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\begin{document} |
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\title{Real space electrostatics for multipoles. III. Dielectric Properties} |
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University |
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of Notre Dame, Notre Dame, IN 46556} |
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\author{Thomas Parsons} |
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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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\author{Kathie E. Newman} |
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\affiliation{Department of Physics, University |
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of Notre Dame, Notre Dame, IN 46556} |
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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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\begin{abstract} |
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In the first two papers in this series, we developed new shifted |
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potential (SP), gradient shifted force (GSF), and Taylor shifted |
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force (TSF) real-space methods for multipole interactions in |
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condensed phase simulations. Here, we discuss the dielectric |
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properties of fluids that emerge from simulations using these |
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methods. Most electrostatic methods (including the Ewald sum) |
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require correction to the conducting boundary fluctuation formula |
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for the static dielectric constants, and we discuss the derivation |
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of these corrections for the new real space methods. For quadrupolar |
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fluids, the analogous material property is the quadrupolar |
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susceptibility. As in the dipolar case, the fluctuation formula for |
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the quadrupolar susceptibility has corrections that depend on the |
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electrostatic method being utillized. One of the most important |
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effects measured by both the static dielectric and quadrupolar |
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susceptibility is the ability to screen charges embedded in the |
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fluid. We use potentials of mean force between solvated ions to |
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discuss how geometric factors can lead to distance-dependent |
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screening in both quadrupolar and dipolar fluids. |
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\end{abstract} |
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\maketitle |
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\section{Introduction} |
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Over the past several years, there has been increasing interest in |
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pairwise or ``real space'' methods for computing electrostatic |
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interactions in condensed phase |
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simulations.\cite{Wolf99,Zahn02,Kast03,Beckd05,Ma05,Wu:2005nr,Fennell06,Fukuda:2013uq,Wang:2016kx} |
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These techniques were initially developed by Wolf {\it et al.} in |
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their work towards an $\mathcal{O}(N)$ Coulombic sum.\cite{Wolf99} |
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Wolf's method of using cutoff neutralization and electrostatic damping |
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is able to obtain excellent agreement with Madelung energies in ionic |
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crystals.\cite{Wolf99} |
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|
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Zahn \textit{et al.}\cite{Zahn02} and Fennell and Gezelter extended |
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this method using shifted force approximations at the cutoff distance |
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in order to conserve total energy in molecular dynamics |
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simulations.\cite{Fennell06} Other recent advances in real-space |
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methods for systems of point charges have included explicit |
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elimination of the net multipole moments inside the cutoff sphere |
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around each charge site.\cite{Fukuda:2013uq,Wang:2016kx} |
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|
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In the previous two papers in this series, we developed three |
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generalized real space methods: shifted potential (SP), gradient |
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shifted force (GSF), and Taylor shifted force (TSF).\cite{PaperI, |
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PaperII} These methods evaluate electrostatic interactions for |
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charges and higher order multipoles using a finite-radius cutoff |
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sphere. The neutralization and damping of local moments within the |
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cutoff sphere is a multipolar generalization of Wolf's sum. In the |
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GSF and TSF methods, additional terms are added to the potential |
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energy so that forces and torques also vanish smoothly at the cutoff |
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radius. This ensures that the total energy is conserved in molecular |
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dynamics simulations. |
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|
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One of the most stringent tests of any new electrostatic method is the |
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fidelity with which that method can reproduce the bulk-phase |
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polarizability or equivalently, the dielectric properties of a |
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fluid. Before the advent of computer simulations, Kirkwood and Onsager |
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developed fluctuation formulae for the dielectric properties of |
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dipolar fluids.\cite{Kirkwood39,Onsagar36} Along with projections of |
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the frequency-dependent dielectric to zero frequency, these |
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fluctuation formulae are now widely used to predict the static |
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dielectric constant of simulated materials. |
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|
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If we consider a system of dipolar or quadrupolar molecules under the |
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influence of an external field or field gradient, the net polarization |
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of the system will largely be proportional to the applied |
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perturbation.\cite{Chitanvis96, Stern-Feller03, SalvchovI14, |
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SalvchovII14} In simulations, the net polarization of the system is |
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also determined by the interactions \textit{between} the |
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molecules. Therefore the macroscopic polarizablity obtained from a |
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simulation depends on the details of the electrostatic interaction |
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methods that were employed in the simulation. To determine the |
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relevant physical properties of the multipolar fluid from the system |
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fluctuations, the interactions between molecules must be incorporated |
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into the formalism for the bulk properties. |
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|
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In most simulations, bulk materials are treated using periodic |
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replicas of small regions, and this level of approximation requires |
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corrections to the fluctuation formulae that were derived for the bulk |
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fluids. In 1983 Neumann proposed a general formula for evaluating |
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dielectric properties of dipolar fluids using both Ewald and |
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real-space cutoff methods.\cite{NeumannI83} Steinhauser and Neumann |
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used this formula to evaluate the corrected dielectric constant for |
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the Stockmayer fluid using two different methods: Ewald-Kornfield (EK) |
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and reaction field (RF) methods.\cite{NeumannII83} |
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|
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Zahn \textit{et al.}\cite{Zahn02} utilized this approach and evaluated |
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the correction factor for using damped shifted charge-charge |
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kernel. This was later generalized by Izvekov \textit{et |
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al.},\cite{Izvekov08} who noted that the expression for the |
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dielectric constant reduces to widely-used conducting boundary formula |
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for real-space methods that have first derivatives that vanish at the |
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cutoff radius. |
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One of the primary topics of this paper is the derivation of |
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correction factors for the three new real space methods. We find that |
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the correction formulae for dipolar molecules depends not only on the |
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methodology being used, but also on whether the molecular dipoles are |
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treated using point charges or point dipoles. We derive correction |
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factors for both cases. |
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|
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In quadrupolar fluids, the relationship between quadrupolar |
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susceptibility and the dielectric constant is not as straightforward |
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as in the dipolar case. The effective dielectric constant depends on |
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the geometry of the external (or internal) field |
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perturbation.\cite{Ernst92} Significant efforts have been made to |
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increase our understanding the dielectric properties of these |
