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\begin{document} |
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\preprint{AIP/123-QED} |
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\title[Taylor-shifted and Gradient-shifted electrostatics for multipoles] |
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{Real space alternatives to the Ewald |
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Sum. I. Taylor-shifted and Gradient-shifted electrostatics for multipoles} |
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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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We have extended the original damped-shifted force (DSF) |
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electrostatic kernel and have been able to derive two new |
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electrostatic potentials for higher-order multipoles that are based |
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on truncated Taylor expansions around the cutoff radius. For |
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multipole-multipole interactions, we find that each of the distinct |
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orientational contributions has a separate radial function to ensure |
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that the overall forces and torques vanish at the cutoff radius. In |
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this paper, we present energy, force, and torque expressions for the |
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new models, and compare these real-space interaction models to exact |
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results for ordered arrays of multipoles. |
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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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There has been increasing interest in real-space methods for |
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calculating electrostatic interactions in computer simulations of |
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condensed molecular |
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systems.\cite{Wolf99,Zahn02,Kast03,BeckD.A.C._bi0486381,Ma05,Fennell:2006zl,Chen:2004du,Chen:2006ii,Rodgers:2006nw,Denesyuk:2008ez,Izvekov:2008wo} |
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The simplest of these techniques was developed by Wolf {\it et al.} |
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in their work towards an $\mathcal{O}(N)$ Coulombic sum.\cite{Wolf99} |
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For systems of point charges, Fennell and Gezelter showed that a |
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simple damped shifted force (DSF) modification to Wolf's method could |
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give nearly quantitative agreement with smooth particle mesh Ewald |
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(SPME)\cite{Essmann95} configurational energy differences as well as |
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atomic force and molecular torque vectors.\cite{Fennell:2006zl} |
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|
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The computational efficiency and the accuracy of the DSF method are |
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surprisingly good, particularly for systems with uniform charge |
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density. Additionally, dielectric constants obtained using DSF and |
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similar methods where the force vanishes at $R_\textrm{c}$ are |
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essentially quantitative.\cite{Izvekov:2008wo} The DSF and other |
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related methods have now been widely investigated,\cite{Hansen:2012uq} |
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and DSF is now used routinely in a diverse set of chemical |
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environments.\cite{doi:10.1021/la400226g,McCann:2013fk,kannam:094701,Forrest:2012ly,English:2008kx,Louden:2013ve,Tokumasu:2013zr} |
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DSF electrostatics provides a compromise between the computational |
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speed of real-space cutoffs and the accuracy of fully-periodic Ewald |
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treatments. |
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One common feature of many coarse-graining approaches, which treat |
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entire molecular subsystems as a single rigid body, is simplification |
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of the electrostatic interactions between these bodies so that fewer |
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site-site interactions are required to compute configurational |
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energies. The coarse-graining approaches of Ren \& |
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coworkers,\cite{Golubkov06} and Essex \& |
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coworkers,\cite{ISI:000276097500009,ISI:000298664400012} both utilize |
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point multipoles to model electrostatics for entire molecules or |
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functional groups. |
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|
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Ichiye and coworkers have recently introduced a number of very fast |
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water models based on a ``sticky'' multipole model which are |
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qualitatively better at reproducing the behavior of real water than |
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the more common point-charge models (SPC/E, |
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TIPnP).\cite{Chowdhuri:2006lr,Te:2010rt,Te:2010ys,Te:2010vn} The SSDQO |
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model requires the use of an approximate multipole expansion (AME) as |
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the exact multipole expansion is quite expensive (particularly when |
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handled via the Ewald sum).\cite{Ichiye:2006qy} |
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Another particularly important use of point multipoles (and multipole |
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polarizability) is in the very high-quality AMOEBA water model and |
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related force fields.\cite{Ponder:2010fk,schnieders:124114,Ren:2011uq} |
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Higher-order multipoles present a peculiar issue for molecular |
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dynamics. Multipolar interactions are inherently short-ranged, and |
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should not need the relatively expensive Ewald treatment. However, |
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real-space cutoff methods are normally applied in an orientation-blind |
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fashion so multipoles which leave and then re-enter a cutoff sphere in |
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a different orientation can cause energy discontinuities. |
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|
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This paper outlines an extension of the original DSF electrostatic |
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kernel to point multipoles. We describe two distinct real-space |
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interaction models for higher-order multipoles based on two truncated |
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Taylor expansions that are carried out at the cutoff radius. We are |
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calling these models {\bf Taylor-shifted} and {\bf Gradient-shifted} |
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electrostatics. Because of differences in the initial assumptions, |
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the two methods yield related, but different expressions for energies, |
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forces, and torques. |
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In this paper we outline the new methodology and give functional forms |
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for the energies, forces, and torques up to quadrupole-quadrupole |
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order. We also compare the new methods to analytic energy constants |
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for periodic arrays of point multipoles. In the following paper, we |
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provide numerical comparisons to Ewald-based electrostatics in common |
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simulation enviornments. |
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\section{Methodology} |
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\subsection{Self-neutralization, damping, and force-shifting} |
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The DSF and Wolf methods operate by neutralizing the total charge |
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contained within the cutoff sphere surrounding each particle. This is |
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accomplished by shifting the potential functions to generate image |
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charges on the surface of the cutoff sphere for each pair interaction |
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computed within $R_\textrm{c}$. Damping using a complementary error |
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function is applied to the potential to accelerate convergence. The |
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potential for the DSF method, where $\alpha$ is the adjustable damping |
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parameter, is given by |
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\begin{equation*} |
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V_\mathrm{DSF}(r) = C_a C_b \Biggr{[} |
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\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} |
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- \frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}} + \left(\frac{\mathrm{erfc}\left(\alpha R_\mathrm{c}\right)}{R_\mathrm{c}^2} |
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+ \frac{2\alpha}{\pi^{1/2}} |
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\frac{\exp\left(-\alpha^2R_\mathrm{c}^2\right)}{R_\mathrm{c}} |
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\right)\left(r_{ij}-R_\mathrm{c}\right)\ \Biggr{]} |
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\label{eq:DSFPot} |
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\end{equation*} |
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Note that in this potential and in all electrostatic quantities that |
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follow, the standard $4 \pi \epsilon_{0}$ has been omitted for |
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clarity. |
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To insure net charge neutrality within each cutoff sphere, an |
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additional ``self'' term is added to the potential. This term is |
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constant (as long as the charges and cutoff radius do not change), and |
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exists outside the normal pair-loop for molecular simulations. It can |
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be thought of as a contribution from a charge opposite in sign, but |
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equal in magnitude, to the central charge, which has been spread out |
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over the surface of the cutoff sphere. A portion of the self term is |
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identical to the self term in the Ewald summation, and comes from the |
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utilization of the complimentary error function for electrostatic |
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damping.\cite{deLeeuw80,Wolf99} |
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|
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There have been recent efforts to extend the Wolf self-neutralization |
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method to zero out the dipole and higher order multipoles contained |
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within the cutoff |
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sphere.\cite{Fukuda:2011jk,Fukuda:2012yu,Fukuda:2013qv} |
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|
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In this work, we extend the idea of self-neutralization for the point |
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multipoles by insuring net charge-neutrality and net-zero moments |
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within each cutoff sphere. In Figure \ref{fig:shiftedMultipoles}, the |
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central dipolar site $\mathbf{D}_i$ is interacting with point dipole |
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$\mathbf{D}_j$ and point quadrupole, $\mathbf{Q}_k$. The |
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self-neutralization scheme for point multipoles involves projecting |
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opposing multipoles for sites $j$ and $k$ on the surface of the cutoff |
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sphere. There are also significant modifications made to make the |
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forces and torques go smoothly to zero at the cutoff distance. |
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\begin{figure} |
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\includegraphics[width=3in]{SM} |
