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.

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.

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.