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\documentclass[aps,jcp,preprint,showpacs,superscriptaddress,groupedaddress]{revtex4} % for double-spaced preprint |
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\usepackage{times} |
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\usepackage[version=3]{mhchem} |
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\usepackage{lineno} |
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\usepackage{gensymb} |
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\begin{document} |
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\title{Supporting Information for: Interfacial Thermal Conductance of Thiolate-Protected |
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Gold Nanospheres} |
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\author{Kelsey M. Stocker} |
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\author{Suzanne M. Neidhart} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu} |
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\affiliation{Department of Chemistry and Biochemistry, University of |
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Notre Dame, Notre Dame, IN 46556} |
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\maketitle |
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\vfill |
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Gold -- gold interactions were described by the quantum Sutton-Chen |
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(QSC) model.\cite{Qi:1999ph} The hexane solvent is described by the |
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TraPPE united atom model,\cite{TraPPE-UA.alkanes} where sites are |
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located at the carbon centers for alkyl groups. Bonding interactions |
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were used for intra-molecular sites closer than 3 bonds. Effective |
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Lennard-Jones potentials were used for non-bonded interactions. |
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The TraPPE-UA force field includes parameters for thiol |
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molecules\cite{TraPPE-UA.thiols} which were used for the |
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alkanethiolate molecules in our simulations. To derive suitable |
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parameters for butanethiolate adsorbed on Au(111) surfaces, we adopted |
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the S parameters from Luedtke and Landman\cite{landman:1998} and |
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modified the parameters for the CTS atom to maintain charge neutrality |
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in the molecule. |
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To describe the interactions between metal (Au) and non-metal atoms, |
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potential energy terms were adapted from an adsorption study of alkyl |
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thiols on gold surfaces by Vlugt, \textit{et |
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al.}\cite{vlugt:cpc2007154} They fit an effective pair-wise |
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Lennard-Jones form of potential parameters for the interaction between |
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Au and pseudo-atoms CH$_x$ and S based on a well-established and |
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widely-used effective potential of Hautman and Klein for the Au(111) |
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surface.\cite{hautman:4994} |
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\begin{table}[h] |
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\centering |
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\caption{Properties of the United atom sites. \label{tab:atypes}} |
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\begin{tabular}{ c|cccc } |
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\toprule |
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atom type & mass (amu)& $\epsilon$ (kcal/mol) & $\sigma$ (\AA) & source \\ |
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\colrule |
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CH3 & 15.04 & 0.1947 & 3.75 & \\ |
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CH2 & 14.03 & 0.09141 & 3.95 & \\ |
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CH & 13.02 & 0.01987 & 4.68 & \\ |
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CHene & 13.02 & 0.09340 & 3.73 & \\ |
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CH2ene & 14.03 & 0.16891 & 3.675 & \\ |
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S & 32.0655 & 0.2504 & 4.45 & Refs. \protect\cite{landman:1998} ($\sigma$) and \protect\cite{vlugt:cpc2007154} ($\epsilon$) \\ |
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CHar & 13.02 & 0.1004 & 3.695 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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CH2ar & 14.03 & 0.1004 & 3.695 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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\botrule |
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\end{tabular} |
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\end{table} |
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Parameters not found in the TraPPE-UA force field for the |
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intramolecular interactions of the conjugated and the penultimate |
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alkenethiolate ligands were calculated using constrained geometry |
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scans using the B3LYP functional~\cite{Becke:1993kq,Lee:1988qf} and |
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the 6-31G(d,p) basis set. Structures were scanned starting at the |
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minimum energy gas phase structure for small ($C_4$) ligands. Only |
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one degree of freedom was constrained for any given scan -- all other |
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atoms were allowed to minimize subject to that constraint. The |
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resulting potential energy surfaces were fit to a harmonic potential |
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for the bond stretching, |
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\begin{equation} |
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V_\mathrm{bond} = \frac{k_\mathrm{bond}}{2} \left( r - r_0 \right)^2, |
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\end{equation} |
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and angle bending potentials, |
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\begin{equation} |
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V_\mathrm{bend} = \frac{k_\mathrm{bend}}{2} \left(\theta - \theta_0\right)^2. |
