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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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Note: This manuscript is a work in progress. |
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We report on the dielectric properties of the shifted potential |
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(SP), gradient shifted force (GSF), and Taylor shifted force (TSF) |
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real-space methods for multipole interactions that were developed in |
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the first two papers in this series. We find that some subtlety is |
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required for computing dielectric properties with the real-space |
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methods, particularly when using the common fluctuation formulae. |
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Three distinct methods for computing the dielectric constant are |
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investigated, including the standard fluctuation formulae, |
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potentials of mean force between solvated ions, and direct |
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measurement of linear solvent polarization in response to applied |
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fields and field gradients. |
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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 methods for correcting electrostatic interactions in computer |
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simulations of condensed molecular |
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systems.\cite{Wolf99,Zahn02,Kast03,Beckd.A.C._Bi0486381,Ma05,Fennell06} |
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These techniques were initially developed from the observations and efforts of |
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Wolf {\it et al.} and their work towards an $\mathcal{O}(N)$ |
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Coulombic sum.\cite{Wolf99} Wolf's method of cutoff neutralization is |
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able to obtain excellent agreement with Madelung energies in ionic |
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crystals.\cite{Wolf99} Later, Zahn \textit{et al.} and Fennell and Gezelter extended this method which incorporates Wolf's electrostatic energy and modified it to conserve the total energy in molecular dynamic simulation.\cite{Zahn02, Fennell06} We have also previously developed three generalized real space methods: Shifted potential (SP), Gradeint shifted force (GSF), and Taylor shifted force (TSF) for evaluating electrostatic interactions for higher order multipoles (dipoles and quadrupoles) in the previous two papers in the series.\cite{PaperI, PaperII} These real space methods consider a finite cutoff sphere with the neutralization of the electrostatic moment within the cutoff sphere. Furthermore, extra terms have been added to the potential energy to vanish force and torque smoothly at the cutoff radius which ensures the conservation of the total energy in the molecular dynamic simulation. |
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One of the most difficult tests of any new electrostatic method is the fidelity with which that method can reproduce the bulk-phase polarizability or equivalently, the dielectric properties of a fluid. Since dielectric properties are macroscopic properties, all interactions between molecules in an entire system are significantly important. But it is computationally infeasible to consider every interactions between molecules in the macroscopically large system. Therefore small molecular system with periodic boundary condition and finite cutoff region of interactions is usually considered in computer simulations. While calculating dielectric properties, the formula should be modified in such a way so that it can accommodate behaviour of electrostatic neutrality and smoothness of energy, force and torque at the cutoff radius. Previously many studies have been conducted to calculate dipolar and quadrupolar dielectric properties using computer simulations. \cite{Kirkwood39, Onsagar36,LoganI81, LoganII82, LoganIII82} But these methods do not specifically take account of the cutoff behavior common in real-space electrosatic methods. In 1983 Neumann proposed a general formula for evaluating dielectric properties for dipolar fluid using real-space cutoff methods. \cite{Neumann83} In the same year Steinhauser and Neumann used this formula to evaluate the correct dielectric constant for the Stockmayer fluid using two different methods: Ewald-Kornfield (EK) and reaction field (RF) methods. \cite{Neumann-Steinhauser83} This formula contains a correction factor which is equal to $\frac{3}{4 \pi} $ times volume integral of the dipole-dipole interactions for a given electrostatic cutoff method (See equation \ref{dipole-diopleTensor}).\cite{Neumann83} Similarly Zahn \textit{et al.}\cite{Zahn02} also evaluated correction factor for dipole-dipole interaction using damped shifted charge-charge kernel (see equation \ref{dipole-chargeTensor}). This later generalized by Izvekove \textit{et al.}, which is equal to $\frac{3}{4 \pi} $. \cite{Izvekov:2008wo} When the correction factor is equal to $\frac{3}{4 \pi} $, the expression for the dielectric constant reduces to widely-used \textit{conducting boundary} formula (see equation (\ref{correctionFormula})). Many studies have also been conducted to understand solvation theory using dielectric properties of quadrupolar fluid.\cite{JeonI03, JeonII03, Chitanvis96}. But these studies do not use correction factor straight forwardly to evaluate correct dielectric properties for quadrupolar fluid. |
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In this paper we are proposing analogous general formula for calculating dielectric properties for quadrupolar fluid. Furthermore we have also evaluated the correction factor for SP, GSF, and TSF method for both dipolar and quadrupolar fluid considering charge-charge, dipole-dipole or quadrupole-quadrupole interactions. The relation between quadrupolar susceptibility and dielectric constant is not straight forward for quadrupolar fluid as in the dipolar case. The dielectric constant depends on the geometry of the external field perturbation.\cite{Ernst92} We have also calculated the geometrical factor for two ions immersed quadrupolar system to evaluate dielectric constant from the quadrupolar susceptibility. We have used three different methods: fluctuation formula, external field perturbation and the potential of mean force, to study dielectric properties of the dipolar and quadrupolar system. The fluctuation formula observes the time average fluctuation of the multipolar moment as a function of temperature. The average fluctuation value of the system is determined by the multipole-multipole interactions between molecules at a given temperature. In the external field perturbation, the net polarization of the system is observed as a linear response of the applied field perturbation, where proportionality constant is determined by the electrostatic interaction between the electrostatic multipoles at a given temperature. Since the expression of the electrostatic interaction energy, force, and torque in the real space electrostatic methods are slightly modified from their original definition therefore both fluctuation and external field perturbation formula should also be modified accordingly.The potential of mean force method calculates dielectric constant from the potential energy between ions before and after dielectric material is introduced. All of these different methods for calculating dielectric properties will be discussed in detail in the following sections: \ref{sec:perturbation}, \ref{sec:fluctuation}, and \ref{sec:PMF}. |
