Perfect Biradicals

The energy diagram may be derived equally well starting with any orbital choice, but the three canonical ones are the simplest to use. We shall outline the derivation starting with the most localized orbitals. If for the moment spin is ignored, the four configurations if (l) if (2), if (l)Xfc(2), and 

Starting from delocalized orbitals and and introducing and K n = (i -I- J22V2 — y,2 as the corresponding electron repulsion integrals. 

The overlap density q q) is smaller for the localized orbitals Xa and Xb which occupy as distinct parts of space as at all possible, than for the delocalized orbitals and so 

The wave function [ ffl(l) f (2) - t ( I ) fa(2)]/V2 is antisymmetric with respect to an interchange of electrons 1 and 2 and needs to be multiplied with one of the three symmetric two-electron spin functions to describe a triplet state T. The other three functions are symmetric, need to be multiplied with the antisymmetric two-electron spin function, and describe three singlet states, So, S, and Sj. The amusing isomorphism of the two-electron ordinary and spin functional space has been discussed elsewhere (Michl, 1991, 1992). 

POTENTIAL ENERGY SURFACES BARRIERS, MINIMA, AND FUNNELS 

POTENTIAL ENERGY SUKIACES BAKRIEK.S, MINIMA, AND FLINNELS 

Biradicals for which K = K, are referred to as axial hiradicals and in these. So and S, are degenerate. This condition is normally enforced by the presence of a threefold or higher axis of symmetry (e.g., in O2 and the pentagonal cyclopentadienyl cation), but alone, this is not sufficient (e.g., square cyclobutadiene is almost a pair biradical). 

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Figure 4.19. Wave functions and energy levels of a perfect biradical (center), constructed from the most localized orbitals x nnd Xi> tind from the most delocalized orbitals < > and (right) (adapted from BonaCiC-Kouteck et al., 1987).

First of all, we consider the significance of the presence of C-, +, the coefficient of the in-phase combination of the two hole-pair functions in the So wave function, in Equation 4.12. In the simple model for a perfect biradical, this in-phase combination is exactly equal to the wave function of the Sj state, and it does not enter into those of the So and S, states at all. Thus, in this approximation. So does not spin-orbit couple to the triplet. The same is true in weakly heterosymmetric biradicaloids (0 < 5 < S ), in which the in-phase hole-pair character is shared by S, and Sj, but not Sg, and the former two spin-orbit couple to T, but So does not. In strongly heterosymmetric... 

Figure 6.12. The effect of diagonal inieraciions on the energy of the degenerate localized nonhonding orbitals of a cyclobiitadiene-likc perfect biradical.

Xi - Xi + Xt)- The most stable MO , = j( f, + Xi Xi X ) >s neglected in this example it is always doubly occupied and forms a nonpolar-izable core. At square or nearly square geometries the system is a perfect biradical or a homosymmetric biradicaloid, respectively, and the energy ordering of the three singlet states may be obtained from Figure 4.19 or 4.20 ... 

In homosymmetric biradicaioids, a bonding interaction between the most localized orbitals is the only perturbation ( covalent perturbation ). Its magnitude is described by y, approximately equal to twice the resonance integral of simple theories. Its presence causes a mixing of the state functions (XaXb) and (jd + of the perfect biradical, which stabilizes its ground... 

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. 

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