The Spin-Pair Compositions as Local Indicators of the Nature of the Bonding

The spin-pair compositions at a given point of the position space are defined as the following ratios: that of the parallel spin-pair concentration to the antiparallel spin-pair concentration D anti ( r ), to the total spin-pair concentration D tot ( r ), and to the single spin-pair concentration D s ( r ). The spin-pair concentrations are calculated by the integration of the pair functions π αα ( r 1, r 2 ), π αβ ( r 1, r 2 ), π βα ( r 1, r 2 ), and π ββ ( r 1, r 2 ) over an arbitrary volume V ( r ) around the reference point r . Because the numbers of spin pairs are not proportional to the sampling volume, the dependence of the D ( r ) functions upon the sample size V ( r ) has been studied. It is shown that all D ( r ) functions depend on the population N̄ ( r ) of the sample. For N̄ (( r ) ≤ 10 -3, all D ( r ) functions behave as N̄ 2/3 ( r ); therefore, size-independent spin-pair composition functions are defined as c π ( r ) = N̄ 2/3 ( r ) D ( r ). Approximate expressions of the c π ( r ) functions are proposed, which enable the recovery of the electron localization function of Becke and Edgecombe.