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\begin_body \begin_layout Title Derivation of the 1D and 2D lattice sums for the 3D Helmholtz equation with general lattice offset \end_layout \begin_layout Author Marek Nečada \end_layout \begin_layout Section Periodic Green's functions vs. VSWF lattice sums \end_layout \begin_layout Standard \begin_inset FormulaMacro \newcommand{\ud}{\mathrm{d}} \end_inset \begin_inset FormulaMacro \newcommand{\abs}[1]{\left|#1\right|} \end_inset \begin_inset FormulaMacro \newcommand{\vect}[1]{\mathbf{#1}} \end_inset \begin_inset FormulaMacro \newcommand{\uvec}[1]{\hat{\mathbf{#1}}} \end_inset \lang english \begin_inset FormulaMacro \newcommand{\ush}[2]{Y_{#1}^{#2}} \end_inset \begin_inset FormulaMacro \newcommand{\ushD}[2]{Y'_{#1}^{#2}} \end_inset \end_layout \begin_layout Standard \begin_inset FormulaMacro \newcommand{\vsh}{\vect A} \end_inset \begin_inset FormulaMacro \newcommand{\vshD}{\vect{A'}} \end_inset \begin_inset FormulaMacro \newcommand{\wfkc}{\vect y} \end_inset \begin_inset FormulaMacro \newcommand{\wfkcout}{\vect u} \end_inset \begin_inset FormulaMacro \newcommand{\wfkcreg}{\vect v} \end_inset \begin_inset FormulaMacro \newcommand{\wckcreg}{a} \end_inset \begin_inset FormulaMacro \newcommand{\wckcout}{f} \end_inset \end_layout \begin_layout Subsection Some definitions and useful relations \end_layout \begin_layout Standard \begin_inset Formula \[ \mathcal{H}_{l}^{m}\left(\vect d\right)\equiv h_{l}^{+}\left(\left|\vect d\right|\right)\ush lm\left(\uvec d\right), \] \end_inset \begin_inset Formula \[ \mathcal{J}_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ush lm\left(\uvec d\right). \] \end_inset Dual spherical harmonics and waves: \end_layout \begin_layout Standard \begin_inset Formula \[ \int\ush lm\ushD{l'}{m'}\,\ud\Omega=\delta_{l,l'}\delta_{m,m'}, \] \end_inset \begin_inset Formula \[ \mathcal{J}'_{l}^{m}\left(\vect d\right)\equiv j_{l}\left(\left|\vect d\right|\right)\ushD lm\left(\uvec d\right). \] \end_inset Expansion of a plane wave: \begin_inset Note Note status open \begin_layout Plain Layout (CHECKME whether this is really convention-independent, but it seems so) \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ e^{i\kappa\vect r\cdot\uvec r'}=4\pi\sum_{l,m}i^{n}\mathcal{J}'_{l}^{m}\left(\kappa\vect r\right)\ush lm\left(\uvec r'\right)=4\pi\sum_{l,m}i^{n}\mathcal{J}{}_{l}^{m}\left(\kappa\vect r\right)\ushD lm\left(\uvec r'\right). \] \end_inset This one is also convention independent (similarly for \begin_inset Formula $\mathcal{H}_{l}^{m}$ \end_inset ): \begin_inset Formula \[ \mathcal{J}_{l}^{m}\left(-\vect r\right)=\left(-1\right)^{l}\mathcal{J}_{l}^{m}\left(\vect r\right). \] \end_inset \end_layout \begin_layout Subsection Helmholtz equation and Green's functions (in 3D) \end_layout \begin_layout Standard Note that the notation does not follow Linton's (where the wavenumbers are often implicit) \end_layout \begin_layout Standard \begin_inset Formula \[ \left(\nabla^{2}+\kappa^{2}\right)G^{(\kappa)}\left(\vect x,\vect x_{0}\right)=\delta\left(\vect x-\vect x_{0}\right), \] \end_inset \begin_inset Formula \begin{align*} G_{0}^{(\kappa)}\left(\vect x,\vect x_{0}\right) & =G_{0}^{(\kappa)}\left(\vect x-\vect x_{0}\right)==-\frac{e^{i\kappa\left|\vect x-\vect x_{0}\right|}}{4\pi\left|\vect x-\vect x_{0}\right|}=-\frac{i\kappa}{4\pi}h_{0}^{+}\left(\kappa\left|\vect x-\vect x_{0}\right|\right)=-\frac{i\kappa}{\sqrt{4\pi}}\mathcal{H}_{0}^{0}\left(\kappa\left|\vect x-\vect