The V state of ethylene: valence bond theory takes up the challenge

Wu, W.; Zhang, H.; Braïda, B.; Shaik, S.; Hiberty, P., Theor. Chem. Acc. 2014, 133, 1-13
Highlighted by Mario Barbatti

The computation of the excitation from the ground state into the first singlet ππ* state of ethylene - or the N→V transition as it is known - has been a challenge for decades.^1-4 Conventional molecular-orbital (MO) theories tend to overshoot the transition energy by far too much. A CASSCF (complete active space self-consistent field) computation in a 2-electrons/2-orbitals space with a modest basis set as the 6-31G*, for instance, predicts this transition to be 2.5 eV larger than the experimental value, 7.88 eV.

Far from being a simple curiosity restricted to ethylene, the difficulty for describing this type of state is overspread everywhere in organic photochemistry. The prediction of the Soret band in porphyrin, for example, will suffer from the same deficiencies, for the same reasons.

Two main factors contribute to the problem:

• Lack of proper electron dynamic correlation: usually, dynamic correlation is computed after the CASSCF step via perturbative or configuration interaction (CI) methods. In the case of the N→V transition, the convergence is anomalously slow. Some hardcore calculations have employed 80 million configurations in the CI expansion to obtain acceptable energies.²       
• Basis set effects: due to the ionic character of the V state, diffuse functions are essential to describe the molecular orbitals. However, the inclusion of diffuse functions also induces an artificial mixing of the V state with Rydberg states, which brings the the excitation energy closer to the experimental value by wrong reasons.

For decades, theoreticians have investigated the V state of ethylene with the most diverse methods. In 2009, Angeli showed that the dynamic response of the σ framework to the fluctuation of the π electrons is of central importance.³ It explains the reason underlying the poor convergence of the CI methods: in conventional methods, dynamic correlation is computed with a pre-optimized set of MOs, which does not include such σ-π dynamic mixing.

In this highlighted paper,⁴ Wu and co-workers systematically investigate the N→V transition using valence-bond (VB) methods. Two properties are taken into account, the transition energy and the spatial extension of the V state.With a clever sequence of calculations using different methods and levels, they showed the impact of several different factors to the predictions of the V state. 

Most impressive, they also showed that a compact set of only 5 valence bond structures (Fig. 1) is already enough to obtain a quantitatively accurate description of the N→V transition, just 0.13 eV above the experimental value. To achieve that result, however, each VB structure must have its own set of orbitals.

Fig. 1 - Set of VB structures for the computation of the N and V states in the (2,2) space. Reproduced from Ref.1.

References

(1) Bender, C. F.; Dunning Jr, T. H.; Schaefer III, H. F.; Goddard Iii, W. A.; Hunt, W. J., "Multiconfiguration wavefuntions for the lowest (ππ*) excited states of ethylene". Chem. Phys. Lett. 1972, 15, 171-178. doi: 10.1016/0009-2614(72)80143-X
(2) Müller, T.; Dallos, M.; Lischka, H., "The ethylene 1¹B_1u V state revisited". J. Chem. Phys. 1999, 110, 7176-7184. doi: 10.1063/1.478621
(3) Angeli, C., "On the Nature of the π → π* Ionic Excited States: The V State of Ethene as a Prototype". J. Comput. Chem. 2009, 30, 1319-1333. doi: 10.1002/jcc.21155
(4) Wu, W.; Zhang, H.; Braïda, B.; Shaik, S.; Hiberty, P., "The V state of ethylene: valence bond theory takes up the challenge". Theor. Chem. Acc. 2014, 133, 1-13.doi: 10.1007/s00214-013-1441-x



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Quadruple bonding in C2 and analogous eight-valence electron species

Sason Shaik, David Danovich, Wei Wu, Peifeng Su, Henry S. Rzepa, Philippe C. Hiberty, Nature Chemistry 2012, 4, 195 (Paywall)
Contributed by Steven Bachrach.
Reposted from Computational Organic Chemistry with permission

Inspired by a blog post of Henry Rzepa (see here) Shaik and co-workers examined the C₂ species with an eye towards the nature of the bond between the two carbon atoms. Using both a valence bond approach and a full CI approach, they end up at the same place: there is a quadruple bond here!

The argument rests largely on a definition of of an in situ bond energy. For the VB approach, this requires choosing as a reference a non-bonding interaction between the atoms with regards to a pair of electrons. For the CI approach, the bond energy is half the energy of the singlet-triplet gap. So, for C₂, the VB/6-31G* estimate of the bond energy of the putative fourth bond is 14.3 kcal mol^-1. For the full CI/6-31G* computations of the singlet-triplet gap, the bond energy estimate is 14.8 kcal mol^-1, and using the experimental value of the gap, the estimate is 13.2 kcal mol^-1. Not a strong bond, but certainly meaningful!

In the VB approach, the fourth bond is a weighted sum of the antibonding 2σ_u and bonding 3σ_g orbitals – a combination that gives rise to small constructive overlap between the two C atoms. In the CI model, the wavefunction is dominated by the first two configurations; the first configuration, with a coefficient of C₀=0.828 has 2σ_u doubly occupied and the second coefficient, with C_D=0.324, has the 3σ_g orbital doubly occupied. Considering that 3σ_g is a bonding orbital, the significant contribution of this configuration gives rise to the fourth bond.



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