F Second Order Co Tunneling Rate at Zero Temperature

At zero temperature the Fermi functions are either zero or one and may therefore be moved in the boundaries of the integrals.

The integration over is eliminated by the -function. The condition demanded by the -function and the integration regions of give the inequalities

for the new upper boundaries of integration.

Integrating over results in

Substituting gives

The integrand can be changed to a partial sum

And finally collecting the logarithms yields