Working on 1D ewald sums
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Marek Nečada
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notes/ewald.lyx
437
notes/ewald.lyx
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@ -141,6 +141,12 @@ theorems-starred
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\sgn}{\operatorname{sgn}}
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{\mathrm{sgn}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2}
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\end_inset
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@ -646,11 +652,15 @@ W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\bas
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where both sums should converge nicely.
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\end_layout
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\begin_layout Standard
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\begin_inset Note Note
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status collapsed
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\begin_layout Section
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Finding a good decomposition
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Finding a good decomposition – deprecated
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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The remaining challenge is therefore finding a suitable decomposition
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\begin_inset Formula $S^{\textup{L}}+S^{\textup{S}}$
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\end_inset
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@ -690,7 +700,7 @@ reference "eq:W Long definition"
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absolutely convergent.
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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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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@ -724,9 +734,26 @@ where
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\end_layout
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\begin_layout Standard
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The spherical Hankel functions can be expressed analytically as (REF DLMF
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10.49.6, 10.49.1)
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\begin_layout Plain Layout
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The spherical Hankel functions can be expressed analytically as
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\begin_inset CommandInset citation
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LatexCommand cite
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after "10.49.6, 10.49.1"
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key "NIST:DLMF"
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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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(REF DLMF 10.49.6, 10.49.1)
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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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h_{n}^{(1)}(r)=e^{ir}\sum_{k=0}^{n}\frac{i^{k-n-1}}{r^{k+1}}\frac{\left(n+k\right)!}{2^{k}k!\left(n-k\right)!},\label{eq:spherical Hankel function series}
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@ -753,7 +780,7 @@ 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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2d
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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Assume that all scatterers are placed in the plane
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\begin_inset Formula $\vect z=0$
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\end_inset
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@ -769,7 +796,7 @@ reference "eq:Fourier v. Hankel tf 2d"
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, reads
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{multline*}
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\uaft{S_{l',m',t'\leftarrow l,m,t}^{\textup{L}}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
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@ -803,7 +830,7 @@ Here
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sum, but might be higher in order to speed the convergence up.
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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Obviously, all the terms
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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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@ -826,7 +853,7 @@ reference "eq:spherical Hankel function series"
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, as they decay fast enough.
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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The remaining task is therefore to find a suitable decomposition of
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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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@ -896,7 +923,7 @@ The remaining task is therefore to find a suitable decomposition of
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.
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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The electrostatic Ewald summation uses regularisation with
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\begin_inset Formula $1-e^{-cr^{2}}$
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\end_inset
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@ -923,7 +950,7 @@ reference "eq:2d long range regularisation problem statement"
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might lead to satisfactory results; see below.
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Note
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status open
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@ -983,7 +1010,7 @@ s_{q}^{\textup{L}}\left(x\right)\equiv s_{q}(x)\tilde{\rho}_{\kappa,č}(x)=e^{ix
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Note
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status open
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@ -1019,7 +1046,7 @@ name "sub:Hankel-transforms-binom-reg"
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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Let
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\begin_inset Note Note
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status open
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@ -1043,7 +1070,7 @@ reference "eq:binom regularisation function"
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\end_layout
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\begin_layout Standard
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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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\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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@ -1053,7 +1080,25 @@ reference "eq:binom regularisation function"
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\end_inset
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From [REF DLMF 10.22.49] one digs
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From
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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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[REF DLMF 10.22.49]
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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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after "10.22.49"
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key "NIST:DLMF"
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\end_inset
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one digs
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\begin_inset Note Note
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status open
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@ -1095,10 +1140,28 @@ status open
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\end_inset
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[REF DLMF 14.9.5]
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\begin_inset CommandInset citation
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LatexCommand cite
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after "14.9.5"
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key "NIST:DLMF"
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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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[REF DLMF 14.9.5]
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\end_layout
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\begin_layout Standard
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Note Note
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status collapsed
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@ -1609,7 +1672,7 @@ reference "eq:2D Hankel transform of regularized outgoing wave, decomposition"
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.
