An improved semiclassical theory of radical pair recombination reactions

We present a practical semiclassical method for computing the electron spin dynamics of a radical in which the electron spin is hyperfine coupled to a large number of nuclear spins. This can be used to calculate the singlet and triplet survival probabilities and quantum yields of radical recombination reactions in the presence of magnetic fields. Our method differs from the early semiclassical theory of Schulten and Wolynes [J. Chem. Phys. 68, 3292 (1978)] in allowing each individual nuclear spin to precess around the electron spin, rather than assuming that the hyperfine coupling-weighted sum of nuclear spin vectors is fixed in space. The downside of removing this assumption is that one can no longer obtain a simple closed-form expression for the electron spin correlation tensor: our method requires a numerical calculation. However, the computational effort increases only linearly with the number of nuclear spins, rather than exponentially as in an exact quantum mechanical calculation. The method is therefore applicable to arbitrarily large radicals. Moreover, it approaches quantitative agreement with quantum mechanics as the number of nuclear spins increases and the environment of the electron spin becomes more complex, owing to the rapid quantum decoherence in complex systems. Unlike the Schulten-Wolynes theory, the present semiclassical theory predicts the correct long-time behaviour of the electron spin correlation tensor, and it therefore correctly captures the low magnetic field effect in the singlet yield of a radical recombination reaction with a slow recombination rate.