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%\usepackage[mathlines]{lineno}% Enable numbering of text and display math |
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\begin{document} |
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\title[Real space electrostatics for multipoles. III. Dielectric Properties] |
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{Supplemental Material for: Real space electrostatics for multipoles. III. Dielectric Properties} |
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\title{Supplemental Material for: Real space electrostatics for |
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multipoles. III. Dielectric Properties} |
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|
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University |
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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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|
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\begin{abstract} |
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This document includes useful relationships for computing the |
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interactions between fields and field gradients and point multipolar |
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representations of molecular electrostatics. We also provide |
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explanatory derivations of a number of relationships used in the |
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main text. This includes the Boltzmann averages of quadrupole |
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orientations, and the interaction of a quadrupole density with the |
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self-generated field gradient. This last relationship is assumed to |
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be zero in the main text but is explicitly shown to be zero here. |
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\end{abstract} |
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|
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\maketitle |
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|
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\newpage |
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\section{Generating Uniform Field Gradients} |
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One important task in carrying out the simulations mentioned in the |
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main text was to generate uniform electric field gradients. To do |
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this, we relied heavily on both the notation and results from Torres |
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del Castillo and Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} |
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In this work, tensors were expressed in Cartesian components, using at |
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times a dyadic notation. This proves quite useful for computer |
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simulations that make use of toroidal boundary conditions. |
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|
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An alternative formalism uses the theory of angular momentum and |
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spherical harmonics and is common in standard physics texts such as |
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Jackson,\cite{Jackson98} Morse and Feshbach,\cite{Morse:1946zr} and |
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Stone.\cite{Stone:1997ly} Because this approach has its own |
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advantages, relationships are provided below comparing that |
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terminology to the Cartesian tensor notation. |
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|
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\section{Boltzmann averages for orientational polarization} |
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The dielectric properties of the system is mainly arise from two |
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different ways: i) the applied field distort the charge distributions |
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so it produces an induced multipolar moment in each molecule; and ii) |
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the applied field tends to line up originally randomly oriented |
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molecular moment towards the direction of the applied field. In this |
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study, we basically focus on the orientational contribution in the |
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dielectric properties. If we consider a system of molecules in the |
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presence of external field perturbation, the perturbation experienced |
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by any molecule will not be only due to external field or field |
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gradient but also due to the field or field gradient produced by the |
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all other molecules in the system. In the following subsections |
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\ref{subsec:boltzAverage-Dipole} and \ref{subsec:boltzAverage-Quad}, |
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we will discuss about the molecular polarization only due to external |
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field perturbation. The contribution of the field or field gradient |
