| 82 |
|
arrays, while the Taylor-Shifted Force (TSF) is too severe an |
| 83 |
|
approximation to provide accurate convergence to lattice energies. |
| 84 |
|
Because real-space methods can be made to scale linearly with system |
| 85 |
< |
size, the SP and GSF are attractive options for large Monte Carlo |
| 86 |
< |
and molecular dynamics simulations. |
| 85 |
> |
size, SP and GSF are attractive options for large Monte |
| 86 |
> |
Carlo and molecular dynamics simulations, respectively. |
| 87 |
|
\end{abstract} |
| 88 |
|
|
| 89 |
|
%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy |
| 143 |
|
a different orientation can cause energy discontinuities. |
| 144 |
|
|
| 145 |
|
This paper outlines an extension of the original DSF electrostatic |
| 146 |
< |
kernel to point multipoles. We describe two distinct real-space |
| 147 |
< |
interaction models for higher-order multipoles based on two truncated |
| 148 |
< |
Taylor expansions that are carried out at the cutoff radius. We are |
| 149 |
< |
calling these models {\bf Taylor-shifted} and {\bf Gradient-shifted} |
| 146 |
> |
kernel to point multipoles. We describe three distinct real-space |
| 147 |
> |
interaction models for higher-order multipoles based on truncated |
| 148 |
> |
Taylor expansions that are carried out at the cutoff radius. We are |
| 149 |
> |
calling these models {\bf Taylor-shifted} (TSF), {\bf |
| 150 |
> |
gradient-shifted} (GSF) and {\bf shifted potential} (SP) |
| 151 |
|
electrostatics. Because of differences in the initial assumptions, |
| 152 |
< |
the two methods yield related, but somewhat different expressions for |
| 153 |
< |
energies, forces, and torques. |
| 152 |
> |
the two methods yield related, but distinct expressions for energies, |
| 153 |
> |
forces, and torques. |
| 154 |
|
|
| 155 |
|
In this paper we outline the new methodology and give functional forms |
| 156 |
|
for the energies, forces, and torques up to quadrupole-quadrupole |
| 455 |
|
$Q_{{\bf a}\alpha\beta}Q_{{\bf b}\gamma\delta}$, the derivatives are |
| 456 |
|
$\partial^4/\partial r_\alpha \partial r_\beta \partial |
| 457 |
|
r_\gamma \partial r_\delta$, with implied summation combining the |
| 458 |
< |
space indices. |
| 458 |
> |
space indices. Appendix \ref{radialTSF} contains details on the |
| 459 |
> |
radial functions. |
| 460 |
|
|
| 461 |
|
In the formulas presented in the tables below, the placeholder |
| 462 |
|
function $f(r)$ is used to represent the electrostatic kernel (either |
| 479 |
|
orders: |
| 480 |
|
\begin{equation} |
| 481 |
|
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 482 |
< |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{r} |
| 483 |
< |
\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert _{r_c} \right] |
| 482 |
> |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) |
| 483 |
> |
\hat{\mathbf{r}} \cdot \nabla U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 484 |
|
\label{generic2} |
| 485 |
|
\end{equation} |
| 486 |
< |
where the sum describes a separate force-shifting that is applied to |
| 487 |
< |
each orientational contribution to the energy. Both the potential and |
| 488 |
< |
the gradient for force shifting are evaluated for an image multipole |
| 489 |
< |
projected onto the surface of the cutoff sphere (see fig |
| 490 |
< |
\ref{fig:shiftedMultipoles}). The image multipole retains the |
| 486 |
> |
where $\hat{\mathbf{r}}$ is the unit vector pointing between the two |
| 487 |
> |
multipoles, and the sum describes a separate force-shifting that is |
| 488 |
> |
applied to each orientational contribution to the energy. Both the |
| 489 |
> |
potential and the gradient for force shifting are evaluated for an |
| 490 |
> |
