Photoionization of atoms and ions from endohedral anions

The quintessence of the model is as follows.

Firstly, a neutral C_N cage is modeled by a $U_{\rm C}(r)$ spherical annular, i.e., rectangular in the radial coordinate potential of a certain inner radius, $r_{0}$, thickness, $\Delta$, and depth, $U_{0}$:

$\displaystyle U_{\rm C}(r)=\left\{\begin{array}[]{ll}-U_{0},&\mbox{if $r_{0}\leq r\leq r_{0}+\Delta$}\\ 0,&\mbox{otherwise.}\end{array}\right.$

(3)

Such modeling of a C_N cage was apparently first proposed by Puska and Nieminen PuskaPRA93 . Later, it was widely used in numerous theoretical studies on photoionization and other elementary processes involving endohedral fullerenes (also referred to as endohedral atoms/ions in this paper), $A$@C_N. The reader is referred to the works cited above, as well as, for example, to a relatively recent review DeshmukhEPJD2021 on these topics).

Secondly, following the work DolmMansPRA2006 , the effect of the charged shell C${}_{N}^{q}$ on elementary atomic processes involving fullerene anions, $A$@C${}_{N}^{q<0}$, is taken into account by a uniform distribution of the excess charge $q$ over the entire outer surface of the C_N cage. This leads to the appearance of an additional Coulomb spherical potential, $V_{q}(r)$, both inside and outside the fullerene cage:

$\displaystyle V_{q}(r)=\left\{\begin{array}[]{ll}\frac{q}{r_{0}+\Delta},&\mbox{if $0\leq r\leq r_{0}+\Delta$}\\ \frac{q}{r},&\mbox{otherwise.}\end{array}\right.$

(6)

Consequently, the entire model potential, $U_{\rm C}^{q}(r)$, for a charged C${}_{N}^{q}$ cage, is taken as the sum of the two potentials: $U_{\rm C}(r)$ [Equation (3)], and $V_{q}(r)$ [Equation (6)]. That is,

$\displaystyle U_{\rm C}^{q}(r)=U_{\rm C}(r)+V_{q}(r).$

(7)

Next, to address the calculation of the structure and spectra of the encaged atom/ion, $A$, in the endohedral fullerene anion, $A$@C${}_{N}^{q}$, the $V_{q}(r)$ potential is added to the Shrödinger equation (for a single-electron atom), or to the Hartree-Fock, or the relativistic Dirac-Fock equations, or to the equations of other kinds of approximations for a multielectron atom.

For instance, the thus modified non-relativistic Hartree-Fock equation, applicable to $A$@C${}_{N}^{q}$, is as follows:

	$\displaystyle\left[-\frac{\Delta}{2}-\frac{Z}{r}+U_{\rm C}^{q}(r)\right]\psi_{i}({\vec{r}})+\sum_{j=1}^{N_{A}}\int{\frac{\psi^{*}_{j}({\vec{r}}^{\prime})}{|{\vec{r}}-{\vec{r}}^{\prime}|}}$
	$\displaystyle\times[\psi_{j}({\vec{r}}^{\prime})\psi_{i}({\vec{r}})-\psi_{i}({\vec{r}}^{\prime})\psi_{j}({\vec{r}})]d{\vec{r}}^{\prime}=\epsilon_{i}\psi_{i}({\vec{r}}).$		(8)

Here, $N_{A}$ is the number of electrons in the multielectron atom $A$, $i$ (and, similar, $j$) denotes the set of the principal ($n_{i}$), orbital ($\ell_{i}$), magnetic ($m_{\ell_{i}}$) and spin-magnetic ($m_{s_{i}}$) quantum numbers for the $i$-th atomic electron in $A$@C${}_{N}^{q}$. For the continuum spectrum of the $i$’s electron, $\epsilon_{i}$ is to be replaced by the energy of the continuum spectrum, $\epsilon$. Furthermore, the function $\psi_{n_{i}\ell_{i}m_{\ell_{i}}}({\vec{r}})$ of the $i$’s electron admits a separable representation: $\psi_{n_{i}\ell_{i}m_{\ell_{i}}}({\vec{r}})=r^{-1}P_{n_{i}\ell_{i}}(r)Y_{\ell_{i}m_{\ell_{i}}}(\theta,\phi)$.

