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\begin{document} |
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|
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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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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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|
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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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\maketitle |
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|
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This document contains derivations of useful relationships for |
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electric field gradients and their interactions with point multipoles. |
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We rely heavily on both the notation and results from Torres del |
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Castillo and Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} In |
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this work, tensors are expressed in Cartesian components, using at |
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times a dyadic notation. This proves quite useful for our work as we |
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employ toroidal boundary conditions in our simulations, and these are |
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easily implemented in Cartesian coordinate systems. |
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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, and Baym. Because this |
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approach has its own advantages, relationships are provided below |
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comparing that terminology to the Cartesian tensor notation. |
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|
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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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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) =\; -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) . |
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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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An alternative to this notation is to write an electrostatic potential |
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that generates a uniform gradient using the notation of Morse and |
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Feshbach, |
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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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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. 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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which 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} |
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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} |
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0 & 0 & 0\\ |
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0 & 0 & 1\\ |
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0 & 1 & 0\\ |
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\end{pmatrix} |
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+6a_{22}^e |
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\begin{pmatrix} |
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0 & 1 & 0\\ |
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1 & 0 & 0\\ |
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0 & 0 & 0\\ |
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\end{pmatrix} |
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+6a_{22}^o |
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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 & 0\\ |
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\end{pmatrix} |
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\end{aligned} |
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\label{eq:intro_tensors} |
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\end{gather} |
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The five matrices in the expression above represent five different |
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symmetric traceless tensors of rank 2. The trace corresponds to |
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$\nabla \cdot \mathbf{E} = 0$, consistent with being in a charge-free |
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region. Using the Cartesian notation of |
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Eq. (\ref{eq:appliedPotential}), this tensor is written: |
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\begin{equation} |
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\mathsf{G} =\nabla\bf{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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|
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It is useful to find vectors $\mathbf a$ and $\mathbf b$ that generate |
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the five types of tensors shown in Eq. (\ref{eq:intro_tensors}). If |
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the two vectors are co-linear, e.g., $\psi=0$, $\mathbf{a}=(0,0,1)$ and |
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$\mathbf{b}=(0,0,1)$, then |
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\begin{equation*} |
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\mathsf{G} = \frac{g_0}{3} |
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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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\end{equation*} |
