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\begin{document} |
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\preprint{AIP/123-QED} |
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\title[Generalization of the Shifted-Force Potential to Higher-Order Potentials] |
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{Generalization of the Shifted-Force Potential to Higher-Order Potentials} |
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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{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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\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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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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\begin{abstract} |
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Over the past several years, there has been increasing interest |
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in real space methods for calculating electrostatic interactions |
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in computer simulations of condensed molecular systems. We |
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have extended our original damped-shifted force (DSF) |
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electrostatic kernel and have been able to derive a set of |
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interaction models for higher-order multipoles based on |
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truncated Taylor expansions around the cutoff. For multipolemultipole |
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interactions, we find that each of the distinct |
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orientational contributions has a separate radial function to |
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ensure that the overall forces and torques vanish at the cutoff |
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radius. |
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\end{abstract} |
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\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
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% Classification Scheme. |
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\keywords{Suggested keywords}%Use showkeys class option if keyword |
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%display desired |
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\maketitle |
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\section{Introduction} |
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The Coulomb electrostatic interaction is of importance in a number of physical chemistry problems |
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[background needed, do we mention gases, liquids, solids?]. |
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[...] |
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The methods that we develop in this paper are meant specifically for problems involving interacting rigid molecules which will be described |
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in terms of classical mechanics and electrodynamics. From mechanics, the molecule's mass distribution |
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determines its total mass and moment of inertia tensor. |
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From electrostatics, its charge distribution is conveniently described using multipoles. |
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Our goal is to advance methods for handling inter-molecular interactions in molecular dynamics simulations. |
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To do this, we must develop consistent approximate equations for |
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interaction energies, forces, and torques. |
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[...] |
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This paper extends the shifted-force potential method |
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of Fennel and Gezelter to higher-order multipole interactions. [describe?] |
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Extending an idea from Wolf, multipole images are placed on the surface of a |
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``cutoff'' sphere of radius $r_c$. These images are used to make all interaction energies, forces, and torques |
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be zero for $r < r_c$. |
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Two such methods have been developed, both based on Taylor-series expansions. |
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The first is applied to the Coulomb kernel of the multipole expansion. The second is |
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applied to individual terms for interaction energies in the multipole expansion. |
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Because of differences in the initial assumptions, the two methods yield different results. |
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Also explored here is the effect of replacing the bare Coulomb kernel with that of a smeared |
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charge distribution. Thus four methods are compared in this paper: |
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(1) Shifted force, Coulomb, method 1; |
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(2) Sihfted force, Coulomb, method 2; |
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(3) Shifted force, smeared charge, method 1; and |
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(4) Shifted force, smeared charge, method 2. |
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The last of these methods is our preferred method and is called the Extended Shifted Force Method. |
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Subsequent papers will apply this method to various problems of physical and chemical interest. |
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[...] |
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\section{Development of the Methods} |
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\subsection{Multipole Expansion} |
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Consider two discrete rigid collections of atoms and ions, denoted as objects $\bf a$ and $\bf b$. |
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In the following, we assume |
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that the two objects only interact via electrostatics and describe those interactions in terms of |
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a multipole expansion. Putting the origin of the coordinate system at the center of mass of $\bf a$, we use |
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vectors $\mathbf{r}_k$ to denote the positions of all charges $q_k$ in $\bf a$. |
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Then the electrostatic potential of object $\bf a$ at $\mathbf{r}$ is given by |
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\begin{equation} |
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V_a(\mathbf r) = \frac{1}{4\pi\epsilon_0} |
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\sum_{k \, \text{in \bf a}} \frac{q_k}{\lvert \mathbf{r} - \mathbf{r}_k \rvert}. |
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\end{equation} |
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We write the Taylor series expansion in $r$ using an implied summation notation, |
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Greek indices are used to indicate space coordinates $x$, $y$, $z$ and the subscripts |
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$k$ and $j$ are reserved for labelling specific charges in $\bf a$ and $\bf b$ respectively, and find: |
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\begin{equation} |
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\frac{1}{\lvert \mathbf{r} - \mathbf{r}_k \rvert} = |
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\left( 1 |
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- r_{k\alpha} \frac{\partial}{\partial r_{\alpha}} |
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+ \frac{1}{2} r_{k\alpha} r_{k\beta} \frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} +\dots |
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\right) |
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\frac{1}{r} . |
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\end{equation} |
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We then follow Smith in defining an operator for the expansion: |
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\begin{equation} |
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V_{\bf a}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0}\hat{M}_{\bf a} \frac{1}{r} |
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\end{equation} |
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where |
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\begin{equation} |
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\hat{M}_{\bf a} = C_{\bf a} - D_{{\bf a}\alpha} \frac{\partial}{\partial r_{\alpha}} |
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+ Q_{{\bf a}\alpha\beta} |
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\frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} + \dots |
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\end{equation} |
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and the charge $C_{\bf a}$, dipole $D_{{\bf a}\alpha}$, |
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and quadrupole $Q_{{\bf a}\alpha\beta}$ are defined by |
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\begin{equation} |
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C_{\bf a}=\sum_{k \, \text{in \bf a}} q_k , |
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\end{equation} |
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\begin{equation} |
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D_{{\bf a}\alpha} = \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} , |
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\end{equation} |
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\begin{equation} |
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Q_{{\bf a}\alpha\beta} = \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} r_{k\beta} . |
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\end{equation} |
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It is convenient to locate charges $q_j$ relative to the center of mass of $\bf b$. Then with $\bf{r}$ pointing from |
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$\bf a$ to $\bf b$ ($\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $), the interaction energy is given by |
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\begin{eqnarray} |
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U_{\bf{ab}}(r) =&& \frac{1}{4\pi \epsilon_0} |
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\sum_{k \, \text{in \bf a}} \sum_{j \, \text{in \bf b}} |
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\frac{q_k q_j}{\vert \bf{r}_k - (\bf{r}+\bf{r}_j) \vert} \nonumber\\ |
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=&& \frac{1}{4\pi \epsilon_0} |
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\sum_{k \, \text{in \bf a}} \sum_{j \, \text{in \bf b}} |
