Branching plane vector

Eq. (A.9) shows how the branching-plane vectors are calculated in practice from Cl difference and transition densities as well as the CSF first-derivative density ... 

In Fig. 5 we plot the branching plane vectors (Xi and X2) at the conical intersection of 1. It is apparent that in this molecule Xi and X2 describe either a local torsional deformation of the central segment of the molecule (this second mode can be more rigorously described as coupled... 

Deformation of the Cl geometry of 3db-PSB11 along the branching plane vectors X] and X2... 

At a conical intersection, the branching plane is invariant through any unitary transformation within the two electronic states and any such combination of degenerate states is still a solution. Thus, the precise definition of the two vectors in (A.9) or (A. 11) is not unique and depends on an arbitrary rotation within the space of the Cl coefficients (i.e., between the generalized crude adiabatic states), unless the states have different symmetries (then xi is totally symmetric and X2 breaks the symmetry). 

The pair (x, y) define the branching plane or g-h plane. The remainder of the intersection adapted coordinate system, w , i = l-(Ar " — 2), spans the seam space. These — 2 mutually orthonormal vectors need only be orthogonal to the branching space. It is also convenient to define... 

Fig. 5. Branching (or g, h) plane vectors for the Cl structure of Fig. 3. The Xi and X2 vectors correspond to the derivative coupling (or non-adiabatic coupling) and gradient difference vectors between the and So states.

Fig. 22. The S i/5 o conical intersection mediating the photochemical ring-opening of benzopyran together with the Xi and X2 vectors, (right) Schematic representation of the branching plane structure. The values of the relevant structural parameters are given in A and degrees. Data from Ref. 59.

Figure 3.16 The coordinates X X2) corresponding to the branching plane and reaction path. The black and red arrows indicate the opposite phase if one were to treat the branching-space vectors as vibrations. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this book.) Adapted from Figure 3.12 in Robb et ai. ...

Maeda S, Ohno K, Morokuma K. Updated branching plane for finding conical intersections without coupling derivative vectors. J Chem Theory Comput. 2010 6 1538-1545. 

Figure 3 The g and h vectors defining the branching plane (a) for OHOH, (b) for uracil (reproduced with permission from Ref. 56).

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.

It is, however, more revealing in the context of monodromy to allow/(s, ) to pass from one Riemann sheet to the next, at the branch cut, a procedure that leads to the construction in Fig. 4, due to Sadovskii and Zhilinskii [2], by which a unit cell of the quantum lattice, with sides defined here by unit changes in k and v, is transported from one cell to the next on a path around the critical point at the center of the lattice. Note, in particular, that the lattice is locally regular in any region of the [k, s) plane that excludes the critical point and that any vector in the unit cell such as the base vector, marked by arrows, rotates as the cell is transported around the cycle. Consequently, the transported dashed cell differs from that of the original quantized lattice. 

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