Two of the Hermitian eigenvalue equations resulting from the separation of the three-dimensional Schroedinger equation for H 2 + in spheroidals are solved perturbatively for the ground state by expanding the action in positive powers of the internuclear distance R near the united atom He + . The dispersion relations between the separation constants A and E e are seen to have rigorous analytic solutions, the third-order equation leading to an exact expansion for the inner determinantal equation up to R 10 . The explicit form for the expansion coefficients is determined up to n = 10, and is seen to contain up to the third power of (γ + ln 4R) logarithmic terms. Even if the general range of validity of the short-range R n -expansion is expected to be smaller than the corresponding long-range R −n -expansion, it is important to stress that such higher expansion coefficients are calculated exactly for the first time. These formulae give extremely accurate numerical results up to R ∼ = 0.3a 0 .

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An exact calculation of the diffusion controlled first-order intramolecular rate constant k1 is given for the harmonic spring model and for any type of reaction sink in the frame work of the Wilemski–Fixman theory. For a one parameter spherically symmetric sink the resulting exact expansion for k−11 in terms of the sink parameter γ, k−11 = (√π/2Λγ)+(ln2−1)/Λ+0.5(√πγ/Λ)+⋅⋅⋅, gives the same γ→0 behavior of other known asymptotic calculations obtained for simpler diffusion controlled reaction theories. Also, the mean first passage time for end-to-end contact may be evaluated and results coincident to the third term in the expansion with the exact result given by the first passage time approach theory. These results support the vicinity of the Wilemski–Fixman closure approximation.

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