Electron pairs in solid state chemistry

Chemists like the concept of electron pairs since the early days of Lewis. Bonding and nonbonding electron pairs are the basis to argue about shapes of molecules and characteristic coordinations in solids. In the following we tell two short stories demonstrating the limits of that concept.

Figure 1: Structure of the cluster unit $[Th$₁₂N₆ X₄₁]^13-: Capping of an octahedron or condensation of six tetrahedra?

Stabilization of a complex cluster unit by a single multi-center electron pair - this is, in condensed form, the essential type of chemical bonding in the recently discovered cluster compounds $ATh$₁₂N₆X₂₉ (A = Li,...,Rb; X = Cl, Br). While all of them contain a hitherto unknown cluster type that is built up from twelve metal atoms, the geometrical arrangement of the latter may be described (Fig. 1) by two complementary topological models:

Figure 2: The highest occupied molecular orbital (HOMO) of the $\{[Th$₁₂N₆ X₄₁ }^13- cluster unit.

The first one relies on ($i$) six nitrogen-centered thorium tetrahedra $[Th$₄N] which are condensed via edges in a cyclic manner, while the second one focuses on ($ii$) an empty octahedron of thorium atoms $[Th$₆] that is capped by six additional thorium atoms but leaving two opposite faces open. The resulting tetrahedral voids are filled by nitrogen atoms. In any case, this star-shaped cluster is coordinated by a total of 41 halogen atoms in a rather complex way, furthermore interconnecting the individual cluster units in a variety of different and complex modes.

The solid state structures of chlorides and bromides exhibit remarkable differences with respect to both lattice symmetry as well as halide substructure, however, the cluster units [Th₁₂N₆ X₄₁]^13- themselves are almost identical. One reason for this finding is already indicated by a rather primitive electron count for the "ionic limit" according to $A+(Th4+)$₁₂ (N^3-)₆ (X^-)₂₉, yielding a surplus of $(1+48-18-29=)$2 electrons that are available for metal-metal bonding.

Molecular orbital calculations both confirm and further clarify this description. In addition to strongly ionic Th-N and Th-(Cl, Br) bonds, there is a significant contribution of covalent bonding between thorium atoms detectable, and it is associated with an energetically well-separated highest occupied molecular orbital (HOMO) of $a$_1g symmetry. This wave function is depicted in Fig. 2.

It is evident that only $6d$ atomic orbitals from the central thorium atoms mix strongly into this molecular orbital, incorporating the two ``surplus'' electrons and thus stabilizing the entire cluster unit by the formation of a six-center two-electron bond. Besides this multi-center orbital, the electronic structure of these clusters parallels the pattern found in other nitride halide compounds with similar tetrahedral arrangements. The cyclic condensation of the tetrahedra inevitably leads to the formation of this additional orbital (and vice versa), being essential for the stability of these compounds.

Also, an application of topological counting rules illustrates this finding. Such empirical rules are surprisingly successful for the description of molecular cluster compounds (and thus widely used in the chemical literature). They provide simple formulae to estimate so-called ``magic'' numbers for optimum stability of various geometrical fragments. For example, if the central octahedron of the cluster unit is not explicitly accounted for, e.g. by only counting the edges or formally decomposing the cluster into tetrahedra, the total electron count is two electrons lower compared to the description of a capped octahedron. In other words, the recognition of the central octahedron as the host for multi-center metal-metal bonding is a prerequisite for an adequate designation of the chemical bonding in this cluster type.

Figure 3: Left: Coordination of the Jahn-Teller instable monovalent indium cation in In$$₄Br$$₇ at room temperature. Right: Difference fourier synthesis around this atom at low temperature (90 K) with contours in multiples of e/Å$3$.

Stabilization of a complex halide structure via a Jahn-Teller reduction of an excess electron pair's antibonding character--that is, on the other hand, the idea of chemical bonding inside the newly characterized phase $In$₄Br₇.

$In$₄Br₇ is a member of the large family of complex indium halides, materials that are characterized by an astonishing structural variety which may be traced back to the indium atom's multi-valency (mono-, di-, and trivalent indium) and the possibility to mix these valencies in varying combinations in the solid state. The more reduced (bromide) phases deserve special attention since they are all containing In$+$ cations that show fairly uncommon, strongly distorted coordination polyhedra (7-12-fold coordination with strongly widened In$+$-Br$-$ bond lengths). A naive solid-state chemical interpretation would call for "directed electron lone-pairs" being responsible for this finding, however, band structure investigations reveal the cause in a strongly antibonding combination between almost doubly filled, non-directional indium 5s functions and neighboring bromine 4p hybrids. Any coordination polyhedron giving sufficient space for In$+$, aside from repelling bromine-bromine interactions, is almost equally appropriate.

Because of that antibonding excess electron pair in the indium 5s atomic orbital, the overall bonding is weak in reduced indium bromides (and halides in general). The resulting soft crystal potential for In$+$ is nicely reflected in significantly enlarged displacement factors in all known crystal structure investigations. Moreover, energy hypersurface calculations for the In$+$/Br$-$ combination indicate that the latter one is a very good candidate for a second-order Jahn-Teller instability.

The last prediction has now been experimentally verified for the case of trigonal $In$₄Br₇; its crystal-chemical formulation may be described as ($In3+)$₃(In⁺)₅(Br^-)₁₄ = $In$₈Br₁₄. On the macroscopic level, $In$₄Br₇ exhibits an extraordinarily high sensitivity (and chemical reactivity), not only when brought into contact with air or humidity but also when subjected to visible light and mechanical stress, followed by rapid decay.

There are, of course, the overall weak In$+$-Br$-$ interactions to be accounted for this high sensitivity; however, band structure calculations reveal that one out of three positions for monovalent indium (Fig. 3, left), characterized by a 6+6=12-fold coordination by bromine anions, is the source of an unusual electronic instability, thereby being able to localize $In$₄Br₇'s high reactivity to a good extent from crystallography.

The key to understanding lies in the lowest crystal orbital (LUCO, with almost pure $5p$_x,y character) that is centered on this particular monovalent indium cation. Because of the LUCO's small energetic distance from the occupied band levels, it may mix with the highest bands (antibonding indium 5s regime) upon atomic displacement, by that reducing the total energy. Such an effect, equivalent to a second-order Jahn-Teller distortion, manifests at low temperature (90 K) from the measured electron density (Fig. 3, right), showing that indium no longer resides on the crystallographic high-symmetry position but has moved away from it by roughly 40 pm. Note that classifying this effect by assuming a nonbonding ``directed electron lone-pair'' is completely erroneous. On the contrary, the atomic dislocation results from a bonding contribution that slightly weakens the overall antibonding In$+$-Br$-$ interactions close to the Fermi energy.

(Th. P. Braun, A. Simon and R.Dronskowski)

From the yearbook of the institute ("Wissenschaftlicher Tätigkeitsbericht") 1994

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