--- trunk/5cb/5CB.tex	2014/02/03 21:46:13	4018
+++ trunk/5cb/5CB.tex	2014/02/19 19:48:47	4033
@@ -40,7 +40,7 @@
 
 
 \title{Nitrile vibrations as reporters of field-induced phase
-  transitions in liquid crystals}  
+  transitions in 4-cyano-4'-pentylbiphenyl (5CB)}  
 \author{James M. Marr} 
 \author{J. Daniel Gezelter} 
 \email{gezelter@nd.edu} 
@@ -58,173 +58,350 @@ Notre Dame, Indiana 46556}
 \begin{doublespace}
 
 \begin{abstract}
-  The behavior of the spectral lineshape of the nitrile group in
-  4-Cyano-4'-pentylbiphenyl (5CB) in response to an applied electric
-  field has been simulated using both classical molecular dynamics
-  simulations and {\it ab initio} quantum mechanical calculations of
-  liquid-like clusters.  This nitrile group is a well-known reporter
-  of local field effects in other condensed phase settings, and our
-  simulations suggest that there is a significant response when 5CB
-  liquids are exposed to a relatively large external field.  However,
-  this response is due largely to the field-induced phase transition.
-  We observe a peak shift to the red of nearly 40
-  cm\textsuperscript{-1}. These results indicate that applied fields
-  can play a role in the observed peak shape and position even if
-  those fields are significantly weaker than the local electric fields
-  that are normally felt within polar liquids.
+  4-cyano-4'-pentylbiphenyl (5CB) is a liquid-crystal-forming compound
+  with a terminal nitrile group aligned with the long axis of the
+  molecule.  Simulations of condensed-phase 5CB were carried out both
+  with and without applied electric fields to provide an understanding
+  of the the Stark shift of the terminal nitrile group.  A
+  field-induced isotropic-nematic phase transition was observed in the
+  simulations, and the effects of this transition on the distribution
+  of nitrile frequencies were computed. Classical bond displacement
+  correlation functions exhibit a $\sim 40 \mathrm{~cm}^{-1}$ red
+  shift of a portion of the main nitrile peak, and this shift was
+  observed only when the fields were large enough to induce
+  orientational ordering of the bulk phase.  Our simulations appear to
+  indicate that phase-induced changes to the local surroundings are a
+  larger contribution to the change in the nitrile spectrum than
+  direct field contributions.
 \end{abstract}
 