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fluids,\cite{JeonI03,JeonII03,Chitanvis96} although a general |
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correction formula has not yet been developed. |
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|
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In this paper we derive general formulae for calculating the |
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quadrupolar susceptibility of quadrupolar fluids. We also evaluate the |
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correction factor for SP, GSF, and TSF methods for quadrupolar fluids |
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interacting via point charges, point dipoles or directly through |
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quadrupole-quadrupole interactions. |
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We also calculate the screening behavior for two ions immersed in |
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multipolar fluids to estimate the distance dependence of charge |
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screening in both dipolar and quadrupolar fluids. We use three |
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distinct methods to compare our analytical results with computer |
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simulations: |
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\begin{enumerate} |
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\item responses of the fluid to external perturbations, |
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\item fluctuations of system multipole moments, and |
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\item potentials of mean force between solvated ions, |
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\end{enumerate} |
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Under the influence of weak external fields, the bulk polarization of |
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the system is primarily a linear response to the perturbation, where |
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proportionality constant depends on the electrostatic interactions |
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between the multipoles. The fluctuation formulae connect bulk |
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properties of the fluid to equilibrium fluctuations in the system |
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multipolar moments during a simulation. These fluctuations also depend |
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on the form of the electrostatic interactions between molecules. |
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Therefore, the connections between the actual bulk properties and both |
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the computed fluctuation and external field responses must be modified |
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accordingly. |
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|
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The potential of mean force (PMF) allows calculation of an effective |
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dielectric constant or screening factor from the potential energy |
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between ions before and after dielectric material is introduced. |
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Computing the PMF between embedded point charges is an additional |
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check on the bulk properties computed via the other two methods. |
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\section{The Real-Space Methods} |
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\section{Dipolar Fluids and the Dielectric Constant} |
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Dielectric properties of a fluid arise mainly from responses of the |
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fluid to either applied fields or transient fields internal to the |
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fluid. In response to an applied field, the molecules have electronic |
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polarizabilities, changes to internal bond lengths, and reorientations |
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towards the direction of the applied field. There is an added |
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complication that in the presence of external field, the perturbation |
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experienced by any single molecule is not only due to external field |
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but also to the fields produced by the all other molecules in the |
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system. |
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|
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\subsection{Response to External Perturbations} |
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In the presence of uniform electric field $\mathbf{E}$, an individual |
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molecule with a permanent dipole moment $p_o$ will realign along the |
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direction of the field with an average polarization given by |
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\begin{equation} |
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\braket{\mathbf{p}} = \epsilon_0 \alpha_p \mathbf{E}, |
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\end{equation} |
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where $\alpha_p = {p_o}^2 / 3 \epsilon_0 k_B T$ is the contribution to |
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molecular polarizability due solely to reorientation dynamics. |
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Because the applied field must overcome thermal motion, the |
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orientational polarization depends inversely on the temperature. |
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|
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Likewise, a condensed phase system of permanent dipoles will also |
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polarize along the direction of an applied field. The polarization |
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density of the system is |
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\begin{equation} |
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\textbf{P} = \epsilon_o \alpha_{D} \mathbf{E}. |
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\label{pertDipole} |
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\end{equation} |
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The constant $\alpha_D$ is the macroscopic polarizability, which is an |
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emergent property of the dipolar fluid. Note that the polarizability, |
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$\alpha_D$ is distinct from the dipole susceptibility, $\chi_D$, |
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which is the quantity most directly related to the static dielectric |
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constant, $\epsilon = 1 + \chi_D$. |
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|
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\subsection{Fluctuation Formula} |
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% Using the Ewald sum under tin-foil boundary conditions, $\epsilon$ can |
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% be calculated for systems of orientationally-polarized molecules, |
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% \begin{equation} |
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% \epsilon = 1 + \frac{\langle M^2\rangle}{3\epsilon_0k_\textrm{B}TV}, |
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% \label{eq:staticDielectric} |
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% \end{equation} |
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% where $\epsilon_0$ is the permittivity of free space and |
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% $\langle M^2\rangle$ is the fluctuation of the system dipole |
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% moment.\cite{Allen89} The numerator in the fractional term in |
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% equation (\ref{eq:staticDielectric}) is identical to the quantity |
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% calculated in the finite-system Kirkwood $g$ factor ($G_k$): |
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% \begin{equation} |
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% G_k = \frac{\langle M^2\rangle}{N\mu^2}, |
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% \label{eq:KirkwoodFactor} |
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% \end{equation} |
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% where $\mu$ is the dipole moment of a single molecule of the |
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% homogeneous system.\cite{NeumannI83,NeumannII83,Neumann84,Neumann85} The |
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% fluctuation term in both equation (\ref{eq:staticDielectric}) and |
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% (\ref{eq:KirkwoodFactor}) is calculated as follows, |
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% \begin{equation} |
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% \begin{split} |
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% \langle M^2\rangle &= \langle\bm{M}\cdot\bm{M}\rangle |
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% - \langle\bm{M}\rangle\cdot\langle\bm{M}\rangle \\ |
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% &= \langle M_x^2+M_y^2+M_z^2\rangle |
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% - (\langle M_x\rangle^2 + \langle M_x\rangle^2 |
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% + \langle M_x\rangle^2). |
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% \end{split} |
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% \label{eq:fluctBoxDipole} |
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% \end{equation} |
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% This fluctuation term can be accumulated during a simulation; however, |
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% it converges rather slowly, thus requiring multi-nanosecond simulation |
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% times.\cite{Horn04} In the case of tin-foil boundary conditions, the |
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% dielectric/surface term of the Ewald summation is equal to zero. |
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|
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For a system of dipolar molecules at thermal equilibrium, we can |
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define both a system dipole moment, $\mathbf{M} = \sum_i \mathbf{p}_i$ |
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as well as a dipole polarization density, |
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$\mathbf{P} = \braket{\mathbf{M}}/V$. |
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|
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In the presence of applied field $\mathbf{E}$, the polarization |
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density can be expressed in terms of fluctuations in the net dipole |
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moment, |
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\begin{equation} |
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\mathbf{P} = \epsilon_o \frac{\braket{\mathbf{M}^2}-{\braket{\mathbf{M}}}^2}{3 \epsilon_o V k_B T}\bf{E} |
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\label{flucDipole} |
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\end{equation} |
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This has structural similarity with the Boltzmann average for the |
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polarization of a single molecule. Here |
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$ \braket{\mathbf{M}^2}-{\braket{\mathbf{M}}}^2$ measures fluctuations |
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in the net dipole moment, |
