# How do the Hückel and Baird Rules Fade away in Annulenes?

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Aromaticity Indices

#### 2.1.1. The Aromatic Fluctuation Index: FLU

#### 2.1.2. The Bond-Length and Bond-Order Alternation Indices

#### 2.1.3. A Many-Center Electron Delocalization Index: ${I}_{\mathrm{ring}}$

#### 2.1.4. AV1245 and ${\mathrm{AV}}_{\mathrm{min}}$

#### 2.2. Hückel Molecular Orbital Method

## 3. Results

#### 3.1. Aromaticity from the HMO Method

#### 3.2. Geometrical Relaxation

#### 3.2.1. $4n+2$ Annulenes

#### 3.2.2. $4n$ Annulenes

#### 3.3. Aromaticity from DFAs

#### 3.4. The Delocalization Error in DFAs

## 4. Materials and Methods

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AOM | Atomic overlaps matrix |

AV1245 | Aromaticity index for large rings [55] |

${\mathrm{AV}}_{\mathrm{min}}$ | Minimal value of 12-45 delocalizations [56] |

BLA | Bond-length alternation |

BOA | Bond-order alternation |

DFA | Density Functional Approximation |

DI | Delocalization index [28] |

FLU | Fluctuation aromaticity index [27] |

HF | Hartree-Fock |

HMO | Hückel Molecular Orbital |

HOMA | Harmonic Oscillator Model of Aromaticity [33] |

LDA | Local density approximation [109] |

MCI | Multicenter index [43] |

RE | Resonance energy |

TREPE | Topological resonance energy per $\pi $ electron |

${I}_{\mathrm{ring}}$ | Giambiagi’s multicenter index [42] |

$\overline{{I}_{\mathrm{ring}}}$ | Approximation to ${I}_{\mathrm{ring}}$ (Equation (8)) |

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**Figure 1.**Values of ${I}_{\mathrm{ring}}$${}^{1/N}$ for the annulenes series against the number of C atoms (N) for different singlets and triplets. The species have been divided according to the number of electrons (4n and 4n + 2) and the spin multiplicity (singlet and triplet).

**Figure 3.**Values of $\overline{{I}_{\mathrm{ring}}}$ ${}^{1/N}$ for the annulenes series in terms of the number of C atoms (N) for the lowest-lying singlets and triplets. The species have been divided according to the number of electrons (4n and 4$n+2$) and the spin multiplicity (singlet and triplet). Calculations were performed with CAM-B3LYP/6-311G(d,p).

**Figure 4.**Values of ${\mathrm{AV}}_{\mathrm{min}}$ for the annulenes series (lowest-lying singlets and triplets) in terms of the number of C atoms (N). The species have been divided according to the number of electrons (4n and 4$n+2$) and the spin multiplicity (singlet and triplet). Calculations were performed with CAM-B3LYP/6-311G(d,p).

**Figure 5.**Values of $\overline{{I}_{\mathrm{ring}}}$ ${}^{1/N}$ for the lowest-lying states of the studied annulenes obtained with methods using a different percentage of Hartree Fock (HF) exchange: HF (100%), M06-2X (54%), CAM-B3LYP (19–65%) and B3LYP (19%).

**Table 1.**Aromaticity indices for $4n+2$ annulenes calculated with HF, B3LYP, CAM-B3LYP, and M06-2X and the 6-311G(d,p) basis set. ${}^{a}$${I}_{\mathrm{ring}}$ values were too small for an accurate calculation of ${I}_{\mathrm{ring}}$

^{1/N}.

