Adiabatic state representation

The above expressions are defined in terms of adiabatic potentials E R) and E2(R). The treatment in the adiabatic state representation is generally more accurate than the diabatic state representation. Since the diabatic state representation is, however, sometimes more convenient, the corresponding expressions in terms of the diabatic states are also given here for convenience ... 

The Zhu-Nakamura formulas can also be applied to multichannel problems, since the nonadiabatic transition is well localized at avoided crossing in the adiabatic state representation [48,109]. This was also tested for various potential systems. One numerical example is depicted in Figure 5.6 with use of the potential system shown in Figure 5.7 (see (3) of Reference [109]). These potentials are given as follows (in atomic imits) ... 

X + P )/4, which by construction varies between 0 (system is in /i)) and 1 (system is in /2)). Describing, as usual, the nuclear motion through the position X, the vibronic PO can then be drawn in the (A dia,v) plane. Here, the subscript dia emphasizes that we refer to the population of the diabatic states which are used to define the molecular Hamiltonian H. For inteipretational purposes, on the other hand, it is often advantageous to change to the adiabatic electronic representation. Introducing the adiabatic population A ad. where Nad = 0 corresponds to the lower and A ad = 1 to the upper adiabatic electronic state, the vibronic PO can be viewed in the (Nad,x) plane. Alternatively, one may represent the vibronic PO as a curve N d i + (1 — A ad)IFi between the... 

M. Lombardi What is not needed is the validity of the adiabatic approximation, that is, that there is no transition between adiabatic states. But the geometric phase is defined by following states along a path in parameter space (here nuclear coordinates) with some continuity condition. In the diabatic representation, there is no change of basis at all and thus the geometric phase is identically zero. Do not confuse adiabatic basis (which is required) and adiabatic approximation (which may not be valid). 

The coupling between the adiabatic states is provided by the off-diagonal elements of the matrix representation of Tnu ( kinetic coupling) while the coupling between the diabatic states arises from the off-diagonal elements of the matrix representation of Hei (potential coupling). 

Diabatic electronic states (previously termed crude adiabatic states ) are defined as slowly varying functions of the nuclear geometry in the vicinity of the reference geometry [9-11]. The final vibronic-coupling Hamiltonian is obtained by adding the nuclear kinetic-energy operator which is assumed to be diagonal in the diabatic representation. 

It is clear from (A.8) and (A.9) that the gradient difference and derivative coupling in the adiabatic representation can be related to Hamiltonian derivatives in a quasidiabatic representation. In the two-level approximation used in Section 2, the crude adiabatic states are trivial diabatic states. In practice (see (A.9)), the fully frozen states at Qo are not convenient because the CSF basis set l Q) is not complete and the states may not be expanded in a CSF basis set evaluated at another value of Q (this would require an infinite number of states). However, generalized crude adiabatic states are introduced for multiconfiguration methods by freezing the expansion coefficients but letting the CSFs relax as in the adiabatic states ... 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

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. 

In the adiabatic representation, Hei is diagonal and the transition among the adiabatic states of the same electronic symmetry is induced by the nonadiabatic radial coupling... 

Finally, the terms adiabatic, diabatic, and nonadiabatic are explained from the author s viewpoint. First of all, when we talk about transition in the case of curvecrossing, we have to specify the representation (adiabatic or diabatic). Nonadiabatic should be used basically to mean coupling. Thus nonadiabatic transition means a transition between adiabatic states induced by nonadiabatic coupling nonadiabatic state is not an appropriate expression. On the other hand, adiabatic should be used to mean state, and adiabatic transition should not be used. Diabatic should also basically mean state, but diabatic transition can be used to mean a transition due to diabatic coupling in the diabatic state representation. 

---