Lamé spheroconal harmonics in atoms and molecules

摘要

Spheroconal harmonics are the natural basis for the description of asymmetric-molecule rotations (Kramers and Ittmann, Zeitschrift für Physik, 1929, 53, 553; Pi?a, J Mol Str 1999, 493, 159) and also an alternative to the familiar spherical harmonics as the angular part of the Schr?dringer equation eigenfunctions for central potentials (Kalnins et al. SIAM J Appl Math 1976, 30, 360).We have dealt with their properties and matrix evaluation in connection with the rotations of asymmetric molecules (Ley-Koo and Méndez-Fragoso, Rev Mex Fís 2008, 54, 162) and the construction of a generating function for the complete wave functions of the Hydrogen atom (Ley-Koo and Góngora, Int J Quantum Chem 2009, 109, 790). For these cases, the spheroconal harmonics are products of Lamé polynomials Δ~A_(n1)(X1)Δ~B_(n2)(X2) in the respective angular coordinates X1, X2, with n1 + n2 = l, the angular momentum label. More recently during the investigations of the Hydrogen atom (Méndez-Fragoso and Ley-Koo, Int J Quantum Chem. In press) and the rotations of asymmetric molecules (Méndez-Fragoso and Ley-Koo, to be submitted), confined in elliptical cones associated with the spheroconal coordinates in which the respective Schr?dinger equations are separable, we have recognized the need to use and to construct quasi-periodic Lamé functions. In fact, the new boundary conditions require that the angular momentum label becomes noninteger l → λ, and the respective Lamé functions become infinite series. This contribution contains details about the evaluation of the polynomial and quasi-periodic Lamé functions, and their applications in the free particle confined by an elliptical cone with a spherical cap and the harmonic oscillator confined by an elliptical cone.