Electronic, vibrational and optical properties of two-electron atoms and ions trapped in small fullerene-like cages

The fully optimized geometries of the considered systems at the B3LYP/6-31G(d, p) level of calculation are displayed in figure 1. As visually discernible, not all endohedral fullerenes contain the atom/ion in the center. This is clear in the cases of Be⁺⁺@C₂₀, Be⁺⁺@C₂₀H₂₀ and Li⁺@C₂₀H₂₀, for example. The calculated equilibrium distances, r_C–C, r_C–X and r_C–H, and also the average diameters, $\bar{D}$, of all these structures are reported in table 1, where we indicate the smallest and largest (in parentheses) distances calculated. Additional informations about the rotational constants and the SP values, obtained from equation (1), are given in table 2.

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Table 1. Calculated distances between atoms and cage average diameters (in Å) for the optimized geometries of the systems, at the B3LYP/6-31G(d, p) level of theory. Our results are compared with those obtained by Zhang et al [63]^a and Moran et al [45]^b .

Systems	${r}_{\mathrm{C}-\mathrm{C}}^{{< }}$ (${r}_{\mathrm{C}-\mathrm{C}}^{{ >}}$)	${r}_{\mathrm{C}-\mathrm{H}}^{{< }}$ (${r}_{\mathrm{C}-\mathrm{H}}^{{ >}}$)	${r}_{\mathrm{X}-\mathrm{C}}^{{< }}$ (${r}_{\mathrm{X}-\mathrm{C}}^{{ >}}$)	$\bar{D}$
C20	1.400 (1.536)	—	—	4.06
	1.41(1.52)a		—
He@C20	1.408 (1.535)	—	1.984 (2.105)	4.09
	1.41 (1.53)a	—	—
Li+@C20	1.409 (1.533)	—	1.964 (2.146)	4.09
Be++@C20	1.360 (1.398)	—	1.799 (2.434)	4.08
C20H20	1.556	1.094	—	4.36
	1.555b	1.092b	—
He@C20H20	1.564	1.094	2.192	4.43
	1.563b	1.092b	2.190b
Li+@C20H20	1.556 (1.580)	1.094 (1.105)	2.064 (2.347)	4.42
	1.570b	1.087b	2.200b
Be++@C20H20	1.563 (1.640)	1.084 (1.090)	1.722 (2.180)	4.44
	1.571b	1.087b	2.202b

We observe that our calculated values of r_C–C for C₂₀ and of r_C–C and r_C–H for C₂₀H₂₀ are in good agreement with those reported in the literature [45, 63]. In order to determine the average diameters of the cavities, we have determined the distance between two diametrically opposite carbon atoms. So, our calculated values are $\bar{D}\left({\text{C}}_{20}\right)=4.05$ Å and $\bar{D}\left({\text{C}}_{20}{\text{H}}_{20}\right)=4.36$ Å, which are also comparable to the values determined by others [45, 63]. Notice that these values are higher than the van der Waals radii of the atoms and also their ionic radii, which indicate that some atoms/ions can be favorably encapsulated in these cavities. In C₂₀ case, our calculated $\bar{D}$ exhibits an interesting decreasing order, i.e. $\bar{D}\left(\text{He}@{\text{C}}_{20}\right)=\bar{D}\left({\text{Li}}^{+}@{\mathrm{C}}_{20}\right){ >}\bar{D}\left({\text{Be}}^{++}@{\mathrm{C}}_{20}\right){ >}\bar{D}\left({\text{C}}_{20}\right)$, which is consistent with the number of electrons and ionization state in each guest atom. However, the same decreasing order does not hold for C₂₀H₂₀, and that is, in this case, $\bar{D}\left({\text{Be}}^{++}@{\text{C}}_{20}{\text{H}}_{20}\right){ >}\bar{D}\left(\text{He}@{\text{C}}_{20}{\text{H}}_{20}\right){ >}\bar{D}\left({\text{Li}}^{+}@{\text{C}}_{20}{\text{H}}_{20}\right){ >}\bar{D}\left({\text{C}}_{20}{\text{H}}_{20}\right)$.

