  In local effective potential theories of electronic structure, the electron
correlations due to the Pauli exclusion principle, Coulomb repulsion, and
correlation-kinetic effects, are all incorporated in the local
electron-interaction potential $v_{ee}({\bf r})$. In previous work, it has been
shown that for spherically symmetric or sphericalized systems, the asymptotic
near nucleus expansion of this potential is $v_{ee}(r) = v_{ee}(0) + \beta r +
O(r^2)$, with $v_{ee}(0)$ being finite. By assuming that the Schr\"odinger and
local effective potential theory wave functions are analytic near the nucleus
of atoms, we prove the following via Quantal density functional theory (Q-DFT):
(i) correlations due to the Pauli principle and Coulomb correlations do not
contribute to the linear structure; (ii) these Pauli and Coulomb correlations
contribute quadratically; (iii) the linear structure is {\em solely} due to
correlation-kinetic effects, the contributions of these effects being
determined analytically. We also derive by application of adiabatic coupling
constant perturbation theory via Q-DFT (iv) the asymptotic near nucleus
expansion of the Hohenberg-Kohn-Sham theory exchange $v_x({\bf r})$ and
correlation $v_c({\bf r})$ potentials. These functions also approach the
nucleus linearly with the linear term of $v_x({\bf r})$ being {\em solely} due
to the lowest-order correlation kinetic effects, and the linear term of
$v_c({\bf r})$ being due {\em solely} to the higher-order correlation kinetic
contributions. The above conclusions are equally valid for systems of arbitrary
symmetry, provided spherical averages of the properties are employed.
