Polar Coordinate Representation of Hb(rc) versus (ℏ2/8m)▽2ρb(rc) at BCP in AIM Analysis: Classification and Evaluation of Weak to Strong Interactions

Polar coordinate (R, θ) representation is proposed for the plot of Hb(rc) versus (ℏ2/8m)▽2ρb(rc) in AIM analysis to classify, evaluate, and understand weak to strong interactions in a unified way and in more detail; Hb(rc) and ▽2ρb(rc) are total electron energy densities and the Laplacian of ρb(rc) at bond critical points (BCPs: rc), respectively, where ρb(rc) are electron densities at rc. Both the x- and y-axes of the plot are expressed in the common unit of energy since Hb(rc) = Gb(rc) + Vb(rc) and (ℏ2/8m)▽2ρb(rc) = Hb(rc) − Vb(rc)/2 (= Gb(rc) + Vb(rc)/2), where Gb(rc) and Vb(rc) 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 = ro + wao, where ro are the full-optimized distances, ao is the Bohr radius, and w = ±0.1 and ±0.2. The plot draws a helical stream starting from near origin (Hb(rc) = (ℏ2/8m)▽2ρb(rc) = 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 Vb(rc)/Gb(rc) (= 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 θp and κp; θp corresponds to the tangent line measured from the y-direction, and κp is the curvature of the plot at w = 0. The polar coordinate (R, θ) representation with (θp, κp) helps us to classify, evaluate, and understand the nature of weak to strong interactions in a unified way.