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\frac{q_k q_j}{\vert \bf{r}+ (\bf{r}_j-\bf{r}_k) \vert} \nonumber\\ |
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=&&\frac{1}{4\pi \epsilon_0} \sum_{j \, \text{in \bf b}} q_j V_a(\bf{r}+\bf{r}_j) \nonumber\\ |
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=&&\frac{1}{4\pi \epsilon_0} \hat{M}_a \sum_{j \, \text{in \bf b}} \frac {q_j}{\vert \bf{r}+\bf{r}_j \vert} . |
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\end{eqnarray} |
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The last expression can also be expanded as a Taylor series in $r$. Using a notation similar to before, we define |
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\begin{equation} |
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\hat{M}_{\bf b} = C_{\bf b} + D_{{\bf b}\alpha} \frac{\partial}{\partial r_{\alpha}} |
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+ Q_{{\bf b}\alpha\beta} |
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\frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} + \dots |
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\end{equation} |
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and |
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\begin{equation} |
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U_{\bf{ab}}(r)=\frac{\hat{M}_{\bf a} \hat{M}_{\bf b}}{4\pi \epsilon_0} \frac{1}{r} \label{kernel}. |
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\end{equation} |
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Note the ease of separting out the respective energies of interaction of the charge, dipole, and quadrupole of $\bf a$ from those of $\bf b$. |
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\subsection{Bare Coulomb versus smeared charge} |
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With the four types of methods being considered here, it is desirable to list the approximations in as transparent a form |
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as possible. First, one may use the bare Coulomb potential, with radial dependence $1/r$, |
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as shown in Eq.~(\ref{kernel}). Alternatively, one may use |
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a smeared charge distribution, then the``kernel'' $1/r$ of the expansion is replaced with a function: |
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\begin{equation} |
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B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r} |
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\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
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\end{equation} |
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We develop equations using a function $f(r)$ to represent either $1/r$ or $B_0(r)$, dependent on the the type of approach being considered. |
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Smith's convenient functions $B_l(r)$ are summarized in Appendix A. |
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\subsection{Shifting the force, method 1} |
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As discussed in the introduction, it is desirable to cutoff the electrostatic energy at a radius |
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$r_c$. For consistency in approximation, we want the interaction energy as well as the force and |
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torque to go to zero at $r=r_c$. |
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To describe how this goal may be met using a radial approximation, we use two examples, charge-charge |
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and charge-dipole, using the bare Coulomb kernel $f(r)=1/r$ to explain the idea. |
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In the shifted-force approximation, the interaction energy $U_{\bf{ab}}(r_c)=0$ |
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for two charges $C_{\bf a}$ and $C_{\bf b}$ separated by a distance $r$ is written: |
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\begin{equation} |
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U_{C_{\bf a}C_{\bf b}}(r)=\frac{1}{4\pi \epsilon_0} C_{\bf a} C_{\bf b} |
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\left({ \frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
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\right) . |
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\end{equation} |
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Two shifting terms appear in this equations because we want the force to |
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also be shifted due to an image charge located at a distance $r_c$. |
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Since one derivative of the interaction energy is needed for the force, we want a term |
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linear in $(r-r_c)$ in the interaction energy, that is: |
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\begin{equation} |
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\frac{d\,}{dr} |
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\left( {\frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
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\right) = \left(- \frac{1}{r^2} + \frac{1}{r_c^2} |
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\right) . |
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\end{equation} |
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This demonstrates the need of the third term in the brackets of the energy expression, but leads to the question, how does this idea generalize for higher-order multipole energies? |
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In method 1, the procedure that we follow is based on the number of derivatives need for each energy, force, or torque. That is, |
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a quadrupole-quadrupole interaction energy will have four derivatives, |
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$\partial^4/\partial r_\alpha \partial r_\beta \partial r_\gamma \partial r_\delta$, |
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and the force or torque will bring in yet another derivative. |
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We thus want shifted energy expressions to include terms so that all energies, forces, and torques |
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are zero at $r=r_c$. In each case, we will subtract off a function $f_n^{\text{shift}}(r)$ from the |
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kernel $f(r)=1/r$. The index $n$ indicates the number of derivatives to be taken when |
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deriving a given multipole energy. |
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We choose a function with guaranteed smooth derivatives --- a truncated Taylor series of the function |
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$f(r)$, e.g., |
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% |
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\begin{equation} |
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f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)} \Big \lvert _{r_c} . |
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\end{equation} |
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The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$. |
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Thus, for $f(r)=1/r$, we find |
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\begin{equation} |
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f_1(r)=\frac{1}{r}- \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} - \frac{(r-r_c)^2}{r_c^3} . |
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\end{equation} |
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% |
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Continuing with the example of a charge $\bf a$ interacting with a dipole $\bf b$, we write |
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\begin{equation} |
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U_{C_{\bf a}D_{\bf b}}(r)= |
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\frac{C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} \frac {\partial f_1(r) }{\partial r_\alpha} |
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=\frac{ C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} |
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\frac {r_\alpha}{r} \frac {\partial f_1(r)}{\partial r} . |
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\end{equation} |
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The force that dipole $\bf b$ puts on charge $\bf a$ is |
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\begin{equation} |
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F_{C_{\bf a}D_{\bf b}\beta} =\frac{ C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} |
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\left[ \frac{\delta_{\alpha\beta}}{r} \frac {\partial}{\partial r} + |
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\frac{r_\alpha r_\beta}{r^2} |
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\left( -\frac{1}{r} \frac {\partial} {\partial r} |
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+ \frac {\partial ^2} {\partial r^2} \right) \right] f_1(r) . |
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\end{equation} |
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% |
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For $f(r)=1/r$, we find |
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\begin{equation} |
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F_{C_{\bf a}D_{\bf b}\beta} = |
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\frac{C_{\bf a} D_{{\bf b}\beta} }{4\pi \epsilon_0r} |
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\left[ -\frac{1}{r^2}+\frac{1}{r_c^2}-\frac{2(r-r_c)}{r_c^3} \right] |
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+\frac{C_{\bf a} D_{{\bf b}\alpha}r_\alpha r_\beta }{4\pi \epsilon_0} |
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\left[ \frac{3}{r^5}-\frac{3}{r^3r_c^2} \right] . |
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\end{equation} |
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% |
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This expansion shows the expected $1/r^3$ dependence of the force. |
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In general, we write |
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\begin{equation} |
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U=\frac{1}{4\pi \epsilon_0} (\text{prefactor}) (\text{derivatives}) f_n(r) |
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\label{generic} |
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\end{equation} |
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% |
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where $n=0$ for charge-charge, $n=1$ for charge-dipole, $n=2$ for charge-quadrupole |
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and dipole-dipole, $n=3$ for dipole-quadrupole, and $n=4$ for quadrupole-quadrupole. |
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An example is the case of quadrupole-quadrupole for which the $\text{prefactor}$ is |
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$Q_{{\bf a}\alpha\beta}Q_{{\bf b}\gamma\delta}$ and the derivatives are |
