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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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|
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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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|
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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{self} |
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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 self-interaction term ($\sum_i |
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V_i^\mathrm{self}$) 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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|
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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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|
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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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|
308 |
gezelter |
3982 |
\subsection{Damped Coulomb interactions} |
309 |
|
|
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 |
gezelter |
3989 |
Although objects $\bf a$ and $\bf b$ rotate during a molecular |
475 |
|
|
dynamics (MD) simulation, their multipole tensors remain fixed in |
476 |
|
|
body-frame coordinates. While deriving force and torque expressions, |
477 |
|
|
it is therefore convenient to write the energies, forces, and torques |
478 |
|
|
in intermediate forms involving the vectors of the rotation matrices. |
479 |
|
|
We denote body axes for objects $\bf a$ and $\bf b$ using unit vectors |
480 |
|
|
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$. |
481 |
|
|
In a typical simulation , the initial axes are obtained by |
482 |
|
|
diagonalizing the moment of inertia tensors for the objects. (N.B., |
483 |
|
|
the body axes are generally {\it not} the same as those for which the |
484 |
|
|
quadrupole moment is diagonal.) The rotation matrices are then |
485 |
|
|
propagated during the simulation. |
486 |
gezelter |
3906 |
|
487 |
gezelter |
3989 |
The rotation matrices $\hat{\mathbf {a}}$ and $\hat{\mathbf {b}}$ can be |
488 |
gezelter |
3985 |
expressed using these unit vectors: |
489 |
gezelter |
3906 |
\begin{eqnarray} |
490 |
|
|
\hat{\mathbf {a}} = |
491 |
|
|
\begin{pmatrix} |
492 |
|
|
\hat{a}_1 \\ |
493 |
|
|
\hat{a}_2 \\ |
494 |
|
|
\hat{a}_3 |
495 |
gezelter |
3989 |
\end{pmatrix}, \qquad |
496 |
gezelter |
3906 |
\hat{\mathbf {b}} = |
497 |
|
|
\begin{pmatrix} |
498 |
|
|
\hat{b}_1 \\ |
499 |
|
|
\hat{b}_2 \\ |
500 |
|
|
\hat{b}_3 |
501 |
|
|
\end{pmatrix} |
502 |
|
|
\end{eqnarray} |
503 |
|
|
% |
504 |
gezelter |
3985 |
These matrices convert from space-fixed $(xyz)$ to body-fixed $(123)$ |
505 |
gezelter |
3989 |
coordinates. |
506 |
|
|
|
507 |
|
|
Allen and Germano,\cite{Allen:2006fk} following earlier work by Price |
508 |
|
|
{\em et al.},\cite{Price:1984fk} showed that if the interaction |
509 |
|
|
energies are written explicitly in terms of $\hat{r}$ and the body |
510 |
|
|
axes ($\hat{a}_m$, $\hat{b}_n$) : |
511 |
gezelter |
3906 |
% |
512 |
gezelter |
3985 |
\begin{equation} |
513 |
gezelter |
3989 |
U(r, \{\hat{a}_m \cdot \hat{r} \}, |
514 |
|
|
\{\hat{b}_n\cdot \hat{r} \}, |
515 |
|
|
\{\hat{a}_m \cdot \hat{b}_n \}) . |
516 |
|
|
\label{ugeneral} |
517 |
|
|
\end{equation} |
518 |
|
|
% |
519 |
|
|
the forces come out relatively cleanly, |
520 |
|
|
% |
521 |
|
|
\begin{equation} |
522 |
|
|
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
523 |
|
|
= \frac{\partial U}{\partial r} \hat{r} |
524 |
|
|
+ \sum_m \left[ |
525 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
526 |
|
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
527 |
|
|
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
528 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
529 |
|
|
\right] \label{forceequation}. |
530 |
|
|
\end{equation} |
531 |
|
|
|
532 |
|
|
The torques can also be found in a relatively similar |
533 |
|
|
manner, |
534 |
|
|
% |
535 |
|
|
\begin{eqnarray} |
536 |
|
|
\mathbf{\tau}_{\bf a} = |
537 |
|
|
\sum_m |
538 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
539 |
|
|
( \hat{r} \times \hat{a}_m ) |
540 |
|
|
-\sum_{mn} |
541 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
542 |
|
|
(\hat{a}_m \times \hat{b}_n) \\ |
543 |
|
|
% |
544 |
|
|
\mathbf{\tau}_{\bf b} = |
545 |
|
|
\sum_m |
546 |
|
|
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
547 |
|
|
( \hat{r} \times \hat{b}_m) |
548 |
|
|
+\sum_{mn} |
549 |
|
|
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
550 |
|
|
(\hat{a}_m \times \hat{b}_n) . |
551 |
|
|
\end{eqnarray} |
552 |
|
|
|
553 |
|
|
Note that our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ |
554 |
|
|
is opposite in sign to that of Allen and Germano.\cite{Allen:2006fk} |
555 |
|
|
We also made use of the identities, |
556 |
|
|
% |
557 |
|
|
\begin{align} |
558 |
|
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
559 |
|
|
=& \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
560 |
|
|
\right) \\ |
561 |
|
|
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
562 |
|
|
=& \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
563 |
|
|
\right) . |
564 |
|
|
\end{align} |
565 |
|
|
|
566 |
|
|
Many of the multipole contractions required can be written in one of |
567 |
|
|
three equivalent forms using the unit vectors $\hat{r}$, $\hat{a}_m$, |
568 |
|
|
and $\hat{b}_n$. In the torque expressions, it is useful to have the |
569 |
|
|
angular-dependent terms available in all three fashions, e.g. for the |
570 |
|
|
dipole-dipole contraction: |
571 |
|
|
% |
572 |
|
|
\begin{equation} |
573 |
gezelter |
3906 |
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
574 |
gezelter |
3985 |
= D_{\bf {a}\alpha} D_{\bf {b}\alpha} = |
575 |
|
|
\sum_{mn} {D_{\mathbf{a}m} \hat{a}_m \cdot \hat{b}_n D_{\mathbf{b}n}} |
576 |
|
|
\end{equation} |
577 |
gezelter |
3906 |
% |
578 |
gezelter |
3985 |
The first two forms are written using space coordinates. The first |
579 |
|
|
form is standard in the chemistry literature, while the second is |
580 |
|
|
expressed using implied summation notation. The third form shows |
581 |
|
|
explicit sums over body indices and the dot products now indicate |
582 |
|
|
contractions using space indices. |
583 |
gezelter |
3906 |
|
584 |
gezelter |
3989 |
In computing our force and torque expressions, we carried out most of |
585 |
|
|
the work in body coordinates, and have transformed the expressions |
586 |
|
|
back to space-frame coordinates, which are reported below. Interested |
587 |
|
|
readers may consult the supplemental information for this paper for |
588 |
|
|
the intermediate body-frame expressions. |
589 |
gezelter |
3906 |
|
590 |
gezelter |
3982 |
\subsection{The Self-Interaction \label{sec:selfTerm}} |
591 |
|
|
|
592 |
gezelter |
3985 |
In addition to cutoff-sphere neutralization, the Wolf |
593 |
|
|
summation~\cite{Wolf99} and the damped shifted force (DSF) |
594 |
|
|
extension~\cite{Fennell:2006zl} also included self-interactions that |
595 |
|
|
are handled separately from the pairwise interactions between |
596 |
|
|
sites. The self-term is normally calculated via a single loop over all |
597 |
|
|
sites in the system, and is relatively cheap to evaluate. The |
598 |
|
|
self-interaction has contributions from two sources. |
599 |
|
|
|
600 |
|
|
First, the neutralization procedure within the cutoff radius requires |
601 |
|
|
a contribution from a charge opposite in sign, but equal in magnitude, |
602 |
|
|
to the central charge, which has been spread out over the surface of |
603 |
|
|
the cutoff sphere. For a system of undamped charges, the total |
604 |
|
|
self-term is |
605 |
gezelter |
3980 |
