Spin-Correlation Following Cross Recombination

Supposing radical pairs 1, 2 and 3,4 are spin correlated, then the spin-dynamics of correlated pair 1,2 can be described by wavefunction ijfn and correlated pair (3,4 by wavefunction t/r34. Each individual one-pair spin wavefunction evolves independently according to its Hamiltonian. Following a cross recombination, it is necessary to construct the total wavefunction V tot = V i2V 34 and collapse the wave-function t/f(ot onto either a singlet or triplet state depending on the spin-dynamics of the encountering pair. It is then necessary to extract the coefficients for the surviving radical pair and reconstruct the one pair wavefunction. From hereon, this is referred to as the decomposition method. 

Although a simple two pair spur can be simulated entirely using tot, it becomes necessary to use the decomposition method for a large number of radical pairs because of the increasing size of the basis set. 

Singlet encounter Assuming the encounter of the disjoint radical pair 1,3 is found to be in a singlet state, the algorithm constructs the one pair wavefunction for the radical pair 2,4 by first collapsing V tot in which the radical pair 1,3 is in a pure singlet state. V tot can be expressed more formally as 

The second step involves constructing V 24, by extracting all the coefficients from V tot that contain the state a1fi3) (with subscripts representing the spin on the radical) and populating back the one pair basis set. Obviously the state fi1a3) could also be chosen, in which case the coefficients would be equal but opposite. 

Triplet encounter For a triplet encounter, the process of extracting the coefficients is similar to that described above. At the moment of encounter the wavefunction collapses onto a triplet state and the probability of each triplet substate is calculated from the coefficients of the wavefunction. A uniformly distributed random number is generated between (0,1] which is used to decide in which triplet substate the encountering pair is formed. Next, the coefficients belonging to the encountering pair from V tot are used to repopulate the coefficients of the disjoint pair. 

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