Lamé spheroconal harmonics in atoms and molecules

Spheroconal harmonics are the natural basis for the description of asymmetric‐molecule rotations (Kramers and Ittmann, Zeitschrift für Physik, 1929, 53, 553; Piña, J Mol Str 1999, 493, 159) and also an alternative to the familiar spherical harmonics as the angular part of the Schrödringer 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 Λ   n   1A (χ 1 )Λ   n   2B (χ 2 ) in the respective angular coordinates χ 1 , χ 2 , with n 1 + n 2 = 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 Schrödinger 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. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010