Fixed nuclear approximation

The fixed nuclear approximation is extremely useful, not only for the bound states, but also for the treatment of electronic scattering by molecules. For instance, stationary-state scattering theories within the fixed nuclear approximation have been extensively developed for molecular photoionization [18, 250, 336, 516] and electron scattering from polyatomic molecules [210, 418, 419]. These scattering phenomena are quite important in that they are widely foimd in nature as elementary processes and even in industrial applications using plasma processes. These scatterings may be referred to as stationary-state electron dynamics in fixed nuclei approximation. 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

Background Philosophy. Within the framework of the Born-Oppenheimer approximation (JJ ), the solutions of the Schroedin-ger equation, Hf = Ef, introduce the concept of molecular structure and, thereby, the total energy hyperspace provided that the electronic wave function varies only slowly with the nuclear coordinates, electronic energies can be calculated for sets of fixed nuclear positions. The total energies i.e. the sums of electronic energy and the energy due to the electrostatic re-... 

The adiabatic approximation means the neglect of the nuclear motion in the Schrodinger equation. The electronic structure is thus calculated for a set of fixed nuclear coordinates. This approach can in principle be exact if one uses the set of wave functions for fixed nuclear coordinates as a basis set for the full Schrodinger equation, and solves the nuclear motion on this basis. The adiabatic approximation stops at the step before. (The Born-Oppenheimer approximation assumes a specific classical behavior of the nuclei and hence it is more approximate than the adiabatic approximation.)... 

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. 

One approach to the approximate representation of molecular bodies is based on molecular isodensity contours, MIDCOs, defined with respect to some fixed nuclear configuration K and some electron density threshold a. A MIDCO G(a,K) is defined (in the fixed nuclear configuration approximation) as the collection of all those points r of the three-dimensional space where the electronic density is equal to the threshold a ... 

Such a separation is exact for atoms. For molecules, only the translational motion of the whole system can be rigorously separated, while their kinetic energy includes all kinds of motion, vibration and rotation as well as translation. First, as in the case of atoms, the translational motion of the molecule is isolated. Then a two-step approximation can be introduced. The first is the separation of the rotation of the molecule as a whole, and thus the remaining equation describes only the internal motion of the system. The second step is the application of the Born-Oppenheimer approximation, in order to separate the electronic and the nuclear motion. Since the relatively heavy nuclei move much more slowly than the electrons, the latter can be assumed to move about a fixed nuclear arrangement. Accordingly, not only the translation and rotation of the whole molecular system but also the internal motion of the nuclei is ignored. The molecular wave function is written as a product of the nuclear and electronic wave functions. The electronic wave function depends on the positions of both nuclei and electrons but it is solved for the motion of the electrons only. 

In the fixed cavity approximation derivation of equation (51) with respect to a second nuclear displacement, y, gives... 

All of the above-mentioned investigations on the hydrogen atom were performed in the fixed-nuclei approximation, i.e. for the assumption of an infinitely heavy nucleus. In the absence of a magnetic field the center of mass (CM) and electronic motion separate exactly, and the influence of the finite nuclear mass on the behavior and properties of the atom can be taken into... 

The first step in simplifying the SOS-PT formulas is to apply the clamped nucleus approximation for the states K 0 [16]. In this approximation the energy denominators b are replaced by the difference in electronic energies at a fixed nuclear configuration, i.e. by E K, R) - (0, R). The consequences of this approximation have been investigated and were found to be negligible [16]. 

The polaron transformation, executed on the Hamiltonian (12.8)-( 12.10) was seen to yield a new Hamiltonian, Eq. (12.15), in which the interstate coupling is renormalized or dressed by an operator that shifts the position coordinates associated with the boson field. This transformation is well known in the solid-state physics literature, however in much of the chemical literature a similar end is achieved via a different route based on the Bom-Oppenheimer (BO) theory of molecular vibronic stmcture (Section 2.5). In the BO approximation, molecular vibronic states are of the form (/) (r,R)x ,v(R) where r and R denote electronic and nuclear coordinates, respectively, R) are eigenfunctions of the electronic Hamiltonian (with corresponding eigenvalues E r ) ) obtained at fixed nuclear coordinates R and... 

One of the most important characteristics of molecular systems is their behavior as a function of the nuclear coordinates. The most important molecular property is total energy of the system which, as a function of the nuclear coordinates, is called the potential energy (hyper)surface, an obvious generalization of the potential energy curve in diatomics. Other expectation values as functions of the nuclear coordinates are frequently called property surfaces. The notion of the total molecular energy in a given electronic state, which depends only parametrically on the nuclear coordinates, is based on the fixed-nuclei approximation. In most cases (e.g. closed-shell molecules in the ground electronic state and in a low vibrational state) this is an excellent approximation. Even when it breaks down, the most convenient treatment is based on the fixed-nuclei picture, i.e. on the assumption that the nuclear mass is infinite compared with the electronic mass. 

We consider a cluster of AT He atoms of mass m and radius ro, together with a single excess electron. The subsystem of the helium atoms will be treated by the density functional formalism [113, 247]. The excess electron will be treated quantum mechanically. The energetics and charge distribution of the electron were calculated within the framework of the adiabatic approximation for each fixed nuclear configuration. 

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... 

Solid State Physical Methods. - The theoretical treatment of a molecule or a polymer in the presence of an electric field or more generally of a laser beam presents a formidable problem. Here we shall remain first within the framework of the Born-Oppenheimer approximation and shall not consider the change of the phonons in the presence of an electric field because we shall work in a fixed nuclear (framework). Further, first we shall not take into account the effect of the interaction between the linear polymers on their polarizabilities and hyperpolarizabilities either although both effects are non-neglible.110-1,2 They will be treated subsequently. 

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