Many-particle Hamiltonian Born-Oppenheimer approximation

Up to now, we have been discussing many-particle molecular systems entirely in the abstract. In fact, accurate wave functions for such systems are extremely difficult to express because of the correlated motions of particles. That is, the Hamiltonian in Eq. (4.3) contains pairwise attraction and repulsion tenns, implying that no particle is moving independently of all of the others (the term correlation is used to describe this interdependency). In order to simplify the problem somewhat, we may invoke the so-called Born-Oppenheimer approximation. This approximation is described with more rigor in Section 15.5, but at this point we present the conceptual aspects without delving deeply into the mathematical details. 

Yaspatial positions rj of the N molecules yields a set of energy eigenvalues ( rj ), which can be interpreted as the effective Al-particle potential in the single-channel many-body Hamiltonian (Equation 12.1). The dependence of Vgl ( r ) on the electric fields E provides the basis for the engineering of the many body interactions in (Equation 12.2). The validity of this adiabatic approximation and of the associated decoupling of the Born-Oppenheimer channels will be discussed below. 

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