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Note
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status collapsed
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@ -1692,7 +1755,7 @@ end{russian}
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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In fact, Mathematica is usually able to calculate the transforms for specific
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values of
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\begin_inset Formula $\kappa,q,n$
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@ -1720,7 +1783,7 @@ reference "tab:Asymptotical-behaviour-Mathematica"
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Float table
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wide false
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sideways false
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@ -2747,11 +2810,11 @@ Case
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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As shown in a separate note,
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Formula
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\[
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\pht 0{s_{2,k_{0}}^{\textup{L}\kappa,c}}\left(k\right)=-\sum_{\sigma=0}^{\kappa}\left(-1\right)^{\sigma}\binom{\kappa}{\sigma}\frac{1}{k_{0}^{2}}\sinh^{-1}\left(\frac{\sigma c-ik_{0}}{k}\right)
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@ -2778,7 +2841,7 @@ Case
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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As shown in separate note (check whether copied correctly)
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\begin_inset Formula
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\[
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@ -2806,7 +2869,7 @@ Case
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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As shown in separate note (check whether copied correctly)
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\lang finnish
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@ -2843,7 +2906,7 @@ for
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Note Note
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status open
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@ -3009,7 +3072,7 @@ reference "eq:prudnikov2 eq 2.12.9.14"
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3d (TODO)
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\end_layout
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\begin_layout Standard
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{multline*}
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\uaft{S_{l',m',t'\leftarrow l,m,t}\left(\vect{\bullet}\leftarrow\vect 0\right)}(\vect k)=\\
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@ -3019,6 +3082,11 @@ reference "eq:prudnikov2 eq 2.12.9.14"
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Section
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@ -3027,11 +3095,34 @@ Exponentially converging decompositions
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\begin_layout Standard
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(As in Linton, Thompson, Journal of Computational Physics 228 (2009) 1815–1829
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[LT].)
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[LT]
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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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\end_inset
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.)
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\end_layout
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\begin_layout Standard
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[LT] offers a better decomposition than above.
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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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[LT]
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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 "linton_one-_2009"
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\end_inset
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offers an exponentially convergent decomposition.
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Let
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\begin_inset Formula
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\begin{eqnarray*}
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@ -3070,15 +3161,66 @@ convention independent
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):
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\begin_inset Formula
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\[
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\sigma_{n}^{m(2)}=-\frac{2^{n+1}i}{k^{n+1}\sqrt{\pi}}\sum_{\vect R\in\Lambda}^{'}\left|\vect R\right|^{n}e^{i\vect{\beta}\cdot\vect R}Y_{n}^{m}\left(\vect R\right)\int_{\eta}^{\infty}e^{-\left|\vect R\right|^{2}\xi^{2}}e^{-k/4\xi^{2}}\xi^{2n}\ud\xi.
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\]
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\begin{equation}
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\sigma_{n}^{m(2)}=-\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.\label{eq:Ewald in 3D short-range part}
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\end{equation}
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\end_inset
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However the other parts in [LT] are convention dependend, so let me fix
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it here.
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[LT] use the convention [LT(A.7)]
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However the other parts in
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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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\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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[LT]
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\end_layout
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\end_inset
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are convention dependend, so let me fix it here.
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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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[LT]
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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 "linton_one-_2009"
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\end_inset
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use the convention
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\begin_inset CommandInset citation
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LatexCommand cite
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after "(A.7)"
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key "linton_one-_2009"
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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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[LT(A.7)]
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{eqnarray*}
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P_{n}^{m}\left(0\right) & = & \frac{\left(-1\right)^{\left(n-m\right)/2}\left(n+m\right)!}{2^{n}\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!}\qquad n+m\mbox{ even,}\\
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@ -3091,7 +3233,25 @@ noting that the former formula is valid also for negative
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\begin_inset Formula $m$
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\end_inset
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(as can be checked by substituting [LT(A.4)]).
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(as can be checked by substituting
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\begin_inset CommandInset citation
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LatexCommand cite
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after "(A.4)"
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key "linton_one-_2009"
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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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[LT(A.4)]
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\end_layout
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\end_inset
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).
