**1. Introduction**

The photoionization/photodetachment of various neutral (*q* = 0) and charged (*q* = 0) fullerenes, C±*<sup>q</sup> N* , and their endohedral counterparts, *A*@C±*<sup>q</sup> N* (where *A* is the atom encapsulated inside C±*<sup>q</sup> N* cage), has been the subject of experimental (see, e.g., [1–7] and references therein) as well as intense systematic theoretical studies for many years now (see, e.g., a recent review paper [8] with an abundance of references therein). In particular, Professor M.Y. Amusia, to the legacy of whom this Special Issue of *Atoms* is devoted to, has contributed vastly to the study of the interaction of particles and light with fullerenes and endo-fullerenes, see, e.g., [9–18], to name a few.

Although the research on the interaction of fullerenes and endo-fullerenes with light has also touched upon the subject of fullerene anions (see, e.g., [1–4,8,9,19–24] and references therein), yet, to the best of the authors' knowledge, the subject of photodetachment of giant fullerenes anions [(<sup>C</sup>*N*)<sup>−</sup> with *N* 60] as well as of nested fullerene anions, (<sup>C</sup>*n*@C*m*>*n*@ ...)<sup>−</sup>, referred to as fullerene onion-anions in the present paper, has not been studied. Given the current strong interest in studying various elementary processes of basic importance involving fullerene formations, it is appealing to fill in this gap in the present state of knowledge. The present paper remedies the situation by presenting a first

**Citation:** Dolmatov, V.K.; Manson, S.T. A Glimpse into Photodetachment Spectra of Giant and Nested Fullerene Anions. *Atoms* **2022**, *10*, 99. https://doi.org/ 10.3390/atoms10040099

Academic Editor: Eugene T. Kennedy

Received: 24 August 2022 Accepted: 19 September 2022 Published: 22 September 2022

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insight into the phenomenon of photodetachment of both giant fullerene and fullerene onion-anions.

In general, elementary processes involving fullerene formations present a formidable multifaceted problem for theorists, thereby requiring the investment of considerable efforts to comprehensibly address all facets of the problem as well as the interaction(s) between them. Therefore, before investing such efforts in a comprehensive study, a kind of roadmap is needed as a guide to the subsequent comprehensive study of this multifaceted problem. Thus, the main narrow goal of the present study is to gain insight, using the simplest reasonable approximation, for modifications of the photodetachment cross sections of giant and nested fullerene anions owing to changes in their geometry induced by stuffing of a larger bare fullerene with smaller and smaller fullerenes: (C240)<sup>−</sup>, (C540)<sup>−</sup>, (C60@C240)<sup>−</sup>, (C60@C540)<sup>−</sup>, (C240@C540)− and (C60@C240@C540)<sup>−</sup>. To meet this goal, we approximate each individual fullerene cage by a rigid potential sphere of a certain inner radius *r*in, thickness Δ and potential depth *U*0, as in many earlier model studies of fullerene-involved processes cited above. Within the framework of this approximation, we detail how the photodetachment of the fullerene onion-anions differs crucially from the photodetachment of the largest bare (i.e., single-cage) fullerene anion owing to the differences in geometries between the fullerene formations.

To label the discrete states occupied by the attached electron in a fullerene anion, we adopt, just as a matter of labeling, the traditional notation used for atoms, i.e., the *n*-notation, where is the orbital quantum number and *n* ≥ + 1. Thus, in our notations, the first *s*-state of the attached electron is 1*s*, the next *s*-state is a 2*s* state, the first *p*-state is a 2*p* state, the next *p*-state is a 3*p* state, and so on.

Finally, atomic units (a.u.) (|*e*| = *h*¯ = *m* = 1, where *e* and *m* are the electron's charge and mass, respectively, and *h*¯ is a reduced Planck's constant) are used throughout the paper unless stated otherwise.

## **2. Review of Theory**

We model an individual C *N* cage ( *N* being the number of carbon atoms in the cage) by a *<sup>U</sup>*C*N* (*r*) spherical annular potential of the inner radius, *r*in, finite thickness, Δ, and depth, *U*0:

$$\mathcal{U}L\_{\mathbb{C}\_N}(r) = \begin{cases} -\mathcal{U}\_{0\prime} & r\_{\text{in}} \le r \le r\_{\text{in}} + \Delta \\ 0, & \text{otherwise.} \end{cases} \tag{1}$$

Such modeling of a C *N* cage was suggested in the early work by Puska and Nieminen [25] and, since then, has found an extensive use in numerous studies to date; the reader is referred to [5,13,16,18,21,22,25–27] and to the review paper [8] for many more references on the subject, as well as, e.g., to references [6,8–10,18,21,22,29,31,46] from [21] (and references therein).