 \newpage
 
 \section{Introduction}
 
+Nitrile groups can serve as very precise electric field reporters via
+their distinctive Raman and IR signatures.\cite{Boxer:2009xw} The
+triple bond between the nitrogen and the carbon atom is very sensitive
+to local field changes and has been observed to have a direct impact
+on the peak position within the spectrum.  The Stark shift in the
+spectrum can be quantified and mapped into a field value that is
+impinging upon the nitrile bond. This has been used extensively in
+biological systems like proteins and
+enzymes.\cite{Tucker:2004qq,Webb:2008kn}
+
+The response of nitrile groups to electric fields has now been
+investigated for a number of small molecules,\cite{Andrews:2000qv} as
+well as in biochemical settings, where nitrile groups can act as
+minimally invasive probes of structure and
+dynamics.\cite{Lindquist:2009fk,Fafarman:2010dq} The vibrational Stark
+effect has also been used to study the effects of electric fields on
+nitrile-containing self-assembled monolayers at metallic
+interfaces.\cite{Oklejas:2002uq,Schkolnik:2012ty}
+
+Recently 4-cyano-4'-pentylbiphenyl (5CB), a liquid crystalline
+molecule with a terminal nitrile group, has seen renewed interest as
+one way to impart order on the surfactant interfaces of
+nanodroplets,\cite{Moreno-Razo:2012rz} or to drive surface-ordering
+that can be used to promote particular kinds of
+self-assembly.\cite{PhysRevLett.111.227801} The nitrile group in 5CB
+is a particularly interesting case for studying electric field
+effects, as 5CB exhibits an isotropic to nematic phase transition that
+can be triggered by the application of an external field near room
+temperature.\cite{Gray:1973ca,Hatta:1991ee} This presents the
+possiblity that the field-induced changes in the local environment
+could have dramatic effects on the vibrations of this particular CN
+bond.  Although the infrared spectroscopy of 5CB has been
+well-investigated, particularly as a measure of the kinetics of the
+phase transition,\cite{Leyte:1997zl} the 5CB nitrile group has not yet
+seen the detailed theoretical treatment that biologically-relevant
+small molecules have
+received.\cite{Lindquist:2008bh,Lindquist:2008qf,Oh:2008fk,Choi:2008cr,Waegele:2010ve}
+ 
 The fundamental characteristic of liquid crystal mesophases is that
 they maintain some degree of orientational order while translational
 order is limited or absent. This orientational order produces a
 complex direction-dependent response to external perturbations like
-electric fields and mechanical distortions.  The anisotropy of the
+electric fields and mechanical distortions. The anisotropy of the
 macroscopic phases originates in the anisotropy of the constituent
 molecules, which typically have highly non-spherical structures with a
-significant degree of internal rigidity.  In nematic phases, rod-like
+significant degree of internal rigidity. In nematic phases, rod-like
 molecules are orientationally ordered with isotropic distributions of
-molecular centers of mass, while in smectic phases, the molecules
-arrange themselves into layers with their long (symmetry) axis normal
-($S_{A}$) or tilted ($S_{C}$) with respect to the layer planes.
+molecular centers of mass. For example, 5CB has a solid to nematic
+phase transition at 18C and a nematic to isotropic transition at
+35C.\cite{Gray:1973ca}
 
-The behavior of the $S_{A}$ phase can be partially explained with
-models mainly based on geometric factors and van der Waals
-interactions.  However, these simple models are insufficient to
-describe liquid crystal phases which exhibit more complex polymorphic
-nature.  X-ray diffraction studies have shown that the ratio between
-lamellar spacing ($s$) and molecular length ($l$) can take on a wide
-range of values.\cite{Gray:1984hc,Leadbetter:1976vf,Hardouin:1980yq}
-Typical $S_{A}$ phases have $s/l$ ratios on the order of $0.8$, while
-for some compounds, e.g. the 4-alkyl-4'-cyanobiphenyls, the $s/l$
-ratio is on the order of $1.4$.  Molecules which form $S_{A}$ phases
-can exhibit a wide variety of subphases like monolayers ($S_{A1}$),
-uniform bilayers ($S_{A2}$), partial bilayers ($S_{\tilde A}$) as well
-as interdigitated bilayers ($S_{A_{d}}$), and often have a terminal
-cyano or nitro group.  In particular lyotropic liquid crystals (those
-exhibiting liquid crystal phase transition as a function of water
-concentration) often have polar head groups or zwitterionic charge
-separated groups that result in strong dipolar
-interactions.\cite{Collings97} Because of their versatile polymorphic
-nature, polar liquid crystalline materials have important
-technological applications in addition to their immense relevance to
-biological systems.\cite{Collings97}
+In smectic phases, the molecules arrange themselves into layers with
+their long (symmetry) axis normal ($S_{A}$) or tilted ($S_{C}$) with
+respect to the layer planes. The behavior of the $S_{A}$ phase can be
+partially explained with models mainly based on geometric factors and
+van der Waals interactions. The Gay-Berne potential, in particular,
+has been widely used in the liquid crystal community to describe this
+anisotropic phase
+behavior.~\cite{Gay:1981yu,Berne72,Kushick:1976xy,Luckhurst90,Cleaver:1996rt}
+However, these simple models are insufficient to describe liquid
+crystal phases which exhibit more complex polymorphic nature.
+Molecules which form $S_{A}$ phases can exhibit a wide variety of
+subphases like monolayers ($S_{A1}$), uniform bilayers ($S_{A2}$),
+partial bilayers ($S_{\tilde A}$) as well as interdigitated bilayers
+($S_{A_{d}}$), and often have a terminal cyano or nitro group.  In
+particular, lyotropic liquid crystals (those exhibiting liquid crystal
+phase transition as a function of water concentration), often have
+polar head groups or zwitterionic charge separated groups that result
+in strong dipolar interactions,\cite{Collings:1997rz} and terminal cyano
+groups (like the one in 5CB) can induce permanent longitudinal
+dipoles.\cite{Levelut:1981eu}
 