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\end{equation} |
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Torsional potentials were fit to the TraPPE torsional function, |
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\begin{equation} |
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V_\mathrm{tor} = c_0 + c_1 \left(1 + \cos\phi \right) + c_2 \left(1 - \cos 2\phi \right) + c_3 \left(1 + \cos 3 \phi \right). |
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\end{equation} |
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Say something here about which molecules were used for which scans.... |
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The fit values for the bond, bend, and torsional parameters were in |
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relatively good agreement with similar parameters already present in |
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TraPPE. |
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to find an equilibrium bend angles $\theta_0$ and spring constants, |
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$k$. Torsional parameters were fit to the same part of the |
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penultimate ligand (\(S - CH_{2}- CH-CH)\) |
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for the rotation around the \( CH_{2}- CH\) |
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bond. This potential energy surface was then fit to |
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\begin{table}[h] |
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\centering |
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\caption{Bond parameters. \label{tab:bond}} |
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\begin{tabular}{ cc|lll } |
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\toprule |
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$i$&$j$ & $r_0$ (\AA) & $k (\mathrm{~kcal/mole/\AA}^2)$ & source\\ |
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\colrule |
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CH3 & CH3 & 1.540 & 536 & \\ |
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CH3 & CH2 & 1.540 & 536 & \\ |
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CH3 & CH & 1.540 & 536 & \\ |
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CH2 & CH2 & 1.540 & 536 & \\ |
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CH2 & CH & 1.540 & 536 & \\ |
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CH & CH & 1.540 & 536 & \\ |
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Chene & CHene & 1.330 & 1098 & \\ |
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CH2ene & CHene & 1.330 & 1098 & \\ |
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CH3 & CHene & 1.540 & 634 & \\ |
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CH2 & CHene & 1.540 & 634 & \\ |
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S & CH2 & 1.820 & 444 & \\ |
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CHar & CHar & 1.40 & 938 & \\ |
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CHar & CH2 & 1.540 & 536 & \\ |
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CHar & CH3 & 1.540 & 536 & \\ |
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CH2ar & CHar & 1.40 & 938 & \\ |
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S & CHar & 1.80384 & 527.951 & fit \\ |
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\botrule |
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\end{tabular} |
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\end{table} |
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\begin{table}[h] |
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\centering |
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\caption{Bend angle parameters. The central atom in the bend is atom $j$.\label{tab:bend}} |
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\begin{tabular}{ ccc|lll } |
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\toprule |
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$i$&$j$&$k$ & $\theta_0 (\degree)$ & $k (\mathrm{kcal/mole/rad}^2)$ & source\\ |
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\colrule |
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CH2 & CH2 & S & 114.0 & 124.20& Ref. \protect\cite{TraPPE-UA.thiols}\\ |
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CH3 & CH2 & S & 114.0 & 124.20& Ref. \protect\cite{TraPPE-UA.thiols}\\ |
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CH3 & CH2 & CH3 & 114.0 & 124.20& Ref. \protect\cite{TraPPE-UA.thiols}\\ |
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CH3 & CH2 & CH2 & 114.0 & 124.20& Ref. \protect\cite{TraPPE-UA.thiols}\\ |
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CH2 & CH2 & CH2 & 114.0 & 124.20& Ref. \protect\cite{TraPPE-UA.thiols}\\ |
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CH3 & CH2 & CH & 114.0 & 124.20& Ref. \protect\cite{TraPPE-UA.thiols}\\ |
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CHene & CHene & CH3 & 119.7 & 139.94& Ref. \protect\cite{Maerzke:2009qy}\\ |
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CHene & CHene & CHene & 119.7 & 139.94& Ref. \protect\cite{Maerzke:2009qy}\\ |
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CH2ene & CHene & CH3 & 119.7 & 139.94& Ref. \protect\cite{Maerzke:2009qy}\\ |
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CHene & CHene & CH2 & 119.7 & 139.94& Ref. \protect\cite{Maerzke:2009qy}\\ |
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CH2 & CH2 & CHene & 114.0 & 124.20& Ref. \protect\cite{TraPPE-UA.thiols}\\ |
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CHar & CHar & CHar & 120.0 & 126.0 & Refs. \protect\cite{Maerzke:2009qy} and \\ |
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CHar & CHar & CH2 & 120.0 & 140.0 & Refs. \protect\cite{Maerzke:2009qy} and \\ |
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CHar & CHar & CH3 & 120.0 & 140.0 & Refs. \protect\cite{Maerzke:2009qy} and \\ |
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CHar & CHar & CH2ar & 120.0 & 126.0 & Refs. \protect\cite{Maerzke:2009qy} and \\ |
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S & CH2 & CHene & 109.97 & 127.37 & fit \\ |
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S & CH2 & CHar & 109.97 & 127.37 & fit \\ |
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S & CHar & CHar & 123.911 & 138.093 & fit \\ |
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\botrule |
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\end{tabular} |
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\end{table} |
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The conjugated system was fit to a bond, bend, and torsion. The |
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terminal bond for the shortest conjugated ligand \(CH-CH_2\) |
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was fit to a potential energy surface to find an equilibrium bond |
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length of 1.4 \AA and a spring constant of 938 kcal/mol using the |
