Surface crossings conical intersection

Class Diabatic reactionss proceed directly from the excited state to the ground state of the photoproduct via geometries, at which the two surfaces cross (conical intersections, Special Topic 2.5) or nearly cross (avoided crossings, funnels). In such reactions, no intermediates other than the excited state reached by absorption can usually be detected, because the molecule arrives on or near to a cusp of the ground-state surface and immediately proceeds to a stable product. 

Pig. 12. Representative cut through the potential energy surfaces of the X E g — E B2u states of the benzene radical cation. An effective coordinate has been used, which is a linear combination of all linearly active normal modes and has been chosen so as to reveal the various low-energy curve crossings (conical intersections) qualitatively correctly. These are indicated by the open circles and comprise JT intersections (at Qejf = 0) as well as PJT intersections (at Qeff 0). 

Figure B3.4.16. A generic example of crossing 2D potential surfaces. Note that, upon rotating around the conic intersection point, the phase of the wavefunction need not return to its original value.

Conical intersections can be broadly classified in two topological types peaked and sloped [189]. These are sketched in Figure 6. The peaked case is the classical theoretical model from Jahn-Teller and other systems where the minima in the lower surface are either side of the intersection point. As indicated, the dynamics of a system through such an intersection would be expected to move fast from the upper to lower adiabatic surfaces, and not return. In contrast, the sloped form occurs when both states have minima that lie on the same side of the intersection. Here, after crossing from the upper to lower surfaces, recrossing is very likely before relaxation to the ground-state minimum can occur. 

A conical intersection needs at least two nuclear degrees of freedom to form. In a ID system states of different symmetry will cross as Wy = 0 for i j and so when Wu = 0 the surfaces are degenerate. There is, however, no coupling between the states. States of the same symmetry in contrast cannot cross, as both Wij and Wu are nonzero and so the square root in Eq. (68) is always nonzero. This is the basis of the well-known non-crossing rule. 

The phase-change nale, also known as the Ben phase [101], the geometric phase effect [102,103] or the molecular Aharonov-Bohm effect [104-106], was used by several authors to verify that two near-by surfaces actually cross, and are not repelled apart. This point is of particular relevance for states of the same symmetry. The total electronic wave function and the total nuclear wave function of both the upper and the lower states change their phases upon being bansported in a closed loop around a point of conical intersection. Any one of them may be used in the search for degeneracies. 

The emphasis in our previous studies was on isolated two-state conical intersections. Here, we would like to refer to cases where at a given point three (or more) states become degenerate. This can happen, for example, when two (line) seams cross each other at a point so that at this point we have three surfaces crossing each other. The question is How do we incorporate this situation into our theoretical framework ... 

Figure 14. Classical trajectories for the H + H2(v = l,j = 0) reaction representing a 1-TS (a-d) and a 2-TS reaction path (e-h). Both trajectories lead to H2(v = 2,/ = 5,k = 0) products and the same scattering angle, 0 = 50°. (a-c) 1-TS trajectory in Cartesian coordinates. The positions of the atoms (Ha, solid circles Hb, open circles He, dotted circles) are plotted at constant time intervals of 4.1 fs on top of snapshots of the potential energy surface in a space-fixed frame centered at the reactant HbHc molecule. The location of the conical intersection is indicated by crosses (x). (d) 1-TS trajectory in hyperspherical coordinates (cf. Fig. 1) showing the different H - - H2 arrangements (open diamonds) at the same time intervals as panels (a-c) the potential energy contours are for a fixed hyperradius of p = 4.0 a.u. (e-h) As above for the 2-TS trajectory. Note that the 1-TS trajectory is deflected to the nearside (deflection angle 0 = +50°), whereas the 2-TS trajectory proceeds via an insertion mechanism and is deflected to the farside (0 = —50°).

The closer the trajectory approaches the conical intersection, the smaller Cy becomes. Since the nonadiabatic transitions are expected to take place in the close vicinity of the conical intersection, the nonadiabatic transition direction can be approximated by the eigenvector of the Hessian d AV/dRidRj corresponding to its maximum eigenvalue. Similar arguments hold for nonadiabatic transitions near the crossing seam surface, in which case the nondiagonal elements of the diabatic Hamiltonian of Eq. (1) should be taken as nonzero constant. 

The other aspect of a conical intersection that we have tried to emphasize is that there is a relationship between the valence bond structures associated with the ground state or the excited state and the position of the surface crossing. In any mechanistic study this is also very interesting because it provides information that can be used to think intuitively about mechanisms. We will try to emphasize this point of view in the rationalization of all the examples we will look at. 

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