Rewrite Ewald intro.

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Marek Nečada 2020-06-22 16:29:11 +03:00
parent f756592bc5
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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

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\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.+\\