# A unifying polarization formalism for electric- and magnetic-multipole interactions

R Casini, R. Manso Sainz, A. Lopez Ariste, N Kaikati

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R Casini, R. Manso Sainz, A. Lopez Ariste, N Kaikati. A unifying polarization formalism for electric-and magnetic-multipole interactions. The Astrophysical Journal, 2025, 980 (1), pp.67. 10.3847/1538-4357/ad7677. hal-04701046

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# A Unifying Polarization Formalism for Electric and Magnetic Multipole Interactions

R. Casini¹, R. Manso Sainz², A. López Ariste³, and N. Kaikati³

¹NSF NCAR High Altitude Observatory, P.O. Box 3000, Boulder, CO 80307-3000, USA

²Third Institute of Physics, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

³IRAP, Université de Toulouse, CNRS, CNES, UPS, 14, Av. E. Belin, 31400 Toulouse, France

Received 2024 July 18; revised 2024 August 28; accepted 2024 August 31; published 2025 February 6

## Abstract

We extend the spherical tensorial formalism for polarization to the treatment of electric and magnetic multipole transitions of any order. We rely on the spherical-wave expansion to derive the tensor form of the operator describing the interaction of the atomic system with a polarized radiation field, which naturally leads to the introduction of spherical tensors describing the polarization properties of the interacting field. As a direct application, the formalism is used to model the radiation anisotropy affecting the scattering of radiation in an electric quadrupole transition and the associated Hanle effect in the presence of a magnetic field.

*Unified Astronomy Thesaurus concepts: Radiative transfer (1335)*

## 1. Introduction

The study of the polarization of the electromagnetic radiation we receive from astrophysical objects is at the basis of any diagnostics of nonisotropic physical processes in the emitting gas. Typical examples are the presence of deterministic magnetic and electric fields that break the symmetry of the atom-photon interaction process and nonisotropic excitation—both radiative and collisional—of plasma, which can lead to significant departures of the plasma conditions from local thermodynamic equilibrium (LTE). The importance of non-LTE conditions in astrophysical plasmas for the production and transfer of polarized radiation in spectral lines has long been recognized. For example, a comprehensive understanding of non-LTE effects is critical to our ability to model polarized signals in the solar spectrum and how these are modified by the presence of magnetic and electric fields in the solar atmosphere. This is at the basis of most solar magnetism diagnostics (e.g., R. Casini & E. Landi Degl'Innocenti 2008).

The largest majority of spectral lines from neutral and ionized atomic species observed in the solar atmosphere and other astrophysical plasmas are permitted transitions, which originate in the interaction of the radiation field with the induced electric dipole of the optical electron in the ion. For this reason, the modeling of these lines has received most of the attention of the heliophysics community since the very inception of solar magnetism investigations (e.g., G. E. Hale 1908). As these studies even predate the development of quantum mechanics, the modeling of polarized line formation was originally based on Lorentz's classical theory of the bound optical electron as a damped, three-dimensional oscillator forced by the radiation field (H. A. Lorentz 1952) and how the spectral shape and polarization of the radiated energy are affected by the presence of a magnetic field via the Zeeman effect. The complexity of atomic spectra observed in both laboratory gases and astrophysical plasmas was largely responsible for the development of the quantum mechanical description of atomic transitions (e.g., A. Sommerfeld 1923) and how magnetic and electric fields modify the absorbed and reemitted radiation (see, e.g., H. A. Bethe & E. E. Salpeter 1957).

However, along with the prevalent permitted lines, forbidden transitions are also often observed in astrophysical contexts. In fact, they were discovered in the spectra of planetary nebulae, the solar corona, and also in Earth's auroras, even before being observed in laboratory plasmas (I. S. Bowen 1936; S. Pasternack 1940; R. H. Garstang 1968; M. Eidelsberg et al. 1981). This was possible because, in the typical rarefied state of those environments, the upper levels of transitions with relatively low oscillator strengths can often survive long enough for spontaneous radiative decay to occur. The ensuing lines are thus optically thin and play an important role for diagnosing chemical abundances, temperature, density, and velocity fields in various astrophysical contexts, including the Sun (G. A. Doschek et al. 1975; G. D. Sandlin et al. 1977; C. C. Cheng et al. 1979), young stars (I. Appenzeller et al. 1984; F. Hamann 1994; M. Corcoran & T. P. Ray 1997), stellar winds (P. N. Safier 1993; M. N. Simon et al. 2016), nebulae (M. J. Seaton 1953; Y. G. Tsamis et al. 2003), and active galactic nuclei (AGNs; A. Pedlar et al. 1985). In environments with intense nearby radiation sources, such as the solar corona, gaseous nebulae, and AGNs, the incoming radiation may be anisotropically scattered, producing distinctive line polarization signatures that can probe the geometry of unresolved plasma structures, large-scale mass motions, and magnetic fields via the Hanle effect (H.-W. Lee 1994; P. G. Judge 1998; P. G. Judge et al. 2001; R. Casini et al. 2017). As an example, magnetic dipole emission lines from highly ionized atomic species are regularly observed in the solar corona (P. G. Judge 1998) and commonly used for magnetism diagnostics (P. Charvin 1965; L. L. House 1974; S. Sahal-Bréchot 1977; R. Casini & P. G. Judge 1999; H. Li et al. 2017). However, the contribution of electric quadrupole and higher multipoles, although less significant, can be important in certain cases of astrophysical interest (M. J. Seaton 1955; R. H. Garstang 1957, 1962a; P. Quinet et al. 1996). Some electric quadrupole lines have also been predicted in the relatively denser environment of the solar photosphere (R. H. Garstang 1962b), although detection conditions are exceedingly difficult in that case. Another notable example is represented by the transitions among the lowest energy levels of O I, O II, and O III, which are exclusively of the magnetic dipole and electric quadrupole types. These atomic species are important tracers of the dynamics of H II star-forming regions (e.g., B. Balick et al. 1974; S. N. Wilson et al. 2024) and have formerly

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Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further
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been used (J. L. Leroy & J. F. Le Borgne 1987) or suggested (H. Yan & A. Lazarian 2006, 2008, 2012) as valuable tools for the determination of magnetic fields in these regions. However, the importance of the contribution of the electric quadrupole (E2) transitions of those systems to the final observed transition has escaped attention. Hα linear spectropolarimetry has been a common diagnostic tool in Herbig Ae/Be and T Tauri stars (J. S. Vink et al. 2002, 2005). The very same atomic lines of the O I, O II, and O III systems are visible in these objects, although linear polarimetry of these lines is scarce (K. M. Ababakr et al. 2016). Future polarimetry of these objects, often associated with the presence of disks, requires a correct understanding of the polarization of E2 lines. In order to exploit the diagnostic potential of these forbidden transitions, a unified theoretical approach becomes necessary to model self-consistently the various multipolar contributions to the scattered radiation.

The application of quantum mechanical methods to the description of polarized line formation in permitted transitions is nowadays well established (F. K. Lamb & D. Ter Haar 1971; E. Landi Degl'Innocenti 1983; J. O. Stenflo 1994; E. Landi Degl'Innocenti & M. Landolfi 2004). In the context of astrophysical plasmas, the complex nonlocality of the radiation transfer problem in an optically thick medium represents a difficult numerical challenge. This is the well-known problem of non-LTE, which in its most benign form leads to the departure of the atomic-level populations from the Maxwell-Boltzmann statistical distribution in response to the out-of-equilibrium, ambient radiation field, but which in its most general form also leads to the manifestation of quantum interference and population imbalances among the magnetic levels, producing a complex variety of polarization signatures. While in the first case a description of the magnetic effects with the interacting atom-photon system can be limited to the Zeeman and the (incomplete) Paschen-Back effects, the most general case—often dubbed the “non-LTE problem of the 2nd kind” (E. Landi Degl'Innocenti 1983)—fully manifests the quantum mechanical nature of the problem, requiring a complete grasp of the density matrix description of the partially polarized statistical ensemble of interacting atoms and photons.

The standard formalism that describes this problem in its generality, with applications to the study of astrophysical plasmas, is exhaustively presented in the monograph by Landi Degl'Innocenti & Landolfi (2004, hereafter LL04), with the notable physical restriction that the frequency of the incident photon is assumed to be completely redistributed (CRD hypothesis) during the interaction process. Through this assumption, the scattering of radiation can be described as the incoherent succession of single-photon absorption and reemission (e.g., J. J. Sakurai 1967). Such a formalism relies on the framework of the irreducible spherical tensors (U. Fano 1957), which allows one to describe the process of atom-photon interaction in the form of a multipolar expansion of products of tensorial components of the atomic density matrix (accounting for the excitation state of the atomic system) and the radiation field (describing the geometric distribution and polarization of the incident and scattered radiation). Such an approach was ushered into the field of magnetism diagnostics of astrophysical objects in the 1970s and early 1980s (F. K. Lamb & D. Ter Haar 1971; S. Sahal-Bréchot 1977; V. Bommier 1980; E. Landi Degl'Innocenti 1983), and it has now become mainstream in solar physics. This formalism provides a very elegant, compact, and physically insightful description of the polarized scattering of radiation by a magnetized plasma. LL04 gives a comprehensive description of the formalism and of many astrophysical applications, but it is predominantly focused on permitted, electric dipole transitions.<fnref n="4" />

In order to generalize the current CRD theory of polarized line formation to the treatment of electric and magnetic multipole radiative transitions of order *l* (for brevity, hereafter we refer to these simply as *E*/*l* and *M*/*l* transitions), it is necessary to extend such a tensorial formalism to atomic transitions of any multipolar order. In order to do so, we first summarize the method of spherical-wave expansion of the interaction Hamiltonian. While this derivation is not new, exhaustive presentations of the problem are scarce in the literature, and they consistently rely on personal choices of notation that make the comparison of results quite difficult. We hope to provide here a unifying formalism that is going to be more accessible to scientists working in the field of solar and stellar spectropolarimetry, being based on the standard notation and formalism of LL04. We show how this expansion naturally leads to the introduction of new generalized spherical tensors (in both reducible and irreducible forms) necessary to model the polarized line formation of multipolar transitions. We also rely on the monograph by Varshalovich et al. (1988, hereafter VMK88) for many definitions and on the exhaustive monograph by Shore & Menzel (1968, hereafter SM68) for the justification of several key approximations.

From the outset, we will limit our study to the so-called long-wavelength regime, $2\pi a_0/\lambda \ll 1$, where $a_0$ is Bohr's radius. This enables applications of the presented formalism for $\lambda \gtrsim 10$ nm, spanning most of the spectral range of interest for the magnetic diagnostics of solar and astrophysical plasmas.

## 2. General Formalism

The entire formalism for the modeling of *E*/*l* and *M*/*l* transitions is based on the derivation of the matrix element of the interaction Hamiltonian of the atom with a polarized radiation field. One can start from the expression of the interaction Hamiltonian operator given by LL04 (see Equation (6.18)) between a multielectron atom, with the *i*th optical electron of position $\mathbf{r}_i$ and momentum $\mathbf{p}_i$, and a generally polarized transverse field of angular frequency $\omega$, propagation vector $\mathbf{k} = (\omega/c)\hat{\mathbf{k}}$, and (circular) polarization state $\beta = \pm 1$ on the plane normal to $\hat{\mathbf{k}} \equiv (\vartheta, \varphi)$ (see Figure 1),

$$Q(\omega, \hat{\mathbf{k}}, \beta) = d_{\omega} \sum_{i} \exp(i \mathbf{k} \cdot \mathbf{r}_i) e_{\beta}(\hat{\mathbf{k}}) \cdot \mathbf{p}_i, (\beta = \pm 1) \quad (1)$$

<footnote n="4">⁴ Because of the applicability of the same tensorial formalism to the case of magnetic dipole transitions, LL04 also briefly consider the line formation problem of the corresponding forbidden transitions (see Section 6.8 of that monograph).</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Geometry describing radiation scattering in the presence of a magnetic field. All the relevant angles for the transformation between the solar frame $S \equiv (O; xyz)$ and the magnetic frame $B \equiv (O; x'y'z')$ are shown. The scattered radiation field is specified in the observer's frame $(\xi, \eta, \hat{k})$, where $\xi$ identifies the reference direction for linear polarization. We note that this geometry corresponds to the choice $\varphi = 0$ for the triplet $(\vartheta, \varphi, \gamma)$ needed to specify the polarization tensors $T_Q^{l:K}(i, \hat{k})_s$ in the solar frame (see Table 1), as the line of sight lies in the $x-z$ plane of that reference frame.</figcaption></figure>

where, using standard notation (we assume $e > 0$ throughout),

$$d_{\omega} \equiv \frac{e}{m_e} \left( \frac{2\pi\hbar}{\omega V} \right)^{\frac{1}{2}} \equiv \frac{e}{m_e} c_{\omega}. \qquad (2)$$

Here $V$ is the volume of the quantization box, which ultimately is replaced by the volumetric density of the emitter.

It is important to remark that expression (1) only considers the $A \cdot p$ part of the photon-atom interaction Hamiltonian. This is completely adequate to treat $E/l$ radiative transitions in the long-wavelength approximation (see discussions in LL04, after Equation (6.15), and in SM68, after Equation (10.(7.10))). On the other hand, in treating $M/l$ transitions, the interaction of the electron spin $S$ with the magnetic component of the radiation field is comparable to that of the electron's orbital angular momentum (similarly to the case of the Zeeman effect in a static magnetic field), and the interaction Hamiltonian must take into account such an additional contribution, which has the form $\hbar S \cdot \text{curl } A$ (see LL04, Equation (6.15); see also SM68, Equation (10.(4.16))).

