diff --git a/notes/ewald.lyx b/notes/ewald.lyx index 12b3ead..6e0fff4 100644 --- a/notes/ewald.lyx +++ b/notes/ewald.lyx @@ -3209,7 +3209,7 @@ For the short-range part \end_inset , the radially varying part reads -\begin_inset Formula $f_{\eta}^{\mathrm{L}}\left(R_{pq}\right)\equiv R_{pq}^{n}\int_{\eta}^{\infty}e^{-R_{pq}^{2}\xi^{2}}e^{k^{2}/4\xi^{2}}\xi^{2n}\ud\xi$ +\begin_inset Formula $f_{\eta}^{\mathrm{S}}\left(R_{pq}\right)\equiv R_{pq}^{n}\int_{\eta}^{\infty}e^{-R_{pq}^{2}\xi^{2}}e^{k^{2}/4\xi^{2}}\xi^{2n}\ud\xi$ \end_inset and for its integral as in @@ -3222,7 +3222,7 @@ reference "eq:lsum_bound" we have \begin_inset Formula \begin{eqnarray*} -B_{R_{\mathrm{s}}}\left[f_{\eta}^{\mathrm{L}}\right] & = & \int_{R_{\mathrm{s}}}^{\infty}r^{n+1}\int_{\eta}^{\infty}e^{-r^{2}\xi^{2}}e^{k^{2}/4\xi^{2}}\xi^{2n}\ud\xi\,\ud r\\ +B_{R_{\mathrm{s}}}\left[f_{\eta}^{\mathrm{S}}\right] & = & \int_{R_{\mathrm{s}}}^{\infty}r^{n+1}\int_{\eta}^{\infty}e^{-r^{2}\xi^{2}}e^{k^{2}/4\xi^{2}}\xi^{2n}\ud\xi\,\ud r\\ & \le & e^{k^{2}/4\eta^{2}}\int_{R_{\mathrm{s}}}^{\infty}\int_{\eta}^{\infty}r^{n+1}e^{-r^{2}\xi^{2}}\xi^{2n}\ud\xi\,\ud r\\ & = & e^{k^{2}/4\eta^{2}}\frac{\eta^{2n+1}R_{\mathrm{s}}^{2+n}\left(E_{\frac{1}{2}-n}\left(\eta^{2}R_{\mathrm{s}}^{2}\right)-E_{-\frac{n}{2}}\left(\eta^{2}R_{\mathrm{s}}^{2}\right)\right)}{2\left(n-1\right)}\\ & = & e^{k^{2}/4\eta^{2}}\frac{\eta^{2n+1}R_{\mathrm{s}}^{2+n}\left(\left(\eta R_{\mathrm{s}}\right)^{-2n-1}\Gamma\left(n+\frac{1}{2},\eta^{2}R_{\mathrm{s}}^{2}\right)-\left(\eta R_{\mathrm{s}}\right)^{-n-2}\Gamma\left(\frac{n}{2}+1,\eta^{2}R_{\mathrm{s}}^{2}\right)\right)}{2\left(n-1\right)}\\ @@ -3258,8 +3258,48 @@ Apparently, this expression is problematic for \begin_inset Formula $_{2}F_{2}$ \end_inset +, resulting in: +\begin_inset Formula +\[ +B_{R_{\mathrm{s}}}\left[f_{\eta}^{\mathrm{S}}\right]\le e^{k^{2}/4\eta^{2}}\left(\frac{\eta R}{2}{}_{2}F_{2}\left(\begin{array}{cc} +\frac{1}{2}, & \frac{1}{2}\\ +\frac{3}{2}, & \frac{3}{2} +\end{array};-\eta^{2}R_{\mathrm{s}}^{2}\right)-\frac{\sqrt{\pi}}{8}\left(\gamma_{\mathrm{E}}-2\mathrm{erfc}\left(\eta R_{\mathrm{s}}\right)+2\log\left(2\eta R_{\mathrm{s}}\right)\right)\right). +\] + +\end_inset + +The problem is that evaluation of the +\begin_inset Formula $_{2}F_{2}$ +\end_inset + + for large argument is very problematic. + However, Mathematica says that the value of the right parenthesis drops + below DBL_EPSILON for +\begin_inset Formula $\eta R_{\mathrm{s}}>6$ +\end_inset + . - Hence it might make sense to take a rougher estimate using (for +\end_layout + +\begin_layout Standard +Also the expression for +\begin_inset Formula $n\ne1$ +\end_inset + + decreases very fast, so as long as the value of +\begin_inset Formula $e^{k^{2}/4\eta^{2}}$ +\end_inset + +is reasonably low, there should not be much trouble. +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Maybe it might make sense to take a rougher estimate using (for \begin_inset