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% ****** Start of file aipsamp.tex ****** |
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% This file is part of the AIP files in the AIP distribution for REVTeX 4. |
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% Version 4.1 of REVTeX, October 2009 |
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% Copyright (c) 2009 American Institute of Physics. |
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% 1) latex aipsamp |
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% 2) bibtex aipsamp |
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% 3) latex aipsamp |
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% 4) latex aipsamp |
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% Use this file as a source of example code for your aip document. |
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% Use the file aiptemplate.tex as a template for your document. |
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\documentclass[% |
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aip,jcp, |
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amsmath,amssymb, |
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preprint,% |
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% reprint,% |
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%author-year,% |
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%author-numerical,% |
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jcp]{revtex4-1} |
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\usepackage{graphicx}% Include figure files |
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\usepackage{dcolumn}% Align table columns on decimal point |
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%\usepackage{bm}% bold math |
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\usepackage{times} |
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\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions |
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\usepackage{url} |
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\usepackage{rotating} |
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%\usepackage[mathlines]{lineno}% Enable numbering of text and display math |
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%\linenumbers\relax % Commence numbering lines |
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\begin{document} |
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|
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\title[Taylor-shifted and Gradient-shifted electrostatics for multipoles] |
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{Supplemental Material for: Real space alternatives to the Ewald |
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Sum. I. Shifted Electrostatics for Multipoles} |
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|
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University |
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of Notre Dame, Notre Dame, IN 46556} |
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|
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu.} |
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\affiliation{Department of Chemistry and Biochemistry, University |
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of Notre Dame, Notre Dame, IN 46556} |
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|
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\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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|
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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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|
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\maketitle |
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|
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\section{Interaction Energies in body-frame coordiantes} |
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% |
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% |
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%Energy in body coordinate form --------------------------------------------------------------- |
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% |
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Although they are not as widely used as space-frame coordinates, the |
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body-frame versions may occasionally prove useful. In this section, |
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we list the interaction energies, forces, and torques in terms of the |
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body coordinates for both the Taylor-Shifted and Gradient-Shifted |
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approximations. The radial functions ($v_{ij}(r)$ and $w_{\alpha}(r)$) |
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are given in the Tables I and II in the paper. These functions depend |
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on the choice of electrostatic kernel as well as the approximation |
