1.5 KiB
VSWF expansions in terms of SSWF
From \cite necada_multiple-scattering_2021, eq. (2.19) \f[ \wfkcout_{\tau lm}\left(\kappa (\vect r - \vect r_1) \right) = \sum_{\tau'l'm'} \tropSr{\kappa(\vect r_2 - \vect r_1)}{\tau l m;\tau'l'm} \wfkcreg{\tau'l'm'}(\vect r -\vect r_2), \qquad |\vect r -\vect r_2| < |\vect r_1 - \vect r_2|, \f] setting \f$ \vect r = \vect r_2\f$ and considering that \f$ \wfkcreg_{\tau'l'm'}(\vect 0) \ne \vect 0 \f$ only for electric dipole waves (\f$ \tau = \mathrm{E}, l=1 \f$), we have \f[ \wfkcout_{\tau lm}\left(\kappa (\vect r - \vect r_1) \right) = \sum_{m'} \tropSr{\kappa(\vect r - \vect r_1)}{\tau l m;\mathrm{E}1m} \wfkcreg{\mathrm{E}1m'}(\vect 0), \qquad \vect r \ne \vect r_1 . \f] Combining this with \cite necada_multiple-scattering_2021, eq. (2.25) \f[ \tropSr{\vect d}{\tau l m; \tau' l' m'} = \sum{\lambda =|l-l'|+|\tau-\tau'|}^{l+l'} C^{\lambda}{\tau l m;\tau' l'm'} \underbrace{ \spharm{\lambda}{m-m'}(\uvec d) h\lambda^{(1)}(d)}{\sswfout\lambda^{m-m'}(\vect d)}, \f] we get \f[ \wfkcout_{\tau lm}(\vect d) = \sum_{m'=-1}^1 \wfkcreg_{\mathrm{E}1m'}(\vect 0) \sum_{\lambda=l-1+|\tau-\tau'|}^{l+1} C^\lambda_{\tau l m;\mathrm{E}1m'} \sswfout_\lambda^{m-m'}(\vect d). \f] Note that the VSWF components in this expression are given in global "cartesian" basis, not the local orthonormal basis derived from spherical coordinates. (This is mostly desirable, because in lattices we need to work with flat coordinates anyway.)