Born-Oppenheimer separation, electronic potential

Within the Born-Oppenheimer approximation, we still need to know that the nuclear position parameters really correspond to the distances and angles of a classical molecular framework. Our choice of the Coulomb gauge ensures this—the nuclear positions only appear in the electron-nucleus interaction terms, and the derivation of this potential from relativistic field theory shows us that it is indeed the quantities of normal 3-space that appear here. Thus, any potential surface that we might calculate on the basis of the Born-Oppenheimer-separated electronic molecular Dirac equation is indeed spanned by the variations of molecular structural parameters in the usual meaning. 

In the vibrational treatment we assumed, as usually done, that the Born-Oppenheimer separation is possible and that the electronic energy as a function of the internuclear variables can be taken as a potential in the equation of the internal motions of the nuclei. The vibrational anharmonic functions are obtained by means of a variational treatment in the basis of the harmonic solutions for the vibration considered (for more details about the theory see Pauzat et al [20]). 

We now introduce an excess electron into the bubble, which is located in the center of the helium cluster at a fixed nuclear configuration of the helium balloon. The electronic energy of the excess electron will be calculated within the Born-Oppenheimer separability approximation. We modified the nonlocal effective potential developed by us for surface excess electron states on helium clusters [178-180] for the case of an excess electron in a bubble of radius Rb... 

The development of electronic geometric phase factors is governed by an adiabatic vector potential induced in the nuclear kinetic energy when we extend the Born-Oppenheimer separation to the degenerate pair of states [1, 23-25]. To see how the induced vector potential appears, we consider the family of transformations which diagonalize the excited state electronic coupling in the form... 

The great majority of reactions in fluid media are best treated in terms of the vibronically coupled crossings between potential-energy surfaces of reactant and product electronic configurations. Thus, presuming the preassembly of reactants and a Born-Oppenheimer separation of electronic and nuclear motions, the electron-transfer rate constant can usually be represented as in equation (1), where... 

Potential energies for the nuclear motions in a polyatomic system can be obtained from the Born-Oppenheimer separation of electronic and nuclear motions, for each adiabatic electronic state. Their values E can be separated into asymptotic contributions giving internal potential energies and Vg, and a remainder term V describing the interaction potential. 

The eigenfunctions depend parametrically on the choice of intemuclear separation R = R — Rb. The K-dependent eigenvalues k(R) act as potential energy functions for nuclear vibrational motion when the Born-Oppenheimer separation of nuclear and electronic motions is valid. 

The first basic approximation of quantum chemistry is the Born-Oppenheimer Approximation (also referred to as the clamped-nuclei approximation). The Born-Oppenheimer Approximation is used to define and calculate potential energy surfaces. It uses the heavier mass of nuclei compared with electrons to separate the... 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

Consider a stable diatomic molecule with nuclei denoted as A and B. The Born-Oppenheimer potential V for such a molecule will depend on the internuclear distance rAB and will typically have the form shown in Fig. 3.1. The potential energy has a minimum at r0, which is often referred to as the equilibrium internuclear distance. As the distance rAB increases, the potential V increases and finally reaches a limiting value where the molecule is now better described as two separated atoms (or depending on the electronic state of the system, two separated atomic species one or both of which may be ions). The difference in energy between the two separated atoms and the minimum of the potential is the dissociation energy De of the molecule. As the internuclear distance of the diatomic molecule is decreased... 

Fig. 1. The Marcus parabolic free energy surfaces corresponding to the reactant electronic state of the system (DA) and to the product electronic state of the system (D A ) cross (become resonant) at the transition state. The curves which cross are computed with zero electronic tunneling interaction and are known as the diabatic curves, and include the Born-Oppenheimer potential energy of the molecular system plus the environmental polarization free energy as a function of the reaction coordinate. Due to the finite electronic coupling between the reactant and charge separated states, a fraction k l of the molecular systems passing through the transition state region will cross over onto the product surface this electronically controlled fraction k l thus enters directly as a factor into the electron transfer rate constant...

Representing the molecular potential energy as an analytic function of the nuclear coordinates in this fashion implicitly invokes the Born-Oppenheimer approximation in separating the very fast electronic motions from the much slower ones of the nuclei. 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

In addition, we assume, for the systems of interest here, that the electronic motion is fast relative to the kinetic motion of the nuclei and that the total wave functions can be separated into a product form, with one term depending on the electronic motion and parametric in the nuclear coordinates and a second term describing the nuclear motion in terms of adiabatic potential hypersurfaces. This separation, based on the relative mass and velocity of an electron as compared with the nucleus mass and velocity, is known as the Born-Oppenheimer approximation. 

This is so because no coupling between electronic and nuclear motion is assumed within the Born-Oppenheimer approximation, which in the classical limit leads to separate conservation of the instantaneous heavy-particle motion. Denoting by EA(R,) and (/ ,) the instantaneous kinetic energy at the moment of transition in the upper- and lower potential curve,... 

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