Fullerenes Pose a Strain on Hybrid Density Functional Theory.

The computational modeling of fullerenes plays a fundamental role in designing low-dimension carbon nanostructures. Nevertheless, the relative energies of fullerenes larger than C 20 and C 24 have not been comprehensively examined by means of highly accurate ab initio methods, for example, the CCSD(T) method. Here we report such an investigation for a diverse set of 29 C 40 isomers. We calculate the energies of the C 40 fullerenes using the G4(MP2) composite ab initio method, which approximates the CCSD(T) energy in conjunction with a triple-ζ-quality basis set (CCSD(T)/TZ). The CCSD(T)/TZ isomerization energies span 43.1-763.3 kJ mol -1 . We find a linear correlation ( R 2 = 0.96) between the CCSD(T)/TZ isomerization energies and the fullerene pentagon signatures ( P 1 index), which reflect the strain associated with fused pentagon-pentagon rings. Using the reference CCSD(T)/TZ isomerization energies, we examine the relationship between the percentage of exact Hartree-Fock (HF) exchange in hybrid density functional theory (DFT) methods and the pentagon-pentagon strain energies. We find that the performance of hybrid DFT methods deteriorates with the pentagon-pentagon strain energy. This deterioration in performance becomes more pronounced with the inclusion of high amounts of HF exchange. For example, for B3LYP (20% HF exchange), the root-mean-square deviation (RMSD) relative to G4(MP2) increases from 8.9 kJ mol -1 for the low-strain isomers ( P 1 = 11) to 18.0 kJ mol -1 for the high-strain isomers ( P 1 > 13). However, for BH&HLYP (50% HF exchange) the RMSD increases from 23.0 ( P 1 = 11) to 113.2 ( P 1 > 13) kJ mol -1 . A similar trend is observed for the M06/M06-2X pair of functionals. Namely, for M06 (27% HF exchange) the RMSD increases from 0.8 ( P 1 = 11) to 21.0 ( P 1 > 1 3) kJ mol -1 , whereas for M06-2X (54% HF exchange) the RMSD increases from 16.7 ( P 1 = 11) to 77.7 ( P 1 > 13) kJ mol -1 . Overall, we find that the strain associated with pentagon adjacency is an inherently challenging problem for hybrid DFT methods involving high amounts of HF exchange and that there is an inverse relationship between the optimal percentage of HF exchange and the pentagon-pentagon strain energy. For example, for BLYP the optimal percentages of HF exchange are 13% ( P 1 = 11), 10% ( P 1 = 12), 7.5% ( P 1 = 13), and 6% ( P 1 > 13).