| 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} |
| 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}(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 , |
| 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 has linear-scaling with the number of |
| 169 |
|
particles. Here $\Omega_i$ and $\Omega_j$ represent orientational |
| 170 |
|
coordinates of the two sites. The computational efficiency, energy |
| 171 |
|
conservation, and even some physical properties of a simulation can |
| 471 |
|
the radial factors differ between the two methods. |
| 472 |
|
|
| 473 |
|
\subsection{\label{sec:level2}Body and space axes} |
| 474 |
< |
|
| 475 |
< |
[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} |
| 474 |
> |
Although objects $\bf a$ and $\bf b$ rotate during a molecular |
| 475 |
> |
dynamics (MD) simulation, their multipole tensors remain fixed in |
| 476 |
> |
body-frame coordinates. While deriving force and torque expressions, |
| 477 |
> |
it is therefore convenient to write the energies, forces, and torques |
| 478 |
> |
in intermediate forms involving the vectors of the rotation matrices. |
| 479 |
> |
We denote body axes for objects $\bf a$ and $\bf b$ using unit vectors |
| 480 |
> |
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$. |
| 481 |
> |
In a typical simulation , the initial axes are obtained by |
| 482 |
> |
diagonalizing the moment of inertia tensors for the objects. (N.B., |
| 483 |
> |
the body axes are generally {\it not} the same as those for which the |
| 484 |
> |
quadrupole moment is diagonal.) The rotation matrices are then |
| 485 |
> |
propagated during the simulation. |
| 486 |
|
|
| 487 |
< |
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 |
| 487 |
> |
The rotation matrices $\hat{\mathbf {a}}$ and $\hat{\mathbf {b}}$ can be |
| 488 |
|
expressed using these unit vectors: |
| 489 |
|
\begin{eqnarray} |
| 490 |
|
\hat{\mathbf {a}} = |
| 492 |
|
\hat{a}_1 \\ |
| 493 |
|
\hat{a}_2 \\ |
| 494 |
|
\hat{a}_3 |
| 495 |
< |
\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}\\ |
| 495 |
> |
\end{pmatrix}, \qquad |
| 496 |
|
\hat{\mathbf {b}} = |
| 497 |
|
\begin{pmatrix} |
| 498 |
|
\hat{b}_1 \\ |
| 499 |
|
\hat{b}_2 \\ |
| 500 |
|
\hat{b}_3 |
| 501 |
|
\end{pmatrix} |
| 520 |
– |
= |
| 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} . |
| 502 |
|
\end{eqnarray} |
| 503 |
|
% |
| 504 |
|
These matrices convert from space-fixed $(xyz)$ to body-fixed $(123)$ |
| 505 |
< |
coordinates. All contractions of prefactors with derivatives of |
| 506 |
< |
functions can be written in terms of these matrices. It proves to be |
| 507 |
< |
equally convenient to just write any contraction in terms of unit |
| 508 |
< |
vectors $\hat{r}$, $\hat{a}_m$, and $\hat{b}_n$. In the torque |
| 509 |
< |
expressions, it is useful to have the angular-dependent terms |
| 510 |
< |
available in three different fashions, e.g. for the dipole-dipole |
| 535 |
< |
contraction: |
| 505 |
> |
coordinates. |
| 506 |
> |
|
| 507 |
> |
Allen and Germano,\cite{Allen:2006fk} following earlier work by Price |
| 508 |
> |
{\em et al.},\cite{Price:1984fk} showed that if the interaction |
| 509 |
> |
energies are written explicitly in terms of $\hat{r}$ and the body |
| 510 |
> |
axes ($\hat{a}_m$, $\hat{b}_n$) : |
| 511 |
|
% |
| 512 |
|
\begin{equation} |
| 513 |
+ |
U(r, \{\hat{a}_m \cdot \hat{r} \}, |
| 514 |
+ |
\{\hat{b}_n\cdot \hat{r} \}, |
| 515 |
+ |
\{\hat{a}_m \cdot \hat{b}_n \}) . |
| 516 |
+ |
\label{ugeneral} |
| 517 |
+ |
\end{equation} |
| 518 |
+ |
% |
| 519 |
+ |
the forces come out relatively cleanly, |
| 520 |
+ |
% |
| 521 |
+ |
\begin{equation} |
| 522 |
+ |
\mathbf{F}_{\bf a}=-\mathbf{F}_{\bf b} = \frac{\partial U}{\partial \mathbf{r}} |
| 523 |
+ |
= \frac{\partial U}{\partial r} \hat{r} |
