| 152 |
|
An efficient real-space electrostatic method involves the use of a |
| 153 |
|
pair-wise functional form, |
| 154 |
|
\begin{equation} |
| 155 |
< |
V = \sum_i \sum_{j>i} V_\mathrm{pair}(r_{ij}, \Omega_i, \Omega_j) + |
| 156 |
< |
\sum_i V_i^\mathrm{correction} |
| 155 |
> |
V = \sum_i \sum_{j>i} V_\mathrm{pair}(\mathbf{r}_{ij}, \Omega_i, \Omega_j) + |
| 156 |
> |
\sum_i V_i^\mathrm{self} |
| 157 |
|
\end{equation} |
| 158 |
|
that is short-ranged and easily truncated at a cutoff radius, |
| 159 |
|
\begin{equation} |
| 160 |
< |
V_\mathrm{pair}(r_{ij}, \Omega_i, \Omega_j) = \left\{ |
| 160 |
> |
V_\mathrm{pair}(\mathbf{r}_{ij},\Omega_i, \Omega_j) = \left\{ |
| 161 |
|
\begin{array}{ll} |
| 162 |
< |
V_\mathrm{approx} (r_{ij}, \Omega_i, \Omega_j) & \quad r \le r_c \\ |
| 163 |
< |
0 & \quad r > r_c , |
| 162 |
> |
V_\mathrm{approx} (\mathbf{r}_{ij}, \Omega_i, \Omega_j) & \quad \left| \mathbf{r}_{ij} \right| \le r_c \\ |
| 163 |
> |
0 & \quad \left| \mathbf{r}_{ij} \right| > r_c , |
| 164 |
|
\end{array} |
| 165 |
|
\right. |
| 166 |
|
\end{equation} |
| 167 |
< |
along with an easily computed correction term ($\sum_i |
| 168 |
< |
V_i^\mathrm{correction}$) which has linear-scaling with the number of |
| 167 |
> |
along with an easily computed self-interaction term ($\sum_i |
| 168 |
> |
V_i^\mathrm{self}$) which scales linearly with the number of |
| 169 |
|
particles. Here $\Omega_i$ and $\Omega_j$ represent orientational |
| 170 |
< |
coordinates of the two sites. The computational efficiency, energy |
| 170 |
> |
coordinates of the two sites, and $\mathbf{r}_{ij}$ is the vector |
| 171 |
> |
between the two sites. The computational efficiency, energy |
| 172 |
|
conservation, and even some physical properties of a simulation can |
| 173 |
|
depend dramatically on how the $V_\mathrm{approx}$ function behaves at |
| 174 |
|
the cutoff radius. The goal of any approximation method should be to |
| 181 |
|
contained within the cutoff sphere surrounding each particle. This is |
| 182 |
|
accomplished by shifting the potential functions to generate image |
| 183 |
|
charges on the surface of the cutoff sphere for each pair interaction |
| 184 |
< |
computed within $r_c$. Damping using a complementary error |
| 185 |
< |
function is applied to the potential to accelerate convergence. The |
| 186 |
< |
potential for the DSF method, where $\alpha$ is the adjustable damping |
| 186 |
< |
parameter, is given by |
| 184 |
> |
computed within $r_c$. Damping using a complementary error function is |
| 185 |
> |
applied to the potential to accelerate convergence. The interaction |
| 186 |
> |
for a pair of charges ($C_i$ and $C_j$) in the DSF method, |
| 187 |
|
\begin{equation*} |
| 188 |
|
V_\mathrm{DSF}(r) = C_i C_j \Biggr{[} |
| 189 |
|
\frac{\mathrm{erfc}\left(\alpha r_{ij}\right)}{r_{ij}} |
| 193 |
|
\right)\left(r_{ij}-r_c\right)\ \Biggr{]} |
| 194 |
|
\label{eq:DSFPot} |
| 195 |
|
\end{equation*} |
| 196 |
< |
Note that in this potential and in all electrostatic quantities that |
| 197 |
< |
follow, the standard $1/4 \pi \epsilon_{0}$ has been omitted for |
| 198 |
< |
clarity. |
| 196 |
> |
where $\alpha$ is the adjustable damping parameter. Note that in this |
| 197 |
> |
potential and in all electrostatic quantities that follow, the |
| 198 |
> |
standard $1/4 \pi \epsilon_{0}$ has been omitted for clarity. |
| 199 |
|
|
| 200 |
|
To insure net charge neutrality within each cutoff sphere, an |
| 201 |
|
additional ``self'' term is added to the potential. This term is |
| 250 |
|
The Taylor expansion in $r$ can be written using an implied summation |
| 251 |
|
notation. Here Greek indices are used to indicate space coordinates |
| 252 |
|
($x$, $y$, $z$) and the subscripts $k$ and $j$ are reserved for |
| 253 |
< |
labelling specific charges in $\bf a$ and $\bf b$ respectively. The |
| 253 |
> |
labeling specific charges in $\bf a$ and $\bf b$ respectively. The |
| 254 |
|
Taylor expansion, |
| 255 |
|
\begin{equation} |
| 256 |
|
\frac{1}{\lvert \mathbf{r} - \mathbf{r}_k \rvert} = |
| 276 |
|
a}\alpha\beta}$, respectively. These are the primitive multipoles |
| 277 |
|
which can be expressed as a distribution of charges, |
| 278 |
|
\begin{align} |
| 279 |
< |
C_{\bf a} =&\sum_{k \, \text{in \bf a}} q_k , \\ |
| 280 |
< |
D_{{\bf a}\alpha} =&\sum_{k \, \text{in \bf a}} q_k r_{k\alpha} ,\\ |
| 281 |
< |
Q_{{\bf a}\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k r_{k\alpha} r_{k\beta} . |
| 279 |
> |
C_{\bf a} =&\sum_{k \, \text{in \bf a}} q_k , \label{eq:charge} \\ |
| 280 |
> |
D_{{\bf a}\alpha} =&\sum_{k \, \text{in \bf a}} q_k r_{k\alpha}, \label{eq:dipole}\\ |
| 281 |
> |
Q_{{\bf a}\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k |
| 282 |
> |
r_{k\alpha} r_{k\beta} . \label{eq:quadrupole} |
| 283 |
|
\end{align} |
| 284 |
|
Note that the definition of the primitive quadrupole here differs from |
| 285 |
|
the standard traceless form, and contains an additional Taylor-series |
| 286 |
< |
based factor of $1/2$. |
| 286 |
> |
based factor of $1/2$. We are essentially treating the mass |
| 287 |
> |
distribution with higher priority; the moment of inertia tensor, |
| 288 |
> |