In order to express the interaction Hamiltonian as a series of multipole orders beyond the electric dipole approximation, it is advantageous to exploit a spherical-wave expansion of the phase factor $\exp(ik \cdot r)$ in expression (1), as this leads to a unified formalism for treating both electric and magnetic multipoles of any order. This derivation is given in detail in Appendix C. On the other hand, if one is only interested in deriving the magnetic dipole (M1) and E2 contributions, a physically more insightful approach consists in decomposing the dyadic product $(k \cdot r)(e_\beta(\hat{k}) \cdot p)$, arising from the first-order Taylor expansion of the phase factor, as the sum of two contributions: a symmetric one, corresponding to the electric quadrupole term, and an antisymmetric one, corresponding to the magnetic dipole term (e.g., J. J. Sakurai 1967, LL04). Such a decomposition is readily achieved using the Binet-Cauchy identity, and it is presented in Appendix D for explicitly deriving the electric quadrupole term. The same decomposition approach, while in principle still possible, becomes exceedingly cumbersome and less intuitive for deriving the subsequent terms of the
multipolar expansion of the interaction Hamiltonian, hence the need to rely on a unified approach, which also naturally leads to the development of the tensorial formalism necessary to treat polarized line formation for multipolar transitions.

The main result of the spherical-wave expansion of the interaction Hamiltonian is represented by the following matrix elements for the electric and magnetic contributions to the interaction operator (1) describing the radiative transition between two atomic states $|m\rangle$ and $|n\rangle$, in the presence of a monochromatic radiation field of propagation vector $k$, and state of polarization $\beta = \pm 1$ (see Appendix C, Equations (C.1) and (C23)),

$$[Q(\omega, \hat{k}, \beta)_E]_{mn} \approx -i c_\omega \omega_{mn} \sum_K \frac{i^{K-1} k^{K-1}}{(2K-1)!!} \left(\frac{K+1}{2K}\right)^{\frac{1}{2}} \sum_Q (-1)^Q (\mathcal{Q}_{-Q}^K)_{mn} D_{\beta Q}^K(\hat{R}), \quad (3a)$$

$$[Q(\omega, \hat{k}, \beta)_M]_{mn} \approx c_\omega \omega_{mn} \beta \sum_K \frac{i^{K-1} k^{K-1}}{(2K-1)!!} \left(\frac{K+1}{2K}\right)^{\frac{1}{2}} \sum_Q (-1)^Q (\mathcal{M}_{-Q}^K)_{mn} D_{\beta Q}^K(\hat{R}), \quad (3b)$$

where $\hat{R} = \hat{R}_z(-\gamma)\hat{R}_y(-\vartheta)\hat{R}_z(-\varphi) \equiv (-\gamma, -\vartheta, -\varphi)$ is the rotation from the observer's frame to the solar frame (see Figure 1), and $\mathcal{Q}_Q^K$ and $\mathcal{M}_Q^K$ are the electric and magnetic multipole components of order $K$, expressed in terms of spherical harmonics $Y_Q^K$ (see Appendix C, Equations (C15) and (C24); see also SM68, Equations (10.(7.11)) and (10.(6.8a))),

$$\mathcal{Q}_Q^K = - \left( \frac{4\pi}{2K+1} \right)^{\frac{1}{2}} e \sum_i r_i^K Y_Q^K(\hat{r}_i), \quad (4a)$$

$$\mathcal{M}_Q^K = - \left( \frac{4\pi}{2K+1} \right)^{\frac{1}{2}} \mu_0 \sum_i \mathbf{grad}_i \{ r_i^K Y_Q^K(\hat{r}_i) \} \cdot \left( \frac{2\mathbf{L}_i}{K+1} + 2\mathbf{S}_i \right), \quad (4b)$$

where $\mu_0 = e\hbar/(2m_e c)$ is Bohr's magneton. We note how expressions (3a) and (3b) are formally identical, the only difference being the presence of an additional factor $i\beta$ in the magnetic contribution.

It is instructive to derive the expressions of these multipole operators to the lowest orders $l=1, 2$. For the electric multipole case, we immediately find from Equation (4a) the well-known expressions

$$\mathcal{Q}_Q^1 = -e \sum_i r_i(\hat{r}_i)_Q = -e r_Q, \quad (5a)$$

$$\mathcal{Q}_Q^2 = - \left( \frac{4\pi}{5} \right)^{\frac{1}{2}} e \sum_i r_i^2 Y_Q^2(\hat{r}_i). \quad (5b)$$

For calculating the magnetic multipole case, we can use Equation (5.8.3.(9)) of VMK88 to evaluate the $\mathbf{grad}_i\{\}$ factors in Equation (4b),

$$\mathbf{grad}\{r^K Y_Q^K(\hat{r}_i)\} = [K(2K+1)]^{\frac{1}{2}} r^{K-1} Y_{KQ}^{K-1}(\hat{r}_i),$$

where $Y_{KQ}^l$ are vector spherical harmonics (see Appendix C, Equation (C4) for the definition). We thus find

$$\mathcal{M}_Q^1 = -(4\pi)^{\frac{1}{2}} \mu_0 \sum_i Y_{1Q}^0(\hat{r}_i) \cdot (\mathbf{L}_i + 2\mathbf{S}_i) = -\mu_0 \sum_i e_Q^* \cdot (\mathbf{L}_i + 2\mathbf{S}_i) = -\mu_0 (\mathbf{L} + 2\mathbf{S})_Q, \quad (6a)$$

$$\begin{aligned} \mathcal{M}_Q^2 &= -\left(\frac{4\pi}{5}\right)^{\frac{1}{2}} \mu_0 \sum_i \sqrt{10} r_i Y_{2Q}^1(\hat{r}_i) \cdot \left(\frac{2}{3}\mathbf{L}_i + 2\mathbf{S}_i\right) = -(8\pi)^{\frac{1}{2}} \mu_0 \sum_i r_i \sum_{pq} (1p, 1q|2Q) Y_p^1(\hat{r}_i) \left(\frac{2}{3}\mathbf{L}_i + 2\mathbf{S}_i\right)_q \\ &= -\left(\frac{8}{3}\right)^{\frac{1}{2}} \mu_0 \sum_i r_i \left(\frac{4\pi}{3}\right)^{\frac{1}{2}} [Y^1(\hat{r}_i) \otimes (\mathbf{L}_i + 3\mathbf{S}_i)]_Q^2 \end{aligned} \quad (6b)$$

(see also SM68, Equation (10.(6.9b))).

## 2.1. Spherical Tensors for Multipolar Interactions

The expressions (3a) and (3b) show that the calculation of the matrix element of the multipolar operator $\mathcal{O}^l$ of order $l$, between two atomic states $|a\rangle$ and $|b\rangle$, generally implies the evaluation of products of the forms

$$\sum_q (-1)^q (\mathcal{O}_{-q}^l)_{ab} D_{\beta q}^l(\hat{R}), \quad i\beta \sum_q (-1)^q (\mathcal{O}_{-q}^l)_{ab} D_{\beta q}^l(\hat{R}).$$
The calculation of the corresponding atomic transition probabilities thus involves products of the forms

$$ \sum_{pq} (-1)^{p+q} (C_{-p}^l ab C_{-q}^l)^* D_{\alpha p}^l (\hat{R}) D_{\beta q}^l (\hat{R})^* = \sum_{pq} (-1)^{p+q} (C_{-p}^l ab C_{-q}^l)^* \mathcal{E}_{pq}^l (\alpha, \beta; \hat{k})_{\text{E}}, \\ \alpha \beta \sum_{pq} (-1)^{p+q} (C_{-p}^l ab C_{-q}^l)^* D_{\alpha p}^l (\hat{R}) D_{\beta q}^l (\hat{R})^* = \sum_{pq} (-1)^{p+q} (C_{-p}^l ab C_{-q}^l)^* \mathcal{E}_{pq}^l (\alpha, \beta; \hat{k})_{\text{M}}, $$

where we introduced the dyadic tensors for the electric and magnetic multipolar cases (see LL04, Equation (5.142)),

$$ \mathcal{E}_{pq}^{l}(\alpha, \beta; \hat{k})_{\text{E}} = D_{\alpha p}^{l}(\hat{R}) D_{\beta q}^{l}(\hat{R})^{*}, \qquad (7a) $$

$$ \mathcal{E}_{pq}^{l}(\alpha, \beta; \hat{k})_{\text{M}} = \alpha \beta D_{\alpha p}^{l}(\hat{R}) D_{\beta q}^{l}(\hat{R})^{*}. \qquad (7b) $$

Since $\alpha, \beta = \pm 1$, evidently

$$ \mathcal{E}_{pq}^{l}(\beta, \beta; \hat{k})_{\text{E}} \equiv \mathcal{E}_{pq}^{l}(\beta, \beta; \hat{k})_{\text{M}}. $$

These dyadic tensors satisfy conjugation and trace properties similar to those given by LL04 for the dipole case $l=1$. In particular, the conjugation relation Equation (5.145) remains identical, as well as the trace over the spherical-basis components $(p, q)$, Equation (5.144), since

$$ \sum_{q} \mathcal{E}_{qq}^{l}(\alpha, \beta; \hat{k})_{\text{M}} = \alpha \beta \delta_{\alpha}^{\beta} = \alpha^{2} \delta_{\alpha}^{\beta} = \delta_{\alpha}^{\beta} = \sum_{q} \mathcal{E}_{qq}^{l}(\alpha, \beta; \hat{k})_{\text{E}}. \qquad (8) $$

Instead, the trace over the radiation modes, Equation (5.143), generalizes to the following:

$$ \sum_{\beta=\pm 1} \oint \frac{d\hat{k}}{4\pi} \mathcal{E}_{pq}^{l}(\beta, \beta; \hat{k})_{\text{E,M}} = \frac{2}{2l+1} \delta_{p}^{q}. \qquad (9) $$

Both sum rules (8) and (9) follow directly from the orthogonality properties of the rotation matrices (e.g., VMK88, Equation (4.1. (6))). Analogously to Equation (5.146) of LL04, we can introduce a reducible set of polarization tensors

$$ T_{pq}^{l}(i, \hat{k})_{\text{E,M}} = \frac{1}{2} \sum_{\alpha\beta=\pm 1} (\sigma_{i})_{\alpha\beta} \mathcal{E}_{pq}^{l}(\beta, \alpha; \hat{k})_{\text{E,M}}, \qquad (10) $$

where $\sigma_i$, $i=0, 1, 2, 3$ are the Pauli matrices, with $\sigma_0$ the $2 \times 2$ identity. Again, we follow the convention adopted by LL04 for the definition of the Pauli matrices (see Equation (5.128)). The polarization tensors $T_{pq}^l$ satisfy the orthogonality property

$$ \sum_{pq} T_{pq}^{l}(i, \hat{k})_{\text{E,M}}^{*} T_{pq}^{l}(j, \hat{k})_{\text{E,M}} = \frac{1}{2} \delta_{j}^{i}, \qquad (11) $$

for all orders $l$. This follows immediately from definitions (7b) and trace property (8), for which

$$ \sum_{pq} \mathcal{E}_{pq}^{l}(\alpha, \beta; \hat{k})_{\text{E,M}}^{*} \mathcal{E}_{pq}^{l}(\gamma, \delta; \hat{k})_{\text{E,M}} = \sum_{pq} \mathcal{E}_{pp}^{l}(\alpha, \gamma; \hat{k})_{\text{E,M}}^{*} \mathcal{E}_{qq}^{l}(\beta, \delta; \hat{k})_{\text{E,M}} = \delta_{\alpha}^{\gamma} \delta_{\beta}^{\delta}, \quad \forall l \in \mathbb{N}, $$

and from the orthogonality of the Pauli matrices,

$$ \sum_{\alpha\beta=\pm 1} (\sigma_{i}^{*})_{\alpha\beta} (\sigma_{j})_{\alpha\beta} = 2 \delta_{i}^{j}. $$

Finally, through the polarization tensors $T_{pq}^l$ and the Stokes parameters $S_i$ of the radiation field, we can introduce the radiation tensors of rank $2l+1$ (see LL04, Equation (5.153); see also Appendix B)

$$ J_{pq}^{l}(\omega)_{\text{E,M}} = \sum_{i=0}^{3} \oint \frac{d\hat{k}}{4\pi} T_{pq}^{l}(i, \hat{k})_{\text{E,M}} S_{i}(\omega, \hat{k}). \qquad (12) $$

These reducible tensors generalize the usual ones of rank 3 that enter the expressions of the transition and relaxation rates for radiative processes in the statistical equilibrium of electric (and magnetic) dipole transitions (see LL04, Ch. 7).

