--- trunk/5cb/5CB.tex	2014/02/19 21:25:27	4034
+++ trunk/5cb/5CB.tex	2014/02/20 14:58:52	4035
@@ -286,17 +286,17 @@ The vibrational frequency of the nitrile bond in 5CB i
 
 \section{Sampling the CN bond frequency}
 
-The vibrational frequency of the nitrile bond in 5CB is assumed to
-depend on features of the local solvent environment of the individual
-molecules as well as the bond's orientation relative to the applied
-field.  Therefore, the primary quantity of interest is the
-distribution of vibrational frequencies exhibited by the 5CB nitrile
-bond under the different electric fields.  Three distinct methods for
-mapping classical simulations onto vibrational spectra were brought to
-bear on these simulations:
+The vibrational frequency of the nitrile bond in 5CB depends on
+features of the local solvent environment of the individual molecules
+as well as the bond's orientation relative to the applied field.  The
+primary quantity of interest for interpreting the condensed phase
+spectrum of this vibration is the distribution of frequencies
+exhibited by the 5CB nitrile bond under the different electric fields.
+Three distinct methods for mapping classical simulations onto
+vibrational spectra were brought to bear on these simulations:
 \begin{enumerate}
 \item Isolated 5CB molecules and their immediate surroundings were
-  extracted from the simulations, their nitrile bonds were stretched
+  extracted from the simulations.  These nitrile bonds were stretched
   and single-point {\em ab initio} calculations were used to obtain
   Morse-oscillator fits for the local vibrational motion along that
   bond.
@@ -324,69 +324,82 @@ midpoint of the target nitrile bond had its own molecu
 ``body'' (the two phenyl rings and the nitrile bond) and ``tail'' (the
 alkyl chain).  Any molecule which had a body atom within 6~\AA of the
 midpoint of the target nitrile bond had its own molecular body (the
-4-cyano-4'-pentylbiphenyl moiety) included in the configuration.  For
-the alkyl tail, the entire tail was included if any tail atom was
-within 4~\AA of the target nitrile bond.  If tail atoms (but no body
-atoms) were included within these distances, only the tail was
-included as a capped propane molecule.  
+4-cyano-biphenyl moiety) included in the configuration.  For the alkyl
+tail, the entire tail was included if any tail atom was within 4~\AA
+of the target nitrile bond.  If tail atoms (but no body atoms) were
+included within these distances, only the tail was included as a
+capped propane molecule.
 
 \begin{figure}[H]
   \includegraphics[width=\linewidth]{Figure2}
   \caption{Cluster calculations were performed on randomly sampled 5CB
-    molecules from each of the simualtions. Surrounding molecular
-    bodies were included if any body atoms were within 6 \AA\ of the
-    target nitrile bond, and tails were included if they were within 4
-    \AA.  The CN bond on the target molecule was stretched and
-    compressed (left), and the resulting single point energies were
-    fit to Morse oscillators to obtain frequency distributions.}
+    molecules (shown in red) from each of the simulations. Surrounding
+    molecular bodies were included if any body atoms were within 6
+    \AA\ of the target nitrile bond, and tails were included if they
+    were within 4 \AA.  Included portions of these molecules are shown
+    in green.  The CN bond on the target molecule was stretched and
+    compressed, and the resulting single point energies were fit to
+    Morse oscillators to obtain frequency distributions.}
   \label{fig:cluster}
 \end{figure}
 
-Inferred hydrogen atom locations were generated, and cluster
-geometries were created that stretched the nitrile bond along from
-0.87 to 1.52~\AA at increments of 0.05~\AA. This generated 13 single
-point energies to be calculated per gas phase cluster. Energies were
-computed with the B3LYP functional and 6-311++G(d,p) basis set.  For
-the cluster configurations that had been generated with applied
-fields, a field strength of 5 atomic units in the $z$ direction was
-applied to match the molecular dynamics runs.
+Inferred hydrogen atom locations were added to the cluster geometries,
+and the nitrile bond was stretched from 0.87 to 1.52~\AA\ at
+increments of 0.05~\AA. This generated 13 configurations per gas phase
+cluster. Single-point energies were computed using the B3LYP
+functional~\cite{Becke:1993kq,Lee:1988qf} and the 6-311++G(d,p) basis
+set.  For the cluster configurations that had been generated from
+molecular dynamics running under applied fields, the density
+functional calculations had a field of $5 \times 10^{-4}$ atomic units
+($E_h / (e a_0)$) applied in the $+z$ direction in order to match the
+molecular dynamics simulations.
 
-The relative energies for the stretched and compressed nitrile bond
-were used to fit a Morse oscillator, and the frequencies were obtained
-from the $0 \rightarrow 1$ transition for the exact energies. To
-obtain a spectrum, each of the frequencies was convoluted with a
-Lorentzian lineshape with a width of 1.5 $\mathrm{cm}^{-1}$.  Our
-available computing resources limited us to 67 clusters for the
-zero-field spectrum, and 59 for the full field.
+The energies for the stretched / compressed nitrile bond in each of
+the clusters were used to fit Morse oscillators, and the frequencies
+were obtained from the $0 \rightarrow 1$ transition for the energy
+levels for this potential.\cite{Morse:1929xy} To obtain a spectrum,
+each of the frequencies was convoluted with a Lorentzian lineshape
+with a width of 1.5 $\mathrm{cm}^{-1}$.  Available computing resources
+limited the sampling to 67 clusters for the zero-field spectrum, and
+59 for the full field.  Comparisons of the quantum mechanical spectrum
+to the classical are shown in figure \ref{fig:spectrum}.
 