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\subsection{Boltzmann average of orientational polarization - dipole} |
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Consider a system of molecules with permenent dipole moment $p_o$. In the absense of external field, thermal agitation makes dipole randomly oriented therefore there is no net dipole moment. But external field tends them to line up in the direction of applied field. Therefore the total Hamiltonian of the molecule is,\cite{Jackson98} |
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\begin{equation} |
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H = H_o - \bf{p_o} .\bf{E}, |
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\end{equation} |
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where $H_o$ is a function of the internal coordinates of the molecule. Now Boltzmann average of the dipole moment is given by, |
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\begin{equation} |
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\braket{p_{mol}} = \frac{\displaystyle\int d\Omega\; p_o\; cos\theta\; e^{\frac{p_oE\; cos\theta}{k_B T}}}{\displaystyle\int d\Omega\; e^{\frac{p_oE\;cos\theta}{k_B T}}}, |
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\end{equation} |
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where $\bf{E}$ is selected along z-axis. If we consider applied field is small i.e. $\frac{p_oE\; cos\theta}{k_B T} << 1$ then we get, |
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\begin{equation} |
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\braket{p_{mol}} \approx \frac{1}{3}\frac{{p_o}^2}{k_B T}E, |
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\end{equation} |
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where $ \alpha_p = \frac{1}{3}\frac{{p_o}^2}{k_B T}$ is a molecular polarizability. The orientational polarization depends inversely on the temperature and applied field must overcome the thermal agitation. |
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\label{boltzAverage-dipole} |
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\subsection{Boltzmann average of orientational polarization - quadrupole} |
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Consider a system of molecules with permanent quadrupole moment $q_{\alpha\beta} $. The average quadrupole moment at temperature T in the presence of uniform applied field gradient is given by,\cite{AduGyamfi78, AduGyamfi81} |
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\begin{equation} |
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\braket{q_{\alpha\beta}} \;=\; \frac{\displaystyle\int d\Omega\; e^{-\frac{H}{k_B T}}q_{\alpha\beta}}{\displaystyle\int d\Omega\; e^{-\frac{H}{k_B T}}} \;=\; \frac{\displaystyle\int d\Omega\; e^{\frac{q_{\mu\nu}\;\partial_\nu E_\mu}{k_B T}}q_{\alpha\beta}}{\displaystyle\int d\Omega\; e^{\frac{q_{\mu\nu}\;\partial_\nu E_\mu}{k_B T}}}, |
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\label{boltzQuad} |
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\end{equation} |
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where $\int d\Omega = \int_0^{2\pi} \int_0^\pi \int_0^{2\pi} |
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sin\theta\; d\theta\ d\phi\ d\psi$ is the integration over Euler |
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angles, $ H = H_o -q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of |
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a quadrupole in the gradient of the |
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applied field and $ H_o$ is a function of internal coordinates of the molecule. The energy and quadrupole moment can be transformed into body frame using following relation, |
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\begin{equation} |
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\begin{split} |
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&q_{\alpha\beta} = \eta_{\alpha\alpha'}{q}^* _{\alpha'\beta'}\eta_{\beta'\beta} \\ |
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&H = H_o - q:\vec{\nabla}\vec{E} = H_o - q_{\mu\nu}\;\partial_\nu E_\mu = H_o -\eta_{\mu\mu'}{q}^*_{\mu'\nu'}\eta_{\nu'\nu}\;\partial_\nu E_\mu. |
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\end{split} |
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\label{energyQuad} |
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\end{equation} |
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Here the starred tensors are the components in the body fixed |
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frame. Substituting equation (\ref{energyQuad}) in the equation (\ref{boltzQuad}) |
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and taking linear terms in the expansion we get, |
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\begin{equation} |
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\braket{q_{\alpha\beta}} = \frac{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}{q}^*_{\mu'\nu'}\eta_{\nu'\nu}\;\partial_\nu E_\mu }{k_B T}\right)q_{\alpha\beta}}{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}{q}^*_{\mu'\nu'}\eta_{\nu'\nu}\;\partial_\nu E_\mu }{k_B T}\right)}, |
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\end{equation} |
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where $\eta_{\alpha\alpha'}$ is the rotation matrix that transforms |
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the body fixed to the space fixed frame, |
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\[\eta_{\alpha\alpha'} |
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= \left(\begin{array}{ccc} |
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cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & cos\theta\; cos\psi\; sin\phi + cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
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-cos\psi\; sin\phi - cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & cos\phi\; cos\theta \\ |
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sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
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\end{array} \right).\] |
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Integration of 1st and 2nd terms in the denominator gives $8 \pi^2$ |
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and $8 \pi^2 /3\;\vec{\nabla}.\vec{E}\; Tr(q^*) $ respectively. The |