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\begin{equation} |
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\langle \mathbf{M}^2 \rangle - \langle \mathbf{M} \rangle^2 = |
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\langle M_x^2+M_y^2+M_z^2 \rangle - \left( \langle M_x \rangle^2 + |
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\langle M_y \rangle^2 + |
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\langle M_z \rangle^2 \right). |
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\label{eq:flucDip} |
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\end{equation} |
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For the limiting case $\textbf{E} \rightarrow 0 $, the ensemble |
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average of both the net dipole moment $\braket{\mathbf{M}}$ and |
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dipolar polarization $\bf{P}$ tends to vanish but |
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$\braket{\mathbf{M}^2}$ does not. The macroscopic dipole |
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polarizability can therefore be written as, |
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\begin{equation} |
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\alpha_D = \frac{\braket{\mathbf{M}^2}-{\braket{\mathbf{M}}}^2}{3 \epsilon_o V k_B T} |
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\end{equation} |
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This relationship between a macroscopic property of dipolar material |
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and microscopic fluctuations is true even when the applied field |
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$ \textbf{E} \rightarrow 0 $. |
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|
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\subsection{Correction Factors} |
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|
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In the presence of a uniform external field $ \mathbf{E}^\circ$, the |
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total electric field at $\mathbf{r}$ depends on the polarization |
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density at all other points in the system,\cite{NeumannI83} |
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\begin{equation} |
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\mathbf{E}(\mathbf{r}) = \mathbf{E}^\circ(\mathbf{r}) + |
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\frac{1}{4\pi\epsilon_o} \int d\mathbf{r}^\prime |
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\mathbf{T}(\mathbf{r}-\mathbf{r}^\prime)\cdot |
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{\mathbf{P}(\mathbf{r}^\prime)}. |
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\label{eq:localField} |
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\end{equation} |
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$\mathbf{T}$ is the dipole interaction tensor connecting dipoles at |
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$\mathbf{r}^\prime$ with the point of interest ($\mathbf{r}$). |
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|
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In simulations of dipolar fluids, the molecular dipoles may be |
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represented either by closely-spaced point charges or by |
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point dipoles (see 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} |
320 |
|
|
\end{figure} |
321 |
|
|
In the case where point charges are interacting via an electrostatic |
322 |
|
|
kernel, $v(r)$, the effective {\it molecular} dipole tensor, |
323 |
|
|
$\mathbf{T}$ is obtained from two successive applications of the |
324 |
|
|
gradient operator to the electrostatic kernel, |
325 |
gezelter |
4400 |
\begin{align} |
326 |
|
|
\mathbf{T}_{\alpha \beta}(r) =& \nabla_\alpha \nabla_\beta |
327 |
|
|
\left(v(r)\right) \\ |
328 |
|
|
=& \delta_{\alpha \beta} |
329 |
mlamichh |
4353 |
\left(\frac{1}{r} v^\prime(r) \right) + \frac{r_{\alpha} |
330 |
|
|
r_{\beta}}{r^2} \left( v^{\prime \prime}(r) - \frac{1}{r} |
331 |
|
|
v^{\prime}(r) \right) |
332 |
|
|
\label{dipole-chargeTensor} |
333 |
gezelter |
4400 |
\end{align} |
334 |
mlamichh |
4353 |
where $v(r)$ may be either the bare kernel ($1/r$) or one of the |
335 |
|
|
modified (Wolf or DSF) kernels. This tensor describes the effective |
336 |
gezelter |
4400 |
interaction between molecular dipoles ($\mathbf{D}$) in Gaussian units |
337 |
|
|
as $-\mathbf{D} \cdot \mathbf{T} \cdot \mathbf{D}$. |
338 |
mlamichh |
4353 |
|
339 |
gezelter |
4400 |
When utilizing any of the three new real-space methods for point |
340 |
|
|
\textit{dipoles}, the tensor is explicitly constructed, |
341 |
mlamichh |
4353 |
\begin{equation} |
342 |
|
|
\mathbf{T}_{\alpha \beta}(r) = \delta_{\alpha \beta} v_{21}(r) + |
343 |
|
|
\frac{r_{\alpha} r_{\beta}}{r^2} v_{22}(r) |
344 |
|
|
\label{dipole-diopleTensor} |
345 |
|
|
\end{equation} |
346 |
|
|
where the functions $v_{21}(r)$ and $v_{22}(r)$ depend on the level of |
347 |
gezelter |
4400 |
the approximation.\cite{PaperI,PaperII} Although the Taylor-shifted |
348 |
|
|
(TSF) and gradient-shifted (GSF) models produce to the same $v(r)$ |
349 |
|
|
function for point charges, they have distinct forms for the |
350 |
|
|
dipole-dipole interaction. |
351 |
mlamichh |
4353 |
|
352 |
gezelter |
4400 |
Using a constitutive relation, the polarization density |
353 |
|
|
$\mathbf{P}(\mathbf{r})$ is given by, |
354 |
mlamichh |
4353 |
\begin{equation} |
355 |
gezelter |
4400 |
\mathbf{P}(\mathbf{r}) = \epsilon_o \chi_D |
356 |
|
|
\left(\mathbf{E}^\circ(\mathbf{r}) + \frac{1}{4\pi\epsilon_o} \int |
357 |
|
|
d\mathbf{r}^\prime \mathbf{T}(\mathbf{r}-\mathbf{r}^\prime ) \cdot \mathbf{P}(\mathbf{r}^\prime)\right). |
358 |
mlamichh |
4353 |
\label{constDipole} |
359 |
|
|
\end{equation} |
360 |
gezelter |
4400 |
Note that $\chi_D$ explicitly depends on the details of the dipole |
361 |
|
|
interaction tensor. Neumann \textit{et al.} |
362 |
|
|
\cite{NeumannI83,NeumannII83,Neumann84,Neumann85} derived an elegant |
363 |
|
|
way to modify the fluctuation formula to correct for approximate |
364 |
|
|
interaction tensors. This correction was derived using a Fourier |
365 |
|
|
representation of the interaction tensor, |
366 |
|
|
$\tilde{\mathbf{T}}(\mathbf{k})$, and involves the quantity, |
367 |
mlamichh |
4353 |
\begin{equation} |
368 |
gezelter |
4400 |
A = \frac{3}{4\pi}\tilde{\mathbf{T}}(0) = \frac{3}{4\pi} \int_V |
369 |
|
|
d\mathbf{r} \mathbf{T}(\mathbf{r}) |
370 |
mlamichh |
4353 |
\end{equation} |
371 |
gezelter |
4400 |
which is the $k \rightarrow 0$ limit of $\tilde{\mathbf{T}}$. Using |
372 |
|
|
this quantity (originally called $Q$ in |
373 |
|
|
refs. \onlinecite{NeumannI83,NeumannII83,Neumann84,Neumann85}), the |
374 |
|
|
dielectric constant can be computed |
375 |
mlamichh |
4353 |
\begin{equation} |
376 |
gezelter |
4400 |
\epsilon = \frac{3+(A + 2)(\epsilon_{CB}-1)}{3 + (A -1)(\epsilon_{CB}-1)} |
377 |
|
|
\label{correctionFormula} |
378 |
mlamichh |
4353 |
\end{equation} |
379 |
gezelter |
4400 |
where $\epsilon_{CB}$ is the widely-used conducting boundary |
380 |
|
|
expression for the dielectric constant, |
381 |
mlamichh |
4353 |
\begin{equation} |
382 |
gezelter |
4400 |
\epsilon_{CB} = 1 + \frac{\braket{\bf{M}^2}-{\braket{\bf{M}}}^2}{3 |
383 |
|
|
\epsilon_o V k_B T} |
384 |
mlamichh |
4353 |
\label{correctionFormula} |
385 |
|
|
\end{equation} |
386 |
gezelter |
4400 |
Equation (\ref{correctionFormula}) allows estimation of the static |
387 |
|
|
dielectric constant from fluctuations easily computed directly from |
388 |
|
|
simulations. |
389 |
|
|
|
390 |
|
|
|
391 |
|
|
% Here $\chi^*_D$ is a dipolar susceptibility can be expressed in terms of dielectric constant as $ \chi^*_D = \epsilon - 1$ which different than macroscopic dipolar polarizability $\alpha_D$ in the sections \ref{subsec:perturbation} and \ref{subsec:fluctuation}. We can split integral into two parts: singular part i.e $|\textbf{r}-\textbf{r}'|\rightarrow 0 $ and non-singular part i.e $|\textbf{r}-\textbf{r}'| > 0 $ . The singular part of the integral can be written as,\cite{NeumannI83, Jackson98} |
392 |
|
|
% \begin{equation} |
393 |
|
|
% \frac{1}{4\pi\epsilon_o} \int_{|\textbf{r}-\textbf{r}'| \rightarrow 0} d^3r'\; \textbf{T}(\textbf{r}-\textbf{r}')\cdot {\textbf{P}(\textbf{r}')} = - \frac{\textbf{P}(\textbf{r})}{3\epsilon_o} |
394 |
|
|
% \label{singular} |
395 |
|
|
% \end{equation} |
396 |
|
|
% Substituting equation (\ref{singular}) in the equation (\ref{constDipole}) we get, |
397 |
|
|
% \begin{equation} |
398 |
|
|
% \textbf{P}(\textbf{r}) = 3 \epsilon_o\; \frac{\chi^*_D}{\chi^*_D + 3} \left(\textbf{E}^o(\textbf{r}) + \frac{1}{4\pi\epsilon_o} \int_{|\textbf{r}-\textbf{r}'| > 0} d^3r'\; \textbf{T}(\textbf{r}-\textbf{r}')\cdot {\textbf{P}(\textbf{r}')}\right). |
399 |
|
|
% \end{equation} |
400 |
|
|
% For both polarization and electric field homogeneous, this can be easily solved using Fourier transformation, |
401 |
|
|
% \begin{equation} |
402 |
|
|
% \textbf{P}(\mathbf{k}) = 3 \epsilon_o\; \frac{\chi^*_D}{\chi^*_D + 3} \left(1- \frac{3}{4\pi}\;\frac{\chi^*_D}{\chi^*_D + 3}\; \textbf{T}(\mathbf{k})\right)^{-1}\textbf{E}^o(\mathbf{k}). |
403 |
|
|
% \end{equation} |
404 |
|
|
% For homogeneous applied field Fourier component is non-zero only if $\kappa = 0$. Therefore, |
405 |
|
|
% \begin{equation} |
406 |
|
|
% \textbf{P}(0) = 3 \epsilon_o\; \frac{\chi^*_D}{\chi^*_D + 3} \left(1- \frac{\chi^*_D}{\chi^*_D + 3}\; A_{dipole})\right)^{-1}\textbf{E}^o({0}). |
407 |
|
|
% \label{fourierDipole} |
408 |
|
|
% \end{equation} |
409 |
|
|
% where $A_{dipole}=\frac{3}{4\pi}T(0) = \frac{3}{4\pi} \int_V d^3r\;T(r)$. Now equation (\ref{fourierDipole}) can be compared with equation (\ref{flucDipole}). Therefore, |
410 |
|
|
% \begin{equation} |
411 |
|
|
% \frac{\braket{\bf{M}^2}-{\braket{\bf{M}}}^2}{3 \epsilon_o V k_B T} = \frac{3\;\chi^*_D}{\chi^*_D + 3} \left(1- \frac{\chi^*_D}{\chi^*_D + 3}\; A_{dipole})\right)^{-1} |
412 |
|
|
% \end{equation} |
413 |
|
|
% Substituting $\chi^*_D = \epsilon-1$ and $ \frac{\braket{\bf{M}^2}-{\braket{\bf{M}}}^2}{3 \epsilon_o V k_B T} = \epsilon_{CB}-1 = \alpha_D$ in above equation we get, |
414 |
|
|
% \begin{equation} |
415 |
|
|
% \epsilon = \frac{3+(A_{dipole} + 2)(\epsilon_{CB}-1)}{3+(A_{dipole} -1)(\epsilon_{CB}-1)} = \frac{3+(A_{dipole} + 2)\alpha_D}{3+(A_{dipole} -1)\alpha_D} |
416 |
|
|
% \label{correctionFormula} |
417 |
|
|
% \end{equation} |
418 |
|
|
|
419 |
|
|
|
420 |
|
|
We have utilized the Neumann \textit{et al.} approach for thre three |
421 |
|
|
new real-space methods, and obtain method-dependent correction |
422 |
|
|
factors. The expression for the correction factor also depends on |
423 |
|
|
whether the simulation involves point charges or point dipoles to |
424 |
|
|
represent the molecular dipoles. These corrections factors are |
425 |
|
|
listed in Table \ref{tab:A}. |
426 |
mlamichh |
4353 |
\begin{table} |
427 |
gezelter |
4400 |
\caption{Expressions for the dipolar correction factor ($A$) for the |
428 |
|
|
real-space electrostatic methods in terms of the damping parameter |
429 |
mlamichh |
4353 |
($\alpha$) and the cutoff radius ($r_c$). The Ewald-Kornfeld result |
430 |
gezelter |
4400 |
derived in Refs. \onlinecite{Adams80,Adams81,NeumannI83} is shown |
431 |
|
|
for comparison using the Ewald convergence parameter ($\kappa$) |
432 |
|
|
and the real-space cutoff value ($r_c$). } |
433 |
mlamichh |
4353 |
\label{tab:A} |
434 |
|
|
{% |
435 |
gezelter |
4400 |
\begin{tabular}{l|c|c|c} |
436 |
mlamichh |
4353 |
|
437 |
gezelter |
4400 |
Method & point charges & point dipoles \\ |
438 |
mlamichh |
4353 |
\hline |
439 |
|
|
Spherical Cutoff (SC) & $\mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $\mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ \\ |
440 |
|
|
Shifted Potental (SP) & $ \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}}\left(1+\frac{2\alpha^2 {r_c}^2}{3} \right)e^{-\alpha^2{r_c}^2} $\\ |
441 |
|
|
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} $ \\ |
442 |
|
|
Taylor-shifted (TSF) & 1 & 1 \\ |
443 |
gezelter |
4400 |
Ewald-Kornfeld (EK) & $\mathrm{erf}(r_c \kappa) - \frac{2 \kappa |
444 |
|
|
r_c}{\sqrt{\pi}} e^{-\kappa^2 r_c^2}$ & \\ |
445 |
mlamichh |
4353 |
\end{tabular}% |
446 |
|
|
} |
447 |
|
|
\end{table} |
448 |
gezelter |
4400 |
Note that for point charges, the GSF and TSF methods produce estimates |
