10.1021/jp903622a.s001
Waro Nakanishi
Waro
Nakanishi
Satoko Hayashi
Satoko
Hayashi
Kenji Narahara
Kenji
Narahara
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
American Chemical Society
2009
κ p
density
Gb
BCP
Hb
interaction
energy densities
Polar Coordinate Representation
θ p
electron energy densities
AIM
helical stream
Vb
representation
2009-09-17 00:00:00
Journal contribution
https://acs.figshare.com/articles/journal_contribution/Polar_Coordinate_Representation_of_i_H_i_sub_b_sub_i_r_i_sub_c_sub_versus_sup_2_sup_8_i_m_i_sup_2_sup_sub_b_sub_i_r_i_sub_c_sub_at_BCP_in_AIM_Analysis_Classification_and_Evaluation_of_Weak_to_Strong_Interactions/2827291
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.