The Ground-State Energy to First-Order of Heliumlike Systems

Therefore, r-i contributes to (f) with a coefficient about the magiutude of the contribution from jri. The sign of C32 is positive and a sketch of 1/ 2 plus a small amount of V 3 will demonstrate that this causes charge shifting to the left in accord with our general statement above. 

Since 1/ 4 contributes nothing (by symmetry) and 1/ 5 is fairly distant in energy, we will neglect all contributions to above 1/ 3. 

We have, then, two sizable contributions to each favoring charge shifts in 

W2 is the difference between energy contributions of opposite sign. The net result (energy increases) comes about in this case because ipi contributes more heavily than 1/ 3. 

Because the unperturbed energies of the particle in the box increase as n, any state (except the lowest) is closer in energy to states below than to states above. Hence, for at least some kinds of perturbation, we might expect these states to feel the effects of states at lower energies more strongly and to rise in energy (as far as IF is concerned). This is what happens in this example. There are no states below the lowest, and so IF cannot be positive, but IF is positive and it turns out that IF is positive for all higher i as well. 

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Section 12-4 The Ground-State Energy to First-Order of Heliumlike Systems... 

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