| 286 |
|
|
| 287 |
|
\section{Sampling the CN bond frequency} |
| 288 |
|
|
| 289 |
< |
The vibrational frequency of the nitrile bond in 5CB is assumed to |
| 290 |
< |
depend on features of the local solvent environment of the individual |
| 291 |
< |
molecules as well as the bond's orientation relative to the applied |
| 292 |
< |
field. Therefore, the primary quantity of interest is the |
| 293 |
< |
distribution of vibrational frequencies exhibited by the 5CB nitrile |
| 294 |
< |
bond under the different electric fields. Three distinct methods for |
| 295 |
< |
mapping classical simulations onto vibrational spectra were brought to |
| 296 |
< |
bear on these simulations: |
| 289 |
> |
The vibrational frequency of the nitrile bond in 5CB depends on |
| 290 |
> |
features of the local solvent environment of the individual molecules |
| 291 |
> |
as well as the bond's orientation relative to the applied field. The |
| 292 |
> |
primary quantity of interest for interpreting the condensed phase |
| 293 |
> |
spectrum of this vibration is the distribution of frequencies |
| 294 |
> |
exhibited by the 5CB nitrile bond under the different electric fields. |
| 295 |
> |
Three distinct methods for mapping classical simulations onto |
| 296 |
> |
vibrational spectra were brought to bear on these simulations: |
| 297 |
|
\begin{enumerate} |
| 298 |
|
\item Isolated 5CB molecules and their immediate surroundings were |
| 299 |
< |
extracted from the simulations, their nitrile bonds were stretched |
| 299 |
> |
extracted from the simulations. These nitrile bonds were stretched |
| 300 |
|
and single-point {\em ab initio} calculations were used to obtain |
| 301 |
|
Morse-oscillator fits for the local vibrational motion along that |
| 302 |
|
bond. |
| 324 |
|
``body'' (the two phenyl rings and the nitrile bond) and ``tail'' (the |
| 325 |
|
alkyl chain). Any molecule which had a body atom within 6~\AA of the |
| 326 |
|
midpoint of the target nitrile bond had its own molecular body (the |
| 327 |
< |
4-cyano-4'-pentylbiphenyl moiety) included in the configuration. For |
| 328 |
< |
the alkyl tail, the entire tail was included if any tail atom was |
| 329 |
< |
within 4~\AA of the target nitrile bond. If tail atoms (but no body |
| 330 |
< |
atoms) were included within these distances, only the tail was |
| 331 |
< |
included as a capped propane molecule. |
| 327 |
> |
4-cyano-biphenyl moiety) included in the configuration. For the alkyl |
| 328 |
> |
tail, the entire tail was included if any tail atom was within 4~\AA |
| 329 |
> |
of the target nitrile bond. If tail atoms (but no body atoms) were |
| 330 |
> |
included within these distances, only the tail was included as a |
| 331 |
> |
capped propane molecule. |
| 332 |
|
|
| 333 |
|
\begin{figure}[H] |
| 334 |
|
\includegraphics[width=\linewidth]{Figure2} |
| 335 |
|
\caption{Cluster calculations were performed on randomly sampled 5CB |
| 336 |
< |
molecules from each of the simualtions. Surrounding molecular |
| 337 |
< |
bodies were included if any body atoms were within 6 \AA\ of the |
| 338 |
< |
target nitrile bond, and tails were included if they were within 4 |
| 339 |
< |
\AA. The CN bond on the target molecule was stretched and |
| 340 |
< |
compressed (left), and the resulting single point energies were |
| 341 |
< |
fit to Morse oscillators to obtain frequency distributions.} |
| 336 |
> |
molecules (shown in red) from each of the simulations. Surrounding |
| 337 |
> |
molecular bodies were included if any body atoms were within 6 |
| 338 |
> |
\AA\ of the target nitrile bond, and tails were included if they |
| 339 |
> |
were within 4 \AA. Included portions of these molecules are shown |
| 340 |
> |
in green. The CN bond on the target molecule was stretched and |
| 341 |
> |
compressed, and the resulting single point energies were fit to |
| 342 |
> |
Morse oscillators to obtain frequency distributions.} |
| 343 |
|
\label{fig:cluster} |
| 344 |
|
\end{figure} |
| 345 |
|
|
| 346 |
< |
Inferred hydrogen atom locations were generated, and cluster |
| 347 |
< |