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due to all other molecules will be taken into account while |
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calculating correction factor in the paper. |
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The gradient of the electric field, |
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\begin{equation*} |
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\mathsf{G}(\mathbf{r}) = -\nabla \nabla \Phi(\mathbf{r}), |
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\end{equation*} |
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where $\Phi(\mathbf{r})$ is the electrostatic potential. In a |
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charge-free region of space, $\nabla \cdot \mathbf{E}=0$, and |
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$\mathsf{G}$ is a symmetric traceless tensor. From symmetry |
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arguments, we know that this tensor can be written in terms of just |
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five independent components. |
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|
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\subsection{Dipoles} |
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\label{subsec:boltzAverage-Dipole} |
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Consider a system of molecules, each with permanent dipole moment |
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$p_o$. In the absense of external field, thermal agitation orients the |
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dipoles randomly, reducing the system moment to zero. External fields |
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will tend to line up the dipoles in the direction of applied field. |
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Here we have considered net field from all other molecules is |
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considered to be zero. Therefore the total Hamiltonian of each |
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molecule is,\cite{Jackson98} |
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Following Torres del Castillo and Mend\'{e}z Garido's notation, the |
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gradient of the electric field may also be written in terms of two |
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vectors $\mathbf{a}$ and $\mathbf{b}$, |
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\begin{equation*} |
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G_{ij}=\frac{1}{2} (a_i b_j + a_j b_i) - \frac{1}{3}(\mathbf a \cdot \mathbf b) \delta_{ij} . |
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\end{equation*} |
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If the vectors $\mathbf{a}$ and $\mathbf{b}$ are unit vectors, the |
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electrostatic potential that generates a uniform gradient may be |
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written: |
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\begin{align} |
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\Phi(x, y, z) =\; -\frac{g_o}{2} & \left(\left(a_1b_1 - |
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\frac{cos\psi}{3}\right)\;x^2+\left(a_2b_2 |
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- \frac{cos\psi}{3}\right)\;y^2 + |
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\left(a_3b_3 - |
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\frac{cos\psi}{3}\right)\;z^2 \right. \nonumber \\ |
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& + (a_1b_2 + a_2b_1)\; xy + (a_1b_3 + a_3b_1)\; xz + (a_2b_3 + a_3b_2)\; yz \bigg) . |
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\label{eq:appliedPotential} |
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\end{align} |
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Note $\mathbf{a}\cdot\mathbf{a} = \mathbf{b} \cdot \mathbf{b} = 1$, |
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$\mathbf{a} \cdot \mathbf{b}=\cos \psi$, and $g_0$ is the overall |
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strength of the potential. |
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|
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Taking the gradient of Eq. (\ref{eq:appliedPotential}), we find the |
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field due to this potential, |
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\begin{equation} |
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H = H_o - \bf{p_o}\cdot \mathbf{E}, |
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\end{equation} |
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where $H_o$ is a function of the internal coordinates of the molecule. |
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The Boltzmann average of the dipole moment is given by, |
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\mathbf{E} = -\nabla \Phi |
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=\frac{g_o}{2} \left(\begin{array}{ccc} |
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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 \\ |
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(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 \\ |
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(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 |
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\end{array} \right), |
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\label{eq:CE} |
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\end{equation} |
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while the gradient of the electric field in this form, |
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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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\mathsf{G} = \nabla\mathbf{E} |
