Dominance and separability in posets, their application to isoelectronic species with equal total nuclear charge

We developed the dominance and separability degrees as two new mathematical tools measuring the amount of comparabilities and incomparabilities among pairs of disjoint subposets in a parent poset and we have related them through a theorem. Their mathematical properties when these measures are constrained to be higher than 0.5 have been studied. We have shown that variations of dominance and separability degrees from values in the real interval (0.5, 1] permit to “tune” the level of detail on the comparabilites and incomparabilities among the subsets studied. The lack of transitivity of dominance and separability degrees is established, along with the special requirement, needed on the poset, to have a transitivity of these measures. As a chemical application, the Hasse diagram of Born-Oppenheimer molecular total energies of the complete set of isoelectronic species with total nuclear charge 10 in their minimum energy configurations has been studied. We partition this set into 10 subsets, each one containing all the species with the same number of nuclei. By the calculation of the dominance and separability degrees a relation between the number of atoms in any ensemble and the Born-Oppenheimer energies is established.

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