We will frequently refer to such model of the fullerene anion as “structureless” model or approximation in this paper, to underline that it ignores possible quantum states of the excess electronic charge on the cage.

Next, the dipole photoionization cross section, $\sigma_{n\ell}$, of a $n\ell$-subshell of the encapsulated atom is calculated using the standard formula (see, e.g., Amusia_book ):

$\displaystyle\sigma_{n\ell}(\omega)=\frac{4\pi{{}^{2}}\alpha N_{n\ell}}{3(2l+1)}\omega[lD_{\ell-1}^{2}+(\ell+1)D_{\ell+1}^{2}].$

(9)

Here, $\omega$ is the photon energy, $\alpha$ is the fine structure constant, $N_{n\ell}$ is the number of electrons in the $n\ell$ subshell, $D_{l\pm 1}$ is a radial dipole photoionization amplitude,

$\displaystyle D_{\ell\pm 1}=\int_{0}^{\infty}{P_{\epsilon\ell\pm 1}(r)rP_{n\ell}(r)dr}.$

(10)

In our modeling of A@C${}_{N}^{q}$, the extra electron(s) in the A@C${}_{N}^{q}$ fullerene anion is(are) bound by the central field which is the sum of the fields of the atom and neutral C_N cage. Therefore, the excess electronic states are characterized by quantum numbers $n$ and $\ell$. Let us designate such endohedral fullerene anions as A@C${}_{N}^{q}(n\ell)$ or A@C${}_{N}^{q}(n\ell,n^{\prime}\ell^{\prime})$ and so on, depending on how many excess electrons exist and how many different $n\ell$-states they occupy in the fullerene anion. For example, He@C${}_{N}^{-1}(2s)$ means that the single attached electron resides in a $2s$ state on the singly-charged fullerene shell, He@C${}_{N}^{-2}(2s2p)$ indicates that one of the two attached electrons in the doubly-charged anion is in a $2s$ state and the other electron is in a $2p$ state, and so on.

To account for the $n\ell$-structure of the excess electronic charge in the anion and, at the same time, retain the spirit of the semi-empirical framework for modeling the fullerene cage, we follow the methodology outlined in Dolm2020 . The essence of the latter is as follows.

The A@C${}_{N}^{q}(n\ell,...)$ system is a complete (or “single”) system in the sense that the state(s) of the attached electron(s) is(are) affected by the field of the encapsulated atom, whereas the states of the atomic electrons, in turn, are affected by both the field of the cage itself, $U_{\rm C}(r)$, and by the field(s) of the attached electron(s). To account for the mutuality of the states of the atomic electrons and those of the excess electrons on the fullerene cage, we simply solve, simultaneously, a system of the $N_{tot}$ Hartree-Fock equations for the “atom + fullerene anion” system, where $N_{tot}=N_{A}+N_{q}$, with $N_{q}$ being the number of the excess electrons on the fullerene cage, and $N_{A}$ the number of the electrons in the encapsulated atom.

	$\displaystyle\left[-\frac{\Delta}{2}-\frac{Z}{r}+U_{\rm C}(r)\right]\psi_{i}({\vec{r}})+\sum_{j=1}^{N_{\rm tot}}\int{\frac{\psi^{*}_{j}({\vec{r}}^{\prime})}{|{\bm{r}}-{\vec{r}}^{\prime}|}}$
	$\displaystyle\times[\psi_{j}({\vec{r}}^{\prime})\psi_{i}({\vec{r}})-\psi_{i}({\vec{r}}^{\prime})\psi_{j}({\vec{r}})]d{\vec{r}}^{\prime}=\epsilon_{i}\psi_{i}({\vec{r}}).$		(11)

Here, $i$ and $j$ now run from unity to $N_{\rm tot}$: $i,j=1,...,N_{\rm tot}$. This system of equations differs from Equation (8) in that the $N$ number is replaced by $N_{\rm tot}$ and the $U_{\rm C}^{q}$ potential is replaced by $U_{\rm C}$, [Equation (3)]. It allows one to calculate the needed energies, $\epsilon_{i}$, and the wavefunctions, $\psi_{i}({\vec{r}})$, of both the atomic and excess electrons in A@C${}_{N}^{q}(n\ell,...)$ and apply them to calculations of the photoionization cross sections of the thus confined atom A.