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\caption{Reversed multipoles are projected onto the surface of the |
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cutoff sphere. The forces, torques, and potential are then smoothly |
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shifted to zero as the sites leave the cutoff region.} |
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\label{fig:shiftedMultipoles} |
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\end{figure} |
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As in the point-charge approach, there is a contribution from |
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self-neutralization of site $i$. The self term for multipoles is |
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described in section \ref{sec:selfTerm}. |
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\subsection{The multipole expansion} |
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Consider two discrete rigid collections of point charges, denoted as |
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$\bf a$ and $\bf b$. In the following, we assume that the two objects |
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interact via electrostatics only and describe those interactions in |
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terms of a standard multipole expansion. Putting the origin of the |
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coordinate system at the center of mass of $\bf a$, we use vectors |
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$\mathbf{r}_k$ to denote the positions of all charges $q_k$ in $\bf |
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a$. Then the electrostatic potential of object $\bf a$ at |
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$\mathbf{r}$ is given by |
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\begin{equation} |
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V_a(\mathbf r) = |
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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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The Taylor expansion in $r$ can be written using an implied summation |
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notation. Here Greek indices are used to indicate space coordinates |
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($x$, $y$, $z$) and the subscripts $k$ and $j$ are reserved for |
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labelling specific charges in $\bf a$ and $\bf b$ respectively. The |
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Taylor expansion, |
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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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can then be used to express the electrostatic potential on $\bf a$ in |
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terms of multipole operators, |
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\begin{equation} |
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V_{\bf a}(\mathbf{r}) =\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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Here, the point charge, dipole, and quadrupole for object $\bf a$ are |
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given by $C_{\bf a}$, $D_{{\bf a}\alpha}$, and $Q_{{\bf |
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a}\alpha\beta}$, respectively. These are the primitive multipoles |
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which can be expressed as a distribution of charges, |
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\begin{align} |
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C_{\bf a} =&\sum_{k \, \text{in \bf a}} q_k , \\ |
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D_{{\bf a}\alpha} =&\sum_{k \, \text{in \bf a}} q_k r_{k\alpha} ,\\ |
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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{align} |
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Note that the definition of the primitive quadrupole here differs from |
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the standard traceless form, and contains an additional Taylor-series |
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based factor of $1/2$. |
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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{equation} |
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U_{\bf{ab}}(r) |
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= \hat{M}_a \sum_{j \, \text{in \bf b}} \frac {q_j}{\vert \bf{r}+\bf{r}_j \vert} . |
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\end{equation} |
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This can also be expanded as a Taylor series in $r$. Using a notation |
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similar to before to define the multipoles on object {\bf b}, |
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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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we arrive at the multipole expression for the total interaction energy. |
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\begin{equation} |
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U_{\bf{ab}}(r)=\hat{M}_{\bf a} \hat{M}_{\bf b} \frac{1}{r} \label{kernel}. |
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\end{equation} |
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This form has the benefit of separating out the energies of |
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interaction into contributions from the charge, dipole, and quadrupole |
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of $\bf a$ interacting with the same multipoles $\bf b$. |
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\subsection{Damped Coulomb interactions} |
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In the standard multipole expansion, one typically uses the bare |
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Coulomb potential, with radial dependence $1/r$, as shown in |
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Eq.~(\ref{kernel}). It is also quite common to use a damped Coulomb |
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interaction, which results from replacing point charges with Gaussian |
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distributions of charge with width $\alpha$. In damped multipole |
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electrostatics, the kernel ($1/r$) of the expansion is replaced with |
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the 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 below using the function $f(r)$ to represent |
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either $1/r$ or $B_0(r)$, and all of the techniques can be applied |
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either to bare or damped Coulomb kernels as long as derivatives of |
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these functions are known. Smith's convenient functions $B_l(r)$ are |
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summarized in Appendix A. |
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The main goal of this work is to smoothly cut off the interaction |
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energy as well as forces and torques as $r\rightarrow r_c$. To |
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|
|
describe how this goal may be met, we use two examples, charge-charge |
312 |
|
|
and charge-dipole, using the bare Coulomb kernel $f(r)=1/r$ to explain |
313 |
|
|
the idea. |
314 |
gezelter |
3906 |
|
315 |
gezelter |
3984 |
\subsection{Shifted-force methods} |
316 |
gezelter |
3982 |
In the shifted-force approximation, the interaction energy for two |
317 |
|
|
charges $C_{\bf a}$ and $C_{\bf b}$ separated by a distance $r$ is |
318 |
|
|
written: |
319 |
gezelter |
3906 |
\begin{equation} |
320 |
gezelter |
3985 |
U_{C_{\bf a}C_{\bf b}}(r)= C_{\bf a} C_{\bf b} |
321 |
gezelter |
3906 |
\left({ \frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
322 |
|
|
\right) . |
323 |
|
|
\end{equation} |
324 |
gezelter |
3982 |
Two shifting terms appear in this equations, one from the |
325 |
gezelter |
3984 |
neutralization procedure ($-1/r_c$), and one that causes the first |
326 |
|
|
derivative to vanish at the cutoff radius. |
327 |
gezelter |
3982 |
|
328 |
|
|
Since one derivative of the interaction energy is needed for the |
329 |
|
|
force, the minimal perturbation is a term linear in $(r-r_c)$ in the |
330 |
|
|
interaction energy, that is: |
331 |
gezelter |
3906 |
\begin{equation} |
332 |
|
|
\frac{d\,}{dr} |
333 |
|
|
\left( {\frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
334 |
|
|
\right) = \left(- \frac{1}{r^2} + \frac{1}{r_c^2} |
335 |
|
|
\right) . |
336 |
|
|
\end{equation} |
337 |
gezelter |
3985 |
which clearly vanishes as the $r$ approaches the cutoff radius. There |
338 |
|
|
are a number of ways to generalize this derivative shift for |
339 |
gezelter |
3984 |
higher-order multipoles. Below, we present two methods, one based on |
340 |
|
|
higher-order Taylor series for $r$ near $r_c$, and the other based on |
341 |
|
|
linear shift of the kernel gradients at the cutoff itself. |
342 |
gezelter |
3906 |
|
343 |
gezelter |
3984 |
\subsection{Taylor-shifted force (TSF) electrostatics} |
344 |
gezelter |
3982 |
In the Taylor-shifted force (TSF) method, the procedure that we follow |
345 |
|
|
is based on a Taylor expansion containing the same number of |
346 |
|
|
derivatives required for each force term to vanish at the cutoff. For |
347 |
|
|
example, the quadrupole-quadrupole interaction energy requires four |
348 |
|
|
derivatives of the kernel, and the force requires one additional |
349 |
|
|
derivative. We therefore require shifted energy expressions that |
350 |
|
|
include enough terms so that all energies, forces, and torques are |
351 |
|
|
zero as $r \rightarrow r_c$. In each case, we will subtract off a |
352 |
|
|
function $f_n^{\text{shift}}(r)$ from the kernel $f(r)=1/r$. The |
353 |
gezelter |
3984 |
subscript $n$ indicates the number of derivatives to be taken when |
354 |
gezelter |
3982 |
deriving a given multipole energy. We choose a function with |
355 |
|
|
guaranteed smooth derivatives --- a truncated Taylor series of the |
356 |
|
|
function $f(r)$, e.g., |
357 |
gezelter |
3906 |
% |
358 |
|
|
\begin{equation} |
359 |
gezelter |
3984 |
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)}(r_c) . |
360 |
gezelter |
3906 |
\end{equation} |
361 |
|
|
% |
362 |
|
|
The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$. |
363 |
|
|
Thus, for $f(r)=1/r$, we find |
364 |
|
|
% |
365 |
|
|
\begin{equation} |
366 |
|
|
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} . |
367 |
|
|
\end{equation} |
368 |
|
|
% |
369 |
gezelter |
3982 |
Continuing with the example of a charge $\bf a$ interacting with a |
370 |
|
|
dipole $\bf b$, we write |
371 |
gezelter |
3906 |
% |
372 |
|
|
\begin{equation} |
373 |
|
|
U_{C_{\bf a}D_{\bf b}}(r)= |
374 |
gezelter |
3985 |
C_{\bf a} D_{{\bf b}\alpha} \frac {\partial f_1(r) }{\partial r_\alpha} |
375 |
|
|
= C_{\bf a} D_{{\bf b}\alpha} |
376 |
gezelter |
3906 |
\frac {r_\alpha}{r} \frac {\partial f_1(r)}{\partial r} . |
377 |
|
|
\end{equation} |
378 |
|
|
% |
379 |
gezelter |
3984 |
The force that dipole $\bf b$ exerts on charge $\bf a$ is |
380 |
gezelter |
3906 |
% |
381 |
|
|
\begin{equation} |
382 |
gezelter |
3985 |
F_{C_{\bf a}D_{\bf b}\beta} = C_{\bf a} D_{{\bf b}\alpha} |
383 |
gezelter |
3906 |
\left[ \frac{\delta_{\alpha\beta}}{r} \frac {\partial}{\partial r} + |
384 |
|
|
\frac{r_\alpha r_\beta}{r^2} |
385 |
|
|
\left( -\frac{1}{r} \frac {\partial} {\partial r} |
386 |
|
|
+ \frac {\partial ^2} {\partial r^2} \right) \right] f_1(r) . |
387 |
|
|
\end{equation} |
388 |
|
|
% |
389 |
gezelter |
3984 |
For undamped coulombic interactions, $f(r)=1/r$, we find |
390 |
gezelter |
3906 |
% |
391 |
|
|
\begin{equation} |
392 |
|
|
F_{C_{\bf a}D_{\bf b}\beta} = |
393 |
gezelter |
3985 |
\frac{C_{\bf a} D_{{\bf b}\beta}}{r} |
394 |
gezelter |
3906 |
\left[ -\frac{1}{r^2}+\frac{1}{r_c^2}-\frac{2(r-r_c)}{r_c^3} \right] |
395 |
gezelter |
3985 |
+C_{\bf a} D_{{\bf b}\alpha}r_\alpha r_\beta |
396 |
gezelter |
3906 |
\left[ \frac{3}{r^5}-\frac{3}{r^3r_c^2} \right] . |
397 |
|
|
\end{equation} |
398 |
|
|
% |
399 |
|
|
This expansion shows the expected $1/r^3$ dependence of the force. |
400 |
|
|
|
401 |
gezelter |
3984 |
In general, we can write |
402 |
gezelter |
3906 |
% |
403 |
|
|
\begin{equation} |
404 |
gezelter |
3985 |
U= (\text{prefactor}) (\text{derivatives}) f_n(r) |
405 |
gezelter |
3906 |
\label{generic} |
406 |
|
|
\end{equation} |
407 |
|
|
% |
408 |
gezelter |
3985 |
with $n=0$ for charge-charge, $n=1$ for charge-dipole, $n=2$ for |
409 |
|
|
charge-quadrupole and dipole-dipole, $n=3$ for dipole-quadrupole, and |
410 |
|
|
$n=4$ for quadrupole-quadrupole. For example, in |
411 |
|
|
quadrupole-quadrupole interactions for which the $\text{prefactor}$ is |
412 |
|
|
$Q_{{\bf a}\alpha\beta}Q_{{\bf b}\gamma\delta}$, the derivatives are |
413 |
|
|
$\partial^4/\partial r_\alpha \partial r_\beta \partial |
414 |
|
|
r_\gamma \partial r_\delta$, with implied summation combining the |
415 |
|
|
space indices. |
416 |
gezelter |
3906 |
|
417 |
gezelter |
3984 |
In the formulas presented in the tables below, the placeholder |
418 |
|
|
function $f(r)$ is used to represent the electrostatic kernel (either |
419 |
|
|
damped or undamped). The main functions that go into the force and |
420 |
gezelter |
3985 |
torque terms, $g_n(r), h_n(r), s_n(r), \mathrm{~and~} t_n(r)$ are |
421 |
|
|
successive derivatives of the shifted electrostatic kernel, $f_n(r)$ |
422 |
|
|
of the same index $n$. The algebra required to evaluate energies, |
423 |
|
|
forces and torques is somewhat tedious, so only the final forms are |
424 |
|
|
presented in tables XX and YY. |
425 |
gezelter |
3906 |
|
426 |
gezelter |
3982 |
\subsection{Gradient-shifted force (GSF) electrostatics} |
427 |
gezelter |
3985 |
The second, and conceptually simpler approach to force-shifting |
428 |
|
|
maintains only the linear $(r-r_c)$ term in the truncated Taylor |
429 |
|
|
expansion, and has a similar interaction energy for all multipole |
430 |
|
|
orders: |
431 |
gezelter |
3906 |
\begin{equation} |
432 |
gezelter |
3985 |
U^{\text{shift}}(r)=U(r)-U(r_c)-(r-r_c)\hat{r}\cdot \nabla U(r) \Big |
433 |
|
|
\lvert _{r_c} . |
434 |
|
|
\label{generic2} |
435 |
gezelter |
3906 |
\end{equation} |
436 |
gezelter |
3985 |