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second term vanishes for charge free space |
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(i.e. $\vec{\nabla}.\vec{E} \; = \; 0)$. Similarly integration of the |
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1st term in the numerator produces |
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$8 \pi^2 /3\; Tr(q^*)\delta_{\alpha\beta}$ and the 2nd term produces |
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$8 \pi^2 /15k_B T (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
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{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta$, |
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if $\vec{\nabla}.\vec{E} \; = \; 0$, |
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$ \partial_\alpha E_\beta = \partial_\beta E_\alpha$ and |
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${q}^*_{\alpha'\beta'}= {q}^*_{\beta'\alpha'}$. Therefore the |
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Boltzmann average of a quadrupole moment can be written as, |
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\begin{equation} |
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\braket{q_{\alpha\beta}}\; = \; \frac{1}{3} Tr(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
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\end{equation} |
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where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupolarizablity and ${\bar{q_o}}^2= |
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3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. |
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\section{External field perturbation} |
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In the presence of uniform electric field $\textbf{E}^o$ (or field gradient $\partial_\alpha E^o_\beta$), a system of dipolar (or quadrupolar) molecules polarizes along the direction of the applied field (or field gradient). Therefore the net dipolar polarization $ \textbf{P}$ of the system is, |
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\begin{equation} |
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\textbf{P} = \epsilon_o \alpha_{D}\; \textbf{E}^o. |
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\label{pertDipole} |
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\end{equation} |
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The constant $\alpha_D$ is a macroscopic polarizability, which is a property of the dipolar fluid in a given density and temperature. Similarly, the net quadrupolar polarization of the system can be given by, |
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\begin{equation} |
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\begin{split} |
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& {Q}_{\alpha\beta} = \frac{1}{3}\; Tr({Q})\; \delta_{\alpha\beta} + \epsilon_o\; \alpha_Q \; \partial_{\alpha} E^o_{\beta} |
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\\ & or \\ |
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& \frac{1}{3}\;\Theta_{\alpha\beta} = \epsilon_o\; \alpha_Q \; \partial_{\alpha} E^o_{\beta} |
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\end{split} |
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\label{pertQuad} |
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\end{equation} |
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where $Q_{\alpha\beta}$ is a tensor component of the traced quadrupolar moment of the system, $ \alpha_Q$ is a macroscopic quadrupolarizability has a dimension of $length^{-2}$, and $\Theta_{\alpha\beta} = 3Q_{\alpha\beta}-Tr(Q) $ is the traceless component of the quadrupole moment. |
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\label{sec:perturbation} |
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\section{Fluctuation formula} |
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For a system of molecules with net dipolar moment $\bf{M}$ at thermal equilibrium of temperature T in the presence of applied field $\bf{E}^o$, the average dipolar polarization can be expressed in terms of fluctuation of the net dipole moment as below,\cite{Stern03} |
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\begin{equation} |
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\braket{\bf{P}} = \epsilon_o \frac{\braket{\bf{M}^2}-{\braket{\bf{M}}}^2}{3 \epsilon_o V k_B T}\bf{E}^o |
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\label{flucDipole} |
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\end{equation} |
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This is similar to the formula for boltzmann average of single dipolar molecule in the subsection \ref{boltzAverage-dipole}. Here $\braket{\bf{P}}$ is average polarization and $ \braket{\textbf{M}^2}-{\braket{\textbf{M}}}^2$ is the net dipole fluctuation at temperature T. For the limiting case $\textbf{E}^o \rightarrow 0 $, ensemble average of both net dipole moment $\braket{\textbf{M}}$ and dipolar polarization $\braket{\bf{P}}$ tends to vanish but $\braket{\bf{M}^2}$ will still be non-zero. The dipolar macroscopic polarizability can be written as, |
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\begin{equation} |
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\alpha_D = \frac{\braket{\bf{M}^2}-{\braket{\bf{M}}}^2}{3 \epsilon_o V k_B T} |
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\end{equation} |
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This is a macroscopic property of dipolar material which is true even if applied field $ \textbf{E}^o \rightarrow 0 $. |
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Analogous formula can also be written for a system with quadrupolar molecules, |
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\begin{equation} |
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\braket{Q_{\alpha\beta}} = \frac{1}{3} Tr(\textbf{Q})\; \delta_{\alpha\beta} + \epsilon_o \frac{\braket{\textbf{Q}^2}-{\braket{\textbf{Q}}}^2}{15 \epsilon_o V k_B T}{\partial_\alpha E^o_\beta} |
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\label{flucQuad} |
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\end{equation} |
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where $Q_{\alpha\beta}$ is a component of system quadrupole moment, $\bf{Q}$ is net quadrupolar moment which can be expressed as $\textbf{Q}^2 =3Q_{\alpha\beta}Q_{\alpha\beta}-(Tr\textbf{Q})^2 $. The macroscopic quadrupolarizability is given by, |
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\begin{equation} |
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\alpha_Q = \frac{\braket{\textbf{Q}^2}-{\braket{\textbf{Q}}}^2}{15 \epsilon_o V k_B T} |
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\label{propConstQuad} |
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\end{equation} |
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\label{sec:fluctuation} |
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\section{Potential of mean force} |