Structure | Multiplicity | Functional | FLU | $\overline{{\mathit{I}}_{\mathbf{ring}}}$ ^{1/N} | BOA | BLA | ${\mathit{I}}_{\mathbf{ring}}$ ^{1/N} | |${\mathbf{AV}}_{\mathbf{min}}$ | |
---|---|---|---|---|---|---|---|---|

C${}_{6}$H${}_{6}$ | S | HF | 0.000 | 0.624 | 0.000 | 0.000 | 0.597 | 10.25 |

B3LYP | 0.000 | 0.625 | 0.000 | 0.000 | 0.603 | 10.72 | ||

CAM-B3LYP | 0.000 | 0.628 | 0.000 | 0.000 | 0.603 | 10.71 | ||

M06-2X | 0.000 | 0.626 | 0.000 | 0.000 | 0.603 | 10.73 | ||

C${}_{6}$H${}_{6}$ | T | HF | 0.024 | 0.393 | 0.246 | 0.089 | 0.341 | 0.39 |

B3LYP | 0.025 | 0.399 | 0.275 | 0.090 | 0.353 | 1.51 | ||

CAM-B3LYP | 0.025 | 0.408 | 0.276 | 0.091 | 0.363 | 1.20 | ||

M06-2X | 0.041 | 0.467 | 0.281 | 0.056 | 0.380 | 0.28 | ||

C${}_{10}$H${}_{10}$ (twist) | S | HF | 0.068 | 0.421 | 0.728 | 0.157 | 0.339 | 0.00 |

B3LYP | 0.052 | 0.480 | 0.639 | 0.128 | 0.322 | 0.03 | ||

CAM-B3LYP | 0.059 | 0.466 | 0.677 | 0.137 | 0.341 | 0.03 | ||

M06-2X | 0.058 | 0.463 | 0.670 | 0.136 | 0.325 | 0.05 | ||

C${}_{10}$H${}_{10}$ (heart) | S | HF | 0.065 | 0.436 | 0.712 | 0.153 | 0.380 | 0.01 |

B3LYP | 0.000 | 0.610 | 0.007 | 0.009 | 0.579 | 5.19 | ||

CAM-B3LYP | 0.000 | 0.614 | 0.009 | 0.010 | 0.579 | 5.13 | ||

M06-2X | 0.000 | 0.611 | 0.010 | 0.010 | 0.579 | 5.11 | ||

C${}_{10}$H${}_{10}$ (naphthalene) | T | HF | 0.030 | 0.470 | 0.364 | 0.085 | 0.353 | 0.14 |

B3LYP | 0.023 | 0.531 | 0.328 | 0.064 | 0.460 | 0.91 | ||

CAM-B3LYP | 0.027 | 0.524 | 0.364 | 0.072 | 0.446 | 0.68 | ||

M06-2X | 0.028 | 0.522 | 0.367 | 0.073 | 0.444 | 0.63 | ||

C${}_{10}$H${}_{10}$ (twist) | T | HF | 0.020 | 0.478 | 0.266 | 0.068 | 0.295 | 0.05 |

