KEY POINT Molecular orbitals are constructed from SALCs and atomic orbitals of the same symmetry species.
We have seen in Example 3.10 that the SALC ϕ1 of H1s orbitals in NH3 has A1 symmetry. The N2s and N2pz orbit- als also have A1 symmetry in this molecule, so all three can
EXAMPLE 3.10 Identifying the symmetry species of a SALC Identify the symmetry species of the SALCs that may be constructed from the H1s orbitals of NH3.
Answer We start by establishing how the set of H1s orbitals transforms under the operations of the appropriate symmetry group of the molecule. An NH3 molecule has symmetry C3v and the three H1s orbitals all remain unchanged under the identity operation E.
None of the H1s orbitals remains unchanged under a C3 rotation, and only one remains unchanged under a vertical reflection σv. As a set they therefore span a representation with the characters
E 2C3 3σv 3 0 1
We now need to reduce this set of characters, and by inspection of the character table in Table 3.5 we can see that they correspond to A1+ E (1,1,1 and 2,−1,0). It follows that the three H1s orbitals contribute two SALCs, one with A1 symmetry and the other with E symmetry. The C3v character table has both x and y components in the fourth column for the E symmetry. Therefore, the SALC with E symmetry has two members of the same energy (Fig. 3.21). In more complicated examples the reduction might not be obvious and we use the systematic procedure discussed in Section 3.10.
Self-test 3.10 What is the symmetry label of the SALC ϕ = ψA1s + ψB1s+ ψC1s+ ψD1s in CH4, where ψJ1s is an H1s orbital on atom J?
+
+ +
+
+
– –
– A1
E
FIGURE 3.21 The (a) A1 and (b) E symmetry-adapted linear combinations of H1s orbitals in NH3.
EXAMPLE 3.11 Identifying the symmetry species of SALCs Identify the symmetry species of the SALC φ ψ ψ= ′ − ′′0 0 in the C2v molecule NO2, where ψ ′0 is a 2px orbital on one O atom and ψ ′′0 is a 2px orbital on the other O atom.
Answer To establish the symmetry species of a SALC we need to see how it transforms under the symmetry operations of the group. A picture of the SALC is shown in Fig. 3.22, and we can see that under C2, the SALC, ϕ, changes into itself, implying a character of 1. Under σv, both atomic orbitals change sign, so ϕ is transformed into −ϕ, implying a character of −1. The SALC also changes sign under σ ′v so the character for this operation is also −1. The characters are therefore
E C2 σv σ ′v
1 1 −1 −1
Inspection of the character table for C2v shows that these characters correspond to symmetry species A2.
Self-test 3.11 Identify the symmetry species of the combination ϕ = ψA1s − ψB1s + ψC1s − ψD1s for a square-planar (D4h) array of H atoms A, B, C, D.
C2 N
O
O
2px 2px +
+
– –
FIGURE 3.22 The combination of O2px orbitals referred to in Example 3.11.
contribute to the same molecular orbitals. The symmetry species of these molecular orbitals will be A1, like their com- ponents, and they are called a1 orbitals. Note that the labels for molecular orbitals are lowercase versions of the symme- try species of the orbital. Three such molecular orbitals are possible, each of the form
c c c
1 N2s 2 N2pz 3 1
ψ = ψ + ψ + φ
with ci coefficients that are found by computational meth- ods and can be positive or negative in sign. They are labelled 1a1, 2a1, and 3a1 in order of increasing energy (the order of increasing number of internuclear nodes), and correspond to bonding, nonbonding, and antibonding combinations (Fig. 3.23).
We have also seen (and can confirm by referring to Resource section 5) that in a C3v molecule the SALCs ϕ2 and
ϕ3 of the H1s orbitals have E symmetry. The C3v character table shows that the same is true of the N2px and N2py orbitals (Fig. 3.24). It follows that ϕ2 and ϕ3 can combine with these two N2p orbitals to give doubly degenerate bonding and antibonding orbitals of the form
4 N2p 5 2and 6 N2p 7 3
c c c c
x y
ψ = ψ + φ ψ + φ
These molecular orbitals have E symmetry and are there- fore called e orbitals. The pair of lower energy, denoted 1e, are bonding (the coefficients have the same sign) and the upper pair, 2e, are antibonding (the coefficients have oppo- site sign).
