Born-Oppenheimers Adiabatic Approach

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 total Hamiltonian of the electron nuclear system is described as  

the electron wave function J/M(r R) is the eigen function of Hamiltonian 

There is no operator of differentiation on R in Hamiltonian (3), and hence these coordinates are the parameters. The wave function i///i(r R) obeys the following Schrodinger equation 

The index /i numbers the electron states and Efl(R) is the /fs electron term of the system. 

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In previous section, by considering the electrostatic energy of the quantum dot charging we have determined the tunnel curves using the phenomenological approach. A strict definition of the tunnel curves as total electronic energy at a fixed dot location between leads is implied by the Born-Oppenheimer adiabatic strategy. For the quantum-mechanical computation of the tunnel curves, the information about (1) the spatial profile of electrostatic potential and (2) the electron and ion distributions of the SET is required as an input. 

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.)...

The effects of deviations from the Born-Oppenheimer approximation (BOA) due to the interaction of the electron in the sub-barrier region with the local vibrations of the donor or the acceptor were considered for electron transfer processes in Ref. 68. It was shown that these effects are of importance for long-distance electron transfer since in this case the time when the electron is in the sub-barrier region may be long as compared to the period of the local vibration.68 A similar approach has been used in Ref. 65 to treat non-adiabatic effects in the sub-barrier region in atom transfer processes. However, nonadiabatic effects in the classically attainable region may also be of importance in atom transfer processes. In the harmonic approximation, when these effects are taken into account exactly, they manifest themselves in the noncoincidence of the... 

Each of the approaches is based on the premise that it makes sense to focus on the Born Oppenheimer potential for the OH stretch for fixed bath variables. Such a potential has vibrational eigenvalues, and for example h times the transition frequency of the fundamental is simply the difference between the first excited and ground state eigenvalues. Thus in essence this is an adiabatic approximation the assumption is that the vibrational chromophore is sufficiently fast compared to the bath coordinates. To the extent that the h times frequency of the chromophore is large compared to kT, and those of the bath are small compared to kT, this separation of time scales exists and so this should be a reasonable approximation. For water, as discussed earlier, some of the bath variables (librations) have frequencies somewhat larger than kT/h, and... 

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 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 adiabatic approximation means the neglect of the nuclear motion in the Schrodinger equation. The electronic structure is thus calculated for a set of fixed nuclear coordinates. This approach can in principle be exact if one uses the set of wave functions for fixed nuclear coordinates as a basis set for the full Schrodinger equation, and solves the nuclear motion on this basis. The adiabatic approximation stops at the step before. (The Born-Oppenheimer approximation assumes a specific classical behavior of the nuclei and hence it is more approximate than the adiabatic approximation.)... 

The concept of the static dot is justified by the fact that the dot s translational and rotational degrees of freedom vary slowly as compared to the fast motion of the electrons. The parameters Edot and I of the scattering matrix S(E) represent adiabatic variables, validated by the Born-Oppenheimer approach. An instant conductivity of a movable dot is to be computed in a phase space of the NEM oscillator s coordinates x and momenta p. The x, p point plays the role of a partial scattering channel in which the Wigner delay time of electron tunneling r = dS (E) /dE is the shortest time scale. The marginal delay r... 

Let us consider in detail, for example, kfK Applying the second-order perturbation approach to the adiabatic Born—Oppenheimer Hamiltonian, H, of a given diatomic molecule, we obtain the following expressions for k(2 (Byers-Brown, 1958 Byers-Brown and Steiner, 1962 Murrell, 1960 Salem, 1963b) ... 

In difference to normal ground state thermal chemistry (ignoring chemiluminescence and bioluminescence), which is usually well described by the Born-Oppenheimer approximation, photochemistry usually require a non-adiabatic description for a qualitative and quantitative model to be possible. A number of techniques have been developed to address this problem. Out of these we find the semi-classical trajectory surface hopping (TSH) approach or more sophisticated approaches based on a nuclear... 

In this approach, the nuclei are simulated at some finite temperature, T, which ultimately dictates the kinetic energy of the nuclei. The electronic structure, however, is kept close to the Born-Oppenheimer surface. The fictitious temperature of the electrons must therefore be close to zero. In simulating the dynamics for a specific system, the electrons must remain cold while the atoms must remain hot and thus maintain a nearly adiabatic system. The fictitious mass of the electron and the time step>s for the dynamics must be carefully structured so as to prevent energy transfer from the hot nuclei into the cold electrons. The Verlet algorithm is typically used to integrate these equations. 

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