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= \frac{g_o}{2}\left(\begin{array}{ccc} |
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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) \\ |
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(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) \\ |
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(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}) |
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\end{array} \right), |
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\label{eq:GC} |
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\end{equation} |
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is uniform over the entire space. Therefore, to describe a uniform |
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gradient in this notation, two unit vectors ($\mathbf{a}$ and |
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$\mathbf{b}$) as well as a potential strength, $g_0$, must be |
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specified. As expected, this requires five independent parameters. |
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|
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The common alternative to the Cartesian notation expresses the |
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electrostatic potential using the notation of Morse and |
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Feshbach,\cite{Morse:1946zr} |
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\begin{equation} \label{eq:quad_phi} |
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\Phi(x,y,z) = -\left[ a_{20} \frac{2 z^2 -x^2 - y^2}{2} |
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+ 3 a_{21}^e \,xz + 3 a_{21}^o \,yz |
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+ 6a_{22}^e \,xy + 3 a_{22}^o (x^2 - y^2) \right]. |
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\end{equation} |
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where $\bf{E}$ is selected along z-axis. If we consider that the |
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applied field is small, \textit{i.e.} $\frac{p_oE\; cos\theta}{k_B T} << 1$, |
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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 |
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polarizability. The orientational polarization depends inversely on |
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the temperature and applied field must overcome the thermal agitation. |
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Here we use the standard $(l,m)$ form for the $a_{lm}$ coefficients, |
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with superscript $e$ and $o$ denoting even and odd, respectively. |
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This form makes the functional analogy to ``d'' atomic states |
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apparent. |
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|
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\subsection{Quadrupoles} |
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\label{subsec:boltzAverage-Quad} |
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Consider a system of molecules with permanent quadrupole moment |
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$q_{\alpha\beta}$. The average quadrupole moment at temperature T in |
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the presence of uniform applied field gradient is given |
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by,\cite{AduGyamfi78, AduGyamfi81} |
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Applying the gradient operator to Eq. (\ref{eq:quad_phi}) the electric |
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field due to this potential, |
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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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\mathbf{E} = -\nabla \Phi |
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= \left(\begin{array}{ccc} |
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\left( 6a_{22}^o -a_{20} \right)\; x &+\; 6a_{22}^e\; y &+\; 3a_{21}^e\; z \\ |
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6a_{22}^e\; x & -\; (a_{20} + 6a_{22}^o)\; y & +\; 3a_{21}^o\; z \\ |
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3a_{21}^e\; x & +\; 3a_{21}^o\; y & +\; 2a_{20}\; z |
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\end{array} \right), |
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\label{eq:MFE} |
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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'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
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&H = H_o - q:{\nabla}\mathbf{E} = H_o - q_{\mu\nu}\;\partial_\nu E_\mu = H_o -\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu. |
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\end{split} |
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\label{energyQuad} |
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while the gradient of the electric field in this form is: |
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\begin{equation} \label{eq:grad_e2} |
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\mathsf{G} = |
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\begin{pmatrix} |
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6 a_{22}^o - a_{20} & 6a_{22}^e & 3a_{21}^e\\ |
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6a_{22}^e & -(a_{20}+6a_{22}^o) & 3a_{21}^o \\ |
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3a_{21}^e & 3a_{21}^o & 2a_{20} \\ |
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\end{pmatrix} \\ |