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Harmonic Model: \(V_{bond} = \frac{k}{2} (b - b_0)^2\). |
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A bend parameter for the beginning the longer conjugated ligands |
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(\(S - CH_2- CH)\), |
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was approximated to be equal to the shortest penultimate ligand |
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parameters found. For the shortest conjugated ligand the first bend |
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(\(S - CH- CH)\) |
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was fit a potential energy surface in the same manor as the |
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penultimate bend. The torsion for the first four atoms of the two |
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longer conjugated systems is equal to the torsion calculated for the |
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penultimate system. |
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\begin{table}[h] |
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\centering |
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\caption{Torsion parameters. The central atoms are atoms $j$ and $k$, and wildcard atom types are denoted by ``X''. All $c_n$ parameters have units of kcal/mol. \label{tab:torsion}} |
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\begin{tabular}{ cccc|lllll } |
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\toprule |
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$i$&$j$&$k$&$l$& $c_0$&$c_1$& $c_2$ & $c_3$ & source\\ |
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\colrule |
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CH3 & CH2 & CH2 & CH3 & 0.0 & 0.7055 & -0.13551 & 1.5725 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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CH3 & CH2 & CH2 & CH2 & 0.0 & 0.7055 & -0.13551 & 1.5725 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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CH3 & CH2 & CH2 & CH & 0.0 & 0.7055 & -0.13551 & 1.5725 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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CH2 & CH2 & CH2 & CH2 & 0.0 & 0.7055 & -0.13551 & 1.5725 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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CH2 & CH2 & CH2 & S & 0.0 & 0.7055 & -0.13551 & 1.5725 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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CH3 & CH2 & CH2 & S & 0.0 & 0.7055 & -0.13551 & 1.5725 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ \colrule |
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X & CHene & CHene & X & \multicolumn{4}{c}{\multirow{2}{*}{$V = \frac{0.008112}{2} (\phi - 180.0)^2$}} & \multirow{2}{*}{Ref. \protect\cite{TraPPE-UA.alkylbenzenes}} \\ |
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X & CHar & CHar & X & & & & & \\ \colrule |
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CH2 & CH2 & CHene & CHene & 1.368 & 0.1716 & -0.2181 & -0.56081 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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CH2 & CH2 & CH2 & CHene & 0.0 & 0.7055 & -0.13551 & 1.5725 & Ref. \protect\cite{TraPPE-UA.alkylbenzenes}\\ |
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CHene & CHene & CH2 & S & 3.20753 & 0.207417& -0.912929& -0.958538 & fit \\ |
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CHar & CHar & CH2 & S & 3.20753 & 0.207417& -0.912929& -0.958538 & fit \\ |
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\botrule |
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\end{tabular} |
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\end{table} |
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The conjugated system was fit to a bond, bend, and torsion. The |
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terminal bond for the shortest conjugated ligand \(CH-CH_2\) |
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was fit to a potential energy surface to find an equilibrium bond |
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length of 1.4 \AA and a spring constant of 938 kcal/mol using the |
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Harmonic Model: \(V_{bond} = \frac{k}{2} (b - b_0)^2\). |
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A bend parameter for the beginning the longer conjugated ligands |
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(\(S - CH_2- CH)\), |
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was approximated to be equal to the shortest penultimate ligand |
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parameters found. For the shortest conjugated ligand the first bend |
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(\(S - CH- CH)\) |
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was fit a potential energy surface in the same manor as the |
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penultimate bend. The torsion for the first four atoms of the two |
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longer conjugated systems is equal to the torsion calculated for the |
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penultimate system. |
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\begin{table}[h] |
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\centering |
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\caption{Non-bonded cross interaction parameters between gold atoms and the united atom sites\label{tab:nb}} |
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\begin{tabular}{ cc|ccc } |
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\toprule |
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$i$&$j$ & $\sigma$ (\AA)& $\epsilon$ $(kcal/mol)$ & source \\ |
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\colrule |
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Au &CH3 &3.54 &0.2146& Ref. \protect\cite{vlugt:cpc2007154}\\ |
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Au &CH2 &3.54 &0.1749& Ref. \protect\cite{vlugt:cpc2007154}\\ |
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Au &CHene &3.4625 &0.1680& Ref. \protect\cite{vlugt:cpc2007154}\\ |
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Au &CHar &3.4625 &0.1680& Ref. \protect\cite{vlugt:cpc2007154}\\ |
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Au &CH2ar &3.4625 &0.1680& Ref. \protect\cite{vlugt:cpc2007154}\\ |
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Au &S &2.40 &8.465& Ref. \protect\cite{vlugt:cpc2007154}\\ |
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\botrule |
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\end {tabular} |
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\end{table} |
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\newpage |
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\bibliographystyle{aip} |
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\bibliography{NPthiols} |
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\end{document} |