Comparing the values of the smallest and largest distances between two neighbor atoms in C₂₀ and C₂₀H₂₀, it is noticed that in C₂₀ the difference reaches 13.6%, whereas in C₂₀H₂₀ the difference is only of 0.1%. As a consequence, C₂₀ exhibits a less symmetrical structure than C₂₀H₂₀, which can be verified by the calculated SP values. We obtain 0.1215 and 0.000 GHz⁻¹ for C₂₀ and C₂₀H₂₀, respectively (see table 2).

The confinement of atoms/ions X = He, Li⁺, and Be⁺⁺ in the C₂₀ and C₂₀H₂₀ cavities is performed by geometry optimization of the centered complexes without symmetry constraints. In this case, when He atom is placed in the center of the carbon cages, we notice a small increase of 0.8% in the r_C–C distances, in both cases, whereas the r_H–C distances exhibit a small decrease of around 0.1%, in the case of He@C₂₀H₂₀. The increase of the r_C–C distances leads to an increase of the average diameter of the cavities, while the He atom remains in the center. As it can be seen in table 2, the presence of the He atom does not alter the symmetry of C₂₀H₂₀, while in C₂₀ there is a decrease in the value of SP, i.e. SP(He@C₂₀) < SP(C₂₀).

For confined Li⁺ and Be⁺⁺ cases, the ions in the relaxed complexes exhibit off-center positions, and Be⁺⁺ is more off-center in both types of cages. Moreover, although these ions are both two-electron systems, they deform more the cages than the He atom, what leads to a significantly less spherical structure in the case of X@C₂₀H₂₀, X = Li⁺ and Be⁺⁺. Considering the endohedral systems, X@C₂₀ and X@C₂₀H₂₀, the behavior in the SP values are similar, i.e. there is an increase in SP values. However, when comparing with pure systems, we noticed that SP(C₂₀) > SP (X@C₂₀) while SP (C₂₀H₂₀) < SP (X@C₂₀H₂₀).

At this point, we analyze the dynamic stability and the inclusion energy of the atoms/ions in the cages. Our results are reported in table 3. The calculated vibrational frequencies of these systems indicate that all these structures have found their stationary points as a true minimum. Thus, we report in table 3 only the lowest vibrational frequency (ω₁) of each system, while vibrational spectra are discussed later. We realized that the inclusion of an He atom or a Li⁺ ion in the C₂₀ cavity yields to a systematic blue shift in ω₁ of 27 and 72 cm⁻¹ respectively, whereas the inclusion of a Be⁺⁺ ion yields to a red shift of 5 cm⁻¹. On the other hand, in the case of X@C₂₀H₂₀, only the spatial confinement of an He atom yields a blue shift of 22 cm⁻¹ in this vibrational mode. Now, the inclusion of Li⁺ and Be⁺⁺ yields to red shifts of 5 and 295 cm⁻¹. We compare our calculated values with those obtained by Moran et al [45].

Naturally, systems that can be superimposed on a dodecahedron have I_h symmetry type, and two good examples are fullerenes C₂₀ and C₂₀H₂₀. However, in the case of endohedral systems, it must be observed whether the confined atom is located on a point or axis of symmetry. Among the systems presented by Moran et al [45], the one that contains encapsulated Be⁺⁺, has no minimum with I_h symmetry (imaginary frequency, see table 3). The minimum is obtained considering the symmetry C_5v , with the encapsulated atom located on the axis of rotation, close and centered to one of the C₂₀H₂₀ faces, with distances ${r}_{\mathrm{X}-\mathrm{C}}=1.735\enspace \circ{A}$. In our calculations, as we do not restrict the symmetry of the system, we have obtained a local minimum with Be⁺⁺ offset from the center of the cavity, and slightly offset from the center of the nearest face. This difference in methodology leads us to different values for the fundamental frequency ω₁, for both Be⁺⁺@C₂₀H₂₀ and Li⁺@C₂₀H₂₀.

