Rewrite the Ewald summation part.

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Marek Nečada 2019-08-07 06:55:47 +03:00
parent 46b651d97f
commit 9aff527bd9
4 changed files with 343 additions and 297 deletions

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@ -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}
}

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@ -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

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@ -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

View File

@ -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