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\begin{document} |
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\title{Notes on the Dielectric} |
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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 |
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of Notre Dame, Notre Dame, IN 46556} |
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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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\maketitle |
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\section{Dielectric constants} |
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The new real-space methods require some care when computing dielectric |
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properties because the interaction between molecular dipoles depends |
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on the level of approximation being utilized. Consider the cases of |
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Stockmayer (dipolar) soft spheres that are represented either by two |
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closely-spaced point charges or by a single point dipole (see |
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Fig. \ref{fig:stockmayer}). |
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\begin{figure} |
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\includegraphics[width=3in]{DielectricFigure} |
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\caption{With the real-space electrostatic methods, the effective |
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dipole tensor, $\mathbf{T}$, governing interactions between |
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molecular dipoles is not the same for charge-charge interactions as |
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for point dipoles.} |
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\label{fig:stockmayer} |
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\end{figure} |
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In the case where point charges are interacting via an electrostatic |
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kernel, $v(r)$, the effective {\it molecular} dipole tensor, |
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$\mathbf{T}$ is obtained from two successive applications of the |
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gradient operator to the electrostatic kernel, |
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\begin{equation} |
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\mathbf{T}_{\alpha \beta}(r) = \nabla_\alpha \nabla_\beta \left(v(r)\right) = \delta_{\alpha \beta} |
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\left(\frac{1}{r} v^\prime(r) \right) + \frac{r_{\alpha} |
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r_{\beta}}{r^2} \left( v^{\prime \prime}(r) - \frac{1}{r} |
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v^{\prime}(r) \right) |
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\end{equation} |
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where $v(r)$ may be either the bare kernel ($1/r$) or one of the |
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modified (Wolf or DSF) kernels. This tensor describes the effective |
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interaction between molecular dipoles ($\mathbf{D}$) in Gaussian |
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units as $-\mathbf{D} \cdot \mathbf{T} \cdot \mathbf{D}$. |
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When utilizing the new real-space methods for point dipoles, the |
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tensor is explicitly constructed, |
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\begin{equation} |
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\mathbf{T}_{\alpha \beta}(r) = \delta_{\alpha \beta} v_{21}(r) + |
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\frac{r_{\alpha} r_{\beta}}{r^2} v_{22}(r) |
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\end{equation} |
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where the functions $v_{21}(r)$ and $v_{22}(r)$ depend on the level of |
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the approximation. Although the Taylor-shifted (TSF) and |
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gradient-shifted (GSF) models produce to the same $v(r)$ function for |
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point charges, they have distinct forms for the dipole-dipole |
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interactions. |
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Neumann and Steinhauser showed~\cite{Neumann:1983mz,Neumann:1983yq} |
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that the relative dielectric permittivity $\epsilon$ is given by the |
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general fluctuation formula, |
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\begin{equation} |
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\frac{\epsilon - 1}{\epsilon +2} = \frac{4 \pi}{3} \frac{\left<M^2\right>}{3 V k_B |
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T} \left( 1 - \frac{3}{4 \pi} \frac{\epsilon -1}{\epsilon + 2} |
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\tilde{T}(0) \right) |
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\end{equation} |
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where $\left<M^2\right>$ is the mean fluctuation of the square of the |
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net dipole moment of the simulation cell, $V$ is the volume of the |
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cell, and $\tilde{T}(0) = \tilde{T̃}_{xx}(0) = \tilde{T̃}_{yy}(0) = |
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\tilde{T̃}_{zz}(0)$ is the $\mathbf{k} = 0$ limit of Fourier transform |
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of the diagonal term of the effective molecular dipolar tensor, |
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\begin{equation} |
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\tilde{T}(0) = \int_V \mathbf{T}(\mathbf{r}) d\mathbf{r} |
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\end{equation} |
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where the integral is carried out over the relevant geometry for the |
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interaction. For the real-space methods, the integration can be done |
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easily up to the imposed cutoff radius, while for the Ewald sum, the |
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results also depend on details of the $k$-space |
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calculation.\cite{Neumann:1983yq} |
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For molecules composed of point charges interacting via the bare |
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$(1/r)$ kernel, the simple cutoff (SC) and minimum image (MI) |
