Bom and Oppenheimer

Electronic transitions fexcitations or deexcitations) can take place during the course of a chemical reaction and have important consequences for its dynamics. The motion of electrons and nuclei were first analyzed in a quantum mechanical framework by Bom and Oppenheimer [1], who separated the... 

An alternative approximation scheme, also proposed by Bom and Oppenheimer [5-7], employed the straightforward perturbation method. To tell the difference between these two different BO approximation, we call the latter the crude BOA (CBOA). A main purpose of this chapter is to study the original BO approximation, which is often referred to as the crude BO approximation and to develop this approximation into a practical method for computing potential energy suifaces of molecules. 

Bom and Oppenheimer tackled the problem quantum-mechanically in 1927 their treatment is pretty involved, but the basic physical idea is as outlined above. To simplify the notation, I will write the total Hamiltonian as follows ... 

Bom and Oppenheimer showed that, to a very good approximation, these extra terms were of the order of m /M and so the motions of the electron and the nuclei could indeed he considered separately for many purposes. 

In the early days of quanttrm mechanics Bom and Oppenheimer[18] showed that the energy and motion of the nuclei and electrons could be separated approximately. This was accomplished using a perturbation treatment in which the perturbation parameter is In actuality, the term Bom-Oppenheimer approximation ... 

The validity conditions for the semiclassic adiabatic approach in the description of the systems with orbitally non-degenerate levels are elucidated in the basic works of Bom and Oppenheimer (comprehensive discussion can be found in Refs. [6,7]). In these systems, the slow nuclear motion can be separated from the fast electronic one. The situation is quite different in the JT systems where, in general, this separation is impossible due to hybridization of the electronic and vibrational states. Nevertheless, in many important cases the adiabatic approach can serve as a relatively simple and at the same time powerful tool for the theoretical study of the JT systems giving accurate quantitative results and clear insight on the physical nature of the physical phenomena. 

Bom and Oppenheimer expanded the molecular Hamiltonian in terms of a parameter k given by the ratio of a typical nuclear displacement to the internuclear distance R. Simple order-of-magnitude arguments showed that... 

This separation of the total wavefunction into an electronic wavefunction y/(r) and a nuclear wavefunction 0(R) means that the positions of the nuclei can be fixed, leaving it only necessary to solve for the electronic part. This approximation was proposed by Bom and Oppenheimer and is valid for the vast majority of organic molecules. 

The wavefunction i (r) depends on the coordinates of all of the electrons in the molecule. Hartree proposed the idea, reminiscent of the separation of variables used by Bom and Oppenheimer, that the electronic wavefunction can be separated into a product of functions that depend only on one electron. 

In view of the discussion in the arguments advanced by Bom and Oppenheimer [11], it may be helpful to express H t) in terms of two sets of coordinates.3 One set consists of A — 1 translationally invariant coordinates t" expressed entirely in terms of the coordinates used originally to describe the nuclei, rf,... 

Having evaluated the characteristic electronic energy Un(() as a function of the nuclear coordinates for a given set of values of the electronic quantum numbers n by solving the wave equation 34-3 for various nuclear configurations, we next obtain expressions for the nuclear wave functions n,v(f). It was shown by Bom and Oppenheimer that these functions are... 

Equation (49) indicates that the electronic energy acts as the potential function for the motion of the nuclei. Indeed, Bom and Oppenheimer have shown that such an equation as Eq. (49) is valid to the fourth order in x, i.e.,... 

The notion of a (reactive) potential energy surface for the nuclear motion is key to the development of the theory. While the idea was introduced by Bom and Heisenbeig (1924) and Bom and Oppenheimer (1927) it may be argued that the idea of a potential energy surface only reached its full fruition with the development of a theory of chemical reaction rates. The idea of a potential energy surface is part of Wigner s three threes (see below). 

Wigner s three steps are (WSl) The determination of potential energy surfaces, which gives, in the words of Wigner, the behaviour of all molecules present in the system during the reaction, how they will move, and which products they will yield when colliding with definite velocities, etc. (p. 29). The solution of this problem requires the calculation of a potential energy surface, which is a quantum chemistry problem that was solved, somewhat unsatisfactorily, by Bom and Oppenheimer(1927). 

We will therefore base our discussion on a general Hamiltonian for a molecule with n electrons and N nuclei formulated within the Bom-Oppenheimer approximation (Bom and Oppenheimer 1927), and restrict it to one- and two-particle terms. 

This is the general equation for the nuclei obtained by Bom and Oppenheimer. For a single (K = 1) nuclear coordinate R (for example, diatomic molecules), where R is the distance between the atoms, we have... 

Bom and Oppenheimer s 1927 paper justifying the Bom-Oppenheimer approximation is seriously lacking in rigor. Subsequent work has better justified the Bora-Oppenheimer approximation, but significant questions stiU remain the problem of the coupling of nuclear and electronic motions is, at the moment, without a sensible solution and. .. is an area where much future woik can and must be done [B. T. Sutcliffe, J. Chem. Soc. Faraday Trans., 89, 2321 (1993) see also B. T. Sutcliffe and R. G. Woolley, Phys. Chem. Chem. Phys., 7, 3664 (2005), and Sutcliffe and Woolley, arxiv.org/abs/1206.4239]. 

A brief description of the fundamental concepts underlying the ab initio molecular orbital computations is presented here. The reader is requested to refer to some of the textbooks providing comprehensive discussion of molecnlar orbital theory (Atkins 1991 Atkins and Friedman 1997 Cook 1998 Levine 1983 Simons and Nichols 1997). In electronic structure theory, given the position of atomic nuclei, R [under the Bom-Oppenheimer approximation (Bom and Oppenheimer 1927)], the Schrodinger equation for motion of electrons (r) is solved as ... 

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