Normal modes (2)

Now I am used to the fact that almost everything in theoretical chemistry comes down to eigenvalues and -vectors. But it kind of struck me when I first heard that about normal modes some time ago. In short: normal mode frequencies are the eigenvalues and normal modes the eigenvectors of the Hessian matrix of the energy in mass-weighted coordinates.

Here's the math behind it because it's always fun to make some formulas in LaTex.

We look at the effective potential energy V(R) for the nuclei of a molecule with N atoms in the Born-Oppenheimer picture [1] which is a function of the nuclear positions R=(x₁, y₂, ..., z_N)^T. This function is expanded into a Taylor-Series up to second order which you can write like this (where a suitable origin 0 is chosen):

The first term is called the gradient, defined according to

The second term, the Hessian matrix

That was actually just kind of a warm up as we are interested in the energy gradient which we can expand to first order in the following way:

With normal mode analysis we can describe the motions at a local minimum. At a local minimum the gradient is 0 and all the eigenvalues of the Hessian are greater or equal to zero. The first condition gives:

Now we have a convenient description of the gradient that we can plug into Newton's second axiom. We get a differential equation system with an equation for every coordinate. This can of course be represented by a matrix equation. First it is convenient to introduce a diagonal matrix M that contains the masses (each of them 3 times because there are x,y, and z coordinates for every atom):

Then the equation system looks like this:

If you write out the matrix equations then you have a system of coupled ordinary differential equations. And actually the part that I wanted to show is how you can uncouple this system by taking proper care of the mass and diagonalizing the Hessian. Well next time ...

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[1] That means electronic kinetic energy and all potential energy terms.

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.