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$\partial^4/\partial r_\alpha \partial r_\beta \partial r_\gamma \partial r_\delta$, with |
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implied summation combining the space indices. |
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To apply this method to the smeared-charge approach, |
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we write $f(r)=\text{erfc}(\alpha r)/r$. By using one function $f(r)$ for both |
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approaches, we simplify the tabulation of equations used. Because |
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of the many derivatives that are taken, the algebra is tedious and are summarized |
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in Appendices A and B. |
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\subsection{Shifting the force, method 2} |
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Note the method used in the previous subsection to shift a force is basically that of using |
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a truncated Taylor Series in the radius $r$. An alternate method exists, best explained by |
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writing one shifted formula for all interaction energies $U(r)$: |
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\begin{equation} |
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U^{\text{shift}}(r)=U(r)-U(r_c)-(r-r_c)\hat{r}\cdot \nabla U(r) \Big \lvert _{r_c} . |
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\end{equation} |
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Note that this method uses only the linear term, $(r-r_c)$ in the Taylor series, no higher order terms |
314 |
|
|
$(r-r_c)^n$ appear. The primary difference between methods 1 and 2 originates |
315 |
|
|
with the stage in the derivation where the Taylor Series is applied; in method 1, it is applied to the |
316 |
|
|
kernel. In method 2, it is applied to individual interaction energies of the multipole expansion. |
317 |
|
|
Terms from this method thus have the general form: |
318 |
|
|
\begin{equation} |
319 |
|
|
U=\frac{1}{4\pi \epsilon_0}\sum (\text{angular factor}) (\text{radial factor}). |
320 |
|
|
\label{generic2} |
321 |
|
|
\end{equation} |
322 |
|
|
|
323 |
|
|
Results for both methods can be summarized using the form of Eq.~(\ref{generic2}) |
324 |
|
|
and are listed in Table I below. |
325 |
|
|
|
326 |
|
|
\subsection{\label{sec:level2}Body and space axes} |
327 |
|
|
|
328 |
|
|
Up to this point, all energies and forces have been written in terms of fixed space |
329 |
|
|
coordinates $x$, $y$, $z$. Interaction energies are computed from the generic formulas Eq.~(\ref{generic}) and ~(\ref{generic2}) which |
330 |
|
|
combine prefactors with radial functions. But because objects |
331 |
|
|
$\bf a$ and $\bf b$ both translate and rotate as part of a MD simulation, |
332 |
|
|
it is desirable to contract all $r$-dependent terms with dipole and quadrupole |
333 |
|
|
moments expressed in terms of their body axes. |
334 |
|
|
Since the interaction energy expressions will be used to derive both forces and torques, |
335 |
|
|
we follow here the development of Allen and Germano, which was itself based on an |
336 |
|
|
earlier paper by Price \em et al.\em |
337 |
|
|
|
338 |
|
|
Denote body axes for objects $\bf a$ and $\bf b$ by unit vectors |
339 |
|
|
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$ referring to a convenient |
340 |
|
|
set of inertial body axes. (Note, these body axes are generally not the same as those for which the |
341 |
|
|
quadrupole moment is diagonal.) Then, |
342 |
|
|
% |
343 |
|
|
\begin{eqnarray} |
344 |
|
|
\hat{a}_m= a_{mx}\hat{x} + a_{my}\hat{y} + a_{mz}\hat{z} \\ |
345 |
|
|
\hat{b}_m= b_{mx}\hat{x} + b_{my}\hat{y} + b_{mz}\hat{z} . |
346 |
|
|
\end{eqnarray} |
347 |
|
|
Allen and Germano define matrices $\hat{\mathbf {a}}$ |
348 |
|
|
and $\hat{\mathbf {b}}$ using these unit vectors: |
349 |
|
|
\begin{eqnarray} |
350 |
|
|
\hat{\mathbf {a}} = |
351 |
|
|
\begin{pmatrix} |
352 |
|
|
\hat{a}_1 \\ |
353 |
|
|
\hat{a}_2 \\ |
354 |
|
|
\hat{a}_3 |
355 |
|
|
\end{pmatrix} |
356 |
|
|
= |
357 |
|
|
\begin{pmatrix} |
358 |
|
|
a_{1x} \quad a_{1y} \quad a_{1z} \\ |
359 |
|
|
a_{2x} \quad a_{2y} \quad a_{2z} \\ |
360 |
|
|
a_{3x} \quad a_{3y} \quad a_{3z} |
361 |
|
|
\end{pmatrix}\\ |
362 |
|
|
\hat{\mathbf {b}} = |
363 |
|
|
\begin{pmatrix} |
364 |
|
|
\hat{b}_1 \\ |
365 |
|
|
\hat{b}_2 \\ |
366 |
|
|
\hat{b}_3 |
367 |
|
|
\end{pmatrix} |
368 |
|
|
= |
369 |
|
|
\begin{pmatrix} |
370 |
|
|
b_{1x}\quad b_{1y} \quad b_{1z} \\ |
371 |
|
|
b_{2x} \quad b_{2y} \quad b_{2z} \\ |
372 |
|
|
b_{3x} \quad b_{3y} \quad b_{3z} |
373 |
|
|
\end{pmatrix} . |
374 |
|
|
\end{eqnarray} |
375 |
|
|
% |
376 |
|
|
These matrices convert from space-fixed $(xyz)$ to object-fixed $(123)$ coordinates. |
377 |
|
|
All contractions of prefactors with derivatives of functions can be written in terms of these matrices. |
378 |
|
|
It proves to be equally convenient to just write any contraction in terms of unit vectors |
379 |
|
|
$\hat{r}$, $\hat{a}_m$, and $\hat{b}_n$. |
380 |
|
|
We have found it useful to write angular-dependent terms in three different fashions, |
381 |
|
|
illustrated by the following |
382 |
|
|
three examples from the interaction-energy expressions: |
383 |
|
|
% |
384 |
|
|
\begin{eqnarray} |
385 |
|
|
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
386 |
|
|
=D_{\bf {a}\alpha} D_{\bf {b}\alpha}= |
387 |
|
|
\sum_{mn} {D_{\mathbf{a}m} \hat{a}_m \cdot \hat{b}_n D_{\mathbf{b}n}} \\ |
388 |
|
|
r^2 \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right)= |
389 |
|
|
r_\alpha Q_{\bf b \alpha \beta} r_\beta = r^2 |
390 |
|
|
\sum_{mn}(\hat{r} \cdot \hat{b}_m) Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \\ |
391 |
|
|
r ( \mathbf{D}_{\mathbf{a}} \cdot |
392 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r})= |
393 |
|
|
D_{\bf {a}\alpha} Q_{\bf b \alpha \beta} r_\beta |
394 |
|
|
=r \sum_{lmn} D_{\mathbf{a}l} (\hat{a}_l \cdot \hat{b}_m ) Q_{\mathbf{b}mn} |
395 |
|
|
(\hat{b}_n \cdot \hat{r}) . |
396 |
|
|
\end{eqnarray} |
397 |
|
|
% |
398 |
|
|
[Dan, perhaps there are better examples to show here.] |
399 |
|
|
|
400 |
|
|
In each line, the first two terms are written using space coordinates. The first form is standard |
401 |
|
|
in the chemistry literature, and the second is ``physicist style'' using implied summation notation. The third |
402 |
|
|
form shows explicitly sums over body indices and the dot products now indicate contractions using space indices. |
403 |
|
|
We find the first form to be useful in writing equations prior to converting to computer code. The second form is helpful |
404 |
|
|
in derivations of the interaction energy expressions. The third one is specifically helpful when deriving forces and torques, as will |
405 |
|
|
be discussed below. |
406 |
|
|
|
407 |
|
|
\section{Energies, forces, and torques} |
408 |
|
|
\subsection{Interaction energies} |
409 |
|
|
|
410 |
|
|
We now list multipole interaction energies for the four types of approximation. |
411 |
|
|
A ``generic'' set of radial functions is introduced so to be able to present the results in Table I. This set of |
412 |
|
|
equations is written in terms of space coordinates: |
413 |
|
|
|
414 |
|
|
% Energy in space coordinate form ---------------------------------------------------------------------------------------------- |
415 |
|
|
% |
416 |
|
|
% |
417 |
|
|
% u ca cb |
418 |
|
|
% |
419 |
|
|
\begin{equation} |
420 |
|
|
U_{C_{\bf a}C_{\bf b}}(r)= |
421 |
|
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) \label{uchch} |
422 |
|
|
\end{equation} |
423 |
|
|
% |
424 |
|
|
% u ca db |
425 |
|
|
% |
426 |
|
|
\begin{equation} |
427 |
|
|
U_{C_{\bf a}D_{\bf b}}(r)= |
428 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} \left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) v_{11}(r) |
429 |
|
|
\label{uchdip} |
430 |
|
|
\end{equation} |
431 |
|
|
% |
432 |
|
|
% u ca qb |
433 |
|
|
% |
434 |
|
|
\begin{equation} |
435 |
|
|
U_{C_{\bf a}Q_{\bf b}}(r)= |
436 |
|
|
\frac{C_{\bf a }}{4\pi \epsilon_0} \Bigl[ \text{Tr}Q_{\bf b} v_{21}(r) |
437 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
438 |
|
|
\label{uchquad} |
439 |
|
|
\end{equation} |
440 |
|
|
% |
441 |
|
|
% u da cb |
442 |
|
|
% |
443 |
|
|
\begin{equation} |
444 |
|
|
U_{D_{\bf a}C_{\bf b}}(r)= |
445 |
|
|
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
446 |
|
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) v_{11}(r) \label{udipch} |
447 |
|
|
\end{equation} |
448 |
|
|
% |
449 |
|
|
% u da db |
450 |
|
|
% |
451 |
|
|
\begin{equation} |
452 |
|
|
U_{D_{\bf a}D_{\bf b}}(r)= |
453 |
|
|
-\frac{1}{4\pi \epsilon_0} \Bigr[ \left( \mathbf{D}_{\mathbf {a}} \cdot |
454 |
|
|
\mathbf{D}_{\mathbf{b}} \right) v_{21}(r) |
455 |
|
|
+\left( \mathbf{D}_{\mathbf {a}} \cdot \hat{r} \right) |
456 |
|
|
\left( \mathbf{D}_{\mathbf {b}} \cdot \hat{r} \right) |
457 |
|
|
v_{22}(r) \Bigr] |
458 |
|
|
\label{udipdip} |
459 |
|
|
\end{equation} |
460 |
|
|
% |
461 |
|
|
% u da qb |
462 |
|
|
% |
463 |
|
|
\begin{equation} |
464 |
|
|
\begin{split} |
465 |
|
|
% 1 |
466 |
|
|
U_{D_{\bf a}Q_{\bf b}}(r)&= |
467 |
|
|
-\frac{1}{4\pi \epsilon_0} \Bigl[ |
468 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
469 |
|
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
470 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
471 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r} ) \Bigr] v_{31}(r) \\ |
472 |
|
|
% 2 |
473 |
|
|
&-\frac{1}{4\pi \epsilon_0} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
474 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{32}(r) |
475 |
|
|
\label{udipquad} |
476 |
|
|
\end{split} |
477 |
|
|
\end{equation} |
478 |
|
|
% |
479 |
|
|
% u qa cb |
480 |
|
|
% |
481 |
|
|
\begin{equation} |
482 |
|
|
U_{Q_{\bf a}C_{\bf b}}(r)= |
483 |
|
|
\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\bf a} v_{21}(r) |
484 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
485 |
|
|
\label{uquadch} |
486 |
|
|
\end{equation} |
487 |
|
|
% |
488 |
|
|
% u qa db |
489 |
|
|
% |
490 |
|
|
\begin{equation} |
491 |
|
|
\begin{split} |
492 |
|
|
%1 |
493 |
|
|
U_{Q_{\bf a}D_{\bf b}}(r)&= |
494 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
495 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} |
496 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
497 |
|
|
+2 ( \mathbf{D}_{\mathbf{b}} \cdot |
498 |
|
|
\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
499 |
|
|
% 2 |
500 |
|
|
+\frac{1}{4\pi \epsilon_0} |
501 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
502 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{32}(r) |
503 |
|
|
\label{uquaddip} |
504 |
|
|
\end{split} |
505 |
|
|
\end{equation} |
506 |
|
|
% |
507 |
|
|
% u qa qb |
508 |
|
|
% |
509 |
|
|
\begin{equation} |
510 |
|
|
\begin{split} |
511 |
|
|
%1 |
512 |
|
|
U_{Q_{\bf a}Q_{\bf b}}(r)&= |
513 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
514 |
|
|
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
515 |
|
|
+2 \text{Tr} \left( |
516 |
|
|
\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \right) \Bigr] v_{41}(r) |
517 |
|
|
\\ |
518 |
|
|
% 2 |
519 |
|
|
&+ \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
520 |
|
|
\left( \hat{r} \cdot |
521 |
|
|
\mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) |
522 |
|
|
+\text{Tr}\mathbf{Q}_{\mathbf{b}} |
523 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} |
524 |
|
|
\cdot \hat{r} \right) +4 (\hat{r} \cdot |
525 |
|
|
\mathbf{Q}_{{\mathbf a}}\cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
