Born-Oppenheimer approximation, local

The effects of deviations from the Born-Oppenheimer approximation (BOA) due to the interaction of the electron in the sub-barrier region with the local vibrations of the donor or the acceptor were considered for electron transfer processes in Ref. 68. It was shown that these effects are of importance for long-distance electron transfer since in this case the time when the electron is in the sub-barrier region may be long as compared to the period of the local vibration.68 A similar approach has been used in Ref. 65 to treat non-adiabatic effects in the sub-barrier region in atom transfer processes. However, nonadiabatic effects in the classically attainable region may also be of importance in atom transfer processes. In the harmonic approximation, when these effects are taken into account exactly, they manifest themselves in the noncoincidence of the... 

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. 

A minimum, local or global, on a potential energy surface. Any small change of the nuclear coordinates on the (3 AT - 6)-dimensional surface will lead to an increase in strain energy, i.e., there is always a force driving the molecule back to this minimum (see energy surface, Born-Oppenheimer approximations, saddle point). 

All aspects of molecular shape and size are fully reflected by the molecular electron density distribution. A molecule is an arrangement of atomic nuclei surrounded by a fuzzy electron density cloud. Within the Born-Oppenheimer approximation, the location of the maxima of the density function, the actual local maximum values, and the shape of the electronic density distribution near these maxima are fully sufficient to deduce the type and relative arrangement of the nuclei within the molecule. Consequently, the electronic density itself contains all information about the molecule. As follows from the fundamental relationships of quantum mechanics, the electronic density and, in a less spectacular way, the nuclear distribution are both subject to the Heisenberg uncertainty relationship. The profound influence of quantum-mechanical uncertainty at the molecular level raises important questions concerning the legitimacy of using macroscopic analogies and concepts for the description of molecular properties. ... 

The structure of approximate reasoning is not simple. Consider the Born-Oppenheim approximation (separability of electronic and nuclear motions due to extreme mass difference), which in application produces "fixed nuclei" Hamiltonians for individual molecules. In assuming a nuclear skeleton, the idealization neatly corresponds to classical conceptions of a molecule containing localized bonds and definite structure. All early quantum calculations, and the vast majority to date, invoke the approximation. In 1978, following decades of quiet assumption, Cambridge chemist R. G. Woolley asserted ... 

Density Functional Theory and the Local Density Approximation Even in light of the insights afforded by the Born-Oppenheimer approximation, our problem remains hopelessly complex. The true wave function of the system may be written as i/f(ri, T2, T3,. .., Vf ), where we must bear in mind, N can be a number of Avogadrian proportions. Furthermore, if we attempt the separation of variables ansatz, what is found is that the equation for the i electron depends in a nonlinear way upon the single particle wave functions of all of the other electrons. Though there is a colorful history of attempts to cope with these difficulties, we skip forth to the major conceptual breakthrough that made possible a systematic approach to these problems. 

The modem theory of chemical reaction is based on the concept of the potential energy surface, which assumes that the Born-Oppenheimer adiabatic approximation [16] is obeyed. However, in systems subjected to the Jahn-Teller effect, adiabatic potentials have the physical meaning of the potential energy of nuclei only under the condition that non-adiabatic corrections are small [28]. In the vicinity of the locally symmetric intermediate, these corrections will be very large. The complete description of nuclear motion, i.e. of the mechanism of the chemical reaction, can be obtained only from Schroedinger s equation without applying the Born-Oppenheimer approximation in the vicinity of the locally... 

As most of the electronic structure simulation methods, we start with the Born-Oppenheimer approximation to decouple the ionic and electronic degrees of freedom. The ions are treated classically, while the electrons are described by quantum mechanics. The electronic wavefunctions are solved in the instantaneous potential created by the ions, and are assumed to evolve adiabatically during the ionic dynamics, so as to remain on the Born-Oppenheimer surface. Beyond this, the most basic approximations of the method concern the treatment of exchange and correlation (XC) and the use of pseudopotentials. XC is treated within Kohn-Sham DFT [3]. Both the local (spin) density approximation (LDA/LSDA) [16] and the generalized gradients approximation (GGA) [17] are implemented. The pseudopotentials are standard norm-conserving [18, 19], treated in the fully non-local form proposed by Kleinman and Bylander [20]. 