We emphasize that, with regard to C60, such model has been proven [5,18,26] (and references therein) to produce results in a reasonable agreemen<sup>t</sup> with the experimental photoionization spectrum of endohedral Xe@C+60 [5] and a qualitative and even semiquantitative agreemen<sup>t</sup> with experimental differential elastic electron scattering off C60 [6]. Such modeling was also shown [27] to result in a semi-quantitative agreemen<sup>t</sup> with some of the most prominent features of the e<sup>−</sup> − C60 total elastic electron scattering cross section predicted by a far more sophisticated ab initio molecular-Hartree–Fock approximation [27]. This lays out a supporting background for a reasonable suitability of such modelling of C60 for the application to photodetachment of a C60 fullerene anion as well. Furthermore, our model replaces the earlier fullerene-anion-photodetachment approximations [9,19], which utilized the idea of an infinitesimally thin fullerene wall, by a more realistic finite-widthwall approximation, which is certainly an improvement to the cited approximations.

A fullerene onion, then, is modeled by a potential which is a linear combination of the corresponding *<sup>U</sup>*C*N* (*r*) potentials, as in [28]:

$$\mathcal{U}\mathcal{L}\_{\mathbb{C}\_N} \otimes \mathbb{C}\_M \otimes \dots = \mathcal{U}\_{\mathbb{C}\_{t0}} + \mathcal{U}\_{\mathbb{C}\_{240}} + \dots \tag{2}$$

The parameters *rin*, *U*0 and Δ of the individual C60, C240 and C540 fullerene cages in fullerene onion-anions are assumed to be the same as for the corresponding isolated bare (singlecage) fullerenes. In the present paper, we take the values for *r*in, Δ and *U*0 for C60/C240/C540 from [28]: *r*in = 5.8/12.6/18.8, Δ = 1.9/1.9/1.9and *U*0 = 8.22/10/12 eV, respectively.

A fullerene anion, C− *N*, or a fullerene onion-anion, (C *N*@C *M* @ ...)<sup>−</sup>, then, is formed by binding of an external electron into a *s*-state or a *p*-state in the field of corresponding *<sup>U</sup>*CN (*r*) or *<sup>U</sup>*CN@CM@... potential, respectively. Thus, the bound, *Pn*, and continuum, *P* , radial wavefunctions for the attached electron in a corresponding fullerene anion satisfy the radial Schrödinger equation:

$$-\frac{1}{2}\frac{d^2P\_{n/c\ell}}{dr^2} + \left[\mathcal{U}\_{\mathbb{C}} + \frac{\ell(\ell+1)}{2r^2}\right]P\_{n/c\ell}(r) = E\_{n/c\ell}P\_{n/c\ell}(r). \tag{3}$$

Here, *n* and are is the principal and orbital quantum numbers, respectively,  is the photoelectron energy and *U*C is the fullerene potential determined by Equations (1) or (2), respectively.

This equation is solved with the following boundary conditions for the discrete and continuum states:

$$P\_{n\ell}(r)|\_{r\to 0,\infty} = 0,\text{ whereas }P\_{\kappa\ell}(r)|\_{r\gg 1} \to \sqrt{\frac{2}{k\pi}}\sin\left(kr - \frac{\pi\ell}{2} + \delta\_{\ell}(\varepsilon)\right). \tag{4}$$

Here, *<sup>δ</sup>*() is the phase of the continuum state wavefunction and *k* is the photoelectron momentum.

Note that such model of fullerene anion photodetachment is similar in spirit to the one suggested earlier [9,19], albeit there is a Dirac-bubble potential, rather than the spherical annular potential, was used to approximate the C60 cage.

The photodetachment cross sections, *<sup>σ</sup>n*<sup>→</sup>,±1, as well as the oscillator strengths of the discrete, *fn*→*n*,±1, and continuum, *fn*<sup>→</sup>,±1, spectra of fullerene anions, were calculated using well-known formulas, see, e.g., [29]:

$$
\sigma\_{n\ell \to \epsilon, \ell \pm 1} = \frac{4}{3} \pi^2 a \frac{N\_{n\ell}}{2\ell + 1} \omega d\_{\ell \pm 1\prime}^2 \tag{5}
$$

$$f\_{n\ell \to n', \ell \pm 1} = \frac{N\_{n\ell}}{3(2\ell + 1)} \omega d\_{\ell \pm 1'}^2 \tag{6}$$

$$f\_{n\ell \to c,\ell \pm 1} = \frac{1}{2\pi^2 a} \int\_0^\infty \sigma\_{n\ell \to c,\ell \pm 1} d\omega. \tag{7}$$

Here, *α* is the fine-structure constant, *ω* is the photon energy, *Nn* is the number of electron in the *n* state (a single electron in our case), and *d*±1 is the reduced radial matrix element for the transition from the *n* state to a *n* (), ± 1 final state.

