The Basic Ideas of Molecular Orbital Methods

The formalism that applies to the molecular orbital method will be illustrated by considering a hydrogen molecule, but a more detailed description of both H2+ and H2 will be given in the next section. 

In order to describe the hydrogen molecule by quantum mechanical methods, it is necessary to make use of the principles given in Chapter 2. It was shown that a wave function provided the starting point for application of the methods that permitted the calculation of values for the dynamical variables. It is with a wave function that we must again begin our treatment of the H2 molecule by the molecular orbital method. But what wave function do we need The answer is that we need a wave function for the H2 molecule, and that wave function is constructed from the atomic wave functions. The technique used to construct molecular wave functions is known as the linear combination of atomic orbitals (abbreviated as LCAO-MO). The linear combination of atomic orbitals can be written mathematically as 

In this equation, f is the molecular wave function, f is an atomic wave function, and a is a weighting coefficient that gives the relative weight in the mix of the atomic wave functions. The summation is 

Although the combination has been written as a sum, the difference is also an acceptable linear combination. The weighting coefficients are variables that must be determined. 

In Chapter 2, it was shown that in order to calculate the average value for a dynamical variable a whose operator is a, it is necessary to make use of the relationship 

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