Adiabatic approach

In the nonrelativistic case much has been, and continues to be, learned about the outcome of nonadiabatic processes from the locus and topography of seams of conical intersection. It will now be possible to describe nonadiabatic processes driven by conical intersections, for which the spin-orbit interaction cannot be neglected, on the same footing that has been so useful in the nonrelativistic case. This fully adiabatic approach offers both conceptual and potential computational... 

The polytropic process is mathematically easier to handle than the adiabatic approach for the following (1) determination of the discharge temperature (see later discussion under Temperature Rise During Compression ) and (2) advantage of the polytropic efficiency ... 

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. 

However, if the PES are multidimensional, as is the case for reactions in the condensed phase, the adiabatic approach is inconvenient for practical calculations, especially for nonadiabatic reactions. 

The diabatic approach will be mainly used below, since it is more convenient for nonadiabatic reactions. However, in Section VI adiabatic reactions will be also considered using the adiabatic approach. 

E. Theoretical Connection Between the Standard and the Adiabatic Approaches... 

Now, let us look at the autocorrelation function of the dipole moment operator within the adiabatic approach. In the representation I it is... 

In Fig. 4 we compare the adiabatic (dotted line) and the stabilized standard spectral densities (continuous line) for three values of the anharmonic coupling parameter and for the same damping parameter. Comparison shows that for a0 1, the adiabatic lineshapes are almost the same as those obtained by the exact approach. For aG = 1.5, this lineshape escapes from the exact one. That shows that for ac > 1, the adiabatic corrections becomes sensitive. However, it may be observed by inspection of the bottom spectra of Fig. 4, that if one takes for the adiabatic approach co0o = 165cm 1 and aG = 1.4, the adiabatic lineshape simulates sensitively the standard one obtained with go,, = 150 cm-1 and ac = 1.5. 

Now, it may be of interest to look at the connection between the autocorrelation functions appearing in the standard and the adiabatic approaches. Clearly, it is the representation I of the adiabatic approach which is the most narrowing to that of the standard one [see Eqs. (43) and (17)] because both are involving the diagonalization of the matricial representation of Hamiltonians, within the product base built up from the bases of the quantum harmonic oscillators corresponding to the separate slow and fast modes. However, among the... 

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. 

Finally, we should still mention work by Makram-Ebeid and Lannoo (1982a,b) that calculates (in the adiabatic approach) the effect of an electric field in enhancing, with phonon assistance, the tunneling rate. Very good... 

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. 

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. 

Hancock et al. [1989] used a version of the small curvature semiclassical adiabatic approach introduced by Truhlar et al. [1982] to calculate the temperature dependence of the rate constant, as shown in Figure 6.29. Variations in k(T) below the crossover point (25-30 K) are due to changes in the prefactor due to zero-point vibrations of the H atom in the crystal. Obviously, the gas-phase model does not take these into account. The absolute values of the rate constant differ by 1-2 orders of magnitude from the experimental ones for the same reason. 

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 question of the applicability of the semiclassic adiabatic approach to the vibronic problems in the JT systems is rather complicated in general and the thorough answer can be done with regard to a particular problem. In our brief discussion of this question we will refer to two areas - thermodynamic properties and resonance (optical) problems. In Ref. [9] the magnetic properties of mixed-valence dimeric and trimeric clusters are considered in the framework of... 

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. 

However, if Ed ErD, the adiabatic approach proves valid even at a sufficiently large distance r. This inequality permits us to expand the exponent in formula (56) on the small value ErDj ED. Then, the exponent... 

The physical sense of the condition of the violation of the adiabatic approach in the representation of the wave functions can be easily understood if we expand the exponent in the expression (64) on e and turn to the usual units ... 

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 ... 

As for the special situations involved in our above adiabatic approach and dealing with the CEL method, we may write the complex energy levels as... 

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