-Experimental studies by Levelut {\it et al.}~\cite{Levelut:1981eu}
-revealed that terminal cyano or nitro groups usually induce permanent
-longitudinal dipole moments on the molecules.
+Macroscopic electric fields applied using electrodes on opposing sides
+of a sample of 5CB have demonstrated the phase change of the molecule
+as a function of electric field.\cite{Lim:2006xq} These previous
+studies have shown the nitrile group serves as an excellent indicator
+of the molecular orientation within the applied field. Lee {\it et
+  al.}~showed a 180 degree change in field direction could be probed
+with the nitrile peak intensity as it changed along with molecular
+alignment in the field.\cite{Lee:2006qd,Leyte:1997zl}
 
-Liquid crystalline materials with dipole moments located at the ends
-of the molecules have important applications in display technologies
-in addition to their relevance in biological systems.\cite{LCapp}
-
-Many of the technological applications of the lyotropic mesogens
-manipulate the orientation and structuring of the liquid crystal
-through application of local electric fields.\cite{?}
-Macroscopically, this restructuring is visible in the interactions the
-bulk phase has with scattered or transmitted light.\cite{?}  
-
-4-Cyano-4'-pentylbiphenyl (5CB), has been a model for field-induced
-phase changes due to the known electric field response of the liquid
-crystal\cite{Hatta:1991ee}.  It was discovered (along with three
-similar compounds) in 1973 in an effort to find a LC that had a
-melting point near room temperature.\cite{Gray:1973ca} It's known to
-have a crystalline to nematic phase transition at 18 C and a nematic
-to isotropic transition at 35 C.\cite{Gray:1973ca}
-
-Nitrile groups can serve as very precise electric field reporters via
-their distinctive Raman and IR signatures.\cite{Boxer:2009xw} The
-triple bond between the nitrogen and the carbon atom is very sensitive
-to local field changes and is observed to have a direct impact on the
-peak position within the spectrum.  The Stark shift in the spectrum
-can be quantified and mapped into a field value that is impinging upon
-the nitrile bond. This has been used extensively in biological systems
-like proteins and enzymes.\cite{Tucker:2004qq,Webb:2008kn} 
-
-To date, the nitrile electric field response of
-4-Cyano-4'-n-alkylbiphenyl liquid crystals has not been investigated.
-While macroscopic electric fields applied across macro electrodes show
-the phase change of the molecule as a function of electric
-field,\cite{Lim:2006xq} the effect of a microscopic field application
-has not been probed. These previous studies have shown the nitrile
-group serves as an excellent indicator of the molecular orientation
-within the field applied. Blank showed the 180 degree change in field
-direction could be probed with the nitrile peak intensity as it
-decreased and increased with molecule alignment in the
-field.\cite{Lee:2006qd,Leyte:97}
-
-While these macroscopic fields worked well at showing the bulk
+While these macroscopic fields work well at indicating the bulk
 response, the atomic scale response has not been studied. With the
 advent of nano-electrodes and coupling them with atomic force
 microscopy, control of electric fields applied across nanometer
-distances is now possible\cite{citation1}. This application of
-nanometer length is interesting in the case of a nitrile group on the
-molecule. While macroscopic fields are insufficient to cause a Stark
-effect, small fields across nanometer-sized gaps are of sufficient
-strength. If one were to assume a gap of 5 nm between a lower
-electrode having a nanoelectrode placed near it via an atomic force
-microscope, a field of 1 V applied across the electrodes would
-translate into a field of 2x10\textsuperscript{8} $\frac{V}{M}$. This
-field is theoretically strong enough to cause a phase change from
-isotropic to nematic, as well as Stark tuning of the nitrile
-bond. This should be readily visible experimentally through Raman or
-IR spectroscopy.
+distances is now possible.\cite{citation1} While macroscopic fields
+are insufficient to cause a Stark effect without dielectric breakdown
+of the material, small fields across nanometer-sized gaps may be of
+sufficient strength.  For a gap of 5 nm between a lower electrode
+having a nanoelectrode placed near it via an atomic force microscope,
+a potential of 1 V applied across the electrodes is equivalent to a
+field of 2x10\textsuperscript{8} $\frac{V}{M}$.  This field is
+certainly strong enough to cause the isotropic-nematic phase change
+and as well as Stark tuning of the nitrile bond.  This should be
+readily visible experimentally through Raman or IR spectroscopy.
 
