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\begin{document} |
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%\preprint{AIP/123-QED} |
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\title{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 $r_{c}$ 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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|
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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. To do this, the interactions between coarse-grained sites |
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are typically taken to be point |
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multipoles.\cite{Golubkov06,ISI:000276097500009,ISI:000298664400012} |
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Water, in particular, has been modeled recently with point multipoles |
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up to octupolar |
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order.\cite{Chowdhuri:2006lr,Te:2010rt,Te:2010ys,Te:2010vn} For |
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maximum efficiency, these models require the use of an approximate |
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multipole expansion as the exact multipole expansion can become quite |
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expensive (particularly when handled via the Ewald |
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sum).\cite{Ichiye:2006qy} Point multipoles and multipole |
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polarizability have also been utilized in the 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 somewhat different expressions for |
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energies, 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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An efficient real-space electrostatic method involves the use of a |
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pair-wise functional form, |
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\begin{equation} |
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V = \sum_i \sum_{j>i} V_\mathrm{pair}(r_{ij}, \Omega_i, \Omega_j) + |
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\sum_i V_i^\mathrm{correction} |
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\end{equation} |
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that is short-ranged and easily truncated at a cutoff radius, |
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\begin{equation} |
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V_\mathrm{pair}(r_{ij}, \Omega_i, \Omega_j) = \left\{ |
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\begin{array}{ll} |
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V_\mathrm{approx} (r_{ij}, \Omega_i, \Omega_j) & \quad r \le r_c \\ |
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0 & \quad r > r_c , |
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\end{array} |
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\right. |
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\end{equation} |
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along with an easily computed correction term ($\sum_i |
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V_i^\mathrm{correction}$) which has linear-scaling with the number of |
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particles. Here $\Omega_i$ and $\Omega_j$ represent orientational |
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coordinates of the two sites. The computational efficiency, energy |
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conservation, and even some physical properties of a simulation can |
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depend dramatically on how the $V_\mathrm{approx}$ function behaves at |
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the cutoff radius. The goal of any approximation method should be to |
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mimic the real behavior of the electrostatic interactions as closely |
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as possible without sacrificing the near-linear scaling of a cutoff |
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method. |
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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 $r_c$. 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_i C_j \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_c\right)}{r_c} + \left(\frac{\mathrm{erfc}\left(\alpha r_c\right)}{r_c^2} |
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+ \frac{2\alpha}{\pi^{1/2}} |
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\frac{\exp\left(-\alpha^2r_c^2\right)}{r_c} |
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\right)\left(r_{ij}-r_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 $1/4 \pi \epsilon_{0}$ has been omitted for |
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clarity. |
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|
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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} There have also been recent efforts to |
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extend the Wolf self-neutralization method to zero out the dipole and |
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higher order multipoles contained 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 an additional contribution |
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from 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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|
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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 on $\bf b$. |
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|
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\subsection{Damped Coulomb interactions} |
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|
|
In the standard multipole expansion, one typically uses the bare |
310 |
|
|
Coulomb potential, with radial dependence $1/r$, as shown in |
311 |
|
|
Eq.~(\ref{kernel}). It is also quite common to use a damped Coulomb |
312 |
|
|
interaction, which results from replacing point charges with Gaussian |
313 |
|
|
distributions of charge with width $\alpha$. In damped multipole |
314 |
|
|
electrostatics, the kernel ($1/r$) of the expansion is replaced with |
315 |
|
|
the function: |
316 |
gezelter |
3906 |
\begin{equation} |
317 |
|
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r} |
318 |
|
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
319 |
|
|
\end{equation} |
320 |
gezelter |
3982 |
We develop equations below using the function $f(r)$ to represent |
321 |
gezelter |
3986 |
either $1/r$ or $B_0(r)$, and all of the techniques can be applied to |
322 |
|
|
bare or damped Coulomb kernels (or any other function) as long as |
323 |
|
|
derivatives of these functions are known. Smith's convenient |
324 |
|
|
functions $B_l(r)$ are summarized in Appendix A. |
325 |
gezelter |
3906 |
|
326 |
gezelter |
3982 |
The main goal of this work is to smoothly cut off the interaction |
327 |
|
|
energy as well as forces and torques as $r\rightarrow r_c$. To |
328 |
|
|
describe how this goal may be met, we use two examples, charge-charge |
329 |
gezelter |
3986 |
and charge-dipole, using the bare Coulomb kernel, $f(r)=1/r$, to |
330 |
|
|
explain the idea. |
331 |
gezelter |
3906 |
|
332 |
gezelter |
3984 |
\subsection{Shifted-force methods} |
333 |
gezelter |
3982 |
In the shifted-force approximation, the interaction energy for two |
334 |
|
|
charges $C_{\bf a}$ and $C_{\bf b}$ separated by a distance $r$ is |
335 |
|
|
written: |
336 |
gezelter |
3906 |
\begin{equation} |
337 |
gezelter |
3985 |
U_{C_{\bf a}C_{\bf b}}(r)= C_{\bf a} C_{\bf b} |
338 |
gezelter |
3906 |
\left({ \frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
339 |
|
|
\right) . |
340 |
|
|
\end{equation} |
341 |
gezelter |
3982 |
Two shifting terms appear in this equations, one from the |
342 |
gezelter |
3984 |
neutralization procedure ($-1/r_c$), and one that causes the first |
343 |
|
|
derivative to vanish at the cutoff radius. |
344 |
gezelter |
3982 |
|
345 |
|
|
Since one derivative of the interaction energy is needed for the |
346 |
|
|
force, the minimal perturbation is a term linear in $(r-r_c)$ in the |
347 |
|
|
interaction energy, that is: |
348 |
gezelter |
3906 |
\begin{equation} |
349 |
|
|
\frac{d\,}{dr} |
350 |
|
|
\left( {\frac{1}{r} - \frac{1}{r_c} + (r - r_c) \frac{1}{r_c^2} } |
351 |
|
|
\right) = \left(- \frac{1}{r^2} + \frac{1}{r_c^2} |
352 |
|
|
\right) . |
353 |
|
|
\end{equation} |
354 |
gezelter |
3985 |
which clearly vanishes as the $r$ approaches the cutoff radius. There |
355 |
|
|
are a number of ways to generalize this derivative shift for |
356 |
gezelter |
3984 |
higher-order multipoles. Below, we present two methods, one based on |
357 |
|
|
higher-order Taylor series for $r$ near $r_c$, and the other based on |
358 |
|
|
linear shift of the kernel gradients at the cutoff itself. |
359 |
gezelter |
3906 |
|
360 |
gezelter |
3984 |
\subsection{Taylor-shifted force (TSF) electrostatics} |
361 |
gezelter |
3982 |
In the Taylor-shifted force (TSF) method, the procedure that we follow |
362 |
|
|
is based on a Taylor expansion containing the same number of |
363 |
|
|
derivatives required for each force term to vanish at the cutoff. For |
364 |
|
|
example, the quadrupole-quadrupole interaction energy requires four |
365 |
|
|
derivatives of the kernel, and the force requires one additional |
366 |
gezelter |
3986 |
