--- trunk/5cb/5CB.tex	2014/02/06 23:20:32	4025
+++ trunk/5cb/5CB.tex	2014/02/18 21:48:35	4029
@@ -40,7 +40,7 @@
 
 
 \title{Nitrile vibrations as reporters of field-induced phase
-  transitions in liquid crystals}  
+  transitions in 4-cyano-4'-pentylbiphenyl}  
 \author{James M. Marr} 
 \author{J. Daniel Gezelter} 
 \email{gezelter@nd.edu} 
@@ -58,168 +58,263 @@ Nitrile Stark shift repsonses to electric fields have 
 \begin{doublespace}
 
 \begin{abstract}
-Nitrile Stark shift repsonses to electric fields have been used
-extensively in biology for the probing of local internal fields of
-structures like proteins and DNA. Intigration of these probes into
-different areas of interest are important for studing local structure
-and fields within confined, nanoscopic
-systems. 4-Cyano-4'-pentylbiphenyl (5CB) is a liquid crystal with a known
-macroscopic structure reordering from the isotropic to nematic
-phase with the application of an external
-field and as the name suggests has an inherent nitrile group. Through
-simulations of this molecule where application of
-large, nanoscale external fields were applied, the nitrile was invenstigated
-as a local field sensor. It was
-found that while most computational methods for nitrile spectral
-calculations rely on a correlation between local electric field and
-the nitrile bond, 5CB did not have an easily obtained
-correlation. Instead classical calculation through correlation of the
-cyanide bond displacement in time use enabled to show a spectral
-change in the formation of a red
-shifted peak from the main peak as an external field was applied. This indicates
-that local structure has a larger impact on the nitrile frequency then
-does the local electric field. By better understanding how nitrile
-groups respond to local and external fields it will help
-nitrile groups branch out beyond their biological
-applications to uses in electronics and surface sciences.
+  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 external 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} Recently it has
-seen new life with the application of droplets of the molecule in
-water being used to study defect sites and nanoparticle
-strcuturing.\cite{PhysRevLett.111.227801}
-
-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 nanoscopic 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. Lee {\it et al.}~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 of 5CB with external
-fields applied. These simulations are then coupled with ab intio and
-classical spectrum calculations to predict changes. These changes are
-easily varifiable with experiments and should be able to replicated
-experimentally.
+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\ . 
 
+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.
+\begin{figure}
+  \includegraphics[trim = 5mm 10mm 3mm 10mm, clip, width=4in]{P2}
+  \caption{Evolution of the orientational order parameter 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.}
+  \label{fig:orderParameter}
+\end{figure}
+
+The field-induced isotropic-nematic transition can be visualized in
+figure \ref{fig:Cigars}, where each molecule has been represented
+using and ellipsoids aligned along the long-axis of each molecule.
+Both the zero field and partial field simulations appear isotropic,
+while the full field simulations has been orientationally ordered
+\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) Each image was created from the final configuration
+    of each 60 ns equilibration run.}
+  \label{fig:Cigars}
+\end{figure}
+
+\section{Sampling the CN bond frequency}
+
+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
+  work done 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}
+
 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
@@ -243,6 +338,49 @@ Classical nitrile bond frequencies were found by repla
 field spectrum, 67 frequencies were used and for the full field, 59
 frequencies were used.
 
+\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
@@ -265,59 +403,7 @@ switched back to a Morse oscillator and a short produc
 with a rigid bond followed by an equilibrium run with the bond
 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 change 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.
-\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}
-
-In the figure below, this phase change is represented nicely as
-ellipsoids that are created by the vector formed between the nitrogen
-of the nitrile group and the tail terminal carbon atom. These
-illistrate the change from isotropic phase to nematic change. Both the
-zero field and partial field images look mostly disordered. The
-partial field does look somewhat orded but without any overall order
-of the entire system. This is most likely a high point in the ordering
-for the trajectory. The full field image on the other hand looks much
-more ordered when compared to the two lower field simulations.
-\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) Each image was created by taking the final
-    snapshot of each 60 ns run}
-  \label{fig:Cigars}
-\end{figure}
-
 This change in phase was followed by two courses of further
 analysis. First was the replacement of the static nitrile bond with a
 morse oscillator bond. This was then simulated for a period of time
@@ -345,44 +431,6 @@ Before Gaussian silumations were carried out, it was a
   \label{fig:twoSpectra}
 \end{figure}
 
-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.
 
 Due to this, Gaussian calculations were performed in lieu of this
 method. A set of snapshots for the zero and full field simualtions,
