Conic intersections vector potentials

State basis in the molecule consists of more than one component. This situation also characterizes the conical intersections between potential surfaces, as already mentioned. In Section V, we show how an important theorem, originally due to Baer [72], and subsequently used in several equivalent forms, gives some new insight to the nature and source of these YM fields in a molecular (and perhaps also in a particle field) context. What the above theorem shows is that it is the truncation of the BO set that leads to the YM fields, whereas for a complete BO set the field is inoperative for molecular vector potentials. 

As mentioned in the introduction, the simplest way of approximately accounting for the geomehic or topological effects of a conical intersection incorporates a phase factor in the nuclear wave function. In this section, we shall consider some specific situations where this approach is used and furthermore give the vector potential that can be derived from the phase factor. 

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

The relative shift of the peak position of the rotational distiibution in the presence of a vector potential thus confirms the effect of the geometric phase for the D + H2 system displaying conical intersections. The most important aspect of our calculation is that we can also see this effect by using classical mechanics and, with respect to the quantum mechanical calculation, the computer time is almost negligible in our calculation. This observation is important for heavier systems, where the quantum calculations ai e even more troublesome and where the use of classical mechanics is also more justified. 

In this chapter, we discussed the significance of the GP effect in chemical reactions, that is, the influence of the upper electronic state(s) on the reactive and nonreactive transition probabilities of the ground adiabatic state. In order to include this effect, the ordinary BO equations are extended either by using a HLH phase or by deriving them from first principles. Considering the HLH phase due to the presence of a conical intersection between the ground and the first excited state, the general fomi of the vector potential, hence the effective... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

The gradient of v l with respect to Jacobi coordinates (the vector potential) considering the physical region of the conical intersection, is obtained by using Eqs. (C.6-C.8) and after some simplification ( ) we get,... 

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