An exact model calculation of the diffusion controlled intramolecular rate constant

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Abstract
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.