Efficient evaluation of one-center three-electron Gaussian integrals

 An algorithm is presented for the efficient evaluation of two types of one-center three-electron Gaussian integrals. These integrals are required to avoid the resolution-of-identity (RI) approximation in explicitly correlated linear R12 methods. Without the RI approximation, it is possible to enforce rigorously the strong orthogonality of the second-order M?ller–Plesset R12 ansatz. A test calculation is performed using atomic Gaussian-type orbitals of the neon atom. Received: 21 November 2000 / Accepted: 6 April 2001 / Published online: 9 August 2001     

Atomic three-electron integrals for correlated Slater-type s orbitals

A compact series expansion method is described for evaluation atomic three-electron integrals which involve odd powers of the three interelectronic distances and Slater-type s orbitals. Only one dimensional integrals appear in the final expression, and these are readily amenable to machine computation. Convergence of the series is discussed.     

Numerical evaluation of three- and four-center bielectronic integrals using exponential-type atomic orbitals

Three- and four-center Slater orbital bielectronic integrals are evaluated by means of a complete function set. The method provides a series to approximate the bielectronic integrals. Their corresponding partial sums are analyzed in detail for 1s orbitals. The comparison with the Fourier transform–based method brings forth encouraging perspectives for the present approach. © 1995 John Wiley & Sons, Inc.     

Evaluation of two-center,two- and three-electron integrals involving correlation factors over Slater-type orbitals. I. Basic integrals

The modified and extended version of the Neumann expansion of the interrelectronic distance function r for u = ?1, 0, 1, 2, using the set of orthogonal polynomials normalized to unity, is presented. This expansion has been utilized to obtain analytical expressions for evaluating two-center two- and three-electron integrals in the Slater orbital basis occurring if variational correlated functions are used.     

The use of improved atomic orbitals in the evaluation of zero-field splitting integrals

The effect of improving the 2p _z-atomic orbital representation on the values of molecular zero-field splitting integrals is assessed on the example of the two-center Coulomb integral involving the (r ²–3z²)/r⁵ operator in the cases of nitrogen and carbon. The results suggest that the use of the classical Slater orbital or its Gaussian equivalent may be misleading.

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Zusammenfassung Ein spezielles, bei der Berechnung der molekularen Nullfeldaufspaltung von Tripletts auftretendes Zweizentrenintegral wird in AbhÄngigkeit von der Güte der 2p _z-Orbitale des Kohlenstoff- bzw. Stickstoffatoms untersucht. Offenbar sind die üblichen Slaterorbitale und deren Approximation durch Gau\funktionen ungeeignet.

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Résumé On a étudié l'effet de l'amélioration de la base d'orbitales atomiques 2p _z sur les valeurs des intégrales qui interviennent dans le calcul de la séparation des niveaux d'un triplet moléculaire en l'absence de champ magnétique. L'utilisation de l'orbitale de Slater classique ou de son équivalent en Gaussienne peut conduire à des conclusions appréciablement erronées.

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This work was supported by grant GM 12289-02 of the United States Public Health Service (National Institute of General Medical Sciences).     

Second-order Møller–Plesset perturbation theory with terms linear in the interelectronic coordinates and exact evaluation of three-electron integrals

 The second-order correlation energy of M?ller–Plesset perturbation theory is computed for the neon atom using a wave function that depends explicitly on the interelectronic coordinates (MP2-R12). The resolution-of-identity (RI) approximation, which is invoked in the standard formulation of MP2-R12 theory, is largely avoided by rigorously computing the necessary three-electron integrals. The basis-set limit for the second-order correlation energy is reached to within 0.1 mE _h. A comparison with the conventional RI-based MP2-R12 method shows that only three-electron integrals over s and p orbitals need to be computed exactly, indicating that the RI approximation can be safely used for integrals involving orbitals of higher angular momentum. Received: 9 May 2001 / Accepted: 31 October 2001 / Published online: 9 January 2002     

On the reliability of atomic transition integrals

A simple procedure due to Layzer and Garstang for estimating the reliability of calculated dipole-length transition integrals is re-examined. Some recent Hartree—Fock f-values of Ghoshal, Dutta and Sengupta for helium-sequence transitions are analyzed and corrected, and are shown to be of generally high accuracy.     

Evaluation of Hylleraas-CI atomic integrals. III. Two-electron kinetic energy integrals

This paper is the part III of a series about the evaluation of Hylleraas-Configuration Interaction (Hy-CI) integrals by the method of direct integration over the interelectronic coordinates. The two-electron kinetic-energy integrals have been derived using the Hamiltonian in Hylleraas coordinates. We have improved the algorithm used in part II of this series and obtained general expressions. The method used for the two-electron integrals can be used in the same fashion for the evaluation of the three-electron ones. The formulas shown here have been tested in actual Hy-CI calculations of two-electron systems. The two-electron kinetic energy integrals values agree with the ones obtained using the Kolos and Roothaan transformation. The effectiveness of the different methods is discussed.     