Here the gradient for force shifting is evaluated for an image |
437 |
|
|
multipole on the surface of the cutoff sphere (see fig |
438 |
|
|
\ref{fig:shiftedMultipoles}). No higher order terms $(r-r_c)^n$ |
439 |
|
|
appear. The primary difference between the TSF and GSF methods is the |
440 |
|
|
stage at which the Taylor Series is applied; in the Taylor-shifted |
441 |
|
|
approach, it is applied to the kernel itself. In the Gradient-shifted |
442 |
|
|
approach, it is applied to individual radial interactions terms in the |
443 |
|
|
multipole expansion. Energies from this method thus have the general |
444 |
|
|
form: |
445 |
gezelter |
3906 |
\begin{equation} |
446 |
gezelter |
3985 |
U= \sum (\text{angular factor}) (\text{radial factor}). |
447 |
|
|
\label{generic3} |
448 |
gezelter |
3906 |
\end{equation} |
449 |
|
|
|
450 |
gezelter |
3985 |
Functional forms for both methods (TSF and GSF) can be summarized |
451 |
|
|
using the form of Eq.~(\ref{generic3}). The basic forms for the |
452 |
|
|
energy, force, and torque expressions are tabulated for both shifting |
453 |
|
|
approaches below - for each separate orientational contribution, only |
454 |
|
|
the radial factors differ between the two methods. |
455 |
gezelter |
3906 |
|
456 |
|
|
\subsection{\label{sec:level2}Body and space axes} |
457 |
|
|
|
458 |
gezelter |
3985 |
[XXX Do we need this section in the main paper? or should it go in the |
459 |
|
|
extra materials?] |
460 |
|
|
|
461 |
gezelter |
3984 |
So far, all energies and forces have been written in terms of fixed |
462 |
gezelter |
3985 |
space coordinates. Interaction energies are computed from the generic |
463 |
|
|
formulas Eq.~(\ref{generic}) and ~(\ref{generic2}) which combine |
464 |
|
|
orientational prefactors with radial functions. Because objects $\bf |
465 |
gezelter |
3984 |
a$ and $\bf b$ both translate and rotate during a molecular dynamics |
466 |
|
|
(MD) simulation, it is desirable to contract all $r$-dependent terms |
467 |
|
|
with dipole and quadrupole moments expressed in terms of their body |
468 |
gezelter |
3985 |
axes. To do so, we have followed the methodology of Allen and |
469 |
|
|
Germano,\cite{Allen:2006fk} which was itself based on earlier work by |
470 |
|
|
Price {\em et al.}\cite{Price:1984fk} |
471 |
gezelter |
3906 |
|
472 |
gezelter |
3984 |
We denote body axes for objects $\bf a$ and $\bf b$ by unit vectors |
473 |
|
|
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$ |
474 |
|
|
referring to a convenient set of inertial body axes. (N.B., these |
475 |
|
|
body axes are generally not the same as those for which the quadrupole |
476 |
|
|
moment is diagonal.) Then, |
477 |
gezelter |
3906 |
% |
478 |
|
|
\begin{eqnarray} |
479 |
|
|
\hat{a}_m= a_{mx}\hat{x} + a_{my}\hat{y} + a_{mz}\hat{z} \\ |
480 |
|
|
\hat{b}_m= b_{mx}\hat{x} + b_{my}\hat{y} + b_{mz}\hat{z} . |
481 |
|
|
\end{eqnarray} |
482 |
gezelter |
3985 |
Rotation matrices $\hat{\mathbf {a}}$ and $\hat{\mathbf {b}}$ can be |
483 |
|
|
expressed using these unit vectors: |
484 |
gezelter |
3906 |
\begin{eqnarray} |
485 |
|
|
\hat{\mathbf {a}} = |
486 |
|
|
\begin{pmatrix} |
487 |
|
|
\hat{a}_1 \\ |
488 |
|
|
\hat{a}_2 \\ |
489 |
|
|
\hat{a}_3 |
490 |
|
|
\end{pmatrix} |
491 |
|
|
= |
492 |
|
|
\begin{pmatrix} |
493 |
|
|
a_{1x} \quad a_{1y} \quad a_{1z} \\ |
494 |
|
|
a_{2x} \quad a_{2y} \quad a_{2z} \\ |
495 |
|
|
a_{3x} \quad a_{3y} \quad a_{3z} |
496 |
|
|
\end{pmatrix}\\ |
497 |
|
|
\hat{\mathbf {b}} = |
498 |
|
|
\begin{pmatrix} |
499 |
|
|
\hat{b}_1 \\ |
500 |
|
|
\hat{b}_2 \\ |
501 |
|
|
\hat{b}_3 |
502 |
|
|
\end{pmatrix} |
503 |
|
|
= |
504 |
|
|
\begin{pmatrix} |
505 |
gezelter |
3985 |
b_{1x} \quad b_{1y} \quad b_{1z} \\ |
506 |
gezelter |
3906 |
b_{2x} \quad b_{2y} \quad b_{2z} \\ |
507 |
|
|
b_{3x} \quad b_{3y} \quad b_{3z} |
508 |
|
|
\end{pmatrix} . |
509 |
|
|
\end{eqnarray} |
510 |
|
|
% |
511 |
gezelter |
3985 |
These matrices convert from space-fixed $(xyz)$ to body-fixed $(123)$ |
512 |
|
|
coordinates. All contractions of prefactors with derivatives of |
513 |
|
|
functions can be written in terms of these matrices. It proves to be |
514 |
|
|
equally convenient to just write any contraction in terms of unit |
515 |
|
|
vectors $\hat{r}$, $\hat{a}_m$, and $\hat{b}_n$. In the torque |
516 |
|
|
expressions, it is useful to have the angular-dependent terms |
517 |
|
|
available in three different fashions, e.g. for the dipole-dipole |
518 |
|
|
contraction: |
519 |
gezelter |
3906 |
% |
520 |
gezelter |
3985 |
\begin{equation} |
521 |
gezelter |
3906 |
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
522 |
gezelter |
3985 |
= D_{\bf {a}\alpha} D_{\bf {b}\alpha} = |
523 |
|
|
\sum_{mn} {D_{\mathbf{a}m} \hat{a}_m \cdot \hat{b}_n D_{\mathbf{b}n}} |
524 |
|
|
\end{equation} |
525 |
gezelter |
3906 |
% |
526 |
gezelter |
3985 |
The first two forms are written using space coordinates. The first |
527 |
|
|
form is standard in the chemistry literature, while the second is |
528 |
|
|
expressed using implied summation notation. The third form shows |
529 |
|
|
explicit sums over body indices and the dot products now indicate |
530 |
|
|
contractions using space indices. |
531 |
gezelter |
3906 |
|
532 |
|
|
|
533 |
gezelter |
3982 |
\subsection{The Self-Interaction \label{sec:selfTerm}} |
534 |
|
|
|
535 |
gezelter |
3985 |
In addition to cutoff-sphere neutralization, the Wolf |
536 |
|
|
summation~\cite{Wolf99} and the damped shifted force (DSF) |
537 |
|
|
extension~\cite{Fennell:2006zl} also included self-interactions that |
538 |
|
|
are handled separately from the pairwise interactions between |
539 |
|
|
sites. The self-term is normally calculated via a single loop over all |
540 |
|
|
sites in the system, and is relatively cheap to evaluate. The |
541 |
|
|
self-interaction has contributions from two sources. |
542 |
|
|
|
543 |
|
|
First, the neutralization procedure within the cutoff radius requires |
544 |
|
|
a contribution from a charge opposite in sign, but equal in magnitude, |
545 |
|
|
to the central charge, which has been spread out over the surface of |
546 |
|
|
the cutoff sphere. For a system of undamped charges, the total |
547 |
|
|
self-term is |
548 |
gezelter |
3980 |
\begin{equation} |
549 |
|
|
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
550 |
|
|
\label{eq:selfTerm} |
551 |
|
|
\end{equation} |
552 |
gezelter |
3985 |
|
553 |
|
|
Second, charge damping with the complementary error function is a |
554 |
|
|
partial analogy to the Ewald procedure which splits the interaction |
555 |
|
|
into real and reciprocal space sums. The real space sum is retained |
556 |
|
|
in the Wolf and DSF methods. The reciprocal space sum is first |
557 |
|
|
minimized by folding the largest contribution (the self-interaction) |
558 |
|
|
into the self-interaction from charge neutralization of the damped |
559 |
|
|
potential. The remainder of the reciprocal space portion is then |
560 |
|
|
discarded (as this contributes the largest computational cost and |
561 |
|
|
complexity to the Ewald sum). For a system containing only damped |
562 |
|
|
charges, the complete self-interaction can be written as |
563 |
gezelter |
3980 |
\begin{equation} |
564 |
|
|
V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
565 |
|
|
\frac{\alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N |
566 |
|
|
C_{\bf a}^{2}. |
567 |
|
|
\label{eq:dampSelfTerm} |
568 |
|
|
\end{equation} |
569 |
|
|
|
570 |
|
|
The extension of DSF electrostatics to point multipoles requires |
571 |
|
|
treatment of {\it both} the self-neutralization and reciprocal |
572 |
|
|
contributions to the self-interaction for higher order multipoles. In |
573 |
|
|
this section we give formulae for these interactions up to quadrupolar |
574 |
|
|
order. |
575 |
|
|
|
576 |
|
|
The self-neutralization term is computed by taking the {\it |
577 |
|
|
non-shifted} kernel for each interaction, placing a multipole of |
578 |
|
|
equal magnitude (but opposite in polarization) on the surface of the |
579 |
|
|
cutoff sphere, and averaging over the surface of the cutoff sphere. |
580 |
|
|
Because the self term is carried out as a single sum over sites, the |
581 |
|
|
reciprocal-space portion is identical to half of the self-term |
582 |
|
|
obtained by Smith and Aguado and Madden for the application of the |
583 |
|
|
Ewald sum to multipoles.\cite{Smith82,Smith98,Aguado03} For a given |
584 |
|
|
site which posesses a charge, dipole, and multipole, both types of |
585 |
|
|
contribution are given in table \ref{tab:tableSelf}. |
586 |
|
|
|
587 |
|
|
\begin{table*} |
588 |
|
|
\caption{\label{tab:tableSelf} Self-interaction contributions for |
589 |
|
|
site ({\bf a}) that has a charge $(C_{\bf a})$, dipole |
590 |
|
|
$(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$} |
591 |
|
|
\begin{ruledtabular} |
592 |
|
|
\begin{tabular}{lccc} |
593 |
|
|
Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline |
594 |
|
|
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
595 |
|
|
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
596 |
|
|
\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
597 |
|
|
Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
598 |
|
|
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
599 |
|
|
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
600 |
|
|
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
601 |
|
|
h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$ \\ |
602 |
|
|
\end{tabular} |
603 |
|
|
\end{ruledtabular} |
604 |
|
|
\end{table*} |
605 |
|
|
|
606 |
|
|
For sites which simultaneously contain charges and quadrupoles, the |
607 |
|
|
self-interaction includes a cross-interaction between these two |
608 |
|
|
multipole orders. Symmetry prevents the charge-dipole and |
609 |
|
|
dipole-quadrupole interactions from contributing to the |
610 |
|
|
self-interaction. The functions that go into the self-neutralization |
611 |
gezelter |
3985 |
terms, $g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive |
612 |
|
|
derivatives of the electrostatic kernel, $f(r)$ (either the undamped |
613 |
|
|
$1/r$ or the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that |
614 |
|
|
have been evaluated at the cutoff distance. For undamped |
615 |
|
|
interactions, $f(r_c) = 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For |
616 |
|
|
damped interactions, $f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so |
617 |
|
|
on. Appendix \ref{SmithFunc} contains recursion relations that allow |
618 |
|
|
rapid evaluation of these derivatives. |
619 |
gezelter |
3980 |
|
620 |
gezelter |
3985 |
\section{Interaction energies, forces, and torques} |
621 |
|
|
The main result of this paper is a set of expressions for the |
622 |
|
|
energies, forces and torques (up to quadrupole-quadrupole order) that |
623 |
|
|
work for both the Taylor-shifted and Gradient-shifted approximations. |
624 |
|
|
These expressions were derived using a set of generic radial |
625 |
|
|
functions. Without using the shifting approximations mentioned above, |
626 |
|
|
some of these radial functions would be identical, and the expressions |
627 |
|
|
coalesce into the familiar forms for unmodified multipole-multipole |
628 |
|
|
interactions. Table \ref{tab:tableenergy} maps between the generic |
629 |
|
|
functions and the radial functions derived for both the Taylor-shifted |
630 |
|
|
and Gradient-shifted methods. The energy equations are written in |
631 |
|
|
terms of lab-frame representations of the dipoles, quadrupoles, and |
632 |
|
|
the unit vector connecting the two objects, |
633 |
gezelter |
3906 |
|
634 |
|
|
% Energy in space coordinate form ---------------------------------------------------------------------------------------------- |
635 |
|
|
% |
636 |
|
|
% |
637 |
|
|
% u ca cb |
638 |
|
|
% |
639 |
gezelter |
3983 |
\begin{align} |
640 |
|
|
U_{C_{\bf a}C_{\bf b}}(r)=& |
641 |
gezelter |
3985 |
C_{\bf a} C_{\bf b} v_{01}(r) \label{uchch} |
642 |
gezelter |
3983 |
\\ |
643 |
gezelter |
3906 |
% |
644 |
|
|
% u ca db |
645 |
|
|
% |
646 |
gezelter |
3983 |
U_{C_{\bf a}D_{\bf b}}(r)=& |
647 |
gezelter |
3985 |
C_{\bf a} \left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) v_{11}(r) |
648 |
gezelter |
3906 |
\label{uchdip} |
649 |
gezelter |
3983 |
\\ |
650 |
gezelter |
3906 |
% |
651 |
|
|
% u ca qb |
652 |
|
|
% |
653 |
gezelter |
3985 |
U_{C_{\bf a}Q_{\bf b}}(r)=& C_{\bf a } \Bigl[ \text{Tr}Q_{\bf b} |
654 |
|
|
v_{21}(r) + \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot |
655 |
|
|
\hat{r} \right) v_{22}(r) \Bigr] |
656 |
gezelter |
3906 |
\label{uchquad} |
657 |
gezelter |
3983 |
\\ |
658 |
gezelter |
3906 |
% |
659 |
|
|
% u da cb |
660 |
|
|
% |
661 |
gezelter |
3983 |
%U_{D_{\bf a}C_{\bf b}}(r)=& |
662 |
|
|
%-\frac{C_{\bf b}}{4\pi \epsilon_0} |
663 |
|
|
%\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) v_{11}(r) \label{udipch} |
664 |
|
|
%\\ |
665 |
gezelter |
3906 |
% |
666 |
|
|
% u da db |
667 |
|
|
% |
668 |
gezelter |
3983 |
U_{D_{\bf a}D_{\bf b}}(r)=& |
669 |
gezelter |
3985 |
-\Bigr[ \left( \mathbf{D}_{\mathbf {a}} \cdot |
670 |
gezelter |
3906 |
\mathbf{D}_{\mathbf{b}} \right) v_{21}(r) |
671 |
|
|
+\left( \mathbf{D}_{\mathbf {a}} \cdot \hat{r} \right) |
672 |
|
|
\left( \mathbf{D}_{\mathbf {b}} \cdot \hat{r} \right) |
673 |
|
|
v_{22}(r) \Bigr] |
674 |
|
|
\label{udipdip} |
675 |
gezelter |
3983 |
\\ |
676 |
gezelter |
3906 |
% |
677 |
|
|
% u da qb |
678 |
|
|
% |
679 |
|
|
\begin{split} |
680 |
|
|
% 1 |
681 |
gezelter |
3983 |
U_{D_{\bf a}Q_{\bf b}}(r) =& |
682 |
gezelter |
3985 |
-\Bigl[ |
683 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
684 |
|
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
685 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
686 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r} ) \Bigr] v_{31}(r) \\ |
687 |
|
|
% 2 |
688 |
gezelter |
3985 |
&- \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
689 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{32}(r) |
690 |
|
|
\label{udipquad} |
691 |
|
|
\end{split} |
692 |
gezelter |
3983 |
\\ |
693 |
gezelter |
3906 |
% |
694 |
|
|
% u qa cb |
695 |
|
|
% |
696 |
gezelter |
3983 |
%U_{Q_{\bf a}C_{\bf b}}(r)=& |
697 |
|
|
%\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\bf a} v_{21}(r) |
698 |
|
|
%\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
699 |
|
|
%\label{uquadch} |
700 |
|
|
%\\ |
701 |
gezelter |
3906 |
% |
702 |
|
|
% u qa db |
703 |
|
|
% |
704 |
gezelter |
3983 |
%\begin{split} |
705 |
gezelter |
3906 |
%1 |
706 |
gezelter |
3983 |
%U_{Q_{\bf a}D_{\bf b}}(r)=& |
707 |
|
|
%\frac{1}{4\pi \epsilon_0} \Bigl[ |
708 |
|
|
%\text{Tr}\mathbf{Q}_{\mathbf{a}} |
709 |
|
|
%\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
710 |
|
|
