Extensions to the simple Heitler-London treatment

In our discussion we have merely given the expressions for the five integrals that appear in the energy. Those interested in the problem of evaluation are referred to Slater[27]. In practice, these expressions are neither very important nor useful. They are essentially restricted to the discussion of this simplest case of the H2 molecule and a few other diatomic systems. The use of AOs written as sums of Gaussian functions has become universal except for single-atom calculations. We, too, will use the Gaussian scheme for most of this book. The present discussion, included for historical reasons, is an exception. 

In the last section, our calculation used only the function of Eq. (2.9), what is now called the covalent bonding function. According to our discussion of linear variation functions, we should see an improvement in the energy if we perform a two-state calculation that also includes the ionic function. 

When this is done we obtain the curve labeled (d) in Fig. 2.1, which, we see, represents a small improvement in the energy. 

At the calculated energy minimum (optimum a) the total wave function is found to be 

The relative values of the coefficients indicate that the variation theorem thinks better of the covalent function, but the other appears fairly high at first glance. If, however, we apply the EGSO process described in Section 1.4.2, we obtain 0.996 50 Vc + 0.083 54 where, of course, the covalent function is unchanged, 

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