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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'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)q_{\alpha\beta}}{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\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 inverse of the rotation matrix that transforms |
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the body fixed co-ordinates to the space co-ordinates, |
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\[\eta_{\alpha\alpha'} |
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= \left(\begin{array}{ccc} |
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cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
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cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
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sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
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\end{array} \right).\] |
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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\;{\nabla}.\mathbf{E}\; Tr(q^*) $ respectively. The |
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second term vanishes for charge free space, ${\nabla}.\mathbf{E} \; = \; 0$. 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 ${\nabla}.\mathbf{E} \; = \; 0$, |
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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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which is also uniform over the entire space. This form for the |
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gradient can be factored as |
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\begin{gather} |
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\begin{aligned} |
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\mathsf{G} = a_{20} |
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\begin{pmatrix} |
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-1 & 0 & 0\\ |
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0 & -1 & 0\\ |
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0 & 0 & 2\\ |
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> |
\end{pmatrix} |
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> |
+3a_{21}^e |
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> |
\begin{pmatrix} |
| 203 |
> |
0 & 0 & 1\\ |
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> |
0 & 0 & 0\\ |
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> |
1 & 0 & 0\\ |
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> |
\end{pmatrix} |
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+3a_{21}^o |
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> |
\begin{pmatrix} |
| 209 |
> |
0 & 0 & 0\\ |
| 210 |
> |
0 & 0 & 1\\ |
| 211 |
> |
0 & 1 & 0\\ |
| 212 |
> |
\end{pmatrix} |
| 213 |
> |
+6a_{22}^e |
| 214 |
> |
\begin{pmatrix} |
| 215 |
> |
0 & 1 & 0\\ |
| 216 |
> |
1 & 0 & 0\\ |
| 217 |
> |
0 & 0 & 0\\ |
| 218 |
> |
\end{pmatrix} |
| 219 |
> |
+6a_{22}^o |
| 220 |
> |
\begin{pmatrix} |
| 221 |
> |
1 & 0 & 0\\ |
| 222 |
> |
0 & -1 & 0\\ |
| 223 |
> |
0 & 0 & 0\\ |
| 224 |
> |
\end{pmatrix} |
| 225 |
> |
\end{aligned} |
| 226 |
> |
\label{eq:intro_tensors} |
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> |
\end{gather} |
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> |
The five matrices in the expression above represent five different |
| 229 |
> |
symmetric traceless tensors of rank 2. |
| 230 |
|
|
| 231 |
< |
\begin{equation} |
| 232 |
< |
\braket{q_{\alpha\beta}}\; = \; \frac{1}{3} Tr(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
| 233 |
< |
\end{equation} |
| 234 |
< |
where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupole polarizablity and ${\bar{q_o}}^2= |
| 235 |
< |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. |
| 236 |
< |
|
| 237 |
< |
\section{External application of a uniform field gradient} |
| 238 |
< |
\label{Ap:fieldOrGradient} |
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< |
|
| 240 |
< |
To satisfy the condition $ \nabla \cdot \mathbf{E} = 0 $, within the box of molecules we have taken electrostatic potential in the following form |
| 241 |
< |
\begin{equation} |
| 242 |
< |
\begin{split} |
| 243 |
< |
\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. \\ |
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< |
& \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), |
| 245 |
< |
\end{split} |
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< |
\label{eq:appliedPotential} |
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< |
\end{equation} |
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< |
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, |
| 249 |
< |
\[\mathbf{E} |
| 250 |
< |
= \frac{g_o}{2} \left(\begin{array}{ccc} |
| 251 |
< |
2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+ (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
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< |
(a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y + (a_2\; b_3 \;+ a_3\; b_2)\;z \\ |
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(a_3\; b_1 \;+ a_3\; b_2)\;x + (a_3\; b_2 \;+ a_2\; b_3)y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
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< |
\end{array} \right).\] |
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< |
The gradient of the applied field derived from the potential can be written in the following form, |
| 256 |
< |
\[\nabla\mathbf{E} |
| 257 |
< |
= \frac{g_o}{2}\left(\begin{array}{ccc} |
| 258 |
< |
2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
| 259 |
< |
(a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_2) \\ |
| 260 |
< |
(a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
| 261 |
< |
\end{array} \right).\] |
| 231 |
> |