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In this method, we will measure the interaction between a positive and negative charge at varying distances after introducing a dipolar (or quadrupolar) material between them. The potential of mean force (PMF) between two ions in a liquid is obtained by constraining their distance and measuring the mean constraint force required to hold them at a fixed distance $r.$ The PMF is obtained from a sequence of simulations as, |
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\begin{equation} |
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w(r) = \int_{\inf}^{r}\braket{\frac{\partial f}{\partial r'}}dr', |
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\end{equation} |
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where $\braket{\partial f/\partial r'}$ is the mean constraint force. |
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Since the ions have a protecting Lennard-Jones (LJ) potential, |
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\begin{equation} |
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w(r) = w_{LJ}(r) + \frac{q_iq_j}{4\pi \epsilon_o \epsilon(r)}U_{method}(r). |
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\end{equation} |
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Here $w_{LJ}$ is the PMF calculated without electrostatic interactions and $U_{method}(r)$ is the radial function for the charge-charge interaction, which is different for various real space truncation methods. |
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The quadrupole molecule can only couple with the gradient of the electric field and the region between two opposite point charges has both an electric field and a gradient of the electric field present. Therefore, this methodology should be usable to determine the dielectric constant for both the dipolar and quadrupolar fluid. |
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\label{sec:PMF} |
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\section{Correction factor} |
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Since equations (\ref{pertDipole}, \ref{pertQuad}, \ref{flucDipole}, and \ref{flucQuad}) provide relation between polarization (dipolar or quadrupolar) and applied field (uniform field or field gradient), $\chi_d$ (or $ \chi_q$) is actually a macroscopic polarizability (or quadrupolarizability), which is different than the dipolar (or quadrupolar) susceptibility of the fluid. Actual constitutive relation should have a relation between polarization and Maxwell field (or field gradient) at different point in the sample. We can obtain susceptibility of the fluid from its macroscopic polarizability using correction factor evaluated below. |
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\subsection{Dipolar system} |
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In the presence of an external field $ \textbf{E}$ polarization $\textbf{E}$ will be induced in a dipolar system. The total electrostatic field (or Maxwell electric field) at point $\bf{r}$ in a system is,\cite{Neumann83} |
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\begin{equation} |
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\textbf{E}(\textbf{r}) = \textbf{E}^o(\textbf{r}) + \frac{1}{4\pi\epsilon_o} \int d^3r' \textbf{T}(\textbf{r}-\textbf{r}')\cdot {\textbf{P}(\textbf{r}')}. |
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\end{equation} |
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We can consider the cases of Stockmayer (dipolar) soft spheres that are represented either by two closely-spaced point charges or by a single point dipole (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} |
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\end{figure} |
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In the case where point charges are interacting via an electrostatic |
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kernel, $v(r)$, the effective {\it molecular} dipole tensor, |
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$\mathbf{T}$ is obtained from two successive applications of the |
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gradient operator to the electrostatic kernel, |
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\begin{equation} |
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\mathbf{T}_{\alpha \beta}(r) = \nabla_\alpha \nabla_\beta \left(v(r)\right) = \delta_{\alpha \beta} |
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\left(\frac{1}{r} v^\prime(r) \right) + \frac{r_{\alpha} |
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r_{\beta}}{r^2} \left( v^{\prime \prime}(r) - \frac{1}{r} |
| 227 |
|
|
v^{\prime}(r) \right) |
| 228 |
|
|
\label{dipole-chargeTensor} |
| 229 |
|
|
\end{equation} |
| 230 |
|
|
where $v(r)$ may be either the bare kernel ($1/r$) or one of the |
| 231 |
|
|
modified (Wolf or DSF) kernels. This tensor describes the effective |
| 232 |
|
|
interaction between molecular dipoles ($\mathbf{D}$) in Gaussian |
| 233 |
|
|
units as $-\mathbf{D} \cdot \mathbf{T} \cdot \mathbf{D}$. |
| 234 |
|
|
|
| 235 |
|
|
When utilizing the new real-space methods for point dipoles, the |
| 236 |
|
|
tensor is explicitly constructed, |
| 237 |
|
|
\begin{equation} |
| 238 |
|
|
\mathbf{T}_{\alpha \beta}(r) = \delta_{\alpha \beta} v_{21}(r) + |
| 239 |
|
|
\frac{r_{\alpha} r_{\beta}}{r^2} v_{22}(r) |
| 240 |
|
|
\label{dipole-diopleTensor} |
| 241 |
|
|
\end{equation} |
| 242 |
|
|
where the functions $v_{21}(r)$ and $v_{22}(r)$ depend on the level of |
| 243 |
|
|
the approximation. Although the Taylor-shifted (TSF) and |
| 244 |
|
|
gradient-shifted (GSF) models produce to the same $v(r)$ function for |
| 245 |
|
|
point charges, they have distinct forms for the dipole-dipole |
| 246 |
|
|
interactions. |
| 247 |
|
|
|
| 248 |
|
|
Using constitutive relation, the polarization density $\textbf{P}(\textbf{r})$ is given by, |
| 249 |
|
|
\begin{equation} |
| 250 |
|
|
\textbf{P}(\textbf{r}) = \epsilon_o\; \chi^*_D \left(\textbf{E}^o(\textbf{r}) + \frac{1}{4\pi\epsilon_o} \int d^3r' \textbf{T}(\textbf{r}-\textbf{r}')\cdot {\textbf{P}(\textbf{r}')}\right). |
| 251 |
|
|
\label{constDipole} |
| 252 |
|
|
\end{equation} |
| 253 |
|
|
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{sec:perturbation} and \ref{sec: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{Neumann83, Jackson98} |
| 254 |
|
|
\begin{equation} |
| 255 |
|
|
\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} |
| 256 |
|
|
\label{singular} |
| 257 |
|
|
\end{equation} |
| 258 |
|
|
Substituting equation (\ref{singular}) in the equation (\ref{constDipole}) we get, |
| 259 |
|
|
\begin{equation} |
| 260 |
|
|
\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). |
| 261 |
|
|
\end{equation} |
| 262 |
|
|
For both polarization and electric field homogeneous, this can be easily solved using Fourier transformation, |
| 263 |
|
|
\begin{equation} |
| 264 |
|
|
\textbf{P}(\kappa) = 3 \epsilon_o\; \frac{\chi^*_D}{\chi^*_D + 3} \left(1- \frac{3}{4\pi}\;\frac{\chi^*_D}{\chi^*_D + 3}\; \textbf{T}({\kappa})\right)^{-1}\textbf{E}^o({\kappa}). |
| 265 |
|
|
\end{equation} |
| 266 |
|
|
For homogeneous applied field Fourier component is non-zero only if $\kappa = 0$. Therefore, |
| 267 |
|
|
\begin{equation} |
| 268 |
|
|