x_{0}\right|\right). \end{align*} \end_inset In case of wacky conventions, \begin_inset Formula $G_{0}^{(\kappa)}\left(\vect x,\vect x_{0}\right)=-\frac{i\kappa}{\ush 00}\mathcal{H}_{0}^{0}\left(\kappa\left|\vect x-\vect x_{0}\right|\right)$ \end_inset . \end_layout \begin_layout Standard Lattice GF \begin_inset CommandInset citation LatexCommand cite after "(2.3)" key "linton_lattice_2010" literal "false" \end_inset : \begin_inset Formula \begin{equation} G_{\Lambda}^{(\kappa)}\left(\vect s,\vect k\right)\equiv\sum_{\vect R\in\Lambda}G_{0}^{\kappa}\left(\vect s-\vect R\right)e^{i\vect k\cdot\vect R}\label{eq:Lattice GF} \end{equation} \end_inset \end_layout \begin_layout Subsection GF expansion and lattice sum definition \end_layout \begin_layout Standard Let's define \begin_inset Formula \[ \sigma_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}_{l}^{m}\left(\kappa\left(\vect s+\vect R\right)\right)e^{i\vect k\cdot\vect R}, \] \end_inset and also its dual version \begin_inset Formula \[ \sigma'_{l}^{m}\left(\vect s,\vect k\right)=\sum_{\vect R\in\Lambda}\mathcal{H}'_{l}^{m}\left(\kappa\left(\vect s+\vect R\right)\right)e^{i\vect k\cdot\vect R}. \] \end_inset \end_layout \begin_layout Standard Inspired by \begin_inset CommandInset citation LatexCommand cite after "(4.1)" key "linton_lattice_2010" literal "false" \end_inset : assuming that \begin_inset Formula $\vect s\notin\Lambda$ \end_inset , let's expand the lattice Green's function around \begin_inset Formula $\vect s$ \end_inset : \end_layout \begin_layout Standard \begin_inset Formula \[ G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)=-i\kappa\sum_{l,m}\tau_{l}^{m}\left(\vect s,\vect k\right)\mathcal{J}_{l}^{m}\left(\kappa\vect r\right), \] \end_inset and multiply with a dual SH + integrate \begin_inset Formula \begin{align} \int\ud\Omega_{\vect r}\,G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)\ushD{l'}{m'}\left(\uvec r\right) & =-i\kappa\sum_{l,m}\tau_{l}^{m}\left(\vect s,\vect k\right)j_{l}\left(\kappa\left|\vect r\right|\right)\delta_{ll'}\delta_{mm'}\nonumber \\ & =-i\kappa\tau_{l'}^{m'}\left(\vect s,\vect k\right)j_{l'}\left(\kappa\left|\vect r\right|\right).\label{eq:tau extraction} \end{align} \end_inset The expansion coefficients \begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$ \end_inset is then typically extracted by taking the limit \begin_inset Formula $\left|\vect r\right|\to0$ \end_inset . \end_layout \begin_layout Standard The relation between \begin_inset Formula $\sigma_{l}^{m}\left(\vect s,\vect k\right)$ \end_inset and \begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$ \end_inset can be obtained e.g. from the addition theorem for scalar spherical wavefunctions \begin_inset CommandInset citation LatexCommand cite after "(C.3)" key "linton_lattice_2010" literal "false" \end_inset , \begin_inset Formula \[ \mathcal{H}_{l}^{m}\left(\vect a+\vect b\right)=\sum_{l'm'}S_{ll'}^{mm'}\left(\vect b\right)\mathcal{J}_{l'}^{m'}\left(\vect a\right),\quad\left|\vect a\right|<\left|\vect b\right|, \] \end_inset where for the zeroth degree and order one has \begin_inset CommandInset citation LatexCommand cite after "(C.3)" key "linton_lattice_2010" literal "false" \end_inset \begin_inset Foot status open \begin_layout Plain Layout In a totally convention-independent version probably looks like \begin_inset Formula $S_{0l'}^{0m'}\left(\vect b\right)=\ush 00\mathcal{H}'_{l'}^{m'}\left(-\vect b\right)$ \end_inset , but the \begin_inset Formula $Y_{0}^{0}$ \end_inset will