449 |
|
|
of the dielectric that need no correction, and the TSF method likewise |
450 |
|
|
needs no correction for point dipoles. |
451 |
gezelter |
4399 |
|
452 |
|
|
\section{Quadrupolar Fluids and the Quadrupolar Susceptibility} |
453 |
|
|
\subsection{Response to External Perturbations} |
454 |
|
|
|
455 |
gezelter |
4400 |
A molecule with a permanent quadrupole, $\mathsf{q}$, will align in |
456 |
|
|
the presence of an electric field gradient $\nabla\mathbf{E}$. The |
457 |
|
|
anisotropic polarization of the quadrupole is given |
458 |
|
|
by,\cite{AduGyamfi78,AduGyamfi81} |
459 |
gezelter |
4399 |
\begin{equation} |
460 |
gezelter |
4400 |
\braket{\mathsf{q}} - \frac{\mathbf{I}}{3} |
461 |
|
|
\mathrm{Tr}(\mathsf{q}) = \epsilon_o \alpha_q \nabla\mathbf{E}, |
462 |
gezelter |
4399 |
\end{equation} |
463 |
gezelter |
4400 |
where $\alpha_q = q_o^2 / 15 \epsilon_o k_B T $ is a molecular quadrupole |
464 |
|
|
polarizablity and $q_o$ is an effective quadrupole moment for the molecule, |
465 |
|
|
\begin{equation} |
466 |
|
|
q_o^2 = 3 \mathsf{q}:\mathsf{q} - \mathrm{Tr}(\mathsf{q})^2. |
467 |
|
|
\end{equation} |
468 |
gezelter |
4399 |
|
469 |
gezelter |
4400 |
In the presence of an external field gradient, a system of quadrupolar |
470 |
|
|
molecules also organizes with an anisotropic polarization, |
471 |
gezelter |
4399 |
\begin{equation} |
472 |
gezelter |
4400 |
\mathsf{Q} - \frac{\mathbf{I}}{3} \mathrm{Tr}(\mathsf{Q}) = \epsilon_o |
473 |
|
|
\alpha_Q \nabla\mathbf{E} |
474 |
|
|
\end{equation} |
475 |
|
|
where $\mathsf{Q}$ is the traced quadrupole density of the system and |
476 |
|
|
$\alpha_Q$ is a macroscopic quadrupole polarizability which has |
477 |
|
|
dimensions of $\mathrm{length}^{-2}$. Equivalently, the traceless form |
478 |
|
|
may be used, |
479 |
|
|
\begin{equation} |
480 |
|
|
\mathsf{\Theta} = 3 \epsilon_o \alpha_Q \nabla\mathbf{E}, |
481 |
gezelter |
4399 |
\label{pertQuad} |
482 |
|
|
\end{equation} |
483 |
gezelter |
4400 |
where |
484 |
|
|
$\mathsf{\Theta} = 3\mathsf{Q} - \mathbf{I} \mathrm{Tr}(\mathsf{Q})$ |
485 |
|
|
is the traceless tensor that also describes the system quadrupole |
486 |
|
|
density. It is this tensor that will be utilized to derive correction |
487 |
|
|
factors below. |
488 |
|
|
|
489 |
gezelter |
4399 |
\subsection{Fluctuation Formula} |
490 |
gezelter |
4400 |
As in the dipolar case, we may define a system quadrupole moment, |
491 |
|
|
$\mathsf{M}_Q = \sum_i \mathsf{q}_i$ and the traced quadrupolar |
492 |
|
|
density, $\mathsf{Q} = \mathsf{M}_Q / V$. A fluctuation formula can |
493 |
|
|
be written for a system comprising quadrupolar |
494 |
|
|
molecules,\cite{LoganI81,LoganII82,LoganIII82} |
495 |
gezelter |
4399 |
\begin{equation} |
496 |
gezelter |
4400 |
\mathsf{Q} - \frac{\mathbf{I}}{3} \mathrm{Tr}(\mathsf{Q}) = \epsilon_o |
497 |
|
|
\frac{\braket{\mathsf{M}_Q^2}-{\braket{\mathsf{M}_Q}}^2}{15 \epsilon_o |
498 |
|
|
V k_B T} \cdot \nabla\mathbf{E}. |
499 |
gezelter |
4399 |
\label{flucQuad} |
500 |
|
|
\end{equation} |
501 |
gezelter |
4400 |
Some care is needed in the definitions of the averaged quantities. These |
502 |
|
|
refer to the effective quadrupole moment of the system, and they are |
503 |
|
|
computed as follows, |
504 |
|
|
\begin{align} |
505 |
|
|
\braket{\mathsf{M}_Q^2} &= \braket{3 \mathsf{M}_Q:\mathsf{M}_Q - |
506 |
|
|
\mathrm{Tr}(\mathsf{M}_Q)^2 }\\ |
507 |
|
|
\braket{\mathsf{M}_Q}^2 &= 3 \braket{\mathsf{M}_Q}:\braket{\mathsf{M}_Q} - |
508 |
|
|
\mathrm{Tr}(\braket{\mathsf{M}_Q})^2 |
509 |
gezelter |
4401 |
\label{eq:flucQuad} |
510 |
gezelter |
4400 |
\end{align} |
511 |
|
|
The macroscopic quadrupolarizability is given by, |
512 |
gezelter |
4399 |
\begin{equation} |
513 |
gezelter |
4400 |
\alpha_Q = \frac{\braket{\mathsf{M}_Q^2}-{\braket{\mathsf{M}_Q}}^2}{15 \epsilon_o V k_B T} |
514 |
gezelter |
4399 |
\label{propConstQuad} |
515 |
|
|
\end{equation} |
516 |
gezelter |
4400 |
This relationship between a macroscopic property of a quadrupolar |
517 |
|
|
fluid and microscopic fluctuations should be true even when the |
518 |
|
|
applied field gradient $\nabla \mathbf{E} \rightarrow 0$. |
519 |
gezelter |
4399 |
|
520 |
|
|
\subsection{Correction Factors} |
521 |
gezelter |
4400 |
In this section we generalize the treatment of Neumann \textit{et al.} |
522 |
|
|
for quadrupolar fluids. Interactions involving multiple quadrupoles |
523 |
|
|
are rank 4 tensors, and we therefore describe quantities in this |
524 |
|
|
section using Einstein notation. |
525 |
|
|
|
526 |
|
|
In the presence of a uniform external field gradient, |
527 |
|
|
$\partial_\alpha {E}^\circ_\beta $, the total field gradient at |
528 |
|
|
$\mathbf{r}$ depends on the quadrupole polarization density at all |
529 |
|
|
other points in the system, |
530 |
mlamichh |
4353 |
\begin{equation} |
531 |
gezelter |
4400 |
\partial_\alpha E_\beta(\mathbf{r}) = \partial_\alpha |
532 |
|
|
{E}^\circ_\beta(\mathbf{r}) + \frac{1}{4\pi \epsilon_o}\int |
533 |
|
|
T_{\alpha\beta\gamma\delta}({\mathbf{r}-\mathbf{r}^\prime}) |
534 |
|
|
Q_{\gamma\delta}(\mathbf{r}^\prime) d\mathbf{r}^\prime |
535 |
mlamichh |
4353 |
\label{gradMaxwell} |
536 |
|
|
\end{equation} |
537 |
gezelter |
4400 |
where and $T_{\alpha\beta\gamma\delta}$ is the quadrupole interaction |
538 |
|
|
tensor connecting quadrupoles at $\mathbf{r}^\prime$ with the point of |
539 |
|
|
interest ($\mathbf{r}$). |
540 |
mlamichh |
4353 |
|
541 |
gezelter |
4400 |
\begin{figure} |
542 |
|
|
\includegraphics[width=3in]{QuadrupoleFigure} |
543 |
|
|
\caption{With the real-space electrostatic methods, the molecular |
544 |
|
|
quadrupole tensor, $\mathbf{T}_{\alpha\beta\gamma\delta}(r)$, |
545 |
|
|
governing interactions between molecules is not the same for |
546 |
|
|
quadrupoles represented via sets of charges, point dipoles, or by |
547 |
|
|
single point quadrupoles.} |
548 |
|
|
\label{fig:quadrupolarFluid} |
549 |
|
|
\end{figure} |
550 |
|
|
|
551 |
|
|
In simulations of quadrupolar fluids, the molecular quadrupoles may be |
552 |
|
|
represented by closely-spaced point charges, by multiple point |
553 |
|
|
dipoles, or by a single point quadrupole (see |
554 |
|
|
Fig. \ref{fig:quadrupolarFluid}). In the case where point charges are |
555 |
|
|
interacting via an electrostatic kernel, $v(r)$, the effective |
556 |
|
|
molecular quadrupole tensor can obtained from four successive |
557 |
|
|
applications of the gradient operator to the electrostatic kernel, |
558 |
|
|
\begin{eqnarray} |
559 |
|
|
T_{\alpha\beta\gamma\delta}(\mathbf{r}) &=& \nabla_\alpha \nabla_\beta |
560 |
|
|
\nabla_\gamma \nabla_\delta v(r) \\ |
561 |
|
|
&=& \left(\delta_{\alpha\beta}\delta_{\gamma\delta} + |
562 |
|
|
\delta_{\alpha\gamma}\delta_{\beta\delta}+ |
563 |
|
|
\delta_{\alpha\delta}\delta_{\beta\gamma}\right)\left(-\frac{v^\prime(r)}{r^3} |
564 |
|
|
+ \frac{v^{\prime \prime}(r)}{r^2}\right) \nonumber \\ |
565 |
|
|
& &+ \left(\delta_{\alpha\beta} r_\gamma r_\delta + 5 \mathrm{~permutations} |
566 |
|
|
\right) \left(\frac{3v^\prime(r)}{r^5}-\frac{3v^{\prime \prime}(r)}{r^4} + |
567 |
|
|
\frac{v^{\prime \prime \prime}(r)}{r^3}\right) \nonumber \\ |
568 |
|
|
& &+ r_\alpha r_\beta r_\gamma r_\delta |
569 |
|
|
\left(-\frac{15v^\prime(r)}{r^7}+\frac{15v^{\prime \prime}(r)}{r^6}-\frac{6v^{\prime |
570 |
|
|
\prime \prime}(r)}{r^5} + \frac{v^{\prime \prime \prime \prime}(r)}{r^4}\right), |
571 |
mlamichh |
4353 |
\label{quadCharge} |
572 |
gezelter |
4400 |
\end{eqnarray} |
573 |
|
|
where $v(r)$ can either be the electrostatic kernel ($1/r$) or one of |
574 |
|
|
the modified (Wolf or DSF) kernels. |
575 |
mlamichh |
4353 |
|
576 |
gezelter |
4400 |
Similarly, when representing quadrupolar molecules with multiple point |
577 |
|
|
dipoles, the molecular quadrupole interaction tensor can be obtained |
578 |
|
|
using two successive applications of the gradient operator to the |
579 |
|
|
dipole interaction tensor, |
580 |
|
|
\begin{eqnarray} |
581 |
|
|
T_{\alpha\beta\gamma\delta}(\mathbf{r}) &=& \nabla_\alpha \nabla_\beta |
582 |
|
|
v_{\gamma\delta}(\mathbf{r}) \\ |
583 |
|
|
& = & \delta_{\alpha\beta}\delta_{\gamma\delta} \frac{v^\prime_{21}(r)}{r} + |
584 |
|
|
\left(\delta_{\alpha\gamma}\delta_{\beta\delta}+ |
585 |
|
|
\delta_{\alpha\delta}\delta_{\beta\gamma}\right)\frac{v_{22}(r)}{r^2} |
586 |
|
|
\nonumber\\ |
587 |
|
|
& &+ \delta_{\gamma\delta} r_\alpha r_\beta |
588 |
|
|
\left(\frac{v^{\prime \prime}_{21}(r)}{r^2}-\frac{v^\prime_{21}(r)}{r^3} \right)\nonumber \\ |
589 |
|
|
& &+\left(\delta_{\alpha\beta} r_\gamma r_\delta + |
590 |
|
|
\delta_{\alpha\gamma} r_\beta r_\delta +\delta_{\alpha\delta} |
591 |
|
|
r_\gamma r_\beta + \delta_{\beta\gamma} r_\alpha r_\delta |
592 |
|
|
+\delta_{\beta\delta} r_\alpha r_\gamma \right) |
593 |
|
|
\left(\frac{v^\prime_{22}(r)}{r^3}-\frac{2v_{22}(r)}{r^4}\right) |
594 |
|
|
\nonumber \\ |
595 |
|
|
& &+ r_\alpha r_\beta r_\gamma r_\delta |
596 |
|
|
\left(\frac{v^{\prime |
597 |
|
|
\prime}_{22}(r)}{r^4}-\frac{5v^\prime_{22}(r)}{r^5}+\frac{8v_{22}(r)}{r^6}\right), |
598 |
mlamichh |
4353 |
\label{quadDip} |
599 |
gezelter |
4400 |
\end{eqnarray} |
600 |
|
|
where $v_{\gamma\delta}(\mathbf{r})$ is a dipole-dipole interaction |
601 |
|
|
tensor that depends on the level of the |
602 |
|
|
approximation.\cite{PaperI,PaperII} Similarly $v_{21}(r)$ and |
603 |
|
|
$v_{22}(r)$ are the radial function for different real space cutoff |
604 |
|
|
methods defined in the first paper in this series.\cite{PaperI} |
605 |
|
|
|
606 |
|
|
For quadrupolar liquids modeled using point quadrupoles, the |
607 |
|
|
interaction tensor can be constructed as, |
608 |
|
|
\begin{eqnarray} |
609 |
|
|
T_{\alpha\beta\gamma\delta}(\mathbf{r}) & = & |
610 |
|
|
\left(\delta_{\alpha\beta}\delta_{\gamma\delta} |
611 |
|
|
+ |
612 |
|
|
\delta_{\alpha\gamma}\delta_{\beta\delta}+ |
613 |
|
|
\delta_{\alpha\delta}\delta_{\beta\gamma}\right)v_{41}(r) |
614 |
|
|
+ \delta_{\gamma\delta} |
615 |
|
|
r_\alpha r_\beta |
616 |
|
|
\frac{v_{42}(r)}{r^2} |
617 |
|
|
\nonumber \\ |
618 |
|
|
& & + r_\alpha r_\beta r_\gamma r_\delta \left(\frac{v_{43}(r)}{r^4}\right), |
619 |
mlamichh |
4353 |
\label{quadRadial} |
620 |
gezelter |
4400 |
\end{eqnarray} |
621 |
|
|
where again $v_{41}(r)$, $v_{42}(r)$, and $v_{43}(r)$ are radial |
622 |
|
|
functions defined in Paper I of the series. \cite{PaperI} Note that |
623 |
|
|
these radial functions have different functional forms depending on |
624 |
|
|
the level of approximation being employed. |
625 |
|
|
|
626 |
|
|
The integral in equation (\ref{gradMaxwell}) can be divided into two |
627 |
|
|
parts, $|\mathbf{r}-\mathbf{r}^\prime|\rightarrow 0 $ and |
628 |
|
|
$|\mathbf{r}-\mathbf{r}^\prime|> 0$. Since the self-contribution to |
629 |
|
|
the field gradient vanishes at the singularity (see Appendix |
630 |
|
|
\ref{singularQuad}), equation (\ref{gradMaxwell}) can be written as, |
631 |
mlamichh |
4353 |
\begin{equation} |
632 |
gezelter |
4400 |
\partial_\alpha E_\beta(\mathbf{r}) = \partial_\alpha {E}^\circ_\beta(\mathbf{r}) + |
633 |
|
|
\frac{1}{4\pi \epsilon_o}\int\limits_{|\mathbf{r}-\mathbf{r}^\prime|> 0 } |
634 |
|
|
T_{\alpha\beta\gamma\delta}(\mathbf{r}-\mathbf{r}^\prime) |
635 |
|
|
{Q}_{\gamma\delta}(\mathbf{r}^\prime) d\mathbf{r}^\prime. |
636 |
mlamichh |
4353 |
\end{equation} |
637 |
gezelter |
4400 |
If $\mathbf{r} = \mathbf{r}^\prime$ is excluded from the integration, |
638 |
|
|
the total gradient can be most easily expressed in terms of |
639 |
|
|
traceless quadrupole density as below,\cite{LoganI81} |
640 |
mlamichh |
4353 |
\begin{equation} |
641 |
gezelter |
4400 |
\partial_\alpha E_\beta(\mathbf{r}) = \partial_\alpha |
642 |
|
|
{E}^\circ_\beta(\mathbf{r}) + \frac{1}{12\pi |
643 |
|
|
\epsilon_o}\int\limits_{|\mathbf{r}-\mathbf{r}^\prime|> 0 } |
644 |
|
|
T_{\alpha\beta\gamma\delta}(\mathbf{r}-\mathbf{r}^\prime) \Theta_{\gamma\delta}(\mathbf{r}') d\mathbf{r}^\prime, |
645 |
mlamichh |
4353 |
\end{equation} |
646 |
gezelter |
4400 |
where |
647 |
|
|
$\Theta_{\alpha\beta} = 3Q_{\alpha\beta} - \delta_{\alpha\beta}Tr(Q)$ |
648 |
|
|
is the traceless quadrupole density. In analogy to |
649 |
|
|
Eq. (\ref{pertQuad}) above, the quadrupolarization density may now be |
650 |
|
|
related to the actual quadrupolar susceptibility, $\chi_Q$, |
651 |
mlamichh |
4353 |
\begin{equation} |
652 |
gezelter |
4400 |
\frac{1}{3}{\Theta}_{\alpha\beta}(\mathbf{r}) = \epsilon_o {\chi}_Q |
653 |
|
|
\left(\partial_\alpha {E}^\circ_\beta(\mathbf{r}) + \frac{1}{12\pi |
654 |
|
|
\epsilon_o}\int\limits_{|\mathbf{r}-\mathbf{r}^\prime|> 0 } |
655 |
|
|