%+2 ( \mathbf{D}_{\mathbf{b}} \cdot |
711 |
|
|
%\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r)\\ |
712 |
gezelter |
3906 |
% 2 |
713 |
gezelter |
3983 |
%&+\frac{1}{4\pi \epsilon_0} |
714 |
|
|
%\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
715 |
|
|
%\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{32}(r) |
716 |
|
|
%\label{uquaddip} |
717 |
|
|
%\end{split} |
718 |
|
|
%\\ |
719 |
gezelter |
3906 |
% |
720 |
|
|
% u qa qb |
721 |
|
|
% |
722 |
|
|
\begin{split} |
723 |
|
|
%1 |
724 |
gezelter |
3983 |
U_{Q_{\bf a}Q_{\bf b}}(r)=& |
725 |
gezelter |
3985 |
\Bigl[ |
726 |
gezelter |
3906 |
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
727 |
|
|
+2 \text{Tr} \left( |
728 |
|
|
\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \right) \Bigr] v_{41}(r) |
729 |
|
|
\\ |
730 |
|
|
% 2 |
731 |
gezelter |
3985 |
&+\Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
732 |
gezelter |
3906 |
\left( \hat{r} \cdot |
733 |
|
|
\mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) |
734 |
|
|
+\text{Tr}\mathbf{Q}_{\mathbf{b}} |
735 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} |
736 |
|
|
\cdot \hat{r} \right) +4 (\hat{r} \cdot |
737 |
|
|
\mathbf{Q}_{{\mathbf a}}\cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
738 |
|
|
\Bigr] v_{42}(r) |
739 |
|
|
\\ |
740 |
|
|
% 4 |
741 |
gezelter |
3985 |
&+ |
742 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) |
743 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{43}(r). |
744 |
|
|
\label{uquadquad} |
745 |
|
|
\end{split} |
746 |
gezelter |
3983 |
\end{align} |
747 |
gezelter |
3985 |
% |
748 |
gezelter |
3983 |
Note that the energies of multipoles on site $\mathbf{b}$ interacting |
749 |
|
|
with those on site $\mathbf{a}$ can be obtained by swapping indices |
750 |
|
|
along with the sign of the intersite vector, $\hat{r}$. |
751 |
gezelter |
3906 |
|
752 |
|
|
% |
753 |
|
|
% |
754 |
|
|
% TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
755 |
|
|
% |
756 |
|
|
|
757 |
gezelter |
3985 |
\begin{sidewaystable} |
758 |
|
|
\caption{\label{tab:tableenergy}Radial functions used in the energy |
759 |
|
|
and torque equations. The $f, g, h, s, t, \mathrm{and} u$ |
760 |
|
|
functions used in this table are defined in Appendices B and C.} |
761 |
|
|
\begin{tabular}{|c|c|l|l|} \hline |
762 |
|
|
Generic&Bare Coulomb&Taylor-Shifted&Gradient-Shifted |
763 |
gezelter |
3906 |
\\ \hline |
764 |
|
|
% |
765 |
|
|
% |
766 |
|
|
% |
767 |
|
|
%Ch-Ch& |
768 |
|
|
$v_{01}(r)$ & |
769 |
|
|
$\frac{1}{r}$ & |
770 |
|
|
$f_0(r)$ & |
771 |
|
|
$f(r)-f(r_c)-(r-r_c)g(r_c)$ |
772 |
|
|
\\ |
773 |
|
|
% |
774 |
|
|
% |
775 |
|
|
% |
776 |
|
|
%Ch-Di& |
777 |
|
|
$v_{11}(r)$ & |
778 |
|
|
$-\frac{1}{r^2}$ & |
779 |
|
|
$g_1(r)$ & |
780 |
|
|
$g(r)-g(r_c)-(r-r_c)h(r_c)$ \\ |
781 |
|
|
% |
782 |
|
|
% |
783 |
|
|
% |
784 |
|
|
%Ch-Qu/Di-Di& |
785 |
|
|
$v_{21}(r)$ & |
786 |
|
|
$-\frac{1}{r^3} $ & |
787 |
|
|
$\frac{g_2(r)}{r} $ & |
788 |
|
|
$\frac{g(r)}{r}-\frac{g(r_c)}{r_c} -(r-r_c) |
789 |
|
|
\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right)$ \\ |
790 |
|
|
$v_{22}(r)$ & |
791 |
|
|
$\frac{3}{r^3} $ & |
792 |
|
|
$\left(-\frac{g_2(r)}{r} + h_2(r) \right)$ & |
793 |
|
|
$\left(-\frac{g(r)}{r}+h(r) \right) |
794 |
gezelter |
3985 |
-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right)$ \\ |
795 |
|
|
&&& $ ~~~-(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$ |
796 |
gezelter |
3906 |
\\ |
797 |
|
|
% |
798 |
|
|
% |
799 |
|
|
% |
800 |
|
|
%Di-Qu & |
801 |
|
|
$v_{31}(r)$ & |
802 |
|
|
$\frac{3}{r^4} $ & |
803 |
|
|
$\left(-\frac{g_3(r)}{r^2} + \frac{h_3(r)}{r} \right)$ & |
804 |
|
|
$\left( -\frac{g(r)}{r^2}+\frac{h(r)}{r} \right) |
805 |
|
|
-\left(-\frac{g(r_c)}{r_c^2}+\frac{h(r_c)}{r_c} \right) $\\ |
806 |
gezelter |
3985 |
&&&$ ~~~ -(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)$ |
807 |
gezelter |
3906 |
\\ |
808 |
|
|
% |
809 |
|
|
$v_{32}(r)$ & |
810 |
|
|
$-\frac{15}{r^4} $ & |
811 |
|
|
$\left( \frac{3g_3(r)}{r^2} - \frac{3h_3(r)}{r} + s_3(r) \right)$ & |
812 |
|
|
$\left( \frac{3g(r)}{r^2} - \frac{3h(r)}{r} + s(r) \right) |
813 |
|
|
- \left( \frac{3g(r_c)}{r_c^2} - \frac{3h(r_c)}{r_c} + s(r_c) \right)$ \\ |
814 |
gezelter |
3985 |
&&&$ ~~~ -(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)$ |
815 |
gezelter |
3906 |
\\ |
816 |
|
|
% |
817 |
|
|
% |
818 |
|
|
% |
819 |
|
|
%Qu-Qu& |
820 |
|
|
$v_{41}(r)$ & |
821 |
|
|
$\frac{3}{r^5} $ & |
822 |
|
|
$\left(-\frac{g_4(r)}{r^3} +\frac{h_4(r)}{r^2} \right) $ & |
823 |
|
|
$\left( -\frac{g(r)}{r^3} + \frac{h(r)}{r^2} \right) |
824 |
|
|
- \left( -\frac{g(r_c)}{r_c^3} + \frac{h(r_c)}{r_c^2} \right)$ \\ |
825 |
gezelter |
3985 |
&&&$ ~~~ -(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)$ |
826 |
gezelter |
3906 |
\\ |
827 |
|
|
% 2 |
828 |
|
|
$v_{42}(r)$ & |
829 |
|
|
$- \frac{15}{r^5} $ & |
830 |
|
|
$\left( \frac{3g_4(r)}{r^3} - \frac{3h_4(r)}{r^2}+\frac{s_4(r)}{r} \right)$ & |
831 |
|
|
$\left( \frac{3g(r)}{r^3} - \frac{3h(r)}{r^2}+\frac{s(r)}{r} \right) |
832 |
|
|
-\left( \frac{3g(r_c)}{r_c^3} - \frac{3h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ \\ |
833 |
gezelter |
3985 |
&&&$ ~~~ -(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
834 |
gezelter |
3906 |
-\frac{4s(r_c)}{r_c^2} + \frac{t(r_c)}{r_c}\right)$ |
835 |
|
|
\\ |
836 |
|
|
% 3 |
837 |
|
|
$v_{43}(r)$ & |
838 |
|
|
$ \frac{105}{r^5} $ & |
839 |
|
|
$\left(-\frac{15g_4(r)}{r^3}+\frac{15h_4(r)}{r^2}-\frac{6s_4(r)}{r} + t_4(r)\right) $ & |
840 |
|
|
$\left(-\frac{15g(r)}{r^3}+\frac{15h(r)}{r^2}-\frac{6s(r)}{r} + t(r)\right)$ \\ |
841 |
gezelter |
3985 |
&&&$~~~ -\left(-\frac{15g(r_c)}{r_c^3}+\frac{15h(r_c)}{r_c^2}-\frac{6s(r_c)}{r_c} + t(r_c)\right)$ \\ |
842 |
|
|
&&&$~~~ -(r-r_c)\left(\frac{45g(r_c)}{r_c^4}-\frac{45h(r_c)}{r_c^3}+\frac{21s(r_c)}{r_c^2} |
843 |
|
|
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ \hline |
844 |
gezelter |
3906 |
\end{tabular} |
845 |
gezelter |
3985 |
\end{sidewaystable} |
846 |
gezelter |
3906 |
% |
847 |
|
|
% |
848 |
|
|
% FORCE TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
849 |
|
|
% |
850 |
|
|
|
851 |
gezelter |
3985 |
\begin{sidewaystable} |
852 |
gezelter |
3906 |
\caption{\label{tab:tableFORCE}Radial functions used in the force equations.} |
853 |
gezelter |
3985 |
\begin{tabular}{|c|c|l|l|} \hline |
854 |
|
|
Function&Definition&Taylor-Shifted&Gradient-Shifted |
855 |
gezelter |
3906 |
\\ \hline |
856 |
|
|
% |
857 |
|
|
% |
858 |
|
|
% |
859 |
|
|
$w_a(r)$& |
860 |
gezelter |
3985 |
$\frac{d v_{01}}{dr}$& |
861 |
|
|
$g_0(r)$& |
862 |
|
|
$g(r)-g(r_c)$ \\ |
863 |
gezelter |
3906 |
% |
864 |
|
|
% |
865 |
|
|
$w_b(r)$ & |
866 |
gezelter |
3985 |
$\frac{d v_{11}}{dr} - \frac{v_{11}(r)}{r} $& |
867 |
|
|
$\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
868 |
|
|
$h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
869 |
gezelter |
3906 |
% |
870 |
|
|
$w_c(r)$ & |
871 |
gezelter |
3985 |
$\frac{v_{11}(r)}{r}$ & |
872 |
|
|
$\frac{g_1(r)}{r} $ & |
873 |
|
|
$\frac{v_{11}(r)}{r}$\\ |
874 |
gezelter |
3906 |
% |
875 |
|
|
% |
876 |
|
|
$w_d(r)$& |
877 |
gezelter |
3985 |
$\frac{d v_{21}}{dr}$& |
878 |
|
|
$\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
879 |
|
|
$\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
880 |
|
|
-\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $ \\ |
881 |
gezelter |
3906 |
% |
882 |
|
|
$w_e(r)$ & |
883 |
gezelter |
3985 |
$\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
884 |
|
|
$\frac{v_{22}(r)}{r}$ & |
885 |
gezelter |
3906 |
$\frac{v_{22}(r)}{r}$ \\ |
886 |
|
|
% |
887 |
|
|
% |
888 |
|
|
$w_f(r)$& |
889 |
gezelter |
3985 |
$\frac{d v_{22}}{dr} - \frac{2v_{22}(r)}{r}$& |
890 |
|
|
$\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
891 |
|
|
$ \left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) $ \\ |
892 |
|
|
&&& $ ~~~- \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) |
893 |
|
|
\right)-\frac{2v_{22}(r)}{r}$\\ |
894 |
gezelter |
3906 |
% |
895 |
|
|
$w_g(r)$& |
896 |
gezelter |
3985 |
$\frac{v_{31}(r)}{r}$& |
897 |
|
|
$ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
898 |
gezelter |
3906 |
$\frac{v_{31}(r)}{r}$\\ |
899 |
|
|
% |
900 |
|
|
$w_h(r)$ & |
901 |
gezelter |
3985 |
$\frac{d v_{31}}{dr} -\frac{v_{31}(r)}{r}$& |
902 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
903 |
|
|
$ \left(\frac{2g(r)}{r^3} -\frac{2h(r)}{r^2} +\frac{s(r)}{r} \right) - \left(\frac{2g(r_c)}{r_c^3} -\frac{2h(r_c)}{r_c^2} +\frac{s(r_c)}{r_c} \right) $ \\ |
904 |
|
|
&&& $ ~~~ -\frac{v_{31}(r)}{r}$ \\ |
905 |
gezelter |
3906 |
% 2 |
906 |
|
|
$w_i(r)$ & |
907 |
gezelter |
3985 |
$\frac{v_{32}(r)}{r}$ & |
908 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
909 |
|
|
$\frac{v_{32}(r)}{r}$\\ |
910 |
gezelter |
3906 |
% |
911 |
|
|
$w_j(r)$ & |
912 |
gezelter |
3985 |
$\frac{d v_{32}}{dr} - \frac{3v_{32}}{r}$& |
913 |
|
|
$\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
914 |
|
|
$\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right)$ \\ |
915 |
|
|
&&& $~~~-\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} |
916 |
|
|
-\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
917 |
gezelter |
3906 |
% |
918 |
|
|
$w_k(r)$ & |
919 |
gezelter |
3985 |
$\frac{d v_{41}}{dr} $ & |
920 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
921 |
|
|
$\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right) |
922 |
|
|
-\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
923 |
gezelter |
3906 |
% |
924 |
|
|
$w_l(r)$ & |
925 |
gezelter |
3985 |
$\frac{d v_{42}}{dr} -\frac{2v_{42}(r)}{r}$ & |
926 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
927 |
|
|
$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
928 |
|
|
&&& $~~~ -\left(-\frac{9g(r_c)}{r_c^4} +\frac{9h(r_c)}{r_c^3} -\frac{4s(r_c)}{r_c^2} +\frac{t(r_c)}{r_c} \right) |
929 |
|
|
-\frac{2v_{42}(r)}{r}$\\ |
930 |
gezelter |
3906 |
% |
931 |
|
|
$w_m(r)$ & |
932 |
gezelter |
3985 |
$\frac{d v_{43}}{dr} -\frac{4v_{43}(r)}{r}$& |
933 |
|
|
$\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
934 |
|
|
$\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$\\ |
935 |
|
|
&&& $~~~- \left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
936 |
|
|
+\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $\\ |
937 |
|
|
&&& $~~~-\frac{4v_{43}(r)}{r}$ \\ |
938 |
gezelter |
3906 |
% |
939 |
|
|
$w_n(r)$ & |
940 |
gezelter |
3985 |
$\frac{v_{42}(r)}{r}$ & |
941 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
942 |
|
|
$\frac{v_{42}(r)}{r}$\\ |
943 |
gezelter |
3906 |
% |
944 |
|
|
$w_o(r)$ & |
945 |
gezelter |
3985 |
$\frac{v_{43}(r)}{r}$& |
946 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
947 |
|
|
$\frac{v_{43}(r)}{r}$ \\ \hline |
948 |
gezelter |
3906 |
% |
949 |
|
|
|
950 |
|
|
\end{tabular} |
951 |
gezelter |
3985 |
\end{sidewaystable} |
952 |
gezelter |
3906 |
% |
953 |
|
|
% |
954 |
|
|
% |
955 |
|
|
|
956 |
|
|
\subsection{Forces} |
957 |
gezelter |
3985 |
The force on object $\bf{a}$, $\mathbf{F}_{\bf a}$, due to object |
958 |
|
|
$\bf{b}$ is the negative of the force on $\bf{b}$ due to $\bf{a}$. For |
959 |
|
|
a simple charge-charge interaction, these forces will point along the |
960 |
|
|
$\pm \hat{r}$ directions, where $\mathbf{r}=\mathbf{r}_b - |
961 |
|
|
\mathbf{r}_a $. Thus |
962 |
gezelter |
3906 |
% |
963 |
|
|
\begin{equation} |
964 |
|
|
F_{\bf a \alpha} = \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}}{\partial r} |
965 |
|
|
\quad \text{and} \quad F_{\bf b \alpha} |
966 |
|
|
= - \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}} {\partial r} . |
967 |
|
|
\end{equation} |
968 |
|
|
% |
969 |
gezelter |
3985 |
Obtaining the force from the interaction energy expressions is the |
970 |
|
|
same for higher-order multipole interactions -- the trick is to make |
971 |
|
|
sure that all $r$-dependent derivatives are considered. This is |
972 |
|
|
straighforward if the interaction energies are written explicitly in |
973 |
|
|
terms of $\hat{r}$ and the body axes ($\hat{a}_m$, |
974 |
|
|
$\hat{b}_n$) : |
975 |
gezelter |
3906 |
% |
976 |
|
|
\begin{equation} |
977 |
|
|
U(r,\{\hat{a}_m \cdot \hat{r} \}, |
978 |
gezelter |
3985 |
\{\hat{b}_n\cdot \hat{r} \}, |
979 |
gezelter |
3906 |
\{\hat{a}_m \cdot \hat{b}_n \}) . |
980 |
|
|
\label{ugeneral} |
981 |
|
|
\end{equation} |
982 |
|
|
% |
983 |
gezelter |
3985 |
Allen and Germano,\cite{Allen:2006fk} showed that if the energy is |
984 |
|
|
written in this form, the forces come out relatively cleanly, |
985 |
gezelter |
3906 |
% |
986 |
|
|
\begin{equation} |
987 |
|
|
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
988 |
|
|
= \frac{\partial U}{\partial r} \hat{r} |
989 |
|
|
+ \sum_m \left[ |
990 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
991 |
|
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
992 |
|
|
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
993 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
994 |
|
|
\right] \label{forceequation}. |
995 |
|
|
\end{equation} |
996 |
|
|
% |
997 |
gezelter |
3985 |
Note that our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ |
998 |
|
|
is opposite in sign to that of Allen and Germano.\cite{Allen:2006fk} |
999 |
|
|
In simplifying the algebra, we have also used: |
1000 |
gezelter |
3906 |
% |
1001 |
gezelter |
3985 |
\begin{align} |
1002 |
gezelter |
3906 |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
1003 |
gezelter |
3985 |
=& \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
1004 |
gezelter |
3906 |
\right) \\ |
1005 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
1006 |
gezelter |
3985 |
=& \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
1007 |
gezelter |
3906 |
\right) . |
1008 |
gezelter |
3985 |
\end{align} |
1009 |
gezelter |
3906 |
% |
1010 |
gezelter |
3985 |
We list below the force equations written in terms of lab-frame |
1011 |
|
|
coordinates. The radial functions used in the two methods are listed |
1012 |
|
|
in Table \ref{tab:tableFORCE} |
1013 |
gezelter |
3906 |
% |
1014 |
gezelter |
3985 |
%SPACE COORDINATES FORCE EQUATIONS |
1015 |
gezelter |
3906 |
% |
1016 |
|
|
% ************************************************************************** |
1017 |
|
|
% f ca cb |
1018 |
|
|
% |
1019 |
gezelter |
3985 |
\begin{align} |
1020 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} =& |
1021 |
|
|
C_{\bf a} C_{\bf b} w_a(r) \hat{r} \\ |
1022 |
gezelter |
3906 |
% |
1023 |
|
|
% |
1024 |
|
|
% |
1025 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} =& |
1026 |
|
|
C_{\bf a} \Bigl[ |
1027 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{b}} \right) |
1028 |
|
|
w_b(r) \hat{r} |
1029 |
gezelter |
3985 |
+ \mathbf{D}_{\mathbf{b}} w_c(r) \Bigr] \\ |
1030 |
gezelter |
3906 |
% |
1031 |
|
|
% |
1032 |
|
|
% |
