Born-Oppenheimer complex approximation

For further discussion, we must first consider some aspects of quantum mechanics, associated with the transitions of systems from one state to another. Complex polyatomic systems consist of nuclei and electrons. An absolutely accurate description of such systems even in the stationary state is quite complicated. For this reason, an approximate method is employed, which gives a fairly high accuracy in a majority of cases (we are speaking about the Born-Oppenheimer adiabatic approximation). The physical meaning of this method was in fact explained in the previous section. Since the characteristic velocities of motion of nuclei and electrons differ considerably, we can consider the motion of electrons in the field created by fixed nuclei. Incidentally, it is clear from this that the Franck-Condon principle is a direct corollary of the adiabatic approximation. 

The proper quantumdynamical treatment of fast electronic transfer reactions and reactions involving electronically excited states is very complex, not only because the Born-Oppenheimer approximation brakes down but... 

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

Under the Born-Oppenheimer approximation, two major methods exist to determine the electronic structure of molecules The valence bond (VB) and the molecular orbital (MO) methods (Atkins, 1986). In the valence bond method, the chemical bond is assumed to be an electron pair at the onset. Thus, bonds are viewed to be distinct atom-atom interactions, and upon dissociation molecules always lead to neutral species. In contrast, in the MO method the individual electrons are assumed to occupy an orbital that spreads the entire nuclear framework, and upon dissociation, neutral and ionic species form with equal probabilities. Consequently, the charge correlation, or the avoidance of one electron by others based on electrostatic repulsion, is overestimated by the VB method and is underestimated by the MO method (Atkins, 1986). The MO method turned out to be easier to apply to complex systems, and with the advent of computers it became a powerful computational tool in chemistry. Consequently, we shall concentrate on the MO method for the remainder of this section. 

The most extensive potential obtained so far with experimental confirmation is that of Le Roy and Van Kranendonk for the Hj — rare gas complexes 134). These systems have been found to be very amenable to an adiabatic model in which there is an effective X—Hj potential for each vibrational-rotational state of (c.f. the Born Oppenheimer approximation of a vibrational potential for each electronic state). The situation for Ar—Hj is shown in Fig. 14, and it appears that although the levels with = 1) are in the dissociation continuum they nevertheless are quasi bound and give spectroscopically sharp lines. 

Electronic and optical properties of complex systems are now accessible thanks to the impressive development of theoretical approaches and of computer power. Surfaces, nanostructures, and even biological systems can now be studied within ab-initio methods [53,54]. In principle within the Born-Oppenheimer approximation to decouple the ionic and electronic dynamics, the equation that governs the physics of all those systems is the many-body equation ... 

This chapter is the first step in the study of the magnetic behavior of the complexes of 5d-ions. The model is based on the Born-Oppenheimer version of the adiabatic approximation, i.e., only the minima of the lower adiabatic surfaces are taken into account while calculating g-factors and TIP. 

Electrons are much lighter than nuclei (a proton is 1836 times heavier than an electron) and thereby move much faster. In the time it takes the electrons of a molecule to explore the entire space around all the nuclei, these have essentially remained at standstill. This means that when calculating the energies of the electrons, we may treat the nuclei as fixed. This Born-Oppenheimer approximation greatly reduces the problem complexity by removing the motion of the nuclei. 

So far, this discussion of selection rules has considered only the electronic component of the transition. For molecular species, vibrational and rotational structure is possible in the spectrum, although for complex molecules, especially in condensed phases where collisional line broadening is important, the rotational lines, and sometimes the vibrational bands, may be too close to be resolved. Where the structure exists, however, certain transitions may be allowed or forbidden by vibrational or rotational selection rules. Such rules once again use the Born-Oppenheimer approximation, and assume that the wavefunctions for the individual modes may be separated. Quite apart from the symmetry-related selection rules, there is one further very important factor that determines the intensity of individual vibrational bands in electronic transitions, and that is the geometries of the two electronic states concerned. Relative intensities of different vibrational components of an electronic transition are of importance in connection with both absorption and emission processes. The populations of the vibrational levels obviously affect the relative intensities. In addition, electronic transitions between given vibrational levels in upper and lower states have a specific probability, determined in part... 

The equation is used to describe the behaviour of an atom or molecule in terms of its wave-like (or quantum) nature. By trying to solve the equation the energy levels of the system are calculated. However, the complex nature of multielectron/nuclei systems is simplified using the Born-Oppenheimer approximation. Unfortunately it is not possible to obtain an exact solution of the Schrddinger wave equation except for the simplest case, i.e. hydrogen. Theoretical chemists have therefore established approaches to find approximate solutions to the wave equation. One such approach uses the Hartree-Fock self-consistent field method, although other approaches are possible. Two important classes of calculation are based on ab initio or semi-empirical methods. Ah initio literally means from the beginning . The term is used in computational chemistry to describe computations which are not based upon any experimental data, but based purely on theoretical principles. This is not to say that this approach has no scientific basis - indeed the approach uses mathematical approximations to simplify, for example, a differential equation. In contrast, semi-empirical methods utilize some experimental data to simplify the calculations. As a consequence semi-empirical methods are more rapid than ab initio. 

Solving for the dynamics of such system is a highly complex problem. The first simplification amounts to assume an adiabatic separations of nuclei and electrons motions. In this approximation a partial factorization of the total system wavefunction is performed and we consider the ions fixed from the point of view of the electrons. This will lead us to the Born-Oppenheimer approximation. 

Quantum mechanics (QM) is a field of quantum chemistry that uses mathematical basis to study chemical phenomena at a molecular level. It uses a complex mathematical expression called as a wave function with which energy and properties of atoms and molecules can be computed. For simple model systems wave functions can be analytically determined, while for complex systems such as those that involve molecular modeling, approximations have to be made. One of the commonly employed approximation methods is that of Born and Oppenheimer. This approximation exploits the idea that does not necessitate the development of a wave function description for both the electrons and the nuclei at the same time. The nuclei are heavier, and move much more slowly than the electrons, and therefore can be regarded as stationary, while electronic wave function is computed. By computing the QM of the electronic motion, the energy changes for different chemical processes, vibrations and chemical reactions can be understood.142... 

In the very same way as the Born-Oppenheimer approximation allows the definition of a potential energy surface for a Van der Waals molecule, it enables, too, the concept of an interaction tensor field. This is a field dependent on the relative coordinates of the monomers and transforming as a tensor under rotation of the complex as a whole. (The potential energy surface is an example of a rank zero interaction tensor field). In the case of tensor fields it is also convenient to base the theory on irreducible tensors and to use an expansion in terms of a complete set of functions of the five angular coordinates describing a Van der Waals dimer. 

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 ability to predict accurate potential surfaces means that we are in a position to investigate the nature of the kinetic barriers, including the activated complexes, of key surface processes. In fact, Born-Oppenheimer potential surfaces can be used not only with the transition-state approach to kinetics but also with the much more general and exact collision theory (e.g., scattering S-matrix ilieory). While methods based on collision and scattering theory have pointed out deficiencies in the traditional transition-state theory (TST), they have also served to uphold many of TST s simple claims. In turn, new generalized transition state theories have been born. For complex systems, the transition-dale approach, while admittedly approximate, has been well established. 

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