## Friday 28 September 2007

### CESTC 2007

CESTC 2007 is over. It was pretty nice, of course only for scientific reasons ... Anyway what I am going to talk about is only the scientific part. I want to cover a few lectures during the next days.

The first one has the advantage of being mostly mathematical and you don't need to know a lot about theoretical chemistry, but you need linear algebra. I always thought that in theoretical chemistry everything was hermitian or unitary and you didn't need to worry about singular value decomposition. But I guess I don't have an excuse anymore to skip that in my linear algebra script when I will do the exam.

It goes like this: When using configuration interaction singles we have to use a nocc x nvirt (numbers of occupied and virtual orbitals) matrix C that represents the weights of all possible excitations.

where cij represents the weight of the component where excitation goes from the ith occupied to the jth virtual orbital.

This is not a very handy information especially when you try to visualise the orbitals.

But we know that for every real matrix orthogonal matrices U and V exist with the following property:

with

We only have to consider n=min(nocc, nvirt) excitations any more. The orbitals they are taken from and given to, are the columns in U and V respectively. Typically only one singular value λi deviates much from zero. And the corresponding MOs are localised on the chromophores.

Give it up for linear algebra!

If you want some homework, you can think about how this works for a 4-dimensional CISD excitation tensor.

Peter Surjan said...

This is interesting that you picked up it! Congratulations! Here are some
follow-up works:

P. R. Surjan, Natural orbitals in CIS and singular-value decomposition
Chem.Phys.Letters 439 393–394 (2007)

and

I. Mayer, Chem.Phys.Letters, 443 420 (2007)

Best,
Peter Surjan

Felix said...

i kind of like this idea of using some abstract math where i did not think it had much practical application
your paper seems interesting. i'll take a look at it when i have some time