526 |
|
|
\Bigr] v_{42}(r) |
527 |
|
|
\\ |
528 |
|
|
% 4 |
529 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
530 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) |
531 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{43}(r). |
532 |
|
|
\label{uquadquad} |
533 |
|
|
\end{split} |
534 |
|
|
\end{equation} |
535 |
|
|
|
536 |
|
|
|
537 |
|
|
% |
538 |
|
|
% |
539 |
|
|
% TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
540 |
|
|
% |
541 |
|
|
|
542 |
|
|
\begin{table*} |
543 |
|
|
\caption{\label{tab:tableenergy}Radial functions used in the energy and torque equations. Functions |
544 |
|
|
used in this table are defined in Appendices B and C.} |
545 |
|
|
\begin{ruledtabular} |
546 |
|
|
\begin{tabular}{cccc} |
547 |
|
|
Generic&Coulomb&Method 1&Method 2 |
548 |
|
|
\\ \hline |
549 |
|
|
% |
550 |
|
|
% |
551 |
|
|
% |
552 |
|
|
%Ch-Ch& |
553 |
|
|
$v_{01}(r)$ & |
554 |
|
|
$\frac{1}{r}$ & |
555 |
|
|
$f_0(r)$ & |
556 |
|
|
$f(r)-f(r_c)-(r-r_c)g(r_c)$ |
557 |
|
|
\\ |
558 |
|
|
% |
559 |
|
|
% |
560 |
|
|
% |
561 |
|
|
%Ch-Di& |
562 |
|
|
$v_{11}(r)$ & |
563 |
|
|
$-\frac{1}{r^2}$ & |
564 |
|
|
$g_1(r)$ & |
565 |
|
|
$g(r)-g(r_c)-(r-r_c)h(r_c)$ \\ |
566 |
|
|
% |
567 |
|
|
% |
568 |
|
|
% |
569 |
|
|
%Ch-Qu/Di-Di& |
570 |
|
|
$v_{21}(r)$ & |
571 |
|
|
$-\frac{1}{r^3} $ & |
572 |
|
|
$\frac{g_2(r)}{r} $ & |
573 |
|
|
$\frac{g(r)}{r}-\frac{g(r_c)}{r_c} -(r-r_c) |
574 |
|
|
\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right)$ \\ |
575 |
|
|
$v_{22}(r)$ & |
576 |
|
|
$\frac{3}{r^3} $ & |
577 |
|
|
$\left(-\frac{g_2(r)}{r} + h_2(r) \right)$ & |
578 |
|
|
$\left(-\frac{g(r)}{r}+h(r) \right) |
579 |
|
|
-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right) $ \\ |
580 |
|
|
&&&$ -(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$ |
581 |
|
|
\\ |
582 |
|
|
% |
583 |
|
|
% |
584 |
|
|
% |
585 |
|
|
%Di-Qu & |
586 |
|
|
$v_{31}(r)$ & |
587 |
|
|
$\frac{3}{r^4} $ & |
588 |
|
|
$\left(-\frac{g_3(r)}{r^2} + \frac{h_3(r)}{r} \right)$ & |
589 |
|
|
$\left( -\frac{g(r)}{r^2}+\frac{h(r)}{r} \right) |
590 |
|
|
-\left(-\frac{g(r_c)}{r_c^2}+\frac{h(r_c)}{r_c} \right) $\\ |
591 |
|
|
&&&$ -(r-r_c) \left(\frac{2g(r_c)}{r_c^3}-\frac{2h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ |
592 |
|
|
\\ |
593 |
|
|
% |
594 |
|
|
$v_{32}(r)$ & |
595 |
|
|
$-\frac{15}{r^4} $ & |
596 |
|
|
$\left( \frac{3g_3(r)}{r^2} - \frac{3h_3(r)}{r} + s_3(r) \right)$ & |
597 |
|
|
$\left( \frac{3g(r)}{r^2} - \frac{3h(r)}{r} + s(r) \right) |
598 |
|
|
- \left( \frac{3g(r_c)}{r_c^2} - \frac{3h(r_c)}{r_c} + s(r_c) \right)$ \\ |
599 |
|
|
&&&$ -(r-r_c) \left( \frac{-6g(r_c)}{r_c^3}+\frac{6h(r_c)}{r_c^2}-\frac{3s(r_c)}{r_c}+t(r_c) \right)$ |
600 |
|
|
\\ |
601 |
|
|
% |
602 |
|
|
% |
603 |
|
|
% |
604 |
|
|
%Qu-Qu& |
605 |
|
|
$v_{41}(r)$ & |
606 |
|
|
$\frac{3}{r^5} $ & |
607 |
|
|
$\left(-\frac{g_4(r)}{r^3} +\frac{h_4(r)}{r^2} \right) $ & |
608 |
|
|
$\left( -\frac{g(r)}{r^3} + \frac{h(r)}{r^2} \right) |
609 |
|
|
- \left( -\frac{g(r_c)}{r_c^3} + \frac{h(r_c)}{r_c^2} \right)$ \\ |
610 |
|
|
&&&$ -(r-r_c) \left( \frac{3g(r_c)}{r_c^4}-\frac{3h(r_c)}{r_c^3}+\frac{s(r_c)}{r_c^2} \right)$ |
611 |
|
|
\\ |
612 |
|
|
% 2 |
613 |
|
|
$v_{42}(r)$ & |
614 |
|
|
$- \frac{15}{r^5} $ & |
615 |
|
|
$\left( \frac{3g_4(r)}{r^3} - \frac{3h_4(r)}{r^2}+\frac{s_4(r)}{r} \right)$ & |
616 |
|
|
$\left( \frac{3g(r)}{r^3} - \frac{3h(r)}{r^2}+\frac{s(r)}{r} \right) |
617 |
|
|
-\left( \frac{3g(r_c)}{r_c^3} - \frac{3h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ \\ |
618 |
|
|
&&&$ -(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
619 |
|
|
-\frac{4s(r_c)}{r_c^2} + \frac{t(r_c)}{r_c}\right)$ |
620 |
|
|
\\ |
621 |
|
|
% 3 |
622 |
|
|
$v_{43}(r)$ & |
623 |
|
|
$ \frac{105}{r^5} $ & |
624 |
|
|
$\left(-\frac{15g_4(r)}{r^3}+\frac{15h_4(r)}{r^2}-\frac{6s_4(r)}{r} + t_4(r)\right) $ & |
625 |
|
|
$\left(-\frac{15g(r)}{r^3}+\frac{15h(r)}{r^2}-\frac{6s(r)}{r} + t(r)\right)$ \\ |
626 |
|
|
&&&$ -\left(-\frac{15g(r_c)}{r_c^3}+\frac{15h(r_c)}{r_c^2}-\frac{6s(r_c)}{r_c} + t(r_c)\right)$ \\ |
627 |
|
|
&&&$ -(r-r_c)\left(\frac{45g(r_c)}{r_c^4}-\frac{45h(r_c)}{r_c^3}+\frac{21s(r_c)}{r_c^2} |
628 |
|
|
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ |
629 |
|
|
\end{tabular} |
630 |
|
|
\end{ruledtabular} |
631 |
|
|
\end{table*} |
632 |
|
|
% |
633 |
|
|
% |
634 |
|
|
% FORCE TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
635 |
|
|
% |
636 |
|
|
|
637 |
|
|
\begin{table} |
638 |
|
|
\caption{\label{tab:tableFORCE}Radial functions used in the force equations.} |
639 |
|
|
\begin{ruledtabular} |
640 |
|
|
\begin{tabular}{cc} |
641 |
|
|
Generic&Method 1 or Method 2 |
642 |
|
|
\\ \hline |
643 |
|
|
% |
644 |
|
|
% |
645 |
|
|
% |
646 |
|
|
$w_a(r)$& |
647 |
|
|
$\frac{d v_{01}}{dr}$ \\ |
648 |
|
|
% |
649 |
|
|
% |
650 |
|
|
$w_b(r)$ & |
651 |
|
|
$\frac{d v_{11}}{dr} - \frac{v_{11}(r)}{r} $ \\ |
652 |
|
|
% |
653 |
|
|
$w_c(r)$ & |
654 |
|
|
$\frac{v_{11}(r)}{r}$ \\ |
655 |
|
|
% |
656 |
|
|
% |
657 |
|
|
$w_d(r)$& |
658 |
|
|
$\frac{d v_{21}}{dr}$ \\ |
659 |
|
|
% |
660 |
|
|
$w_e(r)$ & |
661 |
|
|
$\frac{v_{22}(r)}{r}$ \\ |
662 |
|
|
% |
663 |
|
|
% |
664 |
|
|
$w_f(r)$& |
665 |
|
|
$\frac{d v_{22}}{dr} - \frac{2v_{22}(r)}{r}$\\ |
666 |
|
|
% |
667 |
|
|
$w_g(r)$& |
668 |
|
|
$\frac{v_{31}(r)}{r}$\\ |
669 |
|
|
% |
670 |
|
|
$w_h(r)$ & |
671 |
|
|
$\frac{d v_{31}}{dr} -\frac{v_{31}(r)}{r}$\\ |
672 |
|
|
% 2 |
673 |
|
|
$w_i(r)$ & |
674 |
|
|
$\frac{v_{32}(r)}{r}$ \\ |
675 |
|
|
% |
676 |
|
|
$w_j(r)$ & |
677 |
|
|
$\frac{d v_{32}}{dr} - \frac{3v_{32}}{r}$ \\ |
678 |
|
|
% |
679 |
|
|
$w_k(r)$ & |
680 |
|
|
$\frac{d v_{41}}{dr} $ \\ |
681 |
|
|
% |
682 |
|
|
$w_l(r)$ & |
683 |
|
|
$\frac{d v_{42}}{dr} -\frac{2v_{42}(r)}{r}$ \\ |
684 |
|
|
% |
685 |
|
|
$w_m(r)$ & |
686 |
|
|
$\frac{d v_{43}}{dr} -\frac{4v_{43}(r)}{r}$ \\ |
687 |
|
|
% |
688 |
|
|
$w_n(r)$ & |
689 |
|
|
$\frac{v_{42}(r)}{r}$ \\ |
690 |
|
|
% |
691 |
|
|
$w_o(r)$ & |
692 |
|
|
$\frac{v_{43}(r)}{r}$ \\ |
693 |
|
|
% |
694 |
|
|
|
695 |
|
|
\end{tabular} |
696 |
|
|
\end{ruledtabular} |
697 |
|
|
\end{table} |
698 |
|
|
% |
699 |
|
|
% |
700 |
|
|
% |
701 |
|
|
|
702 |
|
|
\subsection{Forces} |
703 |
|
|
|
704 |
|
|
The force $\mathbf{F}_{\bf a}$ on $\bf{a}$ due to $\bf{b}$ is the negative of |
705 |
|
|
the force $\mathbf{F}_{\bf b}$ on $\bf{b}$ due to $\bf{a}$. For a simple charge-charge |
706 |
|
|
interaction, these forces will point along the $\pm \hat{r}$ directions, where |
707 |
|
|
$\mathbf{r}=\mathbf{r}_b - \mathbf{r}_a $. Thus |
708 |
|
|
% |
709 |
|
|
\begin{equation} |
710 |
|
|
F_{\bf a \alpha} = \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}}{\partial r} |
711 |
|
|
\quad \text{and} \quad F_{\bf b \alpha} |
712 |
|
|
= - \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}} {\partial r} . |
713 |
|
|
\end{equation} |
714 |
|
|
% |
715 |
|
|
The concept of obtaining a force from an energy by taking a gradient is the same for |
716 |
|
|
higher-order multipole interactions, the trick is to make sure that all |
717 |
|
|
$r$-dependent derivatives are considered. |
718 |
|
|
As is pointed out by Allen and Germano, this is straightforward if the |
719 |
|
|
interaction energies are written recognizing explicit |
720 |
|
|
$\hat{r}$ and body axes ($\hat{a}_m$, $\hat{b}_n$) dependences: |
721 |
|
|
% |
722 |
|
|
\begin{equation} |
723 |
|
|
U(r,\{\hat{a}_m \cdot \hat{r} \}, |
724 |
|
|
\{\hat{b}_n\cdot \hat{r} \} |
725 |
|
|
\{\hat{a}_m \cdot \hat{b}_n \}) . |
726 |
|
|
\label{ugeneral} |
727 |
|
|
\end{equation} |
728 |
|
|
% |
729 |
|
|
Then, |
730 |
|
|
% |
731 |
|
|
\begin{equation} |
732 |
|
|
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
733 |
|
|
= \frac{\partial U}{\partial r} \hat{r} |
734 |
|
|
+ \sum_m \left[ |
735 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
736 |
|
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
737 |
|
|
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
738 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
739 |
|
|
\right] \label{forceequation}. |
740 |
|
|
\end{equation} |
741 |
|
|
% |
742 |
|
|
Note our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ is opposite |
743 |
|
|
that of Allen and Germano. In simplifying the algebra, we also use: |
744 |
|
|
% |
745 |
|
|
\begin{eqnarray} |
746 |
|
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
747 |
|
|
= \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
748 |
|
|
\right) \\ |
749 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
750 |
|
|
= \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
751 |
|
|
\right) . |
752 |
|
|
\end{eqnarray} |
753 |
|
|
% |
754 |
|
|
We list below the force equations written in terms of space coordinates. The |
755 |
|
|
radial functions used in the two methods are listed in Table II. |
756 |
|
|
% |
757 |
|
|
%SPACE COORDINATES FORCE EQUTIONS |
758 |
|
|
% |
759 |
|
|
% ************************************************************************** |
760 |
|
|
% f ca cb |
761 |
|
|
% |
762 |
|
|
\begin{equation} |
763 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} = |
764 |
|
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} w_a(r) \hat{r} |
765 |
|
|
\end{equation} |
766 |
|
|
% |
767 |
|
|
% |
768 |
|
|
% |
769 |
|
|
\begin{equation} |
770 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} = |
771 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} \Bigl[ |
772 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{b}} \right) |
773 |
|
|
w_b(r) \hat{r} |
774 |
|
|
+ \mathbf{D}_{\mathbf{b}} w_c(r) \Bigr] |
775 |
|
|
\end{equation} |
776 |
|
|
% |
777 |
|
|
% |
778 |
|
|
% |
779 |
|
|
\begin{equation} |
780 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} = |
781 |
|
|
\frac{C_{\bf a }}{4\pi \epsilon_0} \Bigr[ |
782 |
|
|
\text{Tr}\mathbf{Q}_{\bf b} w_d(r) \hat{r} |
783 |
|
|
+ 2 \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} w_e(r) |
784 |
|
|
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
785 |
|
|
\end{equation} |
786 |
|
|
% |
787 |
|
|
% |
788 |
|
|
% |
789 |
|
|
\begin{equation} |
790 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
791 |
|
|
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} \Bigl[ |
792 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) w_b(r) \hat{r} |
793 |
|
|
+ \mathbf{D}_{\mathbf{a}} w_c(r) \Bigr] |
794 |
|
|
\end{equation} |
795 |
|
|
% |
796 |
|
|
% |
797 |
|
|
% |
798 |
|
|
\begin{equation} |
799 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} = |
800 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
801 |
|
|
- \mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} w_d(r) \hat{r} |
802 |
|
|
+ \left( \mathbf{D}_{\mathbf {a}} |
803 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
804 |
|
|
+ \mathbf{D}_{\mathbf {b}} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) \right) w_e(r) |
805 |
|
|
% 2 |
806 |
|
|
- \left( \hat{r} \cdot \mathbf{D}_{\mathbf {a}} \right) |
807 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf {b}} \right) w_f(r) \hat{r} \Bigr] |
808 |
|
|
\end{equation} |
809 |
|
|
% |
810 |
|
|
% |
811 |
|
|
% |
812 |
|
|
\begin{equation} |
813 |
|
|
\begin{split} |
814 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
815 |
|
|
& - \frac{1}{4\pi \epsilon_0} \Bigl[ |
816 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} \mathbf{ D}_{\mathbf{a}} |
817 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \cdot |
818 |
|
|
\mathbf{Q}_{\mathbf{b}} \Bigr] w_g(r) |
819 |
|
|
- \frac{1}{4\pi \epsilon_0} \Bigl[ |
820 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
821 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) |
822 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
823 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