$\overleftrightarrow{\mathsf I}$, is diagonalized to obtain body-fixed |
| 289 |
> |
axes, and the charge distribution may result in a quadrupole tensor |
| 290 |
> |
that is not necessarily diagonal in the body frame. Additional |
| 291 |
> |
reasons for utilizing the primitive quadrupole are discussed in |
| 292 |
> |
section \ref{sec:damped}. |
| 293 |
|
|
| 294 |
|
It is convenient to locate charges $q_j$ relative to the center of mass of $\bf b$. Then with $\bf{r}$ pointing from |
| 295 |
|
$\bf a$ to $\bf b$ ($\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $), the interaction energy is given by |
| 313 |
|
of $\bf a$ interacting with the same multipoles on $\bf b$. |
| 314 |
|
|
| 315 |
|
\subsection{Damped Coulomb interactions} |
| 316 |
+ |
\label{sec:damped} |
| 317 |
|
In the standard multipole expansion, one typically uses the bare |
| 318 |
|
Coulomb potential, with radial dependence $1/r$, as shown in |
| 319 |
|
Eq.~(\ref{kernel}). It is also quite common to use a damped Coulomb |
| 329 |
|
either $1/r$ or $B_0(r)$, and all of the techniques can be applied to |
| 330 |
|
bare or damped Coulomb kernels (or any other function) as long as |
| 331 |
|
derivatives of these functions are known. Smith's convenient |
| 332 |
< |
functions $B_l(r)$ are summarized in Appendix A. |
| 332 |
> |
functions $B_l(r)$ are summarized in Appendix A. (N.B. there is one |
| 333 |
> |
important distinction between the two kernels, which is the behavior |
| 334 |
> |
of $\nabla^2 \frac{1}{r}$ compared with $\nabla^2 B_0(r)$. The former |
| 335 |
> |
is zero everywhere except for a delta function evaluated at the |
| 336 |
> |
origin. The latter also has delta function behavior, but is non-zero |
| 337 |
> |
for $r \neq 0$. Thus the standard justification for using a traceless |
| 338 |
> |
quadrupole tensor fails for the damped case.) |
| 339 |
|
|
| 340 |
|
The main goal of this work is to smoothly cut off the interaction |
| 341 |
|
energy as well as forces and torques as $r\rightarrow r_c$. To |
| 432 |
|
In general, we can write |
| 433 |
|
% |
| 434 |
|
\begin{equation} |
| 435 |
< |
U= (\text{prefactor}) (\text{derivatives}) f_n(r) |
| 435 |
> |
U^{\text{TSF}}= (\text{prefactor}) (\text{derivatives}) f_n(r) |
| 436 |
|
\label{generic} |
| 437 |
|
\end{equation} |
| 438 |
|
% |
| 453 |
|
of the same index $n$. The algebra required to evaluate energies, |
| 454 |
|
forces and torques is somewhat tedious, so only the final forms are |
| 455 |
|
presented in tables \ref{tab:tableenergy} and \ref{tab:tableFORCE}. |
| 456 |
+ |
One of the principal findings of our work is that the individual |
| 457 |
+ |
orientational contributions to the various multipole-multipole |
| 458 |
+ |
interactions must be treated with distinct radial functions, but each |
| 459 |
+ |
of these contributions is independently force shifted at the cutoff |
| 460 |
+ |
radius. |
| 461 |
|
|
| 462 |
|
\subsection{Gradient-shifted force (GSF) electrostatics} |
| 463 |
|
The second, and conceptually simpler approach to force-shifting |
| 465 |
|
expansion, and has a similar interaction energy for all multipole |
| 466 |
|
orders: |
| 467 |
|
\begin{equation} |
| 468 |
< |
U^{\text{shift}}(r)=U(r)-U(r_c)-(r-r_c)\hat{r}\cdot \nabla U(r) \Big |
| 469 |
< |
\lvert _{r_c} . |
| 468 |
> |
U^{\text{GSF}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 469 |
> |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) - (r-r_c) \hat{r} |
| 470 |
> |
\cdot \nabla U(\mathbf{r},\hat{\mathbf{a}}, \hat{\mathbf{b}}) \Big \lvert _{r_c} \right] |
| 471 |
|
\label{generic2} |
| 472 |
|
\end{equation} |
| 473 |
< |
Here the gradient for force shifting is evaluated for an image |
| 474 |
< |
multipole projected onto the surface of the cutoff sphere (see fig |
| 475 |
< |
\ref{fig:shiftedMultipoles}). No higher order terms $(r-r_c)^n$ |
| 476 |
< |
appear. The primary difference between the TSF and GSF methods is the |
| 477 |
< |
stage at which the Taylor Series is applied; in the Taylor-shifted |
| 478 |
< |
approach, it is applied to the kernel itself. In the Gradient-shifted |
| 479 |
< |
approach, it is applied to individual radial interactions terms in the |
| 480 |
< |
multipole expansion. Energies from this method thus have the general |
| 481 |
< |
form: |
| 473 |
> |
where the sum describes a separate force-shifting that is applied to |
| 474 |
> |
each orientational contribution to the energy. Both the potential and |
| 475 |
> |
the gradient for force shifting are evaluated for an image multipole |
| 476 |
> |
projected onto the surface of the cutoff sphere (see fig |
| 477 |
> |
\ref{fig:shiftedMultipoles}). The image multipole retains the |
| 478 |
> |
orientation ($\hat{\mathbf{b}}$) of the interacting multipole. No |
| 479 |
> |
higher order terms $(r-r_c)^n$ appear. The primary difference between |
| 480 |
> |
the TSF and GSF methods is the stage at which the Taylor Series is |
| 481 |
> |
applied; in the Taylor-shifted approach, it is applied to the kernel |
| 482 |
> |
itself. In the Gradient-shifted approach, it is applied to individual |
| 483 |
> |
radial interactions terms in the multipole expansion. Energies from |
| 484 |
