Two-electron repulsion integrals over Gaussian s functions

We present an efficient scheme to evaluate the [ 0 ]^(m) integrals that arise in many ab initio quantum chemical two-electron integral algorithms. The total number of floating-point operations (FLOPS ) required by the scheme has been carefully minimized, both for cases where multipole expansions of the integrals are admissable and for cases where this is not so. The algorithm is based on the use of a modified Chebyshev interpolation formula to compute the function exp(?T) and the integral F_m(T) = ∫₀¹u^2mexp(?Tu²) du very cheaply.     

Evaluation of screened nuclear attraction and electron repulsion molecular integrals over Gaussian basis functions

Analytical solutions to the Yukawa-like screened Coulomb nuclear attraction and electron repulsion molecular basic integrals, as well as to the basic integrals required to compute the virial coefficient, over Gaussian basis functions, are derived and cast into a practical closed form, suitable to interface with modern codes for the calculation of molecular electronic structure. © 1997 John Wiley & Sons, Inc.     

Recurrence relations for the evaluation of electron repulsion integrals over spherical Gaussian functions

Recurrence relations are derived for the evaluation of two-electron repulsion integrals (ERIs) over Hermite and spherical Gaussian functions. Through such relations, a generic ERI or ERI derivative may be reduced to “basic” integrals, i.e., true and auxiliary integrals involving only zero angular momentum functions. Extensive use is made of differential operators, in particular, of the spherical tensor gradient ??(?). Spherical Gaussians, being nonseparable in the x, y, and z coordinates, were not included in previous formulations. The advantages of using spherical Gaussians instead of Cartesian or Hermite Gaussians are briefly discussed. © 1993 John Wiley & Sons, Inc.     

Long-Range multicenter integrals with slater functions: Gauss transform-based methods

The separation of the short- and long-range terms in the potentials generated by pairs of Slater functions is reformulated in the context of the Gauss transform method. Analytic expressions of the long-range potentials (in closed form) are derived for equal exponents and generalized (as expansion series) for different exponents. Additionally, the representation of these potentials from small sets of charges or lowest-order multipoles is examined, paying special attention to their values and optimal positions. Finally, numerical tests of the formal developments are presented. It is concluded that the long-range three- and four-center integrals can be calculated with high accuracy in a simple and relatively inexpensive manner. © John Wiley & Sons, Inc.     

Recurrence formulas for molecular integrals over Laguerre Gaussian-type functions

Recurrence formulas for overlap, nuclear attraction, and electron-repulsion integrals over Laguerre Gaussian-type functions are presented. They have been derived using compact recurrence relations for homogeneous solid spherical harmonic operators but are rather lengthy as compared to those over Cartesian Gaussian-type functions. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 273–279, 1998     

Asymptotic formulas for Coulomb and hybrid integrals with slater type functions

Asymptotic expressions are obtained for two-center Coulomb and hybrid integrals with Slater functions. Results of numerical test of these expressions are presented. The use of asymptotic formulas in LCAO-MO calculations of electronic molecular structure substantially reduces the computation time. V.I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 2, pp. 12–15, March–April, 1994. Translated by L. Chernomorskaya     

Recursion formula for electron repulsion integrals over Hermite polynomials

A method for computing electron repulsion integrals over contracted Gaussian functions is described in which intermediate integrals over Hermite polynomials are generated by a “pre‐Hermite” recursion (PHR) step before the conversion to regular integrals. This greatly reduces the floating‐point operation counts inside the contraction loops, where only simple “scaling”‐type operations are required, making the method efficient for contracted Gaussians, particularly of high angular momentum. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Three‐center Coulomb repulsion integrals with Slater functions

The translation method for products of two Slater functions is improved and combined with the short–long range separation in order to develop a robust and efficient algorithm for the evaluation of three‐center Coulomb integrals with Slater functions. Several tests are carried out showing that the algorithm reported here yields integrals with an absolute error below 10^?12 hartree and a computational cost of a few microseconds per integral. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Auxiliary functions for Slater molecular integrals

Summary Some types of recurrence relations are modified to overcome the cases in which their conventional application is unstable in both the forward and backward directions. The original recurrence relations — connecting adjacent elements — are replaced by more general ones, where the non-adjacent elements are connected by coefficients obtained by new sets of relations derived from the original ones. This modification can be helpful for the calculation of the complicated molecular integrals with Slater Type Orbitals (STOs).As a simple test we prove that some auxiliary functions — previously evaluated by expensive expansions — appearing in two-center two-electron integrals can be thus calculated with very low computer cost and high accuracy.     

Orbit-orbit molecular integrals over Gaussian orbitals

Analytical formulas are derived for integrals of the orbit-orbit operator in the Breit-Pauli Hamiltonian. The present method differs from an earlier one in the introduction of a new auxiliary function G_n(a) which is an integral of Shavitt's F_m(t) function. A method for its evaluation is discussed and some numerical examples are given.     

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 mixed basis set of slater and gaussian functions for molecular calculations

A procedure for using simultaneously Slater and Gaussian basis functions in molecular calculations is presented here. The analytic expressions of the integral prototypes involving both Slater and Gaussian functions are explicitly derived.     

Molecular integrals over Gaussian-type geminal basis functions

We present formulas for the evaluation of molecular integrals over basis functions with an explicit Gaussian dependence on interelectronic coordinates. These formulas use expansions in Hermite Gaussian functions and represent an extension to the work of McMurchie and Davidson to two-electron basis functions. Integrals that depend on the coordinates of up to four electrons are discussed explicitly. A key feature of this approach is that it allows full exploitation of the shell structure of the orbital part of the basis. Received: 24 February 1997 / Accepted: 4 March 1997     

Molecular integrals for slater type orbitals using coulomb sturmians

The use of Slater type orbitals in molecular calculations is hindered by the slowness of integral evaluation. In the present paper, we introduce a method for overcoming this problem by expanding STO’s in terms of Coulomb Sturmians, for which the problem of evaluating molecular integrals rapidly has been satisfactorily solved using methods based on the theory of hyperspherical harmonics.     

Parallelization of three-center electron repulsion integrals

The parallelization of the three-center electron repulsion integrals arising from the variational fitting of the Coulomb potential is presented. A scheme for dynamical load balancing of the corresponding loop structure is discussed. The implementation in the density functional theory program deMon using the message passing interface is described. The efficiency of the parallelization is analyzed by selected benchmark calculations     

Horizontal vectorization of electron repulsion integrals

We present an efficient implementation of the Obara–Saika algorithm for the computation of electron repulsion integrals that utilizes vector intrinsics to calculate several primitive integrals concurrently in a SIMD vector. Initial benchmarks display a 2–4 times speedup with AVX instructions over comparable scalar code, depending on the basis set. Speedup over scalar code is found to be sensitive to the level of contraction of the basis set, and is best for quartets when l_D = 0 or , which makes such a vectorization scheme particularly suitable for density fitting. The basic Obara–Saika algorithm, how it is vectorized, and the performance bottlenecks are analyzed and discussed. © 2016 Wiley Periodicals, Inc.     

Two-center overlap integrals in some acetylacetonates of f elements

Overlap integrals of the type 〈2s|f〉, 〈2pσ|f〉 and 〈2pπ|f〉 have been evaluated for certain lanthanide and actinide acetylacetonates. Slater-type orbitals with Burns's exponents, analytical Hartree-Fock wavefunctions and interatomic distances taken from known crystal structure determinations of the same complexes have been used. The results indicate that the decreasing f-electron density with increasing atomic number predominates over the corresponding contribution of the metal ion-oxygen distances.