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Theory and implementation

QHA is a method for obtaining effective modes of vibration from fluctuations calculated by a MD simulation. The underlying principle is that from atomic fluctuations, an effective force field can be calculated relative to the average dynamic structure that yields the same fluctuation matrix as that obtained from a normal mode calculation. Since the fluctuation matrix is inversely proportional to the effective force constant matrix, they have common eigenvectors which correspond to the quasiharmonic modes of vibration. Quasiharmonic modes can be analyzed in the same way as normal modes, and comparison of the results with those from harmonic approximation calculations for the same system is straightforward.

The way to perform a QHA in nMOLDYN is based on the diagonalization of the fluctuation matrix that can be easily retrieved from a MD simulation. Indeed, from a MD simulation, coordinates which define the position of all atoms as a function of time are saved at each step of the MD. From these coordinates, both the average position $\langle \mathbf{x} \rangle$ and the covariance matrix of fluctuations about the average position $\sigma$ can be calculated the latter being defined as:

\begin{displaymath}
\sigma_{ij} = \left\langle \left ( x_{i} - \langle x \rangle \right ) \left ( x_{j} - \langle x \rangle \right )\right\rangle
\end{displaymath} (4.101)

with variances as the diagonal elements and with covariances as the off-diagonal elements.

Once the covariance matrix of fluctuations is obtained, the Quasi-Harmonic modes of vibration $\Delta x$ and their corresponding frequencies $\omega$ are calculated by solving (diagonalizing) the equation:

\begin{displaymath}
\left ( \mathbf{\sigma^{\prime}} - \lambda^{\prime} \mathbf{I}\right )\mathbf{\eta} = 0
\end{displaymath} (4.102)

where $\mathbf{I}$ is the identity matrix and
\begin{displaymath}
\mathbf{\sigma^{\prime}} = \mathbf{M^{1/2}} \mathbf{\sigma} \mathbf{M^{1/2}}
\end{displaymath} (4.103)

$\mathbf{M}$ being the diagonal mass matrix.

The solutions of 4.102 yielding:

\begin{displaymath}
\omega = \left (k_{B}T/\lambda^{\prime} \right )^{1/2}
\end{displaymath} (4.104)

and
\begin{displaymath}
\mathbf{\Delta x} = \mathbf{M}^{-1/2}\mathbf{\eta}
\end{displaymath} (4.105)

Once the normal modes have been obtained, a great variety of analysis can be performed. nMOLDYN proposes some of them namely:

Beside the quantitative analysis described above, one of the best way to understand a normal mode of vibration is through a visual examination. To do this, a trajectory must be created which depicts the molecular system as a function of time from a given starting configuration. For a set of normal modes to view, this trajectory is given by:

\begin{displaymath}
\textbf{x}(t) = \langle \textbf{x} \rangle + \sum_{i=1}^{N_{modes}} \alpha_i M^{-\frac{1}{2}}\Delta x_i cos(\omega_i t)
\end{displaymath} (4.109)

where $\alpha_i$ and $\omega_i$ are respectively the amplitude and frequency of the motion associated to normal mode $i$ and $N_{modes}$ is the number of selected normal modes to visualize. nMOLDYN allows the visualization of normal modes through a specific viewer (see Section 4.3.3).

For more details about QHA and related techniques please refer to Ref. [53]


next up previous contents
Next: Parameters Up: Quasi Harmonic Analysis Previous: Quasi Harmonic Analysis   Contents
pellegrini eric 2009-10-06