Application of cholesky-like matrix decomposition methods to the evaluation of atomic orbital integrals and integral derivatives

When viewed as a square two-indexed matrix, the array of atomic orbital-based, two-electron integrals (ij|kl) is a positive semidefinite array. Beebe and Linderberg showed, in 1977, that actual or near linear dependencies often exist within the types of atomic orbital basis sets employed in conventional quantum chemical calculations. In fact, large (i.e., higher quality) bases were shown to be substantially more redundant than smaller or more spatially separated bases. In situations where there exists significant basis near redundancy, the rank (r) of the (ij|kl) ≡ V_l,J matrix of integrals will be significantly smaller than the matrix dimension M. When this occurs, it proves computationally tractable to decompose the M-dimensional matrix V into components L ( V = LL ^T) which contain all of the information needed to form the full V matrix. The Cholesky algorithm allow such a decomposition to be carried out and forms the basis of the work described here. The method is found to be highly successful in reducing the number of integrals and integral derivatives that must actually be calculated. In particular, results on the C₂ molecule indicate that the algorithm can be superior to traditional methods of integral derivative generation if the orbital basis is large enough to contain appreciable near redundancy. In contrast, results on benzene with a more spatially delocalized basis show that conventional methods are preferred whenever substantial basis (near) redundancy is not present.     

Half-numerical evaluation of pseudopotential integrals

A half-numeric algorithm for the evaluation of effective core potential integrals over Cartesian Gaussian functions is described. Local and semilocal integrals are separated into two-dimensional angular and one-dimensional radial integrals. The angular integrals are evaluated analytically using a general approach that has no limitation for the l-quantum number. The radial integrals are calculated by an adaptive one-dimensional numerical quadrature. For the semilocal radial part a pretabulation scheme is used. This pretabulation simplifies the handling of radial integrals, makes their calculation much faster, and allows their easy reuse for different integrals within a given shell combination. The implementation of this new algorithm is described and its performance is analyzed.     

Highly optimized fourth-order short-time approximation for path integrals

We derive a fourth-order short-time approximation for use in imaginary-time path-integral simulations. The short-time approximation converges for all continuous and bounded-from-below potentials, attains quartic order of convergence for sufficiently smooth potentials, and utilizes statistically independent random variables for its construction. These properties recommend the approximation as a natural replacement of the trapezoidal Trotter-Suzuki approximation for physical systems with continuous distributions.     

Evaluation of two-center,two- and three-electron integrals involving correlation factors over Slater-type orbitals. II. Kinetic and potential energy integrals and examples of numerical results

This article is concerned with the construction of the general algorithm for evaluating two-center, two- and three-electron integrals occurring in matrix elements of one-electron operators in the basis of variational correlated functions. This problem has been solved here in prolate spherical coordinates, using the modified and extended form of the Neumann expansion of the interelectronic distance function r^k_ij derived in Part I of this series for k = ?1, 0, 1, 2. This work expands the method proposed by one of us in the preceding paper for integrals of the types mentioned above. The results of numerical calculations for different types of the two- and three-electron integrals are presented. The problem of convergence of the proposed procedures used is also discussed.     

About evaluation of many-center molecular integrals

Studies of truncated expansions of Slater s-orbitals, in terms of associated Laguerre functions and spherical harmonics at another center, are carried out. The possibility of using such an expansion to calculate many-center molecular integrals is considered. The procedure is applied to solve three-center nuclear attraction integrals; it is shown that in some cases this expansion can provide relatively good results.

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Zusammenfassung Slater-s-Orbitale werden an einem anderen Zentrum in Reihen von Produkten aus Kugelfunktionen und zugeordneten Laguerreschen Funktionen entwickelt. Untersucht wird die Konvergenz der Reihen und ihre Eignung zur Berechnung von Mehrzentren-, insbesondere von Dreizentren-Kernwechselwirkungs-Integralen. Die Ergebnisse sind zum Teil recht gut.

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Résumé On étudie des développements tronqués d'orbitales s de Slater, en série de fonctions associées de Laguerre et d'harmoniques sphériques centrés en un autre point. On examine les possibilités d'employer ce genre de développement dans le calcul des intégrales moléculaires polycentriques. On applique le procédé à la résolution des intégrales tricentriques d'attraction nucléaire et l'on montre que dans certains cas cette expansion peut conduire à des résultats relativement bons.

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A preliminary report of this work was read at the XII. meeting of the Real Sociedad Española de Física y Química, Salamanca, June 1965.     

Covalent bond orders and atomic anisotropies from iterated stockholder atoms

Iterated stockholder atoms are produced by dividing molecular electron densities into sums of overlapping, near-spherical atomic densities. It is shown that there exists a good correlation between the overlap of the densities of two atoms and the order of the covalent bond between the atoms (as given by simple valence rules). Furthermore, iterated stockholder atoms minimise a functional of the charge density, and this functional can be expressed as a sum of atomic contributions, which are related to the deviation of the atomic densities from spherical symmetry. Since iterated stockholder atoms can be obtained uniquely from the electron density, this work gives an orbital-free method for predicting bond orders and atomic anisotropies from experimental or theoretical charge density data.     

Numerical investigations of different orders of relativistic effects in atomic shells

Summary Different orders in ² of the hydrogenic energy levels are discussed. The spurious contribution of – 0.2Z^{5 3} of the first order Douglas-Kroll-Hess (DKH) Hamiltonian is cancelled at the second order DKH level, which recovers the energy up to the order of Z^{6 4} at least to a large extent and includes also a significant part of the higher order terms. Concerning the valence shell energies of many electron atoms, their behavior is more similar to the one of the hydrogen-like 1 s than of the respective nlj levels. The convergence of the ² expansion becomes unsatisfactory for the very heavy elements.