Rewrite the Ewald summation part.
Former-commit-id: 9321d465ed0ba2da536afaab3813873a7ea64ac0
This commit is contained in:
Marek Nečada
parent
46b651d97f
commit
9aff527bd9
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@ -474,4 +474,21 @@
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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}
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}
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@article{ewald_berechnung_1921,
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title = {Die {{Berechnung}} Optischer Und Elektrostatischer {{Gitterpotentiale}}},
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volume = {369},
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copyright = {Copyright \textcopyright{} 1921 WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim},
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issn = {1521-3889},
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language = {en},
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number = {3},
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urldate = {2019-08-07},
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journal = {Annalen der Physik},
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doi = {10.1002/andp.19213690304},
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url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19213690304},
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author = {Ewald, P. P.},
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year = {1921},
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pages = {253-287},
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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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}
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@ -319,6 +319,11 @@ status open
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\sswfoutlm}[2]{\psi_{#1,#2}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\outcoeff}{f}
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\end_inset
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@ -1927,7 +1927,7 @@ m & -m' & m'-m
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\begin_inset Formula
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\begin{multline*}
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\begin{multline}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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@ -1939,8 +1939,8 @@ D_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}.
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\end{multline*}
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\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}
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\end{multline}
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\end_inset
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@ -138,15 +138,6 @@ Topology anoyne?
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scatterer arrays.
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\end_layout
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\begin_layout Subsection
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Notation
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\end_layout
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\begin_layout Standard
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TODO Fourier transforms, Delta comb, lattice bases, reciprocal lattices
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etc.
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\end_layout
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\begin_layout Subsection
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Formulation of the problem
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\end_layout
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@ -171,8 +162,8 @@ noprefix "false"
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\begin_inset Formula $d$
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\end_inset
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can be 1, 2 or 3) lattice embedded into the three-dimensional real space,
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with lattice vectors
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can be 1, 2 or 3) Bravais lattice embedded into the three-dimensional real
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space, with lattice vectors
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\begin_inset Formula $\left\{ \dlv_{i}\right\} _{i=1}^{d}$
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\end_inset
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@ -307,7 +298,7 @@ lattice Fourier transform
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of the translation operator,
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\begin_inset Formula
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\begin{equation}
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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}
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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}
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\end{equation}
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\end_inset
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@ -505,8 +496,8 @@ noprefix "false"
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\end_inset
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, which, for a given geometry, depends only on frequency).
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Therefore, a much more efficient approach to determine the photonic bands
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is to sample the
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Therefore, a much more efficient but not completely robust approach to
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determine the photonic bands is to sample the
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\begin_inset Formula $\vect k$
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\end_inset
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@ -579,6 +570,19 @@ noprefix "false"
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\end_inset
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.
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Another, more robust approach is Beyn's contour integral algorithm
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\begin_inset CommandInset citation
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LatexCommand cite
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key "beyn_integral_2012"
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literal "false"
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\end_inset
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which finds the roots of
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\begin_inset Formula $M\left(\omega,\vect k\right)=0$
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\end_inset
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in a given frequency contour.
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\end_layout
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\begin_layout Subsection
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@ -633,265 +637,69 @@ Note that
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\end_inset
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In electrostatics, this problem can be solved with Ewald summation [TODO
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REF].
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Its basic idea is that if what asymptoticaly decays poorly in the direct
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space, will perhaps decay fast in the Fourier space.
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We use the same idea here, but the technical details are more complicated
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than in electrostatics.
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\end_layout
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The problem of poorly converging lattice sums has been originally solved
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for electrostatic potentials with Ewald summation
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\begin_inset CommandInset citation
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LatexCommand cite
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key "ewald_berechnung_1921"
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literal "false"
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\begin_layout Standard
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Let us re-express the sum in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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in terms of integral with a delta comb
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\begin_inset FormulaMacro
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\renewcommand{\basis}[1]{\mathfrak{#1}}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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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}
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\end{equation}
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\end_inset
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The translation operator
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\begin_inset Formula $S$
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\end_inset
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is now a function defined in the whole 3d space;
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\begin_inset Formula $\vect r_{\alpha},\vect r_{\beta}$
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\end_inset
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are the displacements of scatterers
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\begin_inset Formula $\alpha$
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\end_inset
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and
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\begin_inset Formula $\beta$
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\end_inset
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in a unit cell.
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The arrow notation
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\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})$
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\end_inset
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means
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\begin_inset Quotes eld
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\end_inset
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translation operator for spherical waves originating in
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\begin_inset Formula $\vect r+\vect r_{\beta}$
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\end_inset
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evaluated in
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\begin_inset Formula $\vect r_{\alpha}$
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\end_inset
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\begin_inset Quotes erd
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\end_inset
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and obviously
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\begin_inset Formula $S$
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\end_inset
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is in fact a function of a single 3d argument,
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\begin_inset Formula $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
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\end_inset
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.
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Expression
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W integral"
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\end_inset
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can be rewritten as
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\begin_inset Formula
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\[
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W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect0))\left(\vect k\right)}
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\]
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\end_inset
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where changed the sign of
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\begin_inset Formula $\vect r/\vect{\bullet}$
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\end_inset
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has been swapped under integration, utilising evenness of
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\begin_inset Formula $\dc{\basis u}$
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\end_inset
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.
