From 306cb1bef80e40026fbb8f504ebd64b0db3aa3c6 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= <marek@necada.org>
Date: Mon, 5 Aug 2019 15:47:49 +0300
Subject: [PATCH] Fix representation of spatial inversion.

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+\selectlanguage{finnish}%
+
+\section{Symmetries}\label{sec:Symmetries}
+
+If the system has nontrivial point group symmetries, group theory
+gives additional understanding of the system properties, and can be
+used to reduce the computational costs. 
+
+As an example, if our system has a $D_{2h}$ symmetry and our truncated
+$\left(I-T\trops\right)$ matrix has size $N\times N$, it can be
+block-diagonalized into eight blocks of size about $N/8\times N/8$,
+each of which can be LU-factorised separately (this is due to the
+fact that $D_{2h}$ has eight different one-dimensional irreducible
+representations). This can reduce both memory and time requirements
+to solve the scattering problem (\ref{eq:Multiple-scattering problem block form})
+by a factor of 64.
+
+In periodic systems (problems (\ref{eq:Multiple-scattering problem unit cell block form}),
+(\ref{eq:lattice mode equation})) due to small number of particles
+per unit cell, the costliest part is usually the evaluation of the
+lattice sums in the $W\left(\omega,\vect k\right)$ matrix, not the
+linear algebra. However, the lattice modes can be searched for in
+each irrep separately, and the irrep dimension gives a priori information
+about mode degeneracy.
+
+\subsection{Excitation coefficients under point group operations}
+
+In order to use the point group symmetries, we first need to know
+how they affect our basis functions, i.e. the VSWFs.
+
+Let $g$ be a member of orthogonal group $O(3)$, i.e. a 3D point
+rotation or reflection operation that transforms vectors in $\reals^{3}$
+with an orthogonal matrix $R_{g}$:
+\[
+\vect r\mapsto R_{g}\vect r.
+\]
+Spherical harmonics $\ush lm$, being a basis the $l$-dimensional
+representation of $O(3)$, transform as \cite[???]{dresselhaus_group_2008}
+\[
+\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right)
+\]
+where $D_{m,m'}^{l}\left(g\right)$ denotes the elements of the \emph{Wigner
+matrix} representing the operation $g$. By their definition, vector
+spherical harmonics $\vsh 2lm,\vsh 3lm$ transform in the same way,
+\begin{align*}
+\vsh 2lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
+\vsh 3lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
+\end{align*}
+but the remaining set $\vsh 1lm$ transforms differently due to their
+pseudovector nature stemming from the cross product in their definition:
+\[
+\vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
+\]
+where $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$
+if $g$ is a proper rotation, but for spatial inversion operation
+$i:\vect r\mapsto-\vect r$ we have $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$.
+The transformation behaviour of vector spherical harmonics directly
+propagates to the spherical vector waves, cf. (\ref{eq:VSWF regular}),
+(\ref{eq:VSWF outgoing}):
+\begin{align*}
+\vswfouttlm 1lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\
+\vswfouttlm 2lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right),
+\end{align*}
+(and analogously for the regular waves $\vswfrtlm{\tau}lm$).  For
+convenience, we introduce the symbol $D_{m,m'}^{\tau l}$ that describes
+the transformation of both types (``magnetic'' and ``electric'')
+of waves at once:
+\[
+\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right).
+\]
+Using these, we can express the VSWF expansion (\ref{eq:E field expansion})
+of the electric field around origin in a rotated/reflected system,
+\[
+\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right),
+\]
+which, together with the $T$-matrix definition, (\ref{eq:T-matrix definition})
+can be used to obtain a $T$-matrix of a rotated or mirror-reflected
+particle. Let $T$ be the $T$-matrix of an original particle; the
+$T$-matrix of a particle physically transformed by operation $g\in O(3)$
+is then 
+\begin{equation}
+T'_{\tau lm;\tau'l'm'}=\sum_{\mu=-l}^{l}\sum_{\mu'=-l'}^{l'}\left(D_{\mu,m}^{\tau l}\left(g\right)\right)^{*}T_{\tau l\mu;\tau'l'm'}D_{m',\mu'}^{\tau l}\left(g\right).\label{eq:T-matrix of a transformed particle}
+\end{equation}
+If the particle is symmetric (so that $g$ produces a particle indistinguishable
+from the original one), the $T$-matrix must remain invariant under
+the transformation (\ref{eq:T-matrix of a transformed particle}),
+$T'_{\tau lm;\tau'l'm'}=T{}_{\tau lm;\tau'l'm'}$. Explicit forms
+of these invariance properties for the most imporant point group symmetries
+can be found in \cite{schulz_point-group_1999}.
+
+If the field expansion is done around a point $\vect r_{p}$ different
+from the global origin, as in \ref{eq:E field expansion multiparticle},
+we have\foreignlanguage{english}{
+\begin{align}
+\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin}
+\end{align}
+}
+
+\begin{figure}
+\caption{Scatterer orbits under $D_{2}$ symmetry. Particles $A,B,C,D$ lie
+outside of origin or any mirror planes, and together constitute an
+orbit of the size equal to the order of the group, $\left|D_{2}\right|=4$.
+Particles $E,F$ lie on the $xz$ plane, hence the corresponding reflection
+maps each of them to itself, but the $yz$ reflection (or the $\pi$
+rotation around the $z$ axis) maps them to each other, forming a
+particle orbit of size 2. The particle $O$ in the very origin is
+always mapped to itself, constituting its own orbit.}\label{fig:D2-symmetric structure particle orbits}
+\end{figure}
+
+With these transformation properties in hand, we can proceed to the
+effects of point symmetries on the whole many-particle system. Let
+us have a many-particle system symmetric with respect to a point group
+$G$. A symmetry operation $g\in G$ determines a permutation of the
+particles: $p\mapsto\pi_{g}(p)$, $p\in\mathcal{P}$. For a given
+particle $p$, we will call the set of particles onto which any of
+the symmetries maps the particle $p$, i.e. the set $\left\{ \pi_{g}\left(p\right);g\in G\right\} $,
+as the \emph{orbit} of particle $p$. The whole set $\mathcal{P}$
+can therefore be divided into the different particle orbits; an example
+is in Fig. \ref{fig:D2-symmetric structure particle orbits}. The
+importance of the particle orbits stems from the following: in the
+multiple-scattering problem, outside of the scatterers  one has \foreignlanguage{english}{
+\begin{align}
+\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}(p)}\right)\right)+\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right)\label{eq:rotated E field expansion around outside origin-1}\\
+ & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right).
+\end{align}
+This means that the field expansion coefficients $\rcoeffp p,\outcoeffp p$
+transform as 
+\begin{align}
+\rcoeffptlm p{\tau}lm & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\
+\outcoeffptlm p{\tau}lm & \mapsto\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation}
+\end{align}
+Obviously, the expansion coefficients belonging to particles in different
+orbits do not mix together. As before, we introduce a short-hand block-matrix
+notation for \ref{eq:excitation coefficient under symmetry operation}}
+
+\selectlanguage{english}%
+\begin{align}
+\rcoeff & \mapsto D\left(g\right)a,\nonumber \\
+\outcoeff & \mapsto D\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation block form}
+\end{align}
+
+\selectlanguage{finnish}%
+
+\subsection{Irrep decomposition}
+
+\subsection{Periodic systems}
+
+\selectlanguage{english}%
+