| 695 |
|
the modified (Wolf or DSF) kernels. |
| 696 |
|
|
| 697 |
|
Similarly, when representing quadrupolar molecules with multiple point |
| 698 |
< |
dipoles, the molecular quadrupole interaction tensor can be obtained |
| 699 |
< |
using two successive applications of the gradient operator to the |
| 700 |
< |
dipole interaction tensor, |
| 698 |
> |
\textit{dipoles}, the molecular quadrupole interaction tensor can be |
| 699 |
> |
obtained using two successive applications of the gradient operator to |
| 700 |
> |
the dipole interaction tensor, |
| 701 |
|
\begin{eqnarray} |
| 702 |
|
T_{\alpha\beta\gamma\delta}(\mathbf{r}) &=& \nabla_\alpha \nabla_\beta |
| 703 |
< |
v_{\gamma\delta}(\mathbf{r}) \\ |
| 703 |
> |
T_{\gamma\delta}(\mathbf{r}) \\ |
| 704 |
|
& = & \delta_{\alpha\beta}\delta_{\gamma\delta} \frac{v^\prime_{21}(r)}{r} + |
| 705 |
|
\left(\delta_{\alpha\gamma}\delta_{\beta\delta}+ |
| 706 |
|
\delta_{\alpha\delta}\delta_{\beta\gamma}\right)\frac{v_{22}(r)}{r^2} |
| 718 |
|
\prime}_{22}(r)}{r^4}-\frac{5v^\prime_{22}(r)}{r^5}+\frac{8v_{22}(r)}{r^6}\right), |
| 719 |
|
\label{quadDip} |
| 720 |
|
\end{eqnarray} |
| 721 |
< |
where $v_{\gamma\delta}(\mathbf{r})$ is a dipole-dipole interaction |
| 721 |
> |
where $T_{\gamma\delta}(\mathbf{r})$ is a dipole-dipole interaction |
| 722 |
|
tensor that depends on the level of the |
| 723 |
|
approximation.\cite{PaperI,PaperII} Similarly $v_{21}(r)$ and |
| 724 |
|
$v_{22}(r)$ are the radial function for different real space cutoff |
| 764 |
|
where |
| 765 |
|
$\Theta_{\alpha\beta} = 3Q_{\alpha\beta} - \delta_{\alpha\beta}Tr(Q)$ |
| 766 |
|
is the traceless quadrupole density. In analogy to |
| 767 |
< |
Eq. (\ref{pertQuad}) above, the quadrupolarization density may now be |
| 768 |
< |
related to the actual quadrupolar susceptibility, $\chi_Q$, |
| 767 |
> |
Eq. (\ref{pertQuad}) above, the quadrupole polarization density may |
| 768 |
> |
now be related to the quadrupolar susceptibility, $\chi_Q$, |
| 769 |
|
\begin{equation} |
| 770 |
|
\frac{1}{3}{\Theta}_{\alpha\beta}(\mathbf{r}) = \epsilon_o {\chi}_Q |
| 771 |
|
\left[\partial_\alpha {E}^\circ_\beta(\mathbf{r}) + \frac{1}{24\pi |
| 786 |
|
\tilde{\Theta}_{\gamma\delta}(\mathbf{k})\right]. |
| 787 |
|
\label{fourierQuad} |
| 788 |
|
\end{equation} |
| 789 |
< |
If the applied field gradient is homogeneous over the |
| 790 |
< |
entire volume, ${\partial_ \alpha \tilde{E}^\circ_\beta}(\mathbf{k}) = 0 $ except at $ \mathbf{k} = 0$. Similarly quadrupolar polarization density can also considered to be uniform over entire space. Therefore as in the dipolar case, \cite{NeumannI83} the only relevant contribution of the interaction tensor will also be when $\mathbf{k} = 0$. Therefore equation (\ref{fourierQuad}) can be written as, |
| 789 |
> |
If the applied field gradient is homogeneous over the entire volume, |
| 790 |
> |
${\partial_ \alpha \tilde{E}^\circ_\beta}(\mathbf{k}) = 0 $ except at |
