Slater orbital molecular integrals with numerical fourier transform methods. I. (coplanar) multicenter exchange integrals over 1s orbitals

Slater orbital r₁₂^?1 integrals are calculated with a numerical Fourier-transform method based on a formulation first given by Bonham, Peacher and Cox. Spherical wave expansions are introduced that decouple the Feynman integrations for the charge distribution Fourier transforms. The Feynman integrals are evaluated semianalytically, and their properties are analyzed in detail. The final computational step involves a numerical integration over charge distribution quantities. Results for (coplanar) multicenter exchange integrals over 1s orbitals are given. As long as the charge distributions are overlapping considerably, the method gives good results, even when these distributions are highly asymmetric. The method as presently implemented fails when highly disconnected charge distributions are involved.     

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.     

Nonlinear transformation methods for accelerating the convergence of Coulomb integrals over exponential type functions

It is well known that in any ab initio molecular orbital (MO) calculation, the major task involves the computation of molecular integrals, among which the computation of Coulomb integrals are the most frequently encountered. As the molecular system gets larger, computation of these integrals becomes one of the most laborious and time consuming steps in molecular systems calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. The atomic orbital basis functions chosen in the present work are Slater type functions. These functions can be expressed as finite linear combinations of B functions which are suitable to apply the Fourier transform method. The difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of highly oscillatory semi-infinite integrals. In this work, we present a generalized algorithm based on the nonlinear transformation of Sidi, for a precise and fast numerical evaluation of molecular integrals over Slater type functions and over B functions. Numerical results obtained for the three-center two-electron Coulomb and hybrid integrals over B functions and over Slater type functions. Comparisons with numerical results obtained using alternatives approaches and an existing code are listed.     

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.     

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     

The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Three‐center nuclear attraction integrals over exponential‐type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two‐dimensional integral representations, involving spherical Bessel integral functions. % The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear D transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three‐center nuclear attraction integrals. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. II. Four-electron integrals

An alternative procedure to the classical method for evaluating the four-electron Hylleraas-CI integrals is given. The method consists of direct integration over the r ₁₂ coordinate and integration over the coordinates of one of the electrons, reducing the integrals to lower order. The method based on the earlier work of Calais and L?wdin and of Perkins is extended to the general angular case. In this way it is possible to solve all of the four-electron integrals appearing in the Hylleraas-CI method. The four-electron integrals are expanded in three-electron ones which are in turn expanded in two-electron integrals. Finally the two-electron integrals are expanded into two-electron auxiliary integrals which usually have one negative power. The use of three- and four-electron electron auxiliary integrals is avoided. Some special cases lead to one- and two-electron auxiliary integrals with negative powers which do not converge individually but do converge in combination with others. These relations and their solutions are presented, together with results of various kinds of integrals.     

The accuracy of hyperfine integrals in relativistic NMR computations based on the zeroth-order regular approximation

The accuracy of the hyperfine integrals obtained in relativistic NMR computations based on the zeroth–order regular approximation (ZORA) is investigated. The matrix elements of the Fermi contact operator and its relativistic analogs for s orbitals obtained from numerical nonrelativistic, ZORA, and four–component Hartree–Fock–Slater calculations on atoms are compared. It is found that the ZORA yields very accurate hyperfine integrals for the valence shells of heavy atoms, but performs rather poorly for the innermost core shells. Because the important observables of the NMR experiment—chemical shifts and spin–spin coupling constants—can be understood as valence properties it is concluded that ZORA computations represent a reliable tool for the investigations of these properties. On the other hand, absolute shieldings calculated with the ZORA might be substantially in error. Because applications to molecules have so far exclusively been based on basis set expansions of the molecular orbitals, ZORA hyperfine integrals obtained from atomic Slater-type basis set computations for mercury are compared with the accurate numerical values. It is demonstrated that the core part of the basis set requires functions with Slater exponents only up to 10⁴ in the case where errors in the hyperfine integrals of a few percent are acceptable.     

Molecular integrals by numerical quadrature. I. Radial integration

 This article presents a numerical quadrature intended primarily for evaluating integrals in quantum chemistry programs based on molecular orbital theory, in particular density functional methods. Typically, many integrals must be computed. They are divided up into different classes, on the basis of the required accuracy and spatial extent. Ideally, each batch should be integrated using the minimal set of integration points that at the same time guarantees the required precision. Currently used quadrature schemes are far from optimal in this sense, and we are now developing new algorithms. They are designed to be flexible, such that given the range of functions to be integrated, and the required precision, the integration is performed as economically as possible with error bounds within specification. A standard approach is to partition space into a set of regions, where each region is integrated using a spherically polar grid. This article presents a radial quadrature which allows error control, uniform error distribution and uniform error reduction with increased number of radial grid points. A relative error less than 10⁻¹⁴ for all s-type Gaussian integrands with an exponent range of 14 orders of magnitude is achieved with about 200 grid points. Higher angular l quantum numbers, lower precision or narrower exponent ranges require fewer points. The quadrature also allows controlled pruning of the angular grid in the vicinity of the nuclei. Received: 30 August 2000 / Accepted: 21 December 2000 / Published online: 3 April 2001     

