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Herein, we present analytical solutions for the electronic energy eigenvalues of the hydrogen molecular ion H2+, namely the one-electron two-fixed-center problem. These are given for the homonuclear case for the countable infinity of discrete states when the magnetic quantum number m is zero. In this case, these solutions are the roots of a set of two coupled three-term recurrence relations. The eigensolutions are obtained from an application of EXPERIMENTAL MATHEMATICS using Computer Algebra as its principal tool and are vindicated by numerical and algebraic demonstrations. Finally, the mathematical nature of the eigenenergies is identified. The eigenenergies are related to a generalization of the Lambert W function., This is an analytical breakthrough for a special case of the quantum 3-body problem. The results have been published in Chem. Phys. and an internal report at the Forschungszentrum Juelich (Germany). (see references)

We have analyzed and reduced a general (quantum-mechanical) expression for the atom-atom exchange energy formulated as a five-dimensional surface integral, which arises in studying the charge exchange processes in diatomic molecules. It is shown that this five-dimensional surface integral can be decoupled into a three-dimensional integral and a two-dimensional angular integral which can be solved analytically using a special decomposition. Exact solutions of the two-dimensional angular integrals are presented and generalized. Algebraic aspects, invariance properties and exact solutions of integrals involving Legendre and Chebyshev polynomials are also discussed.

We present a general procedure. based on the Holstein-Herring method. for calculating exactly the leading term in the exponentially small exchange, energy splitting between two asymptotically degenerate states of a diatomic molecule or molecular ion. The general formulae we have derived are shown to reduce correctly to the previously known exact results for the specific cases of the lowest Sigma and Pi states of H-2(+). We then apply our general formulae 2 to calculate the exchange energy splittincys between the lowest states of the diatomic alkali cations K-2(+), Rb-2(+) and Cs-2, which are isovalent to H-2(+). Our results are found to be in very good agreement with the best available experimental data and ab initio calculations.