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% \usepackage[square, comma, sort&compress]{natbib} |
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\usepackage{url} |
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\title{Nitrile vibrations as reporters of field-induced phase |
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transitions in 4-cyano-4'-pentylbiphenyl (5CB)} |
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\title{Nitrile Vibrations as Reporters of Field-induced Phase |
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Transitions in 4-cyano-4'-pentylbiphenyl (5CB)} |
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\author{James M. Marr} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu} |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\keywords{} |
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\begin{document} |
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\begin{tocentry} |
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%\includegraphics[width=9cm]{Elip_3} |
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\includegraphics[width=9cm]{cluster.pdf} |
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V(r_\ce{CN}) = D_e \left(1 - e^{-\beta (r_\ce{CN}-r_e)}\right)^2 |
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\label{eq:morse} |
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\end{equation} |
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where $r_e= 1.157437$ \AA , $D_e = 212.95 \mathrm{~kcal~} / |
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\mathrm{mol}^{-1}$ and $\beta = 2.67566 $\AA~$^{-1}$. These |
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parameters correspond to a vibrational frequency of $2358 |
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\mathrm{~cm}^{-1}$, somewhat higher than the experimental |
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frequency. The flexible nitrile moiety required simulation time steps |
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of 1~fs, so the additional flexibility was introduced only after the |
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rigid systems had come to equilibrium under the applied fields. |
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Whenever time correlation functions were computed from the flexible |
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simulations, statistically-independent configurations (separated in |
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time by 10 ns) were sampled from the last 110 ns of the induced-field |
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runs. These configurations were then equilibrated with the flexible |
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nitrile moiety for 100 ps, and time correlation functions were |
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computed using data sampled from an additional 20 ps of run time |
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where $r_e= 1.157$ \AA (the fixed CN bond length from the force field |
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of Guo {\it et al.}\cite{Zhang:2011hh}), $D_e = 212.95 \mathrm{~kcal~} |
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/ \mathrm{mol}^{-1}$ (the average bond energy for CN triple bonds) and |
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$\beta = 2.526 $\AA~$^{-1}$. These parameters correspond to a |
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vibrational frequency of $\approx 2226 \mathrm{~cm}^{-1}$, which is |
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very close to the frequency of the nitrile peak in the vibrational |
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spectrum of neat 5CB. The flexible nitrile moiety required simulation |
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time steps of 1~fs, so the additional flexibility was introduced only |
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after the rigid systems had come to equilibrium under the applied |
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fields. Whenever time correlation functions were computed from the |
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flexible simulations, statistically-independent configurations |
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(separated in time by 10 ns) were sampled from the last 110 ns of the |
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induced-field runs. These configurations were then equilibrated with |
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the flexible nitrile moiety for 100 ps, and time correlation functions |
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were computed using data sampled from an additional 20 ps of run time |
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carried out in the microcanonical ensemble. |
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\section{Field-induced Nematic Ordering} |
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$S$ takes on values close to 1 in highly ordered (smectic A) phases, |
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but falls to much smaller values ($0 \rightarrow 0.3$) for isotropic |
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fluids. Note that the nitrogen and the terminal chain atom were used |
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to define the vectors for each molecule, so the typical order |
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to define the vector for each molecule, so the typical order |
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parameters are lower than if one defined a vector using only the rigid |
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core of the molecule. In nematic phases, typical values for $S$ are |
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core of the molecule. In nematic phases, typical values for $S$ are |
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close to 0.5. |
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The field-induced phase transition can be clearly seen over the course |
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perturbation theory approach,\cite{Morales:2009fp} the use of an |
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optimized QM/MM approach coupled with the fluctuating frequency |
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approximation,\cite{Lindquist:2008qf} and empirical frequency |
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correlation maps.\cite{Oh:2008fk} Three distinct (and comparatively |
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primitive) methods for mapping classical simulations onto vibrational |
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spectra were brought to bear on the simulations in this work: |
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correlation maps.\cite{Choi:2008cr,Oh:2008fk} Three distinct (and |
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comparatively primitive) methods for mapping classical simulations |
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onto vibrational spectra were brought to bear on the simulations in |
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this work: |
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\begin{enumerate} |
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\item Isolated 5CB molecules and their immediate surroundings were |
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extracted from the simulations. These nitrile bonds were stretched |
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and single-point {\em ab initio} calculations were used to obtain |
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Morse-oscillator fits for the local vibrational motion along that |
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bond. |
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\item A static-field extension of the empirical frequency correlation |
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maps developed by Choi {\it et al.}~\cite{Oh:2008fk} for nitrile |
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moieties in water was attempted. |
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by displacing the nitrogen along the CN bond vector with the carbon |
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atom remaining stationary. Single-point {\em ab initio} calculations |
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were used to obtain Morse-oscillator fits for the local vibrational |
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motion along that bond. |
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\item The empirical frequency correlation maps developed by Choi {\it |
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et al.}~\cite{Choi:2008cr,Oh:2008fk} for nitrile moieties in water |
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were utilized by adding an electric field contribution to the local |
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electrostatic potential. |
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\item Classical bond-length autocorrelation functions were Fourier |
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transformed to directly obtain the vibrational spectrum from |
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molecular dynamics simulations. |
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include steric, electrostatic, and other effects from molecules |
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located near the targeted nitrile group, portions of other molecules |
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nearest to the nitrile group were included in the quantum mechanical |
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calculations. The surrounding solvent molecules were divided into |
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``body'' (the two phenyl rings and the nitrile bond) and ``tail'' (the |
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alkyl chain). Any molecule which had a body atom within 6~\AA\ of the |
