| 26 |
|
% reprint,% |
| 27 |
|
%author-year,% |
| 28 |
|
%author-numerical,% |
| 29 |
< |
]{revtex4-1} |
| 29 |
> |
jcp]{revtex4-1} |
| 30 |
|
|
| 31 |
|
\usepackage{graphicx}% Include figure files |
| 32 |
|
\usepackage{dcolumn}% Align table columns on decimal point |
| 316 |
|
these functions are known. Smith's convenient functions $B_l(r)$ are |
| 317 |
|
summarized in Appendix A. |
| 318 |
|
|
| 319 |
– |
\subsection{Taylor-shifted force (TSF) electrostatics} |
| 319 |
|
|
| 320 |
|
The main goal of this work is to smoothly cut off the interaction |
| 321 |
|
energy as well as forces and torques as $r\rightarrow r_c$. To |
| 323 |
|
and charge-dipole, using the bare Coulomb kernel $f(r)=1/r$ to explain |
| 324 |
|
the idea. |
| 325 |
|
|
| 326 |
+ |
\subsection{Shifted-force methods} |
| 327 |
|
In the shifted-force approximation, the interaction energy for two |
| 328 |
|
charges $C_{\bf a}$ and $C_{\bf b}$ separated by a distance $r$ is |
| 329 |
|
written: |
| 333 |
|
\right) . |
| 334 |
|
\end{equation} |
| 335 |
|
Two shifting terms appear in this equations, one from the |
| 336 |
< |
neutralization procedure ($-1/r_c$), and one that will make the first |
| 337 |
< |
derivative also vanish at the cutoff radius. |
| 336 |
> |
neutralization procedure ($-1/r_c$), and one that causes the first |
| 337 |
> |
derivative to vanish at the cutoff radius. |
| 338 |
|
|
| 339 |
|
Since one derivative of the interaction energy is needed for the |
| 340 |
|
force, the minimal perturbation is a term linear in $(r-r_c)$ in the |
| 346 |
|
\right) . |
| 347 |
|
\end{equation} |
| 348 |
|
There are a number of ways to generalize this derivative shift for |
| 349 |
< |
higher-order multipoles. |
| 349 |
> |
higher-order multipoles. Below, we present two methods, one based on |
| 350 |
> |
higher-order Taylor series for $r$ near $r_c$, and the other based on |
| 351 |
> |
linear shift of the kernel gradients at the cutoff itself. |
| 352 |
|
|
| 353 |
+ |
\subsection{Taylor-shifted force (TSF) electrostatics} |
| 354 |
|
In the Taylor-shifted force (TSF) method, the procedure that we follow |
| 355 |
|
is based on a Taylor expansion containing the same number of |
| 356 |
|
derivatives required for each force term to vanish at the cutoff. For |
| 360 |
|
include enough terms so that all energies, forces, and torques are |
| 361 |
|
zero as $r \rightarrow r_c$. In each case, we will subtract off a |
| 362 |
|
function $f_n^{\text{shift}}(r)$ from the kernel $f(r)=1/r$. The |
| 363 |
< |
index $n$ indicates the number of derivatives to be taken when |
| 363 |
> |
subscript $n$ indicates the number of derivatives to be taken when |
| 364 |
|
deriving a given multipole energy. We choose a function with |
| 365 |
|
guaranteed smooth derivatives --- a truncated Taylor series of the |
| 366 |
|
function $f(r)$, e.g., |
| 367 |
|
% |
| 368 |
|
\begin{equation} |
| 369 |
< |
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)} \Big \lvert _{r_c} . |
| 369 |
> |
f_n^{\text{shift}}(r)=\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} f^{(m)}(r_c) . |
| 370 |
|
\end{equation} |
| 371 |
|
% |
| 372 |
|