image multipole projected onto the surface of the cutoff sphere (see |
| 491 |
> |
fig \ref{fig:shiftedMultipoles}). The image multipole retains the |
| 492 |
|
orientation ($\hat{\mathbf{b}}$) of the interacting multipole. No |
| 493 |
|
higher order terms $(r-r_c)^n$ appear. The primary difference between |
| 494 |
|
the TSF and GSF methods is the stage at which the Taylor Series is |
| 495 |
|
applied; in the Taylor-shifted approach, it is applied to the kernel |
| 496 |
|
itself. In the Gradient-shifted approach, it is applied to individual |
| 497 |
< |
radial interactions terms in the multipole expansion. Energies from |
| 497 |
> |
radial interaction terms in the multipole expansion. Energies from |
| 498 |
|
this method thus have the general form: |
| 499 |
|
\begin{equation} |
| 500 |
|
U= \sum (\text{angular factor}) (\text{radial factor}). |
| 515 |
|
to zero at the cutoff radius, |
| 516 |
|
\begin{equation} |
| 517 |
|
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 518 |
< |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 518 |
> |
U(r_c \hat{\mathbf{r}},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 519 |
|
\label{eq:SP} |
| 520 |
|
\end{equation} |
| 521 |
|
independent of the orientations of the two multipoles. The sum again |
| 687 |
|
\end{equation} |
| 688 |
|
|
| 689 |
|
The extension of DSF electrostatics to point multipoles requires |
| 690 |
< |
treatment of {\it both} the self-neutralization and reciprocal |
| 690 |
> |
treatment of the self-neutralization \textit{and} reciprocal |
| 691 |
|
contributions to the self-interaction for higher order multipoles. In |
| 692 |
|
this section we give formulae for these interactions up to quadrupolar |
| 693 |
|
order. |
| 739 |
|
\section{Interaction energies, forces, and torques} |
| 740 |
|
The main result of this paper is a set of expressions for the |
| 741 |
|
energies, forces and torques (up to quadrupole-quadrupole order) that |
| 742 |
< |
work for both the Taylor-shifted and Gradient-shifted approximations. |
| 743 |
< |
These expressions were derived using a set of generic radial |
| 744 |
< |
functions. Without using the shifting approximations mentioned above, |
| 745 |
< |
some of these radial functions would be identical, and the expressions |
| 746 |
< |
coalesce into the familiar forms for unmodified multipole-multipole |
| 747 |
< |
interactions. Table \ref{tab:tableenergy} maps between the generic |
| 748 |
< |
functions and the radial functions derived for both the Taylor-shifted |
| 749 |
< |
and Gradient-shifted methods. The energy equations are written in |
| 750 |
< |
terms of lab-frame representations of the dipoles, quadrupoles, and |
| 751 |
< |
the unit vector connecting the two objects, |
| 742 |
> |
work for the Taylor-shifted, gradient-shifted, and shifted potential |
| 743 |
> |
approximations. These expressions were derived using a set of generic |
| 744 |
> |
radial functions. Without using the shifting approximations mentioned |
| 745 |
> |
above, some of these radial functions would be identical, and the |
| 746 |
> |
expressions coalesce into the familiar forms for unmodified |
| 747 |
> |
multipole-multipole interactions. Table \ref{tab:tableenergy} maps |
| 748 |
> |
between the generic functions and the radial functions derived for the |
| 749 |
> |
three methods. The energy equations are written in terms of lab-frame |
| 750 |
> |
representations of the dipoles, quadrupoles, and the unit vector |
| 751 |
> |
connecting the two objects, |
| 752 |
|
|
| 753 |
|
% Energy in space coordinate form ---------------------------------------------------------------------------------------------- |
| 754 |
|
% |
| 875 |
|
|
| 876 |
|
\begin{sidewaystable} |