| 791 |
> |
$ \mathbf{k} = 0$. Similarly, the quadrupolar polarization density can |
| 792 |
> |
also considered uniform over entire space. As in the dipolar case, |
| 793 |
> |
\cite{NeumannI83} the only relevant contribution from the interaction |
| 794 |
> |
tensor will also be when $\mathbf{k} = 0$. Therefore equation |
| 795 |
> |
(\ref{fourierQuad}) can be written as, |
| 796 |
|
\begin{equation} |
| 797 |
|
\frac{1}{3}\tilde{\Theta}_{\alpha\beta}(\mathrm{0})= |
| 798 |
|
\epsilon_o {\chi}_Q \left[{\partial_\alpha |
| 801 |
|
\tilde{\Theta}_{\gamma\delta}(\mathrm{0})\right]. |
| 802 |
|
\label{fourierQuadZeroK} |
| 803 |
|
\end{equation} |
| 804 |
< |
The quadrupolar tensor $\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})$ consists of $\mathrm{3^4}$ components. Among them, the components with all same indices, $\tilde{T}_{iiii}(\mathrm{0})$ and the components with two doubly repeated indices, $\mathit{i.e.}$ all possible permutations of $\tilde{T}_{iijj}(\mathrm{0})$ for $i \neq j$, are non zero (See appendix \ref{ap:quadContraction}). Furthermore, for the both diagonal and non-diagonal components of the quadrupolar polarization $\tilde{\Theta}_{\alpha\beta}$, we can contract 2nd term in equation \ref{fourierQuadZeroK} in the following way (Appendix \ref{ap:quadContraction}); |
| 804 |
> |
The quadrupolar tensor |
| 805 |
> |
$\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})$ is a rank-4 tensor |
| 806 |
> |
with 81 elements. The only non-zero elements, however, are those with |
| 807 |
> |
two doubly-repeated indices, \textit{i.e.} |
| 808 |
> |
$\tilde{T}_{aabb}(\mathrm{0})$ and all permutations of these indices. |
| 809 |
> |
The special case of quadruply-repeated indices, |
| 810 |
> |
$\tilde{T}_{aaaa}(\mathrm{0})$ also survives (see appendix |
| 811 |
> |
\ref{ap:quadContraction}). Furthermore, for the both diagonal and |
| 812 |
> |
non-diagonal components of the quadrupolar polarization |
| 813 |
> |
$\tilde{\Theta}_{\alpha\beta}$, we can contract the second term in |
| 814 |
> |
equation \ref{fourierQuadZeroK} (see appendix |
| 815 |
> |
\ref{ap:quadContraction}): |
| 816 |
|
\begin{equation} |
| 817 |
< |
\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})\tilde{\Theta}_{\gamma\delta}(\mathrm{0})= 2\mathrm{B'} \tilde{\Theta}_{\alpha\beta}(\mathrm{0}), |
| 817 |
> |
\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})\tilde{\Theta}_{\gamma\delta}(\mathrm{0})= |
| 818 |
> |
8 \pi \mathrm{B} \tilde{\Theta}_{\alpha\beta}(\mathrm{0}). |
| 819 |
|
\label{quadContraction} |
| 820 |
|
\end{equation} |
| 821 |
< |
here $\mathrm{B'} = \tilde{T}_{ijij}(\mathrm{0})$ for $i \neq j$. Using quadrupolar contraction we can solve equation \ref{quadContraction} as follows |
| 821 |
> |
Here $\mathrm{B} = \tilde{T}_{abab}(\mathrm{0}) / 4 \pi$ for |
| 822 |
> |
$a \neq b$. Using this quadrupolar contraction we can solve equation |
| 823 |
> |
