Conical intersections coordinate origins

Hence, the expression of Eq. (5) indicates that, in a polar coordinate system, Eq. (4) will remain unchanged even if the position of the conical intersection is shifted from the origin of the coordinate system. 

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. 

Reactive State-to-State Transition Probabilities when Calcnladons are Performed Keeping the Position of the Conical Intersection at the Origin of the Coordinates... 

Reactive State-to-State Transition ftobabilides when Calculations are Performed by Shifting the Position of Conical Intersection from the Origin of the Coordinate System... 

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. 

Appendix C On the Single/Multivaluedness of the Adiahatic-to-Diahatic Transformation Matrix Appendix D The Diabatic Representation Appendix E A Numerical Study of a Three-State Model Appendix F The Treatment of a Conical Intersection Removed from the Origin of Coordinates Acknowledgments References... 

Reference [73] presents the first line-integral study between two excited states, namely, between the second and the third states in this series of states. Here, like before, the calculations are done for a fixed value of ri (results are reported for ri = 1.251 A) but in contrast to the previous study the origin of the system of coordinates is located at the point of this particulai conical intersection, that is, the (2,3) conical intersection. Accordingly, the two polar coordinates (adiabatic coupling term i.e. X(p (— C,2 c>(,2/ )) again employing chain rules for the transformation... 

APPENDIX F THE TREATMENT OF A CONICAL INTERSECTION REMOVED FROM THE ORIGIN OF COORDINATES... 

Equation [8] is the equation of an elliptic double cone (i.e., with different axes) with vertex at the origin (it will be a circular cone only for the case k = /). Thus, such crossing points are called conical intersections. Indeed, if we plot the energies of the two intersecting states against the two internal coordinates xx and x2 [whose values at the origin satisfy the two conditions and H1 j = H22 and H12 (= H21) = 0], we obtain a typical double-cone shape (see Figure 5). 

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