Conical intersections branching space

Figure 5, Sketch of a conical intersection. The vectors x and X2 are the GD and DC respectively, that lift the degeneracy of the two adiabatic surfaces, The plane containing these vectors is known as the branching space.

Figure 9.8. A cartoon showing (a) the conical intersection for the [2+2] photocycloaddition of two ethylenes, drawn in the branching space corresponding to the distance between the two ethylenes R (Xj) and a trapezoidal distortion (Xj), and (b) an avoided crossing in a cross-section R (Xi).

Figure 9.12. Potential energy profile along (adapted from reference 10) near the fulvene conical intersection. The branching space consists of stretching and skeletal deformation of the five-membered ring.

To conclude this section, it could be helpful to make a connection between the pictorial discussion we have just given and the type of computation that one can carry out in quantum chemistry. The double cone topology shown in Figure 9.3 can be represented mathematically by Eqs 9.3a and 9.3b. Qx, Qx, are the branching space coordinates. This equation is valid close to the apex of the cone. (A full discussion of the analytical representation of conical intersections can be found in references 9 and 10.)... 

In this section we would like to consider an example which illustrates that one can understand the occurrence of conical intersections—as well as the directions X2 corresponding to the branching space—if one has an understanding of the electronic structure of the two states involved. We address the following two questions ... 

We are now in a position to discuss the reaction profile outlined in Figure 9.17 in the full space of coordinates corresponding to the branching space Xj X2 of a conical intersection and the torsional coordinate X3. This discussion will be focused on four related concepts ... 

The geometry of the S1/S2 conical intersection together with the nature of the Xi X2 branching space, and... 

Fig. 8. Schematic representation of the potential surfaces leading to photoisomerisation of (BQA)PtMe2I from mer to fac isomer via a sloped conical intersection at / -like geometries. Shown to the right are the branching space vectors the gradient difference (gd=x1), and the derivative coupling (dc=x2). The primary orbitals involved in the electronic transition are shown to the left [Adapted from Ref. (110) with permission].

Figure 5 (a) Typical double-cone topology for a conical intersection, (b) Relation between the branching space (xj, x2) and the intersection space (spanning the remainder of the (n - 2)-dimensional space of internal geometric variables. 

Figure 12 Computed branching space vectors (gradient difference vector xx and nonadiabatic coupling vector x2) for Sj/S0 conical intersection of benzene.

Fig. 8 On the left is the Cr(CO)3 potential energy surface topology traversed after photodissociation of Cr(CO)6 to the conical intersection of the S2 and S0 states at D3h symmetry. The right hand side shows the branching space at the D conical intersection. The pseudorotation coordinate is shown connecting the C4y geometries through a C2y transition structure. The D3(i symmetry is indicted by the gray circle, the transition states by open circles while the three equivalent C4v species are indicated are indicted with solid circles. Adapted from [17]...

The predicted quantum yield of CO loss (Oco) should be very high perhaps approaching unity because each of the sequential processes is populated by way of barrierless events from the previous state. The measured co is 0.67 in cyclohexane, considerably less than unity and is further reduced to 0.52 in more viscous media [22, 23], The branching space at the conical intersection allows the Cr(CO)5 fragment to locate in one of three possible C4v species only one of which has an... 

In the many-dimensional case, the situation is more complicated (Figure 6.5). The arrival of a nuclear wave packet into a region of an unavoided or weakly avoided conical intersection (Section 4.1.2) still means that the jump to the lower surface will occur with high probability upon first passage. However, the probability will not be quite 100%. This can be easily understood qualitatively, since the entire wave packet cannot squeeze into the lip of the cone when viewed in the two-dimensional branching space, and some of it is forced to experience a path along a weakly avoided rather than an unavoided crossing, even in the case of a true conical intersection. 

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).

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