Quantum Stress Focusing in Descriptive Chemistry

We show that several important concepts of descriptive chemistry, such as atomic shells, bonding electron pairs, and lone electron pairs, may be described in terms of quantum stress focusing, i.e., the spontaneous formation of high-pressure regions in an electron gas. This description subsumes previous mathematical constructions, such as the Laplacian of the density and the electron localization function, and provides a new tool for visualizing chemical structure. We also show that the full stress tensor, defined as the derivative of the energy with respect to a local deformation, can be easily calculated from density functional theory.

Figure 1

Indicators of electron localization as functions of the radial distance $r$ constructed, respectively, from the noninteracting and interacting pressures $p^{\mathrm{S}}$ and $p$ [Eq. (9)] for the (a) Ne, (b) Ar, (c) Zn, and (d) Xe atoms. The XC part of the pressure is evaluated with the GGA functional of Ref. 18. The maxima of the pressure reveal electronic shells, indicated as usual by the capital letters $K,L,M,\dots$.

Figure 2

Noninteracting pressure ($p^{\mathrm{S}}$) and shear stress (${\pi }^{\mathrm{S}}$) in Ar. The peaks in $p^{\mathrm{S}}$ identify the position of the shells and the peaks in ${\pi }^{\mathrm{S}}$ define the boundary between the shells. Inset: Distribution of stresses in the atom. The gray scale indicates the pressure distribution with dark (light) regions corresponding to high (low) value of the pressure indicator $L$ [Eq. (9)]. The arrows indicate the direction of the shear force $(2/3)\nabla \pi +(2/r)\pi$. Observe how the shells are “squeezed” by shear forces pointing in opposite directions.