geometries were created that stretched the nitrile bond along from |
| 348 |
< |
0.87 to 1.52~\AA at increments of 0.05~\AA. This generated 13 single |
| 349 |
< |
point energies to be calculated per gas phase cluster. Energies were |
| 350 |
< |
computed with the B3LYP functional and 6-311++G(d,p) basis set. For |
| 351 |
< |
the cluster configurations that had been generated with applied |
| 352 |
< |
fields, a field strength of 5 atomic units in the $z$ direction was |
| 353 |
< |
applied to match the molecular dynamics runs. |
| 346 |
> |
Inferred hydrogen atom locations were added to the cluster geometries, |
| 347 |
> |
and the nitrile bond was stretched from 0.87 to 1.52~\AA\ at |
| 348 |
> |
increments of 0.05~\AA. This generated 13 configurations per gas phase |
| 349 |
> |
cluster. Single-point energies were computed using the B3LYP |
| 350 |
> |
functional~\cite{Becke:1993kq,Lee:1988qf} and the 6-311++G(d,p) basis |
| 351 |
> |
set. For the cluster configurations that had been generated from |
| 352 |
> |
molecular dynamics running under applied fields, the density |
| 353 |
> |
functional calculations had a field of $5 \times 10^{-4}$ atomic units |
| 354 |
> |
($E_h / (e a_0)$) applied in the $+z$ direction in order to match the |
| 355 |
> |
molecular dynamics simulations. |
| 356 |
|
|
| 357 |
< |
The relative energies for the stretched and compressed nitrile bond |
| 358 |
< |
were used to fit a Morse oscillator, and the frequencies were obtained |
| 359 |
< |
from the $0 \rightarrow 1$ transition for the exact energies. To |
| 360 |
< |
obtain a spectrum, each of the frequencies was convoluted with a |
| 361 |
< |
Lorentzian lineshape with a width of 1.5 $\mathrm{cm}^{-1}$. Our |
| 362 |
< |
available computing resources limited us to 67 clusters for the |
| 363 |
< |
zero-field spectrum, and 59 for the full field. |
| 357 |
> |
The energies for the stretched / compressed nitrile bond in each of |
| 358 |
> |
the clusters were used to fit Morse oscillators, and the frequencies |
| 359 |
> |
were obtained from the $0 \rightarrow 1$ transition for the energy |
| 360 |
> |
levels for this potential.\cite{Morse:1929xy} To obtain a spectrum, |
| 361 |
> |
each of the frequencies was convoluted with a Lorentzian lineshape |
| 362 |
> |
with a width of 1.5 $\mathrm{cm}^{-1}$. Available computing resources |
| 363 |
> |
limited the sampling to 67 clusters for the zero-field spectrum, and |
| 364 |
> |
59 for the full field. Comparisons of the quantum mechanical spectrum |
| 365 |
> |
to the classical are shown in figure \ref{fig:spectrum}. |
| 366 |
|
|
| 367 |
|
\subsection{CN frequencies from potential-frequency maps} |
| 368 |
< |
Before Gaussian silumations were carried out, it was attempt to apply |
| 369 |
< |
the method developed by Cho {\it et al.}~\cite{Oh:2008fk} This method involves the fitting |
| 370 |
< |
of multiple parameters to Gaussian calculated freuencies to find a |
| 371 |
< |
correlation between the potential around the bond and the |
| 372 |
< |
frequency. This is very similar to work done by Skinner {\it et al.}~with |
| 373 |
< |
water models like SPC/E. The general method is to find the shift in |
| 374 |
< |
the peak position through, |
| 368 |
> |
One approach which has been used to successfully analyze the spectrum |
| 369 |
> |
of nitrile and thiocyanate probes in aqueous environments was |
| 370 |
> |
developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} This |
| 371 |
> |
method involves finding a multi-parameter fit that maps between the |
| 372 |
> |
local electrostatic potential at selected sites surrounding the |
| 373 |
> |
nitrile bond and the vibrational frequency of that bond obtained from |
| 374 |
> |
more expensive {\it ab initio} methods. This approach is similar in |
| 375 |
> |
character to the field-frequency maps developed by Skinner {\it et |
| 376 |
> |
al.} for OH stretches in liquid water.\cite{XXXX} |
| 377 |
> |
|
| 378 |
> |
To use the potential-frequency maps, the local electrostatic |
| 379 |
> |
potential, $\phi$, is computed at 20 sites ($a = 1 \rightarrow 20$) |
| 380 |
