Born-Oppenheimer approach

The initial states are bound eigensolutions in the isolated a electroruc subspace, and 4)" (r, R, t) is propagated in the (3 electroruc state. Thus the calculations involved are similar to the usual calculations in the Born-Oppenheimer approach, and will be briefly described below. 

Some important problems of the theory of multi-phonon electron transition were not touched upon in this chapter. These are, first, the calculation of the expression for the electron matrix element at the tunneling transfer, second, the influence of medium on the electron matrix element, and, finally, the investigation of the applicability of Born-Oppenheimer approach in the electron tunneling transfer. These issues will be considered in the next chapter. 

The issues of the electron transfer theory that should be resolved to make this theory sufficiently complete and closed was formulated in the conclusion of the previous chapter. These issues were concerned with the methods of the calculation of the electron matrix element and the spillover of Born-Oppenheimer approach. It is these problems that will be examined in this chapter. 

The concept of the static dot is justified by the fact that the dot s translational and rotational degrees of freedom vary slowly as compared to the fast motion of the electrons. The parameters Edot and I of the scattering matrix S(E) represent adiabatic variables, validated by the Born-Oppenheimer approach. An instant conductivity of a movable dot is to be computed in a phase space of the NEM oscillator s coordinates x and momenta p. The x, p point plays the role of a partial scattering channel in which the Wigner delay time of electron tunneling r = dS (E) /dE is the shortest time scale. The marginal delay r... 

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. 

A Post-Born-Oppenheimer Approach to Electronuclear Dynamics in External Eields... 

We now demonstrate the calculation of the dimer-dimer scattering length add in the limit of M/m 1 [67]. In the Born-Oppenheimer approach the Schrodinger equation for the relative motion of two molecules reads... 

The trimer states, which in most cases can be called Efimov trimers, are interesting objects. Their existence can be seen from the Born-Oppenheimer picture for two heavy atoms and one light atom in the gerade state. Within the Born-Oppenheimer approach the three-body problem reduces to the calculation of the relative motion of the heavy atoms in the effective potential created by the light atom. For the light atom in the gerade state, this potential is + (/ ), found in the previous subsection. The Schrodinger equation for the wavefunction of the relative motion of the heavy atoms, Xv(R), reads... 

At a larger molecular size or smaller L the trimer formation is possible. The formation rate has been calculated in Ref. [68] by using the hybrid Born-Oppenheimer approach, and here we only present the results and give their qualitative explanation. In order to form a bound trimer state two molecules have to tunnel toward each other at distances / < Kq This can be viewed as tunneling of particles with mass Mif in the repulsive potential UeniR). Therefore, the probability of approaching at... 

The Born-Oppenheimer approach for the three-body system of one hght and two heavy atoms was discussed in Fonseca, A.C., Redish, E.F., and Shanley, P.E., Efimov effect in a solvable model, Nucl. Phys. A, 320, 273,1979. 

The tail of the wave function as is, however, not too well reproduced, since the exponential vanishing of u r) is replaced by a zero of finite order for w(jc) at jc = 1. The main advantage of this method, for our purpose, is its immediate generation to coupled equations. This will be very useful for Faddeev, hyperspherical, or Born-Oppenheimer approaches to the three-body problem. 

Born-Oppenheimer approach T corresponds to the kinetic Hamiltonian for the nuclei described by the wavefunction f t) and V = V R) to the electronic potential as obtained by solving the time-independent electronic Schrbdinger equation at different geometries R. Integrating the time-dependent, nuclear Schrodinger equation [Eq. (8.1)] determines the equations of motion as the action of a propagator on the nuclear wavefunction ... 

The free-energy functional G(R) is defined at a given fixed nuclei position of the molecular solute. Therefore, using the standard Born-Oppenheimer approach, the free energy functional G(R) may be considered as a function of the nuclear... 

The Born-Oppenheimer approach has been put on a rigorous foundation for diatomics with solutions of the form (O Eq. 2.39) in work which is described in a helpful context in Combes and Seiler (1980). For solutions like (O Eq. 2.40), it is possible that more than one vector (coordinate) space can be constructed on it because the transformation is to a manifold. In fact two coordinate spaces are possible on a trivial one and a twisted one, the latter associated with... 

Equation (9) can be derived within the framework of the Born-Oppenheimer approach [21]. This approach is universal and it can be applied to an arbitrary system of interacting nudei and electrons. The total Hamiltonian of such a sy on averaged over fast electronic motion is as follows ... 

Kryachko et al., 1991] E. Kryachko, E. Ludena and V. Mujica. Eormulation of N— and v-representable density functional theory. IV. Non-Born-Oppenheimer approach, Int. Journ. Quant. Chem., 40, 589-604, 1991. 

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