T_{\alpha\beta\gamma\delta}(\mathbf{r}-\mathbf{r}^\prime) |
656 |
|
|
\Theta_{\gamma\delta}(\mathbf{r}^\prime) d\mathbf{r}^\prime \right). |
657 |
mlamichh |
4353 |
\end{equation} |
658 |
gezelter |
4400 |
For toroidal boundaries and with a uniform imposed |
659 |
|
|
$\partial_\alpha E^\circ_\beta$, the quadrupole density |
660 |
|
|
${\Theta}_{\alpha\beta}$ will be uniform over the entire space. After |
661 |
|
|
performing a Fourier transform (see the Appendix in |
662 |
|
|
ref. \onlinecite{NeumannI83}) we obtain, |
663 |
mlamichh |
4353 |
\begin{equation} |
664 |
gezelter |
4400 |
\frac{1}{3}\tilde{\Theta}_{\alpha\beta}(\mathbf{k})= |
665 |
|
|
\epsilon_o {\chi}_Q \left[{\partial_\alpha |
666 |
|
|
\tilde{E}^\circ_\beta}(\mathbf{k})+ \frac{1}{12\pi |
667 |
|
|
\epsilon_o} \tilde{T}_{\alpha\beta\gamma\delta}(\mathbf{k}) |
668 |
|
|
\tilde{\Theta}_{\gamma\delta}(\mathbf{k})\right]. |
669 |
mlamichh |
4353 |
\end{equation} |
670 |
gezelter |
4400 |
|
671 |
|
|
The only surviving components of the quadrupole polarization will be |
672 |
|
|
in the direction of the applied field gradient, so we may simplify |
673 |
|
|
this expression considerably. That is, the only relevant polarization |
674 |
|
|
term is $\tilde{\Theta}_{\alpha\beta}(\mathbf{k})$ and the only relevant |
675 |
|
|
tensor contributions will have doubly-repeated indices, e.g. |
676 |
|
|
$\tilde{T}_{\alpha\beta\alpha\beta}(\mathbf{k})$. Therefore we may |
677 |
|
|
consider each component of the interaction tensor separately in a |
678 |
|
|
scalar expression, |
679 |
|
|
\begin{eqnarray} |
680 |
|
|
\frac{1}{3}\tilde{\Theta}_{\alpha\beta}(\mathbf{k}) &=& \epsilon_o |
681 |
|
|
{\chi}_Q |
682 |
|
|
\left[{\partial_\alpha |
683 |
|
|
\tilde{E}^\circ_\beta}(\mathbf{k})+ |
684 |
|
|
\frac{1}{12\pi |
685 |
|
|
\epsilon_o}\tilde{T}_{\alpha\beta\alpha\beta}(\mathbf{k}) |
686 |
|
|
{\tilde{\Theta}}_{\alpha\beta}(\mathbf{k})\right] |
687 |
|
|
\nonumber \\ |
688 |
|
|
&=& \epsilon_o {\chi}_Q \left(1-\frac{1}{4\pi} {\chi}_Q \tilde{T}_{\alpha\beta\alpha\beta}(\mathbf{k})\right)^{-1} |
689 |
|
|
{\partial_\alpha \tilde{E}^\circ_\beta}(\mathbf{k}) |
690 |
mlamichh |
4353 |
\label{fourierQuad} |
691 |
gezelter |
4400 |
\end{eqnarray} |
692 |
|
|
If the applied field gradient is homogeneous over the |
693 |
|
|
entire volume, ${\partial_ \alpha \tilde{E}^\circ_\beta}(\mathbf{k}) = 0 $ except at |
694 |
|
|
$ \mathbf{k} = 0$. As in the dipolar case, the only relevant |
695 |
|
|
contribution of the interaction tensor will also be when $\mathbf{k} = 0$. |
696 |
|
|
Therefore equation (\ref{fourierQuad}) can be written as, |
697 |
mlamichh |
4353 |
\begin{equation} |
698 |
gezelter |
4400 |
\frac{1}{3}{\tilde{\Theta}}_{\alpha\beta}({0}) = \epsilon_o {\chi}_Q |
699 |
|
|
\left(1-\frac{1}{4\pi} {\chi}_Q \tilde{T}_{\alpha\beta\alpha\beta}({0})\right)^{-1} \partial_\alpha \tilde{E}^\circ_\beta({0}) |
700 |
mlamichh |
4353 |
\label{fourierQuad2} |
701 |
|
|
\end{equation} |
702 |
gezelter |
4400 |
where $\tilde{T}_{\alpha\beta\alpha\beta}({0})$ can be evaluated as, |
703 |
mlamichh |
4353 |
\begin{equation} |
704 |
gezelter |
4400 |
\tilde{T}_{\alpha\beta\alpha\beta}({0}) = \int_V {T}_{\alpha\beta\alpha\beta}(\mathbf{r})d\mathbf{r} |
705 |
mlamichh |
4353 |
\label{realTensorQaud} |
706 |
|
|
\end{equation} |
707 |
gezelter |
4400 |
We may now define a quantity to represent the contribution from the |
708 |
|
|
$\mathbf{k} = 0$ portion of the interaction tensor, |
709 |
|
|
\begin{equation} |
710 |
|
|
B = \frac{1}{4 \pi} \int_V T_{\alpha \beta \alpha \beta}(\mathbf{r}) |
711 |
|
|
d\mathbf{r} |
712 |
|
|
\end{equation} |
713 |
|
|
which has been integrated over the interaction volume $V$ and has |
714 |
|
|
units of $\mathrm{length}^{-2}$. |
715 |
mlamichh |
4353 |
|
716 |
gezelter |
4400 |
In terms of traced quadrupole moment, equation (\ref{fourierQuad2}) |
717 |
|
|
can be written as, |
718 |
mlamichh |
4353 |
\begin{equation} |
719 |
gezelter |
4400 |
\mathsf{Q} - \frac{\mathbf{I}}{3} \mathrm{Tr}(\mathsf{Q}) |
720 |
|
|
= \epsilon_o {\chi}_Q\left(1- {\chi}_Q B \right)^{-1} \nabla \mathbf{E}^\circ |
721 |
mlamichh |
4353 |
\label{tracedConstQuad} |
722 |
|
|
\end{equation} |
723 |
gezelter |
4400 |
Comparing (\ref{tracedConstQuad}) and (\ref{flucQuad}) we obtain, |
724 |
mlamichh |
4353 |
\begin{equation} |
725 |
gezelter |
4400 |
\frac{\braket{\mathsf{M}_Q^2}-{\braket{\mathsf{M}_Q}}^2}{15 \epsilon_o |
726 |
|
|
V k_B T} = \chi_Q \left(1 - \chi_Q B \right)^{-1}, |
727 |
mlamichh |
4353 |
\end{equation} |
728 |
gezelter |
4400 |
or equivalently, |
729 |
mlamichh |
4353 |
\begin{equation} |
730 |
gezelter |
4400 |
\chi_Q = \frac{\braket{\mathsf{M}_Q^2}-{\braket{\mathsf{M}_Q}}^2}{15 \epsilon_o |
731 |
|
|
V k_B T} \left(1 + B \frac{\braket{\mathsf{M}_Q^2}-{\braket{\mathsf{M}_Q}}^2}{15 \epsilon_o |
732 |
|
|
V k_B T} \right)^{-1} |
733 |
|
|
\label{eq:finalForm} |
734 |
mlamichh |
4353 |
\end{equation} |
735 |
gezelter |
4400 |
Eq. (\ref{eq:finalForm}) now expresses a bulk property (the |
736 |
|
|
quadrupolar susceptibility, $\chi_Q$) in terms of a straightforward |
737 |
|
|
fluctuation in the system quadrupole moment and a quadrupolar |
738 |
|
|
correction factor ($B$). The correction factors depend on the cutoff |
739 |
|
|
method being employed in the simulation, and these are listed in Table |
740 |
|
|
\ref{tab:B}. |
741 |
mlamichh |
4353 |
|
742 |
gezelter |
4400 |
In terms of the macroscopic quadrupole polarizability, $\alpha_Q$, |
743 |
|
|
which may be thought of as the ``conducting boundary'' version of the |
744 |
|
|
susceptibility, |
745 |
mlamichh |
4353 |
\begin{equation} |
746 |
gezelter |
4400 |
\chi_Q = \frac{\alpha_Q}{1 + B \alpha_Q}. |
747 |
mlamichh |
4353 |
\end{equation} |
748 |
gezelter |
4400 |
If an electrostatic method produces $B \rightarrow 0$, the computed |
749 |
|
|
quadrupole polarizability and quadrupole susceptibility converge to |
750 |
|
|
the same value. |
751 |
|
|
|
752 |
|
|
\begin{sidewaystable} |
753 |
|
|
\caption{Expressions for the quadrupolar correction factor ($B$) for |
754 |
|
|
the real-space electrostatic methods in terms of the damping |
755 |
|
|
parameter ($\alpha$) and the cutoff radius ($r_c$). The units |
756 |
|
|
of the correction factor are $ \mathrm{length}^{-2}$ for quadrupolar |
757 |
|
|
fluids.} |
758 |
mlamichh |
4353 |
\label{tab:B} |
759 |
gezelter |
4400 |
\begin{tabular}{l|c|c|c|c} |
760 |
mlamichh |
4353 |
|
761 |
gezelter |
4400 |
Method & charges & dipoles & quadrupoles \\\hline |
762 |
mlamichh |
4353 |
Spherical Cutoff (SC) & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $ -\frac{8 {\alpha}^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ \\ |
763 |
|
|
Shifted Potental (SP) & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $- \frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$& $ -\frac{16 \alpha^7 {r_c}^5}{9\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ \\ |
764 |
|
|
Gradient-shifted (GSF) & $- \frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & 0 & $-\frac{4{\alpha}^7{r_c}^5 }{9\sqrt{\pi}}e^{-\alpha^2 r_c^2}(-1+2\alpha ^2 r_c^2)$\\ |
765 |
gezelter |
4400 |
Taylor-shifted (TSF) & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} |
766 |
|
|
e^{-\alpha^2 r_c^2}$ & |
767 |
|
|
$\frac{4\;\mathrm{erfc(\alpha r_c)}}{{r_c}^2} + \frac{8 \alpha}{3\sqrt{\pi}r_c}e^{-\alpha^2 {r_c}^2}\left(3+ 2 \alpha^2 {r_c}^2 + \alpha^4 {r_c}^4\right) $ & $\frac{10\;\mathrm{erfc}(\alpha r_c )}{{r_c}^2} + \frac{4{\alpha}}{9\sqrt{\pi}{r_c}}e^{-\alpha^2 r_c^2}\left(45 + 30\alpha ^2 {r_c}^2 + 12\alpha^4 {r_c}^4 + 3\alpha^6 {r_c}^6 + 2 \alpha^8 {r_c}^8\right)$ \\ |
768 |
|
|
Ewald-Kornfeld (EK) & & & |
769 |
|
|
\end{tabular} |
770 |
|
|
\end{sidewaystable} |
771 |
gezelter |
4399 |
|
772 |
|
|
\section{Screening of Charges by Multipolar Fluids} |
773 |
gezelter |
4401 |
\label{sec:PMF} |
774 |
|
|
In a dipolar fluid, the static dielectric constant is also a measure |
775 |
|
|
of the ability of the fluid to screen charges from one another. A set |
776 |
|
|
of point charges creates an inhomogeneous field in the fluid, and the |
777 |
|
|
fluid responds to this field as if it was created externally or via |
778 |
|
|
local polarization fluctuations. For this reason, the dielectric |
779 |
|
|
constant can be used to estimate an effective potential between two |
780 |
|
|
point charges ($C_i$ and $C_j$) embedded in the fluid, |
781 |
gezelter |
4399 |
\begin{equation} |
782 |
gezelter |
4401 |
U_\mathrm{effective} = \frac{C_i C_j}{4 \pi \epsilon_0 \epsilon |
783 |
|
|
r_{ij}}. |
784 |
|
|
\label{eq:effectivePot} |
785 |
gezelter |
4399 |
\end{equation} |
786 |
gezelter |
4401 |
|
787 |
|
|
The same set of point charges can also create an inhomogeneous field |
788 |
|
|
\textit{gradient}, and this will cause a response in a quadrupolar |
789 |
|
|
fluid that will also cause an effective screening. As discussed above, |
790 |
|
|
however, the relevant phyiscal property in quadrupolar fluids is the |
791 |
|
|
susceptibility, $\chi_Q$. The screening dielectric associated with |
792 |
|
|
the quadrupolar susceptibility is defined as,\cite{Ernst92} |
793 |
gezelter |
4399 |
\begin{equation} |
794 |
gezelter |
4401 |
\epsilon = 1 + \chi_Q G = 1 + G \frac{\alpha_Q}{1 + \alpha_Q B} |
795 |
|
|
\label{eq:dielectricFromQuadrupoles} |
796 |
gezelter |
4399 |
\end{equation} |
797 |
gezelter |
4401 |
where $G$ is a geometrical factor that depends on the geometry of the |
798 |
|
|
field perturbation, |
799 |
|
|
\begin{equation} |
800 |
|
|
G = \frac{\int_V \left| \nabla \mathbf{E}^\circ \right|^2 d\mathbf{r}} |
801 |
|
|
{\int_V \left|\mathbf{E}^\circ\right|^2 d\mathbf{r}} |
802 |
|
|
\end{equation} |
803 |
|
|
integrated over the interaction volume. Note that this geometrical |
804 |
|
|
factor is also required to compute effective dielectric constants even |
805 |
|
|
when the field gradient is homogeneous over the entire sample. |
806 |
gezelter |
4399 |
|
807 |
gezelter |
4401 |
To measure effective screening in a multipolar fluid, we compute an |
808 |
|
|
effective interaction potential, the potential of mean force (PMF), |
809 |
|
|
between positively and negatively charged ions when they screened by |
810 |
|
|
the intervening fluid. The PMF is obtained from a sequence of |
811 |
|
|
simulations in which two ions are constrained to a fixed distance, and |
812 |
|
|
the average constraint force to hold them at a fixed distance $r$ is |
813 |
|
|
collected during a long simulation,\cite{Wilfred07} |
814 |
gezelter |
4400 |
\begin{equation} |
815 |
gezelter |
4401 |
w(r) = \int_{r_o}^{r}\left\langle \frac{\partial f}{\partial r^\prime} |
816 |
|
|
\right\rangle_{r^\prime} dr^\prime + 2k_BT \ln\left(\frac{r}{r_o}\right) + w(r_o), |
817 |
|
|
\label{eq:pmf} |
818 |
gezelter |
4400 |
\end{equation} |
819 |
gezelter |
4401 |
where $\braket{\partial f/\partial r^\prime}_{r^\prime}$ is the mean |
820 |
|
|
constraint force required to hold the ions at distance $r^\prime$, |
821 |
|
|
$2k_BT \log(r/r_o)$ is the Fixman factor,\cite{Fixman:1974fk} and |
822 |
|
|
$r_o$ is a reference position (usually taken as a large separation |
823 |
|
|
betwen the ions). If the dielectric constant is a good measure of the |
824 |
|
|
screening at all inter-ion separations, we would expect $w(r)$ to have |
825 |
|
|
the form in Eq. (\ref{eq:effectivePot}). Because real fluids are not |
826 |
|
|
continuum dielectrics, the effective dielectric constant is a function |
827 |
|
|
of the interionic separation, |
828 |
gezelter |
4400 |
\begin{equation} |
829 |
gezelter |
4401 |
\epsilon(r) = \frac{u_\mathrm{raw}(r) }{w(r)} |
830 |
gezelter |
4400 |
\end{equation} |
831 |
gezelter |
4401 |
where $u_\mathrm{raw}(r)$ is the direct charge-charge interaction |
832 |
|
|
potential that is in use during the simulation. $\epsilon(r)$ may |
833 |
|
|
vary considerably from the bulk estimates at short distances, although |
834 |
|
|
it should converge to the bulk value as the separation between the |
835 |
|
|
ions increases. |
836 |
gezelter |
4400 |
|
837 |
gezelter |
4399 |
\section{Simulation Methodology} |
838 |
gezelter |
4401 |
To test the formalism developed in the preceding sections, we have |
839 |
|
|
carried out computer simulations using three different techniques: i) |
840 |
|
|
simulations in the presence of external fields, ii) equilibrium |
841 |
|
|
calculations of box moment fluctuations, and iii) potentials of mean |
842 |
|
|
force (PMF) between embedded ions. In all cases, the fluids were |
843 |
|
|
composed of point multipoles protected by a Lennard-Jones potential. |
844 |
|
|
The parameters used in the test systems are given in table |
