Symbolic programming language in molecular multicenter integral problem

Abstract

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

The S and G transformations for computing three‐center nuclear attraction integrals

“…The G transformation is introduced in 5, and is extended to G n in 6 and to G in 7. The approximation G n (m) to F(∞) is defined by: where: and where c 1 = 1, c 2 = c 3 = … = c mn + 1 = 0, z j = F ( j − 1) ( x ), and for j = 1,2,… , mn + 1: Considering the works of Safouhi in 19, 21, 22, the integrands in (5) and (7) both satisfy fourth order linear homogeneous differential equations, m = 4, and second order linear homogeneous differential equations, m = 2. The coefficients in the differential equations have asymptotic expansions in inverse powers of x as x → ∞.…”

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

“…We demonstrated previously that ${\cal I}(s)$ satisfies all the conditions to apply D 21. We used a second‐order differential equation, satisfied by the integrand ${\cal F}_s(x)$ 21, 22, to obtain the approximation D for the semi‐infinite integrals ${\cal I}(s)$ 23, which is given by where $g(x) = x^{n_x} {{\hat k}_\nu[R_2 \gamma(s,x)]\over [\gamma(s,x)]^{n_\gamma}}$ and x i for i = 0, 1, 2, …, are the successive positive zeros of j λ ( v x ).…”

Extrapolation methods for improving convergence of spherical Bessel integrals for the two‐center Coulomb integrals

“…By using a symbolic programming language, such as Maple, one can obtain explicitly the linear differential equation satisfied by the integrand ${\cal T}(x)$ 34.…”