Noncrossing rules

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. 

Noncrossing rule, geometric phase theory, 2 Nondemolition measurements, phase interference, 207... 

The photochemical disrotatory closure of butadiene to cyclobutene has been described with a state-correlation diagram, like that shown in Figure 21.4. It is based on the familiar orbital-correlation diagram of Woodward and Hoffmann," from which the intended correlations indicated by the dashed lines can readily be deduced. The solid lines indicate that there is an avoided crossing, which is put in as a result of the quantum mechanical noncrossing rule. It says that two states of the same total symmetry cannot cross. Instead, as they approach each other in energy, they will mix and separate, as the solid lines indicate. 

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... 

Figure 4.6 Noncrossing rule of Teller for potential energy surfaces.

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 ... 

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