Polar coordinate representation of Hb(rc) versus (h2/8m)nabla2rhob(rc) at BCP in AIM analysis: classification and evaluation of weak to strong interactions.

Polar coordinate (R, theta) representation is proposed for the plot of Hb(rc) versus (h2/8m)nabla2rhob(rc) in AIM analysis to classify, evaluate, and understand weak to strong interactions in a unified way and in more detail; Hb(rc) and nabla2rhob(rc) are total electron energy densities and the Laplacian of rhob(rc) at bond critical points (BCPs: rc), respectively, where rhob(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 (h2/8m)nabla2rhob(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 = 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 (Hb(rc) = (h2/8m)nabla2rhob(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, theta); R is given in the energy unit, and theta in degrees is measured from the y-axis. The ratio of Vb(rc)/Gb(rc) (= k) controls theta, of which an acceptable range in the plot is 45.0 < theta < 206.6 degrees. Each plot for an interaction gives a curve, which supplies important information. It is expressed by theta(p) and kappa(p); theta(p) corresponds to the tangent line measured from the y-direction, and kappa(p) is the curvature of the plot at w = 0. The polar coordinate (R, theta) representation with (theta(p), kappa(p)) helps us to classify, evaluate, and understand the nature of weak to strong interactions in a unified way.