| 218 |
|
nitrile bond, |
| 219 |
|
\begin{displaymath} |
| 220 |
|
V(r_\ce{CN}) = D_e \left(1 - e^{-\beta (r_\ce{CN}-r_e)}\right)^2 |
| 221 |
+ |
\label{eq:morse} |
| 222 |
|
\end{displaymath} |
| 223 |
< |
where $r_e= 1.157437$ \AA , $D_e = 212.95 \mathrm{~kca~l} / |
| 223 |
> |
where $r_e= 1.157437$ \AA , $D_e = 212.95 \mathrm{~kcal~} / |
| 224 |
|
\mathrm{mol}^{-1}$ and $\beta = 2.67566 $\AA~$^{-1}$. These |
| 225 |
< |
parameters correspond to a vibrational frequency of $2375 |
| 225 |
> |
parameters correspond to a vibrational frequency of $2358 |
| 226 |
|
\mathrm{~cm}^{-1}$, a bit higher than the experimental frequency. The |
| 227 |
|
flexible nitrile moiety required simulation time steps of 1~fs, so the |
| 228 |
|
additional flexibility was introducuced only after the rigid systems |
| 383 |
|
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{j} |
| 384 |
|
\frac{q_j}{\left|r_{aj}\right|}. |
| 385 |
|
\end{equation} |
| 386 |
< |
Here $q_j$ is the partial site on atom $j$, and $r_{aj}$ is the |
| 387 |
< |
distance between site $a$ and atom $j$. The original map was |
| 388 |
< |
parameterized in liquid water and comprises a set of parameters, |
| 389 |
< |
$l_a$, that predict the shift in nitrile peak frequency, |
| 386 |
> |
Here $q_j$ is the partial site on atom $j$ (residing on a different |
| 387 |
> |
molecule) and $r_{aj}$ is the distance between site $a$ and atom $j$. |
| 388 |
> |
The original map was parameterized in liquid water and comprises a set |
| 389 |
> |
of parameters, $l_a$, that predict the shift in nitrile peak |
| 390 |
> |
frequency, |
| 391 |
|
\begin{equation} |
| 392 |
< |
\delta\tilde{\nu} =\sum^{20}_{a=1} l_{a}\phi_{a} |
| 392 |
> |
\delta\tilde{\nu} =\sum^{20}_{a=1} l_{a}\phi_{a}. |
| 393 |
|
\end{equation} |
| 394 |
|
|
| 395 |
< |
The simulations of 5CB were carried in the presence of external |
| 396 |
< |
electric fields, out without water present, so it is not clear if they |
| 397 |
< |
can be applied to this situation without extensive |
| 398 |
< |
reparameterization. We do, however, suggest a small modification that |
| 399 |
< |
would help |
| 400 |
< |
|
| 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, |
| 395 |
> |
The simulations of 5CB were carried in the presence of |
| 396 |
> |
externally-applied uniform electric fields. Although uniform fields |
| 397 |
> |
exert forces on charge sites, they only contribute to the potential if |
| 398 |
> |
one defines a reference point that can serve as an origin. One simple |
| 399 |
> |
modification to the potential at each of the $a$ sites is to use the |
| 400 |
> |
centroid of the \ce{CN} bond as the origin for that site, |
| 401 |
|
\begin{equation} |
| 402 |
< |
\phi^{5CB}_{a} = \frac{1}{4\pi \epsilon_{0}} \left( \vec{E}\bullet |
| 403 |
< |
\left(\vec{r}_{a}-\vec{r}_{CN}\right) \right) + \phi^{5CB}_{0} |
| 402 |
> |
\phi_a^\prime = \phi_a + \frac{1}{4\pi\epsilon_{0}} \vec{E} \cdot |
| 403 |
> |
\left(\vec{r}_a - \vec{r}_\ce{CN} \right) |
| 404 |
|
\end{equation} |
| 405 |
< |
where $\vec{E}$ is the electric field at each atom, $\left( \vec{r}_{a} - |
| 406 |
< |
\vec{r}_{CN} \right)$ is the vector between the nitrile bond and the |
| 407 |
< |
cooridinates described by Cho around the bond and $\phi^{5CB}_{0}$ is |
| 408 |
< |
the correction factor for the system of parameters. After these |
| 409 |
< |
changes, the correction factor was found for multiple values of an |
| 410 |
< |
external field being applied. However, the factor was no linear and |
| 411 |
< |
was overly large due to the fitting parameters being so small. |
| 405 |
> |
where $\vec{E}$ is the uniform electric field, $\left( \vec{r}_{a} - |
| 406 |
> |
\vec{r}_\ce{CN} \right)$ is the displacement between the |
| 407 |
> |
cooridinates described by Choi {\it et |