Do two helium atoms sense each other when separated by a bridge molecule In the other words, do the intermolecular interactions have local character or not By local character, we mean that their influence is limited to, say, the nearest-neighbor atoms. Let us check this by considering a bridge molecule (two cases butane and butadiene) between two helium atoms. The first helium atom pushes the terminal carbon atom of the butane (the distance Rc He = 1-5 A, the attack is perpendicular to the CC bond). The question is whether the second helium atom in the same position but at the other end of the molecule will feel that pushing or not the positions of all the nuclei do not change, since we are within the Born-Oppenheimer approximation) This means that we are interested in a three-body effect. 

This matrix was introduced by F. T. Smith [25] for the treatment of non-adiabatic (diabatic) couplings in atomic collisions. It is now familiar also in molecular structure problems, to indicate local breakdowns of the Born-Oppenheimer approximation. Within the hyperspherical formalism, it has been introduced in the three-body Coulomb problem [20] and in chemical reactions [21-24], see also Section 3. Also, from equation (A4)... 

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. 

Ground-state baryons with two heavy quarks, QOq, look like a localized colour source QO surrounded by a light quark q. The effect is, however, less and less pronounced for excited states, suggesting instead the use of the Born-Oppenheimer approximation, as discussed in the next section. 

L. A. Nafie and T. H. Walnut, Chem. Phys. Lett., 49, 441 (1977). Vibrarional Circular Dichroism Theory A Localized Molecular Orbital Model. T. H. Walnut and L. A. Nafie, ]. Chem. Phys., 67, 1491 (1977). Infrared Absorption and the Born-Oppenheimer Approximation. I. Vibrational Intensity Expression. T. H. Walnut and L. A. Nafie, J. Chem. Phys., 67, 1501 (1977). Infrared Absorption and the Born-Oppenheimer Approximation. II. Vibrational Circular Dichroism. 

The previous section points out that the Born-Oppenheimer approximation is useful in that electronic parts of wavefunctions can be separated from nuclear parts of wave-functions. However, it does not assist us in determining what the electronic wavefunctions are. Electrons in molecules are described approximately with orbitals just like electrons in atoms are described by orbitals. We have seen how quantum mechanics treats atomic orbitals. How does quantum mechanics treat molecular orbitals Molecular orbital theory is the most popular way to describe electrons in molecules. Rather than being localized on individual atoms, an electron in a molecule has a wave-function that extends over the entire molecule. There are several mathematical procedures for describing molecular orbitals, one of which we consider in this section. (Another perspective on molecular orbitals, called valence bond theory, will be discussed in Chapter 13. Valence bond theory focuses on electrons in the valence shell.)... 

Moreover, we refer to these kinds of concepts as force field calculations (molecular mechanics) which approximate the potential field (Born-Oppenheimer approximation) by "classical energy relations and adjustable parameters. These methods have successfully accompanied and completed the ab initio calculations until now. For the literature covering these methods and their results, we refer to other surveys. Because of the use of analytical potentials, the procedures are not as time-consuming as ab initio methods. However, their importance is placed behind the conceptually stronger ab initio methods, and they are not suited to localize structures between the minimizers on the PES as it is of primary importance for the kinetic characteristic of a chemical reaction. 

The Born-Oppenheimer approximation, whose validity depends on there being a deep enough localized potential well in the electronic energy, has, however, been extensively treated. The mathematical approaches depend upon the theory of fiber bundles and the electronic Hamiltonian in these approaches is defined in terms of a fiber bundle. It is central to these approaches, however, that the fiber bundle should be trivial, that is that the base manifold and the basis for the fibers be describable as a direct product of Cartesian spaces. This is obviously possible with the decomposition choice made for O Eq. 2.42 but not obviously so in the choice made for O Eq. 2.43. 

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