It is useful to find the Cartesian vectors $\mathbf a$ and $\mathbf b$ |
| 232 |
> |
that generate the five types of tensors shown in |
| 233 |
> |
Eq. (\ref{eq:intro_tensors}). If the two vectors are co-linear, e.g., |
| 234 |
> |
$\psi=0$, $\mathbf{a}=(0,0,1)$ and $\mathbf{b}=(0,0,1)$, then |
| 235 |
> |
\begin{equation*} |
| 236 |
> |
\mathsf{G} = \frac{g_0}{3} |
| 237 |
> |
\begin{pmatrix} |
| 238 |
> |
-1 & 0 & 0 \\ |
| 239 |
> |
0 & -1 & 0 \\ |
| 240 |
> |
0 & 0 & 2 \\ |
| 241 |
> |
\end{pmatrix} , |
| 242 |
> |
\end{equation*} |
| 243 |
> |
which is the $a_{20}$ symmetry. |
| 244 |
> |
To generate the $a_{22}^o$ symmetry, we take: |
| 245 |
> |
$\mathbf{a}= (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}},0)$ and |
| 246 |
> |
$\mathbf{b}=(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}},0)$ |
| 247 |
> |
and find: |
| 248 |
> |
\begin{equation*} |
| 249 |
> |
\mathsf{G}=\frac{g_0}{2} |
| 250 |
> |
\begin{pmatrix} |
| 251 |
> |
1 & 0 & 0 \\ |
| 252 |
> |
0 & -1 & 0 \\ |
| 253 |
> |
0 & 0 & 0 \\ |
| 254 |
> |
\end{pmatrix} . |
| 255 |
> |
\end{equation*} |
| 256 |
> |
To generate the $a_{22}^e$ symmetry, we take: |
| 257 |
> |
$\mathbf{a}= (1, 0, 0)$ and $\mathbf{b} = (0,1,0)$ and find: |
| 258 |
> |
\begin{equation*} |
| 259 |
> |
\mathsf{G}=\frac{g_0}{2} |
| 260 |
> |
\begin{pmatrix} |
| 261 |
> |
0 & 1 & 0 \\ |
| 262 |
> |
1 & 0 & 0 \\ |
| 263 |
> |
0 & 0 & 0 \\ |
| 264 |
> |
\end{pmatrix} . |
| 265 |
> |
\end{equation*} |
| 266 |
> |
The pattern is straightforward to continue for the other symmetries. |
| 267 |
|
|
| 268 |
+ |
We find the notation of Ref. \onlinecite{Torres-del-Castillo:2006uo} |
| 269 |
+ |
helpful when creating specific types of constant gradient electric |
| 270 |
+ |
fields in simulations. For this reason, |
| 271 |
+ |
Eqs. (\ref{eq:appliedPotential}), (\ref{eq:GC}), and (\ref{eq:CE}) are |
| 272 |
+ |
implemented in our code. In the simulations using constant applied |
| 273 |
+ |
gradients that are mentioned in the main text, we utilized a field |
| 274 |
+ |
with the $a_{22}^e$ symmetry using vectors, $\mathbf{a}= (1, 0, 0)$ |
| 275 |
+ |
and $\mathbf{b} = (0,1,0)$. |
| 276 |
|
|
| 277 |
|
\section{Point-multipolar interactions with a spatially-varying electric field} |
| 278 |
|
|
| 279 |
< |
We want to derive formulas for the force and torque exerted by an external electric field $\mathbf{E}(\mathbf{r})$ on object $a$. Object a has an embedded collection of charges and in simulations will normally represent a molecule or ion. We describe the charge distributions using primitive monopoles defined in paper I by |
| 280 |
< |
|
| 279 |
> |
This section develops formulas for the force and torque exerted by an |
| 280 |
> |
external electric field, $\mathbf{E}(\mathbf{r})$, on object |
| 281 |
> |
$a$. Object $a$ has an embedded collection of charges and in |
| 282 |
> |
simulations will represent a molecule, ion, or a coarse-grained |
| 283 |
> |
substructure. We describe the charge distributions using primitive |
| 284 |
> |
multipoles defined in Ref. \onlinecite{PaperI} by |
| 285 |
|
\begin{align} |
| 286 |
|
C_a =&\sum_{k \, \text{in }a} q_k , \label{eq:charge} \\ |
| 287 |
|
D_{a\alpha} =&\sum_{k \, \text{in }a} q_k r_{k\alpha}, \label{eq:dipole}\\ |
| 289 |
|
r_{k\alpha} r_{k\beta}, |
| 290 |
|
\label{eq:quadrupole} |
| 291 |
|
\end{align} |
| 292 |
< |
where $\mathbf{r}_k$ is the local coordinate system for the object (for convenience, the real origin is at the "center" of object $a$). Components of vectors and tensors are given using Green indices, using the Einstein repeated summation notation.Note that the definition of the primitive quadrupole here differs from the standard traceless form, and contains an additional Taylor-series based factor of $1/2$. In Paper I \cite{PaperI}, we derived the forces and torques each object exerts on the other objects in the system. |
| 292 |
> |
where $\mathbf{r}_k$ is the local coordinate system for the object |
| 293 |
> |
(usually the center of mass of object $a$). Components of vectors and |
| 294 |
> |
tensors are given using the Einstein repeated summation notation. Note |
| 295 |
> |
that the definition of the primitive quadrupole here differs from the |
| 296 |
> |
standard traceless form, and contains an additional Taylor-series |
| 297 |
> |
based factor of $1/2$. In Ref. \onlinecite{PaperI}, we derived the |
| 298 |
> |
forces and torques each object exerts on the other objects in the |
| 299 |
> |
system. |
| 300 |
|
|
| 301 |
|
Here we must also consider an external electric field that varies in |
| 302 |
|
space: $\mathbf E(\mathbf r)$. Each of the local charges $q_k$ in |
| 303 |
|
object $a$ will then experience a slightly different field. This |
| 304 |
|
electric field can be expanded in a Taylor series around the local |
| 305 |
< |
origin of each object. |
| 306 |
< |
For a particular charge $q_k$, the electric field at that site's |
| 215 |
< |
position is given by: |
| 305 |
> |
origin of each object. For a particular charge $q_k$, the electric |
| 306 |
> |
field at that site's position is given by: |
| 307 |
|
\begin{equation} |
| 308 |
|
\mathbf{E}(\mathbf{r}_k) = E_\gamma|_{\mathbf{r}_k = 0} + \nabla_\delta E_\gamma |_{\mathbf{r}_k = 0} r_{k \delta} |
| 309 |
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma|_{\mathbf{r}_k = 0} r_{k \delta} |
| 310 |
|
r_{k \varepsilon} + ... |
| 311 |
|
\end{equation} |
| 312 |
< |
Note that once one shrinks object $a$ to point size, the ${E}_\gamma$ terms are all evaluated at the center of the object (now a point). Thus later the ${E}_\gamma$ terms can be written using the same global origin for all objects $a, b, c, ...$ in the system. The force exerted on object $a$ by the electric field is given by, |
| 313 |
< |
|
| 312 |
> |
Note that if one shrinks object $a$ to a single point, the |
| 313 |
> |
${E}_\gamma$ terms are all evaluated at the center of the object (now |
| 314 |