The transformation of the reducible tensors $T_{pq}^l$ into their irreducible counterparts $T_Q^{l:K}$ is given by the usual relation (see LL04, Equation (5.124)), apart from a normalization factor that can be arbitrarily set; see discussions in VMK88, Section 3.1, or LL04, after Equation (2.79)),

$$ T_Q^{l:K} = \sum_{pq} (-1)^{l+p} [(2l+1)(2K+1)]^{\frac{1}{2}} \begin{pmatrix} l & l & K \\ p & -q & -Q \end{pmatrix} T_{pq}^{l}, \quad K = 0, \dots, 2l, \qquad (13) $$
together with the inverse relation

$$T_{pq}^{l} = \sum_{KQ} (-1)^{l+p} \left( \frac{2K+1}{2l+1} \right)^{\frac{1}{2}} \begin{pmatrix} l & l & K \\ p & -q & -Q \end{pmatrix} T_{Q}^{l:K}. \quad (14)$$

Our choice of normalization, slightly different from the usual one because of the presence of the $(2l+1)^{1/2}$ factor, has the purpose of making $T_0^{l:0}(0, \hat{k})_{E,M} = 1$ for all $l$. With definition (13), orthogonality relation (11) can be extended to the irreducible tensors as well. In fact, through simple Racah algebra, we find

$$\sum_{KQ} T_{Q}^{l:K}(i, \hat{k})_{E,M} T_{Q}^{l:K}(j, \hat{k})_{E,M} = \frac{2l+1}{2} \delta_{j}^{i}. \quad (15)$$

In strict analogy to Equation (13), we introduce the tensors

$$\mathcal{E}_{Q}^{l:K}(\alpha, \beta; \hat{\mathbf{k}})_{E,M} = \sum_{pq} (-1)^{l+p} [(2l+1)(2K+1)]^{\frac{1}{2}} \begin{pmatrix} l & l & K \\ p & -q & -Q \end{pmatrix} \mathcal{E}_{pq}^{l}(\alpha, \beta; \hat{\mathbf{k}})_{E,M}, \quad (16)$$

in terms of which, recalling Equation (10), we can write directly

$$T_{Q}^{l:K}(i, \hat{\mathbf{k}})_{E,M} = \frac{1}{2} \sum_{\alpha\beta=\pm 1} (\sigma_{i})_{\alpha\beta} \mathcal{E}_{Q}^{l:K}(\beta, \alpha; \hat{\mathbf{k}})_{E,M}. \quad (17)$$

The numerical implementation of Equation (16) is made easier if we rewrite it taking advantage of definitions (7b) and the summation properties of rotation matrices. In particular, for the case of $El$ transitions, we get

$$\begin{align}
\mathcal{E}_{Q}^{l:K}(\alpha, \beta; \hat{\mathbf{k}})_{E} &= (2l + 1)^{\frac{1}{2}} \sum_{pq} (-1)^{l+q} (lp, l-q|KQ) D_{\alpha p}^{l}(\hat{R}) D_{\beta q}^{l}(\hat{R})^{*} \nonumber \\
&= (2l + 1)^{\frac{1}{2}} \sum_{pq} (-1)^{l+q} (-1)^{\beta-q} (lp, l-q|KQ) D_{\alpha p}^{l}(\hat{R}) D_{-\beta-q}^{l}(\hat{R}) \nonumber \\
&= (-1)^{l+\beta} (2l + 1)^{\frac{1}{2}} (l\alpha, l-\beta|K\alpha - \beta) D_{\alpha-\beta Q}^{K}(\hat{R}) \nonumber \\
&\equiv (-1)^{l+\alpha} [(2l + 1)(2K + 1)]^{\frac{1}{2}} \begin{pmatrix} l & l & K \\ \alpha & -\beta & \beta-\alpha \end{pmatrix} D_{\alpha-\beta Q}^{K}(\hat{R}). \tag{18}
\end{align}$$

The case of $Ml$ transitions evidently yields exactly the same expression, apart from an additional factor $\alpha\beta$.

Finally, using Equation (17), we can introduce the irreducible form of the radiation tensors, in strict analogy with Equation (12),

$$J_{Q}^{l:K}(\omega)_{E,M} = \sum_{i=0}^{3} \oint \frac{d\hat{\mathbf{k}}}{4\pi} T_{Q}^{l:K}(i, \hat{\mathbf{k}})_{E,M} S_{i}(\omega, \hat{\mathbf{k}}). \quad (19)$$

This expression will be used in the following section to calculate examples of the E2 radiation tensors in the solar atmosphere.

We want to conclude this presentation of the polarization tensors by pointing out their relation between the electric and magnetic multipole cases. From Equations (10) and (13) and definitions (7b), it follows immediately that

$$T_{pq}^{l}(i, \hat{\mathbf{k}})_{M} = \frac{1}{2} \sum_{\alpha\beta=\pm 1} \alpha\beta (\sigma_{i})_{\alpha\beta} \mathcal{E}_{pq}^{l}(\beta, \alpha; \hat{\mathbf{k}})_{E},$$

$$T_{Q}^{l:K}(i, \hat{\mathbf{k}})_{M} = \frac{1}{2} \sum_{\alpha\beta=\pm 1} \alpha\beta (\sigma_{i})_{\alpha\beta} \mathcal{E}_{Q}^{l:K}(\beta, \alpha; \hat{\mathbf{k}})_{E}.$$

The matrices $\alpha\beta (\sigma_i)_{\alpha\beta}$ are identical to the Pauli matrices for $i=0,3$, since they are purely diagonal (hence $\alpha\beta = 1$), but they change sign for $i=1,2$, since the Pauli matrices are purely off-diagonal in those cases (hence $\alpha\beta = -1$). Thus, the following general relations are true for every multipole order $l$ (see LL04, Section 6.8):

$$T_{pq}^{l}(i, \hat{\mathbf{k}})_{M} = T_{pq}^{l}(i, \hat{\mathbf{k}})_{E}, \quad T_{Q}^{l:K}(i, \hat{\mathbf{k}})_{M} = T_{Q}^{l:K}(i, \hat{\mathbf{k}})_{E}, \quad i = 0, 3. \quad (20a)$$

$$T_{pq}^{l}(i, \hat{\mathbf{k}})_{M} = -T_{pq}^{l}(i, \hat{\mathbf{k}})_{E}, \quad T_{Q}^{l:K}(i, \hat{\mathbf{k}})_{M} = -T_{Q}^{l:K}(i, \hat{\mathbf{k}})_{E}, \quad i = 1, 2. \quad (20b)$$

Table 1 gives the expressions of $T_{Q}^{2:K}(i, \hat{k})$ for the E2 case. The expressions for the electric dipole (E1) case are tabulated by LL04 (Table 5.6).

At a glance, the comparison between the dipolar and quadrupolar tensor components in Table 1 shows that, apart from a numerical factor, the shared multipoles $K=0, 1, 2$ take exactly the same geometric form. In contrast, the quadrupolar tensors carry a more
<table><caption>**Table 1**
The Irreducible Spherical Polarization Tensors $T_Q^{2:K}(i, \hat{k})$, with $i = 0, 1, 2, 3$ Standing for the Four Stokes Parameters $I, Q, U, V$</caption><tbody><tr><td>$T_0^{2:0}(0, \hat{k}) = 1$</td><td>$T_0^{2:2}(1, \hat{k}) = -\frac{3}{2}\sqrt{\frac{5}{14}} \cos 2\gamma \sin^2\vartheta$</td></tr><tr><td>$T_0^{2:2}(0, \hat{k}) = -\frac{1}{2}\sqrt{\frac{5}{14}} (3 \cos^2\vartheta - 1)$</td><td>$T_1^{2:2}(1, \hat{k}) = -\frac{1}{2}\sqrt{\frac{15}{7}} (\cos 2\gamma \cos\vartheta + i \sin 2\gamma) \sin\vartheta e^{i\varphi}$</td></tr><tr><td>$T_1^{2:2}(0, \hat{k}) = \frac{1}{2}\sqrt{\frac{15}{7}} \cos\vartheta \sin\vartheta e^{i\varphi}$</td><td>$T_2^{2:2}(1, \hat{k}) = -\frac{1}{4}\sqrt{\frac{15}{7}} [\cos 2\gamma (1 + \cos^2\vartheta) + i 2 \sin 2\gamma \cos\vartheta] e^{i2\varphi}$</td></tr><tr><td>$T_2^{2:2}(0, \hat{k}) = -\frac{1}{4}\sqrt{\frac{15}{7}} \sin^2\vartheta e^{i2\varphi}$</td><td>$T_0^{2:4}(1, \hat{k}) = \frac{5}{2\sqrt{14}} \cos 2\gamma (7 \cos^2\vartheta - 1) \sin^2\vartheta$</td></tr><tr><td>$T_0^{2:4}(0, \hat{k}) = \frac{1}{2\sqrt{14}} [5 \sin^2\vartheta (7 \cos^2\vartheta + 1) - 8]$</td><td>$T_1^{2:4}(1, \hat{k}) = -\frac{1}{2}\sqrt{\frac{5}{14}} [\cos 2\gamma \cos\vartheta (7 \sin^2\vartheta - 3) - i \sin 2\gamma (7 \cos^2\vartheta - 1)] \sin\vartheta e^{i\varphi}$</td></tr><tr><td>$T_1^{2:4}(0, \hat{k}) = \sqrt{\frac{5}{14}} (7 \cos^2\vartheta - 3) \cos\vartheta \sin\vartheta e^{i\varphi}$</td><td>$T_2^{2:4}(1, \hat{k}) = -\frac{1}{2}\sqrt{\frac{5}{7}} \{\cos 2\gamma [\cos^2\vartheta (7 \sin^2\vartheta - 1) - i \sin 2\gamma \cos\vartheta (7 \sin^2\vartheta - 2)] \sin\vartheta e^{i\varphi}$</td></tr><tr><td>$T_2^{2:4}(0, \hat{k}) = -\frac{1}{2}\sqrt{\frac{5}{7}} (7 \cos^2\vartheta - 1) \sin^2\vartheta e^{i2\varphi}$</td><td>$T_3^{2:4}(1, \hat{k}) = -\frac{1}{2}\sqrt{\frac{5}{2}} [2 \cos 2\gamma \cos^3\vartheta + i \sin 2\gamma (3 \cos^2\vartheta - 1)] \sin\vartheta e^{i3\varphi}$</td></tr><tr><td>$T_3^{2:4}(0, \hat{k}) = \sqrt{\frac{5}{2}} \cos\vartheta \sin^3\vartheta e^{i3\varphi}$</td><td>$T_4^{2:4}(1, \hat{k}) = \frac{\sqrt{5}}{4} [\cos 2\gamma (1 + \cos^2\vartheta) + i 2 \sin 2\gamma \cos\vartheta] \sin^2\vartheta e^{i4\varphi}$</td></tr><tr><td>$T_4^{2:4}(0, \hat{k}) = -\frac{\sqrt{5}}{4} \sin^4\vartheta e^{i4\varphi}$</td><td>$T_0^{2:4}(2, \hat{k}) = T_0^{2:K}(1, \hat{k}) \{\sin 2\gamma \rightarrow \cos 2\gamma, \cos 2\gamma \rightarrow -\sin 2\gamma\}$</td></tr><tr><td>$T_0^{2:1}(3, \hat{k}) = \frac{1}{\sqrt{2}} \cos\vartheta$</td><td></td></tr><tr><td>$T_1^{2:1}(3, \hat{k}) = -\frac{1}{2} \sin\vartheta e^{i\varphi}$</td><td></td></tr><tr><td>$T_0^{2:3}(3, \hat{k}) = -\frac{1}{2\sqrt{2}} (5 \cos 2\vartheta - 1) \cos\vartheta$</td><td></td></tr><tr><td>$T_1^{2:3}(3, \hat{k}) = \frac{1}{4}\sqrt{\frac{3}{2}} (5 \cos 2\vartheta + 3) \sin\vartheta e^{i\varphi}$</td><td></td></tr><tr><td>$T_2^{2:3}(3, \hat{k}) = -\frac{\sqrt{15}}{2} \cos\vartheta \sin^2\vartheta e^{i2\varphi}$</td><td></td></tr><tr><td>$T_3^{2:3}(3, \hat{k}) = \frac{1}{2}\sqrt{\frac{5}{2}} \sin^3\vartheta e^{i3\varphi}$</td><td></td></tr></tbody></table>

Note. This table only lists the nonvanishing components with $Q \ge 0$ for E2 transitions. The components with $Q < 0$ are derived according to $T_Q^{2:K}(i, \hat{k}) = (-1)^Q T_Q^{2:K}(i, \hat{k})^*$. The case of M2 transitions follows from relations (20b). Here $\hat{k} \equiv (\vartheta, \varphi)$ is the propagation direction, while $\gamma$ specifies the reference direction of linear polarization (see Figure 1).

convoluted geometric dependence for the new multipoles $K = 3, 4$, which will manifest itself through new signatures of scattering polarization (see, e.g., Figure 4 below).

## 2.2. Application: Spontaneous Emission Rates

As a straightforward application of the derived formalism, we evaluate the relaxation rate for spontaneous emission (s.e.) from an excited state $m$ toward all lower states $n \prec m$ radiatively bound to $m$. This rate is derived from first principles (e.g., LL04, from Equation (6.48) onward; see also E. Landi Degl'Innocenti 1983) and, using the notation of this paper, is given by

$$R(m)_{\text{s.e.}} = \frac{1}{\hbar^2} \sum_{n \prec m} \sum_k |Q(\omega, \hat{k}, \beta)_{mn}|^2 [\Phi_{nm}(\omega) + \Phi_{nm}^*(\omega)], \quad (21)$$

where $\Phi_{nm}(\omega) = [\epsilon_m + \epsilon_n - i (\omega_{mn} - \omega)]^{-1}$ is the spectral profile of the transition,$^5$ corresponding to the natural line width due to the inverse lifetimes $\epsilon_m$ and $\epsilon_n$ of the transition levels (see R. Casini et al. 2014, Equation (34)). Here $Q(\omega, \hat{k}, \beta)_{mn}$ is one of the matrix elements (3a) or (3b). Further, the sum over all possible radiation modes $k$ implies the following transition to the continuum:$^6$

$$\sum_k \rightarrow \frac{1}{2\pi^2 c^3} \int_0^\infty d\omega \, \omega^2 \oint \frac{d\hat{k}}{4\pi} \sum_\beta \quad (22)$$

For an electric or magnetic multipole transition of order $l$, after substitution of either one of Equation (3a) or Equation (3b) into Equation (21) and using property (9), we find

$$R^l(m)_{\text{s.e.}} = \frac{2}{[(2l-1)!!]^2} \frac{l+1}{2l} \frac{2}{2l+1} \frac{\omega^{2l+1}}{\hbar c^{2l+1}} \sum_{n<m} \sum_q |\langle m | C_q^l | n \rangle|^2, \quad (23)$$

<footnote>$^5$ The line profile arises from the time integral in the master equation of the interacting system (e.g., LL04, Ch. 6, p. 255).</footnote>

<footnote>$^6$ We note that this yields a logarithmically divergent integral of the line profile, which can be treated using standard renormalization techniques, ultimately leading to $(1/\pi) \int_0^\infty d\omega \, \omega [\Phi_{nm}(\omega) + \Phi_{nm}^*(\omega)] \approx 2 \omega_{mn} \approx 2 \omega$.</footnote>
where $C_q^l = \{\mathcal{Q}_q^l, \mathcal{M}_q^l\}$. Specifically for dipolar, quadrupolar, and octupolar interactions, we find, respectively,

$$R^1(m)_{\text{s.e.}} = \frac{4}{3} \frac{\omega^3}{\hbar c^3} \sum_{n<m} \sum_q |\langle m | C_q^1 | n \rangle|^2,$$

$$R^2(m)_{\text{s.e.}} = \frac{1}{15} \frac{\omega^5}{\hbar c^5} \sum_{n<m} \sum_q |\langle m | C_q^2 | n \rangle|^2,$$

$$R^3(m)_{\text{s.e.}} = \frac{8}{4725} \frac{\omega^7}{\hbar c^7} \sum_{n<m} \sum_q |\langle m | C_q^3 | n \rangle|^2,$$

in agreement with the results of SM68, Section 10.6–7.

