Born-Oppenheimer approximation validity

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. 

Assuming the validity of the Born-Oppenheimer approximation, the electrostatic Hellmann-Feynman (H-F) theorem expresses the force F A on a nucleus A, of charge ZA, in a molecule or solid, as... 

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. 

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. 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

If the Born-Oppenheimer approximation is not valid—for example, in the vicinity of surface crossings—nonadiabatic coupling effects (that couple nuclear and electronic motion) need to be taken in account to correctly describe the motion of the molecular system. This is done, for instance, when one needs to describe a jump between two different PESs. In this case, one uses semiclassi-cal theories and the surface-hopping method, which we discuss subsequently. We now discuss in some detail how the region in which nonadiabatic effects become important can be characterized topologically. 

Whereas the quantum-mechanical molecular Hamiltonian is indeed spherically symmetrical, a simplified virial theorem should apply at the molecular level. However, when applied under the Born-Oppenheimer approximation, which assumes a rigid non-spherical nuclear framework, the virial theorem has no validity at all. No amount of correction factors can overcome this problem. All efforts to analyze the stability of classically structured molecules in terms of cleverly modified virial schemes are a waste of time. This stipulation embraces the bulk of modern bonding theories. 

The electronic-transition dipole moment for the G E transition is defined by Mge = ( g A/ ge1 e> where the are the state wave functions and A/ ge is the dilference in dipole moment of the ground and excited states [22]. The intensity of the transition is proportional to Mge - The broad absorption bands usually observed in transition metal systems are composed of progressions in the vibrational modes that correlate with the differences in nuclear coordinates between the vibrationally equilibrated ground and excited state. Since the energy difference between the donor and acceptor is generally solvent-dependent, the distribution of solvent environments that is characteristic of solutions may also contribute to the bandwidth (see further discussion of this point in the sections below). If the validity of the Born Oppenheimer approximation is assumed, the intensity of each of these vibronic components is given by Eq. 11,... 

The above treatment is based upon the traditional Born-Oppenheimer approximation which states that, when nuclei move, the electrons can almost instantaneously adjust to their new positions. Another relevant time frame is the time required to establish the electronic polarization of the medium. To characterize this time frame, Kim and Hynes consider the ratio of Vei, the electron hopping frequency, to Vep, the frequency characteristic of the solvent electronic polarization. The Bom-Oppenheimer-based treatment is valid provided that this ratio is much less than unity, i.e., the time scale for the adjustment of the electronic polarization is much shorter than that for the transferring electron [22-26]. 

To render Eq. (2) tractable for a structure analysis, several approximations are made. The Born-Oppenheimer approximation is invoked, so that the potential energy term in Eq. (3) is just the total electronic energy. It is usually further assumed that the potential for the nuclei is quadratic. In this case it is easy to solve for t(Q, Q 0) in Eq. (3), but Eq. (2) cannot be easily evaluated without further assumptions of the charge density as a function of Q. The practice is to partition p(r, Q) into a sum of charge density functions that are centered on the several nuclei and to further assume that each density function follows the motion of the nucleus on which it is centered. This latter factorization has dubious validity, but nonetheless forms the basis of a working approximation for both structural and charge density analysis of x-ray diffraction data. 

Indeed, Dunham s energy-level formula [Eq. (2.1.1)] is based both on the concept of a potential energy curve, which rests on the separability of electronic and nuclear motions, and on the neglect of certain couplings between the angular momenta associated with nuclear rotation, electron spin, and electron orbital motion. The utility of the potential curve concept is related to the validity of the Born-Oppenheimer approximation, which is discussed in Section 3.1. 

For homonuclear molecules, the g or u symmetry is almost always conserved. Only external electric fields, hyperfine effects (Pique, et al., 1984), and collisions can induce perturbations between g and u states. See Reinhold, et al., (1998) who discuss how several terms that are neglected in the Born-Oppenheimer approximation can give rise to interactions between g and u states in hetero-isotopomers, as in the HD molecule. An additional symmetry will be discussed in Section 3.2.2 parity or, more usefully, the e and / symmetry character of the rotational levels remains well defined for both hetero- and homonuclear diatomic molecules. The matrix elements of Table 3.2 describe direct interactions between basis states. Indirect interactions can also occur and are discussed in Sections 4.2, 4.4.2 and 4.5.1. Even for indirect interactions the A J = 0 and e / perturbation selection rules remain valid (see Section 3.2.2). 

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