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Revision 4163 by mlamichh, Thu May 29 19:08:06 2014 UTC vs.
Revision 4172 by gezelter, Thu Jun 5 15:56:43 2014 UTC

# Line 23 | Line 23 | preprint,%
23   aip,jcp,
24   amsmath,amssymb,
25   preprint,%
26 < %reprint,%
26 > % reprint,%
27   %author-year,%
28   %author-numerical,%
29   jcp]{revtex4-1}
# Line 35 | Line 35 | jcp]{revtex4-1}
35   \usepackage[version=3]{mhchem}  % this is a great package for formatting chemical reactions
36   \usepackage{url}
37   \usepackage{rotating}
38 +
39   %\usepackage[mathlines]{lineno}% Enable numbering of text and display math
40   %\linenumbers\relax % Commence numbering lines
41  
# Line 220 | Line 221 | forces and torques go smoothly to zero at the cutoff d
221   sphere.  There are also significant modifications made to make the
222   forces and torques go smoothly to zero at the cutoff distance.
223  
224 + \begin{figure}
225 + \includegraphics[width=3in]{SM}
226 + \caption{Reversed multipoles are projected onto the surface of the
227 +  cutoff sphere. The forces, torques, and potential are then smoothly
228 +  shifted to zero as the sites leave the cutoff region.}
229 + \label{fig:shiftedMultipoles}
230 + \end{figure}
231  
232   As in the point-charge approach, there is an additional contribution
233   from self-neutralization of site $i$.  The self term for multipoles is
# Line 1410 | Line 1418 | $10^9$) are given in the supplemental information.
1418   constant), $N$ is the number of dipoles per unit volume, and $\mu$ is
1419   the strength of the dipole. Energy constants (converged to 1 part in
1420   $10^9$) are given in the supplemental information.
1413 – \begin{figure}[!htbp]
1421  
1422 < \includegraphics[width=3.5in]{Dipoles_rcutConvergence-crop.pdf}
1423 <
1424 < \includegraphics[width=3.5in]{Quadrupoles_rcutConvergence-crop.pdf}
1425 < \caption{Convergence to the analytic energy constants as a function of
1426 <  cutoff radius (normalized by the lattice constant) for the different
1427 <  real-space methods for (a) dipolar and (b) quadrupolar crystals.The energy constants for hard, SP, GSF, TSF and analytic methods are represented by black sold-circle, red solid-square,green solid-diamond and grey dashed line respectively.
1428 < The left panel shows results for the undamped kernel ($1/r$), while the damped
1429 <  error function kernel, $B_0(r)$ was used in the right panel. }
1430 < \label{fig:Dipoles_rcutCovergence-crop.pdf}
1431 < \label{fig:QuadrupolesrcutCovergence-crop.pdf}
1422 > \begin{figure}
1423 > \includegraphics[width=\linewidth]{Dipoles_rCutNew.pdf}
1424 > \caption{Convergence of the lattice energy constants as a function of
1425 >  cutoff radius (normalized by the lattice constant, $a$) for the new
1426 >  real-space methods.  Three dipolar crystal structures were sampled,
1427 >  and the analytic energy constants for the three lattices are
1428 >  indicated with grey dashed lines.  The left panel shows results for
1429 >  the undamped kernel ($1/r$), while the damped error function kernel,
1430 >  $B_0(r)$ was used in the right panel.}
1431 > \label{fig:Dipoles_rCut}
1432   \end{figure}
1433 +
1434 + \begin{figure}
1435 + \includegraphics[width=\linewidth]{Dipoles_alphaNew.pdf}
1436 + \caption{Convergence to the lattice energy constants as a function of
1437 +  the reduced damping parameter ($\alpha^* = \alpha a$) for the
1438 +  different real-space methods in the same three dipolar crystals in
1439 +  Figure \ref{fig:Dipoles_rCut}.  The left panel shows results for a
1440 +  relatively small cutoff radius ($r_c = 4.5 a$) while a larger cutoff
1441 +  radius ($r_c = 6 a$) was used in the right panel. }
1442 + \label{fig:Dipoles_alpha}
1443 + \end{figure}
1444 +
1445   For the purposes of testing the energy expressions and the
1446   self-neutralization schemes, the primary quantity of interest is the
1447   analytic energy constant for the perfect arrays.  Convergence to these
1448   constants are shown as a function of both the cutoff radius, $r_c$,
1449 < and the damping parameter, $\alpha$ in Figs.
