Born-Oppenheimer approximation general solution

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schradinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

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,... 

In the related work of Kim and Hynes [50], Equations (3.107) and (3.112) have been designated, respectively, by the labels SC (self-consistent or mean field) and BO (where Born-Oppenheimer here refers to timescale separation of solvent and solute electrons). More general timescale analysis has also been reported [50,51], Equation (3.112) is similar in spirit to the so-called direct RF method (DRF) [54-56], The difference between the BO and SC results has been related to electronic fluctuations associated with dispersion interactions [55], Approximate means of separating the full solute electronic densities into an ET-active subspace and the remainder, treated, respectively, at the BO and SC levels, have also been explored [52],... 

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