#### **3. Results and Discussion**

#### *3.1. Single-Cage Fullerene Anions*

As the first step, we scrutinize the 1*s* ground-states and 2*p* excited-states in the bare fullerene anions: <sup>C</sup><sup>−</sup>60, <sup>C</sup><sup>−</sup>240 and <sup>C</sup><sup>−</sup>540. We note that our calculations revealed no *np* excitedstates with *n* > 2 in any of these anions. The corresponding ground-state *<sup>P</sup>*1*s*(*r*) and

excited-state *<sup>P</sup>*2*p*(*r*) radial functions and the corresponding *E*1*s* and *<sup>E</sup>*2*p* energies of the attached electron in the <sup>C</sup><sup>−</sup>60, <sup>C</sup><sup>−</sup>240 and <sup>C</sup><sup>−</sup>540 anions are presented in Figure 1.

**Figure 1.** Calculated ground-state *<sup>P</sup>*1*s*(*r*) and excited- state *<sup>P</sup>*2*p*(*r*) radial functions and corresponding *E*1*s* and *<sup>E</sup>*2*p* binding energies of the attached electron in <sup>C</sup><sup>−</sup>60, <sup>C</sup><sup>−</sup>240, and <sup>C</sup><sup>−</sup>240 anions, as designated.

One can see that the *P*1*s* and *<sup>P</sup>*2*p* functions reach their maxima within the wall of a corresponding fullerene, i.e., within 5.8 < *r* < 7.7 in C60, 12.6 < *r* < 14.5 in C240 and 18.85 < *r* < 20.75 in C540. This, actually, has been expected, for an obvious reason. A strikingly unexpected result (at first glance), though, is that *P*1*s* ≈ *<sup>P</sup>*2*p* to a high degree of approximation, particularly in <sup>C</sup><sup>−</sup>240 and <sup>C</sup><sup>−</sup>540. This seems strange, because the Schrödinger equations for a *s*-state and a *p*-state differ by the presence of a centrifugal potential *<sup>U</sup>*cfg = (+<sup>1</sup>) 2*r*<sup>2</sup> for a *p*-state. Correspondingly, the *P*1*s* function should have differed from the *<sup>P</sup>*2*p* function. To understand why the situation is opposite to the expected one, we depict, for the case of C60, the cage model potential *<sup>U</sup>*C60 (*r*), the centrifugal potential *<sup>U</sup>*cfg = (+<sup>1</sup>) 2*r*<sup>2</sup> = 1 *r*2 for a *p*-electron, and the 2*p* probability density distribution, *ρ*<sup>2</sup>*p* = *P*<sup>2</sup> 2*p*, in Figure 2.

**Figure 2.** Calculated cage model potential *<sup>U</sup>*C60 (*r*), the centrifugal potential *<sup>U</sup>*cfg = (+<sup>1</sup>) 2*r*<sup>2</sup> = 1 *r*2 for a *p*-electron, and the 2*p* probability density distribution, *ρ*2*p* = *P*<sup>2</sup> 2*p*, as designated.

One can see that, inside the hollow interior of C60 (*U*C60 = 0), the *ρ*<sup>2</sup>*p* probability density is practically a zero up to about *r* = 3. Therefore, the presence of the centrifugal potential, however large it is, does not matter in this spatial region. *ρ*<sup>2</sup>*p* starts differ from *ρ*<sup>2</sup>*p* ≈ 0 between approximately 3 < *r* < 5.8. There, however, *<sup>U</sup>*cfg is already small and, additionally, only less than 20% of electronic charge is concentrated in this spatial region. Hence, again, a role of the small *<sup>U</sup>*cfg = 0 is largely obliterated in there. Inside the C60 wall itself *<sup>U</sup>*cfg, on average, is less than 3% of *<sup>U</sup>*C60 = 0.302, whereas outside of the C60 wall, *<sup>U</sup>*cfg *<sup>U</sup>*C60 in addition to a rapidly damping probability density distribution. Thus, it now becomes clear that the presence of the centrifugal potential for a *p*-electron cannot make the solution of the Schrödinger equation to differ any notably from its solution for a *s*-electron. This discussion explains why *P*1*s* differs from *<sup>P</sup>*2*p* only insignificantly, in <sup>C</sup><sup>−</sup>60. Additionally, we believe that the reader can easily extend this discussion of the behavior of *P*1*s* and *<sup>P</sup>*2*p* in <sup>C</sup><sup>−</sup>60 to giant fullerene anions to understand why *P*1*s* and *<sup>P</sup>*2*p* become practically identical in each of the <sup>C</sup><sup>−</sup>240 and <sup>C</sup><sup>−</sup>540 anions.

