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The energy change on each Occupied Molecular Orbital as a function of rotation about the C-C bond in ethane was studied using the B3LYP, mPWB95 functional and MP2 methods with different basis sets. Also, the effect of the ZPE on rotational barrier was analyzed. We have found that σ and π energies contribution stabilize a staggered conformation. The σ_{s} molecular orbital stabilizes the staggered conformation while the _{z} and _{v} and

The existence of a rotational barrier of 2.875 kcal·mol^{−1} about the C-C bond in ethane has been known for many years [

The steric effect has its origin in the fact that atoms in molecules occupy a certain amount of space, resulting in changes in shape, energy, and reactivity. It is an essential concept in chemistry, biochemistry, and pharmacology, influencing rates and energies of chemical reactions, impacting structure, dynamics, and function of enzymes, and to a degree, governing how and at what rate a drug molecule interacts with a receptor. Different approaches have been proposed to quantify the steric effect. For example, Shubin Liu recently proposed an energy partition scheme under the framework of Density Functional Theory (DFT) [

Alternatively, the possible role of hyperconjugation effect in the ethane rotation barrier has been conjectured for many years [^{*} antibonding orbital of one C–H unit with the corresponding occupied σ bonding orbital at the other side. They have used a natural bond orbital (NBO) [^{*} interactions. The conclusion of this method is supported by other similar studies [

Therefore, there is no definite explanation for the driving force of the preferred ethane conformation mainly due to the different approximations used for the calculation of steric and hyperconjugative effects, in addition to the difficulty of their simultaneous calculation and because hyperconjugation, steric repulsion, and possibly some other effects coexist entangled in the ethane molecule. In consequence, different authors obtain different amounts of steric and hyperconjugation effects.

In order to contribute to the understanding of the conformational driving force in ethane, we propose an alternative point of view based on a systematic analysis of its Molecular Orbitals (MOs), the most basic concept in conformation, to assign the different MOs to each of the preferred conformations and estimate the overall net effect by subtracting the molecular orbital energy from the total energy during the rotation about the C–C bond in the ethane. In addition, we propose to study the effect that this theoretical model has on the behavior of energy of the MOs as a function of angle of rotation about C–C bond in ethane.

Based on these considerations, we carried out the analysis of electronic and structural properties of ethane as a function of the C–C angle (φ) rotation. The DFT (B3LYP, mPWB95) and MP2 methods with 6-31G(d, p), 6-31+G(d, p), and 6-31++G(d, p) basis sets were used to evaluate the effects of these models in the ethane rotational molecular orbital energy and the relationship to the ethane-preferred conformation. The B3LYP functional was used because of its wide application to calculate electronic structure, reaction and activation energies. However, there is evidence that the B3LYP method usually underestimates barrier heights [

The aim of this work is to contribute to understanding of the contribution of each of the molecular orbitals in ethane to the rotational barrier and its overall net effect. In addition, we propose analyze the effect that theoretical model has on the behavior of energy of the MOs as a function of angle of rotation about C–C bond in ethane. To the best of our knowledge, this is the first study of energy changes on each orbital in ethane by different methods.

The Kohn-Sham total energy rotation (E_{rot}) for conversion of ethane from staggered to eclipsed conformation was calculated in the gas phase and the geometries were fully optimized. In the applied models the total energy of ethane is calculated as a function of the torsion angle φ, obtaining an energy minimum at the staggered (E_{s}) conformation and a maximum at the eclipsed (E_{e}) conformation. The energy difference between E_{e} and E_{s} (E_{e}−E_{s}) is the calculated rotational barrier (ΔE_{rot}).

Regarding evaluation of the of ZPE effect on the rotational barrier it is important to notice that if ZPE is not included (^{−1}) to the experimental value (2.875 kcal·mol^{−1}). Similar results are observed when ZPE was included for MP2/6-31G(d, p) level of theory, which estimates a rotational barrier of 2.9116 kcal·mol^{−1}.

Calculated values of ΔE_{rot} (kcal·mol^{−1}).

B3G | B3+G | B3++G | MPG | MP+G | MP++G | MP2G | MP2+G | MP2++G | |
---|---|---|---|---|---|---|---|---|---|

2.803 | 2.732 | 2.736 | 2.752 | 2.673 | 2.683 | 3.025 | 2.966 | 2.981 | |

2.541 | 2.466 | 2.472 | 2.504 | 2.422 | 2.435 | 2.912 | 2.799 | 2.761 |

Labels B3, MP correspond to the B3LYP and mPWB95 functionals, MP2 correspond to the Moller-Plesset perturbation theory of order 2 and the symbols G, +G, and ++G represent the basis set 6-31G(d, p), 6-31+G(d, p), and 6-31++G(d, p), respectively.