> |
this method thus have the general form: |
| 485 |
|
\begin{equation} |
| 486 |
|
U= \sum (\text{angular factor}) (\text{radial factor}). |
| 487 |
|
\label{generic3} |
| 494 |
|
the radial factors differ between the two methods. |
| 495 |
|
|
| 496 |
|
\subsection{\label{sec:level2}Body and space axes} |
| 497 |
+ |
Although objects $\bf a$ and $\bf b$ rotate during a molecular |
| 498 |
+ |
dynamics (MD) simulation, their multipole tensors remain fixed in |
| 499 |
+ |
body-frame coordinates. While deriving force and torque expressions, |
| 500 |
+ |
it is therefore convenient to write the energies, forces, and torques |
| 501 |
+ |
in intermediate forms involving the vectors of the rotation matrices. |
| 502 |
+ |
We denote body axes for objects $\bf a$ and $\bf b$ using unit vectors |
| 503 |
+ |
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$. |
| 504 |
+ |
In a typical simulation , the initial axes are obtained by |
| 505 |
+ |
diagonalizing the moment of inertia tensors for the objects. (N.B., |
| 506 |
+ |
the body axes are generally {\it not} the same as those for which the |
| 507 |
+ |
quadrupole moment is diagonal.) The rotation matrices are then |
| 508 |
+ |
propagated during the simulation. |
| 509 |
|
|
| 510 |
< |
[XXX Do we need this section in the main paper? or should it go in the |
| 476 |
< |
extra materials?] |
| 477 |
< |
|
| 478 |
< |
So far, all energies and forces have been written in terms of fixed |
| 479 |
< |
space coordinates. Interaction energies are computed from the generic |
| 480 |
< |
formulas Eq.~(\ref{generic}) and ~(\ref{generic2}) which combine |
| 481 |
< |
orientational prefactors with radial functions. Because objects $\bf |
| 482 |
< |
a$ and $\bf b$ both translate and rotate during a molecular dynamics |
| 483 |
< |
(MD) simulation, it is desirable to contract all $r$-dependent terms |
| 484 |
< |
with dipole and quadrupole moments expressed in terms of their body |
| 485 |
< |
axes. To do so, we have followed the methodology of Allen and |
| 486 |
< |
Germano,\cite{Allen:2006fk} which was itself based on earlier work by |
| 487 |
< |
Price {\em et al.}\cite{Price:1984fk} |
| 488 |
< |
|
| 489 |
< |
We denote body axes for objects $\bf a$ and $\bf b$ by unit vectors |
| 490 |
< |
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$ |
| 491 |
< |
referring to a convenient set of inertial body axes. (N.B., these |
| 492 |
< |
body axes are generally not the same as those for which the quadrupole |
| 493 |
< |
moment is diagonal.) Then, |
| 494 |
< |
% |
| 495 |
< |
\begin{eqnarray} |
| 496 |
< |
\hat{a}_m= a_{mx}\hat{x} + a_{my}\hat{y} + a_{mz}\hat{z} \\ |
| 497 |
< |
\hat{b}_m= b_{mx}\hat{x} + b_{my}\hat{y} + b_{mz}\hat{z} . |
| 498 |
< |
\end{eqnarray} |
| 499 |
< |
Rotation matrices $\hat{\mathbf {a}}$ and $\hat{\mathbf {b}}$ can be |
| 510 |
> |
The rotation matrices $\hat{\mathbf {a}}$ and $\hat{\mathbf {b}}$ can be |
| 511 |
|
expressed using these unit vectors: |
| 512 |
|
\begin{eqnarray} |
| 513 |
|
\hat{\mathbf {a}} = |
| 515 |
|
\hat{a}_1 \\ |
| 516 |
|
\hat{a}_2 \\ |
| 517 |
|
\hat{a}_3 |
| 518 |
< |
\end{pmatrix} |
| 508 |
< |
= |
| 509 |
< |
\begin{pmatrix} |
| 510 |
< |
a_{1x} \quad a_{1y} \quad a_{1z} \\ |
| 511 |
< |
a_{2x} \quad a_{2y} \quad a_{2z} \\ |
| 512 |
< |
a_{3x} \quad a_{3y} \quad a_{3z} |
| 513 |
< |
\end{pmatrix}\\ |
| 518 |
> |
\end{pmatrix}, \qquad |
| 519 |
|
\hat{\mathbf {b}} = |
| 520 |
|
\begin{pmatrix} |
| 521 |
|
\hat{b}_1 \\ |
| 522 |
|
\hat{b}_2 \\ |
| 523 |
|
\hat{b}_3 |
| 524 |
|
\end{pmatrix} |
| 525 |
< |
= |
| 521 |
< |
\begin{pmatrix} |
| 522 |
< |
b_{1x} \quad b_{1y} \quad b_{1z} \\ |
| 523 |
< |
b_{2x} \quad b_{2y} \quad b_{2z} \\ |
| 524 |
< |
b_{3x} \quad b_{3y} \quad b_{3z} |
| 525 |
< |
\end{pmatrix} . |
| 526 |
< |
\end{eqnarray} |
| 525 |
> |
\end{eqnarray} |
| 526 |
|
% |
| 527 |
|
These matrices convert from space-fixed $(xyz)$ to body-fixed $(123)$ |
| 528 |
< |
coordinates. All contractions of prefactors with derivatives of |
| 529 |
< |
functions can be written in terms of these matrices. It proves to be |
| 530 |
< |
equally convenient to just write any contraction in terms of unit |
| 531 |
< |
vectors $\hat{r}$, $\hat{a}_m$, and $\hat{b}_n$. In the torque |
| 532 |
< |
expressions, it is useful to have the angular-dependent terms |
| 533 |
< |
available in three different fashions, e.g. for the dipole-dipole |
| 534 |
< |
contraction: |
| 528 |
> |
coordinates. |
| 529 |
> |
|
| 530 |
> |
Allen and Germano,\cite{Allen:2006fk} following earlier work by Price |
| 531 |
> |
{\em et al.},\cite{Price:1984fk} showed that if the interaction |
| 532 |
> |
energies are written explicitly in terms of $\hat{r}$ and the body |
| 533 |
> |
axes ($\hat{a}_m$, $\hat{b}_n$) : |
| 534 |
> |
% |
| 535 |
> |
\begin{equation} |
| 536 |
> |
U(r, \{\hat{a}_m \cdot \hat{r} \}, |
| 537 |
> |
\{\hat{b}_n\cdot \hat{r} \}, |
| 538 |
> |
\{\hat{a}_m \cdot \hat{b}_n \}) . |
| 539 |
> |
\label{ugeneral} |
| 540 |
> |
\end{equation} |
| 541 |
> |
% |
| 542 |
> |
the forces come out relatively cleanly, |
| 543 |
> |
% |
| 544 |
> |
\begin{equation} |
| 545 |
> |
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
