[ewald]Formulating the decomposition problem.
Former-commit-id: 1bf687085f79f24adac02ef7ac2cb3a7f97c424e
This commit is contained in:
parent
e708c353e9
commit
f3d27e74d8
248
notes/ewald.lyx
248
notes/ewald.lyx
|
@ -152,6 +152,16 @@
|
|||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\ints}{\mathbb{Z}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\begin_inset FormulaMacro
|
||||
\newcommand{\reals}{\mathbb{R}}
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Title
|
||||
|
@ -166,6 +176,24 @@ Accelerating lattice mode calculations with
|
|||
Marek Nečada
|
||||
\end_layout
|
||||
|
||||
\begin_layout Abstract
|
||||
The
|
||||
\begin_inset Formula $T$
|
||||
\end_inset
|
||||
|
||||
-matrix approach is the method of choice for simulating optical response
|
||||
of a reasonably small system of compact linear scatterers on isotropic
|
||||
background.
|
||||
However, its direct utilisation for problems with infinite lattices is
|
||||
problematic due to slowly converging sums over the lattice.
|
||||
Here I develop a way to compute the problematic sums in the reciprocal
|
||||
space, making the
|
||||
\begin_inset Formula $T$
|
||||
\end_inset
|
||||
|
||||
-matrix method very suitable for infinite periodic systems as well.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Formulation of the problem
|
||||
\end_layout
|
||||
|
@ -308,7 +336,7 @@ reference "eq:W definition"
|
|||
\end_inset
|
||||
|
||||
-dimensional lattice.
|
||||
In electrostatics, one can solve this with problem with Ewald summation.
|
||||
In electrostatics, one can solve this problem with Ewald summation.
|
||||
Its basic idea is that if what asymptoticaly decays poorly in the direct
|
||||
space, will perhaps decay fast in the Fourier space.
|
||||
I use the same idea here, but everything will be somehow harder than in
|
||||
|
@ -404,13 +432,21 @@ where changed the sign of
|
|||
\end_inset
|
||||
|
||||
.
|
||||
Fourier transform of product is convolution of Fourier transforms, so
|
||||
Fourier transform of product is convolution of Fourier transforms, so (using
|
||||
formula
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:Dirac comb uaFt"
|
||||
|
||||
\end_inset
|
||||
|
||||
for the Fourier transform of Dirac comb)
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray*}
|
||||
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)\\
|
||||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\rec{\basis u}}^{(d)}\left(\frac{1}{2\pi}\vect{\circ}\right)\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\quad\mbox{(re-check facs)}\\
|
||||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}
|
||||
\end{eqnarray*}
|
||||
\begin{eqnarray}
|
||||
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)\nonumber \\
|
||||
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\left(\dc{\recb{\basis u}}^{(d)}\ast\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\right)\left(\vect k\right)\nonumber \\
|
||||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right).\label{eq:W sum in reciprocal space}
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
@ -434,7 +470,134 @@ Factor
|
|||
|
||||
\end_inset
|
||||
|
||||
As such, this is not extremely helpful because the the
|
||||
\emph on
|
||||
whole
|
||||
\emph default
|
||||
translation operator
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
has singularities in origin, hence its Fourier transform
|
||||
\begin_inset Formula $\uaft S$
|
||||
\end_inset
|
||||
|
||||
will decay poorly.
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
However, Fourier transform is linear, so we can in principle separate
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
in two parts,
|
||||
\begin_inset Formula $S=S^{\textup{L}}+S^{\textup{S}}$
|
||||
\end_inset
|
||||
|
||||
.
|
||||
|
||||
\begin_inset Formula $S^{\textup{S}}$
|
||||
\end_inset
|
||||
|
||||
is a short-range part that decays sufficiently fast with distance so that
|
||||
its direct-space lattice sum converges well;
|
||||
\begin_inset Formula $S^{\textup{S}}$
|
||||
\end_inset
|
||||
|
||||
must as well contain all the singularities of
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
in the origin.
|
||||
The other part,
|
||||
\begin_inset Formula $S^{\textup{L}}$
|
||||
\end_inset
|
||||
|
||||
, will retain all the slowly decaying terms of
|
||||
\begin_inset Formula $S$
|
||||
\end_inset
|
||||
|
||||
but it also has to be smooth enough in the origin, so that its Fourier
|
||||
transform
|
||||
\begin_inset Formula $\uaft{S^{\textup{L}}}$
|
||||
\end_inset
|
||||
|
||||
decays fast enough.
|
||||
(The same idea lies behind the Ewald summation in electrostatics.) Using
|
||||
the linearity of Fourier transform and formulae
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:W definition"
|
||||
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:W sum in reciprocal space"
|
||||
|
||||
\end_inset
|
||||
|
||||
, the operator
|
||||
\begin_inset Formula $W_{\alpha\beta}$
|
||||
\end_inset
|
||||
|
||||
can then be re-expressed as
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray}
|
||||
W_{\alpha\beta}\left(\vect k\right) & = & W_{\alpha\beta}^{\textup{S}}\left(\vect k\right)+W_{\alpha\beta}^{\textup{L}}\left(\vect k\right)\nonumber \\
|
||||
W_{\alpha\beta}^{\textup{S}}\left(\vect k\right) & = & \sum_{\vect R\in\basis u\ints^{d}}S^{\textup{S}}(\vect 0\leftarrow\vect R+\vect r_{\beta}-\vect r_{\alpha})e^{i\vect k\cdot\vect R}\label{eq:W Short definition}\\
|
||||
W_{\alpha\beta}^{\textup{L}}\left(\vect k\right) & = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\sum_{\vect K\in\recb{\basis u}\ints^{d}}\left(\uaft{S^{\textup{L}}(\vect{\bullet}-\vect r_{\beta}+\vect r_{\alpha}\leftarrow\vect 0)}\right)\left(\vect k-\vect K\right)\label{eq:W Long definition}
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
where both sums should converge nicely.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
Finding a good decomposition
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
The remaining challenge is therefore finding a suitable decomposition
|
||||
\begin_inset Formula $S^{\textup{L}}+S^{\textup{S}}$
|
||||
\end_inset
|
||||
|
||||
such that both
|
||||
\begin_inset Formula $S^{\textup{S}}$
|
||||
\end_inset
|
||||
|
||||
and
|
||||
\begin_inset Formula $\uaft{S^{\textup{L}}}$
|
||||
\end_inset
|
||||
|
||||
decay fast enough with distance and are expressable analytically.