The combination of $f(r)$ with the shifted function is denoted $f_n(r)=f(r)-f_n^{\text{shift}}(r)$. |
| 386 |
|
\frac {r_\alpha}{r} \frac {\partial f_1(r)}{\partial r} . |
| 387 |
|
\end{equation} |
| 388 |
|
% |
| 389 |
< |
The force that dipole $\bf b$ puts on charge $\bf a$ is |
| 389 |
> |
The force that dipole $\bf b$ exerts on charge $\bf a$ is |
| 390 |
|
% |
| 391 |
|
\begin{equation} |
| 392 |
|
F_{C_{\bf a}D_{\bf b}\beta} =\frac{ C_{\bf a} D_{{\bf b}\alpha}}{4\pi \epsilon_0} |
| 396 |
|
+ \frac {\partial ^2} {\partial r^2} \right) \right] f_1(r) . |
| 397 |
|
\end{equation} |
| 398 |
|
% |
| 399 |
< |
For $f(r)=1/r$, we find |
| 399 |
> |
For undamped coulombic interactions, $f(r)=1/r$, we find |
| 400 |
|
% |
| 401 |
|
\begin{equation} |
| 402 |
|
F_{C_{\bf a}D_{\bf b}\beta} = |
| 408 |
|
% |
| 409 |
|
This expansion shows the expected $1/r^3$ dependence of the force. |
| 410 |
|
|
| 411 |
< |
In general, we write |
| 411 |
> |
In general, we can write |
| 412 |
|
% |
| 413 |
|
\begin{equation} |
| 414 |
|
U=\frac{1}{4\pi \epsilon_0} (\text{prefactor}) (\text{derivatives}) f_n(r) |
| 422 |
|
$\partial^4/\partial r_\alpha \partial r_\beta \partial r_\gamma \partial r_\delta$, with |
| 423 |
|
implied summation combining the space indices. |
| 424 |
|
|
| 425 |
< |
To apply this method to the smeared-charge approach, |
| 426 |
< |
we write $f(r)=\text{erfc}(\alpha r)/r$. By using one function $f(r)$ for both |
| 427 |
< |
approaches, we simplify the tabulation of equations used. Because |
| 428 |
< |
of the many derivatives that are taken, the algebra is tedious and are summarized |
| 429 |
< |
in Appendices A and B. |
| 425 |
> |
In the formulas presented in the tables below, the placeholder |
| 426 |
> |
function $f(r)$ is used to represent the electrostatic kernel (either |
| 427 |
> |
damped or undamped). The main functions that go into the force and |
| 428 |
> |
torque terms, $f_n(r), g_n(r), h_n(r), s_n(r), \mathrm{~and~} t_n(r)$ |
| 429 |
> |
are successive derivatives of the shifted electrostatic kernel of the |
| 430 |
> |
same index $n$. The algebra required to evaluate energies, forces and |
| 431 |
> |
torques is somewhat tedious and are summarized in Appendices A and B. |
| 432 |
|
|
| 433 |
|
\subsection{Gradient-shifted force (GSF) electrostatics} |
| 434 |
< |
|
| 435 |
< |
Note the method used in the previous subsection to shift a force is basically that of using |
| 436 |
< |
a truncated Taylor Series in the radius $r$. An alternate method exists, best explained by |
| 437 |
< |
writing one shifted formula for all interaction energies $U(r)$: |
| 434 |
> |
Note the method used in the previous subsection to smoothly shift the |
| 435 |
> |
force to zero is a truncated Taylor Series in the radius $r$. The |
| 436 |
> |
second method maintains only the linear $(r-r_c)$ term and has a |
| 437 |
> |
similar interaction energy for all multipole orders: |
| 438 |
|
\begin{equation} |
| 439 |
|
U^{\text{shift}}(r)=U(r)-U(r_c)-(r-r_c)\hat{r}\cdot \nabla U(r) \Big \lvert _{r_c} . |
| 440 |
|
\end{equation} |
| 441 |
< |
Note that this method uses only the linear term, $(r-r_c)$ in the Taylor series, no higher order terms |
