Investigating Istvan Mayer's "improved" definitions of bond orders and free valence for correlated singlet-state wave functions

Chosen from Istvan Mayer's very impressive canon of important work on bond orders and related quantities, we explore a paper from 2012 which introduced a "pseudo spin density matrix" for correlated singlet-state wave functions, leading to "improved" definitions of bond orders and free valence. Examining such "improved" bond orders for the singlet ground states of H-2, N-2 and CH+ we find that all of them exhibit sensible geometry dependences. Mayer's free valence index works well for H-2 and N-2; the asymptotic behavior for CH+ turns out to be slightly more complicated but can easily be explained. Using B2H6 to examine three-center bonding, a simple generalization of Mayer's approach produces numerical results close to those based on the "pseudo spin density matrix." As expected, the various "correction" terms remain small, albeit some of them are larger for the multicenter indices of ground and excited singlet states of benzene, S2N2 and square (D-4h) cyclobutadiene.