derivative. For quadrupole-quadrupole interactions, we therefore |
367 |
|
|
require shifted energy expressions that include up to $(r-r_c)^5$ so |
368 |
|
|
that all energies, forces, and torques are zero as $r \rightarrow |
369 |
|
|
r_c$. In each case, we subtract off a function $f_n^{\text{shift}}(r)$ |
370 |
|
|
from the kernel $f(r)=1/r$. The subscript $n$ indicates the number of |
371 |
|
|
derivatives to be taken when deriving a given multipole energy. We |
372 |
|
|
choose a function with guaranteed smooth derivatives -- a truncated |
373 |
|
|
Taylor series of the function $f(r)$, e.g., |
374 |
gezelter |
3906 |
% |
375 |
|
|
\begin{equation} |
376 |
gezelter |
3984 |
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)}(r_c) . |
377 |
gezelter |
3906 |
\end{equation} |
378 |
|
|
% |
379 |
|
|
The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$. |
380 |
|
|
Thus, for $f(r)=1/r$, we find |
381 |
|
|
% |
382 |
|
|
\begin{equation} |
383 |
|
|
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} . |
384 |
|
|
\end{equation} |
385 |
|
|
% |
386 |
gezelter |
3982 |
Continuing with the example of a charge $\bf a$ interacting with a |
387 |
|
|
dipole $\bf b$, we write |
388 |
gezelter |
3906 |
% |
389 |
|
|
\begin{equation} |
390 |
|
|
U_{C_{\bf a}D_{\bf b}}(r)= |
391 |
gezelter |
3985 |
C_{\bf a} D_{{\bf b}\alpha} \frac {\partial f_1(r) }{\partial r_\alpha} |
392 |
|
|
= C_{\bf a} D_{{\bf b}\alpha} |
393 |
gezelter |
3906 |
\frac {r_\alpha}{r} \frac {\partial f_1(r)}{\partial r} . |
394 |
|
|
\end{equation} |
395 |
|
|
% |
396 |
gezelter |
3984 |
The force that dipole $\bf b$ exerts on charge $\bf a$ is |
397 |
gezelter |
3906 |
% |
398 |
|
|
\begin{equation} |
399 |
gezelter |
3985 |
F_{C_{\bf a}D_{\bf b}\beta} = C_{\bf a} D_{{\bf b}\alpha} |
400 |
gezelter |
3906 |
\left[ \frac{\delta_{\alpha\beta}}{r} \frac {\partial}{\partial r} + |
401 |
|
|
\frac{r_\alpha r_\beta}{r^2} |
402 |
|
|
\left( -\frac{1}{r} \frac {\partial} {\partial r} |
403 |
|
|
+ \frac {\partial ^2} {\partial r^2} \right) \right] f_1(r) . |
404 |
|
|
\end{equation} |
405 |
|
|
% |
406 |
gezelter |
3984 |
For undamped coulombic interactions, $f(r)=1/r$, we find |
407 |
gezelter |
3906 |
% |
408 |
|
|
\begin{equation} |
409 |
|
|
F_{C_{\bf a}D_{\bf b}\beta} = |
410 |
gezelter |
3985 |
\frac{C_{\bf a} D_{{\bf b}\beta}}{r} |
411 |
gezelter |
3906 |
\left[ -\frac{1}{r^2}+\frac{1}{r_c^2}-\frac{2(r-r_c)}{r_c^3} \right] |
412 |
gezelter |
3985 |
+C_{\bf a} D_{{\bf b}\alpha}r_\alpha r_\beta |
413 |
gezelter |
3906 |
\left[ \frac{3}{r^5}-\frac{3}{r^3r_c^2} \right] . |
414 |
|
|
\end{equation} |
415 |
|
|
% |
416 |
|
|
This expansion shows the expected $1/r^3$ dependence of the force. |
417 |
|
|
|
418 |
gezelter |
3984 |
In general, we can write |
419 |
gezelter |
3906 |
% |
420 |
|
|
\begin{equation} |
421 |
gezelter |
3985 |
U= (\text{prefactor}) (\text{derivatives}) f_n(r) |
422 |
gezelter |
3906 |
\label{generic} |
423 |
|
|
\end{equation} |
424 |
|
|
% |
425 |
gezelter |
3985 |
with $n=0$ for charge-charge, $n=1$ for charge-dipole, $n=2$ for |
426 |
|
|
charge-quadrupole and dipole-dipole, $n=3$ for dipole-quadrupole, and |
427 |
|
|
$n=4$ for quadrupole-quadrupole. For example, in |
428 |
|
|
quadrupole-quadrupole interactions for which the $\text{prefactor}$ is |
429 |
|
|
$Q_{{\bf a}\alpha\beta}Q_{{\bf b}\gamma\delta}$, the derivatives are |
430 |
|
|
$\partial^4/\partial r_\alpha \partial r_\beta \partial |
431 |
|
|
r_\gamma \partial r_\delta$, with implied summation combining the |
432 |
|
|
space indices. |
433 |
gezelter |
3906 |
|
434 |
gezelter |
3984 |
In the formulas presented in the tables below, the placeholder |
435 |
|
|
function $f(r)$ is used to represent the electrostatic kernel (either |
436 |
|
|
damped or undamped). The main functions that go into the force and |
437 |
gezelter |
3985 |
torque terms, $g_n(r), h_n(r), s_n(r), \mathrm{~and~} t_n(r)$ are |
438 |
|
|
successive derivatives of the shifted electrostatic kernel, $f_n(r)$ |
439 |
|
|
of the same index $n$. The algebra required to evaluate energies, |
440 |
|
|
forces and torques is somewhat tedious, so only the final forms are |
441 |
gezelter |
3986 |
presented in tables \ref{tab:tableenergy} and \ref{tab:tableFORCE}. |
442 |
gezelter |
3906 |
|
443 |
gezelter |
3982 |
\subsection{Gradient-shifted force (GSF) electrostatics} |
444 |
gezelter |
3985 |
The second, and conceptually simpler approach to force-shifting |
445 |
|
|
maintains only the linear $(r-r_c)$ term in the truncated Taylor |
446 |
|
|
expansion, and has a similar interaction energy for all multipole |
447 |
|
|
orders: |
448 |
gezelter |
3906 |
\begin{equation} |
449 |
gezelter |
3985 |
U^{\text{shift}}(r)=U(r)-U(r_c)-(r-r_c)\hat{r}\cdot \nabla U(r) \Big |
450 |
|
|
\lvert _{r_c} . |
451 |
|
|
\label{generic2} |
452 |
gezelter |
3906 |
\end{equation} |
453 |
gezelter |
3985 |
Here the gradient for force shifting is evaluated for an image |
454 |
gezelter |
3986 |
multipole projected onto the surface of the cutoff sphere (see fig |
455 |
gezelter |
3985 |
\ref{fig:shiftedMultipoles}). No higher order terms $(r-r_c)^n$ |
456 |
|
|
appear. The primary difference between the TSF and GSF methods is the |
457 |
|
|
stage at which the Taylor Series is applied; in the Taylor-shifted |
458 |
|
|
approach, it is applied to the kernel itself. In the Gradient-shifted |
459 |
|
|
approach, it is applied to individual radial interactions terms in the |
460 |
|
|
multipole expansion. Energies from this method thus have the general |
461 |
|
|
form: |
462 |
gezelter |
3906 |
\begin{equation} |
463 |
gezelter |
3985 |
U= \sum (\text{angular factor}) (\text{radial factor}). |
464 |
|
|
\label{generic3} |
465 |
gezelter |
3906 |
\end{equation} |
466 |
|
|
|
467 |
gezelter |
3986 |
Functional forms for both methods (TSF and GSF) can both be summarized |
468 |
gezelter |
3985 |
using the form of Eq.~(\ref{generic3}). The basic forms for the |
469 |
|
|
energy, force, and torque expressions are tabulated for both shifting |
470 |
gezelter |
3986 |
approaches below -- for each separate orientational contribution, only |
471 |
gezelter |
3985 |
the radial factors differ between the two methods. |
472 |
gezelter |
3906 |
|
473 |
|
|
\subsection{\label{sec:level2}Body and space axes} |
474 |
|
|
|
475 |
gezelter |
3985 |
[XXX Do we need this section in the main paper? or should it go in the |
476 |
|
|
extra materials?] |
477 |
|
|
|
478 |
gezelter |
3984 |
So far, all energies and forces have been written in terms of fixed |
479 |
gezelter |
3985 |
space coordinates. Interaction energies are computed from the generic |
480 |
|
|
formulas Eq.~(\ref{generic}) and ~(\ref{generic2}) which combine |
481 |
|
|
orientational prefactors with radial functions. Because objects $\bf |
482 |
gezelter |
3984 |
a$ and $\bf b$ both translate and rotate during a molecular dynamics |
483 |
|
|
(MD) simulation, it is desirable to contract all $r$-dependent terms |
484 |
|
|
with dipole and quadrupole moments expressed in terms of their body |
485 |
gezelter |
3985 |
axes. To do so, we have followed the methodology of Allen and |
486 |
|
|
Germano,\cite{Allen:2006fk} which was itself based on earlier work by |
487 |
|
|
Price {\em et al.}\cite{Price:1984fk} |
488 |
gezelter |
3906 |
|
489 |
gezelter |
3984 |
We denote body axes for objects $\bf a$ and $\bf b$ by unit vectors |
490 |
|
|
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$ |
491 |
|
|
referring to a convenient set of inertial body axes. (N.B., these |
492 |
|
|
body axes are generally not the same as those for which the quadrupole |
493 |
|
|
moment is diagonal.) Then, |
494 |
gezelter |
3906 |
% |
495 |
|
|
\begin{eqnarray} |
496 |
|
|
\hat{a}_m= a_{mx}\hat{x} + a_{my}\hat{y} + a_{mz}\hat{z} \\ |
497 |
|
|
\hat{b}_m= b_{mx}\hat{x} + b_{my}\hat{y} + b_{mz}\hat{z} . |
498 |
|
|
\end{eqnarray} |
499 |
gezelter |
3985 |
Rotation matrices $\hat{\mathbf {a}}$ and $\hat{\mathbf {b}}$ can be |
500 |
|
|
expressed using these unit vectors: |
501 |
gezelter |
3906 |
\begin{eqnarray} |
502 |
|
|
\hat{\mathbf {a}} = |
503 |
|
|
\begin{pmatrix} |
504 |
|
|
\hat{a}_1 \\ |
505 |
|
|
\hat{a}_2 \\ |
506 |
|
|
\hat{a}_3 |
507 |
|
|
\end{pmatrix} |
508 |
|
|
= |
509 |
|
|
\begin{pmatrix} |
510 |
|
|
a_{1x} \quad a_{1y} \quad a_{1z} \\ |
511 |
|
|
a_{2x} \quad a_{2y} \quad a_{2z} \\ |
512 |
|
|
a_{3x} \quad a_{3y} \quad a_{3z} |
513 |
|
|
\end{pmatrix}\\ |
514 |
|
|
\hat{\mathbf {b}} = |
515 |
|
|
\begin{pmatrix} |
516 |
|
|
\hat{b}_1 \\ |
517 |
|
|
\hat{b}_2 \\ |
518 |
|
|
\hat{b}_3 |
519 |
|
|
\end{pmatrix} |
520 |
|
|
= |
521 |
|
|
\begin{pmatrix} |
522 |
gezelter |
3985 |
b_{1x} \quad b_{1y} \quad b_{1z} \\ |
523 |
gezelter |
3906 |
b_{2x} \quad b_{2y} \quad b_{2z} \\ |
524 |
|
|
b_{3x} \quad b_{3y} \quad b_{3z} |
525 |
|
|
\end{pmatrix} . |
526 |
|
|
\end{eqnarray} |
527 |
|
|
% |
528 |
gezelter |
3985 |
These matrices convert from space-fixed $(xyz)$ to body-fixed $(123)$ |
529 |
|
|
coordinates. All contractions of prefactors with derivatives of |
530 |
|
|
functions can be written in terms of these matrices. It proves to be |
531 |
|
|
equally convenient to just write any contraction in terms of unit |
532 |
|
|
vectors $\hat{r}$, $\hat{a}_m$, and $\hat{b}_n$. In the torque |
533 |
|
|
expressions, it is useful to have the angular-dependent terms |
534 |
|
|
available in three different fashions, e.g. for the dipole-dipole |
535 |
|
|
contraction: |
536 |
gezelter |
3906 |
% |
537 |
gezelter |
3985 |
\begin{equation} |
538 |
gezelter |
3906 |
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
539 |
gezelter |
3985 |
= D_{\bf {a}\alpha} D_{\bf {b}\alpha} = |
540 |
|
|
\sum_{mn} {D_{\mathbf{a}m} \hat{a}_m \cdot \hat{b}_n D_{\mathbf{b}n}} |
541 |
|
|