\ref{quadContraction} as follows |
| 824 |
|
|
| 825 |
|
\begin{eqnarray} |
| 826 |
|
\frac{1}{3}\tilde{\Theta}_{\alpha\beta}(\mathrm{0}) &=& \epsilon_o |
| 827 |
|
{\chi}_Q |
| 828 |
|
\left[{\partial_\alpha |
| 829 |
|
\tilde{E}^\circ_\beta}(\mathrm{0})+ |
| 830 |
< |
\frac{\mathrm{B'}}{12\pi |
| 830 |
> |
\frac{\mathrm{B}}{3 |
| 831 |
|
\epsilon_o} |
| 832 |
|
{\tilde{\Theta}}_{\alpha\beta}(\mathrm{0})\right] |
| 833 |
|
\nonumber \\ |
| 834 |
|
&=& \left[\frac{\epsilon_o {\chi}_Q} {1-{\chi}_Q \mathrm{B}}\right] |
| 835 |
< |
{\partial_\alpha \tilde{E}^\circ_\beta}(\mathrm{0}), |
| 835 |
> |
{\partial_\alpha \tilde{E}^\circ_\beta}(\mathrm{0}). |
| 836 |
|
\label{fourierQuad} |
| 837 |
|
\end{eqnarray} |
| 838 |
< |
where |
| 839 |
< |
\begin{eqnarray} |
| 840 |
< |
\mathrm{B} &=& \frac{\mathrm{B'}}{4\pi} \nonumber \\ |
| 841 |
< |
&=& \frac{1}{4 \pi} \int_V {T}_{ijij}(\mathbf{r}) d\mathbf{r}, |
| 823 |
< |
\end{eqnarray} |
| 838 |
> |
In real space, the correction factor, |
| 839 |
> |
\begin{equation} |
| 840 |
> |
\mathrm{B} = \frac{1}{4 \pi} \tilde{T}_{abab}(0) = \frac{1}{4 \pi} \int_V {T}_{abab}(\mathbf{r}) d\mathbf{r}, |
| 841 |
> |
\end{equation} |
| 842 |
|
%If the applied field gradient is homogeneous over the |
| 843 |
|
%entire volume, ${\partial_ \alpha \tilde{E}^\circ_\beta}(\mathbf{k}) = 0 $ except at |
| 844 |
|
%$ \mathbf{k} = 0$. As in the dipolar case, the only relevant |
| 863 |
|
which has been integrated over the interaction volume $V$ and has |
| 864 |
|
units of $\mathrm{length}^{-2}$. |
| 865 |
|
|
| 866 |
< |
In terms of traced quadrupole moment, equation (\ref{fourierQuad}) |
| 867 |
< |
can be written as, |
| 866 |
> |
In terms of the traced quadrupole moment, equation (\ref{fourierQuad}) |
| 867 |
> |
can be written, |
| 868 |
|
\begin{equation} |
| 869 |
|
\mathsf{Q} - \frac{\mathbf{I}}{3} \mathrm{Tr}(\mathsf{Q}) |
| 870 |
|
= \frac{\epsilon_o {\chi}_Q}{1- {\chi}_Q \mathrm{B}} \nabla \mathbf{E}^\circ |
| 883 |
|
\label{eq:finalForm} |
| 884 |
|
\end{equation} |
| 885 |
|
Eq. (\ref{eq:finalForm}) now expresses a bulk property (the |
| 886 |
< |
quadrupolar susceptibility, $\chi_Q$) in terms of a straightforward |
| 887 |
< |
fluctuation in the system quadrupole moment and a quadrupolar |
| 888 |
< |
correction factor ($\mathrm{B}$). The correction factors depend on the cutoff |
| 889 |
< |
method being employed in the simulation, and these are listed in Table |
| 890 |
< |
\ref{tab:B}. |
| 886 |
> |
quadrupolar susceptibility, $\chi_Q$) in terms of a fluctuation in the |
| 887 |
> |
system quadrupole moment and a quadrupolar correction factor |
| 888 |
> |
($\mathrm{B}$). The correction factors depend on the cutoff method |
| 889 |
> |
being employed in the simulation, and these are listed in Table |
| 890 |
> |
\ref{tab:B}. |
| 891 |
|
|
| 892 |
|
In terms of the macroscopic quadrupole polarizability, $\alpha_Q$, |
| 893 |
|
which may be thought of as the ``conducting boundary'' version of the |
