Polar Coordinate Representation of <i>H</i><sub>b</sub>(<i>r</i><sub>c</sub>) versus (ℏ<sup>2</sup>/8<i>m</i>)▽<sup>2</sup>ρ<sub>b</sub>(<i>r</i><sub>c</sub>) at BCP in AIM Analysis: Classification and Evaluation of Weak to Strong Interactions

Polar coordinate (<i>R</i>, θ) representation is proposed for the plot of <i>H</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) versus (ℏ<sup>2</sup>/8<i>m</i>)▽<sup>2</sup>ρ<sub>b</sub>(<b><i>r</i></b><sub>c</sub>) in AIM analysis to classify, evaluate, and understand weak to strong interactions in a unified way and in more detail; <i>H</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) and ▽<sup>2</sup>ρ<sub>b</sub>(<b><i>r</i></b><sub>c</sub>) are total electron energy densities and the Laplacian of ρ<sub>b</sub>(<b><i>r</i></b><sub>c</sub>) at bond critical points (BCPs: <b><i>r</i></b><sub>c</sub>), respectively, where ρ<sub>b</sub>(<b><i>r</i></b><sub>c</sub>) are electron densities at <b><i>r</i></b><sub>c</sub>. Both the <i>x</i>- and <i>y</i>-axes of the plot are expressed in the common unit of energy since <i>H</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) = <i>G</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) + <i>V</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) and (ℏ<sup>2</sup>/8<i>m</i>)▽<sup>2</sup>ρ<sub>b</sub>(<b><i>r</i></b><sub>c</sub>) = <i>H</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) − <i>V</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>)/2 (= <i>G</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) + <i>V</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>)/2), where <i>G</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) and <i>V</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) 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 (<i>r</i>) of <i>r</i> = <i>r</i><sub>o</sub> + <i>wa</i><sub>o</sub>, where <i>r</i><sub>o</sub> are the full-optimized distances, <i>a</i><sub>o</sub> is the Bohr radius, and <i>w</i> = ±0.1 and ±0.2. The plot draws a helical stream starting from near origin (<i>H</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) = (ℏ<sup>2</sup>/8<i>m</i>)▽<sup>2</sup>ρ<sub>b</sub>(<b><i>r</i></b><sub>c</sub>) = 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 (<i>R</i>, θ); <i>R</i> is given in the energy unit, and θ in degrees is measured from the <i>y</i>-axis. The ratio of <i>V</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>)/<i>G</i><sub>b</sub>(<b><i>r</i></b><sub>c</sub>) (= <i>k</i>) 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<sub>p</sub> and κ<sub>p</sub>; θ<sub>p</sub> corresponds to the tangent line measured from the <i>y</i>-direction, and κ<sub>p</sub> is the curvature of the plot at <i>w</i> = 0. The polar coordinate (<i>R</i>, θ) representation with (θ<sub>p</sub>, κ<sub>p</sub>) helps us to classify, evaluate, and understand the nature of weak to strong interactions in a unified way.