Orthogonality relativistic wave functions

Here af and cf for the cases n = l + 1 are found from the variational principle requiring the minimum of the non-relativistic energy, whereas cf (n > l + 1) - form the orthogonality conditions for wave functions. More complex, but more accurate, are the analytical approximations of numerical Hartree-Fock wave functions, presented as the sums of Slater type radial orbitals (28.31), namely... 

One can minimize the energy of just a valence shell to obtain its wave function both in the H.F. part and the i,/s provided the trial functions are kept orthogonal to the inner orbitals [see Eqs. (86) and (99b)]. The relativistic shrinkage of the inner shells and the shell structure then causes the outer electrons to shrink too. Thus, even to calculate the Mon-relativistic valence shell, a knowledge of the inner orbitals is needed. These may probably be approximated by the relativistic H.F. orbitals obtained from the actual free ions corresponding to the cores. The medium potential V is also affected by them, and hence by relativistic effects. However, the m, parts of the outer wave function are not sensitive to changes in V. ... 

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