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Therefore
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\begin_inset Formula
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\begin{eqnarray*}
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@ -3101,22 +3261,57 @@ Y_{n}^{m}\left(\frac{\pi}{2},\phi\right) & = & \left(-1\right)^{m}\sqrt{\frac{\l
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\end_inset
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Let us substitute this into [LT(4.5)]
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Let us substitute this into
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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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[LT(4.5)]
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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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after "(4.5)"
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key "linton_one-_2009"
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\end_inset
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\begin_inset Formula
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\begin{eqnarray*}
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\sigma_{n}^{m(1)} & = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\left(-1\right)^{\left(n+m\right)/2}\sqrt{\left(2n+1\right)\left(n-m\right)!\left(n+m\right)!}\times\\
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& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}e^{im\phi_{\vect{\beta}_{pq}}}\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(\frac{\gamma_{pq}}{2}\right)^{2j-1}\\
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& = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\sqrt{\pi}2^{n+1}\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(0,\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}/2k\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(\frac{\gamma_{pq}}{2}\right)^{2j-1}\\
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& = & -\frac{i^{n+1}}{k\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(0,\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}
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\end{eqnarray*}
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\begin{eqnarray}
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\sigma_{n}^{m(1)} & = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\left(-1\right)^{\left(n+m\right)/2}\sqrt{\left(2n+1\right)\left(n-m\right)!\left(n+m\right)!}\times\nonumber \\
|
||||
& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}e^{im\phi_{\vect{\beta}_{pq}}}\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(\frac{\gamma_{pq}}{2}\right)^{2j-1}\nonumber \\
|
||||
& = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\sqrt{\pi}2^{n+1}\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
|
||||
& & \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}/2k\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(\frac{\gamma_{pq}}{2}\right)^{2j-1}\nonumber \\
|
||||
& = & -\frac{i^{n+1}}{k\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
|
||||
& & \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:2D Ewald in 3D long-range part}
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
which basically replaces an ugly prefactor with another, similarly ugly
|
||||
one.
|
||||
See [LT] for the meanings of the
|
||||
See
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
key "linton_one-_2009"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
[LT]
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
for the meanings of the
|
||||
\begin_inset Formula $pq$
|
||||
\end_inset
|
||||
|
||||
|
|
@ -3135,15 +3330,30 @@ which basically replaces an ugly prefactor with another, similarly ugly
|
|||
\begin_layout Standard
|
||||
To have it complete,
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\sigma_{n}^{m(0)}=\frac{\delta_{n0}\delta_{m0}}{4\pi}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)=\frac{\delta_{n0}\delta_{m0}}{\sqrt{4\pi}}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)Y_{n}^{m}.
|
||||
\]
|
||||
\begin{equation}
|
||||
\sigma_{n}^{m(0)}=\frac{\delta_{n0}\delta_{m0}}{4\pi}\Gamma\left(-\frac{1}{2},-\frac{k}{4\eta^{2}}\right)=\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 origin part}
|
||||
\end{equation}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
N.B.
|
||||
Apparently, the formulae might be valid regardless of coordinate system
|
||||
orientation (then the spherical harmonic arguments would be of course general
|
||||
|
||||
\begin_inset Formula $Y_{n}^{m}\left(\theta,\phi\right)$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $Y_{n}^{m}\left(\theta_{b_{pq}},\phi_{\vect{\beta}_{pq}}\right)$
|
||||
\end_inset
|
||||
|
||||
accordingly; but CHECK).