We will frequently refer to such model of the fullerene anion as “structured” model or approximation in this paper, to underline that it accounts for possible quantum states of the excess electronic charge on the cage.

Now, however, a complication arises due to the existence of various total terms for the A@C${}_{N}^{q}(n\ell,...)$ system, which is now an open-shell system. In the present study, we bypass this complication by utilizing the term-average Hartree-Fock formalism YarzhCher2024 to calculate the energies and wavefunctions of both the atomic and excess electrons in the A@C${}_{N}^{q}(n\ell,...)$ system. To avoid diverting the reader’s attention from the main topic of the paper, a review of the term-average Hartree-Fock formalism is provided in Appendix.

Lastly, in performed calculations for the present study, we used the values for $r_{0}$, $\Delta$ and $U_{0}$ for C₆₀/C₂₄₀ as stated and discussed in DolmBrMans2008 : $r_{0}\approx 5.8/12.6$, $\Delta\approx 1.9/1.9$, $U_{0}\approx 0.302/0.378$ a.u., respectively.

Here, we present and discuss calculated results concerning the photoionization of the ground-state hydrogen atom, H$(1s)$, encapsulated inside the singly-charged fullerene anion ($q=-1$). These results were obtained in the framework of both aforementioned models for the endohedral fullerene anions - the structureless [${\rm H}(1s)@{\rm C}_{N}^{q}$] and structured [${\rm H}(1s)@{\rm C}_{N}^{q}(n\ell)$] models. As case studies, we arbitrarily choose $n\ell=2p$ and $3d$, and we run calculations for C_N fullerene cages with both $N=60$ and $N=240$ carbon atoms in the cage, respectively.

In Figure $1$, the $P_{1s}$ radial function of the encapsulated H atom is depicted. Additionally, the $P_{2p}(r)$ and $P_{3d}(r)$ radial functions are depicted for the $2p$ and $3d$ orbitals of the attached electron, respectively, within the endohedral fullerene anions.

Firstly, it can be concluded from Figure 1 that the $1s$ electron density in the encapsulated atom remains largely unaffected by the excess charge on the fullerene cage, especially in the case of the C₂₄₀ cage.

Secondly, note how the $P_{2p}(r)$ and $P_{3d}(r)$ functions of the excess electron extend, to some degree, into the hollow interior of the C_N shell, thereby overlapping with the $P_{1s}$ function of the encapsulated H atom. This, obviously, should make the $1s$-photoionization cross section, $\sigma_{1s}$, of the ${\rm H}(1s)@{\rm C}_{N}^{-1}(n\ell)$ system differ from that of the structureless ${\rm H}(1s)@{\rm C}_{N}^{-1}$ system. How strong can the difference be? One of the aims of the present study is to answer this question.

Thirdly, one can see that the overlap of the $P_{1s}$ function with the functions for the $2p$ and $3d$ excess electrons is far less significant in the case associated with a C₂₄₀ shell than in the other case. Hence, it is reasonable to expect that any differences in values of $\sigma_{1s}$ between the ${\rm H}(1s)@{\rm C}_{N}^{-1}$ and ${\rm H}(1s)@{\rm C}_{N}^{-1}(n\ell)$ systems will diminish with the increasing size of the fullerene cage.

Fourthly, interestingly, the $2p$ and $3d$ functions of the excess electron differ insignificantly, especially in the case of the giant C₂₄₀ cage. A similar situation was observed and explained for the $P_{n\ell}$ functions of the $n\ell$-electron attached to the empty fullerene cage Dolm2022 . The reason for the noted indistinguishability between the functions with different $\ell$ values (the $P_{2p}$ and $P_{3d}$ functions in our case) arises from the electrons’ primary localization on the cage itself, which has a large radius. Consequently, the centrifugal barrier $\frac{\ell(\ell+1)}{2r^{2}}$, which induces the dependence of $P_{n\ell}$ on $\ell$, becomes so small that the difference between the functions with close $\ell$ values effectively vanishes. The latter is particularly true for the case of the giant C₂₄₀ cage.

