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Electron counting rules and simple bonding descriptions for electron-poor materials

-SiB3

2

Boron – the master of clusters

B4Cl4 B8Cl8B9Br9

Icosahedral clusters in elemental B

-rhombohedral boron

-rhombohedral boron

Bn Clusters in halides and hydrides (boranes)

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Boranes

Hoffmann, R.; Lipscomb, W. N. J. Chem. Phys. 1962, 37, 2872.Wade K. J. Chem. Soc. Chem. Comm. 1971, 792.Wade, K. Inorg. Nucl. Chem. Lett. 1972, 8, 559.

Bonding in boranes

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Constructing MOs:B atoms: The two sets of skeleton bonding combinations (12 basis functions) transform as:T1g, T2g, T1u, T2u

Those combinations correspond already to (triply degenerated) MOs.The two sets of combinations transform as:A1g, Eg, T1u

A1g, Eg, T1u

of which one is skeleton bonding (the set of inward pointing sp hybrid orbitals) and thus already represent MOs.H atoms: One set of type SALCs A1g, Eg, T1u

py

px5

2

3

4

x

y

1

2 345

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H HB

H

ligand bonding

skeleton bonding

Number of electrons: 26Number of basis functions: 30

Point group: Oh

Local coordinate system

Dividing the orbitals:B atoms: two type functions (px and py)

two type orbitals (s, pz or better: two sp hybrid orbitals, one inward and one outward pointing)

H ligand atoms: one type orbital (s)

B6H62-

Use of 12 basis functions and 12 electrons for terminal ligand bonding, six bonding MOs (a1g, eg, t2u). For skeleton bonding 18 basis functions and 14 electrons remain.

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T1g

T2u

T1u

T2g

B- MO diagram B- MO diagram

- and type skeleton MOs with the same symmetry (T1u) interact which leads to a net stabilisation of the borane skeleton.

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Wade’s rules

A closo deltahedral cluster cage (parent poyhedron) with n vertices requires (n+1) pairs of electrons for skeleton bonding.

From a parent closo page with n vertices, a set of more open cages (nido, arachno, hypho) can be derived with a formally unchangedskeleton bonding picture

Thus, for a parent closo deltahedron with n vertices, the related nido-cluster has (n-1) vertices, but still (n+1) skeleton bonding MOs.

Thus, for a parent closo deltahedron with n vertices, the related arachno-cluster has (n-2) vertices, but still (n+1) skeleton bonding MOs.

Thus, for a parent closo deltahedron with n vertices, the related hypho-cluster has (n-3) vertices, but still (n+1) skeleton bonding MOs.

A entity BH in boranes may be replaced by a entity CH (carboranes) or P.

Wade’s rules link cluster geometries to certain electron counts

Alternatively:

Closo deltahedral clusters with n entities (vertices) (BH, CH, P) are stable with (4n+2) electrons.

Nido clusters with n entities are stable with (4n+4) electrons.

Arachno clusters with n entities are stable with (4n+6) electrons.

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-B12Electron counting for -B12

36 electrons per icosahedron

26 for skeleton bonding6 for 2c2e terminal bonding6x2/3 = 4 for 3c2e bonding within layers

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-B12

G. Will et al. (2001)

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-B28Electron counting for -B28

structure unit linkage

B12 4 × 2c2e 8 × 3c2e

26 4 × 2/2 8 × 2/3

B2 2 × 2c2e 4 × 3c2e

2 2 × 2/2 4 × 2/3

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From III-V to II-V semiconductors

Sb

GaSb and ZnSbII III IV V

EN: Sb = 1.7, Ga = 1.7, Zn = 1.6

GaSb ZnSb

EN 0 0.1

Eg [eV] 0.81

direct

0.50

indirect

vec 4 3.5

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Electronic structure of ZnSb

The ZnSb framework has a modest polarity

The optimum electron count is 3.5 e/atom

Non-classical 4c4e bonding within rhomboid rings Zn2Sb2 (localized multicentre bonding)

A. Mikhaylushkin, J. Nylén, U. Häussermann, Chem. Eur. J, 11 (2005), 4912

ZnSb – An electron poor framework semiconductor

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3 [Zn2Sb12/2]

2 [Sb2]

Electronic structure of -Zn4Sb3 (Zn6Sb5 )

Zn6Sb5 =

3 (4 + 4 + 4/2) = 30

2 (4 x 2) = 8

38 e for electron precise conditions (3.454 e/atom);37 e available

electron count

R-3c 36 Zn 18 Sb1 12 Sb2 = Zn36Sb30 (Zn6Sb5 = Zn3.6Sb3)

H. W. Mayer, I. Mikhail, K. Schubert, J. Less-Common Met. 59 (1978), 43.

Less electrons than ZnSb: rhomboid rings condense into chains

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-SiB3

Si42+

B122-

a

b

c

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-boron?