824 |
|
|
% 3 |
825 |
|
|
& - \frac{1}{4\pi \epsilon_0} \Bigl[\mathbf{ D}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
826 |
|
|
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \Bigr] |
827 |
|
|
w_i(r) |
828 |
|
|
% 4 |
829 |
|
|
-\frac{1}{4\pi \epsilon_0} |
830 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) |
831 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) w_j(r) \hat{r} |
832 |
|
|
\end{split} |
833 |
|
|
\end{equation} |
834 |
|
|
% |
835 |
|
|
% |
836 |
|
|
\begin{equation} |
837 |
|
|
\mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
838 |
|
|
\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigr[ |
839 |
|
|
\text{Tr}\mathbf{Q}_{\bf a} w_d(r) \hat{r} |
840 |
|
|
+ 2 \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} w_e(r) |
841 |
|
|
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
842 |
|
|
\end{equation} |
843 |
|
|
% |
844 |
|
|
\begin{equation} |
845 |
|
|
\begin{split} |
846 |
|
|
\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
847 |
|
|
&\frac{1}{4\pi \epsilon_0} \Bigl[ |
848 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} \mathbf{D}_{\mathbf{b}} |
849 |
|
|
+2 \mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} \Bigr] w_g(r) |
850 |
|
|
% 2 |
851 |
|
|
+ \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
852 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
853 |
|
|
+2 (\mathbf{D}_{\mathbf{b}} \cdot |
854 |
|
|
\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
855 |
|
|
% 3 |
856 |
|
|
&+ \frac{1}{4\pi \epsilon_0} \Bigl[ \mathbf{D}_{\mathbf{b}} |
857 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
858 |
|
|
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
859 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \Bigr] w_i(r) |
860 |
|
|
% 4 |
861 |
|
|
+\frac{1}{4\pi \epsilon_0} |
862 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
863 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) w_j(r) \hat{r} |
864 |
|
|
\end{split} |
865 |
|
|
\end{equation} |
866 |
|
|
% |
867 |
|
|
% |
868 |
|
|
% |
869 |
|
|
\begin{equation} |
870 |
|
|
\begin{split} |
871 |
|
|
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
872 |
|
|
+\frac{1}{4\pi \epsilon_0} \Bigl[ |
873 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} \hat{r} |
874 |
|
|
+ 2 \text{Tr} ( \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_k(r) \hat{r} \\ |
875 |
|
|
% 2 |
876 |
|
|
+\frac{1}{4\pi \epsilon_0} \Bigl[ |
877 |
|
|
2\text{Tr}\mathbf{Q}_{\mathbf{b}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
878 |
|
|
+ 2\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) |
879 |
|
|
% 3 |
880 |
|
|
+4 (\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
881 |
|
|
+ 4(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}}) \Bigr] w_n(r) \\ |
882 |
|
|
% 4 |
883 |
|
|
+ \frac{1}{4\pi \epsilon_0} \Bigl[ |
884 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
885 |
|
|
+ \text{Tr}\mathbf{Q}_{\mathbf{b}} |
886 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
887 |
|
|
% 5 |
888 |
|
|
+4 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot |
889 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
890 |
|
|
% |
891 |
|
|
+ \frac{1}{4\pi \epsilon_0} \Bigl[ |
892 |
|
|
+ 2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
893 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
894 |
|
|
%6 |
895 |
|
|
+2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
896 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_o(r) \\ |
897 |
|
|
% 7 |
898 |
|
|
+ \frac{1}{4\pi \epsilon_0} |
899 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
900 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) w_m(r) \hat{r} |
901 |
|
|
\end{split} |
902 |
|
|
\end{equation} |
903 |
|
|
% |
904 |
|
|
% |
905 |
|
|
% TORQUES SECTION ----------------------------------------------------------------------------------------- |
906 |
|
|
% |
907 |
|
|
\subsection{Torques} |
908 |
|
|
|
909 |
|
|
Following again Allen and Germano, when energies are written in the form |
910 |
|
|
of Eq.~({\ref{ugeneral}), then torques can be expressed as: |
911 |
|
|
% |
912 |
|
|
\begin{eqnarray} |
913 |
|
|
\mathbf{\tau}_{\bf a} = |
914 |
|
|
\sum_m |
915 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
916 |
|
|
( \hat{r} \times \hat{a}_m ) |
917 |
|
|
-\sum_{mn} |
918 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
919 |
|
|
(\hat{a}_m \times \hat{b}_n) \\ |
920 |
|
|
% |
921 |
|
|
\mathbf{\tau}_{\bf b} = |
922 |
|
|
\sum_m |
923 |
|
|
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
924 |
|
|
( \hat{r} \times \hat{b}_m) |
925 |
|
|
+\sum_{mn} |
926 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
927 |
|
|
(\hat{a}_m \times \hat{b}_n) . |
928 |
|
|
\end{eqnarray} |
929 |
|
|
% |
930 |
|
|
% |
931 |
|
|
Here we list the torque equations written in terms of space coordinates. |
932 |
|
|
% |
933 |
|
|
% |
934 |
|
|
% |
935 |
|
|
\begin{equation} |
936 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} = |
937 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} (\hat{r} \times \mathbf{D}_{\mathbf{b}}) v_{11}(r) |
938 |
|
|
\end{equation} |
939 |
|
|
% |
940 |
|
|
% |
941 |
|
|
% |
942 |
|
|
\begin{equation} |
943 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} = |
944 |
|
|
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
945 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{22}(r) |
946 |
|
|
\end{equation} |
947 |
|
|
% |
948 |
|
|
% |
949 |
|
|
% |
950 |
|
|
\begin{equation} |
951 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
952 |
|
|
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
953 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{a}}) v_{11}(r) |
954 |
|
|
\end{equation} |
955 |
|
|
% |
956 |
|
|
% |
957 |
|
|
% |
958 |
|
|
\begin{equation} |
959 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} = |
960 |
|
|
\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
961 |
|
|
% 2 |
962 |
|
|
-\frac{1}{4\pi \epsilon_0} |
963 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf {a}} ) |
964 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
965 |
|
|
\end{equation} |
966 |
|
|
% |
967 |
|
|
% |
968 |
|
|
% |
969 |
|
|
\begin{equation} |
970 |
|
|
\mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} = |
971 |
|
|
-\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
972 |
|
|
% 2 |
973 |
|
|
+\frac{1}{4\pi \epsilon_0} |
974 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf {a}} ) |
975 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
976 |
|
|
\end{equation} |
977 |
|
|
% |
978 |
|
|
% |
979 |
|
|
% |
980 |
|
|
\begin{equation} |
981 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
982 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
983 |
|
|
-\text{Tr}\mathbf{Q}_{\mathbf{b}} |
984 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
985 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \times |
986 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
987 |
|
|
\Bigr] v_{31}(r) |
988 |
|
|
% 3 |
989 |
|
|
-\frac{1}{4\pi \epsilon_0} |
990 |
|
|
\ (\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
991 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{32}(r) |
992 |
|
|
\end{equation} |
993 |
|
|
% |
994 |
|
|
% |
995 |
|
|
% |
996 |
|
|
\begin{equation} |
997 |
|
|
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} = |
998 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
999 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \times |
1000 |
|
|
\hat{r} |
1001 |
|
|
-2 \mathbf{D}_{\mathbf{a}} \times |
1002 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1003 |
|
|
\Bigr] v_{31}(r) |
1004 |
|
|
% 2 |
1005 |
|
|
+\frac{2}{4\pi \epsilon_0} |
1006 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}}) |
1007 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}}) \times \hat{r} v_{32}(r) |
1008 |
|
|
\end{equation} |
1009 |
|
|
% |
1010 |
|
|
% |
1011 |
|
|
% |
1012 |
|
|
\begin{equation} |
1013 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1014 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1015 |
|
|
-2 (\mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} ) \times \hat{r} |
1016 |
|
|
+2 \mathbf{D}_{\mathbf{b}} \times |
1017 |
|
|
(\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1018 |
|
|
\Bigr] v_{31}(r) |
1019 |
|
|
% 3 |
1020 |
|
|
- \frac{2}{4\pi \epsilon_0} |
1021 |
|
|
(\hat{r} \cdot \mathbf{D}_{\mathbf{b}} ) |
1022 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}}) \times \hat{r} v_{32}(r) |
1023 |
|
|
\end{equation} |
1024 |
|
|
% |
1025 |
|
|
% |
1026 |
|
|
% |
1027 |
|
|
\begin{equation} |
1028 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} = |
1029 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1030 |
|
|
\text{Tr}\mathbf{Q}_{\mathbf{a}} |
1031 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1032 |
|
|
+2 \mathbf{D}_{\mathbf{b}} \times |
1033 |
|
|
( \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
1034 |
|
|
% 2 |
1035 |
|
|
+\frac{1}{4\pi \epsilon_0} |
1036 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1037 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) v_{32}(r) |
1038 |
|
|
\end{equation} |
1039 |
|
|
% |
1040 |
|
|
% |
1041 |
|
|
% |
1042 |
|
|
\begin{equation} |
1043 |
|
|
\begin{split} |
1044 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
1045 |
|
|
&-\frac{4}{4\pi \epsilon_0} |
1046 |
|
|
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} |
1047 |
|
|
v_{41}(r) \\ |
1048 |
|
|
% 2 |
1049 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1050 |
|
|
\Bigl[-2\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1051 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times \hat{r} |
1052 |
|
|
+4 \hat{r} \times |
1053 |
|
|
( \mathbf{Q}_{{\mathbf a}} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1054 |
|
|
% 3 |
1055 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} )\times |
1056 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} ) \Bigr] v_{42}(r) \\ |
1057 |
|
|
% 4 |
1058 |
|
|
&+ \frac{2}{4\pi \epsilon_0} |
1059 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1060 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) |
1061 |
|
|
\end{split} |
1062 |
|
|
\end{equation} |
1063 |
|
|
% |
1064 |
|
|
% |
1065 |
|
|
% |
1066 |
|
|
\begin{equation} |
1067 |
|
|
\begin{split} |
1068 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} = |
1069 |
|
|
&\frac{4}{4\pi \epsilon_0} |
1070 |
|
|
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} v_{41}(r) \\ |
1071 |
|
|
% 2 |
1072 |
|
|
&+ \frac{1}{4\pi \epsilon_0} \Bigl[- 2\text{Tr}\mathbf{Q}_{\mathbf{a}} |
1073 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \times \hat{r} |
1074 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot |
1075 |
|
|
\mathbf{Q}_{{\mathbf b}} ) \times |
1076 |
|
|
\hat{r} |
1077 |
|
|
+4 ( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times |
1078 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1079 |
|
|
\Bigr] v_{42}(r) \\ |
1080 |
|
|
% 4 |
1081 |
|
|
&+ \frac{2}{4\pi \epsilon_0} |
1082 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1083 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) |
1084 |
|
|
\end{split} |
1085 |
|
|
\end{equation} |
1086 |
|
|
% |
1087 |
|
|
% |
1088 |
|
|
% |
1089 |
|
|
\begin{acknowledgments} |
1090 |
|
|
We wish to acknowledge the support of the author community in using |
1091 |
|
|
REV\TeX{}, offering suggestions and encouragement, testing new versions, |
1092 |
|
|
\dots. |
1093 |
|
|
\end{acknowledgments} |
1094 |
|
|
|
1095 |
|
|
\appendix |
1096 |
|
|
|
1097 |
|
|
\section{Smith's $B_l(r)$ functions for smeared-charge distributions} |
1098 |
|
|
|
1099 |
|
|
The following summarizes Smith's $B_l(r)$ functions and |
1100 |
