Normal and Phonon Modes

A simple explanation of:

• what normal and phonon modes are

• how to compute them

• how to “project” a Molecular Dynamics trajectory onto these modes.

We will first give a mathematical derivation of the normal modes of a molecule. Later on, we will extend this concept to phonon modes of a crystal with a minimal effort.

Normal Modes

The Harmonic Approximation

Let’s consider the system moving around a minimum of the Potential Energy Surface (PES).

The nuclear coordinates of the system will be indicated with \(\mathbf{R}_{eq} = \left( R^1_x, R^1_y, R^1_z, R^2_x, \dots , R^N_z \right)\), which is a vector containing all the degrees of freedom (nuclear coordinates) of the sysyem.

For the sake of simplicity we will indicate the displacement on the nuclei w.r.t. \(\mathbf{R}_{eq}\) just with \(\mathbf{R}\) instead of \(\Delta \mathbf{R} = \mathbf{R} - \mathbf{R}_{eq}\)

Let’s express the potential energy of the system within the harmonic approximation around the equilibrium positions \(\mathbf{R}_{eq}\).

\[U \approx U_{\rm harm } = \frac{1}{2} \mathbf{R}^T \cdot \Phi \cdot \mathbf{R}\]

where \(\Phi\) is called “force constants matrix” and is defined as:

\[\Phi_{ij} = \left. \frac{\partial^2 U}{\partial \mathbf{R}_i \partial \mathbf{R}_j }\right|_{\mathbf{R}=\mathbf{R}_{eq}}\]

where the indices \(i,j\) runs over all the degrees of freedoms, i.e. \(i,j\in[1,\dots,3N]\).

This means that \(\Phi\) is a \(3N\times3N\) matrix.

Note

For reference, usually the harmonic potential energy and the force constant matrix are expressed in the following way (equivalent but more complicated way):

\[\begin{split}\begin{aligned} U_{\text{harm}} & = \frac{1}{2} \sum_{IJ}^{N_{n}} \sum_{\alpha,\beta=x,y,z} \Delta \mathbf{R}^I_\alpha \cdot \Phi_{IJ}^{\alpha\beta} \cdot \Delta \mathbf{R}^J_\beta \\ \Phi_{IJ}^{\alpha\beta} & = \left. \frac{\partial^2 U}{\partial \mathbf{R}^I_\alpha \partial \mathbf{R}^J_\beta}\right|_{\mathbf{R}=\mathbf{R}_{eq}} \end{aligned}\end{split}\]

where the indices \(I,J\in[1,\dots,N]\) label the nuclei and the indices \(\alpha,\beta\in\left\{x,y,z\right\}\) label the Cartesian direction.

However, the previous expression using the indices \(i,j\) are equivalent but much more compact.

We want to find the solutions of the (Newton) equations of motion of following linearized/harmonic hamiltonian:

\[\begin{split}\mathcal{H} = & T + U_{\rm harm } \\ = & \frac{1}{2} \mathbf{v}^T \cdot \mathbf{M} \cdot \mathbf{v} + \frac{1}{2} \mathbf{R}^T \cdot \Phi \cdot \mathbf{R}\end{split}\]

where the kinetic energy \(T\) has been expressed using the same convention previously adopted for the displacement and the potential energy, and \(\mathbf{M}\) is a diagonal \(3N\times3N\) matrix containing the masses of all the atoms (repeated 3 times):

\[{\rm diag} \, \mathbf{M} = \left( M^1, M^1, M^1, M^2, \dots , M^N \right)\]