\end{equation} |
542 |
gezelter |
3906 |
% |
543 |
gezelter |
3985 |
The first two forms are written using space coordinates. The first |
544 |
|
|
form is standard in the chemistry literature, while the second is |
545 |
|
|
expressed using implied summation notation. The third form shows |
546 |
|
|
explicit sums over body indices and the dot products now indicate |
547 |
|
|
contractions using space indices. |
548 |
gezelter |
3906 |
|
549 |
|
|
|
550 |
gezelter |
3982 |
\subsection{The Self-Interaction \label{sec:selfTerm}} |
551 |
|
|
|
552 |
gezelter |
3985 |
In addition to cutoff-sphere neutralization, the Wolf |
553 |
|
|
summation~\cite{Wolf99} and the damped shifted force (DSF) |
554 |
|
|
extension~\cite{Fennell:2006zl} also included self-interactions that |
555 |
|
|
are handled separately from the pairwise interactions between |
556 |
|
|
sites. The self-term is normally calculated via a single loop over all |
557 |
|
|
sites in the system, and is relatively cheap to evaluate. The |
558 |
|
|
self-interaction has contributions from two sources. |
559 |
|
|
|
560 |
|
|
First, the neutralization procedure within the cutoff radius requires |
561 |
|
|
a contribution from a charge opposite in sign, but equal in magnitude, |
562 |
|
|
to the central charge, which has been spread out over the surface of |
563 |
|
|
the cutoff sphere. For a system of undamped charges, the total |
564 |
|
|
self-term is |
565 |
gezelter |
3980 |
\begin{equation} |
566 |
|
|
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
567 |
|
|
\label{eq:selfTerm} |
568 |
|
|
\end{equation} |
569 |
gezelter |
3985 |
|
570 |
|
|
Second, charge damping with the complementary error function is a |
571 |
|
|
partial analogy to the Ewald procedure which splits the interaction |
572 |
|
|
into real and reciprocal space sums. The real space sum is retained |
573 |
|
|
in the Wolf and DSF methods. The reciprocal space sum is first |
574 |
|
|
minimized by folding the largest contribution (the self-interaction) |
575 |
|
|
into the self-interaction from charge neutralization of the damped |
576 |
|
|
potential. The remainder of the reciprocal space portion is then |
577 |
|
|
discarded (as this contributes the largest computational cost and |
578 |
|
|
complexity to the Ewald sum). For a system containing only damped |
579 |
|
|
charges, the complete self-interaction can be written as |
580 |
gezelter |
3980 |
\begin{equation} |
581 |
|
|
V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
582 |
|
|
\frac{\alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N |
583 |
|
|
C_{\bf a}^{2}. |
584 |
|
|
\label{eq:dampSelfTerm} |
585 |
|
|
\end{equation} |
586 |
|
|
|
587 |
|
|
The extension of DSF electrostatics to point multipoles requires |
588 |
|
|
treatment of {\it both} the self-neutralization and reciprocal |
589 |
|
|
contributions to the self-interaction for higher order multipoles. In |
590 |
|
|
this section we give formulae for these interactions up to quadrupolar |
591 |
|
|
order. |
592 |
|
|
|
593 |
|
|
The self-neutralization term is computed by taking the {\it |
594 |
|
|
non-shifted} kernel for each interaction, placing a multipole of |
595 |
|
|
equal magnitude (but opposite in polarization) on the surface of the |
596 |
|
|
cutoff sphere, and averaging over the surface of the cutoff sphere. |
597 |
|
|
Because the self term is carried out as a single sum over sites, the |
598 |
|
|
reciprocal-space portion is identical to half of the self-term |
599 |
|
|
obtained by Smith and Aguado and Madden for the application of the |
600 |
|
|
Ewald sum to multipoles.\cite{Smith82,Smith98,Aguado03} For a given |
601 |
|
|
site which posesses a charge, dipole, and multipole, both types of |
602 |
|
|
contribution are given in table \ref{tab:tableSelf}. |
603 |
|
|
|
604 |
|
|
\begin{table*} |
605 |
|
|
\caption{\label{tab:tableSelf} Self-interaction contributions for |
606 |
|
|
site ({\bf a}) that has a charge $(C_{\bf a})$, dipole |
607 |
|
|
$(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$} |
608 |
|
|
\begin{ruledtabular} |
609 |
|
|
\begin{tabular}{lccc} |
610 |
|
|
Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline |
611 |
|
|
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
612 |
|
|
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
613 |
|
|
\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
614 |
|
|
Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
615 |
|
|
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
616 |
|
|
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
617 |
|
|
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
618 |
|
|
h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$ \\ |
619 |
|
|
\end{tabular} |
620 |
|
|
\end{ruledtabular} |
621 |
|
|
\end{table*} |
622 |
|
|
|
623 |
|
|
For sites which simultaneously contain charges and quadrupoles, the |
624 |
|
|
self-interaction includes a cross-interaction between these two |
625 |
|
|
multipole orders. Symmetry prevents the charge-dipole and |
626 |
|
|
dipole-quadrupole interactions from contributing to the |
627 |
|
|
self-interaction. The functions that go into the self-neutralization |
628 |
gezelter |
3985 |
terms, $g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive |
629 |
|
|
derivatives of the electrostatic kernel, $f(r)$ (either the undamped |
630 |
|
|
$1/r$ or the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that |
631 |
|
|
have been evaluated at the cutoff distance. For undamped |
632 |
|
|
interactions, $f(r_c) = 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For |
633 |
|
|
damped interactions, $f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so |
634 |
|
|
on. Appendix \ref{SmithFunc} contains recursion relations that allow |
635 |
|
|
rapid evaluation of these derivatives. |
636 |
gezelter |
3980 |
|
637 |
gezelter |
3985 |
\section{Interaction energies, forces, and torques} |
638 |
|
|
The main result of this paper is a set of expressions for the |
639 |
|
|
energies, forces and torques (up to quadrupole-quadrupole order) that |
640 |
|
|
work for both the Taylor-shifted and Gradient-shifted approximations. |
641 |
|
|
These expressions were derived using a set of generic radial |
642 |
|
|
functions. Without using the shifting approximations mentioned above, |
643 |
|
|
some of these radial functions would be identical, and the expressions |
644 |
|
|
coalesce into the familiar forms for unmodified multipole-multipole |
645 |
|
|
interactions. Table \ref{tab:tableenergy} maps between the generic |
646 |
|
|
functions and the radial functions derived for both the Taylor-shifted |
647 |
|
|
and Gradient-shifted methods. The energy equations are written in |
648 |
|
|
terms of lab-frame representations of the dipoles, quadrupoles, and |
649 |
|
|
the unit vector connecting the two objects, |
650 |
gezelter |
3906 |
|
651 |
|
|
% Energy in space coordinate form ---------------------------------------------------------------------------------------------- |
652 |
|
|
% |
653 |
|
|
% |
654 |
|
|
% u ca cb |
655 |
|
|
% |
656 |
gezelter |
3983 |
\begin{align} |
657 |
|
|
U_{C_{\bf a}C_{\bf b}}(r)=& |
658 |
gezelter |
3985 |
C_{\bf a} C_{\bf b} v_{01}(r) \label{uchch} |
659 |
gezelter |
3983 |
\\ |
660 |
gezelter |
3906 |
% |
661 |
|
|
% u ca db |
662 |
|
|
% |
663 |
gezelter |
3983 |
U_{C_{\bf a}D_{\bf b}}(r)=& |
664 |
gezelter |
3985 |
C_{\bf a} \left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) v_{11}(r) |
665 |
gezelter |
3906 |
\label{uchdip} |
666 |
gezelter |
3983 |
\\ |
667 |
gezelter |
3906 |
% |
668 |
|
|
% u ca qb |
669 |
|
|
% |
670 |
gezelter |
3985 |
U_{C_{\bf a}Q_{\bf b}}(r)=& C_{\bf a } \Bigl[ \text{Tr}Q_{\bf b} |
671 |
|
|
v_{21}(r) + \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot |
672 |
|
|
\hat{r} \right) v_{22}(r) \Bigr] |
673 |
gezelter |
3906 |
\label{uchquad} |
674 |
gezelter |
3983 |
\\ |
675 |
gezelter |
3906 |
% |
676 |
|
|
% u da cb |
677 |
|
|
% |
678 |
gezelter |
3983 |
%U_{D_{\bf a}C_{\bf b}}(r)=& |
679 |
|
|
%-\frac{C_{\bf b}}{4\pi \epsilon_0} |
680 |
|
|
%\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) v_{11}(r) \label{udipch} |
681 |
|
|
%\\ |
682 |
gezelter |
3906 |
% |
683 |
|
|
% u da db |
684 |
|
|
% |
685 |
gezelter |
3983 |
U_{D_{\bf a}D_{\bf b}}(r)=& |
686 |
gezelter |
3985 |
-\Bigr[ \left( \mathbf{D}_{\mathbf {a}} \cdot |
687 |
gezelter |
3906 |
\mathbf{D}_{\mathbf{b}} \right) v_{21}(r) |
688 |
|
|
+\left( \mathbf{D}_{\mathbf {a}} \cdot \hat{r} \right) |
689 |
|
|
\left( \mathbf{D}_{\mathbf {b}} \cdot \hat{r} \right) |
690 |
|
|
v_{22}(r) \Bigr] |
691 |
|
|
\label{udipdip} |
692 |
gezelter |
3983 |
\\ |
693 |
gezelter |
3906 |
% |
694 |
|
|
% u da qb |
695 |
|
|
% |
696 |
|
|
\begin{split} |
697 |
|
|
% 1 |
698 |
gezelter |
3983 |
U_{D_{\bf a}Q_{\bf b}}(r) =& |
699 |
gezelter |
3985 |
-\Bigl[ |
700 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
701 |
|
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
702 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
703 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r} ) \Bigr] v_{31}(r) \\ |
704 |
|
|
% 2 |
705 |
gezelter |
3985 |
&- \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
706 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{32}(r) |
707 |
|
|
\label{udipquad} |
708 |
|
|
\end{split} |
709 |
gezelter |
3983 |
\\ |
710 |
gezelter |
3906 |
% |
711 |
|
|
% u qa cb |
712 |
|
|
% |
713 |
gezelter |
3983 |
%U_{Q_{\bf a}C_{\bf b}}(r)=& |
714 |
|
|
%\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\bf a} v_{21}(r) |
715 |
|
|
%\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
716 |
|
|
%\label{uquadch} |
717 |
|
|
%\\ |
718 |
gezelter |
3906 |
% |
719 |
|
|
% u qa db |
720 |
|
|
% |
721 |