1033 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} =& |
1034 |
|
|
C_{\bf a } \Bigr[ |
1035 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\bf b} w_d(r) \hat{r} |
1036 |
|
|
+ 2 \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} w_e(r) |
1037 |
gezelter |
3985 |
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} |
1038 |
|
|
\right) w_f(r) \hat{r} \Bigr] \\ |
1039 |
gezelter |
3906 |
% |
1040 |
|
|
% |
1041 |
|
|
% |
1042 |
gezelter |
3985 |
% \begin{equation} |
1043 |
|
|
% \mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1044 |
|
|
% -C_{\bf{b}} \Bigl[ |
1045 |
|
|
% \left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) w_b(r) \hat{r} |
1046 |
|
|
% + \mathbf{D}_{\mathbf{a}} w_c(r) \Bigr] |
1047 |
|
|
% \end{equation} |
1048 |
gezelter |
3906 |
% |
1049 |
|
|
% |
1050 |
|
|
% |
1051 |
gezelter |
3985 |
\begin{split} |
1052 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
1053 |
gezelter |
3906 |
- \mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} w_d(r) \hat{r} |
1054 |
|
|
+ \left( \mathbf{D}_{\mathbf {a}} |
1055 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
1056 |
gezelter |
3985 |
+ \mathbf{D}_{\mathbf {b}} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) \right) w_e(r)\\ |
1057 |
gezelter |
3906 |
% 2 |
1058 |
gezelter |
3985 |
& - \left( \hat{r} \cdot \mathbf{D}_{\mathbf {a}} \right) |
1059 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf {b}} \right) w_f(r) \hat{r} |
1060 |
|
|
\end{split}\\ |
1061 |
gezelter |
3906 |
% |
1062 |
|
|
% |
1063 |
|
|
% |
1064 |
|
|
\begin{split} |
1065 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} =& - \Bigl[ |
1066 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} \mathbf{ D}_{\mathbf{a}} |
1067 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \cdot |
1068 |
|
|
\mathbf{Q}_{\mathbf{b}} \Bigr] w_g(r) |
1069 |
gezelter |
3985 |
- \Bigl[ |
1070 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1071 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) |
1072 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
1073 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1074 |
|
|
% 3 |
1075 |
gezelter |
3985 |
& - \Bigl[\mathbf{ D}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1076 |
gezelter |
3906 |
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \Bigr] |
1077 |
|
|
w_i(r) |
1078 |
|
|
% 4 |
1079 |
gezelter |
3985 |
- |
1080 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) |
1081 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) w_j(r) \hat{r} \end{split} \\ |
1082 |
gezelter |
3906 |
% |
1083 |
|
|
% |
1084 |
gezelter |
3985 |
% \begin{equation} |
1085 |
|
|
% \mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
1086 |
|
|
% \frac{C_{\bf b }}{4\pi \epsilon_0} \Bigr[ |
1087 |
|
|
% \text{Tr}\mathbf{Q}_{\bf a} w_d(r) \hat{r} |
1088 |
|
|
% + 2 \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} w_e(r) |
1089 |
|
|
% + \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
1090 |
|
|
% \end{equation} |
1091 |
|
|
% % |
1092 |
|
|
% \begin{equation} |
1093 |
|
|
% \begin{split} |
1094 |
|
|
% \mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1095 |
|
|
% &\frac{1}{4\pi \epsilon_0} \Bigl[ |
1096 |
|
|
% \text{Tr}\mathbf{Q}_{\mathbf{a}} \mathbf{D}_{\mathbf{b}} |
1097 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} \Bigr] w_g(r) |
1098 |
|
|
% % 2 |
1099 |
|
|
% + \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
1100 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1101 |
|
|
% +2 (\mathbf{D}_{\mathbf{b}} \cdot |
1102 |
|
|
% \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1103 |
|
|
% % 3 |
1104 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} \Bigl[ \mathbf{D}_{\mathbf{b}} |
1105 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1106 |
|
|
% +2 (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1107 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \Bigr] w_i(r) |
1108 |
|
|
% % 4 |
1109 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1110 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1111 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) w_j(r) \hat{r} |
1112 |
|
|
% \end{split} |
1113 |
|
|
% \end{equation} |
1114 |
gezelter |
3906 |
% |
1115 |
|
|
% |
1116 |
|
|
% |
1117 |
|
|
\begin{split} |
1118 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
1119 |
|
|
\Bigl[ |
1120 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} \hat{r} |
1121 |
|
|
+ 2 \text{Tr} ( \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_k(r) \hat{r} \\ |
1122 |
|
|
% 2 |
1123 |
gezelter |
3985 |
&+ \Bigl[ |
1124 |
gezelter |
3906 |
2\text{Tr}\mathbf{Q}_{\mathbf{b}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
1125 |
|
|
+ 2\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) |
1126 |
|
|
% 3 |
1127 |
|
|
+4 (\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1128 |
|
|
+ 4(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}}) \Bigr] w_n(r) \\ |
1129 |
|
|
% 4 |
1130 |
gezelter |
3985 |
&+ \Bigl[ |
1131 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1132 |
|
|
+ \text{Tr}\mathbf{Q}_{\mathbf{b}} |
1133 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1134 |
|
|
% 5 |
1135 |
|
|
+4 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot |
1136 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
1137 |
|
|
% |
1138 |
gezelter |
3985 |
&+ \Bigl[ |
1139 |
gezelter |
3906 |
+ 2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
1140 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1141 |
|
|
%6 |
1142 |
|
|
+2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1143 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_o(r) \\ |
1144 |
|
|
% 7 |
1145 |
gezelter |
3985 |
&+ |
1146 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1147 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) w_m(r) \hat{r} \end{split} |
1148 |
|
|
\end{align} |
1149 |
|
|
Note that the forces for higher multipoles on site $\mathbf{a}$ |
1150 |
|
|
interacting with those of lower order on site $\mathbf{b}$ can be |
1151 |
|
|
obtained by swapping indices in the expressions above. |
1152 |
|
|
|
1153 |
gezelter |
3906 |
% |
1154 |
gezelter |
3985 |
% Torques SECTION ----------------------------------------------------------------------------------------- |
1155 |
gezelter |
3906 |
% |
1156 |
|
|
\subsection{Torques} |
1157 |
gezelter |
3985 |
When energies are written in the form of Eq.~({\ref{ugeneral}), then |
1158 |
|
|
torques can be found in a relatively straightforward |
1159 |
|
|
manner,\cite{Allen:2006fk} |
1160 |
gezelter |
3906 |
% |
1161 |
|
|
\begin{eqnarray} |
1162 |
|
|
\mathbf{\tau}_{\bf a} = |
1163 |
|
|
\sum_m |
1164 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
1165 |
|
|
( \hat{r} \times \hat{a}_m ) |
1166 |
|
|
-\sum_{mn} |
1167 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
1168 |
|
|
(\hat{a}_m \times \hat{b}_n) \\ |
1169 |
|
|
% |
1170 |
|
|
\mathbf{\tau}_{\bf b} = |
1171 |
|
|
\sum_m |
1172 |
|
|
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
1173 |
|
|
( \hat{r} \times \hat{b}_m) |
1174 |
|
|
+\sum_{mn} |
1175 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
1176 |
|
|
(\hat{a}_m \times \hat{b}_n) . |
1177 |
|
|
\end{eqnarray} |
1178 |
|
|
% |
1179 |
|
|
% |
1180 |
gezelter |
3985 |
The torques for both the Taylor-Shifted as well as Gradient-Shifted |
1181 |
|
|
methods are given in space-frame coordinates: |
1182 |
gezelter |
3906 |
% |
1183 |
|
|
% |
1184 |
gezelter |
3985 |
\begin{align} |
1185 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} =& |
1186 |
|
|
C_{\bf a} (\hat{r} \times \mathbf{D}_{\mathbf{b}}) v_{11}(r) \\ |
1187 |
gezelter |
3906 |
% |
1188 |
|
|
% |
1189 |
|
|
% |
1190 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} =& |
1191 |
|
|
2C_{\bf a} |
1192 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{22}(r) \\ |
1193 |
gezelter |
3906 |
% |
1194 |
|
|
% |
1195 |
|
|
% |
1196 |
gezelter |
3985 |
% \begin{equation} |
1197 |
|
|
% \mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1198 |
|
|
% -\frac{C_{\bf b}}{4\pi \epsilon_0} |
1199 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{a}}) v_{11}(r) |
1200 |
|
|
% \end{equation} |
1201 |
gezelter |
3906 |
% |
1202 |
|
|
% |
1203 |
|
|
% |
1204 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
1205 |
|
|
\mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
1206 |
gezelter |
3906 |
% 2 |
1207 |
gezelter |
3985 |
- |
1208 |
gezelter |
3906 |
(\hat{r} \times \mathbf{D}_{\mathbf {a}} ) |
1209 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{D}_{\mathbf {b}} ) v_{22}(r)\\ |
1210 |
gezelter |
3906 |
% |
1211 |
|
|
% |
1212 |
|
|
% |
1213 |
gezelter |
3985 |
% \begin{equation} |
1214 |
|
|
% \mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} = |
1215 |
|
|
% -\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
1216 |
|
|
% % 2 |
1217 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1218 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf {a}} ) |
1219 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
1220 |
|
|
% \end{equation} |
1221 |
gezelter |
3906 |
% |
1222 |
|
|
% |
1223 |
|
|
% |
1224 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} =& |
1225 |
|
|
\Bigl[ |
1226 |
gezelter |
3906 |
-\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1227 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
1228 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \times |
1229 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1230 |
|
|
\Bigr] v_{31}(r) |
1231 |
|
|
% 3 |
1232 |
gezelter |
3985 |
- (\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
1233 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{32}(r)\\ |
1234 |
gezelter |
3906 |
% |
1235 |
|
|
% |
1236 |
|
|
% |
1237 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} =& |
1238 |
|
|
\Bigl[ |
1239 |
gezelter |
3906 |
+2 ( \mathbf{D}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \times |
1240 |
|
|
\hat{r} |
1241 |
|
|
-2 \mathbf{D}_{\mathbf{a}} \times |
1242 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1243 |
|
|
\Bigr] v_{31}(r) |
1244 |
|
|
% 2 |
1245 |
gezelter |
3985 |
+ |
1246 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}}) |
1247 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}}) \times \hat{r} v_{32}(r)\\ |
1248 |
gezelter |
3906 |
% |
1249 |
|
|
% |
1250 |
|
|
% |
1251 |
gezelter |
3985 |
% \begin{equation} |
1252 |
|
|
% \mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1253 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1254 |
|
|
% -2 (\mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} ) \times \hat{r} |
1255 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \times |
1256 |
|
|
% (\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1257 |
|
|
% \Bigr] v_{31}(r) |
1258 |
|
|
% % 3 |
1259 |
|
|
% - \frac{2}{4\pi \epsilon_0} |
1260 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}} ) |
1261 |
|
|
% (\hat{r} \cdot \mathbf |
1262 |
|
|
% {Q}_{{\mathbf a}}) \times \hat{r} v_{32}(r) |
1263 |
|
|
% \end{equation} |
1264 |
gezelter |
3906 |
% |
1265 |
|
|
% |
1266 |
|
|
% |
1267 |
gezelter |
3985 |
% \begin{equation} |
1268 |
|
|
% \mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} = |
1269 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1270 |
|
|
% \text{Tr}\mathbf{Q}_{\mathbf{a}} |
1271 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1272 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \times |
1273 |
|
|
% ( \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
1274 |
|
|
% % 2 |
1275 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1276 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1277 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) v_{32}(r) |
1278 |
|
|
% \end{equation} |
1279 |
gezelter |
3906 |
% |
1280 |
|
|
% |
1281 |
|
|
% |
1282 |
|
|
\begin{split} |
1283 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
1284 |
|
|
-4 |
1285 |
gezelter |
3906 |
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} |
1286 |
|
|
v_{41}(r) \\ |
1287 |
|
|
% 2 |
1288 |
gezelter |
3985 |
&+ |
1289 |
gezelter |
3906 |
\Bigl[-2\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1290 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times \hat{r} |
1291 |
|
|
+4 \hat{r} \times |
1292 |
|
|
( \mathbf{Q}_{{\mathbf a}} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1293 |
|
|
% 3 |
1294 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} )\times |
1295 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} ) \Bigr] v_{42}(r) \\ |
1296 |
|
|
% 4 |
1297 |
gezelter |
3985 |
&+ 2 |
1298 |
gezelter |
3906 |
\hat{r} \times ( \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1299 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) \end{split}\\ |
1300 |
gezelter |
3906 |
% |
1301 |
|
|
% |
1302 |
|
|
% |
1303 |
|
|
\begin{split} |
1304 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} = |
1305 |
gezelter |
3985 |
&4 |
1306 |
gezelter |
3906 |
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} v_{41}(r) \\ |
1307 |
|
|
% 2 |
1308 |
gezelter |
3985 |
&+ \Bigl[- 2\text{Tr}\mathbf{Q}_{\mathbf{a}} |
1309 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \times \hat{r} |
1310 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot |
1311 |
|
|
\mathbf{Q}_{{\mathbf b}} ) \times |
1312 |
|
|
\hat{r} |
1313 |
|
|
+4 ( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times |
1314 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1315 |
|
|
\Bigr] v_{42}(r) \\ |
1316 |
|
|
% 4 |
1317 |
gezelter |
3985 |
&+2 |
1318 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1319 |
gezelter |
3985 |
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r)\end{split} |
1320 |
|
|
\end{align} |
1321 |
|
|
% |
1322 |
|
|
Here, we have defined the matrix cross product in an identical form |
1323 |
|
|
as in Ref. \onlinecite{Smith98}: |
1324 |
|
|
\begin{equation} |
1325 |
|
|
\left[\mathbf{A} \times \mathbf{B}\right]_\alpha = \sum_\beta |
1326 |
|
|
\left[\mathbf{A}_{\alpha+1,\beta} \mathbf{B}_{\alpha+2,\beta} |
1327 |
|
|
-\mathbf{A}_{\alpha+2,\beta} \mathbf{B}_{\alpha+2,\beta} |
1328 |
|
|
\right] |
1329 |
gezelter |
3906 |
\end{equation} |
1330 |
gezelter |
3985 |
where $\alpha+1$ and $\alpha+2$ are regarded as cyclic |
1331 |
|
|
permuations of the matrix indices. |
1332 |
gezelter |
3980 |
|
1333 |
gezelter |
3985 |
All of the radial functions required for torques are identical with |
1334 |
|
|
the radial functions previously computed for the interaction energies. |
1335 |
|
|
These are tabulated for both shifted force methods in table |
1336 |
|
|
\ref{tab:tableenergy}. The torques for higher multipoles on site |
1337 |
|
|
$\mathbf{a}$ interacting with those of lower order on site |
1338 |
|
|
$\mathbf{b}$ can be obtained by swapping indices in the expressions |
1339 |
|
|
above. |
1340 |
|
|
|
1341 |
gezelter |
3980 |
