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Addition theorems can be constructed by doing three-dimensional Taylor expansions according to f (r + r ′ ) = exp(r ′ · ∇)f (r). Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form exp(x ′ ∂/∂x) exp(y ′ ∂/∂y) exp(z ′ ∂/∂z) would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator exp(r ′ · ∇) involving powers of the Laplacian ∇ 2 and spherical tensor gradient operators Y (2000)]. The application of the translation operator in its spherical form is particularly simple in the case of B functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slatertype functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of B functions, the addition theorems for these functions can be written down immediately.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

Weniger

2002

“…The mechanized evaluation of molecular integrals actually dates back to two of the earliest papers on computer algebra [62,63]. Recent work on Gaussian and Slater integrals for energy and scattering calculations has used axiom [64], maple [65][66][67][68], mathematica [7,8,69,72] and reduce [10,70]. We look forward to the mechanized pooling of the results of different software systems, and their incorporation in the further steps of quantum chemical studies that are supported by ca.…”

Section: Discussionmentioning

confidence: 99%

Molecular integrals and information processing

Barnett

2003

ABSTRACT:We report new applications of computer algebra (ca) to the evaluation of molecular integrals over STOs. Closed formulas were tabulated for all the overlap integrals with principal quantum numbers 1 to 12. These provide unrestricted precision results that are compared with earlier numerical work of other authors. All the auxiliary functions needed to compute two-and three-center integrals have been reduced to closed form. We describe novel features of the software infra-structure that have wider application in theoretical chemistry and computational science.

“…In previous work 14–20, we have demonstrated the efficiency of the nonlinear D due to Levin and Sidi 21 and $\bar{D}$ due to Sidi 22, 23 transformations in evaluating this kind of semi‐infinite integrals. These nonlinear transformations are highly efficient for improving convergence of semi‐infinite integrals whose integrands satisfy linear differential equation with coefficients having asymptotic power series in the sense of Poincaré 24 and should satisfy some limit conditions.…”

Section: Introductionmentioning

confidence: 97%

Symbolic programming language in molecular multicenter integral problem

Safouhi

Bouferguène

2005

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ABSTRACT: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 three-center nuclear attraction and Coulomb integrals is the most frequently encountered. As the molecular system becomes 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 further development in computational studies of large molecular systems. To develop fast and accurate algorithms for the numerical evaluation of these integrals over B functions, we used nonlinear transformations for improving convergence of highly oscillatory integrals. These methods form the basis of new methods for solving various problems that were unsolvable otherwise and have many applications as well. To apply these nonlinear transformations, the integrands should satisfy linear differential equations with coefficients having asymptotic power series in the sense of Poincaré, which in their turn should satisfy some limit conditions. These differential equations are very difficult to obtain explicitly. In the case of molecular integrals, we used a symbolic programming language (MAPLE) to demonstrate that all the conditions required to apply these nonlinear transformation methods are satisfied. Differential equations are obtained explicitly, allowing us to demonstrate that the limit conditions are also satisfied. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem 106: 65-78, 2006