Ewald sums WIP
Former-commit-id: c22a5be53375ed175a6a68639887c4b617785a8a
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Marek Nečada
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a6e90b43ae
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068e491662
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@ -712,6 +712,12 @@ Consistent notation of balls.
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Abstract.
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\end_layout
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\begin_layout Itemize
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Translation operators: explicit expression, also in sph.
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harm.
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convention independent form.
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\end_layout
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\begin_layout Itemize
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Truncation notation.
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\end_layout
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@ -136,7 +136,8 @@ 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 etc.
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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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@ -277,12 +278,12 @@ noprefix "false"
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can be rewritten as follows
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\begin_inset Formula
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\begin{align*}
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}},\\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m-\vect n}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect 0,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\beta\in\mathcal{P}}W_{\alpha\beta}\left(\vect k\right)\outcoeffp{\vect 0,\beta}\left(\vect k\right) & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),
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\end{align*}
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\begin{align}
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}}-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect n,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect n}},\nonumber \\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect n,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m-\vect n,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m-\vect n}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\nonumber \\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\left(\vect m,\beta\right)\in\mathcal{P}\backslash\left\{ \left(\vect 0,\alpha\right)\right\} }\tropsp{\vect 0,\alpha}{\vect m,\beta}\outcoeffp{\vect 0,\beta}\left(\vect k\right)e^{i\vect k\cdot\vect R_{\vect m}} & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\nonumber \\
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\outcoeffp{\vect 0,\alpha}\left(\vect k\right)-T_{\alpha}\sum_{\beta\in\mathcal{P}}W_{\alpha\beta}\left(\vect k\right)\outcoeffp{\vect 0,\beta}\left(\vect k\right) & =T_{\alpha}\rcoeffincp{\vect 0,\alpha}\left(\vect k\right),\label{eq:Multiple-scattering problem unit cell}
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\end{align}
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\end_inset
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@ -299,16 +300,35 @@ 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}\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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evaluation of which is possible but quite non-trivial due to the infinite
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lattice sum, so we explain it separately in Sect.
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "subsec:W operator evaluation"
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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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.
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\end_layout
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\begin_layout Subsection
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Computing the Fourier sum of the translation operator
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\begin_inset CommandInset label
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LatexCommand label
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name "subsec:W operator evaluation"
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -576,7 +596,178 @@ W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\bas
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\end_inset
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where both sums should converge nicely.
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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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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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\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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\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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\end_layout
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\end_inset
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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\{ \vect 0\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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\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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operator as well.
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\end_layout
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\begin_layout Standard
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TODO ADD MOROZ AND OTHER REFS HERE.
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\begin_inset CommandInset citation
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LatexCommand cite
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key "linton_one-_2009"
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literal "true"
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\end_inset
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offers an exponentially convergent Ewald-type summation method for
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\begin_inset Formula $\sigma_{\nu}^{\mu}\left(\vect k\right)=\sigma_{\nu}^{\mu(\mathrm{S})}\left(\vect k\right)+\sigma_{\nu}^{\mu(\mathrm{L})}\left(\vect k\right)$
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\end_inset
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.
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Here we rewrite them in a form independent on the convention used for spherical
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harmonics (as long as they are complex
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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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lepší formulace
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\end_layout
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\end_inset
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).
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The short-range part reads (UNIFY INDEX NOTATION)
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\begin_inset Formula
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\begin{multline}
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\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\\
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+\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}
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\end{multline}
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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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NEPATŘÍ TAM NĚJAKÁ DELTA FUNKCE K PŮVODNÍMU
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\begin_inset Formula $\sigma_{n}^{m(0)}$
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\end_inset
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?
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\end_layout
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\end_inset
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and the long-range part (FIXME, this is the 2D version; include the 1D and
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3D lattice expressions as well)
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\begin_inset Formula
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\begin{multline}
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\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\\
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\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}
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\end{multline}
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\end_inset
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where
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\begin_inset Formula $\xi$
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\end_inset
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is TODO,
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\begin_inset Formula $\beta_{pq}$
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\end_inset
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is TODO,
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\begin_inset Formula $\Gamma_{j,pq}$
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\end_inset
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is TODO and
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\begin_inset Formula $\eta$
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\end_inset
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is a real parameter that determines the pace of convergence of both parts.
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The larger
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\begin_inset Formula $\eta$
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\end_inset
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is, the faster
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\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$
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\end_inset
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converges but the slower
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\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$
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\end_inset
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converges.
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Therefore (based on the lattice geometry) it has to be adjusted in a way
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that a reasonable amount of terms needs to be evaluated numerically from
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both
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\begin_inset Formula $\sigma_{n}^{m(\mathrm{S})}\left(\vect{\beta}\right)$
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\end_inset
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and
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\begin_inset Formula $\sigma_{n}^{m(\mathrm{L})}\left(\vect{\beta}\right)$
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\end_inset
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.
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Generally, a good choice for
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\begin_inset Formula $\eta$
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\end_inset
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is TODO; in order to achieve accuracy TODO, one has to evaluate the terms
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on TODO lattice points.
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(I HAVE SOME DERIVATIONS OF THE ESTIMATES IN MY NOTES; SHOULD I INCLUDE
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THEM?)
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\end_layout
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\begin_layout Standard
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In practice, the integrals in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:Ewald in 3D short-range part"
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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 easily evaluated by numerical quadrature and the incomplete
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\begin_inset Formula $\Gamma$
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\end_inset
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-functions using the series TODO and TODO from DLMF.
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\end_layout
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\end_body
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