\begin{equation} |
606 |
|
|
V_\textrm{self} = - \frac{1}{r_c} \sum_{{\bf a}=1}^N C_{\bf a}^{2} |
607 |
|
|
\label{eq:selfTerm} |
608 |
|
|
\end{equation} |
609 |
gezelter |
3985 |
|
610 |
|
|
Second, charge damping with the complementary error function is a |
611 |
|
|
partial analogy to the Ewald procedure which splits the interaction |
612 |
|
|
into real and reciprocal space sums. The real space sum is retained |
613 |
|
|
in the Wolf and DSF methods. The reciprocal space sum is first |
614 |
|
|
minimized by folding the largest contribution (the self-interaction) |
615 |
|
|
into the self-interaction from charge neutralization of the damped |
616 |
|
|
potential. The remainder of the reciprocal space portion is then |
617 |
|
|
discarded (as this contributes the largest computational cost and |
618 |
|
|
complexity to the Ewald sum). For a system containing only damped |
619 |
|
|
charges, the complete self-interaction can be written as |
620 |
gezelter |
3980 |
\begin{equation} |
621 |
|
|
V_\textrm{self} = - \left(\frac{\textrm{erfc}(\alpha r_c)}{r_c} + |
622 |
|
|
\frac{\alpha}{\sqrt{\pi}} \right) \sum_{{\bf a}=1}^N |
623 |
|
|
C_{\bf a}^{2}. |
624 |
|
|
\label{eq:dampSelfTerm} |
625 |
|
|
\end{equation} |
626 |
|
|
|
627 |
|
|
The extension of DSF electrostatics to point multipoles requires |
628 |
|
|
treatment of {\it both} the self-neutralization and reciprocal |
629 |
|
|
contributions to the self-interaction for higher order multipoles. In |
630 |
|
|
this section we give formulae for these interactions up to quadrupolar |
631 |
|
|
order. |
632 |
|
|
|
633 |
|
|
The self-neutralization term is computed by taking the {\it |
634 |
|
|
non-shifted} kernel for each interaction, placing a multipole of |
635 |
|
|
equal magnitude (but opposite in polarization) on the surface of the |
636 |
|
|
cutoff sphere, and averaging over the surface of the cutoff sphere. |
637 |
|
|
Because the self term is carried out as a single sum over sites, the |
638 |
|
|
reciprocal-space portion is identical to half of the self-term |
639 |
|
|
obtained by Smith and Aguado and Madden for the application of the |
640 |
|
|
Ewald sum to multipoles.\cite{Smith82,Smith98,Aguado03} For a given |
641 |
|
|
site which posesses a charge, dipole, and multipole, both types of |
642 |
|
|
contribution are given in table \ref{tab:tableSelf}. |
643 |
|
|
|
644 |
|
|
\begin{table*} |
645 |
|
|
\caption{\label{tab:tableSelf} Self-interaction contributions for |
646 |
|
|
site ({\bf a}) that has a charge $(C_{\bf a})$, dipole |
647 |
|
|
$(\mathbf{D}_{\bf a})$, and quadrupole $(\mathbf{Q}_{\bf a})$} |
648 |
|
|
\begin{ruledtabular} |
649 |
|
|
\begin{tabular}{lccc} |
650 |
|
|
Multipole order & Summed Quantity & Self-neutralization & Reciprocal \\ \hline |
651 |
|
|
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
652 |
|
|
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
653 |
|
|
\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
654 |
gezelter |
3989 |
Quadrupole & $2 \mathbf{Q}_{\bf a}:\mathbf{Q}_{\bf a} + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
655 |
gezelter |
3980 |
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
656 |
|
|
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
657 |
|
|
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
658 |
|
|
h(r_c) + \frac{2 g(r_c)}{r_c} \right)$& $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$ \\ |
659 |
|
|
\end{tabular} |
660 |
|
|
\end{ruledtabular} |
661 |
|
|
\end{table*} |
662 |
|
|
|
663 |
|
|
For sites which simultaneously contain charges and quadrupoles, the |
664 |
|
|
self-interaction includes a cross-interaction between these two |
665 |
|
|
multipole orders. Symmetry prevents the charge-dipole and |
666 |
|
|
dipole-quadrupole interactions from contributing to the |
667 |
|
|
self-interaction. The functions that go into the self-neutralization |
668 |
gezelter |
3985 |
terms, $g(r), h(r), s(r), \mathrm{~and~} t(r)$ are successive |
669 |
|
|
derivatives of the electrostatic kernel, $f(r)$ (either the undamped |
670 |
|
|
$1/r$ or the damped $B_0(r)=\mathrm{erfc}(\alpha r)/r$ function) that |
671 |
|
|
have been evaluated at the cutoff distance. For undamped |
672 |
|
|
interactions, $f(r_c) = 1/r_c$, $g(r_c) = -1/r_c^{2}$, and so on. For |
673 |
|
|
damped interactions, $f(r_c) = B_0(r_c)$, $g(r_c) = B_0'(r_c)$, and so |
674 |
|
|
on. Appendix \ref{SmithFunc} contains recursion relations that allow |
675 |
|
|
rapid evaluation of these derivatives. |
676 |
gezelter |
3980 |
|
677 |
gezelter |
3985 |
\section{Interaction energies, forces, and torques} |
678 |
|
|
The main result of this paper is a set of expressions for the |
679 |
|
|
energies, forces and torques (up to quadrupole-quadrupole order) that |
680 |
|
|
work for both the Taylor-shifted and Gradient-shifted approximations. |
681 |
|
|
These expressions were derived using a set of generic radial |
682 |
|
|
functions. Without using the shifting approximations mentioned above, |
683 |
|
|
some of these radial functions would be identical, and the expressions |
684 |
|
|
coalesce into the familiar forms for unmodified multipole-multipole |
685 |
|
|
interactions. Table \ref{tab:tableenergy} maps between the generic |
686 |
|
|
functions and the radial functions derived for both the Taylor-shifted |
687 |
|
|
and Gradient-shifted methods. The energy equations are written in |
688 |
|
|
terms of lab-frame representations of the dipoles, quadrupoles, and |
689 |
|
|
the unit vector connecting the two objects, |
690 |
gezelter |
3906 |
|
691 |
|
|
% Energy in space coordinate form ---------------------------------------------------------------------------------------------- |
692 |
|
|
% |
693 |
|
|
% |
694 |
|
|
% u ca cb |
695 |
|
|
% |
696 |
gezelter |
3983 |
\begin{align} |
697 |
|
|
U_{C_{\bf a}C_{\bf b}}(r)=& |
698 |
gezelter |
3985 |
C_{\bf a} C_{\bf b} v_{01}(r) \label{uchch} |
699 |
gezelter |
3983 |
\\ |
700 |
gezelter |
3906 |
% |
701 |
|
|
% u ca db |
702 |
|
|
% |
703 |
gezelter |
3983 |
U_{C_{\bf a}D_{\bf b}}(r)=& |
704 |
gezelter |
3985 |
C_{\bf a} \left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) v_{11}(r) |
705 |
gezelter |
3906 |
\label{uchdip} |
706 |
gezelter |
3983 |
\\ |
707 |
gezelter |
3906 |
% |
708 |
|
|
% u ca qb |
709 |
|
|
% |
710 |
gezelter |
3985 |
U_{C_{\bf a}Q_{\bf b}}(r)=& C_{\bf a } \Bigl[ \text{Tr}Q_{\bf b} |
711 |
|
|
v_{21}(r) + \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot |
712 |
|
|
\hat{r} \right) v_{22}(r) \Bigr] |
713 |
gezelter |
3906 |
\label{uchquad} |
714 |
gezelter |
3983 |
\\ |
715 |
gezelter |
3906 |
% |
716 |
|
|
% u da cb |
717 |
|
|
% |
718 |
gezelter |
3983 |
%U_{D_{\bf a}C_{\bf b}}(r)=& |
719 |
|
|
%-\frac{C_{\bf b}}{4\pi \epsilon_0} |
720 |
|
|
%\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) v_{11}(r) \label{udipch} |
721 |
|
|
%\\ |
722 |
gezelter |
3906 |
% |
723 |
|
|
% u da db |
724 |
|
|
% |
725 |
gezelter |
3983 |
U_{D_{\bf a}D_{\bf b}}(r)=& |
726 |
gezelter |
3985 |
-\Bigr[ \left( \mathbf{D}_{\mathbf {a}} \cdot |
727 |
gezelter |
3906 |
\mathbf{D}_{\mathbf{b}} \right) v_{21}(r) |
728 |
|
|
+\left( \mathbf{D}_{\mathbf {a}} \cdot \hat{r} \right) |
729 |
|
|
\left( \mathbf{D}_{\mathbf {b}} \cdot \hat{r} \right) |
730 |
|
|
v_{22}(r) \Bigr] |
731 |
|
|
\label{udipdip} |
732 |
gezelter |
3983 |
\\ |
733 |
gezelter |
3906 |
% |
734 |
|
|
% u da qb |
735 |
|
|
% |
736 |
|
|
\begin{split} |
737 |
|
|
% 1 |
738 |
gezelter |
3983 |
U_{D_{\bf a}Q_{\bf b}}(r) =& |
739 |
gezelter |
3985 |
-\Bigl[ |
740 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
741 |
|
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
742 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
743 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r} ) \Bigr] v_{31}(r) \\ |
744 |
|
|
% 2 |
745 |
gezelter |
3985 |