In this context, we must remember that the imposition of symmetry represents a constraint in the variational process and, consequently, an increase in energy [64]. Thus, if we do not impose symmetry constraint on the geometry optimization process, we can obtain a lower energy and, therefore, a better variational value, being necessary, however, to verify if the obtained wave functions describe the spectroscopic states well. Note, however, that when placing the X atom or ion in the cage, the final configuration can change the system symmetry, and in this case, the optimization without the symmetry restriction can give this new configuration. In fact, for C₂₀H₂₀ and He@C₂₀H₂₀ systems cases, we could observe by comparing the values of the first vibration frequency ω₁, a good agreement between our values and those obtained by Moran et al [45]; in particular, for the He@C₂₀H₂₀ case, the optimized structure without the symmetry restriction meets the He atom in the center of the cavity, which coincides with the optimized system using the I_h symmetry restriction. For Li⁺ and Be⁺⁺ ions, our values differ from those presented by Moran et al [45] precisely because, in our geometry optimization process, these ions are located outside a point or axis of symmetry. It is also important to note that for the results obtained by Moran et al for Be⁺⁺ using the I_h and C_5v symmetry constraints (see table 3), the values of ω₁ already differ considerably, and the I_h symmetry constraint does not lead to a global minimum structure.

To evaluate the inclusion energy of the atoms/ions inside the cavities, we have employed equation (2), taking into account both the BSSE effect and the ZPVE correction. From these results, we obtain that He@C₂₀, He@C₂₀H₂₀ and Li⁺@C₂₀ should not be energetically favorable. In these two series, only Be⁺⁺@C₂₀, Li⁺@C₂₀H₂₀, and Be⁺⁺@C₂₀H₂₀ appear to be favorably formed, since their inclusion energies are, respectively, −200.7, −15.2, and −254.5 kcal mol⁻¹. As we can see, Be⁺⁺ exhibits the highest inclusion energies in both cavities, with values in the same order of typical ionic bonds.

Using equation (3), we have calculated the deformation energies (see table 3). We observed that in both structures there is an increase in E_def with the atomic number of the inserted atoms/ions. We can highlight that the He atom in C₂₀H₂₀, has the lowest E_def value, indicating a small deformation in the structure, as previously discussed, while Be⁺⁺ in C₂₀H₂₀ presents the highest deformation energy among the studied systems, indicating a greater deformation in the structure.

To examine the ground state electronic structure of the studied systems, we analyze first the frontier molecular orbitals energies − the highest occupied (HOMO) and the lowest unoccupied (LUMO) − and also the HOMO–LUMO energy differences (_HL), for each system, as reported in table 4. In addition, as an important parameter to investigate the electronic density of these quasi-spherical systems, we calculate the absolute hardness (η) [65], which can be estimated from these frontier energies and is also reported in table 4.

Table 4. Frontier energies (HOMO, LUMO and _HL, in eV) and hardness η (in eV) of the X@C₂₀ and X@C₂₀H₂₀ systems with X = He, Li⁺ and Be⁺⁺, at the B3LYP/6-31G(d, p) level of theory.

Systems	HOMO	LUMO	HL	η
C20	−4.97	−3.02	1.95	0.98
He@C20	−4.99	−3.16	1.83	0.92
Li+@C20	−10.00	−8.22	1.78	0.89
Be++@C20	−14.89	−13.13	1.76	0.88
C20H20	−6.99	0.98	7.97	3.99
He@C20H20	−7.01	2.07	9.08	4.54
Li+@C20H20	−11.27	−2.92	8.35	4.18
Be++@C20H20	−15.25	−9.54	5.71	2.86

It is interesting to notice that the inclusion of an He atom in C₂₀ reduces _HL by 0.1 eV, while in C₂₀H₂₀ it raises by 1.1 eV. Similarly, Li⁺ also reduces _HL by ∼0.2 eV in C₂₀ and increases by ∼0.4 in C₂₀H₂₀. On the other hand, the presence of Be⁺⁺ reduces _HL by ∼0.2 and ∼2.3 eV in both cages.

