Spheroconal coordinates

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. 

The change from spherical coordinafes to spheroconal coordinates is contained in the transformation equations ... 

Niven established the connections between the ellipsoidal harmonics, expressed in cartesian coordinates, and the spheroconal harmonics, expressed in spheroconal coordinates, in the respective factors of Eq. (18), by requiring that the eigenfunctions h satisfy the Laplace equation [18]. The application of the Laplace operator on the eigenfunctions with the condition of vanishing leads to the zeros 0, of the respective polynomials, which are real and different in their respective domains ccartesian coordinates leads to the corresponding condition for its being harmonic... 

Concerning the spheroconal coordinates expressed through Jacobian elliptic functions used in Ref. [3], its Eq. (50) is reproduced here... 

If is inferesfing that Refs. [2-4] did nof include fhe explicit expressions for fhe carfesian components of fhe angular momentum operator in their respective spheroconal coordinates. The Appendix of Ref. [6] does have such expressions for the spheroconal coordinates in the Jacobi elliptic function representation. The interested reader is invited to see them and understand their structure. 

The exact nature of the formulations of the problem of rotations of symmetric molecules in spheroconal coordinates, in Sections 2.1 and 2.2, is emphasized, recognizing also their points of contact and differences. The... 

The spheroconal coordinates (r, xi, Xi) are defined by their transformation equations to cartesian coordinates ... 

Notice the common radial coordinate and scale factor for both spherical and spheroconal coordinates, Equations (45) and (46). Fixed values of Xi correspond to elliptical cones with a common vertex at the origin, and an axis along the z-axis for i = 1 and along the x-axis for i = 2. 

This type of confinement is alternative to that by a circular cone in [22] and Section 4.2, using spheroconal coordinates instead of spherical coordinates. The radial coordinate is common to both sets, but the angular coordinates are different. Here we first identify the set of cones by fixed values of the coordinate xi with their common vertex as the origin, their common axis as the z-axis, and elliptical cross sections as determined next. In fact, by using the relationships... 

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