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 | 1266 | 2 |

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 bond length alternation starts in terms of ring size.

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.