In figures 2 and 3, we display the calculated total state density (DOS) for the two series of systems, i.e. X@C₂₀ and X@C₂₀H₂₀. We have presented a comparison of the DOS of the C₂₀ and C₂₀H₂₀ systems (in pink) with the DOS of the compounds X@C₂₀ and X@C₂₀H₂₀ where (a) X = He (in black), (b) X = Li⁺ (in blue) and (c) X = Be⁺⁺ (in green). The dotted vertical lines indicate the positions of the HOMO and LUMO orbitals. For the He@C₂₀ system it is noted that the pattern of the DOS graph changes little when compared to the graph of C₂₀: the energy of the HOMO orbital is practically the same as that of C₂₀ (0.4% lower) and the energy of the LUMO orbital shows a ∼4.6% decrease; as a result, we have a _HL = 1.83 eV. In the case of the two systems Li⁺@C₂₀ and Be⁺⁺@C₂₀ it is observed that there is a displacement of the HOMO and LUMO orbitals for smaller energy values; it is also observed that the value of _HL decreases in both systems with _HL = 1.78 eV for Li⁺@C₂₀ and _HL = 1.76 eV for Be⁺⁺@C₂₀. These _HL values indicate that the compounds tend to be more reactive than C₂₀.

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When we compare C₂₀ and C₂₀H₂₀ we note that _HL of C₂₀H₂₀ is four times greater than _HL of C₂₀, that is, C₂₀H₂₀ is less reactive than C₂₀; it is also noted that the ionization potential of C₂₀H₂₀ is smaller than the ionization potential of C₂₀ indicating a greater difficulty in donating electrons. For the He@C₂₀H₂₀ system, we have obtained that _HL is 13.9% greater than in C₂₀H₂₀ case; this indicates that He@C₂₀H₂₀ is less reactive than C₂₀H₂₀. The energy of the HOMO orbital is practically the same as that of C₂₀H₂₀; the LUMO orbital, however, in the He@C₂₀H₂₀ system has energy 1.09 eV above the value found in the C₂₀H₂₀ system. In the case of compounds Li⁺@C₂₀H₂₀ and Be⁺⁺@C₂₀H₂₀ the HOMO energies show a decrease of 4.28 eV and 9.26 eV, respectively, when compared to the value for C₂₀H₂₀; an increase in the ionization potential is thus noted with the increase of the atomic number Z of the ion. Regarding the value of _HL, we notice that _HL(Li⁺@C₂₀H₂₀) < _HL(He@C₂₀H₂₀), _HL(Li⁺@C₂₀H₂₀) > _HL(C₂₀H₂₀) and Be⁺⁺@C₂₀H₂₀ has the smallest _HL of the studied systems.

The absolute hardness η is a very useful concept to understand the reactivity of a system. The higher its value is less reactive the system will be, thus being more stable. Our results indicate that, for compounds X@C₂₀, the reactivity increases with the atomic number Z of the confined atom/ion, whereas for X@C₂₀H₂₀ systems, those with X = He and Li⁺ are less reactive than C₂₀H₂₀. In this series, the system with Be⁺⁺ is the most reactive.

We have also calculated the vibrational frequencies and presented infrared spectra (IR) graphs as a function of frequency in figures 4 and 5. As a result, we can observe how the infrared spectrum profile is modified due to the insertion of the atom or ion in the C₂₀ and C₂₀H₂₀ cavities; the rotational constants A, B, C, the moments of inertia I_A , I_B , I_C and the SP are shown in table 2. As we can see in figure 4, the peaks located in the frequency range 1.200–1.500 cm⁻¹ are attenuated for increasing Z of the atom/ion inserted in C₂₀ cavity; this result may be associated with the increase in the moment of inertia of the studied systems (see table 2). A fact to note is that A ≠ B ≠ C which implies that the structure of X@C₂₀ is deformed, i.e. the lengths r_C–C are different.

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In figure 5 we can analyze the behavior of the frequency when the atom/ion is inserted into the C₂₀H₂₀ cavity. The frequency range 3.000–3.500 cm⁻¹ refers to stretch vibration of hydrogen atoms in X@C₂₀H₂₀ system; we can observe that there is a decrease in the infrared spectrum only when the Li⁺ and Be⁺⁺ ions are inserted; the peaks that appear in the frequency range 0–1.500 cm⁻¹ refer to the oscillations of the Li⁺ and Be⁺⁺ ions inside the cavity. By analyzing the data in table 2, it can be seen that the values of the moment of inertia of X@C₂₀H₂₀ are higher than in C₂₀H₂₀, and considering the constants A, B, C and SP values we have that there is structural deformation for Li⁺@C₂₀H₂₀ and Be⁺⁺@C₂₀H₂₀ systems.

