Four-center integrals for Gaussian and exponential functions

Rico, J. Fernández; Fernández, J.F.; Ema, I.; López, Rafael; Ramı́rez, G.

doi:10.1002/1097-461x(2001)81:1<16::aid-qua5>3.0.co;2-a

Cited by 41 publications

(17 citation statements)

References 43 publications

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“…(5) and (6) are expressed in terms of overlap integrals so that we limit our comments to these latter. We developed algorithms for these in the context of the Gauss transform,17 one‐center two‐range expansions16, 18 (Barnett and Coulson's α function method19) a one‐center one‐range expansion technique of our own20, 21 and, recently, with the shift‐operators technique 10, 22. All these algorithms allow us a very efficient calculation of the overlap integrals, and no important differences in their performance are found.…”

Section: Algorithms For Integrals: Efficiencymentioning

confidence: 99%

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

et al. 2004

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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.

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Section: Algorithms For Integrals: Efficiencymentioning

confidence: 99%

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

et al. 2004

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“…These include the Coulomb Sturmians introduced by Shull and Löwdin [46] and used by some other authors [47][48][49][50]. The shift operator technique [51][52][53] is a very elegant method which generates integrals with arbitrary STOs starting with the simplest integrals with 1s functions by application of the so-called shift operator. Gill et al [54,55] introduced the Coulomb resolution techniques where the interaction potential is expanded in terms of the so-called potential functions resulting from the the Poisson equation.…”

Section: Introductionmentioning

confidence: 99%

Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. I. Coulomb and hybrid integrals

Lesiuk

Moszyński

2014

Phys. Rev. E

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In this paper, which constitutes the first part of the series, we consider calculation of two-center Coulomb and hybrid integrals over Slater-type orbitals. General formulas for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters. Direct integration over the coordinates of one of the electrons leaves us with the set of overlaplike integrals which are evaluated by using two distinct methods. The first one is based on the transformation to the ellipsoidal coordinates system and the second utilizes a recursive scheme for consecutive increase of the angular momenta in the integrand. In both methods simple one-dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulas. It is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well-behaved functions, calculated recursively with decent speed. Special attention is paid to the numerical stability of the algorithms. Applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including, at most, 7i-type functions (l=6).

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“…In quantum chemistry research, the mathematical description of atomic orbitals have been studied mainly with the use of Slater [25, 26] and Gaussian functions [27–29]. The Slater function has the correct cusp at the origin and the physical exponential decay at long range [30, 31], expected by the orbital representation, nevertheless, when a many‐electron system is taken into account, the matrix elements become very hard to calculate in the coordinate space [32].…”

Section: Introductionmentioning

confidence: 99%

Electron and positron elastic scattering by non‐central potentials: Matrix elements and symmetry properties in the first Born approximation

Barp

Arretche

2022

Int J of Quantum Chemistry

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The Fourier transform of Cartesian Gaussian functions product is presented in the light of positron and electron elastic scattering. The calculation of this class of integrals is crucial in order to obtain the scattering amplitude in the first Born approximation framework for an ab initio method recently proposed. A general solution to the scattering amplitude is given to a molecular target with no restriction due to symmetry. Moreover, symmetry relations are presented with the purpose of identifying terms that do not contribute to the calculation for the molecules in the D∞h point group optimizing the computational effort.

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Four-center integrals for Gaussian and exponential functions

Cited by 41 publications

References 43 publications

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

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

Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. I. Coulomb and hybrid integrals

Electron and positron elastic scattering by non‐central potentials: Matrix elements and symmetry properties in the first Born approximation

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