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We applied density functional theory (DFT) to study three polycyclic aromatic compounds (PAHs), corannulene, coronene, and circulene, for the preparation of twelve new buckyballs with molecular dimensions of less than a nanometer. The results showed that the corannulene molecule is bowl-shaped, the coronene molecule is planar, and the circulene molecule has a unique saddle-shaped structure. Cyclic polymerization of the three molecules can be used to prepare new buckyballs, and this process produces hydrogen molecules. The most symmetric buckyball is also the most stable based on the values of the HOMO energy levels and has the most efficient gap energy, making it potentially useful for solar cell applications.

Polycyclic aromatic hydrocarbons (PAHs) are a class of unique compounds that consist of fused, conjugated aromatic rings that do not contain heteroatoms or carry substituents [_{53}. Additionally, the solid state packing behavior of corannulene is interesting [

To calculate ground-state geometries, Gaussian 03, Revision C.01 [_{HOMO}), Lowest Unoccupied Molecular Orbital Energies (E_{LUMO}), and physical properties for the molecules in this study.

Previous studies have shown that not all polycyclic aromatic hydrocarbons (PAHs) are flat molecules. We selected aromatic compounds known as circulenes for this study. The circulenes include 5-circulene (corannulene), 6-circulene (coronene), and 7-circulene (circulene). According to Density Function Theory (DFT) calculations, not all of these molecules are flat, as shown in Figure

B3LYP/6-31G optimized structures of corannulene, coronene, and circulene.

A DFT calculation introduces an additional step to each major phase of a Hartree-Fock calculation. This additional step is a numerical integration of the functional (or various derivatives of the functional). Thus, in addition to the sources of numerical error in Hartree-Fock calculations (integral accuracy, SCF convergence, and CPHF convergence), the accuracy of DFT calculations also depends on the number of points used in the numerical integration. The “fine” integration grid is the default in Gaussian 03. This grid greatly enhances the calculation accuracy at minimal additional cost. We do not recommend using a smaller grid in production DFT calculations. It is important to use the same grid for all calculations when energies will be compared (e.g., computing energy differences, heats of formation, and so forth). Larger grids are available, for example, for tight optimization of certain systems. An alternate grid may be selected in the route section [

To prepare new buckyballs, a fundamental understanding of the polymerization of polycyclic aromatic hydrocarbons (PAHs) is important. The polymerization process for new buckyballs revealed the production of hydrogen molecules [

Scheme _{60}H_{2} (I) by forming five butagons, three pentagons, three hexagons, and two decagon cycles from three corannulene molecules. Reaction (2) produced a new buckyball C_{60}H_{2} (II) through the formation of three butagons, five pentagons, three hexagons, and two nonagon cycles. Reaction (3) produced a new buckyball C_{60}H_{2} (III) by forming three butagons, three pentagons, five hexagons, and two octagon cycles. Reaction (4) produced a new buckyball C_{60}H_{2} (IV) by forming eleven pentagons and two nonagon cycles. Thirteen different cycles were formed in these four reactions. All of these reactions are spontaneous and exothermic according to the values of the entropy change (_{HOMO} (the Energy of High Occupied Molecular Orbital) and the total energy for the four reaction products in Table _{60}H_{2} (IV) is the most stable among the four, and that the increase in the E_{HOMO} for C_{60}H_{2} (IV) is (−0.4530 eV), (−0.6177 eV), and (−0.6236 eV) relative to C_{60}H_{2} (I), C_{60}H_{2} (II), and C_{60}H_{2} (III) respectively. Additionally, the increase in the total energy is −0.1978 a.u or −124.121 KCal^{−1}, −0.1365 a.u or −85.655 KCal^{−1}, and −0.2260 a.u or −141.817 KCal^{−1} for C_{60}H_{2} (I), C_{60}H_{2} (II), and C_{60}H_{2} (III), respectively. The structures of the four new buckyballs are shown in Figure

Physical values of all new buckyballs were calculated with B3LYP/6-31G.

