MULLIKEN Symbols

The character tables contain various information, for instance the number and kind of irreducible representations for a specific point group. Usually these irreducible representations are denoted using a scheme suggested by Robert S. Mulliken (1896-1986, awarded with the Nobel prize in 1966) in the early 1930s. Here, an overview on the meaning of Mulliken's symbols is given.

1. The dimension of characters are denoted with one of the following capital letters:

dimension	Mulliken symbol
1	A and B
2	E
3	T
4	G
5	H

In vibrational spectroscopy, F replaces T. The very common groups C_nv, D_nh and D_nd possess only characters of the dimension 1 and 2.

Though renouncing on an austere mathematical definition of the dimension of a character, we want to mention that there is a direct connection to the dimension of a vector space. With reference to quantum mechanics, there is an analogy with the degree of degeneracy. Thus, states characterized as A or B are not degenerated, whereas E and T show a two- and threefold degeneracy respectively. In other words, there are two or three wavefunctions for one single energy state.

Sometimes in this context, the minuscules a, b, e, t,... are used to describe systems with one electron (atomic orbitals, wave functions) whereas A, B, E, T,... characterize multielectronic systems.

2. If C_n represents rotation about the principal axis, the one-dimensional characters are denoted dependent of the value obtained for χ(C_n).

χ(C_n)	denoted as
+1	A
−1	B

A and B thus indicate whether rotation of a wave function about an axis causes the sign to change (B) or to remain constant (A).

3. Indices reflect an additional classification of symmetry. If the molecule possesses an axis C₂ or a plane of reflection σ or σ_d perpendicular to the principal axis C_n, the values for function ψ change or keep their sign and are therefore regarded as symmetric and antisymmetric respectively.

function ψ	Index
sign unaffected	1
change of sign	2

Analogous indices exist for E and T, but the underlying rules are more complicated.

4. Dependent on the effect of inversion i, Mulliken's symbols take the indices g for gerade and u for ungerade.

χ(i)	Index
+1	g
−1	u

5. The way reflection on a horizontal plane affects a function is denoted by primed or doubly primed symbols.

χ(σ_h)	Indicated by
+1	'
−1	"

Example

In the character table of point group O_h, the irreducible representations are denoted by the use of Mulliken's symbols.

O_h E 8C₃ 6C'₂ 6C₄ 3C₂(=C²₄) i 6S₄ 8S₆ 3&sigma_h σ_d

A_1g 1 1 1 1 1 1 1 1 1 1

A_2g 1 1 -1 -1 1 1 -1 1 1 -1

E_g 2 -1 0 0 2 2 0 -1 2 0

T_1g 3 0 -1 1 -1 3 1 0 -1 -1

T_2g 3 0 1 -1 -1 3 -1 0 -1 1

A_1u 1 1 1 1 1 -1 -1 -1 -1 -1

A_2u 1 1 -1 -1 1 -1 1 -1 -1 1

E_u 2 -1 0 0 2 -2 0 1 -2 0

T_1u 3 0 -1 1 -1 -3 -1 0 1 1

T_2u 3 0 1 -1 -1 -3 1 0 1 -1

O_h	E	8C₃	6C'₂	6C₄	3C₂(=C²₄)	i	6S₄	8S₆	3&sigma_h	σ_d
A_1g	1	1	1	1	1	1	1	1	1	1
A_2g	1	1	-1	-1	1	1	-1	1	1	-1
E_g	2	-1	0	0	2	2	0	-1	2	0
T_1g	3	0	-1	1	-1	3	1	0	-1	-1
T_2g	3	0	1	-1	-1	3	-1	0	-1	1
A_1u	1	1	1	1	1	-1	-1	-1	-1	-1
A_2u	1	1	-1	-1	1	-1	1	-1	-1	1
E_u	2	-1	0	0	2	-2	0	1	-2	0
T_1u	3	0	-1	1	-1	-3	-1	0	1	1
T_2u	3	0	1	-1	-1	-3	1	0	1	-1

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