*3.2. Fullerene Onion-Anions*

We now move to the discussion of the wavefunctions of the valence electron in the fullerene onion-anions (C60@C240)− , (C60@C540)<sup>−</sup>, (C240@C540)− and (C60@C240@C540)<sup>−</sup>. We note first that the potentials of these fullerene onions are, obviously, either double-well or triple-well potentials. Correspondingly, one can expect a greater number of bound states available to the attached electron in these fullerene onion-anions. In our case, the calculations predicted the existence of only two discrete *p*-states—the 2*p* and 3*p* excited states—, in contrast to only the 2*p* excited-state in the bare fullerene anions. The corresponding *P*1*s*, *<sup>P</sup>*2*p* and *<sup>P</sup>*3*p* functions are plotted in Figure 3 where a number of new features are exhibited.

The most striking discovery relates to the behavior of the *<sup>P</sup>*3*p* excited-state wave functions. Their highly peculiar behavior is completely different from the behavior of the *P*1*s* and *<sup>P</sup>*2*p* functions in any of these fullerene onion-anions. Indeed, we find that a significant part of the *<sup>P</sup>*3*p* function and, thus, the electron density of the attached electron, is packed inside the wall of the inner cage directly adjacent to a larger fullerene cage in each of these double- and triple-cage fullerene onion-anions. Although this is evident from Figure 3, this is also supported by looking at the mean radii, *<sup>r</sup>*¯3*p*, of the 3*p* orbitals in these fullerene onion-anions as well. The calculated *<sup>r</sup>*¯3*p*'s are: *<sup>r</sup>*¯3*p* ≈ 7 in both (C60@C540)−

and (C60@C240)− (thus, the 3*p* orbital falls into the C60 potential well), whereas *<sup>r</sup>*¯3*p* ≈ 14 in both (C240@C540)− and (C60@C240@C540)− (thus, *<sup>r</sup>*¯3*p* falls into the C240 potential well). This is in a sharp contrast to the *P*1*s* and *<sup>P</sup>*2*p* functions that are mainly packed in the potential well of the largest fullerene cage (C540, in our case) in corresponding fullerene onion-anions, respectively, as is evident from Figure 3 (also, *<sup>r</sup>*¯1*s* ≈ *<sup>r</sup>*¯2*p* ≈ 20 in all fullerene onion-anions under discussion). Especially surprising is the behavior of the *<sup>P</sup>*3*p* function in (C60@C540)<sup>−</sup>, where its probability density is almost entirely located inside C60, despite the size of C60 being significantly smaller than C540, so that the C60 potential well should not have affected the attached valence electron at all, as in the case of the *P*1*s* and *<sup>P</sup>*2*p* functions (see Figure 3c,d). In any case, the behavior of *<sup>P</sup>*3*p* in these fullerene onion-anions is extraordinary, a complete break with conventional wisdom.

**Figure 3.** Calculated radial ground-state *<sup>P</sup>*1*s*(*r*) and excited-state *<sup>P</sup>*2*p* and *<sup>P</sup>*3*p* (due to the 1*s* → *np* transitions, *n* = 2, 3) of the attached electron in fullerene onion-anions: (**a**) (C60@C240)<sup>−</sup>, (**b**) (C240@C540)<sup>−</sup>, (**c**) (C60@C540)<sup>−</sup>, (**d**) (C60@C240@C540)<sup>−</sup>: solid, *<sup>P</sup>*2*p*; dashed, *P*1*s*; dash–dot, *<sup>P</sup>*3*p*. Also plotted are the *P*1*s* (dash–dot–dot) functions in bare <sup>C</sup><sup>−</sup>240 and <sup>C</sup><sup>−</sup>540, as designated, for comparison purposes. Note, to avoid any confusion, that the graphs for the *P*1*s* and *<sup>P</sup>*2*p* functions in the fullerene onion-anions tightly overlap with each other and are practically indistinguishable from each other with some exception in the case of (C60@C240)<sup>−</sup>.

