  Electronic correlation is a complex many-body effect and the correlation
energy depends on the specific electronic structure and spatial distribution of
electrons in each atom and molecule. Although the total correlation energy in
an atom can be decomposed into different components such as inter-orbital and
intra-orbital pair-correlation energies (PCE), it is generally believed that
the PCEs in different atoms cannot be the same. In this work, we investigate
the correlation energies of the atoms in the first three rows of the periodic
table (He to Ar). It is found that when the correlation energy is defined as
the difference between the exact ground-state energy and the unrestricted
Hartree-Fock (UHF) energy, the inter- and intra-orbital PECs associated with
the valence electrons of the atoms in the same row of the periodic table have
the same values. These PCEs are not entangled and their values depend only on
the electron orbitals. For two specific orbitals, the inter-orbital correlation
energy is the same between two electrons of parallel spins or anti-parallel
spins. We also show that the effects of orbital relaxation on the correlation
energy are surprisingly small.
