Molecular vibrations

Let’s calculate the vibrational modes of H₂O.

Consider the molecule at its equilibrium positions. If we displace the atoms slightly, the energy \(E\) will increase, and restoring forces will make the atoms oscillate in some pattern around the equilibrium.

We can Taylor expand the energy with respect to the 9 coordinates (generally \(3N\) coordinates for a molecule with \(N\) atoms), \(u_i\):

\[E = E_0 + \frac{1}{2}\sum_{i}^{3N} \sum_{j}^{3N} \frac{\partial^2 E}{\partial u_{i}\partial u_{j}}\bigg\rvert_0 (u_i - u_{i0}) (u_j - u_{j0}) + \cdots\]

Since we are expanding around the equilibrium positions, the energy should be stationary and we can omit linear contributions.

The matrix of all the second derivatives is called the Hessian, \(\mathbf H\), and it expresses a linear system of differential equations

\[\mathbf{Hu}_k = \omega_k^2\mathbf{Mu}_k\]

for the vibrational eigenmodes \(u_k\) and their frequencies \(\omega_k\) that will characterise the collective movement of the atoms. In short, we need the eigenvalues and eigenvectors of the Hessian.

The elements of the Hessian can be approximated as

\[H_{ij} = \frac{\partial^2 E}{\partial u_{i}\partial u_{j}}\bigg\rvert_0 = -\frac{\partial F_{j}}{\partial u_{i}},\]

where \(F_j\) are the forces. Hence we calculate the derivative of the forces using finite differences. We need to displace each atom back and forth along each Cartesian direction, calculating forces at each configuration to establish \(H_{ij} \approx \Delta F_{j} / \Delta u_{i}\), then get eigenvalues and vectors of that.

ASE provides the Vibrations class for this purpose. Note how the linked documentation contains an example for the N₂ molecule, which means we almost don’t have to do any work ourselves. We just scavenge the online ASE documentation like we always do, then hack as necessary until the thing runs.

Exercise

Calculate the vibrational frequencies of H₂O using GPAW in LCAO mode, saving the modes to trajectory files. What are the frequencies, and what do the eigenmodes look like?

Since there are nine coordinates, we get nine eigenvalues and corresponding modes. However the three translational and three rotational degrees of freedom will contribute six “modes” that do not correspond to true vibrations. In principle there are no restoring forces if we translate or rotate the molecule, but these will nevertheless have different energies (often imaginary) because of various artifacts of the simulation such as the grid used to represent the density, or effects of the simulation box size.

A solution and other comments to this exercise can be found on the GPAW web page:

https://wiki.fysik.dtu.dk/gpaw/exercises/vibrations/vibrations.html