We interpret the predicted behavior of the *<sup>P</sup>*3*p* excited-state functions in the fullerene onion-anions as being due to both the multi-well nature of the fullerene onion-anion potentials and the fact that, in contrast to the nodeless *P*1*s* and *<sup>P</sup>*2*p* functions, the *<sup>P</sup>*3*p* function has one node. That is, the *<sup>P</sup>*3*p* function is distinctly split into an inner part (before the node) and an outer part (beyond the node). It appears that the inner part of the *<sup>P</sup>*3*p* function falls into the potential well associated with a fullerene cage adjacent to the outermost cage in the fullerene onion-anion. Thus, the attached electron partially resides in the inner well.

We note, though, that the behavior of the *<sup>P</sup>*3*p* function in the fullerene onion-anions is somewhat reminiscent of the behavior of the excited *P*3*d* and *P*4*d* functions, excited from the 3*p* subshell, in endohedral calcium, Ca@C60 [30]. There, a significant transfer of the 4*d*, but not 3*d*, electron density into the inner space of C60 was demonstrated. That resulted in a significant increase in the amplitude of the *P*4*d* orbital in the inner space of C60. Consequently, the mean radius of the 4*d* orbital was reduced from *<sup>r</sup>*¯4*d* ≈ 14 in free Ca to only *<sup>r</sup>*¯4*d* ≈ 4.3 < *rc* = 5.8 in Ca@C60 [30]. That situation, in turn, was commented on to be somewhat reminiscent of the behavior of the excited 4 *f* and 5 *f* orbitals in Ba<sup>+</sup> [31,32] that was shown to be due to the double-well nature of the potential of Ba<sup>+</sup> that caused partial orbital collapse of 5 *f* into the inner well, thereby causing 5 *f* , rather than 4 *f* , to have the greater amplitude near *r* = 0.

#### *3.3. Oscillator Strengths and Photodetachment Cross Sections*

Calculated oscillator strengths, *f*<sup>1</sup>*s*→*np*, of the <sup>C</sup><sup>−</sup>60, <sup>C</sup><sup>−</sup>240, and <sup>C</sup><sup>−</sup>540 bare fullerene anions as well as the (C60@C240)<sup>−</sup>, (C60@C540)<sup>−</sup>, (C240 @ C540)<sup>−</sup>, and (C60@C240@C540)− fullerene onion-anions are/were listed in Table 1 which contains a wealth of information. Since a principal goal of this work is to explore the spectral distribution of oscillator strength, we focus on a comparison among the total oscillator strengths of the discrete spectra, i.e., *f*<sup>1</sup>*s*→2*p* + *f*<sup>1</sup>*s*→3*p* ≡ *f*(<sup>2</sup>*p*+3*p*), for the single and nested fullerene cages Thus, as we transition from <sup>C</sup><sup>−</sup>240 → (C60@C240)<sup>−</sup>, the *f*(<sup>2</sup>*p*+3*p*) oscillator strength is increased. The same change in *f*(<sup>2</sup>*p*+3*p*) is characteristic along all other transition paths as well: <sup>C</sup><sup>−</sup>540 → (C240@C540)<sup>−</sup>, <sup>C</sup><sup>−</sup>540 → (C60@C240@C540)<sup>−</sup>, and, in principle, <sup>C</sup><sup>−</sup>540 → (C60@C540)<sup>−</sup>, too. Hence, we have unraveled a general tendency: stuffing of a bigger fullerene cage with a smaller fullerene cage, as well as progressively stuffing the biggest fullerene cage with several smaller fullerene cages, results in the transfer of a part of the oscillator strength from a continuum spectrum into a discrete spectrum.

Now, how does the discovered tendency affect the <sup>1</sup>*s*-photodetachment cross section, *<sup>σ</sup>*1*s*? Obviously, the total area under the graph for *σ*1*s* should be decreasing along the discussed fullerene transition paths. This may result in the disappearance of some of the resonance structures in *σ*1*s*, or making them narrower, or decreasing their heights, or lowering the values of other parts of *σ*1*s*, or all of the above cumulatively. It is, therefore, extremely interesting to study the modifications in *<sup>σ</sup>*1*s*'s on a comparative one-to-one basis for different fullerene anions.