| 524 |
+ |
+ \sum_m \left[ |
| 525 |
+ |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 526 |
+ |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 527 |
+ |
+ \frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 528 |
+ |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 529 |
+ |
\right] \label{forceequation}. |
| 530 |
+ |
\end{equation} |
| 531 |
+ |
|
| 532 |
+ |
The torques can also be found in a relatively similar |
| 533 |
+ |
manner, |
| 534 |
+ |
% |
| 535 |
+ |
\begin{eqnarray} |
| 536 |
+ |
\mathbf{\tau}_{\bf a} = |
| 537 |
+ |
\sum_m |
| 538 |
+ |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{r})} |
| 539 |
+ |
( \hat{r} \times \hat{a}_m ) |
| 540 |
+ |
-\sum_{mn} |
| 541 |
+ |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 542 |
+ |
(\hat{a}_m \times \hat{b}_n) \\ |
| 543 |
+ |
% |
| 544 |
+ |
\mathbf{\tau}_{\bf b} = |
| 545 |
+ |
\sum_m |
| 546 |
+ |
\frac{\partial U}{\partial (\hat{b}_m \cdot \hat{r})} |
| 547 |
+ |
( \hat{r} \times \hat{b}_m) |
| 548 |
+ |
+\sum_{mn} |
| 549 |
+ |
\frac{\partial U}{\partial (\hat{a}_m \cdot \hat{b}_n)} |
| 550 |
+ |
(\hat{a}_m \times \hat{b}_n) . |
| 551 |
+ |
\end{eqnarray} |
| 552 |
+ |
|
| 553 |
+ |
Note that our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ |
| 554 |
+ |
is opposite in sign to that of Allen and Germano.\cite{Allen:2006fk} |
| 555 |
+ |
We also made use of the identities, |
| 556 |
+ |
% |
| 557 |
+ |
\begin{align} |
| 558 |
+ |
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 559 |
+ |
=& \frac{1}{r} \left( \hat{a}_m - (\hat{a}_m \cdot \hat{r})\hat{r} |
| 560 |
+ |
\right) \\ |
| 561 |
+ |
\frac { \partial (\hat{b}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 562 |
+ |
=& \frac{1}{r} \left( \hat{b}_m - (\hat{b}_m \cdot \hat{r})\hat{r} |
| 563 |
+ |
\right) . |
| 564 |
+ |
\end{align} |
| 565 |
+ |
|
| 566 |
+ |
Many of the multipole contractions required can be written in one of |
| 567 |
+ |
three equivalent forms using the unit vectors $\hat{r}$, $\hat{a}_m$, |
| 568 |
+ |
and $\hat{b}_n$. In the torque expressions, it is useful to have the |
| 569 |
+ |
angular-dependent terms available in all three fashions, e.g. for the |
| 570 |
+ |
dipole-dipole contraction: |
| 571 |
+ |
% |
| 572 |
+ |
\begin{equation} |
| 573 |
|
\mathbf{D}_{\mathbf {a}} \cdot \mathbf{D}_{\mathbf{b}} |
| 574 |
|
= D_{\bf {a}\alpha} D_{\bf {b}\alpha} = |
| 575 |
|
\sum_{mn} {D_{\mathbf{a}m} \hat{a}_m \cdot \hat{b}_n D_{\mathbf{b}n}} |
| 581 |
|
explicit sums over body indices and the dot products now indicate |
| 582 |
|
contractions using space indices. |
| 583 |
|
|
| 584 |
+ |
In computing our force and torque expressions, we carried out most of |
| 585 |
+ |
the work in body coordinates, and have transformed the expressions |
| 586 |
+ |
back to space-frame coordinates, which are reported below. Interested |
| 587 |
+ |
readers may consult the supplemental information for this paper for |
| 588 |
+ |
the intermediate body-frame expressions. |
| 589 |
|
|
| 590 |
|
\subsection{The Self-Interaction \label{sec:selfTerm}} |
| 591 |
|
|
| 651 |
|
Charge & $C_{\bf a}^2$ & $-f(r_c)$ & $-\frac{\alpha}{\sqrt{\pi}}$ \\ |
| 652 |
|
Dipole & $|\mathbf{D}_{\bf a}|^2$ & $\frac{1}{3} \left( h(r_c) + |
| 653 |
|
\frac{2 g(r_c)}{r_c} \right)$ & $-\frac{2 \alpha^3}{3 \sqrt{\pi}}$\\ |
| 654 |
< |
Quadrupole & $2 \text{Tr}(\mathbf{Q}_{\bf a}^2) + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
| 654 |
> |
Quadrupole & $2 \mathbf{Q}_{\bf a}:\mathbf{Q}_{\bf a} + \text{Tr}(\mathbf{Q}_{\bf a})^2$ & |
| 655 |
|
$- \frac{1}{15} \left( t(r_c)+ \frac{4 s(r_c)}{r_c} \right)$ & |
| 656 |
|
$-\frac{4 \alpha^5}{5 \sqrt{\pi}}$ \\ |
| 657 |
|
Charge-Quadrupole & $-2 C_{\bf a} \text{Tr}(\mathbf{Q}_{\bf a})$ & $\frac{1}{3} \left( |
| 781 |
|
U_{Q_{\bf a}Q_{\bf b}}(r)=& |