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsection
|
||||
Error estimates
|
||||
\end_layout
|
||||
|
|
@ -3240,7 +3450,25 @@ where the integral is according to mathematica and the error functions were
|
|||
\begin_inset Formula $\Gamma\left(1-n,z\right)=z^{1-n}E_{n}\left(z\right)$
|
||||
\end_inset
|
||||
|
||||
from [DLMF(8.4.13)].
|
||||
from
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
after "8.4.13"
|
||||
key "NIST:DLMF"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
[DLMF(8.4.13)]
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Therefore, the upper estimate for the short-range sum error is
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
|
|
@ -3388,7 +3616,24 @@ The only diverging factor here is apparently
|
|||
\begin_inset Formula $\left(\beta_{pq}/k\right)^{n}$
|
||||
\end_inset
|
||||
|
||||
; Mathematica and [DMLF] say
|
||||
; Mathematica and
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
key "NIST:DLMF"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
[DMLF]
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
say
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
\int_{B_{\mathrm{s}}}^{\infty}e^{-\frac{\beta^{2}}{4\eta^{2}}}\beta^{n}\beta\ud\beta & = & \frac{B_{\mathrm{s}}^{n+2}}{2}E_{-\frac{n}{2}}\left(\frac{B_{\mathrm{s}}^{2}}{4\eta^{2}}\right)\\
|
||||
|
|
@ -3406,6 +3651,80 @@ The only diverging factor here is apparently
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
For 1D chains, one can use almost the same formulae as above – the main
|
||||
difference is that there are different exponents in some terms of the long-rang
|
||||
e part so that
|
||||
\begin_inset Formula $\sigma_{n[1\mathrm{d}]}^{m(1)}/\sigma_{n[2\mathrm{d}]}^{m(1)}=k\gamma_{pq}/2\sqrt{\pi}$
|
||||
\end_inset
|
||||
|
||||
(see
|
||||
\begin_inset CommandInset citation
|
||||
LatexCommand cite
|
||||
after "(4.62)"
|
||||
key "linton_lattice_2010"
|
||||
|
||||
\end_inset
|
||||
|
||||
), so
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray}
|
||||
\sigma_{n}^{m(1)} & = & -\frac{i^{n+1}}{2k\sqrt{\pi}\mathscr{A}}\left(-1\right)^{\left(n+m\right)/2}\sqrt{\left(2n+1\right)\left(n-m\right)!\left(n+m\right)!}\times\nonumber \\
|
||||
& & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}e^{im\phi_{\vect{\beta}_{pq}}}\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(\frac{\gamma_{pq}}{2}\right)^{2j}\nonumber \\
|
||||
& = & -\frac{i^{n+1}}{2k\mathscr{A}}2^{n+1}\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
|
||||
& & \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}/2k\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(\frac{\gamma_{pq}}{2}\right)^{2j}\nonumber \\
|
||||
& = & -\frac{i^{n+1}}{\mathscr{A}}\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
|
||||
& & \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}\label{eq:1D Ewald in 3D long-range part}
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
and of course, in this case the unit cell
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
volume
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset Formula $\mathscr{A}$
|
||||
\end_inset
|
||||
|
||||
has the dimension of length instead of
|
||||
\begin_inset Formula $\mbox{length}^{2}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
Eqs.
|
||||
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:Ewald in 3D short-range part"
|
||||
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:Ewald in 3D origin part"
|
||||
|
||||
\end_inset
|
||||
|
||||
for
|
||||
\begin_inset Formula $\sigma_{n}^{m(2)},\sigma_{n}^{m(0)}$
|
||||
\end_inset
|
||||
|
||||
can be used directly without modifications.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
One-dimensional lattice sums are provided in [REF LT, sect.
|
||||
3].
|
||||
However, these are the
|
||||
|
|
@ -3431,6 +3750,11 @@ where we used
|
|||
\begin_inset Formula $P_{n}^{m}\left(\pm1\right)=\left(\pm1\right)^{n}\delta_{m0}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
|
|
@ -3989,6 +4313,17 @@ reference "eq:1D Dirac comb Ft ordinary freq"
|
|||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset CommandInset bibtex
|
||||
LatexCommand bibtex
|
||||
bibfiles "Ewald summation,Tables"
|
||||
options "plain"
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_body
|
||||
|
|
|
|||
26
qpms/ewald.h
26
qpms/ewald.h
|
|
@ -42,6 +42,8 @@ typedef struct {
|
|||
* what the multipliers from translations.c count with.