&- \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
746 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{32}(r) |
747 |
|
|
\label{udipquad} |
748 |
|
|
\end{split} |
749 |
gezelter |
3983 |
\\ |
750 |
gezelter |
3906 |
% |
751 |
|
|
% u qa cb |
752 |
|
|
% |
753 |
gezelter |
3983 |
%U_{Q_{\bf a}C_{\bf b}}(r)=& |
754 |
|
|
%\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\bf a} v_{21}(r) |
755 |
|
|
%\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
756 |
|
|
%\label{uquadch} |
757 |
|
|
%\\ |
758 |
gezelter |
3906 |
% |
759 |
|
|
% u qa db |
760 |
|
|
% |
761 |
gezelter |
3983 |
%\begin{split} |
762 |
gezelter |
3906 |
%1 |
763 |
gezelter |
3983 |
%U_{Q_{\bf a}D_{\bf b}}(r)=& |
764 |
|
|
%\frac{1}{4\pi \epsilon_0} \Bigl[ |
765 |
|
|
%\text{Tr}\mathbf{Q}_{\mathbf{a}} |
766 |
|
|
%\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
767 |
|
|
%+2 ( \mathbf{D}_{\mathbf{b}} \cdot |
768 |
|
|
%\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r)\\ |
769 |
gezelter |
3906 |
% 2 |
770 |
gezelter |
3983 |
%&+\frac{1}{4\pi \epsilon_0} |
771 |
|
|
%\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
772 |
|
|
%\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{32}(r) |
773 |
|
|
%\label{uquaddip} |
774 |
|
|
%\end{split} |
775 |
|
|
%\\ |
776 |
gezelter |
3906 |
% |
777 |
|
|
% u qa qb |
778 |
|
|
% |
779 |
|
|
\begin{split} |
780 |
|
|
%1 |
781 |
gezelter |
3983 |
U_{Q_{\bf a}Q_{\bf b}}(r)=& |
782 |
gezelter |
3985 |
\Bigl[ |
783 |
gezelter |
3906 |
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
784 |
gezelter |
3989 |
+2 |
785 |
|
|
\mathbf{Q}_{\mathbf{a}} : \mathbf{Q}_{\mathbf{b}} \Bigr] v_{41}(r) |
786 |
gezelter |
3906 |
\\ |
787 |
|
|
% 2 |
788 |
gezelter |
3985 |
&+\Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
789 |
gezelter |
3906 |
\left( \hat{r} \cdot |
790 |
|
|
\mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) |
791 |
|
|
+\text{Tr}\mathbf{Q}_{\mathbf{b}} |
792 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} |
793 |
|
|
\cdot \hat{r} \right) +4 (\hat{r} \cdot |
794 |
|
|
\mathbf{Q}_{{\mathbf a}}\cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
795 |
|
|
\Bigr] v_{42}(r) |
796 |
|
|
\\ |
797 |
|
|
% 4 |
798 |
gezelter |
3985 |
&+ |
799 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) |
800 |
|
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{43}(r). |
801 |
|
|
\label{uquadquad} |
802 |
|
|
\end{split} |
803 |
gezelter |
3983 |
\end{align} |
804 |
gezelter |
3985 |
% |
805 |
gezelter |
3983 |
Note that the energies of multipoles on site $\mathbf{b}$ interacting |
806 |
|
|
with those on site $\mathbf{a}$ can be obtained by swapping indices |
807 |
|
|
along with the sign of the intersite vector, $\hat{r}$. |
808 |
gezelter |
3906 |
|
809 |
|
|
% |
810 |
|
|
% |
811 |
|
|
% TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
812 |
|
|
% |
813 |
|
|
|
814 |
gezelter |
3985 |
\begin{sidewaystable} |
815 |
|
|
\caption{\label{tab:tableenergy}Radial functions used in the energy |
816 |
|
|
and torque equations. The $f, g, h, s, t, \mathrm{and} u$ |
817 |
|
|
functions used in this table are defined in Appendices B and C.} |
818 |
|
|
\begin{tabular}{|c|c|l|l|} \hline |
819 |
|
|
Generic&Bare Coulomb&Taylor-Shifted&Gradient-Shifted |
820 |
gezelter |
3906 |
\\ \hline |
821 |
|
|
% |
822 |
|
|
% |
823 |
|
|
% |
824 |
|
|
%Ch-Ch& |
825 |
|
|
$v_{01}(r)$ & |
826 |
|
|
$\frac{1}{r}$ & |
827 |
|
|
$f_0(r)$ & |
828 |
|
|
$f(r)-f(r_c)-(r-r_c)g(r_c)$ |
829 |
|
|
\\ |
830 |
|
|
% |
831 |
|
|
% |
832 |
|
|
% |
833 |
|
|
%Ch-Di& |
834 |
|
|
$v_{11}(r)$ & |
835 |
|
|
$-\frac{1}{r^2}$ & |
836 |
|
|
$g_1(r)$ & |
837 |
|
|
$g(r)-g(r_c)-(r-r_c)h(r_c)$ \\ |
838 |
|
|
% |
839 |
|
|
% |
840 |
|
|
% |
841 |
|
|
%Ch-Qu/Di-Di& |
842 |
|
|
$v_{21}(r)$ & |
843 |
|
|
$-\frac{1}{r^3} $ & |
844 |
|
|
$\frac{g_2(r)}{r} $ & |
845 |
|
|
$\frac{g(r)}{r}-\frac{g(r_c)}{r_c} -(r-r_c) |
846 |
|
|
\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right)$ \\ |
847 |
|
|
$v_{22}(r)$ & |
848 |
|
|
$\frac{3}{r^3} $ & |
849 |
|
|
$\left(-\frac{g_2(r)}{r} + h_2(r) \right)$ & |
850 |
|
|
$\left(-\frac{g(r)}{r}+h(r) \right) |
851 |
gezelter |
3985 |
-\left(-\frac{g(r_c)}{r_c}+h(r_c) \right)$ \\ |
852 |
|
|
&&& $ ~~~-(r-r_c) \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)$ |
853 |
gezelter |
3906 |
\\ |
854 |
|
|
% |
855 |
|
|
% |
856 |
|
|
% |
857 |
|
|
%Di-Qu & |
858 |
|
|
$v_{31}(r)$ & |
859 |
|
|
$\frac{3}{r^4} $ & |
860 |
|
|
$\left(-\frac{g_3(r)}{r^2} + \frac{h_3(r)}{r} \right)$ & |
861 |
|
|
$\left( -\frac{g(r)}{r^2}+\frac{h(r)}{r} \right) |
862 |
|
|
-\left(-\frac{g(r_c)}{r_c^2}+\frac{h(r_c)}{r_c} \right) $\\ |
863 |
gezelter |
3985 |
&&&$ ~~~ -(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)$ |
864 |
gezelter |
3906 |
\\ |
865 |
|
|
% |
866 |
|
|
$v_{32}(r)$ & |
867 |
|
|
$-\frac{15}{r^4} $ & |
868 |
|
|
$\left( \frac{3g_3(r)}{r^2} - \frac{3h_3(r)}{r} + s_3(r) \right)$ & |
869 |
|
|
$\left( \frac{3g(r)}{r^2} - \frac{3h(r)}{r} + s(r) \right) |
870 |
|
|
- \left( \frac{3g(r_c)}{r_c^2} - \frac{3h(r_c)}{r_c} + s(r_c) \right)$ \\ |
871 |
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)$ |
872 |
gezelter |
3906 |
\\ |
873 |
|
|
% |
874 |
|
|
% |
875 |
|
|
% |
876 |
|
|
%Qu-Qu& |
877 |
|
|
$v_{41}(r)$ & |
878 |
|
|
$\frac{3}{r^5} $ & |
879 |
|
|
$\left(-\frac{g_4(r)}{r^3} +\frac{h_4(r)}{r^2} \right) $ & |
880 |
|
|
$\left( -\frac{g(r)}{r^3} + \frac{h(r)}{r^2} \right) |
881 |
|
|
- \left( -\frac{g(r_c)}{r_c^3} + \frac{h(r_c)}{r_c^2} \right)$ \\ |
882 |
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)$ |
883 |
gezelter |
3906 |
\\ |
884 |
|
|
% 2 |
885 |
|
|
$v_{42}(r)$ & |
886 |
|
|
$- \frac{15}{r^5} $ & |
887 |
|
|
$\left( \frac{3g_4(r)}{r^3} - \frac{3h_4(r)}{r^2}+\frac{s_4(r)}{r} \right)$ & |
888 |
|
|
$\left( \frac{3g(r)}{r^3} - \frac{3h(r)}{r^2}+\frac{s(r)}{r} \right) |
889 |
|
|
-\left( \frac{3g(r_c)}{r_c^3} - \frac{3h(r_c)}{r_c^2}+\frac{s(r_c)}{r_c} \right)$ \\ |
890 |
gezelter |
3985 |
&&&$ ~~~ -(r-r_c) \left(- \frac{9g(r_c)}{r_c^4}+\frac{9h(r_c)}{r_c^3} |
891 |
gezelter |
3906 |
-\frac{4s(r_c)}{r_c^2} + \frac{t(r_c)}{r_c}\right)$ |
892 |
|
|
\\ |
893 |
|
|
% 3 |
894 |
|
|
$v_{43}(r)$ & |
895 |
|
|
$ \frac{105}{r^5} $ & |
896 |
|
|
$\left(-\frac{15g_4(r)}{r^3}+\frac{15h_4(r)}{r^2}-\frac{6s_4(r)}{r} + t_4(r)\right) $ & |
897 |
|
|
$\left(-\frac{15g(r)}{r^3}+\frac{15h(r)}{r^2}-\frac{6s(r)}{r} + t(r)\right)$ \\ |
898 |
gezelter |
3985 |
&&&$~~~ -\left(-\frac{15g(r_c)}{r_c^3}+\frac{15h(r_c)}{r_c^2}-\frac{6s(r_c)}{r_c} + t(r_c)\right)$ \\ |
899 |
|
|
&&&$~~~ -(r-r_c)\left(\frac{45g(r_c)}{r_c^4}-\frac{45h(r_c)}{r_c^3}+\frac{21s(r_c)}{r_c^2} |
900 |
|
|
-\frac{6t(r_c)}{r_c}+u(r_c) \right)$ \\ \hline |
901 |
gezelter |
3906 |
\end{tabular} |
902 |
gezelter |
3985 |
\end{sidewaystable} |
903 |
gezelter |
3906 |
% |
904 |
|
|
% |
905 |
|
|
% FORCE TABLE of radial functions ---------------------------------------------------------------------------------------------------------------- |
906 |
|
|
% |
907 |
|
|
|
908 |
gezelter |
3985 |
\begin{sidewaystable} |
909 |
gezelter |
3906 |
\caption{\label{tab:tableFORCE}Radial functions used in the force equations.} |
910 |
gezelter |
3985 |
\begin{tabular}{|c|c|l|l|} \hline |
911 |
|
|
Function&Definition&Taylor-Shifted&Gradient-Shifted |
912 |
gezelter |
3906 |
\\ \hline |
913 |
|
|
% |
914 |
|
|
% |
915 |
|
|
% |
916 |
|
|
$w_a(r)$& |
917 |
gezelter |
3985 |
$\frac{d v_{01}}{dr}$& |
918 |
|
|
$g_0(r)$& |
919 |
|
|
$g(r)-g(r_c)$ \\ |
920 |
gezelter |
3906 |
% |
921 |
|
|
% |
922 |
|
|
$w_b(r)$ & |
923 |
gezelter |
3985 |
$\frac{d v_{11}}{dr} - \frac{v_{11}(r)}{r} $& |
924 |
|
|
$\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
925 |
|
|
$h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
926 |
gezelter |
3906 |
% |
927 |
|
|
$w_c(r)$ & |
928 |
gezelter |
3985 |
$\frac{v_{11}(r)}{r}$ & |
929 |
|
|
$\frac{g_1(r)}{r} $ & |
930 |
