Predicting structures of cluster compounds Polyhedral skeletal electron pair theory

1 predicting structures of cluster compounds

1.1 4n rules
1.2 5n rules
1.3 6n rules
1.4 isolobal vertex units

predicting structures of cluster compounds

different rules (4n, 5n, or 6n) invoked depending on number of electrons per vertex.

the 4n rules reasonably accurate in predicting structures of clusters having 4 electrons per vertex, case many boranes , carboranes. such clusters, structures based on deltahedra, polyhedra in every face triangular. 4n clusters classified closo-, nido-, arachno- or hypho-, based on whether represent complete (closo-) deltahedron, or deltahedron missing 1 (nido-), 2 (arachno-) or 3 (hypho-) vertices.

however, hypho clusters relatively uncommon due fact electron count high enough start fill antibonding orbitals , destabilize 4n structure. if electron count close 5 electrons per vertex, structure changes 1 governed 5n rules, based on 3-connected polyhedra.

as electron count increases further, structures of clusters 5n electron counts become unstable, 6n rules can implemented. 6n clusters have structures based on rings.

a molecular orbital treatment can used rationalize bonding of cluster compounds of 4n, 5n, , 6n types.

the structure of butterfly cluster [re4(co)12] conforms predictions of psept.


4n rules

the following polyhedra closo polyhedra, , basis 4n rules; each of these have triangular faces. number of vertices in cluster determines polyhedron structure based on.

using electron count, predicted structure can found. n number of vertices in cluster. 4n rules enumerated in following table.

pb2−

10

when counting electrons each cluster, number of valence electrons enumerated. each transition metal present, 10 electrons subtracted total electron count. example, in rh6(co)16 total number of electrons 6 × 9 + 16 × 2 − 6 × 10 = 86 – 6 × 10 = 26. therefore, cluster closo polyhedron because n = 6, 4n + 2 = 26.

s2+

4

other rules may considered when predicting structure of clusters:

in general, closo structures n vertices n-vertex polyhedra.

to predict structure of nido cluster, closo cluster n + 1 vertices used starting point; if cluster composed of small atoms high connectivity vertex removed, while if cluster composed of large atoms low connectivity vertex removed.

to predict structure of arachno cluster, closo polyhedron n + 2 vertices used starting point, , n + 1 vertex nido complex generated following rule above; second vertex adjacent first removed if cluster composed of small atoms, second vertex not adjacent first removed if cluster composed of large atoms.

os6(co)18, carbonyls omitted

example: pb2−

10

electron count: 10 × pb + 2 (for negative charge) = 10 × 4 + 2 = 42 electrons.
since n = 10, 4n + 2 = 42, cluster closo bicapped square antiprism.

example: s2+

4

electron count: 4 × s – 2 (for positive charge) = 4 × 6 – 2 = 22 electrons.
since n = 4, 4n + 6 = 22, cluster arachno.
starting octahedron, vertex of high connectivity removed, , non-adjacent vertex removed.

example: os6(co)18

electron count: 6 × os + 18 × co – 60 (for 6 osmium atoms) = 6 × 8 + 18 × 2 – 60 = 24
since n = 6, 4n = 24, cluster capped closo.
starting trigonal bipyramid, face capped. carbonyls have been omitted clarity.


b

5h4−

5, hydrogens omitted

example: b

5h4−

5

electron count: 5 × b + 5 × h + 4 (for negative charge) = 5 × 3 + 5 × 1 + 4 = 24
since n = 5, 4n + 4 = 24, cluster nido.
starting octahedron, 1 of vertices removed.

the rules useful in predicting structure of carboranes. example: c2b7h13

electron count = 2 × c + 7 × b + 13 × h = 2 × 4 + 3 × 7 + 13 × 1 = 42
since n in case 9, 4n + 6 = 42, cluster arachno.

the bookkeeping deltahedral clusters carried out counting skeletal electrons instead of total number of electrons. skeletal orbital (electron pair) , skeletal electron counts 4 types of deltahedral clusters are:

n-vertex closo: n + 1 skeletal orbitals, 2n + 2 skeletal electrons
n-vertex nido: n + 2 skeletal orbitals, 2n + 4 skeletal electrons
n-vertex arachno: n + 3 skeletal orbitals, 2n + 6 skeletal electrons
n-vertex hypho: n + 4 skeletal orbitals, 2n + 8 skeletal electrons

the skeletal electron counts determined summing total of following number of electrons:

2 each bh unit
3 each ch unit
1 each additional hydrogen atom (over , above ones on bh , ch units)
the anionic charge electrons

5n rules

as discussed previously, 4n rule deals clusters electron counts of 4n + k, in approximately 4 electrons on each vertex. more electrons added per vertex, number of electrons per vertex approaches 5. rather adopting structures based on deltahedra, 5n-type clusters have structures based on different series of polyhedra known 3-connected polyhedra, in each vertex connected 3 other vertices. 3-connected polyhedra duals of deltahedra. common types of 3-connected polyhedra listed below.

5n cluster: p4



5n + 3 cluster: p4s3



5n + 6 cluster: p4o6

the 5n rules follows.

example: p4

electron count: 4 × p = 4 × 5 = 20
it 5n structure n = 4, tetrahedral

example: p4s3

electron count 4 × p + 3 × s = 4 × 5 + 3 × 6 = 38
it 5n + 3 structure n = 7. 3 vertices inserted edges

example: p4o6

electron count 4 × p + 6 × o = 4 × 5 + 6 × 6 = 56
it 5n + 6 structure n = 10. 6 vertices inserted edges

6n rules

as more electrons added 5n cluster, number of electrons per vertex approaches 6. instead of adopting structures based on 4n or 5n rules, clusters tend have structures governed 6n rules, based on rings. rules 6n structures follows.

s8 crown

example: s8

electron count = 8 × s = 8 × 6 = 48 electrons.
since n = 8, 6n = 48, cluster 8-membered ring.


6n + 2 cluster: hexane

hexane (c6h14)

electron count = 6 × c + 14 × h = 6 × 4 + 14 × 1 = 38
since n = 6, 6n = 36 , 6n + 2 = 38, cluster 6-membered chain.

isolobal vertex units

provided vertex unit isolobal bh can, in principle @ least, substituted bh unit, though bh , ch not isoelectronic. ch unit isolobal, hence rules applicable carboranes. can explained due frontier orbital treatment. additionally there isolobal transition-metal units. example, fe(co)3 provides 2 electrons. derivation of briefly follows:

fe has 8 valence electrons.
each carbonyl group net 2 electron donor after internal σ- , π-bonding taken account making 14 electrons.
3 pairs considered involved in fe–co σ-bonding , 3 pairs involved in π-backbonding fe co reducing 14 2.




^ cotton, f. albert; wilkinson, geoffrey; murillo, carlos a.; bochmann, manfred (1999), advanced inorganic chemistry (6th ed.), new york: wiley-interscience, isbn 0-471-19957-5 
^ cotton, albert (1990). chemical applications of group theory. john wiley & sons. pp. 205–251. isbn 0-471-51094-7.