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method being utilized. Again, all energy, force, and torque equations |
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have an an implied factor of $1/4\pi \epsilon_0$: |
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% |
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% u ca cb |
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% |
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\begin{align} |
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U_{C_{\bf a}C_{\bf b}}(r)=& |
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C_{\bf a} C_{\bf b} v_{01}(r) |
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\\ |
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% |
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% u ca db |
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% |
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U_{C_{\bf a}D_{\bf b}}(r)=& |
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C_{\bf a} |
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\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
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\\ |
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% |
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% u ca qb |
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% |
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U_{C_{\bf a}Q_{\bf b}}(r)=& |
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C_{\bf a }\text{Tr}Q_{\bf b} |
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v_{21}(r) +C_{\bf a} |
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\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) |
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v_{22}(r) \\ |
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% |
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% u da cb |
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% |
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U_{D_{\bf a}C_{\bf b}}(r)=& |
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-C_{\bf b} |
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\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
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\\ |
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% |
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% u da db |
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% |
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% 1 |
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U_{D_{\bf a}D_{\bf b}}(r)=& |
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- \sum_{mn} D_{\mathbf {a}m} |
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(\hat{a}_m \cdot \hat{b}_n) |
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D_{\mathbf{b}n} v_{21}(r) \nonumber \\ |
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% 2 |
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&- |
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\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
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\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} |
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v_{22}(r) |
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\\ |
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% |
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% u da qb |
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% |
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% 1 |
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U_{D_{\bf a}Q_{\bf b}}(r)=& |
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-\left( |
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\text{Tr}Q_{\mathbf{b}} |
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\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
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+2\sum_{lmn}D_{\mathbf{a}l} |
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(\hat{a}_l \cdot \hat{b}_m) |
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Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
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\right) v_{31}(r) \nonumber \\ |
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% 2 |
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&- |
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\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
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\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
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Q_{{\mathbf b}mn} |
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(\hat{b}_n \cdot \hat{r}) v_{32}(r) |
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\\ |
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% |
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% u qa cb |
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% |
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U_{Q_{\bf a}C_{\bf b}}(r)=& |
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C_{\bf b }\text{Tr}Q_{\bf a} v_{21}(r) |
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+C_{\bf b} |