| 442 |
< |
$(r-r_c)^n$ appear. The primary difference between methods 1 and 2 originates |
| 443 |
< |
with the stage in the derivation where the Taylor Series is applied; in method 1, it is applied to the |
| 444 |
< |
kernel. In method 2, it is applied to individual interaction energies of the multipole expansion. |
| 441 |
> |
No higher order terms $(r-r_c)^n$ appear. The primary difference |
| 442 |
> |
between the TSF and GSF methods is the stage at which the Taylor |
| 443 |
> |
Series is applied; in the Taylor-shifted approach, it is applied to |
| 444 |
> |
the kernel itself. In the Gradient-shifted approach, it is applied to |
| 445 |
> |
individual radial interactions terms in the multipole expansion. |
| 446 |
|
Terms from this method thus have the general form: |
| 447 |
|
\begin{equation} |
| 448 |
|
U=\frac{1}{4\pi \epsilon_0}\sum (\text{angular factor}) (\text{radial factor}). |
| 449 |
|
\label{generic2} |
| 450 |
|
\end{equation} |
| 451 |
|
|
| 452 |
< |
Results for both methods can be summarized using the form of Eq.~(\ref{generic2}) |
| 453 |
< |
and are listed in Table I below. |
| 452 |
> |
Results for both methods can be summarized using the form of |
| 453 |
> |
Eq.~(\ref{generic2}) and are listed in Table I below. |
| 454 |
|
|
| 455 |
|
\subsection{\label{sec:level2}Body and space axes} |
| 456 |
|
|
| 457 |
< |
Up to this point, all energies and forces have been written in terms of fixed space |
| 458 |
< |
coordinates $x$, $y$, $z$. Interaction energies are computed from the generic formulas Eq.~(\ref{generic}) and ~(\ref{generic2}) which |
| 459 |
< |
combine prefactors with radial functions. But because objects |
| 460 |
< |
$\bf a$ and $\bf b$ both translate and rotate as part of a MD simulation, |
| 461 |
< |
it is desirable to contract all $r$-dependent terms with dipole and quadrupole |
| 462 |
< |
moments expressed in terms of their body axes. |
| 463 |
< |
Since the interaction energy expressions will be used to derive both forces and torques, |
| 464 |
< |
we follow here the development of Allen and Germano, which was itself based on an |
| 465 |
< |
earlier paper by Price \em et al.\em |
| 457 |
> |
So far, all energies and forces have been written in terms of fixed |
| 458 |
> |
space coordinates $x$, $y$, $z$. Interaction energies are computed |
| 459 |
> |
from the generic formulas Eq.~(\ref{generic}) and ~(\ref{generic2}) |
| 460 |
> |
which combine prefactors with radial functions. Because objects $\bf |
| 461 |
> |
a$ and $\bf b$ both translate and rotate during a molecular dynamics |
| 462 |
> |
(MD) simulation, it is desirable to contract all $r$-dependent terms |
| 463 |
> |
with dipole and quadrupole moments expressed in terms of their body |
| 464 |
> |
axes. To do so, we follow the methodology of Allen and |
| 465 |
> |
Germano,\cite{Allen:2006fk} which was itself based on an earlier paper |
| 466 |
> |
by Price {\em et al.}\cite{Price:1984fk} |
| 467 |
|
|
| 468 |
< |
Denote body axes for objects $\bf a$ and $\bf b$ by unit vectors |
| 469 |
< |
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$ referring to a convenient |
