Molecular orbitals The Fock and Roothaan equations

In principle, a description of the electronic structure of many-electron atoms and of polyatomic molecules requires a solution of a Schrodinger equation for stationary states quite similar to equation 3.36 [2]. Even for a simple molecule like, say, methane, however, such an equation would be enormously more complicated, because the hamiltonian operator would include kinetic energy terms for all electrons, plus coulombic terms for the electrostatic interaction of all electrons with all nuclei and of all electrons with all other electrons. The QM hamiltonian operator for the electrons in a molecule reads  

The single gaussians 3.40 decay to zero as r tends to infinity because the inverse exponential decreases faster than the (n — 1 )-th power increases, but the whole function 

For a single atom the nucleus is fixed at the origin of coordinates. For molecules, a big simplification results from the fact that electrons rearrange so much faster than nuclei that the positional coordinates of the nuclei can be kept fixed in the calculation of electronic energies, and the molecular wavefiinction depends only on the coordinates of electrons. This is called the Born-Oppenheimer assumption (there are no potential energy terms for nuclei in the hamiltonian 3.39). The total electronic energy is the expectation value of the hamiltonian operator, equation 3.8  

In an obvious conceptual development, molecular orbitals should be obtained as some form of combination of atomic orbitals [3], A convenient approximation is to take a product of many functions, each of which contains the coordinates of one electron only, i.e. a product of one-electron orbitals  

This is a very severe approximation, because this wavefiinction cannot describe the effects of electron correlation, or the simultaneous displacement, or other simultaneous change in properties, of many electrons at a time. 

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