## Thursday, 23 July 2009

### Marcus-Levich-Hush equation

Here is an interesting piece of math:

$\lim_{T\rightarrow 0}\frac{H_{AD}^{2}}{\hbar}\sqrt{\frac{\pi}{\lambda R T}} \exp \left( \frac{-(\Delta G^o+\lambda)^{2}}{4 \lambda R T} \right) =0$

$\lambda >0,\Delta G^o+\lambda \neq 0$

Or in other words show that in the Marcus-Levich-Hush theory, electron transfer stops when you approach zero temperature. Mathematica tells me that this is the case but I could not really show it. The problem has the following type:

$\lim_{x\rightarrow 0}\sqrt{\frac{1}{x}} \exp \left( \frac{-1}{x} \right)=0$

It is not an apparent l'Hospital application and I do not really know what else to do. Anyway, I believe Mathematica and it is also the thing you'd expect physically.

In the full quantum picture you actually have a non-zero transfer rate even when approaching zero temperature that comes from nuclear tunneling (as was derived here and is reviewed here).

Interestingely it is quite easy to find derivations of this quantum formula. But I did not find much about the semi-classical formula, that is shown above. Only that it is some kind of application of Fermi's golden rule which apparently gives this prefactor for the rate equation.

Echiral said...

The formula looks cool. As for the limit, let 1/sqrt(x) = u and go from there.

Felix said...

oh, yes then it works :)
thanks

Bejugam Vinith said...
This comment has been removed by the author.
Felix said...

hi, i think you could do some molecular modelling on the side. it is almost like playing computer games ;)

there is lots of very nice free software around. you could check out my category: http://chemical-quantum-images.blogspot.com/search/label/chemical%20software
in particular: http://chemical-quantum-images.blogspot.com/2007/04/molecular-modelling-tutorial.html

and this blog: http://molecularmodelingbasics.blogspot.com/

Bejugam Vinith said...

Thank you so much

northbeggar said...

i like mathematical modelling but not very good in using the software.