| 782 |
|
\Bigl[ |
| 783 |
|
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
| 784 |
< |
+2 \text{Tr} \left( |
| 785 |
< |
\mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} \right) \Bigr] v_{41}(r) |
| 784 |
> |
+2 |
| 785 |
> |
\mathbf{Q}_{\mathbf{a}} : \mathbf{Q}_{\mathbf{b}} \Bigr] v_{41}(r) |
| 786 |
|
\\ |
| 787 |
|
% 2 |
| 788 |
|
&+\Bigl[ \text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 1021 |
|
F_{\bf a \alpha} = \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}}{\partial r} |
| 1022 |
|
\quad \text{and} \quad F_{\bf b \alpha} |
| 1023 |
|
= - \hat{r}_\alpha \frac{\partial U_{C_{\bf a}C_{\bf b}}} {\partial r} . |
| 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} |
| 1024 |
|
\end{equation} |
| 1025 |
|
% |
| 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 |
– |
% |
| 1026 |
|
We list below the force equations written in terms of lab-frame |
| 1027 |
|
coordinates. The radial functions used in the two methods are listed |
| 1028 |
|
in Table \ref{tab:tableFORCE} |
| 1133 |
|
\begin{split} |
| 1134 |
|
\mathbf{F}_{{\bf a}Q_{\bf a}Q_{\bf b}} =& |
| 1135 |
|
\Bigl[ |
| 1136 |
< |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} \hat{r} |
| 1137 |
< |
+ 2 \text{Tr} ( \mathbf{Q}_{\mathbf{a}} \cdot \mathbf{Q}_{\mathbf{b}} ) \Bigr] w_k(r) \hat{r} \\ |
| 1136 |
> |
\text{Tr}\mathbf{Q}_{\mathbf{a}} \text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 1137 |
> |
+ 2 \mathbf{Q}_{\mathbf{a}} : \mathbf{Q}_{\mathbf{b}} \Bigr] w_k(r) \hat{r} \\ |
| 1138 |
|
% 2 |
| 1139 |
|
&+ \Bigl[ |
| 1140 |
|
2\text{Tr}\mathbf{Q}_{\mathbf{b}} (\hat{r} \cdot \mathbf{Q}_{\mathbf{a}} ) |
| 1170 |
|
% Torques SECTION ----------------------------------------------------------------------------------------- |
| 1171 |
|
% |
| 1172 |
|
\subsection{Torques} |
| 1173 |
< |
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} |
| 1177 |
< |
% |
| 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} |
| 1173 |
> |
|
| 1174 |
|
% |
| 1196 |
– |
% |
| 1175 |
|
The torques for both the Taylor-Shifted as well as Gradient-Shifted |
| 1176 |
|
methods are given in space-frame coordinates: |
| 1177 |
|
% |
| 1361 |
|
\begin{table*}[h] |
| 1362 |
|
\centering{ |
| 1363 |
|
\caption{Luttinger \& Tisza arrays and their associated |
| 1364 |
< |
energy constants. Type "A" arrays have nearest neighbor strings of |
| 1365 |
< |
antiparallel dipoles. Type "B" arrays have nearest neighbor |
| 1364 |
> |
energy constants. Type ``A'' arrays have nearest neighbor strings of |
| 1365 |
> |
antiparallel dipoles. Type ``B'' arrays have nearest neighbor |
| 1366 |
|
strings of antiparallel dipoles if the dipoles are contained in a |
| 1367 |
|
plane perpendicular to the dipole direction that passes through |
| 1368 |
|
the dipole.} |
| 1409 |
|
truncated at the cutoff radius), as well as a shifted potential (SP) |
| 1410 |
|
form which includes a potential-shifting and self-interaction term, |
| 1411 |
|
but does not shift the forces and torques smoothly at the cutoff |
| 1412 |
< |
radius. |
| 1412 |
> |
radius. The SP method is essentially an extension of the original |
| 1413 |
> |
Wolf method for multipoles. |
| 1414 |
|
|
| 1415 |
< |
\begin{figure} |
| 1415 |
> |
\begin{figure}[!htbp] |
| 1416 |
|
\includegraphics[width=4.5in]{energyConstVsCutoff} |
| 1417 |
|
\caption{Convergence to the analytic energy constants as a function of |
| 1418 |
|
cutoff radius (normalized by the lattice constant) for the different |
| 1588 |
|
\begin{equation} |
| 1589 |
|
u_4(r)=B_0^{(5)}(r) - B_0^{(5)}(r_c) . |
| 1590 |
|
\end{equation} |
| 1591 |
< |
|
| 1591 |
> |
% The functions |
| 1592 |
> |
% needed are listed schematically below: |
| 1593 |
> |
% % |
| 1594 |
> |
% \begin{eqnarray} |
| 1595 |
> |
% f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
