8.8 KiB
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 D2h symmetry and our truncated $\left(I-T\trops\right)$ matrix has size N × N, it can be block-diagonalized into eight blocks of size about N/8 × N/8, each of which can be LU-factorised separately (this is due to the fact that D2h has eight different one-dimensional irreducible representations). This can reduce both memory and time requirements to solve the scattering problem ([eq:Multiple-scattering problem block form]) by a factor of 64.
In periodic systems (problems ([eq:Multiple-scattering problem unit cell block form]), ([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.
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 Rg:
$$\vect r\mapsto R_{g}\vect r.$$
Spherical harmonics $\ush lm$, being a basis the l-dimensional representation of O(3), transform as
$$\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right)$$
where Dm, m′l(g) denotes the elements of the Wigner matrix representing the operation g. By their definition, vector spherical harmonics $\vsh 2lm,\vsh 3lm$ transform in the same way,
$$\begin{aligned}
\vsh 2lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
\vsh 3lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),\end{aligned}$$
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. ([eq:VSWF regular]), ([eq:VSWF outgoing]):
$$\begin{aligned}
\vswfouttlm 1lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\
\vswfouttlm 2lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right),\end{aligned}$$
(and analogously for the regular waves $\vswfrtlm{\tau}lm$). For convenience, we introduce the symbol Dm, m′τl 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 ([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, ([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 ∈ O(3) is then
$$T'_{\tau lm;\tau'l'm'}=\sum_{\mu=-l}^{l}\sum_{\mu'=-l'}^{l'}\left(D_{\mu,m}^{\tau l}\left(g\right)\right)^{*}T_{\tau l\mu;\tau'l'm'}D_{m',\mu'}^{\tau l}\left(g\right).\label{eq:T-matrix of a transformed particle}$$
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 ([eq:T-matrix of a transformed particle]), T′τlm; τ′l′m′ = Tτlm; τ′l′m′. Explicit forms of these invariance properties for the most imporant point group symmetries can be found in .
If the field expansion is done around a point $\vect r_{p}$ different from the global origin, as in [eq:E field expansion multiparticle], we have
$$\begin{aligned}
\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)+\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin}\end{aligned}$$
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 ∈ G determines a permutation of the particles: p ↦ πg(p), p ∈ 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 {πg(p);g∈G}, as the orbit of particle p. The whole set 𝒫 can therefore be divided into the different particle orbits; an example is in Fig. [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
$$\begin{aligned}
\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}(p)}\right)\right)+\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right)\label{eq:rotated E field expansion around outside origin-1}\\
& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)+\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right).\end{aligned}$$
This means that the field expansion coefficients $\rcoeffp p,\outcoeffp p$ transform as
$$\begin{aligned}
\rcoeffptlm p{\tau}lm & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\
\outcoeffptlm p{\tau}lm & \mapsto\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation}\end{aligned}$$
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 [eq:excitation coefficient under symmetry operation]
$$\begin{aligned}
\rcoeff & \mapsto D\left(g\right)a,\nonumber \\
\outcoeff & \mapsto D\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation block form}\end{aligned}$$