| 546 |
> |
= \frac{\partial U}{\partial r} \hat{r} |
| 547 |
> |
+ \sum_m \left[ |
| 548 |
> |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 549 |
> |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 550 |
> |
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 551 |
> |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 552 |
> |
\right] \label{forceequation}. |
| 553 |
> |
\end{equation} |
| 554 |
> |
|
| 555 |
> |
The torques can also be found in a relatively similar |
| 556 |
> |
manner, |
| 557 |
> |
% |
| 558 |
> |
\begin{eqnarray} |
| 559 |
> |
\mathbf{\tau}_{\bf a} = |
| 560 |
> |
\sum_m |
| 561 |
> |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 562 |
> |
( \hat{r} \times \hat{a}_m ) |
| 563 |
> |
-\sum_{mn} |
| 564 |
> |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 565 |
> |
(\hat{a}_m \times \hat{b}_n) \\ |
| 566 |
> |
% |
| 567 |
> |
\mathbf{\tau}_{\bf b} = |
| 568 |
> |
\sum_m |
| 569 |
> |
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 570 |
> |
( \hat{r} \times \hat{b}_m) |
| 571 |
> |
+\sum_{mn} |
| 572 |
> |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 573 |
> |
(\hat{a}_m \times \hat{b}_n) . |
| 574 |
> |
\end{eqnarray} |
| 575 |
> |
|
| 576 |
> |
Note that our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ |
| 577 |
> |
is opposite in sign to that of Allen and Germano.\cite{Allen:2006fk} |
| 578 |
> |
We also made use of the identities, |
| 579 |
> |
% |
| 580 |
> |
\begin{align} |
| 581 |
> |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 582 |
> |
=& \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
| 583 |
> |
\right) \\ |
| 584 |
> |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 585 |
> |
=& \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
| 586 |
> |
\right) . |
| 587 |
> |
\end{align} |
| 588 |
> |
|
| 589 |
> |
Many of the multipole contractions required can be written in one of |
| 590 |
> |
three equivalent forms using the unit vectors $\hat{r}$, $\hat{a}_m$, |
| 591 |
> |
and $\hat{b}_n$. In the torque expressions, it is useful to have the |
| 592 |
> |
angular-dependent terms available in all three fashions, e.g. for the |
| 593 |
> |
dipole-dipole contraction: |
| 594 |
|
% |
| 595 |
|
\begin{equation} |
| 596 |
|
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
| 604 |
|
explicit sums over body indices and the dot products now indicate |
| 605 |
|
contractions using space indices. |
| 606 |
|
|
| 607 |
+ |
In computing our force and torque expressions, we carried out most of |
| 608 |
+ |
the work in body coordinates, and have transformed the expressions |
| 609 |
+ |
back to space-frame coordinates, which are reported below. Interested |
| 610 |
+ |
readers may consult the supplemental information for this paper for |
| 611 |
+ |
the intermediate body-frame expressions. |
| 612 |
|
|
| 613 |
|
\subsection{The Self-Interaction \label{sec:selfTerm}} |
| 614 |
|
|
| 615 |
|
In addition to cutoff-sphere neutralization, the Wolf |
| 616 |
|
summation~\cite{Wolf99} and the damped shifted force (DSF) |
| 617 |
< |
extension~\cite{Fennell:2006zl} also included self-interactions that |
| 617 |
> |
extension~\cite{Fennell:2006zl} also include self-interactions that |
| 618 |
|
are handled separately from the pairwise interactions between |
| 619 |
|
sites. The self-term is normally calculated via a single loop over all |
| 620 |
|
sites in the system, and is relatively cheap to evaluate. The |
| 661 |
|
reciprocal-space portion is identical to half of the self-term |
| 662 |
|
obtained by Smith and Aguado and Madden for the application of the |
| 663 |
|
Ewald sum to multipoles.\cite{Smith82,Smith98,Aguado03} For a given |
| 664 |
< |
site which posesses a charge, dipole, and multipole, both types of |
| 664 |
> |
site which posesses a charge, dipole, and quadrupole, both types of |
| 665 |
|
contribution are given in table \ref{tab:tableSelf}. |
| 666 |
|
|
| 667 |
|
\begin{table*} |
| 674 |
|
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
| 675 |
|
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
| 676 |
|
\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
| 677 |
< |
Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
| 677 |
> |
Quadrupole & $2 \mathbf{Q}_{\bf a}:\mathbf{Q}_{\bf a} + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
| 678 |
|
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
| 679 |
|
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
| 680 |
|
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
| 804 |
|
U_{Q_{\bf a}Q_{\bf b}}(r)=& |
| 805 |
|
\Bigl[ |
| 806 |
|
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
| 807 |
< |
+2 \text{Tr} \left( |
| 808 |
< |
\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \right) \Bigr] v_{41}(r) |
| 807 |
> |
+2 |
| 808 |
> |
\mathbf{Q}_{\mathbf{a}} : \mathbf{Q}_{\mathbf{b}} \Bigr] v_{41}(r) |
| 809 |
|
\\ |
| 810 |
|
% 2 |
| 811 |
|
&+\Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 1044 |
|
F_{\bf a \alpha} = \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}}{\partial r} |
| 1045 |
|
\quad \text{and} \quad F_{\bf b \alpha} |
| 1046 |
|
= - \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}} {\partial r} . |