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Fourier transform of product is convolution of Fourier transforms, so (using
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formula
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Dirac comb uaFt"
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\end_inset
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\end_layout
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\end_inset
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(REF?) for the Fourier transform of Dirac comb)
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\begin_inset Formula
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\begin{eqnarray}
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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 \\
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& = & \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 \\
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& = & \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}\\
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& = & \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
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\end{eqnarray}
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\end_inset
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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Factor
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\begin_inset Formula $\left(2\pi\right)^{\frac{d}{2}}$
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\end_inset
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cancels out with the
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\begin_inset Formula $\left(2\pi\right)^{-\frac{d}{2}}$
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\end_inset
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factor appearing in the convolution/product formula in the unitary angular
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momentum convention.
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\end_layout
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\end_inset
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As such, this is not extremely helpful because the the
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\emph on
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whole
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\emph default
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translation operator
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\begin_inset Formula $S$
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\end_inset
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has singularities in origin, hence its Fourier transform
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\begin_inset Formula $\uaft S$
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\end_inset
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will decay poorly.
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Its basic idea is to decompose the divide the lattice-summed function in
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two parts: a short-range part that decays fast and can be summed directly,
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and a long-range part which decays poorly but is fairly smooth everywhere,
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so that its Fourier transform decays fast enough, and to deal with the
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long range part by Poisson summation over the reciprocal lattice.
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The same idea can be used also in this case case of linear electrodynamic
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problems, just the technical details are more complicated than in electrostatic
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s.
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\end_layout
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\begin_layout Standard
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However, Fourier transform is linear, so we can in principle separate
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\begin_inset Formula $S$
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\end_inset
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In eq.
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in two parts,
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\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
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\end_inset
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.
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\begin_inset Formula $S^{\textup{S}}$
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\end_inset
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is a short-range part that decays sufficiently fast with distance so that
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its direct-space lattice sum converges well;
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\begin_inset Formula $S^{\textup{S}}$
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\end_inset
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must as well contain all the singularities of
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\begin_inset Formula $S$
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\end_inset
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in the origin.
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The other part,
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\begin_inset Formula $S^{\textup{L}}$
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\end_inset
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, will retain all the slowly decaying terms of
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\begin_inset Formula $S$
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\end_inset
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but it also has to be smooth enough in the origin, so that its Fourier
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transform
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\begin_inset Formula $\uaft{S^{\textup{L}}}$
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\end_inset
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decays fast enough.
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(The same idea lies behind the Ewald summation in electrostatics.) Using
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the linearity of Fourier transform and formulae
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W definition"
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\end_inset
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and
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:W sum in reciprocal space"
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\end_inset
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, the operator
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\begin_inset Formula $W_{\alpha\beta}$
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\end_inset
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can then be re-expressed as
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\begin_inset Formula
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\begin{eqnarray}
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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 \\
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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}\\
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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}
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\end{eqnarray}
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\end_inset
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where both sums expected to converge nicely.
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We note that the elements of the translation operators
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\begin_inset Formula $\tropr,\trops$
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\end_inset
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in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:translation operator"
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reference "eq:translation operator singular"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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can be rewritten as linear combinations of expressions
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\begin_inset Formula $\ush{\nu}{\mu}\left(\uvec d\right)j_{n}\left(d\right),\ush{\nu}{\mu}\left(\uvec d\right)h_{n}^{(1)}\left(d\right)$
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we demonstratively expressed the translation operator elements as linear
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combinations of (outgoing) scalar spherical wavefunctions
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\begin_inset Formula $\sswfoutlm lm\left(\vect r\right)=h_{l}^{\left(1\right)}\left(r\right)\ush lm\left(\uvec r\right)$
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\end_inset
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(TODO WRITE THEM EXPLICITLY IN THIS FORM), respectively, hence if we are
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able evaluate the lattice sums sums
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, because for them, fortunately, exponentially convergent Ewald-type summation
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formulae have been already developed
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS
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add refs
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\end_layout
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\end_inset
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\begin_inset CommandInset citation
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LatexCommand cite
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key "moroz_quasi-periodic_2006,linton_one-_2009,linton_lattice_2010"
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literal "false"
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\end_inset
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and can be applied to our case.
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If we formally label
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\begin_inset Marginal
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status open
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\begin_layout Plain Layout
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Check signs.
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\end_layout
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\end_inset
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@ -899,50 +707,132 @@ CHECK THE FOLLOWING EXPRESSION FOR CORRECT FUNCTION ARGUMENTS
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\begin_inset Formula
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\begin{equation}
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\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}
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\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}
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\end{equation}
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\end_inset
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then by linearity, we can get the
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\begin_inset Formula $W_{\alpha\beta}\left(\vect k\right)$
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\end_inset
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we see from eqs.
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|
||||
operator as well.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
TODO ADD MOROZ AND OTHER REFS HERE.
|
||||
|
||||
\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
|
||||
|
||||
.
|
||||
\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.
|
||||
\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
|
||||
|
|
|
|||
Loading…
Reference in New Issue