Endohedral beryllium atoms in germanium clusters with eight and fewer vertices: how small can a cluster be and still encapsulate a central atom?

Structures of the beryllium-centered germanium clusters Be@Ge(n)(z) (n = 8, 7, 6; z = -4, -2, 0, +2) have been investigated by density functional theory to provide some insight regarding the smallest metal cluster that can encapsulate an interstitial atom. The lowest energy structures of the eight-vertex Be@Ge(8)(z) clusters (z = -4, -2, 0, +2) all have the Be atom at the center of a closed polyhedron, namely, a D(4d) square antiprism for Be@Ge(8)(4-), a D(2d) bisdisphenoid for Be@Ge(8)(2-), an ideal O(h) cube for Be@Ge(8), and a C(2v) distorted cube for Be@Ge(8)(2+). The Be-centered cubic structures predicted for Be@Ge(8) and Be@Ge(8)(2+) differ from the previously predicted lowest energy structures for the isoelectronic Ge(8)(2-) and Ge(8). This appears to be related to the larger internal volume of the cube relative to other closed eight-vertex polyhedra. The lowest energy structures for the smaller seven- and six-vertex clusters Be@Ge(n)(z) (n = 7, 6; z = -4, -2, 0, +2) no longer have the Be atom at the center of a closed Ge(n) polyhedron. Instead, either the Ge(n) polyhedron has opened up to provide a larger volume for the Be atom or the Be atom has migrated to the surface of the polyhedron. However, higher energy structures are found in which the Be atom is located at the center of a Ge(n) (n = 7, 6) polyhedron. Examples of such structures are a centered C(2v) capped trigonal prismatic structure for Be@Ge(7)(2-), a centered D(5h) pentagonal bipyramidal structure for Be@Ge(7), a centered D(3h) trigonal prismatic structure for Be@Ge(6)(4-), and a centered octahedral structure for Be@Ge(6). Cluster buildup reactions of the type Be@Ge(n)(z) + Ge(2) → Be@Ge(n+2)(z) (n = 6, 8; z = -4, -2, 0, +2) are all predicted to be highly exothermic. This suggests that interstitial clusters having an endohedral atom inside a bare post transition element polyhedron with eight or fewer vertices are less than the optimum size. This is consistent with the experimental observation of several types of 10-vertex polyhedral bare post transition element clusters with interstitial atoms but the failure to observe such clusters with external polyhedra having eight or fewer vertices.