| 1177 |
|
$\alpha = 0.2$~\AA$^{-1}$ and $0.3$~\AA$^{-1}$ and large separation |
| 1178 |
|
between ions, the screening factor does indeed approach the correct |
| 1179 |
|
dielectric constant. |
| 1180 |
– |
|
| 1181 |
– |
REVISIT THESE THREE PARAGRAPHS: |
| 1180 |
|
|
| 1181 |
+ |
REVISIT AFTER EWALD RESULTS: |
| 1182 |
|
It is also notable that the TSF method again displays smaller |
| 1183 |
|
perturbations away from the correct dielectric screening behavior. We |
| 1184 |
|
also observe that for TSF method yields high dielectric screening even |
| 1349 |
|
On the basis of symmetry, the 81 elements can be placed in four |
| 1350 |
|
different groups: $\tilde{T}_{aaaa}$, $\tilde{T}_{aaab}$, |
| 1351 |
|
$\tilde{T}_{aabb}$, and $\tilde{T}_{aabc}$, where $a$, $b$, and $c$, |
| 1352 |
< |
and can take on distinct values from the set: $x$, $y$ and $z$. The |
| 1353 |
< |
elements belonging to each of these groups can be obtained using |
| 1354 |
< |
permutations of the indices. Only the tensor groups with indices |
| 1355 |
< |
${aaaa}$ and ${aabb}$ are non-zero. |
| 1352 |
> |
and can take on distinct values from the set $\left\{x, y, z\right\}$. |
| 1353 |
> |
The elements belonging to each of these groups can be obtained using |
| 1354 |
> |
permutations of the indices. Integration of all of the elements shows |
| 1355 |
> |
that only the groups with indices ${aaaa}$ and ${aabb}$ are non-zero. |
| 1356 |
|
|
| 1357 |
|
We can derive values of the components of $\tilde{T}_{aaaa}$ and |
| 1358 |
|
$\tilde{T}_{aabb}$ as follows; |
| 1359 |
|
\begin{eqnarray} |
| 1360 |
|
\tilde{T}_{xxxx}(0) &=& |
| 1361 |
|
\int_{\textrm{V}} |
| 1362 |
< |
\big [ 3v_{41}(R)+6x^2v_{42}(r)/r^2 + x^4\,v_{43}(r)/r^4 \big] d^3r \nonumber \\ |
| 1362 |
> |
\left[ 3v_{41}(R)+6x^2v_{42}(r)/r^2 + x^4\,v_{43}(r)/r^4 \right] d\mathbf{r} \nonumber \\ |
| 1363 |
|
&=&12\pi \int_0^{r_c} |
| 1364 |
< |
\big [ v_{41}(r)+\frac{2}{3} v_{42}(r) + \frac{1}{15}v_{43}(r) \big] r^2\,dr = |
| 1364 |
> |
\left[ v_{41}(r)+\frac{2}{3} v_{42}(r) + \frac{1}{15}v_{43}(r) \right] r^2\,dr = |
| 1365 |
|
\mathrm{12 \pi B} |
| 1366 |
|
\end{eqnarray} |
| 1367 |
|
and |
| 1368 |
|
\begin{eqnarray} |
| 1369 |
|
\tilde{T}_{xxyy}(0)&=& |
| 1370 |
|
\int_{\textrm{V}} |
| 1371 |
< |
\big [ v_{41}(R)+(x^2+y^2) v_{42}(r)/r^2 + x^2 y^2\,v_{43}(r)/r^4 \big] d^3r \nonumber \\ |
| 1371 |
> |
\left[ v_{41}(R)+(x^2+y^2) v_{42}(r)/r^2 + x^2 y^2\,v_{43}(r)/r^4 \right] d\mathbf{r} \nonumber \\ |
| 1372 |
|
&=&4\pi \int_0^{r_c} |
| 1373 |
< |
\big [ v_{41}(r)+\frac{2}{3} v_{42}(r) + \frac{1}{15}v_{43}(r) \big] r^2\,dr = |
| 1373 |
> |
\left[ v_{41}(r)+\frac{2}{3} v_{42}(r) + \frac{1}{15}v_{43}(r) \right] r^2\,dr = |
| 1374 |
|
\mathrm{4 \pi B}. |
| 1375 |
|
\end{eqnarray} |
| 1376 |
|
These integrals yield the same values for all permutations of the |
| 1416 |
|
molecular quadrupoles represented by point dipoles, the symmetry of |
| 1417 |
|
the quadrupolar tensor must be handled separately (compare |
| 1418 |
|
equations~\ref{quadDip} and~\ref{quadRadial}). Although there is a |