| 470 |
< |
set of inertial body axes. (Note, these body axes are generally not the same as those for which the |
| 471 |
< |
quadrupole moment is diagonal.) Then, |
| 468 |
> |
We denote body axes for objects $\bf a$ and $\bf b$ by unit vectors |
| 469 |
> |
$\hat{a}_m$ and $\hat{b}_m$, respectively, with the index $m=(123)$ |
| 470 |
> |
referring to a convenient set of inertial body axes. (N.B., these |
| 471 |
> |
body axes are generally not the same as those for which the quadrupole |
| 472 |
> |
moment is diagonal.) Then, |
| 473 |
|
% |
| 474 |
|
\begin{eqnarray} |
| 475 |
|
\hat{a}_m= a_{mx}\hat{x} + a_{my}\hat{y} + a_{mz}\hat{z} \\ |
| 933 |
|
The concept of obtaining a force from an energy by taking a gradient is the same for |
| 934 |
|
higher-order multipole interactions, the trick is to make sure that all |
| 935 |
|
$r$-dependent derivatives are considered. |
| 936 |
< |
As is pointed out by Allen and Germano, this is straightforward if the |
| 936 |
> |
As is pointed out by Allen and Germano,\cite{Allen:2006fk} this is straightforward if the |
| 937 |
|
interaction energies are written recognizing explicit |
| 938 |
|
$\hat{r}$ and body axes ($\hat{a}_m$, $\hat{b}_n$) dependences: |
| 939 |
|
% |
| 958 |
|
\end{equation} |
| 959 |
|
% |
| 960 |
|
Note our definition of $\mathbf{r}=\mathbf{r}_b - \mathbf{r}_b $ is opposite |
| 961 |
< |
that of Allen and Germano. In simplifying the algebra, we also use: |
| 961 |
> |
that of Allen and Germano.\cite{Allen:2006fk} In simplifying the algebra, we also use: |
| 962 |
|
% |
| 963 |
|
\begin{eqnarray} |
| 964 |
|
\frac { \partial (\hat{a}_m \cdot \hat{r})}{\partial \mathbf{r}} |
| 1124 |
|
% |
| 1125 |
|
\subsection{Torques} |
| 1126 |
|
|
| 1127 |
< |
Following again Allen and Germano, when energies are written in the form |
| 1127 |
> |
Following again Allen and Germano,\cite{Allen:2006fk} when energies are written in the form |
| 1128 |
|
of Eq.~({\ref{ugeneral}), then torques can be expressed as: |
| 1129 |
|
% |
| 1130 |
|
\begin{eqnarray} |
| 1408 |
|
Dame. |
| 1409 |
|
\end{acknowledgments} |
| 1410 |
|
|
| 1411 |
+ |
\newpage |
| 1412 |
|
\appendix |
| 1413 |
|
|
| 1414 |
< |
\section{Smith's $B_l(r)$ functions for smeared-charge distributions} |
| 1414 |
> |
\section{Smith's $B_l(r)$ functions for damped-charge distributions} |
| 1415 |
|
|
| 1416 |
< |
The following summarizes Smith's $B_l(r)$ functions and |
| 1417 |
< |
includes formulas given in his appendix. |
| 1418 |
< |
|
| 1410 |
< |
The first function $B_0(r)$ is defined by |
| 1416 |
> |
The following summarizes Smith's $B_l(r)$ functions and includes |
| 1417 |
> |
formulas given in his appendix.\cite{Smith98} The first function |
| 1418 |
> |
$B_0(r)$ is defined by |
| 1419 |
|
% |
| 1420 |
|
\begin{equation} |
| 1421 |
|
B_0(r)=\frac{\text{erfc}(\alpha r)}{r} = \frac{2}{\sqrt{\pi}r}= |
| 1429 |
|
-\frac{2\alpha}{r\sqrt{\pi}}\text{e}^{-{\alpha}^2r^2} |
| 1430 |
|
\end{equation} |
| 1431 |
|
% |
| 1432 |
< |
and can be rewritten in terms of a function $B_1(r)$: |
| 1432 |
> |
which can be used to define a function $B_1(r)$: |
| 1433 |
|
% |
| 1434 |
|
\begin{equation} |
