Kimball's tables as a general description
for Symmetry and Bonding in molecules AB_n and AB_mC_n

 
 

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 AB_n with equivalent atoms B, i.e. AB₂ linear (D_∞h), AB₂ angular (C_2v), AB₃ planar (D_3h), AB₃ pyramidal (C_3v), AB₄ planar quadratic (D_4h), AB₄ tetrahedral (T_d), AB₆ octahedral (O_h), AB₈ cubic (O_h), AB₈ antiprismatic (D_4d), AB₈ dodecahedral (D_2d), AB₁₂ icosaedric (I_h). We have omitted the rare cases of AB₄ rectangular (D_2h), AB₄ pyramidal (C_4v), AB₄ diedric (D_2d), AB₆ trigonal prismatic (D_3h).

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 AB_mC_n.

Above, we also presented Kimball's tables for the most important structures AB_mC_n, i.e. AB₂C₂ diedric (C_2v), 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 I_h.)

For a number of symmetry point groups like C_2v, D₂, D_2h 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 AB_N. 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.

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 D_4h or another belonging to conformation T_d 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.

If we assume point group D_4h, there are three bonding orbitals (e_u 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 T_d.

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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