Zeroth-order eigenfunctions expansion

An alternate treatment, due to Dirac, uses the variation of the coefficients to discuss a system of many levels, not just two levels. In analogy to the above, use the perturbation expansion, Eq. (3.34.2), and the zeroth-order eigenfunctions vF (r, t) as follows ... 

Although the function M4>0 would seem a natural zeroth-order approximation for a perturbation expansion of the interaction energies, it unfortunately is not an eigenfunction ofy/0. J)ne could try to introduce a new partitionmg of the Hamiltonian, H = H0 + V, such that is an eigenfunction of H0, but no... 

A further difficulty arises with the zeroth-order functions used for the perturbation expansion. The natural choice is the set of product functions but these are not antisymmetric. If antisymmetrized they become an overcomplete set, though this is not a problem with the finite basis sets used in practical calculations. Whether antisymmetrized or not, they are not orthogonal in the short-range region. Consequently they cannot be eigenfunctions of any Hermitian operator, so that a zeroth-order... 

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