| 1435 |
|
B_1(r)=-\frac{1}{r}\frac{dB_0(r)}{dr} |
| 1436 |
|
\end{equation} |
| 1437 |
|
% |
| 1438 |
< |
In general, |
| 1438 |
> |
In general, the recurrence relation, |
| 1439 |
|
\begin{equation} |
| 1440 |
|
B_l(r)=-\frac{1}{r}\frac{dB_{l-1}(r)}{dr} |
| 1441 |
|
= \frac{1}{r^2} \left[ (2l-1)B_{l-1}(r) + \frac {(2\alpha^2)^l}{\alpha \sqrt{\pi}} |
| 1442 |
|
\text{e}^{-{\alpha}^2r^2} |
| 1443 |
< |
\right] . |
| 1443 |
> |
\right] , |
| 1444 |
|
\end{equation} |
| 1445 |
+ |
is very useful for building up higher derivatives. Using these |
| 1446 |
+ |
formulas, we find: |
| 1447 |
|
% |
| 1448 |
< |
Using these formulas, we find |
| 1448 |
> |
\begin{align} |
| 1449 |
> |
\frac{dB_0}{dr}=&-rB_1(r) \\ |
| 1450 |
> |
\frac{d^2B_0}{dr^2}=& - B_1(r) + r^2 B_2(r) \\ |
| 1451 |
> |
\frac{d^3B_0}{dr^3}=& 3 r B_2(r) - r^3 B_3(r) \\ |
| 1452 |
> |
\frac{d^4B_0}{dr^4}=& 3 B_2(r) - 6 r^2 B_3(r) + r^4 B_4(r) \\ |
| 1453 |
> |
\frac{d^5B_0}{dr^5}=& - 15 r B_3(r) + 10 r^3 B_4(r) - r^5 B_5(r) . |
| 1454 |
> |
\end{align} |
| 1455 |
|
% |
| 1456 |
< |
\begin{eqnarray} |
| 1457 |
< |
\frac{dB_0}{dr}=-rB_1(r) \\ |
| 1442 |
< |
\frac{d^2B_0}{dr^2}=-B_1(r) + r^2B_2(r) \\ |
| 1443 |
< |
\frac{d^3B_0}{dr^3}=3rB_2(r) - r^3B_3(r) \\ |
| 1444 |
< |
\frac{d^4B_0}{dr^4}=3B_2(r) - 6r^2B_3(r)+r^4B_4(r) \\ |
| 1445 |
< |
\frac{d^5B_0}{dr^5}=-15rB_3(r) + 10r^3B_4(r) -r^5B_5(r) . |
| 1446 |
< |
\end{eqnarray} |
| 1456 |
> |
As noted by Smith, it is possible to approximate the $B_l(r)$ |
| 1457 |
> |
functions, |
| 1458 |
|
% |
| 1448 |
– |
As noted by Smith, |
| 1449 |
– |
% |
| 1459 |
|
\begin{equation} |
| 1460 |
|
B_l(r)=\frac{(2l)!}{l!2^lr^{2l+1}} - \frac {(2\alpha^2)^{l+1}}{(2l+1)\alpha \sqrt{\pi}} |
| 1461 |
|
+\text{O}(r) . |
| 1462 |
|
\end{equation} |
| 1463 |
+ |
\newpage |
| 1464 |
+ |
\section{The $r$-dependent factors for TSF electrostatics} |
| 1465 |
|
|
| 1455 |
– |
\section{Method 1, the $r$-dependent factors} |
| 1456 |
– |
|
| 1466 |
|
Using the shifted damped functions $f_n(r)$ defined by: |
| 1467 |
|
% |
| 1468 |
|
\begin{equation} |
| 1469 |
< |
f_n(r)= B_0 \Big \lvert _r -\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} B_0^{(m)} \Big \lvert _{r_c} , |
| 1469 |
> |
f_n(r)= B_0(r) -\sum_{m=0}^{n+1} \frac {(r-r_c)^m}{m!} B_0^{(m)}(r_c) , |
| 1470 |
|
\end{equation} |
| 1471 |
|
% |
| 1472 |
< |
we first provide formulas for successive derivatives of this function. (If there is |
| 1473 |
< |
no damping, then $B_0(r)$ is replaced by $1/r$, as discussed in Section~\ref{damped???}.) First, we find: |
| 1472 |
> |
where the superscript $(m)$ denotes the $m^\mathrm{th}$ derivative. In |
| 1473 |
> |
this Appendix, we provide formulas for successive derivatives of this |
| 1474 |
> |
function. (If there is no damping, then $B_0(r)$ is replaced by |
| 1475 |
> |
$1/r$.) First, we find: |
| 1476 |
|
% |
| 1477 |
|
\begin{equation} |
| 1478 |
|
\frac{\partial f_n}{\partial r_\alpha}=\hat{r}_\alpha \frac{d f_n}{d r} . |
| 1479 |
|
\end{equation} |
| 1480 |
|
% |
| 1481 |
< |
This formula clearly brings in derivatives of Smith's $B_0(r)$ function, motivating us to |
| 1482 |
< |
define higher-order derivatives as follows: |