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which is the $a_{20}$ symmetry. |
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To generate the $a_{22}^o$ symmetry, we take: |
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$\mathbf{a}= (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}},0)$ and |
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$\mathbf{b}=(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}},0)$ |
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and find: |
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\begin{equation*} |
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\mathsf{G}=\frac{g_0}{2} |
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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 & 0 \\ |
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\end{pmatrix} . |
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\end{equation*} |
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To generate the $a_{22}^e$ symmetry, we take: |
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$\mathbf{a}= (1, 0, 0)$ and $\mathbf{b} = (0,1,0)$ and find: |
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\begin{equation*} |
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\mathsf{G}=\frac{g_0}{2} |
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\begin{pmatrix} |
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0 & 1 & 0 \\ |
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1 & 0 & 0 \\ |
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0 & 0 & 0 \\ |
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\end{pmatrix} . |
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\end{equation*} |
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The pattern is straightforward to continue for the other symmetries. |
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|
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Using Eq. (\ref{eq:quad_phi}) the electric field is written: |
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\begin{equation} |
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\mathbf{E} |
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= \left(\begin{array}{ccc} |
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\left(-a_{20} + 6a_{22}^o \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}^e 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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while using Eq. (\ref{eq:appliedPotential}), we find: |
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\begin{equation} |
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\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})\;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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We find the notation of Ref. \onlinecite{Torres-del-Castillo:2006uo} |
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to be helpful when creating specific types of constant gradient |
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electric fields in simulations. For this reason, |
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Eqs. (\ref{eq:appliedPotential}), (\ref{eq:GC}), and (\ref{eq:CE}) are |
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used in our code. |
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|
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\section{Point-multipolar interactions with a spatially-varying electric field} |
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|
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This section develops formulas for the force and torque exerted by an |
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external electric field, $\mathbf{E}(\mathbf{r})$, on object |
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$a$. Object $a$ has an embedded collection of charges and in |
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simulations will represent a molecule, ion, or a coarse-grained |
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substructure. We describe the charge distributions using primitive |
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multipoles defined in Ref. \onlinecite{PaperI} by |
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\begin{align} |
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C_a =&\sum_{k \, \text{in }a} q_k , \label{eq:charge} \\ |
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D_{a\alpha} =&\sum_{k \, \text{in }a} q_k r_{k\alpha}, \label{eq:dipole}\\ |
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Q_{a\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in } a} q_k |
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r_{k\alpha} r_{k\beta}, |
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\label{eq:quadrupole} |
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\end{align} |
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where $\mathbf{r}_k$ is the local coordinate system for the object |
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(usually the center of mass of object $a$). Components of vectors and |
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tensors are given using Green indices, using the Einstein repeated |
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summation notation. Note that the definition of the primitive |
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quadrupole here differs from the standard traceless form, and contains |
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an additional Taylor-series based factor of $1/2$. In Ref. |
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\onlinecite{PaperI}, we derived the forces and torques each object |
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exerts on the other objects in the system. |
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|
