# The Different Story of π Bonds

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Trans-Bending

_{2}Si=SiHR disilenes with R=Li, BeH, BH

_{2}, CH

_{3}, SiH

_{3}, NH

_{2}, OH and F are plotted in Figure 3 as functions of the sum of triplet–singlet energy difference, $\sum \phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}{E}_{TS}$, of the building fragments, i.e., SiH${}_{2}$ and SiHR, at different level of theory (density functional); also shown in the same graphs are the results for H

_{2}Si=SiF

_{2}and FHSi=SiHF. In general, all the functionals considered agree well with each other (and with the original MP3/MP4 results by Karni and Apeloig [58]), except M06HF. The latter tends to underestimate the disilene pyramidalization, and predicts many disilenes to be planar. This makes clear that electron correlation plays a major role in angular distortions, as already observed in reference [35] for several systems. Table 1 shows for instance the buckling height computed for different “silicene flakes” or “Si dots” (see Figure 4) with the same functionals above, including bare HF results which are seen to severely underestimate pyramidalization.

_{2}, OH and F, in fact, the largest distortions are induced in the mono-substituted case; on the other hand, the electropositive and $\pi $-accepting Li, BeH and BH

_{2}substituents drastically reduce the pyramidalization angles. Out of the two effects, $\pi $-donation or acceptation abilities seems to most affect distortions: Pyramidalization of the unsubstituted silicon increases in the F, OH, NH

_{2}series, the latter inducing the highest pyramidalization angles in mono-substituted disilenes (${\theta}_{H}={69}^{\circ}$ and ${74}^{\circ}$ according to B3LYP and M062X, respectively), whereas when R=BeH and BH

_{2}disilenes turn out to be flat. It is thus clear that electron density in the ${\pi}^{*}$ MO orbital (or $\pi $ on-site repulsion in the Hubbard bond description) is particularly effective on distortion. This is confirmed by the fact that the substituted silicon decreases its distortion in the said series, as $n\to {\pi}^{*}$ interactions give rise to a partial negative charge in the other silicon, while the substituted one is less affected. Here Hydrogen, being slightly more electronegative than silicon, should be considered mildly distortive. As a matter of fact, disilene (H

_{2}Si=SiH

_{2}) possess higher pyramidalization angles than H

_{2}Si=SiH(SiH

_{3}). In the doubly substituted disilene H

_{2}Si=SiF

_{2}the distortion induced to the unsubstituted silicon is even larger, the pyramidalization angle reaching the largest value of ${86}^{\circ}$, at the M062X level of theory.

_{2}C=CF

_{2}the binding energy ${E}_{BE}$ is smaller than the triplet–singlet energy difference (${E}_{BE}$ = 75.0 kcal mol

^{−1}vs. $\sum \phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}{E}_{TS}$ = 142.5 according to M062X), but the molecule is found to be planar, at odds with the rule. In fact, in reference [35] it was argued that the correlation between pyramidalization and $\sum \phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}{E}_{TS}$ is somewhat incidental: The same factors favoring distortion through $\sigma $-strengthening/$\pi $-weakening determine an increase of the triplet–singlet separation. For instance, electronegative substituents stabilize the n-like state in SiY${}_{2}$, while they do not affect the p-like one, and hence increase $\mathsf{\Delta}{E}_{TS}$. Likewise, $\pi $-donors destabilize the triplet state by introducing Coulomb repulsion in the p-like orbital.

_{2}series. However, no trend appears in less distortive substituents. As a matter of fact, the computed bond lengths turn out to be very similar in both flat and considerably pyramidalized disilenes. M062X even predicts that the shortest Si=Si bond is in H

_{2}Si=SiH(SiH

_{3}), for which ${\theta}_{H}={17}^{\circ}$ and ${\theta}_{R}={13}^{\circ}$, rather than in a flat disilene. Thus, a correlation between angular distortion and bond length stretching cannot be safely assumed.

_{3}SiSiR isomer and the double bridged RSi(${\eta}_{2}$-R

_{2})SiR structure. For instance, at the M062X level of theory we find that in tetra-substituted disilenes Si${}_{2}$R${}_{4}$ the energy of the single-bonded (double bridge) structure—referenced to the double bonded one—is −12.90 (−15.13), −11.50 (−11.96) and −10.51 (−6.85) kcal mol${}^{-1}$ for R = NH${}_{2}$, OH and F, respectively.