gezelter |
3983 |
%\begin{split} |
722 |
gezelter |
3906 |
%1 |
723 |
gezelter |
3983 |
%U_{Q_{\bf a}D_{\bf b}}(r)=& |
724 |
|
|
%\frac{1}{4\pi \epsilon_0} \Bigl[ |
725 |
|
|
%\text{Tr}\mathbf{Q}_{\mathbf{a}} |
726 |
|
|
%\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
727 |
|
|
%+2 ( \mathbf{D}_{\mathbf{b}} \cdot |
728 |
|
|
%\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r)\\ |
729 |
gezelter |
3906 |
% 2 |
730 |
gezelter |
3983 |
%&+\frac{1}{4\pi \epsilon_0} |
731 |
|
|
%\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
732 |
|
|
%\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{32}(r) |
733 |
|
|
%\label{uquaddip} |
734 |
|
|
%\end{split} |
735 |
|
|
%\\ |
736 |
gezelter |
3906 |
% |
737 |
|
|
% u qa qb |
738 |
|
|
% |
739 |
|
|
\begin{split} |
740 |
|
|
%1 |
741 |
gezelter |
3983 |
U_{Q_{\bf a}Q_{\bf b}}(r)=& |
742 |
gezelter |
3985 |
\Bigl[ |
743 |
gezelter |
3906 |
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
744 |
|
|
+2 \text{Tr} \left( |
745 |
|
|
\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \right) \Bigr] v_{41}(r) |
746 |
|
|
\\ |
747 |
|
|
% 2 |
748 |
gezelter |
3985 |
&+\Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
749 |
gezelter |
3906 |
\left( \hat{r} \cdot |
750 |
|
|
\mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) |
751 |
|
|
+\text{Tr}\mathbf{Q}_{\mathbf{b}} |
752 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} |
753 |
|
|
\cdot \hat{r} \right) +4 (\hat{r} \cdot |
754 |
|
|
\mathbf{Q}_{{\mathbf a}}\cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
755 |
|
|
\Bigr] v_{42}(r) |
756 |
|
|
\\ |
757 |
|
|
% 4 |
758 |
gezelter |
3985 |
&+ |
759 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) |
760 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{43}(r). |
761 |
|
|
\label{uquadquad} |
762 |
|
|
\end{split} |
763 |
gezelter |
3983 |
\end{align} |
764 |
gezelter |
3985 |
% |
765 |
gezelter |
3983 |
Note that the energies of multipoles on site $\mathbf{b}$ interacting |
766 |
|
|
with those on site $\mathbf{a}$ can be obtained by swapping indices |
767 |
|
|
along with the sign of the intersite vector, $\hat{r}$. |
768 |
gezelter |
3906 |
|
769 |
|
|
% |
770 |
|
|
% |
771 |
|
|
% TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
772 |
|
|
% |
773 |
|
|
|
774 |
gezelter |
3985 |
\begin{sidewaystable} |
775 |
|
|
\caption{\label{tab:tableenergy}Radial functions used in the energy |
776 |
|
|
and torque equations. The $f, g, h, s, t, \mathrm{and} u$ |
777 |
|
|
functions used in this table are defined in Appendices B and C.} |
778 |
|
|
\begin{tabular}{|c|c|l|l|} \hline |
779 |
|
|
Generic&Bare Coulomb&Taylor-Shifted&Gradient-Shifted |
780 |
gezelter |
3906 |
\\ \hline |
781 |
|
|
% |
782 |
|
|
% |
783 |
|
|
% |
784 |
|
|
%Ch-Ch& |
785 |
|
|
$v_{01}(r)$ & |
786 |
|
|
$\frac{1}{r}$ & |
787 |
|
|
$f_0(r)$ & |
788 |
|
|
$f(r)-f(r_c)-(r-r_c)g(r_c)$ |
789 |
|
|
\\ |
790 |
|
|
% |
791 |
|
|
% |
792 |
|
|
% |
793 |
|
|
%Ch-Di& |
794 |
|
|
$v_{11}(r)$ & |
795 |
|
|
$-\frac{1}{r^2}$ & |
796 |
|
|
$g_1(r)$ & |
797 |
|
|
$g(r)-g(r_c)-(r-r_c)h(r_c)$ \\ |
798 |
|
|
% |
799 |
|
|
% |
800 |
|
|
% |
801 |
|
|
%Ch-Qu/Di-Di& |
802 |
|
|
$v_{21}(r)$ & |
803 |
|
|
$-\frac{1}{r^3} $ & |
804 |
|
|
$\frac{g_2(r)}{r} $ & |
805 |
|
|
$\frac{g(r)}{r}-\frac{g(r_c)}{r_c} -(r-r_c) |
806 |
|
|
\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right)$ \\ |
807 |
|
|
$v_{22}(r)$ & |
808 |
|
|
$\frac{3}{r^3} $ & |
809 |
|
|
$\left(-\frac{g_2(r)}{r} + h_2(r) \right)$ & |
810 |
|
|
$\left(-\frac{g(r)}{r}+h(r) \right) |
811 |
gezelter |
3985 |
-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right)$ \\ |
812 |
|
|
&&& $ ~~~-(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$ |
813 |
gezelter |
3906 |
\\ |
814 |
|
|
% |
815 |
|
|
% |
816 |
|
|
% |
817 |
|
|
%Di-Qu & |
818 |
|
|
$v_{31}(r)$ & |
819 |
|
|
$\frac{3}{r^4} $ & |
820 |
|
|
$\left(-\frac{g_3(r)}{r^2} + \frac{h_3(r)}{r} \right)$ & |
821 |
|
|
$\left( -\frac{g(r)}{r^2}+\frac{h(r)}{r} \right) |
822 |
|
|
-\left(-\frac{g(r_c)}{r_c^2}+\frac{h(r_c)}{r_c} \right) $\\ |
823 |
gezelter |
3985 |
&&&$ ~~~ -(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)$ |
824 |
gezelter |
3906 |
\\ |
825 |
|
|
% |
826 |
|
|
$v_{32}(r)$ & |
827 |
|
|
$-\frac{15}{r^4} $ & |
828 |
|
|
$\left( \frac{3g_3(r)}{r^2} - \frac{3h_3(r)}{r} + s_3(r) \right)$ & |
829 |
|
|
$\left( \frac{3g(r)}{r^2} - \frac{3h(r)}{r} + s(r) \right) |
830 |
|
|
- \left( \frac{3g(r_c)}{r_c^2} - \frac{3h(r_c)}{r_c} + s(r_c) \right)$ \\ |
831 |
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)$ |
832 |
gezelter |
3906 |
\\ |
833 |
|
|
% |
834 |
|
|
% |
835 |
|
|
% |
836 |
|
|
%Qu-Qu& |
837 |
|
|
$v_{41}(r)$ & |
838 |
|
|
$\frac{3}{r^5} $ & |
839 |
|
|
$\left(-\frac{g_4(r)}{r^3} +\frac{h_4(r)}{r^2} \right) $ & |
840 |
|
|
$\left( -\frac{g(r)}{r^3} + \frac{h(r)}{r^2} \right) |
841 |
|
|
- \left( -\frac{g(r_c)}{r_c^3} + \frac{h(r_c)}{r_c^2} \right)$ \\ |
842 |
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)$ |
843 |
gezelter |
3906 |
\\ |
844 |
|
|
% 2 |
845 |
|
|
$v_{42}(r)$ & |
846 |
|
|
$- \frac{15}{r^5} $ & |
847 |
|
|
$\left( \frac{3g_4(r)}{r^3} - \frac{3h_4(r)}{r^2}+\frac{s_4(r)}{r} \right)$ & |
848 |
|
|
$\left( \frac{3g(r)}{r^3} - \frac{3h(r)}{r^2}+\frac{s(r)}{r} \right) |
849 |
|
|
-\left( \frac{3g(r_c)}{r_c^3} - \frac{3h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ \\ |
850 |
gezelter |
3985 |
&&&$ ~~~ -(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
851 |
gezelter |
3906 |
-\frac{4s(r_c)}{r_c^2} + \frac{t(r_c)}{r_c}\right)$ |
852 |
|
|
\\ |
853 |
|
|
% 3 |
854 |
|
|
$v_{43}(r)$ & |
855 |
|
|
$ \frac{105}{r^5} $ & |
856 |
|
|
$\left(-\frac{15g_4(r)}{r^3}+\frac{15h_4(r)}{r^2}-\frac{6s_4(r)}{r} + t_4(r)\right) $ & |
857 |
|
|
$\left(-\frac{15g(r)}{r^3}+\frac{15h(r)}{r^2}-\frac{6s(r)}{r} + t(r)\right)$ \\ |
858 |
gezelter |
3985 |
&&&$~~~ -\left(-\frac{15g(r_c)}{r_c^3}+\frac{15h(r_c)}{r_c^2}-\frac{6s(r_c)}{r_c} + t(r_c)\right)$ \\ |
859 |
|
|
&&&$~~~ -(r-r_c)\left(\frac{45g(r_c)}{r_c^4}-\frac{45h(r_c)}{r_c^3}+\frac{21s(r_c)}{r_c^2} |
860 |
|
|
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ \hline |
861 |
gezelter |
3906 |
\end{tabular} |
862 |
gezelter |
3985 |
\end{sidewaystable} |
863 |
gezelter |
3906 |
% |
864 |
|
|
% |
865 |
|
|
% FORCE TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
866 |
|
|
% |
867 |
|
|
|
868 |
gezelter |
3985 |
\begin{sidewaystable} |
869 |
gezelter |
3906 |
\caption{\label{tab:tableFORCE}Radial functions used in the force equations.} |
870 |
gezelter |
3985 |
\begin{tabular}{|c|c|l|l|} \hline |
871 |
|
|
Function&Definition&Taylor-Shifted&Gradient-Shifted |
872 |
gezelter |
3906 |
\\ \hline |
873 |
|
|
% |
874 |
|
|
% |
875 |
|
|
% |
876 |
|
|
$w_a(r)$& |
877 |
gezelter |
3985 |
$\frac{d v_{01}}{dr}$& |
878 |
|
|
$g_0(r)$& |
879 |
|
|
$g(r)-g(r_c)$ \\ |
880 |
gezelter |
3906 |
% |
881 |
|
|
% |
882 |
|
|
$w_b(r)$ & |
883 |
gezelter |
3985 |
$\frac{d v_{11}}{dr} - \frac{v_{11}(r)}{r} $& |
884 |
|
|
$\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
885 |
|
|
$h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
886 |
gezelter |
3906 |
% |
887 |
|
|
$w_c(r)$ & |
888 |
gezelter |
3985 |
$\frac{v_{11}(r)}{r}$ & |
889 |
|
|
$\frac{g_1(r)}{r} $ & |
890 |
|
|
$\frac{v_{11}(r)}{r}$\\ |
891 |
gezelter |
3906 |
% |
892 |
|
|
% |
893 |
|
|
$w_d(r)$& |
894 |
gezelter |
3985 |
$\frac{d v_{21}}{dr}$& |
895 |
|
|
$\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
896 |
|
|
$\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
897 |
|
|
-\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $ \\ |
898 |
gezelter |
3906 |
% |
899 |
|
|
$w_e(r)$ & |
900 |
gezelter |
3985 |
$\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
901 |
|
|
$\frac{v_{22}(r)}{r}$ & |
902 |
gezelter |
3906 |
$\frac{v_{22}(r)}{r}$ \\ |
903 |
|
|
% |
904 |
|
|
% |
905 |
|
|
$w_f(r)$& |
906 |
gezelter |
3985 |
$\frac{d v_{22}}{dr} - \frac{2v_{22}(r)}{r}$& |
907 |
|
|
$\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
908 |
|
|
$ \left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) $ \\ |
909 |
|
|
&&& $ ~~~- \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) |
910 |
|
|
\right)-\frac{2v_{22}(r)}{r}$\\ |
911 |
gezelter |
3906 |
% |
912 |
|
|
$w_g(r)$& |
913 |
gezelter |
3985 |
$\frac{v_{31}(r)}{r}$& |
914 |
|
|
$ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
915 |
gezelter |
3906 |
$\frac{v_{31}(r)}{r}$\\ |
916 |
|
|
% |
917 |
|
|
$w_h(r)$ & |
918 |
gezelter |
3985 |
$\frac{d v_{31}}{dr} -\frac{v_{31}(r)}{r}$& |
919 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
920 |
|
|
$ \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) $ \\ |
921 |
|
|
&&& $ ~~~ -\frac{v_{31}(r)}{r}$ \\ |
922 |
gezelter |
3906 |
% 2 |
923 |
|
|
$w_i(r)$ & |
924 |
gezelter |
3985 |
$\frac{v_{32}(r)}{r}$ & |
925 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
926 |
|
|
$\frac{v_{32}(r)}{r}$\\ |
927 |
gezelter |
3906 |
% |
928 |
|
|
$w_j(r)$ & |
929 |
gezelter |
3985 |
$\frac{d v_{32}}{dr} - \frac{3v_{32}}{r}$& |
930 |
|
|
$\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
931 |
|
|
$\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right)$ \\ |
932 |
|
|
&&& $~~~-\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} |
933 |
|
|
-\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
934 |
gezelter |
3906 |
% |
935 |
|
|
$w_k(r)$ & |
936 |
gezelter |
3985 |
$\frac{d v_{41}}{dr} $ & |
937 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
938 |
|
|
$\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right) |
939 |
|
|
-\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
940 |
gezelter |
3906 |
% |
941 |
|
|
$w_l(r)$ & |
942 |
gezelter |
3985 |
$\frac{d v_{42}}{dr} -\frac{2v_{42}(r)}{r}$ & |
943 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
