From 9aff527bd9b2a4e71f4c1d7734d15f43b10bf9af Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Wed, 7 Aug 2019 06:55:47 +0300 Subject: [PATCH] Rewrite the Ewald summation part. Former-commit-id: 9321d465ed0ba2da536afaab3813873a7ea64ac0 --- lepaper/Tmatrix.bib | 17 ++ lepaper/arrayscat.lyx | 5 + lepaper/finite.lyx | 6 +- lepaper/infinite.lyx | 612 ++++++++++++++++++++++-------------------- 4 files changed, 343 insertions(+), 297 deletions(-) diff --git a/lepaper/Tmatrix.bib b/lepaper/Tmatrix.bib index 34c9e30..2bcaa7f 100644 --- a/lepaper/Tmatrix.bib +++ b/lepaper/Tmatrix.bib @@ -474,4 +474,21 @@ file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/WTJU82S7/beyn2012.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/XSR5YIQM/Beyn - 2012 - An integral method for solving nonlinear eigenvalu.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/D24EDI64/S0024379511002540.html} } +@article{ewald_berechnung_1921, + title = {Die {{Berechnung}} Optischer Und Elektrostatischer {{Gitterpotentiale}}}, + volume = {369}, + copyright = {Copyright \textcopyright{} 1921 WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim}, + issn = {1521-3889}, + language = {en}, + number = {3}, + urldate = {2019-08-07}, + journal = {Annalen der Physik}, + doi = {10.1002/andp.19213690304}, + url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19213690304}, + author = {Ewald, P. P.}, + year = {1921}, + pages = {253-287}, + file = {/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/TL9NGJTR/Ewald - 1921 - Die Berechnung optischer und elektrostatischer Git.pdf;/u/46/necadam1/unix/.mozilla/firefox/6m8fw48s.default/zotero/storage/HXX7A93Q/andp.html} +} + diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index 663d9ed..9861623 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -319,6 +319,11 @@ status open \end_inset +\begin_inset FormulaMacro +\newcommand{\sswfoutlm}[2]{\psi_{#1,#2}} +\end_inset + + \begin_inset FormulaMacro \newcommand{\outcoeff}{f} \end_inset diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 5e83f82..12c7140 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -1927,7 +1927,7 @@ m & -m' & m'-m \begin_inset Formula -\begin{multline*} +\begin{multline} C_{lm;l'm'}^{\lambda}=\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\right)}}\times\\ \times\begin{pmatrix}l & l' & \lambda\\ 0 & 0 & 0 @@ -1939,8 +1939,8 @@ D_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix}l & l' & \lambda\\ m & -m' & m'-m -\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}. -\end{multline*} +\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}.\label{eq:translation operator constant factors} +\end{multline} \end_inset diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx index 5c93352..872340e 100644 --- a/lepaper/infinite.lyx +++ b/lepaper/infinite.lyx @@ -138,15 +138,6 @@ Topology anoyne? scatterer arrays. \end_layout -\begin_layout Subsection -Notation -\end_layout - -\begin_layout Standard -TODO Fourier transforms, Delta comb, lattice bases, reciprocal lattices - etc. -\end_layout - \begin_layout Subsection Formulation of the problem \end_layout @@ -171,8 +162,8 @@ noprefix "false" \begin_inset Formula $d$ \end_inset - can be 1, 2 or 3) lattice embedded into the three-dimensional real space, - with lattice vectors + can be 1, 2 or 3) Bravais lattice embedded into the three-dimensional real + space, with lattice vectors \begin_inset Formula $\left\{ \dlv_{i}\right\} _{i=1}^{d}$ \end_inset @@ -307,7 +298,7 @@ lattice Fourier transform of the translation operator, \begin_inset Formula \begin{equation} -W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}},\label{eq:W definition} +W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\delta_{\vect m\vect 0}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}},\label{eq:W definition} \end{equation} \end_inset @@ -505,8 +496,8 @@ noprefix "false" \end_inset , which, for a given geometry, depends only on frequency). - Therefore, a much more efficient approach to determine the photonic bands - is to sample the + Therefore, a much more efficient but not completely robust approach to + determine the photonic bands is to sample the \begin_inset Formula $\vect k$ \end_inset @@ -579,6 +570,19 @@ noprefix "false" \end_inset . + Another, more robust approach is Beyn's contour integral algorithm +\begin_inset CommandInset citation +LatexCommand cite +key "beyn_integral_2012" +literal "false" + +\end_inset + + which finds the roots of +\begin_inset Formula $M\left(\omega,\vect k\right)=0$ +\end_inset + + in a given frequency contour. \end_layout \begin_layout Subsection @@ -633,265 +637,69 @@ Note that \end_inset - In electrostatics, this problem can be solved with Ewald summation [TODO - REF]. - Its basic idea is that if what asymptoticaly decays poorly in the direct - space, will perhaps decay fast in the Fourier space. - We use the same idea here, but the technical details are more complicated - than in electrostatics. -\end_layout + The problem of poorly converging lattice sums has been originally solved + for electrostatic potentials with Ewald summation +\begin_inset CommandInset citation +LatexCommand cite +key "ewald_berechnung_1921" +literal "false" -\begin_layout Standard -Let us re-express the sum in -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:W definition" - -\end_inset - - in terms of integral with a delta comb -\begin_inset FormulaMacro -\renewcommand{\basis}[1]{\mathfrak{#1}} -\end_inset - - -\end_layout - -\begin_layout Standard -\begin_inset Formula -\begin{equation} -W_{\alpha\beta}(\vect k)=\int\ud^{d}\vect r\dc{\basis u}(\vect r)S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})e^{i\vect k\cdot\vect r}.\label{eq:W integral} -\end{equation} - -\end_inset - -The translation operator -\begin_inset Formula $S$ -\end_inset - - is now a function defined in the whole 3d space; -\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$ -\end_inset - - are the displacements of scatterers -\begin_inset Formula $\alpha$ -\end_inset - - and -\begin_inset Formula $\beta$ -\end_inset - - in a unit cell. - The arrow notation -\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$ -\end_inset - - means -\begin_inset Quotes eld -\end_inset - -translation operator for spherical waves originating in -\begin_inset Formula $\vect r+\vect r_{\beta}$ -\end_inset - - evaluated in -\begin_inset Formula $\vect r_{\alpha}$ -\end_inset - - -\begin_inset Quotes erd -\end_inset - - and obviously -\begin_inset Formula $S$ -\end_inset - - is in fact a function of a single 3d argument, -\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$ \end_inset . - Expression -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:W integral" - -\end_inset - - can be rewritten as -\begin_inset Formula -\[ -W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0))\left(\vect k\right)} -\] - -\end_inset - -where changed the sign of -\begin_inset Formula $\vect r/\vect{\bullet}$ -\end_inset - - has been swapped under integration, utilising evenness of -\begin_inset Formula $\dc{\basis u}$ -\end_inset - -. - Fourier transform of product is convolution of Fourier transforms, so (using - formula -\begin_inset Note Note -status open - -\begin_layout Plain Layout -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:Dirac comb uaFt" - -\end_inset - - -\end_layout - -\end_inset - -(REF?) for the Fourier transform of Dirac comb) -\begin_inset Formula -\begin{eqnarray} -W_{\alpha\beta}(\vect k) & = & \left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)(\vect k)\nonumber \\ - & = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\right)\left(\vect k\right)\nonumber \\ - & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W sum in reciprocal space}\\ - & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}e^{i\left(\vect k-\vect K\right)\cdot\left(-\vect r_{\beta}+\vect r_{\alpha}\right)}\left(\uaft{S(\vect{\bullet}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\nonumber -\end{eqnarray} - -\end_inset - - -\begin_inset Note Note -status open - -\begin_layout Plain Layout -Factor -\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$ -\end_inset - - cancels out with the -\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$ -\end_inset - - factor appearing in the convolution/product formula in the unitary angular - momentum convention. - -\end_layout - -\end_inset - -As such, this is not extremely helpful because the the -\emph on -whole -\emph default - translation operator -\begin_inset Formula $S$ -\end_inset - - has singularities in origin, hence its Fourier transform -\begin_inset Formula $\uaft S$ -\end_inset - - will decay poorly. - + Its basic idea is to decompose the divide the lattice-summed function in + two parts: a short-range part that decays 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 same idea can be used also in this case case of