|
||||
*/
|
||||
// gsl_sf_legendre_t legendre0_type;
|
||||
double *legendre_plus1; // needed? TODO; in any case, nonzero only for m=0
|
||||
double *legendre_minus1; // needed? TODO; in any case, nonzero only for m=0
|
||||
int legendre0_csphase; /* 1 or -1; csphase of the Legendre polynomials saved in legendre0.
|
||||
This is because I dont't actually consider this fixed in
|
||||
translations.c */
|
||||
|
|
@ -152,5 +154,29 @@ int ewald32_sigma_short_points_rordered(//NI
|
|||
);
|
||||
|
||||
|
||||
// 1D sums aligned along z-axis
|
||||
int ewald31z_sigma_long_points_and_shift (
|
||||
complex double *target_sigmalr_y, // must be c->nelem_sc long
|
||||
double *target_sigmalr_y_err, // must be c->nelem_sc long or NULL
|
||||
const qpms_ewald32_constants_t *c,
|
||||
double eta, double k, double unitcell_area,
|
||||
size_t npoints, const double *Kpoints,
|
||||
double beta,
|
||||
double particle_shift
|
||||
);
|
||||
int ewald31z_sigma_short_points_and_shift(
|
||||
complex double *target_sigmasr_y, // must be c->nelem_sc long
|
||||
double *target_sigmasr_y_err, // must be c->nelem_sc long or NULL
|
||||
const qpms_ewald32_constants_t *c, // N.B. not too useful here
|
||||
double eta, double k,
|
||||
size_t npoints, const double *Rpoints,
|
||||
double beta,
|
||||
double particle_shift
|
||||
);
|
||||
int ewald31z_sigma0(complex double *result, double *err,
|
||||
const qpms_ewald32_constants_t *c,
|
||||
double eta, double k
|
||||
); // exactly the same as ewald32_sigma0
|
||||
|
||||
|
||||
#endif //EWALD_H
|
||||
|
|
|
|||
|
|
@ -1296,6 +1296,121 @@ int qpms_trans_calculator_e32_short_points_and_shift(const qpms_trans_calculator
|
|||
#endif // 0
|
||||
#endif // LATTICESUMS32
|
||||
|
||||
#ifdef LATTICESUMS31
|
||||
int qpms_trans_calculator_get_AB_arrays_e31z_both_points_and_shift(const qpms_trans_calculator *c,
|
||||
complex double * const Adest, double * const Aerr,
|
||||
complex double * const Bdest, double * const Berr,
|
||||
const ptrdiff_t deststride, const ptrdiff_t srcstride,
|
||||
/* qpms_bessel_t J*/ // assume QPMS_HANKEL_PLUS
|
||||
const double eta, const double k, const double unitcell_area,
|
||||
const size_t nRpoints, const cart2_t *Rpoints, // n.b. can't contain 0; TODO automatic recognition and skip
|
||||
const size_t nKpoints, const cart2_t *Kpoints,
|
||||
const double beta,//DIFF21
|
||||
const double particle_shift//DIFF21
|
||||
)
|
||||
{
|
||||
|
||||
const qpms_y_t nelem2_sc = qpms_lMax2nelem_sc(c->e32c->lMax);
|
||||
//const qpms_y_t nelem = qpms_lMax2nelem(c->lMax);
|
||||
const bool doerr = Aerr || Berr;
|
||||
const bool do_sigma0 = (particle_shift == 0)//DIFF21((particle_shift.x == 0) && (particle_shift.y == 0)); // FIXME ignoring the case where particle_shift equals to lattice vector
|
||||
|
||||