\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}). |
| 269 |
|
|
\label{fourierDipole} |
| 270 |
|
|
\end{equation} |
| 271 |
|
|
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, |
| 272 |
|
|
\begin{equation} |
| 273 |
|
|
\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} |
| 274 |
|
|
\end{equation} |
| 275 |
|
|
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$ in above equation we get, |
| 276 |
|
|
\begin{equation} |
| 277 |
|
|
\epsilon = \frac{3+(A_{dipole} + 2)(\epsilon_{CB}-1)}{3+(A_{dipole} -1)(\epsilon_{CB}-1)} |
| 278 |
|
|
\label{correctionFormula} |
| 279 |
|
|
\end{equation} |
| 280 |
|
|
where $\epsilon_{CB}$ is dielectric constant obtained from conducting boundary condition. Equation (\ref{correctionFormula}) calculates actual dielectric constant from the dielectric constant obtained from the conducting boundary condition (which can be obtained directly from the simulation) using correction factor. The correction factor is different for different real-space cutoff methods. The expression for correction factor assuming a single point dipole or two closely spaced point charges for SP, GSF, and TSF method is listed in Table \ref{tab:A}. |
| 281 |
|
|
\begin{table} |
| 282 |
|
|
\caption{Expressions for the dipolar correction factor ($A$) for the real-space electrostatic methods in terms of the damping parameter |
| 283 |
|
|
($\alpha$) and the cutoff radius ($r_c$). The Ewald-Kornfeld result |
| 284 |
|
|
derived in Refs. \onlinecite{Adams:1980rt,Adams:1981fr,Neumann83} is shown for comparison using the Ewald |
| 285 |
|
|
convergence parameter ($\kappa$) and the real-space cutoff value ($r_c$). } |
| 286 |
|
|
\label{tab:A} |
| 287 |
|
|
{% |
| 288 |
|
|
\begin{tabular}{l|c|c|c|} |
| 289 |
|
|
|
| 290 |
|
|
Method & $A_\mathrm{charges}$ & $A_\mathrm{dipoles}$ \\ |
| 291 |
|
|
\hline |
| 292 |
|
|
Spherical Cutoff (SC) & $\mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $\mathrm{erf}(r_c \alpha) - \frac{2 \alpha r_c}{\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ \\ |
| 293 |
|
|
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} $\\ |
| 294 |
|
|
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} $ \\ |
| 295 |
|
|
Taylor-shifted (TSF) & 1 & 1 \\ |
| 296 |
|
|
Ewald-Kornfeld (EK) & $\mathrm{erf}(r_c \kappa) - \frac{2 \kappa r_c}{\sqrt{\pi}} e^{-\kappa^2 r_c^2}$ & $\mathrm{erf}(r_c \kappa) - \frac{2 \kappa r_c}{\sqrt{\pi}} e^{-\kappa^2 r_c^2}$ \\\hline |
| 297 |
|
|
\end{tabular}% |
| 298 |
|
|
} |
| 299 |
|
|
\end{table} |
| 300 |
|
|
\subsection{Quadrupolar system} |
| 301 |
|
|
In the presence of the field gradient $\partial_\alpha {E}_\beta $, a |
| 302 |
|
|
non-vanishing quadrupolar polarization (quadrupole moment per unit |
| 303 |
|
|
volume) $\bar{Q}_{\alpha\beta}$ will be induced in the entire volume |
| 304 |
|
|
of a sample. The total field at any point $\vec{r}$ in the sample is |
| 305 |
|
|
given by, |
| 306 |
|
|
\begin{equation} |
| 307 |
|
|
\partial_\alpha E_\beta(\textbf{r}) = \partial_\alpha {E^o}_\beta(\textbf{r}) + \frac{1}{4\pi \epsilon_o}\int T_{\alpha\beta\gamma\delta}(|{\textbf{r}-\textbf{r}'}|)\;{Q}_{\gamma\delta}(\textbf{r}')\; d^3r' |
| 308 |
|
|
\label{gradMaxwell} |
| 309 |
|
|
\end{equation} |
| 310 |
|
|
where $\partial_\alpha {E^o}_\beta$ is the applied field gradient and $ T_{\alpha\beta\gamma\delta}$ is the quadrupole-quadrupole interaction tensor. We can represent quadrupole as a group of four closely spaced charges, two closely spaced point dipoles or single point quadrupole (see Fig. \ref{fig:quadrupolarFluid}). The quadrupole-quadrupole interaction tensor from the charge representation can obtained from the application of the four successive gradient operator to the electrostatic kernel $v(r)$. |
| 311 |
|
|
|
| 312 |
|
|
\begin{equation} |
| 313 |
|
|
\begin{split} |
| 314 |
|
|
T_{\alpha\beta\gamma\delta}(r) &=\nabla_\alpha \nabla_\beta \nabla_\gamma \nabla_\delta\;v(r) |
| 315 |
|
|
\\ &= \left(\delta_{\alpha\beta}\delta_{\gamma\delta} + \delta_{\alpha\gamma}\delta_{\beta\delta}+ \delta_{\alpha\delta}\delta_{\beta\gamma}\right)\left(-\frac{v'(r)}{r^3} + \frac{v''(r)}{r^2}\right) |
| 316 |
|
|
\\ &+ \left(\delta_{\alpha\beta} r_\gamma r_\delta + 5 \; permutations \right) \left(\frac{3v'(r)}{r^5}-\frac{3v''(r)}{r^4} + \frac{v'''(r)}{r^3}\right) |
| 317 |
|
|
\\ &+ r_\alpha r_\beta r_\gamma r_\delta\; \left(-\frac{15v'(r)}{r^7}+\frac{15v''(r)}{r^6}-\frac{6v'''(r)}{r^5} + \frac{v''''(r)}{r^4}\right), |
| 318 |
|
|
\end{split} |
| 319 |
|
|
\label{quadCharge} |
| 320 |
|
|
\end{equation} |
| 321 |
|
|
where $v(r)$ can either be electrostatic kernel for spherical truncation or one of the modified (Wolf or DSF) method. Similarly in point dipole representation the qaudrupole-quadrupole interaction tensor can be obtained from the applications of the two successive gradient in the dipole-dipole interaction tensor, |
| 322 |
|
|
|
| 323 |
|
|
\begin{equation} |
| 324 |
|
|
\begin{split} |
| 325 |
|
|
T_{\alpha\beta\gamma\delta}(r) &=\nabla_\alpha \nabla_\beta \;v_{\gamma\delta}(r) |
| 326 |
|
|
\\ &= \delta_{\alpha\beta}\delta_{\gamma\delta} \frac{v'_{21}(r)}{r} + \left(\delta_{\alpha\gamma}\delta_{\beta\delta}+ \delta_{\alpha\delta}\delta_{\beta\gamma}\right)\frac{v_{22}(r)}{r^2} |
| 327 |
|
|
\\ &+ \delta_{\gamma\delta} r_\alpha r_\beta \left(\frac{v''_{21}(r)}{r^2}-\frac{v'_{21}(r)}{r^3} \right) |
| 328 |
|
|
\\ &+\left(\delta_{\alpha\beta} r_\gamma r_\delta + \delta_{\alpha\gamma} r_\beta r_\delta +\delta_{\alpha\delta} r_\gamma r_\beta + \delta_{\beta\gamma} r_\alpha r_\delta +\delta_{\beta\delta} r_\alpha r_\gamma \right) \left(\frac{v'_{22}(r)}{r^3}-\frac{2v_{22}(r)}{r^4}\right) |
| 329 |
|
|
\\ &+ r_\alpha r_\beta r_\gamma r_\delta\; \left(\frac{v''_{22}(r)}{r^4}-\frac{5v'_{22}(r)}{r^5}+\frac{8v_{22}(r)}{r^6}\right), |
| 330 |
|
|
\end{split} |
| 331 |
|
|
\label{quadDip} |
| 332 |
|
|
\end{equation} |
| 333 |
|
|
where $v_{\gamma\delta}(r)$ is the electrostatic dipole-dipole interaction tensor, which is different for different electrostatic cut off methods. Similarly $v_{21}(r) \;and\; v_{22}(r)$ are the radial function for different real space cutoff methods defined in Paper I of the series.\cite{PaperI} Using point quadrupole representation the quadrupole-quadrupole interaction can be constructed as, |
| 334 |
|
|
\begin{equation} |
| 335 |
|
|
\begin{split} |
| 336 |
|
|
T_{\alpha\beta\gamma\delta}(r) &= \left(\delta_{\alpha\beta}\delta_{\gamma\delta} + \delta_{\alpha\gamma}\delta_{\beta\delta}+ \delta_{\alpha\delta}\delta_{\beta\gamma}\right)v_{41}(r) + \delta_{\gamma\delta} r_\alpha r_\beta \frac{v_{42}(r)}{r^2} \\ &+ r_\alpha r_\beta r_\gamma r_\delta\; \left(\frac{v_{43}(r)}{r^4}\right), |
| 337 |
|
|
\end{split} |
| 338 |
|
|
\label{quadRadial} |
| 339 |
|
|
\end{equation} |
| 340 |
|
|
where $v_{41}(r),\; v_{42}(r), \; \text{and} \; v_{43}(r)$ are defined in Paper I of the series. \cite{PaperI} They have different functional forms for different electrostatic cutoff methods. |
| 341 |
|
|
\begin{figure} |
| 342 |
|
|
\includegraphics[width=3in]{QuadrupoleFigure} |
| 343 |
|
|
\caption{With the real-space electrostatic methods, the effective |
| 344 |
|
|