B3LYP | 0.020 | 0.517 | 0.305 | 0.066 | 0.352 | 0.08 | ||

CAM-B3LYP | 0.022 | 0.513 | 0.324 | 0.071 | 0.347 | 0.07 | ||

M06-2X | 0.022 | 0.511 | 0.328 | 0.072 | 0.337 | 0.10 | ||

C${}_{14}$H${}_{14}$ | S | HF | 0.050 | 0.497 | 0.626 | 0.136 | - ${}^{a}$ | 0.49 |

B3LYP | 0.001 | 0.605 | 0.010 | 0.008 | - ${}^{a}$ | 4.24 | ||

CAM-B3LYP | 0.026 | 0.561 | 0.449 | 0.091 | - ${}^{a}$ | 1.80 | ||

M06-2X | 0.025 | 0.561 | 0.438 | 0.088 | - ${}^{a}$ | 1.89 | ||

C${}_{14}$H${}_{14}$ (TS) | S | CAM-B3LYP | 0.000 | 0.609 | 0.007 | 0.007 | - | 4.29 |

M06-2X | 0.001 | 0.606 | 0.007 | 0.007 | - ${}^{a}$ | 4.27 | ||

C${}_{14}$H${}_{14}$ | T | HF | 0.021 | 0.502 | 0.282 | 0.069 | - ${}^{a}$ | 0.01 |

B3LYP | 0.017 | 0.554 | 0.302 | 0.060 | - ${}^{a}$ | 0.13 | ||

CAM-B3LYP | 0.023 | 0.544 | 0.348 | 0.071 | - ${}^{a}$ | 0.08 | ||

M06-2X | 0.022 | 0.544 | 0.349 | 0.071 | - ${}^{a}$ | 0.04 | ||

C${}_{18}$H${}_{18}$ | S | HF | 0.049 | 0.504 | 0.616 | 0.133 | 0.472 | 0.57 |

B3LYP | 0.001 | 0.606 | 0.026 | 0.011 | 0.573 | 4.27 | ||

CAM-B3LYP | 0.026 | 0.563 | 0.446 | 0.090 | 0.530 | 1.80 | ||

M06-2X | 0.025 | 0.561 | 0.444 | 0.089 | 0.529 | 1.81 | ||

C${}_{18}$H${}_{18}$ (TS) | S | CAM-B3LYP | 0.001 | 0.609 | 0.022 | 0.010 | 0.572 | 4.29 |

M06-2X | 0.001 | 0.607 | 0.022 | 0.010 | 0.572 | 4.28 | ||

C${}_{18}$H${}_{18}$ | T | HF | 0.018 | 0.514 | 0.257 | 0.060 | 0.411 | 0.12 |

B3LYP | 0.013 | 0.570 | 0.265 | 0.053 | 0.533 | 0.58 | ||

CAM-B3LYP | 0.019 | 0.559 | 0.324 | 0.066 | 0.513 | 0.30 | ||

M06-2X | 0.019 | 0.559 | 0.324 | 0.065 | 0.517 | 0.46 |

**Table 2.**Aromaticity indices for the studied $4n$ annulenes calculated with HF, B3LYP, CAM-B3LYP, and M06-2X and the 6-311G(d,p) basis set. ${}^{a}$${I}_{\mathrm{ring}}$ values were too small for an accurate calculation of ${I}_{\mathrm{ring}}$

^{1/N}. ${}^{b}$${\mathrm{AV}}_{\mathrm{min}}$ cannot be calculated for rings with less than six members.

Structure | Multiplicity | Method | FLU | $\overline{{\mathit{I}}_{\mathbf{ring}}}$ ^{1/N} | BOA | BLA | ${\mathit{I}}_{\mathbf{ring}}$ ^{1/N} | |${\mathbf{AV}}_{\mathbf{min}}$ | |
---|---|---|---|---|---|---|---|---|

