Born-Oppenheimer approximation vibration-rotation Hamiltonians

Within the Born-Oppenheimer approximation, we assume the nuclei are held fixed while the electrons move really fast around them, (note Mp/Me 1840.) In this case, nuclear motion and electronic motion are seperated. The last two terms can be removed from the total hamiltonian to give the electronic hamiltonian, He, since Vnn = K, and = 0. The nuclear motion is handled in a rotational/vibrational analysis. We will be working within the B-0 approximation, so realizing... 

The Born-Oppenheimer approximation permits the molecular Hamiltonian H to be separated into a component H, that depends only on the coordinates of the electrons relative to the nuclei, plus a component depending upon the nuclear coordinates. This in turn can be wriuen as a sum Hr + H, of terms for vibrational and rotational motion of ihe nuclei. 

To determine an effective dressed Hamiltonian characterizing a molecule excited by strong laser fields, we have to apply the standard construction of the free effective Hamiltonian (such as the Born-Oppenheimer approximation), taking into account the interaction with the field nonperturbatively (if resonances occur). This leads to four different time scales in general (i) for the motion of the electrons, (ii) for the vibrations of the nuclei, (iii) for the rotation of the nuclei, and (iv) for the frequency of the interacting field. It is well known that it is a good strategy to take into account the time scales from the fastest to the slowest one. 

Abstract When considering the work of Carl Ballhausen on vibrational spectra, it is suggested that his use of the Born-Oppenheimer approximation is capable of some refinement and extension in the light of later developments. A consideration of the potential energy surface in the context of a full Coulomb Schrodinger Hamiltonian in which translational and rotational motions are explicitly considered would seem to require a reformulation of the Born-Oppenheimer approach. The resulting potential surface for vibrational motion should be treated, allowing for the rotational motion and the nuclear permutational symmetry of the molecule. 

There are numerous interactions which are ignored by invoking the Born-Oppenheimer approximation, and these interactions can lead to terms that couple different adiabatic electronic states. The full Hamiltonian, H, for the molecule is the sum of the electronic Hamiltonian, the nuclear kinetic energy operator, Tf, the spin-orbit interaction, H, and all the remaining relativistic and hyperfine correction terms. The adiabatic Born-Oppenheimer approximation assumes that the wavefunctions of the system can be written in terms of a product of an electronic wavefunction, (r, R), a vibrational wavefunction, Xni( )> rotational wavefunction, and a spin wavefunction, Xspin- However, such a product wave-function is not an exact eigenfunction of the full Hamiltonian for the... 

Bunker, P.R., Moss, R.E. Breakdown of Born-Oppenheimer approximation—effective vibration-rotation Hamiltonian for a diatomic molecule. Mol. Phys. 1977,33,417-24 ... 

The classical dynamics of molecular models is generated by Hamilton s (or Newton s) equations of motion. In the absence of external, time-dependent forces, and within the Born-Oppenheimer approximation, the dynamics of molecular vibrations, rotations, and reactions conserves the total energy . We therefore restrict our attention in the nonlinear dynamics literature to energy-conserving systems, which are technically referred to as Hamiltonian systems. For the purposes of the present discussion, we restrict our attention to Hamiltonian systems with two degrees of freedom ... 

One learns directly that the converted Bom-Handy formula leads to a curiosity, viz. the hydrogen molecule does not move and does not rotate. Nevertheless the final Born-Handy formula contains contributions from vibrational as well as from translational and rotational degrees of freedom in contrast to our previous theory, based on the quantization of the electron-vibrational Hamiltonian, which contained solely the contributions from the vibrational degrees. As it will be shown below, this understanding has a profound significance for aU systems and phenomena beyond the Born-Oppenheimer approximation. Moreover, the interpretation of the... 

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