944 |
|
|
$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
945 |
|
|
&&& $~~~ -\left(-\frac{9g(r_c)}{r_c^4} +\frac{9h(r_c)}{r_c^3} -\frac{4s(r_c)}{r_c^2} +\frac{t(r_c)}{r_c} \right) |
946 |
|
|
-\frac{2v_{42}(r)}{r}$\\ |
947 |
gezelter |
3906 |
% |
948 |
|
|
$w_m(r)$ & |
949 |
gezelter |
3985 |
$\frac{d v_{43}}{dr} -\frac{4v_{43}(r)}{r}$& |
950 |
|
|
$\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
951 |
|
|
$\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$\\ |
952 |
|
|
&&& $~~~- \left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
953 |
|
|
+\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $\\ |
954 |
|
|
&&& $~~~-\frac{4v_{43}(r)}{r}$ \\ |
955 |
gezelter |
3906 |
% |
956 |
|
|
$w_n(r)$ & |
957 |
gezelter |
3985 |
$\frac{v_{42}(r)}{r}$ & |
958 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
959 |
|
|
$\frac{v_{42}(r)}{r}$\\ |
960 |
gezelter |
3906 |
% |
961 |
|
|
$w_o(r)$ & |
962 |
gezelter |
3985 |
$\frac{v_{43}(r)}{r}$& |
963 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
964 |
|
|
$\frac{v_{43}(r)}{r}$ \\ \hline |
965 |
gezelter |
3906 |
% |
966 |
|
|
|
967 |
|
|
\end{tabular} |
968 |
gezelter |
3985 |
\end{sidewaystable} |
969 |
gezelter |
3906 |
% |
970 |
|
|
% |
971 |
|
|
% |
972 |
|
|
|
973 |
|
|
\subsection{Forces} |
974 |
gezelter |
3985 |
The force on object $\bf{a}$, $\mathbf{F}_{\bf a}$, due to object |
975 |
|
|
$\bf{b}$ is the negative of the force on $\bf{b}$ due to $\bf{a}$. For |
976 |
|
|
a simple charge-charge interaction, these forces will point along the |
977 |
|
|
$\pm \hat{r}$ directions, where $\mathbf{r}=\mathbf{r}_b - |
978 |
|
|
\mathbf{r}_a $. Thus |
979 |
gezelter |
3906 |
% |
980 |
|
|
\begin{equation} |
981 |
|
|
F_{\bf a \alpha} = \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}}{\partial r} |
982 |
|
|
\quad \text{and} \quad F_{\bf b \alpha} |
983 |
|
|
= - \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}} {\partial r} . |
984 |
|
|
\end{equation} |
985 |
|
|
% |
986 |
gezelter |
3985 |
Obtaining the force from the interaction energy expressions is the |
987 |
|
|
same for higher-order multipole interactions -- the trick is to make |
988 |
|
|
sure that all $r$-dependent derivatives are considered. This is |
989 |
|
|
straighforward if the interaction energies are written explicitly in |
990 |
|
|
terms of $\hat{r}$ and the body axes ($\hat{a}_m$, |
991 |
|
|
$\hat{b}_n$) : |
992 |
gezelter |
3906 |
% |
993 |
|
|
\begin{equation} |
994 |
|
|
U(r,\{\hat{a}_m \cdot \hat{r} \}, |
995 |
gezelter |
3985 |
\{\hat{b}_n\cdot \hat{r} \}, |
996 |
gezelter |
3906 |
\{\hat{a}_m \cdot \hat{b}_n \}) . |
997 |
|
|
\label{ugeneral} |
998 |
|
|
\end{equation} |
999 |
|
|
% |
1000 |
gezelter |
3985 |
Allen and Germano,\cite{Allen:2006fk} showed that if the energy is |
1001 |
|
|
written in this form, the forces come out relatively cleanly, |
1002 |
gezelter |
3906 |
% |
1003 |
|
|
\begin{equation} |
1004 |
|
|
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
1005 |
|
|
= \frac{\partial U}{\partial r} \hat{r} |
1006 |
|
|
+ \sum_m \left[ |
1007 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
1008 |
|
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
1009 |
|
|
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
1010 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
1011 |
|
|
\right] \label{forceequation}. |
1012 |
|
|
\end{equation} |
1013 |
|
|
% |
1014 |
gezelter |
3985 |
Note that our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ |
1015 |
|
|
is opposite in sign to that of Allen and Germano.\cite{Allen:2006fk} |
1016 |
|
|
In simplifying the algebra, we have also used: |
1017 |
gezelter |
3906 |
% |
1018 |
gezelter |
3985 |
\begin{align} |
1019 |
gezelter |
3906 |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
1020 |
gezelter |
3985 |
=& \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
1021 |
gezelter |
3906 |
\right) \\ |
1022 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
1023 |
gezelter |
3985 |
=& \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
1024 |
gezelter |
3906 |
\right) . |
1025 |
gezelter |
3985 |
\end{align} |
1026 |
gezelter |
3906 |
% |
1027 |
gezelter |
3985 |
We list below the force equations written in terms of lab-frame |
1028 |
|
|
coordinates. The radial functions used in the two methods are listed |
1029 |
|
|
in Table \ref{tab:tableFORCE} |
1030 |
gezelter |
3906 |
% |
1031 |
gezelter |
3985 |
%SPACE COORDINATES FORCE EQUATIONS |
1032 |
gezelter |
3906 |
% |
1033 |
|
|
% ************************************************************************** |
1034 |
|
|
% f ca cb |
1035 |
|
|
% |
1036 |
gezelter |
3985 |
\begin{align} |
1037 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} =& |
1038 |
|
|
C_{\bf a} C_{\bf b} w_a(r) \hat{r} \\ |
1039 |
gezelter |
3906 |
% |
1040 |
|
|
% |
1041 |
|
|
% |
1042 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} =& |
1043 |
|
|
C_{\bf a} \Bigl[ |
1044 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{b}} \right) |
1045 |
|
|
w_b(r) \hat{r} |
1046 |
gezelter |
3985 |
+ \mathbf{D}_{\mathbf{b}} w_c(r) \Bigr] \\ |
1047 |
gezelter |
3906 |
% |
1048 |
|
|
% |
1049 |
|
|
% |
1050 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} =& |
1051 |
|
|
C_{\bf a } \Bigr[ |
1052 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\bf b} w_d(r) \hat{r} |
1053 |
|
|
+ 2 \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} w_e(r) |
1054 |
gezelter |
3985 |
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} |
1055 |
|
|
\right) w_f(r) \hat{r} \Bigr] \\ |
1056 |
gezelter |
3906 |
% |
1057 |
|
|
% |
1058 |
|
|
% |
1059 |
gezelter |
3985 |
% \begin{equation} |
1060 |
|
|
% \mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1061 |
|
|
% -C_{\bf{b}} \Bigl[ |
1062 |
|
|
% \left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) w_b(r) \hat{r} |
1063 |
|
|
% + \mathbf{D}_{\mathbf{a}} w_c(r) \Bigr] |
1064 |
|
|
% \end{equation} |
1065 |
gezelter |
3906 |
% |
1066 |
|
|
% |
1067 |
|
|
% |
1068 |
gezelter |
3985 |
\begin{split} |
1069 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
1070 |
gezelter |
3906 |
- \mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} w_d(r) \hat{r} |
1071 |
|
|
+ \left( \mathbf{D}_{\mathbf {a}} |
1072 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
1073 |
gezelter |
3985 |
+ \mathbf{D}_{\mathbf {b}} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) \right) w_e(r)\\ |
1074 |
gezelter |
3906 |
% 2 |
1075 |
gezelter |
3985 |
& - \left( \hat{r} \cdot \mathbf{D}_{\mathbf {a}} \right) |
1076 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf {b}} \right) w_f(r) \hat{r} |
1077 |
|
|
\end{split}\\ |
1078 |
gezelter |
3906 |
% |
1079 |
|
|
% |
1080 |
|
|
% |
1081 |
|
|
\begin{split} |
1082 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} =& - \Bigl[ |
1083 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} \mathbf{ D}_{\mathbf{a}} |
1084 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \cdot |
1085 |
|
|
\mathbf{Q}_{\mathbf{b}} \Bigr] w_g(r) |
1086 |
gezelter |
3985 |
- \Bigl[ |
1087 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1088 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) |
1089 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
1090 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1091 |
|
|
% 3 |
1092 |
gezelter |
3985 |
& - \Bigl[\mathbf{ D}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1093 |
gezelter |
3906 |
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \Bigr] |
1094 |
|
|
w_i(r) |
1095 |
|
|
% 4 |
1096 |
gezelter |
3985 |
- |
1097 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) |
1098 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) w_j(r) \hat{r} \end{split} \\ |
1099 |
gezelter |
3906 |
% |
1100 |
|
|
% |
1101 |
gezelter |
3985 |
% \begin{equation} |
1102 |
|
|
% \mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
1103 |
|
|
% \frac{C_{\bf b }}{4\pi \epsilon_0} \Bigr[ |
1104 |
|
|
% \text{Tr}\mathbf{Q}_{\bf a} w_d(r) \hat{r} |
1105 |
|
|
% + 2 \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} w_e(r) |
1106 |
|
|
% + \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
1107 |
|
|
% \end{equation} |
1108 |
|
|
% % |
1109 |
|
|
% \begin{equation} |
1110 |
|
|
% \begin{split} |
1111 |
|
|
% \mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1112 |
|
|
% &\frac{1}{4\pi \epsilon_0} \Bigl[ |
1113 |
|
|
% \text{Tr}\mathbf{Q}_{\mathbf{a}} \mathbf{D}_{\mathbf{b}} |
1114 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} \Bigr] w_g(r) |
1115 |
|
|
% % 2 |
1116 |
|
|
% + \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
1117 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1118 |
|
|
% +2 (\mathbf{D}_{\mathbf{b}} \cdot |
1119 |
|
|
% \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1120 |
|
|
% % 3 |
1121 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} \Bigl[ \mathbf{D}_{\mathbf{b}} |
1122 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1123 |
|
|
% +2 (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1124 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \Bigr] w_i(r) |
1125 |
|
|
% % 4 |
1126 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1127 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1128 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) w_j(r) \hat{r} |
1129 |
|
|
% \end{split} |
1130 |
|
|
% \end{equation} |
1131 |
gezelter |
3906 |
% |
1132 |
|
|
% |
1133 |
|
|
% |
1134 |
|
|
\begin{split} |
1135 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
1136 |
|
|
\Bigl[ |
1137 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} \hat{r} |
1138 |
|
|
+ 2 \text{Tr} ( \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_k(r) \hat{r} \\ |
1139 |
|
|
% 2 |
1140 |
gezelter |
3985 |
