| 620 |
|
\section{Energies, forces, and torques} |
| 621 |
|
\subsection{Interaction energies} |
| 622 |
|
|
| 623 |
< |
We now list multipole interaction energies for the four types of approximation. |
| 624 |
< |
A ``generic'' set of radial functions is introduced so to be able to present the results in Table I. This set of |
| 625 |
< |
equations is written in terms of space coordinates: |
| 623 |
> |
We now list multipole interaction energies using a set of generic |
| 624 |
> |
radial functions. Table \ref{tab:tableenergy} maps between the |
| 625 |
> |
generic functions and the radial functions derived for both the |
| 626 |
> |
Taylor-shifted and Gradient-shifted methods. This set of equations is |
| 627 |
> |
written in terms of space coordinates: |
| 628 |
|
|
| 629 |
|
% Energy in space coordinate form ---------------------------------------------------------------------------------------------- |
| 630 |
|
% |
| 631 |
|
% |
| 632 |
|
% u ca cb |
| 633 |
|
% |
| 634 |
< |
\begin{equation} |
| 635 |
< |
U_{C_{\bf a}C_{\bf b}}(r)= |
| 634 |
> |
\begin{align} |
| 635 |
> |
U_{C_{\bf a}C_{\bf b}}(r)=& |
| 636 |
|
\frac{C_{\bf a} C_{\bf b}}{4\pi \epsilon_0} v_{01}(r) \label{uchch} |
| 637 |
< |
\end{equation} |
| 637 |
> |
\\ |
| 638 |
|
% |
| 639 |
|
% u ca db |
| 640 |
|
% |
| 641 |
< |
\begin{equation} |
| 640 |
< |
U_{C_{\bf a}D_{\bf b}}(r)= |
| 641 |
> |
U_{C_{\bf a}D_{\bf b}}(r)=& |
| 642 |
|
\frac{C_{\bf a}}{4\pi \epsilon_0} \left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) v_{11}(r) |
| 643 |
|
\label{uchdip} |
| 644 |
< |
\end{equation} |
| 644 |
> |
\\ |
| 645 |
|
% |
| 646 |
|
% u ca qb |
| 647 |
|
% |
| 648 |
< |
\begin{equation} |
| 648 |
< |
U_{C_{\bf a}Q_{\bf b}}(r)= |
| 648 |
> |
U_{C_{\bf a}Q_{\bf b}}(r)=& |
| 649 |
|
\frac{C_{\bf a }}{4\pi \epsilon_0} \Bigl[ \text{Tr}Q_{\bf b} v_{21}(r) |
| 650 |
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
| 651 |
|
\label{uchquad} |
| 652 |
< |
\end{equation} |
| 652 |
> |
\\ |
| 653 |
|
% |
| 654 |
|
% u da cb |
| 655 |
|
% |
| 656 |
< |
\begin{equation} |
| 657 |
< |
U_{D_{\bf a}C_{\bf b}}(r)= |
| 658 |
< |
-\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 659 |
< |
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) v_{11}(r) \label{udipch} |
| 660 |
< |
\end{equation} |
| 656 |
> |
%U_{D_{\bf a}C_{\bf b}}(r)=& |
| 657 |
> |
%-\frac{C_{\bf b}}{4\pi \epsilon_0} |
| 658 |
> |
%\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) v_{11}(r) \label{udipch} |
| 659 |
> |
%\\ |
| 660 |
|
% |
| 661 |
|
% u da db |
| 662 |
|
% |
| 663 |
< |
\begin{equation} |
| 665 |
< |
U_{D_{\bf a}D_{\bf b}}(r)= |
| 663 |
> |
U_{D_{\bf a}D_{\bf b}}(r)=& |
| 664 |
|
-\frac{1}{4\pi \epsilon_0} \Bigr[ \left( \mathbf{D}_{\mathbf {a}} \cdot |
| 665 |
|
\mathbf{D}_{\mathbf{b}} \right) v_{21}(r) |
| 666 |
|
+\left( \mathbf{D}_{\mathbf {a}} \cdot \hat{r} \right) |
| 667 |
|
\left( \mathbf{D}_{\mathbf {b}} \cdot \hat{r} \right) |
| 668 |
|
v_{22}(r) \Bigr] |
| 669 |
|
\label{udipdip} |
| 670 |
< |
\end{equation} |
| 670 |
> |
\\ |
| 671 |
|
% |
| 672 |
|
% u da qb |
| 673 |
|
% |
| 676 |
– |
\begin{equation} |
| 674 |
|
\begin{split} |
| 675 |
|
% 1 |
| 676 |
< |
U_{D_{\bf a}Q_{\bf b}}(r)&= |
| 676 |
> |
U_{D_{\bf a}Q_{\bf b}}(r) =& |
| 677 |
|
-\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 678 |
|
\text{Tr}\mathbf{Q}_{\mathbf{b}} |
| 679 |
|
\left( \mathbf{D}_{\mathbf{a}} \cdot \hat{r} \right) |
| 684 |
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{32}(r) |
