Electronically adiabatic approach

Fig. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

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. 

For typical outer-sphere exchanges at ordinary temperatures, it seems probable that the original assumption of Hush and of Marcus that barrier penetration is a comparatively minor effect is correct. Moreover> it is, in a particular case, quite simple to calculate. The more general questions to which we do not yet have an answer are how adequate is the Golden Rule approach in discussing tunnelling, and, in particular, what would be expected for systems strictly remaining on one surface (electronically adiabatic) A number of fundamental issues involved here have been discussed in a recent series of papers (42-45). 

We have considered the case of vibrational motion of the photofragments accompanied by slow relative motion. We have developed the adiabatic approach to evaluate the nuclear wave-function (Jp and obtained eqs. 74 and 96. Note, that instead of a system of electrons and nuclei (Born-Oppenheimer approximation), we considered here only nuclear motion of a polyatomic system with several degrees of freedom, one of which is "fast" relative to the others. 

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... 

Here, He(j) is Hamiltonian of a free electron, V,-(r) is Coulomb s interaction of the electron with the donor ion residue, Hlv( q ) is Hamiltonian of the vibration subsystem depending on the set of the vibration coordinates qj that corresponds to the movement of nuclei without taking into account the interaction of the electron with the vibrations. The short-range (on r) potential Ui(r, q ) describes the electron interaction with the donor ion residue and with the nuclear oscillations. The wave function of the system donor + electron may be represented in MREL in the adiabatic approach (see Section 2 of Chapter 2) ... 

The deviation from the adiabatic approach in the asymptotics is particularly remarkable when the energy ED is small and comparable with the reorganization energy in the vibration system ErD at the transition from the neutral donor to the positive ion. Such smallness oflEhlis possible, if the donor electron level in a crystal (see Section 3) lies near the conductivity or valence bands (compare with expression (23)), and hE plays the role of Ed. 

What is the criterion of the applicability of the adiabatic approach for the description of the asymptotics of the wave function of an electron-nuclear system ... 

Within the quasiclassical approach the nuclei are considered to be subject to a classical motion in the field of force, the potential of which is given by the energy pertinent to one of the eigenstates of the electronic subsystem. In the case of electronically adiabatic processes, the field of force for the nuclear motion is determined by a single potential energy surface (pertinent to a single electronic state). 

Considerable use continues to be made of classical trajectory calculations in relating the experimentally determined attributes of electronically adiabatic reactions to the features in the potential energy surface that determine these properties. However, over the past 3 or 4 years, considerable progress has been made with semiclassical and quantum mechanical calculations with the result that it is now possible to predict with some degree of confidence the situations in which a purely classical approach to the collision dynamics will give acceptable results. Application of the semiclassical method, which utilises classical dynamics plus the superposition of probability amplitudes [456], has been pioneered by Marcus [457-466] and by Miller [456, 467-476],... 

We have developed an unified adiabatic approach allowing one to tackle transport problems in traps of different geometry. The magnetic and electrical fields, charge screening, and other factors (a spin-orbit interaction, hyperfine structure, etc) can influence the quantum dot paths within an easily tractable Breit-Wigner-resonance approximation for the electron scattering. The utility... 

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. 

The main conclusion is that the success of this field is due to a close interconnection between analytical and computational approaches. The paper has clearly demonstrated that we need to take into account electron-phonon interactions. In other terms, there exists a timely need to include translational, rotational, vibrational contributions and non-adiabatic approaches in our models. Then the road will be open to complete non-equilibrium interpretations of the electronic properties of macromolecules [89]. 

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). 

An alternative approach to the selection of A(q) is to consider an electronically adiabatic expansion truncated at a small number X of terms and require A(q) to be an (X X >T)-dimensional matrix. In this case, neither the adiabatic nor the diabatic electronic basis set is complete, but we assume that the adiabatic expansion in X terms is sufficiently accurate for our purposes. We now wish to select this A(q) so as to minimize the effect of the term in (75) containing W(1)ad(R). Ideally, we would like to force this matrix to vanish identically. Unfortunately, this is not always possible, as we shall now show. 

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