-Herein, we show the computational investigation of these electric field effects through atomistic simulations. These are then coupled with ab intio and classical spectrum calculations to predict changes experiments should be able to replicate.
+In the sections that follow, we outline a series of coarse-grained
+classical molecular dynamics simulations of 5CB that were done in the
+presence of static electric fields. These simulations were then
+coupled with both {\it ab intio} calculations of CN-deformations and
+classical bond-length correlation functions to predict spectral
+shifts. These predictions made should be easily varifiable with
+scanning electrochemical microscopy experiments.
 
 \section{Computational Details}
-The force field was mainly taken from Guo et al.\cite{Zhang:2011hh} A
-deviation from this force field was made to create a rigid body from
-the phenyl rings. Bond distances within the rigid body were taken from
-equilibrium bond distances. While the phenyl rings were held rigid,
-bonds, bends, torsions and inversion centers still included the rings.
+The force field used for 5CB was taken from Guo {\it et
+  al.}\cite{Zhang:2011hh} However, for most of the simulations, each
+of the phenyl rings was treated as a rigid body to allow for larger
+time steps and very long simulation times.  The geometries of the
+rigid bodies were taken from equilibrium bond distances and angles.
+Although the phenyl rings were held rigid, bonds, bends, torsions and
+inversion centers that involved atoms in these substructures (but with
+connectivity to the rest of the molecule) were still included in the
+potential and force calculations.
 
-Simulations were with boxes of 270 molecules locked at experimental
-densities with periodic boundaries. The molecules were thermalized
-from 0 kelvin to 300 kelvin. To equilibrate, each was first run in NVT
-for 1 ns. This was followed by NVE for simulations used in the data
-collection.
+Periodic simulations cells containing 270 molecules in random
+orientations were constructed and were locked at experimental
+densities.  Electrostatic interactions were computed using damped
+shifted force (DSF) electrostatics.\cite{Fennell:2006zl} The molecules
+were equilibrated for 1~ns at a temperature of 300K.  Simulations with
+applied fields were carried out in the microcanonical (NVE) ensemble
+with an energy corresponding to the average energy from the canonical
+(NVT) equilibration runs.  Typical applied-field runs were more than
+60ns in length.
 
-External electric fields were applied in a simplistic charge-field
-interaction. Forces were calculated by multiplying the electric field
-being applied by the charge at each atom. For the potential, the
-origin of the box was used as a point of reference. This allows for a
-potential value to be added to each atom as the molecule move in space
-within the box. Fields values were applied in a manner representing
-those that would be applable in an experimental set-up. The assumed
-electrode seperation was 5 nm and the field was input as
-$\frac{V}{\text{\AA}}$. The three field environments were, 1) no field
-  applied, 2) 0.01 $\frac{V}{\text{\AA}}$ (0.5 V) and 3) 0.024
-    $\frac{V}{\text{\AA}}$ (1.2 V). Each field was applied in the
-    Z-axis of the simulation box. For the simplicity of this paper,
-    each field will be called zero, partial and full, respectively.
+Static electric fields with magnitudes similar to what would be
+available in an experimental setup were applied to the different
+simulations.  With an assumed electrode seperation of 5 nm and an
+electrostatic potential that is limited by the voltage required to
+split water (1.23V), the maximum realistic field that could be applied
+is $\sim 0.024$ V/\AA.  Three field environments were investigated:
+(1) no field applied, (2) partial field = 0.01 V/\AA\ , and (3) full
+field = 0.024 V/\AA\ . 
 
