On the Ionisation Tolerance of C₂₀ Fullerene in Ground and Excited Electronic States in Planetary Nebulae

1. Introduction

The formation of the smallest member of the fullerene family C${}_{20}$ (with 12 pentagon rings) in the gas phase has been reported by Pinzach et.al in 2000 [1,2] via chemo-synthetic procedure starting from C${}_{20}$H${}_{20}$, the fully hydrogenated form of C${}_{20}$. Shortly after that synthesis of C${}_{20}$ in a solid state was achieved under laboratory conditions by Wang et al. [3] and Iqbal et al[4]. Recently Torrisi et al [5] identified the formation of this species in the hot plasma of carbon atoms generated by laser, a condition that resembles the central stars in planetary nebulae. These interesting astronomical sources [6] are one of the sources of mysterious Unidentified Infrared Emission (UIE) bands attributed to the formation of complex organic molecules [7,8] including fullerenes [9,10] and fulleranes [11,12]. Although in terms of its chemical properties such as molecular structure, chemical bonding, chemical reactivity, and lifetime, C${}_{20}$ fullerene is a focus for both theoretical [13] and experimental studies [14], in astrochemistry of fullerenes in space its status looks like a Darwinian lost species. Any knowledge about the stability of C${}_{20}$ (here we mean the lifetime of a molecule in space before undergoing destruction by radiation) would not only significantly assist the efforts of detecting it but also shed light on the formation processes of other fullerenes such as C${}_{60}$ and C${}_{70}$ in astronomical sources [10]. The presence of C${}_{20}$ alongside other small members of the fullerene family such as C${}_{24}$ and C${}_{26}$ has been speculated in planetary nebulae using theoretically calculated IR spectra [13,15]. A glance at the latest census of confirmed interstellar molecules [16] reveals that only a small number of pure carbon clusters such as C${}_2$, C${}_3$, C${}_5$, C${}_{60}$, C${}_{60}$${}^{+}$ and C${}_{70}$ have been detected. Other carbon clusters and members of the fullerene family including C${}_{20}$ still to be found. In comparison to C${}_{60}$ with the positive heat of formation of 2520.0 ± 20.7 kJ.mol${}^{-1}$[17] C${}_{20}$ fullerene also show positive heat of formation (2358.2 ± 8.0 kJ. mol${}^{-1}$[18], however it is less positive than C${}_{60}$ by 161.8 kJ.mol${}^{-1}$. This delivers slightly more thermodynamic stability to C${}_{20}$ that can potentially favor its formation in astronomical sources. Theoretically, the C${}_{20}$ fullerene shows high thermal stability in the gas and solid phases up to T= 3000 K before starting to convert to other isomers [19,20]. This is contrary to the existence of classical strain energy in this molecule [19]. This implies that such molecular species could survive in the outer regions of PN envelopes where the central stars UV radiation field is diminished and where dust and other molecular species in the form of globules and condensations can act as a shade mechanism. In the context of the electronic structure, it was found that C${}_{60}$ fullerene shows an exceptional characteristic for rearrangement of its electron density that led to the survival of a network of chemical bondings between carbon atoms at a high degree of ionization [22]. In laboratory conditions and theoretical calculations, C${}_{60}$ cations with a positive charge of as high as $+12$[21] and $+26$[22] have been reported respectively. In the case of C${}_{20}$ we anticipate qualitatively lower tolerance toward ionization and excitation. However, we also look at scenarios that C${}_{20}$ undergoes a low, moderate, and high degree of electronic excitation without ionization. In this way, we explore the structural stability of this molecule under the extremely high electronic multiplicity. A comprehensive report on the high thermal, mechanical stabilities and other interesting molecular properties such as superconductivity of neutral C${}_{20}$ has been published by Fei Lin et al [23]. Considering all these backgrounds the present work aims to examine molecular structure responses of C${}_{20}$ to extreme radiation conditions of planetary nebulae via first principle methods of quantum chemistry.