## 4. Bond-Length Alternation

#### 4.1. n-Annulenes

#### 4.2. $\pi $-Distortivity from Hubbard Calculations

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

## Abbreviations

DFT | Density Functional Theory |

AO | Atomic Orbital |

MO | Molecular Orbital |

NBO | Natural Bond Orbital |

BLA | Bond Length Alternation |

AE | Atomization Energy |

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**Figure 1.**DFT energetics of disilene (H${}_{2}$SiSiH${}_{2}$) along the path devised by Trinquier and Malrieu [27], sketched on top of the graphs, along with some important energy values: the triplet-singlet separation ($\mathsf{\Delta}{E}_{TS}$) of the fragments and the strength of the double bond (${E}_{\sigma +\pi}$). Energy is referenced to the pair of singlet fragments in their equilibrium geometry (${\alpha}_{0}$). Shown also are the lowest lying singlet excited states (solid and dotted lines in purple), and the relevant (occupied) Natural Bonding Orbitals for selected structures along the path, as indicated.

**Figure 3.**Angles of pyramidalization vs. $\sum \mathsf{\Delta}{E}_{TS}$ of different mono- and di-substituted disilenes computed with B3LYP, M06HF, M06L, M06X, M062X, PBE functionals, along with their linear regressions and the corresponding (squared) correlation coefficients.

**Figure 4.**The “silicene flakes” considered in Table 1. From left to right, Si${}_{10}$–naphatelene, Si${}_{54}$–circumcoronene and hexasilabenzene.

**Figure 6.**From left to right: The atomization energy per C atom ($AE$), the average CC bond length ($\overline{R}$) and the bond length alternation (BLA) for the C${}_{n}$H${}_{n}$ structures exemplified in Figure 5, as functions of $1/n$, on a linear-log scale.

**Figure 7.**Bending stiffness of the structural sequences defined in Figure 5, on a linear-log scale.

**Figure 8.**

**Left**and

**central**panel: Ground-state energy of the $n=4,6$ Hubbard models for different values of $U/t$ as functions of $\Delta t/t$. From

**bottom**to

**top**$U/t=2,4,6,\dots $.

**Right**panel: distortion energy ${E}_{\delta =2}-{E}_{\delta =0}$ per site as a function of $U/t$ for different number of sites.

**Figure 9.**

**Upper**panels: Intrinsic distortivities—${\kappa}_{-}$ in a log-linear (

**a**) and linear–linear (

**b**) scale as functions of $U/t$ for different values of n. (

**c**) panel: Size-dependence of the distortivity in the weakly ($U/t=2$) and strongly ($U/t=10$) interacting limits. (

**d**): Distortivity for the ground (GS) and the twin (TS) states for $n=4,6$. Notice that ${\kappa}_{-}>0$ (green colored area) means that the structure is stable against the bond alternation distortion.

**Table 1.**Buckling height (h) in some Silicene “dots” (Figure 4), as obtained with different density functionals and a 6-31++G${}^{**}$ basis set. In hexasilabenzene h was determined from the heights of the Si atoms above and below the natural plane, which is midway between the planes defined by up- and down-Si atoms. For Si${}_{10}$–naphatelene and Si${}_{54}$–circumcoronene h was defined similarly but at the center of the molecule only.

h/Å | Si${}_{6}$H${}_{6}$ | Si${}_{10}$H${}_{8}$ | Si${}_{54}$H${}_{18}$ |
---|---|---|---|

HF | 0.18 | 0.17 | 0.18 |

PBE | 0.45 | 0.44 | 0.46 |

B3LYP | 0.43 | 0.41 | 0.42 |

M06L | 0.40 | 0.38 | 0.38 |

M06 | 0.48 | 0.49 | 0.51 |

M062X | 0.37 | 0.40 | 0.42 |

M06HF | 0.33 | 0.43 | 0.50 |

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Cappelletti, M.; Leccese, M.; Cococcioni, M.; Proserpio, D.M.; Martinazzo, R. The Different Story of *π* Bonds. *Molecules* **2021**, *26*, 3805.
https://doi.org/10.3390/molecules26133805

**AMA Style**

Cappelletti M, Leccese M, Cococcioni M, Proserpio DM, Martinazzo R. The Different Story of *π* Bonds. *Molecules*. 2021; 26(13):3805.
https://doi.org/10.3390/molecules26133805

**Chicago/Turabian Style**

Cappelletti, Marco, Mirko Leccese, Matteo Cococcioni, Davide M. Proserpio, and Rocco Martinazzo. 2021. "The Different Story of *π* Bonds" *Molecules* 26, no. 13: 3805.
https://doi.org/10.3390/molecules26133805