Calculated $\sigma_{1s}$ photoionization cross sections of H@C${}_{N}^{-1}$ and H@C${}_{N}^{-1}(n\ell)$ with $n\ell=2p$ and $3d$ are depicted in Figure $2$.

One can see that some differences between these $\sigma_{1s}$ cross sections are seen only in the low-energy region. There, the sharp, intense maxima in $\sigma_{1s}$’s are somewhat shifted along the energy scale relative to each other without a significant change in their magnitudes. Thus, it is reasonable to summarize that the change in $\sigma_{1s}$’s between the H@C${}_{60}^{-1}$, and H@C${}_{N}(n\ell)$ systems are minimal. This reveals the first indication that the photoionization cross section of the atom from an endohedral fullerene anion depends little on the state of the excess charge on the anion.

And, as expected, the differences between the discussed $\sigma_{1s}$’s are diminishing with the increasing size of the fullerene cage. This is obvious from comparing calculated data plotted in Figure $2$a (for $C_{60}$) with those plotted in Figure $2$b for the case of a C₂₄₀ cage. Note that a would-be larger shift in energy between the low-energy maxima in $\sigma_{1s}$’s, plotted in Figure $2$b, compared to the energy shift in Figure $2$a is deceptive. This is because the energy-scale range is less than $10$ eV in Figure $2$b, but is much broader, about $30$ eV, in Figure $2$a.

Finally, let us comment on the origin and nature of all the maxima in $\sigma_{1s}$’s displayed in Figure $2$. To meet the goal, we also plotted calculated $\sigma_{1s}$ photoionization cross sections of the hydrogen atom confined inside a neutral $C_{N}$ cage, designated as “H$@{\rm C}_{N}$ (neutral)” in the figure.

Firstly, the calculated $\sigma_{1s}$ values, plotted in Figure $2a$ or $2b$, are nearly identical down to about $14$ eV of photon energy. Secondly, for photon energies down to about $14$ eV, all $\sigma_{1s}$’s exhibit a maximum around $22$ eV for $N=60$ (though this maximum is poorly developed) and around $17$ eV for $N=240$. Therefore, the higher-energy maxima in $\sigma_{1s}$ for endohedral fullerene anions have the same origin as the maximum in $\sigma_{1s}$ for the neutral H@C_N endohedral fullerene. The origin of such maxima in the $\sigma_{n\ell}$ for atoms in neutral A@C_N endohedral fullerenes has been extensively studied in previous works, e.g., PuskaPRA93 ; DeshmukhEPJD2021 ; Baltenkov ; ConDolmMans ; DolmAQC (and references therein). These maxima arise from interference between the outgoing photoelectron wave from the encapsulated atom and the photoelectron waves reflected from the fullerene cage boundaries. In ConDolmMans , these resonances were termed confinement resonances. This term has become common in the literature, and we use it in the present paper as well. Thus, the observed higher-energy resonances in $\sigma_{1s}$ for H@C${}_{60}^{-1}$, H@C${}_{N}^{-1}(n\ell)$, and neutral H@C_N are confinement resonances.

Next, in the photon energy region below about $14$ eV, all calculated $\sigma_{1s}$ values for fullerene anions show a sharp, strong resonance, whereas the $\sigma_{1s}$ for neutral H@C_N does not extend into this region. Thus, these sharp lower-energy resonances originate from a different source compared to the confinement resonances. Such resonances in the photoionization cross sections of atoms encapsulated within fullerene anions were predicted and studied in detail in DolmMansPRA2006 . It was demonstrated that these resonances arise due to an additional Coulomb potential barrier created by the excess charge on the fullerene cage. In DolmMansPRA2006 , these novel resonances in $\sigma_{n\ell}$ values were termed Coulomb confinement resonances. To date, Coulomb confinement resonances have been explored in various contexts with different levels of detail in numerous theoretical studies (see, e.g., KumarVarmaJPCS2015 ; KumarVarmaPRA2016 ; VarmaPRA2020 ; VarmaPhysSCR2021 ; VarmaPRA2023 and references therein). Therefore, the lower-energy resonance structures in $\sigma_{1s}$ for H@C${N}^{-1}$ and H@C${N}^{-1}(n\ell)$ are Coulomb confinement resonances.

