137 ExSpherium

A quasi-exactly solvable model is one for which it is possible to solve the Schrödinger equation exactly for a finite portion of the energy spectrum [1]. In quantum chemistry, a famous example of this is the Hooke’s law atom [2–5], which consists of a pair of electrons, repelling Coulombically but trapped in a harmonic external potential. This model and others [6–12] have been used extensively to test various approximations [13–20] within density functional theory (DFT) [21–23] and explicitly correlated methods [24–28]. We have recently discovered [29] that a pair electrons, repelling Coulombically but constrained to remain on the surface of a D-sphere of radius R [17,30–37] is also quasi-exactly solvable and we have called this system D-spherium. (We adopt the convention that a D-sphere is the surface of a (Dþ 1)-dimensional ball.) We have shown that the Schrödinger equation for the S and the P states of D-spherium can be solved exactly for a countably infinite set of R values and that the resulting wave functions are polynomials in the interelectronic distance 0! u# jr1$ r2j! 2R. In this article, we extend our earlier results [29] to higher angular momentum (up to L1⁄4 3) states of D-spherium (D& 2) for both the singlet and triplet manifolds. The D1⁄4 1 case is anomalous and, for brevity, is not discussed here. We use atomic units throughout. 2. Wave function The Hamiltonian of D-spherium is Ĥ 1⁄4 $ 1 2 r 1 þ r 2 2 ! " þ 1 u , ð1Þ