From 07fe3a46458a7edbf8ab21c172376c992b04ec43 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Sun, 7 Jun 2020 21:53:59 +0300 Subject: [PATCH] Adjustment of Ewald parameter. Former-commit-id: 45810dae1a715a12ece882200501c2304528d0a1 --- lepaper/infinite.lyx | 154 +++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 150 insertions(+), 4 deletions(-) diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index 0c84cae..4f5683b 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -912,7 +912,12 @@ Note that fast and can be summed directly, and a long-range part which decays poorly but is fairly smooth everywhere, so that its Fourier transform decays fast enough, and to deal with the long range part by Poisson summation over - the reciprocal lattice. + the reciprocal lattice; these two parts put together shall give an analytical + continuation of the original sum for +\begin_inset Formula $\Im\kappa\le0$ +\end_inset + +. This idea dates back to Ewald \begin_inset CommandInset citation LatexCommand cite @@ -1615,7 +1620,7 @@ and if the normal component \begin_inset Formula \begin{multline} \sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{d}\mathcal{A}}\pi^{2+\left(3-d\right)/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\ -\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(\frac{d-1}{2}-j,\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\times\\ +\times\sum_{\vect K\in\Lambda^{*}}\underbrace{e^{i\vect K\cdot\vect s}}_{\text{nemá tu být \ensuremath{\vect{k\cdot s}?}}}\ush lm\left(\vect k+\vect K\right)\sum_{j=0}^{\left[\left(l-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\left|\vect k+\vect K\right|/\kappa\right)^{l-2j}\Gamma\left(\frac{d-1}{2}-j,\frac{\kappa^{2}\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)^{2}}{4\eta^{2}}\right)}{j!\left(\frac{1}{2}\left(l-m\right)-j\right)!\left(\frac{1}{2}\left(l+m\right)-j\right)!}\times\\ \times\left(\gamma\left(\left|\vect k+\vect K\right|/\kappa\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D z = 0} \end{multline} @@ -1951,8 +1956,149 @@ Whatabout different geometries? \end_inset - However, at larger wavelengths, TODO BLA BLA BLA which is detrimental for - accuracy in floating point arithmetics. + However, in floating point arithmetics, the magnitude of the summands must + be taken into account as well in order to maintain accuracy. + There is a particular problem with the +\begin_inset Quotes eld +\end_inset + +central +\begin_inset Quotes erd +\end_inset + + reciprocal lattice points in the long-range sums for which the real part + of +\begin_inset Formula $\left|\vect k+\vect K\right|^{2}-\kappa^{2}$ +\end_inset + + is negative: the incomplete +\begin_inset Formula $\Gamma$ +\end_inset + + function present in the sum (either explicitly or in the expansions of + +\family roman +\series medium +\shape up +\size normal +\emph off +\nospellcheck off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $\Delta_{j}$ +\end_inset + +) +\family default +\series default +\shape default +\size default +\emph default +\nospellcheck default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit + grows exponentially with respect to the negative second argument, with + asymptotic behaviour +\begin_inset Formula $\Gamma\left(a,z\right)\sim e^{-z}z^{a-1}$ +\end_inset + + . + Therefore for higher frequencies, the parameter +\begin_inset Formula $\eta$ +\end_inset + + needs to be adjusted in a way that keeps the value of +\begin_inset Formula $\Gamma\left(a,\frac{\left|\vect k+\vect K\right|^{2}-\kappa^{2}}{4\eta^{2}}\right)$ +\end_inset + + within reasonable bounds. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Setting a target maximum magnitude for +\begin_inset Formula $M$ +\end_inset + +, so that +\begin_inset Formula $\left|\Gamma\left(a,-\left|z\right|\right)\right|\lesssim M$ +\end_inset + +, using the asymptotic estimate +\begin_inset Formula $\Gamma\left(a,-\left|z\right|\right)\sim e^{-\left|z\right|}\left|z\right|^{a-1}$ +\end_inset + +, we get +\begin_inset Formula +\begin{align*} +e^{-\left|z\right|}\left|z\right|^{a-1} & \lesssim M,\\ +-\left|z\right|\left(a-1\right)\log\left|z\right| & \lesssim\log M. +\end{align*} + +\end_inset + + +\end_layout + +\end_inset + +If we assume that +\begin_inset Formula $\vect k$ +\end_inset + + lies in the first Brillouin zone, the minimum real part of the second argument + of the +\begin_inset Formula $\Gamma$ +\end_inset + + function will be +\begin_inset Formula $\left(\left|\vect k\right|^{2}-\kappa^{2}\right)/4\eta^{2}$ +\end_inset + +, so setting +\begin_inset Formula $\eta\ge\sqrt{\left|\kappa\right|^{2}-\left|\vect k\right|^{2}}/2\log M$ +\end_inset + + eliminates the exponential growth in the incomplete +\begin_inset Formula $\Gamma$ +\end_inset + + function, where the constant +\begin_inset Formula $M$ +\end_inset + + is chosen to represent the (rough) maximum tolerated magnitude of the summand + with regard to target accuracy. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +\begin_inset Formula +\[ +-\frac{\left|\left|\vect k\right|^{2}-\kappa^{2}\right|}{4\eta^{2}}\left(a-1\right)2\log\frac{\left|\left|\vect k\right|^{2}-\kappa^{2}\right|^{\frac{1}{2}}}{2\eta}\lesssim\log M +\] + +\end_inset + + +\end_layout + +\end_inset + + \end_layout \begin_layout Standard