Molecules | Total energy a.u. | Enthalpy (^{−1} |
Entropy (^{−1}·K^{−1} |
E_{HOMO} eV |
E_{LUMO} eV |
Gap energy ( |
---|---|---|---|---|---|---|

Hydrogen molecule (H_{2}) |
−1.1755 | 7.847 | 31.132 | −11.8086 | +2.7235 | 14.5321 |

Corannulene | −767.9696 | 154.105 | 103.100 | −6.0170 | −1.5783 | 4.4387 |

Buckyball C_{60}H_{2} (I) |
−2286.0419 | 273.343 | 144.707 | −5.4453 | −3.9097 | 1.5356 |

Buckyball C_{60}H_{2} (II) |
−2286.1032 | 274.146 | 141.115 | −5.2806 | −3.9288 | 1.3518 |

Buckyball C_{60}H_{2} (III) |
−2286.0137 | 271.613 | 143.457 | −5.2747 | −4.2510 | 1.0237 |

Buckyball C_{60}H_{2} (IV) |
−2286.2397 | 277.261 | 139.157 | −5.8983 | −3.7968 | 2.1015 |

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Coronene | −921.6885 | 185.772 | 115.845 | −5.4880 | −1.4049 | 4.0831 |

Buckyball C_{27} (I) |
−2741.7592 | 304.289 | 157.722 | −4.9922 | −3.9154 | 1.0768 |

Buckyball C_{27} (II) |
−2742.1237 | 315.328 | 151.394 | −5.2425 | −4.1043 | 1.1382 |

Buckyball C_{27} (III) |
−2741.9284 | 316.535 | 149.671 | −5.0902 | −4.2567 | 0.8335 |

Buckyball C_{27} (IV) |
−2742.1856 | 316.159 | 150.231 | −5.7024 | −3.3212 | 2.3812 |

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Circulene | −1075.2072 | 216.027 | 132.547 | −5.1430 | −1.4348 | 3.7082 |

Buckyball C_{84}H_{2} (I) |
−3199.7795 | 372.282 | 180.179 | −5.2175 | −3.6338 | 1.5837 |

Buckyball C_{84}H_{2} (II) |
−3199.9262 | 374.865 | 176.142 | −5.2404 | −3.7780 | 1.4624 |

Buckyball C_{84}H_{2} (III) |
−3199.9163 | 374.917 | 174.315 | −5.0417 | −4.0741 | 0.9676 |

Buckyball C_{84}H_{2} (IV) |
−3200.2067 | 381.333 | 170.827 | −5.7372 | −3.8110 | 1.9262 |

The reaction of formation of buckyball C_{60}H_{2} from three molecules of corannulene and the values of the change of enthalpy (

B3LYP/6-31G optimized structures of four new buckyballs: C_{60}H_{2} (I), C_{60}H_{2} (II), C_{60}H_{2} (III), and C_{60}H_{2} (IV).

The molecular dimensions of the four new buckyballs are as follows: C_{60}H_{2} (I)—(0.85 _{60}H_{2} (II)—(0.83 _{60}H_{2} (III)—(0.84 _{60}H_{2} (IV)—(0.84

Scheme _{72} (I) and forms nine butagons, six hexagons, and two nonagon cycles from the three coronene molecules. Reaction (2) produced a new buckyball C_{72} (II) by forming three butagons, ten pentagons, two hexagons, and two octagon cycles. Reaction (3) produced a new buckyball C_{72} (III) and formed two butagons, ten pentagons, three hexagons, and two heptagon cycles. Reaction (4) produced a new buckyball C_{72} (IV) and formed six butagons, and eleven hexagon cycles. Seventeen cycles were formed in each reaction. Every reaction is spontaneous and exothermic according to the values of _{HOMO} and the total energy for the four reaction products in Table _{72} (IV) is the most stable among the four, and that the increase in E_{HOMO} for C_{72} (IV) is (−0.7102 eV), (−0.4599 eV), and (−0.6122 eV) relative to new buckyballs C_{72} (I), C_{72} (II) and C_{72} (III), respectively. Additionally, the increases in total energy are −0.4264 a.u or −267.570 KCal^{−1}, −0.0619 a.u or −38.843 KCal^{−1}, and −0.2572 a.u or −161.395 KCal^{−1} for new buckyballs C_{72} (I), C_{72} (II), and C_{72} (III), respectively. The structures of the four new buckyballs are shown in Figure

The reaction of formation of buckyball C_{72} from three molecules of coronene and the values of

B3LYP/6-31G optimized structures of new buckyballs C_{72} (I), C_{72} (II), C_{72} (III), and C_{72} (IV).

