Spin Evolution and Relaxation The Wavefunction Approach

To evolve the wavefunction with a certain Hamiltonian (which is time-independent), it is necessary to solve the time dependent Schrddinger equation of the form 

Using this expression for H, the time dependent Schrodinger equation becomes 

The chemical system under study requires a description of spin-relaxation on the hydroxyl radicals, as it is this parameter which is hypothesised to produce the observed polarisation phase. For the purpose of this work a new algorithm has been developed to model this effect using the wavefunction of the system (as first suggested by B. Brocklehurst, unpublished), which requires far fewer computational resources than a traditional density matrix calculation would typically utilise, allowing many more realisations to be computed (required to obtain acceptable statistics in the spin polarisation). This has not been previously attempted with a random flights or IRT simulation. 

To illustrate the algorithm, consider a wavefunction set up as 1 2 1 2) (with n, 2, s, 62 representing the nuclear and electron spins on radicals 1 and 2 respectively) 

In the random flights simulation, at time (t+St) arandom number 17(0, 1] is generated from a uniform distribution and is compared to (l-e / ), where Tju is the relaxation time of the radical i. If U (0, 1] (1 ), then spin relaxation takes 

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