Born-Oppenheimer frame

In the light of the chapter on special relativity (chapter 2), it is apparent that there is a possible problem in performing this separation of the space and time variables, because the Lorentz transformation mixes them. The separation would have to be performed in a particular frame of reference, and only be valid in this frame of reference. If we want results in another frame of reference, we must perform a Lorentz transformation to that frame, and there is no guarantee that we will still have a stationary state. However, if our Hamiltonian is Lorentz invariant, the choice of the frame of reference is arbitrary, and, as we saw above, the probability density is independent of time and of the frame of reference. We may therefore choose the frame that is most convenient. In molecules (and in atoms) the Born-Oppenheimer frame is the most convenient frame of reference for electronic stmcture calculations because the nuclear potential is then simply the static Coulomb potential. Regardless of whether the Hamiltonian is Lorentz invariant or not, it is this frame that we work in from here on. 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

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]. 

Derivatives of the dipole moment with respect to Qj can be expressed within a Cartesian reference frame via a similarity transformation, introducing atomic polar tensors (APTs) [13, 14], The connection between the latter and the electric shielding is obtained by means of the Hellmann-Feynman theorem. Within the Born-Oppenheimer approximation and allowing for the dipole length formalism, the perturbed Hamiltonian in the presence of a static external electric field E is given by Eqs. (6) and (35). 

Towards a Rigged Born-Oppenheimer Electronic Theory of Chemical Processes 117 function in the laboratory frame is given by Fik(r,Q) so that... 

The Born-Oppenheimer diagonal correction is given in Eq. (2a). In that equation, the gradients refer to space fixed frame (SFF) coordinates. For diatomic molecules, considerable savings result from a transformation to body fixed frame (BFF) coordinates. This transformation is accomplished in two steps. The SFF coordinates are transformed to center of mass fixed frame (CMFF) coordinates and then the CMFF coordinates are transformed to BFF coordinates. The details of the transformation are beyond the scope of this review. Here we sketch the ideas involved. A detailed treatment, based on the pioneering work of Kronig, can be found in Ref. 7. In particular, first the rigorously removable center of mass of the nuclei and... 

When very accurate dipole moments are deduced, it is proper to query the significance of a breakdown of the Bom-Oppenheimer approximation. This approximation justifies the assignment of molecular property tensors, such as dipole moments and polarizabilities, to specific directions in a molecule-fixed frame and supports the use of a property function or surface representing the variation of the property with nuclear position. The dipole moment of HD (5.85 X 1(T4 D)26 arises solely from the breakdown of the approximation and may have the sense H D-.27-29 In HC1 and DC1, there is an isotope effect on the dipole moment that has been attributed to a violation of the Born-Oppen-heimer approximation30 there is an apparent difference of 0.0010 0.0002 D between the dipole functions of HC1 and DC1, with HC1 having the bigger moment. This result is in accord with a recent theoretical analysis by Bunker.31... 

It might also be hoped that a first approximation to a solution of (16) could be constructed in terms of product functions in which one portion of each product was obtained from a problem in which the electronic motion was treated as primary, and the other portion described the nuclear motion in the electronic field derived from the first part of the product. This is the standard technique for treating a system of coupled differential equations in which one group of equations represent fast motions and another slow motions. Most chemists get their familiarity with this technique by considering the kinetics of sequential chemical reactions. It is this technique that underlies the Born and Oppenheimer program in which the electronic motion is approached in a frame fixed in the laboratory with an electronic Hamiltonian in which the nuclear motion is at first ignored. Thus, it is natural with the present coordinate choice to hope that functions of the form... 

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