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Chemical MFEs arising from the RPM depend critically on the manner in which the overall RP spin

state evolves. They derive from a detailed interplay of the interactions of the unpaired electron spins and

of the kinetics of the radical reactions. We discuss first the electron spin evolution.

The unpaired electron spins and their selective reactivity are inherently quantum mechanical. In

full generality, we ought to describe them by a wavefunction Ψ(ri,si,t) evolving under the influence of

a Hamiltonian H ˆ (t), where ri and si are the spatial and spin coordinates of the ith electron. Of course,

the unpaired electrons would interact with all the other, paired electrons in each radical giving rise to

electron correlation effects. Solving the Schrödinger equation for an RP described at this level of detail

would be a formidable undertaking. Even after making the Born–Oppenheimer approximation, we

would need to complete a full time-dependent quantum chemical calculation for every diffusive step. It

is doubtful whether such a calculation would be feasible for any radicals of chemical interest.

Fortunately, it is not necessary to work in such generality. As in so much of science, the key to

developing a useful theory of MFEs is to keep things as simple as possible, making as many simplifying

assumptions as are permitted by experimental data. The first, and most essential, approximation that

we make is to treat the time evolution of the electron spin and spatial coordinates separately. We do this

by converting the full Hamiltonian into a spin Hamiltonian, which contains a number of empirical parameters,

and the full wavefunction into a wavefunction containing only spin variables [38,39]. Since the

spin Hamiltonian approach is ubiquitous in magnetic resonance, the form of the contributions arising

from different magnetic interactions are known (see [40,41]). Furthermore, modern quantum chemical

methods can calculate accurately many of the spin Hamiltonian parameters that we require.

state evolves. They derive from a detailed interplay of the interactions of the unpaired electron spins and

of the kinetics of the radical reactions. We discuss first the electron spin evolution.

The unpaired electron spins and their selective reactivity are inherently quantum mechanical. In

full generality, we ought to describe them by a wavefunction Ψ(ri,si,t) evolving under the influence of

a Hamiltonian H ˆ (t), where ri and si are the spatial and spin coordinates of the ith electron. Of course,

the unpaired electrons would interact with all the other, paired electrons in each radical giving rise to

electron correlation effects. Solving the Schrödinger equation for an RP described at this level of detail

would be a formidable undertaking. Even after making the Born–Oppenheimer approximation, we

would need to complete a full time-dependent quantum chemical calculation for every diffusive step. It

is doubtful whether such a calculation would be feasible for any radicals of chemical interest.

Fortunately, it is not necessary to work in such generality. As in so much of science, the key to

developing a useful theory of MFEs is to keep things as simple as possible, making as many simplifying

assumptions as are permitted by experimental data. The first, and most essential, approximation that

we make is to treat the time evolution of the electron spin and spatial coordinates separately. We do this

by converting the full Hamiltonian into a spin Hamiltonian, which contains a number of empirical parameters,

and the full wavefunction into a wavefunction containing only spin variables [38,39]. Since the

spin Hamiltonian approach is ubiquitous in magnetic resonance, the form of the contributions arising

from different magnetic interactions are known (see [40,41]). Furthermore, modern quantum chemical

methods can calculate accurately many of the spin Hamiltonian parameters that we require.

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