 \subsection{CN frequencies from potential-frequency maps}
-Before Gaussian silumations were carried out, it was attempt to apply
-the method developed by Cho  {\it et al.}~\cite{Oh:2008fk} This method involves the fitting
-of multiple parameters to Gaussian calculated freuencies to find a
-correlation between the potential around the bond and the
-frequency. This is very similar to work done by Skinner  {\it et al.}~with
-water models like SPC/E. The general method is to find the shift in
-the peak position through,
+One approach which has been used to successfully analyze the spectrum
+of nitrile and thiocyanate probes in aqueous environments was
+developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} This
+method involves finding a multi-parameter fit that maps between the
+local electrostatic potential at selected sites surrounding the
+nitrile bond and the vibrational frequency of that bond obtained from
+more expensive {\it ab initio} methods. This approach is similar in
+character to the field-frequency maps developed by Skinner {\it et
+  al.} for OH stretches in liquid water.\cite{XXXX}
+
+To use the potential-frequency maps, the local electrostatic
+potential, $\phi$, is computed at 20 sites ($a = 1 \rightarrow 20$)
+that surround the nitrile bond,
 \begin{equation}
-\delta\tilde{\nu} =\sum^{n}_{a=1} l_{a}\phi^{water}_{a}
+\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{j}
+\frac{q_j}{\left|r_{aj}\right|}.
 \end{equation}
-where $l_{a}$ are the fitting parameters and $\phi^{water}_{a}$ is the
-potential from the surrounding water cluster. This $\phi^{water}_{a}$
-takes the form,
+Here $q_j$ is the partial site on atom $j$, and $r_{aj}$ is the
+distance between site $a$ and atom $j$.  The original map was
+parameterized in liquid water and comprises a set of parameters,
+$l_a$, that predict the shift in nitrile peak frequency,
 \begin{equation}
-\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{m} \sum_{j}
-\frac{C^{H_{2}O}_{j \left(m \right) }}{r_{aj \left(m\right)}}
+\delta\tilde{\nu} =\sum^{20}_{a=1} l_{a}\phi_{a}
 \end{equation}
-where $C^{H_{2}O}_{j \left(m \right) }$ indicates the partial charge
-on the $j$th site of the $m$th water molecule and $r_{aj \left(m\right)}$
-is the distance between the site $a$ of the nitrile molecule and the $j$th
-site of the $m$th water molecule. However, since these simulations
-are done under the presence of external fields and in the
-absence of water, the equations need a correction factor for the shift
-caused by the external field. The equation is also reworked to use
-electric field site data instead of partial charges from surrounding
-atoms. So by modifing the original
-$\phi^{water}_{a}$ to $\phi^{5CB}_{a}$ we get, 
+
+The simulations of 5CB were carried in the presence of external
+electric fields, out without water present, so it is not clear if they
+can be applied to this situation without extensive
+reparameterization.  We do, however, suggest a small modification that
+would help 
+
+, so the equations need to be corrected
+for the frequency shift caused by the electric field. We attempted to
+make small modifications the original $\phi^{water}_{a}$ to
+$\phi^{5CB}_{a}$ we get,
 \begin{equation}
 \phi^{5CB}_{a} = \frac{1}{4\pi \epsilon_{0}} \left( \vec{E}\bullet
   \left(\vec{r}_{a}-\vec{r}_{CN}\right) \right) + \phi^{5CB}_{0}
@@ -403,14 +416,14 @@ cyanide bond with a flexible Morse oscillator bond
 \subsection{CN frequencies from bond length autocorrelation functions}
 
 Classical nitrile bond frequencies were found by replacing the rigid
-cyanide bond with a flexible Morse oscillator bond
-($r_0= 1.157437$ \AA , $D_0 = 212.95$ and
-$\beta = 2.67566$) . Once replaced, the
-systems were allowed to re-equilibrate in NVT for 100 ps. After
-re-equilibration, the system was run in NVE for 20 ps with a snapshot
-spacing of 1 fs. These snapshot were then used in bond correlation
-calculation to find the decay structure of the bond in time using the
-average bond displacement in time,
+cyanide bond with a flexible Morse oscillator bond ($r_0= 1.157437$
+\AA , $D_0 = 212.95$ and $\beta = 2.67566$). Once replaced, the
+systems were allowed to re-equilibrate in the canonical (NVT) ensemble
+for 100 ps. After re-equilibration, the system was run in the
+microcanonical (NVE) ensemble for 20 ps.  Configurations sampled every
+fs were then used to compute bond-length autocorrelation functions to
+find the decay structure of the bond in time using the average bond
+displacement in time,
 \begin{equation}
 C(t) = \langle \left(r(t) - r_0 \right) \cdot \left(r(0) - r_0 \right) \rangle
 \end{equation}
