eads the torus. Fig. 3 is analogous to Fig. 1 as described above with the notable exception that the $\nu$ splitting is non-trivial (hence fewer distinct states are shown), and level crossings occur at larger values. There is no plot analogous to Fig. 2 for larger values of $\tau_1$ with $\tau_0 = 0$ because $\phi$ dependence increases rapidly in the eigenstates even for small $\tau_1$; a contour plot is preferable. Figs. 4 and 5 show contour plot results for two states at three field strengths. Note that there is slightly more dependence in $\theta$ when $\tau_1 = 0$ for the state displayed in Fig. 6 than in Fig. 5; the integration measure acts to cancel the angular variation of the state displayed in Fig. 5.

Extension to Coulomb integrals
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The two-electron problem on $T^2$ is complicated by the inability (at least by the author) to find a transformation that decouples the relative electron motion from their center of mass motion as is easily done on $R^2$ [@qdhe]. The obvious transformations do not lead to any advantage over the method adopted by workers long used in atomic and molecular physics, which is to evaluate the two-body matrix elements (supressing spin indices and physical constants) $$\int \int \Phi^*_i({\bf r}_1)\Phi^*_j({\bf
r}_2)V({\bf r}_1,{\bf r}_2)(1-P_{12})\Phi_k({\bf r}_1)\Phi_l({\bf
r}_2)d^3{\bf r}_1d^3{\bf r}_2$$ with $$V({\bf r}_1,{\bf r}_2) = 4\pi\sum_{L,M}{1\over 2L+1} {r^l_{<}
\over r^{l+1}_{>}}
Y^*_{LM}(\theta_1,\phi_1)Y^*_{LM}(\theta_2,\phi_2).$$ Eq. (20) can be adopted on $T^2$ subject to some peculiarities which are due to the restriction of ${\bf r}_1,{\bf
r}_2$ to a surface. Eq. (20) on $T^2$ with the notation employed in section 3 becomes $$\notag
 \int_0^{2\pi}... \int_0^{2\pi}
\Psi^*_{P{\nu_1}}(\theta_1,\phi_1)\Psi^*_{Q{\nu_2}}(\theta_2,\phi_2)V({\bf
r}_1,{\bf r}_2)(1-P_{12})$$ $$\Psi_{R{\nu_3}}(\theta_1,\phi_1)\Psi_{S{\nu_4}}(\theta_2,\phi_2)F(\theta_1)F(\theta_2)d\theta_1
 d\theta_2 d\phi_1 d\phi_2.$$ Consider the direct term; in terms of a sph