&+ \Bigl[ |
1141 |
gezelter |
3906 |
2\text{Tr}\mathbf{Q}_{\mathbf{b}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
1142 |
|
|
+ 2\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) |
1143 |
|
|
% 3 |
1144 |
|
|
+4 (\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1145 |
|
|
+ 4(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}}) \Bigr] w_n(r) \\ |
1146 |
|
|
% 4 |
1147 |
gezelter |
3985 |
&+ \Bigl[ |
1148 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1149 |
|
|
+ \text{Tr}\mathbf{Q}_{\mathbf{b}} |
1150 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1151 |
|
|
% 5 |
1152 |
|
|
+4 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot |
1153 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
1154 |
|
|
% |
1155 |
gezelter |
3985 |
&+ \Bigl[ |
1156 |
gezelter |
3906 |
+ 2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
1157 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1158 |
|
|
%6 |
1159 |
|
|
+2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1160 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_o(r) \\ |
1161 |
|
|
% 7 |
1162 |
gezelter |
3985 |
&+ |
1163 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1164 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) w_m(r) \hat{r} \end{split} |
1165 |
|
|
\end{align} |
1166 |
|
|
Note that the forces for higher multipoles on site $\mathbf{a}$ |
1167 |
|
|
interacting with those of lower order on site $\mathbf{b}$ can be |
1168 |
|
|
obtained by swapping indices in the expressions above. |
1169 |
|
|
|
1170 |
gezelter |
3906 |
% |
1171 |
gezelter |
3985 |
% Torques SECTION ----------------------------------------------------------------------------------------- |
1172 |
gezelter |
3906 |
% |
1173 |
|
|
\subsection{Torques} |
1174 |
gezelter |
3985 |
When energies are written in the form of Eq.~({\ref{ugeneral}), then |
1175 |
|
|
torques can be found in a relatively straightforward |
1176 |
|
|
manner,\cite{Allen:2006fk} |
1177 |
gezelter |
3906 |
% |
1178 |
|
|
\begin{eqnarray} |
1179 |
|
|
\mathbf{\tau}_{\bf a} = |
1180 |
|
|
\sum_m |
1181 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
1182 |
|
|
( \hat{r} \times \hat{a}_m ) |
1183 |
|
|
-\sum_{mn} |
1184 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
1185 |
|
|
(\hat{a}_m \times \hat{b}_n) \\ |
1186 |
|
|
% |
1187 |
|
|
\mathbf{\tau}_{\bf b} = |
1188 |
|
|
\sum_m |
1189 |
|
|
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
1190 |
|
|
( \hat{r} \times \hat{b}_m) |
1191 |
|
|
+\sum_{mn} |
1192 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
1193 |
|
|
(\hat{a}_m \times \hat{b}_n) . |
1194 |
|
|
\end{eqnarray} |
1195 |
|
|
% |
1196 |
|
|
% |
1197 |
gezelter |
3985 |
The torques for both the Taylor-Shifted as well as Gradient-Shifted |
1198 |
|
|
methods are given in space-frame coordinates: |
1199 |
gezelter |
3906 |
% |
1200 |
|
|
% |
1201 |
gezelter |
3985 |
\begin{align} |
1202 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} =& |
1203 |
|
|
C_{\bf a} (\hat{r} \times \mathbf{D}_{\mathbf{b}}) v_{11}(r) \\ |
1204 |
gezelter |
3906 |
% |
1205 |
|
|
% |
1206 |
|
|
% |
1207 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} =& |
1208 |
|
|
2C_{\bf a} |
1209 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{22}(r) \\ |
1210 |
gezelter |
3906 |
% |
1211 |
|
|
% |
1212 |
|
|
% |
1213 |
gezelter |
3985 |
% \begin{equation} |
1214 |
|
|
% \mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1215 |
|
|
% -\frac{C_{\bf b}}{4\pi \epsilon_0} |
1216 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{a}}) v_{11}(r) |
1217 |
|
|
% \end{equation} |
1218 |
gezelter |
3906 |
% |
1219 |
|
|
% |
1220 |
|
|
% |
1221 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
1222 |
|
|
\mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
1223 |
gezelter |
3906 |
% 2 |
1224 |
gezelter |
3985 |
- |
1225 |
gezelter |
3906 |
(\hat{r} \times \mathbf{D}_{\mathbf {a}} ) |
1226 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{D}_{\mathbf {b}} ) v_{22}(r)\\ |
1227 |
gezelter |
3906 |
% |
1228 |
|
|
% |
1229 |
|
|
% |
1230 |
gezelter |
3985 |
% \begin{equation} |
1231 |
|
|
% \mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} = |
1232 |
|
|
% -\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
1233 |
|
|
% % 2 |
1234 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1235 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf {a}} ) |
1236 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
1237 |
|
|
% \end{equation} |
1238 |
gezelter |
3906 |
% |
1239 |
|
|
% |
1240 |
|
|
% |
1241 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} =& |
1242 |
|
|
\Bigl[ |
1243 |
gezelter |
3906 |
-\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1244 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
1245 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \times |
1246 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1247 |
|
|
\Bigr] v_{31}(r) |
1248 |
|
|
% 3 |
1249 |
gezelter |
3985 |
- (\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
1250 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{32}(r)\\ |
1251 |
gezelter |
3906 |
% |
1252 |
|
|
% |
1253 |
|
|
% |
1254 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} =& |
1255 |
|
|
\Bigl[ |
1256 |
gezelter |
3906 |
+2 ( \mathbf{D}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \times |
1257 |
|
|
\hat{r} |
1258 |
|
|
-2 \mathbf{D}_{\mathbf{a}} \times |
1259 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1260 |
|
|
\Bigr] v_{31}(r) |
1261 |
|
|
% 2 |
1262 |
gezelter |
3985 |
+ |
1263 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}}) |
1264 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}}) \times \hat{r} v_{32}(r)\\ |
1265 |
gezelter |
3906 |
% |
1266 |
|
|
% |
1267 |
|
|
% |
1268 |
gezelter |
3985 |
% \begin{equation} |
1269 |
|
|
% \mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1270 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1271 |
|
|
% -2 (\mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} ) \times \hat{r} |
1272 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \times |
1273 |
|
|
% (\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1274 |
|
|
% \Bigr] v_{31}(r) |
1275 |
|
|
% % 3 |
1276 |
|
|
% - \frac{2}{4\pi \epsilon_0} |
1277 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}} ) |
1278 |
|
|
% (\hat{r} \cdot \mathbf |
1279 |
|
|
% {Q}_{{\mathbf a}}) \times \hat{r} v_{32}(r) |
1280 |
|
|
% \end{equation} |
1281 |
gezelter |
3906 |
% |
1282 |
|
|
% |
1283 |
|
|
% |
1284 |
gezelter |
3985 |
% \begin{equation} |
1285 |
|
|
% \mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} = |
1286 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1287 |
|
|
% \text{Tr}\mathbf{Q}_{\mathbf{a}} |
1288 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1289 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \times |
1290 |
|
|
% ( \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
1291 |
|
|
% % 2 |
1292 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1293 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1294 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) v_{32}(r) |
1295 |
|
|
% \end{equation} |
1296 |
gezelter |
3906 |
% |
1297 |
|
|
% |
1298 |
|
|
% |
1299 |
|
|
\begin{split} |
1300 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
1301 |
|
|
-4 |
1302 |
gezelter |
3906 |
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} |
1303 |
|
|
v_{41}(r) \\ |
1304 |
|
|
% 2 |
1305 |
gezelter |
3985 |
&+ |
1306 |
gezelter |
3906 |
\Bigl[-2\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1307 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times \hat{r} |
1308 |
|
|
+4 \hat{r} \times |
1309 |
|
|
( \mathbf{Q}_{{\mathbf a}} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1310 |
|
|
% 3 |
1311 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} )\times |
1312 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} ) \Bigr] v_{42}(r) \\ |
1313 |
|
|
% 4 |
1314 |
gezelter |
3985 |
&+ 2 |
1315 |
gezelter |
3906 |
\hat{r} \times ( \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1316 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) \end{split}\\ |
1317 |
gezelter |
3906 |
% |
1318 |
|
|
% |
1319 |
|
|
% |
1320 |
|
|
\begin{split} |
1321 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} = |
1322 |
gezelter |
3985 |
&4 |
1323 |
gezelter |
3906 |
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} v_{41}(r) \\ |
1324 |
|
|
% 2 |
1325 |
gezelter |
3985 |
&+ \Bigl[- 2\text{Tr}\mathbf{Q}_{\mathbf{a}} |
1326 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \times \hat{r} |
1327 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot |
1328 |
|
|
\mathbf{Q}_{{\mathbf b}} ) \times |
1329 |
|
|
\hat{r} |
1330 |
|
|
+4 ( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times |
1331 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1332 |
|
|
\Bigr] v_{42}(r) \\ |
1333 |
|
|
% 4 |
1334 |
gezelter |
3985 |
&+2 |
1335 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1336 |
gezelter |
3985 |
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r)\end{split} |
1337 |
|
|
\end{align} |
1338 |
|
|
% |
1339 |
|
|
Here, we have defined the matrix cross product in an identical form |
1340 |
|
|
as in Ref. \onlinecite{Smith98}: |
1341 |
|
|
\begin{equation} |
1342 |
|
|
\left[\mathbf{A} \times \mathbf{B}\right]_\alpha = \sum_\beta |
1343 |
|
|
\left[\mathbf{A}_{\alpha+1,\beta} \mathbf{B}_{\alpha+2,\beta} |
1344 |
|
|
-\mathbf{A}_{\alpha+2,\beta} \mathbf{B}_{\alpha+2,\beta} |
1345 |
|
|
\right] |
1346 |
gezelter |
3906 |
\end{equation} |
1347 |
gezelter |
3985 |
where $\alpha+1$ and $\alpha+2$ are regarded as cyclic |
1348 |
|
|
permuations of the matrix indices. |
1349 |
gezelter |
3980 |
|
1350 |
gezelter |
3985 |