| 984 |
– |
\end{equation} |
| 985 |
– |
% |
| 986 |
– |
Obtaining the force from the interaction energy expressions is the |
| 987 |
– |
same for higher-order multipole interactions -- the trick is to make |
| 988 |
– |
sure that all $r$-dependent derivatives are considered. This is |
| 989 |
– |
straighforward if the interaction energies are written explicitly in |
| 990 |
– |
terms of $\hat{r}$ and the body axes ($\hat{a}_m$, |
| 991 |
– |
$\hat{b}_n$) : |
| 992 |
– |
% |
| 993 |
– |
\begin{equation} |
| 994 |
– |
U(r,\{\hat{a}_m \cdot \hat{r} \}, |
| 995 |
– |
\{\hat{b}_n\cdot \hat{r} \}, |
| 996 |
– |
\{\hat{a}_m \cdot \hat{b}_n \}) . |
| 997 |
– |
\label{ugeneral} |
| 1047 |
|
\end{equation} |
| 1048 |
|
% |
| 1000 |
– |
Allen and Germano,\cite{Allen:2006fk} showed that if the energy is |
| 1001 |
– |
written in this form, the forces come out relatively cleanly, |
| 1002 |
– |
% |
| 1003 |
– |
\begin{equation} |
| 1004 |
– |
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
| 1005 |
– |
= \frac{\partial U}{\partial r} \hat{r} |
| 1006 |
– |
+ \sum_m \left[ |
| 1007 |
– |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 1008 |
– |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 1009 |
– |
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 1010 |
– |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 1011 |
– |
\right] \label{forceequation}. |
| 1012 |
– |
\end{equation} |
| 1013 |
– |
% |
| 1014 |
– |
Note that our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ |
| 1015 |
– |
is opposite in sign to that of Allen and Germano.\cite{Allen:2006fk} |
| 1016 |
– |
In simplifying the algebra, we have also used: |
| 1017 |
– |
% |
| 1018 |
– |
\begin{align} |
| 1019 |
– |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 1020 |
– |
=& \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
| 1021 |
– |
\right) \\ |
| 1022 |
– |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 1023 |
– |
=& \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
| 1024 |
– |
\right) . |
| 1025 |
– |
\end{align} |
| 1026 |
– |
% |
| 1049 |
|
We list below the force equations written in terms of lab-frame |
| 1050 |
|
coordinates. The radial functions used in the two methods are listed |
| 1051 |
|
in Table \ref{tab:tableFORCE} |
| 1156 |
|
\begin{split} |
| 1157 |
|
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
| 1158 |
|
\Bigl[ |
| 1159 |
< |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} \hat{r} |
| 1160 |
< |
+ 2 \text{Tr} ( \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_k(r) \hat{r} \\ |
| 1159 |
> |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 1160 |
> |
+ 2 \mathbf{Q}_{\mathbf{a}} : \mathbf{Q}_{\mathbf{b}} \Bigr] w_k(r) \hat{r} \\ |
| 1161 |
|
% 2 |
| 1162 |
|
&+ \Bigl[ |
| 1163 |
|
2\text{Tr}\mathbf{Q}_{\mathbf{b}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
| 1193 |
|
% Torques SECTION ----------------------------------------------------------------------------------------- |
| 1194 |
|
% |
| 1195 |
|
\subsection{Torques} |
| 1196 |
< |
When energies are written in the form of Eq.~({\ref{ugeneral}), then |
| 1175 |
< |
torques can be found in a relatively straightforward |
| 1176 |
< |
manner,\cite{Allen:2006fk} |
| 1196 |
> |
|
| 1197 |
|
% |
| 1178 |
– |
\begin{eqnarray} |
| 1179 |
– |
\mathbf{\tau}_{\bf a} = |
| 1180 |
– |
\sum_m |
| 1181 |
– |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 1182 |
– |
( \hat{r} \times \hat{a}_m ) |
| 1183 |
– |
-\sum_{mn} |
| 1184 |
– |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 1185 |
– |
(\hat{a}_m \times \hat{b}_n) \\ |
| 1186 |
– |
% |
| 1187 |
– |
\mathbf{\tau}_{\bf b} = |
| 1188 |
– |
\sum_m |
| 1189 |
– |
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 1190 |
– |
( \hat{r} \times \hat{b}_m) |
| 1191 |
– |
+\sum_{mn} |
| 1192 |
– |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 1193 |
– |
(\hat{a}_m \times \hat{b}_n) . |
| 1194 |
– |
\end{eqnarray} |
| 1195 |
– |
% |
| 1196 |
– |
% |
| 1198 |
|
The torques for both the Taylor-Shifted as well as Gradient-Shifted |
| 1199 |
|
methods are given in space-frame coordinates: |
| 1200 |
|
% |
| 1356 |
|
$\mathbf{b}$ can be obtained by swapping indices in the expressions |
| 1357 |
|
above. |
| 1358 |
|
|
| 1359 |
+ |
\section{Related real-space methods} |
| 1360 |
+ |
One can also formulate an extension of the Wolf approach for point |
| 1361 |
+ |
multipoles by simply projecting the image multipole onto the surface |
| 1362 |
+ |
of the cutoff sphere, and including the interactions with the central |
| 1363 |
+ |
multipole and the image. This effectively shifts the total potential |
| 1364 |
+ |
to zero at the cutoff radius, |
| 1365 |
+ |
\begin{equation} |
| 1366 |
+ |
U^{\text{SP}} = \sum \left[ U(\mathbf{r}, \hat{\mathbf{a}}, \hat{\mathbf{b}}) - |
| 1367 |
+ |
U(\mathbf{r}_c,\hat{\mathbf{a}}, \hat{\mathbf{b}}) \right] |
| 1368 |
+ |
\label{eq:SP} |
| 1369 |
+ |
\end{equation} |
| 1370 |
+ |
where the sum describes separate potential shifting that is applied to |
| 1371 |
+ |
each orientational contribution to the energy. |
| 1372 |
+ |
|
| 1373 |
+ |
The energies and torques for the shifted potential (SP) can be easily |
