From 46138df4fe30a765726c0c704e684ac2ed35fbbf Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Wed, 9 Aug 2017 11:09:38 +0000 Subject: [PATCH] =?UTF-8?q?[ewald]=20dudopr=C3=A1ce?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Former-commit-id: 798838a72c453bc8c30764373b36d52156500251 --- notes/ewald.lyx | 92 +++++++++++++++++++++++++++++++++++++++++-------- 1 file changed, 78 insertions(+), 14 deletions(-) diff --git a/notes/ewald.lyx b/notes/ewald.lyx index a51bccb..3ace3ae 100644 --- a/notes/ewald.lyx +++ b/notes/ewald.lyx @@ -656,12 +656,76 @@ where \begin_inset Formula $z_{p}^{(J)}\left(r\right)$ \end_inset - some of the Bessel or Hankel functions (TODO) and + some of the Bessel or Hankel functions (probably +\begin_inset Formula $h_{p}^{(1)}$ +\end_inset + + in the meaningful cases; TODO) and \begin_inset Formula $c_{p}^{l,m,t\leftarrow l',m',t'}$ \end_inset are some ugly but known coefficients (REF Xu 1996, eqs. 76,77). + +\end_layout + +\begin_layout Standard +The spherical Hankel functions can be expressed analytically as (REF DLMF + 10.49.6, 10.49.1) +\begin_inset Formula +\[ +h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!}, +\] + +\end_inset + + so if we find a way to deal with the radial functions +\begin_inset Formula $s_{q}(r)=e^{ir}r^{-q}$ +\end_inset + +, +\begin_inset Formula $q=1,2$ +\end_inset + + in 2d case or +\begin_inset Formula $q=1,2,3$ +\end_inset + + in 3d case, we get absolutely convergent summations in the direct space. +\end_layout + +\begin_layout Subsection +2d +\end_layout + +\begin_layout Standard + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\frac{\pi}{2},0\right)e^{i(m'-m)\phi}i^{m'-m}\pht{m'-m}{z_{p}^{(J)}}\left(\left|\vect k\right|\right) +\] + +\end_inset + + +\end_layout + +\begin_layout Subsection +3d +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\sum_{p}c_{p}^{l',m',t'\leftarrow l,m,t}\ush p{m'-m}\left(\theta_{\vect k},\phi_{\vect k}\right)\left(-i\right)^{p}\usht p{z_{p}^{(J)}}\left(\left|\vect k\right|\right) +\] + +\end_inset + + \end_layout \begin_layout Section @@ -735,9 +799,9 @@ so it is not unitary. \begin_layout Standard An unitary convention would look like this: \begin_inset Formula -\[ -\usht lg(k)\equiv\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\ud r\, r^{2}g(r)j_{l}\left(kr\right). -\] +\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} +\end{equation} \end_inset @@ -747,10 +811,10 @@ Then and the unitary, angular-momentum Fourier transform reads \begin_inset Formula -\begin{eqnarray*} -\uaft f(\vect k) & = & \frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sqrt{\frac{\pi}{2}}\sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)\\ - & = & \sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right). -\end{eqnarray*} +\begin{eqnarray} +\uaft f(\vect k) & = & \frac{4\pi}{\left(2\pi\right)^{\frac{3}{2}}}\sqrt{\frac{\pi}{2}}\sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right)\nonumber \\ + & = & \sum_{l,m}\left(-i\right)^{l}\left(\usht l{f_{l,m}}\right)\left(\left|\vect k\right|\right)\ush lm\left(\theta_{\vect k},\phi_{\vect k}\right).\label{eq:Fourier v. Hankel tf 3d} +\end{eqnarray} \end_inset @@ -777,9 +841,9 @@ f\left(\vect r\right)=\sum_{m}f_{m}\left(\left|\vect r\right|\right)e^{im\phi_{\ its Fourier transform is then (CHECK this, it is taken from the Wikipedia article on Hankel transform) \begin_inset Formula -\[ -\uaft f\left(\vect k\right)=\sum_{m}i^{m}e^{im\theta_{\vect k}}\pht mf\left(\left|\vect k\right|\right) -\] +\begin{equation} +\uaft f\left(\vect k\right)=\sum_{m}i^{m}e^{im\phi_{\vect k}}\pht mf_{m}\left(\left|\vect k\right|\right)\label{eq:Fourier v. Hankel tf 2d} +\end{equation} \end_inset @@ -789,9 +853,9 @@ where the Hankel transform of order is defined as \begin_inset Formula -\[ -\pht mg\left(k\right)=\int_{0}^{\infty}\ud r\, J_{m}(kr)r -\] +\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} +\end{equation} \end_inset