| 877 |
|
\caption{\label{tab:tableenergy}Radial functions used in the energy |
| 878 |
< |
and torque equations. The $f, g, h, s, t, \mathrm{and} u$ |
| 879 |
< |
functions used in this table are defined in Appendices B and C. |
| 880 |
< |
Gradient shifted (GSF) functions are constructed using the shifted |
| 881 |
< |
potential (SP) functions.} |
| 878 |
> |
and torque equations. The $f, g, h, s, t, \mathrm{and~} u$ |
| 879 |
> |
functions used in this table are defined in Appendices |
| 880 |
> |
\ref{radialTSF} and \ref{radialGSF}. The gradient shifted (GSF) |
| 881 |
> |
functions include the shifted potential (SP) |
| 882 |
> |
contributions (\textit{cf.} Eqs. \ref{generic2} and |
| 883 |
> |
\ref{eq:SP}).} |
| 884 |
|
\begin{tabular}{|c|c|l|l|l|} \hline |
| 885 |
|
Generic&Bare Coulomb&Taylor-Shifted (TSF)&Shifted Potential (SP)&Gradient-Shifted (GSF) |
| 886 |
|
\\ \hline |
| 1107 |
|
\end{equation} |
| 1108 |
|
% |
| 1109 |
|
We list below the force equations written in terms of lab-frame |
| 1110 |
< |
coordinates. The radial functions used in the two methods are listed |
| 1110 |
> |
coordinates. The radial functions used in the three methods are listed |
| 1111 |
|
in Table \ref{tab:tableFORCE} |
| 1112 |
|
% |
| 1113 |
|
%SPACE COORDINATES FORCE EQUATIONS |
| 1255 |
|
\subsection{Torques} |
| 1256 |
|
|
| 1257 |
|
% |
| 1258 |
< |
The torques for both the Taylor-Shifted as well as Gradient-Shifted |
| 1259 |
< |
methods are given in space-frame coordinates: |
| 1258 |
> |
The torques for the three methods are given in space-frame |
| 1259 |
> |
coordinates: |
| 1260 |
|
% |
| 1261 |
|
% |
| 1262 |
|
\begin{align} |
| 1411 |
|
|
| 1412 |
|
All of the radial functions required for torques are identical with |
| 1413 |
|
the radial functions previously computed for the interaction energies. |
| 1414 |
< |
These are tabulated for both shifted force methods in table |
| 1414 |
> |
These are tabulated for all three methods in table |
| 1415 |
|
\ref{tab:tableenergy}. The torques for higher multipoles on site |
| 1416 |
|
$\mathbf{a}$ interacting with those of lower order on site |
| 1417 |
|
$\mathbf{b}$ can be obtained by swapping indices in the expressions |
| 1423 |
|
computer simulations, it is vital to test against established methods |
| 1424 |
|
for computing electrostatic interactions in periodic systems, and to |
| 1425 |
|
evaluate the size and sources of any errors that arise from the |
| 1426 |
< |
real-space cutoffs. In this paper we test both TSF and GSF |
| 1426 |
> |
real-space cutoffs. In this paper we test SP, TSF, and GSF |
| 1427 |
|
electrostatics against analytical methods for computing the energies |
| 1428 |
|
of ordered multipolar arrays. In the following paper, we test the new |
| 1429 |
|
methods against the multipolar Ewald sum for computing the energies, |
| 1487 |
|
constants are shown as a function of both the cutoff radius, $r_c$, |
| 1488 |
|
and the damping parameter, $\alpha$ in Figs.\ref{fig:Dipoles_rCut} |
| 1489 |
|
and \ref{fig:Dipoles_alpha}. We have simultaneously tested a hard |
| 1490 |
< |
cutoff (where the kernel is simply truncated at the cutoff radius), as |
| 1491 |
< |
well as a shifted potential (SP) form which includes a |
| 1487 |
< |
potential-shifting and self-interaction term, but does not shift the |
| 1488 |
< |
forces and torques smoothly at the cutoff radius. The SP method is |
| 1489 |
< |
essentially an extension of the original Wolf method for multipoles. |
| 1490 |
> |
cutoff (where the kernel is simply truncated at the cutoff radius) in |
| 1491 |
> |
addition to the three new methods. |
| 1492 |
|
|
| 1493 |
|
The hard cutoff exhibits oscillations around the analytic energy |
| 1494 |