EXAMPLE 3.12 Constructing molecular orbitals from SALCs
The two SALCs of H1s orbitals in the C2v molecule H2O are ϕ1 = ψA1s + ψB1s (18) and ϕ2= ψA1s− ψB1s (19). Which oxygen orbitals can be used to form molecular orbitals with them?
+ +
ΨA1s ΨB1s
18 ϕ1 = ψA1s + ψB1s
+ –
ΨA1s ΨB1s
19 ϕ2 = ψA1s − ψB1s
Answer We start by establishing how the SALCs transform under the symmetry operations of the group (C2v). Under E neither SALC changes sign, so their characters are 1. Under C2, ψ1 does not change sign but ψ2 does; their characters are therefore 1 and −1, respectively. Under σv the combination ψ1 does not change sign but ψ2 does change sign, so their characters are again +1 and −1, respectively. Under the reflection σv′ neither SALC changes sign, so their characters are 1. The characters are therefore
E C2 σv σ ′v
ψ1 1 1 1 1
ψ2 1 −1 −1 1
We now consult the character table and identify their symmetry labels as A1 and B2, respectively. The same conclusion could have been obtained more directly by referring to Resource section 5.
According to the entries on the right of the character table, the O2s and O2pz orbitals also have A1 symmetry; O2py has B2 symmetry. The linear combinations that can be formed are therefore
ψ ψ ψ φ
ψ ψ φ
= + +
= +
c c c
c c
a b
1 1 O2s 2 O2p 3 1
2 4 O2p 5 2
z
y
The three a1 orbitals are bonding, intermediate, and antibonding in character according to the relative signs of the coefficients c1, FIGURE 3.23 The three a1 molecular orbitals of NH3 as computed
by molecular modelling software.
1a1 2a1 3a1
+
+ –
+
+
– –
–
–
FIGURE 3.24 The two bonding e orbitals of NH3 as schematic diagrams and as computed by molecular modelling software.
c2, and c3, respectively. Similarly, depending on the relative signs of the coefficients c4 and c5, one of the two b2 orbitals is bonding and the other is antibonding.
Self-test 3.12 The four SALCs built from Cl3s orbitals in the square planar (D4h) [PtCl4]2− anion have symmetry species A1g, B1g, and Eu. Which Pt atomic orbitals can combine with which of these SALCs?
A symmetry analysis has nothing to say about the ener- gies of orbitals other than to identify degeneracies. To calculate the energies, and even to arrange the orbitals in order, it is necessary to use quantum mechanics; to assess them experimentally it is necessary to use techniques such as photoelectron spectroscopy. In simple cases, however, we can use the general rules set out in Section 2.8 to judge the relative energies of the orbitals. For example, in NH3, the 1a1 orbital, containing the low-lying N2s orbital, can be expected to lie lowest in energy, and its antibonding part- ner, 3a1, will probably lie highest, with the nonbonding 2a1 approximately halfway between. The 1e bonding orbital is next-higher in energy after 1a1, and the 2e correspondingly lower in energy than the 3a1 orbital. This qualitative analy- sis leads to the energy-level scheme shown in Fig. 3.25.
These days, there is no difficulty in using one of the widely available software packages to calculate the energies of the orbitals directly by either an ab initio or a semi-empirical procedure; the relative energies shown in Fig. 3.25 have in fact been calculated in this way. Nevertheless, the ease of achieving computed values should not be seen as a rea- son for disregarding the understanding of the energy-level order that comes from investigating the structures of the orbitals.
The general procedure for constructing a molecular orbital scheme for a reasonably simple molecule can now be summarized as follows:
1. Assign a point group to the molecule.
2. Look up the shapes of the SALCs in Resource section 5.
3. Arrange the SALCs of each molecular fragment in increasing order of energy, first noting whether they stem from s, p, or d orbitals (and putting them in the order s < p < d), and then their number of internuclear nodes.
4. Combine SALCs of the same symmetry type from the two fragments, and from N SALCs form N molecular orbitals.
5. Estimate the relative energies of the molecular orbitals from considerations of overlap and relative energies of the parent orbitals, and draw the levels on a molecular orbital energy-level diagram (showing the origin of the orbitals).
6. Confirm, correct, and revise this qualitative order by car- rying out a molecular orbital calculation by using appro- priate software.