# Equilibrium Geometries, Adiabatic Excitation Energies and Intrinsic C=C/C–H Bond Strengths of Ethylene in Lowest Singlet Excited States Described by TDDFT

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## Abstract

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## 1. Introduction

- Does bond strengthening exist for the excited states of ethylene?
- To what extent can the bond strengthening/weakening be for the excited states of ethylene?
- How can we rationalize and interpret the result of bond strength changes determined with the local mode analysis?

## 2. Computational Details

## 3. Results and Discussion

## 4. Conclusions

- Among the several commonly used density functionals for describing the Rydberg states, $\omega $B97X-D, CAM-B3LYP and $\omega $hPBE0 give relatively low rms deviations (∼0.3 eV) against the experimental excitation energies for ethylene.
- The equilibrium geometries for 11 out of 17 excited states of ethylene were successfully modeled at the CAM-B3LYP/aug-cc-pVTZ level. Six of them are twisted in contrast to the planar ground-state structure.
- For the first time, the local vibrational mode theory has been applied to molecules in excited states to characterize the intrinsic bond strength which can be directly compared with the ground-state counterpart. In this study on ethylene, the majority of the 11 excited states with optimized geometries exhibit weakened C=C/C–H bonds as characterized by the local stretching force constant ${k}^{a}$, which is consistent with the assumption that a molecule in its excited states with higher electronic energy has in general smaller bond (dissociation) energies. However, the results from the local mode analysis show that 3 excited states have stronger C=C bonds with BSO values of 2.4, which almost reaches the midpoint between ethylene C=C (BSO = 2.0) and ethyne C≡C (BSO = 3.0). Meanwhile, 2 excited states show only marginally stronger C–H bond strength the ground state. Overall, the local stretching force constants for these 11 excited states of ethylene show a larger range of variation for the C=C bond compared to the ground state (variation of −48 % to +27 %) than for C–H bonds (variation of −72 % to +1 %). This leads to the important conclusion that the C=C bond of ethylene is more tunable by electronic excitation, while this does not hold for the C–H bond.
- The NTO analysis plays an important role in helping interpret the result of local stretching force constants for C=C/C–H bonds of ethylenes in excited states.
- When the particle NTO is the $\pi $ bonding orbital, the C=C bond is always weakened in terms of local stretching force constant.
- When the particle NTO is the ${\pi}_{y}^{\prime}$ orbital arising from the $\sigma $-backbone of ethylene, the C=C bond is in general strengthened except when the hole NTO is the ${\pi}^{*}$ anti-bonding orbital. Surprisingly, when the hole NTO orbital is 3p${}_{z}$ the C=C bond is the strongest and this hole NTO has ${\sigma}^{*}$(C=C) anti-bonding character.
- The excited state leading to slightly increased C–H bond strength has its hole NTO as 4d${}_{yz}$ which consists of 4 C–H bonding orbitals.
- When the hole NTO has significant C–H anti-bonding character, the C–H bonds are greatly weakened.

- Computational modeling of molecular excited states is generally more challenging than for the ground state. Therefore, we used ethylene as a simple theoretical model system to provide the proof of concept that fine-tuning chemical bond strength via electronic excitation is feasible, although the calculations performed in this work can be further improved by more advanced wavefunction-based correlation methods other than TDDFT [5,27]. It is also possible to find the equilibrium geometries of more excited states with dedicated algorithms [43], which we will test in the future.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TDDFT | Time-Dependent Density-Functional Theory |

NTO | Natural Transition Orbitals |

BSO | Bond Strength Order |

BO | Bond Order |

VEE | Vertical Excitation Energy |

AEE | Adiabatic Excitation Energy |

rms | root mean square |

MSA | Mean Signed Average |

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**Figure 1.**Root mean square (rms) and mean signed average (MSA) differences (in eV) between vertical excitation energies calculated theoretically and experimentally measured excitation energies for the ethylene molecule. Eight singlet excited states are included in comparison.

**Figure 2.**Relationship between local stretching force constant ${k}^{a}$ and bond length r for C=C bond of ethylene in different excited-state equilibrium geometries. Linear fitting was employed for 10 data points with R${}^{2}$ = 0.966 while S5 and S7 identified as outliers were excluded for fitting.