|
constants, and converges to incorrect energies when the complementary |
| 1571 |
|
respectively. All of the methods (except for TSF) have excellent |
| 1572 |
|
behavior for the undamped or weakly-damped cases. All of the methods |
| 1573 |
|
can be forced to converge by increasing the value of $\alpha$, which |
| 1574 |
< |
effectively shortens the overall range of the potential, but which |
| 1575 |
< |
equalizing the truncation effects on the different orientational |
| 1576 |
< |
contributions. In the second paper in the series, we discuss how |
| 1577 |
< |
large values of $\alpha$ can perturb the force and torque vectors, but |
| 1578 |
< |
both weakly-damped or over-damped electrostatics appear to generate |
| 1574 |
> |
effectively shortens the overall range of the potential by equalizing |
| 1575 |
> |
the truncation effects on the different orientational contributions. |
| 1576 |
> |
In the second paper in the series, we discuss how large values of |
| 1577 |
> |
$\alpha$ can perturb the force and torque vectors, but both |
| 1578 |
> |
weakly-damped or over-damped electrostatics appear to generate |
| 1579 |
|
reasonable values for the total electrostatic energies under both the |
| 1580 |
|
SP and GSF approximations. |
| 1581 |
|
|
| 1657 |
|
\text{e}^{-{\alpha}^2r^2} |
| 1658 |
|
\right] , |
| 1659 |
|
\end{equation} |
| 1660 |
< |
is very useful for building up higher derivatives. Using these |
| 1661 |
< |
formulas, we find: |
| 1660 |
> |
is very useful for building up higher derivatives. As noted by Smith, |
| 1661 |
> |
it is possible to approximate the $B_l(r)$ functions, |
| 1662 |
|
% |
| 1661 |
– |
\begin{align} |
| 1662 |
– |
\frac{dB_0}{dr}=&-rB_1(r) \\ |
| 1663 |
– |
\frac{d^2B_0}{dr^2}=& - B_1(r) + r^2 B_2(r) \\ |
| 1664 |
– |
\frac{d^3B_0}{dr^3}=& 3 r B_2(r) - r^3 B_3(r) \\ |
| 1665 |
– |
\frac{d^4B_0}{dr^4}=& 3 B_2(r) - 6 r^2 B_3(r) + r^4 B_4(r) \\ |
| 1666 |
– |
\frac{d^5B_0}{dr^5}=& - 15 r B_3(r) + 10 r^3 B_4(r) - r^5 B_5(r) . |
| 1667 |
– |
\end{align} |
| 1668 |
– |
% |
| 1669 |
– |
As noted by Smith, it is possible to approximate the $B_l(r)$ |
| 1670 |
– |
functions, |
| 1671 |
– |
% |
| 1663 |
|
\begin{equation} |
| 1664 |
|
B_l(r)=\frac{(2l)!}{l!2^lr^{2l+1}} - \frac {(2\alpha^2)^{l+1}}{(2l+1)\alpha \sqrt{\pi}} |
| 1665 |
|
+\text{O}(r) . |
| 1666 |
|
\end{equation} |
| 1667 |
|
\newpage |
| 1668 |
|
\section{The $r$-dependent factors for TSF electrostatics} |
| 1669 |
+ |
\label{radialTSF} |
| 1670 |
|
|
| 1671 |
|
Using the shifted damped functions $f_n(r)$ defined by: |
| 1672 |
|
% |
| 1761 |
|
% |
| 1762 |
|
\newpage |
| 1763 |
|
\section{The $r$-dependent factors for GSF electrostatics} |
| 1764 |
+ |
\label{radialGSF} |
| 1765 |
|
|
| 1766 |
|
In Gradient-shifted force electrostatics, the kernel is not expanded, |
| 1767 |
< |
rather the individual terms in the multipole interaction energies. |
| 1768 |
< |
For damped charges, this still brings into the algebra multiple |
| 1769 |
< |
derivatives of the Smith's $B_0(r)$ function. To denote these terms, |
| 1770 |
< |
we generalize the notation of the previous appendix. For either |
| 1771 |
< |
$f(r)=1/r$ (undamped) or $f(r)=B_0(r)$ (damped), |
| 1767 |
> |
and the expansion is carried out on the individual terms in the |
| 1768 |
> |
multipole interaction energies. For damped charges, this still brings |
| 1769 |
> |
multiple derivatives of the Smith's $B_0(r)$ function into the |
| 1770 |
> |
algebra. To denote these terms, we generalize the notation of the |
| 1771 |
> |
previous appendix. For either $f(r)=1/r$ (undamped) or $f(r)=B_0(r)$ |
| 1772 |
> |
(damped), |
| 1773 |
|
% |
| 1774 |
|
\begin{align} |
| 1775 |
|
g(r) &= \frac{df}{d r} && &&=-\frac{1}{r^2} |