| 1481 |
> |
This formula clearly brings in derivatives of Smith's $B_0(r)$ |
| 1482 |
> |
function, and we define higher-order derivatives as follows: |
| 1483 |
|
% |
| 1484 |
< |
\begin{eqnarray} |
| 1485 |
< |
g_n(r)= \frac{d f_n}{d r} = |
| 1486 |
< |
B_0^{(1)} \Big \lvert _r -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)} \Big \lvert _{r_c} \\ |
| 1487 |
< |
h_n(r)= \frac{d^2f_n}{d r^2} = |
| 1488 |
< |
B_0^{(2)} \Big \lvert _r -\sum_{m=0}^{n-1} \frac {(r-r_c)^m}{m!} B_0^{(m+2)} \Big \lvert _{r_c} \\ |
| 1489 |
< |
s_n(r)= \frac{d^3f_n}{d r^3} = |
| 1490 |
< |
B_0^{(3)} \Big \lvert _r -\sum_{m=0}^{n-2} \frac {(r-r_c)^m}{m!} B_0^{(m+3)} \Big \lvert _{r_c} \\ |
| 1491 |
< |
t_n(r)= \frac{d^4f_n}{d r^4} = |
| 1492 |
< |
B_0^{(4)} \Big \lvert _r -\sum_{m=0}^{n-3} \frac {(r-r_c)^m}{m!} B_0^{(m+4)} \Big \lvert _{r_c} \\ |
| 1493 |
< |
u_n(r)= \frac{d^5f_n}{d r^5} = |
| 1494 |
< |
B_0^{(5)} \Big \lvert _r -\sum_{m=0}^{n-4} \frac {(r-r_c)^m}{m!} B_0^{(m+5)} \Big \lvert _{r_c} . |
| 1495 |
< |
\end{eqnarray} |
| 1484 |
> |
\begin{align} |
| 1485 |
> |
g_n(r)=& \frac{d f_n}{d r} = |
| 1486 |
> |
B_0^{(1)}(r) -\sum_{m=0}^{n} \frac {(r-r_c)^m}{m!} B_0^{(m+1)}(r_c) \\ |
| 1487 |
> |
h_n(r)=& \frac{d^2f_n}{d r^2} = |
| 1488 |
> |
B_0^{(2)}(r) -\sum_{m=0}^{n-1} \frac {(r-r_c)^m}{m!} B_0^{(m+2)}(r_c) \\ |
| 1489 |
> |
s_n(r)=& \frac{d^3f_n}{d r^3} = |
| 1490 |
> |
B_0^{(3)}(r) -\sum_{m=0}^{n-2} \frac {(r-r_c)^m}{m!} B_0^{(m+3)}(r_c) \\ |
| 1491 |
> |
t_n(r)=& \frac{d^4f_n}{d r^4} = |
| 1492 |
> |
B_0^{(4)}(r) -\sum_{m=0}^{n-3} \frac {(r-r_c)^m}{m!} B_0^{(m+4)}(r_c) \\ |
| 1493 |
> |
u_n(r)=& \frac{d^5f_n}{d r^5} = |
| 1494 |
> |
B_0^{(5)}(r) -\sum_{m=0}^{n-4} \frac {(r-r_c)^m}{m!} B_0^{(m+5)}(r_c) . |
| 1495 |
> |
\end{align} |
| 1496 |
|
% |
| 1497 |
< |
We note that the last function needed (for quadrupole-quadrupole) is |
| 1497 |
> |
We note that the last function needed (for quadrupole-quadrupole interactions) is |
| 1498 |
|
% |
| 1499 |
|
\begin{equation} |
| 1500 |
< |
u_4(r)=B_0^{(5)} \Big \lvert _r - B_0^{(5)} \Big \lvert _{r_c} . |
| 1500 |
> |
u_4(r)=B_0^{(5)}(r) - B_0^{(5)}(r_c) . |
| 1501 |
|
\end{equation} |
| 1502 |
|
|
| 1503 |
< |
The functions $f_n(r)$ to $u_n(r)$ are recursively computed and stored for values of $r$ |
| 1504 |
< |
from $0$ to $r_c$. The functions needed are listed schematically below: |
| 1503 |
> |
The functions $f_n(r)$ to $u_n(r)$ can be computed recursively and |
| 1504 |
> |
stored on a grid for values of $r$ from $0$ to $r_c$. The functions |
| 1505 |
> |
needed are listed schematically below: |
| 1506 |
|
% |
| 1507 |
|
\begin{eqnarray} |
| 1508 |
|
f_0 \quad f_1 \qquad \qquad \quad & \nonumber \\ |
| 1515 |
|
|
| 1516 |
|
Using these functions, we find |
| 1517 |
|
% |
| 1518 |
< |
\begin{equation} |
| 1519 |
< |
\frac{\partial f_n}{\partial r_\alpha} =r_\alpha \frac {g_n}{r} |
| 1520 |
< |
\end{equation} |
| 1521 |
< |
% |
| 1522 |
< |
\begin{equation} |
| 1511 |
< |
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =\delta_{\alpha \beta}\frac {g_n}{r} |
| 1512 |
< |
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) |
| 1513 |