The molecular dimensions of all four new C_{72} buckyballs are as follows: C_{72} (I)—(0.82 _{72} (II)—(0.83 _{72} (III)—(0.88 _{72} (IV)—(0.82

Scheme _{84}H_{2} (I) by forming seven butagons, five pentagons, five hexagons, and two decagon cycles from three circulene molecules. Reaction (2) produced a new buckyball C_{84}H_{2} (II) by forming five butagons, seven pentagons, five hexagons, and two nonagon cycles. Reaction (3) produced a new buckyball C_{84}H_{2} (III) and formed five butagons, five pentagons, seven hexagons, and two octagon cycles. Reaction (4) produced a new buckyball C_{84}H_{2} (IV), seventeen pentagons, and two nonagon cycles. Nineteen different cycles were formed in each reaction. All of these reactions are spontaneous and exothermic according to the values of the change of entropy (_{HOMO}) and the total energy for the reaction products in Table _{84}H_{2} (IV), most stable among the four new buckyballs, was C_{84}H_{2} (IV) with an increase in the E_{HOMO}, that is (−0.5197 eV), (−0.4968 eV), and (−0.6955 eV) relative to C_{84}H_{2} (I), C_{84}H_{2} (II), and C_{84}H_{2} (III), respectively. Additionally, the increases in total energy are −0.4272 a.u or −268.072 KCal^{−1}, −0.2805 a.u or −176.016 KCal^{−1}, and −0.2904 a.u or −182.228 KCal^{−1} relative to C_{84}H_{2} (I), C_{84}H_{2} (II) and C_{84}H_{2} (III), respectively. The structures of the four new C_{84}H_{2} buckyballs are shown in Figure _{84}H_{2} buckyballs are as follows: C_{84}H_{2} (I)—(0.86 _{84}H_{2} (II)—(0.90 _{84}H_{2} (III)—(0.89 _{84}H_{2} (IV)—(0.90

The reaction of formation of C_{84}H_{2} buckyballs from three molecules of circulene and the values of

B3LYP/6-31G optimized structures of new buckyballs C_{84}H_{2} (I), buckyballs C_{84}H_{2} (II), C_{84}H_{2} (III), and C_{84}H_{2} (IV).

The energy gap, which is also called the band gap, is an energy range in a solid where no electron states can exist. The gap energy generally refers to the energy difference (in electron volts) between the Lowest Unoccupied Molecular Orbital (LUMO) and the Highest Occupied Molecular Orbital (HOMO) in insulators and semiconductors. This gap energy is equivalent to the energy required to free an outer shell electron from its orbit about the nucleus to become a mobile charge carrier that moves freely within the solid material. The band gap is a major factor that determines the electrical conductivity of a solid. Substances with large gap energies are generally insulators, materials with smaller gap energies are semiconductors, and conducting materials have very small or no gaps energies. The Shockley-Queisser limit gives the maximum possible efficiency of single junction solar cells under unconcentrated sunlight as a function of the semiconductor band gap. If the band gap is too high, then the material cannot absorb most daylight photons; if the band gap is too low, then most photons have much more energy than is necessary to excite electrons across the band gap, and the rest is wasted. The semiconductors that are used commonly in commercial solar cells have band gaps near the peak of this curve shown in Figure

The Shockley-Queisser limit for the energy efficiency of a gap.

In Table

A quantum chemistry calculation is performed using the Density Function Theory (DFT) method to study the preparation of twelve new buckyballs from the cyclic polymerization of three polycyclic aromatic hydrocarbons (PAHs), corannulene, coronene, and circulene, and the production of hydrogen molecules. The results obtained for the new buckyballs show that the most symmetric buckyball is the most stable, depending on the values of E_{HOMO}. The molecular dimensions of all the new buckyballs are less than a nanometer, and the new buckyballs are characterized by the high efficiency of their energy gaps.

The authors declare that there is no conflict of interests regarding the publication of this paper.

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