In this section we will show how the ideas developed in the previous sections may be used to study the electronic structure of symmetrical molecules by considering theπ system of regular cyclobutadiene. Let us point out that the procedure we will follow here will be completely equivalent to the approach that we will follow when determining the orbitals describing the behaviour of
an electron in a periodic system (see Section 3.1 of Chapter 3), or when study- ing related systems like polyacetylene (Chapter 5) or polyacene (Chapter 7).
2.3.1 Decomposition of the ( pz) basis
Let us consider the basis for a reducible representation of dimension 4, (pz)= {2pzi,i =1, . . . ,4}containing the four atomic orbitals involved in the π system of regular cyclobutadiene. We will reduce this basis within the D4h group. To this end, we need to consider the character table for this group (Table 2.3) and decompose the representation (pz). We thus need to calculate the character of this representation for the different symmetry operations of the group. For example, theC4operation does not leave invariant any 2pzi (i =1,2,3,4),as shown in Figs 2.5 and 2.7. The character of(pz) for this operation is 0.
In contrast, the symmetry operationC2a transforms the 2pz2and the 2pz4to their opposites. At the same time it transforms the two orbitals 2pz1 and 2pz3
in−2pz3 and−2pz1, respectively. Consequently, the character of (pz)for the C2a rotation is−2 (Fig. 2.7). Proceeding in this way, we can obtain the characters of the (pz)representation, which are reported in the last line of Table 2.3.
Table 2.3 Character table for theD4h group. The last line contains the characters of the {2pzi,i=1, . . . ,4}representation. By convention, theC2axes and the symmetry planesσv go through two carbon atoms whereas theC2axes and the symmetry planesσdpass in between the atoms.
D4h E 2C4 C2 2C2 2C2 i 2S4 σh 2σv 2σd
A1g 1 1 1 1 1 1 1 1 1 1 x2+y2,z2
A2g 1 1 1 −1 −1 1 1 1 −1 −1
B1g 1 −1 1 1 −1 1 −1 1 1 −1 x2−y2
B2g 1 −1 1 −1 1 1 −1 1 −1 1 xy
Eg 2 0 −2 0 0 2 0 −2 0 0 (xz,yz)
A1u 1 1 1 1 1 −1 −1 −1 −1 −1
A2u 1 1 1 −1 −1 −1 −1 −1 1 1 z
B1u 1 −1 1 1 −1 −1 1 −1 −1 1
B2u 1 −1 1 −1 1 −1 1 −1 1 −1
Eu 2 0 −2 0 0 −2 0 2 0 0 (x,y)
(pz) 4 0 0 −2 0 0 0 −4 2 0
Fig. 2.7
Action of theC4andC2a rotations on the 2pzi (i=1,2,3,4)orbitals.
h Rk
Thus, for instance, nA1g = 1
16(1ã(1ã4)+2ã(1ã0)+1ã(1ã0)+2ã(1ã(−2))+2ã(1ã0) +1ã(1ã0)+2ã(1ã0)+1ã(1ã(−4))+2ã(1ã2)+2ã(1ã0))=0 Repeating the same calculation for every irreducible representation we obtain the decomposition:
(pz)=Eg+A2u+B2u (2.18)
2.3.2 Determination of the basis elements for different irreducible representations
What are the linear combinations of 2pz atomic orbitals that form a basis for the irreducible representationsEg,A2u, andB2u? We will answer this question in two different ways: direct and indirect. The direct approach consists of applying the projectors (see eqn (2.14)) for the D4hgroup. This method can, however, be tedious when the symmetry group possesses a large number of symmetry operations, and an indirect approach may reach equivalent results without the need for long calculations, simply using the character tables.
Direct determination
The linear combinations of 2pzi orbitals of A2u and B2u symmetry may be obtained by application of the projector operatorsPA2u andPB2u (eqn (2.14)) onto, for instance, the 2pz1 function. For theEg representation, the operator PEg must be applied onto two functions, for instance the 2pz1 and 2pz2, to obtain a basis for this doubly degenerate representation. Shown in Table 2.4 are the results of the action of the different symmetry operations of the D4h
group on the 2pz1 and 2pz2 functions.
We can obtain the different (non-normalised) basis elements of the different irreducible representations by means of eqn (2.14):
A2u =NA2u pz1 +pz2+pz3 +pz4
B2u =NB2u pz1−pz2 +pz3−pz4
Eg
1=NEg1 pz1−pz3
Eg
2=NEg2 pz2−pz4
Table 2.4 Action of the different symmetry operations of theD4hgroup on thepz1and pz2 functions.