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\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) v_{22}(r) |
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\\ |
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% |
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% u qa db |
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% |
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%1 |
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U_{Q_{\bf a}D_{\bf b}}(r)=& |
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\left( |
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\text{Tr}Q_{\mathbf{a}} |
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\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
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+2\sum_{lmn}D_{\mathbf{b}l} |
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(\hat{b}_l \cdot \hat{a}_m) |
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Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
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\right) v_{31}(r) \nonumber \\ |
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% 2 |
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&+ |
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\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
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\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
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Q_{{\mathbf a}mn} |
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(\hat{a}_n \cdot \hat{r}) v_{32}(r) |
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\end{align} |
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|
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\begin{align} |
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% |
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% u qa qb |
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% |
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%1 |
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U_{Q_{\bf a}Q_{\bf b}}(r)=& |
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\Bigl[ |
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\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
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+2\sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
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Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
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(\hat{a}_m \cdot \hat{b}_n) \Bigr] |
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v_{41}(r) \nonumber \\ |
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% 2 |
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&+ |
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\Bigl[ \text{Tr}Q_{\mathbf{a}} |
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\sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
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Q_{{\mathbf b}lm} |
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(\hat{b}_m \cdot \hat{r}) |
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+\text{Tr}Q_{\mathbf{b}} |
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\sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
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Q_{{\mathbf a}lm} |
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(\hat{a}_m \cdot \hat{r}) \nonumber \\ |
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% 3 |
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&+4 \sum_{lmnp} |
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(\hat{r} \cdot \hat{a}_l ) |
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Q_{{\mathbf a}lm} |
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(\hat{a}_m \cdot \hat{b}_n) |
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Q_{{\mathbf b}np} |
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(\hat{b}_p \cdot \hat{r}) |
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\Bigr] v_{42}(r) \nonumber \\ |
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% 4 |
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&+ |
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\sum_{lm} (\hat{r} \cdot \hat{a}_l) |
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Q_{{\mathbf a}lm} |
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(\hat{a}_m \cdot \hat{r}) |
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\sum_{np} (\hat{r} \cdot \hat{b}_n) |
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Q_{{\mathbf b}np} |
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(\hat{b}_p \cdot \hat{r}) v_{43}(r). |
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\end{align} |
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|
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|
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% BODY coordinates force equations -------------------------------------------- |
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% |
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% |
