lpha^2 F^2 \over 4}
\bigg ({\rm sin^2}\phi + {\alpha^2 \ {\rm sin^2}\theta \over
F^2}\bigg) +{\tau_0 \tau_1 \alpha^3 F \over 2}\rm sin\theta
cos\phi
  \bigg] \Psi = \varepsilon\Psi\end{aligned}$$ $$\Rightarrow H_\tau\ \Psi = \varepsilon \Psi.$$

Calculational scheme
====================

To proceed with a basis set expansion, Gram-Schmidt (GS) functions orthogonal over the integration measure $ F = 1 + \alpha \  \rm
cos \theta$ must be generated. Fortunately, it is possible to construct such functions almost trivially. The method for doing so has been described elsewhere [@gst2], so only the salient results will be presented below.

The $\tau_1 = 0, \theta \rightarrow -\theta$ invariance of $H_\tau$ suggests that the solutions of the Schrodinger equation be split into even and odd functions, and the primitive basis set can be taken to possess this property; $$u_n(\theta) = {1 \over \sqrt
\pi} {\rm cos}[n\theta], \qquad v_n(\theta) = {1 \over \sqrt \pi}
{\rm sin}[n\theta].$$ The GS functions will take the form $$\psi^{\pm}_{K}(\theta) = \sum_{m}c^{\pm}_{Km} \left ( \begin{array}{c} u_m(\theta) \\
v_m(\theta) \end{array} \right)$$ with the $c_{Km}$ given by (momentarily supressing the parity superscripts [@parnote]) $$c_{Km}=(-)^{K+m}N_K(N_{K-1}\beta N_{K-1})(N_{K-2}\beta
N_{K-2})...(N_{m}\beta N_{m})$$ and the normalization factors $N_K$ determined from $$N^2_{k+1}={1 \over {1-\beta^2 N^2_{k}}}$$ starting from $N_0  = \sqrt{1/2}$ for positive parity states and $N_1 = 1$ for negative parity states. The $K\nu^{th}$ basis state is attained by appending azimuthal eigenfunctions onto the GS functions described above, $$\Psi^{\pm}_{K\nu}(\theta,\phi) = {1 \over \sqrt {2
\pi}}\sum_{m}c^{\pm}_{Km} \left ( \begin{array}{c} u_m(\theta) \\
v_m(\theta) \end{array} \right)
 e^{i\nu\phi}.$$ The matrix $$H_{\tau \bar{K} K \bar{\nu} \nu}^{pq}= \big < \bar{K}^{\pm}
\bar{\nu} |H_\tau| K^{\pm} \nu \big >$$ is then easily constructed since the matrix elements can all be written in closed form (see the Appendix), and the eigenvalues a