Conical intersections pairing

Nonlinearities That Lead to Multiple Degeneracies A. Conical Intersection Pairs... 

A chemical reaction takes place on a potential surface that is determined by the solution of the electronic Schrddinger equation. In Section, we defined an anchor by the spin-pairing scheme of the electrons in the system. In the discussion of conical intersections, the only important reactions are those that are accompanied by a change in the spin pairing, that is, interanchor reactions. We limit the following discussion to these class of reactions. 

A given pair of anchors may be part of several loops, containing different conical intersections. A systematic search for the third anchor is conducted by considering the electrons that are to be re-paired (i.e., that form the chemical bonds that are created in the reaction), A pragmatic and systematic way of doing this is by considering first the re-pairing of the smallest possible number of... 

Although this reaction appears to involve only two electrons, it was shown by Mulder [57] that in fact two jc and two ct elections are required to account for this system. The three possible spin pairings become clear when it is realized that a pair of carbene radicals are formally involved. Figure 14. In practice, the conical intersection defined by the loop in Figme 14 is high-lying, so that often other conical intersections are more important in ethylene photochemistry. Flydrogen-atom shift products are observed [58]. This topic is further detailed in Section VI. 

We illustrate the method for the relatively complex photochemistry of 1,4-cyclohexadiene (CHDN), a molecule that has been extensively studied [60-64]. There are four it electrons in this system. They may be paired in three different ways, leading to the anchors shown in Figure 17. The loop is phase inverting (type i ), as every reaction is phase inverting), and therefore contains a conical intersection Since the products are highly strained, the energy of this conical intersection is expected to be high. Indeed, neither of the two expected products was observed experimentally so far. 

The exchange of two pairs of a electrons is expected to lead to a high-lying conical intersection that is not likely to be important in the UV photochemistry of CHDN. This winds up the possibilities of loops involving two-election pair exchanges only. 

The next simplest loop would contain at least one reaction in which three electron pairs are re-paired. Inspection of the possible combinations of two four-electron reactions and one six-electron reaction starting with CHDN reveals that they all lead to phase preseiwing i p loops that do not contain a conical intersection. It is therefore necessary to examine loops in which one leg results in a two electron-pair exchange, and the other two legs involve three elechon-pair exchanges fip loops). As will be discussed in Section VI, all reported products (except the helicopter-type elimination of H2) can be understood on the basis of four-electron loops. We therefore proceed to discuss the unique helicopter... 

The system provides an opportunity to test our method for finding the conical intersection and the stabilized ground-state structures that are formed by the distortion. Recall that we focus on the distinction between spin-paired structures, rather than true minima. A natural choice for anchors are the two C2v stmctures having A2 and B, symmetry shown in Figures 21 and 22 In principle, each set can serve as the anchors. The reaction converting one type-I structirre to another is phase inverting, since it transforms one allyl structure to another (Fig. 12). 

Structures III and IV that have different spin-pairing schemes are expected to be higher in energy than type-I because of the strain introduced by the cyclopropyl rings. They may be anchors for secondary conical intersections around the most symmetric one. 

The transformation of ethylene to the carbene requires the re-pairing of three electron pairs. It is a phase-preserving reaction, so that the loop is an ip one. The sp -hybridized carbon atom formed upon H transfer is a chiral center consequently, there are two equivalent loops, and thus conical intersections, leading to two enantiomers. 

UNSUBSTITUTED BUTADIENE. Butadiene anchors were presented in Figures 1(3) and 13. The basic tetrahedral character of the conical intersection (as for H4) is expected to be maintained, when considering the re-pairing of four electrons. Flowever, the situation is more complicated (and the photochemistiy much richer), since here p electrons are involved rather than s electrons as in H4. It is therefore necessary to consider the consequences of the p-orbital rotation, en route to a new sigma bond. 

The nonbonding electrons of the nitrogen atom are important in determining spin re-pairing, and thus the conical intersections. This is the physical origin of the topicity concept developed by Salem and co-workers [2,30]. Two different spin... 

Conical intersections are important in molecular photochemistry, according to the current consensus, which is based on the combination of experimental and theoretical data. In this chapter, we tried to show that the location and approximate structure of conical intersections may be deduced by simple considerations of the changes in spin-pairing accompanying a reaction. We have also shown how these ideas may be put to practical computational application. 

If a contour in a given plane surrounds two conical intersections belonging to two different (adjacent) pairs of states, only two eigenfunctions flip sign—the one that belongs to the lowest state and the one that belongs to the highest one. 

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