# The first densest lattice of chemical supertetrahedra?

In one of the last issues of the *Angewandte,* Wang and coworkers [1] presented an extremely interesting structure, in which metal-organic polyhedra (MOP) are assembled in different ways. These polyhedra consist of polyoxovanadate metal clusters and bridging dicarboxylate linkers. The authors call them **VMOP**. The overall shape of the MOP can be described as tetrahedron, or to be precise, as truncated tetrahedron see **Fig. 1**.

**Fig. 1**: VMOP, in atomistic representation (left) and simplified as a truncated tetrahedron (right).

In fact, the authors obtained two different phases, which differ in the kind of packing: in the **VMOP-alpha** isomer (a very low-density phase, which is thermodynamically less stable) each truncated tetrahedron makes perfect contact with four neighbors via the (small) four trigonal faces (the truncated faces), thus leading to a **dia**-like framework (corner-connected tetrahedra).

However, in** VMOP-beta** the MOPs are packed in a corner-to-face fashion, *i.e.* each

truncated tetrahedron has contact with eight neighboring tetrahedra (trigonal-to-hexagonal face-to-face-connection), see **Fig. 2**.

**Fig. 2**: The eight direct neighbors of a central truncated tetrahedron (in green) in the packing of **VMOP-beta**.

This packing mode of regular (non-truncated) tetrahedra is known as the densest *lattice* packing (*i.e.* only translations are allowed) of tetrahedra, which was proven in 1969 by Hoylman [2]. The resulting packing density is 18/49 ≈ 36.73 %. Here, each tetrahedron is in contact with 14 others (4 corners + 4 faces + 6 edges). However, in **VMOP-beta** the 6 edge-edge connections are a bit further apart.

I think, the structure of **VMOP-beta** is very remarkable and I am not aware of any analogous chemical structure – do you?

PS: I would like to thank Ahmad Rafsanjani Abbasi (ETH Zürich) for bringing this structure to my attention.

References:

[1] Y. Gong, Y. Zhang, C. Qin, C. Sun, X. Wang, Z. Su, *Angew. Chem. Int. Ed.* **2019**, *58*, 780.

https://doi.org/10.1002/anie.201811027

[2] D.J. Hoylman, *Bull. Amer. Math. Soc.* **1970, ***76,* 135.

https://doi.org/10.1090/S0002-9904-1970-12400-4