|
|
$\frac{v_{11}(r)}{r}$\\ |
931 |
gezelter |
3906 |
% |
932 |
|
|
% |
933 |
|
|
$w_d(r)$& |
934 |
gezelter |
3985 |
$\frac{d v_{21}}{dr}$& |
935 |
|
|
$\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
936 |
|
|
$\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
937 |
|
|
-\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $ \\ |
938 |
gezelter |
3906 |
% |
939 |
|
|
$w_e(r)$ & |
940 |
gezelter |
3985 |
$\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
941 |
|
|
$\frac{v_{22}(r)}{r}$ & |
942 |
gezelter |
3906 |
$\frac{v_{22}(r)}{r}$ \\ |
943 |
|
|
% |
944 |
|
|
% |
945 |
|
|
$w_f(r)$& |
946 |
gezelter |
3985 |
$\frac{d v_{22}}{dr} - \frac{2v_{22}(r)}{r}$& |
947 |
|
|
$\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
948 |
|
|
$ \left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) $ \\ |
949 |
|
|
&&& $ ~~~- \left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) |
950 |
|
|
\right)-\frac{2v_{22}(r)}{r}$\\ |
951 |
gezelter |
3906 |
% |
952 |
|
|
$w_g(r)$& |
953 |
gezelter |
3985 |
$\frac{v_{31}(r)}{r}$& |
954 |
|
|
$ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
955 |
gezelter |
3906 |
$\frac{v_{31}(r)}{r}$\\ |
956 |
|
|
% |
957 |
|
|
$w_h(r)$ & |
958 |
gezelter |
3985 |
$\frac{d v_{31}}{dr} -\frac{v_{31}(r)}{r}$& |
959 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
960 |
|
|
$ \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) $ \\ |
961 |
|
|
&&& $ ~~~ -\frac{v_{31}(r)}{r}$ \\ |
962 |
gezelter |
3906 |
% 2 |
963 |
|
|
$w_i(r)$ & |
964 |
gezelter |
3985 |
$\frac{v_{32}(r)}{r}$ & |
965 |
|
|
$\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
966 |
|
|
$\frac{v_{32}(r)}{r}$\\ |
967 |
gezelter |
3906 |
% |
968 |
|
|
$w_j(r)$ & |
969 |
gezelter |
3985 |
$\frac{d v_{32}}{dr} - \frac{3v_{32}}{r}$& |
970 |
|
|
$\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
971 |
|
|
$\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right)$ \\ |
972 |
|
|
&&& $~~~-\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} |
973 |
|
|
-\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
974 |
gezelter |
3906 |
% |
975 |
|
|
$w_k(r)$ & |
976 |
gezelter |
3985 |
$\frac{d v_{41}}{dr} $ & |
977 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
978 |
|
|
$\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right) |
979 |
|
|
-\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
980 |
gezelter |
3906 |
% |
981 |
|
|
$w_l(r)$ & |
982 |
gezelter |
3985 |
$\frac{d v_{42}}{dr} -\frac{2v_{42}(r)}{r}$ & |
983 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
984 |
|
|
$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
985 |
|
|
&&& $~~~ -\left(-\frac{9g(r_c)}{r_c^4} +\frac{9h(r_c)}{r_c^3} -\frac{4s(r_c)}{r_c^2} +\frac{t(r_c)}{r_c} \right) |
986 |
|
|
-\frac{2v_{42}(r)}{r}$\\ |
987 |
gezelter |
3906 |
% |
988 |
|
|
$w_m(r)$ & |
989 |
gezelter |
3985 |
$\frac{d v_{43}}{dr} -\frac{4v_{43}(r)}{r}$& |
990 |
|
|
$\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
991 |
|
|
$\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$\\ |
992 |
|
|
&&& $~~~- \left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
993 |
|
|
+\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $\\ |
994 |
|
|
&&& $~~~-\frac{4v_{43}(r)}{r}$ \\ |
995 |
gezelter |
3906 |
% |
996 |
|
|
$w_n(r)$ & |
997 |
gezelter |
3985 |
$\frac{v_{42}(r)}{r}$ & |
998 |
|
|
$\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
999 |
|
|
$\frac{v_{42}(r)}{r}$\\ |
1000 |
gezelter |
3906 |
% |
1001 |
|
|
$w_o(r)$ & |
1002 |
gezelter |
3985 |
$\frac{v_{43}(r)}{r}$& |
1003 |
|
|
$\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
1004 |
|
|
$\frac{v_{43}(r)}{r}$ \\ \hline |
1005 |
gezelter |
3906 |
% |
1006 |
|
|
|
1007 |
|
|
\end{tabular} |
1008 |
gezelter |
3985 |
\end{sidewaystable} |
1009 |
gezelter |
3906 |
% |
1010 |
|
|
% |
1011 |
|
|
% |
1012 |
|
|
|
1013 |
|
|
\subsection{Forces} |
1014 |
gezelter |
3985 |
The force on object $\bf{a}$, $\mathbf{F}_{\bf a}$, due to object |
1015 |
|
|
$\bf{b}$ is the negative of the force on $\bf{b}$ due to $\bf{a}$. For |
1016 |
|
|
a simple charge-charge interaction, these forces will point along the |
1017 |
|
|
$\pm \hat{r}$ directions, where $\mathbf{r}=\mathbf{r}_b - |
1018 |
|
|
\mathbf{r}_a $. Thus |
1019 |
gezelter |
3906 |
% |
1020 |
|
|
\begin{equation} |
1021 |
|
|
F_{\bf a \alpha} = \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}}{\partial r} |
1022 |
|
|
\quad \text{and} \quad F_{\bf b \alpha} |
1023 |
|
|
= - \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}} {\partial r} . |
1024 |
|
|
\end{equation} |
1025 |
|
|
% |
1026 |
gezelter |
3985 |
We list below the force equations written in terms of lab-frame |
1027 |
|
|
coordinates. The radial functions used in the two methods are listed |
1028 |
|
|
in Table \ref{tab:tableFORCE} |
1029 |
gezelter |
3906 |
% |
1030 |
gezelter |
3985 |
%SPACE COORDINATES FORCE EQUATIONS |
1031 |
gezelter |
3906 |
% |
1032 |
|
|
% ************************************************************************** |
1033 |
|
|
% f ca cb |
1034 |
|
|
% |
1035 |
gezelter |
3985 |
\begin{align} |
1036 |
|
|
\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} =& |
1037 |
|
|
C_{\bf a} C_{\bf b} w_a(r) \hat{r} \\ |
1038 |
gezelter |
3906 |
% |
1039 |
|
|
% |
1040 |
|
|
% |
1041 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} =& |
1042 |
|
|
C_{\bf a} \Bigl[ |
1043 |
gezelter |
3906 |
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{b}} \right) |
1044 |
|
|
w_b(r) \hat{r} |
1045 |
gezelter |
3985 |
+ \mathbf{D}_{\mathbf{b}} w_c(r) \Bigr] \\ |
1046 |
gezelter |
3906 |
% |
1047 |
|
|
% |
1048 |
|
|
% |
1049 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} =& |
1050 |
|
|
C_{\bf a } \Bigr[ |
1051 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\bf b} w_d(r) \hat{r} |
1052 |
|
|
+ 2 \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} w_e(r) |
1053 |
gezelter |
3985 |
+ \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} |
1054 |
|
|
\right) w_f(r) \hat{r} \Bigr] \\ |
1055 |
gezelter |
3906 |
% |
1056 |
|
|
% |
1057 |
|
|
% |
1058 |
gezelter |
3985 |
% \begin{equation} |
1059 |
|
|
% \mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1060 |
|
|
% -C_{\bf{b}} \Bigl[ |
1061 |
|
|
% \left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) w_b(r) \hat{r} |
1062 |
|
|
% + \mathbf{D}_{\mathbf{a}} w_c(r) \Bigr] |
1063 |
|
|
% \end{equation} |
1064 |
gezelter |
3906 |
% |
1065 |
|
|
% |
1066 |
|
|
% |
1067 |
gezelter |
3985 |
\begin{split} |
1068 |
|
|
\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
1069 |
gezelter |
3906 |
- \mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} w_d(r) \hat{r} |
1070 |
|
|
+ \left( \mathbf{D}_{\mathbf {a}} |
1071 |
|
|
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
1072 |
gezelter |
3985 |
+ \mathbf{D}_{\mathbf {b}} \left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) \right) w_e(r)\\ |
1073 |
gezelter |
3906 |
% 2 |
1074 |
gezelter |
3985 |
& - \left( \hat{r} \cdot \mathbf{D}_{\mathbf {a}} \right) |
1075 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf {b}} \right) w_f(r) \hat{r} |
1076 |
|
|
\end{split}\\ |
1077 |
gezelter |
3906 |
% |
1078 |
|
|
% |
1079 |
|
|
% |
1080 |
|
|
\begin{split} |
1081 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} =& - \Bigl[ |
1082 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} \mathbf{ D}_{\mathbf{a}} |
1083 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \cdot |
1084 |
|
|
\mathbf{Q}_{\mathbf{b}} \Bigr] w_g(r) |
1085 |
gezelter |
3985 |
- \Bigl[ |
1086 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1087 |
|
|
\left( \hat{r} \cdot \mathbf{D}_{\mathbf{a}} \right) |
1088 |
|
|
+2 ( \mathbf{D}_{\mathbf{a}} \cdot |
1089 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1090 |
|
|
% 3 |
1091 |
gezelter |
3985 |
& - \Bigl[\mathbf{ D}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1092 |