quadrupolar tensor, $\mathbf{T}_{\alpha\beta\gamma\delta}(r)$, governing interactions between molecular quadrupoles can be represented by interaction of charges, point dipoles or single point quadrupoles.} |
| 345 |
|
|
\label{fig:quadrupolarFluid} |
| 346 |
|
|
\end{figure} |
| 347 |
|
|
The integral in equation (\ref{gradMaxwell}) can be divided into two parts, $|\textbf{r}-\textbf{r}'|\rightarrow 0 $ and $|\textbf{r}-\textbf{r}'|> 0$. Since the total |
| 348 |
|
|
field gradient due to quadrupolar fluid vanishes at the singularity (see Appendix \ref{singularQuad}), equation (\ref{gradMaxwell}) can be written as, |
| 349 |
|
|
\begin{equation} |
| 350 |
|
|
\partial_\alpha E_\beta(\textbf{r}) = \partial_\alpha {E^o}_\beta(\textbf{r}) + |
| 351 |
|
|
\frac{1}{4\pi \epsilon_o}\int\limits_{|\textbf{r}-\textbf{r}'|> 0 } |
| 352 |
|
|
T_{\alpha\beta\gamma\delta}(|\textbf{r}-\textbf{r}'|)\;{Q}_{\gamma\delta}(\textbf{r}')\; |
| 353 |
|
|
d^3r'. |
| 354 |
|
|
\end{equation} |
| 355 |
|
|
If $\textbf{r} = \textbf{r}'$ is excluded from the integration, the gradient of the electric can be expressed in terms of traceless quadrupole moment as below, \cite{LoganI81} |
| 356 |
|
|
\begin{equation} |
| 357 |
|
|
\partial_\alpha E_\beta(\textbf{r}) = \partial_\alpha {E^o}_\beta(\textbf{r}) + \frac{1}{12\pi \epsilon_o}\int\limits_{|\textbf{r}-\textbf{r}'|> 0 } T_{\alpha\beta\gamma\delta}(|\textbf{r}-\textbf{r}'|)\;{\Theta}_{\gamma\delta}(\textbf{r}')\; d^3r', |
| 358 |
|
|
\end{equation} |
| 359 |
|
|
where $\Theta_{\alpha\beta} = 3Q_{\alpha\beta} - \delta_{\alpha\beta}Tr(Q)$ |
| 360 |
|
|
is the traceless quadrupole moment. The total quadrupolar polarization is written as, |
| 361 |
|
|
\begin{equation} |
| 362 |
|
|
{Q}_{\alpha\beta}(\textbf{r}) = \frac{1}{3}\delta_{\alpha\beta}\;Tr({Q})+\epsilon_o {\chi}^*_Q\;\partial_\alpha E_\beta(\textbf{r}), |
| 363 |
|
|
\label{constQaud} |
| 364 |
|
|
\end{equation} |
| 365 |
|
|
In the equation (\ref{constQaud}), $\partial_{\alpha}E_{\beta}$ is Maxwell field gradient and ${\chi}^*_Q$ is the actual quadrupolar susceptibility of the fluid which is different than the proportionality constant $\chi_q $ in the equation (\ref{propConstQuad}). In terms of traceless quadrupole moment, equation (\ref{constQaud}) can be written as, |
| 366 |
|
|
\begin{equation} |
| 367 |
|
|
\frac{1}{3}{\Theta}_{\alpha\beta}(\textbf{r}) = \epsilon_o {\chi}^*_Q \; \partial_\alpha E_\beta (\textbf{r})= \epsilon_o {\chi}^*_Q \left(\partial_\alpha {E^o}_\beta(\textbf{r}) + \frac{1}{12\pi \epsilon_o}\int\limits_{|\textbf{r}-\textbf{r}'|> 0 } T_{\alpha\beta\gamma\delta}(|\textbf{r}-\textbf{r}'|)\;{\Theta}_{\gamma\delta}(\textbf{r}')\; d^3r'\right) |
| 368 |
|
|
\end{equation} |
| 369 |
|
|
For toroidal boundary conditions, both $\partial_\alpha E_\beta$ and |
| 370 |
|
|
$\bar{\Theta}_{\alpha\beta}$ are uniform over the entire space. After |
| 371 |
|
|
performing a Fourier transform (see the Appendix in the Neumann's Paper \cite{Neumann83}) we get, |
| 372 |
|
|
\begin{equation} |
| 373 |
|
|
\frac{1}{3}{{\Theta}}_{\alpha\beta}({\kappa})= |
| 374 |
|
|
\epsilon_o {\chi}^*_Q \;\left[{\partial_\alpha |
| 375 |
|
|
{E^o}_\beta}({\kappa})+ \frac{1}{12\pi |
| 376 |
|
|
\epsilon_o}\;{T}_{\alpha\beta\gamma\delta}({\kappa})\; |
| 377 |
|
|
{{\Theta}}_{\gamma\delta}({\kappa})\right] |
| 378 |
|
|
\end{equation} |
| 379 |
|
|
Since the quadrupolar polarization is in the direction of the applied |
| 380 |
|
|
field, we can write |
| 381 |
|
|
${{\Theta}}_{\gamma\delta}({\kappa}) = |
| 382 |
|
|
{{\Theta}}_{\alpha\beta}({\kappa})$ |
| 383 |
|
|
and |
| 384 |
|
|
${T}_{\alpha\beta\gamma\delta}({\kappa}) = |
| 385 |
|
|
{T}_{\alpha\beta\alpha\beta}({\kappa})$ |
| 386 |
|
|
(note: This is true for orientational polarization but might not be |
| 387 |
|
|
true for polarization due to changes in the atomic distribution. For |
| 388 |
|
|
the orientational case, we only need to work with individual |
| 389 |
|
|
components of the matrix). |
| 390 |
|
|
\begin{equation} |
| 391 |
|
|
\begin{split} |
| 392 |
|
|
\frac{1}{3}{{\Theta}}_{\alpha\beta}({\kappa}) &= \epsilon_o {\chi}^*_Q \left[{\partial_\alpha E^o_\beta}({\kappa})+ \frac{1}{12\pi \epsilon_o}{T}_{\alpha\beta\alpha\beta}({\kappa})\;{{\Theta}}_{\alpha\beta}({\kappa})\right] \\ |
| 393 |
|
|
&= \epsilon_o {\chi}^*_Q\;\left(1-\frac{1}{4\pi} {\chi}^*_Q\; |
| 394 |
|
|
{T}_{\alpha\beta\alpha\beta}({\kappa})\right)^{-1} |
| 395 |
|
|
{\partial_\alpha E^o_\beta}({\kappa}) |
| 396 |
|
|
\end{split} |
| 397 |
|
|
\label{fourierQuad} |
| 398 |
|
|
\end{equation} |
| 399 |
|
|
Matrix contraction of quadrupole-quadrupole tensor and quadrupole |
| 400 |
|
|
moment gives the identity matrix. Therefore each component can be |
| 401 |
|
|
treated as scalar. If the field gradient is homogeneous over the |
| 402 |
|
|
entire volume, ${\partial_ \alpha E_\beta}({\kappa}) = 0 $ except at |
| 403 |
|
|
$ {\kappa} = 0$, hence it is sufficient to know |
| 404 |
|
|
${T}_{\alpha\beta\alpha\beta}({\kappa})$ at $ {\kappa} = |
| 405 |
|
|
0$. Therefore equation (\ref{fourierQuad}) can be written as, |
| 406 |
|
|
\begin{equation} |
| 407 |
|
|
\begin{split} |
| 408 |
|
|
\frac{1}{3}{{\Theta}}_{\alpha\beta}({0}) &= \epsilon_o {\chi}^*_Q\; \left(1-\frac{1}{4\pi} {\chi}^*_Q\;{T}_{\alpha\beta\alpha\beta}({0})\right)^{-1} \partial_\alpha E^o_\beta({0}) |
| 409 |
|
|
\end{split} |
| 410 |
|
|
\label{fourierQuad2} |
| 411 |
|
|
\end{equation} |
| 412 |
|
|
where $ {T}_{\alpha\beta\alpha\beta}({0})$ can be evaluated as, |
| 413 |
|
|
\begin{equation} |
| 414 |
|
|
{T}_{\alpha\beta\alpha\beta}({0}) = \int {T}_{\alpha\beta\alpha\beta}\;(\textbf{r})d^3r |
| 415 |
|
|
\label{realTensorQaud} |
| 416 |
|
|
\end{equation} |
| 417 |
|
|
|
| 418 |
|
|
In terms of traced quadrupole moment equation (\ref{fourierQuad2}) can be written as, |
| 419 |
|
|
\begin{equation} |
| 420 |
|
|
{{Q}}_{\alpha\beta} = \frac{1}{3}\delta_{\alpha\beta}\;Tr({Q}) + \epsilon_o\; {\chi}^*_Q\left(1-\frac{1}{4\pi} {\chi}^*_Q\;{T}_{\alpha\beta\alpha\beta}({0})\right)^{-1}\; \partial_\alpha E^o_\beta |
| 421 |
|
|
\label{tracedConstQuad} |
| 422 |
|
|
\end{equation} |
| 423 |
|
|
Comparing (\ref{tracedConstQuad}) and (\ref{flucQuad}) we get, |
| 424 |
|
|
\begin{equation} |
| 425 |
|
|
\begin{split} |
| 426 |
|
|
&\frac{\braket{{Q^2}} - \braket{Q}^2}{15 \epsilon_o Vk_BT}\; =\; {\chi}^*_Q\;\left(1-\frac{1}{4\pi} {\chi}^*_Q\;{T}_{\alpha\beta\alpha\beta}({0})\right)^{-1}, \\ |
| 427 |
|
|
&{\chi}^*_Q \;=\; \frac{\braket{{Q^2}} - \braket{Q}^2}{15 \epsilon_o Vk_BT}\left(1 + \frac{1}{4\pi} \frac{\braket{{Q^2}} - \braket{Q}^2}{15 \epsilon_o Vk_BT}\;{T}_{\alpha\beta\alpha\beta}({0})\right)^{-1} |
| 428 |
|
|
\end{split} |
| 429 |
|
|
\end{equation} |
| 430 |
|
|
Finally the quadrupolar susceptibility cab be written in terms of quadrupolar correction factor ($A_{quad}$) as below, |
| 431 |
|
|
\begin{equation} |
| 432 |
|
|
{\chi}^*_Q \;=\; \frac{\braket{{Q^2}} - \braket{Q}^2}{15 \epsilon_o Vk_BT}\left(1 + \frac{\braket{{Q^2}} - \braket{Q}^2}{15 \epsilon_o Vk_BT}\; A_{quad}\right)^{-1} |
| 433 |
|
|
\end{equation} |
| 434 |
|
|
where $A_{quad} = \frac{1}{4\pi}\int {T}_{\alpha\beta\alpha\beta}\;(\textbf{r})d^3r $ has dimension of the $length^{-2}$ is different for different cutoff methods which is listed in Table \ref{tab:B}. The dielectric constant associated with the quadrupolar susceptibility is defined as,\cite{Ernst92} |
