In the previous post, I looked at the so-called Kekulé vibration of cyclo[18]carbon using various quantum methods and basis sets. Because some of these procedures can take a very long time, I could not compare them using the same high-quality consistent atom basis set for the carbon (Def2-TZVPP). Here I try to start to do this using the smaller six and ten carbon rings to see what trends might emerge. FAIR data are at DOI: 10.14469/hpc/6069

Method | C-C bond length | Kekulé mode, cm^{-1} |
Number of -ve force constants |
---|---|---|---|

Cyclo[6]carbon | |||

B3LYP+GD3BJ | 1.302 | 1300 | 2 |

wB97XD | 1.298 | 1262 | 2 |

PBEQIDH | 1.302 | 1332 | 1 |

MP2 | 1.318 | 1428 | 0 |

MP3 | 1.303 | 923 | 0 |

MP4(SDQ) | 1.308 | 943 | 0 |

CCSD | 1.308 | 1020 | 2 |

CCSD(T) | 1.320 | 1189 | 2 |

Cyclo[10]carbon | |||

B3LYP+GD3BJ | 1.282 | 1334 | 1 |

wB97XD | 1.279 | 975 | 1 |

PBEQIDH | 1.282 | 1578 | 0 |

MP2 | 1.295 | 2829 | 0 |

MP3 | 1.283 | -1946 | 1 |

MP4(SDQ) | 1.285 | -1003 | 2 |

CCSD | 1.286 | -781 | 3 |

CCSD(T) | 1.295 | run |

The conclusions can be summarised as:

- For six carbons, all the methods agree that the Kekulé vibration is real (+ve force constant), but there are distinct signs that the MP expansion may not be fully converged, with MP2 and MP3 differing significantly
- For ten carbons, we already see that MP3 and MP4 differ from MP2 in predicting a -ve force constant.

Multi-configuration calculations are problematic with these species. For C_{10} for example, 20 electrons (10 π and 10 σ) in an active orbital space of 20 is required; a CASSCF(20,20) is beyond the scope of most quantum programs. And there is a need to evaluate the second derivatives of such a wavefunction in order to get the force constant for the required vibration.

So the onset of bond length alternation in these cyclo-carbons could be as early as 10 atoms. But until higher level calculations can be performed in a systematic manner on these rings, the jury should perhaps still remain out as to when BLE starts.

Here are some simple MCSCF calculations on a fixed geometry of C10, indicating the % contribution of the doubly-occupied Hartree-Fock configuration.

CASSCF(10,10): 80.4%

CASSCF(12,12): 73.3%

CASSCF(14,14): 65.9%

One might presume that for a more complete CASSCF(20,20) expansion, the % of the HF single configuration would be < 65%, indicating that ignoring other configurations in the overall calculation might be unsound.