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calculations. Steric interactions are generally shorter ranged than |
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electrostatic interactions, so portions of surrounding molecules that |
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cause electrostatic perturbations to the central nitrile (e.g. the |
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biphenyl core and nitrile moieties) must be included if they fall |
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anywhere near the CN bond. Portions of these molecules that interact |
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primarily via dispersion and steric repulsion (e.g. the alkyl tails) |
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can be truncated at a shorter distance. |
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The surrounding solvent molecules were therefore divided into ``body'' |
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(the two phenyl rings and the nitrile bond) and ``tail'' (the alkyl |
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chain). Any molecule which had a body atom within 6~\AA\ of the |
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midpoint of the target nitrile bond had its own molecular body (the |
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4-cyano-biphenyl moiety) included in the configuration. Likewise, the |
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entire alkyl tail was included if any tail atom was within 4~\AA\ of |
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were obtained from the $0 \rightarrow 1$ transition for the energy |
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levels for this potential.\cite{Morse:1929xy} To obtain a spectrum, |
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each of the frequencies was convoluted with a Lorentzian line shape |
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with a width of 1.5 $\mathrm{cm}^{-1}$. Available computing resources |
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limited the sampling to 100 clusters for both the no-field and |
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full-field spectra. Comparisons of the quantum mechanical spectrum to |
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the classical are shown in figure \ref{fig:spectra}. The mean |
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frequencies obtained from the distributions give a field-induced red |
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shift of $2.68~\mathrm{cm}^{-1}$. |
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with a width of 1.5 $\mathrm{cm}^{-1}$. This linewidth corresponds to |
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a vibrational lifetime of $\sim 3.5$ ps, which is within the reported |
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ranges ($\sim 1 - 5$ ps) for CN stretching vibrational lifetimes in |
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other molecules.\cite{Ghosh:2009qf,Ha:2009xy,Waegele:2010ve}. |
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Available computing resources limited the sampling to 100 clusters for |
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both the no-field and full-field spectra. Comparisons of the quantum |
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mechanical spectrum to the classical are shown in figure |
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\ref{fig:spectra}. The mean frequencies obtained from the |
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distributions give a field-induced red shift of |
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$2.68~\mathrm{cm}^{-1}$. |
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\subsection{CN frequencies from potential-frequency maps} |
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developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} are quite |
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symmetric around the \ce{CN} centroid, and even at large uniform field |
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values we observed nearly-complete cancellation of the potential |
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contributions from the uniform field. In order to utilize the |
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potential-frequency maps for this problem, one would therefore need |
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extensive reparameterization of the maps to include explicit |
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contributions from the external field. This reparameterization is |
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outside the scope of the current work, but would make a useful |
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addition to the potential-frequency map approach. |
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contributions from the uniform field. |
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The frequency shifts were computed for 4000 configurations sampled |
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every 1 ps after the systems had equilibrated. The potential |
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frequency map produces a small blue shift of 0.34 cm$^{-1}$, and the |
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frequency shifts are quite narrowly distributed. However, the |
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parameters for the potential frequency maps were derived for nitrile |
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bonds in aqueous solutions, where the magnitudes of the local fields |
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and electrostatic potentials are much larger than they would be in |
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neat 5CB. |
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We note that in 5CB there does not appear to be a particularly strong |
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correlation between the electric field strengths observed at the |
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nitrile centroid and the calculated vibrational frequencies. In |
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\begin{equation} |
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I(\omega) = \int_{-\infty}^{\infty} C(t) f(t) e^{-i \omega t} dt. |
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\end{equation} |
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The sample-averaged classical nitrile spectrum can be seen in Figure |
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\ref{fig:spectra}. Note that the Morse oscillator parameters listed |
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above yield a natural frequency of 2358 $\mathrm{cm}^{-1}$, somewhat |
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higher than the experimental peak near 2226 $\mathrm{cm}^{-1}$. This |
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shift does not effect the ability to qualitatively compare peaks from |
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the classical and quantum mechanical approaches, so the classical |
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spectra are shown as a shift relative to the natural oscillation of |
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the Morse bond. The quantum cluster values are referenced to the |
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actual experimental vibrational frequency. |
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This time constant was chosen to match the Lorentzian linewidth that |
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was used for computing the quantum mechanical spectra, and falls |
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within the range of reported lifetimes for CN vibrations in other |
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nitrile-containing molecules. The sample-averaged classical nitrile |
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spectrum can be seen in Figure \ref{fig:spectra}. The Morse oscillator |
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parameters listed above yield a natural frequency of 2226 |
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$\mathrm{cm}^{-1}$ (close to the experimental value). To compare peaks |
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from the classical and quantum mechanical approaches, both are |
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displayed on an axis centered on the experimental nitrile frequency. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{spectra.pdf} |
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mean frequencies for each of the distributions. The cluster |
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calculations exhibit a $2.68~\mathrm{cm}^{-1}$ field-induced red |
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shift, while the classical correlation functions predict a red |
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shift of $3.05~\mathrm{cm}^{-1}$.} |
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shift of $2.29~\mathrm{cm}^{-1}$.} |
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\label{fig:spectra} |
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\end{figure} |
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quantum calculations are quite narrowly peaked around the experimental |
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nitrile frequency. Although the spectra are quite noisy, the main |
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effect seen in both distributions is a moderate shift to the red |
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($3.05~\mathrm{cm}^{-1}$ classical and $2.68~\mathrm{cm}^{-1}$ |
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($2.29~\mathrm{cm}^{-1}$ classical and $2.68~\mathrm{cm}^{-1}$ |
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quantum) after the electrostatic field had induced the nematic phase |
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transition. |
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