| 408 |
> |
al.}~\cite{Choi:2008cr,Oh:2008fk} and the \ce{CN} bond centroid. |
| 409 |
> |
$\phi_a^\prime$ then contains an effective potential contributed by |
| 410 |
> |
the uniform field in addition to the local potential contributions |
| 411 |
> |
from other molecules. |
| 412 |
|
|
| 413 |
+ |
The sites $\{\vec{r}_a\}$ and weights $\left\{l_a \right\}$ developed by |
| 414 |
+ |
Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} are quite symmetric |
| 415 |
+ |
around the \ce{CN} centroid, and even at large uniform field values we |
| 416 |
+ |
observed nearly-complete cancellation of the potenial contributions |
| 417 |
+ |
from the uniform field. In order to utilize the potential-frequency |
| 418 |
+ |
maps for this problem, one would therefore need extensive |
| 419 |
+ |
reparameterization of the maps to include explicit contributions from |
| 420 |
+ |
the external field. This reparameterization is outside the scope of |
| 421 |
+ |
the current work, but would make a useful addition to the |
| 422 |
+ |
potential-frequency map approach. |
| 423 |
|
|
| 424 |
|
\subsection{CN frequencies from bond length autocorrelation functions} |
| 425 |
|
|
| 426 |
< |
Classical nitrile bond frequencies were found by replacing the rigid |
| 427 |
< |
cyanide bond with a flexible Morse oscillator bond ($r_0= 1.157437$ |
| 428 |
< |
\AA , $D_0 = 212.95$ and $\beta = 2.67566$). Once replaced, the |
| 429 |
< |
systems were allowed to re-equilibrate in the canonical (NVT) ensemble |
| 430 |
< |
for 100 ps. After re-equilibration, the system was run in the |
| 431 |
< |
microcanonical (NVE) ensemble for 20 ps. Configurations sampled every |
| 432 |
< |
fs were then used to compute bond-length autocorrelation functions to |
| 433 |
< |
find the decay structure of the bond in time using the average bond |
| 434 |
< |
displacement in time, |
| 426 |
> |
The distributions of nitrile vibrational frequencies can also be found |
| 427 |
> |
using classical time correlation functions. This was done by |
| 428 |
> |
replacing the rigid \ce{CN} bond with a flexible Morse oscillator |
| 429 |
> |
described in Eq. \ref{eq:morse}. Since the systems were perturbed by |
| 430 |
> |
the addition of a flexible high-frequency bond, they were allowed to |
| 431 |
> |
re-equilibrate in the canonical (NVT) ensemble for 100 ps with 1 fs |
| 432 |
> |
timesteps. After equilibration, each configuration was run in the |
| 433 |
> |
microcanonical (NVE) ensemble for 20 ps. Configurations sampled every |
| 434 |
> |
fs were then used to compute bond-length autocorrelation functions, |
| 435 |
|
\begin{equation} |
| 436 |
< |
C(t) = \langle \left(r(t) - r_0 \right) \cdot \left(r(0) - r_0 \right) \rangle |
| 436 |
> |
C(t) = \langle \delta r(t) \cdot \delta r(0) ) \rangle |
| 437 |
|
\end{equation} |
| 438 |
|
% |
| 439 |
< |
where $r_0$ is the equilibrium bond distance and $r(t)$ is the |
| 440 |
< |
instantaneous bond displacement at time $t$. Once calculated, |
| 441 |
< |
smoothing was applied by adding an exponential decay on top of the |
| 442 |
< |
decay with a $\tau$ of 6000. Further smoothing |
| 443 |
< |
was applied by padding 20,000 zeros on each side of the symmetric |
| 436 |
< |
data. This was done five times by allowing the systems to run 1 ns |
| 437 |
< |
with a rigid bond followed by an equilibrium run with the bond |
| 438 |
< |
switched back to a Morse oscillator and a short production run of 20 ps. |
| 439 |
< |
|
| 439 |
> |
where $\delta r(t) = r(t) - r_0$ is the deviation from the equilibrium |
| 440 |
> |
bond distance at time $t$. Ten statistically-independent correlation |
| 441 |
> |
functions were obtained by allowing the systems to run 10 ns with |
| 442 |
> |
rigid \ce{CN} bonds followed by 120 ps equilibration and data |
| 443 |
> |