### 3. Hanle Effect of Electric Quadrupole Transitions in a Two-level Atom

To demonstrate the formalism developed in Section 2, we model the Hanle effect of the scattering polarization from an E2 transition in the presence of a magnetic field.

We consider a collisionless plasma of two-level atoms, illuminated by an unpolarized radiation field with cylindrical symmetry around the local vertical through the scattering point (Figure 1). We assume that we can neglect the effects of stimulated emission, as well as the polarization of the lower level $J_l$. Under these conditions, we can adapt expression (10.27) of LL04 for the density matrix of the upper level $J_u$, expressed in the magnetic reference frame, B, having the z-axis (quantization axis) along the B vector (z'-axis in Figure 1). Thus, we can write for the normalized density matrix tensor $\rho_Q^K(J_u) = \rho_Q^K(J_u)/\rho_0^K(J_l)$,

$$\rho_Q^K(J_u) = \left(\frac{2J_l + 1}{2J_u + 1}\right)^{\frac{1}{2}} \frac{B_{J_l J_u}}{A_{J_u J_l} + i Q g_{J_u} \omega_B} w_{J_u J_l}^{(2:K)} (-1)^Q J_{-Q}^{2:K}(\omega_{ul})_B, \quad K = 0, \dots, 2J_u, \quad (24)$$

where $A_{J_u J_l}$ and $B_{J_l J_u}$ are, respectively, the Einstein coefficients for spontaneous emission and absorption of the E2 transition, $\omega_B = eB/(2m_e c)$ is the Larmor frequency corresponding to the field strength B, $g_{J_u}$ is the Landé factor of the excited level, and finally we introduced the polarizability factor of the E2 transition

$$w_{J_u J_l}^{(2:K)} = \left\{ \begin{matrix} 2 & 2 & K \\ J_u & J_u & J_l \end{matrix} \right\} / \left\{ \begin{matrix} 2 & 2 & 0 \\ J_u & J_u & J_l \end{matrix} \right\}, \quad K = 0, \dots, 4. \quad (25)$$

We note that the nonnullity condition of the polarizability factor automatically restricts the highest possible multipole order K in Equation (24) to min{4, $2J_u$}.

Because the incident radiation field is assumed to be unpolarized and cylindrically symmetric around the local vertical, the radiation tensors in the B-frame entering Equation (24) are simply given by

$$J_Q^{2:K}(\omega_{ul})_B = D_{0Q}^K(\hat{R}_{BS}) J_0^{2:K}(\omega_{ul})_S, \quad (26)$$

where $\hat{R}_{BS}$ is the rotation operator that transforms the solar frame, S (z-axis along the local vertical; see Figure 1), into the B-frame. In addition, because of the assumed symmetry, all odd multipole orders of the radiation field are identically zero (see Equation (19) and Table 1). This property extends directly to the density matrix solution, as seen from Equation (24).

Density matrix solution (24) can be used to derive the spectrally integrated, fractional linear$^7$ polarization of the emission line, $p_i(\hat{k}) = \varepsilon_i(\hat{k})/\varepsilon_0(\hat{k})$, for $i=1, 2$. Using Equation (10.31) of LL04 and Equations (24) and (26) above, we can write, for $i=0, 1, 2$,

$$\begin{align}
\varepsilon_i(\hat{k}) &\propto \sum_{KQ} w_{J_u J_l}^{(2:K)} \mathcal{T}_Q^{2:K}(i, \hat{k})_B \frac{\rho_Q^K(J_u)}{\rho_0^K(J_u)} \nonumber \\
&\propto \sum_{KQ} [w_{J_u J_l}^{(2:K)}]^2 \mathcal{T}_Q^{2:K}(i, \hat{k})_B \frac{(-1)^Q D_{0-Q}^K(\hat{R}_{BS})}{1 + i Q g_{J_u} \omega_B / A_{J_u J_l}} \frac{J_0^{2:K}(\omega_{ul})_S}{J_0^{2:0}(\omega_{ul})_S} \nonumber \\
&\propto \sum_K [w_{J_u J_l}^{(2:K)}]^2 \frac{J_0^{2:K}(\omega_{ul})_S}{J_0^{2:0}(\omega_{ul})_S} \sum_P \mathcal{T}_P^{2:K}(i, \hat{k})_S \sum_Q \frac{(-1)^Q D_{PQ}^K(\hat{R}_{BS}) D_{0-Q}^K(\hat{R}_{BS})}{1 + i Q g_{J_u} \omega_B / A_{J_u J_l}}, \tag{27}
\end{align}$$

where in the last line we used

$$\mathcal{T}_Q^{2:K}(i, \hat{k})_B = \sum_P D_{PQ}^K(\hat{R}_{BS}) \mathcal{T}_P^{2:K}(i, \hat{k})_S. \quad (28)$$

If we indicate with $\hat{b} \equiv (\vartheta_B, \varphi_B)$ the direction of the magnetic field in the solar frame, we may choose the magnetic frame $B \equiv (O; x'y'z')$ such that the transformation $\hat{R}_{BS}$ consists only of a rotation by $\varphi_B$ around the local vertical, followed by a rotation by

<footnote>$^7$ The spectrally integrated circular polarization is always vanishing under the conditions of this example.</footnote>
$\vartheta_B$ around the transformed y-axis (see Figure 1). We then have

$$(-1)^Q D_{0-Q}^K(\hat{R}_{BS}) = d_{0Q}^K(\vartheta_B), \quad D_{PQ}^K(\hat{R}_{BS}) = \exp(-i P \varphi_B) d_{PQ}^K(\vartheta_B),$$

where $d_{PQ}^K(\beta)$ is the reduced rotation matrix (VMK88, Section 4.3). Thus, Equation (27) becomes explicitly

$$\varepsilon_i(\hat{k}) \propto \sum_K [w_{J_u J_l}^{(2:K)}]^2 \frac{J_0^{2:K}(\omega_{ul})_S}{J_0^{2:0}(\omega_{ul})_S} \sum_P \mathcal{T}_P^{2:K}(i, \hat{k})_S \exp(-i P \varphi_B) \sum_Q \frac{d_{PQ}^K(\vartheta_B) d_{0Q}^K(\vartheta_B)}{1 + i Q g_{J_u} \omega_B / A_{J_u J_l}}, \quad i = 0, 1, 2. \quad (29)$$

A typical system described by such a model (apart from the additional assumption of negligible stimulated emission) is the transition $(J_l, J_u) \equiv (0, 2)$, for which

$$w_{20}^{(2:2)} = w_{20}^{(2:4)} = 1. \quad (30)$$

For a transition $(J_l, J_u) \equiv (1/2, 5/2)$, which also has a nonpolarizable lower level by radiation anisotropy, we have instead

$$w_{\frac{5}{2} \frac{5}{2}}^{(2:2)} = 2/\sqrt{5}, \quad w_{\frac{5}{2} \frac{5}{2}}^{(2:4)} = 1/\sqrt{3}. \quad (31)$$

Both types of transitions require the calculation of the radiation anisotropies in the solar frame for the orders $K=2,4$, which we undertake in the next section (see Equation (39) below).

### 3.1. Radiation Anisotropy Tensors

Here we explicitly compute the radiation tensors $J_Q^{1:K}(\omega)_S$ generalized to the multipole order $K=4$, which are necessary to describe the scattering polarization in the E2 transition $(J_l, J_u) \equiv (0, 2)$. Because we are considering the case of unpolarized incident radiation, we remark that the following derivation applies identically to both E2 and M2 cases.

While the incident radiation is further assumed to be cylindrically symmetric around the local vertical to the solar surface, we are taking into account the presence of a wavelength-dependent center-to-limb variation (CLV), represented through the usual polynomial expression of C. W. Allen (1973),

$$I(\lambda, \mu) = I_0(\lambda)[1 - u(\lambda)(1 - \mu) - v(\lambda)(1 - \mu^2)], \quad \mu = \cos \vartheta, \quad (32)$$

where $\vartheta$ is the angle between the line of sight and the local vertical through the observed point (see Figure 1). Following Landi Degl'Innocenti (1982; see also LL04, Section 12.3), the irreducible radiation tensors relevant to dipolar transitions can be written in the form

$$J_0^{1:0} = \frac{1}{2} I_0 (a_1 + a_2 u + a_3 v), \quad (33a)$$

$$J_0^{1:2} = \frac{1}{4\sqrt{2}} I_0 (b_1 + b_2 u + b_3 v), \quad (33b)$$

and are obtained by integrating Equation (32) within the radiation cone between 0 (disk center) and $\vartheta_0$ (limb edge for a given height $h$, defined to be 1 at the solar surface; see Figure 1), weighted by the proper polarization tensors $T_0^{1:K}$ for the intensity (see Equation (13)). We find (see E. Landi Degl'Innocenti 1982)

$$a_1 = 1 - \mu_0, \quad (34a)$$

$$a_2 = -a_1 + \frac{1}{2} \left( 1 + \frac{\mu_0^2 l_0}{\sqrt{1 - \mu_0^2}} \right), \quad (34b)$$

$$a_3 = \frac{1}{3} \left( \mu_0 - \frac{2}{1 + \mu_0} \right), \quad (34c)$$

and

$$b_1 = \mu_0(1 - \mu_0^2), \quad (35a)$$

$$b_2 = -b_1 - \frac{1}{8} \left[ (3\mu_0^2 - 4) \left( 1 - \frac{\mu_0^2 l_0}{\sqrt{1 - \mu_0^2}} \right) + 2 \right], \quad (35b)$$

$$b_3 = \frac{1}{15} \left[ \mu_0 (9\mu_0^2 - 11) + \frac{4}{1 + \mu_0} \right], \quad (35c)$$
where we defined

$$ \mu_0 = \sqrt{1 - \frac{1}{h^2}}, \quad l_0 = \ln \frac{\mu_0}{1 + \sqrt{1 - \mu_0^2}} = \frac{1}{2} \ln \frac{h-1}{h+1}. \qquad (36) $$

We can proceed similarly to derive the radiation tensors relevant to E2 transitions. Because of the relations between the corresponding polarization tensors,

$$ \mathcal{T}_{0}^{2:0}(i, \hat{k}) = \mathcal{T}_{0}^{1:0}(i, \hat{k}) = \delta_{i0}, \quad \mathcal{T}_{0}^{2:2}(i, \hat{k}) = -\frac{\sqrt{5}}{7} \mathcal{T}_{0}^{1:2}(i, \hat{k}), \quad i = 0, 1, 2, 3, $$

we choose to define

$$ J_0^{2:0} = \frac{1}{2} I_0 (A_1 + A_2 u + A_3 v), \qquad (37a) $$

$$ J_0^{2:2} = -\frac{1}{4} \sqrt{\frac{5}{14}} I_0 (B_1 + B_2 u + B_3 v), \qquad (37b) $$

$$ J_0^{2:4} = -\frac{1}{4\sqrt{14}} I_0 (C_1 + C_2 u + C_3 v), \qquad (37c) $$

so we have at once

$$ (A_1, A_2, A_3) \equiv (a_1, a_2, a_3), \quad (B_1, B_2, B_3) \equiv (b_1, b_2, b_3). $$

Thus, we must only calculate ($C_1, C_2, C_3$) anew, for which we find

$$ C_1 = \mu_0(1 - \mu_0^2)(7\mu_0^2 - 3), \qquad (38a) $$

$$ C_2 = \left(7\mu_0^5 - 10\mu_0^3 - \frac{35}{24}\mu_0^2 + 3\mu_0 + \frac{4}{3}\right) - \frac{1}{16}\left(35\mu_0^4 - 60\mu_0^2 + 24\right)\left(1 - \frac{\mu_0^2 l_0}{\sqrt{1 - \mu_0^2}}\right), \qquad (38b) $$

$$ C_3 = -\mu_0(1 - \mu_0^2)(5\mu_0^2 - 3). \qquad (38c) $$

We choose to define the first- and second-order anisotropies for the quadrupolar case as follows:

$$ W_1 = \sqrt{\frac{14}{5}} \frac{J_0^{2:2}}{J_0^{2:0}}, \quad W_2 = \frac{\sqrt{14}}{4} \frac{J_0^{2:4}}{J_0^{2:0}}, \qquad (39) $$

both of which have a lower asymptotic bound at $-1$ for $\mu_0 \to 1$. In particular, with these definitions, $W_1$ happens to be identical in magnitude but opposite in sign to the usual radiation anisotropy for the dipolar case. This is demonstrated by the $W_1$ plots shown in Figures 2 and 3. Consequently, we can anticipate that the linear polarization arising from an E2 transition will have the opposite sign with respect to an E1 transition formed under the same physical conditions, similarly to what happens in the case of M1 transitions (see, e.g., R. Casini & P. G. Judge 1999, LL04). However, while in M1 transitions the polarization signatures only undergo a sign change with respect to their E1 counterparts, in the case of E2 transitions, new polarization patterns will arise as a consequence of the line polarizability $W_2$ shown in Figure 3. This is clearly demonstrated by the examples discussed in the next section (see Figure 4).