Formula $n=1$ \end_inset @@ -3285,8 +3325,7 @@ symmetric \begin_inset Formula $R_{\mathrm{s}}\leftrightarrow\eta$ \end_inset -, so we can write either TODO; dammit, I should implement the hypergeometric - fn instead. +, so we can write either \begin_inset Formula \[ B_{R_{\mathrm{s}}}\left[f_{\eta}^{\mathrm{L}}\right]\le e^{k^{2}/4\eta^{2}}\int_{R_{\mathrm{s}}}^{\infty}\int_{\eta}^{\infty}e^{-r^{2}\xi^{2}}r^{2}\xi^{2}\ud\xi\,\ud r @@ -3295,6 +3334,11 @@ B_{R_{\mathrm{s}}}\left[f_{\eta}^{\mathrm{L}}\right]\le e^{k^{2}/4\eta^{2}}\int_ \end_inset +\end_layout + +\end_inset + + \end_layout \begin_layout Subsubsection @@ -3334,7 +3378,22 @@ For \begin{eqnarray*} \left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}\left(\gamma_{pq}\right)^{2j-1} & = & \left(\beta_{pq}/k\right)^{n-2j}\Gamma\left(\frac{1}{2}-j,\frac{\beta_{pq}^{2}-k^{2}}{4\eta^{2}}\right)\left(\frac{\beta_{pq}^{2}}{k^{2}}-1\right)^{j-\frac{1}{2}}\\ & \le & \left(\beta_{pq}/k\right)^{n-2j}\left(\frac{\beta_{pq}^{2}-k^{2}}{4\eta^{2}}\right)^{-j-\frac{1}{2}}e^{-\frac{\beta_{pq}^{2}-k^{2}}{4\eta^{2}}}\left(\frac{\beta_{pq}^{2}}{k^{2}}-1\right)^{j-\frac{1}{2}}\\ - & & TODO + & = & \left(2\eta\right)^{2j+1}e^{-\frac{\beta_{pq}^{2}-k^{2}}{4\eta^{2}}}k^{-n-1}\beta_{pq}^{n-2j}\left(\frac{\beta_{pq}^{2}}{k^{2}}-1\right)^{-1}\\ + & = & e^{-\frac{\beta_{pq}^{2}-k^{2}}{4\eta^{2}}}\left(\frac{\beta_{pq}}{k}\right)^{n}\frac{2\eta}{k}\left(\frac{2\eta}{\beta_{pq}}\right)^{2j}\left(\frac{\beta_{pq}^{2}}{k^{2}}-1\right)^{-1}. +\end{eqnarray*} + +\end_inset + +The only diverging factor here is apparently +\begin_inset Formula $\left(\beta_{pq}/k\right)^{n}$ +\end_inset + +; Mathematica and [DMLF] say +\begin_inset Formula +\begin{eqnarray*} +\int_{B_{\mathrm{s}}}^{\infty}e^{-\frac{\beta^{2}}{4\eta^{2}}}\beta^{n}\beta\ud\beta & = & \frac{B_{\mathrm{s}}^{n+2}}{2}E_{-\frac{n}{2}}\left(\frac{B_{\mathrm{s}}^{2}}{4\eta^{2}}\right)\\ + & = & \frac{B_{\mathrm{s}}^{n+2}}{2}\left(\frac{B_{\mathrm{s}}^{2}}{4\eta^{2}}\right)^{-1-\frac{n}{2}}\Gamma\left(1+\frac{n}{2},\frac{B_{\mathrm{s}}^{2}}{4\eta^{2}}\right)\\ + & = & \frac{\left(2\eta\right)^{n+2}}{2}\Gamma\left(1+\frac{n}{2},\frac{B_{\mathrm{s}}^{2}}{4\eta^{2}}\right). \end{eqnarray*} \end_inset @@ -3426,7 +3485,7 @@ 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 @@ -3436,7 +3495,7 @@ 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 @@ -3449,7 +3508,7 @@ 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 @@ -3503,8 +3562,8 @@ where the Hankel transform of order is defined as \begin_inset Formula \begin{eqnarray} -\pht mg\left(k\right) & = & \int_{0}^{\infty}\ud r\, g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}\\ - & = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\, g(r)J_{-m}(kr)r +\pht mg\left(k\right) & = & \int_{0}^{\infty}\ud r\,g(r)J_{m}(kr)r\label{eq:unitary 2d Hankel tf definition}\\ + & = & \left(-1\right)^{m}\int_{0}^{\infty}\ud r\,g(r)J_{-m}(kr)r \end{eqnarray} \end_inset