Quantum-Mechanical Calculations of Static Polarizabilities

The first step in our quantum-mechanical calculation is the construction of an appropriate molecular Hamiltonian. For a weak constant static field E we have 

I will write A = —pe E as appropriate in the following few pages, for the sake of compactness. 

Examination of the energy expression 17.7 shows that the polarizability components are the second-order energies and a little analysis shows that (for example) 

This expression is correct, but not useful because we need information about the excited states of the molecule. Such information is hard to come by. In the old days, we would try to make progress by taking some average energy difference Ae between the excited states and the ground state. We can then write 

The sum runs over all excited states but not the ground state. It can also be written as a sum over all states (including the ground state) as 

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