The Linear Model for Conical Intersection

The linear model of a conical intersection [58, 66, 77-79] is obtained by neglecting terms of order higher than one in the expansion of the matrix elements Hj- around the apex of the cone (r = 0, Oc = 71.6°)  

Note that depends on the value of q= l/2(J i + J 2)- The same applies to the quantities F and F. This leads to a particular simple model of conical intersections, the features of which are as follows. 

The coupling matrix elements and gr along cross sections parallel to the symmetry lowering a and symmetry conserving r axes derived from Equation 1.63 have the form 

Equation 1.67 represents graphically in a coordinate system with the axes (a, r) a series of straight lines around the apex of the cone for the locus of constant 0 in conformity with the result of numerical integration of (1.64) [70]. In particular, the line of intersection between diabatic states corresponds to the locus Hu = H22 or 0 = ir/4 + kji/2. It thus follows that as 0 — Jt/4, the intersection coincides precisely with axis r at a = 71.6°. At a complete rotation around the apex of the cone, the angle 0 increases from 0 to jt only. According to Equation 1.68, the closer the cross section lies to the apex of the cone, the sharper the Lorentzian. Therefore, Equation 1.68 correctly describes the nonradiative coupling matrix elements g(, well in accordance with the numerical calculations of the g-function cited above. 

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