cancel with the expression for GF anyways, so no harm to the final result. \end_layout \end_inset \begin_inset Formula \[ S_{0l'}^{0m'}\left(\vect b\right)=\sqrt{4\pi}\mathcal{H}'_{l'}^{m'}\left(-\vect b\right). \] \end_inset From the lattice GF definition \begin_inset CommandInset ref LatexCommand eqref reference "eq:Lattice GF" plural "false" caps "false" noprefix "false" \end_inset \begin_inset Formula \begin{align*} G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right) & \equiv\frac{-i\kappa}{\sqrt{4\pi}}\sum_{\vect R\in\Lambda}\mathcal{H}_{0}^{0}\left(\kappa\left(\vect s+\vect r-\vect R\right)\right)e^{i\vect k\cdot\vect R}\\ & =\frac{-i\kappa}{\sqrt{4\pi}}\sum_{\vect R\in\Lambda}\mathcal{H}_{0}^{0}\left(\kappa\left(\vect s+\vect r-\vect R\right)\right)e^{i\vect k\cdot\vect R}\\ & =\frac{-i\kappa}{\sqrt{4\pi}}\sum_{\vect R\in\Lambda}\sum_{l'm'}S_{0l'}^{0m'}\left(\kappa\left(\vect s-\vect R\right)\right)\mathcal{J}_{l'}^{m'}\left(\kappa\vect r\right)e^{i\vect k\cdot\vect R}\\ & =-i\kappa\sum_{\vect R\in\Lambda}\sum_{lm}\mathcal{H}'_{l}^{m}\left(-\kappa\left(\vect s-\vect R\right)\right)\mathcal{J}_{l}^{m}\left(\kappa\vect r\right)e^{i\vect k\cdot\vect R}, \end{align*} \end_inset and mutliplying with a dual SH and integrating \begin_inset Formula \begin{align*} \int\ud\Omega_{\vect r}\,G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)\ushD{l'}{m'}\left(\uvec r\right) & =-i\kappa\sum_{\vect R\in\Lambda}\sum_{lm}\mathcal{H}'_{l}^{m}\left(-\kappa\left(\vect s-\vect R\right)\right)j_{l}\left(\kappa\left|\vect r\right|\right)\delta_{ll'}\delta_{mm'}e^{i\vect k\cdot\vect R}\\ & =-i\kappa\sum_{\vect R\in\Lambda}\mathcal{H}'_{l'}^{m'}\left(\kappa\left(-\vect s+\vect R\right)\right)j_{l'}\left(\kappa\left|\vect r\right|\right)e^{i\vect k\cdot\vect R}\\ & =-i\kappa\sigma'_{l'}^{m'}\left(-\vect s,\vect k\right)j_{l'}\left(\kappa\left|\vect r\right|\right), \end{align*} \end_inset and comparing with \begin_inset CommandInset ref LatexCommand eqref reference "eq:tau extraction" plural "false" caps "false" noprefix "false" \end_inset we have \begin_inset Formula \[ \tau_{l}^{m}\left(\vect s,\vect k\right)=\sigma'_{l}^{m}\left(-\vect s,\vect k\right). \] \end_inset \begin_inset Note Note status open \begin_layout Plain Layout TODO maybe also define some \begin_inset Formula $\tau'_{l}^{m}$ \end_inset as expansion coefficients of GF into dual regular SSWFs. \end_layout \end_inset \end_layout \begin_layout Section Derivation of the 1D and 2D lattice sum \end_layout \begin_layout Standard With \begin_inset CommandInset citation LatexCommand cite key "linton_lattice_2010" literal "false" \end_inset in hand, the short-range part is rather easy. Let's get the long-range part. \end_layout \begin_layout Standard We first need to find the long-range part of the expansion coefficient \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{i}{\kappa j_{l'}\left(\kappa\left|\vect r\right|\right)}\int\ud\Omega_{\vect r}\,G_{\Lambda}^{(\kappa)}\left(\vect s+\vect r,\vect k\right)\ushD{l'}{m'}\left(\uvec r\right).