|
||||
With these requirements, I do not expect to find gaussian asymptotics as
|
||||
in the electrostatic Ewald formula—having
|
||||
\begin_inset Formula $\sim x^{-t}$
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset Formula $t>d$
|
||||
\end_inset
|
||||
|
||||
asymptotics would be nice, making the sums in
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:W Short definition"
|
||||
|
||||
\end_inset
|
||||
|
||||
,
|
||||
\begin_inset CommandInset ref
|
||||
LatexCommand eqref
|
||||
reference "eq:W Long definition"
|
||||
|
||||
\end_inset
|
||||
|
||||
absolutely convergent.
|
||||
\end_layout
|
||||
|
||||
\begin_layout Section
|
||||
|
@ -528,7 +691,7 @@ we have
|
|||
|
||||
\end_inset
|
||||
|
||||
we have
|
||||
we have (CHECK)
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\uaft{Ш_{T}}(\omega)=\frac{\sqrt{2\pi}}{T}Ш_{\frac{2\pi}{T}}(\omega)=\frac{1}{\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-i\omega nT}
|
||||
|
@ -644,7 +807,8 @@ From the scaling property of delta function,
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
From the book:
|
||||
From the Osgood's book (p.
|
||||
375):
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
|
@ -722,22 +886,19 @@ reference "eq:Dirac comb factorisation"
|
|||
\end_layout
|
||||
|
||||
\begin_layout Subsubsection
|
||||
Fourier transform
|
||||
Fourier transform (OK)
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
From the book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf, p.
|
||||
From the Osgood's book https://see.stanford.edu/materials/lsoftaee261/chap8.pdf,
|
||||
p.
|
||||
379
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
(CHECK THIS)
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\uoft{\dc A}\left(\vect{\xi}\right)=\left|\det A^{-T}\right|\dc{}^{(d)}\left(A^{-T}\vect{\xi}\right).
|
||||
\uoft{\dc{\basis u}}\left(\vect{\xi}\right)=\left|\det\rec{\basis u}\right|\dc{\rec{\basis u}}^{(d)}\left(\vect{\xi}\right).
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
@ -746,29 +907,48 @@ And consequently, for unitary/angular frequency it is
|
|||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
(CHECK THIS)
|
||||
\begin_inset Formula
|
||||
\begin{eqnarray}
|
||||
\uaft{\dc{\basis u}}\left(\vect k\right) & = & \frac{1}{\left(2\pi\right)^{\frac{d}{2}}}\uoft{\dc{\basis u}}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
|
||||
& = & \frac{\left|\det\rec{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\rec{\basis u}}^{(d)}\left(\frac{\vect k}{2\pi}\right)\nonumber \\
|
||||
& = & \left(2\pi\right)^{\frac{d}{2}}\left|\det\rec{\basis u}\right|\dc{\recb{\basis u}}\left(\vect k\right)\nonumber \\
|
||||
& = & \frac{\left|\det\recb{\basis u}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{\recb{\basis u}}\left(\vect k\right).\label{eq:Dirac comb uaFt}
|
||||
\end{eqnarray}
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Note Note
|
||||
status open
|
||||
|
||||
\begin_layout Plain Layout
|
||||
On the third line, we used the stretch theorem, getting
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\dc{\recb{\basis u}}\left(\vect k\right)=\dc{2\pi\rec{\basis u}}\left(\vect k\right)=\left(2\pi\right)^{-d}\dc{\rec{\basis u}}\left(\frac{\vect k}{2\pi}\right)
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\end_inset
|
||||
|
||||
|
||||
\end_layout
|
||||
|
||||
\begin_layout Subsubsection
|
||||
Convolution
|
||||
\end_layout
|
||||
|
||||
\begin_layout Standard
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\uaft{\dc A}\left(\vect{\xi}\right)=\frac{\left|\det A^{-T}\right|}{\left(2\pi\right)^{\frac{d}{2}}}\dc{}^{(d)}\left(\frac{1}{2\pi}A^{-T}\vect{\xi}\right).
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
||||
Using my own
|
||||
\begin_inset Quotes eld
|
||||
\end_inset
|
||||
|
||||
basis notation
|
||||
\begin_inset Quotes erd
|
||||
\end_inset
|
||||
|
||||
TODO
|
||||
\begin_inset Formula
|
||||
\[
|
||||
\uoft{\dc{\basis u}}\left(\vect{\xi}\right)=\left|\det\rec{\basis u}\right|\dc{}^{(d)}\left(A^{-T}\vect{\xi}\right).
|
||||
\left(f\ast\dc{\basis u}\right)(\vect x)=\sum_{\vect t\in\basis u\ints^{d}}f(\vect x-\vect t)
|
||||
\]
|
||||
|
||||
\end_inset
|
||||
|
|
Loading…
Reference in New Issue