Conical intersections gradient difference vector

For states of different symmetry, to first order the terms AW and W[2 are independent. When they both go to zero, there is a conical intersection. To connect this to Section III.C, take Qq to be at the conical intersection. The gradient difference vector in Eq. f75) is then a linear combination of the symmetric modes, while the non-adiabatic coupling vector inEq. (76) is a linear combination of the appropriate nonsymmetric modes. States of the same symmetry may also foiiti a conical intersection. In this case it is, however, not possible to say a priori which modes are responsible for the coupling. All totally symmetric modes may couple on- or off-diagonal, and the magnitudes of the coupling determine the topology. 

Figure 21.5. A schematic representation of a conical intersection between two electronic states of a molecule. Coordinates qi and 52 ars the nonadiahatic coupling vector and the gradient difference vector, along which the degeneracy between the states is lifted.

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

Figure 6.12 The S -Sq conical intersection of 2ff-azirine. (a) Geometry (distances in A, angles in degrees), (b) gradient difference vector and (c) nonadiabatic coupling vector X2-...

Figure 3.20 VB structures and branching space (X, GDV and X2 DCV) for fulvene conical intersection. GDV, gradient difference vector DCV, derivative coupling vector.

Figure 3.22 Branching space for the conical intersection of azulene. Xi (the gradient difference vector) is dominated by the change in the transannular bond X2, (the derivative coupling vector) is dominated by the re-aromatization of the rings (similar to benzene).

In these last equations, g = <5 is the gradient difference vector and h = A is the linear derivative coupling vector. The space spanned by these two vectors is called the - ft space or branching space whereas the space orthogonal to the branching space is the intersection space, also called conical intersection seam. Thus, a conical intersection is a subspace of the nuclear configuration space of dimension 3N-8, where N denotes the number of atoms of the system (the space of the nuclear configurations is of dimension 3N-6). 

Robb, Bemaidi, and Olivucci (RBO) [37] developed a method based on the idea that a conical intersection can be found if one moves in a plane defined by two vectors xi and X2, defined in the adiabatic basis of the molecular Hamiltonian H. The direction of Xi corresponds to the gradient difference... 

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].

The X1 and X2 vectors presented in Fig. 2 are known as the gradient difference and derivative coupling vectors, which are special internal coordinates that lift the degeneracy to first order in nuclear motion. The remaining 3N-8 internal coordinates do not lift the degeneracy at first-order, and they span the intersection space, which is a hyperUne cmisisting of an infinite number of conical intersection points (known as the seam). 

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