The lengths of the C–C and C–H bonds were obtained as a function of φ. It is interesting to note that when the conformation goes from eclipsed (φ = 0°) to staggered (φ = 60°) the C–C bond length decreases slightly, while the C–H bonds increase slightly. The percent decrease of C–C bond length varies from 0.9 when the calculation is performed with B3LYP/6-31G(d,p) to 0.85 if the MP2/6-31G(d,p) method is applied. For the C–H bond length increase the percent variation is from 0.089 [MP2/6-31G(d,p)) to 0.1 (B3LYP/6-31G(d,p)]. Thus, rotation about the C–C bond generates a small effect over the geometry of ethane.

Ethane consists of two carbon and six hydrogen atoms sharing nine filled MOs. The valence molecular orbital configuration appropriate to the D_{3d} symmetry staggered conformation is _{1g})^{2}(2a_{2u})^{2}(e_{g})^{4}(3a_{2u})^{2}(e_{u})^{4} while that of the D_{3h} symmetry of the eclipsed conformer is _{1})^{2}(2a´´_{2})^{2}(e´´)^{4}(3a´´_{2})^{2}(e´)^{4}_{s}, _{v}, π_{z}, σ_{x}, _{v}, π_{z}, _{s}, _{x} molecular orbitals at B3LYP/6-31+G(d, p) level of theory.

Filled molecular orbitals of ethane calculated with B3LYP/6-31+G(d, p) theoretical model.

Along the potential energy surface (PES) the MOs are changing, but, for simplicity we retain the labels of the MOs in its evolution as a function of angle of rotation. The energy changes in the MOs, calculated as a function of φ are shown in ^{−1} (HF/6-31G(d,p)) down to 1.626 kcal·mol^{−1} (mPWB95/6-31+G(d,p)).

Energy of the core,

These results show the importance of the ^{−1} [HF/6-31G(d,p)] down to −0.9538 kcal·mol^{−1} [mPWB95/6-31+G(d,p)]. The negative sign of the value reveals that the

The bonding π_{v} and π_{z}, and antibonding _{z}, π_{v}, ^{−1} [mPWB95/6-31+G(d,p)] to 1.267 [HF/6-31G(d,p)], 1.794 [HF/6-31G(d,p)], 1.2989 [B3LYP/6-31G(d,p)] and 1.3114 kcal·mol^{−1} [B3LYP/6-31+G(d,p)], respectively (

Energy of the π_{v}, π_{z}, σ_{x} and

Energy of the

The calculated energy change for the studied MOs was equivalent to ΔE_{rot}, demonstrating the role of these orbitals for the rotational barrier of ethane. However, for all models, the bonding and antibonding π_{v} and _{v} and

The total energy of the π_{z} and _{z} and

Finally, the molecular orbital σ_{x}, exhibited large H(1^{−1} [HF/6-31G(d,p)] to 0.5396 kcal·mol^{−1} [mPWB95/6-31G(d,p)] (_{x} orbital.

From _{x}, π_{z}, and

Calculated values of _{rot} and

B3G | B3+G | B3++G | MPG | MP+G | MP++G | MP2G | MP2+G | MP2++G | |
---|---|---|---|---|---|---|---|---|---|

2.803 | 2.732 | 2.736 | 2.752 | 2.673 | 2.683 | 3.025 | 2.966 | 2.981 | |

2.541 | 2.466 | 2.472 | 2.504 | 2.422 | 2.435 | 2.912 | 2.799 | 2.761 | |

12.27 | 9.400 | 9.79 | 11.170 | 7.919 | 8.559 | 14.92 | 12.75 | 13.29 | |

3.539 | 1.694 | 1.983 | 2.548 | 0.502 | 0.904 | 5.095 | 3.640 | 3.953 | |

8.735 | 7.706 | 7.806 | 8.622 | 7.417 | 7.656 | 9.827 | 9.111 | 9.337 |

Labels B3, MP correspond to the B3LYP and mPWB95 functionals, MP2 correspond to the HF theory and the symbols G, +G, and ++G represent the basis set 6-31G(d, p), 6-31+G(d, p), and 6-31++G(d, p), respectively.

The value of total σ (

The quantum chemical calculation was performed using the GAUSSIAN 09 code [_{3} geometries of the ethane were carried out in all calculations.

The calculated rotational energy barriers at different levels show that it is not necessary to incorporate diffusion functions for an accurate description of the energetic barrier in ethane. It is important to note that the B3LYP/6-31G(d,p) model underestimates the value of the rotational barrier in ethane, while the MP2/6-31G(d,p) model overestimates it. In addition, the functional mPWB95 predict the worst values for rotational barrier and MP2/6-31+G(d, p) predicts the higher energy changes. We have found that the π_{v} and _{z} and _{z} and _{x} MO, the DFT energy changes contribute to stabilize the staggered conformation and shows irregular behavior. In addition, we found that for all models if

The financial support was provided by Facultad de Química of Universidad Autónoma de Yucatán, México (PIFI-FOMES 2007). The authors wish to thank to Leovigildo Quijano for the critical review and the help for document translation, and to Manuel Flores-Arce for English revision of the manuscript.