From a5cf8505f727f404cbec2619ef883c8b0f8d513a Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek@necada.org>
Date: Mon, 22 Jun 2020 16:29:11 +0300
Subject: [PATCH] Rewrite Ewald intro.

Former-commit-id: df9c666b9cb34b45fab00155b0ee2a89f0c7c1e8
---
 lepaper/infinite.lyx   | 32 +++++++++++++++++++++++++-------
 lepaper/symmetries.lyx |  9 +++++++++
 2 files changed, 34 insertions(+), 7 deletions(-)

diff --git a/lepaper/infinite.lyx b/lepaper/infinite.lyx
index 1485270..c882b8f 100644
--- a/lepaper/infinite.lyx
+++ b/lepaper/infinite.lyx
@@ -953,8 +953,8 @@ literal "false"
  basic idea can be used as well, resulting in exponentially convergent summation
  formulae, but the technical details are considerably more complicated than
  in electrostatics.
- For the scalar Helmholtz equation in three dimensions, the formulae were
- developed by Ham & Segall 
+ For the scalar Helmholtz equation in three dimensions, the formulae for
+ lattice Green's functions were developed by Ham & Segall 
 \begin_inset CommandInset citation
 LatexCommand cite
 key "ham_energy_1961"
@@ -988,10 +988,28 @@ literal "false"
 \end_inset
 
 .
- We will not rederive the formulae here, but for reference, we restate the
- results in a form independent upon the normalisation and phase conventions
- for spherical harmonic bases (pointing out some errors in the aforementioned
- literature) and discuss some practical aspects of the numerical evaluation.
+ 
+\end_layout
+
+\begin_layout Standard
+For our purposes we do not need directly the lattice Green's functions but
+ rather the related lattice sums of spherical wavefunctions defined in 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "eq:sigma lattice sums"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, which can be derived by an analogous procedure.
+ Below, we state the results in a form independent upon the normalisation
+ and phase conventions for spherical harmonic bases (pointing out some errors
+ in the aforementioned literature) and discuss some practical aspects of
+ the numerical evaluation.
+ The derivation of the somewhat more complicated 1D and 2D periodicities
+ is provided in the Supplementary Material.
 \begin_inset Note Note
 status open
 
@@ -1001,7 +1019,7 @@ Tady ještě upřesnit, co vlastně dělám.
 
 \end_inset
 
-
+ 
 \end_layout
 
 \begin_layout Standard
diff --git a/lepaper/symmetries.lyx b/lepaper/symmetries.lyx
index 4b8beaa..90f1fc4 100644
--- a/lepaper/symmetries.lyx
+++ b/lepaper/symmetries.lyx
@@ -596,6 +596,10 @@ noprefix "false"
 \end_inset
 
 , we have
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
 \begin_inset Marginal
 status open
 
@@ -607,6 +611,11 @@ Check this carefully.
 \end_inset
 
 
+\end_layout
+
+\end_inset
+
+
 \begin_inset Formula 
 \begin{multline}
 \left(\groupop g\vect E\right)\left(\omega,\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(\kappa\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\