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Here we must also consider an external electric field that varies in |
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space: $\mathbf E(\mathbf r)$. Each of the local charges $q_k$ in |
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object $a$ will then experience a slightly different field. This |
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electric field can be expanded in a Taylor series around the local |
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origin of each object. For a particular charge $q_k$, the electric |
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field at that site's position is given by: |
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\begin{equation} |
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\mathbf{E}(\mathbf{r}_k) = E_\gamma|_{\mathbf{r}_k = 0} + \nabla_\delta E_\gamma |_{\mathbf{r}_k = 0} r_{k \delta} |
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+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma|_{\mathbf{r}_k = 0} r_{k \delta} |
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r_{k \varepsilon} + ... |
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\end{equation} |
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Note that once one shrinks object $a$ to point size, the ${E}_\gamma$ |
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terms are all evaluated at the center of the object (now a |
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point). Thus later the ${E}_\gamma$ terms can be written using the |
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same global origin for all objects $a, b, c, ...$ in the system. The |
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force exerted on object $a$ by the electric field is given by, |
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|
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\begin{align} |
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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} |
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+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
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r_{k \varepsilon} + ... \rbrace \\ |
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&= C_a E_\gamma + D_{a \delta} \nabla_\delta E_\gamma |
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+ Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma + |
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... |
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\end{align} |
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Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. |
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|
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Similarly, the torque exerted by the field on $a$ can be expressed as |
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\begin{align} |
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\tau^a_\alpha &= \sum_{k \textrm{~in~} a} (\mathbf r_k \times q_k \mathbf E)_\alpha \\ |
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& = \sum_{k \textrm{~in~} a} \epsilon_{\alpha \beta \gamma} q_k |
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r_{k\beta} E_\gamma(\mathbf r_k) \\ |
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& = \epsilon_{\alpha \beta \gamma} D_\beta E_\gamma |
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+ 2 \epsilon_{\alpha \beta \gamma} Q_{\beta \delta} \nabla_\delta |
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E_\gamma + ... |
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\end{align} |
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We note that the Levi-Civita symbol can be eliminated by utilizing the matrix cross product as defined in Ref. \onlinecite{Smith98}: |
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\begin{equation} |
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\left[\mathsf{A} \times \mathsf{B}\right]_\alpha = \sum_\beta |
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\left[\mathsf{A}_{\alpha+1,\beta} \mathsf{B}_{\alpha+2,\beta} |
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-\mathsf{A}_{\alpha+2,\beta} \mathsf{B}_{\alpha+1,\beta} |
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\right] |
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\label{eq:matrixCross} |
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\end{equation} |
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where $\alpha+1$ and $\alpha+2$ are regarded as cyclic permuations of |
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the matrix indices. Finally, the interaction energy $U^a$ of object $a$ with the external field is given by, |
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\begin{equation} |
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U^a = \sum_{k~in~a} q_k \phi_k (\mathrm{r}_k) |
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\end{equation} |
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Performing another Taylor series expansion about the local body origin, |
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\begin{equation} |
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\phi({\mathbf{r}_k}) = \phi|_{\mathbf{r}_k = 0 } + r_{k \alpha} \nabla_\alpha \phi_\alpha|_{\mathbf{r}_k = 0 } + \frac{1}{2} r_{k\alpha}r_{k\beta}\nabla_\alpha \nabla_\beta \phi|_{\mathbf{r}_k = 0} + ... |
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\end{equation} |
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Writing this in terms of the global origin $\mathrm{r}$, we find |
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\begin{equation} |
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U(\mathbf{r}) = \mathrm{C} \phi(\mathbf{r}) - \mathrm{D}_\alpha \mathrm{E}_\alpha - \mathrm{Q}_{\alpha\beta}\nabla_\alpha \mathrm{E}_\beta + ... |
