The role of radial nodes of atomic orbitals for chemical bonding and the periodic table

Abstract

The role of radial nodes, or of their absence, in valence orbitals for chemical bonding and periodic trends is discussed from a unified viewpoint. In particular, we emphasize the special role of the absence of a radial node whenever a shell with angular quantum number l is occupied for the first time (lack of ''primogenic repulsion''), as with the 1s, 2p, 3d, and 4f shells. Although the consequences of the very compact 2p shell (e.g. good isovalent hybridization, multiple bonding, high electronegativity, lone-pair repulsion, octet rule) are relatively well known, it seems that some of the aspects of the very compact 3d shell in transition-metal chemistry are less well appreciated, e.g., the often weakened and stretched bonds at equilibrium structure, the frequently colored complexes, and the importance of nondynamical electron-correlation effects in bonding.

Strong Chemical Bonds

“…The 1s wave function is nodeless, whereas the 2s orbital has one radial node; the 3s orbital has two radial nodes, and so on. In analogy, the 2p orbitals have only a trivial node at r = 0, whereas the higher p orbitals have radial nodes at finite r. The 3d orbitals have a double node at r 2 = 0, but no nodes at finite r and, similarly, the 4f orbitals exhibit only a trivial triple node at r 3 = 0, but no further radial nodes at finite r. The number of radial nodes increases with principal quantum number n. More radial nodes will move the outermost maximum further away from the nucleus [64].…”

“…Consideration of 1s, 2p, 3d, or 4f shells requires special treatment because these orbitals lack a core shell of the same angular momentum and thus do not have a radial node [64]. Nodes are intrinsic features of atomic and molecular wave functions, and they are closely coupled to the orthogonality between the wave functions for different solutions of the Schrödinger equation.…”

“…While it is impossible to exactly solve the Schrödinger equation for multielectron atoms, approximate solutions may be separated conveniently into an angular part consisting of spherical harmonics, and a radial part, which can be solved numerically to any desired accuracy, or approximated as a basis set expansion [64]. The concept of atomic orbitals is based on a single-determinant approximation to the wave function, where the concept of radial nodes is preserved in closed analogy to the one-electron atom.…”

“…The elements H and He are very different from all other elements because of the special properties of the 1s shell [64]. The shielding of nuclear charge by a 1s shell is almost total, so that effects of incomplete screening that lead to relatively strong binding of valence electrons present in 2p, 3d, or 4f shells are virtually absent.…”

“…Bonds involving the resulting nonorthogonal hybrids tend to be relatively weak [67], and it becomes thus more favorable for the particular element to remain in the lower oxidation state. The valence s-character is concentrated in a free (unshared) electron pair at the central atom, and the bonds acquire predominantly the p-character [64].…”