ISSN:
0020-7608
Keywords:
Computational Chemistry and Molecular Modeling
;
Atomic, Molecular and Optical Physics
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Chemistry and Pharmacology
Notes:
The radial one-electron Schrödinger equation can be written as a nonlinear first-order differential equation by making a suitable logarithmic transformation. The resulting Riccati equation has the equivalent Hammerstein integral representation [1], \documentclass{article}\pagestyle{empty}\begin{document}$$ \beta (r) = \int_{r' = 0}^\infty P(r') N(r,r')dr' \quad 0\buildrel{〈}\over{=} r 〈 \infty $$\end{document} where the kernel, N(r, r′) is \documentclass{article}\pagestyle{empty}\begin{document}$$ N\left( {r,\,r\prime} \right) = H\left( {r,\,r\prime} \right)\exp \left\{ {\int_{\xi = r\prime}^r {R\left( \xi \right)\beta \left( \xi \right)d\xi } } \right\} $$\end{document} and H(r, r′) is the Heaviside unit step function. This kernel is a more general one than that developed in ref. [1]. Both kernels apply in cases where the Riccati equation corresponds to a Sturm-Liouville problem.It is shown that this integral equation can be integrated by parts so that, for any local potential, the integrand decreases as the cyclic folding procedure is applied. During this cyclic folding, the kernel generates an equation that contains only coefficients of β(r)0 and β(r)1. Consequently, after truncating at the end of the nth cycle, it is possible to write down a Padé-type approximation to the logarithmic derivative as a known function of the independent variable. All coefficients in this approximation can be evaluated as simple algebraic formulations of P(r), R(r), and integrals over P(r).
Additional Material:
1 Ill.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/qua.560080505