In order to study the optical absorption of C₂₀ and X@C₂₀ systems, we have performed the calculation of the first 150 excitations, while for C₂₀H₂₀ and X@C₂₀H₂₀ systems, we have performed the calculation of the first 200 excitations.

Our results are presented in tables 5 and 6 for C₂₀ series and in table 7 for C₂₀H₂₀. In these tables, we have presented the main transitions with the transition energy ΔE, the λ wavelength, oscillator strength f and the transitions with the respective weight (in percentages).

For C₂₀ and X@C₂₀ systems, the absorption spectra have been shown in figure 6. In C₂₀ spectrum, we have observed that, in the studied wavelength range, we can highlight 5 absorption peaks with allowed transitions; the highest intensity peak (f = 0.5113) occurs for λ = 231.2 nm and ΔE = 5.362 eV; we will adopt this peak as a reference. Introducing the He atom into the cavity, we can still identify 5 main peaks but shifted to longer wavelengths, that is, with smaller energies; in fact, the peak of greatest intensity in He@C₂₀ appears at λ = 235.5 nm, with f = 0.4659 and ΔE = 5.264 eV. In the Li⁺ ion case, the peak displacement is for shorter wavelengths, i.e. for larger energies; thus, in the compound Li⁺@C₂₀, the peak of absorption of greater intensity occurs for λ = 228.3 nm and ΔE = 5.431 eV. When the Be⁺⁺ ion is introduced in the cavity, the absorption pattern changes more significantly and a new transition is allowed for λ = 350.0 nm, ΔE = 3.542 eV and f = 0.0221; the peak of greatest intensity in the Be⁺⁺@C₂₀ system has λ = 224.9 nm and ΔE = 5.512 eV. Comparing C₂₀ with Be⁺⁺@C₂₀, in Be⁺⁺@C₂₀ the peak occurs for shorter wavelength values and therefore greater energy.

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The spectra for the C₂₀ and X@C₂₀ systems, show absorption in the ultraviolet (UV) range. The highest intensities occur in the UVC and UVB regions. With the encapsulation of atoms and ions, we can notice an increase in absorption in the UVA region, due to the atomic number Z. The less intense peaks occur in the vacuum UV region. Highlight for Be⁺⁺@C₂₀, which presents a significant peak, in the UVA region, close to the visible region (λ = 350 nm and f = 0.021).

Figure 7 shows that, in the studied wavelength range, the C₂₀H₂₀ system presents two absorption peaks with allowed transitions: one with λ = 112.3 nm, ΔE = 11.03 eV and f = 0.2125 and the other with λ = 110.9 nm, ΔE = 11.181 eV and f = 0.1164. When inserting the X = He atom in the cavity, the absorption spectrum remains with two peaks but the peaks are slightly shifted to values of greater wavelength; then we have, in He@C₂₀H₂₀, λ = 112.8 nm, ΔE = 10.99 eV, f = 0.2039 and λ = 111.2 nm, ΔE = 11.153 eV and f = 0.1028 respectively. We can associate this behavior of the systems with the similarity observed when comparing the electronic structure of C₂₀H₂₀ and He@C₂₀H₂₀ i.e. the state distributions (see figure 3).

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The insertion of X = Li⁺ into the cavity shows a change in the absorption pattern; there is in Li⁺@C₂₀H₂₀ a peak of lower intensity for λ = 125.4 nm, ΔE = 9.88 eV, f = 0.184 and another with a higher intensity for λ = 120.4 nm, ΔE = 10.31 eV and f = 0.184. For the Be⁺⁺@C₂₀H₂₀ system, there are 6 absorption peaks: two peaks with partially allowed transitions (λ = 167.3 nm, ΔE = 7.41 eV, f = 0.0095 and λ = 119.6 nm, ΔE = 10.36 eV, f = 0.0045) and the others highlighted in the figure 7 with permitted transitions. A fact to note is that in the Be⁺⁺@C₂₀H₂₀ system we have absorption processes with lower intensities than in other systems, but the range in which this compound absorbs energy and presents transitions is greater. We can see in our calculations that, for the C₂₀H₂₀ and X@C₂₀H₂₀ systems, the absorption spectrum is found in vacuum UV region.