|
|
includes formulas given in his appendix. |
1101 |
|
|
|
1102 |
|
|
The first function $B_0(r)$ is defined by |
1103 |
|
|
% |
1104 |
|
|
\begin{equation} |
1105 |
|
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r}= |
1106 |
|
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
1107 |
|
|
\end{equation} |
1108 |
|
|
% |
1109 |
|
|
The first derivative of this function is |
1110 |
|
|
% |
1111 |
|
|
\begin{equation} |
1112 |
|
|
\frac{dB_0(r)}{dr}=-\frac{1}{r^2}\text{erfc}(\alpha r) |
1113 |
|
|
-\frac{2\alpha}{r\sqrt{\pi}}\text{e}^{-{\alpha}^2r^2} |
1114 |
|
|
\end{equation} |
1115 |
|
|
% |
1116 |
|
|
and can be rewritten in terms of a function $B_1(r)$: |
1117 |
|
|
% |
1118 |
|
|
\begin{equation} |
1119 |
|
|
B_1(r)=-\frac{1}{r}\frac{dB_0(r)}{dr} |
1120 |
|
|
\end{equation} |
1121 |
|
|
% |
1122 |
|
|
In general, |
1123 |
|
|
\begin{equation} |
1124 |
|
|
B_l(r)=-\frac{1}{r}\frac{dB_{l-1}(r)}{dr} |
1125 |
|
|
= \frac{1}{r^2} \left[ (2l-1)B_{l-1}(r) + \frac {(2\alpha^2)^l}{\alpha \sqrt{\pi}} |
1126 |
|
|
\text{e}^{-{\alpha}^2r^2} |
1127 |
|
|
\right] . |
1128 |
|
|
\end{equation} |
1129 |
|
|
% |
1130 |
|
|
Using these formulas, we find |
1131 |
|
|
% |
1132 |
|
|
\begin{eqnarray} |
1133 |
|
|
\frac{dB_0}{dr}=-rB_1(r) \\ |
1134 |
|
|
\frac{d^2B_0}{dr^2}=-B_1(r) + r^2B_2(r) \\ |
1135 |
|
|
\frac{d^3B_0}{dr^3}=3rB_2(r) - r^3B_3(r) \\ |
1136 |
|
|
\frac{d^4B_0}{dr^4}=3B_2(r) - 6r^2B_3(r)+r^4B_4(r) \\ |
1137 |
|
|
\frac{d^5B_0}{dr^5}=-15rB_3(r) + 10r^3B_4(r) -r^5B_5(r) . |
1138 |
|
|
\end{eqnarray} |
1139 |
|
|
% |
1140 |
|
|
As noted by Smith, |
1141 |
|
|
% |
1142 |
|
|
\begin{equation} |
1143 |
|
|
B_l(r)=\frac{(2l)!}{l!2^lr^{2l+1}} - \frac {(2\alpha^2)^{l+1}}{(2l+1)\alpha \sqrt{\pi}} |
1144 |
|
|
+\text{O}(r) . |
1145 |
|
|
\end{equation} |
1146 |
|
|
|
1147 |
|
|
\section{Method 1, the $r$-dependent factors} |
1148 |
|
|
|
1149 |
|
|
Using the shifted damped functions $f_n(r)$ defined by: |
1150 |
|
|
% |
1151 |
|
|
\begin{equation} |
1152 |
|
|
f_n(r)= B_0 \Big \lvert _r -\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} B_0^{(m)} \Big \lvert _{r_c} , |
1153 |
|
|
\end{equation} |
1154 |
|
|
% |
1155 |
|
|
we first provide formulas for successive derivatives of this function. (If there is |
1156 |
|
|
no damping, then $B_0(r)$ is replaced by $1/r$, as discussed in Section~\ref{damped???}.) First, we find: |
1157 |
|
|
% |
1158 |
|
|
\begin{equation} |
1159 |
|
|
\frac{\partial f_n}{\partial r_\alpha}=\hat{r}_\alpha \frac{d f_n}{d r} . |
1160 |
|
|
\end{equation} |
1161 |
|
|
% |
1162 |
|
|
This formula clearly brings in derivatives of Smith's $B_0(r)$ function, motivating us to |
1163 |
|
|
define higher-order derivatives as follows: |
1164 |
|
|
% |
1165 |
|
|
\begin{eqnarray} |
1166 |
|
|
g_n(r)= \frac{d f_n}{d r} = |
1167 |
|
|
B_0^{(1)} \Big \lvert _r -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)} \Big \lvert _{r_c} \\ |
1168 |
|
|
h_n(r)= \frac{d^2f_n}{d r^2} = |
1169 |
|
|
B_0^{(2)} \Big \lvert _r -\sum_{m=0}^{n-1} \frac {(r-r_c)^m}{m!} B_0^{(m+2)} \Big \lvert _{r_c} \\ |
1170 |
|
|
s_n(r)= \frac{d^3f_n}{d r^3} = |
1171 |
|
|
B_0^{(3)} \Big \lvert _r -\sum_{m=0}^{n-2} \frac {(r-r_c)^m}{m!} B_0^{(m+3)} \Big \lvert _{r_c} \\ |
1172 |
|
|
t_n(r)= \frac{d^4f_n}{d r^4} = |
1173 |
|
|
B_0^{(4)} \Big \lvert _r -\sum_{m=0}^{n-3} \frac {(r-r_c)^m}{m!} B_0^{(m+4)} \Big \lvert _{r_c} \\ |
1174 |
|
|
u_n(r)= \frac{d^5f_n}{d r^5} = |
1175 |
|
|
B_0^{(5)} \Big \lvert _r -\sum_{m=0}^{n-4} \frac {(r-r_c)^m}{m!} B_0^{(m+5)} \Big \lvert _{r_c} . |
1176 |
|
|
\end{eqnarray} |
1177 |
|
|
% |
1178 |
|
|
We note that the last function needed (for quadrupole-quadrupole) is |
1179 |
|
|
% |
1180 |
|
|
\begin{equation} |
1181 |
|
|
u_4(r)=B_0^{(5)} \Big \lvert _r - B_0^{(5)} \Big \lvert _{r_c} . |
1182 |
|
|
\end{equation} |
1183 |
|
|
|
1184 |
|
|
The functions $f_n(r)$ to $u_n(r)$ are recursively computed and stored for values of $r$ |
1185 |
|
|
from $0$ to $r_c$. The functions needed are listed schematically below: |
1186 |
|
|
% |
1187 |
|
|
\begin{eqnarray} |
1188 |
|
|
f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
1189 |
|
|
g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
1190 |
|
|
h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
1191 |
|
|
s_2 \quad s_3 \quad &s_4 \nonumber \\ |
1192 |
|
|
t_3 \quad &t_4 \nonumber \\ |
1193 |
|
|
&u_4 \nonumber . |
1194 |
|
|
\end{eqnarray} |
1195 |
|
|
|
1196 |
|
|
Using these functions, we find |
1197 |
|
|
% |
1198 |
|
|
\begin{equation} |
1199 |
|
|
\frac{\partial f_n}{\partial r_\alpha} =r_\alpha \frac {g_n}{r} |
1200 |
|
|
\end{equation} |
1201 |
|
|
% |
1202 |
|
|
\begin{equation} |
1203 |
|
|
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =\delta_{\alpha \beta}\frac {g_n}{r} |
1204 |
|
|
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) |
1205 |
|
|
\end{equation} |
1206 |
|
|
% |
1207 |
|
|
\begin{equation} |
1208 |
|
|
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} = |
1209 |
|
|
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
1210 |
|
|
\delta_{ \beta \gamma} r_\alpha \right) |
1211 |
|
|
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) |
1212 |
|
|
+ r_\alpha r_\beta r_\gamma |
1213 |
|
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) |
1214 |
|
|
\end{equation} |
1215 |
|
|
% |
1216 |
|
|
\begin{eqnarray} |
1217 |
|
|
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} = |
1218 |
|
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
1219 |
|
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
1220 |
|
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
1221 |
|
|
\left( - \frac{g_n}{r^3} + \frac{h_n}{r^2} \right) \nonumber \\ |
1222 |
|
|
+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
1223 |
|
|
+ \text{5 perm} |
1224 |
|
|
\right) \left( \frac{3 g_n}{r^5} - \frac{3h_n}{r^4} + \frac{s_n}{r^3} |
1225 |
|
|
\right) \nonumber \\ |
1226 |
|
|
+ r_\alpha r_\beta r_\gamma r_\delta |
1227 |
|
|
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
1228 |
|
|
+ \frac{t_n}{r^4} \right) |
1229 |
|
|
\end{eqnarray} |
1230 |
|
|
% |
1231 |
|
|
\begin{eqnarray} |
1232 |
|
|
\frac{\partial^5 f_n} |
1233 |
|
|
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} = |
1234 |
|
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
1235 |
|
|
+ \text{14 perm} \right) |
1236 |
|
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
1237 |
|
|
+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
1238 |
|
|
+ \text{9 perm} |
1239 |
|
|
\right) \left(- \frac{15g_n}{r^7}+\frac{15h_n}{r^7} -\frac{6s_n}{r^5} +\frac{t_n}{r^4} |
1240 |
|
|
\right) \nonumber \\ |
1241 |
|
|
+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
1242 |
|
|
\left( \frac{105g_n}{r^9} - \frac{105h_n}{r^8} + \frac{45s_n}{r^7} |
1243 |
|
|
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) |
1244 |
|
|
\end{eqnarray} |
1245 |
|
|
% |
1246 |
|
|
% |
1247 |
|
|
% |
1248 |
|
|
\section{Method 2, the $r$-dependent factors} |
1249 |
|
|
|
1250 |
|
|
In method 2, the kernel is not expanded, rather the individual terms in the multipole interaction energies, |
1251 |
|
|
see Eq. (20?). For a smeared-charge distribution, this still brings into the algebra multiple derivatives |
1252 |
|
|
of the kernel $B_0(r)$. To denote these terms, we generalize the notation of the previous appendix. |
1253 |
|
|
For $f(r)=1/r$ (bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
1254 |
|
|
% |
1255 |
|
|
\begin{eqnarray} |
1256 |
|
|
g(r)= \frac{df}{d r}\\ |
1257 |
|
|
h(r)= \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
1258 |
|
|
s(r)= \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
1259 |
|
|
t(r)= \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
1260 |
|
|
u(r)= \frac{dt}{d r} =\frac{d^5f}{d r^5} . |
1261 |
|
|
\end{eqnarray} |
1262 |
|
|
% |
1263 |
|
|
For $f(r)=1/r$, Table I lists these derivatives under the column ``Bare Coulomb.'' Checks of algebra can be made by using limiting forms |
1264 |
|
|
of equations, e.g., the leading term in the function $g_n(r)$ has $r$ dependence given by $g(r)$. Equations (B9) to B(13) |
1265 |
|
|
are correct for method 2 if one just eliminates the subscript $n$. |
1266 |
|
|
|
1267 |
|
|
\section{Extra Material} |
1268 |
|
|
% |
1269 |
|
|
% |
1270 |
|
|
%Energy in body coordinate form --------------------------------------------------------------- |
1271 |
|
|
% |
1272 |
|
|
Here are the interaction energies written in terms of the body coordinates: |
1273 |
|
|
|
1274 |
|
|
% |
1275 |
|
|
% u ca cb |
1276 |
|
|
% |
1277 |
|
|
\begin{equation} |
1278 |
|
|
U_{C_{\bf a}C_{\bf b}}(r)= |
1279 |
|
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) |
1280 |
|
|
\end{equation} |
1281 |
|
|
% |
1282 |
|
|
% u ca db |
1283 |
|
|
% |
1284 |
|
|
\begin{equation} |
1285 |
|
|
U_{C_{\bf a}D_{\bf b}}(r)= |
1286 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} |
1287 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
1288 |
|
|
\end{equation} |
1289 |
|
|
% |
1290 |
|
|
% u ca qb |
1291 |
|
|
% |
1292 |
|
|
\begin{equation} |
1293 |
|
|
U_{C_{\bf a}Q_{\bf b}}(r)= |
1294 |
|
|
\frac{C_{\bf a }\text{Tr}Q_{\bf b}}{4\pi \epsilon_0} |
1295 |
|
|
v_{21}(r) \nonumber \\ |
1296 |
|
|
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
1297 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) |
1298 |
|
|
v_{22}(r) |
1299 |
|
|
\end{equation} |
1300 |
|
|
% |
1301 |
|
|
% u da cb |
1302 |
|
|
% |
1303 |
|
|
\begin{equation} |
1304 |
|
|
U_{D_{\bf a}C_{\bf b}}(r)= |
1305 |
|
|
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
1306 |
|
|
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
1307 |
|
|
\end{equation} |
1308 |
|
|
% |
1309 |
|
|
% u da db |
1310 |
|
|
% |
1311 |
|
|
\begin{equation} |
1312 |
|
|
\begin{split} |
1313 |
|
|
% 1 |
1314 |
|
|
U_{D_{\bf a}D_{\bf b}}(r)&= |
1315 |
|
|
-\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
1316 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
1317 |
|
|
D_{\mathbf{b}n} v_{21}(r) \\ |
1318 |
|
|
% 2 |
1319 |
|
|
&-\frac{1}{4\pi \epsilon_0} |
1320 |
|
|
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
1321 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} |
1322 |
|
|
v_{22}(r) |
1323 |
|
|
\end{split} |
1324 |
|
|
\end{equation} |
1325 |
|
|
% |
1326 |
|
|
% u da qb |
1327 |
|
|
% |
1328 |
|
|
\begin{equation} |
1329 |
|
|
\begin{split} |
1330 |
|
|
% 1 |
1331 |
|
|
U_{D_{\bf a}Q_{\bf b}}(r)&= |
1332 |
|
|
-\frac{1}{4\pi \epsilon_0} \left( |
1333 |
|
|
\text{Tr}Q_{\mathbf{b}} |
1334 |
|
|
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
1335 |
|
|
+2\sum_{lmn}D_{\mathbf{a}l} |
1336 |
|
|
(\hat{a}_l \cdot \hat{b}_m) |
1337 |
|
|
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
1338 |
|
|
\right) v_{31}(r) \\ |
1339 |
|
|
% 2 |
1340 |
|
|
&-\frac{1}{4\pi \epsilon_0} |
1341 |
|
|
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
1342 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1343 |
|
|
Q_{{\mathbf b}mn} |
1344 |
|
|
(\hat{b}_n \cdot \hat{r}) v_{32}(r) |
1345 |
|
|
\end{split} |
1346 |
|
|
\end{equation} |
1347 |
|
|
% |
1348 |
|
|
% u qa cb |
1349 |
|
|
% |
1350 |
|
|
\begin{equation} |
1351 |
|
|
U_{Q_{\bf a}C_{\bf b}}(r)= |
1352 |
|
|
\frac{C_{\bf b }\text{Tr}Q_{\bf a}}{4\pi \epsilon_0} v_{21}(r) |
1353 |
|
|
+\frac{C_{\bf b}}{4\pi \epsilon_0} |
1354 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) v_{22}(r) |
1355 |
|
|
\end{equation} |
1356 |
|
|
% |
1357 |
|
|
% u qa db |
1358 |
|
|
% |
1359 |
|
|
\begin{equation} |
1360 |
|
|
\begin{split} |
1361 |
|
|
%1 |
1362 |
|
|
U_{Q_{\bf a}D_{\bf b}}(r)&= |
1363 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
1364 |
|
|
\text{Tr}Q_{\mathbf{a}} |
1365 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