| 435 |
|
|
|
| 436 |
|
|
\begin{equation} |
| 437 |
|
|
\epsilon = 1 + \chi^*_Q\; G, |
| 438 |
|
|
\end{equation} |
| 439 |
|
|
where $G = \frac{\displaystyle\int_V |\partial_\alpha E^o_\beta|^2 d^3r}{\displaystyle\int_V{|E^o|}^2 d^3r}$ is a geometrical factor depends on the nature of the external field perturbation. This is true when the quadrupolar fluid is homogeneous over the sample. Since quadrupolar molecule couple with the gradient of the field, the distribution of the quadrupoles is inhomogeneous for varying field gradient. Hence the distribution function should also be taken into account to calculate actual geometrical factor in the presence of non-uniform gradient field. Therefore, |
| 440 |
|
|
\begin{equation} |
| 441 |
|
|
G = \frac{\displaystyle\int_V\; g(r, \theta, \phi)\; |\partial_\alpha E^o_\beta|^2 d^3r}{\displaystyle\int_V{|E^o|}^2 d^3r} |
| 442 |
|
|
\end{equation} |
| 443 |
|
|
where $g(r,\theta, \phi)$ is a distribution function of the quadrupoles in with respect to origin at the center of line joining two probe charges. |
| 444 |
|
|
\begin{table} |
| 445 |
|
|
\caption{Expressions for the quadrupolar correction factor ($A$) for the real-space electrostatic methods in terms of the damping parameter |
| 446 |
|
|
($\alpha$) and the cutoff radius ($r_c$). The dimension of the correction factor is $ length^{-2}$ in case of quadrupolar fluid.} |
| 447 |
|
|
\label{tab:B} |
| 448 |
|
|
\centering |
| 449 |
|
|
\resizebox{\columnwidth}{!}{% |
| 450 |
|
|
|
| 451 |
|
|
\begin{tabular}{l|c|c|c|c|} |
| 452 |
|
|
|
| 453 |
|
|
Method & $A_\mathrm{charges}$ & $A_\mathrm{dipoles}$ &$A_\mathrm{quadrupoles}$ \\\hline |
| 454 |
|
|
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}$ \\ |
| 455 |
|
|
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}$ \\ |
| 456 |
|
|
Gradient-shifted (GSF) & $- \frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & 0 & $-\frac{4{\alpha}^7{r_c}^5 }{9\sqrt{\pi}}e^{-\alpha^2 r_c^2}(-1+2\alpha ^2 r_c^2)$\\ |
| 457 |
|
|
Taylor-shifted (TSF) & $ -\frac{8 \alpha^5 {r_c}^3}{3\sqrt{\pi}} e^{-\alpha^2 r_c^2}$ & $\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) $ & $\frac{10\;\mathrm{erfc}(\alpha r_c )}{{r_c}^2} + \frac{4{\alpha}}{9\sqrt{\pi}{r_c}}e^{-\alpha^2 r_c^2}\left(45 + 30\alpha ^2 {r_c}^2 + 12\alpha^4 {r_c}^4 + 3\alpha^6 {r_c}^6 + 2 \alpha^8 {r_c}^8\right)$ \\\hline |
| 458 |
|
|
\end{tabular}% |
| 459 |
|
|
} |
| 460 |
|
|
\end{table} |
| 461 |
|
|
\section{Methodology} |
| 462 |
|
|
|
| 463 |
|
|
\section{Result} |
| 464 |
|
|
|
| 465 |
|
|
\section{Conclusion} |
| 466 |
|
|
|
| 467 |
|
|
\newpage |
| 468 |
|
|
|
| 469 |
|
|
\appendix |
| 470 |
|
|
\section{Point-multipolar interactions with a spatially-varying electric field} |
| 471 |
|
|
|
| 472 |
|
|
We can treat objects $a$, $b$, and $c$ containing embedded collections |
| 473 |
|
|
of charges. When we define the primitive moments, we sum over that |
| 474 |
|
|
collections of charges using a local coordinate system within each |
| 475 |
|
|
object. The point charge, dipole, and quadrupole for object $a$ are |
| 476 |
|
|
given by $C_a$, $\mathbf{D}_a$, and $\mathsf{Q}_a$, respectively. |
| 477 |
|
|
These are the primitive multipoles which can be expressed as a |
| 478 |
|
|
distribution of charges, |
| 479 |
|
|
\begin{align} |
| 480 |
|
|
C_a =&\sum_{k \, \text{in }a} q_k , \label{eq:charge} \\ |
| 481 |
|
|
D_{a\alpha} =&\sum_{k \, \text{in }a} q_k r_{k\alpha}, \label{eq:dipole}\\ |
| 482 |
|
|
Q_{a\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in } a} q_k |
| 483 |
|
|
r_{k\alpha} r_{k\beta} . \label{eq:quadrupole} |
| 484 |
|
|
\end{align} |
| 485 |
|
|
Note that the definition of the primitive quadrupole here differs from |
| 486 |
|
|
the standard traceless form, and contains an additional Taylor-series |
| 487 |
|
|
based factor of $1/2$. In Paper 1, we derived the forces and torques |
| 488 |
|
|
each object exerts on the others. |
| 489 |
|
|
|
| 490 |
|
|
Here we must also consider an external electric field that varies in |
| 491 |
|
|
space: $\mathbf E(\mathbf r)$. Each of the local charges $q_k$ in |
| 492 |
|
|
object $a$ will then experience a slightly different field. This |
| 493 |
|
|
electric field can be expanded in a Taylor series around the local |
| 494 |
|
|
origin of each object. A different Taylor series expansion is carried |
| 495 |
|
|
out for each object. |
| 496 |
|
|
|
| 497 |
|
|
For a particular charge $q_k$, the electric field at that site's |
| 498 |
|
|
position is given by: |
| 499 |
|
|
\begin{equation} |
| 500 |
|
|
E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
| 501 |
|
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
| 502 |
|
|
r_{k \varepsilon} + ... |
| 503 |
|
|
\end{equation} |
| 504 |
|
|
Note that the electric field is always evaluated at the origin of the |
| 505 |
|
|
objects, and treating each object using point multipoles simplifies |
| 506 |
|
|
this greatly. |
| 507 |
|
|
|
| 508 |
|
|
To find the force exerted on object $a$ by the electric field, one |
| 509 |
|
|
takes the electric field expression, and multiplies it by $q_k$, and |
| 510 |
|
|
then sum over all charges in $a$: |
| 511 |
|
|
|
| 512 |
|
|
\begin{align} |
| 513 |
|
|
F_\gamma &= \sum_{k \textrm{~in~} a} q_k \lbrace E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
| 514 |
|
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
| 515 |
|
|
r_{k \varepsilon} + ... \rbrace \\ |
| 516 |
|
|
&= C_a E_\gamma + D_{a \delta} \nabla_\delta E_\gamma |
| 517 |
|
|
+ Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma + |
| 518 |
|
|
... |
| 519 |
|
|
\end{align} |
| 520 |
|
|
|
| 521 |
|
|
Similarly, the torque exerted by the field on $a$ can be expressed as |
| 522 |
|
|
\begin{align} |
| 523 |
|
|
\tau_\alpha &= \sum_{k \textrm{~in~} a} (\mathbf r_k \times q_k \mathbf E)_\alpha \\ |
| 524 |
|
|
& = \sum_{k \textrm{~in~} a} \epsilon_{\alpha \beta \gamma} q_k |
| 525 |
|
|
r_{k\beta} E_\gamma(\mathbf r_k) \\ |
| 526 |
|
|
& = \epsilon_{\alpha \beta \gamma} D_\beta E_\gamma |
| 527 |
|
|
+ 2 \epsilon_{\alpha \beta \gamma} Q_{\beta \delta} \nabla_\delta |
| 528 |
|
|
E_\gamma + ... |
| 529 |
|
|
\end{align} |
| 530 |
|
|
|
| 531 |
|
|
The last term is essentially identical with form derived by Torres del |
| 532 |
|
|
Castillo and M\'{e}ndez Garrido,\cite{Torres-del-Castillo:2006uo} although their derivation |
| 533 |
|
|
utilized a traceless form of the quadrupole that is different than the |
| 534 |
|
|
primitive definition in use here. We note that the Levi-Civita symbol |
| 535 |
|
|
can be eliminated by utilizing the matrix cross product in an |
| 536 |
|
|
identical form as in Ref. \onlinecite{Smith98}: |
| 537 |
|
|
\begin{equation} |
| 538 |
|
|
\left[\mathsf{A} \times \mathsf{B}\right]_\alpha = \sum_\beta |
| 539 |
|
|
\left[\mathsf{A}_{\alpha+1,\beta} \mathsf{B}_{\alpha+2,\beta} |
| 540 |
|
|
-\mathsf{A}_{\alpha+2,\beta} \mathsf{B}_{\alpha+1,\beta} |
| 541 |
|
|
\right] |
| 542 |
|
|
\label{eq:matrixCross} |
| 543 |
|
|
\end{equation} |
| 544 |
|
|
where $\alpha+1$ and $\alpha+2$ are regarded as cyclic permuations of |
| 545 |
|
|
the matrix indices. In table \ref{tab:UFT} we give compact |
