Kimball's tables as a general description
for Symmetry and Bonding in molecules ABn and ABmCn



 
 
Table 1: Irreducible representations for atomic orbitals of the central atom A and ligands B within molecules ABn and ABnCm according to Kimball.
1. AB2 linear
D∞h σg σu πg πu δg δu

A
 
s
p
d
1
0
1
0
1
0
0
0
1
0
1
0
0
0
1
0
0
0
B2 σ
π
1
0
1
0
0
1
0
1
0
0
0
0
2. AB2 angular (atoms B in plane yz, atom A within axis z)
C2v a1 a2 b1 b2

A
 
s
p
d
1
1
2
0
0
1
0
1
1
0
1
1
B2 σ
π
1
1
0
1
0
1
1
1
3. AB3 planar, symmetric
D3h a1' a1'' a2' a2'' e' e''

A
 
s
p
d
1
0
1
0
0
0
0
0
0
0
1
0
0
1
1
0
0
1
B3 σ
π
1
0
0
0
0
1
0
1
1
1
0
1
4. AB3 pyramidal
C3v a1 a2 e
A s
p
d
1
1
1
0
0
0
0
1
2
B3 σ
π
1
1
0
1
1
2
5. AB4 tetrahedral
Td a1 a2 e t1 t2
A s
p
d
1
0
0
0
0
0
0
0
1
0
0
0
0
1
1
B4 σ
π
1
0
0
0
0
1
0
1
1
1
6. AB4 planar tetragonal
D4h a1g a1u a2g a2u b1g b1u b2g b2u eg eu

A
 
s
p
d
1
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
B4 σ
π
1
0
0
0
0
1
0
1
0
1
0
1
1
0
0
0
0
1
1
1
7. AB6 octahedral
Oh a1g a1u a2g a2u eg eu t1g t1u t2g t2u
A s
p
d
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
B6 σ
π
1
0
0
0
0
0
0
0
1
0
0
0
0
1
1
1
0
1
0
1
8. AB8 cubic
Oh a1g a1u a2g a2u eg eu t1g t1u t2g t2u
A s
p
d
f
1
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
1
B8 σ
π
1
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
1
0
1
9. AB8 tetragonal antiprismatic
D4d a1 a2 b1 b2 e1 e2 e3

A
 
s
p
d
1
0
1
0
0
0
0
0
0
0
1
0
0
1
0
0
0
1
0
0
1
B8 σ
π
1
1
0
1
0
1
1
1
1
2
1
2
1
2
10. AB8 dodecahedral
D2d a1 a2 b1 b2 e
A s
p
d
1
0
1
0
0
0
0
0
1
0
1
1
0
1
1
B8 σ
π
2
2
0
2
0
2
2
2
2
4
11. AB12 icosaedric
Ih ag au t1g t1u t2g t2u ug uu vg vu
A s
p
d
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
B12 σ
π
1
0
0
0
0
1
1
1
0
0
1
0
0
1
0
1
1
1
0
1
12. AB2C2 diedric,     (atoms B within plane yz, atoms C within plane xz, atom A on axis z)
C2v a1 a2 b1 b2
A s
p
d
1
1
2
0
0
1
0
1
1
0
1
1
B2 σ
π
1
1
0
1
0
1
1
1
C2 σ
π
1
1
0
1
1
1
0
1
13. AB4C tetragonal pyramidal
C4v a1 a2 b1 b2 e
A s
p
d
1
1
1
0
0
0
0
0
1
0
0
1
0
1
1
B4 σ
π
1
1
0
1
0
1
1
1
1
2
C σ
π
1
0
0
0
0
0
0
0
0
1
14. AB3C2 trigonal bipyramidal
D3h a1' a1'' a2' a2'' e' e''
A s
p
d
1
0
1
0
0
0
0
0
0
0
1
0
0
1
1
0
0
1
B3
 
σ
π
1
0
0
0
0
1
0
1
1
1
0
1
C2 σ
π
1
0
0
0
0
0
1
0
0
1
0
1
Table published by G.E. Kimball in J. Chem. Phys. 8, 188 (1940) and corrected by W. Kutzelnigg.

The table above presents the Kimball's tables for the most important molecules ABn with equivalent atoms B, i.e. AB2 linear (D∞h), AB2 angular (C2v), AB3 planar (D3h), AB3 pyramidal (C3v), AB4 planar quadratic (D4h), AB4 tetrahedral (Td), AB6 octahedral (Oh), AB8 cubic (Oh), AB8 antiprismatic (D4d), AB8 dodecahedral (D2d), AB12 icosaedric (Ih). We have omitted the rare cases of AB4 rectangular (D2h), AB4 pyramidal (C4v), AB4 diedric (D2d), AB6 trigonal prismatic (D3h).