| 1596 |
> |
% g_0 \quad g_1 \quad g_2 \quad g_3 \quad &g_4 \nonumber \\ |
| 1597 |
> |
% h_1 \quad h_2 \quad h_3 \quad &h_4 \nonumber \\ |
| 1598 |
> |
% s_2 \quad s_3 \quad &s_4 \nonumber \\ |
| 1599 |
> |
% t_3 \quad &t_4 \nonumber \\ |
| 1600 |
> |
% &u_4 \nonumber . |
| 1601 |
> |
% \end{eqnarray} |
| 1602 |
|
The functions $f_n(r)$ to $u_n(r)$ can be computed recursively and |
| 1603 |
< |
stored on a grid for values of $r$ from $0$ to $r_c$. The functions |
| 1604 |
< |
needed are listed schematically below: |
| 1616 |
< |
% |
| 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 |
| 1603 |
> |
stored on a grid for values of $r$ from $0$ to $r_c$. Using these |
| 1604 |
> |
functions, we find |
| 1605 |
|
% |
| 1606 |
|
\begin{align} |
| 1607 |
|
\frac{\partial f_n}{\partial r_\alpha} =&r_\alpha \frac {g_n}{r} \label{eq:b9}\\ |
| 1608 |
|
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =&\delta_{\alpha \beta}\frac {g_n}{r} |
| 1609 |
|
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) \\ |
| 1610 |
< |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} =& |
| 1610 |
> |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta \partial r_\gamma} =& |
| 1611 |
|
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
| 1612 |
|
\delta_{ \beta \gamma} r_\alpha \right) |
| 1613 |
< |
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) |
| 1614 |
< |
+ r_\alpha r_\beta r_\gamma |
| 1613 |
> |
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) \nonumber \\ |
| 1614 |
> |
& + r_\alpha r_\beta r_\gamma |
| 1615 |
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \\ |
| 1616 |
< |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} =& |
| 1616 |
> |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta \partial |
| 1617 |
> |
r_\gamma \partial r_\delta} =& |
| 1618 |
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
| 1619 |
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
| 1620 |
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
| 1627 |
|
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
| 1628 |
|
+ \frac{t_n}{r^4} \right)\\ |
| 1629 |
|
\frac{\partial^5 f_n} |
| 1630 |
< |
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} =& |
| 1630 |
> |
{\partial r_\alpha \partial r_\beta \partial r_\gamma \partial |
| 1631 |
> |
r_\delta \partial r_\epsilon} =& |
| 1632 |
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
| 1633 |
|
+ \text{14 permutations} \right) |
| 1634 |
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
| 1650 |
|
rather the individual terms in the multipole interaction energies. |
| 1651 |
|
For damped charges , this still brings into the algebra multiple |
| 1652 |
|
derivatives of the Smith's $B_0(r)$ function. To denote these terms, |
| 1653 |
< |
we generalize the notation of the previous appendix. For $f(r)=1/r$ |
| 1654 |
< |
(bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
| 1653 |
> |
we generalize the notation of the previous appendix. For either |
| 1654 |
> |
$f(r)=1/r$ (undamped) or $f(r)=B_0(r)$ (damped), |
| 1655 |
|
% |
| 1656 |
|
\begin{align} |
| 1657 |
|
g(r)=& \frac{df}{d r}\\ |
| 1661 |
|
u(r)=& \frac{dt}{d r} = \frac{d^5f}{d r^5} . |
| 1662 |
|
\end{align} |
| 1663 |
|
% |
| 1664 |
< |
For undamped charges, $f(r)=1/r$, Table I lists these derivatives |
| 1665 |
< |
under the column ``Bare Coulomb.'' Equations \ref{eq:b9} to |
| 1666 |
< |
\ref{eq:b13} are still correct for GSF electrostatics if the subscript |
| 1687 |
< |
$n$ is eliminated. |
| 1664 |
> |
For undamped charges Table I lists these derivatives under the column |
| 1665 |
> |
``Bare Coulomb.'' Equations \ref{eq:b9} to \ref{eq:b13} are still |
| 1666 |
> |
correct for GSF electrostatics if the subscript $n$ is eliminated. |
| 1667 |
|
|
| 1668 |
|
\newpage |
| 1669 |
|
|