\section{Comparison to known multipolar energies} |
1342 |
|
|
|
1343 |
|
|
To understand how these new real-space multipole methods behave in |
1344 |
|
|
computer simulations, it is vital to test against established methods |
1345 |
|
|
for computing electrostatic interactions in periodic systems, and to |
1346 |
|
|
evaluate the size and sources of any errors that arise from the |
1347 |
|
|
real-space cutoffs. In this paper we test Taylor-shifted and |
1348 |
|
|
Gradient-shifted electrostatics against analytical methods for |
1349 |
|
|
computing the energies of ordered multipolar arrays. In the following |
1350 |
|
|
paper, we test the new methods against the multipolar Ewald sum for |
1351 |
|
|
computing the energies, forces and torques for a wide range of typical |
1352 |
|
|
condensed-phase (disordered) systems. |
1353 |
|
|
|
1354 |
|
|
Because long-range electrostatic effects can be significant in |
1355 |
|
|
crystalline materials, ordered multipolar arrays present one of the |
1356 |
|
|
biggest challenges for real-space cutoff methods. The dipolar |
1357 |
|
|
analogues to the Madelung constants were first worked out by Sauer, |
1358 |
|
|
who computed the energies of ordered dipole arrays of zero |
1359 |
|
|
magnetization and obtained a number of these constants.\cite{Sauer} |
1360 |
|
|
This theory was developed more completely by Luttinger and |
1361 |
|
|
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays and |
1362 |
|
|
other periodic structures. We have repeated the Luttinger \& Tisza |
1363 |
|
|
series summations to much higher order and obtained the following |
1364 |
|
|
energy constants (converged to one part in $10^9$): |
1365 |
|
|
\begin{table*} |
1366 |
|
|
\centering{ |
1367 |
|
|
\caption{Luttinger \& Tisza arrays and their associated |
1368 |
|
|
energy constants. Type "A" arrays have nearest neighbor strings of |
1369 |
|
|
antiparallel dipoles. Type "B" arrays have nearest neighbor |
1370 |
|
|
strings of antiparallel dipoles if the dipoles are contained in a |
1371 |
|
|
plane perpendicular to the dipole direction that passes through |
1372 |
|
|
the dipole.} |
1373 |
|
|
} |
1374 |
|
|
\label{tab:LT} |
1375 |
|
|
\begin{ruledtabular} |
1376 |
|
|
\begin{tabular}{cccc} |
1377 |
|
|
Array Type & Lattice & Dipole Direction & Energy constants \\ \hline |
1378 |
|
|
A & SC & 001 & -2.676788684 \\ |
1379 |
|
|
A & BCC & 001 & 0 \\ |
1380 |
|
|
A & BCC & 111 & -1.770078733 \\ |
1381 |
|
|
A & FCC & 001 & 2.166932835 \\ |
1382 |
|
|
A & FCC & 011 & -1.083466417 \\ |
1383 |
|
|
|
1384 |
|
|
* & BCC & minimum & -1.985920929 \\ |
1385 |
|
|
|
1386 |
|
|
B & SC & 001 & -2.676788684 \\ |
1387 |
|
|
B & BCC & 001 & -1.338394342 \\ |
1388 |
|
|
B & BCC & 111 & -1.770078733 \\ |
1389 |
|
|
B & FCC & 001 & -1.083466417 \\ |
1390 |
|
|
B & FCC & 011 & -1.807573634 |
1391 |
|
|
\end{tabular} |
1392 |
|
|
\end{ruledtabular} |
1393 |
|
|
\end{table*} |
1394 |
|
|
|
1395 |
|
|
In addition to the A and B arrays, there is an additional minimum |
1396 |
|
|
energy structure for the BCC lattice that was found by Luttinger \& |
1397 |
|
|
Tisza. The total electrostatic energy for an array is given by: |
1398 |
|
|
\begin{equation} |
1399 |
|
|
E = C N^2 \mu^2 |
1400 |
|
|
\end{equation} |
1401 |
|
|
where $C$ is the energy constant given above, $N$ is the number of |
1402 |
|
|
dipoles per unit volume, and $\mu$ is the strength of the dipole. |
1403 |
|
|
|
1404 |
|
|
{\it Quadrupolar} analogues to the Madelung constants were first worked out by Nagai and Nakamura who |
1405 |
|
|
computed the energies of selected quadrupole arrays based on |
1406 |
|
|
extensions to the Luttinger and Tisza |
1407 |
|
|
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
1408 |
|
|
energy constants for the lowest energy configurations for linear |
1409 |
|
|
quadrupoles shown in table \ref{tab:NNQ} |
1410 |
|
|
|
1411 |
|
|
\begin{table*} |
1412 |
|
|
\centering{ |
1413 |
|
|
\caption{Nagai and Nakamura Quadurpolar arrays}} |
1414 |
|
|
\label{tab:NNQ} |
1415 |
|
|
\begin{ruledtabular} |
1416 |
|
|
\begin{tabular}{ccc} |
1417 |
|
|
Lattice & Quadrupole Direction & Energy constants \\ \hline |
1418 |
|
|
SC & 111 & -8.3 \\ |
1419 |
|
|
BCC & 011 & -21.7 \\ |
1420 |
|
|
FCC & 111 & -80.5 |
1421 |
|
|
\end{tabular} |
1422 |
|
|
\end{ruledtabular} |
1423 |
|
|
\end{table*} |
1424 |
|
|
|
1425 |
|
|
In analogy to the dipolar arrays, the total electrostatic energy for |
1426 |
|
|
the quadrupolar arrays is: |
1427 |
|
|
\begin{equation} |
1428 |
|
|
E = C \frac{3}{4} N^2 Q^2 |
1429 |
|
|
\end{equation} |
1430 |
|
|
where $Q$ is the quadrupole moment. |
1431 |
|
|
|
1432 |
gezelter |
3985 |
\section{Conclusion} |
1433 |
|
|
We have presented two efficient real-space methods for computing the |
1434 |
|
|
interactions between point multipoles. These methods have the benefit |
1435 |
|
|
of smoothly truncating the energies, forces, and torques at the cutoff |
1436 |
|
|
radius, making them attractive for both molecular dynamics (MD) and |
1437 |
|
|
Monte Carlo (MC) simulations. We find that the Gradient-Shifted Force |
1438 |
|
|
(GSF) and the Shifted-Potential (SP) methods converge rapidly to the |
1439 |
|
|
correct lattice energies for ordered dipolar and quadrupolar arrays, |
1440 |
|
|
while the Taylor-Shifted Force (TSF) is too severe an approximation to |
1441 |
|
|
provide accurate convergence to lattice energies. |
1442 |
gezelter |
3980 |
|
1443 |
gezelter |
3985 |
In most cases, GSF can obtain nearly quantitative agreement with the |
1444 |
|
|
lattice energy constants with reasonably small cutoff radii. The only |
1445 |
|
|
exception we have observed is for crystals which exhibit a bulk |
1446 |
|
|
macroscopic dipole moment (e.g. Luttinger \& Tisza's $Z_1$ lattice). |
1447 |
|
|
In this particular case, the multipole neutralization scheme can |
1448 |
|
|
interfere with the correct computation of the energies. We note that |
1449 |
|
|
the energies for these arrangements are typically much larger than for |
1450 |
|
|
crystals with net-zero moments, so this is not expected to be an issue |
1451 |
|
|
in most simulations. |
1452 |
gezelter |
3980 |
|
1453 |
gezelter |
3985 |
In large systems, these new methods can be made to scale approximately |
1454 |
|
|
linearly with system size, and detailed comparisons with the Ewald sum |
1455 |
|
|
for a wide range of chemical environments follows in the second paper. |
1456 |
gezelter |
3980 |
|
1457 |
gezelter |
3906 |
\begin{acknowledgments} |
1458 |
gezelter |
3985 |
JDG acknowledges helpful discussions with Christopher |
1459 |
|
|
Fennell. Support for this project was provided by the National |
1460 |
|
|
Science Foundation under grant CHE-0848243. Computational time was |
1461 |
|
|
provided by the Center for Research Computing (CRC) at the |
1462 |
|
|
University of Notre Dame. |
1463 |
gezelter |
3906 |
\end{acknowledgments} |
1464 |
|
|
|
1465 |
gezelter |
3984 |
\newpage |
1466 |
gezelter |
3906 |
\appendix |
1467 |
|
|
|
1468 |
gezelter |
3984 |
\section{Smith's $B_l(r)$ functions for damped-charge distributions} |
1469 |
gezelter |
3985 |
\label{SmithFunc} |
1470 |
gezelter |
3984 |
The following summarizes Smith's $B_l(r)$ functions and includes |
1471 |
|
|
formulas given in his appendix.\cite{Smith98} The first function |
1472 |
|
|
$B_0(r)$ is defined by |
1473 |
gezelter |
3906 |
% |
1474 |
|
|
\begin{equation} |
1475 |
|
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r}= |
1476 |
|
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
1477 |
|
|
\end{equation} |
1478 |
|
|
% |
1479 |
|
|
The first derivative of this function is |
1480 |
|
|
% |
1481 |
|
|
\begin{equation} |
1482 |
|
|
\frac{dB_0(r)}{dr}=-\frac{1}{r^2}\text{erfc}(\alpha r) |
1483 |
|
|
-\frac{2\alpha}{r\sqrt{\pi}}\text{e}^{-{\alpha}^2r^2} |
1484 |
|
|
\end{equation} |
1485 |
|
|
% |
1486 |
gezelter |
3984 |
which can be used to define a function $B_1(r)$: |
1487 |
gezelter |
3906 |
% |
1488 |
|
|
\begin{equation} |
1489 |
|
|
B_1(r)=-\frac{1}{r}\frac{dB_0(r)}{dr} |
1490 |
|
|
\end{equation} |
1491 |
|
|
% |
1492 |
gezelter |
3984 |
In general, the recurrence relation, |
1493 |
gezelter |
3906 |
\begin{equation} |
1494 |
|
|
B_l(r)=-\frac{1}{r}\frac{dB_{l-1}(r)}{dr} |
1495 |
|
|
= \frac{1}{r^2} \left[ (2l-1)B_{l-1}(r) + \frac {(2\alpha^2)^l}{\alpha \sqrt{\pi}} |
1496 |
|
|
\text{e}^{-{\alpha}^2r^2} |
1497 |
gezelter |
3984 |
\right] , |
1498 |
gezelter |
3906 |
\end{equation} |
1499 |
gezelter |
3984 |
is very useful for building up higher derivatives. Using these |
1500 |
|
|
formulas, we find: |
1501 |
gezelter |
3906 |
% |
1502 |
gezelter |
3984 |
\begin{align} |
1503 |
|
|
\frac{dB_0}{dr}=&-rB_1(r) \\ |
1504 |
|
|
\frac{d^2B_0}{dr^2}=& - B_1(r) + r^2 B_2(r) \\ |
1505 |
|
|
\frac{d^3B_0}{dr^3}=& 3 r B_2(r) - r^3 B_3(r) \\ |
1506 |
|
|
\frac{d^4B_0}{dr^4}=& 3 B_2(r) - 6 r^2 B_3(r) + r^4 B_4(r) \\ |
1507 |
|
|
\frac{d^5B_0}{dr^5}=& - 15 r B_3(r) + 10 r^3 B_4(r) - r^5 B_5(r) . |
1508 |
|
|
\end{align} |
1509 |
gezelter |
3906 |
% |
1510 |
gezelter |
3984 |
As noted by Smith, it is possible to approximate the $B_l(r)$ |
1511 |
|
|
functions, |
1512 |
gezelter |
3906 |
% |
1513 |
|
|
\begin{equation} |
1514 |
|
|
B_l(r)=\frac{(2l)!}{l!2^lr^{2l+1}} - \frac {(2\alpha^2)^{l+1}}{(2l+1)\alpha \sqrt{\pi}} |
1515 |
|
|
+\text{O}(r) . |
1516 |
|
|
\end{equation} |
1517 |
gezelter |
3984 |
\newpage |
1518 |
|
|
\section{The $r$-dependent factors for TSF electrostatics} |
1519 |
gezelter |
3906 |
|
1520 |
|
|
Using the shifted damped functions $f_n(r)$ defined by: |
1521 |
|
|
% |
1522 |
|
|
\begin{equation} |
1523 |
gezelter |
3984 |
f_n(r)= B_0(r) -\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} B_0^{(m)}(r_c) , |
1524 |
gezelter |
3906 |
\end{equation} |
1525 |
|
|
% |
1526 |
gezelter |
3984 |
where the superscript $(m)$ denotes the $m^\mathrm{th}$ derivative. In |
1527 |
|
|
this Appendix, we provide formulas for successive derivatives of this |
1528 |
|
|
function. (If there is no damping, then $B_0(r)$ is replaced by |
1529 |
|
|
$1/r$.) First, we find: |
1530 |
gezelter |
3906 |
% |
1531 |
|
|
\begin{equation} |
1532 |
|
|
\frac{\partial f_n}{\partial r_\alpha}=\hat{r}_\alpha \frac{d f_n}{d r} . |
1533 |
|
|
\end{equation} |
1534 |
|
|
% |
1535 |
gezelter |
3984 |
This formula clearly brings in derivatives of Smith's $B_0(r)$ |
1536 |
|
|
function, and we define higher-order derivatives as follows: |
1537 |
gezelter |
3906 |
% |
1538 |
gezelter |
3984 |
\begin{align} |
1539 |
|
|
g_n(r)=& \frac{d f_n}{d r} = |
1540 |
|
|
B_0^{(1)}(r) -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)}(r_c) \\ |
1541 |
|
|
h_n(r)=& \frac{d^2f_n}{d r^2} = |
1542 |
|
|
B_0^{(2)}(r) -\sum_{m=0}^{n-1} \frac {(r-r_c)^m}{m!} B_0^{(m+2)}(r_c) \\ |
1543 |
|
|
s_n(r)=& \frac{d^3f_n}{d r^3} = |
1544 |
|
|
B_0^{(3)}(r) -\sum_{m=0}^{n-2} \frac {(r-r_c)^m}{m!} B_0^{(m+3)}(r_c) \\ |
1545 |
|
|
t_n(r)=& \frac{d^4f_n}{d r^4} = |
1546 |
|
|
B_0^{(4)}(r) -\sum_{m=0}^{n-3} \frac {(r-r_c)^m}{m!} B_0^{(m+4)}(r_c) \\ |
1547 |
|
|
u_n(r)=& \frac{d^5f_n}{d r^5} = |
1548 |
|
|
B_0^{(5)}(r) -\sum_{m=0}^{n-4} \frac {(r-r_c)^m}{m!} B_0^{(m+5)}(r_c) . |
1549 |
|
|
\end{align} |
1550 |
gezelter |
3906 |
% |
1551 |
gezelter |
3984 |
We note that the last function needed (for quadrupole-quadrupole interactions) is |
1552 |
gezelter |
3906 |
% |
1553 |
|
|
\begin{equation} |
1554 |
gezelter |
3984 |
u_4(r)=B_0^{(5)}(r) - B_0^{(5)}(r_c) . |
1555 |
gezelter |
3906 |
\end{equation} |
1556 |
|
|
|
1557 |
gezelter |
3984 |
The functions $f_n(r)$ to $u_n(r)$ can be computed recursively and |
1558 |
|
|
stored on a grid for values of $r$ from $0$ to $r_c$. The functions |
1559 |
|
|
needed are listed schematically below: |
1560 |
gezelter |
3906 |
% |
1561 |
|
|
\begin{eqnarray} |
1562 |
|
|
f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
1563 |
|
|
g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
1564 |
|
|
h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
1565 |
|
|
s_2 \quad s_3 \quad &s_4 \nonumber \\ |
1566 |
|
|
t_3 \quad &t_4 \nonumber \\ |
1567 |
|
|
&u_4 \nonumber . |
1568 |
|
|
\end{eqnarray} |
1569 |
|
|
|
1570 |
|
|
Using these functions, we find |
1571 |
|
|
% |
1572 |
gezelter |
3984 |
\begin{align} |
1573 |
|
|
\frac{\partial f_n}{\partial r_\alpha} =&r_\alpha \frac {g_n}{r} \label{eq:b9}\\ |
1574 |
|
|
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =&\delta_{\alpha \beta}\frac {g_n}{r} |
1575 |
|
|
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) \\ |
1576 |
|
|
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} =& |
1577 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
1578 |
|
|
\delta_{ \beta \gamma} r_\alpha \right) |
1579 |
|
|
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) |
1580 |
|
|
+ r_\alpha r_\beta r_\gamma |
1581 |
gezelter |
3984 |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \\ |
1582 |
|
|
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} =& |
1583 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
1584 |
|
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
1585 |
|
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
1586 |
|
|
\left( - \frac{g_n}{r^3} + \frac{h_n}{r^2} \right) \nonumber \\ |
1587 |
gezelter |
3984 |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
1588 |
|
|
+ \text{5 permutations} |
1589 |
gezelter |
3906 |
\right) \left( \frac{3 g_n}{r^5} - \frac{3h_n}{r^4} + \frac{s_n}{r^3} |
1590 |
|
|
\right) \nonumber \\ |
1591 |
gezelter |
3984 |
&+ r_\alpha r_\beta r_\gamma r_\delta |
1592 |
gezelter |
3906 |
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
1593 |
gezelter |
3984 |
+ \frac{t_n}{r^4} \right)\\ |
1594 |
gezelter |
3906 |
\frac{\partial^5 f_n} |
1595 |
gezelter |
3984 |
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} =& |
1596 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
1597 |
gezelter |
3984 |
+ \text{14 permutations} \right) |
1598 |
gezelter |
3906 |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
1599 |
gezelter |
3984 |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
1600 |
|
|
+ \text{9 permutations} |
1601 |
gezelter |
3906 |
\right) \left(- \frac{15g_n}{r^7}+\frac{15h_n}{r^7} -\frac{6s_n}{r^5} +\frac{t_n}{r^4} |
1602 |
|
|
\right) \nonumber \\ |
1603 |
gezelter |
3984 |
&+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
1604 |
gezelter |
3906 |
\left( \frac{105g_n}{r^9} - \frac{105h_n}{r^8} + \frac{45s_n}{r^7} |
1605 |
gezelter |
3984 |
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) \label{eq:b13} |
1606 |
|
|
\end{align} |
1607 |
gezelter |
3906 |
% |
1608 |
|
|
% |
1609 |
|
|
% |
1610 |
gezelter |
3984 |
\newpage |
1611 |
|
|
\section{The $r$-dependent factors for GSF electrostatics} |
1612 |
gezelter |
3906 |
|
1613 |
gezelter |
3984 |
In Gradient-shifted force electrostatics, the kernel is not expanded, |
