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