| 546 |
|
|
expressions for how the multipole sites interact with an external |
| 547 |
|
|
field that has exhibits spatial variations. |
| 548 |
|
|
|
| 549 |
|
|
\begin{table} |
| 550 |
|
|
\caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque |
| 551 |
|
|
$(\mathbf{\tau})$ expressions for a multipolar site embedded in an |
| 552 |
|
|
electric field with spatial variations, $\mathbf{E}(\mathbf{r})$. |
| 553 |
|
|
\label{tab:UFT}} |
| 554 |
|
|
\begin{tabular}{r|ccc} |
| 555 |
|
|
& Charge & Dipole & Quadrupole \\ \hline |
| 556 |
|
|
$U$ & $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\ |
| 557 |
|
|
$\mathbf{F}$ & $C \mathbf{E}(\mathbf{r})$ & $+\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ & $+\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\ |
| 558 |
|
|
$\mathbf{\tau}$ & & $\mathbf{D} \times \mathbf{E}(\mathbf{r})$ & $+2 \mathsf{Q} \times \nabla \mathbf{E}(\mathbf{r})$ |
| 559 |
|
|
\end{tabular} |
| 560 |
|
|
\end{table} |
| 561 |
|
|
\section{Gradient of the field due to quadrupolar polarization} |
| 562 |
|
|
\label{singularQuad} |
| 563 |
|
|
In this section, we will discuss the gradient of the field produced by |
| 564 |
|
|
quadrupolar polarization. For this purpose, we consider a distribution |
| 565 |
|
|
of charge ${\rho}(r)$ which gives rise to an electric field |
| 566 |
|
|
$\vec{E}(r)$ and gradient of the field $\vec{\nabla} \vec{E}(r)$ |
| 567 |
|
|
throughout space. The total gradient of the electric field over volume |
| 568 |
|
|
due to the all charges within the sphere of radius $R$ is given by |
| 569 |
|
|
(cf. Jackson equation 4.14): |
| 570 |
|
|
\begin{equation} |
| 571 |
|
|
\int_{r<R} \vec{\nabla}\vec{E}\;d^3r = -\int_{r=R} R^2 \vec{E}\;\hat{n}\; d\Omega |
| 572 |
|
|
\label{eq:8} |
| 573 |
|
|
\end{equation} |
| 574 |
|
|
where $d\Omega$ is the solid angle and $\hat{n}$ is the normal vector |
| 575 |
|
|
of the surface of the sphere which is equal to |
| 576 |
|
|
$sin[\theta]cos[\phi]\hat{x} + sin[\theta]sin[\phi]\hat{y} + |
| 577 |
|
|
cos[\theta]\hat{z}$ |
| 578 |
|
|
in spherical coordinates. For the charge density ${\rho}(r')$, the |
| 579 |
|
|
total gradient of the electric field can be written as (cf. Jackson |
| 580 |
|
|
equation 4.16), |
| 581 |
|
|
\begin{equation} |
| 582 |
|
|
\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 |
| 583 |
|
|
\label{eq:9} |
| 584 |
|
|
\end{equation} |
| 585 |
|
|
The radial function in the equation (\ref{eq:9}) can be expressed in |
| 586 |
|
|
terms of spherical harmonics as (cf. Jackson equation 3.70), |
| 587 |
|
|
\begin{equation} |
| 588 |
|
|
\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) |
| 589 |
|
|
\label{eq:10} |
| 590 |
|
|
\end{equation} |
| 591 |
|
|
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, |
| 592 |
|
|
\begin{equation} |
| 593 |
|
|
\begin{split} |
| 594 |
|
|
\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 \\ |
| 595 |
|
|
&= -\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 |
| 596 |
|
|
' |
| 597 |
|
|
\end{split} |
| 598 |
|
|
\label{eq:11} |
| 599 |
|
|
\end{equation} |
| 600 |
|
|
The gradient of the product of radial function and spherical harmonics |
| 601 |
|
|
is given by (cf. Arfken, p.811 eq. 16.94): |
| 602 |
|
|
\begin{equation} |
| 603 |
|
|
\begin{split} |
| 604 |
|
|
\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 |
| 605 |
|
|
{\partial}{\partial r}+\frac{l}{r} \right]f(r)\; Y_{l, l-1, m}(\theta, \phi). |
| 606 |
|
|
\end{split} |
| 607 |
|
|
\label{eq:12} |
| 608 |
|
|
\end{equation} |
| 609 |
|
|
Using equation (\ref{eq:12}) we get, |
| 610 |
|
|
\begin{equation} |
| 611 |
|
|
\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}}, |
| 612 |
|
|
\label{eq:13} |
| 613 |
|
|
\end{equation} |
| 614 |
|
|
where $ Y_{l,l+1,m}(\theta, \phi)$ is the vector spherical harmonics |
| 615 |
|
|
which can be expressed in terms of spherical harmonics as shown in |
| 616 |
|
|
below (cf. Arfkan p.811), |
| 617 |
|
|
\begin{equation} |
| 618 |
|
|
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}, |
| 619 |
|
|
\label{eq:14} |
| 620 |
|
|
\end{equation} |
| 621 |
|
|
where $C(l+1,1,l|m_1,m_2,m)$ is a Clebsch-Gordan coefficient and |
| 622 |
|
|
$\hat{e}_{m_2}$ is a spherical tensor of rank 1 which can be expressed |
| 623 |
|
|
in terms of Cartesian coordinates, |
| 624 |
|
|
\begin{equation} |
| 625 |
|
|
{\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}} |
| 626 |
|
|
\label{eq:15} |
| 627 |
|
|
\end{equation} |
| 628 |
|
|
The normal vector $\hat{n} $ can be expressed in terms of spherical tensor of rank 1 as shown in below, |
| 629 |
|
|
\begin{equation} |
| 630 |
|
|
\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) |
| 631 |
|
|
\label{eq:16} |
| 632 |
|
|
\end{equation} |
| 633 |
|
|
The surface integral of the product of $\hat{n}$ and |
| 634 |
|
|
${Y_{l+1}}^{m_1}(\theta, \phi)$ gives, |
| 635 |
|
|
\begin{equation} |
| 636 |
|
|
\begin{split} |
| 637 |
|
|
\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 \\ |
| 638 |
|
|
&= \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 \\ |
| 639 |
|
|
&= \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), |
| 640 |
|
|
\end{split} |
| 641 |
|
|
\label{eq:17} |
| 642 |
|
|
\end{equation} |
| 643 |
|
|
where ${Y_{l}}^{-m} = (-1)^m\;{{Y_{l}}^{m}}^* $ and |
| 644 |
|
|
$ \int {{Y_{l}}^{m}}^*\;{Y_{l'}}^{m'}\;d\Omega = |
| 645 |
|
|
\delta_{ll'}\delta_{mm'} $. |
| 646 |
|
|
Non-vanishing values of equation \ref{eq:17} require $l = 0$, |
| 647 |
|
|
therefore the value of $ m = 0 $. Since the values of $ m_1$ are -1, |
| 648 |
|
|
1, and 0 then $m_2$ takes the values 1, -1, and 0, respectively |
| 649 |
|
|
provided that $m = m_1 + m_2$. Equation \ref{eq:11} can therefore be |
| 650 |
|
|
modified, |
| 651 |
|
|
\begin{equation} |
| 652 |
|
|
\begin{split} |
| 653 |
|
|
\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( |
| 654 |
|
|
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d^3r'. |
| 655 |
|
|
\end{split} |
| 656 |
|
|
\label{eq:18} |
| 657 |
|
|
\end{equation} |
| 658 |
|
|
After substituting ${Y^*}_{00} = \frac{1}{\sqrt{4\pi}} $ and using the |
| 659 |
|
|
values of the Clebsch-Gorden coefficients: $ C(1, 1, 0|-1,1,0) = |
| 660 |
|
|
\frac{1}{\sqrt{3}}, \; C(1, 1, 0|-1,1,0)= \frac{1}{\sqrt{3}}$ and $ |
| 661 |
|
|
C(1, 1, 0|0,0,0) = -\frac{1}{\sqrt{3}}$ in equation \ref{eq:18} we |
| 662 |
|
|
obtain, |
| 663 |
|
|
\begin{equation} |
| 664 |
|
|
\begin{split} |
| 665 |
|
|
\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)\\ |
| 666 |
|
|
&= - \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). |
| 667 |
|
|
\end{split} |
| 668 |
|
|
\label{eq:19} |
| 669 |
|
|
\end{equation} |
| 670 |
|
|
Equation (\ref{eq:19}) gives the total gradient of the field over a |
| 671 |
|
|
sphere due to the distribution of the charges. For quadrupolar fluids |
| 672 |
|
|
the total charge within a sphere is zero, therefore |
| 673 |
|
|
$ \int_{r<R} \vec{\nabla}\vec{E}\;d^3r = 0 $. Hence the quadrupolar |
| 674 |
|
|
polarization produces zero net gradient of the field inside the |
| 675 |
|
|
sphere. |
| 676 |
|
|
|
| 677 |
|
|
|
| 678 |
|
|
|
| 679 |
|
|
\newpage |
| 680 |
|
|
|
| 681 |
|
|
\bibliography{multipole} |
| 682 |
|
|
|
| 683 |
|
|
\end{document} |