D4h E C4 C43 C2 C2a C2b C2a C2b
(pz)1 (pz)1 (pz)2 (pz)4 (pz)3 −(pz)3 −(pz)1 −(pz)4 −(pz)2 (pz)2 (pz)2 (pz)3 (pz)1 (pz)4 −(pz)2 −(pz)4 −(pz)3 −(pz)1
D4h i S4 S43 σh σva σvb σda σdb
(pz)1 −(pz)3 −(pz)2 −(pz)4 −(pz)1 (pz)3 (pz)1 (pz)4 (pz)2 (pz)2 −(pz)4 −(pz)3 −(pz)1 −(pz)2 (pz)2 (pz)4 (pz)3 (pz)1
Now we can normalise these functions using, for instance, the H¨uckel approach:
A2u = 1
2 pz1+pz2+pz3+pz4
(2.19a)
B2u = 1
2 pz1−pz2+pz3−pz4
(2.19b)
Eg
1= 1
2 pz1 −pz3
(2.19c)
Eg
2= 1
2 pz2 −pz4
(2.19d)
Indirect determination using the information in the right-hand column of the character table
It is sometimes possible to avoid the use of eqn (2.14) by bearing in mind the information found in the right-hand column of the character table. This information assumes that (i) the coordinate system O x y is centred at the O point, which is invariant after all symmetry operations, and (ii) the z-axis coincides with the main axis of this group. Thus for cyclobutadiene the O point lies at the centre of the cycle and thez-axis is the axis perpendicular to the plane of the molecule. Thex- and they-axes may be taken as going through the carbon atoms labelled 2 and 3 in Fig. 2.4. The information found at the right- hand column of the character tables consists of examples of functions that are bases for different irreducible representations of the group. In particular, the functions f(x,y,z)equal tox2+y2+z2,x,y,z,x y,x z,yz, 2z2−x2−y2, and x2−y2 are always given. These functions possess the same symmetry properties as orbitalsns,npx,npy,npz,ndx y,ndx z,ndyz,ndz2, andndx2−y2, respectively, centred at the point O , which is left invariant by the different symmetry operations of the group.
Consequently, the linear combination of the 2pzi (i =1,2,3,4) orbitals basis of the A2u irreducible representation must exhibit the same symmetry as an npz orbital sitting at the centre of cyclobutadiene. Thus, as shown in Fig. 2.8, it is easy to guess the linear combination of A2u symmetry simply
orbital lying at the centre of butadiene.
These two functions have the same symmetry properties inD4h.
Fig. 2.9
Construction of the linear combinations of 2pzorbitals of regular cyclobutadiene withEgsymmetry using the fact that they must possess the same symmetry properties as twodx zanddyzorbitals lying at the centre of the cycle.
by looking at the symmetry properties of this fictitious 2pzorbital lying at the centre of the cycle.
It is also possible to find the two linear combinations that are a basis of an irreducible representation of the type Eg by considering the symmetry properties ofdx zanddyz orbitals lying at the centre of the cycle (see Fig. 2.9).
It is possible to determine the linear combination ofB2usymmetry, for which we do not have any information in the character table, by imposing the orthog- onality condition with respect to the three functions already determined. Thus, in this favourable example, we can avoid the use of the projectors just by looking at the standard character table. The linear combinations obtained by this procedure may be normalised through the use of the H¨uckel approach and are given in eqn (2.20).
A2u = 1
2 pz1+pz2+pz3+pz4
(2.20a)
B2u = 1
2 pz1−pz2+pz3−pz4
(2.20b)
Eg
1= 1
2 pz1−pz3
(2.20c)
Eg
2= 1
2 pz2−pz4
(2.20d)
These linear combinations are identical to those obtained from the projection operator (see eqn (2.19))6. As usually happens, this indirect approach leads 6
The indirect method could have led to a different but equivalent basis for the doubly degenerate representationEg.
more easily to linear combinations of atomic orbitals adapted to the symmetry of theD4hgroup.
Fig. 2.10
Molecular orbital diagram for theπ system of regular cyclobutadiene.
2.3.3 Molecular orbital diagram of the π system of regular cyclobutadiene
The different linear combinations of 2pzatomic orbitals obtained cannot inter- act among themselves because they are bases of different irreducible represen- tations of the symmetry group of the molecule. These bases may thus be taken as theπmolecular orbitals of regular cyclobutadiene. If we now consider the bonding or antibonding character of the interaction between adjacent atomic orbitals in these molecular orbitals, we can propose the energy diagram in Fig. 2.10. Since the π system of the molecule contains four electrons, the ground-state configuration of the system is (a2u)2(e1g)2. The delocalised π bonds of regular cyclobutadiene are thus described by three molecular orbitals populated with four electrons, which are delocalised through the molecule.
Thus, in this very favourable example, with high symmetry and a small number of orbitals, group theory can provide the detailed expression of the molecular orbitals of theπsystem in a direct and very simple manner .