Introduction orthogonality of wave functions

This chapter is not intended to provide a rigorous treatment of MO theory. Its purpose is to help the reader who has some elementary acquaintance with the subject to appreciate the MO arguments likely to be encountered in the study of descriptive inorganic chemistry, and to emphasise the points which, the reader should look out for in more detailed expositions of MO theory. 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2. which satisfy the Schrodinger equation  

Analytic, exact solutions cannot be obtained except for the simplest systems, i.e. hydrogen-like atoms with just one electron and one nucleus. Good approximate solutions can be found by means of the self-consistent field (SCF) method, the details of which need not concern us. If all the electrons have been explicitly considered in the Hamiltonian, the wave functions V, will be many-electron functions V, will contain the coordinates of all the electrons, and a complete electron density map can be obtained by plotting Vf. The associated energies E, are the energy states of the molecule (see Section 2.6) the lowest will be the ground state , and the calculated energy differences En — El should match the spectroscopic transitions in the electronic spectrum. 

One of the constraints to be imposed (as for AOs) is that the set of MOs for a given system must be linearly independent, i.e. it should not be possible to express any member of the set as a linear combination of the others. The fact that MOs y, y2. y satisfy the Schrodinger equation under the one-electron Hamiltonian leads to the conclusion that any linear combination ya  

It is easy to show that if any member of the set is a linear combination of the others, it will not be orthogonal to any function which has a non-zero coefficient in the linear combination. If all the functions are mutually orthogonal, they must be linearly independent. Integrals of this type are often called overlhp integrals, because they provide a numerical measure of the extent to which y, and y, overlap with each other in space. Two functions which overlap are non-orthogonal. 

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