| 1419 |
< |
difference in symmetry, the final result (Eq.~\ref{contract}) holds |
| 1420 |
< |
true for dipolar representations as well. |
| 1419 |
> |
difference in symmetry, the final result (Eq.~\ref{contract}) also holds |
| 1420 |
> |
true for dipolar representations. |
| 1421 |
|
|
| 1422 |
< |
\section{Quadrupolar correction factor for Ewald-Kornfeld (EK) method} |
| 1423 |
< |
The quadrupolar-quadrupolar interaction tensor for Ewald method can be expressed in the following form, \cite{Smith82,NeumannII83 } |
| 1424 |
< |
\begin{eqnarray} |
| 1425 |
< |
{T}_{\alpha\beta\gamma\delta}(\mathbf{r}) = &+& \frac{4\pi}{V }\sum_{k\neq0}^{\infty} \frac{\exp(-k^2)/4\kappa^2}{k^2} r_\alpha r_\beta k_\delta k_\gamma \exp(-i\mathbf{k}\cdot \mathbf{r}) \nonumber \\ |
| 1426 |
< |
&+& \left(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}\right) |
| 1427 |
< |
v_{41}(r) \nonumber \\ |
| 1428 |
< |
&+& \left(\delta_{\gamma\delta} r_\alpha r_\beta + \mathrm{ 5\; permutations}\right) \frac{v_{42}(r)}{r^2} \nonumber \\ |
| 1429 |
< |
&+& r_\alpha r_\beta r_\gamma r_\delta \left(\frac{v_{43}(r)}{r^4}\right) |
| 1430 |
< |
\label{ewaldTensor} |
| 1431 |
< |
\end{eqnarray} |
| 1432 |
< |
Since the correction factor $B = 1/4\pi \tilde{T}_{ijij} $ and $k \neq 0 $ is excluded from the sum, 1st term does not contribute anything to the correction factor.\cite{NeumannII83} Remaining terms are due to the contribution for the real space parts and can be considered as contribution due to spherical truncation.\cite{Adams76,Adams80, Adams81} Using value of radial functions; $v_{41}(r)$, $v_{42}(r)$, and $v_{43}(r)$ for spherical truncation, we get; |
| 1433 |
< |
\begin{eqnarray} |
| 1434 |
< |
v_{41}(r) &=& \frac{3}{r^5} \left(\frac{2r\kappa \exp(-r^2 \kappa^2)}{\sqrt{\pi}}+\frac{4r^3\kappa^3 \exp(-r^2 \kappa^2)}{3\sqrt{\pi}}+\mathrm{erfc(\kappa r)} \right) \\ |
| 1435 |
< |
v_{42}(r) &=& \frac{15}{r^5}\left(\frac{2r\kappa \exp(-r^2 \kappa^2)}{\sqrt{\pi}}+\frac{4r^3\kappa^3 \exp(-r^2 \kappa^2)}{3\sqrt{\pi}}+\frac{8r^5\kappa^5 \exp(-r^2 \kappa^2)}{15\sqrt{\pi}}+\mathrm{erfc(\kappa r)} \right) \\ |
| 1436 |
< |
v_{43}(r) &=& \frac{105}{r^5}\left(\frac{2r\kappa \exp(-r^2 \kappa^2)}{\sqrt{\pi}}+\frac{4r^3\kappa^3 \exp(-r^2 \kappa^2)}{3\sqrt{\pi}}+\frac{8r^5\kappa^5 \exp(-r^2 \kappa^2)}{15\sqrt{\pi}} \right. \nonumber \\ |
| 1437 |
< |
&+& \left. \frac{16r^7\kappa^7 \exp(-r^2 \kappa^2)}{105\sqrt{\pi}} + \mathrm{erfc(\kappa r)} \right) |
| 1438 |
< |
\end{eqnarray} |
| 1439 |
< |
Substituting above values of the radial functions in the real space part of the quadrupolar tensor, we can ${T}_{\alpha\beta\gamma\delta}(\mathbf{r})$ can be divided into two parts as follows; |
| 1440 |
< |
\begin{eqnarray} |
| 1441 |
< |
& & {T}_{\alpha\beta\gamma\delta}(\mathbf{r}) = {T^I}_{\alpha\beta\gamma\delta}(\mathbf{r}) + {T^{II}}_{\alpha\beta\gamma\delta}(\mathbf{r}) |
| 1442 |
< |
\end{eqnarray} |
| 1443 |
< |
where |
| 1444 |
< |
\begin{eqnarray} |
| 1445 |
< |
{T^I}_{\alpha\beta\gamma\delta}(\mathbf{r}) &=& {T}_{\alpha\beta\gamma\delta}(\mathbf{r})|_{\kappa \rightarrow 0} \nonumber \\ |