Although the energies are generally more sensitive quantities to parameters in the Schrödinger equation than the wavefunctions, the difference between *E*1*s* and *<sup>E</sup>*2*p* binding energies is more noticeable than between the wavefunctions, although still small: it is about 25% of *E*1*s* for <sup>C</sup><sup>−</sup>60, 4% for <sup>C</sup><sup>−</sup>240, and 1.6% for <sup>C</sup><sup>−</sup>540. Note that the difference between *E*1*s* and *<sup>E</sup>*2*p* is decreasing with increasing size of the fullerene anion. The largest energy difference 25% is in <sup>C</sup><sup>−</sup>60, as is the largest difference between *P*1*s* and *<sup>P</sup>*2*p* (see Figure 1). This is because the <sup>2</sup>*p*-centrifugal potential energy in <sup>C</sup><sup>−</sup>60 is larger than in other fullerene anions, owing to a significantly smaller size of the C60 cage as compared to the other two.

Because *P*1*s* ≈ *<sup>P</sup>*2*p*, the corresponding *f*<sup>1</sup>*s*→2*p* oscillator strengths in the <sup>C</sup><sup>−</sup>60, <sup>C</sup><sup>−</sup>240 and <sup>C</sup><sup>−</sup>540 anions must be large. Our calculations show that *f*<sup>1</sup>*s*→2*p* ≈ 0.807, 0.962, and 0.679 in <sup>C</sup><sup>−</sup>60, <sup>C</sup><sup>−</sup>240and <sup>C</sup><sup>−</sup>540, respectively (see Table 1 for more details).

**Table 1.** Calculated *E*1*s* ground-state energies, *<sup>ω</sup>*2*p* and *<sup>ω</sup>*3*p* energies of the 1*s* → 2*p*&3*p* transitions (all in eV), discrete *f*<sup>1</sup>*s*→2*p*&3*p*, and continuum, *f*<sup>1</sup>*s*<sup>→</sup> *<sup>p</sup>*, oscillator strengths in the single-cage and multi-cage fullerene anions. Note, our calculations showed no existence of the *np* excited states with *n* > 2 in the single-cage fullerene anions.


We note that the oscillator strength *f*<sup>1</sup>*s*→2*p* is decreasing with increasing size of the fullerene cage. At first glance this is strange, because the approximate equality *P*1*s* ≈ *<sup>P</sup>*2*p* is getting only stronger with increasing size of the fullerene cage, as discussed above. Thus, the overlap between *P*1*s* and *<sup>P</sup>*2*p* is increasing and so should have been *f*<sup>1</sup>*s*→2*p* as well, with increasing size of the anion. However, the *<sup>ω</sup>*1*s*→2*p* ≡ *<sup>ω</sup>*2*p* excitation energy (see Table 1), is decreasing with the increasing size of the fullerene cage. This counterbalances the increase in the overlap between *P*1*s* and *<sup>P</sup>*2*p*, thereby resulting in a smaller *f*<sup>1</sup>*s*→2*p* (which is proportional to *<sup>ω</sup>np*) in a bigger fullerene anion. This decrease in the *f*<sup>1</sup>*s*→2*p* oscillator strength with increase in the fullerene size leads to an important conclusion. Namely, we conclude there is an increasing transfer of oscillator strength of a fullerene anion from a discrete spectrum to continuum with increasing size of the fullerene cage, as clearly follows from the oscillator strength sum rule: *f*<sup>1</sup>*s*<sup>→</sup> *p* = 1 − *f*<sup>1</sup>*s*→2*p*. Calculated *f*<sup>1</sup>*s*<sup>→</sup> *p*'s are presented in Table 1 as well. At this point it is important to emphasize that the continuum oscillator strengths, presented in Table 1, were calculated using Equation (7) rather than as 1 − *f*<sup>1</sup>*s*→2*p* from the sum rule. The fact that the independently calculated *f*<sup>1</sup>*s*<sup>→</sup> *p* and the *f*<sup>1</sup>*s*<sup>→</sup> *p* = 1 − *f*<sup>1</sup>*s*→2*p* are equal to a high degree of approximation speaks about the adequacy of the calculated photodetachment cross sections themselves, discussed later in the paper.
