A digression on nuclear potential surfaces

In the non-adiabatic limit, where the a b transition probabihty is small, the system may oscillate in the reactant well a for a relatively long time, occasionally passing through the transition point (or lower dimensionahty subspace) E, that is, through a configuration in which the b transition probability may be signific- 

This is a crude description. As we have seen, the probability Pb-( a depends on the crossing speed, and a proper thermal averaging must be taken. We will come back to these issues in Sections 14.3.5 and 16.2. 

The basis for separating the electronic and nuclear dynamics of, say, a molecular system is the Bom-Oppenheimer (BO) approximation. A system of electrons and nuclei is described by the Hamiltonian 

Quantum dynamics using the time-dependent Schrojinger equation 

Diabatic states are obtained from a similar approach, except that additional term (or terms) in the Hamiltonian are disregarded in order to adopt a specific physical picture. For example, suppose we want to describe a process where an electron e is transferred between two centers of attraction, A and B, of a molecular systems. We may choose to work in a basis of vibronic states obtained for the e-A system in the absence of e-B attraction, and for the e-B system in the absence of the e-A attraction. To get these vibronic states we again use a Bom-Oppenheimer procedure as described above. The potential surfaces for the nuclear motion obtained in this approximation are the corresponding diabatic potentials. By the nature of the approximation made, these potentials will correspond to electronic states that describe an electron localized on A or on B, and electron transfer between centers A and B implies that the system has crossed from one diabatic potential surface to the other. 

Here /Tel is the Hamiltonian of the electronic subsystem, —that of the nuclear subsystem (each a sum of kinetic energy and potential energy operators) and kei-N(r, R) is the electrons-nuclei (electrostatic) interaction that depends on the electronic coordinates r and the nuclear coordinates R. The BO approximation relies on the large mass difference between electron and nuclei that in turn implies that electrons move on a much faster timescale than nuclei. Exploring this viewpoint leads one to look for solutions for eigenstates of H of the form iA ,v(r,R) = (r,R)/ R), or a linear combination of such products. Here / (r, R) are solutions of the electronic Schrodinger equation in which the nuclear configuration Ris taken constant 

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