Born-Oppenheimer states

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

The interest aroused by the field of radiationless transitions in recent years has been enormous, and several reviews have been published 72-74) Basically, the ideas of Robinson and Frosch 75) who used the concepts on non-stationary molecular states and time-dependent perturbation theory to calculate the rate of transitions between Born-Oppenheimer states, are still valid, although they have been extended and refined. The nuclear kinetic energy leads to an interaction between different Born-Oppenheimer states and the rate of radiationless transitions is given by... 

A correct representation of the molecular eigenstates, zero-order, Born-Oppenheimer states,6... 

We now consider the special case of one discrete Born-Oppenheimer state in resonance with a quasicontinuum of other BO states. Let the... 

The wavefunction developed in the preceding subsection is, of course, only a first approximation to the true molecular continuum wavefunction. We have thus far considered only the nonvibrating molecule, and treated the resulting Born-Oppenheimer states as if they were the true eigenstates. In a real system vibrational motion of the nuclei, configuration interaction between vibronic states, etc., must be included in the description. 

The vector g(Q) is the coupling term that gives the magnitude of the coupling between the Born-Oppenheimer states described by Cj and C2 as a function of the nuclear motion along Q. 

According to Equation (2.29), in the adiabatic representation (index a) one expands the total molecular wavefunction F(R, r, q) in terms of the Born-Oppenheimer states Ej (q R, r) which solve the electronic Schrodinger equation (2.30) for fixed nuclear configuration (R,r). In this representation, the electronic Hamiltonian is diagonal,... 

Apart from the selection rules for the electronic coupling matrix element, spin-forbidden and spin-allowed nonradiative transitions are treated completely analogously. Nonradiative transitions caused by spin-orbit interaction are mostly calculated in the basis of pure spin Born-Oppenheimer states. With respect to spin-orbit coupling, this implies a diabatic behavior, meaning that curve crossings may occur in this approach. The nuclear Schrodinger equation is first solved separately for each electronic state, and the rovibronic states are spin-orbit coupled then in a second step. 

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... 

The full multiple spawning (FMS) method has been developed as a genuine quantum mechanical method based on semiclassical considerations. The FMS method can be seen as an extension of semiclassical methods that brings back quantum character to the nuclear motion. Indeed, the nuclear wave function is not reduced to a product of delta functions centered on the nuclear positions but retains a minimum uncertainty relationship. The nuclear wave function is expressed as a sum of Born-Oppenheimer states ... 

C. A. Mead, Electronic Hamiltonian, wave functions, and energies, and derivative coupling between Born-Oppenheimer states in the vicinity of a conical intersection, J. Chem. Phys. 78 807 (1983). 

A new perturbation expansion has been derived and applied to the coupling between non-adiabatic Born-Oppenheimer states in polyatomic molecules, for which previous methods are often unreliable.466 Franck-Condon computations on excited states of simple ABa polyatomics, including S02, CS2, and many... 

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch." In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Born-Oppenheimer states. Henry and Kasha > and Jortner and co-workers< > have pointed out that the Born-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

Under the condition that there is a nontrivial MAB effect, let us ask what is its physical nature in relation to the standard AB effect Let us first address the issue of locality. As already indicated, the standard AB effect, which may occur when a charged particle encircles a line of magnetic flux, is nonlocal as it fulfills the criteria (Nl) and (N2) the effect arises although the particle experiences no physical field and no exchange of physical quantity takes place along the particle s path. Could the same be said about the MAB effect One way to address this question is to note that since the electronic Born-Oppenheimer states are eigenstates of it may be tempting to... 

The initial, D), and final, A), states represent the electron localized on D and A, respectively. Within the Born-Oppenheimer approximation the electronic and nuclear degrees of freedom are described by the Born-Oppenheimer states. 

Within the Born-Oppenheimer approximation the electronic and nuclear degrees of freedom are described by the Born-Oppenheimer states,... 

The wave equation for an electronic Born-Oppenheimer state n is written ... 

Figure 7.15 General problem for nonradiative decay of an excited Born-Oppenheimer state with energy in electronic state A, prepared by photon excitation of a level with energy Eg in the electronic ground state. The prepared state is connected by perturbations (spin-orbit coupling, nonradiative coupling, etc.) to a set of Born-Oppenheimer states 1, > with energies , in electronic state B. The states )/7,> are not accessible by El transitions from the ground state.

---