Electronic states Born-Oppenheimer approximation

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. 

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.

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

To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H20)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunc-tions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

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