The Noncrossing Rule

Neumann and Wigner proved, in their seminal work in 1929 that, for a molecular system with N internal nuclear coordinates (N = 3N — 6), two electronic surfaces become degenerate in a subspace of dimension N — 2. To illustrate this dimensionality rule, consider two intersecting adiabatic electronic states, /i and 2- These two states are expanded in terms of two diabatic states t)i and ( i, which are diagonal to all the remaining electronic states and to each other,  

For the eigenvalues of this matrix to be degenerate, two conditions must be satisfied 

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In this chapter, recent advances in the theory of conical intersections for molecules with an odd number of electrons are reviewed. Section II presents the mathematical basis for these developments, which exploits a degenerate perturbation theory previously used to describe conical intersections in nonrelativistic systems [11,12] and Mead s analysis of the noncrossing rule in molecules with an odd number of electrons [2], Section III presents numerical illustrations of the ideas developed in Section n. Section IV summarizes and discusses directions for future work. 

In a concerted reaction, orbital and state symmetry is conserved throughout the course of the reaction. Thus a symmetric orbital in butadiene must transform into a symmetric orbital in cyclobutene and an antisymmetric orbital must transform into an antisymmetric orbital. In drawing the correlation diagram, molecular orbitals of one symmetry on one side of the diagram are connected to orbitals of the same symmetry on the other side, while observing the noncrossing rule. 

So what s the error in this description It turns out to be the invocation of the noncrossing rule. It has been known for many decades that PE hypersurfaces for states of the same total symmetry actually can cross.Quite why this information... 

It is now possible to draw a correlation diagram between the states, as shown in Figure 7.13. The crucial feature to note here is that the Ax to Ax correlations which would seem to follow from direct orbital,correlations cannot and do not actually occur, because of what is called the noncrossing rule. Two states of the same symmetry cannot cross, in the manner indicated by the dotted lines, because of electron repulsion. Instead, as they approach they turn away from each other so that the lowest 4, states on each side are correlated with each other as shown by the full lines. The repulsive interaction is similar in essence to that involved in configuration interaction in naphthalene, as discussed in Section 7.6. Indeed, the noncrossing rule is no more than a special but straightforward instance of configuration interaction. 

The second principle, which has been used earlier (page 194) in constructing the correlation diagrams for the Woodward-Hoffmann rules, and which has its ultimate origin in the phenomenon of configuration interaction (page 179) is called the noncrossing rule ... 

The Ay state arising from 3B1 and the Ax state arising from 1A1 may now mix, since the noncrossing rule applies.128 Thus in the presence of spin interactions there is an avoided crossing as indicated in Figure 5. This noncrossing implies that in the isolated CH2 molecule there is a finite transition probability between the nonstationary zero-order states. If we... 

This equation is similar to eq. (11-4) except that here [7] stands for either the 3Elu or 3B2u states. Noting that the spin interaction matrix element of eq. (11-10) is zero at Q = Qe because of eqs. (11-1) and (11-2), we assume that this Q vibrational coordinate is one of those which destroys the symmetry of the molecule so that the noncrossing rule applies. We may then approximate the spin interaction matrix element, to lowest order in Q, as varying as Q. Then... 

Figure 2-16 The noncrossing rule two energy levels with the same symmetry properties repel each other and therefore never cross. With = H22 and S = 0, we have...

The lines joining reactant and product orbitals in Figure 11.8 are referred to as correlation lines, and the entire diagram is an orbital correlation diagram. It will be noted that since there are two orbitals of each symmetry type on each side, there is an alternative way the correlation might have been made, namely 77-i to a, 7r2 to 7T, 7t3 to 77, 7r4 to (j. This alternative is eliminated by the noncrossing rule Orbitals of the same symmetry do not cross. 

Although the noncrossing rule may in many instances be relied upon to determine the correlation pattern where alternatives exist, it is not infallible. In order to avoid difficulties in constructing correlation diagrams, Woodward and Hoffmann cite three precautions that should be observed.19... 

Show that a correlation diagram for the v2s + n2s cycloaddition constructed by analysis with respect to either the C2 axis or the third mirror plane, neither of which bisect bonds made or broken, and by establishing correlations from the noncrossing rule leads to the incorrect conclusion that the reaction is allowed. How is the picture modified if correlation is established by preserving orbital nodal structure rather than by using the noncrossing rule ... 

Teller19 was the first to point out that in a polyatomic molecule the noncrossing rule, which is rigorously valid for diatomics, fails. Rather, two electronic states, even if they have the same symmetry, are allowed to cross at a conical intersection. Accordingly, radiationless decay from the upper to the lower intersecting state can occur within a single vibrational period when the... 

This requires the existence of at least two independently variable nuclear coordinates. Since in a diatomic molecule there is only one variable coordinate—the interatomic distance—so the noncrossing rule can be stated as follows ... 

The state-symmetry correlation also indicates that electrocyclic radical interconversion favors a conrotatory path from the first excited state and a disrotatory path from the second excited state. Because of the proximity of the energy levels and the violations of the noncrossing rule, it is probable that the excited state process will not be highly stereoselective. The same detailed considerations must be applied to the five-atom five-electron system and yield the results given in Table 1. Differences between the stereochemical predictions of Table 1 and those of others (Woodward and Hoffmann, 1965a Fukui and Fujimoto, 1966b Zimmerman, 1966) tend to be limited to the excited-state reactions of odd-atom radicals. 

The seam must be determined. A reasonable approximation is to choose for seams those regions of M where the interacting potential energy surfaces are degenerate, or nearly degenerate (cf. the noncrossing rules). 

An obvious connection between states that possess the same electronic configuration would be the one indicated by dashed lines in Figure 7-13. This does not occur, however, because states of the same symmetry cannot cross. This is, again, a realization of the noncrossing rule, which applies to electronic states as well as to orbitals. Instead of crossing, when two states are coming too close to each other they will turn away, and so the two ground states, both of Ag symmetry and also two Ag symmetry excited states will each mutually correlate. 

Although the state correlation diagram is physically more meaningful than the orbital correlation diagram, usually the latter is used because of its simplicity. This is similar to the kind of approximation made when the electronic wave function is replaced by the products of one-electron wave functions in MO theory. The physical basis for the rule that only orbitals of the same symmetry can correlate is that only in this case can constructive overlap occur. This again has its analogy in the construction of molecular orbitals. The physical basis for the noncrossing rule is electron repulsion. It is important that this applies to orbitals—or states—of the same symmetry only. Orbitals of different symmetry cannot interact anyway, so their correlation lines are allowed to cross. 

For two terms of the same symmetry, this condition is always satisfied at low velocity as, r /v and the noncrossing rule implies that w 0. Terms of different symmetry may cross, but the most efficient nonadiabatic coupling need not be localized at the crossing point, though this is usually the case. If v is a small parameter, a perturbation treatment of equation (2) can be attempted. It was shown by Landau [10] that for noncrossing terms the matrix elements Nkm are exponentially small, that is, Nkm exp (—a>kmTkm). As for the preexponential factor Bkm, more detailed treatment shows two possibilities Either Bkm depends on velocity v and is small for low v, or it is independent of v. Only the first case, which is realized for terms of different symmetries, can be treated by the usual perturbation method. Thus, to first... 

From the data in Figs. 10 and 11 and from the values of the localization energies72 and superdelocalizabilities it is apparent that all indices lead to the same order of the reactivity centers, so that the noncrossing rule 106 is satisfied. The theoretical prediction of the reactivity centers is presented in Fig. 12. This holds over a wide range... 

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