> |
a point). Thus later the ${E}_\gamma$ terms can be written using the |
| 315 |
> |
same (molecular) origin for all point charges in the object. The force |
| 316 |
> |
exerted on object $a$ by the electric field is given by,\cite{Raab:2004ve} |
| 317 |
|
\begin{align} |
| 318 |
|
F^a_\gamma = \sum_{k \textrm{~in~} a} E_\gamma(\mathbf{r}_k) &= \sum_{k \textrm{~in~} a} q_k \lbrace E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
| 319 |
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
| 322 |
|
+ Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma + |
| 323 |
|
... |
| 324 |
|
\end{align} |
| 325 |
< |
Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. |
| 325 |
> |
Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. |
| 326 |
|
|
| 327 |
|
Similarly, the torque exerted by the field on $a$ can be expressed as |
| 328 |
|
\begin{align} |
| 354 |
|
\begin{equation} |
| 355 |
|
U(\mathbf{r}) = \mathrm{C} \phi(\mathbf{r}) - \mathrm{D}_\alpha \mathrm{E}_\alpha - \mathrm{Q}_{\alpha\beta}\nabla_\alpha \mathrm{E}_\beta + ... |
| 356 |
|
\end{equation} |
| 357 |
< |
The results has been summarized in Table I. |
| 357 |
> |
These results have been summarized in Table \ref{tab:UFT}. |
| 358 |
|
|
| 359 |
|
\begin{table} |
| 360 |
|
\caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque |
| 361 |
< |
$(\mathbf{\tau})$ expressions for a multipolar site $\mathrm{r}$ in an |
| 362 |
< |
electric field, $\mathbf{E}(\mathbf{r})$. |
| 363 |
< |
\label{tab:UFT}} |
| 364 |
< |
\begin{tabular}{r|ccc} |
| 361 |
> |
$(\mathbf{\tau})$ expressions for a multipolar site at $\mathbf{r}$ in an |
| 362 |
> |
electric field, $\mathbf{E}(\mathbf{r})$ using the definitions of the multipoles in Eqs. (\ref{eq:charge}), (\ref{eq:dipole}) and (\ref{eq:quadrupole}). |
| 363 |
> |
\label{tab:UFT}} |
| 364 |
> |
\begin{tabular}{r|C{3cm}C{3cm}C{3cm}} |
| 365 |
|
& Charge & Dipole & Quadrupole \\ \hline |
| 366 |
|
$U(\mathbf{r})$ & $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\ |
| 367 |
|
$\mathbf{F}(\mathbf{r})$ & $C \mathbf{E}(\mathbf{r})$ & $\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ & $\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\ |
| 368 |
|
$\mathbf{\tau}(\mathbf{r})$ & & $\mathbf{D} \times \mathbf{E}(\mathbf{r})$ & $2 \mathsf{Q} \times \nabla \mathbf{E}(\mathbf{r})$ |
| 369 |
|
\end{tabular} |
| 370 |
|
\end{table} |
| 371 |
+ |
|
| 372 |
+ |
\section{Boltzmann averages for orientational polarization} |
| 373 |
+ |
If we consider a collection of molecules in the presence of external |
| 374 |
+ |
field, the perturbation experienced by any one molecule will include |
| 375 |
+ |
contributions to the field or field gradient produced by the all other |
| 376 |
+ |
molecules in the system. In subsections |
| 377 |
+ |
\ref{subsec:boltzAverage-Dipole} and \ref{subsec:boltzAverage-Quad}, |
| 378 |
+ |
we discuss the molecular polarization due solely to external field |
| 379 |
+ |
perturbations. This illustrates the origins of the polarizability |
| 380 |
+ |
equations (Eqs. 6, 20, and 21) in the main text. |
| 381 |
+ |
|
| 382 |
+ |
\subsection{Dipoles} |
| 383 |
+ |
\label{subsec:boltzAverage-Dipole} |
| 384 |
+ |
Consider a system of molecules, each with permanent dipole moment |
| 385 |
+ |
$p_o$. In the absense of an external field, thermal agitation orients |
| 386 |
+ |
the dipoles randomly, and the system moment, $\mathbf{P}$, is zero. |
| 387 |
+ |
External fields will line up the dipoles in the direction of applied |
| 388 |
+ |
field. Here we consider the net field from all other molecules to be |
| 389 |
+ |
zero. Therefore the total Hamiltonian acting on each molecule |
| 390 |
+ |
is,\cite{Jackson98} |
| 391 |
+ |
\begin{equation} |
| 392 |
+ |
H = H_o - \mathbf{p}_o \cdot \mathbf{E}, |
| 393 |
+ |
\end{equation} |
| 394 |
+ |
where $H_o$ is a function of the internal coordinates of the molecule. |
| 395 |
+ |
The Boltzmann average of the dipole moment in the direction of the |
| 396 |
+ |
field is given by, |
| 397 |
+ |
\begin{equation} |
| 398 |
+ |
\langle p_{mol} \rangle = \frac{\displaystyle\int p_o \cos\theta |
| 399 |
+ |
e^{~p_o E \cos\theta /k_B T}\; d\Omega}{\displaystyle\int e^{~p_o E \cos\theta/k_B |
| 400 |
+ |
T}\; d\Omega}, |
| 401 |
+ |
\end{equation} |
| 402 |
+ |
where the $z$-axis is taken in the direction of the applied field, |
| 403 |
+ |
$\bf{E}$ and |
| 404 |
+ |
$\int d\Omega = \int_0^\pi \sin\theta\; d\theta \int_0^{2\pi} d\phi |
| 405 |
+ |
\int_0^{2\pi} d\psi$ |
| 406 |
+ |
is an integration over Euler angles describing the orientation of the |
| 407 |
+ |
molecule. |
| 408 |
+ |
|
| 409 |
+ |
If the external fields are small, \textit{i.e.} |
| 410 |
+ |
$p_oE \cos\theta / k_B T << 1$, |
| 411 |
+ |
\begin{equation} |
| 412 |
+ |
\langle p_{mol} \rangle \approx \frac{{p_o}^2}{3 k_B T}E, |
| 413 |
+ |
\end{equation} |
| 414 |
+ |
where $ \alpha_p = \frac{{p_o}^2}{3 k_B T}$ is the molecular |
| 415 |
+ |
polarizability. The orientational polarization depends inversely on |
| 416 |
+ |
the temperature as the applied field must overcome thermal agitation |
| 417 |
+ |
to orient the dipoles. |
| 418 |
+ |
|
| 419 |
+ |
\subsection{Quadrupoles} |
| 420 |
+ |
\label{subsec:boltzAverage-Quad} |
| 421 |
+ |
If instead, our system consists of molecules with permanent |
| 422 |
+ |
\textit{quadrupole} tensor $q_{\alpha\beta}$. The average quadrupole |
| 423 |
+ |
at temperature $T$ in the presence of uniform applied field gradient |
| 424 |
+ |
is given by,\cite{AduGyamfi78, AduGyamfi81} |
| 425 |
+ |
\begin{equation} |
| 426 |
+ |
\langle q_{\alpha\beta} \rangle \;=\; \frac{\displaystyle\int |
| 427 |
+ |
q_{\alpha\beta}\; e^{-H/k_B T}\; d\Omega}{\displaystyle\int |
| 428 |
+ |
e^{-H/k_B T}\; d\Omega} \;=\; \frac{\displaystyle\int |
| 429 |
+ |
q_{\alpha\beta}\; e^{~q_{\mu\nu}\;\partial_\nu E_\mu /k_B T}\; |
| 430 |
+ |
d\Omega}{\displaystyle\int e^{~q_{\mu\nu}\;\partial_\nu E_\mu /k_B |
