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 n

_{occ}x n

_{virt}(numbers of occupied and virtual orbitals) matrix C that represents the weights of all possible excitations.

where c

_{ij}represents the weight of the component where excitation goes from the i

^{th}occupied to the j

^{th}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(n

_{occ}, n

_{virt}) 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 λ

_{i}deviates much from zero. And the corresponding MOs are localised on the chromophores.

Give it up for linear algebra!

For more information read I. Mayer's article.

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

## 2 comments:

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

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

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