-For quantum calculation of the nitrile bond frequency, Gaussian 09 was
-used. A single 5CB molecule was selected for the center of the
-cluster. For effects from molecules located near the chosen nitrile
-group, parts of molecules nearest to the nitrile group were
-included. For the body not including the tail, any atom within 6~\AA 
-of the midpoint of the nitrile group was included. For the tail
-structure, the whole tail was included if a tail atom was within 4~\AA 
-of the midpoint. If the tail did not include any atoms from the ring
-structure, it was considered a propane molecule and included as
-such. Once the clusters were generated, input files were created that
-stretched the nitrile bond along its axis from 0.87 to 1.52~\AA at
-increments of 0.05~\AA. This generated 13 single point energies to be
-calculated. The Gaussian files were run with B3LYP/6-311++G(d,p) with
-no other keywords for the zero field simulation. Simulations with
-fields applied included the keyword ''Field=Z+5'' to match the
-external field applied in molecular dynamic runs. Once completed, the central nitrile bond frequency
-was calculated with a Morse fit. Using this fit and the solved energy
-levels for a Morse oscillator, the frequency was found. Each set of
-frequencies were then convolved together with a guassian spread
-function over each value. The width value used was 1.5. For the zero
-field spectrum, 67 frequencies were used and for the full field, 59
-frequencies were used.
+After the systems had come to equilibrium under the applied fields,
+additional simulations were carried out with a flexible (Morse)
+nitrile bond,
+\begin{displaymath}
+V(r_\ce{CN}) = D_e \left(1 - e^{-\beta (r_\ce{CN}-r_e)}\right)^2
+\end{displaymath}
+where $r_e= 1.157437$ \AA , $D_e = 212.95 \mathrm{~kca~l} /
+\mathrm{mol}^{-1}$ and $\beta = 2.67566 $\AA~$^{-1}$.  These
+parameters correspond to a vibrational frequency of $2375
+\mathrm{~cm}^{-1}$, a bit higher than the experimental frequency. The
+flexible nitrile moiety required simulation time steps of 1~fs, so the
+additional flexibility was introducuced only after the rigid systems
+had come to equilibrium under the applied fields.  Whenever time
+correlation functions were computed from the flexible simulations,
+statistically-independent configurations were sampled from the last ns
+of the induced-field runs.  These configurations were then
+equilibrated with the flexible nitrile moiety for 100 ps, and time
+correlation functions were computed using data sampled from an
+additional 200 ps of run time carried out in the microcanonical
+ensemble.
+
+\section{Field-induced Nematic Ordering}
+
+In order to characterize the orientational ordering of the system, the
+primary quantity of interest is the nematic (orientational) order
+parameter. This was determined using the tensor
+\begin{equation}
+Q_{\alpha \beta} = \frac{1}{2N} \sum_{i=1}^{N} \left(3 \hat{e}_{i
+    \alpha} \hat{e}_{i \beta} - \delta_{\alpha \beta} \right)
+\end{equation}
+where $\alpha, \beta = x, y, z$, and $\hat{e}_i$ is the molecular
+end-to-end unit vector for molecule $i$. The nematic order parameter
+$S$ is the largest eigenvalue of $Q_{\alpha \beta}$, and the
+corresponding eigenvector defines the director axis for the phase.
+$S$ takes on values close to 1 in highly ordered (smectic A) phases,
+but falls to zero for isotropic fluids.  Note that the nitrogen and
+the terminal chain atom were used to define the vectors for each
+molecule, so the typical order parameters are lower than if one
+defined a vector using only the rigid core of the molecule.  In
+nematic phases, typical values for $S$ are close to 0.5.
 