| 1446 |
< |
& & \left(\frac{2r\kappa \exp(-r^2 \kappa^2)}{\sqrt{\pi}}+\frac{4r^3\kappa^3\exp(-r^2 \kappa^2)}{3\sqrt{\pi}}+\frac{8r^5\kappa^5 \exp(-r^2 \kappa^2)}{15\sqrt{\pi}} + \right. \nonumber \\ |
| 1447 |
< |
& & \left. \frac{16r^7\kappa^7 \exp(-r^2 \kappa^2)}{105\sqrt{\pi}} + \mathrm{erfc(\kappa r)} \right) |
| 1422 |
> |
\section{Quadrupolar correction factor for the Ewald-Kornfeld (EK) |
| 1423 |
> |
method} |
| 1424 |
> |
The interaction tensor between two point quadrupoles in the Ewald |
| 1425 |
> |
method may be expressed,\cite{Smith98,NeumannII83} |
| 1426 |
> |
\begin{align} |
| 1427 |
> |
{T}_{\alpha\beta\gamma\delta}(\mathbf{r}) = &\frac{4\pi}{V |
| 1428 |
> |
}\sum_{k\neq0}^{\infty} |
| 1429 |
> |
e^{-k^2 / 4 |
| 1430 |
> |
\kappa^2} e^{-i\mathbf{k}\cdot |
| 1431 |
> |
\mathbf{r}} \left(\frac{r_\alpha r_\beta k_\delta k_\gamma}{k^2}\right) \nonumber \\ |
| 1432 |
> |
&+ \left(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}\right) |
| 1433 |
> |
B_2(r) \nonumber \\ |
| 1434 |
> |
&- \left(\delta_{\gamma\delta} r_\alpha r_\beta + \mathrm{ 5\; permutations}\right) B_3(r) \nonumber \\ |
| 1435 |
> |
&+ \left(r_\alpha r_\beta r_\gamma r_\delta \right) B_4(r) |
| 1436 |
> |
\label{ewaldTensor} |
| 1437 |
> |
\end{align} |
| 1438 |
> |
where $B_n(r)$ are radial functions defined in reference |
| 1439 |
> |
\onlinecite{Smith98}, |
| 1440 |
> |
\begin{align} |
| 1441 |
> |
B_2(r) =& \frac{3}{r^5} \left(\frac{2r\kappa e^{-r^2 \kappa^2}}{\sqrt{\pi}}+\frac{4r^3\kappa^3 e^{-r^2 \kappa^2}}{3\sqrt{\pi}}+\mathrm{erfc(\kappa r)} \right) \\ |
| 1442 |
> |
B_3(r) =& - \frac{15}{r^7}\left(\frac{2r\kappa e^{-r^2 \kappa^2}}{\sqrt{\pi}}+\frac{4r^3\kappa^3 e^{-r^2 \kappa^2}}{3\sqrt{\pi}}+\frac{8r^5\kappa^5 e^{-r^2 \kappa^2}}{15\sqrt{\pi}}+\mathrm{erfc(\kappa r)} \right) \\ |
| 1443 |
> |
B_4(r) =& \frac{105}{r^9}\left(\frac{2r\kappa e^{-r^2 |
| 1444 |
> |
\kappa^2}}{\sqrt{\pi}}+\frac{4r^3\kappa^3 e^{-r^2 \kappa^2}}{3\sqrt{\pi}}+\frac{8r^5\kappa^5 e^{-r^2 \kappa^2}}{15\sqrt{\pi}} |
| 1445 |
> |
+ \frac{16r^7\kappa^7 e^{-r^2 \kappa^2}}{105\sqrt{\pi}} + \mathrm{erfc(\kappa r)} \right) |
| 1446 |
> |
\end{align} |
| 1447 |
> |
|
| 1448 |
> |
We can divide ${T}_{\alpha\beta\gamma\delta}(\mathbf{r})$ into three |
| 1449 |
> |
parts: |
| 1450 |
> |
\begin{eqnarray} |
| 1451 |
> |
& & \mathbf{T}(\mathbf{r}) = |
| 1452 |
> |
\mathbf{T}^\mathrm{K}(\mathbf{r}) + |
| 1453 |
> |
\mathbf{T}^\mathrm{R1}(\mathbf{r}) + |
| 1454 |
> |
\mathbf{T}^\mathrm{R2}(\mathbf{r}) |
| 1455 |
|
\end{eqnarray} |
| 1456 |
+ |
where the first term is the reciprocal space portion. Since the |
| 1457 |
+ |
quadrupolar correction factor $B = \tilde{T}_{abab}(0) / 4\pi$ and |
| 1458 |
+ |
$\mathbf{k} \neq 0 $ is excluded from the reciprocal space sum, |
| 1459 |
+ |
$\mathbf{T}^\mathrm{K}$ will not contribute.\cite{NeumannII83} The |
| 1460 |
+ |
remaining terms, |
| 1461 |
+ |
\begin{equation} |
| 1462 |
+ |
\mathbf{T}^\mathrm{R1}(\mathbf{r}) = \mathbf{T}^\mathrm{bare}(\mathbf{r}) \left(\frac{2r\kappa e^{-r^2 |