To conclude, we have unraveled the first indication that the photoionization cross section of the atom within the endohedral fullerene anion is largely independent of the quantum structure of the excess charge on the anion. The most noticeable differences (albeit insignificant) in the photoionization cross sections between the structureless and structured fullerene anions primarily occur in the region of Coulomb confinement resonances.

As another case study, we examine the photoionization of the He$(1s^{2})$ atom and its ion, He${}^{+}(1s^{1})$, confined within the structureless fullerene anion cage, C${}_{N}^{q}$, compared to when they are encapsulated inside the structured fullerene anion cages, ${\rm C}_{N}^{q}(n\ell)$ or ${\rm C}_{N}^{q}(n\ell,n^{\prime}\ell^{\prime})$. Specifically, we arbitrarily choose the He@C${}_{N}^{-1}(2p)$ and He@C${}_{N}^{-1}(3d)$ singly-charged endohedral fullerene anions, as well as the He${}^{+}@{\rm C}_{N}^{-2}(2s2p)$ and He${}^{+}@{\rm C}_{N}^{-2}(2s3d)$ doubly-charged anions, with $N=60$ and $240$.

The corresponding $P_{1s}$, $P_{2s}$, $P_{2p}$ and $P_{3d}$ electronic functions are plotted in Figure 3.

The behavior of the plotted functions follows the same trends as in the previously discussed case of the H atom inside the fullerene anion. Therefore, we believe that they are self-explanatory without new elements in the behavior, and we leave it to the reader to draw their own conclusions regarding these functions. It is important to stress, however, that, as earlier, there is overlap between the $P_{1s}$ atomic function and each of the corresponding $P_{2s}$, $P_{2p}$, and $P_{3d}$ functions. Accordingly, we explore below how this overlap between the electronic functions affects the $\sigma_{1s}$ photoionization cross section of He and He⁺ from the structured fullerene anions compared to the structureless fullerene anions.

The correspondingly calculated $\sigma_{1s}$ photoionization cross sections are depicted in Figure $4$.

Firstly, similar to the hydrogen atom case discussed earlier, Figure 4 reveals that $\sigma_{1s}$ for the encapsulated helium atom and its ion exhibit a pronounced, low-energy sharp resonance when photoionization occurs from the fullerene anions. This resonance is absent in the photoionization of He and He⁺ encapsulated inside the neutral C₆₀ cage. Thus, as in the case of the hydrogen atom, these observed features are Coulomb confinement resonances. As photon energy increases, these Coulomb confinement resonances are succeeded by less pronounced resonances in both the charged and neutral endohedral fullerene systems. These latter features are, thus, ordinary confinement resonances.

Secondly, a key finding, however, is that the differences in $\sigma_{1s}$ between photoionization of He or He⁺ from the structureless and various structured endohedral fullerene anions, respectively, are primarily observed in the Coulomb confinement resonances. However, these differences are relatively minor, with no significant qualitative or strong quantitative variations. We, thus, have uncovered one more indication that the photoionization cross sections of the atom or its ion confined inside the endohedral fullerene anion depend little on the quantum structure of the excess charge on the fullerene cage.

Next, we present the results for calculated $\sigma_{1s}$ for He and He⁺ confined within the structureless and various structured fullerene anions with a giant C₂₄₀ carbon cage. These calculated $\sigma_{1s}$’s are shown in Figure 5.

The trends observed in the calculated $\sigma_{1s}$’s for the structureless and structured giant fullerene anions mirror those just discussed above for He and He⁺ confined within the fullerene anions associated with the C₆₀ fullerene cage. It is worth noting, however, that the differences in calculated $\sigma_{1s}$’s are significantly smaller in giant fullerene cases compared to the previous ones, which was anticipated.

The results presented in this section uncover that the photoionization cross sections of both the neutral atom and its ion encapsulated inside the fullerene anion cage are affected only little by the quantum structure of the excess charge on the cage.