> |
that surround the nitrile bond, |
| 381 |
|
\begin{equation} |
| 382 |
< |
\delta\tilde{\nu} =\sum^{n}_{a=1} l_{a}\phi^{water}_{a} |
| 382 |
> |
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{j} |
| 383 |
> |
\frac{q_j}{\left|r_{aj}\right|}. |
| 384 |
|
\end{equation} |
| 385 |
< |
where $l_{a}$ are the fitting parameters and $\phi^{water}_{a}$ is the |
| 386 |
< |
potential from the surrounding water cluster. This $\phi^{water}_{a}$ |
| 387 |
< |
takes the form, |
| 385 |
> |
Here $q_j$ is the partial site on atom $j$, and $r_{aj}$ is the |
| 386 |
> |
distance between site $a$ and atom $j$. The original map was |
| 387 |
> |
parameterized in liquid water and comprises a set of parameters, |
| 388 |
> |
$l_a$, that predict the shift in nitrile peak frequency, |
| 389 |
|
\begin{equation} |
| 390 |
< |
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{m} \sum_{j} |
| 378 |
< |
\frac{C^{H_{2}O}_{j \left(m \right) }}{r_{aj \left(m\right)}} |
| 390 |
> |
\delta\tilde{\nu} =\sum^{20}_{a=1} l_{a}\phi_{a} |
| 391 |
|
\end{equation} |
| 392 |
< |
where $C^{H_{2}O}_{j \left(m \right) }$ indicates the partial charge |
| 393 |
< |
on the $j$th site of the $m$th water molecule and $r_{aj \left(m\right)}$ |
| 394 |
< |
is the distance between the site $a$ of the nitrile molecule and the $j$th |
| 395 |
< |
site of the $m$th water molecule. However, since these simulations |
| 396 |
< |
are done under the presence of external fields and in the |
| 397 |
< |
absence of water, the equations need a correction factor for the shift |
| 398 |
< |
caused by the external field. The equation is also reworked to use |
| 399 |
< |
electric field site data instead of partial charges from surrounding |
| 400 |
< |
atoms. So by modifing the original |
| 401 |
< |
$\phi^{water}_{a}$ to $\phi^{5CB}_{a}$ we get, |
| 392 |
> |
|
| 393 |
> |
The simulations of 5CB were carried in the presence of external |
| 394 |
> |
electric fields, out without water present, so it is not clear if they |
| 395 |
> |
can be applied to this situation without extensive |
| 396 |
> |
reparameterization. We do, however, suggest a small modification that |
| 397 |
> |
would help |
| 398 |
> |
|
| 399 |
> |
, so the equations need to be corrected |
| 400 |
> |
for the frequency shift caused by the electric field. We attempted to |
| 401 |
> |
make small modifications the original $\phi^{water}_{a}$ to |
| 402 |
> |
$\phi^{5CB}_{a}$ we get, |
| 403 |
|
\begin{equation} |
| 404 |
|
\phi^{5CB}_{a} = \frac{1}{4\pi \epsilon_{0}} \left( \vec{E}\bullet |
| 405 |
|
\left(\vec{r}_{a}-\vec{r}_{CN}\right) \right) + \phi^{5CB}_{0} |
| 416 |
|
\subsection{CN frequencies from bond length autocorrelation functions} |
| 417 |
|
|
| 418 |
|
Classical nitrile bond frequencies were found by replacing the rigid |
| 419 |
< |
cyanide bond with a flexible Morse oscillator bond |
| 420 |
< |
($r_0= 1.157437$ \AA , $D_0 = 212.95$ and |
| 421 |
< |
$\beta = 2.67566$) . Once replaced, the |
| 422 |
< |
systems were allowed to re-equilibrate in NVT for 100 ps. After |
| 423 |
< |
re-equilibration, the system was run in NVE for 20 ps with a snapshot |
| 424 |
< |
spacing of 1 fs. These snapshot were then used in bond correlation |
| 425 |
< |
calculation to find the decay structure of the bond in time using the |
| 426 |
< |
average bond displacement in time, |
| 419 |
> |
cyanide bond with a flexible Morse oscillator bond ($r_0= 1.157437$ |
| 420 |
> |
\AA , $D_0 = 212.95$ and $\beta = 2.67566$). Once replaced, the |
| 421 |
> |
systems were allowed to re-equilibrate in the canonical (NVT) ensemble |
| 422 |
> |
for 100 ps. After re-equilibration, the system was run in the |
| 423 |
> |
microcanonical (NVE) ensemble for 20 ps. Configurations sampled every |
| 424 |
> |
fs were then used to compute bond-length autocorrelation functions to |
| 425 |
> |
find the decay structure of the bond in time using the average bond |
| 426 |
> |
displacement in time, |
| 427 |
|
\begin{equation} |
| 428 |
|
C(t) = \langle \left(r(t) - r_0 \right) \cdot \left(r(0) - r_0 \right) \rangle |
| 429 |
|
\end{equation} |