1614 |
|
|
rather the individual terms in the multipole interaction energies. |
1615 |
|
|
For damped charges , this still brings into the algebra multiple |
1616 |
|
|
derivatives of the Smith's $B_0(r)$ function. To denote these terms, |
1617 |
|
|
we generalize the notation of the previous appendix. For $f(r)=1/r$ |
1618 |
|
|
(bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
1619 |
gezelter |
3906 |
% |
1620 |
gezelter |
3984 |
\begin{align} |
1621 |
|
|
g(r)=& \frac{df}{d r}\\ |
1622 |
|
|
h(r)=& \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
1623 |
|
|
s(r)=& \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
1624 |
|
|
t(r)=& \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
1625 |
|
|
u(r)=& \frac{dt}{d r} = \frac{d^5f}{d r^5} . |
1626 |
|
|
\end{align} |
1627 |
gezelter |
3906 |
% |
1628 |
gezelter |
3984 |
For undamped charges, $f(r)=1/r$, Table I lists these derivatives |
1629 |
|
|
under the column ``Bare Coulomb.'' Equations \ref{eq:b9} to |
1630 |
|
|
\ref{eq:b13} are still correct for GSF electrostatics if the subscript |
1631 |
|
|
$n$ is eliminated. |
1632 |
gezelter |
3906 |
|
1633 |
gezelter |
3985 |
% \section{Extra Material} |
1634 |
|
|
% % |
1635 |
|
|
% % |
1636 |
|
|
% %Energy in body coordinate form --------------------------------------------------------------- |
1637 |
|
|
% % |
1638 |
|
|
% Here are the interaction energies written in terms of the body coordinates: |
1639 |
gezelter |
3906 |
|
1640 |
gezelter |
3985 |
% % |
1641 |
|
|
% % u ca cb |
1642 |
|
|
% % |
1643 |
|
|
% \begin{equation} |
1644 |
|
|
% U_{C_{\bf a}C_{\bf b}}(r)= |
1645 |
|
|
% \frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) |
1646 |
|
|
% \end{equation} |
1647 |
|
|
% % |
1648 |
|
|
% % u ca db |
1649 |
|
|
% % |
1650 |
|
|
% \begin{equation} |
1651 |
|
|
% U_{C_{\bf a}D_{\bf b}}(r)= |
1652 |
|
|
% \frac{C_{\bf a}}{4\pi \epsilon_0} |
1653 |
|
|
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
1654 |
|
|
% \end{equation} |
1655 |
|
|
% % |
1656 |
|
|
% % u ca qb |
1657 |
|
|
% % |
1658 |
|
|
% \begin{equation} |
1659 |
|
|
% U_{C_{\bf a}Q_{\bf b}}(r)= |
1660 |
|
|
% \frac{C_{\bf a }\text{Tr}Q_{\bf b}}{4\pi \epsilon_0} |
1661 |
|
|
% v_{21}(r) \nonumber \\ |
1662 |
|
|
% +\frac{C_{\bf a}}{4\pi \epsilon_0} |
1663 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) |
1664 |
|
|
% v_{22}(r) |
1665 |
|
|
% \end{equation} |
1666 |
|
|
% % |
1667 |
|
|
% % u da cb |
1668 |
|
|
% % |
1669 |
|
|
% \begin{equation} |
1670 |
|
|
% U_{D_{\bf a}C_{\bf b}}(r)= |
1671 |
|
|
% -\frac{C_{\bf b}}{4\pi \epsilon_0} |
1672 |
|
|
% \sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
1673 |
|
|
% \end{equation} |
1674 |
|
|
% % |
1675 |
|
|
% % u da db |
1676 |
|
|
% % |
1677 |
|
|
% \begin{equation} |
1678 |
|
|
% \begin{split} |
1679 |
|
|
% % 1 |
1680 |
|
|
% U_{D_{\bf a}D_{\bf b}}(r)&= |
1681 |
|
|
% -\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
1682 |
|
|
% (\hat{a}_m \cdot \hat{b}_n) |
1683 |
|
|
% D_{\mathbf{b}n} v_{21}(r) \\ |
1684 |
|
|
% % 2 |
1685 |
|
|
% &-\frac{1}{4\pi \epsilon_0} |
1686 |
|
|
% \sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
1687 |
|
|
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} |
1688 |
|
|
% v_{22}(r) |
1689 |
|
|
% \end{split} |
1690 |
|
|
% \end{equation} |
1691 |
|
|
% % |
1692 |
|
|
% % u da qb |
1693 |
|
|
% % |
1694 |
|
|
% \begin{equation} |
1695 |
|
|
% \begin{split} |
1696 |
|
|
% % 1 |
1697 |
|
|
% U_{D_{\bf a}Q_{\bf b}}(r)&= |
1698 |
|
|
% -\frac{1}{4\pi \epsilon_0} \left( |
1699 |
|
|
% \text{Tr}Q_{\mathbf{b}} |
1700 |
|
|
% \sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
1701 |
|
|
% +2\sum_{lmn}D_{\mathbf{a}l} |
1702 |
|
|
% (\hat{a}_l \cdot \hat{b}_m) |
1703 |
|
|
% Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
1704 |
|
|
% \right) v_{31}(r) \\ |
1705 |
|
|
% % 2 |
1706 |
|
|
% &-\frac{1}{4\pi \epsilon_0} |
1707 |
|
|
% \sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
1708 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1709 |
|
|
% Q_{{\mathbf b}mn} |
1710 |
|
|
% (\hat{b}_n \cdot \hat{r}) v_{32}(r) |
1711 |
|
|
% \end{split} |
1712 |
|
|
% \end{equation} |
1713 |
|
|
% % |
1714 |
|
|
% % u qa cb |
1715 |
|
|
% % |
1716 |
|
|
% \begin{equation} |
1717 |
|
|
% U_{Q_{\bf a}C_{\bf b}}(r)= |
1718 |
|
|
% \frac{C_{\bf b }\text{Tr}Q_{\bf a}}{4\pi \epsilon_0} v_{21}(r) |
1719 |
|
|
% +\frac{C_{\bf b}}{4\pi \epsilon_0} |
1720 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) v_{22}(r) |
1721 |
|
|
% \end{equation} |
1722 |
|
|
% % |
1723 |
|
|
% % u qa db |
1724 |
|
|
% % |
1725 |
|
|
% \begin{equation} |
1726 |
|
|
% \begin{split} |
1727 |
|
|
% %1 |
1728 |
|
|
% U_{Q_{\bf a}D_{\bf b}}(r)&= |
1729 |
|
|
% \frac{1}{4\pi \epsilon_0} \left( |
1730 |
|
|
% \text{Tr}Q_{\mathbf{a}} |
1731 |
|
|
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
1732 |
|
|
% +2\sum_{lmn}D_{\mathbf{b}l} |
1733 |
|
|
% (\hat{b}_l \cdot \hat{a}_m) |
1734 |
|
|
% Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
1735 |
|
|
% \right) v_{31}(r) \\ |
1736 |
|
|
% % 2 |
1737 |
|
|
% &+\frac{1}{4\pi \epsilon_0} |
1738 |
|
|
% \sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
1739 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1740 |
|
|
% Q_{{\mathbf a}mn} |
1741 |
|
|
% (\hat{a}_n \cdot \hat{r}) v_{32}(r) |
1742 |
|
|
% \end{split} |
1743 |
|
|
% \end{equation} |
1744 |
|
|
% % |
1745 |
|
|
% % u qa qb |
1746 |
|
|
% % |
1747 |
|
|
% \begin{equation} |
1748 |
|
|
% \begin{split} |
1749 |
|
|
% %1 |
1750 |
|
|
% U_{Q_{\bf a}Q_{\bf b}}(r)&= |
1751 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1752 |
|
|
% \text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
1753 |
|
|
% +2\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
1754 |
|
|
% Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
1755 |
|
|
% (\hat{a}_m \cdot \hat{b}_n) \Bigr] |
1756 |
|
|
% v_{41}(r) \\ |
1757 |
|
|
% % 2 |
1758 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} |
1759 |
|
|
% \Bigl[ \text{Tr}Q_{\mathbf{a}} |
1760 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
1761 |
|
|
% Q_{{\mathbf b}lm} |
1762 |
|
|
% (\hat{b}_m \cdot \hat{r}) |
1763 |
|
|
% +\text{Tr}Q_{\mathbf{b}} |
1764 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
1765 |
|
|
% Q_{{\mathbf a}lm} |
1766 |
|
|
% (\hat{a}_m \cdot \hat{r}) \\ |
1767 |
|
|
% % 3 |
1768 |
|
|
% &+4 \sum_{lmnp} |
1769 |
|
|
% (\hat{r} \cdot \hat{a}_l ) |
1770 |
|
|
% Q_{{\mathbf a}lm} |
1771 |
|
|
% (\hat{a}_m \cdot \hat{b}_n) |
1772 |
|
|
% Q_{{\mathbf b}np} |
1773 |
|
|
% (\hat{b}_p \cdot \hat{r}) |
1774 |
|
|
% \Bigr] v_{42}(r) \\ |
1775 |
|
|
% % 4 |
1776 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} |
1777 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) |
1778 |
|
|
% Q_{{\mathbf a}lm} |
1779 |
|
|
% (\hat{a}_m \cdot \hat{r}) |
1780 |
|
|
% \sum_{np} (\hat{r} \cdot \hat{b}_n) |
1781 |
|
|
% Q_{{\mathbf b}np} |
1782 |
|
|
% (\hat{b}_p \cdot \hat{r}) v_{43}(r). |
1783 |
|
|
% \end{split} |
1784 |
|
|
% \end{equation} |
1785 |
|
|
% % |
1786 |
gezelter |
3906 |
|
1787 |
|
|
|
1788 |
gezelter |
3985 |
% % BODY coordinates force equations -------------------------------------------- |
1789 |
|
|
% % |
1790 |
|
|
% % |
1791 |
|
|
% Here are the force equations written in terms of body coordinates. |
1792 |
|
|
% % |
1793 |
|
|
% % f ca cb |
1794 |
|
|
% % |
1795 |
|
|
% \begin{equation} |
1796 |
|
|
% \mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} = |
1797 |
|
|
% \frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} w_a(r) \hat{r} |
1798 |
|
|
% \end{equation} |
1799 |
|
|
% % |
1800 |
|
|
% % f ca db |
1801 |
|
|
% % |
1802 |
|
|
% \begin{equation} |
1803 |
|
|
% \mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} = |
1804 |
|
|
% \frac{C_{\bf a}}{4\pi \epsilon_0} |
1805 |
|
|
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} w_b(r) \hat{r} |
1806 |
|
|
% +\frac{C_{\bf a}}{4\pi \epsilon_0} |
1807 |
|
|
% \sum_n D_{\mathbf{b}n} \hat{b}_n w_c(r) |
1808 |
|
|
% \end{equation} |
1809 |
|
|
% % |
1810 |
|
|
% % f ca qb |
1811 |
|
|
% % |
1812 |
|
|
% \begin{equation} |
1813 |
|
|
% \begin{split} |
1814 |
|
|
% % 1 |
1815 |
|
|
% \mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} = |
1816 |
|
|
% \frac{1}{4\pi \epsilon_0} |
1817 |
|
|
% C_{\bf a }\text{Tr}Q_{\bf b} w_d(r) \hat{r} |
1818 |
|
|
% + 2C_{\bf a } \sum_l \hat{b}_l Q_{{\mathbf b}ln} (\hat{b}_n \cdot \hat{r}) w_e(r) \\ |
1819 |
|
|
% % 2 |
1820 |
|
|
% +\frac{C_{\bf a}}{4\pi \epsilon_0} |
1821 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) w_f(r) \hat{r} |
1822 |
|
|
% \end{split} |
1823 |
|
|
% \end{equation} |
1824 |
|
|
% % |
1825 |
|
|
% % f da cb |
1826 |
|
|
% % |
1827 |
|
|
% \begin{equation} |
1828 |
|
|
% \mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1829 |
|
|
% -\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
1830 |
|
|
% \sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} w_b(r) \hat{r} |
1831 |
|
|
% -\frac{C_{\bf{b}}}{4\pi \epsilon_0} |
1832 |
|
|
% \sum_n D_{\mathbf{a}n} \hat{a}_n w_c(r) |
1833 |
|
|
% \end{equation} |
1834 |
|
|
% % |
1835 |
|
|
% % f da db |
1836 |
|
|
% % |
1837 |
|
|
% \begin{equation} |
1838 |
|
|
% \begin{split} |
1839 |
|
|
% % 1 |
1840 |
|
|
% \mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
1841 |
|
|
% -\frac{1}{4\pi \epsilon_0} |
1842 |
|
|
% \sum_{mn} D_{\mathbf {a}m} |
1843 |
|
|
% (\hat{a}_m \cdot \hat{b}_n) |
1844 |
|
|
% D_{\mathbf{b}n} w_d(r) \hat{r} |
1845 |
|
|
% -\frac{1}{4\pi \epsilon_0} |
1846 |
|
|
% \sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
1847 |
|
|
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} w_f(r) \hat{r} \\ |
1848 |
|
|
% % 2 |
1849 |
|
|
% & \quad + \frac{1}{4\pi \epsilon_0} |
1850 |
|
|
% \Bigl[ \sum_m D_{\mathbf {a}m} |
1851 |
|
|
% \hat{a}_m \sum_n D_{\mathbf{b}n} |
1852 |
|
|
% (\hat{b}_n \cdot \hat{r}) |
1853 |
|
|
% + \sum_m D_{\mathbf {b}m} |
1854 |
|
|
% \hat{b}_m \sum_n D_{\mathbf{a}n} |
1855 |
|
|
% (\hat{a}_n \cdot \hat{r}) \Bigr] w_e(r) \\ |
1856 |
|
|
% \end{split} |
1857 |
|
|
% \end{equation} |
1858 |
|
|
% % |
1859 |
|
|
% % f da qb |
1860 |
|
|
% % |
1861 |
|
|
% \begin{equation} |
1862 |
|
|
% \begin{split} |
1863 |
|
|
% % 1 |
1864 |
|
|
% &\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} = |
1865 |
|
|
% - \frac{1}{4\pi \epsilon_0} \Bigl[ |
1866 |
|
|
% \text{Tr}Q_{\mathbf{b}} |
1867 |
|
|
% \sum_l D_{\mathbf{a}l} \hat{a}_l |
1868 |
|
|
% +2\sum_{lmn} D_{\mathbf{a}l} |
1869 |
|
|
% (\hat{a}_l \cdot \hat{b}_m) |
1870 |
|
|
% Q_{\mathbf{b}mn} \hat{b}_n \Bigr] w_g(r) \\ |
1871 |
|
|
% % 3 |
1872 |
|
|
% & - \frac{1}{4\pi \epsilon_0} \Bigl[ |
1873 |
|
|
% \text{Tr}Q_{\mathbf{b}} |
1874 |
|
|
% \sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
1875 |
|
|
% +2\sum_{lmn}D_{\mathbf{a}l} |
1876 |
|
|
% (\hat{a}_l \cdot \hat{b}_m) |
1877 |
|
|
% Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1878 |
|
|
% % 4 |
1879 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} |
1880 |
|
|
% \Bigl[\sum_l D_{\mathbf{a}l} \hat{a}_l |
1881 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1882 |
|
|
% Q_{{\mathbf b}mn} |
1883 |
|
|
% (\hat{b}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{a}_l) |
1884 |
|
|
% D_{\mathbf{a}l} |
1885 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1886 |
|
|
% Q_{{\mathbf b}mn} \hat{b}_n \Bigr] w_i(r)\\ |
1887 |
|
|
% % 6 |
1888 |
|
|
% & -\frac{1}{4\pi \epsilon_0} |
1889 |
|
|
% \sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
1890 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
1891 |
|
|
% Q_{{\mathbf b}mn} |
1892 |
|
|
% (\hat{b}_n \cdot \hat{r}) w_j(r) \hat{r} |
1893 |
|
|
% \end{split} |
1894 |
|
|
% \end{equation} |
1895 |
|
|
% % |
1896 |
|
|
% % force qa cb |
1897 |
|
|
% % |
1898 |
|
|
% \begin{equation} |
1899 |
|
|
% \begin{split} |
1900 |
|
|
% % 1 |
1901 |
|
|
% \mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} &= |
1902 |
|
|
% \frac{1}{4\pi \epsilon_0} |
1903 |
|
|
% C_{\bf b }\text{Tr}Q_{\bf a} \hat{r} w_d(r) |
1904 |
|
|
% + \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) \\ |
1905 |
|
|
% % 2 |
1906 |
|
|
% & +\frac{C_{\bf b}}{4\pi \epsilon_0} |
1907 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) w_f(r) \hat{r} |
1908 |
|
|
% \end{split} |
1909 |
|
|
% \end{equation} |
1910 |
|
|
% % |
1911 |
|
|
% % f qa db |
1912 |
|
|
% % |
1913 |
|
|
% \begin{equation} |
1914 |
|
|
% \begin{split} |
1915 |
|
|
% % 1 |
1916 |
|
|
% &\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1917 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1918 |
|
|
% \text{Tr}Q_{\mathbf{a}} |
1919 |
|
|
% \sum_l D_{\mathbf{b}l} \hat{b}_l |
1920 |
|
|
% +2\sum_{lmn} D_{\mathbf{b}l} |
1921 |
|
|
% (\hat{b}_l \cdot \hat{a}_m) |
1922 |
|
|
% Q_{\mathbf{a}mn} \hat{a}_n \Bigr] |
1923 |
|
|
% w_g(r)\\ |
1924 |
|
|
% % 3 |
1925 |
|
|
% & + \frac{1}{4\pi \epsilon_0} \Bigl[ |
1926 |
|
|
% \text{Tr}Q_{\mathbf{a}} |
1927 |
|
|
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
1928 |
|
|
% +2\sum_{lmn}D_{\mathbf{b}l} |
1929 |
|
|
% (\hat{b}_l \cdot \hat{a}_m) |
1930 |
|
|
% Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1931 |
|
|
% % 4 |
1932 |
|
|
% & + \frac{1}{4\pi \epsilon_0} \Bigl[ \sum_l D_{\mathbf{b}l} \hat{b}_l |
1933 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1934 |
|
|
% Q_{{\mathbf a}mn} |
1935 |
|
|
% (\hat{a}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{b}_l) |
1936 |
|
|
% D_{\mathbf{b}l} |
1937 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1938 |
|
|
% Q_{{\mathbf a}mn} \hat{a}_n \Bigr] w_i(r) \\ |
1939 |
|
|
% % 6 |
1940 |
|
|
% & +\frac{1}{4\pi \epsilon_0} |
1941 |
|
|
% \sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
1942 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
1943 |
|
|
% Q_{{\mathbf a}mn} |
1944 |
|
|
% (\hat{a}_n \cdot \hat{r}) w_j(r) \hat{r} |
1945 |
|
|
% \end{split} |
1946 |
|
|
% \end{equation} |
1947 |
|
|
% % |
1948 |
|
|
% % f qa qb |
1949 |
|
|
% % |
1950 |
|
|
% \begin{equation} |
1951 |
|
|
% \begin{split} |
1952 |
|
|
% &\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} = |
1953 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1954 |
|
|
% \text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
1955 |
|
|
% + 2 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
1956 |
|
|
% Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
1957 |
|
|
% (\hat{a}_m \cdot \hat{b}_n) \Bigr] w_k(r) \hat{r}\\ |
1958 |
|
|
% &+\frac{1}{4\pi \epsilon_0} \Bigl[ |
1959 |
|
|
% 2\text{Tr}Q_{\mathbf{b}} \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
1960 |
|
|
% + 2\text{Tr}Q_{\mathbf{a}} \sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} \hat{b}_m \\ |
1961 |
|
|
% &+ 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}) |
1962 |
|
|
% + 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 |