C${}_{4}$H${}_{4}$ | S | HF | 0.101 | 0.391 | 0.888 | 0.249 | 0.262 | - ${}^{b}$ |

B3LYP | 0.104 | 0.416 | 0.900 | 0.247 | 0.266 | - ${}^{b}$ | ||

CAM-B3LYP | 0.103 | 0.398 | 0.898 | 0.245 | 0.264 | - ${}^{b}$ | ||

M06-2X | 0.103 | 0.405 | 0.898 | 0.242 | 0.268 | - ${}^{b}$ | ||

C${}_{4}$H${}_{4}$ | T | HF | 0.010 | 0.507 | 0.000 | 0.000 | 0.433 | - ${}^{b}$ |

B3LYP | 0.012 | 0.499 | 0.000 | 0.000 | 0.440 | - ${}^{b}$ | ||

CAM-B3LYP | 0.011 | 0.504 | 0.000 | 0.000 | 0.439 | - ${}^{b}$ | ||

M06-2X | 0.011 | 0.505 | 0.000 | 0.000 | 0.438 | - ${}^{b}$ | ||

C${}_{8}$H${}_{8}$ | S | HF | 0.067 | 0.436 | 0.726 | 0.156 | 0.406 | 0.30 |

B3LYP | 0.056 | 0.477 | 0.664 | 0.134 | 0.441 | 0.72 | ||

CAM-B3LYP | 0.061 | 0.468 | 0.693 | 0.140 | 0.428 | 0.52 | ||

M06-2X | 0.062 | 0.460 | 0.694 | 0.141 | 0.427 | 0.51 | ||

C${}_{8}$H${}_{8}$ | T | HF | 0.001 | 0.590 | 0.000 | 0.000 | 0.534 | 4.07 |

B3LYP | 0.001 | 0.589 | 0.000 | 0.000 | 0.540 | 4.31 | ||

CAM-B3LYP | 0.001 | 0.593 | 0.000 | 0.000 | 0.539 | 4.29 | ||

M06-2X | 0.001 | 0.591 | 0.000 | 0.000 | 0.539 | 4.29 | ||

C${}_{12}$H${}_{12}$ | S | HF | 0.063 | 0.445 | 0.698 | 0.153 | - ${}^{a}$ | 0.04 |

B3LYP | 0.042 | 0.511 | 0.565 | 0.115 | - ${}^{a}$ | 0.01 | ||

CAM-B3LYP | 0.050 | 0.494 | 0.624 | 0.128 | - ${}^{a}$ | 0.02 | ||

M06-2X | 0.050 | 0.488 | 0.624 | 0.128 | - ${}^{a}$ | 0.06 | ||

C${}_{12}$H${}_{12}$ | T | HF | 0.021 | 0.487 | 0.280 | 0.067 | - ${}^{a}$ | 0.07 |

B3LYP | 0.002 | 0.590 | 0.033 | 0.012 | - ${}^{a}$ | 0.07 | ||

CAM-B3LYP | 0.015 | 0.562 | 0.288 | 0.056 | - ${}^{a}$ | 0.21 | ||

M06-2X | 0.008 | 0.577 | 0.208 | 0.039 | - ${}^{a}$ | 0.13 | ||

C${}_{16}$H${}_{16}$ (S${}_{4}$) | S | HF | 0.054 | 0.486 | 0.651 | 0.139 | 0.440 | 0.33 |

B3LYP | 0.029 | 0.551 | 0.476 | 0.095 | 0.513 | 0.96 | ||

CAM-B3LYP | 0.041 | 0.529 | 0.564 | 0.113 | 0.484 | 0.64 | ||

M06-2X | 0.040 | 0.526 | 0.562 | 0.113 | 0.484 | 0.63 | ||

C${}_{16}$H${}_{16}$ (C${}_{1}$) | S | HF | 0.053 | 0.488 | 0.643 | 0.139 | 0.452 | 0.25 |

B3LYP | 0.029 | 0.548 | 0.474 | 0.096 | 0.512 | 0.78 | ||

CAM-B3LYP | 0.040 | 0.530 | 0.555 | 0.113 | 0.487 | 0.55 | ||

M06-2X | 0.039 | 0.526 | 0.553 | 0.112 | 0.487 | 0.58 | ||

C${}_{16}$H${}_{16}$ (C${}_{s}$) | T | HF | 0.001 | 0.598 | 0.013 | 0.005 | - ${}^{a}$ | 1.08 |

B3LYP | 0.002 | 0.596 | 0.057 | 0.013 | - ${}^{a}$ | 1.16 | ||

CAM-B3LYP | 0.015 | 0.568 | 0.294 | 0.059 | - ${}^{a}$ | 0.35 | ||

M06-2X | 0.011 | 0.576 | 0.252 | 0.050 | - ${}^{a}$ | 0.74 |

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**MDPI and ACS Style**

Casademont-Reig, I.; Ramos-Cordoba, E.; Torrent-Sucarrat, M.; Matito, E.
How do the Hückel and Baird Rules Fade away in Annulenes? *Molecules* **2020**, *25*, 711.
https://doi.org/10.3390/molecules25030711

**AMA Style**

Casademont-Reig I, Ramos-Cordoba E, Torrent-Sucarrat M, Matito E.
How do the Hückel and Baird Rules Fade away in Annulenes? *Molecules*. 2020; 25(3):711.
https://doi.org/10.3390/molecules25030711

**Chicago/Turabian Style**

Casademont-Reig, Irene, Eloy Ramos-Cordoba, Miquel Torrent-Sucarrat, and Eduard Matito.
2020. "How do the Hückel and Baird Rules Fade away in Annulenes?" *Molecules* 25, no. 3: 711.
https://doi.org/10.3390/molecules25030711