Adiabatic representation intersections

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. 

Single surface calculations with proper phase treatment in the adiabatic representation with shifted conical intersection has been performed in polai coordinates. For this calculation, the initial adiabatic wave function tad(9, 4 > o) is obtained by mapping t, to) ittlo polai space using the relations,... 

Yarkony DR (2001) Nuclear dynamics near conical intersections in the adiabatic representation. I. The effects of local topography on interstate transition. J Chem Phys 114 2601... 

Single surface calculations with proper phase treatment in the adiabatic representation with shifted conical intersection has been performed in polar coordinates. For this calculation, the initial adiabatic wave function bad(< , to) is obtained by mapping 4 a and R Rq = qcas < x At this point, it is necessary to mention that in all the above cases the initial wave function is localized at the positive end of the R coordinate where the negative and positive ends of the R coordinate are considered as reactive and nonreactive channels. 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

In this section, we describe an approach which, in addition to the implicit inclusion of the GP, takes into account all the non-adiabatic couplings between the two conically intersecting electronic surfaces. In this case, within the adiabatic representation, the nuclear Hamiltonian has the following form ... 

Under the conditions of validity of the two-electronically-adiabatic-state approximation it is possible to change from the i]/al,ad(r q) (n = i, j) electronically adiabatic representation to a diabatic one 1,ad(r q) (n = i, j) for which the VR Xn(R) terms in the corresponding diabatic nuclear motion equations are significantly smaller than in the adiabatic equation or, for favorable conditions, vanish [24-26]. Such an electronically diabatic representation is usually more convenient for scattering calculations involving two electronically adiabatic PESs, but not for those involving a single adiabatic PES. This matter will be further discussed in Sec. III.B.3 for the case in which a conical intersection between the E ad(q) and Ejad(q) PESs occurs. 

DETERMINATION OF POTENTIAL ENERGY SURFACE INTERSECTIONS AND DERIVATIVE COUPLINGS IN THE ADIABATIC REPRESENTATION... 

A point on the surface of conical intersection can be located as follows. Assume X is sufficiently close to a point of conical intersection that Eq. (8) is valid. In the adiabatic representation... 

We refer to Chapter 4 for a detailed discussion on the definition and explicit construction of diabatic states. The diabatic representation is generally advantageous for the computational treatment of the nuclear dynamics if the adiabatic potential-energy surfaces exhibit degeneracies such as conical intersections. Moreover, the diabatic representation often reflects more clearly than the Born ppenheimer adiabatic representation the essential physics of curve crossing problems and is thus very useful for the construction of appropriate model Hamiltonians for polyatomic systems. 

We particularly focused on a conically intersecting manifold of two electronic states and described the quantum flux operator formalism within a time-dependent WP approach to calculate the initial state-specific and energy resolved reaction probabilities. The flux operator is represented in the two-state diabatic as well as adiabatic representation. While the... 

To give an impression of the virtues and shortcomings of the QCL approach and to study the performance of the method when applied to the dynamics at conical intersections, in the following we briefly introduce the QCL working equation in the adiabatic representation, describe a recently proposed stochastic trajectory implementation of the resulting QCL equation, and apply this numerical scheme to Model I, which describes the S2 Si internal conversion in pjo azine. 

Thus, the angle a, which relates a diabatic representation to the adiabatic representation, is related to the angle 0 defined from the intersection-adapted coordinates. 

World Scientific, Singapore, 2004, pp. 129-174. Determination of Potential Energy Surface Intersections and Derivative Couplings in the Adiabatic Representation. 

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