Branching plane

In Figure 14b we show the potential energy surface for a model elliptic conical intersection46 plotted in the branching plane (xt, x2). Because, as stated earlier, the cone is elliptic in the linear approximation (i.e., the base of the cone is an ellipse rather than a circle), there are two steep sides of the ground state cone surface and two ridges . There are two preferred directions for downhill motion located on the steep sides of the ground state cone surface. A simple... 

Figure 14 Illustration of the general procedure used to locate the initial relaxation direction (IRD) toward the possible decay products, (a) General photochemical relaxation path leading (via conical intersection decay) to three different final structures, (b) Potential energy surface for a model elliptic conical intersection plotted in the branching plane, (c) Corresponding energy profile (as a function of the angle a) along a circular cross section centered on the conical intersection point and with radius d.

Here, a topology-adapted representation [55] was chosen, where (Xi,X2) lift the degeneracy at the intersection and thus span the branching plane [74], These modes are obtained by orthogonalizing the modes (X, Xa) of Eq. (10). The third mode Xg is in turn orthogonal to (Xi, ) and carries information on the intersection space, i.e., the X+ component of Eqs. (9)-(10). Alternative construction schemes are possible in particular, the bilinear coupling terms can be eliminated within the three-mode subspace [54,72]. 

The detailed derivation of iTeff is given in Ref. [53]. Here, two of the six modes (i.e., X and X2) are chosen as topology-adapted modes that span the branching plane for a chosen pair of electronic states (here, states 1 and 2). Each /th-order residual term now also comprises 6 modes,... 

Fig. 10 For the 3-state model of Sec. 5.2., projections of the coupled diabatic XT, CT and IS potential surfaces (E configuration) onto the XT-CT branching plane are shown. The white and black circles indicate the conical intersection and Franck-Condon geometry, respectively. Reproduced from Ref. [52]. Copyright 2008 by the American Physical Society.

Neglecting bilinear couplings between branching-plane and intersection-space coordinates as in the approximation used in the solution of (25) leads to... 

Fig. 9 Eigenvalues of the energy-difference Hessian computed at the Franck-Condon point of benzene in the 28-dimensional space orthogonal to the pseudo-branching plane. The labels refer to the most similar normal modes of So benzene (Wilson s convention). The dominant local motions are indicated in boxes (reprinted with permission from [31])...

Quantum dynamics simulations were run within a nine-dimensional model subspace including the nine most important modes displayed on Fig. 10 and a fivedimensional model including only the pseudo-branching-plane modes 1 and 15, and the three out-of-plane photoactive modes 4, 16x, and 16 j [31,53]. The results were interpreted with regard to the topological features of the extended seam of conical intersection and their influence on the photoreactivity. This is illustrated with Fig. 11. 

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 ... 

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 elliptic cone model of the potential energy surface at a conical intersection point discussed above is not general enough to give a correct description of the relaxation in realistic molecules. First, more than two possible IRDs may originate from the tip of the cone. Second, the first-order approximation (i.e., elliptic cone) may break down at larger distances, and some IRDs may lie out of the branching plane because the real... 

Born-Oppenheimer approximation (p. 272) branching plane (p. 312) branching space (p. 311) clamped nuclei Hamiltonian (p. 264) conical intersection (p. 312) continuum states (p. 297)... 

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... 

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... 

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