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Understanding the Uniqueness of 2p Elements in Periodic Tables - PubMed

  • ️Wed Jan 01 2020

Understanding the Uniqueness of 2p Elements in Periodic Tables

Zhen-Ling Wang et al. Chemistry. 2020.

Abstract

The Periodic Table, and the unique chemical behavior of the first element in a column (group), were discovered simultaneously one and a half centuries ago. Half a century ago, this unique chemistry of the light homologs was correlated to the then available atomic orbital (AO) radii. The radially nodeless 1s, 2p, 3d, 4f valence AOs are particularly compact. The similarity of r(2s)≈r(2p) leads to pronounced sp-hybrid bonding of the light p-block elements, whereas the heavier p elements with n≥3 exhibit r(ns) ≪ r(np) of approximately -20 to -30 %. Herein, a comprehensive physical explanation is presented in terms of kinetic radial and angular, as well as potential nuclear-attraction and electron-screening effects. For hydrogen-like atoms and all inner shells of the heavy atoms, r(2s) ≫ r(2p) by +20 to +30 %, whereas r(3s)≳r(3p)≳r(3d), since in Coulomb potentials radial motion is more radial orbital expanding than angular motion. However, the screening of nuclear attraction by inner core shells is more efficient for s than for p valence shells. The uniqueness of the 2p AO is explained by this differential shielding. Thereby, the present work paves the way for future physical explanations of the 3d, 4f, and 5g cases.

Keywords: bond theory; orbital radii; periodic table; quantum chemistry; radial node effect; sp hybridization.

© 2020 The Authors. Published by Wiley-VCH GmbH.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Figure 1
Figure 1

Radial densities r 2 φ(r)2 of valence AOs φ (in atomic units e/Bohr) versus nuclear distance r (in Å). The dashed (red in the electronic version) and solid (blue) curves, respectively, refer to p and s AOs. The bold numbers specify the change of orbital radii from p to s in % (referring to the density maxima at r max; in parentheses for the ⟨r⟩ average values; the trends of both are similar and pictured by the bold (blue) arrows). Left: H‐like atoms/ions (here Be3+(nℓ), without any core shells); 2s is significantly more extended than 2p (ca. +30 and +20 %), whereas 3s and 3p are less different (<+10 %). Right: C and Si: r max of C 2s and C 2p are similar, ⟨r⟩ of C 2s is a little more compact (ca. −10 %); Si 3s is approximately −20 % smaller than Si 3p.

Figure 2
Figure 2

Three‐dimensional spherical potentials V(r) with different shapes. Left: Green: The electrostatic Coulomb potential ∼−r −1 is narrow at short range and flattish at long range, therefore r rad ≫ r ang and Q max=r rad/r ang>1. Blue: The linear interaction ∼+r +1 is the border case with Q max=1. Right: Potentials are wide at low energies with a steep rise, yielding r rad ≪ r ang and Q max<1. Lilac: Harmonic oscillator potential ∼+r +2. Red: Typical screened Coulomb potential. Brown: Spherical box potential ∼+r +∞. For details, see section S.2b and Table S1 in the Supporting Information.

Figure 3
Figure 3

Inverse radii ⟨nℓ|r|nℓ⟩−1 (in nm−1) corresponding to Z eff of the atomic nℓ valence orbitals versus nuclear charge number Z (with electronic core–hole configurations (see Supporting Information, S.4–5). The straight lines (full for s, dotted for p) guide the eyes from H* n(sp)1 through Be** ns2 np2 to C 1s2–2s22p2 or, respectively, Si 1s22s22p6–3s23p2. The smaller slopes for np versus the steeper for ns from Be onward to C or Si show that the np valence orbitals are better shielded from the (increasing) nuclear charge by the (increasing number of) core electrons than the ns orbitals. Note the change of order from H* and Be** (r s>r p) to C or Si (r s<r p).

Figure 4
Figure 4

Variation of the ratios Qn=r ns/r np of valence orbital radii, upon strong differential core–(s,p) valence inter‐shell screening (Δcore≈−0.40, Red); upon weak (s,p) valence intra‐shell screening (Δval≈−0.05, Blue); and the double‐screening cross term of opposite sign (δ(c,v)≈+0.15, Lilac).

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