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Here are the force equations written in terms of body coordinates. |
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% |
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% f ca cb |
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% |
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\begin{align} |
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\mathbf{F}_{{\bf a}C_{\bf a}C_{\bf b}} =& |
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C_{\bf a} C_{\bf b} w_a(r) \hat{r} |
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\\ |
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% |
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% f ca db |
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% |
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\mathbf{F}_{{\bf a}C_{\bf a}D_{\bf b}} =& |
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C_{\bf a} |
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\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} w_b(r) \hat{r} |
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+C_{\bf a} |
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\sum_n D_{\mathbf{b}n} \hat{b}_n w_c(r) |
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\\ |
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% |
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% f ca qb |
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% |
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% 1 |
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\mathbf{F}_{{\bf a}C_{\bf a}Q_{\bf b}} =& |
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C_{\bf a }\text{Tr}Q_{\bf b} w_d(r) \hat{r} |
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+ 2C_{\bf a } \sum_l \hat{b}_l Q_{{\mathbf b}ln} (\hat{b}_n \cdot |
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\hat{r}) w_e(r) \nonumber \\ |
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% 2 |
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&+C_{\bf a} |
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\sum_{mn} (\hat{r} \cdot \hat{b}_m) Q_{{\mathbf b}mn} (\hat{b}_n \cdot \hat{r}) w_f(r) \hat{r} \\ |
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% |
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% f da cb |
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% |
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\mathbf{F}_{{\bf a}D_{\bf a}C_{\bf b}} =& |
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-C_{\bf{b}} |
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\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} w_b(r) \hat{r} |
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-C_{\bf{b}} |
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\sum_n D_{\mathbf{a}n} \hat{a}_n w_c(r) |
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\\ |
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% |
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% f da db |
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% |
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% 1 |
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\mathbf{F}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
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-\sum_{mn} D_{\mathbf {a}m} |
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(\hat{a}_m \cdot \hat{b}_n) |
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D_{\mathbf{b}n} w_d(r) \hat{r} |
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-\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
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\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} w_f(r) \hat{r} |
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\nonumber \\ |
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% 2 |
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& + |
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\Bigl[ \sum_m D_{\mathbf {a}m} |
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\hat{a}_m \sum_n D_{\mathbf{b}n} |
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(\hat{b}_n \cdot \hat{r}) |
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+ \sum_m D_{\mathbf {b}m} |
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\hat{b}_m \sum_n D_{\mathbf{a}n} |
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(\hat{a}_n \cdot \hat{r}) \Bigr] w_e(r) \\ |
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% |
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% f da qb |
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% |
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% 1 |
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\mathbf{F}_{{\bf a}D_{\bf a}Q_{\bf b}} =& |
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- \Bigl[ |
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\text{Tr}Q_{\mathbf{b}} |
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\sum_l D_{\mathbf{a}l} \hat{a}_l |
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+2\sum_{lmn} D_{\mathbf{a}l} |
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(\hat{a}_l \cdot \hat{b}_m) |
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Q_{\mathbf{b}mn} \hat{b}_n \Bigr] w_g(r) \nonumber \\ |