845 |
|
|
\ref{Tab:C}. |
846 |
mlamichh |
4353 |
|
847 |
mlamichh |
4389 |
\begin{table} |
848 |
gezelter |
4401 |
\caption{\label{Tab:C}The parameters used in simulations to evaluate |
849 |
|
|
the dielectric response of the new real-space methods.} |
850 |
mlamichh |
4389 |
\begin{tabularx}{\textwidth}{r|cc|YYccc|Yccc} \hline |
851 |
|
|
& \multicolumn{2}{c|}{LJ parameters} & |
852 |
|
|
\multicolumn{5}{c|}{Electrostatic moments} & & & & \\ |
853 |
|
|
Test system & $\sigma$& $\epsilon$ & $C$ & $D$ & |
854 |
|
|
$Q_{xx}$ & $Q_{yy}$ & $Q_{zz}$ & mass & $I_{xx}$ & $I_{yy}$ & |
855 |
|
|
$I_{zz}$ \\ \cline{6-8}\cline{10-12} |
856 |
|
|
& (\AA) & (kcal/mol) & (e) & (debye) & \multicolumn{3}{c|}{(debye \AA)} & (amu) & \multicolumn{3}{c}{(amu |
857 |
|
|
\AA\textsuperscript{2})} \\ \hline |
858 |
gezelter |
4401 |
Dipolar fluid & 3.41 & 0.2381 & - & 1.4026 &-&-&-& 39.948 & 11.613 & 11.613 & 0.0 \\ |
859 |
|
|
Quadrupolar fluid & 2.985 & 0.265 & - & - & 0.0 & 0.0 &-2.139 & 18.0153 & 43.0565 & 43.0565 & 0.0 \\ |
860 |
mlamichh |
4389 |
\ce{q+} & 1.0 & 0.1 & +1 & - & - & - & - & 22.98 & - & - & - \\ |
861 |
|
|
\ce{q-} & 1.0 & 0.1 & -1 & - & - & - & - & 22.98 & - & - & - \\ \hline |
862 |
|
|
\end{tabularx} |
863 |
|
|
\end{table} |
864 |
mlamichh |
4363 |
|
865 |
gezelter |
4401 |
The first of the test systems consists entirely of fluids of point |
866 |
|
|
dipolar or quadrupolar molecules in the presence of constant field or |
867 |
|
|
field gradients. Since there are no isolated charges within the |
868 |
|
|
system, the divergence of the field will be zero, \textit{i.e.} |
869 |
|
|
$\vec{\nabla} \cdot \mathbf{E} = 0$. This condition can be satisfied |
870 |
|
|
by using the relatively simple applied potential as described in the |
871 |
|
|
supporting information. |
872 |
mlamichh |
4353 |
|
873 |
gezelter |
4401 |
When a constant electric field or field gradient is applied to the |
874 |
|
|
system, the molecules align along the direction of the applied field, |
875 |
|
|
and polarize to a degree determined both by the strength of the field |
876 |
|
|
and the fluid's polarizibility. We have calculated ensemble averages |
877 |
|
|
of the box dipole and quadrupole moments as a function of the strength |
878 |
|
|
of the applied fields. If the fields are sufficiently weak, the |
879 |
|
|
response is linear in the field strength, and one can easily compute |
880 |
|
|
the polarizability directly from the simulations. |
881 |
mlamichh |
4389 |
|
882 |
gezelter |
4401 |
The second set of test systems consists of equilibrium simulations of |
883 |
|
|
fluids of point dipolar or quadrupolar molecules simulated in the |
884 |
|
|
absence of any external perturbation. The fluctuation of the ensemble |
885 |
|
|
averages of the box multipolar moment was calculated for each of the |
886 |
|
|
multipolar fluids. The box multipolar moments were computed as simple |
887 |
|
|
sums over the instantaneous molecular moments, and fluctuations in |
888 |
|
|
these quantities were obtained from Eqs. (\ref{eq:flucDip}) and |
889 |
|
|
(\ref{eq:flucQuad}). The macroscopic polarizabilities of the system at |
890 |
|
|
a were derived using Eqs.(\ref{flucDipole}) and (\ref{flucQuad}). |
891 |
mlamichh |
4389 |
|
892 |
gezelter |
4401 |
The final system consists of dipolar or quadrupolar fluids with two |
893 |
|
|
oppositely charged ions embedded within the fluid. These ions are |
894 |
|
|
constrained to be at fixed distance throughout a simulation, although |
895 |
|
|
they are allowed to move freely throughout the fluid while satisfying |
896 |
|
|
that constraint. Separate simulations were run at a range of |
897 |
|
|
constraint distances. A dielectric screening factor was computed using |
898 |
|
|
the ratio between the potential between the two ions in the absence of |
899 |
|
|
the fluid medium and the PMF obtained from the simulations. |
900 |
mlamichh |
4389 |
|
901 |
gezelter |
4401 |
We carried out these simulations for all three of the new real-space |
902 |
|
|
electrostatic methods (SP, GSF, and TSF) that were developed in the |
903 |
|
|
first paper (Ref. \onlinecite{PaperI}) in the series. The radius of |
904 |
|
|
the cutoff sphere was taken to be 12~\AA. Each of the real space |
905 |
|
|
methods also depends on an adjustable damping parameter $\alpha$ (in |
906 |
|
|
units of $\mathrm{length}^{-1}$). We have selected ten different |
907 |
|
|
values of damping parameter: 0.0, 0.05, 0.1, 0.15, 0.175, 0.2, 0.225, |
908 |
|
|
0.25, 0.3, and 0.35~\AA$^{-1}$ in our simulations of the dipolar |
909 |
|
|
liquids, while four values were chosen for the quadrupolar fluids: |
910 |
|
|
0.0, 0.1, 0.2, and 0.3~\AA$^{-1}$. |
911 |
mlamichh |
4389 |
|
912 |
gezelter |
4401 |
For each of the methods and systems listed above, a reference |
913 |
|
|
simulation was carried out using a multipolar implementation of the |
914 |
|
|
Ewald sum.\cite{Smith82,Smith98} A default tolerance of |
915 |
|
|
$1 \times 10^{-8}$~kcal/mol was used in all Ewald calculations, |
916 |
|
|
resulting in Ewald coefficient 0.3119~\AA$^{-1}$ for a cutoff radius |
917 |
|
|
of 12~\AA. All of the electrostatics and constraint methods were |
918 |
|
|
implemented in our group's open source molecular simulation program, |
919 |
|
|
OpenMD,\cite{Meineke05,openmd} which was used for all calculations in |
920 |
|
|
this work. |
921 |
|
|
|
922 |
|
|
Dipolar systems contained 2048 Lennard-Jones-protected point dipolar |
923 |
|
|
(Stockmayer) molecules with reduced density $\rho^* = 0.822$, |
924 |
|
|
temperature $T^* = 1.15$, moment of inertia $I^* = 0.025$, and dipole |
925 |
|
|
moment $\mu^* = \sqrt{3.0}$. These systems were equilibrated for |
926 |
|
|
0.5~ns in the canonical (NVT) ensemble. Data collection was carried |
927 |
|
|
out over a 1~ns simulation in the microcanonical (NVE) ensemble. Box |
928 |
|
|
dipole moments were sampled every fs. For simulations with external |
929 |
|
|
perturbations, field strengths ranging from |
930 |
|
|
$0 - 10 \times 10^{-4} \mathrm{~V/\r{A}}$ with increments of |
931 |
|
|
$ 10^{-4} \mathrm{~V/\r{A}}$ were carried out for each system. |
932 |
|
|
|
933 |
|
|
Quadrupolar systems contained 4000 linear point quadrupoles with a |
934 |
|
|
density $2.338 \mathrm{~g/cm}^3$ at a temperature of 500~K. These |
935 |
|
|
systems were equilibrated for 200~ps in a canonical (NVT) ensemble. |
936 |
|
|
Data collection was carried out over a 500~ps simulation in the |
937 |
|
|
microcanonical (NVE) ensemble. Components of box quadrupole moments |
938 |
|
|
were sampled every 100 fs. For quadrupolar simulations with external |
939 |
|
|
field gradients, field strengths ranging from |
940 |
|
|
$0 - 9 \times 10^{-2} \mathrm{~V/\r{A}}^2$ with increments of |
941 |
|
|
$10^{-2} \mathrm{~V/\r{A}^2}$ were carried out for each system. |
942 |
|
|
|
943 |
|
|
To carry out the PMF simulations, two of the multipolar molecules in |
944 |
|
|
the test system were converted into \ce{q+} and \ce{q-} ions and |
945 |
|
|
constrained to remain at a fixed distance for the duration of the |
946 |
|
|
simulation. The constrained distance was then varied from |
947 |
|
|
$5 - 12 \mathrm{~\r{A}}$. In the PMF calculations, all simulations were |
948 |
|
|
equilibrated for 500 ps in the NVT ensemble and run for 5 ns in the |
949 |
|
|
microcanonical (NVE) ensemble. Constraint forces were sampled every |
950 |
|
|
20~fs. |
951 |
mlamichh |
4389 |
|
952 |
|
|
\section{Results} |
953 |
mlamichh |
4397 |
\subsection{Dipolar fluid} |
954 |
|
|
\begin{figure} |
955 |
gezelter |
4401 |
\includegraphics[width=5in]{dielectricFinal_Dipole.pdf} |
956 |
|
|
\caption{The polarizability ($\alpha_D$), correction factor ($A$), and |
957 |
|
|
dielectric constant ($\epsilon$) for the test dipolar fluid. The |
958 |
|
|
left panels were computed using external fields, and those on the |
959 |
|
|
right are the result of equilibrium fluctuations. In the GSF and SP |
960 |
|
|
methods, the corrections are large in with small values of $\alpha$, |
961 |
|
|
and a optimal damping coefficient is evident around 0.25 \AA$^{-1}$. |
962 |
|
|
The dashed lines in the upper panel indicate back-calculation of the |
963 |
|
|
polarizability using the Ewald estimate |
964 |
|
|
(Ref. \onlinecite{NeumannI83}) for the dielectric constant. NOTE: |
965 |
|
|
Change $A_{dipole}$ to $A$.} |
966 |
mlamichh |
4397 |
\label{fig:dielectricDipole} |
967 |
|
|
\end{figure} |
968 |
gezelter |
4401 |
The macroscopic polarizability ($\alpha_D$) for the dipolar fluid is |
969 |
|
|
shown in the upper panels in Fig. \ref{fig:dielectricDipole}. The |
970 |
|
|
polarizability obtained from the both perturbation and fluctuation |
971 |
|
|
approaches are in excellent agreement with each other. The data also |
972 |
|
|
show a stong dependence on the damping parameter for both the Shifted |
973 |
|
|
Potential (SP) and Gradient Shifted force (GSF) methods, while Taylor |
974 |
|
|
shifted force (TSF) is largely independent of the damping |
975 |
|
|
parameter. |
976 |
mlamichh |
4397 |
|
977 |
gezelter |
4401 |
The calculated correction factors discussed in section |
978 |
|
|
\ref{sec:corrFactor} are shown in the middle panels. Because the TSF |
979 |
|
|
method has $A = 1$ for all values of the damping parameter, the |
980 |
|
|
computed polarizabilities need no correction for the dielectric |
981 |
|
|
calculation. The value of $A$ varies with the damping parameter in |
982 |
|
|
both the SP and GSF methods, and inclusion of the correction yields |
983 |
|
|
dielectric estimates (shown in the lower panel) that are generally too |
984 |
|
|
large until the damping reaches $\sim 0.25 \mathrm{~\r{A}}^{-1}$. |
985 |
|
|
Above this value, the dielectric constants are generally in good |
986 |
|
|
agreement with previous simulation results.\cite{NeumannI83} |
987 |
mlamichh |
4397 |
|
988 |
gezelter |
4401 |
Figure \ref{fig:dielectricDipole} also contains back-calculations of |
989 |
|
|
the polarizability using the reference (Ewald) simulation |
990 |
|
|
results.\cite{NeumannI83} These are indicated with dashed lines in the |
991 |
|
|
upper panels. It is clear that the expected polarizability for the SP |
992 |
|
|
and GSF methods are quite close to results obtained from the |
993 |
|
|
simulations. This indicates that the correction formula for the |
994 |
|
|
dipolar fluid (Eq. \ref{correctionFormula}) is quite sensitive when |
995 |
|
|
the value of $\mathrm{A}$ deviates significantly from unity. |
996 |
|
|
|
997 |
|
|
These results also suggest an optimal value for the damping parameter |
998 |
|
|
of ($\alpha \sim 0.2-0.3 \mathrm{\r{A}}^{-1}$ when evaluating |
999 |
|
|
dielectric constants of point dipolar fluids using either the |
1000 |
|
|
perturbation and fluctuation approaches for the new real-space |
1001 |
|
|
methods. |
1002 |
|
|
|
1003 |
mlamichh |
4397 |
\begin{figure} |
1004 |
gezelter |
4401 |
\includegraphics[width=4in]{ScreeningFactor_Dipole.pdf} |
1005 |
|
|
\caption{The distance-dependent screening factor, $\epsilon(r)$, |
1006 |
|
|
between two ions immersed in the dipolar fluid. The new methods are |
1007 |
|
|
shown in separate panels, and different values of the damping |
1008 |
|
|
parameter ($\alpha$) are indicated with different symbols. All of |
1009 |
|
|
the methods appear to be converging to the bulk dielectric constant |
1010 |
|
|
($\sim 65$) at large ion separations. CHECK PLOT} |
1011 |
mlamichh |
4397 |
\label{fig:ScreeningFactor_Dipole} |
1012 |
|
|
\end{figure} |
1013 |
gezelter |
4401 |
We have also evaluated the distance-dependent screening factor, |
1014 |
|
|
$\epsilon(r)$, between two oppositely charged ions when they are |
1015 |
|
|
placed in the dipolar fluid. These results were computed using |
1016 |
|
|
Eq. \ref{eq:pmf} and are shown in Fig. |
1017 |
|
|
\ref{fig:ScreeningFactor_Dipole}. |
1018 |
mlamichh |
4397 |
|
1019 |
gezelter |
4401 |
The screening factor is similar to the dielectric constant, but |
1020 |
|
|
measures a local property of the ions in the fluid and depends on both |
1021 |
|
|
ion-dipole and dipole-dipole interactions. These interactions depend |
1022 |
|
|
on the distance between ions as well as the electrostatic interaction |
1023 |
|
|
methods utilized in the simulations. The screening should converge to |
1024 |
|
|
the dielectric constant when the field due to ions is small. This |
1025 |
|
|
occurs when the ions are separated (or when the damping parameter is |
1026 |
|
|
large). In Fig. \ref{fig:ScreeningFactor_Dipole} we observe that for |
1027 |
|
|
the higher value of damping alpha \textit{i.e.} |
1028 |
|
|