| 910 |
|
\begin{tabular}{l|c|c|c} |
| 911 |
|
\toprule |
| 912 |
|
Method & charges & dipoles & quadrupoles \\\colrule |
| 913 |
< |
Spherical Cutoff (SC) & $ -\frac{8 \alpha^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ & $ -\frac{8 \alpha^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ & $ -\frac{8 {\alpha}^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ \\ |
| 913 |
> |
Spherical Cutoff (SC) & \multicolumn{3}{c}{$ -\frac{8 \alpha^5 |
| 914 |
> |
{r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $}\\ \colrule |
| 915 |
|
Shifted Potental (SP) & $ -\frac{8 \alpha^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ & $- \frac{3 \mathrm{erfc(r_c\alpha)}}{5{r_c}^2}- \frac{2 \alpha e^{-\alpha^2 r_c^2}(9+6\alpha^2 r_c^2+4\alpha^4 r_c^4)}{15{\sqrt{\pi}r_c}}$& $ -\frac{16 \alpha^7 {r_c}^5 e^{-\alpha^2 r_c^2 }}{45\sqrt{\pi}}$ \\ |
| 916 |
|
Gradient-shifted (GSF) & $- \frac{8 \alpha^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ & 0 & $-\frac{4{\alpha}^7{r_c}^5 e^{-\alpha^2 r_c^2}(-1+2\alpha ^2 r_c^2)}{45\sqrt{\pi}}$\\ |
| 917 |
|
Taylor-shifted (TSF) & $ -\frac{8 \alpha^5 {r_c}^3e^{-\alpha^2 r_c^2}}{15\sqrt{\pi}} $ & $\frac{4\;\mathrm{erfc(\alpha r_c)}}{{5r_c}^2} + \frac{8\alpha e^{-\alpha^2{r_c}^2}\left(3+ 2\alpha^2 {r_c}^2 +\alpha^4{r_c}^4 \right)}{15\sqrt{\pi}r_c}$ & $\frac{2\;\mathrm{erfc}(\alpha r_c )}{{r_c}^2} + \frac{4{\alpha}e^{-\alpha^2 r_c^2}\left(45 + 30\alpha ^2 {r_c}^2 + 12\alpha^4 {r_c}^4 + 3\alpha^6 {r_c}^6 + 2 \alpha^8 {r_c}^8\right)}{45\sqrt{\pi}{r_c}}$ \\ |
| 1338 |
|
utilized to compute bulk properties of fluids. |
| 1339 |
|
|
| 1340 |
|
\appendix |
| 1341 |
< |
\section{Contraction of quadrupolar tensor with the traceless quadrupole moment } |
| 1341 |
> |
\section{Contraction of the quadrupolar tensor with the traceless |
| 1342 |
> |
quadrupole moment } |
| 1343 |
|
\label{ap:quadContraction} |
| 1344 |
< |
If we express quadrupolar molecule in the point quadrupole representation then quadrupole-quadrupole interaction tensor can be written as, |
| 1345 |
< |
\begin{eqnarray} |
| 1346 |
< |
T_{\alpha\beta\gamma\delta}(\mathbf{r}) &=& |
| 1327 |
< |
\left(\delta_{\alpha\beta}\delta_{\gamma\delta} |
| 1328 |
< |
+ |
| 1329 |
< |
\delta_{\alpha\gamma}\delta_{\beta\delta}+ |
| 1330 |
< |
\delta_{\alpha\delta}\delta_{\beta\gamma}\right)v_{41}(r) |
| 1331 |
< |
+ \left(\delta_{\gamma\delta} r_\alpha r_\beta + \mathrm{ 5\; permutations}\right) \frac{v_{42}(r)}{r^2} \nonumber \\ |
| 1332 |
< |
& & + r_\alpha r_\beta r_\gamma r_\delta \left(\frac{v_{43}(r)}{r^4}\right) |
| 1333 |
< |
\end{eqnarray} |
| 1334 |
< |
The Fourier transformation of quadrupolar tensor in equation \ref{quadRadial} for $ \mathbf{\kappa} = 0$ is expressed as, |
| 1344 |
> |
For quadrupolar liquids modeled using point quadrupoles, the |
| 1345 |
> |
interaction tensor is shown in Eq. (\ref{quadRadial}). The Fourier |
| 1346 |
> |
