Continuing notes on Ewald summation
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Marek Nečada
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notes/ewald.lyx
142
notes/ewald.lyx
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@ -316,6 +316,122 @@ reference "eq:W definition"
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\end_layout
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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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\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 $S(\vect r_{\alpha}\leftarrow\vect r+\vect r_{\beta})=S(\vect 0\leftarrow\vect r+\vect r_{\beta}-\vect r_{\alpha})=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)=S(-\vect r-\vect r_{\beta}+\vect r_{\alpha})$
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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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\[
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W_{\alpha\beta}(\vect k)=\left(2\pi\right)^{\frac{d}{2}}\uaft{(\dc{\basis u}S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0))\left(\vect k\right)}
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\]
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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
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\begin_inset Formula
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\[
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W_{\alpha\beta}(\vect k)=\left(\left(\uaft{\dc{\basis u}}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)(\vect k)
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\]
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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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\end_layout
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@ -405,7 +521,7 @@ we have
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and with unitary angular frequency Ft., i.e.
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\begin_inset Formula
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\[
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\uaft f(\vect k)\equiv\frac{1}{\left(2\pi\right)^{n}}\int_{\mathbb{R}^{n}}f(\vect x)e^{-2\pi i\vect x\cdot\vect k}\ud^{n}\vect x
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\uaft f(\vect k)\equiv\frac{1}{\left(2\pi\right)^{n/2}}\int_{\mathbb{R}^{n}}f(\vect x)e^{-i\vect x\cdot\vect k}\ud^{n}\vect x
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\]
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\end_inset
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@ -608,13 +724,33 @@ Fourier transform
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\end_layout
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\begin_layout Standard
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From the book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf
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From the book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf, p.
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379
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\end_layout
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\begin_layout Standard
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(CHECK THIS)
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\uoft{\dc A}\left(\vect{\xi}\right)=\dc{}^{(d)}\left(A^{T}\vect{\xi}\right).
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\uoft{\dc A}\left(\vect{\xi}\right)=\left|\det A^{-T}\right|\dc{}^{(d)}\left(A^{-T}\vect{\xi}\right).
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\]
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\end_inset
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And consequently, for unitary/angular frequency it is
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\end_layout
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\begin_layout Standard
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(CHECK THIS)
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\uaft{\dc A}\left(\vect{\xi}\right)=\frac{\left|\det A^{-T}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{}^{(d)}\left(\frac{1}{2\pi}A^{-T}\vect{\xi}\right).
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\]
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\end_inset
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