gezelter |
3906 |
+2 (\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) (\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \Bigr] |
1093 |
|
|
w_i(r) |
1094 |
|
|
% 4 |
1095 |
gezelter |
3985 |
- |
1096 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}} ) |
1097 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) w_j(r) \hat{r} \end{split} \\ |
1098 |
gezelter |
3906 |
% |
1099 |
|
|
% |
1100 |
gezelter |
3985 |
% \begin{equation} |
1101 |
|
|
% \mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} = |
1102 |
|
|
% \frac{C_{\bf b }}{4\pi \epsilon_0} \Bigr[ |
1103 |
|
|
% \text{Tr}\mathbf{Q}_{\bf a} w_d(r) \hat{r} |
1104 |
|
|
% + 2 \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} w_e(r) |
1105 |
|
|
% + \left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) w_f(r) \hat{r} \Bigr] |
1106 |
|
|
% \end{equation} |
1107 |
|
|
% % |
1108 |
|
|
% \begin{equation} |
1109 |
|
|
% \begin{split} |
1110 |
|
|
% \mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1111 |
|
|
% &\frac{1}{4\pi \epsilon_0} \Bigl[ |
1112 |
|
|
% \text{Tr}\mathbf{Q}_{\mathbf{a}} \mathbf{D}_{\mathbf{b}} |
1113 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} \Bigr] w_g(r) |
1114 |
|
|
% % 2 |
1115 |
|
|
% + \frac{1}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
1116 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1117 |
|
|
% +2 (\mathbf{D}_{\mathbf{b}} \cdot |
1118 |
|
|
% \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] w_h(r) \hat{r} \\ |
1119 |
|
|
% % 3 |
1120 |
|
|
% &+ \frac{1}{4\pi \epsilon_0} \Bigl[ \mathbf{D}_{\mathbf{b}} |
1121 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1122 |
|
|
% +2 (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1123 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \Bigr] w_i(r) |
1124 |
|
|
% % 4 |
1125 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1126 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}}) |
1127 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) w_j(r) \hat{r} |
1128 |
|
|
% \end{split} |
1129 |
|
|
% \end{equation} |
1130 |
gezelter |
3906 |
% |
1131 |
|
|
% |
1132 |
|
|
% |
1133 |
|
|
\begin{split} |
1134 |
gezelter |
3985 |
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
1135 |
|
|
\Bigl[ |
1136 |
gezelter |
3989 |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} |
1137 |
|
|
+ 2 \mathbf{Q}_{\mathbf{a}} : \mathbf{Q}_{\mathbf{b}} \Bigr] w_k(r) \hat{r} \\ |
1138 |
gezelter |
3906 |
% 2 |
1139 |
gezelter |
3985 |
&+ \Bigl[ |
1140 |
gezelter |
3906 |
2\text{Tr}\mathbf{Q}_{\mathbf{b}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
1141 |
|
|
+ 2\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) |
1142 |
|
|
% 3 |
1143 |
|
|
+4 (\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1144 |
|
|
+ 4(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}}) \Bigr] w_n(r) \\ |
1145 |
|
|
% 4 |
1146 |
gezelter |
3985 |
&+ \Bigl[ |
1147 |
gezelter |
3906 |
\text{Tr}\mathbf{Q}_{\mathbf{a}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1148 |
|
|
+ \text{Tr}\mathbf{Q}_{\mathbf{b}} |
1149 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1150 |
|
|
% 5 |
1151 |
|
|
+4 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot |
1152 |
|
|
\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \\ |
1153 |
|
|
% |
1154 |
gezelter |
3985 |
&+ \Bigl[ |
1155 |
gezelter |
3906 |
+ 2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
1156 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1157 |
|
|
%6 |
1158 |
|
|
+2 (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1159 |
|
|
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_o(r) \\ |
1160 |
|
|
% 7 |
1161 |
gezelter |
3985 |
&+ |
1162 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1163 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) w_m(r) \hat{r} \end{split} |
1164 |
|
|
\end{align} |
1165 |
|
|
Note that the forces for higher multipoles on site $\mathbf{a}$ |
1166 |
|
|
interacting with those of lower order on site $\mathbf{b}$ can be |
1167 |
|
|
obtained by swapping indices in the expressions above. |
1168 |
|
|
|
1169 |
gezelter |
3906 |
% |
1170 |
gezelter |
3985 |
% Torques SECTION ----------------------------------------------------------------------------------------- |
1171 |
gezelter |
3906 |
% |
1172 |
|
|
\subsection{Torques} |
1173 |
gezelter |
3989 |
|
1174 |
gezelter |
3906 |
% |
1175 |
gezelter |
3985 |
The torques for both the Taylor-Shifted as well as Gradient-Shifted |
1176 |
|
|
methods are given in space-frame coordinates: |
1177 |
gezelter |
3906 |
% |
1178 |
|
|
% |
1179 |
gezelter |
3985 |
\begin{align} |
1180 |
|
|
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} =& |
1181 |
|
|
C_{\bf a} (\hat{r} \times \mathbf{D}_{\mathbf{b}}) v_{11}(r) \\ |
1182 |
gezelter |
3906 |
% |
1183 |
|
|
% |
1184 |
|
|
% |
1185 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} =& |
1186 |
|
|
2C_{\bf a} |
1187 |
|
|
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{22}(r) \\ |
1188 |
gezelter |
3906 |
% |
1189 |
|
|
% |
1190 |
|
|
% |
1191 |
gezelter |
3985 |
% \begin{equation} |
1192 |
|
|
% \mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} = |
1193 |
|
|
% -\frac{C_{\bf b}}{4\pi \epsilon_0} |
1194 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{a}}) v_{11}(r) |
1195 |
|
|
% \end{equation} |
1196 |
gezelter |
3906 |
% |
1197 |
|
|
% |
1198 |
|
|
% |
1199 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
1200 |
|
|
\mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
1201 |
gezelter |
3906 |
% 2 |
1202 |
gezelter |
3985 |
- |
1203 |
gezelter |
3906 |
(\hat{r} \times \mathbf{D}_{\mathbf {a}} ) |
1204 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{D}_{\mathbf {b}} ) v_{22}(r)\\ |
1205 |
gezelter |
3906 |
% |
1206 |
|
|
% |
1207 |
|
|
% |
1208 |
gezelter |
3985 |
% \begin{equation} |
1209 |
|
|
% \mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} = |
1210 |
|
|
% -\frac{1}{4\pi \epsilon_0} \mathbf{D}_{\mathbf {a}} \times \mathbf{D}_{\mathbf{b}} v_{21}(r) |
1211 |
|
|
% % 2 |
1212 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1213 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf {a}} ) |
1214 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf {b}} ) v_{22}(r) |
1215 |
|
|
% \end{equation} |
1216 |
gezelter |
3906 |
% |
1217 |
|
|
% |
1218 |
|
|
% |
1219 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} =& |
1220 |
|
|
\Bigl[ |
1221 |
gezelter |
3906 |
-\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1222 |
|
|
(\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
1223 |
|
|
+2 \mathbf{D}_{\mathbf{a}} \times |
1224 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1225 |
|
|
\Bigr] v_{31}(r) |
1226 |
|
|
% 3 |
1227 |
gezelter |
3985 |
- (\hat{r} \times \mathbf{D}_{\mathbf{a}} ) |
1228 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{32}(r)\\ |
1229 |
gezelter |
3906 |
% |
1230 |
|
|
% |
1231 |
|
|
% |
1232 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} =& |
1233 |
|
|
\Bigl[ |
1234 |
gezelter |
3906 |
+2 ( \mathbf{D}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \times |
1235 |
|
|
\hat{r} |
1236 |
|
|
-2 \mathbf{D}_{\mathbf{a}} \times |
1237 |
|
|
(\mathbf{Q}_{\mathbf{b}} \cdot \hat{r}) |
1238 |
|
|
\Bigr] v_{31}(r) |
1239 |
|
|
% 2 |
1240 |
gezelter |
3985 |
+ |
1241 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{D}_{\mathbf{a}}) |
1242 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{\mathbf{b}}) \times \hat{r} v_{32}(r)\\ |
1243 |
gezelter |
3906 |
% |
1244 |
|
|
% |
1245 |
|
|
% |
1246 |
gezelter |
3985 |
% \begin{equation} |
1247 |
|
|
% \mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = |
1248 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1249 |
|
|
% -2 (\mathbf{D}_{\mathbf{b}} \cdot \mathbf{Q}_{\mathbf{a}} ) \times \hat{r} |
1250 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \times |
1251 |