| 431 |
+ |
T}\; d\Omega }, |
| 432 |
+ |
\label{boltzQuad} |
| 433 |
+ |
\end{equation} |
| 434 |
+ |
where $H = H_o - q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of a |
| 435 |
+ |
quadrupole in the gradient of the applied field and $H_o$ is a |
| 436 |
+ |
function of internal coordinates of the molecule. The energy and |
| 437 |
+ |
quadrupole moment can be transformed into the body frame using a |
| 438 |
+ |
rotation matrix $\mathsf{\eta}^{-1}$, |
| 439 |
+ |
\begin{align} |
| 440 |
+ |
q_{\alpha\beta} &= \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
| 441 |
+ |
H &= H_o - q:{\nabla}\mathbf{E} \\ |
| 442 |
+ |
&= H_o - q_{\mu\nu}\;\partial_\nu E_\mu \\ |
| 443 |
+ |
&= H_o |
| 444 |
+ |
-\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu |
| 445 |
+ |
E_\mu. \label{energyQuad} |
| 446 |
+ |
\end{align} |
| 447 |
+ |
Here the starred tensors are the components in the body fixed |
| 448 |
+ |
frame. Substituting equation (\ref{energyQuad}) in the equation |
| 449 |
+ |
(\ref{boltzQuad}) and taking linear terms in the expansion we obtain, |
| 450 |
+ |
\begin{equation} |
| 451 |
+ |
\braket{q_{\alpha\beta}} = \frac{\displaystyle \int q_{\alpha\beta} \left(1 + |
| 452 |
+ |
\frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu |
| 453 |
+ |
E_\mu }{k_B T}\right)\; d\Omega}{\displaystyle \int \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)\; d\Omega}. |
| 454 |
+ |
\end{equation} |
| 455 |
+ |
Recall that $\eta_{\alpha\alpha'}$ is the inverse of the rotation |
| 456 |
+ |
matrix that transforms the body fixed co-ordinates to the space |
| 457 |
+ |
co-ordinates. |
| 458 |
+ |
% \[\eta_{\alpha\alpha'} |
| 459 |
+ |
% = \left(\begin{array}{ccc} |
| 460 |
+ |
% cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
| 461 |
+ |
% cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
| 462 |
+ |
% sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
| 463 |
+ |
% \end{array} \right).\] |
| 464 |
|
|
| 465 |
+ |
Integration of the first and second terms in the denominator gives |
| 466 |
+ |
$8 \pi^2$ and |
| 467 |
+ |
$8 \pi^2 ({\nabla} \cdot \mathbf{E}) \mathrm{Tr}(q^*) / 3 $ |
| 468 |
+ |
respectively. The second term vanishes for charge free space (where |
| 469 |
+ |
${\nabla} \cdot \mathbf{E}=0$). Similarly, integration of the first |
| 470 |
+ |
term in the numerator produces |
| 471 |
+ |
$8 \pi^2 \delta_{\alpha\beta} \mathrm{Tr}(q^*) / 3$ while the second |
| 472 |
+ |
produces |
| 473 |
+ |
$8 \pi^2 (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
| 474 |
+ |
{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta / |
| 475 |
+ |
15 k_B T $. |
| 476 |
+ |
Therefore the Boltzmann average of a quadrupole moment can be written |
| 477 |
+ |
as, |
| 478 |
+ |
\begin{equation} |
| 479 |
+ |
\langle q_{\alpha\beta} \rangle = \frac{1}{3} \mathrm{Tr}(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
| 480 |
+ |
\end{equation} |
| 481 |
+ |
where $\alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular |
| 482 |
+ |
quadrupole polarizablity and |
| 483 |
+ |
${\bar{q_o}}^2= |
| 484 |
+ |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ |
| 485 |
+ |
is the square of the net quadrupole moment of a molecule. |
| 486 |
+ |
|
| 487 |
|
\section{Gradient of the field due to quadrupolar polarization} |
| 488 |
|
\label{singularQuad} |
| 489 |
< |
In this section, we will discuss the gradient of the field produced by |
| 490 |
< |
quadrupolar polarization. For this purpose, we consider a distribution |
| 491 |
< |
of charge ${\rho}(\mathbf r)$ which gives rise to an electric field |
| 492 |
< |
$\mathbf{E}(\mathbf r)$ and gradient of the field ${\nabla} \mathbf{E}(\mathbf r)$ |
| 493 |
< |
throughout space. The total gradient of the electric field over volume |
| 494 |
< |
due to the all charges within the sphere of radius $R$ is given by |
| 495 |
< |
(cf. Jackson equation 4.14): |
| 489 |
> |
In section IV.C of the main text, we stated that for quadrupolar |
| 490 |
> |
fluids, the self-contribution to the field gradient vanishes at the |
| 491 |
> |
singularity. In this section, we prove this statement. For this |
| 492 |
> |
purpose, we consider a distribution of charge $\rho(\mathbf{r})$ which |
| 493 |
> |
gives rise to an electric field $\mathbf{E}(\mathbf{r})$ and gradient |
| 494 |
> |
of the field $\nabla\mathbf{E}(\mathbf{r})$ throughout space. The |
| 495 |
> |
gradient of the electric field over volume due to the charges within |
| 496 |
> |
the sphere of radius $R$ is given by (cf. Ref. \onlinecite{Jackson98}, |
| 497 |
> |
equation 4.14): |
| 498 |
|
\begin{equation} |
| 499 |
< |
\int_{r<R} {\nabla}\mathbf{E}\;d^3r = -\int_{r=R} R^2 \mathbf{E}\;\hat{n}\; d\Omega |
| 499 |
> |
\int_{r<R} \nabla\mathbf{E} d\mathbf{r} = -\int_{r=R} R^2 \mathbf{E}\;\hat{n}\; d\Omega |
| 500 |
|
\label{eq:8} |
| 501 |
|
\end{equation} |
| 502 |
|
where $d\Omega$ is the solid angle and $\hat{n}$ is the normal vector |
| 503 |
|
of the surface of the sphere, |
| 293 |
– |
$\hat{n} = sin[\theta]cos[\phi]\hat{x} + sin[\theta]sin[\phi]\hat{y} + |
| 294 |
– |
cos[\theta]\hat{z}$ |
| 295 |
– |
in spherical coordinates. For the charge density ${\rho}(\mathbf r')$, the |
| 296 |
– |
total gradient of the electric field can be written as, ~\cite{Jackson98} |
| 504 |
|
\begin{equation} |
| 505 |
< |
\int_{r<R} {\nabla}\mathbf {E}\; d^3r=-\int_{r=R} R^2\; {\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; {\nabla}\;\left(\int \frac{\rho(\mathbf r')}{|\mathbf{r}-\mathbf{r}'|}\;d^3r'\right) \hat{n}\; d\Omega |
| 505 |
> |
\hat{n} = \sin\theta\cos\phi\; \hat{x} + \sin\theta\sin\phi\; \hat{y} + |
| 506 |
> |
\cos\theta\; \hat{z} |
| 507 |
> |
\end{equation} |
| 508 |
> |
in spherical coordinates. For the charge density $\rho(\mathbf{r}')$, the |
| 509 |
> |
total gradient of the electric field can be written as,\cite{Jackson98} |
| 510 |
> |
\begin{equation} |
