Application of Laguerre Polynomials to Evaluation of Two-Center, Two- and Three-Electron Integrals in the Basis of Slater-Type Orbitals - Algorithm and Numerical Results

Budziński, Jan; Firszt, M.; Woźnicki, W.

doi:10.12693/aphyspola.86.883

Cited by 4 publications

(1 citation statement)

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“…Unfortunately, the problem of evaluating the three‐ and six‐dimensional molecular multicenter integrals effectively and reliably is, in spite of all efforts, not yet solved in a completely satisfactory way, in particular, if one wants to use the physically better motivated exponentially declining basis functions like Slater‐type orbitals. Moreover, in recent years, atomic and molecular calculations using explicitly correlated basis functions have become popular (see, e.g., 17–37 and references therein). In this context, extremely difficult three‐and four‐electron integrals have to be evaluated.…”

Section: Introductionmentioning

confidence: 99%

Addition theorems as three-dimensional Taylor expansions

Weniger

2000

Int. J. Quant. Chem.

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A principal tool for the construction of the addition theorem of a r Ји١Ž . function f is the translation operator e , which generates f r q rЈ by doing a threedimensional Taylor expansion of f around r. In atomic and molecular quantum mechanics, one is usually interested in irreducible spherical tensors. In such a case, the application of the translation operator in its Cartesian form e xЈѨ rѨ x e yЈѨ rѨ y e zЈѨ rѨ z leads to serious technical problems. A much more promising approach consists of the use of an operator expansion for e r Ји١ which contains exclusively irreducible spherical tensors. This expansion contains as differential operators the Laplacian ٌ 2 and the spherical tensor m Ž . gradient operator Y Y ٌ , which is an irreducible spherical tensor of rank l. Thus, if l m Ž . Y Y ٌ is applied to another spherical tensor, the angular part of the resulting expression l is completely determined by angular momentum coupling, whereas the radial part is obtained by differentiating the radial part of the spherical tensor with respect to r. Consequently, the systematic use of the spherical tensor gradient operator leads to a considerable technical simplification. In this way, the originally three-dimensional expansion problem in x, y, and z is reduced to an essentially one-dimensional expansion problem in r. The practical usefulness of this approach is demonstrated by constructing the Laplace expansion of the Coulomb potential as well as the addition theorems of the regular and irregular solid harmonics and of the Yukawa potential.

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Section: Introductionmentioning

confidence: 99%