Born-Oppenheimer approximation states

The Born-Oppenheimer approximation states the vibrational and rotational energies of a molecule can be separated and the individual terms added. The overall energy E can thus be considered as a function of the vibrational quantum number v (taking values 0, 1, 2,. ..) and the rotational quantum number J (which independently takes values 0, 1, 2,...). For a diatomic molecule it can be approximated by the function... 

The Born-Oppenheimer approximation states that the electrons are able to adjust themselves instantaneously to tlie motions of the nuclei. The motions of the nuclei are in this approximation therefore not able to induce electronic transitions, an assumption that is also known as tire adiabatic approximation. The electrons thus create an effective electronic potential in which the nuclei move, and for a given electronic state tire valuation in the electronic energy with respect to the nuclear configuration defines a potential energy surface for the electronic state. The electronic Schrodinger equation can be written as... 

The Born-Oppenheimer approximation states that a diatomic s electronic energy depends only on the internuclear separation. Use this information to sketch and explain the relative location of the first few vibrational levels for H2 and D2. 

The molecular mechanics method is used to calculate molecular structures, conformational energies, and other molecular properties using concepts from classical mechanics. Electrons are not explicitly included in the molecular mechanics method, which is justified on the basis of the Born-Oppenheimer approximation stating that the movements of electrons and the nuclei can be separated. Thus, the nuclei may be viewed as moving in an average electronic potential field, and the molecular mechanics method attempts to describe this field by its force field. ... 

Because the nuclei are several orders of magnitude more massive than the electrons and will therefore move more slowly than the electrons, the Born-Oppenheimer approximation states that the nuclear and electronic motions of the molecule can be treated separately. Essentially, we can assume that the nuclei in the molecule are stationary and solve the equation solely for the electronic motion. This causes the nuclear or first term in Equation (10.13) to drop out. 

The proper quantumdynamical treatment of fast electronic transfer reactions and reactions involving electronically excited states is very complex, not only because the Born-Oppenheimer approximation brakes down but... 

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

Chemical reactions of molecules at metal surfaces represent a fascinating test of the validity of the Born-Oppenheimer approximation in chemical reactivity. Metals are characterized by a continuum of electronic states with many possible low energy excitations. If metallic electrons are transferred between electronic states as a result of the interactions they make with molecular adsorbates undergoing reaction at the surface, the Born-Oppenheimer approximation is breaking down. 

How important the breakdown of the Born-Oppenheimer approximation is in limiting our ability to carry out ab initio simulations of chemical reactivity at metal surfaces is the central topic of this review. Stated more provocatively, do we have the correct theoretical picture of heterogeneous catalysis. This review will restrict itself to a consideration of experiments that have begun to shed light on this important question. The reader is directed to other recent review articles, where aspects of this field of research not mentioned in this article are more fully addressed.10-16... 

Perhaps the first evidence for the breakdown of the Born-Oppenheimer approximation for adsorbates at metal surfaces arose from the study of infrared reflection-absorption line-widths of adsorbates on metals, a topic that has been reviewed by Hoffmann.17 In the simplest case, one considers the mechanism of vibrational relaxation operative for a diatomic molecule that has absorbed an infrared photon exciting it to its first vibrationally-excited state. Although the interpretation of spectral line-broadening experiments is always fraught with problems associated with distinguishing... 

Fig. 4. Accumulating evidence is starting to show that molecules which undergo large amplitude vibration can interact strongly with metallic electrons in collisions and reactions at metal surfaces. This suggests that the Born-Oppenheimer approximation may be suspect near transition states of reactions at metal surfaces.

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

Fluctuational Preparation of the Barrier and Role Played by the Excited Vibrational States in the Born-Oppenheimer Approximation... 

Below we will restrict ourselves to the Born-Oppenheimer approximation and, unlike Refs. 62, 64, and 65, we will take into account the contribution from the excited vibrational states of the tunneling particle and consider the role played by the transverse quantum vibrations of the tunneling particle itself in the preparation of the potential barrier.48... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

Fig. 5.2 Radial distribution curves, Pv

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