Phenomenological model of the thermodynamic functions for energetically most stable isomers of fullerenes
In order to reveal their structural, thermodynamic and spectral regularities, we selected the most energetically favourable fullerenes C <sub>n</sub> with carbon number n from 20 to 720 (n being an even number) among a large variety of hypothetically possible closed carbon clusters using the quantum-chemical approximation PBE /2ζ. The resultant set of molecules included fullerenes which did not obey the isolated pentagon rule (C<sub>20</sub>-I <sub>h</sub> (D<sub>3d</sub>), C<sub>24</sub>-D<sub>6 d </sub> (D<sub>2</sub>), <sup>5</sup>C<sub>26</sub>-D<sub>3 h </sub> (C<sub>2 v </sub>), <sup>5</sup>C<sub>28</sub>-T <sub>d</sub> (C<sub>1</sub>), C<sub>30</sub>-C<sub>2 v </sub>-3, C<sub>32</sub>-D<sub>3</sub>-6, C<sub>34</sub>-C<sub>2</sub>-5, C<sub>36</sub>-D<sub>6 h </sub>-15, C<sub>38</sub>-C<sub>2</sub>-17, C<sub>40</sub>-D<sub>2</sub>-38, C<sub>42</sub>-D<sub>3</sub>-45, C<sub>44</sub>-D<sub>2</sub>-75, C<sub>46</sub>-C<sub>2</sub>-109, C<sub>48</sub>-C<sub>2</sub>-171, C<sub>50</sub>-D<sub>3</sub>-270, and C<sub>52</sub>-C<sub>2</sub>-422) and those which complied with this rule (C<sub>60</sub>-I <sub>h</sub>, C<sub>70</sub>-D<sub>5 h </sub>, C<sub>72</sub>-D<sub>6 d </sub>, <sup>3</sup>C<sub>74</sub>-D<sub>3 h </sub>, C<sub>76</sub>-D<sub>2</sub>, C<sub>78</sub>-C<sub>2 v </sub>-3, C<sub>80</sub>-D<sub>5 d </sub>-1, C<sub>82</sub>-C<sub>2</sub>-3, C<sub>84</sub>-D<sub>2</sub>-22, C<sub>86</sub>-C<sub>2</sub>-17, C<sub>90</sub>-C<sub>2</sub>-45, C<sub>92</sub>-D<sub>3</sub>-28, C<sub>94</sub>-C<sub>2</sub>-43, C<sub>96</sub>-C<sub>2</sub>-181, C<sub>98</sub>-C<sub>3</sub>-148, C<sub>100</sub>-C<sub>1</sub>-173, C<sub>120</sub>-T <sub>d</sub>, C<sub>180</sub>-I <sub>h</sub>, C<sub>240</sub>-I <sub>h</sub>, <sup>3</sup>C<sub>260</sub>-C<sub>3</sub>, <sup>3</sup>C<sub>320</sub>-D<sub>3 d </sub>, <sup>3</sup>C<sub>500</sub>-D<sub>3 d </sub>, C<sub>540</sub>-I <sub>h</sub>, and C<sub>720</sub>-I <sub>h</sub> ) (all the structures were denoted as in the Atlas of Fullerenes and the multiplicity of the ground state was indicated if different from the singlet one). The validity of this series of compounds selected for further theoretical studies was confirmed by the fact that it appeared to include a whole family of fullerenes already obtained and experimentally identified. In addition, the sampled set included the experimentally confirmed C<sub>62</sub>-C<sub>2 v </sub> structure which satisfied the isolated penta- and tetragon rule. Detailed studies were carried out for the energetically most stable or a trifle less stable C<sub>20</sub>-C<sub>720</sub> fullerene isomers using a more complete and flexible triple split-valence basis set in the PBE / 3ζ approximation. The values of some thermodynamic functions (Gibbs energy, enthalpy, entropy) for standard conditions were calculated. For the molecules with a singlet state as the ground one, a phenomenological model was constructed for the dependence of the standard enthalpy of formation on the number of carbon atoms described fairly well by the function Δ <sub>f</sub>H <sup>0</sup><sub>298</sub>(C <sub>n</sub> ) = (-8.826+43.888·e<sup>-9.046·</sup><sup>α</sup>+6.836·e<sup>-9.331·</sup><sup>α</sup>+19.165·e<sup>-2.236·</sup><sup>β</sup>)· n. The parameters of the model α = n / 180 and β = n / 1500 can be interpreted as the degree of steric strain of the fullerene framework. Analysis of the constants of the phenomenological model allowed us to conclude that fullerenes C<sub>60</sub>, C<sub>70</sub>, C<sub>180</sub> and C<sub>1500</sub> hold a special place among all hypothetically possible energetically stable structures. Thus, the effect of the steric factor α reduces substantially in the range between C<sub>60</sub> and C<sub>70</sub> and becomes vanishingly small starting from n = 180. The relevance of the second factor β extends to approximately n = 1500. With a further increase in the number of atoms n, one should expect the energy characteristics of fullerenes to obey the simple additive model. The energy of singlet-triplet splitting was calculated for the entire set of selected energetically favourable fullerenes. As is shown, the electronic ground state of C<sub>260</sub>-C<sub>3</sub>, C<sub>320</sub>-D<sub>3 d </sub>, and C<sub>500</sub>-D<sub>3 d </sub> structures is of triplet multiplicity probably indicative of their low kinetic stability.