1366 |
|
|
+2\sum_{lmn}D_{\mathbf{b}l} |
1367 |
|
|
(\hat{b}_l \cdot \hat{a}_m) |
1368 |
|
|
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
1369 |
|
|
\right) v_{31}(r) \\ |
1370 |
|
|
% 2 |
1371 |
|
|
&+\frac{1}{4\pi \epsilon_0} |
1372 |
|
|
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
1373 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1374 |
|
|
Q_{{\mathbf a}mn} |
1375 |
|
|
(\hat{a}_n \cdot \hat{r}) v_{32}(r) |
1376 |
|
|
\end{split} |
1377 |
|
|
\end{equation} |
1378 |
|
|
% |
1379 |
|
|
% u qa qb |
1380 |
|
|
% |
1381 |
|
|
\begin{equation} |
1382 |
|
|
\begin{split} |
1383 |
|
|
%1 |
1384 |
|
|
U_{Q_{\bf a}Q_{\bf b}}(r)&= |
1385 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1386 |
|
|
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
1387 |
|
|
+2\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
1388 |
|
|
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
1389 |
|
|
(\hat{a}_m \cdot \hat{b}_n) \Bigr] |
1390 |
|
|
v_{41}(r) \\ |
1391 |
|
|
% 2 |
1392 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1393 |
|
|
\Bigl[ \text{Tr}Q_{\mathbf{a}} |
1394 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
1395 |
|
|
Q_{{\mathbf b}lm} |
1396 |
|
|
(\hat{b}_m \cdot \hat{r}) |
1397 |
|
|
+\text{Tr}Q_{\mathbf{b}} |
1398 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
1399 |
|
|
Q_{{\mathbf a}lm} |
1400 |
|
|
(\hat{a}_m \cdot \hat{r}) \\ |
1401 |
|
|
% 3 |
1402 |
|
|
&+4 \sum_{lmnp} |
1403 |
|
|
(\hat{r} \cdot \hat{a}_l ) |
1404 |
|
|
Q_{{\mathbf a}lm} |
1405 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
1406 |
|
|
Q_{{\mathbf b}np} |
1407 |
|
|
(\hat{b}_p \cdot \hat{r}) |
1408 |
|
|
\Bigr] v_{42}(r) \\ |
1409 |
|
|
% 4 |
1410 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1411 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) |
1412 |
|
|
Q_{{\mathbf a}lm} |
1413 |
|
|
(\hat{a}_m \cdot \hat{r}) |
1414 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) |
1415 |
|
|
Q_{{\mathbf b}np} |
1416 |
|
|
(\hat{b}_p \cdot \hat{r}) v_{43}(r). |
1417 |
|
|
\end{split} |
1418 |
|
|
\end{equation} |
1419 |
|
|
% |
1420 |
|
|
|
1421 |
|
|
|
1422 |
|
|
% BODY coordinates force equations -------------------------------------------- |
1423 |
|
|
% |
1424 |
|
|
% |
1425 |
|
|
Here are the force equations written in terms of body coordinates. |
1426 |
|
|
% |
1427 |
|
|
% f ca cb |
1428 |
|
|
% |
1429 |
|
|
\begin{equation} |
1430 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} = |
1431 |
|
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} w_a(r) \hat{r} |
1432 |
|
|
\end{equation} |
1433 |
|
|
% |
1434 |
|
|
% f ca db |
1435 |
|
|
% |
1436 |
|
|
\begin{equation} |
1437 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} = |
1438 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} |
1439 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} w_b(r) \hat{r} |
1440 |
|
|
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
1441 |
|
|
\sum_n D_{\mathbf{b}n} \hat{b}_n w_c(r) |
1442 |
|
|
\end{equation} |
1443 |
|
|
% |
1444 |
|
|
% f ca qb |
1445 |
|
|
% |
1446 |
|
|
\begin{equation} |
1447 |
|
|
\begin{split} |
1448 |
|
|
% 1 |
1449 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} = |
1450 |
|
|
\frac{1}{4\pi \epsilon_0} |
1451 |
|
|
C_{\bf a }\text{Tr}Q_{\bf b} w_d(r) \hat{r} |
1452 |
|
|
+ 2C_{\bf a } \sum_l \hat{b}_l Q_{{\mathbf b}ln} (\hat{b}_n \cdot \hat{r}) w_e(r) \\ |
1453 |
|
|
% 2 |
1454 |
|
|
+\frac{C_{\bf a}}{4\pi \epsilon_0} |
1455 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) w_f(r) \hat{r} |
1456 |
|
|
\end{split} |
1457 |
|
|
\end{equation} |
1458 |
|
|
% |
1459 |
|
|
% f da cb |
1460 |
|
|
% |
1461 |
|
|
\begin{equation} |
1462 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1463 |
|
|
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
1464 |
|
|
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} w_b(r) \hat{r} |
1465 |
|
|
-\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
1466 |
|
|
\sum_n D_{\mathbf{a}n} \hat{a}_n w_c(r) |
1467 |
|
|
\end{equation} |
1468 |
|
|
% |
1469 |
|
|
% f da db |
1470 |
|
|
% |
1471 |
|
|
\begin{equation} |
1472 |
|
|
\begin{split} |
1473 |
|
|
% 1 |
1474 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
1475 |
|
|
-\frac{1}{4\pi \epsilon_0} |
1476 |
|
|
\sum_{mn} D_{\mathbf {a}m} |
1477 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
1478 |
|
|
D_{\mathbf{b}n} w_d(r) \hat{r} |
1479 |
|
|
-\frac{1}{4\pi \epsilon_0} |
1480 |
|
|
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
1481 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} w_f(r) \hat{r} \\ |
1482 |
|
|
% 2 |
1483 |
|
|
& \quad + \frac{1}{4\pi \epsilon_0} |
1484 |
|
|
\Bigl[ \sum_m D_{\mathbf {a}m} |
1485 |
|
|
\hat{a}_m \sum_n D_{\mathbf{b}n} |
1486 |
|
|
(\hat{b}_n \cdot \hat{r}) |
1487 |
|
|
+ \sum_m D_{\mathbf {b}m} |
1488 |
|
|
\hat{b}_m \sum_n D_{\mathbf{a}n} |
1489 |
|
|
(\hat{a}_n \cdot \hat{r}) \Bigr] w_e(r) \\ |
1490 |
|
|
\end{split} |
1491 |
|
|
\end{equation} |
1492 |
|
|
% |
1493 |
|
|
% f da qb |
1494 |
|
|
% |
1495 |
|
|
\begin{equation} |
1496 |
|
|
\begin{split} |
1497 |
|
|
% 1 |
1498 |
|
|
&\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
1499 |
|
|
- \frac{1}{4\pi \epsilon_0} \Bigl[ |
1500 |
|
|
\text{Tr}Q_{\mathbf{b}} |
1501 |
|
|
\sum_l D_{\mathbf{a}l} \hat{a}_l |
1502 |
|
|
+2\sum_{lmn} D_{\mathbf{a}l} |
1503 |
|
|
(\hat{a}_l \cdot \hat{b}_m) |
1504 |
|
|
Q_{\mathbf{b}mn} \hat{b}_n \Bigr] w_g(r) \\ |
1505 |
|
|
% 3 |
1506 |
|
|
& - \frac{1}{4\pi \epsilon_0} \Bigl[ |
1507 |
|
|
\text{Tr}Q_{\mathbf{b}} |
1508 |
|
|
\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
1509 |
|
|
+2\sum_{lmn}D_{\mathbf{a}l} |
1510 |
|
|
(\hat{a}_l \cdot \hat{b}_m) |
1511 |
|
|
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1512 |
|
|
% 4 |
1513 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1514 |
|
|
\Bigl[\sum_l D_{\mathbf{a}l} \hat{a}_l |
1515 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1516 |
|
|
Q_{{\mathbf b}mn} |
1517 |
|
|
(\hat{b}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{a}_l) |
1518 |
|
|
D_{\mathbf{a}l} |
1519 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1520 |
|
|
Q_{{\mathbf b}mn} \hat{b}_n \Bigr] w_i(r)\\ |
1521 |
|
|
% 6 |
1522 |
|
|
& -\frac{1}{4\pi \epsilon_0} |
1523 |
|
|
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
1524 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1525 |
|
|
Q_{{\mathbf b}mn} |
1526 |
|
|
(\hat{b}_n \cdot \hat{r}) w_j(r) \hat{r} |
1527 |
|
|
\end{split} |
1528 |
|
|
\end{equation} |
1529 |
|
|
% |
1530 |
|
|
% force qa cb |
1531 |
|
|
% |
1532 |
|
|
\begin{equation} |
1533 |
|
|
\begin{split} |
1534 |
|
|
% 1 |
1535 |
|
|
\mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} &= |
1536 |
|
|
\frac{1}{4\pi \epsilon_0} |
1537 |
|
|
C_{\bf b }\text{Tr}Q_{\bf a} \hat{r} w_d(r) |
1538 |
|
|
+ \frac{2C_{\bf b }}{4\pi \epsilon_0} \sum_l \hat{a}_l Q_{{\mathbf a}ln} (\hat{a}_n \cdot \hat{r}) w_e(r) \\ |
1539 |
|
|
% 2 |
1540 |
|
|
& +\frac{C_{\bf b}}{4\pi \epsilon_0} |
1541 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) w_f(r) \hat{r} |
1542 |
|
|
\end{split} |
1543 |
|
|
\end{equation} |
1544 |
|
|
% |
1545 |
|
|
% f qa db |
1546 |
|
|
% |
1547 |
|
|
\begin{equation} |
1548 |
|
|
\begin{split} |
1549 |
|
|
% 1 |
1550 |
|
|
&\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1551 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1552 |
|
|
\text{Tr}Q_{\mathbf{a}} |
1553 |
|
|
\sum_l D_{\mathbf{b}l} \hat{b}_l |
1554 |
|
|
+2\sum_{lmn} D_{\mathbf{b}l} |
1555 |
|
|
(\hat{b}_l \cdot \hat{a}_m) |
1556 |
|
|
Q_{\mathbf{a}mn} \hat{a}_n \Bigr] |
1557 |
|
|
w_g(r)\\ |
1558 |
|
|
% 3 |
1559 |
|
|
& + \frac{1}{4\pi \epsilon_0} \Bigl[ |
1560 |
|
|
\text{Tr}Q_{\mathbf{a}} |
1561 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
1562 |
|
|
+2\sum_{lmn}D_{\mathbf{b}l} |
1563 |
|
|
(\hat{b}_l \cdot \hat{a}_m) |
1564 |
|
|
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1565 |
|
|
% 4 |
1566 |
|
|
& + \frac{1}{4\pi \epsilon_0} \Bigl[ \sum_l D_{\mathbf{b}l} \hat{b}_l |
1567 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1568 |
|
|
Q_{{\mathbf a}mn} |
1569 |
|
|
(\hat{a}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{b}_l) |
1570 |
|
|
D_{\mathbf{b}l} |
1571 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1572 |
|
|
Q_{{\mathbf a}mn} \hat{a}_n \Bigr] w_i(r) \\ |
1573 |
|
|
% 6 |
1574 |
|
|
& +\frac{1}{4\pi \epsilon_0} |
1575 |
|
|
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
1576 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1577 |
|
|
Q_{{\mathbf a}mn} |
1578 |
|
|
(\hat{a}_n \cdot \hat{r}) w_j(r) \hat{r} |
1579 |
|
|
\end{split} |
1580 |
|
|
\end{equation} |
1581 |
|
|
% |
1582 |
|
|
% f qa qb |
1583 |
|
|
% |
1584 |
|
|
\begin{equation} |
1585 |
|
|
\begin{split} |
1586 |
|
|
&\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
1587 |
|
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
1588 |
|
|
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
1589 |
|
|
+ 2 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
1590 |
|
|
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
1591 |
|
|
(\hat{a}_m \cdot \hat{b}_n) \Bigr] w_k(r) \hat{r}\\ |
1592 |
|
|
&+\frac{1}{4\pi \epsilon_0} \Bigl[ |
1593 |
|
|
2\text{Tr}Q_{\mathbf{b}} \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
1594 |
|
|
+ 2\text{Tr}Q_{\mathbf{a}} \sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} \hat{b}_m \\ |
1595 |
|
|
&+ 4\sum_{lmnp} \hat{a}_l Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) |
1596 |
|
|
+ 4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_p |
1597 |
|
|
\Bigr] w_n(r) \\ |
1598 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1599 |
|
|
\Bigl[ \text{Tr}Q_{\mathbf{a}} |
1600 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} (\hat{b}_m \cdot \hat{r}) |
1601 |
|
|
+ \text{Tr}Q_{\mathbf{b}} |
1602 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) \\ |
1603 |
|
|
&+4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) |
1604 |
|
|
Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
1605 |
|
|
% |
1606 |
|
|
&+\frac{1}{4\pi \epsilon_0} \Bigl[ |
1607 |
|
|
2\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
1608 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_n \cdot \hat{r}) \\ |
1609 |
|
|
&+2 \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
1610 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_n \Bigr] w_o(r) \hat{r} \\ |
1611 |
|
|
& + \frac{1}{4\pi \epsilon_0} |
1612 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
1613 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) w_m(r) \hat{r} |
1614 |
|
|
\end{split} |
1615 |
|
|
\end{equation} |
1616 |
|
|
% |
1617 |
|
|
Here we list the form of the non-zero damped shifted multipole torques showing |
1618 |
|
|
explicitly dependences on body axes: |
1619 |
|
|
% |
1620 |
|
|
% t ca db |
1621 |
|
|
% |
1622 |
|
|
\begin{equation} |
1623 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} = |
1624 |
|
|
\frac{C_{\bf a}}{4\pi \epsilon_0} |
1625 |
|
|
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
1626 |
|
|
\end{equation} |
1627 |
|
|
% |
1628 |
|
|
% t ca qb |
1629 |
|
|
% |
1630 |
|
|
\begin{equation} |
1631 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} = |
1632 |
|
|
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
1633 |
|
|
\sum_{lm} (\hat{r} \times \hat{b}_l) Q_{{\mathbf b}lm} (\hat{b}_m \cdot \hat{r}) v_{22}(r) |
1634 |
|
|
\end{equation} |
1635 |
|
|
% |
1636 |
|
|
% t da cb |
1637 |
|
|
% |
1638 |
|
|
\begin{equation} |
1639 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1640 |
|
|
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
1641 |
|
|
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