| 685 |
|
\label{udipquad} |
| 686 |
|
\end{split} |
| 687 |
< |
\end{equation} |
| 687 |
> |
\\ |
| 688 |
|
% |
| 689 |
|
% u qa cb |
| 690 |
|
% |
| 691 |
< |
\begin{equation} |
| 692 |
< |
U_{Q_{\bf a}C_{\bf b}}(r)= |
| 693 |
< |
\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\bf a} v_{21}(r) |
| 694 |
< |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
| 695 |
< |
\label{uquadch} |
| 699 |
< |
\end{equation} |
| 691 |
> |
%U_{Q_{\bf a}C_{\bf b}}(r)=& |
| 692 |
> |
%\frac{C_{\bf b }}{4\pi \epsilon_0} \Bigl[ \text{Tr}\mathbf{Q}_{\bf a} v_{21}(r) |
| 693 |
> |
%\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{22}(r) \Bigr] |
| 694 |
> |
%\label{uquadch} |
| 695 |
> |
%\\ |
| 696 |
|
% |
| 697 |
|
% u qa db |
| 698 |
|
% |
| 699 |
< |
\begin{equation} |
| 704 |
< |
\begin{split} |
| 699 |
> |
%\begin{split} |
| 700 |
|
%1 |
| 701 |
< |
U_{Q_{\bf a}D_{\bf b}}(r)&= |
| 702 |
< |
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 703 |
< |
\text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 704 |
< |
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
| 705 |
< |
+2 ( \mathbf{D}_{\mathbf{b}} \cdot |
| 706 |
< |
\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r) |
| 701 |
> |
%U_{Q_{\bf a}D_{\bf b}}(r)=& |
| 702 |
> |
%\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 703 |
> |
%\text{Tr}\mathbf{Q}_{\mathbf{a}} |
| 704 |
> |
%\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
| 705 |
> |
%+2 ( \mathbf{D}_{\mathbf{b}} \cdot |
| 706 |
> |
%\mathbf{Q}_{\mathbf{a}} \cdot \hat{r}) \Bigr] v_{31}(r)\\ |
| 707 |
|
% 2 |
| 708 |
< |
+\frac{1}{4\pi \epsilon_0} |
| 709 |
< |
\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
| 710 |
< |
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{32}(r) |
| 711 |
< |
\label{uquaddip} |
| 712 |
< |
\end{split} |
| 713 |
< |
\end{equation} |
| 708 |
> |
%&+\frac{1}{4\pi \epsilon_0} |
| 709 |
> |
%\left( \mathbf{D}_{\mathbf{b}} \cdot \hat{r} \right) |
| 710 |
> |
%\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf a}} \cdot \hat{r} \right) v_{32}(r) |
| 711 |
> |
%\label{uquaddip} |
| 712 |
> |
%\end{split} |
| 713 |
> |
%\\ |
| 714 |
|
% |
| 715 |
|
% u qa qb |
| 716 |
|
% |
| 722 |
– |
\begin{equation} |
| 717 |
|
\begin{split} |
| 718 |
|
%1 |
| 719 |
< |
U_{Q_{\bf a}Q_{\bf b}}(r)&= |
| 719 |
> |
U_{Q_{\bf a}Q_{\bf b}}(r)=& |
| 720 |
|
\frac{1}{4\pi \epsilon_0} \Bigl[ |
| 721 |
|
\text{Tr} \mathbf{Q}_{\mathbf{a}} \text{Tr} \mathbf{Q}_{\mathbf{b}} |
| 722 |
|
+2 \text{Tr} \left( |
| 738 |
|
\left( \hat{r} \cdot \mathbf{Q}_{{\mathbf b}} \cdot \hat{r} \right) v_{43}(r). |
| 739 |
|
\label{uquadquad} |
| 740 |
|
\end{split} |
| 741 |
< |
\end{equation} |
| 741 |
> |
\end{align} |
| 742 |
|
|
| 743 |
+ |
Note that the energies of multipoles on site $\mathbf{b}$ interacting |
| 744 |
+ |
with those on site $\mathbf{a}$ can be obtained by swapping indices |
| 745 |
+ |
along with the sign of the intersite vector, $\hat{r}$. |
| 746 |
|
|
| 747 |
|
% |
| 748 |
|
% |
| 753 |
|
\caption{\label{tab:tableenergy}Radial functions used in the energy and torque equations. Functions |
| 754 |
|
used in this table are defined in Appendices B and C.} |
| 755 |
|
\begin{ruledtabular} |
| 756 |
< |
\begin{tabular}{cccc} |
| 757 |
< |
Generic&Coulomb&Method 1&Method 2 |
| 756 |
> |
\begin{tabular}{|l|c|l|l} |
| 757 |
> |
Generic&Coulomb&Taylor-Shifted&Gradient-Shifted |
| 758 |
|
\\ \hline |
| 759 |
|
% |
| 760 |
|
% |