## 3.2. Scattering Polarization and Hanle Effect of the 2–0 Atom

As an application of these results, we model the scattering of polarized radiation in the E2 transition ($J_l, J_u \equiv (0, 2)$). We assume for simplicity a height $h = 1$ ($\mu_0 = 0$) over the solar surface, so the only radiation anisotropy comes from the CLV of the solar atmosphere. Because we assume an unpolarized radiation with cylindrical symmetry around the local vertical, all odd multipole orders of the density matrix solution vanish identically, so the corresponding summation in Equation (29) is limited to $K = 0, 2, 4$. Additionally, we assume the scattering direction in the local vertical frame of reference (z-axis along the local vertical, and x–z plane coinciding with the Sun’s central meridian) to be $\hat{k} \equiv (\vartheta, \varphi) \equiv (90^\circ, 0^\circ)$. Finally, we assume the tangent to the limb as the reference direction for positive linear polarization ($\gamma = 90^\circ$). The tensors $T_p^{2:K}(i, \hat{k})_S$ in Equation (29) can then be calculated from Table 1.
<figure><img src="image_4.png" /><figcaption>Figure 2. Plots of the $J_0^{1:0}(\lambda)$ radiation tensor (left) and radiation anisotropy $W_1 = \sqrt{2} J_0^{1:2}(\lambda)/J_0^{1:0}(\lambda)$ (right), between 200 nm and 10 µm, for $h = 1 R_⊙$ (solar surface), adopting the CLV data of C. W. Allen (1973). These are the radiation tensors necessary for the modeling of polarized line formation in dipolar transition systems.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3. Plots of the radiation anisotropies $W_1$ (left) and $W_2$ (right), as defined by Equation (39). Atmospheric conditions and wavelength range are the same as for Figure 2. These radiation tensors need to be included, along with the dipolar tensors (noting the condition $J_0^{2:0} \equiv J_0^{1:0}$), in order to model the scattering polarization from E2 transitions in multilevel atomic systems where the atomic levels are also populated via E1 transitions.</figcaption></figure>

With the above assumptions, we also have

$$ (A_1, A_2, A_3) \equiv \left(1, -\frac{1}{2}, -\frac{2}{3}\right), $$

$$ (B_1, B_2, B_3) \equiv \left(0, \frac{1}{4}, \frac{4}{15}\right), $$

$$ (C_1, C_2, C_3) \equiv \left(0, -\frac{1}{6}, 0\right), $$

and assuming a wavelength of 550 nm, for which $u \approx 0.93$ and $v \approx -0.23$ according to C. W. Allen (1973), we derive

$$ \frac{J_0^{2:2}}{J_0^{2:0}} \approx -0.0743, \quad \frac{J_0^{2:4}}{J_0^{2:0}} \approx 0.0301, $$

or alternatively (see Figure 3),

$$ W_1 \approx -0.1243, \quad W_2 \approx 0.0281. $$

Figure 4 shows the Hanle polarization diagrams calculated using Equation (29), with $p_Q = p_1(\hat{k})$ and $p_U = p_2(\hat{k})$. We display both forms of these diagrams as parametric curves ($p_Q, p_U$) (left panels) and ($P, \chi$) (right panels), where $P = (p_Q^2 + p_U^2)^{1/2}$ is the linear polarization degree and $\chi = (1/2)\arctan(p_U/p_Q)$ gives the “azimuth” in the plane of sky of the linear polarization direction. Two different inclinations of the magnetic field from the local vertical, $\vartheta_B = 30^\circ, 90^\circ$, are shown. The 12 colored curves in each plot correspond to different field azimuths $\varphi_B$ in the interval $[-180^\circ, 180^\circ)$ in steps of $30^\circ$. The points on each curve correspond to different field strengths, such that $\omega_B/A_{J_u J_f}$ varies in the interval $[0.1, 10]$ in a decimal-log scale. The series of diagrams at the top are
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Hanle diagrams for the 90° scattering of resonant radiation in the E2 transition ($J_l, J_u \equiv (0, 2)$ (top row) and in the E1 transition ($J_l, J_u \equiv (0, 1)$ (bottom row). We considered two inclinations of the magnetic field from the local vertical, $\vartheta_B = 30^\circ$ (left column) and $\vartheta_B = 90^\circ$ (right column). For each of the four plots, the Hanle diagrams are given as parametric curves in both forms of ($p_Q, p_U$) (left panels) and ($P, \chi$) (right panels), where $P = (p_Q^2 + p_U^2)^{1/2}$ and $\chi = (1/2)\arctan(p_U/p_Q)$ (see text). The 12 colored curves are for different field azimuths $\varphi_B \in [-180^\circ, 180^\circ)$ in steps of 30°, and on each curve the symbols mark the field strength $\omega_B \in [0.1, 10]$ $A_{J_u J_l}$ in a decimal-log scale (21 points). The anisotropy conditions of the incident radiation are given in the text.</figcaption></figure>

for the E2 transition ($J_l, J_u \equiv (0, 2)$ discussed in this application. For comparison, the series of diagrams at the bottom show the corresponding diagrams for the E1 transition ($J_l, J_u \equiv (0, 1)$, under exactly the same physical conditions.

Apart from the obvious difference in the sign of the scattering polarization across the Hanle sensitivity field regime—a distinguishing characteristic also shared with M1 transitions (see Equation (20b))—the diagrams of Figure 4 demonstrate a higher degree of degeneracy of the E2 polarization signature to the magnetic field geometry, under similar observing conditions and signal-to-noise ratios, especially for the frequently occurring case of near-right-angle scattering (panels to the right). A full exploration of the diagnostic potential and limitations of these transitions is therefore necessary, especially on how they can best be utilized along with permitted transitions, which goes beyond the scope of the formalism description of this work.

4. Conclusions

The extension to quadrupolar (E2) radiative transitions of the CRD theory of polarized line formation including nonisotropic excitation mechanisms (as presented in LL04) is an important addition to the theoretical tools necessary to model resonance scattering polarization in complex atoms. Such an extension becomes straightforward when we adopt the generalized tensorial formalism presented here. The first key ingredient for such a generalization is the adoption of the irreducible spherical representation of the tensorial quantities that describe the excitation state of the atomic system (density matrix) and the interacting radiation field (polarized radiation tensors), as presented by LL04. The second one, enabling this generalization to be extended to multipole terms of arbitrary order, is the use of the spherical-wave expansion of the interaction Hamiltonian. This formalism can be utilized, for example, to extend the theoretical framework of the polarized line formation in complex atoms with both fine and hyperfine structures, presented by R. Casini & R. Manso Sainz (2005), to include E2 transitions as well. Such an extension will be the subject of a future paper.

While the scattering polarization of M1 transitions is a well-established diagnostic of the solar corona (see cited references in the Introduction), E2 transitions in the solar spectrum are elusive, and their study may be presumed to be irrelevant for diagnostic purposes. The situation is dramatically different in colder and rarefied gas clouds such as the circumstellar environment of T Tauri stars, or in H II regions. In these astrophysical objects, the excitation state of relatively abundant atomic species, such as O, N, and S, strongly departs from LTE, being completely dominated by the ambient (and naturally anisotropic) radiation field from nearby stellar sources. Because collisional processes are essentially negligible, forbidden radiative transitions—mostly of the M1 and E2 types—stand out in the spectra emitted by these objects. Thus, any polarimetric study aimed at diagnosing their magnetic properties, as well as investigating possibly unresolved geometric structures (e.g., anisotropic stellar winds), necessitates a proper modeling of the multipolar interactions, which ultimately must be based on a self-consistent formalism such as the one presented in this work. However, even in solar plasmas, forbidden transitions can play a role in the statistical equilibrium of atomic species that dominate the polarized solar spectrum, e.g., by relaxing the metastability of low-energy levels (e.g., R. Manso Sainz & J. Trujillo Bueno 2003).

The simple examples of scattering polarization signals from E2 transitions presented in this work reveal striking differences from the much more familiar signatures from permitted transitions. These can be regarded as new diagnostic opportunities, but they also come with potential limitations that need to be thoroughly investigated. We hope that this introductory work can open a new
diagnostic window relying on the use of the Hanle effect for the investigation of weakly magnetized plasmas in all branches of astrophysics where forbidden transitions are routinely observed.

Multipolar transitions are also relevant to the laboratory investigation of the quantum theory of the orbital angular momentum of light (e.g., S. M. Barnett et al. 2022, and references therein), particularly with regard to how the additional unit of momentum contributed by it needs to be taken into account in considering the parity and angular momentum conservation laws of atom-photon interactions. We leave to future work the study of how the formalism presented here could be directly adapted or expanded to the treatment of those interactions where both photon spin and orbital angular momentum must play a role.

Acknowledgments

The authors are deeply indebted to the anonymous referee, who painstakingly checked many if not all of the calculations, spotted several typos, and urged us to clarify various obscure points, significantly improving the overall presentation and the potential impact of this work. Thank you!

This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement No. 1852977.

Appendix A
Spherical Bases

Following the definitions of VMK88, the spherical representation of a vector $v = \sum_q v_q e_q$, $q = -1, 0, 1$, adopted by LL04 corresponds to using covariant coordinates $v_q$ and contravariant basis vectors $e_q$, with $e_0 \equiv e_z$. We then have

$$v = \sum_q v_q e_q = \sum_q (-1)^q v_q e_{-q}^* = \sum_q (-1)^q v_{-q} e_q^*.$$

Given a different spherical basis $\{e_\alpha\}$, $\alpha = -1, 0, 1$, we have (see LL04, Equation (5.116))

$$v \cdot e_{\alpha} = \sum_{q} (-1)^{q} v_{-q} (e_{q}^{*} \cdot e_{\alpha}) = \sum_{q} (-1)^{q} v_{-q} D_{\alpha q}^{1}(\hat{R}), \quad (A1)$$

where $D_{QQ'}^{K}(\hat{R})$ are rotation matrices (VMK88, Section 4.1), and $\hat{R}$ is the rotation operator that transforms the new basis $\{e_\alpha\}$ into the old basis $\{e_q\}$ (or the vector coordinates from the old basis to the new basis). Thus, if $(\varphi, \vartheta, \gamma)$ are the Euler angles of the rotation that carries the old basis into the new one, $\hat{R} = \hat{R}_z(-\gamma)\hat{R}_y(-\vartheta)\hat{R}_z(-\varphi) \equiv (-\gamma, -\vartheta, -\varphi)$.

Appendix B
Stokes Parameters

The connection between the second quantization of the radiation field and the classical coherency matrix (e.g., L. Mandel & E. Wolf 1995) is provided by the following trace operator involving the creation and annihilation operators of the various radiation modes (see LL04, Ch. 4 and 6):

$$I_{\alpha\beta}(\omega, \hat{k}; t) = \frac{1}{8\pi^3} \frac{\hbar\omega^3}{c^2} \mathrm{Tr}\{\rho(t) a^{\dagger}(\omega, \hat{k}, \alpha) a(\omega, \hat{k}, \beta)\}, \quad (B1)$$

where $\rho(t)$ is the density matrix of the coupled atom-photon system in the interaction picture. The time dependence of these ensemble-average quantities is in the coarse-grained sense (F. K. Lamb & D. Ter Haar 1971), and therefore it can be ignored in the limit of statistical equilibrium of the coupled system.

