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# Polar Coordinate Representation of *H*_{b}(*r*_{c}) versus (ℏ^{2}/8*m*)▽^{2}ρ_{b}(*r*_{c}) at BCP in AIM Analysis: Classification
and Evaluation of Weak to Strong Interactions

journal contribution

posted on 2009-09-17, 00:00 authored by Waro Nakanishi, Satoko Hayashi, Kenji NaraharaPolar coordinate (

*R*, θ) representation is proposed for the plot of*H*_{b}(*r*_{c}) versus (ℏ^{2}/8*m*)▽^{2}ρ_{b}(*r*_{c}) in AIM analysis to classify, evaluate, and understand weak to strong interactions in a unified way and in more detail;*H*_{b}(*r*_{c}) and ▽^{2}ρ_{b}(*r*_{c}) are total electron energy densities and the Laplacian of ρ_{b}(*r*_{c}) at bond critical points (BCPs:*r*_{c}), respectively, where ρ_{b}(*r*_{c}) are electron densities at*r*_{c}. Both the*x*- and*y*-axes of the plot are expressed in the common unit of energy since*H*_{b}(*r*_{c}) =*G*_{b}(*r*_{c}) +*V*_{b}(*r*_{c}) and (ℏ^{2}/8*m*)▽^{2}ρ_{b}(*r*_{c}) =*H*_{b}(*r*_{c}) −*V*_{b}(*r*_{c})/2 (=*G*_{b}(*r*_{c}) +*V*_{b}(*r*_{c})/2), where*G*_{b}(*r*_{c}) and*V*_{b}(*r*_{c}) are kinetic energy densities and potential energy densities, respectively. Data employed for the plot are calculated at BCPs for full-optimized structures and optimized structures with the fixed distances (*r*) of*r*=*r*_{o}+*wa*_{o}, where*r*_{o}are the full-optimized distances,*a*_{o}is the Bohr radius, and*w*= ±0.1 and ±0.2. The plot draws a helical stream starting from near origin (*H*_{b}(*r*_{c}) = (ℏ^{2}/8*m*)▽^{2}ρ_{b}(*r*_{c}) = 0) for very weak interactions and turns to the right as interactions become stronger. The helical stream is well described by the polar coordinate representation with (*R*, θ);*R*is given in the energy unit, and θ in degrees is measured from the*y*-axis. The ratio of*V*_{b}(*r*_{c})/*G*_{b}(*r*_{c}) (=*k*) controls θ, of which an acceptable range in the plot is 45.0 < θ < 206.6°. Each plot for an interaction gives a curve, which supplies important information. It is expressed by θ_{p}and κ_{p}; θ_{p}corresponds to the tangent line measured from the*y*-direction, and κ_{p}is the curvature of the plot at*w*= 0. The polar coordinate (*R*, θ) representation with (θ_{p}, κ_{p}) helps us to classify, evaluate, and understand the nature of weak to strong interactions in a unified way.