| 1374 |
+ |
obtained by zeroing out the $(r-r_c)$ terms in the final column of |
| 1375 |
+ |
table \ref{tab:tableenergy}. Forces for the SP method retain |
| 1376 |
+ |
discontinuities at the cutoff sphere, and can be obtained by |
| 1377 |
+ |
eliminating all functions that depend on $r_c$ in the last column of |
| 1378 |
+ |
table \ref{tab:tableFORCE}. The self-energy contributions for the SP |
| 1379 |
+ |
potential are identical to both the GSF and TSF methods. |
| 1380 |
+ |
|
| 1381 |
|
\section{Comparison to known multipolar energies} |
| 1382 |
|
|
| 1383 |
|
To understand how these new real-space multipole methods behave in |
| 1384 |
|
computer simulations, it is vital to test against established methods |
| 1385 |
|
for computing electrostatic interactions in periodic systems, and to |
| 1386 |
|
evaluate the size and sources of any errors that arise from the |
| 1387 |
< |
real-space cutoffs. In this paper we test Taylor-shifted and |
| 1388 |
< |
Gradient-shifted electrostatics against analytical methods for |
| 1389 |
< |
computing the energies of ordered multipolar arrays. In the following |
| 1390 |
< |
paper, we test the new methods against the multipolar Ewald sum for |
| 1391 |
< |
computing the energies, forces and torques for a wide range of typical |
| 1392 |
< |
condensed-phase (disordered) systems. |
| 1387 |
> |
real-space cutoffs. In this paper we test both TSF and GSF |
| 1388 |
> |
electrostatics against analytical methods for computing the energies |
| 1389 |
> |
of ordered multipolar arrays. In the following paper, we test the new |
| 1390 |
> |
methods against the multipolar Ewald sum for computing the energies, |
| 1391 |
> |
forces and torques for a wide range of typical condensed-phase |
| 1392 |
> |
(disordered) systems. |
| 1393 |
|
|
| 1394 |
|
Because long-range electrostatic effects can be significant in |
| 1395 |
|
crystalline materials, ordered multipolar arrays present one of the |
| 1399 |
|
magnetization and obtained a number of these constants.\cite{Sauer} |
| 1400 |
|
This theory was developed more completely by Luttinger and |
| 1401 |
|
Tisza\cite{LT,LT2} who tabulated energy constants for the Sauer arrays |
| 1402 |
< |
and other periodic structures. We have repeated the Luttinger \& |
| 1380 |
< |
Tisza series summations to much higher order and obtained the energy |
| 1381 |
< |
constants (converged to one part in $10^9$) in table \ref{tab:LT}. |
| 1402 |
> |
and other periodic structures. |
| 1403 |
|
|
| 1404 |
< |
\begin{table*}[h] |
| 1405 |
< |
\centering{ |
| 1406 |
< |
\caption{Luttinger \& Tisza arrays and their associated |
| 1407 |
< |
energy constants. Type "A" arrays have nearest neighbor strings of |
| 1408 |
< |
antiparallel dipoles. Type "B" arrays have nearest neighbor |
| 1409 |
< |
strings of antiparallel dipoles if the dipoles are contained in a |
| 1410 |
< |
plane perpendicular to the dipole direction that passes through |
| 1390 |
< |
the dipole.} |
| 1391 |
< |
} |
| 1392 |
< |
\label{tab:LT} |
| 1393 |
< |
\begin{ruledtabular} |
| 1394 |
< |
\begin{tabular}{cccc} |
| 1395 |
< |
Array Type & Lattice & Dipole Direction & Energy constants \\ \hline |
| 1396 |
< |
A & SC & 001 & -2.676788684 \\ |
| 1397 |
< |
A & BCC & 001 & 0 \\ |
| 1398 |
< |
A & BCC & 111 & -1.770078733 \\ |
| 1399 |
< |
A & FCC & 001 & 2.166932835 \\ |
| 1400 |
< |
A & FCC & 011 & -1.083466417 \\ |
| 1401 |
< |
B & SC & 001 & -2.676788684 \\ |
| 1402 |
< |
B & BCC & 001 & -1.338394342 \\ |
| 1403 |
< |
B & BCC & 111 & -1.770078733 \\ |
| 1404 |
< |
B & FCC & 001 & -1.083466417 \\ |
| 1405 |
< |
B & FCC & 011 & -1.807573634 \\ |
| 1406 |
< |
-- & BCC & minimum & -1.985920929 \\ |
| 1407 |
< |
\end{tabular} |
| 1408 |
< |
\end{ruledtabular} |
| 1409 |
< |
\end{table*} |
| 1410 |
< |
|
| 1411 |
< |
In addition to the A and B arrays, there is an additional minimum |
| 1404 |
> |
To test the new electrostatic methods, we have constructed very large, |
| 1405 |
> |
$N=$ 16,000~(bcc) arrays of dipoles in the orientations described in |
| 1406 |
> |
Ref. \onlinecite{LT}. These structures include ``A'' lattices with |
| 1407 |
> |
nearest neighbor chains of antiparallel dipoles, as well as ``B'' |
| 1408 |
> |
lattices with nearest neighbor strings of antiparallel dipoles if the |
| 1409 |
> |
dipoles are contained in a plane perpendicular to the dipole direction |
| 1410 |
> |
that passes through the dipole. We have also studied the minimum |
| 1411 |
|
energy structure for the BCC lattice that was found by Luttinger \& |
| 1412 |
|
Tisza. The total electrostatic energy for any of the arrays is given |
| 1413 |
|
by: |
| 1414 |
|
\begin{equation} |
| 1415 |
|
E = C N^2 \mu^2 |
| 1416 |
|
\end{equation} |
| 1417 |
< |
where $C$ is the energy constant given in table \ref{tab:LT}, $N$ is |
| 1418 |
< |
the number of dipoles per unit volume, and $\mu$ is the strength of |
| 1419 |
< |
the dipole. |
| 1417 |
> |
where $C$ is the energy constant (equivalent to the Madelung |
| 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. |
| 1421 |
|
|
| 1422 |
– |
To test the new electrostatic methods, we have constructed very large, |
| 1423 |
– |
