The hydrogen molecular ion energy consideration

Now we consider the simplest molecular system, namely the hydrogen molecular ion H . The Schrodinger equation for H , in a.u., is 

Similarly, when the electron resides on atom b instead, we have 

If the two nuclei are allowed to come together, it is reasonable to expect that the molecular orbital xpMo would be some kind of combination of atomic functions (3.1.14) and (3.1.15). This is the basis of the LCAO approximation for describing the chemical bonding in molecular systems. Thus we now have 

The solution of involves the evaluation of six integrals Saa, Sw, Sab, Haa, Hbb, and Hat,. From the normalization relationship, we have 

Integral Sab is called the overlap integral. Physically it represents the common volume of the two atomic orbitals. A pictorial representation of Sab is given in Fig. 3.1.2. It is clear that Sab varies with rab, the internuclear distance when rab = 0, Sab = 1 when rab - oo, Sab 0. Mathematically, it can be shown that, in a.u., 

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