| 511 |
> |
\int_{r<R} {\nabla}\mathbf {E}\; d\mathbf{r}=-\int_{r=R} R^2\; {\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; {\nabla}\;\left(\int \frac{\rho(\mathbf r')}{|\mathbf{r}-\mathbf{r}'|}\;d\mathbf{r}'\right) \hat{n}\; d\Omega |
| 512 |
|
\label{eq:9} |
| 513 |
|
\end{equation} |
| 514 |
|
The radial function in the equation (\ref{eq:9}) can be expressed in |
| 520 |
|
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, |
| 521 |
|
\begin{equation} |
| 522 |
|
\begin{split} |
| 523 |
< |
\int_{r<R} {\nabla}\mathbf{E}\;d^3r &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; {\nabla}\;\left(\int \rho(\mathbf 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 \\ |
| 524 |
< |
&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(\mathbf 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 |
| 523 |
> |
\int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; {\nabla}\;\left(\int \rho(\mathbf 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\mathbf{r}'\right) \hat{n}\; d\Omega \\ |
| 524 |
> |
&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(\mathbf 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\mathbf{r} |
| 525 |
|
' |
| 526 |
|
\end{split} |
| 527 |
|
\label{eq:11} |
| 535 |
|
\end{split} |
| 536 |
|
\label{eq:12} |
| 537 |
|
\end{equation} |
| 538 |
< |
Using equation (\ref{eq:12}) we get, |
| 538 |
> |
where $Y_{l,l+1,m}(\theta, \phi)$ is a vector spherical |
| 539 |
> |
harmonic.\cite{Arfkan} Using equation (\ref{eq:12}) we get, |
| 540 |
|
\begin{equation} |
| 541 |
|
{\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}}, |
| 542 |
|
\label{eq:13} |
| 543 |
|
\end{equation} |
| 544 |
< |
where $ Y_{l,l+1,m}(\theta, \phi)$ is a vector spherical harmonics \cite{Arfkan}. Using Clebsch-Gorden coefficients $C(l+1, 1, l|m_1,m_2,m) $, equation \ref{eq:14} can be written in spherical harmonics, |
| 544 |
> |
Using Clebsch-Gordan coefficients $C(l+1,1,l|m_1,m_2,m)$, the vector |
| 545 |
> |
spherical harmonics can be written in terms of spherical harmonics, |
| 546 |
|
\begin{equation} |
| 547 |
< |
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}. |
| 547 |
> |
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}. |
| 548 |
|
\label{eq:14} |
| 549 |
|
\end{equation} |
| 550 |
|
Here $\hat{e}_{m_2}$ is a spherical tensor of rank 1 which can be expressed |
| 555 |
|
\end{equation} |
| 556 |
|
The normal vector $\hat{n} $ is then expressed in terms of spherical tensor of rank 1 as shown in below, |
| 557 |
|
\begin{equation} |
| 558 |
< |
\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). |
| 558 |
> |
\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). |
| 559 |
|
\label{eq:16} |
| 560 |
|
\end{equation} |
| 561 |
|
The surface integral of the product of $\hat{n}$ and |
| 562 |
< |
${Y_{l+1}}^{m_1}(\theta, \phi)$ gives, |
| 562 |
> |
$Y_{l+1}^{m_1}(\theta, \phi)$ gives, |
| 563 |
|
\begin{equation} |
| 564 |
|
\begin{split} |
| 565 |
< |
\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 \\ |
| 566 |
< |
&= \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 \\ |
| 565 |
> |
\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 \\ |
| 566 |
> |
&= \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 \\ |
| 567 |
|
&= \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), |
| 568 |
|
\end{split} |
| 569 |
|
\label{eq:17} |
| 570 |
|
\end{equation} |
| 571 |
< |
where ${Y_{l}}^{-m} = (-1)^m\;{{Y_{l}}^{m}}^* $ and |
| 572 |
< |
$ \int {{Y_{l}}^{m}}^*\;{Y_{l'}}^{m'}\;d\Omega = |
| 571 |
> |
where $Y_{l}^{-m} = (-1)^m\;{Y_{l}^{m}}^* $ and |
| 572 |
> |
$ \int {Y_{l}^{m}}^* Y_{l'}^{m'}\;d\Omega = |
| 573 |
|
\delta_{ll'}\delta_{mm'} $. |
| 574 |
|
Non-vanishing values of equation \ref{eq:17} require $l = 0$, |
| 575 |
|
therefore the value of $ m = 0 $. Since the values of $ m_1$ are -1, |
| 578 |
|
modified, |
| 579 |
|
\begin{equation} |
| 580 |
|
\begin{split} |
| 581 |
< |
\int_{r<R} {\nabla}\mathbf{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( |
| 582 |
< |
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d^3r' \\ |
| 583 |
< |
&= -\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)\\ |
| 584 |
< |
&= - \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). |
| 581 |
> |
\int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} = &- \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( |
| 582 |
> |
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d\mathbf{r}' \\ |
| 583 |
> |
&= -\sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;d\mathbf{r}'\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right)\\ |
| 584 |
> |
&= - \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\;C_\mathrm{total}\;\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right). |
| 585 |
|
\end{split} |
| 586 |
|
\label{eq:19} |
| 587 |
|
\end{equation} |
| 588 |
< |
In the last step, the charge density was integrated over the sphere, yielding a total charge $\mathrm{C_total}$.Equation (\ref{eq:19}) gives the total gradient of the field over a sphere due to the distribution of the charges. |
| 589 |
< |
For quadrupolar fluids the total charge within a sphere is zero, therefore |
| 590 |
< |
$ \int_{r<R} {\nabla}\mathbf{E}\;d^3r = 0 $. Hence the quadrupolar |
| 588 |
> |
In the last step, the charge density was integrated over the sphere, |
| 589 |
> |
yielding a total charge $C_\mathrm{total}$.Equation (\ref{eq:19}) |
| 590 |
> |
gives the total gradient of the field over a sphere due to the |
| 591 |
> |
distribution of the charges. For quadrupolar fluids the total charge |
| 592 |
> |
within a sphere is zero, therefore |
| 593 |
> |
$ \int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} = 0 $. Hence the quadrupolar |
| 594 |
|
polarization produces zero net gradient of the field inside the |
| 595 |
|
sphere. |
| 596 |
|
|
| 379 |
– |
|
| 597 |
|
\bibliography{dielectric_new} |
| 598 |
|
\end{document} |
| 599 |
|
% |