$N$ = 8,000~(sc), 16,000~(bcc), or 32,000~(fcc) arrays of dipoles in |
| 1424 |
– |
the orientations described in table \ref{tab:LT}. For the purposes of |
| 1425 |
– |
testing the energy expressions and the self-neutralization schemes, |
| 1426 |
– |
the primary quantity of interest is the analytic energy constant for |
| 1427 |
– |
the perfect arrays. Convergence to these constants are shown as a |
| 1428 |
– |
function of both the cutoff radius, $r_c$, and the damping parameter, |
| 1429 |
– |
$\alpha$ in Figs. \ref{fig:energyConstVsCutoff} and XXX. We have |
| 1430 |
– |
simultaneously tested a hard cutoff (where the kernel is simply |
| 1431 |
– |
truncated at the cutoff radius), as well as a shifted potential (SP) |
| 1432 |
– |
form which includes a potential-shifting and self-interaction term, |
| 1433 |
– |
but does not shift the forces and torques smoothly at the cutoff |
| 1434 |
– |
radius. |
| 1435 |
– |
|
| 1422 |
|
\begin{figure} |
| 1423 |
< |
\includegraphics[width=4.5in]{energyConstVsCutoff} |
| 1424 |
< |
\caption{Convergence to the analytic energy constants as a function of |
| 1425 |
< |
cutoff radius (normalized by the lattice constant) for the different |
| 1426 |
< |
real-space methods. The two crystals shown here are the ``B'' array |
| 1427 |
< |
for bcc crystals with the dipoles along the 001 direction (upper), |
| 1428 |
< |
as well as the minimum energy bcc lattice (lower). The analytic |
| 1429 |
< |
energy constants are shown as a grey dashed line. The left panel |
| 1430 |
< |
shows results for the undamped kernel ($1/r$), while the damped |
| 1431 |
< |
error function kernel, $B_0(r)$ was used in the right panel. } |
| 1432 |
< |
\label{fig:energyConstVsCutoff} |
| 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. \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. |
| 1456 |
+ |
|
| 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 |
| 1463 |
|
for obtaining accurate energies. The Taylor-shifted force (TSF) |
| 1464 |
|
approximation appears to perturb the potential too much inside the |
| 1465 |
|
cutoff region to provide accurate measures of the energy constants. |
| 1458 |
– |
|
| 1459 |
– |
|
| 1466 |
|
{\it Quadrupolar} analogues to the Madelung constants were first |
| 1467 |
|
worked out by Nagai and Nakamura who computed the energies of selected |
| 1468 |
|
quadrupole arrays based on extensions to the Luttinger and Tisza |
| 1469 |
|
approach.\cite{Nagai01081960,Nagai01091963} We have compared the |
| 1470 |
|
energy constants for the lowest energy configurations for linear |
| 1471 |
< |
quadrupoles shown in table \ref{tab:NNQ} |
| 1471 |
> |
quadrupoles. |
| 1472 |
|
|
| 1467 |
– |
\begin{table*} |
| 1468 |
– |
\centering{ |
| 1469 |
– |
\caption{Nagai and Nakamura Quadurpolar arrays}} |
| 1470 |
– |
\label{tab:NNQ} |
| 1471 |
– |
\begin{ruledtabular} |
| 1472 |
– |
\begin{tabular}{ccc} |
| 1473 |
– |
Lattice & Quadrupole Direction & Energy constants \\ \hline |
| 1474 |
– |
SC & 111 & -8.3 \\ |
| 1475 |
– |
BCC & 011 & -21.7 \\ |
| 1476 |
– |
FCC & 111 & -80.5 |
| 1477 |
– |
\end{tabular} |
| 1478 |
– |
\end{ruledtabular} |
| 1479 |
– |
\end{table*} |
| 1480 |
– |
|
| 1473 |
|
In analogy to the dipolar arrays, the total electrostatic energy for |
| 1474 |
|
the quadrupolar arrays is: |
| 1475 |
|
\begin{equation} |
| 1476 |
< |
E = C \frac{3}{4} N^2 Q^2 |
| 1476 |
> |
E = C N \frac{3\bar{Q}^2}{4a^5} |
| 1477 |
|
\end{equation} |
| 1478 |
< |
where $Q$ is the quadrupole moment. |
| 1478 |
> |
where $a$ is the lattice parameter, and $\bar{Q}$ is the effective |
| 1479 |
> |
quadrupole moment, |
| 1480 |
> |
\begin{equation} |
| 1481 |
> |
\bar{Q}^2 = 2 \left(3 Q : Q - (\text{Tr} Q)^2 \right) |
| 1482 |
> |
\end{equation} |
| 1483 |
> |
for the primitive quadrupole as defined in Eq. \ref{eq:quadrupole}. |
| 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.pdf} |
| 1489 |
+ |
\caption{Convergence of the lattice energy constants as a function of |
| 1490 |
+ |
cutoff radius (normalized by the lattice constant, $a$) for the new |
| 1491 |
+ |
real-space methods. Three quadrupolar crystal structures were |
| 1492 |
+ |
sampled, and the analytic energy constants for the three lattices |
| 1493 |
+ |
are indicated with grey dashed lines. The left panel shows results |
| 1494 |
+ |
for the undamped kernel ($1/r$), while the damped error function |
| 1495 |
+ |
kernel, $B_0(r)$ was used in the right panel.} |
| 1496 |
+ |
\label{fig:QuadrupolesrcutCovergence} |
| 1497 |
+ |
\end{figure} |
| 1498 |
+ |
|
| 1499 |
+ |
|
| 1500 |
+ |
\begin{figure}[!htbp] |
| 1501 |
+ |
\includegraphics[width=3.5in]{Quadrupoles_alphaConvergence-crop.pdf} |
| 1502 |
+ |
\caption{Convergence to the analytic energy constants as a function of |
| 1503 |
+ |
cutoff damping alpha for the different |
| 1504 |
+ |
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. |
| 1505 |
+ |
The left panel shows results for the undamped kernel ($1/r$), while the damped |
| 1506 |
+ |
error function kernel, $B_0(r)$ was used in the right panel. } |
| 1507 |
+ |
\label{fig:Quadrupoles_alphaCovergence-crop.pdf} |
| 1508 |
+ |
\end{figure} |
| 1509 |
+ |
|
| 1510 |
+ |
|
| 1511 |
|
