| 321 |
|
|
| 322 |
|
\begin{figure} |
| 323 |
|
\centering |
| 324 |
< |
\includegraphics[width = 3.25in]{./nModeFTPlotDot.pdf} |
| 325 |
< |
\caption{Power spectra obtained from the velocity auto-correlation functions of NaCl crystals at 1000 K while using SPME, Shifted-Force ($\alpha$ = 0, 0.1, \& 0.2), and Shifted-Potential ($\alpha$ = 0.2). Apodization of the correlation functions via a cubic switching function between 40 and 50 ps was used to clear up the spectral noise resulting from data truncation, and had no noticeable effect on peak location or magnitude. The inset shows the frequency region below 100 cm$^{-1}$ to highlight where the spectra begin to differentiate.} |
| 324 |
> |
\includegraphics[width = 3.25in]{./spectraSquare.pdf} |
| 325 |
> |
\caption{Power spectra obtained from the velocity auto-correlation functions of NaCl crystals at 1000 K while using SPME, Shifted-Force ($\alpha$ = 0, 0.1, \& 0.2), and Shifted-Potential ($\alpha$ = 0.2). Apodization of the correlation functions via a cubic switching function between 40 and 50 ps was used to clear up the spectral noise resulting from data truncation, and had no noticeable effect on peak location or magnitude. The inset shows the frequency region below 100 cm$^{-1}$ to highlight where the spectra begin to differ.} |
| 326 |
|
\label{fig:methodPS} |
| 327 |
|
\end{figure} |
| 328 |
|
|
| 333 |
|
where $S(r)$ is a switching function that smoothly zeroes the potential at the cutoff radius. Figure \ref{fig:dampInc} shows how the low frequency motions are dependent on the damping used in the direct electrostatic sum. As the damping increases, the peaks drop to lower frequencies. Incidentally, use of an $\alpha$ of 0.25 \AA$^{-1}$ on a simple electrostatic summation results in low frequency correlated dynamics equivalent to a simulation using SPME. When the coefficient lowers to 0.15 \AA$^{-1}$ and below, these peaks shift to higher frequency in exponential fashion. Though not shown, the spectrum for the simple undamped electrostatic potential is blue-shifted such that the lowest frequency peak resides near 325 cm$^{-1}$. In light of these results, the undamped Shifted-Force method producing low-lying motion peaks within 10 cm$^{-1}$ of SPME is quite respectable; however, it appears as though moderate damping is required for accurate reproduction of crystal dynamics. |
| 334 |
|
\begin{figure} |
| 335 |
|
\centering |
| 336 |
< |
\includegraphics[width = 3.25in]{./alphaCompare.pdf} |
| 337 |
< |
\caption{Normal modes for an NaCl crystal at 1000 K when using SPME and a simple damped Coulombic sum with damping coefficients ($\alpha$)ranging from 0.15 to 0.3 \AA$^{-1}$. As $\alpha$ increases, the peaks are red-shifted toward and eventually beyond the values given by SPME. The larger $\alpha$ values weaken the real-space electrostatics, explaining this shift towards less strongly correlated motions in the crystal.} |
| 336 |
> |
\includegraphics[width = 3.25in]{./comboSquare.pdf} |
| 337 |
> |
\caption{Upper: Zoomed inset from figure \ref{fig:methodPS}. As the damping value for the Shifted-Force potential increases, the low-frequency peaks red-shift. Lower: Low-frequency correlated motions for NaCl crystals at 1000 K when using SPME and a simple damped Coulombic sum with damping coefficients ($\alpha$) ranging from 0.15 to 0.3 \AA$^{-1}$. As $\alpha$ increases, the peaks are red-shifted toward and eventually beyond the values given by SPME. The larger $\alpha$ values weaken the real-space electrostatics, explaining this shift towards less strongly correlated motions in the crystal.} |
| 338 |
|
\label{fig:dampInc} |
| 339 |
|
\end{figure} |
| 340 |
|
|