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% 3 |
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& - \Bigl[ |
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\text{Tr}Q_{\mathbf{b}} |
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\sum_n (\hat{r} \cdot \hat{a}_n) D_{\mathbf{a}n} |
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+2\sum_{lmn}D_{\mathbf{a}l} |
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(\hat{a}_l \cdot \hat{b}_m) |
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Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} |
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\nonumber \\ |
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% 4 |
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&+ |
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\Bigl[\sum_l D_{\mathbf{a}l} \hat{a}_l |
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\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
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Q_{{\mathbf b}mn} |
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(\hat{b}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{a}_l) |
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D_{\mathbf{a}l} |
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\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
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Q_{{\mathbf b}mn} \hat{b}_n \Bigr] w_i(r) \nonumber \\ |
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% 6 |
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& - |
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\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
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\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
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Q_{{\mathbf b}mn} |
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(\hat{b}_n \cdot \hat{r}) w_j(r) \hat{r} |
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\\ |
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% |
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% force qa cb |
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% |
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% 1 |
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\mathbf{F}_{{\bf a}Q_{\bf a}C_{\bf b}} =& |
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C_{\bf b }\text{Tr}Q_{\bf a} \hat{r} w_d(r) |
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+ 2C_{\bf b } \sum_l \hat{a}_l Q_{{\mathbf |
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a}ln} (\hat{a}_n \cdot \hat{r}) w_e(r) \nonumber \\ |
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% 2 |
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& +C_{\bf b} |
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\sum_{mn} (\hat{r} \cdot \hat{a}_m) Q_{{\mathbf a}mn} (\hat{a}_n \cdot \hat{r}) w_f(r) \hat{r} |
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\end{align} |
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|
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\begin{align} |
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% |
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% f qa db |
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% |
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% 1 |
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\mathbf{F}_{{\bf a}Q_{\bf a}D_{\bf b}} =& |
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\Bigl[ |
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\text{Tr}Q_{\mathbf{a}} |
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\sum_l D_{\mathbf{b}l} \hat{b}_l |
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+2\sum_{lmn} D_{\mathbf{b}l} |
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(\hat{b}_l \cdot \hat{a}_m) |
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Q_{\mathbf{a}mn} \hat{a}_n \Bigr] |
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w_g(r) \nonumber \\ |
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% 3 |
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& + \Bigl[ |
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\text{Tr}Q_{\mathbf{a}} |
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\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf{b}n} |
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+2\sum_{lmn}D_{\mathbf{b}l} |
| 336 |
(\hat{b}_l \cdot \hat{a}_m) |
| 337 |
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) \Bigr] w_h(r) \hat{r} |
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\nonumber \\ |
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% 4 |
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& + \Bigl[ \sum_l D_{\mathbf{b}l} \hat{b}_l |
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\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
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Q_{{\mathbf a}mn} |
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(\hat{a}_n \cdot \hat{r}) +2 \sum_l (\hat{r} \cdot \hat{b}_l) |
| 344 |
D_{\mathbf{b}l} |
| 345 |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 346 |