**Figure 3.**Distribution of bond strength order (BSO) of C=C bond in ethylene in different excited-state equilibrium geometries. BSO is defined as a cubic function ($y=0.00014718{x}^{3}-0.0082412{x}^{2}+0.26737x$) of the local stretching force constant for CC bonds where the ethane C–C, ethylene C=C and ethyne C≡C bonds takes the BSO values of 1.0, 2.0 and 3.0, respectively as references.

**Figure 4.**Relationship between local stretching force constant ${k}^{a}$ and bond length r for C–H bond of ethylene in different excited-state equilibrium geometries. Linear fitting was employed for 14 data points with R${}^{2}$ = 0.902 while S4 and one S14 identified as outliers were excluded for fitting. In addition, the C–H bond of ethane molecule was included in the linear fitting.

**Figure 5.**Natural transition orbital (NTO) pairs illustrating the singlet excited states of ethylene. The label S${}_{n}$ for each excited state corresponds to the first column of Table 1. The left and right NTO diagrams shown as mesh isosurfaces depict the excited particle (occupied) and empty hole (unoccupied) respectively. The red and blue colors are chosen arbitrarily to indicate two different phases. Below each isosurface is the corresponding isovalue in atomic units (a.u.). Each NTO pair accounts for more than 90% contribution to the target electronic excitation unless otherwise specified.

**Figure 6.**Continued from Figure 5.

**Figure 7.**Continued from Figure 6.

**Table 1.**Summary of the lowest 17 singlet excited states of ethylene calculated at CAM-B3LYP/aug-cc-pVTZ level of theory.

No. | State | Assignment ${}^{\mathit{a}}$ | VEE [exp.] ${}^{\mathit{b}}$ | f ${}^{\mathit{d}}$ | Optimized Geometry | |
---|---|---|---|---|---|---|

Symmetry | AEE ${}^{\mathit{c}}$ | |||||

0 | 1 A${}_{g}$ | Ground State | 0.00 | - | D${}_{2h}$ | – |

1 | 1 B${}_{3u}$ | $\pi $ → 3s | 6.86 [7.11] [25] | 0.0695 | D${}_{2}$ | 6.80 |

2 | 1 B${}_{1u}$ | $\pi $ → ${\pi}^{*}$ | 7.34 [7.63] [25,48] | 0.3421 | n/a | |

3 | 1 B${}_{1g}$ | $\pi $ → 3p${}_{y}$ | 7.47 [7.80] [25] | 0.0000 | n/a | |

4 | 1 B${}_{2g}$ | $\pi $ → 3p${}_{z}$ | 7.53 [8.00] [48] | 0.0000 | D${}_{2}$ | 7.45 |

5 | 2 B${}_{1g}$ | ${\pi}_{y}^{\prime}$ → 3d${}_{xz}$ (${\pi}^{*}$) | 8.01 | 0.0000 | C${}_{2h}$ | 7.80 |

6 | 2 A${}_{g}$ | $\pi $ → 3p${}_{x}$ | 8.36 [8.29] [25] | 0.0000 | D${}_{2}$ | 7.96 |

7 | 1 A${}_{u}$ | $\pi $ → 4d${}_{yz}$ | 8.58 | 0.0000 | D${}_{2h}$ | 8.52 |

8 | 2 B${}_{3u}$ | $\pi $ → 3d${}_{{z}^{2}}$ | 8.72 [8.62] [25] | 0.0043 | D${}_{2}$ | 8.65 |

9 | 1 B${}_{3g}$ | ${\pi}_{y}^{\prime}$ → 3s | 9.18 | 0.0000 | C${}_{2v}$ | 8.69 |

10 | 2 B${}_{2g}$ | ${\pi}_{z}^{\prime}$ → 3d${}_{xz}$ (${\pi}^{*}$) | 9.39 | 0.0000 | n/a | |

11 | 3 B${}_{3u}$ | $\pi $ → 4s | 9.56 | 0.0188 | D${}_{2}$ | 9.49 |

12 | 3 B${}_{1g}$ | $\pi $ → 4p${}_{y}$ | 9.68 [9.34] [25] | 0.0000 | n/a | |

13 | 3 B${}_{2g}$ | $\pi $ → 4p${}_{z}$ | 9.68 | 0.0000 | n/a | |

14 | 2 B${}_{1u}$ | $\pi $ → 4d${}_{xz}$ (30%), ${\pi}_{y}^{\prime}$ → 3p${}_{y}$ (70%) | 9.72 [9.33] [25] | 0.0617 | C${}_{2h}$ | 9.33 |