linear electrodynamic + problems, just the technical details are more complicated than in electrostatic +s. \end_layout \begin_layout Standard -However, Fourier transform is linear, so we can in principle separate -\begin_inset Formula $S$ -\end_inset - - in two parts, -\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$ -\end_inset - -. +In eq. -\begin_inset Formula $S^{\textup{S}}$ -\end_inset - - is a short-range part that decays sufficiently fast with distance so that - its direct-space lattice sum converges well; -\begin_inset Formula $S^{\textup{S}}$ -\end_inset - - must as well contain all the singularities of -\begin_inset Formula $S$ -\end_inset - - in the origin. - The other part, -\begin_inset Formula $S^{\textup{L}}$ -\end_inset - -, will retain all the slowly decaying terms of -\begin_inset Formula $S$ -\end_inset - - but it also has to be smooth enough in the origin, so that its Fourier - transform -\begin_inset Formula $\uaft{S^{\textup{L}}}$ -\end_inset - - decays fast enough. - (The same idea lies behind the Ewald summation in electrostatics.) Using - the linearity of Fourier transform and formulae \begin_inset CommandInset ref LatexCommand eqref -reference "eq:W definition" - -\end_inset - - and -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:W sum in reciprocal space" - -\end_inset - -, the operator -\begin_inset Formula $W_{\alpha\beta}$ -\end_inset - - can then be re-expressed as -\begin_inset Formula -\begin{eqnarray} -W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\ -W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\ -W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition} -\end{eqnarray} - -\end_inset - -where both sums expected to converge nicely. - We note that the elements of the translation operators -\begin_inset Formula $\tropr,\trops$ -\end_inset - - in -\begin_inset CommandInset ref -LatexCommand eqref -reference "eq:translation operator" +reference "eq:translation operator singular" plural "false" caps "false" noprefix "false" \end_inset - can be rewritten as linear combinations of expressions -\begin_inset Formula $\ush{\nu}{\mu}\left(\uvec d\right)j_{n}\left(d\right),\ush{\nu}{\mu}\left(\uvec d\right)h_{n}^{(1)}\left(d\right)$ + we demonstratively expressed the translation operator elements as linear + combinations of (outgoing) scalar spherical wavefunctions +\begin_inset Formula $\sswfoutlm lm\left(\vect r\right)=h_{l}^{\left(1\right)}\left(r\right)\ush lm\left(\uvec r\right)$ \end_inset - (TODO WRITE THEM EXPLICITLY IN THIS FORM), respectively, hence if we are - able evaluate the lattice sums sums +, because for them, fortunately, exponentially convergent Ewald-type summation + formulae have been already developed \begin_inset Note Note status open \begin_layout Plain Layout -CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS +add refs +\end_layout + +\end_inset + + +\begin_inset CommandInset citation +LatexCommand cite +key "moroz_quasi-periodic_2006,linton_one-_2009,linton_lattice_2010" +literal "false" + +\end_inset + + and can be applied to our case. + If we formally label +\begin_inset Marginal +status open + +\begin_layout Plain Layout +Check signs. \end_layout \end_inset @@ -899,50 +707,132 @@ CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS \begin_inset Formula \begin{equation} -\sigma_{\nu}^{\mu}\left(\vect k\right)=\sum_{\vect n\in\ints^{d}\backslash\left\{ \vect0\right\} }e^{i\vect{\vect k}\cdot\vect R_{\vect n}}\ush{\nu}{\mu}\left(\uvec{R_{n}}\right)h_{n}^{(1)}\left(R_{n}\right),\label{eq:sigma lattice sums} +\sigma_{l,m}\left(\vect k,\vect s\right)=\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},\vect s}\right)e^{i\vect{\vect k}\cdot\left(\vect R_{\vect n}-\vect s\right)}\sswfoutlm lm\left(\vect{R_{n}}-\vect s\right),\label{eq:sigma lattice sums} \end{equation} \end_inset -then by linearity, we can get the -\begin_inset Formula $W_{\alpha\beta}\left(\vect k\right)$ -\end_inset - - operator as well. -\end_layout - -\begin_layout Standard -TODO ADD MOROZ AND OTHER REFS HERE. +we see from eqs. -\begin_inset CommandInset citation -LatexCommand cite -key "linton_one-_2009" -literal "true" +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:translation operator singular" +plural "false" +caps "false" +noprefix "false" \end_inset - offers an exponentially convergent Ewald-type summation method for -\begin_inset Formula $\sigma_{\nu}^{\mu}\left(\vect