\section{Conclusion} |
| 1512 |
|
We have presented two efficient real-space methods for computing the |
| 1513 |
|
interactions between point multipoles. These methods have the benefit |
| 1536 |
|
\begin{acknowledgments} |
| 1537 |
|
JDG acknowledges helpful discussions with Christopher |
| 1538 |
|
Fennell. Support for this project was provided by the National |
| 1539 |
< |
Science Foundation under grant CHE-0848243. Computational time was |
| 1539 |
> |
Science Foundation under grant CHE-1362211. Computational time was |
| 1540 |
|
provided by the Center for Research Computing (CRC) at the |
| 1541 |
|
University of Notre Dame. |
| 1542 |
|
\end{acknowledgments} |
| 1632 |
|
\begin{equation} |
| 1633 |
|
u_4(r)=B_0^{(5)}(r) - B_0^{(5)}(r_c) . |
| 1634 |
|
\end{equation} |
| 1635 |
< |
|
| 1635 |
> |
% The functions |
| 1636 |
> |
% needed are listed schematically below: |
| 1637 |
> |
% % |
| 1638 |
> |
% \begin{eqnarray} |
| 1639 |
> |
% f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
| 1640 |
> |
% g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
| 1641 |
> |
% h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
| 1642 |
> |
% s_2 \quad s_3 \quad &s_4 \nonumber \\ |
| 1643 |
> |
% t_3 \quad &t_4 \nonumber \\ |
| 1644 |
> |
% &u_4 \nonumber . |
| 1645 |
> |
% \end{eqnarray} |
| 1646 |
|
The functions $f_n(r)$ to $u_n(r)$ can be computed recursively and |
| 1647 |
< |
stored on a grid for values of $r$ from $0$ to $r_c$. The functions |
| 1648 |
< |
needed are listed schematically below: |
| 1647 |
> |
stored on a grid for values of $r$ from $0$ to $r_c$. Using these |
| 1648 |
> |
functions, we find |
| 1649 |
|
% |
| 1617 |
– |
\begin{eqnarray} |
| 1618 |
– |
f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
| 1619 |
– |
g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
| 1620 |
– |
h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
| 1621 |
– |
s_2 \quad s_3 \quad &s_4 \nonumber \\ |
| 1622 |
– |
t_3 \quad &t_4 \nonumber \\ |
| 1623 |
– |
&u_4 \nonumber . |
| 1624 |
– |
\end{eqnarray} |
| 1625 |
– |
|
| 1626 |
– |
Using these functions, we find |
| 1627 |
– |
% |
| 1650 |
|
\begin{align} |
| 1651 |
|
\frac{\partial f_n}{\partial r_\alpha} =&r_\alpha \frac {g_n}{r} \label{eq:b9}\\ |
| 1652 |
|
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =&\delta_{\alpha \beta}\frac {g_n}{r} |
| 1653 |
|
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) \\ |
| 1654 |
< |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} =& |
| 1654 |
> |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta \partial r_\gamma} =& |
| 1655 |
|
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
| 1656 |
|
\delta_{ \beta \gamma} r_\alpha \right) |
| 1657 |
< |
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) |
| 1658 |
< |
+ r_\alpha r_\beta r_\gamma |
| 1657 |
> |
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) \nonumber \\ |
| 1658 |
> |
& + r_\alpha r_\beta r_\gamma |
| 1659 |
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \\ |
| 1660 |
< |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} =& |
| 1660 |
> |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta \partial |
| 1661 |
> |
r_\gamma \partial r_\delta} =& |
| 1662 |
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
| 1663 |
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
| 1664 |
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
| 1671 |
|
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
| 1672 |
|
+ \frac{t_n}{r^4} \right)\\ |
| 1673 |
|
\frac{\partial^5 f_n} |
| 1674 |
< |
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} =& |
| 1674 |
> |
{\partial r_\alpha \partial r_\beta \partial r_\gamma \partial |
| 1675 |
> |
r_\delta \partial r_\epsilon} =& |
| 1676 |
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
| 1677 |
|
+ \text{14 permutations} \right) |
| 1678 |
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
| 1694 |
|
rather the individual terms in the multipole interaction energies. |
| 1695 |
|
For damped charges , this still brings into the algebra multiple |
| 1696 |
|
derivatives of the Smith's $B_0(r)$ function. To denote these terms, |
| 1697 |
< |
we generalize the notation of the previous appendix. For $f(r)=1/r$ |
| 1698 |
< |
(bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
| 1697 |
> |
we generalize the notation of the previous appendix. For either |
| 1698 |
> |
$f(r)=1/r$ (undamped) or $f(r)=B_0(r)$ (damped), |
| 1699 |
|
% |
| 1700 |
|
\begin{align} |
| 1701 |
|
g(r)=& \frac{df}{d r}\\ |
| 1705 |
|
u(r)=& \frac{dt}{d r} = \frac{d^5f}{d r^5} . |
| 1706 |
|
\end{align} |
| 1707 |
|
% |
| 1708 |
< |
For undamped charges, $f(r)=1/r$, Table I lists these derivatives |
| 1709 |
< |
under the column ``Bare Coulomb.'' Equations \ref{eq:b9} to |
| 1710 |
< |
\ref{eq:b13} are still correct for GSF electrostatics if the subscript |
| 1687 |
< |
$n$ is eliminated. |
| 1708 |
> |
For undamped charges Table I lists these derivatives under the column |
| 1709 |
> |
``Bare Coulomb.'' Equations \ref{eq:b9} to \ref{eq:b13} are still |
| 1710 |
> |
correct for GSF electrostatics if the subscript $n$ is eliminated. |
| 1711 |
|
|
| 1712 |
|
\newpage |
| 1713 |
|
|