From the coherency matrix, we can introduce irreducible spherical radiation tensors

$$\mathcal{I}_{Q}^{l:K}(\omega, \hat{k})_{\mathrm{E},\mathrm{M}} = \sum_{\alpha\beta=\pm 1} I_{\alpha\beta}(\omega, \hat{k}) \mathcal{E}_{Q}^{l:K}(\beta, \alpha; \hat{k})_{\mathrm{E},\mathrm{M}}, \quad (B2)$$

as well as the Stokes parameters (see LL04, Equation (5.129))

$$S_i(\omega, \hat{k}) = \sum_{\alpha\beta=\pm 1} (\sigma_i)_{\alpha\beta} I_{\beta\alpha}(\omega, \hat{k}), \quad (B3)$$

$$I_{\alpha\beta}(\omega, \hat{k}) = \frac{1}{2} \sum_{i=0}^{3} (\sigma_i)_{\alpha\beta} S_i(\omega, \hat{k}), \quad (B4)$$

where $\sigma_i$, $i = 0, 1, 2, 3$ are the Pauli matrices (see Section 2.1). Inserting Equation (B4) into Equation (B2) and recalling Equation (17), we find (see LL04, Equation (5.150))

$$\mathcal{I}_{Q}^{l:K}(\omega, \hat{k})_{\mathrm{E},\mathrm{M}} = \sum_{i=0}^{3} \mathcal{T}_{Q}^{l:K}(i, \hat{k})_{\mathrm{E},\mathrm{M}} S_i(\omega, \hat{k}), \quad (B5)$$
which can be inverted using orthogonality property (15),

$$S_i(\omega, \hat{k}) = \frac{2}{2l+1} \sum_{KQ} \mathcal{T}_{Q}^{l:K}(i, \hat{k})_{\text{E,M}}^* \mathcal{I}_{Q}^{l:K}(\omega, \hat{k})_{\text{E,M}} \quad (B6)$$

From Equation (B5), we see at once that the tensors $J_Q^{l:K}(\omega)$ of Equation (19) are simply the angular average of the radiation tensors $\mathcal{I}_Q^{l:K}(\omega, \hat{k})$, i.e.,

$$J_Q^{l:K}(\omega)_{\text{E,M}} = \oint \frac{d\hat{k}}{4\pi} \mathcal{I}_Q^{l:K}(i, \hat{k})_{\text{E,M}} \quad (B7)$$

## Appendix C
Spherical-wave Expansion of the Interaction Hamiltonian

In order to develop operator (1) as a series of multipole orders, we rely on the expansion of the exponential factor in terms of spherical harmonics and spherical Bessel functions of the first kind (e.g., VMK88, Equations (5.17.(14)) and (5.16.(10))),

$$\exp(i \mathbf{k} \cdot \mathbf{r}) = 4\pi \sum_{l=0}^{\infty} i^l j_l(kr) \sum_m Y_m^l(\hat{k})^* Y_m^l(\hat{r}), \quad (C1)$$

and on the spherical-basis representation of a vector,

$$\mathbf{e}_{\beta}(\hat{\mathbf{k}}) = \sum_{q} D_{\beta q}^{1}(\hat{R}) \mathbf{e}_{q} = \sum_{q} (-1)^{q} D_{\beta -q}^{1}(\hat{R}) \mathbf{e}_{q}^{*}, \quad (C2)$$

where $\hat{R} \equiv (-\gamma, -\vartheta, -\varphi)$ is the rotation operator that transforms the helicity basis $\{\mathbf{e}_{\beta}(\hat{\mathbf{k}})\}$, with $\mathbf{e}_{0}(\hat{\mathbf{k}}) = \hat{\mathbf{k}}$, into the spherical basis $\{\mathbf{e}_{q}\}$, $q=0, \pm 1$ (see Appendix A, and after Equations (3a) and (3b)).<fnref n="8" /> Therefore,

$$\exp(i \mathbf{k} \cdot \mathbf{r}) \mathbf{e}_{\beta}(\hat{\mathbf{k}}) = 4\pi \sum_{l=0}^{\infty} i^l j_l(kr) \sum_{mq} (-1)^q Y_m^l(\hat{\mathbf{k}})^* Y_m^l(\hat{\mathbf{r}}) D_{\beta-q}^1(\hat{R}) \mathbf{e}_q^* \quad (\beta = \pm 1) \quad (C3)$$

This expression is dealt with more efficiently by using the algebra of spherical tensors (VMK88). First, we introduce the so-called vector spherical harmonics (VMK88, Section 7.3),

$$\mathbf{Y}_{KQ}^{l} \equiv [\mathbf{Y}^{l} \otimes \mathbf{e}^{*1}]_{Q}^{K} = \sum_{pq} (lp, 1q|KQ) \mathbf{Y}_{p}^{l} \mathbf{e}_{q}^{*} \quad (C4)$$

where $(a\alpha, b\beta|c\gamma)$ are Clebsch–Gordan (C–G) coefficients (VMK88, Section 8.1). Using the C–G orthogonality properties, we have

$$Y_m^l(\hat{\mathbf{r}}) \mathbf{e}_q^* = \sum_{KQ} (lm, 1q|KQ) Y_{KQ}^l(\hat{\mathbf{r}}).$$

Next, we note that (VMK88, Equation (5.2.7.(37)), along with the symmetry properties of the rotation matrices, Equation (4.4.(2)))

$$Y_m^l(\hat{\mathbf{k}})^* = (-1)^m Y_{-m}^l(\hat{\mathbf{k}}) = \left(\frac{2l+1}{4\pi}\right)^{\frac{1}{2}} (-1)^m D_{0m}^l(\gamma, \vartheta, \varphi) \equiv \left(\frac{2l+1}{4\pi}\right)^{\frac{1}{2}} (-1)^m D_{0-m}^l(\hat{R}).$$

Using these results, we can rewrite Equation (C3) as

$$\begin{align*}
\exp(i \mathbf{k} \cdot \mathbf{r}) \mathbf{e}_{\beta}(\hat{\mathbf{k}}) &= 4\pi \sum_{l=0}^{\infty} i^l (-1)^{l+1} j_l(kr) \left(\frac{2l+1}{4\pi}\right)^{\frac{1}{2}} \sum_{KQ} (-1)^{K+Q} \mathbf{Y}_{KQ}^{l}(\hat{\mathbf{r}}) \sum_{mq} (lm, 1q|K-Q) D_{0m}^{l}(\hat{R}) D_{\beta q}^{1}(\hat{R}) \\
&= 4\pi \sum_{l=0}^{\infty} i^l (-1)^{l+1} j_l(kr) \left(\frac{2l+1}{4\pi}\right)^{\frac{1}{2}} \sum_{KQ} (-1)^{K+Q} \mathbf{Y}_{KQ}^{l}(\hat{\mathbf{r}}) (l0, 1\beta|K\beta) D_{\beta-Q}^{K}(\hat{R}) \\
&= 4\pi \sum_{KQ} (-1)^{K+Q} D_{\beta-Q}^{K}(\hat{R}) \sum_{l=K-1}^{K+1} i^l (-1)^{l+1} j_l(kr) \left(\frac{2l+1}{4\pi}\right)^{\frac{1}{2}} (l0, 1\beta|K\beta) \mathbf{Y}_{KQ}^{l}(\hat{\mathbf{r}}), \quad (C5)
\end{align*}$$

where in the last equivalence we reordered the summations to emphasize that the multipole order $l$ is associated with the value of $K \ge 1$,<fnref n="9" /> and we also used the condition of nonnullity of the C–G coefficient to limit the summation over $l$ to just the nonvanishing

<footnote n="8">⁸ Since we adopt a contravariant spherical basis $\{\mathbf{e}_q\}$, the helicity basis defined by Equation (C2) is also contravariant (see VMK88, Equations (1.1.(53))). Note also that our helicity basis undergoes an additional rotation by $\gamma$ with respect to the one defined in VMK88; compare their Figure 1.2 with Figure 1 in this paper.</footnote>

<footnote n="9">⁹ Since $\beta = \pm 1$ for a transverse field, the possibility of $K=0$ is automatically excluded.</footnote>
contributions. For this last summation, explicitly using the fact that $\beta = \pm 1$ for a transverse field, we find

$$ 
\begin{aligned}
\mathbf{S}_{KQ} & \equiv \sum_{l=K-1}^{K+1} i^l (-1)^{K+l+1} j_l(kr) \left(\frac{2l+1}{4\pi}\right)^{\frac{1}{2}} (l0, 1\beta|K\beta) \mathbf{Y}_{KQ}^l(\hat{\mathbf{r}}) \\
& = i^{K-1} \left(\frac{2K+1}{8\pi}\right)^{\frac{1}{2}} \left\{ \left[ \left(\frac{K+1}{2K+1}\right)^{\frac{1}{2}} j_{K-1}(kr) \mathbf{Y}_{KQ}^{K-1}(\hat{\mathbf{r}}) - \left(\frac{K}{2K+1}\right)^{\frac{1}{2}} j_{K+1}(kr) \mathbf{Y}_{KQ}^{K+1}(\hat{\mathbf{r}}) \right] + i\beta j_K(kr) \mathbf{Y}_{KQ}^K(\hat{\mathbf{r}}) \right\} \\
& \equiv \mathbf{S}_{KQ}^{(\mathrm{E})} + \mathbf{S}_{KQ}^{(\mathrm{M})},
\end{aligned}
\quad (\mathrm{C}6) $$

where with the last equivalence we emphasized that the first contribution within square brackets is associated with the electric (E) multipoles of the atomic system, whereas the second one gives rise to magnetic (M) multipoles. These two contributions have different parity, and they will be considered separately.

### C.1. Electric Multipole Interaction

In the long-wavelength regime, $kr = (\omega/c)r \ll 1$, the following approximations of the spherical Bessel functions hold true:

$$ j_l(kr) \underset{kr \ll 1}{\approx} \frac{(kr)^l}{(2l+1)!!}, \quad j_l'(kr) \underset{kr \ll 1}{\approx} \frac{l}{kr} j_l(kr). \quad (\mathrm{C}7) $$

Then, for a given order $l$, the recursion formula,

$$ j_{l+1}(z) = \frac{l}{z} j_l(z) - j_l'(z), \quad (\mathrm{C}8) $$

shows that $j_{l+1}(z)/j_l(z) \to 0$ in this regime. This allows us to approximate $\mathbf{S}_{KQ}^{(\mathrm{E})}$ in Equation (C6) simply as

$$ \mathbf{S}_{KQ}^{(\mathrm{E})} \approx i^{K-1} \left(\frac{K+1}{8\pi}\right)^{\frac{1}{2}} j_{K-1}(kr) \mathbf{Y}_{KQ}^{K-1}(\hat{\mathbf{r}}) \approx i^{K-1} \left(\frac{K+1}{8\pi}\right)^{\frac{1}{2}} \left(\frac{2K+1}{K}\right)^{\frac{1}{2}} \frac{1}{k} \mathbf{grad}\{j_K(kr) \mathbf{Y}_Q^K(\hat{\mathbf{r}})\}, \quad (\mathrm{C}9) $$

where in the second line we used Equations (7.3.(57–58)) of VMK88, taking again into account the long-wavelength approximation. Thus, the electric multipole contribution to Equation (C5) for $\beta = \pm 1$ is given by

$$ [\exp(i\mathbf{k} \cdot \mathbf{r})e_{\beta}(\hat{\mathbf{k}})]_{\mathrm{E}} \approx (2\pi)^{\frac{1}{2}} \frac{1}{k} \sum_{KQ} i^{K-1} \left[ \frac{(K+1)(2K+1)}{K} \right]^{\frac{1}{2}} (-1)^{Q} D_{\beta-Q}^{K}(\hat{\mathbf{R}}) \mathbf{grad}\{j_{K}(kr) \mathbf{Y}_{Q}^{K}(\hat{\mathbf{r}})\}, \quad (\mathrm{C}10) $$

and the corresponding contribution to interaction operator (1) is

$$ Q(\omega, \hat{\mathbf{k}}, \beta)_{\mathrm{E}} \approx (2\pi)^{\frac{1}{2}} d_{\omega} \frac{1}{k} \sum_{KQ} i^{K-1} \left[ \frac{(K+1)(2K+1)}{K} \right]^{\frac{1}{2}} (-1)^{Q} D_{\beta-Q}^{K}(\hat{\mathbf{R}}) \sum_{i} \mathbf{grad}_{i} \{j_{K}(kr_{i}) \mathbf{Y}_{Q}^{K}(\hat{\mathbf{r}}_{i})\} \cdot \mathbf{p}_{i}. \quad (\mathrm{C}11) $$

In order to further develop Equation (C11), we introduce the quantum mechanical operators

$$ \mathbf{p} = -i\hbar \mathbf{grad}, \quad p^2 = -\hbar^2 \nabla^2, $$

together with the following vector-analysis identity:

$$ 2 \mathbf{grad} \mathcal{X} \cdot \mathbf{grad} \mathcal{Y} = \nabla^2(\mathcal{X}\mathcal{Y}) - \mathcal{X}\nabla^2\mathcal{Y} - \mathcal{Y}\nabla^2\mathcal{X}, \quad (\mathrm{C}12) $$

where $\mathcal{X}$ and $\mathcal{Y}$ are any two functions of $\mathbf{r}$. Letting $\mathcal{X} = F_Q^K \equiv j_K(kr) Y_Q^K(\hat{\mathbf{r}})$, we have

$$ \begin{aligned} i \frac{\hbar}{m_e} \mathbf{grad} F_Q^K \cdot \mathbf{p} \mathcal{Y} &= \frac{\hbar^2}{2m_e} \nabla^2 (F_Q^K \mathcal{Y}) - F_Q^K \frac{\hbar^2}{2m_e} \nabla^2 \mathcal{Y} - \mathcal{Y} \frac{\hbar^2}{2m_e} \nabla^2 F_Q^K \\ &\approx \frac{\hbar^2}{2m_e} \nabla^2 (F_Q^K \mathcal{Y}) - F_Q^K \frac{\hbar^2}{2m_e} \nabla^2 \mathcal{Y} \approx -[\mathcal{H}_A, F_Q^K] \mathcal{Y}, \end{aligned} $$

where, in the second line, we considered that $F_Q^K$ is harmonic in the long-wavelength approximation ($\nabla^2 F_Q^K \approx 0$) and also that $F_Q^K$ approximately commutes with the nonkinetic part of the atomic Hamiltonian.<fnref n="10" /> Thus,

$$ \mathbf{grad}\{j_K(kr) Y_Q^K(\hat{\mathbf{r}})\} \cdot \mathbf{p} \approx i \frac{m_e}{\hbar} [\mathcal{H}_A, j_K(kr) Y_Q^K(\hat{\mathbf{r}})], \quad (\mathrm{C}13) $$

<footnote n="10">¹⁰ This property is rigorously verified for the electrostatic potential, whereas it does not hold in the case of the (typically much smaller) spin-orbit correction, since the spherical harmonics are not eigenfunctions of the $L \cdot S$ operator (see also discussion in SM68, Section 10.7).</footnote>
and after substitution in Equation (C11) and some dimensional manipulation,

$$Q(\omega, \hat{k}, \beta)_E \approx (2\pi)^{\frac{1}{2}} c_\omega \frac{1}{k} \sum_{KQ} i^K \left[ \frac{(K+1)(2K+1)}{K} \right]^{\frac{1}{2}} (-1)^Q D_{\beta-Q}^K(\hat{R}) \frac{1}{\hbar} [\mathcal{H}_A, \sum_i e j_K(kr_i) Y_Q^K(\hat{r}_i)]. \quad (C14)$$