transformation of this tensor for $ \mathbf{k} = 0$ is, |
| 1347 |
|
\begin{equation} |
| 1348 |
|
\tilde{T}_{\alpha\beta\gamma\delta}(0) = \int_V T_{\alpha\beta\gamma\delta}(\mathbf{r}) d \mathbf{r} |
| 1349 |
|
\end{equation} |
| 1350 |
< |
The quadrupole tensor has altogether 81 components. On the basis of symmetry, these can be grouped into five different groups: $\tilde{T}_{iiii}$, $\tilde{T}_{ijjj}$, $\tilde{T}_{iiii}$, $\tilde{T}_{ijki}$, and $\tilde{T}_{ijkl}$, where $i$, $j$, $k$, and $l$ can take values from $1$ to $3$ and $i \neq j \neq k$ for each tensor component. The elements of each group can be obtained using all the possible permutations of the indices. Among them, only tensor groups with indices ${iiii}$ and ${iijj}$ are non-zero. |
| 1351 |
< |
We can derive value of the components of $\tilde{T}_{iiii}$ and $\tilde{T}_{iijj}$ as follows; |
| 1350 |
> |
On the basis of symmetry, the 81 elements can be placed in four |
| 1351 |
> |
different groups: $\tilde{T}_{aaaa}$, $\tilde{T}_{aaab}$, |
| 1352 |
> |
$\tilde{T}_{aabb}$, and $\tilde{T}_{aabc}$, where $a$, $b$, and $c$, |
| 1353 |
> |
and can take on distinct values from the set: $x$, $y$ and $z$. The |
| 1354 |
> |
elements belonging to each of these groups can be obtained using |
| 1355 |
> |
permutations of the indices. Only the tensor groups with indices |
| 1356 |
> |
${aaaa}$ and ${aabb}$ are non-zero. |
| 1357 |
> |
|
| 1358 |
> |
We can derive values of the components of $\tilde{T}_{aaaa}$ and |
| 1359 |
> |
$\tilde{T}_{aabb}$ as follows; |
| 1360 |
|
\begin{eqnarray} |
| 1361 |
< |
\tilde{T}_{1111}(0) &=& |
| 1361 |
> |
\tilde{T}_{xxxx}(0) &=& |
| 1362 |
|
\int_{\textrm{V}} |
| 1363 |
|
\big [ 3v_{41}(R)+6x^2v_{42}(r)/r^2 + x^4\,v_{43}(r)/r^4 \big] d^3r \nonumber \\ |
| 1364 |
|
&=&12\pi \int_0^{r_c} |
| 1365 |
|
\big [ v_{41}(r)+\frac{2}{3} v_{42}(r) + \frac{1}{15}v_{43}(r) \big] r^2\,dr = |
| 1366 |
< |
\mathrm{3B^\prime} |
| 1366 |
> |
\mathrm{12 \pi B} |
| 1367 |
|
\end{eqnarray} |
| 1368 |
|
and |
| 1369 |
|
\begin{eqnarray} |
| 1370 |
< |
\tilde{T}_{1122}(0)&=& |
| 1371 |
< |
\int_{\textrm{V}} |
| 1372 |
< |
\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 \\ |
| 1373 |
< |
&=&4\pi \int_0^{r_c} |
| 1374 |
< |
\big [ v_{41}(r)+\frac{2}{3} v_{42}(r) + \frac{1}{15}v_{43}(r) \big] r^2\,dr = |
| 1375 |
< |
\mathrm{B^\prime}. |
| 1370 |
> |
\tilde{T}_{xxyy}(0)&=& |
| 1371 |
> |
\int_{\textrm{V}} |
| 1372 |
> |
\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 \\ |
| 1373 |
> |
&=&4\pi \int_0^{r_c} |
| 1374 |
> |
\big [ v_{41}(r)+\frac{2}{3} v_{42}(r) + \frac{1}{15}v_{43}(r) \big] r^2\,dr = |
| 1375 |
> |
\mathrm{4 \pi B}. |
| 1376 |
|
\end{eqnarray} |
| 1377 |
< |
These results are true for all values of indices. In equation \ref{fourierQuadZeroK}, for a particular value of quadrupolar polarization $\tilde{\Theta}_{ii}$ we can contract $\tilde{T}_{ii\gamma\delta}(0)$ with $\tilde{\Theta}_{\gamma\delta}$, using traceless properties of quadrupolar moment, as follows (we considered $i$ = $1$ for explanation); |