|
|
% (\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) |
1252 |
|
|
% \Bigr] v_{31}(r) |
1253 |
|
|
% % 3 |
1254 |
|
|
% - \frac{2}{4\pi \epsilon_0} |
1255 |
|
|
% (\hat{r} \cdot \mathbf{D}_{\mathbf{b}} ) |
1256 |
|
|
% (\hat{r} \cdot \mathbf |
1257 |
|
|
% {Q}_{{\mathbf a}}) \times \hat{r} v_{32}(r) |
1258 |
|
|
% \end{equation} |
1259 |
gezelter |
3906 |
% |
1260 |
|
|
% |
1261 |
|
|
% |
1262 |
gezelter |
3985 |
% \begin{equation} |
1263 |
|
|
% \mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} = |
1264 |
|
|
% \frac{1}{4\pi \epsilon_0} \Bigl[ |
1265 |
|
|
% \text{Tr}\mathbf{Q}_{\mathbf{a}} |
1266 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1267 |
|
|
% +2 \mathbf{D}_{\mathbf{b}} \times |
1268 |
|
|
% ( \mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
1269 |
|
|
% % 2 |
1270 |
|
|
% +\frac{1}{4\pi \epsilon_0} |
1271 |
|
|
% (\hat{r} \times \mathbf{D}_{\mathbf{b}} ) |
1272 |
|
|
% (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) v_{32}(r) |
1273 |
|
|
% \end{equation} |
1274 |
gezelter |
3906 |
% |
1275 |
|
|
% |
1276 |
|
|
% |
1277 |
|
|
\begin{split} |
1278 |
gezelter |
3985 |
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
1279 |
|
|
-4 |
1280 |
gezelter |
3906 |
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} |
1281 |
|
|
v_{41}(r) \\ |
1282 |
|
|
% 2 |
1283 |
gezelter |
3985 |
&+ |
1284 |
gezelter |
3906 |
\Bigl[-2\text{Tr}\mathbf{Q}_{\mathbf{b}} |
1285 |
|
|
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times \hat{r} |
1286 |
|
|
+4 \hat{r} \times |
1287 |
|
|
( \mathbf{Q}_{{\mathbf a}} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1288 |
|
|
% 3 |
1289 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} )\times |
1290 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} ) \Bigr] v_{42}(r) \\ |
1291 |
|
|
% 4 |
1292 |
gezelter |
3985 |
&+ 2 |
1293 |
gezelter |
3906 |
\hat{r} \times ( \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1294 |
gezelter |
3985 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r) \end{split}\\ |
1295 |
gezelter |
3906 |
% |
1296 |
|
|
% |
1297 |
|
|
% |
1298 |
|
|
\begin{split} |
1299 |
|
|
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} = |
1300 |
gezelter |
3985 |
&4 |
1301 |
gezelter |
3906 |
\mathbf{Q}_{{\mathbf a}} \times \mathbf{Q}_{{\mathbf b}} v_{41}(r) \\ |
1302 |
|
|
% 2 |
1303 |
gezelter |
3985 |
&+ \Bigl[- 2\text{Tr}\mathbf{Q}_{\mathbf{a}} |
1304 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf b}} ) \times \hat{r} |
1305 |
|
|
-4 (\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot |
1306 |
|
|
\mathbf{Q}_{{\mathbf b}} ) \times |
1307 |
|
|
\hat{r} |
1308 |
|
|
+4 ( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} ) \times |
1309 |
|
|
( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) |
1310 |
|
|
\Bigr] v_{42}(r) \\ |
1311 |
|
|
% 4 |
1312 |
gezelter |
3985 |
&+2 |
1313 |
gezelter |
3906 |
(\hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r}) |
1314 |
gezelter |
3985 |
\hat{r} \times ( \mathbf{Q}_{{\mathbf b}} \cdot \hat{r}) v_{43}(r)\end{split} |
1315 |
|
|
\end{align} |
1316 |
|
|
% |
1317 |
|
|
Here, we have defined the matrix cross product in an identical form |
1318 |
|
|
as in Ref. \onlinecite{Smith98}: |
1319 |
|
|
\begin{equation} |
1320 |
|
|
\left[\mathbf{A} \times \mathbf{B}\right]_\alpha = \sum_\beta |
1321 |
|
|
\left[\mathbf{A}_{\alpha+1,\beta} \mathbf{B}_{\alpha+2,\beta} |
1322 |
|
|
-\mathbf{A}_{\alpha+2,\beta} \mathbf{B}_{\alpha+2,\beta} |
1323 |
|
|
\right] |
1324 |
gezelter |
3906 |
\end{equation} |
1325 |
gezelter |
3985 |
where $\alpha+1$ and $\alpha+2$ are regarded as cyclic |
1326 |
|
|
permuations of the matrix indices. |
1327 |
gezelter |
3980 |
|
1328 |
gezelter |
3985 |
All of the radial functions required for torques are identical with |
1329 |
|
|
the radial functions previously computed for the interaction energies. |
1330 |
|
|
These are tabulated for both shifted force methods in table |
1331 |
|
|
\ref{tab:tableenergy}. The torques for higher multipoles on site |
1332 |
|
|
$\mathbf{a}$ interacting with those of lower order on site |
1333 |
|
|
$\mathbf{b}$ can be obtained by swapping indices in the expressions |
1334 |
|
|
above. |
1335 |
|
|
|
1336 |
gezelter |
3980 |
\section{Comparison to known multipolar energies} |
1337 |
|
|
|
1338 |
|
|
To understand how these new real-space multipole methods behave in |
1339 |
|
|
computer simulations, it is vital to test against established methods |
1340 |
|
|
for computing electrostatic interactions in periodic systems, and to |
1341 |
|
|
evaluate the size and sources of any errors that arise from the |
1342 |
|
|
real-space cutoffs. In this paper we test Taylor-shifted and |
1343 |
|
|
Gradient-shifted electrostatics against analytical methods for |
1344 |
|
|
computing the energies of ordered multipolar arrays. In the following |
1345 |
|
|
paper, we test the new methods against the multipolar Ewald sum for |
1346 |
|
|
computing the energies, forces and torques for a wide range of typical |
1347 |
|
|
condensed-phase (disordered) systems. |
1348 |
|
|
|
1349 |
|
|
Because long-range electrostatic effects can be significant in |
1350 |
|
|
crystalline materials, ordered multipolar arrays present one of the |
1351 |
|
|
biggest challenges for real-space cutoff methods. The dipolar |
1352 |
|
|
analogues to the Madelung constants were first worked out by Sauer, |
1353 |
|
|
who computed the energies of ordered dipole arrays of zero |
1354 |
|
|
magnetization and obtained a number of these constants.\cite{Sauer} |
1355 |
|
|
This theory was developed more completely by Luttinger and |
1356 |
gezelter |
3986 |
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays |
1357 |
|
|
and other periodic structures. We have repeated the Luttinger \& |
1358 |
|
|
Tisza series summations to much higher order and obtained the energy |
1359 |
|
|
constants (converged to one part in $10^9$) in table \ref{tab:LT}. |
1360 |
|
|
|
1361 |
|
|
\begin{table*}[h] |
1362 |
gezelter |
3980 |
\centering{ |
1363 |
|
|
\caption{Luttinger \& Tisza arrays and their associated |
1364 |
gezelter |
3989 |
energy constants. Type ``A'' arrays have nearest neighbor strings of |
1365 |
|
|
antiparallel dipoles. Type ``B'' arrays have nearest neighbor |
1366 |
gezelter |
3980 |
strings of antiparallel dipoles if the dipoles are contained in a |
1367 |
|
|
plane perpendicular to the dipole direction that passes through |
1368 |
|
|
the dipole.} |
1369 |
|
|
} |
1370 |
|
|
\label{tab:LT} |
1371 |
|
|
\begin{ruledtabular} |
1372 |
|
|
\begin{tabular}{cccc} |
1373 |
|
|
Array Type & Lattice & Dipole Direction & Energy constants \\ \hline |
1374 |
|
|
A & SC & 001 & -2.676788684 \\ |
1375 |
|
|
A & BCC & 001 & 0 \\ |
1376 |
|
|
A & BCC & 111 & -1.770078733 \\ |
1377 |
|
|
A & FCC & 001 & 2.166932835 \\ |
1378 |
|
|
A & FCC & 011 & -1.083466417 \\ |
1379 |
gezelter |
3986 |
B & SC & 001 & -2.676788684 \\ |
1380 |
|
|
B & BCC & 001 & -1.338394342 \\ |
1381 |
|
|
B & BCC & 111 & -1.770078733 \\ |
1382 |
|
|
B & FCC & 001 & -1.083466417 \\ |
1383 |
|
|
B & FCC & 011 & -1.807573634 \\ |
1384 |
|
|
-- & BCC & minimum & -1.985920929 \\ |
1385 |
gezelter |
3980 |
\end{tabular} |
1386 |
|
|
\end{ruledtabular} |
1387 |
|
|
\end{table*} |
1388 |
|
|
|
1389 |
|
|
In addition to the A and B arrays, there is an additional minimum |
1390 |
|
|
energy structure for the BCC lattice that was found by Luttinger \& |
1391 |
gezelter |
3986 |
Tisza. The total electrostatic energy for any of the arrays is given |
1392 |
|
|
by: |
1393 |
gezelter |
3980 |
\begin{equation} |
1394 |
|
|
E = C N^2 \mu^2 |
1395 |
|
|
\end{equation} |
1396 |
gezelter |
3986 |
where $C$ is the energy constant given in table \ref{tab:LT}, $N$ is |
1397 |
|
|
the number of dipoles per unit volume, and $\mu$ is the strength of |
1398 |
|
|
the dipole. |
1399 |
gezelter |
3980 |