collection using the flexible \ce{CN} bonds. |
| 444 |
|
|
| 445 |
< |
This change in phase was followed by two courses of further |
| 446 |
< |
analysis. First was the replacement of the static nitrile bond with a |
| 447 |
< |
morse oscillator bond. This was then simulated for a period of time |
| 448 |
< |
and a classical spetrum was calculated. Second, ab intio calcualtions |
| 449 |
< |
were performed to investigate if the phase change caused any change |
| 450 |
< |
spectrum through quantum effects. |
| 445 |
> |
The correlation functions were filtered using exponential apodization |
| 446 |
> |
functions,\cite{FILLER:1964yg} $f(t) = e^{-c |t|}$, with a time constant, $c =$ 6 |
| 447 |
> |
ps, and Fourier transformed to yield a spectrum, |
| 448 |
> |
\begin{equation} |
| 449 |
> |
I(\omega) = \int_{-\infty}^{\infty} C(t) f(t) e^{-i \omega t} dt. |
| 450 |
> |
\end{equation} |
| 451 |
> |
The sample-averaged classical nitrile spectrum can be seen in Figure |
| 452 |
> |
\ref{fig:spectra}. Note that the Morse oscillator parameters listed |
| 453 |
> |
above yield a natural frequency of 2358 $\mathrm{cm}^{-1}$, |
| 454 |
> |
significantly higher than the experimental peak near 2226 |
| 455 |
> |
$\mathrm{cm}^{-1}$. This shift does not effect the ability to |
| 456 |
> |
qualitatively compare peaks from the classical and quantum mechanical |
| 457 |
> |
approaches, so the classical spectra are shown as a shift relative to |
| 458 |
> |
the natural oscillation of the Morse bond. |
| 459 |
|
|
| 448 |
– |
The classical nitrile spectrum can be seen in Figure 2. Most noticably |
| 449 |
– |
is the position of the two peaks. Obviously the experimental peak |
| 450 |
– |
position is near 2226 cm\textsuperscript{-1}. However, in this case |
| 451 |
– |
the peak position is shifted to the blue at a position of 2375 |
| 452 |
– |
cm\textsuperscript{-1}. This shift is due solely to the choice of |
| 453 |
– |
oscillator strength in the Morse oscillator parameters. While this |
| 454 |
– |
shift makes the two spectra differ, it does not affect the ability to |
| 455 |
– |
qualitatively compare peak changes to possible experimental changes. |
| 456 |
– |
With this important fact out of the way, differences between the two |
| 457 |
– |
states are subtle but are very much present. The first and |
| 458 |
– |
most notable is the apperance for a strong band near 2300 |
| 459 |
– |
cm\textsuperscript{-1}. |
| 460 |
|
\begin{figure} |
| 461 |
+ |
\includegraphics[width=3.25in]{Convolved} |
| 462 |
|
\includegraphics[width=3.25in]{2Spectra} |
| 463 |
< |
\caption{The classically calculated nitrile bond spetrum for no |
| 464 |
< |
external field application (black) and full external field |
| 465 |
< |
application (red)} |
| 466 |
< |
\label{fig:twoSpectra} |
| 463 |
> |
\caption{Lorentzian convolved Gaussian frequencies of the zero field |
| 464 |
> |
system (black) and the full field system (red), and the |
| 465 |
> |
classically calculated nitrile bond spectrum for no external field |
| 466 |
> |
application (black) and full external field application (red)} |
| 467 |
> |
\label{fig:spectra} |
| 468 |
|
\end{figure} |
| 469 |
|
|
| 470 |
+ |
Note that due to electrostatic interactions, the classical approach |
| 471 |
+ |
implicitly couples \ce{CN} vibrations to the same vibrational mode on |
| 472 |
+ |
other nearby molecules. This coupling is not handled in the {\it ab |
| 473 |
+ |
initio} cluster approach. |
| 474 |
|
|
| 475 |
+ |
\section{Discussion} |
| 476 |
+ |
|
| 477 |
|
Due to this, Gaussian calculations were performed in lieu of this |
| 478 |
|
method. A set of snapshots for the zero and full field simualtions, |
| 479 |
|
they were first investigated for any dependence on the local, with |
| 489 |
|
frequency and the electric field, it was not a viable route for the |
