## Thursday 1 January 2009

### Normal modes (3)

Ok finally comes the part that I wanted to show. But first a little jmol example to provide you with something to play with.

What you see are the normal modes and frequencies determined as I will describe below. They are pretty much what you would expect from IR spectroscopy. Only that they are a little bit higher which is mostly due to the fact that the harmonic approximation is not quite correct. [1]

You can rotate the molecule by dragging with the left mouse button and switch between the vibrations with the right mouse button and changing the "model".

What we want to do is solve this coupled system of differential equations:

The idea is that if we bring it do diagonal form, the equations are uncoupled. So what can we do? One way would be

But the problem now is that M-1H is unlike H not a Hermitian matrix and we do not know if it can be diagonalized. But you can rewrite it as:

Or if you introduce (square root) mass weighted coordinates S(t) it looks like this.

HS is actually the Hessian matrix in mass-weighted coordinates. The important point is that it is Hermitian and hence an orthogonal basis of its eigenvectors exists, i.e. there is an orthogonal matrix Q and diagonal matrix D with the property

Plug this in, multiply with Q and that's actually the result.

The idea is now that you have uncoupled collective coordinates and you can solve each line of this equation system separately and get some cos function each like in the first post. The eigenvectors of HS (the columns of Q) are the corresponding motions. And the eigenvalues determine the frequencies.

[1] The lowest vibration is kind of out of place. With DFT/B3LYP cytosine is planar at its minimum geometry and the NH2 pyramidalization has a very weak force constant. Actually with MP2 and some other ab-initio methods the amino group is pyramidalized at the minimum. So I would guess that DFT is wrong. It is not a big issue but you can see that DFT is sometimes a problem and it makes sense to check the results a little bit.

Anonymous said...

I've seen something similar w/ DFT vs MP2,it was that DFT also preferred to planarize an amino group in the 1 position in an 9,10 anthraquinone.. In my case i wasn't sure if DFT was giving the result to improve conjugation or the H-Bonding..MP2 slightly pyramidalized the amino group..Do you think this DFT result is general? maybe it is b/c it likes to delocalize more?

Felix said...

that's interesting that you had the same result in anthraquinone.
delocalization sounds good but i don't know

Unknown said...

Have you tried the old faithful hartree fock? The cause of delocallisation in DFT is usually a bad approximation of the exchange energy not cancelling the electron self interaction energy in the coulomb term.

If its wrong in HF, its not wrong because of the DFT propensity to give delocallised states, and we can assign it to the other blanket term "correlation"... not that doing so really helps your results that much.

Felix said...

i just tried hartree-fock. and with hartree fock it is even closer to planarity with only a 0.1° torsion angle (the corresponding torsion is 1.2° in B3LYP and 13° with MP2)

what I was trying to say is that every method has to be used with some care. and that the practice of just using b3lyp without any further checking can be problematic