Introduction to LCAO-MO Theory

The previous section points out that the Born-Oppenheimer approximation is useful in that electronic parts of wavefunctions can be separated from nuclear parts of wave-functions. However, it does not assist us in determining what the electronic wavefunctions are. Electrons in molecules are described approximately with orbitals just like electrons in atoms are described by orbitals. We have seen how quantum mechanics treats atomic orbitals. How does quantum mechanics treat molecular orbitals Molecular orbital theory is the most popular way to describe electrons in molecules. Rather than being localized on individual atoms, an electron in a molecule has a wave-function that extends over the entire molecule. There are several mathematical procedures for describing molecular orbitals, one of which we consider in this section. (Another perspective on molecular orbitals, called valence bond theory, will be discussed in Chapter 13. Valence bond theory focuses on electrons in the valence shell.) 

Consider what happens when a molecule is formed Two (or more) atoms combine to make a molecular system. The individual orbitals of the separate atoms combine to make orbitals that span the entire molecule. Why not use this description as a basis for defining molecular orbitals This is exactly what is done. By using linear variation theory, one can take linear combinations of occupied atomic orbitals and mathematically construct molecular orbitals. This defines the linear combination of atomic orbitals—molecular orbitals (LCAO-MO) theory, sometimes referred to simply as molecular orbital theory. 

In the case of H2, the molecular orbitals can be expressed in terms of the ground-state atomic orbitals of each hydrogen atom  

FIGURE 12.16 Representations of molecular orbitals from linear combinations of hydrogen atomic orbitals. The MO in the middle plot is the sum of the two AOs at the top, with electron density concentrated between the nuclei. The lower MO is the difference of the two AOs, with electron density concentrated more outside the two nuclei. 

Just as in linear variation theory, the coefficients can be determined using a secular determinant. But unlike the earlier examples using secular determinants, in this case some of the integrals are not identically zero or 1 due to orthonormality. In cases where there is an integral in terms of h(i) h(2) or vice versa, we cannot assume that the integral is identically zero. This is because the wavefunctions are centered on different atoms. The orthonormality conditions to this point are only strictly 

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