Quantum Mechanics of Bonding

As we hinted at in the section on the hydrogen atom, when we turn our attention to the origins of bonding in molecules and solids, a useful alternative to the 

In this equation, the total potential experienced by the electron is obtained by 

Rather than trying to find those fimctions which satisfy the Schrodinger equation written above, we adopt a simpler variational approach in which the search for imknown fimctions is replaced by the search for unknown coefficients. The basis for this strategy is a variational argument. We represent the wave function for the molecule as a linear combination of the atomic orbitals centered on the two nuclei, that is. 

For simplicity, we now assume that the two atoms are identical (i.e. E = E2) and further, choose our zero of energy such that 1 = 0. 

If we now minimize this function with respect to the ajS, in particular by enforcing 90/9a = 0 for t = 1,2, we can find the optimal choice for these coefficients. Effecting the minimization suggested above results in two coupled algebraic equation that are more conveniently written in matrix form as 

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