< |
\end{equation} |
| 1514 |
< |
% |
| 1515 |
< |
\begin{equation} |
| 1516 |
< |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} = |
| 1518 |
> |
\begin{align} |
| 1519 |
> |
\frac{\partial f_n}{\partial r_\alpha} =&r_\alpha \frac {g_n}{r} \label{eq:b9}\\ |
| 1520 |
> |
\frac{\partial^2 f_n}{\partial r_\alpha \partial r_\beta} =&\delta_{\alpha \beta}\frac {g_n}{r} |
| 1521 |
> |
+r_\alpha r_\beta \left( -\frac{g_n}{r^3} +\frac{h_n}{r^2}\right) \\ |
| 1522 |
> |
\frac{\partial^3 f_n}{\partial r_\alpha \partial r_\beta r_\gamma} =& |
| 1523 |
|
\left( \delta_{\alpha \beta} r_\gamma + \delta_{\alpha \gamma} r_\beta + |
| 1524 |
|
\delta_{ \beta \gamma} r_\alpha \right) |
| 1525 |
|
\left( -\frac{g_n}{r^3} +\frac{h_n}{r^2} \right) |
| 1526 |
|
+ r_\alpha r_\beta r_\gamma |
| 1527 |
< |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) |
| 1528 |
< |
\end{equation} |
| 1523 |
< |
% |
| 1524 |
< |
\begin{eqnarray} |
| 1525 |
< |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} = |
| 1527 |
> |
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \\ |
| 1528 |
> |
\frac{\partial^4 f_n}{\partial r_\alpha \partial r_\beta r_\gamma r_\delta} =& |
| 1529 |
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} |
| 1530 |
|
+ \delta_{\alpha \gamma} \delta_{\beta \delta} |
| 1531 |
|
+\delta_{ \beta \gamma} \delta_{\alpha \delta} \right) |
| 1532 |
|
\left( - \frac{g_n}{r^3} + \frac{h_n}{r^2} \right) \nonumber \\ |
| 1533 |
< |
+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
| 1534 |
< |
+ \text{5 perm} |
| 1533 |
> |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta |
| 1534 |
> |
+ \text{5 permutations} |
| 1535 |
|
\right) \left( \frac{3 g_n}{r^5} - \frac{3h_n}{r^4} + \frac{s_n}{r^3} |
| 1536 |
|
\right) \nonumber \\ |
| 1537 |
< |
+ r_\alpha r_\beta r_\gamma r_\delta |
| 1537 |
> |
&+ r_\alpha r_\beta r_\gamma r_\delta |
| 1538 |
|
\left( -\frac{15g_n}{r^7} + \frac{15h_n}{r^6} - \frac{6s_n}{r^5} |
| 1539 |
< |
+ \frac{t_n}{r^4} \right) |
| 1537 |
< |
\end{eqnarray} |
| 1538 |
< |
% |
| 1539 |
< |
\begin{eqnarray} |
| 1539 |
> |
+ \frac{t_n}{r^4} \right)\\ |
| 1540 |
|
\frac{\partial^5 f_n} |
| 1541 |
< |
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} = |
| 1541 |
> |
{\partial r_\alpha \partial r_\beta r_\gamma r_\delta r_\epsilon} =& |
| 1542 |
|
\left( \delta_{\alpha \beta} \delta_{\gamma \delta} r_\epsilon |
| 1543 |
< |
+ \text{14 perm} \right) |
| 1543 |
> |
+ \text{14 permutations} \right) |
| 1544 |
|
\left( \frac{3g_n}{r^5}-\frac{3h_n}{r^4} +\frac{s_n}{r^3} \right) \nonumber \\ |
| 1545 |
< |
+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
| 1546 |
< |
+ \text{9 perm} |
| 1545 |
> |
&+ \left( \delta_{\alpha \beta} r_\gamma r_\delta r_\epsilon |
| 1546 |
> |
+ \text{9 permutations} |
| 1547 |
|
\right) \left(- \frac{15g_n}{r^7}+\frac{15h_n}{r^7} -\frac{6s_n}{r^5} +\frac{t_n}{r^4} |
| 1548 |
|
\right) \nonumber \\ |
| 1549 |
< |
+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
| 1549 |
> |
&+ r_\alpha r_\beta r_\gamma r_\delta r_\epsilon |
| 1550 |
|
\left( \frac{105g_n}{r^9} - \frac{105h_n}{r^8} + \frac{45s_n}{r^7} |
| 1551 |
< |
