Monday, 8 December 2008

Normal modes

I am kind of excited because I finally understand how normal mode analysis works. Finding the normal modes and corresponding frequencies (and intensities) of a molecule is what you need to predict or interpret IR/Raman data. Like so many other things it comes down to eigenvalues and eigenvectors.

We start out with Newton's second law

The first consideration is the one dimensional oscillator. We have one variable x, and the force defined according to:

This leads us to

In other words we are looking for a function whose second derivative is the same function with a minus ...
How about sine or cosine?

The following satisfies this equation. The proof is just differentiating it twice.

It is not quite so obvious that with two real numbers A and δ this is also the complete real solution of the differential equation (where A is the amplitude and δ the phase) but also not so important in this case.

The part that has an immediate application is which gives the frequency. k is the force constant and it increases according to: torsion < bend < stretch (single < double < triple). m is the mass. The highest frequencies are for low mass and high force constant, i.e. X-H stretch. Triple bond stretches are high because of high k, then double bond stretches and so on.

This was the one-dimensional case. Next time I want to show how to reduce the general case to isolated one-dimensional equations like shown here. And that's where the Linear Algebra comes in.

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