\label{eq:tau extraction formula} \end{equation} \end_inset \end_layout \begin_layout Standard We take \begin_inset CommandInset citation LatexCommand cite after "(2.24)" key "linton_lattice_2010" literal "false" \end_inset with slightly modified notation \begin_inset Formula $\left(\vect k_{\vect K}\equiv\vect K+\vect k\right)$ \end_inset \begin_inset Formula \[ G_{\Lambda}^{(1;\kappa)}\left(\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect r}\int_{1/\eta}^{\infty e^{i\pi/4}}e^{-\kappa^{2}\gamma^{2}t^{2}/4}e^{-\left|\vect r^{\bot}\right|^{2}/t^{2}}t^{1-d_{c}}\ud t \] \end_inset or, evaluated at point \begin_inset Formula $\vect s+\vect r$ \end_inset instead \begin_inset Formula \[ G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\left(\vect s+\vect r\right)}\int_{1/\eta}^{\infty e^{i\pi/4}}e^{-\kappa^{2}\gamma^{2}t^{2}/4}e^{-\left|\vect s^{\bot}+\vect r^{\bot}\right|^{2}/t^{2}}t^{1-d_{c}}\ud t. \] \end_inset The integral can be by substitutions taken into the form \begin_inset Note Note status open \begin_layout Plain Layout \lang english \begin_inset Formula \[ G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{2\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta}^{\infty\exp\left(i\pi/4\right)}e^{-\kappa^{2}\gamma_{m}^{2}\zeta^{2}/4}e^{-\left|\vect r_{\bot}\right|^{2}/\zeta^{2}}\zeta^{1-d_{c}}\ud\zeta \] \end_inset Try substitution \begin_inset Formula $t=\zeta^{2}$ \end_inset : then \begin_inset Formula $\ud t=2\zeta\,\ud\zeta$ \end_inset ( \begin_inset Formula $\ud\zeta=\ud t/2t^{1/2}$ \end_inset ) and \begin_inset Formula \[ G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\int_{1/\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\kappa^{2}\gamma_{m}^{2}t/4}e^{-\left|\vect r_{\bot}\right|^{2}/t}t^{\frac{-d_{c}}{2}}\ud t \] \end_inset Try subst. \begin_inset Formula $\tau=k^{2}\gamma_{m}^{2}/4$ \end_inset \end_layout \begin_layout Plain Layout \lang english \begin_inset Formula \[ G_{\Lambda}^{\left(1\right)}\left(\vect r\right)=\frac{\pi^{-d_{c}/2}}{4\mathcal{A}}\sum_{\vect K_{m}\in\Lambda^{*}}e^{i\vect K_{m}\cdot\vect r}\left(\frac{\kappa\gamma_{m}}{2}\right)^{d_{c}}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{\frac{-d_{c}}{2}}\ud\tau \] \end_inset \end_layout \end_inset \begin_inset Formula \[ G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\left(\vect s+\vect r\right)}\int_{\kappa^{2}\gamma_{m}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{m}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau. \] \end_inset \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout \begin_inset Foot status open \begin_layout Plain Layout [Linton, (2.25)] with slightly modified notation: \begin_inset Formula \[ G_{\Lambda}^{(1;\kappa)}\left(\vect r\right)=-\frac{1}{\sqrt{4\pi}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect r}\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\left|\vect r^{\bot}\right|^{2j}}{j!}\left(\frac{\kappa\gamma_{\vect{\vect k_{\vect K}}}}{2}\right)^{2j-1}\Gamma_{j\vect k_{\vect K}} \] \end_inset We want to express an expansion in a shifted point, so let's substitute \begin_inset Formula $\vect r\to\vect s+\vect r$ \end_inset \begin_inset Formula \[ G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)=-\frac{1}{\sqrt{4\pi}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\left(\vect s+\vect r\right)}\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}\left|\vect s^{\bot}+\vect r^{\bot}\right|^{2j}}{j!}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j-1}\Gamma_{j\vect k_{\vect K}} \] \end_inset \end_layout \end_inset \end_layout \end_inset Let's do the integration to get \begin_inset Formula $\tau_{l}^{m}\left(\vect s,\vect k\right)$ \end_inset \begin_inset Formula \[ \int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\left(\vect s+\vect r\right)}\int_{\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau. \] \end_inset The \begin_inset Formula $\vect r$ \end_inset -dependent plane wave factor can be also written as \begin_inset Formula \begin{align*} e^{i\vect k_{\vect K}\cdot\vect r} & =e^{i\left|\vect k_{\vect K}\right|\vect r\cdot\uvec{\vect k_{\vect K}}}=4\pi\sum_{lm}i^{l}\mathcal{J}'_{l}^{m}\left(\left|\vect k_{\vect K}\right|\vect r\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\\ & =4\pi\sum_{lm}i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ushD lm\left(\uvec{\vect r}\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right), \end{align*} \end_inset \begin_inset Note Note status open \begin_layout Plain Layout or the other way around \begin_inset Formula \[ e^{i\vect k_{\vect K}\cdot\vect r}=4\pi\sum_{lm}i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect r}\right)\ushD lm\left(\uvec{\vect k_{\vect K}}\right) \] \end_inset \end_layout \end_inset so \begin_inset Formula \begin{multline*} \int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\frac{1}{2\pi\mathcal{A}}\times\\ \times\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ushD lm\left(\uvec r\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\int_{\kappa^{2}\gamma_{\vect{\vect k_{\vect K}}}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect{\vect k_{\vect K}}}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}}\ud\tau. \end{multline*} \end_inset \end_layout \begin_layout Standard We also have \begin_inset Formula \begin{align*} e^{-\left|\vect s_{\bot}+\vect r_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau} & =e^{-\left(\left|\vect s_{\bot}\right|^{2}+\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\\ & =e^{-\left|\vect s_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\sum_{j=0}^{\infty}\frac{1}{j!}\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect K}^{2}}{4\tau}\right)^{j}, \end{align*} \end_inset hence \begin_inset Formula \begin{align*} \int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right) & =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ushD lm\left(\uvec r\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\times\\ & \quad\times\sum_{j=0}^{\infty}\frac{1}{j!}\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect{\vect k_{\vect K}}}^{2}}{4}\right)^{j}\underbrace{\int_{\kappa^{2}\gamma_{\vect K}^{2}/4\eta^{2}}^{\infty\exp\left(i\pi/2\right)}e^{-\tau}e^{-\left|\vect s_{\bot}\right|^{2}\kappa^{2}\gamma_{\vect K}^{2}/4\tau}\tau^{-\frac{d_{c}}{2}-j}\ud\tau}_{\Delta_{j}^{\left(d_{\Lambda}\right)}}\\ & =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\sum_{j=0}^{\infty}\frac{\Delta_{j}^{\left(d_{\Lambda}\right)}}{j!}\times\\ & \quad\times\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(-\frac{\left(\left|\vect r_{\bot}\right|^{2}+2\vect r_{\bot}\cdot\vect s_{\bot}\right)\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}}{4}\right)^{j}\\ & =-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\times\\ & \quad\times\left(\frac{\kappa\gamma_{\vect{\vect k_{\vect K}}}}{2}\right)^{2j}\sum_{k=0}^{j}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left|\vect r_{\bot}\right|^{2(j-k)}\left(2\vect r_{\bot}\cdot\vect s_{\bot}\right)^{k}. \end{align*} \end_inset The integral \begin_inset Formula $\Delta_{j}^{\left(d_{\Lambda}\right)}$ \end_inset is (for the 2D case) equivalent to that in \begin_inset CommandInset citation LatexCommand cite key "kambe_theory_1968" literal "false" \end_inset . \end_layout \begin_layout Standard If we label \begin_inset Formula $\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|\cos\varphi\equiv\vect r_{\bot}\cdot\vect s_{\bot}$ \end_inset , we have \begin_inset Formula \begin{multline*} \int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\times\\ \times\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\sum_{k=0}^{j}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left|\vect r_{\bot}\right|^{2j-k}\left(\cos\varphi\right)^{k}, \end{multline*} \end_inset and if we label \begin_inset Formula $\left|\vect r\right|\sin\vartheta\equiv\left|\vect r_{\bot}\right|$ \end_inset \begin_inset Formula \begin{multline*} \int\ud\Omega_{\vect r}\,G_{\Lambda}^{(1;\kappa)}\left(\vect s+\vect r\right)\ushD{l'}{m'}\left(\uvec r\right)=-\frac{1}{2\pi^{d_{c}/2}\mathcal{A}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\ush lm\left(\uvec{\vect k_{\vect K}}\right)\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\times\\ \times\sum_{k=0}^{j}\left|\vect r\right|^{2j-k}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{2j-k}\left(\cos\varphi\right)^{k}. \end{multline*} \end_inset Now let's put the RHS into \begin_inset CommandInset ref LatexCommand eqref reference "eq:tau extraction formula" plural "false" caps "false" noprefix "false" \end_inset and try eliminating some sum by taking the limit \begin_inset Formula $\left|\vect r\right|\to0$ \end_inset . We have \begin_inset Formula $j_{l}\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)\sim\left(\left|\vect k_{\vect K}\right|\left|\vect r\right|\right)^{l}/\left(2l+1\right)!!$ \end_inset ; the denominator from \begin_inset CommandInset ref LatexCommand eqref reference "eq:tau extraction formula" plural "false" caps "false" noprefix "false" \end_inset behaves like \begin_inset Formula $j_{l'}\left(\kappa\left|\vect r\right|\right)\sim\left(\kappa\left|\vect r\right|\right)^{l'}/\left(2l'+1\right)!!.$ \end_inset The leading terms are hence those with \begin_inset Formula $\left|\vect r\right|^{l-l'+2j-k}$ \end_inset . So \begin_inset Formula \begin{multline*} \tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa^{1+l'}}\left(2l'+1\right)!!\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{lm}4\pi i^{l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\ush lm\left(\uvec{\vect k_{\vect K}}\right)\times\\ \times\sum_{j=0}^{\infty}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\sum_{k=0}^{j}\delta_{l'-l,2j-k}\left(2\left|\vect s_{\bot}\right|\right)^{k}\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{l'-l}\left(\cos\varphi\right)^{k}. \end{multline*} \end_inset Let's now focus on rearranging the sums; we have \begin_inset Formula \[ S(l')\equiv\sum_{l=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\delta_{l'-l,2j-k}f(l',l,j,k)=\sum_{l=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{j}\delta_{l'-l,2j-k}f(l',l,j,2j-l'+l). \] \end_inset We have \begin_inset Formula $0\le k\le j$ \end_inset , hence \begin_inset Formula $0\le2j-l'+l\le j$ \end_inset , hence \begin_inset Formula $-2j\le-l'+l\le-j$ \end_inset , hence also \begin_inset Formula $l'-2j\le l\le l'-j$ \end_inset , which gives the opportunity to swap the \begin_inset Formula $l,j$ \end_inset sums and the \begin_inset Formula $l$ \end_inset -sum becomes finite; so also consuming \begin_inset Formula $\sum_{k=0}^{j}\delta_{l'-l,2j-k}$ \end_inset we get \begin_inset Formula \[ S(l')=\sum_{j=0}^{\infty}\sum_{l=\max(0,l'-2j)}^{l'-j}f(l',l,j,2j-l'+l). \] \end_inset Finally, we see that the interval of valid \begin_inset Formula $l$ \end_inset becomes empty when \begin_inset Formula $l'-j<0$ \end_inset , i.e. \begin_inset Formula $j>l'$ \end_inset ; so we get a finite sum \begin_inset Formula \[ S(l')=\sum_{j=0}^{l'}\sum_{l=\max(0,l'-2j)}^{l'-j}f(l',l,j,2j-l'+l). \] \end_inset Applying rearrangement, \begin_inset Formula \begin{multline*} \tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{j=0}^{l'}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\times\sum_{l=\max\left(0,l'-2j\right)}^{l'-j}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l'+l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\\ \times\sum_{m=-l}^{l}\ush lm\left(\uvec{\vect k_{\vect K}}\right)\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\sin\vartheta\right)^{l'-l}\left(\cos\varphi\right)^{2j-l'+l}, \end{multline*} \end_inset or replacing the angles with their original definition, \begin_inset Formula \begin{multline*} \tau_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\sum_{j=0}^{l'}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect K}}{2}\right)^{2j}\times\sum_{l=\max\left(0,l'-2j\right)}^{l'-j}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l'+l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\\ \times\sum_{m=-l}^{l}\ush lm\left(\uvec K\right)\int\ud\Omega_{\vect r}\,\ushD{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{l'-l}\left(\frac{\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|}\right)^{2j-l'+l}, \end{multline*} \end_inset and if we want a \begin_inset Formula $\sigma_{l'}^{m'}\left(\vect s,\vect k\right)$ \end_inset instead, we reverse the sign of \begin_inset Formula $\vect s$ \end_inset and replace all spherical harmonics with their dual counterparts: \begin_inset Formula \begin{multline*} \sigma_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\sum_{j=0}^{l'}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\sum_{l=\max\left(0,l'-2j\right)}^{l'-j}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l'+l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\times\\ \times\sum_{m=-l}^{l}\ushD lm\left(\uvec{\vect k_{\vect K}}\right)\int\ud\Omega_{\vect r}\,\ush{l'}{m'}\left(\uvec r\right)\ush lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{l'-l}\left(\frac{-\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|}\right)^{2j-l'+l}, \end{multline*} \end_inset and remembering that in the plane wave expansion the \begin_inset Quotes eld \end_inset duality \begin_inset Quotes erd \end_inset is interchangeable, \begin_inset Formula \begin{multline*} \sigma_{l'}^{m'}\left(\vect s,\vect k\right)=\frac{-i}{2\pi^{d_{c}/2}\mathcal{A}\kappa}\frac{\left(2l'+1\right)!!}{\kappa^{l'}}\sum_{\vect K\in\Lambda^{*}}e^{-i\vect k_{\vect K}\cdot\vect s}\sum_{j=0}^{l'}\frac{\left(-1\right)^{j}}{j!}\Delta_{j}^{\left(d_{\Lambda}\right)}\left(\frac{\kappa\gamma_{\vect k_{\vect K}}}{2}\right)^{2j}\sum_{l=\max\left(0,l'-2j\right)}^{l'-j}4\pi i^{l}\left(2\left|\vect s_{\bot}\right|\right)^{2j-l'+l}\frac{\left|\vect k_{\vect K}\right|^{l}}{\left(2l+1\right)!!}\times\\ \times\sum_{m=-l}^{l}\ush lm\left(\uvec{\vect k_{\vect K}}\right)\underbrace{\int\ud\Omega_{\vect r}\,\ush{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{l'-l}\left(\frac{-\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|}\right)^{2j-l'+l}}_{\equiv A_{l',l,m',m,j}^{\left(d_{\Lambda}\right)}}. \end{multline*} \end_inset The angular integral is easier to evaluate when \begin_inset Formula $d_{\Lambda}=2$ \end_inset , because then \begin_inset Formula $\vect r_{\bot}$ \end_inset is parallel (or antiparallel) to \begin_inset Formula $\vect s_{\bot}$ \end_inset , which gives \begin_inset Formula \[ A_{l',l,m',m,j}^{\left(2\right)}=\left(-\frac{\vect r_{\bot}\cdot\vect