+The field-induced phase transition can be clearly seen over the course
+of a 60 ns equilibration runs in figure \ref{fig:orderParameter}.  All
+three of the systems started in a random (isotropic) packing, with
+order parameters near 0.2. Over the course 10 ns, the full field
+causes an alignment of the molecules (due primarily to the interaction
+of the nitrile group dipole with the electric field).  Once this
+system started exhibiting nematic ordering, the orientational order
+parameter became stable for the remaining 50 ns of simulation time.
+It is possible that the partial-field simulation is meta-stable and
+given enough time, it would eventually find a nematic-ordered phase,
+but the partial-field simulation was stable as an isotropic phase for
+the full duration of a 60 ns simulation. Ellipsoidal renderings of the
+final configurations of the runs shows that the full-field (0.024
+V/\AA\ ) experienced a isotropic-nematic phase transition and has
+ordered with a director axis that is parallel to the direction of the
+applied field.
+
+\begin{figure}[H]
+  \includegraphics[width=\linewidth]{Figure1}
+  \caption{Evolution of the orientational order parameters for the
+    no-field, partial field, and full field simulations over the
+    course of 60 ns. Each simulation was started from a
+    statistically-independent isotropic configuration.  On the right
+    are ellipsoids representing the final configurations at three
+    different field strengths: zero field (bottom), partial field
+    (middle), and full field (top)}
+  \label{fig:orderParameter}
+\end{figure}
+
+
+\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:
+\begin{enumerate}
+\item Isolated 5CB molecules and their immediate surroundings were
+  extracted from the simulations, their 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.
+\item The potential - frequency maps developed by Cho {\it et
+    al.}~\cite{Oh:2008fk} for nitrile moieties in water were
+  investigated.  This method involves mapping the electrostatic
+  potential around the bond to the vibrational frequency, and is
+  similar in approach to field-frequency maps that were pioneered by
+  Skinner {\it et al.}\cite{XXXX}
+\item Classical bond-length autocorrelation functions were Fourier
+  transformed to directly obtain the vibrational spectrum from
+  molecular dynamics simulations.
+\end{enumerate}
+
+\subsection{CN frequencies from isolated clusters}
+The size of the periodic condensed phase system prevented direct
+computation of the complete library of nitrile bond frequencies using
+{\it ab initio} methods.  In order to sample the nitrile frequencies
+present in the condensed-phase, individual molecules were selected
+randomly to serve as the center of a local (gas phase) cluster.  To
+include steric, electrostatic, and other effects from molecules
+located near the targeted nitrile group, portions of other molecules
+nearest to the nitrile group were included in the quantum mechanical
+calculations.  The surrounding solvent molecules were divided into
+``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.  
+
+\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.}
+  \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.
+
+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.
+
+\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,
+\begin{equation}
+\delta\tilde{\nu} =\sum^{n}_{a=1} l_{a}\phi^{water}_{a}
+\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,
+\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)}}
+\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, 
+\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}
+\end{equation}
+where $\vec{E}$ is the electric field at each atom, $\left( \vec{r}_{a} -
+  \vec{r}_{CN} \right)$ is the vector between the nitrile bond and the
+cooridinates described by Cho around the bond and $\phi^{5CB}_{0}$ is
+the correction factor for the system of parameters. After these
+changes, the correction factor was found for multiple values of an
+external field being applied. However, the factor was no linear and
+was overly large due to the fitting parameters being so small.
+
+
+\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
@@ -241,89 +418,108 @@ decay with a $\tau$ of 3000 (have to check this). Furt
 where $r_0$ is the equilibrium bond distance and $r(t)$ is the
 instantaneous bond displacement at time $t$. Once calculated,
 smoothing was applied by adding an exponential decay on top of the
-decay with a $\tau$ of 3000 (have to check this). Further smoothing
+decay with a $\tau$ of 6000. Further smoothing
 was applied by padding 20,000 zeros on each side of the symmetric
 data. This was done five times by allowing the systems to run 1 ns
 with a rigid bond followed by an equilibrium run with the bond
-switched back on and the short production run.
+switched back to a Morse oscillator and a short production run of 20 ps.
 
