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

Polar coordinate (R, θ) representation is proposed for the plot of H_b(r_c) versus (ℏ²/8m)▽²ρ_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 ▽²ρ_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 (ℏ²/8m)▽²ρ_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) = (ℏ²/8m)▽²ρ_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.