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\end{equation} |
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The results has been summarized in Table I. |
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|
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\begin{table} |
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\caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque |
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$(\mathbf{\tau})$ expressions for a multipolar site $\mathrm{r}$ in an |
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electric field, $\mathbf{E}(\mathbf{r})$. |
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\label{tab:UFT}} |
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\begin{tabular}{r|ccc} |
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& Charge & Dipole & Quadrupole \\ \hline |
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$U(\mathbf{r})$ & $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\ |
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$\mathbf{F}(\mathbf{r})$ & $C \mathbf{E}(\mathbf{r})$ & $\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ & $\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\ |
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$\mathbf{\tau}(\mathbf{r})$ & & $\mathbf{D} \times \mathbf{E}(\mathbf{r})$ & $2 \mathsf{Q} \times \nabla \mathbf{E}(\mathbf{r})$ |
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\end{tabular} |
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\end{table} |
340 |
|
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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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|
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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} |
369 |
\begin{equation} |
370 |
H = H_o - \mathbf{p_o}\cdot \mathbf{E}, |
371 |
\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, |
374 |
\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}}}, |
376 |
\end{equation} |
377 |
where $\bf{E}$ is selected along z-axis. If we consider that the |
378 |
applied field is small, \textit{i.e.} $\frac{p_oE\; cos\theta}{k_B T} << 1$, |
379 |
\begin{equation} |
380 |
\braket{p_{mol}} \approx \frac{1}{3}\frac{{p_o}^2}{k_B T}E, |
381 |
\end{equation} |
382 |
where $ \alpha_p = \frac{1}{3}\frac{{p_o}^2}{k_B T}$ is a molecular |
383 |
polarizability. The orientational polarization depends inversely on |
384 |
the temperature and applied field must overcome the thermal agitation. |
385 |
|
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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 |
389 |
$q_{\alpha\beta}$. The average quadrupole moment at temperature T in |
390 |
the presence of uniform applied field gradient is given |
391 |
by,\cite{AduGyamfi78, AduGyamfi81} |
392 |
\begin{equation} |
393 |
\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}}}, |
394 |
\label{boltzQuad} |
395 |
\end{equation} |
396 |
where $\int d\Omega = \int_0^{2\pi} \int_0^\pi \int_0^{2\pi} |
397 |
sin\theta\; d\theta\ d\phi\ d\psi$ is the integration over Euler |
398 |
angles, $ H = H_o -q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of |
399 |
a quadrupole in the gradient of the |
400 |
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, |
401 |
\begin{equation} |
402 |
\begin{split} |
403 |
&q_{\alpha\beta} = \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
404 |
&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. |
405 |
\end{split} |
406 |
\label{energyQuad} |
407 |
\end{equation} |
408 |
Here the starred tensors are the components in the body fixed |
409 |
frame. Substituting equation (\ref{energyQuad}) in the equation (\ref{boltzQuad}) |
410 |
and taking linear terms in the expansion we get, |
411 |
\begin{equation} |
412 |
\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)}, |
413 |
\end{equation} |
414 |
where $\eta_{\alpha\alpha'}$ is the inverse of the rotation matrix that transforms |
415 |
the body fixed co-ordinates to the space co-ordinates, |
416 |
\[\eta_{\alpha\alpha'} |
417 |
= \left(\begin{array}{ccc} |
418 |
cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
419 |
cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
420 |
sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
421 |
\end{array} \right).\] |
422 |
Integration of 1st and 2nd terms in the denominator gives $8 \pi^2$ |
423 |
and $8 \pi^2 /3\;{\nabla}.\mathbf{E}\; Tr(q^*) $ respectively. The |
424 |
second term vanishes for charge free space, ${\nabla}.\mathbf{E} \; = \; 0$. Similarly integration of the |
425 |
1st term in the numerator produces |
426 |
$8 \pi^2 /3\; Tr(q^*)\delta_{\alpha\beta}$ and the 2nd term produces |
427 |
$8 \pi^2 /15k_B T (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
428 |
{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta$, |
429 |
if ${\nabla}.\mathbf{E} \; = \; 0$, |
430 |
$ \partial_\alpha E_\beta = \partial_\beta E_\alpha$ and |
431 |
${q}^*_{\alpha'\beta'}= {q}^*_{\beta'\alpha'}$. Therefore the |
432 |
Boltzmann average of a quadrupole moment can be written as, |
433 |
|
434 |
\begin{equation} |
435 |
\braket{q_{\alpha\beta}}\; = \; \frac{1}{3} Tr(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
436 |
\end{equation} |
437 |
where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupole polarizablity and ${\bar{q_o}}^2= |
438 |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. |
439 |
|
440 |
% \section{External application of a uniform field gradient} |
441 |
% \label{Ap:fieldOrGradient} |
442 |
|
443 |
% To satisfy the condition $ \nabla \cdot \mathbf{E} = 0 $, within the box of molecules we have taken electrostatic potential in the following form |
444 |
% \begin{equation} |
445 |
% \begin{split} |
446 |
% \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. \\ |
447 |
% & \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), |