1450 < \ref{fig:energyConstVsCutoff} and XXX. We have simultaneously tested a
1451 < hard cutoff (where the kernel is simply truncated at the cutoff
1452 < radius), as well as a shifted potential (SP) form which includes a
1449 > and the damping parameter, $\alpha$ in Figs.  \ref{fig:Dipoles_rCut}
1450 > and \ref{fig:Dipoles_alpha}. We have simultaneously tested a hard
1451 > cutoff (where the kernel is simply truncated at the cutoff radius), as
1452 > well as a shifted potential (SP) form which includes a
1453   potential-shifting and self-interaction term, but does not shift the
1454   forces and torques smoothly at the cutoff radius.  The SP method is
1455   essentially an extension of the original Wolf method for multipoles.
1437 – \begin{figure}[!htbp]
1456  
1439 – \includegraphics[width=3.5in]{Dipoles_alphaConvergence-crop.pdf}
1440 –
1441 – \includegraphics[width=3.5in]{Quadrupoles_alphaConvergence-crop.pdf}
1442 – \caption{Convergence to the analytic energy constants as a function of
1443 –  cutoff damping alpha for the different
1444 –  real-space methods for (a) dipolar and (b) quadrupolar crystals.The energy constants for hard, SP, GSF, TSF, and analytic methods are represented by the black sold-circle, red solid-square, green solid-diamond, and grey dashed line respectively.
1445 – The left panel shows results for the undamped kernel ($1/r$), while the damped
1446 –  error function kernel, $B_0(r)$ was used in the right panel. }
1447 – \label{fig:Dipoles_alphaCovergence-crop.pdf}
1448 – \label{fig:Quadrupoles_alphaCovergence-crop.pdf}
1449 – \end{figure}
1457   The Hard cutoff exhibits oscillations around the analytic energy
1458   constants, and converges to incorrect energies when the complementary
1459   error function damping kernel is used.  The shifted potential (SP) and
# Line 1477 | Line 1484 | the effective moment, $\bar{Q}^2 = \frac{2}{3} \Theta
1484   (For the traceless quadrupole tensor, $\Theta = 3 Q - \text{Tr} Q$,
1485   the effective moment, $\bar{Q}^2 = \frac{2}{3} \Theta : \Theta$.)
1486  
1487 + \begin{figure}
1488 + \includegraphics[width=\linewidth]{Quadrupoles_rcutConvergence-crop.pdf}
1489 + \caption{Convergence to the analytic energy constants as a function of
1490 +  cutoff radius (normalized by the lattice constant) for the different
1491 +  real-space methods for (a) dipolar and (b) quadrupolar crystals.The energy constants for hard, SP, GSF, TSF and analytic methods are represented by black sold-circle, red solid-square,green solid-diamond and grey dashed line respectively.
1492 + The left panel shows results for the undamped kernel ($1/r$), while the damped
1493 +  error function kernel, $B_0(r)$ was used in the right panel. }
1494 + \label{fig:QuadrupolesrcutCovergence-crop.pdf}
1495 + \end{figure}
1496 +
1497 +
1498 + \begin{figure}[!htbp]  
1499 + \includegraphics[width=3.5in]{Quadrupoles_alphaConvergence-crop.pdf}
1500 + \caption{Convergence to the analytic energy constants as a function of
1501 +  cutoff damping alpha for the different
1502 +  real-space methods for (a) dipolar and (b) quadrupolar crystals.The energy constants for hard, SP, GSF, TSF, and analytic methods are represented by the black sold-circle, red solid-square, green solid-diamond, and grey dashed line respectively.
1503 + The left panel shows results for the undamped kernel ($1/r$), while the damped
1504 +  error function kernel, $B_0(r)$ was used in the right panel. }
1505 + \label{fig:Quadrupoles_alphaCovergence-crop.pdf}
1506 + \end{figure}
1507 +
1508 +
1509   \section{Conclusion}
1510   We have presented two efficient real-space methods for computing the
1511   interactions between point multipoles.  These methods have the benefit
# Line 1505 | Line 1534 | for a wide range of chemical environments follows in t
1534   \begin{acknowledgments}
1535    JDG acknowledges helpful discussions with Christopher
1536    Fennell. Support for this project was provided by the National
1537 <  Science Foundation under grant CHE-0848243. Computational time was
1537 >  Science Foundation under grant CHE-1362211. Computational time was
1538    provided by the Center for Research Computing (CRC) at the
1539    University of Notre Dame.
1540   \end{acknowledgments}
# Line 1678 | Line 1707 | correct for GSF electrostatics if the subscript $n$ is
1707   ``Bare Coulomb.''  Equations \ref{eq:b9} to \ref{eq:b13} are still
1708   correct for GSF electrostatics if the subscript $n$ is eliminated.
1709  
1710 + \newpage
1711 +
1712   \bibliography{multipole}
1713  
1714   \end{document}

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