jp903622a_si_001.pdf (305.33 kB)

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

Download (305.33 kB)
journal contribution
posted on 17.09.2009 by Waro Nakanishi, Satoko Hayashi, Kenji Narahara
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.