Rotational states of asymmetric molecules revisited: matrix evaluation and generating function of Lamé functions
E. Ley–Koo and R. Méndez–Fragoso
Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20 364, 01000 México, D.F., México, e–mail: eleykoo@fisica.unam.mx, rmf@fisica.unam.mx
Recibido el 22 de noviembre de 2007
Aceptado el 18 de febrero de 2008
Abstract
The solutions of the Schrödinger equation describing rotational states of asymmetric molecules are known to be separable as products of Lame functions in spheroconal coordinates. However, the numerical evaluation of such solutions has not been implemented in efficient and practical ways. This paper overcomes this deficiency by presenting a matrix method for evaluating the needed Lame functions, in their two kinds and eight species, to obtain accurate energy eigenvalues and eigenfunctions for the rotational states of molecules with any asymmetry. A generating function for these eigenfunctions, including its series expansions for any asymmetry, is also presented.
Keywords: Asymmetric molecules; rotation spectra; matrix evaluation; spherical harmonics; Lame functions; spheroconal harmonics; generating function.
Resumen
Se sabe que las soluciones de la ecuación de Schrödinger que describen estados rotacionales de moléculas asimétricas son separables como productos de funciones de Lame en coordenadas esferoconales. Sin embargo, la evaluación numérica de tales soluciones no ha sido implementada en formas eficientes y prácticas. Este trabajo elimina tal deficiencia presentando un método matricial para evaluar las funciones de Lame necesarias, en sus dos tipos y ocho especies, para obtener valores precisos de los eigenvalores de la energía y de las eigenfunciones para los estados rotacionales de moléculas con cualquier asimetría. También se presenta una función generadora para tales eigenfunciones incluyendo su desarrollo en serie para cualquier asimetría.
Descriptores: Moléculas asimétricas; espectro rotacional; evaluación matricial; armónicos esféricos; funciones de Lame; armónicos esferoconales; función generadora.
PACS: 33.20Sn; 33.15.Mt: 33.20.–t; 31.15.Hz
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Acknowledges
The authors acknowledge the financial support for this work by Consejo Nacional de Ciencia y Tecnología, SNI–1796.
One of the authors (E.L.K.), on sabatical leave from Universidad Nacional Autónoma de México with support from DGAPA, thanks Professors Hans Volkmer of University of Wisconsin–Milwaukee and Araceli Gongora of Northeastern University, Boston for discussions about the evaluation of Lamé functions.
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