1963 |
|
|
% \Bigr] w_n(r) \\ |
1964 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} |
1965 |
|
|
% \Bigl[ \text{Tr}Q_{\mathbf{a}} |
1966 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} (\hat{b}_m \cdot \hat{r}) |
1967 |
|
|
% + \text{Tr}Q_{\mathbf{b}} |
1968 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) \\ |
1969 |
|
|
% &+4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) |
1970 |
|
|
% Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
1971 |
|
|
% % |
1972 |
|
|
% &+\frac{1}{4\pi \epsilon_0} \Bigl[ |
1973 |
|
|
% 2\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
1974 |
|
|
% \sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_n \cdot \hat{r}) \\ |
1975 |
|
|
% &+2 \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
1976 |
|
|
% \sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_n \Bigr] w_o(r) \hat{r} \\ |
1977 |
|
|
% & + \frac{1}{4\pi \epsilon_0} |
1978 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
1979 |
|
|
% \sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) w_m(r) \hat{r} |
1980 |
|
|
% \end{split} |
1981 |
|
|
% \end{equation} |
1982 |
|
|
% % |
1983 |
|
|
% Here we list the form of the non-zero damped shifted multipole torques showing |
1984 |
|
|
% explicitly dependences on body axes: |
1985 |
|
|
% % |
1986 |
|
|
% % t ca db |
1987 |
|
|
% % |
1988 |
|
|
% \begin{equation} |
1989 |
|
|
% \mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} = |
1990 |
|
|
% \frac{C_{\bf a}}{4\pi \epsilon_0} |
1991 |
|
|
% \sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
1992 |
|
|
% \end{equation} |
1993 |
|
|
% % |
1994 |
|
|
% % t ca qb |
1995 |
|
|
% % |
1996 |
|
|
% \begin{equation} |
1997 |
|
|
% \mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} = |
1998 |
|
|
% \frac{2C_{\bf a}}{4\pi \epsilon_0} |
1999 |
|
|
% \sum_{lm} (\hat{r} \times \hat{b}_l) Q_{{\mathbf b}lm} (\hat{b}_m \cdot \hat{r}) v_{22}(r) |
2000 |
|
|
% \end{equation} |
2001 |
|
|
% % |
2002 |
|
|
% % t da cb |
2003 |
|
|
% % |
2004 |
|
|
% \begin{equation} |
2005 |
|
|
% \mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
2006 |
|
|
% -\frac{C_{\bf b}}{4\pi \epsilon_0} |
2007 |
|
|
% \sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
2008 |
|
|
% \end{equation}% |
2009 |
|
|
% % |
2010 |
|
|
% % |
2011 |
|
|
% % ta da db |
2012 |
|
|
% % |
2013 |
|
|
% \begin{equation} |
2014 |
|
|
% \begin{split} |
2015 |
|
|
% % 1 |
2016 |
|
|
% \mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} &= |
2017 |
|
|
% \frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
2018 |
|
|
% (\hat{a}_m \times \hat{b}_n) |
2019 |
|
|
% D_{\mathbf{b}n} v_{21}(r) \\ |
2020 |
|
|
% % 2 |
2021 |
|
|
% &-\frac{1}{4\pi \epsilon_0} |
2022 |
|
|
% \sum_m (\hat{r} \times \hat{a}_m) D_{\mathbf {a}m} |
2023 |
|
|
% \sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
2024 |
|
|
% \end{split} |
2025 |
|
|
% \end{equation} |
2026 |
|
|
% % |
2027 |
|
|
% % tb da db |
2028 |
|
|
% % |
2029 |
|
|
% \begin{equation} |
2030 |
|
|
% \begin{split} |
2031 |
|
|
% % 1 |
2032 |
|
|
% \mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} &= |
2033 |
|
|
% -\frac{1}{4\pi \epsilon_0} \sum_{mn} D_{\mathbf {a}m} |
2034 |
|
|
% (\hat{a}_m \times \hat{b}_n) |
2035 |
|
|
% D_{\mathbf{b}n} v_{21}(r) \\ |
2036 |
|
|
% % 2 |
2037 |
|
|
% &+\frac{1}{4\pi \epsilon_0} |
2038 |
|
|
% \sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
2039 |
|
|
% \sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
2040 |
|
|
% \end{split} |
2041 |
|
|
% \end{equation} |
2042 |
|
|
% % |
2043 |
|
|
% % ta da qb |
2044 |
|
|
% % |
2045 |
|
|
% \begin{equation} |
2046 |
|
|
% \begin{split} |
2047 |
|
|
% % 1 |
2048 |
|
|
% \mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} &= |
2049 |
|
|
% \frac{1}{4\pi \epsilon_0} \left( |
2050 |
|
|
% -\text{Tr}Q_{\mathbf{b}} |
2051 |
|
|
% \sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} |
2052 |
|
|
% +2\sum_{lmn}D_{\mathbf{a}l} |
2053 |
|
|
% (\hat{a}_l \times \hat{b}_m) |
2054 |
|
|
% Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
2055 |
|
|
% \right) v_{31}(r)\\ |
2056 |
|
|
% % 2 |
2057 |
|
|
% &-\frac{1}{4\pi \epsilon_0} |
2058 |
|
|
% \sum_l (\hat{r} \times \hat{a}_l) D_{\mathbf{a}l} |
2059 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
2060 |
|
|
% Q_{{\mathbf b}mn} |
2061 |
|
|
% (\hat{b}_n \cdot \hat{r}) v_{32}(r) |
2062 |
|
|
% \end{split} |
2063 |
|
|
% \end{equation} |
2064 |
|
|
% % |
2065 |
|
|
% % tb da qb |
2066 |
|
|
% % |
2067 |
|
|
% \begin{equation} |
2068 |
|
|
% \begin{split} |
2069 |
|
|
% % 1 |
2070 |
|
|
% \mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} &= |
2071 |
|
|
% \frac{1}{4\pi \epsilon_0} \left( |
2072 |
|
|
% -2\sum_{lmn}D_{\mathbf{a}l} |
2073 |
|
|
% (\hat{a}_l \cdot \hat{b}_m) |
2074 |
|
|
% Q_{\mathbf{b}mn} (\hat{r} \times \hat{b}_n) |
2075 |
|
|
% -2\sum_{lmn}D_{\mathbf{a}l} |
2076 |
|
|
% (\hat{a}_l \times \hat{b}_m) |
2077 |
|
|
% Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
2078 |
|
|
% \right) v_{31}(r) \\ |
2079 |
|
|
% % 2 |
2080 |
|
|
% &-\frac{2}{4\pi \epsilon_0} |
2081 |
|
|
% \sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
2082 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{b}_m) |
2083 |
|
|
% Q_{{\mathbf b}mn} |
2084 |
|
|
% (\hat{r}\times \hat{b}_n) v_{32}(r) |
2085 |
|
|
% \end{split} |
2086 |
|
|
% \end{equation} |
2087 |
|
|
% % |
2088 |
|
|
% % ta qa cb |
2089 |
|
|
% % |
2090 |
|
|
% \begin{equation} |
2091 |
|
|
% \mathbf{\tau}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
2092 |
|
|
% \frac{2C_{\bf a}}{4\pi \epsilon_0} |
2093 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{{\mathbf a}lm} (\hat{r} \times \hat{a}_m) v_{22}(r) |
2094 |
|
|
% \end{equation} |
2095 |
|
|
% % |
2096 |
|
|
% % ta qa db |
2097 |
|
|
% % |
2098 |
|
|
% \begin{equation} |
2099 |
|
|
% \begin{split} |
2100 |
|
|
% % 1 |
2101 |
|
|
% \mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} &= |
2102 |
|
|
% \frac{1}{4\pi \epsilon_0} \left( |
2103 |
|
|
% 2\sum_{lmn}D_{\mathbf{b}l} |
2104 |
|
|
% (\hat{b}_l \cdot \hat{a}_m) |
2105 |
|
|
% Q_{\mathbf{a}mn} (\hat{r} \times \hat{a}_n) |
2106 |
|
|
% +2\sum_{lmn}D_{\mathbf{b}l} |
2107 |
|
|
% (\hat{a}_l \times \hat{b}_m) |
2108 |
|
|
% Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
2109 |
|
|
% \right) v_{31}(r) \\ |
2110 |
|
|
% % 2 |
2111 |
|
|
% &+\frac{2}{4\pi \epsilon_0} |
2112 |
|
|
% \sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
2113 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
2114 |
|
|
% Q_{{\mathbf a}mn} |
2115 |
|
|
% (\hat{r}\times \hat{a}_n) v_{32}(r) |
2116 |
|
|
% \end{split} |
2117 |
|
|
% \end{equation} |
2118 |
|
|
% % |
2119 |
|
|
% % tb qa db |
2120 |
|
|
% % |
2121 |
|
|
% \begin{equation} |
2122 |
|
|
% \begin{split} |
2123 |
|
|
% % 1 |
2124 |
|
|
% \mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} &= |
2125 |
|
|
% \frac{1}{4\pi \epsilon_0} \left( |
2126 |
|
|
% \text{Tr}Q_{\mathbf{a}} |
2127 |
|
|
% \sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} |
2128 |
|
|
% +2\sum_{lmn}D_{\mathbf{b}l} |
2129 |
|
|
% (\hat{a}_l \times \hat{b}_m) |
2130 |
|
|
% Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
2131 |
|
|
% \right) v_{31}(r)\\ |
2132 |
|
|
% % 2 |
2133 |
|
|
% &\frac{1}{4\pi \epsilon_0} |
2134 |
|
|
% \sum_l (\hat{r} \times \hat{b}_l) D_{\mathbf{b}l} |
2135 |
|
|
% \sum_{mn} (\hat{r} \cdot \hat{a}_m) |
2136 |
|
|
% Q_{{\mathbf a}mn} |
2137 |
|
|
% (\hat{a}_n \cdot \hat{r}) v_{32}(r) |
2138 |
|
|
% \end{split} |
2139 |
|
|
% \end{equation} |
2140 |
|
|
% % |
2141 |
|
|
% % ta qa qb |
2142 |
|
|
% % |
2143 |
|
|
% \begin{equation} |
2144 |
|
|
% \begin{split} |
2145 |
|
|
% % 1 |
2146 |
|
|
% \mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} &= |
2147 |
|
|
% -\frac{4}{4\pi \epsilon_0} |
2148 |
|
|
% \sum_{lmnp} (\hat{a}_l \times \hat{b}_p) |
2149 |
|
|
% Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
2150 |
|
|
% (\hat{a}_m \cdot \hat{b}_n) v_{41}(r) \\ |
2151 |
|
|
% % 2 |
2152 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} |
2153 |
|
|
% \Bigl[ |
2154 |
|
|
% 2\text{Tr}Q_{\mathbf{b}} |
2155 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
2156 |
|
|
% Q_{{\mathbf a}lm} |
2157 |
|
|
% (\hat{r} \times \hat{a}_m) |
2158 |
|
|
% +4 \sum_{lmnp} |
2159 |
|
|
% (\hat{r} \times \hat{a}_l ) |
2160 |
|
|
% Q_{{\mathbf a}lm} |
2161 |
|
|
% (\hat{a}_m \cdot \hat{b}_n) |
2162 |
|
|
% Q_{{\mathbf b}np} |
2163 |
|
|
% (\hat{b}_p \cdot \hat{r}) \\ |
2164 |
|
|
% % 3 |
2165 |
|
|
% &-4 \sum_{lmnp} |
2166 |
|
|
% (\hat{r} \cdot \hat{a}_l ) |
2167 |
|
|
% Q_{{\mathbf a}lm} |
2168 |
|
|
% (\hat{a}_m \times \hat{b}_n) |
2169 |
|
|
% Q_{{\mathbf b}np} |
2170 |
|
|
% (\hat{b}_p \cdot \hat{r}) |
2171 |
|
|
% \Bigr] v_{42}(r) \\ |
2172 |
|
|
% % 4 |
2173 |
|
|
% &+ \frac{2}{4\pi \epsilon_0} |
2174 |
|
|
% \sum_{lm} (\hat{r} \times \hat{a}_l) |
2175 |
|
|
% Q_{{\mathbf a}lm} |
2176 |
|
|
% (\hat{a}_m \cdot \hat{r}) |
2177 |
|
|
% \sum_{np} (\hat{r} \cdot \hat{b}_n) |
2178 |
|
|
% Q_{{\mathbf b}np} |
2179 |
|
|
% (\hat{b}_p \cdot \hat{r}) v_{43}(r)\\ |
2180 |
|
|
% \end{split} |
2181 |
|
|
% \end{equation} |
2182 |
|
|
% % |
2183 |
|
|
% % tb qa qb |
2184 |
|
|
% % |
2185 |
|
|
% \begin{equation} |
2186 |
|
|
% \begin{split} |
2187 |
|
|
% % 1 |
2188 |
|
|
% \mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} &= |
2189 |
|
|
% \frac{4}{4\pi \epsilon_0} |
2190 |
|
|
% \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
2191 |
|
|
% Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
2192 |
|
|
% (\hat{a}_m \times \hat{b}_n) v_{41}(r) \\ |
2193 |
|
|
% % 2 |
2194 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} |
2195 |
|
|
% \Bigl[ |
2196 |
|
|
% 2\text{Tr}Q_{\mathbf{a}} |
2197 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
2198 |
|
|
% Q_{{\mathbf b}lm} |
2199 |
|
|
% (\hat{r} \times \hat{b}_m) |
2200 |
|
|
% +4 \sum_{lmnp} |
2201 |
|
|
% (\hat{r} \cdot \hat{a}_l ) |
2202 |
|
|
% Q_{{\mathbf a}lm} |
2203 |
|
|
% (\hat{a}_m \cdot \hat{b}_n) |
2204 |
|
|
% Q_{{\mathbf b}np} |
2205 |
|
|
% (\hat{r} \times \hat{b}_p) \\ |
2206 |
|
|
% % 3 |
2207 |
|
|
% &+4 \sum_{lmnp} |
2208 |
|
|
% (\hat{r} \cdot \hat{a}_l ) |
2209 |
|
|
% Q_{{\mathbf a}lm} |
2210 |
|
|
% (\hat{a}_m \times \hat{b}_n) |
2211 |
|
|
% Q_{{\mathbf b}np} |
2212 |
|
|
% (\hat{b}_p \cdot \hat{r}) |
2213 |
|
|
% \Bigr] v_{42}(r) \\ |
2214 |
|
|
% % 4 |
2215 |
|
|
% &+ \frac{2}{4\pi \epsilon_0} |
2216 |
|
|
% \sum_{lm} (\hat{r} \cdot \hat{a}_l) |
2217 |
|
|
% Q_{{\mathbf a}lm} |
2218 |
|
|
% (\hat{a}_m \cdot \hat{r}) |
2219 |
|
|
% \sum_{np} (\hat{r} \times \hat{b}_n) |
2220 |
|
|
% Q_{{\mathbf b}np} |
2221 |
|
|
% (\hat{b}_p \cdot \hat{r}) v_{43}(r). \\ |
2222 |
|
|
% \end{split} |
2223 |
|
|
% \end{equation} |
2224 |
gezelter |
3906 |
% |
2225 |
gezelter |
3985 |
% \begin{table*} |
2226 |
|
|
% \caption{\label{tab:tableFORCE2}Radial functions used in the force equations.} |
2227 |
|
|
% \begin{ruledtabular} |
2228 |
|
|
% \begin{tabular}{|l|l|l|} |
2229 |
|
|
% Generic&Taylor-shifted Force&Gradient-shifted Force |
2230 |
|
|
% \\ \hline |
2231 |
|
|
% % |
2232 |
|
|
% % |
2233 |
|
|
% % |
2234 |
|
|
% $w_a(r)$& |
2235 |
|
|
% $g_0(r)$& |
2236 |
|
|
% $g(r)-g(r_c)$ \\ |
2237 |
|
|
% % |
2238 |
|
|
% % |
2239 |
|
|
% $w_b(r)$ & |
2240 |
|
|
% $\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
2241 |
|
|
% $h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
2242 |
|
|
% % |
2243 |
|
|
% $w_c(r)$ & |
2244 |
|
|
% $\frac{g_1(r)}{r} $ & |
2245 |
|
|
% $\frac{v_{11}(r)}{r}$ \\ |
2246 |
|
|
% % |
2247 |
|
|
% % |
2248 |
|
|
% $w_d(r)$& |
2249 |
|
|
% $\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
2250 |
|
|
% $\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
2251 |
|
|
% -\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $\\ |
2252 |
|
|
% % |
2253 |
|
|
% $w_e(r)$ & |
2254 |
|
|
% $\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
2255 |
|
|
% $\frac{v_{22}(r)}{r}$ \\ |
2256 |
|
|
% % |
2257 |
|
|
% % |
2258 |
|
|
% $w_f(r)$& |
2259 |
|
|
% $\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
2260 |
|
|
% $\left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) - $ \\ |
2261 |
|
|
% &&$\left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)-\frac{2v_{22}(r)}{r}$\\ |
2262 |
|
|
% % |
2263 |
|
|
% $w_g(r)$& $ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
2264 |
|
|
% $\frac{v_{31}(r)}{r}$\\ |
2265 |
|
|
% % |
2266 |
|
|
% $w_h(r)$ & |
2267 |
|
|
% $\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
2268 |
|
|
% $\left(\frac{2g(r)}{r^3} -\frac{2h(r)}{r^2} +\frac{s(r)}{r} \right) - $\\ |
2269 |
|
|
% &&$\left(\frac{2g(r_c)}{r_c^3} -\frac{2h(r_c)}{r_c^2} +\frac{s(r_c)}{r_c} \right) $ \\ |
2270 |
|
|
% &&$-\frac{v_{31}(r)}{r}$\\ |
2271 |
|
|
% % 2 |
2272 |
|
|
% $w_i(r)$ & |
2273 |
|
|
% $\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
2274 |
|
|
% $\frac{v_{32}(r)}{r}$ \\ |
2275 |
|
|
% % |
2276 |
|
|
% $w_j(r)$ & |
2277 |
|
|
% $\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
2278 |
|
|
% $\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right) $ \\ |
2279 |
|
|
% &&$\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} -\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
2280 |
|
|
% % |
2281 |
|
|
% $w_k(r)$ & |
2282 |
|
|
% $\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
2283 |
|
|
% $\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right)$ \\ |
2284 |
|
|
% &&$\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
2285 |
|
|
% % |
2286 |
|
|
% $w_l(r)$ & |
2287 |
|
|
% $\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
2288 |
|
|
% $\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
2289 |
|
|
% &&$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right) |
2290 |
|
|
% -\frac{2v_{42}(r)}{r}$ \\ |
2291 |
|
|
% % |
2292 |
|
|
% $w_m(r)$ & |
2293 |
|
|
% $\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
2294 |
|
|
% $\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$ \\ |
2295 |
|
|
% &&$\left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
2296 |
|
|
% +\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $ \\ |
2297 |
|
|
% &&$-\frac{4v_{43}(r)}{r}$ \\ |
2298 |
|
|
% % |
2299 |
|
|
% $w_n(r)$ & |
2300 |
|
|
% $\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
2301 |
|
|
% $\frac{v_{42}(r)}{r}$ \\ |
2302 |
|
|
% % |
2303 |
|
|
% $w_o(r)$ & |
2304 |
|
|
% $\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
2305 |
|
|
% $\frac{v_{43}(r)}{r}$ \\ |
2306 |
|
|
% % |
2307 |
|
|
% \end{tabular} |
2308 |
|
|
% \end{ruledtabular} |
2309 |
|
|
% \end{table*} |
2310 |
gezelter |
3980 |
|
2311 |
|
|
\newpage |
2312 |
|
|
|
2313 |
|
|
\bibliography{multipole} |
2314 |
|
|
|
2315 |
gezelter |
3906 |
\end{document} |
2316 |
|
|
% |
2317 |
|
|
% ****** End of file multipole.tex ****** |