Overlap integrals of B functions

Two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically. The first method consists of an infinite series expansion in terms of overlap integrals with equal scaling parameters [14]. The second method consists of an integral representation for the overlap integral which has to be evaluated numerically. Bhattacharya and Dhabal [13] recommend the use of Gauss-Legendre quadrature for this purpose. However, we show that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number. We also show that the convergence of the infinite series can be improved if suitable convergence accelerators are applied. Since an internal error analysis can be done quite easily in the case of an infinite series even if it is accelerated, whereas it is very costly in the case of Gauss quadratures, the infinite series is probably more efficient than the integral representation. Overlap integrals of all commonly occurring exponentially declining basis functions such as Slater-type functions, can be expressed by finite sums of overlap integrals of B functions, because these basis functions can be represented by linear combinations of B functions.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Regularization method and molecular integrals with singular exponential functions

The regularization principle, which is based on the concept of linearly independent singular functions, makes it possible to calculate many important types of molecular matrix elements arising in the variational LCAO-MO-SCF scheme. This is done using a direct approach that employs reduction of these elements to finite sums of convergent and divergent one-electron integrals. A universal algorithm is developed to calculate two-center one-electron molecular integrals involving both singular and ordinary Slater functions. The numerical stability of the algorithms and the accuracy of the integral calculation are analyzed, and numerical estimates are given. V. I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Struktunoi Khimii, Vol. 35, No. 2, pp. 3–11, March–April, 1994. Translated by L. Chernomorskaya     

Numerical calculation of overlap and kinetic integrals in prolate spheroidal coordinates. II

The efficient algorithm calculating the overlap and the kinetic integrals for the numerical atomic orbitals is presented. On the basis of the prolate spheroidal coordinates, the overlap and the kinetic integral are reduced to the integral over the rectangular domain. The integration over the rectangular domain is performed by the adaptive integration scheme. The developed algorithm is applied to calculate the integrals for the pairs of hydrogen and gallium eigenfunctions. It is demonstrated that high accuracy can be obtained for small number of integrand evaluations what guarantees the efficiency of the presented algorithm. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Transferable integrals in a deformation-density approach to crystal orbital calculations. I

In the usual ab initio method of calculating molecular orbitals, the number of integrals to be evaluated increases as M⁴, where M is the number of basis functions. In this paper, an alternative method is discussed, where the computation time increases much less violently with the number of basis functions. Matrix elements of the deformation potential are evaluated by Fourier transform methods, while matrix elements of the neutral-atom potential are evaluated by means of transferable integrals. The transferable integrals (moments of the neutral-atom potentials) can be evaluated once and for all and incorporated as input data in computer programs. In an appendix to the paper, a general expansion theorem is discussed. This theorem allows an arbitrary spherically symmetric function to be expanded about another center.     

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.     

Semi-asymptotic evaluation of certain three-center two-electron integrals

Two-electron repulsion integrals between a two-center charge distribution and a charge distribution about a third center, which do not appreciably interpenetrate, are shown to be given to useful accuracy by numerical differentiation of certain three-center one-electron integrals. This method also may be used to evaluate integrals of this type for which the Mulliken or Sklar approximations are inapplicable.Supported by Contract SD-102 with the Advanced Research Projects Agency.     

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     

Computation of some new two-electron Gaussian integrals

Summary The evaluation of a new form of two-electron integrals is required if the interelectronic distancer ₁₂ is used as a variable in then-electron functions of electron correlation methods. The McMurchie-Davidson algorithm for the generation of molecular integrals over Gaussian-type functions is ideally suited to this. The new Gaussian integrals are formed from Hermite integrals overr ₁₂ (rather than 1/r ₁₂) by standard techniques. The Hermite integrals overr ₁₂ itself are generated by a simple procedure with negligible computational effort. The key results are discussed in the context of general recursion formulas. On leave from: Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, W-4630 Bochum, Germany     

On the use of spatial symmetry in ab initio calculations. Transformation of the two-electron integrals from atomic orbital to localized molecular orbital basis

Anm ⁵-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.     

The Hylleraas-CI method in molecular calculations: Two-electron integrals

In the Hylleraas-CI method, first proposed by Sims and Hagstrom, correlation factors of the type r are included into the configurations of a CI expansion. The computation of the matrix elements requires the evaluation of different two-, three-, and four-electron integrals. In this article we present formulas for the two-electron integrals over Cartesian Gaussian functions, the most used basis functions in molecular calculations. Most of the integrals have been calculated analytically in closed form (some of them in terms of the incomplete Gamma function), but in one case a numerical integration is required, although the interval for the integration is finite and the integrand well-behaved. We have also reported on partial and preliminary computations for the H₂ molecule using our four-center general formulas; a basis set of s- and p-type functions yielded at R = 1.4001 Å an energy of - 1.174380 a.u. to be compared with Kolos and Wolniewicz value of - 1.174475.     

Nuclear attraction and electron interaction integrals of exponentially decaying functions and the Poisson equation

Summary The technique proposed by O-Ohata and Ruedenberg (J Math Phys 7:547 (1966)) and by Silver and Ruedenberg (J Chem Phys 49:4306 (1968)) of computing nuclear attraction and electron interaction integrals by solving an inhomogeneous Laplace equation can also be applied ifB functions (Filter E, Steinborn EO (1978) Phys Rev A 18:1) are used as basis functions in atomic and molecular calculations. It is shown that because of the remarkable mathematical properties ofB functions the derivation of compact explicit expressions for the multicenter integrals mentioned above is particularly simple. These results are also of interest in the context of other exponentially decaying functions, since all the other commonly occurring exponentially decaying functions as, for instance, Slater functions or bound state hydrogen eigenfunctions can be expressed as simple linear combinations ofB functions. Consequently, their multicenter integrals can also be expressed in terms of multicenter integrals ofB functions.Dedicated to Prof. Klaus Ruedenberg on the occasion of his 70th birthday