Conical intersections Hamiltonian equations

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 most frequently encountered type of conical intersection is partly induced by symmetry in the following sense. Let us consider a molecular system that possesses at lea.st one nontrivial symmetry element, e.g., a plane of reflection (Cj. point group), and thus at least two irreducible representations (A and A" in the case of the C,- point group). Consider two electronic states transforming according to different irreducible representations. In this case, the Hamiltonian of equation (18) takes the form... 

The Hamiltonian equations 15.1-15.4 is applicable to various processes characteristic of molecular systems, including the dynamics at conical intersections (Coin s) [1-4] and excitation energy transfer (EET) processes [5,6,8]. Its simplest realization corresponds to a single system operator, in which case the classical spin-boson Hamiltonian [12,18] is obtained, where the bath coordinates couple to the energy gap operator = a) (a — J3) (J31 of a two-level system (TLS). 

---