Ethylene orthogonally twisted

The simplest way to illustrate physical meaning of these quantities is to consider the perturbations of orthogonally twisted ethylene for which SAB = yAB = <5ab = yab = 0 holds via (1) return to planarity or (2) substitution at one end of the C=C bond. For (1), localized orbitals interact, yAB 0, but their energies are the same, 5AB = 0. Since delocalized orbitals become eventually HOMO and LUMO of planar ethylene, they do not have the same energy, Sab 0, but they do not interact, yab = 0. For (2), orthogonal-substituted ethylene, the situation is different. In the localized basis SAB 0, but the interaction is not present due to the symmetry yAn = 0. (A and 2 S belong to different irreducible representations.) For the delocalized description the energies of these orbitals are the same 5ab = 0 since the orbitals are equally distributed over both carbon atoms. But yab 0, since a and b are not canonical orbitals. 

Figure 4.6. Energies a) of the bonding MO Ji and the antibonding MO n and b) of the -electronic states of ethylene as a function of the twist angle 6. On both sides the states are labeled by the MO configuration dominant at planar geometries in the middle, they are labeled by the VB structure that is dominant at the orthogonally twisted geometry (by permission from Michl and BonaCiC-Koutecky, 1990).

Figure 4.25. Sum-over-atoms factor in the spin-orbit coupling vector // a) in orthogonally twisted ethylene and b) in (0, 90°) twisted trimethylene biradical, using Equation (4.12) and (4.13) most localized orbitals x - Xh and nonvanishing atomic vectorial contributions from Xh (white through-space, black through-bond).

Ethylene. 90°-twisted ethylene is a perfect biradical distortion toward planarity (cj) < 90°) leads to an interaction /= 0 of the localized orbitals A and B, yielding a homosymmetric biradicaloid, and the coefficient Cq of the hole-pair configuration in the singlet ground state increases with increasing y- That is to say, ethylene violates condition (1) but satisfies condition (2) when it is orthogonally twisted, and satisfies condition (1) but violates condition (2) when it is planar. In partially twisted ethylene, however, conditions (1), (2) and (3) are fulfilled. Therefore, SOC is expected to vanish for ( ) = 0 and maximum value for (j) = 45°, as has been pointed out first by Caldwell et al. [29], and is apparent from Figure 3. 

Comparison of the energies of twisted and planar ethylene leads to the results in Fig. 20. The singlet state prefers planar geometry since the overlap of the n and n orbitals leads to a strong pi bond. However, for the triplet state, the 7Ti and TTr orbitals must be orthogonalized to each other, and this state prefers the 90° twisted geometry. 

The criteria of allowedness discussed in the preceding two sections do not require the explicit consideration of orbital symmetry, in the sense that the symmetry elements retained along the reaction path do not enter directly into the analysis consequently, they were not drawn in the figures. However, it is easy to ascertain from Fig. 1.1, for example, that two ethylene molecules in either the coplanar or [s -f s] orientation have three perpendicular mirror planes one common to the four carbon atoms, another reflecting one molecule into the other, and a third bisecting both of them three twofold axes of rotation (one at the intersection of each pair of mirror planes) and a center of inversion at the point where the three rotational axes intersect. After both molecules have been twisted so as to react in the [a + a] mode (Fig. 1.1c), only the rotational axes remain, whereas the off-orthogonal orientation of Fig. 1.4b retains a single twofold rotational axis and no other element of symmetry. 

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