All of the radial functions required for torques are identical with |
1351 |
|
|
the radial functions previously computed for the interaction energies. |
1352 |
|
|
These are tabulated for both shifted force methods in table |
1353 |
|
|
\ref{tab:tableenergy}. The torques for higher multipoles on site |
1354 |
|
|
$\mathbf{a}$ interacting with those of lower order on site |
1355 |
|
|
$\mathbf{b}$ can be obtained by swapping indices in the expressions |
1356 |
|
|
above. |
1357 |
|
|
|
1358 |
gezelter |
3980 |
\section{Comparison to known multipolar energies} |
1359 |
|
|
|
1360 |
|
|
To understand how these new real-space multipole methods behave in |
1361 |
|
|
computer simulations, it is vital to test against established methods |
1362 |
|
|
for computing electrostatic interactions in periodic systems, and to |
1363 |
|
|
evaluate the size and sources of any errors that arise from the |
1364 |
|
|
real-space cutoffs. In this paper we test Taylor-shifted and |
1365 |
|
|
Gradient-shifted electrostatics against analytical methods for |
1366 |
|
|
computing the energies of ordered multipolar arrays. In the following |
1367 |
|
|
paper, we test the new methods against the multipolar Ewald sum for |
1368 |
|
|
computing the energies, forces and torques for a wide range of typical |
1369 |
|
|
condensed-phase (disordered) systems. |
1370 |
|
|
|
1371 |
|
|
Because long-range electrostatic effects can be significant in |
1372 |
|
|
crystalline materials, ordered multipolar arrays present one of the |
1373 |
|
|
biggest challenges for real-space cutoff methods. The dipolar |
1374 |
|
|
analogues to the Madelung constants were first worked out by Sauer, |
1375 |
|
|
who computed the energies of ordered dipole arrays of zero |
1376 |
|
|
magnetization and obtained a number of these constants.\cite{Sauer} |
1377 |
|
|
This theory was developed more completely by Luttinger and |
1378 |
gezelter |
3986 |
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays |
1379 |
|
|
and other periodic structures. We have repeated the Luttinger \& |
1380 |
|
|
Tisza series summations to much higher order and obtained the energy |
1381 |
|
|
constants (converged to one part in $10^9$) in table \ref{tab:LT}. |
1382 |
|
|
|
1383 |
|
|
\begin{table*}[h] |
1384 |
gezelter |
3980 |
\centering{ |
1385 |
|
|
\caption{Luttinger \& Tisza arrays and their associated |
1386 |
|
|
energy constants. Type "A" arrays have nearest neighbor strings of |
1387 |
|
|
antiparallel dipoles. Type "B" arrays have nearest neighbor |
1388 |
|
|
strings of antiparallel dipoles if the dipoles are contained in a |
1389 |
|
|
plane perpendicular to the dipole direction that passes through |
1390 |
|
|
the dipole.} |
1391 |
|
|
} |
1392 |
|
|
\label{tab:LT} |
1393 |
|
|
\begin{ruledtabular} |
1394 |
|
|
\begin{tabular}{cccc} |
1395 |
|
|
Array Type & Lattice & Dipole Direction & Energy constants \\ \hline |
1396 |
|
|
A & SC & 001 & -2.676788684 \\ |
1397 |
|
|
A & BCC & 001 & 0 \\ |
1398 |
|
|
A & BCC & 111 & -1.770078733 \\ |
1399 |
|
|
A & FCC & 001 & 2.166932835 \\ |
1400 |
|
|
A & FCC & 011 & -1.083466417 \\ |
1401 |
gezelter |
3986 |
B & SC & 001 & -2.676788684 \\ |
1402 |
|
|
B & BCC & 001 & -1.338394342 \\ |
1403 |
|
|
B & BCC & 111 & -1.770078733 \\ |
1404 |
|
|
B & FCC & 001 & -1.083466417 \\ |
1405 |
|
|
B & FCC & 011 & -1.807573634 \\ |
1406 |
|
|
-- & BCC & minimum & -1.985920929 \\ |
1407 |
gezelter |
3980 |
\end{tabular} |
1408 |
|
|
\end{ruledtabular} |
1409 |
|
|
\end{table*} |
1410 |
|
|
|
1411 |
|
|
In addition to the A and B arrays, there is an additional minimum |
1412 |
|
|
energy structure for the BCC lattice that was found by Luttinger \& |
1413 |
gezelter |
3986 |
Tisza. The total electrostatic energy for any of the arrays is given |
1414 |
|
|
by: |
1415 |
gezelter |
3980 |
\begin{equation} |
1416 |
|
|
E = C N^2 \mu^2 |
1417 |
|
|
\end{equation} |
1418 |
gezelter |
3986 |
where $C$ is the energy constant given in table \ref{tab:LT}, $N$ is |
1419 |
|
|
the number of dipoles per unit volume, and $\mu$ is the strength of |
1420 |
|
|
the dipole. |
1421 |
gezelter |
3980 |
|
1422 |
gezelter |
3988 |
To test the new electrostatic methods, we have constructed very large, |
1423 |
|
|
$N$ = 8,000~(sc), 16,000~(bcc), or 32,000~(fcc) arrays of dipoles in |
1424 |
|
|
the orientations described in table \ref{tab:LT}. For the purposes of |
1425 |
|
|
testing the energy expressions and the self-neutralization schemes, |
1426 |
|
|
the primary quantity of interest is the analytic energy constant for |
1427 |
|
|
the perfect arrays. Convergence to these constants are shown as a |
1428 |
|
|
function of both the cutoff radius, $r_c$, and the damping parameter, |
1429 |
|
|
$\alpha$ in Figs. \ref{fig:energyConstVsCutoff} and XXX. We have |
1430 |
|
|
simultaneously tested a hard cutoff (where the kernel is simply |
1431 |
|
|
truncated at the cutoff radius), as well as a shifted potential (SP) |
1432 |
|
|
form which includes a potential-shifting and self-interaction term, |
1433 |
|
|
but does not shift the forces and torques smoothly at the cutoff |
1434 |
gezelter |
3986 |
radius. |
1435 |
|
|
|
1436 |
gezelter |
3988 |
\begin{figure} |
1437 |
|
|
\includegraphics[width=4.5in]{energyConstVsCutoff} |
1438 |
|
|
\caption{Convergence to the analytic energy constants as a function of |
1439 |
|
|
cutoff radius (normalized by the lattice constant) for the different |
1440 |
|
|
real-space methods. The two crystals shown here are the ``B'' array |
1441 |
|
|
for bcc crystals with the dipoles along the 001 direction (upper), |
1442 |
|
|
as well as the minimum energy bcc lattice (lower). The analytic |
1443 |
|
|
energy constants are shown as a grey dashed line. The left panel |
1444 |
|
|
shows results for the undamped kernel ($1/r$), while the damped |
1445 |
|
|
error function kernel, $B_0(r)$ was used in the right panel. } |
1446 |
|
|
\label{fig:energyConstVsCutoff} |
1447 |
|
|
\end{figure} |
1448 |
|
|
|
1449 |
|
|
The Hard cutoff exhibits oscillations around the analytic energy |
1450 |
|
|
constants, and converges to incorrect energies when the complementary |
1451 |
|
|
error function damping kernel is used. The shifted potential (SP) and |
1452 |
|
|
gradient-shifted force (GSF) approximations converge to the correct |
1453 |
|
|
energy smoothly by $r_c / 6 a$ even for the undamped case. This |
1454 |
|
|
indicates that the correction provided by the self term is required |
1455 |
|
|
for obtaining accurate energies. The Taylor-shifted force (TSF) |
1456 |
|
|
approximation appears to perturb the potential too much inside the |
1457 |
|
|
cutoff region to provide accurate measures of the energy constants. |
1458 |
|
|
|
1459 |
|
|
|
1460 |
gezelter |
3986 |
{\it Quadrupolar} analogues to the Madelung constants were first |
1461 |
|
|
worked out by Nagai and Nakamura who computed the energies of selected |
1462 |
|
|
quadrupole arrays based on extensions to the Luttinger and Tisza |
1463 |
|
|
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
1464 |
gezelter |
3980 |
energy constants for the lowest energy configurations for linear |
1465 |
|
|
quadrupoles shown in table \ref{tab:NNQ} |
1466 |
|
|
|
1467 |
|
|
\begin{table*} |
1468 |
|
|
\centering{ |
1469 |
|
|
\caption{Nagai and Nakamura Quadurpolar arrays}} |
1470 |
|
|
\label{tab:NNQ} |
1471 |
|
|
\begin{ruledtabular} |
1472 |
|
|
\begin{tabular}{ccc} |
1473 |
|
|
Lattice & Quadrupole Direction & Energy constants \\ \hline |
1474 |
|
|
SC & 111 & -8.3 \\ |
1475 |
|
|
BCC & 011 & -21.7 \\ |
1476 |
|
|
FCC & 111 & -80.5 |
1477 |
|
|
\end{tabular} |
1478 |
|
|
\end{ruledtabular} |
1479 |
|
|
\end{table*} |
1480 |
|
|
|
1481 |
|
|
In analogy to the dipolar arrays, the total electrostatic energy for |
1482 |
|
|
the quadrupolar arrays is: |
1483 |
|
|
\begin{equation} |
1484 |
|
|
E = C \frac{3}{4} N^2 Q^2 |
1485 |
|
|
\end{equation} |
1486 |
|
|
where $Q$ is the quadrupole moment. |
1487 |
|
|
|
1488 |
gezelter |
3985 |
\section{Conclusion} |
1489 |
|
|
We have presented two efficient real-space methods for computing the |
1490 |
|
|
interactions between point multipoles. These methods have the benefit |
1491 |
|
|
of smoothly truncating the energies, forces, and torques at the cutoff |
1492 |
|
|
radius, making them attractive for both molecular dynamics (MD) and |
1493 |
|
|
Monte Carlo (MC) simulations. We find that the Gradient-Shifted Force |
1494 |
|
|
(GSF) and the Shifted-Potential (SP) methods converge rapidly to the |
1495 |
|
|
correct lattice energies for ordered dipolar and quadrupolar arrays, |
1496 |
|
|
while the Taylor-Shifted Force (TSF) is too severe an approximation to |
1497 |
|
|
provide accurate convergence to lattice energies. |
1498 |
gezelter |
3980 |
|
1499 |
gezelter |
3985 |
In most cases, GSF can obtain nearly quantitative agreement with the |
1500 |
|
|
lattice energy constants with reasonably small cutoff radii. The only |
1501 |
|
|
exception we have observed is for crystals which exhibit a bulk |
1502 |
|
|
macroscopic dipole moment (e.g. Luttinger \& Tisza's $Z_1$ lattice). |
1503 |
|
|
In this particular case, the multipole neutralization scheme can |
1504 |
|
|
interfere with the correct computation of the energies. We note that |
1505 |
|
|
the energies for these arrangements are typically much larger than for |
1506 |
|
|
crystals with net-zero moments, so this is not expected to be an issue |
1507 |
|
|
in most simulations. |
1508 |
gezelter |
3980 |
|
1509 |
gezelter |
3985 |
In large systems, these new methods can be made to scale approximately |
1510 |
|
|
linearly with system size, and detailed comparisons with the Ewald sum |
1511 |
|
|
for a wide range of chemical environments follows in the second paper. |
1512 |
