[ewald] Pokračování
Former-commit-id: 5c163530c176c0eb5a9e372414898830df65a189
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Marek Nečada
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188
notes/ewald.lyx
188
notes/ewald.lyx
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@ -187,6 +187,16 @@
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\hgfr}{\mathbf{F}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ph}{\mathrm{ph}}
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\end_inset
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\end_layout
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\begin_layout Title
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@ -685,7 +695,7 @@ h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\righ
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\end_inset
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so if we find a way to deal with the radial functions
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\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\begin_inset Formula $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\end_inset
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,
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@ -755,7 +765,7 @@ Here
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\begin_layout Standard
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Obviously, all the terms
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\begin_inset Formula $\propto s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\begin_inset Formula $\propto s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\end_inset
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,
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@ -778,7 +788,7 @@ reference "eq:spherical Hankel function series"
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\begin_layout Standard
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The remaining task is therefore to find a suitable decomposition of
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\begin_inset Formula $s_{q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\begin_inset Formula $s_{k_{0},q}(r)=e^{ik_{0}r}\left(k_{0}r\right)^{-q}$
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\end_inset
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,
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@ -786,16 +796,16 @@ The remaining task is therefore to find a suitable decomposition of
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\end_inset
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into short-range and long-range parts,
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\begin_inset Formula $s_{q}(r)=s_{q}^{\textup{S}}(r)+s_{q}^{\textup{L}}(r)$
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\begin_inset Formula $s_{k_{0},q}(r)=s_{k_{0},q}^{\textup{S}}(r)+s_{k_{0},q}^{\textup{L}}(r)$
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\end_inset
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, such that
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\begin_inset Formula $s_{q}^{\textup{L}}(r)$
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\begin_inset Formula $s_{k_{0},q}^{\textup{L}}(r)$
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\end_inset
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contains all the slowly decaying asymptotics and its Hankel transforms
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decay desirably fast as well,
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\begin_inset Formula $\pht n{s_{q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$
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\begin_inset Formula $\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=o(z^{-Q})$
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\end_inset
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,
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@ -810,7 +820,7 @@ The remaining task is therefore to find a suitable decomposition of
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must be sufficiently smooth in the origin, so that
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\begin_inset Formula
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\begin{equation}
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\pht n{s_{q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement}
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\pht n{s_{k_{0},q}^{\textup{L}}}\left(k\right)=\int_{0}^{\infty}s_{k_{0},q}^{\textup{L}}\left(r\right)rJ_{n}\left(kr\right)\ud r=\int_{0}^{\infty}s_{k_{0},q}\left(r\right)\rho\left(r\right)rJ_{n}\left(kr\right)\ud r\label{eq:2d long range regularisation problem statement}
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\end{equation}
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\end_inset
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@ -852,8 +862,9 @@ The electrostatic Ewald summation uses regularisation with
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\end_inset
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.
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However, such choice does not seem to lead to an analytical solution for
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the current problem
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However, such choice does not seem to lead to an analytical solution (really?
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could not something be dug out of DLMF 10.22.54?) for the current problem
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:2d long range regularisation problem statement"
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@ -876,6 +887,165 @@ leads to satisfactory results, as will be shown below.