complex double *sigmas_short = malloc(sizeof(complex double)*nelem2_sc);
|
||||
complex double *sigmas_long = malloc(sizeof(complex double)*nelem2_sc);
|
||||
complex double *sigmas_total = malloc(sizeof(complex double)*nelem2_sc);
|
||||
double *serr_short, *serr_long, *serr_total;
|
||||
if(doerr) {
|
||||
serr_short = malloc(sizeof(double)*nelem2_sc);
|
||||
serr_long = malloc(sizeof(double)*nelem2_sc);
|
||||
serr_total = malloc(sizeof(double)*nelem2_sc);
|
||||
} else serr_short = serr_long = serr_total = NULL;
|
||||
|
||||
int retval;
|
||||
retval = ewald31z_sigma_long_points_and_shift(sigmas_long, serr_long, //DIFF21
|
||||
c->e32c, eta, k, unitcell_area, nKpoints, Kpoints, beta, particle_shift);
|
||||
if (retval) abort();
|
||||
|
||||
retval = ewald31z_sigma_short_points_and_shift(sigmas_short, serr_short, //DIFF21
|
||||
c->e32c, eta, k, nRpoints, Rpoints, beta, particle_shift);
|
||||
if (retval) abort();
|
||||
|
||||
for(qpms_y_t y = 0; y < nelem2_sc; ++y)
|
||||
sigmas_total[y] = sigmas_short[y] + sigmas_long[y];
|
||||
if (doerr) for(qpms_y_t y = 0; y < nelem2_sc; ++y)
|
||||
serr_total[y] = serr_short[y] + serr_long[y];
|
||||
|
||||
complex double sigma0 = 0; double sigma0_err = 0;
|
||||
if (do_sigma0) {
|
||||
retval = ewald31z_sigma0(&sigma0, &sigma0_err, c->e32c, eta, k);//DIFF21
|
||||
if(retval) abort();
|
||||
const qpms_l_t y = qpms_mn2y_sc(0,0);
|
||||
sigmas_total[y] += sigma0;
|
||||
if(doerr) serr_total[y] += sigma0_err;
|
||||
}
|
||||
|
||||
switch(qpms_normalisation_t_normonly(c->normalisation)) {
|
||||
case QPMS_NORMALISATION_TAYLOR:
|
||||
case QPMS_NORMALISATION_POWER:
|
||||
case QPMS_NORMALISATION_NONE:
|
||||
{
|
||||
ptrdiff_t desti = 0, srci = 0;
|
||||
for (qpms_l_t n = 1; n <= c->lMax; ++n) for (qpms_m_t m = -n; m <= n; ++m) {
|
||||
for (qpms_l_t nu = 1; nu <= c->lMax; ++nu) for (qpms_m_t mu = -nu; mu <= nu; ++mu){
|
||||
const size_t i = qpms_trans_calculator_index_mnmunu(c, m, n, mu, nu);
|
||||
const size_t qmax = c->A_multipliers[i+1] - c->A_multipliers[i] - 1;
|
||||
complex double Asum, Asumc; ckahaninit(&Asum, &Asumc);
|
||||
double Asumerr, Asumerrc; if(Aerr) kahaninit(&Asumerr, &Asumerrc);
|
||||
|
||||
const qpms_m_t mu_m = mu - m;
|
||||
// TODO skip if ... (N.B. skip will be different for 31z and 32)
|
||||
for(qpms_l_t q = 0; q <= qmax; ++q) {
|
||||
const qpms_l_t p = n + nu - 2*q;
|
||||
const qpms_y_t y_sc = qpms_mn2y_sc(mu_m, p);
|
||||
const complex double multiplier = c->A_multipliers[i][q];
|
||||
complex double sigma = sigmas_total[y_sc];
|
||||
ckahanadd(&Asum, &Asumc, multiplier * sigma);
|
||||
if (Aerr) kahanadd(&Asumerr, &Asumerrc, multiplier * serr_total[y_sc]);
|
||||
}
|
||||
|
||||
*(Adest + deststride * desti + srcstride * srci) = Asum;
|
||||
if (Aerr) *(Aerr + deststride * desti + srcstride * srci) = Asumerr;
|
||||
|
||||
// TODO skip if ...