2. Quantum Chemical Computations

In this study, three series of systematic quantum chemical computations have been conducted on C${}_{20}$ fullerene (12 pentagons) and its cations within the density functional theory (DFT) framework. In all computations, molecular geometries were optimized, and stationary points located on the potential energy surfaces were characterized by subsequent calculations of the Hessian matrix (second derivative of energy function). The B3LYP hybrid functional was applied in the restricted (R) and unrestricted (U) framework for singlet and higher electronic multiplicities, respectively. The polarisation-consistent basis set (PC 1) that contains S, P, and D type basis functions for carbon atoms was integrated into the B3LYP framework. The first series of computations were conducted for understanding the ionization tolerance of C${}_{20}$. The general formula of species involved in this series is C${}_{20}$${}^{q+}$ $q=0-14$. Singlet and doublet electronic multiplicities were set for species with even and odd values for q, respectively. The second series of computations was performed on the electronic excitation of local minimum geometries found in the first series to their next higher multiplicity. Species at singlet and doublet electronic states were studied at triplet and quartet excited states, separately. The third series of computations were carried out on neutral C${}_{20}$ and at excited electronic states associated with the multiplicity of 3 to 21 (two to twenty unpaired electrons). All electronic structure computations in this work have been performed using PC-GAMESS 8.2.0 (known as Firefly) Quantum Chemistry (QC) package [24] and without imposing symmetry constraint on geometry optimization. Firefly QC package is partially based on the GAMESS (US) [25] source code. GAMESS stands for General Atomic and Molecular Electronic Structure System. To explore the network of chemical bonding, the electron densities of all local minimum geometries were then analyzed within the framework of the Quantum Theory of Atoms in Molecules (QTAIM). The AIMALL package [26] was utilized for this purpose.

3. Discussion

3.1. Ionisation Tolerance and Electronic Excitation

We have reported previously that the C${}_{60}$ fullerene ionization tolerance limit (ITL) stands at a charge of $+26$[22]. With one-third of the number of carbon atoms, one may estimate the ITL limit for C${}_{20}$ as $+8$ or $+9$. The results of our quantum chemical calculations (at the same level that was applied to C${}_{60}$) combined with chemical bonding analysis reveal that this limit stands at $+13$ for C${}_{20}$. No stationary point on the Potential Energy Surface (PES) of C${}_{20}$${}^{14+}$ at the singlet spin state could be found. Similar to C${}_{60}$, the total electron density in C${}_{20}$ fullerene also shows a remarkable ability to redistribute itself to prevent the disintegration of the molecule at highly positively charged states. In Table 1 electronic multiplicity, dipole moments, and lowest harmonic vibrational frequency of all C${}_{20}$${}^{q+}$ (local minimum geometries) are listed. The chemical bonding analysis via QTAIM confirms that the network of bonding between carbon atoms and thus the cage structure of the molecule is well preserved for all highly positively charged C${}_{20}$ cations (Figure 1). We use the amount of the total electron density at the specific locations within the molecule to explore quantitatively the changes that occur upon ionization. These locations are marked by the formation of specific critical points that show how neighboring atoms are interacting with each other individually (Bond Critical Point-BCP), which group of interacting atoms are forming a ring (Ring Critical Point-RCP), and finally if a group of the rings can form a cage structure (Cage Critical Point-CCP)) (Figure 1). Our calculated values of electron density at these locations show that during the ionization process from C${}_{20}$ to C${}_{20}$${}^{13+}$ the electron density at pentagon rings decreases by 40% on average (0.02 amu/0.05 amu, where amu is atomic mass unit). This value stands around 50% (0.0028 amu/0.0053 amu) when calculated at the cage critical point. This remarkable ability of electron density to maintain the network of bonding and cage structure at the condition of electron density deficiency underlies the high ionization tolerance. It is anticipated that this characteristic is shown by other members of the fullerene family with different sizes. Considering this common behavior, their contribution to carrying elemental carbon to different locations in space would significantly increase. This will lead us to estimate the fractions of carbon mass, produced in PNs and deposited in each of allotropic forms (graphite, diamond, fullerene, and nanotube) in addition to atomic, diatomic, and carbon clusters. Each of these species may be a starting reactant for a certain class of astrochemical reaction(s) (or chain of reactions) and therefore molecular products (some perhaps are bio-molecules) with specific stereochemistry (for instance Cyclic, chain, optically active). Since the synthesis of all-metal fullerene (in form of inorganic complex made of Sb and Au) has been reported recently [27] it is an open question if there are any privileges that have been naturally given to such molecular geometry for carrying the other elements throughout the universe.

