Adiabatic approach states

The simplest approach to simulating non-adiabatic dynamics is by surface hopping [175. 176]. In its simplest fomi, the approach is as follows. One carries out classical simulations of the nuclear motion on a specific adiabatic electronic state (ground or excited) and at any given instant checks whether the diabatic potential associated with that electronic state is mtersectmg the diabatic potential on another electronic state. If it is, then a decision is made as to whedier a jump to the other adiabatic electronic state should be perfomied. 

The CDC-MOVB method is the appropriate computational approach for studying properties associated with the adiabatic ground state such as the reaction barrier for a chemical reaction and the solvent reorganization energy. 

If the probability for the system to jump to the upper PES is small, the reaction is an adiabatic one. The advantage of the adiabatic approach consists in the fact that its application does not lead to difficulties of fundamental character, e.g., to those related to the detailed balance principle. The activation factor is determined here by the energy (or, to be more precise, by the free energy) corresponding to the top of the potential barrier, and the transmission coefficient, k, characterizing the probability of the rearrangement of the electron state is determined by the minimum separation AE of the lower and upper PES. The quantity AE is the same for the forward and reverse transitions. 

The pictures derived from the adiabatic approach are certainly pedagogically useful but they are not necessarily a faithful view of quantum reactive systems. Now, since the adiabatic transition state theory provides the bottom line to describe reaction rates, it is necessary to implement some caveats in order to get a quantum mechanical theory of chemical reactions. 

Due to the large-level density of the lower-lying adiabatic electronic state, the chances of a back transfer of the adiabatic population are quite small for a multidimensional molecular system. To a good approximation, one may therefore assume that subsequent to an electronic transition a random walker will stay on the lower adiabatic potential-energy surface [175]. This observation suggests a physically appealing computational scheme to calculate the time evolution of the system for longer times. First, the initial decay of the adiabatic population is calculated within the QCL approach up to a time to, when the... 

Considering the practical application of the mapping approach, it is most important to note that the quantum correction can also be determined in cases where no reference calculations exist. That is, if we a priori know the long-time limit of an observable, we can use this information to determine the quantum correction. For example, a multidimensional molecular system is for large times expected to completely decay in its adiabatic ground state, that is. 

One can develop a rigorous adiabatic approach in the framework of the nuclear problem which facilitates the evaluation of the nuclear wavefunction for the D state (29,42). Note that the problem of the description of the D state was raised by Landau about 50 years ago in a classic paper (43). According to his development, the correct description of the D state should contain adiabatic coupling of the bound and continuous parts of the wavefunction. 

A means to extrapolate the attractive part of Udip(p) is to remove from the adiabatic potentials the contributions from the arrangement e+ + He+. This may be effected by constructing the basis functions for the adiabatic hyperangular states in Eq. (93) in terms only of the Jacobi coordinates for the arrangement Ps + He. The two lowest potential curves (henceforth denoted by V2(p) and V3(p)) obtained in this way are included in Figure 4.16 as red broken curves. The curve approaching Ps(n = 3), i.e., V3(p), can be... 

The validity conditions for the semiclassic adiabatic approach in the description of the systems with orbitally non-degenerate levels are elucidated in the basic works of Bom and Oppenheimer (comprehensive discussion can be found in Refs. [6,7]). In these systems, the slow nuclear motion can be separated from the fast electronic one. The situation is quite different in the JT systems where, in general, this separation is impossible due to hybridization of the electronic and vibrational states. Nevertheless, in many important cases the adiabatic approach can serve as a relatively simple and at the same time powerful tool for the theoretical study of the JT systems giving accurate quantitative results and clear insight on the physical nature of the physical phenomena. 

The theory of multi-oscillator electron transitions developed in the works [1, 2, 5-7] is based on the Born-Oppenheimer s adiabatic approach where the electron and nuclear variables are divided. Therefore, the matrix element describing the transition is a product of the electron and oscillator matrix elements. The oscillator matrix element depends only on overlapping of the initial and final vibration wave functions and does not depend on the electron transition type. The basic assumptions of the adiabatic approach and the approximate oscillator terms of the nuclear subsystem are considered in the following section. Then, in the subsequent sections, it will be shown that many vibrations take part in the transition due to relative change of the vibration system in the initial and final states. This change is defined by the following factors the displacement of the equilibrium positions in the... 

The consideration of the reactions of the electron tunneling transfer was until now based on Born-Oppenheimer s adiabatic approach (see Section 2 of Chapter 2) that was used for the description of the wave functions of the initial and final states. The electron tunneling interaction V results in the non-adiabatic transition between these states, if the matrix element Vtf... 

The double adiabatic approach provides a convenient starting point for a detpt theory (2i). The principle modification is the treatment of the FC factors for the overlap of the proton initial and final eigenstates, when the final proton state is characterized by a repulsive surface. The sum over final proton states becomes an integration over a continuum of states, and bound-unbound FC factors need to be evaluated. An approach can be formulated with methods that have been used to discuss bond-breaking electron-transfer reactions (22). If the motion along the repulsive surface for the dissociation can treated classically. 

If the diabatic coupling matrix element, He, is -independent, this d/dR matrix element between two adiabatic states must have a Lorentzian H-depen-dence with a full width at half maximum (FWHM) of 46. Evidently, the adiabatic electronic matrix element We(R) is not - independent but is strongly peaked near Rc- Its maximum value occurs at R = Rc and is equal to 1/46 = a/4He. Thus, if the diabatic matrix element He is large, the maximum value of the electronic matrix element between adiabatic curves is small. This is the situation where it is convenient to work with deperturbed adiabatic curves. On the contrary, if He is small, it becomes more convenient to start from diabatic curves. Table 3.5 compares the values of diabatic and adiabatic parameters. The deviation from the relation, We(i )max x FWHM = 1, is due to a slight dependence of He on R and a nonlinear variation of the energy difference between diabatic potentials. When We(R) is a relatively broad curve without a prominent maximum, the adiabatic approach is more convenient. When We (R) is sharply peaked, the diabatic picture is preferable. The first two cases in Table 3.5 would be more convenient to treat from an adiabatic point of view. The description of the last two cases would be simplest in terms of diabatic curves. The third case is intermediate between the two extreme cases and will be examined later (see Table 3.6). 

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