| 1463 |
+ |
\kappa^2}}{\sqrt{\pi}}+\frac{4r^3\kappa^3 e^{-r^2 \kappa^2}}{3\sqrt{\pi}}+\frac{8r^5\kappa^5 e^{-r^2 \kappa^2}}{15\sqrt{\pi}} |
| 1464 |
+ |
+ \frac{16r^7\kappa^7 e^{-r^2 \kappa^2}}{105\sqrt{\pi}} + \mathrm{erfc(\kappa r)} \right) |
| 1465 |
+ |
\end{equation} |
| 1466 |
|
and |
| 1467 |
|
\begin{eqnarray} |
| 1468 |
< |
{T^{II}}_{\alpha\beta\gamma\delta}(\mathbf{r}) = &+& \left(\delta_{\gamma\delta} r_\alpha r_\beta + \mathrm{ 5\; permutations}\right) \frac{16 \kappa^7 \exp(-r^2 \kappa^2)}{7\sqrt{\pi}} \nonumber \\ |
| 1469 |
< |
&+&\left(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}\right) \left(\frac{8 \kappa^5 \exp(-r^2 \kappa^2)}{5\sqrt{\pi}}+ \frac{16 r^2\kappa^7 \exp(-r^2 \kappa^2)}{35\sqrt{\pi} }\right) |
| 1468 |
> |
T^\mathrm{R2}_{\alpha\beta\gamma\delta}(\mathbf{r}) = &+& \left(\delta_{\gamma\delta} r_\alpha r_\beta + \mathrm{ 5\; permutations}\right) \frac{16 \kappa^7 e^{-r^2 \kappa^2}}{7\sqrt{\pi}} \nonumber \\ |
| 1469 |
> |
&-&\left(\delta_{\alpha\beta}\delta_{\gamma\delta}+\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma}\right) \left(\frac{8 \kappa^5 e^{-r^2 \kappa^2}}{5\sqrt{\pi}}+ \frac{16 r^2\kappa^7 e^{-r^2 \kappa^2}}{35\sqrt{\pi} }\right) |
| 1470 |
|
\end{eqnarray} |
| 1471 |
< |
Here ${T}_{\alpha\beta\gamma\delta}(\mathbf{r})|_{\kappa \rightarrow 0}$ is unmodified quadrupolar tensor vanishes on integration over volume due to angular symmetry. Therefore the only term contributing to the correction factor (B) is ${T^{II}}_{\alpha\beta\gamma\delta}(\mathbf{r})$. Now the correction factor for the Ewald-Kornfeld (EK) method can be written as; |
| 1471 |
> |
are contributions from the real space |
| 1472 |
> |
sum.\cite{Adams76,Adams80,Adams81} Here |
| 1473 |
> |
$\mathbf{T}^\mathrm{bare}(\mathbf{r})$ is the unmodified quadrupolar |
| 1474 |
> |
tensor (for undamped quadrupoles). Due to the angular symmetry of the |
| 1475 |
> |
unmodified tensor, the integral of |
| 1476 |
> |
$\mathbf{T}^\mathrm{R1}(\mathbf{r})$ will vanish when integrated over |
| 1477 |
> |
a spherical region. The only term contributing to the correction |
| 1478 |
> |
factor (B) is therefore |
| 1479 |
> |
$T^\mathrm{R2}_{\alpha\beta\gamma\delta}(\mathbf{r})$, which allows us |
| 1480 |
> |
to derive the correction factor for the Ewald-Kornfeld (EK) method, |
| 1481 |
|
\begin{eqnarray} |
| 1482 |
< |
\mathrm{B} &=& \frac{1}{4\pi} \int_V {T^{II}}_{ijij}(\mathbf{r}) \nonumber \\ |
| 1483 |
< |
&=& -\frac{8r_c^3 \kappa^5 \exp(-\kappa^2 r_c^2)}{15\sqrt{\pi}} |
| 1482 |
> |
\mathrm{B} &=& \frac{1}{4\pi} \int_V T^\mathrm{R2}_{abab}(\mathbf{r}) \nonumber \\ |
| 1483 |
> |
&=& -\frac{8r_c^3 \kappa^5 e^{-\kappa^2 r_c^2}}{15\sqrt{\pi}} |
| 1484 |
|
\end{eqnarray} |
| 1485 |
< |
This result is same as the correction factor from the direct spherical cutoff method. |
| 1485 |
> |
which is essentially identical with the correction factor from the |
| 1486 |
> |
direct spherical cutoff (SC) method. |
| 1487 |
|
\newpage |
| 1488 |
|
\bibliography{dielectric_new} |
| 1489 |
|
|