$\alpha = 0.2 \AA^{-1}$ and $0.3 \AA^{-1}$ and large separation |
1029 |
|
|
between ions, the screening factor does indeed approach the correct |
1030 |
|
|
dielectric constant. |
1031 |
mlamichh |
4397 |
|
1032 |
gezelter |
4401 |
REVISIT THESE THREE PARAGRAPHS: |
1033 |
|
|
|
1034 |
|
|
It is also notable that the TSF method again displays smaller |
1035 |
|
|
perturbations away from the correct dielectric screening behavior. We |
1036 |
|
|
also observe that for TSF method yields high dielectric screening even |
1037 |
|
|
for lower values of $\alpha$. |
1038 |
|
|
|
1039 |
|
|
At short distances, the presence of the ions creates a strong local |
1040 |
|
|
field that acts to align nearby dipoles nearly perfectly in opposition |
1041 |
|
|
to the field from the ions. This has the effect of increasing the |
1042 |
|
|
effective screening when the ions are brought close to one another. |
1043 |
|
|
|
1044 |
|
|
This is due to the absence of corrections for dipole-dipole |
1045 |
|
|
interactions as discussed above. However, the TSF remains quite |
1046 |
|
|
perturbative for ion-dipole interactions, an effect that is most |
1047 |
|
|
apparent at small ion separations. |
1048 |
|
|
|
1049 |
mlamichh |
4397 |
\subsection{Quadrupolar fluid} |
1050 |
|
|
\begin{figure} |
1051 |
gezelter |
4401 |
\includegraphics[width=4in]{polarizabilityFinal_Quad.pdf} |
1052 |
|
|
\caption{The quadrupole polarizability ($\alpha_Q$), correction factor |
1053 |
|
|
($B$), and susceptibility ($\chi_Q$) for the test quadrupolar |
1054 |
|
|
fluid. The left panels were computed using external field gradients, |
1055 |
|
|
and those on the right are the result of equilibrium fluctuations. |
1056 |
|
|
The GSF and SP methods allow nearly unmodified use of the |
1057 |
|
|
``conducting boundary'' or polarizability results in place of the |
1058 |
|
|
bulk susceptibility. NOTE: Change $A_{quad}$ to $B$, change $\chi$ |
1059 |
|
|
to $\chi_Q$.} |
1060 |
mlamichh |
4397 |
\label{fig:dielectricQuad} |
1061 |
|
|
\end{figure} |
1062 |
gezelter |
4401 |
The polarizability ($\alpha_Q$), correction factor ($B$), and |
1063 |
|
|
susceptibility ($\chi_Q$) for the quadrupolar fluid is plotted against |
1064 |
|
|
damping parameter Fig. \ref{fig:dielectricQuad}. In quadrupolar |
1065 |
|
|
fluids, both the polarizability and susceptibility have units of |
1066 |
|
|
$\mathrm{length}^2$ Although the susceptibility has dimensionality, it |
1067 |
|
|
is the relevant measure of macroscopic quadrupolar |
1068 |
|
|
properties.\cite{JeonI03, JeonII03}. The left panel in |
1069 |
|
|
Fig. \ref{fig:dielectricQuad} shows results obtained from the applied |
1070 |
|
|
field gradient simulations whereas the results from the equilibrium |
1071 |
|
|
fluctuation formula are plotted in the right panels. |
1072 |
mlamichh |
4397 |
|
1073 |
gezelter |
4401 |
The susceptibility for the quadrupolar fluid is obtained from |
1074 |
|
|
quadrupolarizability and a correction factor using |
1075 |
|
|
Eq. (\ref{eq:quadrupolarSusceptiblity}). The susceptibilities are |
1076 |
|
|
shown in the bottom panels of Fig. \ref{fig:dielectricQuad}. All three |
1077 |
|
|
methods: (SP, GSF, and TSF) produce similar susceptibilities over the |
1078 |
|
|
range of damping parameters. This shows that susceptibility derived |
1079 |
|
|
using the quadrupolarizability and the correction factors are |
1080 |
|
|
essentially independent of the electrostatic method utilized in the |
1081 |
|
|
simulation. |
1082 |
mlamichh |
4397 |
|
1083 |
|
|
\begin{figure} |
1084 |
gezelter |
4401 |
\includegraphics[width=5in]{ScreeningFactor_Quad.pdf} |
1085 |
|
|
\caption{\label{fig:screenQuad} The distance-dependent screening |
1086 |
|
|
factor, $\epsilon(r)$, between two ions immersed in the quadrupolar |
1087 |
|
|
fluid. Results from the perturbation and fluctuation methods are |
1088 |
|
|
shown in right and central panels. Here the susceptibility is |
1089 |
|
|
calculated from the simulation and the geometrical factor is |
1090 |
|
|
evaluated analytically, using the field and field-gradient produced |
1091 |
|
|
by ions. The right hand panel shows the screening factor obtained |
1092 |
|
|
from the PMF calculations.} |
1093 |
mlamichh |
4397 |
\end{figure} |
1094 |
gezelter |
4401 |
A more difficult test of the quadrupolar susceptibility is made by |
1095 |
|
|
comparing with direct calculation of the electrostatic screening using |
1096 |
|
|
the potential of mean force (PMF). Since the effective dielectric |
1097 |
|
|
constant for a quadrupolar fluid depends on the geometry of the field |
1098 |
|
|
and field gradient, this is not a physical property of the quadrupolar |
1099 |
|
|
fluid. |
1100 |
mlamichh |
4397 |
|
1101 |
gezelter |
4401 |
The geometrical factor for embedded ions changes with the ion |
1102 |
|
|
separation distance. It is therefore reasonable to treat the |
1103 |
|
|
dielectric constant as a distance-dependent screening factor. Since |
1104 |
|
|
the quadrupolar molecules couple with the gradient of the field, the |
1105 |
|
|
distribution of the quadrupoles will be inhomogeneously distributed |
1106 |
|
|
around the point charges. Hence the distribution of quadrupolar |
1107 |
|
|
molecules should be taken into account when computing the geometrical |
1108 |
|
|
factors in the presence of this perturbation, |
1109 |
|
|
\begin{eqnarray} |
1110 |
|
|
G &=& \frac{\int_V g(\mathbf{r}) \left|\nabla \mathbf{E}^\circ |
1111 |
|
|
\right|^2 d\mathbf{r}}{\int_V\left|\mathbf{E}^\circ\right|^2 |
1112 |
|
|
d\mathbf{r}} \nonumber \\ \nonumber \\ |
1113 |
|
|
&=& \frac{ 2 \pi \int_{-1}^{1} \int_{0}^{R} r^2 g(r, |
1114 |
|
|
\cos\theta) \left|\nabla \mathbf{E}^\circ \right|^2 dr d\cos\theta }{\int_V\left|\mathbf{E}^\circ\right|^2 d\mathbf{r}} |
1115 |
|
|
\label{eq:geometricalFactor} |
1116 |
|
|
\end{eqnarray} |
1117 |
|
|
where $g(r,\cos\theta)$ is a distribution function for the quadrupoles |
1118 |
|
|
with respect to an origin at midpoint of a line joining the two probe |
1119 |
|
|
charges. |
1120 |
|
|
|
1121 |
|
|
The effective screening factor is plotted against ion separation |
1122 |
|
|
distance in Fig. \ref{fig:screenQuad}. The screening evaluated from |
1123 |
|
|
the perturbation and fluctuation methods are shown in right and |
1124 |
|
|
central panels. Here the susceptibility is calculated from the |
1125 |
|
|
simulation and the geometrical factor is evaluated analytically, using |
1126 |
|
|
the field and field-gradient produced by ions. The right hand panel |
1127 |
|
|
shows the screening factor obtained from the PMF calculations. |
1128 |
|
|
|
1129 |
|
|
We note that the screening factor obtained from both the perturbation |
1130 |
|
|
and fluctuation formula show good agreement with the screening factor |
1131 |
|
|
calculated using PMF method. As there are no large differences in |
1132 |
|
|
quadrupole-quadruople interactions for various real-space |
1133 |
|
|
methods,\cite{PaperI,PaperII} we generally good agreement for the |
1134 |
|
|
screening factors using any of the real space methods. |
1135 |
|
|
|
1136 |
|
|
|
1137 |
gezelter |
4399 |
\section{Discussion} |
1138 |
gezelter |
4401 |
We have used the perturbation and fluctuation formula to evaluate |
1139 |
|
|
dielectric properties for dipolar and quadrupolar fluids implementing |
1140 |
|
|
SP, GSF, and TSF methods in the simulation. Similarly the correction |
1141 |
|
|
factors for the both dipolar and quadrupolar systems have also been |
1142 |
|
|
calculated for all SP, GSF, and TSF methods. We have also derived |
1143 |
|
|
screening factor between two ions immersed in the dipolar and |
1144 |
|
|
quadrupolar fluids using PMF method. The result from the perturbation |
1145 |
|
|
and fluctuation formula are compared with PMF method to test the |
1146 |
|
|
accuracy of the simulation result. |
1147 |
mlamichh |
4353 |
|
1148 |
mlamichh |
4397 |
For the dipolar fluid, the polarizability evaluated using the perturbation and fluctuation methods show excellent agreement with each other. The dielectric constant evaluated using polarizability and correction factor agree with the previous simulation results \cite{NeumannI83} for the damping parameter 0.25 - 0.3$\AA^{-1}$ and the electrostatic interaction methods, SP and GSF implemented in the simulation. Since the correction factor for TSF, $\mathrm{A_{dipole}} = 1$, it produces dielectric constant more closer to previous simulation results \cite{NeumannI83} for all values damping parameter. This method also produces best dielectric constant for damping parameter 0.15 - 0.25$\AA^{-1}$. We have also found that correction formula (equation ~\ref{correctionFormula}) is very sensitive, when the correction factor ($\mathrm{A_{dipole}}$) much away from the 1 for the SP and GSF methods. Therefore it is better to choose damping parameter from 0.25 - 0.3$\AA^{-1}$ to evaluate dielectric constant using SP and GSF methods. The screening factor between ions is found to be closer to dielectric constant when the distance between the ions are far apart ($\sim 9 \AA$) and damping parameter $0.2$ to $0.3 \AA^{-1}$. |
1149 |
mlamichh |
4353 |
|
1150 |
mlamichh |
4397 |
The quadpolarizability evaluated from the both perturbation and fluctuation methods shows excellent agreement with each other for the case of quadrupolar fluid. The susceptibility is calculated from the quadpolarizability and correction factor tends to produce same result for all; SP, GSF, and TSF, methods. Similarly the screening factor calculated using susceptibility and geometrical factor shows excellent agreement with the result obtained from the PMF method. |
1151 |
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|
1152 |
|
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|
1153 |
mlamichh |
4353 |
\appendix |
1154 |
|
|
\section{Point-multipolar interactions with a spatially-varying electric field} |
1155 |
|
|
|
1156 |
|
|
We can treat objects $a$, $b$, and $c$ containing embedded collections |
1157 |
|
|
of charges. When we define the primitive moments, we sum over that |
1158 |
|
|
collections of charges using a local coordinate system within each |
1159 |
|
|
object. The point charge, dipole, and quadrupole for object $a$ are |
1160 |
|
|
given by $C_a$, $\mathbf{D}_a$, and $\mathsf{Q}_a$, respectively. |
1161 |
|
|
These are the primitive multipoles which can be expressed as a |
1162 |
|
|
distribution of charges, |
1163 |
|
|
\begin{align} |
1164 |
|
|
C_a =&\sum_{k \, \text{in }a} q_k , \label{eq:charge} \\ |
1165 |
|
|
D_{a\alpha} =&\sum_{k \, \text{in }a} q_k r_{k\alpha}, \label{eq:dipole}\\ |
1166 |
|
|
Q_{a\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in } a} q_k |
1167 |
|
|
r_{k\alpha} r_{k\beta} . \label{eq:quadrupole} |
1168 |
|
|
\end{align} |
1169 |
|
|
Note that the definition of the primitive quadrupole here differs from |
1170 |
|
|
the standard traceless form, and contains an additional Taylor-series |
1171 |
|
|
based factor of $1/2$. In Paper 1, we derived the forces and torques |
1172 |
|
|
each object exerts on the others. |
1173 |
|
|
|
1174 |
|
|
Here we must also consider an external electric field that varies in |
1175 |
|
|
space: $\mathbf E(\mathbf r)$. Each of the local charges $q_k$ in |
1176 |
|
|
object $a$ will then experience a slightly different field. This |
1177 |
|
|
electric field can be expanded in a Taylor series around the local |
1178 |
|
|
origin of each object. A different Taylor series expansion is carried |
1179 |
|
|
out for each object. |
1180 |
|
|
|
1181 |
|
|
For a particular charge $q_k$, the electric field at that site's |
1182 |
|
|
position is given by: |
1183 |
|
|
\begin{equation} |
1184 |
|
|
E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
1185 |
|
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
1186 |
|
|
r_{k \varepsilon} + ... |
1187 |
|
|
\end{equation} |
1188 |
|
|
Note that the electric field is always evaluated at the origin of the |
1189 |
|
|
objects, and treating each object using point multipoles simplifies |
1190 |
|
|
this greatly. |
1191 |
|
|
|
1192 |
|
|
To find the force exerted on object $a$ by the electric field, one |
1193 |
|
|
takes the electric field expression, and multiplies it by $q_k$, and |
1194 |
|
|
then sum over all charges in $a$: |
1195 |
|
|
|
1196 |
|
|
\begin{align} |
1197 |
|
|
F_\gamma &= \sum_{k \textrm{~in~} a} q_k \lbrace E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
1198 |
|
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
1199 |
|
|
r_{k \varepsilon} + ... \rbrace \\ |
1200 |
|
|
&= C_a E_\gamma + D_{a \delta} \nabla_\delta E_\gamma |
1201 |
|
|
+ Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma + |
1202 |
|
|
... |
1203 |
|
|
\end{align} |
1204 |
|
|
|
1205 |
|
|
Similarly, the torque exerted by the field on $a$ can be expressed as |
1206 |
|
|
\begin{align} |
1207 |
|
|
\tau_\alpha &= \sum_{k \textrm{~in~} a} (\mathbf r_k \times q_k \mathbf E)_\alpha \\ |
1208 |
|
|
& = \sum_{k \textrm{~in~} a} \epsilon_{\alpha \beta \gamma} q_k |
1209 |
|
|
r_{k\beta} E_\gamma(\mathbf r_k) \\ |
1210 |
|
|
& = \epsilon_{\alpha \beta \gamma} D_\beta E_\gamma |
1211 |
|
|
+ 2 \epsilon_{\alpha \beta \gamma} Q_{\beta \delta} \nabla_\delta |
1212 |
|
|
E_\gamma + ... |
1213 |
|
|
\end{align} |
1214 |
|
|
|
1215 |
|
|
The last term is essentially identical with form derived by Torres del |
1216 |
|
|
Castillo and M\'{e}ndez Garrido,\cite{Torres-del-Castillo:2006uo} although their derivation |
1217 |
|
|
utilized a traceless form of the quadrupole that is different than the |
1218 |
|
|
primitive definition in use here. We note that the Levi-Civita symbol |
1219 |
|
|
can be eliminated by utilizing the matrix cross product in an |
1220 |
|
|
identical form as in Ref. \onlinecite{Smith98}: |
1221 |
|
|
\begin{equation} |
1222 |
|
|
\left[\mathsf{A} \times \mathsf{B}\right]_\alpha = \sum_\beta |
1223 |
|
|
\left[\mathsf{A}_{\alpha+1,\beta} \mathsf{B}_{\alpha+2,\beta} |
1224 |
|
|
-\mathsf{A}_{\alpha+2,\beta} \mathsf{B}_{\alpha+1,\beta} |
1225 |
|
|
\right] |
1226 |
|
|
\label{eq:matrixCross} |
1227 |
|
|
\end{equation} |
1228 |
|
|
where $\alpha+1$ and $\alpha+2$ are regarded as cyclic permuations of |
1229 |
|
|
the matrix indices. In table \ref{tab:UFT} we give compact |
1230 |
|
|
expressions for how the multipole sites interact with an external |
1231 |
|
|
field that has exhibits spatial variations. |
1232 |
|
|
|
1233 |
|
|
\begin{table} |
1234 |
|
|
\caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque |
1235 |
|
|
$(\mathbf{\tau})$ expressions for a multipolar site embedded in an |
1236 |
|
|
electric field with spatial variations, $\mathbf{E}(\mathbf{r})$. |
1237 |
|
|
\label{tab:UFT}} |
1238 |
|
|
\begin{tabular}{r|ccc} |
1239 |
|
|
& Charge & Dipole & Quadrupole \\ \hline |
1240 |
|
|
$U$ & $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\ |
1241 |
|
|
$\mathbf{F}$ & $C \mathbf{E}(\mathbf{r})$ & $+\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ & $+\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\ |
1242 |
|
|
$\mathbf{\tau}$ & & $\mathbf{D} \times \mathbf{E}(\mathbf{r})$ & $+2 \mathsf{Q} \times \nabla \mathbf{E}(\mathbf{r})$ |
1243 |
|
|
\end{tabular} |
1244 |
|
|
\end{table} |
1245 |
|
|
\section{Gradient of the field due to quadrupolar polarization} |
1246 |
|
|
\label{singularQuad} |
1247 |
|
|
In this section, we will discuss the gradient of the field produced by |
1248 |
|
|
quadrupolar polarization. For this purpose, we consider a distribution |
1249 |
|
|
of charge ${\rho}(r)$ which gives rise to an electric field |
1250 |
|
|
$\vec{E}(r)$ and gradient of the field $\vec{\nabla} \vec{E}(r)$ |
1251 |
|
|
throughout space. The total gradient of the electric field over volume |
1252 |
|
|
due to the all charges within the sphere of radius $R$ is given by |
1253 |
|
|
(cf. Jackson equation 4.14): |
1254 |
|
|
\begin{equation} |
1255 |
|
|
\int_{r<R} \vec{\nabla}\vec{E}\;d^3r = -\int_{r=R} R^2 \vec{E}\;\hat{n}\; d\Omega |
1256 |
|
|
\label{eq:8} |
1257 |
|
|
\end{equation} |
1258 |
|
|
where $d\Omega$ is the solid angle and $\hat{n}$ is the normal vector |
1259 |
|
|
of the surface of the sphere which is equal to |
1260 |
|
|
$sin[\theta]cos[\phi]\hat{x} + sin[\theta]sin[\phi]\hat{y} + |
1261 |
|
|
cos[\theta]\hat{z}$ |
1262 |
|
|
in spherical coordinates. For the charge density ${\rho}(r')$, the |
1263 |
|
|
total gradient of the electric field can be written as (cf. Jackson |
1264 |
|
|
equation 4.16), |
1265 |
|
|
\begin{equation} |
1266 |
|
|
\int_{r<R} \vec{\nabla}\vec{E}\; d^3r=-\int_{r=R} R^2\; \vec{\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; \vec{\nabla}\;\left(\int \frac{\rho(r')}{|\vec{r}-\vec{r'}|}\;d^3r'\right) \hat{n}\; d\Omega |
1267 |
|
|
\label{eq:9} |
1268 |
|
|
\end{equation} |
1269 |
|
|
The radial function in the equation (\ref{eq:9}) can be expressed in |
1270 |
|
|
terms of spherical harmonics as (cf. Jackson equation 3.70), |
1271 |
|
|
\begin{equation} |
1272 |
|
|
\frac{1}{|\vec{r} - \vec{r'}|} = 4\pi \sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\frac{{r^l_<}}{{r^{l+1}_>}}\;{Y^*}_{lm}(\theta', \phi')\;Y_{lm}(\theta, \phi) |
1273 |
|
|
\label{eq:10} |
1274 |
|
|
\end{equation} |
1275 |
|
|
If the sphere completely encloses the charge density then $ r_< = r'$ and $r_> = R$. Substituting equation (\ref{eq:10}) into (\ref{eq:9}) we get, |
1276 |
|
|
\begin{equation} |
1277 |
|
|
\begin{split} |
1278 |
|
|
\int_{r<R} \vec{\nabla}\vec{E}\;d^3r &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; \vec{\nabla}\;\left(\int \rho(r')\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\frac{{r'^l}}{{R^{l+1}}}\;{Y^*}_{lm}(\theta', \phi')\;Y_{lm}(\theta, \phi)\;d^3r'\right) \hat{n}\; d\Omega \\ |
1279 |
|
|
&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(r')\;{r'^l}\;{Y^*}_{lm}(\theta', \phi')\left(\int_{r=R}\vec{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right)\hat{n}\; d\Omega \right)d^3r |
1280 |
|
|
' |
1281 |
|
|
\end{split} |
1282 |
|
|
\label{eq:11} |
1283 |
|
|
\end{equation} |
1284 |
|
|
The gradient of the product of radial function and spherical harmonics |
1285 |
|
|
is given by (cf. Arfken, p.811 eq. 16.94): |
1286 |
|
|
\begin{equation} |
1287 |
|
|
\begin{split} |
1288 |
|
|
\vec{\nabla}\left[ f(r)\;Y_{lm}(\theta, \phi)\right] = &-\left(\frac{l+1}{2l+1}\right)^{1/2}\; \left[\frac{\partial}{\partial r}-\frac{l}{r} \right]f(r)\; Y_{l, l+1, m}(\theta, \phi)\\ &+ \left(\frac{l}{2l+1}\right)^{1/2}\left[\frac |
1289 |
|
|
{\partial}{\partial r}+\frac{l}{r} \right]f(r)\; Y_{l, l-1, m}(\theta, \phi). |
1290 |
|
|
\end{split} |
1291 |
|
|
\label{eq:12} |
1292 |
|
|
\end{equation} |
1293 |
|
|
Using equation (\ref{eq:12}) we get, |
1294 |
|
|
\begin{equation} |
1295 |
|
|
\vec{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right) = [(l+1)(2l+1)]^{1/2}\; Y_{l,l+1,m}(\theta, \phi) \; \frac{1}{R^{l+2}}, |
1296 |
|
|
\label{eq:13} |
1297 |
|
|
\end{equation} |
1298 |
|
|
where $ Y_{l,l+1,m}(\theta, \phi)$ is the vector spherical harmonics |
1299 |
|
|
which can be expressed in terms of spherical harmonics as shown in |
1300 |
|
|
below (cf. Arfkan p.811), |
1301 |
|
|
\begin{equation} |
1302 |
|
|
Y_{l,l+1,m}(\theta, \phi) = \sum_{m_1, m_2} C(l+1,1,l|m_1,m_2,m)\; {Y_{l+1}}^{m_1}(\theta,\phi)\; \hat{e}_{m_2}, |
1303 |
|
|
\label{eq:14} |
1304 |
|
|
\end{equation} |
1305 |
|
|
where $C(l+1,1,l|m_1,m_2,m)$ is a Clebsch-Gordan coefficient and |
1306 |
|
|
$\hat{e}_{m_2}$ is a spherical tensor of rank 1 which can be expressed |
1307 |
|
|
in terms of Cartesian coordinates, |
1308 |
|
|
\begin{equation} |
1309 |
|
|
{\hat{e}}_{+1} = - \frac{\hat{x}+i\hat{y}}{\sqrt{2}},\quad {\hat{e}}_{0} = \hat{z},\quad and \quad {\hat{e}}_{-1} = \frac{\hat{x}-i\hat{y}}{\sqrt{2}} |
1310 |
|
|
\label{eq:15} |
1311 |
|
|
\end{equation} |
1312 |
|
|
The normal vector $\hat{n} $ can be expressed in terms of spherical tensor of rank 1 as shown in below, |
1313 |
|
|
\begin{equation} |
1314 |
|
|
\hat{n} = \sqrt{\frac{4\pi}{3}}\left(-{Y_1}^{-1}{\hat{e}}_1 -{Y_1}^{1}{\hat{e}}_{-1} + {Y_1}^{0}{\hat{e}}_0 \right) |
1315 |
|
|
\label{eq:16} |
1316 |
|
|
\end{equation} |
1317 |
|
|
The surface integral of the product of $\hat{n}$ and |
1318 |
|
|
${Y_{l+1}}^{m_1}(\theta, \phi)$ gives, |
1319 |
|
|
\begin{equation} |
1320 |
|
|
\begin{split} |
1321 |
|
|
\int \hat{n}\;{Y_{l+1}}^{m_1}\;d\Omega &= \int \sqrt{\frac{4\pi}{3}}\left(-{Y_1}^{-1}{\hat{e}}_1 -{Y_1}^{1}{\hat{e}}_{-1} + {Y_1}^{0}{\hat{e}}_0 \right)\;{Y_{l+1}}^{m_1}\; d\Omega \\ |
1322 |
|
|
&= \int \sqrt{\frac{4\pi}{3}}\left({{Y_1}^{1}}^* {\hat{e}}_1 +{{Y_1}^{-1}}^* {\hat{e}}_{-1} + {{Y_1}^{0}}^* {\hat{e}}_0 \right)\;{Y_{l+1}}^{m_1}\; d\Omega \\ |
1323 |
|
|
&= \sqrt{\frac{4\pi}{3}}\left({\delta}_{l+1, 1}\;{\delta}_{1, m_1}\;{\hat{e}}_1 + {\delta}_{l+1, 1}\;{\delta}_{-1, m_1}\;{\hat{e}}_{-1}+ {\delta}_{l+1, 1}\;{\delta}_{0, m_1} \;{\hat{e}}_0\right), |
1324 |
|
|
\end{split} |
1325 |
|
|
\label{eq:17} |
1326 |
|
|
\end{equation} |
1327 |
|
|
where ${Y_{l}}^{-m} = (-1)^m\;{{Y_{l}}^{m}}^* $ and |
1328 |
|
|
$ \int {{Y_{l}}^{m}}^*\;{Y_{l'}}^{m'}\;d\Omega = |
1329 |
|
|
\delta_{ll'}\delta_{mm'} $. |
1330 |
|
|
Non-vanishing values of equation \ref{eq:17} require $l = 0$, |
1331 |
|
|
therefore the value of $ m = 0 $. Since the values of $ m_1$ are -1, |
1332 |
|
|
1, and 0 then $m_2$ takes the values 1, -1, and 0, respectively |
1333 |
|
|
provided that $m = m_1 + m_2$. Equation \ref{eq:11} can therefore be |
1334 |
|
|
modified, |
1335 |
|
|
\begin{equation} |
1336 |
|
|
\begin{split} |
1337 |
|
|
\int_{r<R} \vec{\nabla}\vec{E}\;d^3r = &- \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;{Y^*}_{00}(\theta', \phi')[ C(1, 1, 0|-1,1,0)\;{\hat{e}_{-1}}{\hat{e}_{1}}\\ &+ C(1, 1, 0|-1,1,0)\;{\hat{e}_{1}}{\hat{e}_{-1}}+C( |
1338 |
|
|
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d^3r'. |
1339 |
|
|
\end{split} |
1340 |
|
|
\label{eq:18} |
1341 |
|
|
\end{equation} |
1342 |
|
|
After substituting ${Y^*}_{00} = \frac{1}{\sqrt{4\pi}} $ and using the |
1343 |
|
|
values of the Clebsch-Gorden coefficients: $ C(1, 1, 0|-1,1,0) = |
1344 |
|
|
\frac{1}{\sqrt{3}}, \; C(1, 1, 0|-1,1,0)= \frac{1}{\sqrt{3}}$ and $ |
1345 |
|
|
C(1, 1, 0|0,0,0) = -\frac{1}{\sqrt{3}}$ in equation \ref{eq:18} we |
1346 |
|
|
obtain, |
1347 |
|
|
\begin{equation} |
1348 |
|
|
\begin{split} |
1349 |
|
|
\int_{r<R} \vec{\nabla}\vec{E}\;d^3r &= -\sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;d^3r'\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right)\\ |
1350 |
|
|
&= - \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\;C_{total}\;\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right). |
1351 |
|
|
\end{split} |
1352 |
|
|
\label{eq:19} |
1353 |
|
|
\end{equation} |
1354 |
|
|
Equation (\ref{eq:19}) gives the total gradient of the field over a |
1355 |
|
|
sphere due to the distribution of the charges. For quadrupolar fluids |
1356 |
|
|
the total charge within a sphere is zero, therefore |
1357 |
|
|
$ \int_{r<R} \vec{\nabla}\vec{E}\;d^3r = 0 $. Hence the quadrupolar |
1358 |
|
|
polarization produces zero net gradient of the field inside the |
1359 |
|
|
sphere. |
1360 |
|
|
|
1361 |
mlamichh |
4389 |
\section{Applied field or field gradient} |
1362 |
|
|
\label{Ap:fieldOrGradient} |
1363 |
mlamichh |
4353 |
|
1364 |
mlamichh |
4389 |
To satisfy the condition $ \nabla . E = 0 $, within the box of molecules we have taken electrostatic potential in the following form |
1365 |
|
|
\begin{equation} |
1366 |
|
|
\begin{split} |
1367 |
|
|
\phi(x, y, z) =\; &-g_o \left(\frac{1}{2}(a_1\;b_1 - \frac{cos\psi}{3})\;x^2+\frac{1}{2}(a_2\;b_2 - \frac{cos\psi}{3})\;y^2 + \frac{1}{2}(a_3\;b_3 - \frac{cos\psi}{3})\;z^2 \right. \\ |
1368 |
|
|
& \left. + \frac{(a_1\;b_2 + a_2\;b_1)}{2} x\;y + \frac{(a_1\;b_3 + a_3\;b_1)}{2} x\;z + \frac{(a_2\;b_3 + a_3\;b_2)}{2} y\;z \right), |
1369 |
|
|
\end{split} |
1370 |
|
|
\label{eq:appliedPotential} |
1371 |
|
|
\end{equation} |
1372 |
|
|
where $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ are basis vectors determine coefficients in x, y, and z direction. And $g_o$ and $\psi$ are overall strength of the potential and angle between basis vectors respectively. The electric field derived from the above potential is, |
1373 |
|
|
\[\bf{E} |
1374 |
|
|
=\frac{g_o}{2} \left(\begin{array}{ccc} |
1375 |
|
|
2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+ (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
1376 |
|
|
(a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y + (a_2\; b_3 \;+ a_3\; b_3)\;z \\ |
1377 |
|
|
(a_3\; b_1 \;+ a_3\; b_2)\;x + (a_3\; b_2 \;+ a_2\; b_3)y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
1378 |
|
|
\end{array} \right).\] |
1379 |
|
|
The gradient of the applied field derived from the potential can be written in the following form, |
1380 |
|
|
\[\nabla\bf{E} |
1381 |
|
|
= \frac{g_o}{2}\left(\begin{array}{ccc} |
1382 |
|
|
2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
1383 |
|
|
(a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_3)\;z \\ |
1384 |
|
|
(a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
1385 |
|
|
\end{array} \right).\] |
1386 |
mlamichh |
4353 |
\newpage |
1387 |
|
|
|
1388 |
mlamichh |
4397 |
\bibliography{dielectric_new} |
1389 |
mlamichh |
4353 |
|
1390 |
|
|
\end{document} |