| 490 |
|
calculation of a nitrile spectrum. Instead, the frequencies were taken |
| 491 |
|
and convolved together with a lorentzian line shape applied around the |
| 492 |
< |
frequency value. These spectra are seen below in Figure |
| 493 |
< |
4. While the spectrum without a field is lower in intensity and is |
| 494 |
< |
almost bimodel in distrobution, the external field spectrum is much |
| 495 |
< |
more unimodel. This tighter clustering has the affect of increasing the |
| 492 |
> |
frequency value. These spectra are seen below in Figure 4. While the |
| 493 |
> |
spectrum without a field is lower in intensity and is almost bimodel |
| 494 |
> |
in distrobution, the external field spectrum is much more |
| 495 |
> |
unimodel. This tighter clustering has the affect of increasing the |
| 496 |
|
intensity around 2226 cm\textsuperscript{-1} where the peak is |
| 497 |
|
centered. The external field also has fewer frequencies of higher |
| 498 |
< |
energy in the spectrum. Unlike the the zero field, where some frequencies |
| 499 |
< |
reach as high as 2280 cm\textsuperscript{-1}. |
| 500 |
< |
\begin{figure} |
| 501 |
< |
\includegraphics[width=3.25in]{Convolved} |
| 494 |
< |
\caption{Lorentzian convolved Gaussian frequencies of the zero field |
| 495 |
< |
system (black) and the full field system (red)} |
| 496 |
< |
\label{fig:Con} |
| 497 |
< |
\end{figure} |
| 498 |
< |
\section{Discussion} |
| 498 |
> |
energy in the spectrum. Unlike the the zero field, where some |
| 499 |
> |
frequencies reach as high as 2280 cm\textsuperscript{-1}. |
| 500 |
> |
|
| 501 |
> |
|
| 502 |
|
Interestingly, the field that is needed to switch the phase of 5CB |
| 503 |
|
macroscopically is larger than 1 V. However, in this case, only a |
| 504 |
|
voltage of 1.2 V was need to induce a phase change. This is impart due |
| 505 |
< |
to the short distance of 5 nm the field is being applied across. At such a small |
| 506 |
< |
distance, the field is much larger than the macroscopic and thus |
| 507 |
< |
easily induces a field dependent phase change. However, this field |
| 508 |
< |
will not cause a breakdown of the 5CB since electrochemistry studies |
| 509 |
< |
have shown that it can be used in the presence of fields as high as |
| 510 |
< |
500 V macroscopically. This large of a field near the surface of the |
| 511 |
< |
elctrode would cause breakdown of 5CB if it could happen. |
| 505 |
> |
to the short distance of 5 nm the field is being applied across. At |
| 506 |
> |
such a small distance, the field is much larger than the macroscopic |
| 507 |
> |
and thus easily induces a field dependent phase change. However, this |
| 508 |
> |
field will not cause a breakdown of the 5CB since electrochemistry |
| 509 |
> |
studies have shown that it can be used in the presence of fields as |
| 510 |
> |
high as 500 V macroscopically. This large of a field near the surface |
| 511 |
> |
of the elctrode would cause breakdown of 5CB if it could happen. |
| 512 |
|
|
| 513 |
|
The absence of any electric field dependency of the freuquency with |
| 514 |
|
the Gaussian simulations is interesting. A large base of research has been |
| 536 |
|
homogeneous distrobution of the local field near the nitrile. |
| 537 |
|
\section{Conclusions} |
| 538 |
|
Field dependent changes |
| 539 |
+ |
|
| 540 |
+ |
\section{Acknowledgements} |
| 541 |
+ |
The authors thank Steven Corcelli for helpful comments and |
| 542 |
+ |
suggestions. Support for this project was provided by the National |
| 543 |
+ |
Science Foundation under grant CHE-0848243. Computational time was |
| 544 |
+ |
provided by the Center for Research Computing (CRC) at the University |
| 545 |
+ |
of Notre Dame. |
| 546 |
+ |
|
| 547 |
|
\newpage |
| 548 |
|
|
| 549 |
|
\bibliography{5CB} |