Ultimately, we are interested in evaluating the matrix element of Equation (C14) between two atomic states $|m\rangle$ and $|n\rangle$. Using again long-wavelength approximation (C7), we find

$$ \begin{aligned} [Q(\omega, \hat{k}, \beta)_E]_{mn} &\approx (2\pi)^{\frac{1}{2}} c_\omega \frac{\omega_{mn}}{k} \sum_{KQ} i^K \left[ \frac{(K+1)(2K+1)}{K} \right]^{\frac{1}{2}} (-1)^Q D_{\beta-Q}^K(\hat{R}) \langle m | \sum_i e \frac{(kr_i)^K}{(2K+1)!!} Y_Q^K(\hat{r}_i) | n \rangle \\ &= i c_\omega \omega_{mn} \sum_{KQ} \frac{i^{K-1} k^{K-1}}{(2K-1)!!} \left( \frac{K+1}{2K} \right)^{\frac{1}{2}} (-1)^Q D_{\beta-Q}^K(\hat{R}) \langle m | \frac{4\pi}{2K+1} \rangle^{\frac{1}{2}} e \sum_i r_i^K Y_Q^K(\hat{r}_i) | n \rangle \\ &= -i c_\omega \omega_{mn} \sum_K \frac{i^{K-1} k^{K-1}}{(2K-1)!!} \left( \frac{K+1}{2K} \right)^{\frac{1}{2}} \sum_Q (-1)^Q D_{\beta-Q}^K(\hat{R}) D_{\beta-Q}^K(\hat{R}), \quad (\beta = \pm 1) \end{aligned} $$

where in the last equivalence we introduced the electric multipole operator of order K,

$$ \mathcal{Q}^K = - \left( \frac{4\pi}{2K+1} \right)^{\frac{1}{2}} e \sum_i r_i^K Y^K(\hat{r}_i). \quad (C15) $$

In the special case of E1 transitions ($K=1$), Equation (C.1) gives at once

$$ [Q(\omega, \hat{k}, \beta)_{E1}]_{mn} = -i c_{\omega} \omega_{mn} \sum_{Q} (-1)^{Q} (\mathcal{Q}_{-Q})_{mn} D_{\beta Q}^{1}(\hat{R}). \quad (C16) $$

For E2 transitions ($K=2$), we find instead

$$ \begin{aligned} [Q(\omega, \hat{k}, \beta)_{E2}]_{mn} &= \frac{1}{2\sqrt{3}} c_{\omega} \omega_{mn} k \sum_{Q} (-1)^{Q} (\mathcal{Q}_{-Q})_{mn} D_{\beta Q}^{2}(\hat{R}) \\ &\approx \frac{1}{2\sqrt{3}} c_{\omega} \frac{\omega_{mn}^{2}}{c} \sum_{Q} (-1)^{Q} (\mathcal{Q}_{-Q})_{mn} D_{\beta Q}^{2}(\hat{R}), \end{aligned} \quad (C17) $$

where in the last approximation we considered that $kc = \omega \approx \omega_{mn}$ near the transition resonance.

## C.2. Magnetic Multipole Interaction

In a similar way, we can evaluate the contribution of magnetic multipoles to Equation (C5). This time, from Equation (C6), we have

$$ \begin{aligned} \boldsymbol{S}_{KQ}^{(M)} &= i^K \left( \frac{2K+1}{8\pi} \right)^{\frac{1}{2}} \beta j_K(kr) \mathbf{Y}_{KQ}^K(\hat{r}) = i^K \left( \frac{2K+1}{8\pi} \right)^{\frac{1}{2}} \frac{\beta}{\sqrt{K(K+1)}} \mathbf{L} \{ j_K(kr) Y_Q^K(\hat{r}) \} \\ &= i^{K-1} \left( \frac{2K+1}{8\pi} \right)^{\frac{1}{2}} \frac{\beta}{\sqrt{K(K+1)}} (\mathbf{r} \times \mathbf{grad}) \{ j_K(kr) Y_Q^K(\hat{r}) \}, \end{aligned} \quad (C18) $$

where we explicitly introduced the quantum operator $\mathbf{L}$ of the orbital angular momentum of the electron, $\hbar\mathbf{L} = \mathbf{r} \times \mathbf{p} = -i\hbar(\mathbf{r} \times \mathbf{grad})$, and we used the properties of the vector spherical harmonics, Equations (7.3.1.(6,9)) of VMK88. This leads to the following expression for the magnetic multipole contribution to interaction operator (1) for $\beta = \pm 1$:

$$ \begin{aligned} Q(\omega, \hat{k}, \beta)_M &= (2\pi)^{\frac{1}{2}} d_\omega \beta \sum_{KQ} i^{K-1} \left[ \frac{2K+1}{K(K+1)} \right]^{\frac{1}{2}} (-1)^Q D_{\beta-Q}^K(\hat{R}) \sum_i (\mathbf{r}_i \times \mathbf{grad}_i) \{ j_K(kr_i) Y_Q^K(\hat{r}_i) \} \cdot \mathbf{p}_i \\ &= - (2\pi)^{\frac{1}{2}} d_\omega \beta \sum_{KQ} i^{K-1} \left[ \frac{2K+1}{K(K+1)} \right]^{\frac{1}{2}} (-1)^Q D_{\beta-Q}^K(\hat{R}) \sum_i \mathbf{grad}_i \{ j_K(kr_i) Y_Q^K(\hat{r}_i) \} \cdot \hbar \mathbf{L}_i, \end{aligned} \quad (C19) $$

where in the second line we used the vector-analysis relation

$$ (\mathbf{r} \times \mathbf{grad}) F_Q^K \cdot \mathbf{p} = -\mathbf{grad} F_Q^K \cdot (\mathbf{r} \times \mathbf{p}) = -\mathbf{grad} F_Q^K \cdot \hbar \mathbf{L}. $$
In particular, using Equation (7.3.6.(58)) of VMK88, we have

$$ \mathbf{grad}\{j_K(kr) Y_Q^K(\hat{\mathbf{r}})\} = k \left(\frac{K}{2K+1}\right)^{\frac{1}{2}} j_{K-1}(kr) Y_{KQ}^{K-1}(\hat{\mathbf{r}}) + k \left(\frac{K+1}{2K+1}\right)^{\frac{1}{2}} j_{K+1}(kr) Y_{KQ}^{K+1}(\hat{\mathbf{r}}). \quad (C20) $$

We must now calculate the contribution to the interaction Hamiltonian associated with the electron spin that has so far been neglected, but which is essential for magnetic multipole transitions (see discussion in the second paragraph of Section 2). For this, we must evaluate **curl A**, which is accomplished by considering Equations (C5) and (C6),

$$ \exp(i \mathbf{k} \cdot \mathbf{r}) e_{\beta}(\hat{\mathbf{k}}) = 4\pi \sum_{KQ} (-1)^Q D_{\beta-Q}^{K}(\hat{\mathbf{R}}) \mathbf{S}_{KQ}, $$

and by using expressions (7.3.6.(60)) of VMK88 to reduce **curl S**<sub>KQ</sub> to its final form. Letting F<sub>KQ</sub><sup>l</sup> ≡ j<sub>l</sub>(kr) Y<sub>KQ</sub><sup>l</sup>(r̂), we have from Equation (C6)

$$ \begin{aligned} \mathbf{curl} \, \mathbf{S}_{KQ} &= \mathbf{curl}[\mathbf{S}_{KQ}^{(E)} + \mathbf{S}_{KQ}^{(M)}] \\ &= i^{K-1} \left(\frac{2K+1}{8\pi}\right)^{\frac{1}{2}} \left\{ \left[ \left(\frac{K+1}{2K+1}\right)^{\frac{1}{2}} \mathbf{curl} \, \mathbf{F}_{KQ}^{K-1} - \left(\frac{K}{2K+1}\right)^{\frac{1}{2}} \mathbf{curl} \, \mathbf{F}_{KQ}^{K+1} \right] + i\beta \, \mathbf{curl} \, \mathbf{F}_{KQ}^{K} \right\} \\ &= k i^{K-1} \left(\frac{2K+1}{8\pi}\right)^{\frac{1}{2}} \left\{ -i \frac{K+1}{2K+1} \mathbf{F}_{KQ}^{K} - i \frac{K}{2K+1} \mathbf{F}_{KQ}^{K} + \beta \left[ \left(\frac{K}{2K+1}\right)^{\frac{1}{2}} \mathbf{F}_{KQ}^{K+1} - \left(\frac{K+1}{2K+1}\right)^{\frac{1}{2}} \mathbf{F}_{KQ}^{K-1} \right] \right\} \\ &= k i^{K-1} \left(\frac{2K+1}{8\pi}\right)^{\frac{1}{2}} \left\{ -i \mathbf{F}_{KQ}^{K} + \beta \left[ \left(\frac{K}{2K+1}\right)^{\frac{1}{2}} \mathbf{F}_{KQ}^{K+1} - \left(\frac{K+1}{2K+1}\right)^{\frac{1}{2}} \mathbf{F}_{KQ}^{K-1} \right] \right\}. \end{aligned} \quad (C21) $$

A comparison of multipole orders between Equations (C20) and (C21) shows that for magnetic multipole transitions we must retain the terms within square brackets in the last line, whereas the first addendum -i F<sub>KQ</sub><sup>K</sup> provides a (typically negligible) spin correction to the electric multipole transitions (see discussion in SM68, after Equation (10.(7.10))). The contributing term is handled by noting that

$$ \begin{aligned} \mathbf{curl} \, \mathbf{S}_{KQ}^{(M)} &= \beta k i^{K-1} \left(\frac{2K+1}{8\pi}\right)^{\frac{1}{2}} \left[ \left(\frac{K}{2K+1}\right)^{\frac{1}{2}} \mathbf{F}_{KQ}^{K+1} - \left(\frac{K+1}{2K+1}\right)^{\frac{1}{2}} \mathbf{F}_{KQ}^{K-1} \right] \\ &= \beta k i^{K-1} \left[ \frac{1}{8\pi} \frac{2K+1}{K(K+1)} \right]^{\frac{1}{2}} \left[ K \left(\frac{K+1}{2K+1}\right)^{\frac{1}{2}} \mathbf{F}_{KQ}^{K+1} - (K+1) \left(\frac{K}{2K+1}\right)^{\frac{1}{2}} \mathbf{F}_{KQ}^{K-1} \right] \\ &= \beta k i^{K-1} \left[ \frac{1}{8\pi} \frac{2K+1}{K(K+1)} \right]^{\frac{1}{2}} \left[ \sqrt{(K+1)(2K+1)} \mathbf{F}_{KQ}^{K+1} - \frac{1}{k}(K+1) \mathbf{grad} \, \mathbf{F}_{Q}^{K} \right], \end{aligned} $$

where in the last equivalence we used Equation (C20) to eliminate F<sub>KQ</sub><sup>K-1</sup>. After taking the inner product of **curl S**<sub>KQ</sub><sup>(M)</sup> with ħS<sub>i</sub> and adding it to the orbital contribution of Equation (C19), as well as some simple transformations, we find for β = ±1

$$ \begin{aligned} Q(\omega, \hat{\mathbf{k}}, \beta)_M = -(2\pi)^{\frac{1}{2}} d_{\omega} \beta \sum_{KQ} i^{K-1} \left[ \frac{(2K+1)(K+1)}{K} \right]^{\frac{1}{2}} (-1)^Q D_{\beta-Q}^{K}(\hat{\mathbf{R}}) \\ \times \sum_i \left[ \mathbf{grad}_i \{j_K(kr_i) Y_Q^K(\hat{\mathbf{r}}_i)\} \cdot \left( \frac{\hbar \mathbf{L}_i}{K+1} + \hbar \mathbf{S}_i \right) - k \left( \frac{2K+1}{K+1} \right)^{\frac{1}{2}} j_{K+1}(kr_i) Y_{KQ}^{K+1}(\hat{\mathbf{r}}_i) \cdot \hbar \mathbf{S}_i \right]. \end{aligned} \quad (C22) $$

Since ultimately we are interested in the long-wavelength approximation, the second term in the curly brackets of Equation (C22) can be neglected with respect to the first term, similarly to the case of the electric multipole transitions (see Equation (C9)). Noting also that, in this approximation,

$$ \mathbf{grad}\{j_K(kr) Y_Q^K(\hat{\mathbf{r}})\} \approx \frac{k^K}{(2K+1)!!} \mathbf{grad}\{r^K Y_Q^K(\hat{\mathbf{r}})\}, $$
the matrix element of the magnetic multipole interaction operator between two atomic states $|m\rangle$ and $|n\rangle$ becomes

$$[Q(\omega, \hat{k}, \beta)_M]_{mn} \approx -(2\pi)^{\frac{1}{2}} d_\omega \beta \sum_{KQ} \frac{i^{K-1}k^K}{(2K-1)!!} \left[ \frac{K+1}{K(2K+1)} \right]^{\frac{1}{2}} (-1)^\Omega D_{\beta-Q}^K(\hat{R}) \\ \times \langle m | \sum_i \mathbf{grad}_i \{ r_i^K Y_Q^K(\hat{r}_i) \} \cdot \left( \frac{\hbar \mathbf{L}_i}{K+1} + \hbar \mathbf{S}_i \right) | n \rangle \\ \approx c_\omega \omega_{mn} \beta \sum_K \frac{i^{K-1}k^{K-1}}{(2K-1)!!} \left( \frac{K+1}{2K} \right)^{\frac{1}{2}} \sum_Q (-1)^\Omega (\mathcal{M}_{-Q}^K)_{mn} D_{\beta Q}^K(\hat{R}), \quad (\beta = \pm 1) \quad (C23)$$

where in the last equivalence we introduced the magnetic multipole operator of order K (see SM68, Equation (10.(6.8a))),

$$\mathcal{M}^K = -\left(\frac{4\pi}{2K+1}\right)^{\frac{1}{2}} \mu_0 \sum_i \mathbf{grad}_i \{r_i^K \mathbf{Y}^K(\hat{r}_i)\} \cdot \left(\frac{2\mathbf{L}_i}{K+1} + 2\mathbf{S}_i\right), \quad (C24)$$

and $\mu_0 = e\hbar/(2m_e c)$ is Bohr's magneton. We note how expressions (C23) and (C.1) are formally identical, the only difference being the presence of an additional factor $i\beta$ in the magnetic multipole case.