| 1377 |
> |
These integrals yield the same values for all permutations of the |
| 1378 |
> |
indices in both tensor element groups. In equation |
| 1379 |
> |
\ref{fourierQuadZeroK}, for a particular value of the quadrupolar |
| 1380 |
> |
polarization $\tilde{\Theta}_{aa}$ we can contract |
| 1381 |
> |
$\tilde{T}_{aa\gamma\delta}(0)$ with $\tilde{\Theta}_{\gamma\delta}$, |
| 1382 |
> |
using the traceless properties of the quadrupolar moment, |
| 1383 |
|
\begin{eqnarray} |
| 1384 |
< |
\tilde{T}_{11\gamma\delta}(0)\tilde{\Theta}_{\gamma\delta}(0) &=& \tilde{T}_{1111}(0)\tilde{\Theta}_{11}(0) + \tilde{T}_{1122}(0)\tilde{\Theta}_{22}(0) + \tilde{T}_{1133}(0)\tilde{\Theta}_{33}(0) \nonumber \\ |
| 1385 |
< |
&=& \mathrm{3B'}\tilde{\Theta}_{11}(0) + \mathrm{B'}\tilde{\Theta}_{22}(0) + \mathrm{B'}\tilde{\Theta}_{33}(0) \nonumber \\ |
| 1386 |
< |
&=& \mathrm{2B'}\tilde{\Theta}_{11}(0) + \mathrm{B'}\left(\tilde{\Theta}_{11}(0)+\tilde{\Theta}_{22}(0) + \tilde{\Theta}_{33}(0)\right) \nonumber \\ |
| 1387 |
< |
&=& \mathrm{2B'}\tilde{\Theta}_{11}(0) |
| 1384 |
> |
\tilde{T}_{xx\gamma\delta}(0)\tilde{\Theta}_{\gamma\delta}(0) &=& \tilde{T}_{xxxx}(0)\tilde{\Theta}_{xx}(0) + \tilde{T}_{xxyy}(0)\tilde{\Theta}_{yy}(0) + \tilde{T}_{xxzz}(0)\tilde{\Theta}_{zz}(0) \nonumber \\ |
| 1385 |
> |
&=& 12 \pi \mathrm{B}\tilde{\Theta}_{xx}(0) + |
| 1386 |
> |
4 \pi \mathrm{B}\tilde{\Theta}_{yy}(0) + |
| 1387 |
> |
4 \pi \mathrm{B}\tilde{\Theta}_{zz}(0) \nonumber \\ |
| 1388 |
> |
&=& 8 \pi \mathrm{B}\tilde{\Theta}_{xx}(0) + 4 \pi |
| 1389 |
> |
\mathrm{B}\left(\tilde{\Theta}_{xx}(0)+\tilde{\Theta}_{yy}(0) + |
| 1390 |
> |
\tilde{\Theta}_{zz}(0)\right) \nonumber \\ |
| 1391 |
> |
&=& 8 \pi \mathrm{B}\tilde{\Theta}_{xx}(0) |
| 1392 |
|
\end{eqnarray} |
| 1393 |
< |
Similarly for a quadrupolar polarization $\tilde{\Theta}_{ij}$ in equation \ref{fourierQuadZeroK}, we can contract $\tilde{T}_{ij\gamma\delta}(0)$ with $\tilde{\Theta}_{\gamma\delta}$ as follows (we considered $i$ = $1$ and $j$ = $2$ for simplicity) |
| 1393 |
> |
Similarly for a quadrupolar polarization $\tilde{\Theta}_{xy}$ in |
| 1394 |
> |
equation \ref{fourierQuadZeroK}, we can contract |
| 1395 |
> |
$\tilde{T}_{xy\gamma\delta}(0)$ with $\tilde{\Theta}_{\gamma\delta}$, |
| 1396 |
> |
using the only surviving terms of the tensor, |
| 1397 |
|
\begin{eqnarray} |
| 1398 |
< |
\tilde{T}_{12\gamma\delta}(0)\tilde{\Theta}_{\gamma\delta}(0) &=& \tilde{T}_{1212}(0)\tilde{\Theta}_{12}(0) + \tilde{T}_{1221}(0)\tilde{\Theta}_{21}(0) \nonumber \\ |
| 1399 |
< |
&=& \mathrm{B'}\tilde{\Theta}_{12}(0) + \mathrm{B'}\tilde{\Theta}_{21}(0) \nonumber \\ |
| 1400 |
< |
&=& \mathrm{2B'}\tilde{\Theta}_{11}(0) |
| 1398 |
> |
\tilde{T}_{xy\gamma\delta}(0)\tilde{\Theta}_{\gamma\delta}(0) &=& \tilde{T}_{xyxy}(0)\tilde{\Theta}_{xy}(0) + \tilde{T}_{xyyx}(0)\tilde{\Theta}_{yx}(0) \nonumber \\ |