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) |
| 1552 |
< |
\end{eqnarray} |
| 1551 |
> |
- \frac{10t_n}{r^6} +\frac{u_n}{r^5} \right) \label{eq:b13} |
| 1552 |
> |
\end{align} |
| 1553 |
|
% |
| 1554 |
|
% |
| 1555 |
|
% |
| 1556 |
< |
\section{Method 2, the $r$-dependent factors} |
| 1556 |
> |
\newpage |
| 1557 |
> |
\section{The $r$-dependent factors for GSF electrostatics} |
| 1558 |
|
|
| 1559 |
< |
In method 2, the kernel is not expanded, rather the individual terms in the multipole interaction energies, |
| 1560 |
< |
see Eq. (20?). For a smeared-charge distribution, this still brings into the algebra multiple derivatives |
| 1561 |
< |
of the kernel $B_0(r)$. To denote these terms, we generalize the notation of the previous appendix. |
| 1562 |
< |
For $f(r)=1/r$ (bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
| 1559 |
> |
In Gradient-shifted force electrostatics, the kernel is not expanded, |
| 1560 |
> |
rather the individual terms in the multipole interaction energies. |
| 1561 |
> |
For damped charges , this still brings into the algebra multiple |
| 1562 |
> |
derivatives of the Smith's $B_0(r)$ function. To denote these terms, |
| 1563 |
> |
we generalize the notation of the previous appendix. For $f(r)=1/r$ |
| 1564 |
> |
(bare Coulomb) or $f(r)=B_0(r)$ (smeared charge) |
| 1565 |
|
% |
| 1566 |
< |
\begin{eqnarray} |
| 1567 |
< |
g(r)= \frac{df}{d r}\\ |
| 1568 |
< |
h(r)= \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
| 1569 |
< |
s(r)= \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
| 1570 |
< |
t(r)= \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
| 1571 |
< |
u(r)= \frac{dt}{d r} =\frac{d^5f}{d r^5} . |
| 1572 |
< |
\end{eqnarray} |
| 1566 |
> |
\begin{align} |
| 1567 |
> |
g(r)=& \frac{df}{d r}\\ |
| 1568 |
> |
h(r)=& \frac{dg}{d r} = \frac{d^2f}{d r^2} \\ |
| 1569 |
> |
s(r)=& \frac{dh}{d r} = \frac{d^3f}{d r^3} \\ |
| 1570 |
> |
t(r)=& \frac{ds}{d r} = \frac{d^4f}{d r^4} \\ |
| 1571 |
> |
u(r)=& \frac{dt}{d r} = \frac{d^5f}{d r^5} . |
| 1572 |
> |
\end{align} |
| 1573 |
|
% |
| 1574 |
< |
For $f(r)=1/r$, Table I lists these derivatives under the column ``Bare Coulomb.'' Checks of algebra can be made by using limiting forms |
| 1575 |
< |
of equations, e.g., the leading term in the function $g_n(r)$ has $r$ dependence given by $g(r)$. Equations (B9) to B(13) |
| 1576 |
< |
are correct for method 2 if one just eliminates the subscript $n$. |
| 1574 |
> |
For undamped charges, $f(r)=1/r$, Table I lists these derivatives |
| 1575 |
> |
under the column ``Bare Coulomb.'' Equations \ref{eq:b9} to |
| 1576 |
> |
\ref{eq:b13} are still correct for GSF electrostatics if the subscript |
| 1577 |
> |
$n$ is eliminated. |
| 1578 |
|
|
| 1579 |
|
\section{Extra Material} |
| 1580 |
|
% |
| 2171 |
|
\begin{table*} |
| 2172 |
|
\caption{\label{tab:tableFORCE2}Radial functions used in the force equations.} |
| 2173 |
|
\begin{ruledtabular} |
| 2174 |
< |
\begin{tabular}{ccc} |
| 2175 |
< |
Generic&Method 1&Method 2 |
| 2174 |
> |
\begin{tabular}{|l|l|l|} |
| 2175 |
> |
Generic&Taylor-shifted Force&Gradient-shifted Force |
| 2176 |
|
\\ \hline |
| 2177 |
|
% |
| 2178 |
|
% |
