2.2 A short review of the theory of symmetry
2.2.3 Basis for an irreducible representation
We will now outline the notions ofbasis for a representation, representation, and irreducible representationof a groupG. These notions will be illustrated by using the symmetry properties of the 2pz atomic orbitals of theπ system of regular cyclobutadiene. The symmetry group of regular cyclobutadiene is D4hwhich contains sixteen symmetry operations denoted{Rj,j =1, . . . ,16}
(h=16).
Notion of basis for a representation
Let us now consider a set of functions f = {f1, f2, . . . , fn}, such that the action of any of the symmetry operations of the group G transforms any of the functions, fi, in a linear combination of the different functions of the f set. Such a set is said to beglobally stableunder the action of the symmetry operations of G and constitutes a basis for a representation of the group G.
From a physical point of view a basis for a representation contains functions equivalent by symmetry. Thus, for instance, the four 2pzorbitals of the carbon atoms of regular cyclobutadiene constitute a basis for a representation of the
D4hgroup as well as of all subgroups ofD4h. Representation of a group
Fig. 2.4
(2pz)iorbitals of the four carbon atoms (i= 1, 2, 3, 4) of cyclobutadiene.
We will now consider the set of matrices M
k representing the action of the symmetry operations Rk (k=1, . . . ,h) on the basis f = {f1, f2, . . . , fn}.
This set of matrices, {M1,M
2, . . . ,M
h}, is called a representation of the group and is denoted. In the case of the basis {2pz1,2pz2,2pz3,2pz4}of the D4h group of regular cyclobutadiene (see Figs 2.1, 2.3, and 2.4), one representation of the group consists of sixteen 4×4 matrices representing the action of the different symmetry operations of the D4h group on the set {2pz1,2pz2,2pz3,2pz4}. Thus, for instance, the matrix associated with rotationC4is that shown in Fig. 2.5.
Fig. 2.5
Matrix representing the action of theC4 rotation on the basis
{2pz1,2pz2,2pz3,2pz4}.
The number of representations of the group is infinite. For instance, starting with one of the bases, f, any unitary matrix U can be used to define another basis, f= {f1, f2, . . . , fn}such that f=U f. The matrix associated with operation Rk in this new basis, noted M
k, is equal to UM
kU−1. The set {M1,M
2, . . . ,M
h}is a new representation of the group.
From a physical perspective, these two representations of the group have the same meaning: we thus need to find some common property characterising both of them. This common property is the trace, i.e. the sum of the diagonal elements of the different matrices M
k. We will denote the trace of the M matrix as χ(Mk),which in group theory is known as the characterof thek
matrix. Using the fact that the products of matricesA BandB Ahave the same trace, it is easy to show that the matricesM
kandM
kalso have the same trace:
The set of numbers{χ(M1), χ(M2), . . . , χ(Mh)}characterises in a unique way the representation{M1,M
2, . . . ,M
h}as well as any physically equivalent representation,{M
1,M
2, . . . ,M
h}. Now that we have learnt how to charac- terise a set of physically equivalent functions by means of their characters, we can tryto find linear combinations of the functions{f1, f2, . . . , fn}adapted by themselves to the symmetry of the group.
Irreducible representation
Let us assume that there is a change of basis allowing the generation of an equivalent basis{f1,f2, . . . , fn}that may be decomposed in several basesi, every one of which is stable with respect to all the symmetry operations of the group. In that case we may write the representationas a direct sum of representations with lower dimension,i:
=1⊕2⊕. . .⊕m (2.12) We can now say that we have decomposed the reducible representation into several representationsi. We can then try to decompose these lower- dimension representations,i. When there is no change of basis allowing the decomposition of the basisiinto several lower-dimension bases, it is said that iis a basis for anirreducible representation. As we will briefly review, group theory makes clear some very useful mathematical properties of irreducible representations.
Properties of irreducible representations
Let us now outline the main properties of irreducible representations of a symmetry group G of order h. In Section 2.3 we will use all these results to study theπ-type orbitals of regular cyclobutadiene.
(i) The number of irreducible representations is equal to the number of classes of the group. For instance, the D4h group has ten irreducible representations.
