Closed Metal Cycles in Clusters With Ligands

Homo: Element Metal Cycles With Ligands

The absence of bridging ligands and high symmetry make three - nuclear carbonyls Ru₃(CO)₁₂, Os₃(CO)₁₂ convenient support compounds for structural and theoretical studies of three - membered homo - element metal cycles. In molecules, each metal atom is associated with four functional groups (Figure 1).

Figure 1.

Shema of three - nuclear carbonyls Ru and Os

It is believed that 12 ligands are arranged so that they form an anti – cube – octahedron as a ligand polyhedron (Mason & Rae, 1968; Benfield & Jonson, 1981; Gubin, 2019). However, evidence of this assumption has not yet been provided. The proof of this assertion could be a concrete construction of an anti – cube – octahedron with a three - link metal cycle enclosed in it, connection by valence bonds of the metal cycle atoms to the vertices of the anti – cube - octahedron. After this, it is required to determine the partition of the anti – cube - octahedron with the constructed valence bonds into elementary three - dimensional cells and the verification of the implementation of the Euler – Poincaré (Poincoré, 1895) equation for the constructed polytope. No such evidence was carried out. In this chapter, this question will be considered as part of the proof of the following theorem:

Theorem 1. The geometric model of three nuclear carbonyls Ru and Os consists of two polytopes of dimension 4, touching each other in a two - dimensional section, containing a three nuclear metal cycle.

Proof. Let us assume that carbonyl ligands form a ligand polyhedron in the form of an anti - cube - octahedron. Then the three nuclear metal cycle contained in an anti - cube - octahedron must be connected by vertices of an anti - cube - octahedron by valence bonds. Each vertex of the metal cycle must be connected with the four nearest vertices of the anti - cube - octahedron. In an arbitrary general form will look like that shown in Figure 2.

Figure 2.

The anti - cube - octahedron with the three nuclear metal cycle

At the vertices 12, 13, 14, belonging to the metal cycle, there are metal atoms, and at the 12 vertices of the anti - cube - octahedron functional groups CO are present. The valence bonds of the metal atoms with each other and the functional groups are indicated by red edges. The edges of the anti - cube - octahedron is denoted by solid black lines. Valence bonds and edges of anti - cube - octahedron already create convex three - dimensional bodies

• 1)

  2 - 3 - 4 - 12,

• 2)

  5 - 6 - 15 - 14,

• 3)

  1 - 9 - 13 - 7,

• 4)

  9 - 12 - 13 - 14 - 15,

• 5)

  1 - 2 - 12 - 9 - 13,

• 6)

  3 - 4 - 12 - 11 - 5 - 15,

• 7)

  4 - 12 - 5 - 15 - 14,

• 8)

  2 - 4 - 9 - 14 - 12,

• 9)

  7 - 9 - 14 - 6 - 13 - 15,

• 10)

  anti - cube - octahedron.