In order to confirm theoretically the assumed electronic multiplicities as the electronic ground states for C${}_{20}$${}^{q+}$ species listed in Table 1, we conducted a series of computations on the next upper electronic multiplicity state. These data are presented in Table 2. All species in Table 2 are characterized as the local minimum geometries at their corresponding electronic states. Cage structure is maintained for all these species as determined by QTAIM calculations. We observed energy crossover between the Potential Energy Surface (PES) of single-triplet and doublet-quartet states for C${}_{20}$${}^{q+}$($q=4-10$). The crossover is between 1-8 kcal/mol in favor of higher multiplicity states i.e. triplet and quartet. We have combined the data in Table 1 and Table 2 to calculate the energetics of step-wise ionization of C${}_{20}$. The calculated step-wise Ionisation Potentials (IPs) corrected to zero point energy, their correspondence spectral region wavelengths, and electronic states are presented in Table 3. Similar to the trend reported for C${}_{60}$ IP values (7.80-82.44 $eV$) [22], C${}_{20}$ values also cover the range between 7.35-65.99 $eV$. This demonstrates that this smallest member of the carbon fullerene family also has a chance to survive vacuum and extreme UV conditions.

3.2. Higher Electronic Multiplicity

High spin states are common phenomena among the inorganic molecular clusters such as transition metals. Here we conducted quantum mechanical calculations for the scenario that led to the electronic states on a neutral C${}_{20}$ with more than two unpaired electrons (triplet states). In this way, information on the responses of this molecule in abnormally high excitation states is explored. The argument may have arisen that such states have extremely short lifetimes and the molecule may undergo disintegration before reaching such states. However, because molecular species are subject to unknown and often extreme conditions in the space environment that can range from low to high levels of temperature, pressure, radiation, density, electric and magnetic fields, etc, the opposite effects can be expected. This is like what we observe for the stability of very reactive species such as C${}_2$ in space. Table 4 presents the results of the theoretical calculations on ultra-high spin electronic states of C${}_{20}$. The QTAIM calculations confirm that all local minimum geometries found from spin multiplicity of 3 to 17(corresponding to 2-16 unpaired electrons) possess the cage structure as C${}_{20}$ (singlet). At the spin multiplicity of 19, the 12 pentagon structure turns into a 10 Pentagon structure with 2 opened/elongated pentagons connected by a long-range weak bonding interaction between two carbon atoms C11-C17 (Figure 2). At a multiplicity of 21 (20 unpaired electrons), the bonding between C11 and C17 is ruptured and as a result, the structure changes into the opened cage with 10 pentagon rings (Figure 2). Despite the significant differences in the spin multiplicity across the species in Table 4, the electron density values at RCPs and CCPs shown in Figure 2 are comparable. Energetics of such processes are listed in Table 4, Their values are comparable to the ionization energies listed in Table 3. This means that the two processes of ionization and excitation are competing with each other within the energy range of 8-23 eV. For instance according to our theoretical values in Table 3 and Table 4 a photon of 7.38 eV can lead to the ionisation of one electron or with slightly higher energy at 8.14 eV to an ultra-high spin C${}_{20}$.

4. Conclusion

The C${}_{20}$ fullerene with its 12 pentagon structure shows a significant tolerance against disintegration upon ionization as well as excitation to high spin states. This is underpinning its abornormal thermal stability [23]. Cage structure is preserved in cationic forms in both the ground and excited potential energy surfaces and up to the charge limit of $q=+13$. In both cationic or radical forms, we found a few cases in which the molecule adopts the dipole moment to be detected by its rotation spectroscopic signature. With its ease of formation in the solid state reported in laboratory conditions and the stability of its electronic structure we anticipate the detection of this smallest member of the fullerene family, its radical or cationic forms in the physical conditions present in planetary nebulea. Our study of changes in molecular bonding and structure at ultra-high spin conditions shows that only a few chemical bonds are broken in C${}_{20}$ at electronic multiplicity higher than 17 (16 unpaired electrons). This gives an extra chance of surviving to C${}_{20}$ in the harsh thermal, turbulence, and radiation conditions in planetary nebulae. The results of our modeling encourage the search for this important and interesting molecule in astrophysical objects.