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Hankel transforms of the long-range parts
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\end_layout
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\begin_layout Standard
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Let
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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{eqnarray}
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\pht n{s_{q,k_{0}}^{\textup{L}\kappa,c}}\left(k\right) & \equiv & \int_{0}^{\infty}\frac{e^{ik_{0}r}}{\left(k_{0}r\right)^{q}}J_{n}\left(kr\right)\left(1-e^{-cr}\right)^{\kappa}r\,\ud r\nonumber \\
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& = & k_{0}^{-q}\int_{0}^{\infty}r^{1-q}J_{n}\left(kr\right)\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}e^{-(\sigma c-ik_{0})r}\ud r\nonumber \\
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& \underset{\equiv}{\textup{form.}} & \sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\pht n{s_{q,k_{0}}^{\textup{L}1,\sigma c}}\left(k\right).\label{eq:2D Hankel transform of regularized outgoing wave, decomposition}
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\end{eqnarray}
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\end_inset
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From [REF DLMF 10.22.49] one digs
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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 Formula
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\begin{eqnarray*}
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\mu & \leftarrow & 2-q\\
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\nu & \leftarrow & n\\
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b & \leftarrow & k\\
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a & \leftarrow & c-ik_{0}
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\end{eqnarray*}
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\end_inset
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{multline}
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\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\hgfr\left(\frac{2-q+n}{2},\frac{3-q+n}{2};1+n;\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right),\\
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\Re\left(2-q+n\right)>0,\Re(c-ik_{0}\pm k)\ge0,\label{eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1}
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\end{multline}
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\end_inset
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and from [REF DLMF 15.9.17]
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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 Formula
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\begin{eqnarray*}
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a & \leftarrow & \frac{2-q+n}{2}\\
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c & \leftarrow & 1+n\\
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z & \leftarrow & \frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}
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\end{eqnarray*}
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\end_inset
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\begin_inset Formula
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\begin{eqnarray*}
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\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right) & = & \frac{k^{n}Γ\left(2-q+n\right)}{2^{n}k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}2^{n}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1-\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right)^{-\frac{2-q+n}{2}+\frac{n}{2}}P_{2-q+n-(1+n)}^{1-(1+n)}\left(\frac{1}{\sqrt{1-\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)}}\right)\\
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& = & \frac{k^{n}Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{1-q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right)
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\end{eqnarray*}
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\end_inset
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\begin_inset Formula
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\[
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\left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|<\pi,\quad\left|\ph\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right|<\pi
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\]
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\end_inset
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in other words, neither
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\begin_inset Formula $-k^{2}/\left(c-ik_{0}\right)^{2}$
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\end_inset
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nor
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\begin_inset Formula $1+k^{2}/\left(c-ik_{0}\right)^{2}$
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\end_inset
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can be non-positive real number.
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For assumed positive
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\begin_inset Formula $k_{0}$
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\end_inset
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and non-negative
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\begin_inset Formula $c$
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\end_inset
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and
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\begin_inset Formula $k$
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\end_inset
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, the former case can happen only if
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\begin_inset Formula $k=0$
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\end_inset
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and the latter only if
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\begin_inset Formula $c=0\wedge k_{0}=k$
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\end_inset
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.
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\begin_inset Formula
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\begin{eqnarray*}
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\left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|<\pi & \Leftrightarrow & \left|\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right|\neq\pi\\
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\varphi & \equiv & \ph\left(c-ik_{0}\right)<0,\\
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\ph k & \equiv & 0\\
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\ph\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}} & = & 2\varphi\\
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\rightsquigarrow\left|\varphi\right| & \neq & \pi/2\\
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\rightsquigarrow c & \neq & k_{0}\\
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\left|\ph\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)\right| & = & \left|-2\varphi+\ph\left(\left(c-ik_{0}\right)^{2}+k^{2}\right)\right|
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\end{eqnarray*}
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\end_inset
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{multline}
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\pht n{s_{q,k_{0}}^{\textup{L}1,c}}\left(k\right)=\frac{k^{n}Γ\left(2-q+n\right)}{k_{0}^{q}\left(c-ik_{0}\right)^{2-q+n}}\left(\frac{-k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{-\frac{n}{2}}\left(1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}\right)^{\frac{q}{2}-1}P_{1-q}^{-n}\left(\frac{1}{\sqrt{1+\frac{k^{2}}{\left(c-ik_{0}\right)^{2}}}}\right),\\
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k>0\wedge k_{0}>0\wedge c\ge0\wedge\lnot\left(c=0\wedge k_{0}=k\right)\label{eq:2D Hankel transform of exponentially suppressed outgoing wave expanded}
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\end{multline}
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\end_inset
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with principal branches of the hypergeometric functions, associated Legendre
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functions, and fractional powers.
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The conditions from
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:2D Hankel transform of exponentially suppressed outgoing wave as 2F1"
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\end_inset
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should hold, but we will use
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:2D Hankel transform of exponentially suppressed outgoing wave expanded"
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\end_inset
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formally even if they are violated, with the hope that the divergences
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eventually cancel in
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:2D Hankel transform of regularized outgoing wave, decomposition"
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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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3d
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\end_layout
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