Born-Oppenheimer electronic theory basis

In the vibrational treatment we assumed, as usually done, that the Born-Oppenheimer separation is possible and that the electronic energy as a function of the internuclear variables can be taken as a potential in the equation of the internal motions of the nuclei. The vibrational anharmonic functions are obtained by means of a variational treatment in the basis of the harmonic solutions for the vibration considered (for more details about the theory see Pauzat et al [20]). 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

The classical idea of molecular structure gained its entry into quantum theory on the basis of the Born Oppenheimer approximation, albeit not as a non-classical concept. The B-0 assumption makes a clear distinction between the mechanical behaviour of atomic nuclei and electrons, which obeys quantum laws only for the latter. Any attempt to retrieve chemical structure quantum-mechanically must therefore be based on the analysis of electron charge density. This procedure is supported by crystallographic theory and the assumption that X-rays are scattered on electrons. Extended to the scattering of neutrons it can finally be shown that the atomic distribution in crystalline solids is identical with molecular structures defined by X-ray diffraction. 

We will discuss an electronic basis, the electronic basis on which the Jahn-Teller theory is based, and compare it with another electronic basis, the electronic basis. Since non-adiabatic coupling in Jahn-Teller effect has a different meaning from that in the Born-Oppenheimer basis, we will also discuss adiabatic approximations in these electronic bases. 

The purpose of this chapter will be to review the fundamentals of ab initio MD. We will consider here Density Functional Theory based ab initio MD, in particular in its Car-Parrinello version. We will start by introducing the basics of Density Functional Theory and the Kohn-Sham method, as the method chosen to perform electronic structure calculation. This will be followed by a rapid discussion on plane wave basis sets to solve the Kohn-Sham equations, including pseudopotentials for the core electrons. Then we will discuss the critical point of ab initio MD, i.e. coupling the electronic structure calculation to the ionic dynamics, using either the Born-Oppenheimer or the Car-Parrinello schemes. Finally, we will extend this presentation to the calculation of some electronic properties, in particular polarization through the modern theory of polarization in periodic systems. 

Within the Born-Oppenheimer approximation, we still need to know that the nuclear position parameters really correspond to the distances and angles of a classical molecular framework. Our choice of the Coulomb gauge ensures this—the nuclear positions only appear in the electron-nucleus interaction terms, and the derivation of this potential from relativistic field theory shows us that it is indeed the quantities of normal 3-space that appear here. Thus, any potential surface that we might calculate on the basis of the Born-Oppenheimer-separated electronic molecular Dirac equation is indeed spanned by the variations of molecular structural parameters in the usual meaning. 

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. 

The exchange intCTaction is an electrostatic interaction between electrons and was first introduced in the theory of the Heitler-London model of a hydrogen molecule. The Hamiltonian for a molecule composed of two hydrogen nuclei (a and b) and two electrons (1 and 2) is given, on the basis of the Born-Oppenheimer approximation, by... 

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