Born Oppenheimer level

The study of McMillan-Mayer level models, in which the solvent coordinates have been averaged over so that only solvent-mediated ion-ion forces need be treated, is relatively well developed. However the real forces at this level are even more poorly known than the forces at the Born-Oppenheim level referred to above. It is found that McMillan-Mayer level models can be brought into good agreement with solution thermodynamic data. 

In this section we consider the possibility of applying the ion association concept to the description of the properties of electrolyte solutions in the ion-molecular or Born-Oppenheimer level approach. The simplest ion-molecular model for electrolyte solution can be represented by the mixture of charged hard spheres and hard spheres with embedded dipoles, the so-called ion-dipolar model. For simplification we consider that ions and solvent molecules are characterized by diameters R and Rs, correspondingly. The model is given by the pair potentials,... 

Born-Oppenheimer level The particles are the water molecules, Na", and Cr. 

Figure 7.18 Eigenvalues and E, (/ =0, 1, 2,...) of the Born-Oppenheimer Hamiltonian, solid lines eigenvalues E n = 0, 1, 2,...) of the perturbed Hamiltonian, dashed lines. The latter eigenvalues are obtained from the graphical solutions in Fig. 7.17. Note the general property that there is one perturbed eigenvalue between every pair of Born-Oppenheimer levels.

Both solvent and solute are treated at the Born-Oppenheimer level, which means separation of nuclear and electronic motions. 

Hagedorn, G. A. Electron energy level crossing in the time-dependent Born-Oppenheimer approximation. Theor. Chim. Acta 67 (1990) 163-190... 

A disadvantage of this technique is that isotopic labeling can cause unwanted perturbations to the competition between pathways through kinetic isotope effects. Whereas the Born-Oppenheimer potential energy surfaces are not affected by isotopic substitution, rotational and vibrational levels become more closely spaced with substitution of heavier isotopes. Consequently, the rate of reaction in competing pathways will be modified somewhat compared to the unlabeled reaction. This effect scales approximately as the square root of the ratio of the isotopic masses, and will be most pronounced for deuterium or... 

In this section the Born-Oppenheimer approximation will be presented in what is necessarily a very simplified form. It has already been introduced without justification in Section 6.5. It is certainly the most important - and most satisfactory - approximation in quantum mechanics, although its rigorous derivation is far beyond the level of this book. Consider, therefore, the Mowing argument... 

Fig. 5.2 Radial distribution curves, Pv

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

Among the functions one can, at least in principle, calculate at the Schroedinger level is the Born-Oppenheimer (BO) potential surface, the potential of the forces among the nuclei assuming that at each nuclear configuration the time-independent Schroedinger equation is satisfied. We may think of this as the electron-averaged potential. Such an N-body potential Ujj often may be adequately represented as a sum of pair potentials... 

The assumption of weak electronic coupling may not be valid for vibrational levels near the region where the reactant and product surfaces intersect. If the extent of electronic coupling is sufficient (tens of cm ), the timescale for electron transfer for vibrational levels near the intersectional region will approach the vibrational timescale, electronic and nuclear motions are coupled, and the Born-Oppenheimer approximation is no longer valid. 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

Fig. 4.5 Schematic projection of the energetics of a reaction. The diagram shows the Born-Oppenheimer energy surface mapped onto the reaction coordinate. The barrier height AE has its zero at the bottom of the reactant well. One of the 3n — 6 vibrational modes orthogonal to the reaction coordinate is shown in the transition state. H and D zero point vibrational levels are shown schematically in the reactant, product, and transition states. The reaction as diagrammed is slightly endothermic, AE > 0. The semiclassical reaction path follows the dash-dot arrows. Alternatively part of the reaction may proceed by tunneling through the barrier from reactants to products with a certain probability as shown with the gray arrow...

Born-Oppenheimer approximation (physchem) The approximation, used in the Born-Oppenheimer method, that the electronic wave functions and energy levels at any instant depend only on the positions of the nuclei at that instant and not on the motions of the nuclei. Also known as adiabatic approximation. born ap an.hT-mar 3,prak s3,ma shan J... 

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