Conical intersection hyperline

At this stage, we wish to emphasize that a point (molecular geometry) on a conical intersection hyperline has a well-defined electronic structure (illustrated in Figure 9.6 or Eq. 9.2 with T = 0) and a well-defined geometry. Of course, the four electrons in four Is orbitals shown in Figure 9.6 is a very simple example, but we believe it is useful in order to be able to appreciate the generality of the conical intersection construct. In more complex systems, the conical intersection hyperline concept persists, but the rationalization may be less obvious. 

Figure 9.9. A cartoon showing the conical intersection hyperline traced out by a degeneracypreserving coordinate X3. The system remains degenerate as one traverses the coordinate X3, but the energy and the shape of the double-cone must change in Xi X2. See color insert.

It is useful to consider the information contained in Figure 9.9 in a different way. In Figure 9.10, we show the conical intersection hyperline traced out by a coordinate... 

Figure 9.10. The conical intersection hyperline traced out by a coordinate X3 plotted in a space containing the coordinate X3 and one coordinate from the degeneracy-lifting space X1X2. See color insert.

The important points on a conical intersection hyperline are those where the reaction path meets with the seam (see Fig. 9.10)... 

Figure 9.22. The geometries in Figure 9.21 located in the cone which changes shape along the conical intersection hyperline (adapted from reference 14). See color insert.

We hope that the preceding discussions have developed the concept of a conical intersection as being as real as many other reactive intermediates. The major difference compared with other types of reactive intermediate is that a conical intersection is really a family of structures, rather than an individual structure. However, the molecular structures corresponding to conical intersections are completely amenable to computation, even if their existence can only be inferred from experimental information. They have a well-defined geometry. Like the transition state, the crucial directions governing dynamics can be determined andX2) even if there are now two such directions rather than one. As for a transition structure, the nature of optimized geometries on the conical intersection hyperline can be determined from second derivative analysis. 

Different topological situations are possible for unavoided crossings between surfaces. One can have intersections between states of different spin multiplicity [an (n - l)-dimensional intersection space in this case, since the interstate coupling vector vanishes by symmetry], or between two singlet surfaces or two triplets [and one has an n - 2)-dimensional conical intersection hyperline in this case]. We have encountered situations in which both types of... 

If the connection between the two surfaces is sloped, the funnel may well be more efficient. In addition, the degeneracy at a crossing point can also be lifted at second order. As a consequence, we can choose a coordinate system in which to mix the branching and intersection space coordinates to remove this splitting and preserve the degeneracy to second order. These new coordinates give the ciu-vature of the conical intersection hyperline and determine whether one has a minimum or a saddle point on it. These studies may also provide the vibrational modes that must be stimulated in order to enhance nonradiative decay because they decrease the energy gap and can lead to a Cl (see Paterson et al. 2004 Sicilia et al. 2007). 

Let us summarize briefly at this stage. We have seen that the point of degeneracy forms an extended hyperline which we have illnstrated in detail for a four electrons in four Is orbitals model. The geometries that lie on the hyperline are predictable for the 4 orbital 4 electron case using the VB bond energy (Eq. 9.1) and the London formula (Eq. 9.2). This concept can be nsed to provide nseful qualitative information in other problems. Thns we were able to rationalize the conical intersection geometry for a [2+2] photochemical cycloaddition and the di-Jt-methane rearrangement. 

We now proceed to look at three examples from recent work in some depth. In the first example, we wish to illustrate that a knowledge of the VB structure or of the states involved in photophysics and photochemistry rationalize the potential surface topology in an intuitively appealing way. We then proceed to look at an example where the extended hyperline concept has interesting mechanistic implications. Finally, we shall look at an example of how conical intersections can control electron transfer problems. 

Figure 6.5. Conical intersection of two potential energy surfaces S, and Sg the coordinates x, and Xj define the branching space, while the touching point corresponds to an (F - 2)-dimensional hyperline. Excitation of reactant R yields R, and passage through the funnel yields products P, and P, (by permission from Klessinger, 1995).

The potential energy surface plotted along Xj and Xg has a conical shape, and therefore crossings of the same multiplicity are called conical intersections. The remaining n-2 coordinates form a hyperline, the intersecting space, which consists of an infinite number of crossing points. (For an illustration, see Figure 2.19 in Section 2.5.)... 

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