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An interesting problem arises as to whether the actual radial distribution of the partly filled shell of a given gaseous ion is not much different from the HFSC behaviour. The latter is defined to give the lowest energy possible of a single, pure configuration; but we saw in eqn. (10) that the orbital energy is fairly insensitive to a moderate variation of the average radius. It might be that the correlation effects in the inner shells decrease their average radii so much that the most loosely-bound, partly filled shell has a weaker central field and consequently is expanded.

The total energy of a normalized wave function is J ^ H ^ where the Hamiltonian function Η partly produces kinetic energy, necessitating in its definition the indication of a wave function before Η and one after Η ; and partly potential energy, which is simply the integration of the product of with a potential. In nearly all cases met in practice, Ρ is only a potential. Thus the diagonal elements of energy are simply the integrals of 1 7 2 P / ! This is the change of energy expected in the classical theory of electricity by putting an extended charge distribution Ψ-f down in a potential Ρ (calculated for electrons, not for positive charges).

37), it is possible 2 to write ζη1 as a quadratic function k^Z^ , where the proportionality constant k nI varies surprisingly little within a given p- or d-shell. In this case, Z # is a linear function of the number of electrons q in the partly filled shell, being Z* = Z 0 + 0-5 + 0-7q for 4p<* and Z* = Z 0 — 0-8 + 0-8q for 4d<*. The factors are k 4P ~ 0 1 kK and k 4d ~ 0Ό29 kK, respectively. Z 0 is the ionic charge plus 1. The 2 fundamental reason why £ nl is proportional to Z^ is not as well understood as why Β is proportional to Z^.

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