Born-Oppenheimer approximation chemistry principles

Based on first principles. Used for rigorous quantum chemistry, i. e., for MO calculations based on Slater determinants. Generally, the Schrodinger equation (Hy/ = Ey/) is solved in the BO approximation (see Born-Oppenheimer approximation) with a large but finite basis set of atomic orbitals (for example, STO-3G, Hartree-Fock with configuration interaction). 

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. 

In Section 2.1, the electronic problem is formulated, i.e., the problem of describing the motion of electrons in the field of fixed nuclear point charges. This is one of the central problems of quantum chemistry and our sole concern in this book. We begin with the full nonrelativistic time-independent Schrodinger equation and introduce the Born-Oppenheimer approximation. We then discuss a general statement of the Pauli exclusion principle called the antisymmetry principle, which requires that many-electron wave functions must be antisymmetric with respect to the interchange of any two electrons. 

Chemical dynamics is the link between the potential energy surface (PES) (or surfaces) and an observable chemical phenomena. In principle the PES comes from an ab initio quantum chemistry calculation (within the Born-Oppenheimer approximation) though in practice it is often constructed by some more approximate model, e.g., semiempirical quantum chemistry or totally empirical force field models. First a brief overview of the present state of the methodology and scope of applications in this area is given. We will concentrate on chemical dynamics in the gas phase, though much of the methodology of this field has carried over to the study of dynamical processes in condensed phases, gas-surface collision processes, and also dynamics in biomolecular systems. 

In the following we shall be concerned with the problem of the electronic structure of molecular systems containing N electrons and M nuclei. The validity of the usual Born-Oppenheimer, or clamped nuclei, approximation will be assumed, that is we shall investigate the distribution of electrons in the field of the fixed nuclei. In principle the approximate solution of the Schrddinger equation of all the electrons provides us with the different electronic states of the molecule, once the position of the nuclei and the number of electrons is given. Essentially this is the procedure followed in everyday routine calculations of ab initio quantum chemistry, where we do not take into account the a priori knowledge about the properties of the different fragments of the total composite system. 

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