Lame spheroconal harmonics

The complete Lame spheroconal harmonic quasi-periodic functions, satisfying the respective boundary conditions in Eq. (91), involve matching the corresponding factors of Table 4.4 and their counterparts. For the boundary... 

Table 4.4 Matching species of Lam6 functions whose products are Lame spheroconal harmonic functions with defined parities...

Section 2.6 recognizes that for the hydrogen atom, its Hamiltonian also commutes with and H correspondingly, it also admits solutions with Lame spheroconal harmonics polynomial eigenfunctions. It also shares the same radial eigenfunction with the familiar solution with spherical harmonics, and additionally both can be obtained from a common generating function and both satisfy the addition theorem. 

Correspondingly, the spheroconal harmonics to be used in the eigenfunctions of the hydrogen atom, Equation (48), are the products of Lame functions in the respective variables, with common values of l and... 

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

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