For molecules like AB5 and AB7, there are no structures with equivalent atoms B. In a few cases, the geometry of these molecules can be described as a planar and regular polygon. In a molecule AB5, we usually have more than one class of atoms B, in consequence, these molecules could be preferentially classified as ABmCn.

Above, we also presented Kimball's tables for the most important structures ABmCn, i.e. AB2C2 diedric (C2v), AB4C tetragonal pyramidal and AB3C2 trigonal bipyramidal.

When using Kimball's tables, keep the aspect of degeneracy in mind. An entry 1 for a representation e indicates two, an entry of 1 for a representation t three atomic orbitals of one energy. (Higher degenerated representations, i.e. fourfold for u and fivefold for v exist only for molecules AB12 of the icosaedric group Ih.)

For a number of symmetry point groups like C2v, D2, D2h it is not clear how a molecule is orientated respective the symmetry elements. In these cases, introduce a Cartesian coordinate system, classify atoms B and C with respect to it. Give an explicit record of your definitions to avoid misunderstandings.

It is amazing, how much information on molecular orbitals can be deduced just from symmetry aspects without calculation. But, a difficulty is to decide which atomic orbitals are involved in bonding. And, making affairs even more complicated, such involvement is not a question of yes or no. Such a reduction proved to be inadequate, especially in the case of d-orbitals of silicium, phosphor, chlorine, etc. The alternative either to transform into a bonding molecular orbital or, to remain nonbonding sometimes turns, due to energy or shape, to a gradual involvement which results in a weakly bonding effect.

Although there are differences between true molecular orbitals and an LCAO-approach using linear combinations of atomic orbitals, there is no flaw for symmetry considerations. We have seen this for diatomic molecules and this is still true for any molecule ABN. Although the LCAO method is fundamental for Kimball's classification of orbitals with respect of symmetry species, the LCAO method underlies more constraints than the information appearing in Kimball's tables.

As we prefer not to spoil a consistent scheme using questionable approaches like a one electron function or a more or less exact matrix element description, we will not introduce the quite exhaustive calculations aiming on values for energy splitting and, eventually, the correct sequence of molecular orbital energies. Let us stress that we only applied the building-up principle whichs states that the orbitals of lowest energy are occupied first and that no reference was made to the total energy of the system or the sum of energies of the present molecular orbitals.

Table 2: Symmetry classification of atomic orbitals of the central atom A for molecules of a given point group.
  D∞h C2v D3h C3v D4h C4v Td Oh Ih
s σg a1 a1' a1 a1g a1 a1 a1g ag
px πu b1 e' e eu e t2 t1u t1u
py πu b2 e' e eu e t2 t1u t1u
pz σu a1 a2'' a1 a2u a1 t2 t1u t1u
dxy δg a2 e' e b1g b1 t2 t2g vg
dyz πg b2 e'' e eg e t2 t2g vg
dxz πg b1 e'' e eg e t2 t2g vg
d σg a1 a1' a1 a1g a1 e eg vg
dx²−y² δg a1 e' e b2g b2 e eg vg

Working with Kimball's tables, we just use the irreducible representation of symmetry adapted linear combinations of orbitals of the involved atoms. No assumption on the shape of these LCAOs is made. In Table 10 "Symmetrieadaptierte Linearkombinationen" of his textbook "Einführung in die theoretische Chemie", Werner Kutzelnigg presents how some of these structures are expressed as linear combination. The respective terms are prerequisite to all further calculations.

So far, we assumed the structure of a molecule as given and did not ask for alternatives. Next we will show how Kimball's tables help to find out the most stable geometry for a molecule. For example, we could imagine methane CH4 to be a planar square or a tetrahedron. We have to ask whether a methane molecule belonging to point group D4h or another belonging to conformation Td is of lower energy. For the two cases, one particular series of molecular orbitals is obtained. (The background color signifies the role for molecular bonding: Blue for bonding, green for non-bonding and red for anti-bonding.

D4h
1a1g 2a1g
1a2u
1b2g
1eu 2eu
T4
1a1 2a1
1t2 2t2

If we assume point group D4h, there are three bonding orbitals (eu is doubly present), two non bonding and three antibonding. In contrast, four bonding (t2 is threefold present) and four antibonding orbitals are expected for a methane belonging to Td.

If we distribute the eight valence electrons, an electronic configuration with four full bonding orbitals is obtained for the CH4 tetrahedron. In a square molecule, the electrons would be harboured in three bonding and one nonbonding orbital. Therefore, we expect the tetrahedron to be the stable conformation of methane.

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