Q_{{\mathbf a}mn} \hat{a}_n \Bigr] w_i(r) \nonumber \\ |
| 347 |
% 6 |
| 348 |
& +\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
| 349 |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 350 |
Q_{{\mathbf a}mn} |
| 351 |
(\hat{a}_n \cdot \hat{r}) w_j(r) \hat{r} |
| 352 |
\\ |
| 353 |
% |
| 354 |
% f qa qb |
| 355 |
% |
| 356 |
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
| 357 |
\Bigl[ |
| 358 |
\text{Tr}Q_{\mathbf{a}} \text{Tr}Q_{\mathbf{b}} |
| 359 |
+ 2 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
| 360 |
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 361 |
(\hat{a}_m \cdot \hat{b}_n) \Bigr] w_k(r) \hat{r} \nonumber \\ |
| 362 |
&+ \Bigl[ |
| 363 |
2\text{Tr}Q_{\mathbf{b}} \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
| 364 |
+ 2\text{Tr}Q_{\mathbf{a}} \sum_{lm} (\hat{r} \cdot \hat{b}_l) |
| 365 |
Q_{\mathbf{b}lm} \hat{b}_m \nonumber \\ |
| 366 |
&+ 4\sum_{lmnp} \hat{a}_l Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) |
| 367 |
+ 4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_p |
| 368 |
\Bigr] w_n(r) \nonumber \\ |
| 369 |
&+ |
| 370 |
\Bigl[ \text{Tr}Q_{\mathbf{a}} |
| 371 |
\sum_{lm} (\hat{r} \cdot \hat{b}_l) Q_{\mathbf{b}lm} (\hat{b}_m \cdot \hat{r}) |
| 372 |
+ \text{Tr}Q_{\mathbf{b}} |
| 373 |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) \\ |
| 374 |
&+4\sum_{lmnp} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{b}_n) |
| 375 |
Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) \Bigr] w_l(r) \hat{r} \nonumber \\ |
| 376 |
% |
| 377 |
&+ \Bigl[ |
| 378 |
2\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} \hat{a}_m |
| 379 |
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_n \cdot |
| 380 |
\hat{r}) \nonumber \\ |
| 381 |
&+2 \sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
| 382 |
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} \hat{b}_n \Bigr] |
| 383 |
w_o(r) \hat{r} \nonumber \\ |
| 384 |
& + |
| 385 |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{\mathbf{a}lm} (\hat{a}_m \cdot \hat{r}) |
| 386 |
\sum_{np} (\hat{r} \cdot \hat{b}_n) Q_{\mathbf{b}np} (\hat{b}_p \cdot \hat{r}) w_m(r) \hat{r} |
| 387 |
\end{align} |
| 388 |
% |
| 389 |
Here we list the form of the non-zero damped shifted multipole torques showing |
| 390 |
explicitly dependences on body axes: |
| 391 |
% |
| 392 |
% t ca db |
| 393 |
% |
| 394 |
\begin{align} |
| 395 |
\mathbf{\tau}_{{\bf b}C_{\bf a}D_{\bf b}} =& |
| 396 |
C_{\bf a} |
| 397 |
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} \, v_{11}(r) |
| 398 |
\\ |
| 399 |
% |
| 400 |
% t ca qb |
| 401 |
% |
| 402 |
\mathbf{\tau}_{{\bf b}C_{\bf a}Q_{\bf b}} =& |
| 403 |
2C_{\bf a} |
| 404 |
\sum_{lm} (\hat{r} \times \hat{b}_l) Q_{{\mathbf b}lm} (\hat{b}_m |
| 405 |
\cdot \hat{r}) v_{22}(r) |
| 406 |
\\ |
| 407 |
% |
| 408 |
% t da cb |
| 409 |
% |
| 410 |
\mathbf{\tau}_{{\bf a}D_{\bf a}C_{\bf b}} =& |
| 411 |
-C_{\bf b} |
| 412 |
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} \, v_{11}(r) |
| 413 |
\\ |
| 414 |
% |
| 415 |
% |
| 416 |
% ta da db |
| 417 |
% |
| 418 |
% 1 |
| 419 |
\mathbf{\tau}_{{\bf a}D_{\bf a}D_{\bf b}} =& |
| 420 |
\sum_{mn} D_{\mathbf {a}m} |
| 421 |
(\hat{a}_m \times \hat{b}_n) |
| 422 |
D_{\mathbf{b}n} v_{21}(r) \nonumber \\ |
| 423 |
% 2 |
| 424 |
&- |
| 425 |
\sum_m (\hat{r} \times \hat{a}_m) D_{\mathbf {a}m} |
| 426 |
\sum_n (\hat{r} \cdot \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
| 427 |
\\ |
| 428 |
% |
| 429 |
% tb da db |
| 430 |
% |
| 431 |
% 1 |
| 432 |
\mathbf{\tau}_{{\bf b}D_{\bf a}D_{\bf b}} =& |
| 433 |
- \sum_{mn} D_{\mathbf {a}m} |
| 434 |
(\hat{a}_m \times \hat{b}_n) |
| 435 |
D_{\mathbf{b}n} v_{21}(r) \nonumber \\ |
| 436 |
% 2 |
| 437 |
&+ |
| 438 |
\sum_m (\hat{r} \cdot \hat{a}_m) D_{\mathbf {a}m} |
| 439 |
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf {b}n} v_{22}(r) |
| 440 |
\\ |
| 441 |
% ta da qb |
| 442 |
% |
| 443 |
% 1 |
| 444 |
\mathbf{\tau}_{{\bf a}D_{\bf a}Q_{\bf b}} =& |
| 445 |
\left( |
| 446 |
-\text{Tr}Q_{\mathbf{b}} |
| 447 |
\sum_n (\hat{r} \times \hat{a}_n) D_{\mathbf{a}n} |
| 448 |
+2\sum_{lmn}D_{\mathbf{a}l} |
| 449 |
(\hat{a}_l \times \hat{b}_m) |
| 450 |
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
| 451 |
\right) v_{31}(r) \nonumber \\ |
| 452 |
% 2 |
| 453 |
&- |
| 454 |
\sum_l (\hat{r} \times \hat{a}_l) D_{\mathbf{a}l} |
| 455 |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 456 |
Q_{{\mathbf b}mn} |
| 457 |
(\hat{b}_n \cdot \hat{r}) v_{32}(r) \\ |
| 458 |
% |
| 459 |
% tb da qb |
| 460 |
% |
| 461 |
% 1 |
| 462 |
\mathbf{\tau}_{{\bf b}D_{\bf a}Q_{\bf b}} =& |
| 463 |
\left( |
| 464 |
-2\sum_{lmn}D_{\mathbf{a}l} |
| 465 |
(\hat{a}_l \cdot \hat{b}_m) |
| 466 |
Q_{\mathbf{b}mn} (\hat{r} \times \hat{b}_n) |
| 467 |
-2\sum_{lmn}D_{\mathbf{a}l} |
| 468 |
(\hat{a}_l \times \hat{b}_m) |
| 469 |
Q_{\mathbf{b}mn} (\hat{b}_n \cdot \hat{r}) |
| 470 |
\right) v_{31}(r) \nonumber \\ |
| 471 |
% 2 |
| 472 |
&-2 |
| 473 |
\sum_l (\hat{r} \cdot \hat{a}_l) D_{\mathbf{a}l} |
| 474 |
\sum_{mn} (\hat{r} \cdot \hat{b}_m) |
| 475 |
Q_{{\mathbf b}mn} |
| 476 |
(\hat{r}\times \hat{b}_n) v_{32}(r) |
| 477 |
\\ |
| 478 |
% |
| 479 |
% ta qa cb |
| 480 |
% |
| 481 |
\mathbf{\tau}_{{\bf a}Q_{\bf a}C_{\bf b}} =& |