### C.3. Alternative Form of the Spherical-wave Expansion

VMK88 provides an expression for the operator of Equation (C3) fully in terms of vector spherical harmonics, which represents the starting point for the derivation of multipolar transition matrix elements as presented by SM68.

In order to derive this expression (VMK88, Equation (7.3.(131))), we consider again Equations (C1) and (C2), rewriting the latter in the explicit form

$$e_{\beta}(\hat{k}) = \sum_{q} (e_{q}^{*} \cdot e_{\beta}(\hat{k})) e_{q} \equiv \sum_{q} (e_{q} \cdot e_{\beta}(\hat{k})) e_{q}^{*},$$

where for the second equivalence we used $e_q^* = (-1)^q e_{-q}$. Then, recalling definition (C4) of the vector spherical harmonics, we find

$$\begin{aligned} \exp(i \mathbf{k} \cdot \mathbf{r}) e_{\beta}(\hat{\mathbf{k}}) &= 4\pi \sum_{l=0}^{\infty} i^l j_l(kr) \sum_{mq} (Y_m^l(\hat{\mathbf{k}})^* e_q) \cdot e_{\beta}(\hat{\mathbf{k}}) (Y_m^l(\hat{\mathbf{r}}) e_q^*) \\ &= 4\pi \sum_{l=0}^{\infty} i^l j_l(kr) \sum_{KQ, K'Q'} Y_{KQ}^l(\hat{\mathbf{k}})^* \cdot e_{\beta}(\hat{\mathbf{k}}) Y_{K'Q'}^l(\hat{\mathbf{r}}) \sum_{mq} (lm, 1q|KQ) (lm, 1q|K'Q') \\ &= 4\pi \sum_{l=0}^{\infty} i^l j_l(kr) \sum_{KQ} Y_{KQ}^l(\hat{\mathbf{k}})^* \cdot e_{\beta}(\hat{\mathbf{k}}) Y_{KQ}^l(\hat{\mathbf{r}}), \end{aligned}$$

which is the result given by VMK88. Comparison of the last equation with Equation (C5) implies

$$Y_{KQ}^{l}(\hat{\mathbf{k}})^{*} \cdot e_{\beta}(\hat{\mathbf{k}}) = (-1)^{K+Q+l+1} \left(\frac{2l+1}{4\pi}\right)^{\frac{1}{2}} (l0, 1\beta|K\beta) D_{\beta-Q}^{K}(\hat{R}). \quad (C25)$$

In order to prove this equivalence, we first recall that our helicity basis $\{e_\beta(\hat{\mathbf{k}})\}$ is contravariant. We then use Equation (7.3.(39)) of VMK88, together with the straightforward generalization of their Equation (7.3.(29)) to the case of $\gamma \neq 0$ (see footnote 5), to write

$$\begin{aligned} Y_{KQ}^l(\hat{\mathbf{k}})^* \cdot e_\beta(\hat{\mathbf{k}}) &= (-1)^{K+Q+l+1} Y_{K-Q}^l(\hat{\mathbf{k}}) \cdot e_\beta(\hat{\mathbf{k}}) \\ &= (-1)^{K+Q+l+1} \left(\frac{2l+1}{4\pi}\right)^{\frac{1}{2}} (l0, 1\beta|K\beta) D_{-\beta Q}^K(\gamma, \vartheta, \varphi). \end{aligned}$$

Since $\hat{R} \equiv (-\gamma, -\vartheta, -\varphi)$, use of the symmetry properties of the rotation matrices (e.g., VMK88, Equations (4.4.(2))) gives

$$D_{-\beta Q}^{K}(\gamma, \vartheta, \varphi) = D_{\beta-Q}^{K}(\hat{R}),$$

which completes the demonstration of Equation (C25).
## Appendix D
### Dyadic Form of the Electric Quadrupole Interaction Operator

The electric quadrupole contribution to the interaction Hamiltonian can also be derived starting from the “dyadic” form presented by LL04 (see Equation (6.38)) and other authors (e.g., J. J. Sakurai 1967):

$$
\begin{align*}
[Q(\omega, \hat{k}, \beta)_{\text{E2}}]_{mn} &= \frac{i}{2} \frac{e}{m_e} c_\omega \sum_i \langle m | [(\mathbf{k} \cdot \mathbf{r}_i)(\mathbf{e}_\beta(\hat{k}) \cdot \mathbf{p}_i) + (\mathbf{k} \cdot \mathbf{p}_i)(\mathbf{e}_\beta(\hat{k}) \cdot \mathbf{r}_i)] | n \rangle \\
&= \frac{i}{2} \frac{e}{m_e} c_\omega \sum_r \sum_i [(m | \mathbf{k} \cdot \mathbf{r}_i | r) \langle r | \mathbf{e}_\beta(\hat{k}) \cdot \mathbf{p}_i | n \rangle + (m | \mathbf{k} \cdot \mathbf{p}_i | r) \langle r | \mathbf{e}_\beta(\hat{k}) \cdot \mathbf{r}_i | n \rangle] \\
&= \frac{i}{2} \frac{e}{m_e} c_\omega k \sum_r \sum_i i \frac{m_e}{\hbar} [(m | \mathbf{e}_0(\hat{k}) \cdot \mathbf{r}_i | r) \langle r | [\mathcal{H}_A, \mathbf{e}_\beta(\hat{k}) \cdot \mathbf{r}_i] | n \rangle \\
&\quad + (m | [\mathcal{H}_A, \mathbf{e}_0(\hat{k}) \cdot \mathbf{r}_i] | r) \langle r | \mathbf{e}_\beta(\hat{k}) \cdot \mathbf{r}_i | n \rangle] \\
&= -\frac{e}{2} c_\omega k \sum_r \sum_i [(\omega_m + \omega_{mr}) (m | \mathbf{e}_0(\hat{k}) \cdot \mathbf{r}_i | r) \langle r | \mathbf{e}_\beta(\hat{k}) \cdot \mathbf{r}_i | n \rangle] \\
&= -\frac{e}{2} c_\omega \omega_{mn} k \sum_r \sum_i \langle m | \mathbf{e}_0(\hat{k}) \cdot \mathbf{r}_i | r \rangle \langle r | \mathbf{e}_\beta(\hat{k}) \cdot \mathbf{r}_i | n \rangle \\
&= -\frac{e}{2} c_\omega \omega_{mn} k \sum_i \langle m | (\mathbf{e}_0(\hat{k}) \cdot \mathbf{r}_i)(\mathbf{e}_\beta(\hat{k}) \cdot \mathbf{r}_i) | n \rangle, \tag{D1}
\end{align*}
$$

where we recalled that $\hat{k} = \mathbf{e}_0(\hat{k})$ and used the well-known relation between the electron momentum $\mathbf{p}$ and the commutator $[\mathcal{H}_A, \mathbf{r}]$ (e.g., J. J. Sakurai 1967, Equation (2.1.24)). Next, we observe that we can rewrite (see VMK88, Equation (1.2.1.(13)); see also Equation (A1))

$$ \mathbf{e}_{\alpha}(\hat{\mathbf{k}}) \cdot \mathbf{r} = \sum_{q} (-1)^q r_{-q} (\mathbf{e}_{\alpha}(\hat{\mathbf{k}}))_q = \left(\frac{4\pi}{3}\right)^{\frac{1}{2}} r \sum_{q} (-1)^q Y_{-q}^{1}(\hat{\mathbf{r}}) D_{\alpha q}^{1}(\hat{\mathbf{R}}), $$

through which Equation (D1) becomes

$$
\begin{align}
[Q(\omega, \hat{k}, \beta)_{\text{E2}}]_{mn} &= -\frac{e}{2} c_{\omega} \omega_{mn} k \sum_{i} \langle m | (\mathbf{e}_0(\hat{\mathbf{k}}) \cdot \mathbf{r}_i)(\mathbf{e}_{\beta}(\hat{\mathbf{k}}) \cdot \mathbf{r}_i) | n \rangle \nonumber \\
&= -\frac{4\pi}{3} \frac{e}{2} c_{\omega} \omega_{mn} k \sum_{i} \langle m | r_i^2 \sum_{qq'} (-1)^{q+q'} Y_{-q}^{1}(\hat{\mathbf{r}}_i) Y_{-q'}^{1}(\hat{\mathbf{r}}_i) D_{0q}^{1}(\hat{\mathbf{R}}) D_{\beta q'}^{1}(\hat{\mathbf{R}}) | n \rangle \nonumber \\
&\equiv -\frac{2\pi}{3} c_{\omega} \omega_{mn} k \sum_{i} \langle m | e r_i^2 C_i(\beta) | n \rangle, \tag{D2}
\end{align}
$$

where in the last equivalence we introduced the operator

$$ C_i(\beta) = \sum_{qq'} (-1)^{q+q'} Y_q^1(\hat{\mathbf{r}}_i) Y_{q'}^1(\hat{\mathbf{r}}_i) D_{0-q}^1(\hat{\mathbf{R}}) D_{\beta-q'}^1(\hat{\mathbf{R}}), $$

which we now evaluate. First, we note that (VMK88, Equation (3.1.(22)))

$$ Y_q^1(\hat{\mathbf{r}}_i) Y_{q'}^1(\hat{\mathbf{r}}_i) = \sum_{KQ} (1q, 1q'|KQ) [\mathbf{Y}^1 \otimes \mathbf{Y}^1]_Q^K(\hat{\mathbf{r}}_i) = \sum_{KQ} (-1)^K (1-q, 1-q'|K-Q) [\mathbf{Y}^1 \otimes \mathbf{Y}^1]_Q^K(\hat{\mathbf{r}}_i), $$

and hence

$$
\begin{align*}
C_i(\beta) &= \sum_{KQ} (-1)^{K+Q} [\mathbf{Y}^1 \otimes \mathbf{Y}^1]_Q^K(\hat{\mathbf{r}}_i) \sum_{qq'} (1q, 1q'|K-Q) D_{0q}^1(\hat{\mathbf{R}}) D_{\beta q'}^1(\hat{\mathbf{R}}) \\
&= \sum_{KQ} (-1)^{K+Q} [\mathbf{Y}^1 \otimes \mathbf{Y}^1]_Q^K(\hat{\mathbf{r}}_i) (10, 1\beta|K\beta) D_{\beta-Q}^K(\hat{\mathbf{R}}).
\end{align*}
$$

Because of the C–G coefficient, the condition $\beta = \pm 1$ for the polarization of a transverse field excludes the possibility $K=0$ (see footnote 6). In addition, by construction, the tensor $[\mathbf{Y}^1 \otimes \mathbf{Y}^1]_Q^K(\hat{\mathbf{r}}_i)$ vanishes identically for $K=1$, being the cross product of $\mathbf{Y}^1$ with itself (e.g., VMK88, Equation (3.1.(29))). Thus, the summation in $C_i(\beta)$ is restricted to just $K=2$. Noting finally that $(10, 1\beta|2\beta) = \sqrt{(4 - \beta^2)/6} = 1/\sqrt{2}$ for $\beta = \pm 1$, we find

$$ C_i(\beta) = \frac{1}{\sqrt{2}} \sum_{Q} (-1)^Q [\mathbf{Y}^1 \otimes \mathbf{Y}^1]_{-Q}^2(\hat{\mathbf{r}}_i) D_{\beta Q}^2(\hat{\mathbf{R}}), $$

which takes the form of Equation (C17) if we can identify the electric quadrupole operator $\mathcal{Q}^2$ with $\sum_i r_i^2 [\mathbf{Y}^1 \otimes \mathbf{Y}^1]^2(\hat{\mathbf{r}}_i)$. Indeed, this is a direct consequence of $[\mathbf{Y}^1 \otimes \mathbf{Y}^1]_Q^2$ being simply proportional to the spherical harmonic $Y_Q^2$ via a factor $\sqrt{3/(10\pi)}$ (e.g., VMK88,
Equation (5.6.2.(14)) and Table 8.11). Taking this into account, we can rewrite the last equation as

$$C_i(\beta) = \frac{1}{2} \left(\frac{3}{5\pi}\right)^{\frac{1}{2}} \sum_{Q} (-1)^Q Y_{-Q}^2(\hat{r}_i) D_{\beta Q}^2(\hat{R}),$$

and after substitution into Equation (D2),

$$[Q(\omega, \hat{k}, \beta)_{E2}]_{mn} = -\frac{1}{2\sqrt{3}} c_{\omega} \omega_{mn} k \sum_{Q} (-1)^Q \langle m | e \left(\frac{4\pi}{5}\right)^{\frac{1}{2}} \sum_{i} r_i^2 Y_{-Q}^2(\hat{r}_i) | n \rangle D_{\beta Q}^2(\hat{R}) \\ = \frac{1}{2\sqrt{3}} c_{\omega} \omega_{mn} k \sum_{Q} (-1)^Q (\mathcal{Q}_{-Q})_{mn} D_{\beta Q}^2(\hat{R}), \qquad (D3)$$

in agreement with Equation (C17).

## ORCID iDs

R. Casini https://orcid.org/0000-0001-6990-513X

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