448 |
% \end{split} |
449 |
% \label{eq:appliedPotential} |
450 |
% \end{equation} |
451 |
% 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, |
452 |
% \[\mathbf{E} |
453 |
% = \frac{g_o}{2} \left(\begin{array}{ccc} |
454 |
% 2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+ (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
455 |
% (a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y + (a_2\; b_3 \;+ a_3\; b_2)\;z \\ |
456 |
% (a_3\; b_1 \;+ a_3\; b_2)\;x + (a_3\; b_2 \;+ a_2\; b_3)y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
457 |
% \end{array} \right).\] |
458 |
% The gradient of the applied field derived from the potential can be written in the following form, |
459 |
% \[\nabla\mathbf{E} |
460 |
% = \frac{g_o}{2}\left(\begin{array}{ccc} |
461 |
% 2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
462 |
% (a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_2) \\ |
463 |
% (a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
464 |
% \end{array} \right).\] |
465 |
|
466 |
|
467 |
|
468 |
\section{Gradient of the field due to quadrupolar polarization} |
469 |
\label{singularQuad} |
470 |
In this section, we will discuss the gradient of the field produced by |
471 |
quadrupolar polarization. For this purpose, we consider a distribution |
472 |
of charge ${\rho}(\mathbf r)$ which gives rise to an electric field |
473 |
$\mathbf{E}(\mathbf r)$ and gradient of the field ${\nabla} \mathbf{E}(\mathbf r)$ |
474 |
throughout space. The total gradient of the electric field over volume |
475 |
due to the all charges within the sphere of radius $R$ is given by |
476 |
(cf. Jackson equation 4.14): |
477 |
\begin{equation} |
478 |
\int_{r<R} {\nabla}\mathbf{E}\;d^3r = -\int_{r=R} R^2 \mathbf{E}\;\hat{n}\; d\Omega |
479 |
\label{eq:8} |
480 |
\end{equation} |
481 |
where $d\Omega$ is the solid angle and $\hat{n}$ is the normal vector |
482 |
of the surface of the sphere, |
483 |
$\hat{n} = sin[\theta]cos[\phi]\hat{x} + sin[\theta]sin[\phi]\hat{y} + |
484 |
cos[\theta]\hat{z}$ |
485 |
in spherical coordinates. For the charge density ${\rho}(\mathbf r')$, the |
486 |
total gradient of the electric field can be written as, ~\cite{Jackson98} |
487 |
\begin{equation} |
488 |
\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 |
489 |
\label{eq:9} |
490 |
\end{equation} |
491 |
The radial function in the equation (\ref{eq:9}) can be expressed in |
492 |
terms of spherical harmonics as,\cite{Jackson98} |
493 |
\begin{equation} |
494 |
\frac{1}{|\mathbf{r} - \mathbf{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) |
495 |
\label{eq:10} |
496 |
\end{equation} |
497 |
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, |
498 |
\begin{equation} |
499 |
\begin{split} |
500 |
\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 \\ |
501 |
&= -\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 |
502 |
' |
503 |
\end{split} |
504 |
\label{eq:11} |
505 |
\end{equation} |
506 |
The gradient of the product of radial function and spherical harmonics |
507 |
is given by:\cite{Arfkan} |
508 |
\begin{equation} |
509 |
\begin{split} |
510 |
{\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 |
511 |
{\partial}{\partial r}+\frac{l}{r} \right]f(r)\; Y_{l, l-1, m}(\theta, \phi). |
512 |
\end{split} |
513 |
\label{eq:12} |
514 |
\end{equation} |
515 |
Using equation (\ref{eq:12}) we get, |
516 |
\begin{equation} |
517 |
{\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}}, |
518 |
\label{eq:13} |
519 |
\end{equation} |
520 |
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, |
521 |
\begin{equation} |
522 |
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}. |
523 |
\label{eq:14} |
524 |
\end{equation} |
525 |
Here $\hat{e}_{m_2}$ is a spherical tensor of rank 1 which can be expressed |
526 |
in terms of Cartesian coordinates, |
527 |
\begin{equation} |
528 |
{\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}}. |
529 |
\label{eq:15} |
530 |
\end{equation} |
531 |
The normal vector $\hat{n} $ is then expressed in terms of spherical tensor of rank 1 as shown in below, |
532 |
\begin{equation} |
533 |
\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). |
534 |
\label{eq:16} |
535 |
\end{equation} |
536 |
The surface integral of the product of $\hat{n}$ and |
537 |
${Y_{l+1}}^{m_1}(\theta, \phi)$ gives, |
538 |
\begin{equation} |
539 |
\begin{split} |
540 |
\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 \\ |
541 |
&= \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 \\ |
542 |
&= \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), |
543 |
\end{split} |
544 |
\label{eq:17} |
545 |
\end{equation} |
546 |
where ${Y_{l}}^{-m} = (-1)^m\;{{Y_{l}}^{m}}^* $ and |
547 |
$ \int {{Y_{l}}^{m}}^*\;{Y_{l'}}^{m'}\;d\Omega = |
548 |
\delta_{ll'}\delta_{mm'} $. |
549 |
Non-vanishing values of equation \ref{eq:17} require $l = 0$, |
550 |
therefore the value of $ m = 0 $. Since the values of $ m_1$ are -1, |
551 |
1, and 0 then $m_2$ takes the values 1, -1, and 0, respectively |
552 |
provided that $m = m_1 + m_2$. Equation \ref{eq:11} can therefore be |
553 |
modified, |
554 |
\begin{equation} |
555 |
\begin{split} |
556 |
\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( |
557 |
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d^3r' \\ |
558 |
&= -\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)\\ |
559 |
&= - \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). |
560 |
\end{split} |
561 |
\label{eq:19} |
562 |
\end{equation} |
563 |
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. |
564 |
For quadrupolar fluids the total charge within a sphere is zero, therefore |
565 |
$ \int_{r<R} {\nabla}\mathbf{E}\;d^3r = 0 $. Hence the quadrupolar |
566 |
polarization produces zero net gradient of the field inside the |
567 |
sphere. |
568 |
|
569 |
|
570 |
\bibliography{dielectric_new} |
571 |
\end{document} |
572 |
% |
573 |
% ****** End of file multipole.tex ****** |