The LiH molecule approximate molecular orbital calculations

We first assume - like Lewis - that the electrons in inner shells are unaffected by bond formation, i.e. that two electrons continue to occupy the Li 1 orbital. Secondly we assume that the two valence electrons occupy bonding molecular orbitals formed by linear combination of the two AOs that contain the valence electrons when the two atoms are far apart, i.e. by combination of the Li 2s and the H l.y orbitals, which we denote by Pi x,y,z) and 

There is, of course, no reason to believe that the two coefficients ci and C2 will be equal. Problem 8.1 Show that the approximate MO, equation (8.1), is normalized if 

Finally we assume that we know the orbital coefficients c and C2 for the MO containing one of the valence electrons electron number 3. The orbital energy of the fourth and last electron may then be written as the sum of its kinetic energy, its Coulomb energy of attraction to both nuclei and its energy of repulsion from the other three electrons. Since we have assumed that we know the wavefunctions of the three other electrons, it should, in principle, be possible to write down the Hamiltonian operator of fourth electron (fi) and compute the orbital energy from 

It may not be obvious, but it can be shown that the integrals Hi 2 and H2,i are equal. Using this result, the orbital energy can now be written as 

Let us assume that we have carried out the calculation of the various integrals in equation (8.4) with R equal to the equilibrium bond distance. According to the variation principle, the calculated (approximate) orbital energy must be higher than the true energy. We now seek those values of the coefficients ci and C2 that yield the minimum value of e by taking the derivative of e with respect to ci and C2 and setting these derivatives equal to zero. 

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