From 83fed81e24611f4f88013c53b32759bf8dd1987e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Wed, 9 Aug 2017 19:53:20 +0300 Subject: [PATCH] =?UTF-8?q?[ewald]=20pokra=C4=8Dov=C3=A1n=C3=AD?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Former-commit-id: c9a2cc5c707dbb90521c93bfa3e894ca956cb5ca --- notes/ewald.lyx | 190 ++++++++++++++++++++++++++++++++++++++++++++---- 1 file changed, 177 insertions(+), 13 deletions(-) diff --git a/notes/ewald.lyx b/notes/ewald.lyx index 3ace3ae..371c868 100644 --- a/notes/ewald.lyx +++ b/notes/ewald.lyx @@ -172,6 +172,11 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\nats}{\mathbb{N}} +\end_inset + + \begin_inset FormulaMacro \newcommand{\reals}{\mathbb{R}} \end_inset @@ -643,7 +648,7 @@ The translation operator for compact scatterers in 3d can be expressed as \begin_inset Formula \[ -S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(\left|\vect r\right|\right) +S_{l',m',t'\leftarrow l,m,t}\left(\vect r\leftarrow\vect 0\right)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect r},\phi_{\vect r}\right)z_{p}^{(J)}\left(k_{0}\left|\vect r\right|\right) \] \end_inset @@ -673,14 +678,14 @@ where The spherical Hankel functions can be expressed analytically as (REF DLMF 10.49.6, 10.49.1) \begin_inset Formula -\[ -h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!}, -\] +\begin{equation} +h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},\label{eq:spherical Hankel function series} +\end{equation} \end_inset so if we find a way to deal with the radial functions -\begin_inset Formula $s_{q}(r)=e^{ir}r^{-q}$ +\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ \end_inset , @@ -699,18 +704,176 @@ h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\righ \end_layout \begin_layout Standard +Assume that all scatterers are placed in the plane +\begin_inset Formula $\vect z=0$ +\end_inset +, so that the 2d Fourier transform of the long-range part of the translation + operator in terms of Hankel transforms, according to +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Fourier v. Hankel tf 2d" + +\end_inset + +, reads \end_layout \begin_layout Standard \begin_inset Formula +\begin{multline*} +\uaft{S_{l',m',t'\leftarrow l,m,t}^{\textup{L}}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\ +\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{h_{p}^{(1)\textup{L}}\left(k_{0}\vect{\bullet}\right)}\left(\left|\vect k\right|\right) +\end{multline*} + +\end_inset + +Here +\begin_inset Formula $h_{p}^{(1)\textup{L}}=h_{p}^{(1)}-h_{p}^{(1)\textup{S}}$ +\end_inset + + is a long range part of a given spherical Hankel function which has to + be found and which contains all the terms with far-field ( +\begin_inset Formula $r\to\infty$ +\end_inset + +) asymptotics proportional to +\begin_inset Formula $\sim e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ +\end_inset + +, +\begin_inset Formula $q\le Q$ +\end_inset + + where +\begin_inset Formula $Q$ +\end_inset + + is at least two in order to achieve absolute convergence of the direct-space + sum, but might be higher in order to speed the convergence up. +\end_layout + +\begin_layout Standard +Obviously, all the terms +\begin_inset Formula $\propto s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ +\end_inset + +, +\begin_inset Formula $q>Q$ +\end_inset + + of the spherical Hankel function +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:spherical Hankel function series" + +\end_inset + + can be kept untouched as part of +\begin_inset Formula $h_{p}^{(1)\textup{S}}$ +\end_inset + +, as they decay fast enough. +\end_layout + +\begin_layout Standard +The remaining task is therefore to find a suitable decomposition of +\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$ +\end_inset + +, +\begin_inset Formula $q\le Q$ +\end_inset + + into short-range and long-range parts, +\begin_inset Formula $s_{q}(r)=s_{q}^{\textup{S}}(r)+s_{q}^{\textup{L}}(r)$ +\end_inset + +, such that +\begin_inset Formula $s_{q}^{\textup{L}}(r)$ +\end_inset + + contains all the slowly decaying asymptotics and its Hankel transforms + decay desirably fast as well, +\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$ +\end_inset + +, +\begin_inset Formula $z\to\infty$ +\end_inset + +. + The latter requirement calls for suitable regularisation functions— +\begin_inset Formula $s_{q}^{\textup{L}}$ +\end_inset + + must be sufficiently smooth in the origin, so that +\begin_inset Formula +\begin{equation} +\pht n{s_{q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement} +\end{equation} + +\end_inset + + exists and decays fast enough. + +\begin_inset Formula $J_{\nu}(r)\sim\left(r/2\right)^{\nu}/\Gamma\left(\nu+1\right)$ +\end_inset + + (REF DLMF 10.7.3) near the origin, so the regularisation function should + be +\begin_inset Formula $\rho(r)=o(r^{q-n-1})$ +\end_inset + + only to make +\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}$ +\end_inset + + converge. + The additional decay speed requirement calls for at least +\begin_inset Formula $\rho(r)=o(r^{q-n+Q-1})$ +\end_inset + +, I guess. + At the same time, +\begin_inset Formula $\rho(r)$ +\end_inset + + must converge fast enough to one for +\begin_inset Formula $r\to\infty$ +\end_inset + +. +\end_layout + +\begin_layout Standard +The electrostatic Ewald summation uses regularisation with +\begin_inset Formula $1-e^{-cr^{2}}$ +\end_inset + +. + However, such choice does not seem to lead to an analytical solution for + the current problem +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:2d long range regularisation problem statement" + +\end_inset + +. + But it turns out that the family of functions +\begin_inset Formula \[ -\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{z_{p}^{(J)}}\left(\left|\vect k\right|\right) +\rho_{\kappa,c}(r)\equiv\left(1-e^{-cr}\right)^{\text{\kappa}},\quad c>0,\kappa\in\nats \] \end_inset +leads to satisfactory results, as will be shown below. +\end_layout +\begin_layout Subsubsection +Hankel transforms of the long-range parts \end_layout \begin_layout Subsection @@ -719,9 +882,10 @@ h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\righ \begin_layout Standard \begin_inset Formula -\[ -\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right) -\] +\begin{multline*} +\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\ +\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right) +\end{multline*} \end_inset @@ -777,7 +941,7 @@ where the spherical Hankel transform 2) \begin_inset Formula \[ -\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right). +\bsht lg(k)\equiv\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right). \] \end_inset @@ -787,7 +951,7 @@ Using this convention, the inverse spherical Hankel transform is given by 3) \begin_inset Formula \[ -g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\, k^{2}\bsht lg(k)j_{l}(k), +g(r)=\frac{2}{\pi}\int_{0}^{\infty}\ud k\,k^{2}\bsht lg(k)j_{l}(k), \] \end_inset @@ -800,7 +964,7 @@ so it is not unitary. An unitary convention would look like this: \begin_inset Formula \begin{equation} -\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition} +\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\,r^{2}g(r)j_{l}\left(kr\right).\label{eq:unitary 3d Hankel tf definition} \end{equation} \end_inset @@ -854,7 +1018,7 @@ where the Hankel transform of order is defined as \begin_inset Formula \begin{equation} -\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition} +\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition} \end{equation} \end_inset