k\right)=\sigma_{\nu}^{\mu(\mathrm{S})}\left(\vect k\right)+\sigma_{\nu}^{\mu(\mathrm{L})}\left(\vect k\right)$ +, +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:W definition" +plural "false" +caps "false" +noprefix "false" + \end_inset -. - Here we rewrite them in a form independent on the convention used for spherical - harmonics (as long as they are complex + that the matrix elements of +\begin_inset Formula $W_{\alpha\beta}(\vect k)$ +\end_inset + + read \begin_inset Note Note status open \begin_layout Plain Layout -lepší formulace +\begin_inset Formula +\[ +W_{\alpha\beta}(\vect k)\equiv\sum_{\vect m\in\ints^{d}}\left(1-\delta_{\alpha\beta}\right)\tropsp{\vect 0,\alpha}{\vect m,\beta}e^{i\vect k\cdot\vect R_{\vect m}}, +\] + +\end_inset + + \end_layout \end_inset -). - The short-range part reads (UNIFY INDEX NOTATION) + +\begin_inset Formula +\begin{align*} +W_{\alpha,\tau lm;\beta,\tau l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\\ +W_{\alpha,\tau lm;\beta,\tau'l'm'}(\vect k) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\sigma_{\lambda,m-m'}\left(\vect k,\vect r_{\beta}-\vect r_{\alpha}\right),\quad\tau'\ne\tau, +\end{align*} + +\end_inset + + +\begin_inset Marginal +status open + +\begin_layout Plain Layout +Check signs +\end_layout + +\end_inset + +where the constant factors are exactly the same as in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:translation operator constant factors" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. +\end_layout + +\begin_layout Standard +For reader's reference, we list the Ewald-type formulae for lattice sums + +\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)$ +\end_inset + + from +\begin_inset CommandInset citation +LatexCommand cite +key "linton_lattice_2010" +literal "false" + +\end_inset + + rewritten in a way that is independent on particular phase or normalisation + conventions of vector spherical harmonics. +\end_layout + +\begin_layout Standard +In all three dimensionality cases, the lattice sums are divided into short-range + and long-range parts, +\begin_inset Formula $\sigma_{l,m}\left(\vect k,\vect s\right)=\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)+\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)$ +\end_inset + + depending on a positive parameter +\begin_inset Formula $\eta$ +\end_inset + +. + The short-range part has in all three cases the same form: +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Check sign of s +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard \begin_inset Formula \begin{multline} -\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)=-\frac{2^{n+1}i}{k^{n+1}\sqrt{\pi}}\sum_{\vect R\in\Lambda}^{'}\left|\vect R\right|^{n}e^{i\vect{\beta}\cdot\vect R}Y_{n}^{m}\left(\vect R\right)\int_{\eta}^{\infty}e^{-\left|\vect R\right|^{2}\xi^{2}}e^{-k/4\xi^{2}}\xi^{2n}\ud\xi\\ -+\frac{\delta_{n0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)Y_{n}^{m},\label{eq:Ewald in 3D short-range part} +\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{2^{l+1}i}{k^{l+1}\sqrt{\pi}}\sum_{\vect n\in\ints^{d}}\left(1-\delta_{\vect{R_{n}},-\vect s}\right)\left|\vect{R_{n}+\vect s}\right|^{l}\ush lm\left(\vect{R_{n}+\vect s}\right)e^{i\vect k\cdot\left(\vect{R_{n}+\vect s}\right)}\int_{\eta}^{\infty}e^{-\left|\vect{R_{n}+\vect s}\right|^{2}\xi^{2}}e^{-k/4\xi^{2}}\xi^{2l}\ud\xi\\ ++\frac{\delta_{l0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)\ush lm\left(\vect{R_{n}+\vect s}\right),\label{eq:Ewald in 3D short-range part} \end{multline} \end_inset @@ -961,67 +851,193 @@ NEPATŘÍ TAM NĚJAKÁ DELTA FUNKCE K PŮVODNÍMU \end_inset -and the long-range part (FIXME, this is the 2D version; include the 1D and - 3D lattice expressions as well) + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Poznámka ohledně zahození radiální části u kulových fcí? +\end_layout + +\end_inset + + +\begin_inset Marginal +status open + +\begin_layout Plain Layout +N.B. + here +\begin_inset Formula $\vect k$ +\end_inset + + is the Bloch vector while +\begin_inset Formula $k$ +\end_inset + + is the wavenumber. + Relabel to make this distinction clear. +\end_layout + +\end_inset + +The long-range part for cases +\begin_inset Formula $d=1,2$ +\end_inset + + reads +\begin_inset Note Note +status open + +\begin_layout Plain Layout +check sign of +\begin_inset Formula $\vect k$ +\end_inset + + +\end_layout + +\end_inset + + \begin_inset Formula \begin{multline} -\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)=-\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\\ -\times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\gamma_{pq}\right)^{2j-1}\label{eq:Ewald in 3D long-range part} +\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{k^{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^{*}}\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|/k\right)^{l-2j}\Gamma\left(-j,\frac{k^{2}\gamma\left(\left|\vect k+\vect K\right|/k\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)!