gezelter |
3980 |
|
1513 |
gezelter |
3906 |
\begin{acknowledgments} |
1514 |
gezelter |
3985 |
JDG acknowledges helpful discussions with Christopher |
1515 |
|
|
Fennell. Support for this project was provided by the National |
1516 |
|
|
Science Foundation under grant CHE-0848243. Computational time was |
1517 |
|
|
provided by the Center for Research Computing (CRC) at the |
1518 |
|
|
University of Notre Dame. |
1519 |
gezelter |
3906 |
\end{acknowledgments} |
1520 |
|
|
|
1521 |
gezelter |
3984 |
\newpage |
1522 |
gezelter |
3906 |
\appendix |
1523 |
|
|
|
1524 |
gezelter |
3984 |
\section{Smith's $B_l(r)$ functions for damped-charge distributions} |
1525 |
gezelter |
3985 |
\label{SmithFunc} |
1526 |
gezelter |
3984 |
The following summarizes Smith's $B_l(r)$ functions and includes |
1527 |
|
|
formulas given in his appendix.\cite{Smith98} The first function |
1528 |
|
|
$B_0(r)$ is defined by |
1529 |
gezelter |
3906 |
% |
1530 |
|
|
\begin{equation} |
1531 |
|
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r}= |
1532 |
|
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
1533 |
|
|
\end{equation} |
1534 |
|
|
% |
1535 |
|
|
The first derivative of this function is |
1536 |
|
|
% |
1537 |
|
|
\begin{equation} |
1538 |
|
|
\frac{dB_0(r)}{dr}=-\frac{1}{r^2}\text{erfc}(\alpha r) |
1539 |
|
|
-\frac{2\alpha}{r\sqrt{\pi}}\text{e}^{-{\alpha}^2r^2} |
1540 |
|
|
\end{equation} |
1541 |
|
|
% |
1542 |
gezelter |
3984 |
which can be used to define a function $B_1(r)$: |
1543 |
gezelter |
3906 |
% |
1544 |
|
|
\begin{equation} |
1545 |
|
|
B_1(r)=-\frac{1}{r}\frac{dB_0(r)}{dr} |
1546 |
|
|
\end{equation} |
1547 |
|
|
% |
1548 |
gezelter |
3984 |
In general, the recurrence relation, |
1549 |
gezelter |
3906 |
\begin{equation} |
1550 |
|
|
B_l(r)=-\frac{1}{r}\frac{dB_{l-1}(r)}{dr} |
1551 |
|
|
= \frac{1}{r^2} \left[ (2l-1)B_{l-1}(r) + \frac {(2\alpha^2)^l}{\alpha \sqrt{\pi}} |
1552 |
|
|
\text{e}^{-{\alpha}^2r^2} |
1553 |
gezelter |
3984 |
\right] , |
1554 |
gezelter |
3906 |
\end{equation} |
1555 |
gezelter |
3984 |
is very useful for building up higher derivatives. Using these |
1556 |
|
|
formulas, we find: |
1557 |
gezelter |
3906 |
% |
1558 |
gezelter |
3984 |
\begin{align} |
1559 |
|
|
\frac{dB_0}{dr}=&-rB_1(r) \\ |
1560 |
|
|
\frac{d^2B_0}{dr^2}=& - B_1(r) + r^2 B_2(r) \\ |
1561 |
|
|
\frac{d^3B_0}{dr^3}=& 3 r B_2(r) - r^3 B_3(r) \\ |
1562 |
|
|
\frac{d^4B_0}{dr^4}=& 3 B_2(r) - 6 r^2 B_3(r) + r^4 B_4(r) \\ |
1563 |
|
|
\frac{d^5B_0}{dr^5}=& - 15 r B_3(r) + 10 r^3 B_4(r) - r^5 B_5(r) . |
1564 |
|
|
\end{align} |
1565 |
gezelter |
3906 |
% |
1566 |
gezelter |
3984 |
As noted by Smith, it is possible to approximate the $B_l(r)$ |
1567 |
|
|
functions, |
1568 |
gezelter |
3906 |
% |
1569 |
|
|
\begin{equation} |
1570 |
|
|
B_l(r)=\frac{(2l)!}{l!2^lr^{2l+1}} - \frac {(2\alpha^2)^{l+1}}{(2l+1)\alpha \sqrt{\pi}} |
1571 |
|
|
+\text{O}(r) . |
1572 |
|
|
\end{equation} |
1573 |
gezelter |
3984 |
\newpage |
1574 |
|
|
\section{The $r$-dependent factors for TSF electrostatics} |
1575 |
gezelter |
3906 |
|
1576 |
|
|
Using the shifted damped functions $f_n(r)$ defined by: |
1577 |
|
|
% |
1578 |
|
|
\begin{equation} |
1579 |
gezelter |
3984 |
f_n(r)= B_0(r) -\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} B_0^{(m)}(r_c) , |
1580 |
gezelter |
3906 |
\end{equation} |
1581 |
|
|
% |
1582 |
gezelter |
3984 |
where the superscript $(m)$ denotes the $m^\mathrm{th}$ derivative. In |
1583 |
|
|
this Appendix, we provide formulas for successive derivatives of this |
1584 |
|
|
function. (If there is no damping, then $B_0(r)$ is replaced by |
1585 |
|
|
$1/r$.) First, we find: |
1586 |
gezelter |
3906 |
% |
1587 |
|
|
\begin{equation} |
1588 |
|
|
\frac{\partial f_n}{\partial r_\alpha}=\hat{r}_\alpha \frac{d f_n}{d r} . |
1589 |
|
|
\end{equation} |
1590 |
|
|
% |
1591 |
gezelter |
3984 |
This formula clearly brings in derivatives of Smith's $B_0(r)$ |
1592 |
|
|
function, and we define higher-order derivatives as follows: |
1593 |
gezelter |
3906 |
% |
1594 |
gezelter |
3984 |
\begin{align} |
1595 |
|
|
g_n(r)=& \frac{d f_n}{d r} = |
1596 |
|
|
B_0^{(1)}(r) -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)}(r_c) \\ |
1597 |
|
|
h_n(r)=& \frac{d^2f_n}{d r^2} = |
1598 |
|
|
B_0^{(2)}(r) -\sum_{m=0}^{n-1} \frac {(r-r_c)^m}{m!} B_0^{(m+2)}(r_c) \\ |
1599 |
|
|
s_n(r)=& \frac{d^3f_n}{d r^3} = |
1600 |
|
|
B_0^{(3)}(r) -\sum_{m=0}^{n-2} \frac {(r-r_c)^m}{m!} B_0^{(m+3)}(r_c) \\ |
1601 |
|
|
t_n(r)=& \frac{d^4f_n}{d r^4} = |
1602 |
|
|
B_0^{(4)}(r) -\sum_{m=0}^{n-3} \frac {(r-r_c)^m}{m!} B_0^{(m+4)}(r_c) \\ |
1603 |
|
|
u_n(r)=& \frac{d^5f_n}{d r^5} = |
1604 |
|
|
B_0^{(5)}(r) -\sum_{m=0}^{n-4} \frac {(r-r_c)^m}{m!} B_0^{(m+5)}(r_c) . |
1605 |
|
|
\end{align} |
1606 |
gezelter |
3906 |
% |
1607 |
gezelter |
3984 |
We note that the last function needed (for quadrupole-quadrupole interactions) is |
1608 |
gezelter |
3906 |
% |
1609 |
|
|
\begin{equation} |
1610 |
gezelter |
3984 |
u_4(r)=B_0^{(5)}(r) - B_0^{(5)}(r_c) . |
1611 |
gezelter |
3906 |
\end{equation} |
1612 |
|
|
|
1613 |
gezelter |
3984 |
The functions $f_n(r)$ to $u_n(r)$ can be computed recursively and |
1614 |
|
|
stored on a grid for values of $r$ from $0$ to $r_c$. The functions |
1615 |
|
|
needed are listed schematically below: |
1616 |
gezelter |
3906 |
% |
1617 |
|
|
\begin{eqnarray} |
1618 |
|
|
f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
1619 |
|
|
g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
1620 |
|
|
h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
1621 |
|
|
s_2 \quad s_3 \quad &s_4 \nonumber \\ |
1622 |
|
|
t_3 \quad &t_4 \nonumber \\ |
1623 |
|
|
&u_4 \nonumber . |
1624 |
|
|
\end{eqnarray} |
1625 |
|
|
|
1626 |
|
|
Using these functions, we find |
1627 |
|
|
% |
1628 |
gezelter |
3984 |
\begin{align} |
1629 |
|
|
\frac{\partial f_n}{\partial r_\alpha} =&r_\alpha \frac {g_n}{r} \label{eq:b9}\\ |
1630 |
|
|
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =&\delta_{\alpha \beta}\frac {g_n}{r} |
1631 |
|
|
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) \\ |
1632 |
|
|
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} =& |
1633 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
1634 |
|
|
\delta_{ \beta \gamma} r_\alpha \right) |
1635 |
|
|
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) |
1636 |
|
|
+ r_\alpha r_\beta r_\gamma |
1637 |
gezelter |
3984 |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \\ |
1638 |
|
|
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} =& |
1639 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
1640 |
|
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
1641 |
|
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
1642 |
|
|
\left( - \frac{g_n}{r^3} + \frac{h_n}{r^2} \right) \nonumber \\ |
1643 |
gezelter |
3984 |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
1644 |
|
|
+ \text{5 permutations} |
1645 |
gezelter |
3906 |
\right) \left( \frac{3 g_n}{r^5} - \frac{3h_n}{r^4} + \frac{s_n}{r^3} |
1646 |
|
|
\right) \nonumber \\ |
1647 |
gezelter |
3984 |
&+ r_\alpha r_\beta r_\gamma r_\delta |
1648 |
gezelter |
3906 |
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
1649 |
gezelter |
3984 |
+ \frac{t_n}{r^4} \right)\\ |
1650 |
gezelter |
3906 |
\frac{\partial^5 f_n} |
1651 |
gezelter |
3984 |
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} =& |
1652 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
1653 |
gezelter |
3984 |
+ \text{14 permutations} \right) |
1654 |
gezelter |
3906 |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
1655 |
gezelter |
3984 |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
1656 |
|
|
+ \text{9 permutations} |
1657 |
gezelter |
3906 |
\right) \left(- \frac{15g_n}{r^7}+\frac{15h_n}{r^7} -\frac{6s_n}{r^5} +\frac{t_n}{r^4} |
1658 |
|
|
\right) \nonumber \\ |
1659 |
gezelter |
3984 |
&+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
1660 |
gezelter |
3906 |
\left( \frac{105g_n}{r^9} - \frac{105h_n}{r^8} + \frac{45s_n}{r^7} |
1661 |
gezelter |
3984 |
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) \label{eq:b13} |
1662 |
|
|
\end{align} |
1663 |
gezelter |
3906 |
% |
1664 |
|
|
% |
1665 |
|
|
% |
1666 |
gezelter |
3984 |
\newpage |
1667 |
|
|
\section{The $r$-dependent factors for GSF electrostatics} |
1668 |
gezelter |
3906 |
|
1669 |
gezelter |
3984 |
In Gradient-shifted force electrostatics, the kernel is not expanded, |
1670 |
|
|
rather the individual terms in the multipole interaction energies. |
1671 |
|
|
For damped charges , this still brings into the algebra multiple |
1672 |
|
|
derivatives of the Smith's $B_0(r)$ function. To denote these terms, |
1673 |
|
|
we generalize the notation of the previous appendix. For $f(r)=1/r$ |
1674 |
|
|
(bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
1675 |
gezelter |
3906 |
% |
1676 |
gezelter |
3984 |
\begin{align} |
1677 |
|
|
g(r)=& \frac{df}{d r}\\ |
1678 |
|
|
h(r)=& \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
1679 |
|
|
s(r)=& \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
1680 |
|
|
t(r)=& \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
1681 |
|
|
u(r)=& \frac{dt}{d r} = \frac{d^5f}{d r^5} . |
1682 |
|
|
\end{align} |
1683 |
gezelter |
3906 |
% |
1684 |
gezelter |
3984 |
For undamped charges, $f(r)=1/r$, Table I lists these derivatives |
1685 |
|
|
under the column ``Bare Coulomb.'' Equations \ref{eq:b9} to |
1686 |
|
|
\ref{eq:b13} are still correct for GSF electrostatics if the subscript |
1687 |
|
|
$n$ is eliminated. |
1688 |
gezelter |
3906 |
|
1689 |
gezelter |
3980 |
\newpage |
1690 |
|
|
|
1691 |
|
|
\bibliography{multipole} |
1692 |
|
|
|
1693 |
gezelter |
3906 |
\end{document} |
1694 |
|
|
% |
1695 |
|
|
% ****** End of file multipole.tex ****** |