| 482 |
2C_{\bf a} |
| 483 |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) Q_{{\mathbf a}lm} (\hat{r} \times \hat{a}_m) v_{22}(r) |
| 484 |
\\ |
| 485 |
% |
| 486 |
% ta qa db |
| 487 |
% |
| 488 |
% 1 |
| 489 |
\mathbf{\tau}_{{\bf a}Q_{\bf a}D_{\bf b}} = & |
| 490 |
\left( |
| 491 |
2\sum_{lmn}D_{\mathbf{b}l} |
| 492 |
(\hat{b}_l \cdot \hat{a}_m) |
| 493 |
Q_{\mathbf{a}mn} (\hat{r} \times \hat{a}_n) |
| 494 |
+2\sum_{lmn}D_{\mathbf{b}l} |
| 495 |
(\hat{a}_l \times \hat{b}_m) |
| 496 |
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
| 497 |
\right) v_{31}(r) \nonumber \\ |
| 498 |
% 2 |
| 499 |
&+2 |
| 500 |
\sum_l (\hat{r} \cdot \hat{b}_l) D_{\mathbf{b}l} |
| 501 |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 502 |
Q_{{\mathbf a}mn} |
| 503 |
(\hat{r}\times \hat{a}_n) v_{32}(r) |
| 504 |
\\ |
| 505 |
% |
| 506 |
% tb qa db |
| 507 |
% |
| 508 |
% 1 |
| 509 |
\mathbf{\tau}_{{\bf b}Q_{\bf a}D_{\bf b}} =& |
| 510 |
\left( |
| 511 |
\text{Tr}Q_{\mathbf{a}} |
| 512 |
\sum_n (\hat{r} \times \hat{b}_n) D_{\mathbf{b}n} |
| 513 |
+2\sum_{lmn}D_{\mathbf{b}l} |
| 514 |
(\hat{a}_l \times \hat{b}_m) |
| 515 |
Q_{\mathbf{a}mn} (\hat{a}_n \cdot \hat{r}) |
| 516 |
\right) v_{31}(r) \nonumber \\ |
| 517 |
% 2 |
| 518 |
& \sum_l (\hat{r} \times \hat{b}_l) D_{\mathbf{b}l} |
| 519 |
\sum_{mn} (\hat{r} \cdot \hat{a}_m) |
| 520 |
Q_{{\mathbf a}mn} |
| 521 |
(\hat{a}_n \cdot \hat{r}) v_{32}(r) |
| 522 |
\end{align} |
| 523 |
|
| 524 |
% |
| 525 |
% ta qa qb |
| 526 |
% |
| 527 |
\begin{align} |
| 528 |
% 1 |
| 529 |
\mathbf{\tau}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
| 530 |
-4 |
| 531 |
\sum_{lmnp} (\hat{a}_l \times \hat{b}_p) |
| 532 |
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 533 |
(\hat{a}_m \cdot \hat{b}_n) v_{41}(r) \nonumber \\ |
| 534 |
% 2 |
| 535 |
&+ |
| 536 |
\Bigl[ |
| 537 |
2\text{Tr}Q_{\mathbf{b}} |
| 538 |
\sum_{lm} (\hat{r} \cdot \hat{a}_l ) |
| 539 |
Q_{{\mathbf a}lm} |
| 540 |
(\hat{r} \times \hat{a}_m) |
| 541 |
+4 \sum_{lmnp} |
| 542 |
(\hat{r} \times \hat{a}_l ) |
| 543 |
Q_{{\mathbf a}lm} |
| 544 |
(\hat{a}_m \cdot \hat{b}_n) |
| 545 |
Q_{{\mathbf b}np} |
| 546 |
(\hat{b}_p \cdot \hat{r}) \nonumber \\ |
| 547 |
% 3 |
| 548 |
&-4 \sum_{lmnp} |
| 549 |
(\hat{r} \cdot \hat{a}_l ) |
| 550 |
Q_{{\mathbf a}lm} |
| 551 |
(\hat{a}_m \times \hat{b}_n) |
| 552 |
Q_{{\mathbf b}np} |
| 553 |
(\hat{b}_p \cdot \hat{r}) |
| 554 |
\Bigr] v_{42}(r) \nonumber \\ |
| 555 |
% 4 |
| 556 |
&+2 |
| 557 |
\sum_{lm} (\hat{r} \times \hat{a}_l) |
| 558 |
Q_{{\mathbf a}lm} |
| 559 |
(\hat{a}_m \cdot \hat{r}) |
| 560 |
\sum_{np} (\hat{r} \cdot \hat{b}_n) |
| 561 |
Q_{{\mathbf b}np} |
| 562 |
(\hat{b}_p \cdot \hat{r}) v_{43}(r)\\ |
| 563 |
% |
| 564 |
% tb qa qb |
| 565 |
% |
| 566 |
% 1 |
| 567 |
\mathbf{\tau}_{{\bf b}Q_{\bf a}Q_{\bf b}} =& |
| 568 |
4 \sum_{lmnp} (\hat{a}_l \cdot \hat{b}_p) |
| 569 |
Q_{\mathbf{a}lm} Q_{\mathbf{b}np} |
| 570 |
(\hat{a}_m \times \hat{b}_n) v_{41}(r) \nonumber \\ |
| 571 |
% 2 |
| 572 |
&+ |
| 573 |
\Bigl[ |
| 574 |
2\text{Tr}Q_{\mathbf{a}} |
| 575 |
\sum_{lm} (\hat{r} \cdot \hat{b}_l ) |
| 576 |
Q_{{\mathbf b}lm} |
| 577 |
(\hat{r} \times \hat{b}_m) |
| 578 |
+4 \sum_{lmnp} |
| 579 |
(\hat{r} \cdot \hat{a}_l ) |
| 580 |
Q_{{\mathbf a}lm} |
| 581 |
(\hat{a}_m \cdot \hat{b}_n) |
| 582 |
Q_{{\mathbf b}np} |
| 583 |
(\hat{r} \times \hat{b}_p) \nonumber \\ |
| 584 |
% 3 |
| 585 |
&+4 \sum_{lmnp} |
| 586 |
(\hat{r} \cdot \hat{a}_l ) |
| 587 |
Q_{{\mathbf a}lm} |
| 588 |
(\hat{a}_m \times \hat{b}_n) |
| 589 |
Q_{{\mathbf b}np} |
| 590 |
(\hat{b}_p \cdot \hat{r}) |
| 591 |
\Bigr] v_{42}(r) \nonumber \\ |
| 592 |
% 4 |
| 593 |
&+2 |
| 594 |
\sum_{lm} (\hat{r} \cdot \hat{a}_l) |
| 595 |
Q_{{\mathbf a}lm} |
| 596 |
(\hat{a}_m \cdot \hat{r}) |
| 597 |
\sum_{np} (\hat{r} \times \hat{b}_n) |
| 598 |
Q_{{\mathbf b}np} |
| 599 |
(\hat{b}_p \cdot \hat{r}) v_{43}(r). |
| 600 |
\end{align} |
| 601 |
|
| 602 |
|
| 603 |
% \begin{table*} |
| 604 |
% \caption{\label{tab:tableFORCE2}Radial functions used in the force equations.} |
| 605 |
% \begin{ruledtabular} |
| 606 |
% \begin{tabular}{|l|l|l|} |
| 607 |
% Generic&Taylor-shifted Force&Gradient-shifted Force |
| 608 |
% \\ \hline |
| 609 |
% % |
| 610 |
% % |
| 611 |
% % |
| 612 |
% $w_a(r)$& |
| 613 |
% $g_0(r)$& |
| 614 |
% $g(r)-g(r_c)$ \\ |
| 615 |
% % |
| 616 |
% % |
| 617 |
% $w_b(r)$ & |
| 618 |
% $\left( -\frac{g_1(r)}{r}+h_1(r) \right)$ & |
| 619 |
% $h(r)- h(r_c) - \frac{v_{11}(r)}{r} $ \\ |
| 620 |
% % |
| 621 |
% $w_c(r)$ & |
| 622 |
% $\frac{g_1(r)}{r} $ & |
| 623 |
% $\frac{v_{11}(r)}{r}$ \\ |
| 624 |
% % |
| 625 |
% % |
| 626 |
% $w_d(r)$& |
| 627 |
% $\left( -\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right) $ & |
| 628 |
% $\left( -\frac{g(r)}{r^2} + \frac{h(r)}{r} \right) |
| 629 |
% -\left( -\frac{g(r_c)}{r_c^2} + \frac{h(r_c)}{r_c} \right) $\\ |
| 630 |
% % |
| 631 |
% $w_e(r)$ & |
| 632 |
% $\left(-\frac{g_2(r)}{r^2} + \frac{h_2(r)}{r} \right)$ & |
| 633 |
% $\frac{v_{22}(r)}{r}$ \\ |
| 634 |
% % |
| 635 |
% % |
| 636 |
% $w_f(r)$& |
| 637 |
% $\left( \frac{3g_2(r)}{r^2}-\frac{3h_2(r)}{r}+s_2(r) \right)$ & |
| 638 |
% $\left( \frac{g(r)}{r^2}-\frac{h(r)}{r}+s(r) \right) - $ \\ |
| 639 |