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approaches give $\tilde{T}(0) = 0$ which means that the |
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Clausius-Mosotti equation, |
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\begin{equation} |
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\frac{4\pi}{3}\frac{\left< M^2 \right>}{3 V k_B T} = \frac{\epsilon |
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-1}{\epsilon+2} |
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\end{equation} |
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is the relevant fluctuation formula for the relative permittivity. |
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Zahn {\it et al.}\cite{Zahn02} showed that for the damped shifted |
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force charge-charge kernel, $\tilde{T}(0) = 4 \pi / 3$. This was later |
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generalized by Izvekov {\it et al.} for all point-charge kernels which |
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have forces that go to zero at a cutoff radius and which maintain a |
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pole of first order at $r=0$.\cite{Izvekov:2008wo} When $\tilde{T}(0) |
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= 4 \pi/ 3$, the expression for the dielectric constant reduces to the |
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widely-used {\it conducting boundary} formula, |
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\begin{equation} |
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\frac{4\pi}{3}\frac{\left< M^2 \right>}{3 V k_B T} = \frac{\epsilon - |
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1}{3} |
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\end{equation} |
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It is convenient to define a quantity $Q = \frac{3}{4 \pi} |
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\tilde{T}(0)$, and to combine all of the fluctuation formulae |
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together: |
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\begin{equation} |
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\frac{4 \pi}{3} \frac{\left< M^2 \right>}{3 V k_B T} = |
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\left\{\begin{array}{ll} |
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\frac{\epsilon-1}{\epsilon+2} \left[1- \frac{\epsilon-1}{\epsilon+2} |
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Q\right]^{-1} & \mathrm{General~Case} \\ |
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\frac{\epsilon-1}{\epsilon+2} & Q \rightarrow 0 \mathrm{~limit} \\ |
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\frac{\epsilon-1}{3} & Q \rightarrow 1 \mathrm{~limit} |
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\end{array}\right. |
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\end{equation} |
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The Clausius-Mossotti $(Q\rightarrow 0)$ approach is subject to noise |
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and error magnification,\cite{Allen:1989fp} and the conducting |
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boundary approach $(Q \rightarrow 1)$ has become widely used as a |
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result. If the electrostatic method being used has $Q < 1$, the |
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relative dielectric permittivity $\epsilon$ can be estimated once the |
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conducting boundary fluctuation result has been found, |
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\begin{equation} |
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\epsilon = \frac{(Q+2) (\epsilon_\text{CB}-1)+3} {(Q-1) (\epsilon_\text{CB}-1)+3} |
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\end{equation} |
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Note that this expression becomes quite numerically sensitive when the |
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value of $Q$ deviates significantly from 1. |
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We have derived a set of expressions for the value of $Q$ for the new |
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real space methods that are shown in Table \ref{tab:Q}. |
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\begin{table} |
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\caption{Expressions for the dielectric correction factor ($Q$) for the |
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real-space electrostatic methods in terms of the damping parameter |
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($\alpha$) and the cutoff radius ($r_c$). The Ewald-Kornfeld result |
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derived in Refs. \onlinecite{Adams:1980rt,Adams:1981fr,Neumann:1983yq} is shown for comparison using the Ewald |
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convergence parameter ($\kappa$) and the real-space cutoff value ($r_c$). } |
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\label{tab:Q} |
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\begin{tabular}{l|c|c|c|c|} |
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& \multicolumn{2}{c|}{damped} & \multicolumn{2}{c|}{undamped} \\ \cline{2-3} \cline{4-5} |
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Method & $Q_\mathrm{charges}$ & $Q_\mathrm{dipoles}$ & $Q_\mathrm{charges}$ & $Q_\mathrm{dipoles}$ \\ |
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\hline |
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Spherical Cutoff (SC) & $\mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $\mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ &0 &0\\ |
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Shifted Potental (SP) & $ \mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $\mathrm{erf}(\alpha r_c)-\frac{2 \alpha r_c }{\sqrt{\pi }} \left(1 + \frac{2 \alpha^2 r_c^2}{3} \right) e^{-\alpha ^2 r_c^2} $ &0 &0\\ |
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Gradient-shifted (GSF) & 1 & $\mathrm{erf}(\alpha r_c)-\frac{2 \alpha r_c}{\sqrt{\pi}} \left(1 + \frac{2 \alpha^2 r_c^2}{3} + \frac{\alpha^4 r_c^4}{3}\right)e^{-\alpha ^2 r_c^2} $ &1 &1\\ |
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Taylor-shifted (TSF) & 1 & 1 & 1 & 1\\ |
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Ewald-Kornfeld (EK) & $\mathrm{erf}(r_c \kappa) - \frac{2 \kappa r_c}{\sqrt{\pi}} e^{-\kappa^2 r_c^2}$ & $\mathrm{erf}(r_c \kappa) - \frac{2 \kappa r_c}{\sqrt{\pi}} e^{-\kappa^2 r_c^2}$ & - & - \\\hline |
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\end{tabular} |
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\end{table} |
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\newpage |
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\bibliography{multipole} |
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\end{document} |
