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.