-\section{Results}
 
-In order to characterize the orientational ordering of the system, the
-primary quantity of interest is the nematic (orientational) order
-parameter. This is determined using the tensor
-\begin{equation}
-Q_{\alpha \beta} = \frac{1}{2N} \sum_{i=1}^{N} \left(3 \hat{e}_{i
-    \alpha} \hat{e}_{i \beta} - \delta_{\alpha \beta} \right)
-\end{equation}
-where $\alpha, \beta = x, y, z$, and $\hat{e}_i$ is the molecular
-end-to-end unit vector for molecule $i$. The nematic order parameter
-$S$ is the largest eigenvalue of $Q_{\alpha \beta}$, and the
-corresponding eigenvector defines the director axis for the phase.
-$S$ takes on values close to 1 in highly ordered phases, but falls to
-zero for isotropic fluids. In the context of 5CB, this value would be
-close to zero for its isotropic phase and raise closer to one as it
-moved to the nematic and crystalline phases.
-
-This value indicates phases changes at temperature boundaries but also
-when a phase changes occurs due to external field applications. In
-Figure 1, this phase change can be clearly seen over the course of 60
-ns. Each system starts with an ordering paramter near 0.1 to 0.2,
-which is an isotropic phase. Over the course 10 ns, the full external field
-causes a shift in the ordering of the system to 0.5, the nematic phase
-of the liquid crystal. This change is consistent over the full 50 ns
-with no drop back into the isotropic phase. This change is clearly
-field induced and stable over a long period of simulation time.
-
-Interestingly, the field that is needed to switch the phase of 5CB
-macroscopically is larger than 1 V. However, in this case, only a
-voltage of 1.2 V was need to induce a phase change. This is impart due
-to the distance the field is being applied across. At such a small
-distance, the field is much larger than the macroscopic and thus
-easily induces a field dependent phase change.
-
 This change in phase was followed by two courses of further
-simulation. First, was replacement of the static nitrile bond with a
+analysis. First was the replacement of the static nitrile bond with a
 morse oscillator bond. This was then simulated for a period of time
-and a classical spetrum was calculated. Second, ab intio calcualtions were performe to investigate
-if the phase change caused any change spectrum from quantum
-effects.
+and a classical spetrum was calculated. Second, ab intio calcualtions
+were performed to investigate if the phase change caused any change
+spectrum through quantum effects.
 