s_{\bot}}{\left|\vect r_{\bot}\cdot\vect s_{\bot}\right|}\right)^{2j-l'+l}\int\ud\Omega_{\vect r}\,\ush{l'}{m'}\left(\uvec r\right)\ushD lm\left(\uvec r\right)\left(\frac{\left|\vect r_{\bot}\right|}{\left|\vect r\right|}\right)^{2j} \] \end_inset and if we set the normal of the lattice correspond to the \begin_inset Formula $z$ \end_inset axis, the azimuthal part of the integral will become zero unless \begin_inset Formula $m'=m$ \end_inset for any meaningful spherical harmonics convention, and the polar part for the only nonzero case has a closed-form expression, see e.g. \begin_inset CommandInset citation LatexCommand cite after "(A.15)" key "linton_lattice_2010" literal "false" \end_inset , so one arrives at an expression similar to \begin_inset CommandInset citation LatexCommand cite after "(3.15)" key "kambe_theory_1968" literal "false" \end_inset \lang english \begin_inset Formula \begin{multline} \sigma_{l,m}^{\left(\mathrm{L},\eta\right)}\left(\vect k,\vect s\right)=-\frac{i^{l+1}}{\kappa^{2}\mathcal{A}}\pi^{3/2}2\left(\left(l-m\right)/2\right)!\left(\left(l+m\right)/2\right)!\times\\ \times\sum_{\vect K\in\Lambda^{*}}e^{i\vect k_{\vect K}\cdot\vect s}\ush lm\left(\vect k_{\vect K}\right)\sum_{j=0}^{l-\left|m\right|}\left(-1\right)^{j}\gamma_{\vect k_{\vect K}}^{2}{}^{2j+1}\times\\ \times\Delta_{j}\left(\frac{\kappa^{2}\gamma_{\vect k_{\vect K}}^{2}}{4\eta^{2}},-i\kappa\gamma_{\vect k_{\vect K}}^{2}s_{\perp}\right)\times\\ \times\sum_{\substack{s\\ j\le s\le\min\left(2j,l-\left|m\right|\right)\\ l-j+\left|m\right|\,\mathrm{even} } }\frac{1}{\left(2j-s\right)!\left(s-j\right)!}\frac{\left(-\kappa s_{\perp}\right)^{2j-s}\left(\left|\vect k_{\vect K}\right|/\kappa\right)^{l-s}}{\left(\frac{1}{2}\left(l-m-s\right)\right)!\left(\frac{1}{2}\left(l+m-s\right)\right)!}\label{eq:Ewald in 3D long-range part 1D 2D-1} \end{multline} \end_inset where \begin_inset Formula $s_{\perp}\equiv\vect s\cdot\uvec z=\vect s_{\bot}\cdot\uvec z$ \end_inset . If \begin_inset Formula $d_{\Lambda}=1$ \end_inset , the angular becomes more complicated to evaluate due to the different behaviour of the \begin_inset Formula $\vect r_{\bot}\cdot\vect s_{\bot}/\left|\vect r_{\bot}\right|\left|\vect s_{\bot}\right|$ \end_inset factor. The choice of coordinates can make most of the terms dissapear: if the lattice is set parallel to the \begin_inset Formula $z$ \end_inset axis, \begin_inset Formula $A_{l',l,m',m,j}^{\left(1\right)}$ \end_inset is zero unless \begin_inset Formula $m=0$ \end_inset , but one still has \begin_inset Formula \[ A_{l',l,m',0,j}^{\left(1\right)}=\pi\delta_{m',l'-l-2j}\lambda'_{l0}\lambda_{l'm'}\int_{-1}^{1}\ud x\,P_{l'}^{m'}\left(x\right)P_{l}^{0}\left(x\right)\left(1-x^{2}\right)^{\frac{l'-l}{2}} \] \end_inset where \begin_inset Formula $\lambda_{lm}$ \end_inset are constants depending on the conventions for spherical harmonics. This does not seem to have such a nice closed-form expression as in the 2D case, but it can be evaluated e.g. using the common recurrence relations for associated Legendre polynomials. Of course when \begin_inset Formula $\vect s=0$ \end_inset , one gets relatively nice closed expressions, such as those in \begin_inset CommandInset citation LatexCommand cite key "linton_lattice_2010" literal "false" \end_inset . \end_layout \begin_layout Standard \lang english \begin_inset CommandInset bibtex LatexCommand bibtex btprint "btPrintCited" bibfiles "Tmatrix" options "siam" encoding "default" \end_inset \end_layout \end_body \end_document