Calculated *<sup>σ</sup>*1*s*'s for <sup>C</sup><sup>−</sup>240 versus (C60@C240)<sup>−</sup>, as well as <sup>C</sup><sup>−</sup>540 versus (C60@C540)<sup>−</sup>, (C240@C540)<sup>−</sup>, and (C60@C240@C540)− are depicted in Figure 4 as functions of the photoelectron momentum *κ*, in order to eliminate the impact of differences in 1*s* ionization potentials between the fullerene anions on details of *<sup>σ</sup>*1*s*'s, for the adequacy of the comparison between these anions.

We first note that the calculated cross sections exhibit the oscillatory behavior versus the photoelectron momentum, *k*. Such resonances have been well understood for both photoionization and photodetachment of, as well as electron scattering by, fullerene and endo-fullerene complexes in a large body of research; we refer the reader to the above references, to the review paper [8] for many more references on the subject, as well as, e.g., to [4–8,10–24,29,31,38,39,46] from [21] (and references therein). Following [33], these resonances are commonly referred to as the *confinement resonances*.

**Figure 4.** Calculated *σ*1*s* photodetachment cross sections of bare fullerene anions and nested fullerene onion-anions, as designated in the figure. Note, on all parts of the figure, the *σ*1*s* of <sup>C</sup><sup>−</sup>60 is represented by a dashed-line.

Secondly, note that the prediction mentioned above on the modification of the photodetachment cross section along the path from the bare fullerene anions to the doubleand triple-cage fullerene onion-anions is seen to be correct. Indeed, we see the disappearance of one or even two confinement resonance structures (near the lower energy end of the spectrum), and the significant decrease in their amplitudes (except for the case of the (C60@C540)− onion-anion, which is an extraordinary case anyway, as was discussed above). Furthermore, it is interesting that the resonances in *σ*1*s* of (C60@C540)− are seen to be shifted towards higher *k*'s, compared to *σ*1*s* of the bare (C540)− anion, whereas in all other nested fullerene onion-anions they shift toward lower *k*'s, compared to corresponding bare counterparts.

Thirdly, it is quite interesting that *<sup>σ</sup>*1*s*'s of all fullerene onion-anions, whether doublecage or triple-cage anions, do not differ much in magnitude from *σ*1*s* of the smallest <sup>C</sup><sup>−</sup>60 anion in this sequence of fullerene anions (except for *σ*1*s* of extraordinary (C60@C540)− at a lower end of the spectrum). To emphasize this, we added *σ*1*s* of <sup>C</sup><sup>−</sup>60 to all plots in Figure 4 to facilitate this comparison. One can see that *<sup>σ</sup>*1*s*'s, associated with the nested

fullerene onion-anions, oscillate around *σ*1*s* of <sup>C</sup><sup>−</sup>60 with average amplitudes that are not much different from *σ*1*s* of <sup>C</sup><sup>−</sup>60.

Lastly, we note that, to check the connection of calculated *<sup>σ</sup>*1*s*'s to calculated oscillator strengths, we calculated the oscillator strength of the continuum spectrum, *f*<sup>1</sup>*s*<sup>→</sup> *<sup>p</sup>*, by appropriately integrating *<sup>σ</sup>*1*s*'s in accordance with Equation (7). These calculated *f*<sup>1</sup>*s*<sup>→</sup> *p*'s are presented in Table 1, and the fact that they have the same values as those obtained from the oscillator sum rule ( *f*<sup>1</sup>*s*<sup>→</sup> *p* = 1 − *f*<sup>1</sup>*s*<sup>→</sup>(<sup>2</sup>*p*+3*p*)) speaks to the accuracy of the calculated *<sup>σ</sup>*1*s*'s.