-In respect to the classical calculations, it was first seen if previous
-studies of nitriles within water and neat simulation done by Cho
-et. al. could be used for the spectrum.
-
-After Gaussian calculations were performed on a set of snapshots, any 
+The classical nitrile spectrum can be seen in Figure 2. Most noticably
+is the position of the two peaks. Obviously the experimental peak
+position is near 2226 cm\textsuperscript{-1}. However, in this case
+the peak position is shifted to the blue at a position of 2375
+cm\textsuperscript{-1}. This shift is due solely to the choice of
+oscillator strength in the Morse oscillator parameters. While this
+shift makes the two spectra differ, it does not affect the ability to
+qualitatively compare peak changes to possible experimental changes.
+With this important fact out of the way, differences between the two
+states are subtle but are very much present. The first and
+most notable is the apperance for a strong band near 2300
+cm\textsuperscript{-1}.
 \begin{figure}
-  \includegraphics[trim = 5mm 10mm 3mm 10mm, clip, width=3.25in]{P2}
-  \caption{Ordering of each external field application over the course
-    of 60 ns with a sampling every 100 ps. Each trajectory was started
-  after equilibration with zero field applied.}
-  \label{fig:orderParameter}
-\end{figure}
-\begin{figure}
   \includegraphics[width=3.25in]{2Spectra}
   \caption{The classically calculated nitrile bond spetrum for no
     external field application (black) and full external field
     application (red)}
   \label{fig:twoSpectra}
 \end{figure}
+
+
+Due to this, Gaussian calculations were performed in lieu of this
+method. A set of snapshots for the zero and full field simualtions,
+they were first investigated for any dependence on the local, with
+external field included, electric field. This was to see if a linear
+or non-linear relationship between the two could be utilized for
+generating spectra. This was done in part because of previous studies
+showing the frequency dependence of nitrile bonds to the electric
+fields generated locally between solvating water. It was seen that
+little to no dependence could be directly shown. This data is not
+shown.
+
+Since no explicit dependence was observed between the calculated
+frequency and the electric field, it was not a viable route for the
+calculation of a nitrile spectrum. Instead, the frequencies were taken
+and convolved together with a lorentzian line shape applied around the
+frequency value. These spectra are seen below in Figure
+4. While the spectrum without a field is lower in intensity and is
+almost bimodel in distrobution, the external field spectrum is much
+more unimodel. This tighter clustering has the affect of increasing the
+intensity around 2226 cm\textsuperscript{-1} where the peak is
+centered. The external field also has fewer frequencies of higher
+energy in the spectrum. Unlike the the zero field, where some frequencies
+reach as high as 2280 cm\textsuperscript{-1}. 
 \begin{figure}
   \includegraphics[width=3.25in]{Convolved}
-  \caption{Gaussian frequencies added together with gaussian }
+  \caption{Lorentzian convolved Gaussian frequencies of the zero field
+  system (black) and the full field system (red)}
   \label{fig:Con}
 \end{figure}
-\begin{figure}
-  \includegraphics[width=7in]{Elip_3}
-  \caption{Ellipsoid reprsentation of 5CB at three different
-          field strengths, Zero Field (Left), Partial Field (Middle), and Full
-        Field (Right)}
-  \label{fig:Cigars}
-\end{figure}
-
 \section{Discussion}
+Interestingly, the field that is needed to switch the phase of 5CB
+macroscopically is larger than 1 V. However, in this case, only a
+voltage of 1.2 V was need to induce a phase change. This is impart due
+to the short distance of 5 nm the field is being applied across. At such a small
+distance, the field is much larger than the macroscopic and thus
+easily induces a field dependent phase change. However, this field
+will not cause a breakdown of the 5CB since electrochemistry studies
+have shown that it can be used in the presence of fields as high as
+500 V macroscopically. This large of a field near the surface of the
+elctrode would cause breakdown of 5CB if it could happen.
 
+The absence of any electric field dependency of the freuquency with
+the Gaussian simulations is interesting. A large base of research has been
+built upon the known tuning of the nitrile bond as the local field
+changes. This difference may be due to the absence of water or a
+molecule that induces a large internal field. Liquid water is known to have a very high internal field which
+is much larger than the internal fields of neat 5CB. Even though the
+application of Gaussian simulations followed by mapping it to
+some classical parameter is easy and straight forward, this system
+illistrates how that 'go to' method can break down.
+
+While this makes the application of nitrile Stark effects in
+simulations without water harder, these data show
+that it is not a deal breaker. The classically calculated nitrile
+spectrum shows changes in the spectra that will be easily seen through
+experimental routes. It indicates a shifted peak lower in energy
+should arise. This peak is a few wavenumbers from the leading edge of
+the larger peak and almost 75 wavenumbers from the center. This
+seperation between the two peaks means experimental results will show
+an easily resolved peak.
+
+The Gaussian derived spectra do indicate an applied field
+and subsiquent phase change does cause a narrowing of freuency
+distrobution. With narrowing, it would indicate an increased
+homogeneous distrobution of the local field near the nitrile. 
 \section{Conclusions}
+Field dependent changes
 \newpage
 
 \bibliography{5CB}