% &&$\left( \frac{g(r_c)}{r_c^2}-\frac{h(r_c)}{r_c}+s(r_c) \right)-\frac{2v_{22}(r)}{r}$\\ |
| 640 |
% % |
| 641 |
% $w_g(r)$& $ \left( -\frac{g_3(r)}{r^3}+\frac{h_3(r)}{r^2} \right)$& |
| 642 |
% $\frac{v_{31}(r)}{r}$\\ |
| 643 |
% % |
| 644 |
% $w_h(r)$ & |
| 645 |
% $\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
| 646 |
% $\left(\frac{2g(r)}{r^3} -\frac{2h(r)}{r^2} +\frac{s(r)}{r} \right) - $\\ |
| 647 |
% &&$\left(\frac{2g(r_c)}{r_c^3} -\frac{2h(r_c)}{r_c^2} +\frac{s(r_c)}{r_c} \right) $ \\ |
| 648 |
% &&$-\frac{v_{31}(r)}{r}$\\ |
| 649 |
% % 2 |
| 650 |
% $w_i(r)$ & |
| 651 |
% $\left(\frac{3g_3(r)}{r^3} -\frac{3h_3(r)}{r^2} +\frac{s_3(r)}{r} \right) $ & |
| 652 |
% $\frac{v_{32}(r)}{r}$ \\ |
| 653 |
% % |
| 654 |
% $w_j(r)$ & |
| 655 |
% $\left(\frac{-15g_3(r)}{r^3} + \frac{15h_3(r)}{r^2} - \frac{6s_3(r)}{r} + t_3(r) \right) $ & |
| 656 |
% $\left(\frac{-6g(r)}{r^3} +\frac{6h(r)}{r^2} -\frac{3s(r)}{r} +t(r) \right) $ \\ |
| 657 |
% &&$\left(\frac{-6g(_cr)}{r_c^3} +\frac{6h(r_c)}{r_c^2} -\frac{3s(r_c)}{r_c} +t(r_c) \right) -\frac{3v_{32}}{r}$ \\ |
| 658 |
% % |
| 659 |
% $w_k(r)$ & |
| 660 |
% $\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
| 661 |
% $\left(\frac{3g(r)}{r^4} -\frac{3h(r)}{r^3} +\frac{s(r)}{r^2} \right)$ \\ |
| 662 |
% &&$\left(\frac{3g(r_c)}{r_c^4} -\frac{3h(r_c)}{r_c^3} +\frac{s(r_c)}{r_c^2} \right)$ \\ |
| 663 |
% % |
| 664 |
% $w_l(r)$ & |
| 665 |
% $\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
| 666 |
% $\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right)$ \\ |
| 667 |
% &&$\left(-\frac{9g(r)}{r^4} +\frac{9h(r)}{r^3} -\frac{4s(r)}{r^2} +\frac{t(r)}{r} \right) |
| 668 |
% -\frac{2v_{42}(r)}{r}$ \\ |
| 669 |
% % |
| 670 |
% $w_m(r)$ & |
| 671 |
% $\left(\frac{105g_4(r)}{r^4} - \frac{105h_4(r)}{r^3} + \frac{45s_4(r)}{r^2} - \frac{10t_4(r)}{r} +u_4(r) \right)$ & |
| 672 |
% $\left(\frac{45g(r)}{r^4} -\frac{45h(r)}{r^3} +\frac{21s(r)}{r^2} -\frac{6t(r)}{r} +u(r) \right)$ \\ |
| 673 |
% &&$\left(\frac{45g(r_c)}{r_c^4} -\frac{45h(r_c)}{r_c^3} |
| 674 |
% +\frac{21s(r_c)}{r_c^2} -\frac{6t(r_c)}{r_c} +u(r_c) \right) $ \\ |
| 675 |
% &&$-\frac{4v_{43}(r)}{r}$ \\ |
| 676 |
% % |
| 677 |
% $w_n(r)$ & |
| 678 |
% $\left(\frac{3g_4(r)}{r^4} -\frac{3h_4(r)}{r^3} +\frac{s_4(r)}{r^2} \right)$ & |
| 679 |
% $\frac{v_{42}(r)}{r}$ \\ |
| 680 |
% % |
| 681 |
% $w_o(r)$ & |
| 682 |
% $\left(-\frac{15g_4(r)}{r^4} +\frac{15h_4(r)}{r^3} -\frac{6s_4(r)}{r^2} +\frac{t_4(r)}{r} \right)$ & |
| 683 |
% $\frac{v_{43}(r)}{r}$ \\ |
| 684 |
% % |
| 685 |
% \end{tabular} |
| 686 |
% \end{ruledtabular} |
| 687 |
% \end{table*} |
| 688 |
% |
| 689 |
% \newpage |
| 690 |
% |
| 691 |
% \bibliography{multipole} |
| 692 |
% |
| 693 |
|
| 694 |
To test the gradient-shifted force (GSF) and Taylor-shifted force |
| 695 |
(TSF) methods against known energies for multipolar crystals, we |
| 696 |
repeated the Luttinger \& Tisza series summations and have obtained |
| 697 |
the energy constants (converged to one part in $10^9$) in table |
| 698 |
\ref{tab:LT}. |
| 699 |
|
| 700 |
\section{} |
| 701 |
\begin{table*}[h] |
| 702 |
\centering{ |
| 703 |
\caption{Luttinger \& Tisza arrays and their associated |
| 704 |
energy constants. Type ``A'' arrays have nearest neighbor strings of |
| 705 |
antiparallel dipoles. Type ``B'' arrays have nearest neighbor |
| 706 |
strings of antiparallel dipoles if the dipoles are contained in a |
| 707 |
plane perpendicular to the dipole direction that passes through |
| 708 |
the dipole.} |
| 709 |
} |
| 710 |
\label{tab:LT} |
| 711 |
\begin{ruledtabular} |
| 712 |
\begin{tabular}{cccc} |
| 713 |
Array Type & Lattice & Dipole Direction & Energy constants \\ \hline |
| 714 |
A & SC & 001 & -2.676788684 \\ |
| 715 |
A & BCC & 001 & 0 \\ |
| 716 |
A & BCC & 111 & -1.770078733 \\ |
| 717 |
A & FCC & 001 & 2.166932835 \\ |
| 718 |
A & FCC & 011 & -1.083466417 \\ |
| 719 |
B & SC & 001 & -2.676788684 \\ |
| 720 |
B & BCC & 001 & -1.338394342 \\ |
| 721 |
B & BCC & 111 & -1.770078733 \\ |
| 722 |
B & FCC & 001 & -1.083466417 \\ |
| 723 |
B & FCC & 011 & -1.807573634 \\ |
| 724 |
-- & BCC & minimum & -1.985920929 \\ |
| 725 |
\end{tabular} |
| 726 |
\end{ruledtabular} |
| 727 |
\end{table*} |
| 728 |
|
| 729 |
We have also tested agains the energy constants for Quadrupolar |
| 730 |
arrays. Nagai and Nakamura computed the energies of selected |
| 731 |
quadrupole arrays based on extensions to the Luttinger and Tisza |
| 732 |
approach. These energy constants are given in table \ref{tab:NNQ}. |
| 733 |
|
| 734 |
\begin{table*} |
| 735 |
\centering{ |
| 736 |
\caption{Nagai and Nakamura Quadrupolar arrays. Note that these |
| 737 |
take into account the factor of two corrections in |
| 738 |
Ref. \onlinecite{Nagai01091963}}} |
| 739 |
\label{tab:NNQ} |
| 740 |
\begin{ruledtabular} |
| 741 |
\begin{tabular}{ccc} |
| 742 |
Lattice & Quadrupole Direction & Energy constants \\ \hline |
| 743 |
SC & 111 & -16.6 \\ |
| 744 |
BCC & 011 & -43.4 \\ |
| 745 |
FCC & 111 & -161 |
| 746 |
\end{tabular} |
| 747 |
\end{ruledtabular} |
| 748 |
\end{table*} |
| 749 |
|
| 750 |
\newpage |
| 751 |
\bibliography{multipole} |
| 752 |
\end{document} |
| 753 |
% |
| 754 |
% ****** End of file multipole.tex ****** |
