Born Oppenheimer approximation classical approach

From a theoretical point of view, in the study of atom-atom or atom-molecule collisions one needs to solve the Schrodinger equation, both for nuclear and electronic motions. When the nuclei move at much lower velocities than those of the electrons inside the atoms or molecules, both motions (nuclear and electronic) can be separated via the Born-Oppenheimer approximation. This approach leads to a wave function for each electronic state, which describes the nuclear motion and enables us to calculate the electronic energy as a function of the intemuclear distance, i.e. the potential energy V r). Therefore, V r) can be obtained by solving the electronic Schrodinger equation for each inter-nuclear distance. As a result, the nuclear motion, which we shall see is the way chemical reactions take place, is a dynamical problem that can be solved by using either quantum or classical mechanics. 

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... 

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.)... 

One has to emphasize that Eqs. (82) and (83) do not involve the Born-Oppenheimer approximation although the nuclear motion is treated classically. This is an important advantage over the quantum molecular dynamics approach [47-54] where the nuclear Newton equations (82) are solved simultaneously with a set of ground-state KS equations at the instantaneous nuclear positions. In spite of the obvious numerical advantages one has to keep in mind that the classical treatment of nuclear motion is justified only if the probability densities (R, t) remain narrow distributions during the whole process considered. The splitting of the nuclear wave packet found, e.g., in pump-probe experiments [55-58] cannot be properly accounted for by treating the nuclear motion classically. In this case, one has to face the complete system (67-72) of coupled TDKS equations for electrons and nuclei. 

Atomic nuclei are much heavier than electrons and can, in general, be treated accurately using a classical approach. Electrons, of course, must be treated quantum mechanically, and they are considered to move via the equations of quantum mechanics within the fixed external potential of the positively charged nuclei. Because of the relative speed of the motion of the electrons compared to that of the nuclei, their motion is, to an excellent approximation, separate from that of the nuclei in what is called the Born-Oppenheimer approximation. Moreover, excited electronic states are usually irrelevant at temperatures of interest to chemical engineers (<10,000 K), so only their ground state (minimum energy state) needs to be considered. (1 do not consider here the interaction of radiation with matter, the treatment of which is not readily possible at this time using Car-Parrinello methods.)... 

The Born-Oppenheimer approximation uncouples electron and nuclear motion. The latter concerns massive (at least, relative to electrons) bodies, and much lower velocities while the formal velocity of an electron may approach the speed of light, a molecule in the gas phase travels at about the speed of a supersonic jet plane. While electronic energies must be calculated by quantum mechanics, nuclear motions are more easily described in a classical framework. 

The force-field method involves the other part of the Born-Oppenheimer approximation, that is the positioning of the nuclei. The electronic system is not considered explicitly, but its effects are of course taken into account indirectly. This method is often referred to as a classical approach, not because the equations and parameters are derived from classical mechanics, but rather because it is assumed that a set of equations exist which are of the form of the classical equations of motion. The problem from this point of view is one of establishing just which equations are necessary, and determining the numerical values for the constants which appear in the equations. In general there is no limit as to what functions may be chosen or what parameters arc to be used, except that the force-field must duplicate the experimental data. 

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 all the schemes proposed at this time for doing molecular calculations, the nuclei were treated as providing a molecular geometry, in terms of which a molecular electronic structure calculation was parameterised. That is, the nuclei were considered as essentially classical particles, which could be clamped at will to construct a molecule with a chosen geometry. This was the approach used by Heitler and London in their pioneering calculation and it became usual to say that it was justified within the Born-Oppenheimer approximation . 

One obvious question is whether the nuclear and electronic motion can be separated in the fashion which is done in most models for molecule surface scattering and also in the above-mentioned treatment of electron-hole pair excitation. The traditional approach is to invoke a Born-Oppenheimer approximation, i.e., one defines adiabatic potential energy surfaces on which the nuclear dynamics is solved — either quantally or classically. In the Bom-Oppenheimer picture the electrons have had enough time to readjust to the nuclear positions. Thus the nuclei are assumed to move infinitely slowly. For finite speed, nonadiabatic corrections therefore have to be introduced. Thus, before comparison with experimental data is carried out we have to consider whether nonadiabatic processes are important. Two types of nonadiabatic processes are possible—one is nonadiabatic transitions in the gas phase from the lower adiabatic to the upper surface (as discussed in Chapter 4). The other is the nonadiabatic excitation of electrons in the metal through electron-hole pair excitation. 

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