Born-Oppenheimer adiabatic approximation

The wavefunctions 4 and t f can be written in the adiabatic Born-Oppenheimer (BO) approximation (see the Appendix) in the form... 

Consider now the influence of the high-frequency fluctuations in the environment only (is 3> B). Since the frequencies of the fluctuations are much higher than the typical spin-dynamics frequencies, one may eliminate these high-frequency fluctuations using the adiabatic (Born-Oppenheimer) approximation, as described, e.g., by Leggett et al. [8]. 

Quantum mechanically, resonance Raman cross-sections can be calculated by the following sum-over-states expression derived from second-order perturbation theory within the adiabatic, Born-Oppenheimer and harmonic approximations... 

In perturbation theory language, one can say that the adiabatic Born-Oppenheimer states are a good zero-order approximation for describing nuclear motion if the non-Born-Oppenheimer corrections are small. This is... 

To proceed further it is necessary to elaborate on the vibronic nature of the states and operators in the scattering tensor. The adiabatic Born-Oppenheimer (ABO) approximation will be employed in which the vibronic states are formed by products of pure vibrational states Ai (Q) with pure electronic states i(Q,r). The latter are referred to some fixed positions of the nuclei and are therefore parametrically dependent on the internuclear distances expressed as a set, Q, of normal coordinates, r is a set of electronic coordinates, and i, u are electronic and vibrational quantum numbers respectively. Thus... 

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... 

If high accuracy is required, vibrational effects must be taken into account. In a proper adiabatic Born-Oppenheimer treatment, the groimd state wave function would be written as a product of an electronic and a vibrational wave function. The response of this wave function should then be computed and subsequently used to construct vibronic response functions. The sum-over-states expressions would include contributions from vibrational states in the electronic groimd and excited states. Since each set of vibrational wave functions is tied to a specific electronic state within the adiabatic Bom-Oppenheimer approximation, this approach is not feasible in practice. Hence, the electronic properties are considered as electronic ground state properties and therefore, averaged in a vibrational state of the electronic ground state. 

Pack R T and Hirschfelder J O 1970 Energy corrections to the Born-Oppenheimer approximation. The best adiabatic approximation J. Chem. Phys. 52 521-34... 

THE CRUDE BORN-OPPENHEIMER ADIABATIC APPROXIMATION OF MOLECULAR POTENTIAL ENERGIES... 

THE CRUDE BORN—OPPENHEIMER ADIABATIC APPROXIMATION 521 Consider the integral... 

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