Born-Oppenheimer separation, electronic

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. 

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

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]). 

Whereas for diatomic molecules the vibration-rotation interaction added only a small correction to the energy, for a number of polyatomic molecules the vibration-rotation interaction leads to relatively large corrections. Similarly, although the Born-Oppenheimer separation of electronic and nuclear motions holds extremely well for diatomic molecules, it occasionally breaks down for polyatomic molecules, leading to substantial interactions between electronic and nuclear motions. 

Usually the Born-Oppenheimer separation of nuclear and electronic coordinates is assumed and small terms in the hamiltonian, such as spin-orbit coupling, are neglected in the first approximation. Perturbation... 

We have already seen in Sec. 3.1 that before the Born-Oppenheimer separation of nuclear and electronic motion is made, the Coulomb Hamiltonian has very high symmetry, but that the clamped-nucleus Hamiltonian has only the spatial symmetry of the nuclear framework. That is, the Hamiltonian... 

The failures of the Born-Oppenheimer separation of the electronic and nuclear motions show up in the spectra of molecules as homogeneous or heterogeneous perturbations in the spectra41. See, e.g. Ref. (42) for an example, a fully ab initio study of the spectrum of the calcium dimer in a coupled manifold of electronic states. Theoretical methods needed to describe the dynamics of molecules in nonadiabatic situations are being developed now. See Ref. (43) for a review. 

In chapter 2 we discussed at length the separation of nuclear and electronic coordinates in the solution of the Schrodinger equation. We described the Born-Oppenheimer approximation which allows us to solve the Schrodinger equation for the motion of the electrons in the electrostatic field produced by fixed nuclear charges. There are certain situations, particularly with polyatomic molecules, when the separation of nuclear and electronic motions cannot be made satisfactorily, but with most diatomic molecules the Born-Oppenheimer separation is acceptable. The discussion of molecular electronic wave functions presented in this chapter is therefore based upon the Born-Oppenheimer approximation. 

For molecules, a common simplification is the Born-Oppenheimer separation in which the slow-moving nuclei and the fast-moving electrons are treated separately. Using this approximation (and suppressing the q dependence), we first solve the electronic Schrodinger equation ... 

We begin with some general considerations of perhaps lesser-known, but important, features of exact electronic wavefunctions. Our motive is to establish a theoretical framework together with a reasonably consistent notation in order to carry through the spin-coupled VB and other expansions of the total wavefunction. We consider an atomic or molecular system consisting of N electrons and A nuclei. We assume the Born-Oppenheimer separation and write the Hamiltonian operator for the motion of the electrons in the form ... 

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... 

We could expect that the Born-Oppenheimer separation of electronic and nuclear motions will provide a not quite satisfactory approximation if the nuclei can move far away from their equilibrium positions which is really the case in their excited vibrational states ... 

This assumption depends upon the accuracy of the Born-Oppenheimer separation of electronic and nuclear coordinates which is described in most standard textbooks on quantum mechanics. The force constants are theoretically determined by the equation for electronic motion, which involves the charges and configuration of the nuclei, but not their masses. 

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. 

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