}\left(\gamma\left(\left|\vect k+\vect K\right|/k\right)\right)^{2j+3-d}\label{eq:Ewald in 3D long-range part 1D 2D} \end{multline} \end_inset -where -\begin_inset Formula $\xi$ +and for +\begin_inset Formula $d=3$ \end_inset - is TODO, -\begin_inset Formula $\beta_{pq}$ + +\begin_inset Formula +\begin{equation} +\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)=\frac{4\pi i^{l+1}}{k\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}\frac{\left(\left|\vect k+\vect K\right|/k\right)^{l}}{k^{2}-\left|\vect k+\vect K\right|^{2}}e^{\left(k^{2}-\left|\vect k+\vect K\right|^{2}\right)/4\eta^{2}}\ush lm\left(\vect k+\vect K\right).\label{eq:Ewald in 3D long-range part 3D} +\end{equation} + \end_inset - is TODO, -\begin_inset Formula $\Gamma_{j,pq}$ +Here +\begin_inset Formula $\mathcal{A}$ \end_inset - is TODO and + is the unit cell volume (or length/area in the 1D/2D lattice cases). + The sums are taken over the reciprocal lattice +\begin_inset Formula $\Lambda^{*}$ +\end_inset + + with lattice vectors +\begin_inset Formula $\left\{ \vect b_{i}\right\} _{i=1}^{d}$ +\end_inset + + satisfying +\begin_inset Formula $\vect a_{i}\cdot\vect b_{j}=\delta_{ij}$ +\end_inset + +. + The function +\begin_inset Formula $\gamma\left(z\right)$ +\end_inset + + used in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Ewald in 3D long-range part 1D 2D" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + is defined as +\begin_inset Formula +\[ +\gamma\left(z\right)=\left(z-1\right)^{\frac{1}{2}}\left(z+1\right)^{\frac{1}{2}},\quad-\frac{3\pi}{2}<\arg\left(z-1\right)<\frac{\pi}{2},-\frac{\pi}{2}<\arg\left(z+1\right)<\frac{3\pi}{2}. +\] + +\end_inset + +The Ewald parameter \begin_inset Formula $\eta$ \end_inset - is a real parameter that determines the pace of convergence of both parts. + determines the pace of convergence of both parts. The larger \begin_inset Formula $\eta$ \end_inset is, the faster -\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$ +\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$ \end_inset converges but the slower -\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$ +\begin_inset Formula $\sigma_{l,m}^{\left(L,\eta\right)}\left(\vect k,\vect s\right)$ \end_inset converges. Therefore (based on the lattice geometry) it has to be adjusted in a way that a reasonable amount of terms needs to be evaluated numerically from both -\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$ +\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{S},\eta\right)}\left(\vect k,\vect s\right)$ \end_inset and -\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$ +\begin_inset Formula $\sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)$ \end_inset . - Generally, a good choice for +\begin_inset Marginal +status open + +\begin_layout Plain Layout +What would be a good choice? +\end_layout + +\end_inset + + +\begin_inset Marginal +status open + +\begin_layout Plain Layout +I have some error estimates derived in my notes. + Should I include them? +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Generally, a good choice for \begin_inset Formula $\eta$ \end_inset is TODO; in order to achieve accuracy TODO, one has to evaluate the terms on TODO lattice points. - (I HAVE SOME DERIVATIONS OF THE ESTIMATES IN MY NOTES; SHOULD I INCLUDE +\end_layout + +\end_inset + + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +(I HAVE SOME DERIVATIONS OF THE ESTIMATES IN MY NOTES; SHOULD I INCLUDE THEM?) \end_layout +\end_inset + + +\end_layout + \begin_layout Standard In practice, the integrals in \begin_inset CommandInset ref @@ -1037,7 +1053,15 @@ noprefix "false" \begin_inset Formula $\Gamma$ \end_inset --functions using the series TODO and TODO from DLMF. +-functions using the series 8.7.3 from +\begin_inset CommandInset citation +LatexCommand cite +key "NIST:DLMF" +literal "false" + +\end_inset + +. \end_layout \end_body