Magnetic field dependence of CIDNP using delta function pseudopotentials in a stochastic Liouville equation

Abstract

A stochastic Liouville radical pair density matrix equation using short-range electron-electron interaction delta function pseudopotentials has been solved by treating the parameters describing radical recombination, λ _K , and Heisenberg exchange, λ _J , as perturbations. The first-order nuclear spin density matrix of a combination product is found to be independent of λ _J for all magnetic fields, in agreement with the conclusions of Adrian's re-encounter averaging approach. The nuclear spin density matrix of a scavenging product is shown to differ in the delta function and Adrian models, however, even to first order in λ _J and λ _K . Analytical expressions have been derived for the nuclear spin level populations in a combination product of the A _N B _M type (where A and B reside on different fragments) in the realistic limiting case where the effective radical pair hamiltonian does not contain explicit interradical interactions. It is shown that the oscillation in the sign of enhancement at low field is a property of the number of equivalent nuclear spins and does not require inclusion of electron-electron interactions. Excellent agreement is obtained between the calculated and observed field dependence of CIDNP from pivaldehyde and chloroform formed as combination products during the photolysis of di-t-butyl ketone in carbon tetrachloride. The influence of chlorine spins in the trichloromethyl radical is shown to be relatively minor in the latter product. The magnetic field dependence of the CIDNP intensity of scavenging product is derived analytically for the case of a radical pair with equal g-factors and a single I = ½ spin in the limit where λ _J , λ _K ≪ 1 and scavenging rates are slow. It is found in this case that the sum of the CIDNP intensities from the combination and scavenging products, I _{z, net}, vanishes at high field but is dominated by the scavenging product at low fields B ∼ A, even when λ _J = 0. Limiting expressions for I _{z, net} are given for B ≫ ≪ A. Expressions are given for the zero field N - 1 multiplet effect for an AX spin system in a combination product for the cases where the radical pair ‘lifetime’ is dominated by a first-order process (the CKO limit) and by diffusion. While the former case leads to Kaptein's qualitative low-field rule, the more realistic latter case does so only when A _A and A _X differ by no more than 0·2 per cent. A general expression is given for the total yield of a combination product derived from an A _N … B _M radical pair. It is estimated, in agreement with previous qualitative predictions, that ‘nuclear spin reaction rate isotope effects’ of the order of 1 per cent are expected for reactions run under the usual CIDNP conditions and that they should exhibit a strong dependence on magnetic field strength.