|
1400 |
gezelter |
3988 |
To test the new electrostatic methods, we have constructed very large, |
1401 |
|
|
$N$ = 8,000~(sc), 16,000~(bcc), or 32,000~(fcc) arrays of dipoles in |
1402 |
|
|
the orientations described in table \ref{tab:LT}. For the purposes of |
1403 |
|
|
testing the energy expressions and the self-neutralization schemes, |
1404 |
|
|
the primary quantity of interest is the analytic energy constant for |
1405 |
|
|
the perfect arrays. Convergence to these constants are shown as a |
1406 |
|
|
function of both the cutoff radius, $r_c$, and the damping parameter, |
1407 |
|
|
$\alpha$ in Figs. \ref{fig:energyConstVsCutoff} and XXX. We have |
1408 |
|
|
simultaneously tested a hard cutoff (where the kernel is simply |
1409 |
|
|
truncated at the cutoff radius), as well as a shifted potential (SP) |
1410 |
|
|
form which includes a potential-shifting and self-interaction term, |
1411 |
|
|
but does not shift the forces and torques smoothly at the cutoff |
1412 |
gezelter |
3989 |
radius. The SP method is essentially an extension of the original |
1413 |
|
|
Wolf method for multipoles. |
1414 |
gezelter |
3986 |
|
1415 |
gezelter |
3989 |
\begin{figure}[!htbp] |
1416 |
gezelter |
3988 |
\includegraphics[width=4.5in]{energyConstVsCutoff} |
1417 |
|
|
\caption{Convergence to the analytic energy constants as a function of |
1418 |
|
|
cutoff radius (normalized by the lattice constant) for the different |
1419 |
|
|
real-space methods. The two crystals shown here are the ``B'' array |
1420 |
|
|
for bcc crystals with the dipoles along the 001 direction (upper), |
1421 |
|
|
as well as the minimum energy bcc lattice (lower). The analytic |
1422 |
|
|
energy constants are shown as a grey dashed line. The left panel |
1423 |
|
|
shows results for the undamped kernel ($1/r$), while the damped |
1424 |
|
|
error function kernel, $B_0(r)$ was used in the right panel. } |
1425 |
|
|
\label{fig:energyConstVsCutoff} |
1426 |
|
|
\end{figure} |
1427 |
|
|
|
1428 |
|
|
The Hard cutoff exhibits oscillations around the analytic energy |
1429 |
|
|
constants, and converges to incorrect energies when the complementary |
1430 |
|
|
error function damping kernel is used. The shifted potential (SP) and |
1431 |
|
|
gradient-shifted force (GSF) approximations converge to the correct |
1432 |
|
|
energy smoothly by $r_c / 6 a$ even for the undamped case. This |
1433 |
|
|
indicates that the correction provided by the self term is required |
1434 |
|
|
for obtaining accurate energies. The Taylor-shifted force (TSF) |
1435 |
|
|
approximation appears to perturb the potential too much inside the |
1436 |
|
|
cutoff region to provide accurate measures of the energy constants. |
1437 |
|
|
|
1438 |
|
|
|
1439 |
gezelter |
3986 |
{\it Quadrupolar} analogues to the Madelung constants were first |
1440 |
|
|
worked out by Nagai and Nakamura who computed the energies of selected |
1441 |
|
|
quadrupole arrays based on extensions to the Luttinger and Tisza |
1442 |
|
|
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
1443 |
gezelter |
3980 |
energy constants for the lowest energy configurations for linear |
1444 |
|
|
quadrupoles shown in table \ref{tab:NNQ} |
1445 |
|
|
|
1446 |
|
|
\begin{table*} |
1447 |
|
|
\centering{ |
1448 |
|
|
\caption{Nagai and Nakamura Quadurpolar arrays}} |
1449 |
|
|
\label{tab:NNQ} |
1450 |
|
|
\begin{ruledtabular} |
1451 |
|
|
\begin{tabular}{ccc} |
1452 |
|
|
Lattice & Quadrupole Direction & Energy constants \\ \hline |
1453 |
|
|
SC & 111 & -8.3 \\ |
1454 |
|
|
BCC & 011 & -21.7 \\ |
1455 |
|
|
FCC & 111 & -80.5 |
1456 |
|
|
\end{tabular} |
1457 |
|
|
\end{ruledtabular} |
1458 |
|
|
\end{table*} |
1459 |
|
|
|
1460 |
|
|
In analogy to the dipolar arrays, the total electrostatic energy for |
1461 |
|
|
the quadrupolar arrays is: |
1462 |
|
|
\begin{equation} |
1463 |
|
|
E = C \frac{3}{4} N^2 Q^2 |
1464 |
|
|
\end{equation} |
1465 |
|
|
where $Q$ is the quadrupole moment. |
1466 |
|
|
|
1467 |
gezelter |
3985 |
\section{Conclusion} |
1468 |
|
|
We have presented two efficient real-space methods for computing the |
1469 |
|
|
interactions between point multipoles. These methods have the benefit |
1470 |
|
|
of smoothly truncating the energies, forces, and torques at the cutoff |
1471 |
|
|
radius, making them attractive for both molecular dynamics (MD) and |
1472 |
|
|
Monte Carlo (MC) simulations. We find that the Gradient-Shifted Force |
1473 |
|
|
(GSF) and the Shifted-Potential (SP) methods converge rapidly to the |
1474 |
|
|
correct lattice energies for ordered dipolar and quadrupolar arrays, |
1475 |
|
|
while the Taylor-Shifted Force (TSF) is too severe an approximation to |
1476 |
|
|
provide accurate convergence to lattice energies. |
1477 |
gezelter |
3980 |
|
1478 |
gezelter |
3985 |
In most cases, GSF can obtain nearly quantitative agreement with the |
1479 |
|
|
lattice energy constants with reasonably small cutoff radii. The only |
1480 |
|
|
exception we have observed is for crystals which exhibit a bulk |
1481 |
|
|
macroscopic dipole moment (e.g. Luttinger \& Tisza's $Z_1$ lattice). |
1482 |
|
|
In this particular case, the multipole neutralization scheme can |
1483 |
|
|
interfere with the correct computation of the energies. We note that |
1484 |
|
|
the energies for these arrangements are typically much larger than for |
1485 |
|
|
crystals with net-zero moments, so this is not expected to be an issue |
1486 |
|
|
in most simulations. |
1487 |
gezelter |
3980 |
|
1488 |
gezelter |
3985 |
In large systems, these new methods can be made to scale approximately |
1489 |
|
|
linearly with system size, and detailed comparisons with the Ewald sum |
1490 |
|
|
for a wide range of chemical environments follows in the second paper. |
1491 |
gezelter |
3980 |
|
1492 |
gezelter |
3906 |
\begin{acknowledgments} |
1493 |
gezelter |
3985 |
JDG acknowledges helpful discussions with Christopher |
1494 |
|
|
Fennell. Support for this project was provided by the National |
1495 |
|
|
Science Foundation under grant CHE-0848243. Computational time was |
1496 |
|
|
provided by the Center for Research Computing (CRC) at the |
1497 |
|
|
University of Notre Dame. |
1498 |
gezelter |
3906 |
\end{acknowledgments} |
1499 |
|
|
|
1500 |
gezelter |
3984 |
\newpage |
1501 |
gezelter |
3906 |
\appendix |
1502 |
|
|
|
1503 |
gezelter |
3984 |
\section{Smith's $B_l(r)$ functions for damped-charge distributions} |
1504 |
gezelter |
3985 |
\label{SmithFunc} |
1505 |
gezelter |
3984 |
The following summarizes Smith's $B_l(r)$ functions and includes |
1506 |
|
|
formulas given in his appendix.\cite{Smith98} The first function |
1507 |
|
|
$B_0(r)$ is defined by |
1508 |
gezelter |
3906 |
% |
1509 |
|
|
\begin{equation} |
1510 |
|
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r}= |
1511 |
|
|
\int_{\alpha r}^{\infty} \text{e}^{-s^2} ds . |
1512 |
|
|
\end{equation} |
1513 |
|
|
% |
1514 |
|
|
The first derivative of this function is |
1515 |
|
|
% |
1516 |
|
|
\begin{equation} |
1517 |
|
|
\frac{dB_0(r)}{dr}=-\frac{1}{r^2}\text{erfc}(\alpha r) |
1518 |
|
|
-\frac{2\alpha}{r\sqrt{\pi}}\text{e}^{-{\alpha}^2r^2} |
1519 |
|
|
\end{equation} |
1520 |
|
|
% |
1521 |
gezelter |
3984 |
which can be used to define a function $B_1(r)$: |
1522 |
gezelter |
3906 |
% |
1523 |
|
|
\begin{equation} |
1524 |
|
|
B_1(r)=-\frac{1}{r}\frac{dB_0(r)}{dr} |
1525 |
|
|
\end{equation} |
1526 |
|
|
% |
1527 |
gezelter |
3984 |
In general, the recurrence relation, |
1528 |
gezelter |
3906 |
\begin{equation} |
1529 |
|
|
B_l(r)=-\frac{1}{r}\frac{dB_{l-1}(r)}{dr} |
1530 |
|
|
= \frac{1}{r^2} \left[ (2l-1)B_{l-1}(r) + \frac {(2\alpha^2)^l}{\alpha \sqrt{\pi}} |
1531 |
|
|
\text{e}^{-{\alpha}^2r^2} |
1532 |
gezelter |
3984 |
\right] , |
1533 |
gezelter |
3906 |
\end{equation} |
1534 |
gezelter |
3984 |
is very useful for building up higher derivatives. Using these |