|
||||
complex double Bsum, Bsumc; ckahaninit(&Bsum, &Bsumc);
|
||||
double Bsumerr, Bsumerrc; if(Berr) kahaninit(&Bsumerr, &Bsumerrc);
|
||||
for(qpms_l_t q = 0; q <= qmax; ++q) {
|
||||
const qpms_l_t p_ = n + nu - 2*q + 1;
|
||||
const qpms_y_t y_sc = qpms_mn2y_sc(mu_m, p_);
|
||||
const complex double multiplier = c->B_multipliers[i][q-BQ_OFFSET];
|
||||
complex double sigma = sigmas_total[y_sc];
|
||||
ckahanadd(&Bsum, &Bsumc, multiplier * sigma);
|
||||
if (Berr) kahanadd(&Bsumerr, &Bsumerrc, multiplier * serr_total[y_sc]);
|
||||
}
|
||||
|
||||
*(Bdest + deststride * desti + srcstride * srci) = Bsum;
|
||||
if (Berr) *(Berr + deststride * desti + srcstride * srci) = Bsumerr;
|
||||
|
||||
++srci;
|
||||
}
|
||||
++desti;
|
||||
srci = 0;
|
||||
}
|
||||
}
|
||||
break;
|
||||
default:
|
||||
abort();
|
||||
}
|
||||
|
||||
free(sigmas_short);
|
||||
free(sigmas_long);
|
||||
free(sigmas_total);
|
||||
if(doerr) {
|
||||
free(serr_short);
|
||||
free(serr_long);
|
||||
free(serr_total);
|
||||
}
|
||||
return 0;
|
||||
}
|
||||
#ifdef LATTICESUMS_OLD
|
||||
|
||||
int qpms_trans_calculator_get_shortrange_AB_arrays_buf(const qpms_trans_calculator *c,
|
||||
|
|
|
|||
|
|
@ -10,11 +10,10 @@
|
|||
#include "bessels.h"
|
||||
#endif
|
||||
|
||||
#ifdef LATTICESUMS32
|
||||
#if defined LATTICESUMS32 || defined LATTICESUMS31
|
||||
#include "ewald.h"
|
||||
#endif
|
||||
|
||||
|
||||
/*
|
||||
* Argument conventions:
|
||||
*
|
||||
|
|
@ -78,7 +77,7 @@ typedef struct qpms_trans_calculator {
|
|||
// TODO
|
||||
#endif
|
||||
|
||||
#ifdef LATTICESUMS32
|
||||
#if defined LATTICESUMS32 || defined LATTICESUMS31
|
||||
qpms_ewald32_constants_t *e32c;
|
||||
#endif
|
||||
#ifdef LATTICESUMS_OLD
|
||||
|
|
@ -188,12 +187,29 @@ int qpms_trans_calculator_get_AB_arrays_e32_both_points_and_shift(const qpms_tra
|
|||
const double eta, const double k,
|
||||
const double unitcell_area,
|
||||
const size_t nRpoints, const cart2_t *Rpoints,
|
||||
const size_t nKpoints, const cart2_t *Kpoinst,
|
||||
const size_t nKpoints, const cart2_t *Kpoints,
|
||||
const cart2_t beta,
|
||||
const cart2_t particle_shift
|
||||
);
|
||||
#endif //LATTICESUMS32
|
||||
|
||||
#ifdef LATTICESUMS31
|
||||
// e31z means that the particles are positioned along the z-axis;
|
||||
// their positions and K-values are then denoted by a single z-coordinate
|
||||
in qpms_trans_calculator_get_AB_arrays_e31z_bost_points_and_shift(const qpms_trans_calculator *c,
|
||||
complex double *Adest, double *Aerr,
|
||||
complex double *Bdest, double *Berr,
|
||||
const ptrdiff_t deststride, const ptrdiff_t srcstride,
|
||||
const double eta, const double k,
|
||||
const double unitcell_area, // just lattice period
|
||||
const size_t nRpoints, const cart2_t *Rpoints,
|
||||
const size_t nKpoints, const cart2_t *Kpoints,
|
||||
const double beta,
|
||||
const double particle_shift
|
||||
);
|
||||
#endif
|
||||
|
||||
|
||||
|
||||
|
||||
#ifdef QPMS_COMPILE_PYTHON_EXTENSIONS
|
||||
|
|
|
|||
Loading…
Reference in New Issue