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.

A final point to be made concerns the symmetry of the molecular system. The branching space vectors in Eqs. (75) and (76) can be obtained by evaluating the derivatives of matrix elements in the adiabatic basis... 

Table I summarizes the differences in the dimension of the branching space. The origin of these differences is the behavior of the wave functions under...

In Eq. (11b), we observed that since the crude adiabatic basis is used S = 0, for kzQ. Therefore the degeneracy is lifted at first order in the Q-space only, which is therefore used to identified the branching space. The first-order result is... 

From the preceding analysis, it is seen that the coordinate space neai R can be usefully partitioned into the branching space described in tenns of intersection adapted coordinates (p, 9, ) or (x,y,z) and its orthogonal complement the seam space spanned by a set of mutually orthonormal set w, = 4 — M . From Eq. (27), spherical radius p is the parameter that lifts the degeneracy linearly in the branching space spanned by x, y, and z. 

We then dehne an internal coordinate <() such that <() = 0 2ti denotes a a path that has described one complete loop around the Cl in the nuclear branching space. Other than this, we need specify no further details about < ). We do not even need to specify whether the complete set of nuclear coordinates give a direct product representation of the space. It is sufficient that closed loop has wound around the CL Using this definition of ((), we can express the effect of the GP on the... 

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... 

Figure 9.25. A cartoon showing the lowest K-K excited states along a proton transfer coordinate and a skeletal deformation coordinate from the branching space (adapted from Figure 4 of reference 16).

We have established an important principle in electron transfer theory that is not present in conventional one-dimensional models. The reaction coordinate is always localizing and corresponds to coordinate Aj. The coordinate X2 corresponds to the direction in which the matrix element between ground and excited states is switched on. If this coordinate has zero length then the branching space becomes one dimensional and an adiabatic reaction path does not exist. We now consider two examples. 

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