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. 

Equations 9.3a and 9.3b give the energy of the upper and lower part of the cone (f/ and f/fi). In Eqs 9.3a and 9.3b, the first term represents Q in Eq. 9.2, while the expression under the square root sign corresponds to Tin Eq. 9.2. is the reference energy at the apex of the cone. The remaining qnantities in these two equations are energy derivatives. The quantity in Eq. 9.3g is the gradient difference vector, while the qnantity in Eq. 9.3h... 

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.

Clearly f will go to zero when E2 = Et, independently of the magnitude of. Note, however, that the gradient will also go to zero if Et is different from E2 but the two surfaces are parallel (i.e., Xj, the gradient difference vector, has zero length). In this case the method would fail. This situation will occur for a Renner-Teller-like degeneracy, for example. Of course, in this case, the geometry can be found by normal unconstrained geometry optimization. 

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-...

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.

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). 

Thus Xi is the gradient difference vector and X2 is the interstate coupling vector that occurs in equation (7). 

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