| 1399 |
> |
&=& 4 \pi \mathrm{B}\tilde{\Theta}_{xy}(0) + |
| 1400 |
> |
4 \pi \mathrm{B}\tilde{\Theta}_{yx}(0) \nonumber \\ |
| 1401 |
> |
&=& 8 \pi \mathrm{B}\tilde{\Theta}_{xy}(0) |
| 1402 |
|
\end{eqnarray} |
| 1403 |
< |
Therefore we can write matrix contraction for $\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})$ and $ \tilde{\Theta}_{\gamma\delta}(\mathrm{0})$ in general the form as below, |
| 1403 |
> |
Here, we have used the symmetry of the quadrupole tensor to combine |
| 1404 |
> |
the symmetric terms. Therefore we can write matrix contraction for |
| 1405 |
> |
$\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})$ and |
| 1406 |
> |
$ \tilde{\Theta}_{\gamma\delta}(\mathrm{0})$ in a general form, |
| 1407 |
|
\begin{equation} |
| 1408 |
< |
\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})\tilde{\Theta}_{\gamma\delta}(\mathrm{0}) = 2\mathrm{B'} \tilde{\Theta}_{\alpha\beta}(\mathrm{0}), |
| 1408 |
> |
\tilde{T}_{\alpha\beta\gamma\delta}(\mathrm{0})\tilde{\Theta}_{\gamma\delta}(\mathrm{0}) |
| 1409 |
> |
= 8 \pi \mathrm{B} \tilde{\Theta}_{\alpha\beta}(\mathrm{0}), |
| 1410 |
|
\label{contract} |
| 1411 |
|
\end{equation} |
| 1412 |
< |
If we consider quadrupole-quadrupole interaction considering charge representation then the symmetry of the quadrupolar tensor is same as in the case of point quadrupolar representation (see equations~\ref{quadCharge} and \ref{quadRadial}. On the other hand, if we consider point dipole representation to evaluate quadrupole-quadrupole interactions then the symmetry of the quadrupolar tensor is slightly different than point quadrupole representation (compare equations~\ref{quadDip} and~\ref{quadRadial}). Although there is difference in symmetry, the final result (equation~\ref{contract}) even holds true for dipolar representations. |
| 1412 |
> |
which is the same as Eq. (\ref{quadContraction}). |
| 1413 |
|
|
| 1414 |
+ |
When the molecular quadrupoles are represented by point charges, the |
| 1415 |
+ |
symmetry of the quadrupolar tensor is same as for point quadrupoles |
| 1416 |
+ |
(see Eqs.~\ref{quadCharge} and \ref{quadRadial}). However, for |
| 1417 |
+ |
molecular quadrupoles represented by point dipoles, the symmetry of |
| 1418 |
+ |
the quadrupolar tensor must be handled separately (compare |
| 1419 |
+ |
equations~\ref{quadDip} and~\ref{quadRadial}). Although there is a |
| 1420 |
+ |
difference in symmetry, the final result (Eq.~\ref{contract}) holds |
| 1421 |
+ |
true for dipolar representations as well. |
| 1422 |
+ |
|
| 1423 |
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\section{Quadrupolar correction factor for Ewald-Kornfeld (EK) method} |
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The quadrupolar-quadrupolar interaction tensor for Ewald method can be expressed in the following form, \cite{Smith82,NeumannII83 } |
| 1425 |
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\begin{eqnarray} |
