Born-Oppenheimer approximation trajectories

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.)...

We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory co t) for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. 

Beam studies have until recently been largely confined to systems in which the dynamics are governed by a single potential surface. The use of classical trajectory studies and adiabatic correlation diagrams in predicting the reaction path are both implicitly founded on the Born-Oppenheimer approximation which allows us to deal with only one electronic state during... 

The first is the Bom-Oppenheimer or fixed-nucleus approximation [32] wherein the more massive nuclei are assumed to be stationary with the electrons moving rapidly about them. The solution of Eq (3) reduces to finding the energies and trajectories of the electrons only, i.e. solution of the so-called many-electron Schrodinger equation. A further simplification which is often assumed, at least initially, is that relativistic effects are negligible. The starting point for so-called ab initio methods is therefore the non-relativistic many-electron Schrodinger equation within the Born-Oppenheimer approximation. 

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... 

It is actually very difficult to solve the entire scheme down to Eq. (6.5) for systems of chemical interest, even if a very good set of >/) is available. (Note that electronic structure theory (quantum chemistry) can handle far larger molecular systems within the Born-Oppenheimer approximation) than the nuclear dynamics based on Eq. (6.5) can do.) This is because the short wavelength natme of nuclear matter wave blocks accurate computation and brings classical nature into the nuclear dynamics, in which path (trajectory) representation is quite often convenient and useful than sticking to the wave representation. Then what do the paths of nuclear dynamics look like on the occasion of nonadiabatic transitions, for which it is known that the nuclear wavepackets bifurcate, reflecting purely quantum nature. 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

This still defines a very complex many-body problem which cannot easily be solved. The next level of approximation is therefore to reduce the many-electron problem down to an effective one-electron problem, i.e. the determination of the energy and trajectory of each electron in the effective potential field generated by the remaining electrons and the nuclei. This then gives rise to the non relativistic, Born-Oppenheimer, one-electron Schrodinger Eq. (5). 

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