15 | 1 B${}_{2u}$ | ${\pi}_{y}^{\prime}$ → 3p${}_{z}$ | 9.80 | 0.0839 | D${}_{2h}$ | 9.18 |

16 | 4 B${}_{3u}$ | $\pi $ → 3d${}_{{x}^{2}-{y}^{2}}$ | 9.95 | 0.2225 | D${}_{2}$ | 9.85 |

17 | 3 B${}_{1u}$ | $\pi $ → 4d${}_{xz}$ (70%), ${\pi}_{y}^{\prime}$ → 3p${}_{y}$ (30%) | 9.99 | 0.0086 | n/a |

**Table 2.**Bond length r (Å) and local stretching force constant ${k}^{a}$ (mdyn/Å) for C=C/C–H bonds in ethylene in different excited-state equilibrium geometries calculated at CAM-B3LYP/aug-cc-pVTZ level.

No. | r_{CC} | ${\mathbf{k}}_{\mathbf{CC}}^{\mathbf{a}}$ | BSO${}_{\mathbf{CC}}$ | r_{CH} | ${\mathbf{k}}_{\mathbf{CH}}^{\mathbf{a}}$ |
---|---|---|---|---|---|

0 | 1.3193 | 10.022 | 2.0 | 1.0817 | 5.575 |

1 | 1.3828 | 5.785 | 1.3 | 1.0852 | 5.412 |

4 | 1.3863 | 5.416 | 1.2 | 1.0868 | 2.462 |

5 | 1.3700 | 5.325 | 1.2 | 1.1218 (C1H2,C4H5) | 4.096 |

1.0843 (C1H3,C4H6) | 4.960 | ||||

6 | 1.3558 | 7.254 | 1.6 | 1.0860 | 5.461 |

7 | 1.4057 | 7.223 | 1.6 | 1.0798 | 5.648 |

8 | 1.3864 | 5.228 | 1.2 | 1.0858 | 5.301 |

9 | 1.2632 | 11.701 | 2.2 | 1.0980 (C1H2,C1H3) | 3.917 |

1.1528 (C4H5,C4H6) | 1.579 | ||||

11 | 1.3991 | 5.653 | 1.3 | 1.0973 | 3.979 |

14 | 1.2577 | 11.348 | 2.2 | 1.0990 (C1H2,C4H5) | 4.219 |

1.1659 (C1H3,C4H6) | 2.666 | ||||

15 | 1.2598 | 12.686 | 2.4 | 1.1313 | 2.864 |

16 | 1.3855 | 6.014 | 1.3 | 1.0812 | 5.583 |

Ethane | 1.5214 | 4.256 | 1.0 | 1.0898 | 5.220 |

Ethyne | 1.1911 | 17.992 | 3.0 | 1.0619 | 6.416 |

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**MDPI and ACS Style**

Tao, Y.; Zhang, L.; Zou, W.; Kraka, E.
Equilibrium Geometries, Adiabatic Excitation Energies and Intrinsic C=C/C–H Bond Strengths of Ethylene in Lowest Singlet Excited States Described by TDDFT. *Symmetry* **2020**, *12*, 1545.
https://doi.org/10.3390/sym12091545

**AMA Style**

Tao Y, Zhang L, Zou W, Kraka E.
Equilibrium Geometries, Adiabatic Excitation Energies and Intrinsic C=C/C–H Bond Strengths of Ethylene in Lowest Singlet Excited States Described by TDDFT. *Symmetry*. 2020; 12(9):1545.
https://doi.org/10.3390/sym12091545

**Chicago/Turabian Style**

Tao, Yunwen, Linyao Zhang, Wenli Zou, and Elfi Kraka.
2020. "Equilibrium Geometries, Adiabatic Excitation Energies and Intrinsic C=C/C–H Bond Strengths of Ethylene in Lowest Singlet Excited States Described by TDDFT" *Symmetry* 12, no. 9: 1545.
https://doi.org/10.3390/sym12091545