     

Evaluation of the auxiliary function $${G_{-ns}^q}$$ using basic overlap integrals over Slater type orbitals

This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs) for the evaluation of auxiliary function which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function is completely general for p _a ≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function is compared with other method.     

Symbolic Calculation of Two‐Center Overlap Integrals Over Slater‐Type Orbitals

Two‐center overlap integrals over Slater type orbitals (STOs) have been expressed in terms of the well‐known Mulliken's integrals B_n(pt) using Rodrigues's formula for normalized associated Legendre functions. A computer program is written in Mathematica 4.0 for the evaluation of two‐center overlap integrals over STOs. Using this computer program, symbolic tables are presented for two‐center overlap integrals up to quantum numbers 1 ≤ n,n′ ≤ 3, 0 ≤ l,l′ ≤ 2, ?2 ≤ m,m′ ≤ 2. Numerical results of this work, for some quantum sets, have also been compared with prior literature and best agreement achieved with recent works of Barnett while some discrepancies were obtained with works of Öztekin et al. and Guseinov et al.     

A modified PCILO method

The perturbative configuration interaction using strictly localized molecular orbitals, called the modified PCILO method, for which the use of the Rayleigh-Schrödinger many-body perturbation theory with the Moller-Plesset Hamiltonian partitioning is characteristic, has been proposed in this communication. On the CNDO/2 and INDO levels of Hamiltonian approximations strictly localized molecular orbitals have been constructed by solving modified Roothaan equations. From the zero and second order energy interatomic distances and harmonic force constants for some diatomic molecules have been calculated. The linear dependence of the correlation energy on the number of valence electrons in the series of the molecules CH₄, CH₃F, CH₂F₂, CHF₃ and CF₄ is perfect.     

Natural spin orbital analysis of diatomic molecular wave functions in terms of generalized diatomic orbitals I. Outline of the method. Results for the ground state of H2

The method of linear combinations of generalized diatomic orbitals (LCGDO) is combined with the method of configuration interaction (CI). CI wave functions obtained in this way are finally submitted to a natural spin orbital analysis; the resulting natural spin orbitals are expansions in terms of generalized diatomic orbitals.For the ground state of H₂, a one-determinantal-approach with a single completely optimized one-electron basis function nearly reproduces the Hartree-Fock-result. The two-determinantal approach with two optimized basis functions of type _g and _u nearly gives the optimized double configuration SCF result.     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. III. Auxiliary functions \documentclass{article}\pagestyle{empty}\begin{document}$Q_{nn'}^{q}$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$G_{-nn'}^{q}$\end{document}

The auxiliary functions $Q_{nn'}^{q}(p,pt)$ and $G_{-nn'}^{q}(p_{a},p,pt)$ which are used in our previous paper [Guseinov, I. I.; Mamedov, B. A. Int J Quantum Chem 2001, 81, 117] for the computation of multicenter electron‐repulsion integrals over Slater‐type orbitals (STOs) are discussed in detail, and the method is given for their numerical computation. The present method is suitable for all values of the parameters p_a, p, and pt. Three‐ and four‐center electron‐repulsion integrals are calculated for extremely large quantum numbers using relations for auxiliary functions obtained in this paper. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Calculation of three-center electric and magnetic multipole moment integrals using translation formulas for Slater-type orbitals

 By the use of translation formulas for the expansion of Slater-type orbitals (STOs) in terms of STOs at a new origin, three-center electric and magnetic multipole moment integrals are expressed in terms of two-center multipole moment integrals for the evaluation of which closed analytical formulas are used. The convergence of the series is tested by calculating concrete cases. Computer results with an accuracy of 10⁻⁷ are obtained for 2^ν– pole electric and magnetic multipole moment integrals for 1≤ν≤5 and for arbitrary values of screening constants of atomic orbitals and internuclear distances. Received: 28 October 1999 / Accepted: 15 February 2000 / Published online: 5 June 2000     

Use of unsymmetrical one-range addition theorems of Slater type orbitals in molecular electronic structure determination

In this work, the applicability of the unsymmetrical one-range addition theorems obtained from the use of complete orthonormal sets of Ψ^α-exponential type orbitals (Ψ^α-ETOs, where α = 1, 0, − 1, − 2, ...) to the study of electronic structure of molecules is demonstrated using minimal basis sets of Slater type orbitals (STOs). As an example of application of unsymmetrical one-range addition expansion method to evaluate the multicenter electronic integrals, the calculation has been performed for the ground state of BH ₃ molecule. The results of computer calculations for the orbital and total energies, and linear combination coefficients of symmetrized molecular orbitals are presented.     

Calculation of Multicenter Nuclear Attraction and Electron Repulsion Integrals over Slater Orbitals by Fourier Transform Method Using Gegenbauer Polynomials

In this study, using complete orthonormal sets of exponential type orbitals (ETOs), a single closed analytical relation is derived for a large number of different expansions of overlap integrals over Slater type orbitals (STOs) with the same screening parameters in terms of Gegenbauer coefficients. The general formula obtained for the overlap integrals is utilized for the evaluation of multicenter nuclear attraction and electron repulsion integrals appearing in the Hartree–Fock–Roothaan equations for molecules. The formulas given in this study for the evaluation of these multicenter integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, scaling parameters of STOs and internuclear distances.     

Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater‐type orbitals with noninteger principal quantum numbers

The expansion formula has been presented for Slater‐type orbitals with noninteger principal quantum numbers (noninteger n‐STOs), which involves conventional STOs (integer n‐STOs) with the same center. By the use of this expansion formula, arbitrary multielectron multicenter molecular integrals over noninteger n‐STOs are expressed in terms of counterpart integrals over integer n‐STOs with a combined infinite series formula. The convergence of the method is tested for two‐center overlap, nuclear attraction, and two‐electron one‐center integrals, due to the scarcity of the literature, and fair uniform convergence and great numerical stability under wide changes in molecular parameters is achieved. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

A steepest-descent method for the calculation of localized orbitals and pseudoorbitals

A method for direct calculation of localized non-orthogonal orbitals, which has been proposed by the authors recently, is extended to cases where the overlap between different subsystems is very large. This is achieved by using a steepest-descent procedure. In addition, a computationally simple treatment of correlation effects is introduced into the method by means of the density functional formalism. Results of the method are given for e.g. LiH, CH₄, Ne₂, CO,(FH)₂.     

Unified treatment of complete orthonormal sets of functions in coordinate,momentum and four-dimensional spaces and their expansion and one-range addition theorems

The simpler formulas are derived for the complete orthonormal sets of exponential- type orbitals, momentum space orbitals and hyperspherical harmonics and their expansion and one-range addition theorems. The continuum states are not properly included in these functions. The analytical formulas are also obtained for the overlap integrals over Ψ^α-ETOs, their extensions to momentum and four-dimensional spaces and STOs with the same screening constants using addition and expansion theorems derived in this paper. The complete orthonormal sets of functions and their expansion and one-range addition theorems obtained can be useful in the study of different quantum mechanical problems when the coordinate, momentum or four-dimensional spaces employed.     

Quadratically convergent calculation of localized molecular orbitals

Two iterative procedures for the transformation of canonical self-consistent field molecular orbitals to intrinsic localized molecular orbitals are proposed. A first-order method based on a series of (n × n) unitary transformations may be applied to orbitals which are far from convergence. The second method, based on Newton's method, yields quadratic convergence. Numerical results based on Boys' criterion are presented for water, carbon monoxide, boron fluoride, nitric oxide, and methylacetylene. A composite method may be used to obtain rapid convergence for large molecules for which it is not practical to calculate the entire hessian matrix. The performance of the composite method is demonstrated by application to the dinitrogen tetroxide molecule. Highly converged localized molecular orbitals may be obtained for most molecules with five to eight first-order iterations followed by three or four iterations based on either the second-order or composite method.     

Extension of the MNDO formalism tod orbitals: Integral approximations and preliminary numerical results

Summary The point charge model for calculating the two-center two-electron integrals in MNDO and related methods is extended tod orbitals. It is suggested to expand these integrals in terms of semiempirical multipole-multipole interactions where all monopoles, dipoles and quadrupoles are included, and all higher multipoles are neglected. The proposed scheme has been implemented, and numerical results for the integrals are reported. A preliminary MNDO parametrization for chlorine indicates that the inclusion ofd orbitals improves the results significantly, compared with the original MNDO and related methods.     

SYMMETRY ADAPTED LIGAND GROUP ORBITALS AND ANALYTICAL COMPOSITIONS OF HYBRID ORBITALS IN ABn TYPE MOLECULES

Abstract

A non-rigorous method of construction of SALC's for AB_n type molecules, based on the sum of the projections of the AB axes on the reference axes of the valence orbitals of the central atom, is presented. The method may be adapted to structures such as TBP where the inequivalence of the equatorial and axial ligands make the existing methods difficult to apply. A convenient method for deriving the analytical compositions of hybrid orbitals is also described.     

Calculation of molecular electric and magnetic multipole moment integrals of integer and noninteger n Slater orbitals using overlap integrals

Closed formulas are established for the magnetic multipole moment integrals of integer and noninteger n Slater‐type orbitals (ISTOs and NISTOs) in terms of electric multipole moment integrals for which the analytic expressions through the overlap integrals with ISTOs and NISTOs are derived. The overlap integrals are evaluated by the use of auxiliary functions. Using the derived expressions the multipole moment integrals, and therefore the electric and magnetic properties of molecules, can be evaluated most efficiently and accurately. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Electric and magnetic multipole integrals in STO basis sets: an analytical approach

A new method for the evaluation of one- and two-centre magnetic and electric multipole integrals for Slater-type functions is presented. The method is strictly analytical in that no approximations of any kind are involved. Two simple functions, ℐ₁ ^aug and ℐ₂ ^aug, are introduced, which employ only functions that are well known in electronic structure theory. With the use of augmentation exponents these functions apply to multipole integrals as well as other one-electron integrals, such as nuclear attraction integrals. The proposed method includes the analytic determination of derivatives of the integrals with respect to atomic displacements. Some illustrative test calculations are presented and compared to results from the literature. Received: 20 April 1998 / Accepted: 13 October 1998 / Published online: 1 February 1999