Next, in order to complete the analysis of the optical properties of the X@C₂₀ and X@C₂₀H₂₀ systems we have used the partial state density (PDOS) graphs, i.e. we have highlighted the contribution of the atom or ion X in the total state density (DOS). In this way, we have identified in which system orbitals the trapped atom or ion contributes more significantly and consequently we have analyzed which transitions involve such orbitals.

The graphs of the PDOS for the X@C₂₀ systems are shown in figure 8. For X = He it is noted that the helium atom contributes more significantly to the densities of the H-14, H-30 and H-40 orbitals; in the range of energy we use, there are no contributions from He to the unoccupied orbitals; comparing the data in figure 8 with those in table 5, the permitted transitions do not involve the orbitals of that atom. For ion X = Li⁺ we can see that the orbitals of that ion contribute to move the orbitals of the compound Li⁺@C₂₀ to lower energies; analyzing figure 8, it can be seen that the contribution of this ion is greater in the density of the H-36 orbital and that the orbitals from H-1 to H-5 also contain contributions from the density of ion states; by analyzing it together with table 6, we can verify that the Li⁺ orbitals contribute directly to the electronic transitions related to ΔE equal to 4.288 eV, 4.849 eV and 5.431 eV. These excitations correspond to electronic transitions between the states of the lithium ion and carbon atoms in C₂₀.

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For Be⁺⁺, we can see that this ion's orbitals contribute to the system's orbitals: H-1, H-5, H13 and H-23. This way, we can verify that, among the six possible transitions presented for the Be⁺⁺@C₂₀ system, three of them present contributions of this ion: ΔE equal to 3.543 eV, 5.512 eV and 5.601 eV. So, we can infer that, there may be electronic transitions between the Be⁺⁺ orbitals and the C₂₀ orbitals.

The PDOS graphs for X@C₂₀H₂₀ systems are shown in figure 9; we can observe that when inserting the atom X = He in C₂₀H₂₀ cavity, the atom contributes more significantly in the densities of the occupied orbitals H-29, H-41, H-50 and in the unoccupied orbital L + 53; we also noticed that in the analyzed energy range there are no permitted transitions involving these orbitals. When the Li⁺ ion is inserted into the cavity, it can be seen that its orbitals contribute mainly to the densities of the unoccupied orbitals LUMO, L + 1, L + 2, L + 3, L + 4, L + 5 and L + 6; it is also noted that in the studied energy range these orbitals are involved in the allowed transitions; the same occurs with the insertion of Be⁺⁺ in the C₂₀H₂₀ cavity: the ion orbitals contribute significantly to the densities of the LUMO, L + 1, L + 2, L + 3, L + 4, L + 5, L + 6, L + 7, L + 8, L + 9, L + 10 and L + 11 orbitals, and these orbitals are involved in the permitted transitions of Be⁺⁺@C₂₀H₂₀. In both cases, transitions from the C₂₀H₂₀ orbitals to the encapsulated ions are possible.

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One way to validate the results obtained for the optical properties discussed here, would be a comparison with spectroscopic data obtained experimentally. However, such data are still scarce in the literature. The comparison between theoretical and experimental results often requires an adaptation in theory, in order to obtain values that can be obtained via experiment. As an example, to obtain UV absorption spectra, we used here the usual TDDFT of the linear response approach (first order) of the system when it is subjected to a weak field, as implemented in GAMESS-US package [54]. Now, some authors [66, 67] have been working to develop methods, from the extension of TDDFT, for the description of non-linear optical phenomena and spectral properties of excited states; in this way, theoretical results can be compared with experiments of transient absorption spectroscopy. This type of analysis, together with an analysis of the electronic population of the states, can elucidate, for example, the behavior of the endohedral systems presented here and similar.