(ii) Matrices associated with symmetry operations of the same class have the same character.
(iii) A reducible representation, , may be decomposed into the different irreducible representations of the group,i, in the following way:
=n11⊕n22⊕. . .⊕nmm
with ni = 1 h
Rk
χi∗(Rk)χ(Rk) (2.13) whereχi(Rk)andχ(Rk)(k=1, . . . ,h) are respectively the characters of the representationsiandfor the symmetry operationsRkof the group
G. The symbol * in eqn (2.13) means that we must take the complex conjugate of χi(Rk)if the character is complex. This formula requires knowledge of the characters of the representation. This calculation is generally very simple.
(iv) Let us assume that the basis{f1, f2, . . . , fn}may be projected onto the irepresentation, i.e.ni =0 in eqn (2.13). Then, a linear combination of {f1, f2, . . . , fn}may result from the action of the projection operatorPi
on one fj function:4
4Here we use a non-normalised opera- tor. The normalisation of the symmetry- adapted functions will be later carried out in the H¨uckel approach. Note that this operator may be non-trivial for a degen- erate representation. [4]
Pi(fj)=
Rk
χi∗(Rk)Rk(fj) (2.14) Application of this formula requires knowledge of how every symmetry operation Rkacts on the function fj.This calculation is not difficult but may be quite tedious.
(v) Ifφiàandφνj are two basis elements of two different irreducible repre- sentations, à and ν, of the symmetry group of a molecule andh isˆ the appropriate one-electron Hamiltonian, the following terms are nil by symmetry:
φià| ˆh|φνj
=0 and
φià|φνj
=0 (2.15)
The functionsφiàandφνj do not interact by symmetry and possess a nil overlap if they generate different irreducible representationsàandν. Application to the determination of the molecular orbitals of a molecule It is essential to bear in mind that any molecular orbital is an element of a basis for an irreducible representation of the symmetry group of the molecule.
As a result, it is possible to find the expression of a molecular orbital using group theory through the following procedure:
(i) Determine the symmetry group of the molecule using Table 2.2.
(ii) List the different atomic orbitals that participate in the relevant molecular orbitals of the system.
(iii) Group the different atomic orbitals into sets of symmetry-equivalent atomic orbitals, i.e. into different bases for a representation.
(iv) Decompose such representations (see eqn (2.13)).
(v) Find a basis for every irreducible representationi (see eqn (2.14)); the different basis elementsφi j (j =1, . . . ,ni)of an irreducible representa- tioniare linear combinations of atomic orbitals adapted to the symmetry of the molecule.
(vi) The molecular orbitals that are a basis for the irreducible representation i result from the interaction of the ni functions φi j (j=1, . . . ,ni).
They may be obtained by solving the secular equations (eqns (2.8) and (2.4)) by considering solely theni functionsφi j (j =1, . . . ,ni)of this symmetry. The functionsφi j now play the role of theχj orbitals in Sec- tion 2.1.3. The same procedure must be followed for all other irreducible representations.
Fig. 2.6
Example of a symmetry lowering.
Symmetry lowering
From the mathematical point of view, a group may be a subgroup of another group of higher order. For instance, groupC4v is a subgroup ofD4h,which itself is a subgroup ofOh:
C4v⊂D4h ⊂Oh (2.16)
A symmetry lowering5corresponds to the passage from a group to one of its 5
When discussing symmetry in com- plexes like these, we will refer to thelocal symmetry around the transition metal atom. In other words, the actual geom- etry of the H2O ligands will be ignored and we will consider the geometry of the MO6group of atoms.
subgroups. This corresponds to the passage from an object with a given sym- metry to another one with a lower symmetry, i.e. one that is invariant through a smaller number of symmetry operations. Thus, an example of the change from OhtoD4hand finally toC4vis provided by the hypothetical distortion of the octahedral compound Cu(H2O)26+through the bond elongation of two ligands (OhtoD4h) and the departure of one of the two apical ligands (D4h toC4v), as shown in Fig. 2.6.
It is important to point out that an irreducible representation for a groupG is a representation that might bereduciblefrom any subgroupSG. Any basis for a representation of the group G is necessarily stable with respect to the symmetry operations of the subgroupSG.