1642 |
|
|
\end{equation}% |
1643 |
|
|
% |
1644 |
|
|
% |
1645 |
|
|
% ta da db |
1646 |
|
|
% |
1647 |
|
|
\begin{equation} |
1648 |
|
|
\begin{split} |
1649 |
|
|
% 1 |
1650 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
1651 |
|
|
\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
1652 |
|
|
(\hat{a}_m \times \hat{b}_n) |
1653 |
|
|
D_{\mathbf{b}n} v_{21}(r) \\ |
1654 |
|
|
% 2 |
1655 |
|
|
&-\frac{1}{4\pi \epsilon_0} |
1656 |
|
|
\sum_m (\hat{r} \times \hat{a}_m) D_{\mathbf {a}m} |
1657 |
|
|
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
1658 |
|
|
\end{split} |
1659 |
|
|
\end{equation} |
1660 |
|
|
% |
1661 |
|
|
% tb da db |
1662 |
|
|
% |
1663 |
|
|
\begin{equation} |
1664 |
|
|
\begin{split} |
1665 |
|
|
% 1 |
1666 |
|
|
\mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} &= |
1667 |
|
|
-\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
1668 |
|
|
(\hat{a}_m \times \hat{b}_n) |
1669 |
|
|
D_{\mathbf{b}n} v_{21}(r) \\ |
1670 |
|
|
% 2 |
1671 |
|
|
&+\frac{1}{4\pi \epsilon_0} |
1672 |
|
|
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
1673 |
|
|
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
1674 |
|
|
\end{split} |
1675 |
|
|
\end{equation} |
1676 |
|
|
% |
1677 |
|
|
% ta da qb |
1678 |
|
|
% |
1679 |
|
|
\begin{equation} |
1680 |
|
|
\begin{split} |
1681 |
|
|
% 1 |
1682 |
|
|
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} &= |
1683 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
1684 |
|
|
-\text{Tr}Q_{\mathbf{b}} |
1685 |
|
|
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} |
1686 |
|
|
+2\sum_{lmn}D_{\mathbf{a}l} |
1687 |
|
|
(\hat{a}_l \times \hat{b}_m) |
1688 |
|
|
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
1689 |
|
|
\right) v_{31}(r)\\ |
1690 |
|
|
% 2 |
1691 |
|
|
&-\frac{1}{4\pi \epsilon_0} |
1692 |
|
|
\sum_l (\hat{r} \times \hat{a}_l) D_{\mathbf{a}l} |
1693 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1694 |
|
|
Q_{{\mathbf b}mn} |
1695 |
|
|
(\hat{b}_n \cdot \hat{r}) v_{32}(r) |
1696 |
|
|
\end{split} |
1697 |
|
|
\end{equation} |
1698 |
|
|
% |
1699 |
|
|
% tb da qb |
1700 |
|
|
% |
1701 |
|
|
\begin{equation} |
1702 |
|
|
\begin{split} |
1703 |
|
|
% 1 |
1704 |
|
|
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} &= |
1705 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
1706 |
|
|
-2\sum_{lmn}D_{\mathbf{a}l} |
1707 |
|
|
(\hat{a}_l \cdot \hat{b}_m) |
1708 |
|
|
Q_{\mathbf{b}mn} (\hat{r} \times \hat{b}_n) |
1709 |
|
|
-2\sum_{lmn}D_{\mathbf{a}l} |
1710 |
|
|
(\hat{a}_l \times \hat{b}_m) |
1711 |
|
|
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
1712 |
|
|
\right) v_{31}(r) \\ |
1713 |
|
|
% 2 |
1714 |
|
|
&-\frac{2}{4\pi \epsilon_0} |
1715 |
|
|
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
1716 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1717 |
|
|
Q_{{\mathbf b}mn} |
1718 |
|
|
(\hat{r}\times \hat{b}_n) v_{32}(r) |
1719 |
|
|
\end{split} |
1720 |
|
|
\end{equation} |
1721 |
|
|
% |
1722 |
|
|
% ta qa cb |
1723 |
|
|
% |
1724 |
|
|
\begin{equation} |
1725 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
1726 |
|
|
\frac{2C_{\bf a}}{4\pi \epsilon_0} |
1727 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{{\mathbf a}lm} (\hat{r} \times \hat{a}_m) v_{22}(r) |
1728 |
|
|
\end{equation} |
1729 |
|
|
% |
1730 |
|
|
% ta qa db |
1731 |
|
|
% |
1732 |
|
|
\begin{equation} |
1733 |
|
|
\begin{split} |
1734 |
|
|
% 1 |
1735 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} &= |
1736 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
1737 |
|
|
2\sum_{lmn}D_{\mathbf{b}l} |
1738 |
|
|
(\hat{b}_l \cdot \hat{a}_m) |
1739 |
|
|
Q_{\mathbf{a}mn} (\hat{r} \times \hat{a}_n) |
1740 |
|
|
+2\sum_{lmn}D_{\mathbf{b}l} |
1741 |
|
|
(\hat{a}_l \times \hat{b}_m) |
1742 |
|
|
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
1743 |
|
|
\right) v_{31}(r) \\ |
1744 |
|
|
% 2 |
1745 |
|
|
&+\frac{2}{4\pi \epsilon_0} |
1746 |
|
|
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
1747 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1748 |
|
|
Q_{{\mathbf a}mn} |
1749 |
|
|
(\hat{r}\times \hat{a}_n) v_{32}(r) |
1750 |
|
|
\end{split} |
1751 |
|
|
\end{equation} |
1752 |
|
|
% |
1753 |
|
|
% tb qa db |
1754 |
|
|
% |
1755 |
|
|
\begin{equation} |
1756 |
|
|
\begin{split} |
1757 |
|
|
% 1 |
1758 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} &= |
1759 |
|
|
\frac{1}{4\pi \epsilon_0} \left( |
1760 |
|
|
\text{Tr}Q_{\mathbf{a}} |
1761 |
|
|
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} |
1762 |
|
|
+2\sum_{lmn}D_{\mathbf{b}l} |
1763 |
|
|
(\hat{a}_l \times \hat{b}_m) |
1764 |
|
|
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
1765 |
|
|
\right) v_{31}(r)\\ |
1766 |
|
|
% 2 |
1767 |
|
|
&\frac{1}{4\pi \epsilon_0} |
1768 |
|
|
\sum_l (\hat{r} \times \hat{b}_l) D_{\mathbf{b}l} |
1769 |
|
|
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1770 |
|
|
Q_{{\mathbf a}mn} |
1771 |
|
|
(\hat{a}_n \cdot \hat{r}) v_{32}(r) |
1772 |
|
|
\end{split} |
1773 |
|
|
\end{equation} |
1774 |
|
|
% |
1775 |
|
|
% ta qa qb |
1776 |
|
|
% |
1777 |
|
|
\begin{equation} |
1778 |
|
|
\begin{split} |
1779 |
|
|
% 1 |
1780 |
|
|
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} &= |
1781 |
|
|
-\frac{4}{4\pi \epsilon_0} |
1782 |
|
|
\sum_{lmnp} (\hat{a}_l \times \hat{b}_p) |
1783 |
|
|
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
1784 |
|
|
(\hat{a}_m \cdot \hat{b}_n) v_{41}(r) \\ |
1785 |
|
|
% 2 |
1786 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1787 |
|
|
\Bigl[ |
1788 |
|
|
2\text{Tr}Q_{\mathbf{b}} |
1789 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
1790 |
|
|
Q_{{\mathbf a}lm} |
1791 |
|
|
(\hat{r} \times \hat{a}_m) |
1792 |
|
|
+4 \sum_{lmnp} |
1793 |
|
|
(\hat{r} \times \hat{a}_l ) |
1794 |
|
|
Q_{{\mathbf a}lm} |
1795 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
1796 |
|
|
Q_{{\mathbf b}np} |
1797 |
|
|
(\hat{b}_p \cdot \hat{r}) \\ |
1798 |
|
|
% 3 |
1799 |
|
|
&-4 \sum_{lmnp} |
1800 |
|
|
(\hat{r} \cdot \hat{a}_l ) |
1801 |
|
|
Q_{{\mathbf a}lm} |
1802 |
|
|
(\hat{a}_m \times \hat{b}_n) |
1803 |
|
|
Q_{{\mathbf b}np} |
1804 |
|
|
(\hat{b}_p \cdot \hat{r}) |
1805 |
|
|
\Bigr] v_{42}(r) \\ |
1806 |
|
|
% 4 |
1807 |
|
|
&+ \frac{2}{4\pi \epsilon_0} |
1808 |
|
|
\sum_{lm} (\hat{r} \times \hat{a}_l) |
1809 |
|
|
Q_{{\mathbf a}lm} |
1810 |
|
|
(\hat{a}_m \cdot \hat{r}) |
1811 |
|
|
\sum_{np} (\hat{r} \cdot \hat{b}_n) |
1812 |
|
|
Q_{{\mathbf b}np} |
1813 |
|
|
(\hat{b}_p \cdot \hat{r}) v_{43}(r)\\ |
1814 |
|
|
\end{split} |
1815 |
|
|
\end{equation} |
1816 |
|
|
% |
1817 |
|
|
% tb qa qb |
1818 |
|
|
% |
1819 |
|
|
\begin{equation} |
1820 |
|
|
\begin{split} |
1821 |
|
|
% 1 |
1822 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} &= |
1823 |
|
|
\frac{4}{4\pi \epsilon_0} |
1824 |
|
|
\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
1825 |
|
|
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
1826 |
|
|
(\hat{a}_m \times \hat{b}_n) v_{41}(r) \\ |
1827 |
|
|
% 2 |
1828 |
|
|
&+ \frac{1}{4\pi \epsilon_0} |
1829 |
|
|
\Bigl[ |
1830 |
|
|
2\text{Tr}Q_{\mathbf{a}} |
1831 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
1832 |
|
|
Q_{{\mathbf b}lm} |
1833 |
|
|
(\hat{r} \times \hat{b}_m) |
1834 |
|
|
+4 \sum_{lmnp} |
1835 |
|
|
(\hat{r} \cdot \hat{a}_l ) |
1836 |
|
|
Q_{{\mathbf a}lm} |
1837 |
|
|
(\hat{a}_m \cdot \hat{b}_n) |
1838 |
|
|
Q_{{\mathbf b}np} |
1839 |
|
|
(\hat{r} \times \hat{b}_p) \\ |
1840 |
|
|
% 3 |
1841 |
|
|
&+4 \sum_{lmnp} |
1842 |
|
|
(\hat{r} \cdot \hat{a}_l ) |
1843 |
|
|
Q_{{\mathbf a}lm} |
1844 |
|
|
(\hat{a}_m \times \hat{b}_n) |
1845 |
|
|
Q_{{\mathbf b}np} |
1846 |
|
|
(\hat{b}_p \cdot \hat{r}) |
1847 |
|
|
\Bigr] v_{42}(r) \\ |
1848 |
|
|
% 4 |
1849 |
|
|
&+ \frac{2}{4\pi \epsilon_0} |
1850 |
|
|
\sum_{lm} (\hat{r} \cdot \hat{a}_l) |
1851 |
|
|
Q_{{\mathbf a}lm} |
1852 |
|
|
(\hat{a}_m \cdot \hat{r}) |
1853 |
|
|
\sum_{np} (\hat{r} \times \hat{b}_n) |
1854 |
|
|
Q_{{\mathbf b}np} |
1855 |
|
|
(\hat{b}_p \cdot \hat{r}) v_{43}(r). \\ |
1856 |
|
|
\end{split} |
1857 |
|
|
\end{equation} |
1858 |
|
|
% |
1859 |
|
|
\begin{table*} |
1860 |
|
|
\caption{\label{tab:tableFORCE2}Radial functions used in the force equations.} |
1861 |
|
|
\begin{ruledtabular} |
1862 |
|
|
\begin{tabular}{ccc} |
1863 |
|
|
Generic&Method 1&Method 2 |
1864 |
|
|
\\ \hline |
1865 |
|
|
% |
1866 |
|
|
% |
1867 |
|
|
% |
1868 |
|
|
$w_a(r)$& |
1869 |
|
|
$g_0(r)$& |
1870 |
|
|
$g(r)-g(r_c)$ \\ |
1871 |
|
|
% |
1872 |
|
|
% |
1873 |
|
|
$w_b(r)$ & |
1874 |
|
|
$\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
1875 |
|
|
$h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
1876 |
|
|
% |
1877 |
|
|
$w_c(r)$ & |
1878 |
|
|
$\frac{g_1(r)}{r} $ & |
1879 |
|
|
$\frac{v_{11}(r)}{r}$ \\ |
1880 |
|
|
% |
1881 |
|
|
% |
1882 |
|
|
$w_d(r)$& |
1883 |
|
|
$\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
1884 |
|
|
$\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
1885 |
|
|
-\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $\\ |
1886 |
|
|
% |
1887 |
|
|
$w_e(r)$ & |
1888 |
|
|
$\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
1889 |
|
|
$\frac{v_{22}(r)}{r}$ \\ |
1890 |
|
|
% |
1891 |
|
|
% |
1892 |
|
|
$w_f(r)$& |
1893 |
|
|
$\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
1894 |
|
|
$\left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) - $ \\ |
1895 |
|
|
&&$\left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)-\frac{2v_{22}(r)}{r}$\\ |
1896 |
|
|
% |
1897 |
|
|
$w_g(r)$& $ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
1898 |
|
|
$\frac{v_{31}(r)}{r}$\\ |
1899 |
|
|
% |
1900 |
|
|
$w_h(r)$ & |
1901 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
1902 |
|
|
$\left(\frac{2g(r)}{r^3} -\frac{2h(r)}{r^2} +\frac{s(r)}{r} \right) - $\\ |
1903 |
|
|
&&$\left(\frac{2g(r_c)}{r_c^3} -\frac{2h(r_c)}{r_c^2} +\frac{s(r_c)}{r_c} \right) $ \\ |
1904 |
|
|
&&$-\frac{v_{31}(r)}{r}$\\ |
1905 |
|
|
% 2 |
1906 |
|
|
$w_i(r)$ & |
1907 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
1908 |
|
|
$\frac{v_{32}(r)}{r}$ \\ |
1909 |
|
|
% |
1910 |
|
|
$w_j(r)$ & |
1911 |
|
|
$\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
1912 |
|
|
$\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right) $ \\ |
1913 |
|
|
&&$\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} -\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
1914 |
|
|
% |
1915 |
|
|
$w_k(r)$ & |
1916 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
1917 |
|
|
$\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right)$ \\ |
1918 |
|
|
&&$\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
1919 |
|
|
% |
1920 |
|
|
$w_l(r)$ & |
1921 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
1922 |
|
|
$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
1923 |
|
|
&&$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right) |
1924 |
|
|
-\frac{2v_{42}(r)}{r}$ \\ |
1925 |
|
|
% |
1926 |
|
|
$w_m(r)$ & |
1927 |
|
|
$\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
1928 |
|
|
$\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$ \\ |
1929 |
|
|
&&$\left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
1930 |
|
|
+\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $ \\ |
1931 |
|
|
&&$-\frac{4v_{43}(r)}{r}$ \\ |
1932 |
|
|
% |
1933 |
|
|
$w_n(r)$ & |
1934 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
1935 |
|
|
$\frac{v_{42}(r)}{r}$ \\ |
1936 |
|
|
% |
1937 |
|
|
$w_o(r)$ & |
1938 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
1939 |
|
|
$\frac{v_{43}(r)}{r}$ \\ |
1940 |
|
|
% |
1941 |
|
|
\end{tabular} |
1942 |
|
|
\end{ruledtabular} |
1943 |
|
|
\end{table*} |
1944 |
|
|
\end{document} |
1945 |
|
|
% |
1946 |
|
|
% ****** End of file multipole.tex ****** |