1535 |
|
|
formulas, we find: |
1536 |
gezelter |
3906 |
% |
1537 |
gezelter |
3984 |
\begin{align} |
1538 |
|
|
\frac{dB_0}{dr}=&-rB_1(r) \\ |
1539 |
|
|
\frac{d^2B_0}{dr^2}=& - B_1(r) + r^2 B_2(r) \\ |
1540 |
|
|
\frac{d^3B_0}{dr^3}=& 3 r B_2(r) - r^3 B_3(r) \\ |
1541 |
|
|
\frac{d^4B_0}{dr^4}=& 3 B_2(r) - 6 r^2 B_3(r) + r^4 B_4(r) \\ |
1542 |
|
|
\frac{d^5B_0}{dr^5}=& - 15 r B_3(r) + 10 r^3 B_4(r) - r^5 B_5(r) . |
1543 |
|
|
\end{align} |
1544 |
gezelter |
3906 |
% |
1545 |
gezelter |
3984 |
As noted by Smith, it is possible to approximate the $B_l(r)$ |
1546 |
|
|
functions, |
1547 |
gezelter |
3906 |
% |
1548 |
|
|
\begin{equation} |
1549 |
|
|
B_l(r)=\frac{(2l)!}{l!2^lr^{2l+1}} - \frac {(2\alpha^2)^{l+1}}{(2l+1)\alpha \sqrt{\pi}} |
1550 |
|
|
+\text{O}(r) . |
1551 |
|
|
\end{equation} |
1552 |
gezelter |
3984 |
\newpage |
1553 |
|
|
\section{The $r$-dependent factors for TSF electrostatics} |
1554 |
gezelter |
3906 |
|
1555 |
|
|
Using the shifted damped functions $f_n(r)$ defined by: |
1556 |
|
|
% |
1557 |
|
|
\begin{equation} |
1558 |
gezelter |
3984 |
f_n(r)= B_0(r) -\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} B_0^{(m)}(r_c) , |
1559 |
gezelter |
3906 |
\end{equation} |
1560 |
|
|
% |
1561 |
gezelter |
3984 |
where the superscript $(m)$ denotes the $m^\mathrm{th}$ derivative. In |
1562 |
|
|
this Appendix, we provide formulas for successive derivatives of this |
1563 |
|
|
function. (If there is no damping, then $B_0(r)$ is replaced by |
1564 |
|
|
$1/r$.) First, we find: |
1565 |
gezelter |
3906 |
% |
1566 |
|
|
\begin{equation} |
1567 |
|
|
\frac{\partial f_n}{\partial r_\alpha}=\hat{r}_\alpha \frac{d f_n}{d r} . |
1568 |
|
|
\end{equation} |
1569 |
|
|
% |
1570 |
gezelter |
3984 |
This formula clearly brings in derivatives of Smith's $B_0(r)$ |
1571 |
|
|
function, and we define higher-order derivatives as follows: |
1572 |
gezelter |
3906 |
% |
1573 |
gezelter |
3984 |
\begin{align} |
1574 |
|
|
g_n(r)=& \frac{d f_n}{d r} = |
1575 |
|
|
B_0^{(1)}(r) -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)}(r_c) \\ |
1576 |
|
|
h_n(r)=& \frac{d^2f_n}{d r^2} = |
1577 |
|
|
B_0^{(2)}(r) -\sum_{m=0}^{n-1} \frac {(r-r_c)^m}{m!} B_0^{(m+2)}(r_c) \\ |
1578 |
|
|
s_n(r)=& \frac{d^3f_n}{d r^3} = |
1579 |
|
|
B_0^{(3)}(r) -\sum_{m=0}^{n-2} \frac {(r-r_c)^m}{m!} B_0^{(m+3)}(r_c) \\ |
1580 |
|
|
t_n(r)=& \frac{d^4f_n}{d r^4} = |
1581 |
|
|
B_0^{(4)}(r) -\sum_{m=0}^{n-3} \frac {(r-r_c)^m}{m!} B_0^{(m+4)}(r_c) \\ |
1582 |
|
|
u_n(r)=& \frac{d^5f_n}{d r^5} = |
1583 |
|
|
B_0^{(5)}(r) -\sum_{m=0}^{n-4} \frac {(r-r_c)^m}{m!} B_0^{(m+5)}(r_c) . |
1584 |
|
|
\end{align} |
1585 |
gezelter |
3906 |
% |
1586 |
gezelter |
3984 |
We note that the last function needed (for quadrupole-quadrupole interactions) is |
1587 |
gezelter |
3906 |
% |
1588 |
|
|
\begin{equation} |
1589 |
gezelter |
3984 |
u_4(r)=B_0^{(5)}(r) - B_0^{(5)}(r_c) . |
1590 |
gezelter |
3906 |
\end{equation} |
1591 |
gezelter |
3989 |
% The functions |
1592 |
|
|
% needed are listed schematically below: |
1593 |
|
|
% % |
1594 |
|
|
% \begin{eqnarray} |
1595 |
|
|
% f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
1596 |
|
|
% g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
1597 |
|
|
% h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
1598 |
|
|
% s_2 \quad s_3 \quad &s_4 \nonumber \\ |
1599 |
|
|
% t_3 \quad &t_4 \nonumber \\ |
1600 |
|
|
% &u_4 \nonumber . |
1601 |
|
|
% \end{eqnarray} |
1602 |
gezelter |
3984 |
The functions $f_n(r)$ to $u_n(r)$ can be computed recursively and |
1603 |
gezelter |
3989 |
stored on a grid for values of $r$ from $0$ to $r_c$. Using these |
1604 |
|
|
functions, we find |
1605 |
gezelter |
3906 |
% |
1606 |
gezelter |
3984 |
\begin{align} |
1607 |
|
|
\frac{\partial f_n}{\partial r_\alpha} =&r_\alpha \frac {g_n}{r} \label{eq:b9}\\ |
1608 |
|
|
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =&\delta_{\alpha \beta}\frac {g_n}{r} |
1609 |
|
|
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) \\ |
1610 |
gezelter |
3989 |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta \partial r_\gamma} =& |
1611 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
1612 |
|
|
\delta_{ \beta \gamma} r_\alpha \right) |
1613 |
gezelter |
3989 |
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) \nonumber \\ |
1614 |
|
|
& + r_\alpha r_\beta r_\gamma |
1615 |
gezelter |
3984 |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \\ |
1616 |
gezelter |
3989 |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta \partial |
1617 |
|
|
r_\gamma \partial r_\delta} =& |
1618 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
1619 |
|
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
1620 |
|
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
1621 |
|
|
\left( - \frac{g_n}{r^3} + \frac{h_n}{r^2} \right) \nonumber \\ |
1622 |
gezelter |
3984 |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
1623 |
|
|
+ \text{5 permutations} |
1624 |
gezelter |
3906 |
\right) \left( \frac{3 g_n}{r^5} - \frac{3h_n}{r^4} + \frac{s_n}{r^3} |
1625 |
|
|
\right) \nonumber \\ |
1626 |
gezelter |
3984 |
&+ r_\alpha r_\beta r_\gamma r_\delta |
1627 |
gezelter |
3906 |
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
1628 |
gezelter |
3984 |
+ \frac{t_n}{r^4} \right)\\ |
1629 |
gezelter |
3906 |
\frac{\partial^5 f_n} |
1630 |
gezelter |
3989 |
{\partial r_\alpha \partial r_\beta \partial r_\gamma \partial |
1631 |
|
|
r_\delta \partial r_\epsilon} =& |
1632 |
gezelter |
3906 |
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
1633 |
gezelter |
3984 |
+ \text{14 permutations} \right) |
1634 |
gezelter |
3906 |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
1635 |
gezelter |
3984 |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
1636 |
|
|
+ \text{9 permutations} |
1637 |
gezelter |
3906 |
\right) \left(- \frac{15g_n}{r^7}+\frac{15h_n}{r^7} -\frac{6s_n}{r^5} +\frac{t_n}{r^4} |
1638 |
|
|
\right) \nonumber \\ |
1639 |
gezelter |
3984 |
&+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
1640 |
gezelter |
3906 |
\left( \frac{105g_n}{r^9} - \frac{105h_n}{r^8} + \frac{45s_n}{r^7} |
1641 |
gezelter |
3984 |
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) \label{eq:b13} |
1642 |
|
|
\end{align} |
1643 |
gezelter |
3906 |
% |
1644 |
|
|
% |
1645 |
|
|
% |
1646 |
gezelter |
3984 |
\newpage |
1647 |
|
|
\section{The $r$-dependent factors for GSF electrostatics} |
1648 |
gezelter |
3906 |
|
1649 |
gezelter |
3984 |
In Gradient-shifted force electrostatics, the kernel is not expanded, |
1650 |
|
|
rather the individual terms in the multipole interaction energies. |
1651 |
|
|
For damped charges , this still brings into the algebra multiple |
1652 |
|
|
derivatives of the Smith's $B_0(r)$ function. To denote these terms, |
1653 |
gezelter |
3989 |
we generalize the notation of the previous appendix. For either |
1654 |
|
|
$f(r)=1/r$ (undamped) or $f(r)=B_0(r)$ (damped), |
1655 |
gezelter |
3906 |
% |
1656 |
gezelter |
3984 |
\begin{align} |
1657 |
|
|
g(r)=& \frac{df}{d r}\\ |
1658 |
|
|
h(r)=& \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
1659 |
|
|
s(r)=& \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
1660 |
|
|
t(r)=& \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
1661 |
|
|
u(r)=& \frac{dt}{d r} = \frac{d^5f}{d r^5} . |
1662 |
|
|
\end{align} |
1663 |
gezelter |
3906 |
% |
1664 |
gezelter |
3989 |
For undamped charges Table I lists these derivatives under the column |
1665 |
|
|
``Bare Coulomb.'' Equations \ref{eq:b9} to \ref{eq:b13} are still |
1666 |
|
|
correct for GSF electrostatics if the subscript $n$ is eliminated. |
1667 |
gezelter |
3906 |
|
1668 |
gezelter |
3980 |
\newpage |
1669 |
|
|
|
1670 |
|
|
\bibliography{multipole} |
1671 |
|
|
|
1672 |
gezelter |
3906 |
\end{document} |
1673 |
|
|
% |
1674 |
|
|
% ****** End of file multipole.tex ****** |