TOPICS
sp 3 Hybridization: a scheme for tetrahedral and related species
4.4 Molecular orbital theory: the ligand group orbital approach and
application to triatomic molecules
Despite its successes, the application of valence bond theory to the bonding in polyatomic molecules leads to conceptual difficulties. The method dictates that bonds are localized and, as a consequence, sets of resonance structures and bonding pictures involving hybridization schemes become rather tedious to establish, even for relatively small molecules (e.g. see Figure 4.10c). We therefore turn our attention to molecular orbital (MO) theory.
Molecular orbital diagrams: moving from a diatomic to polyatomic species
As part of our treatment of the bonding in diatomics in Section 1.13, we constructed MO diagrams such as Figures 1.21, 1.27 and 1.28. In each diagram, the atomic orbitals of the two atoms were represented on the right- and left-hand sides of the diagram with the MOs in the middle. Correlation lines connecting the atomic and molecular orbitals were constructed to produce a readily interpretable diagram.
Now consider the situation for a triatomic molecule such as CO2. The molecular orbitals contain contributions from the atomic orbitals of three atoms, and we are presented with a problem of trying to draw an MO diagram involving four sets of orbitals (three sets of atomic orbitals and one of molecular orbitals). A description of the bonding in CF4 involves five sets of atomic orbitals and one set of molecular orbitals, i.e. a six-component problem. Similarly, SF6 is an eight-component problem. It is obvious that such MO diagrams are complicated and, probably, difficult to both construct and interpret. In order to overcome this difficulty, it is common to resolve the MO description of a polyatomic molecule into a three-component problem, a method known as the ligand group orbital (LGO) approach.
MO approach to the bonding in linear XH
2: symmetry matching by inspection
Initially, we illustrate the ligand group orbital approach by considering the bonding in a linear triatomic XH2 in which the valence orbitals of X are the 2s and 2p atomic orbitals. Let us orient the HXH framework so that it coincides with the z axis as shown in Figure 4.11. Consider the two 1s atomic orbitals of the two H atoms. Each 1s atomic orbital has two possible phases and, when the two 1s orbitals are taken as a group, there are two possible phase combinations. These are called ligand group orbitals (LGOs) and are shown at the right-hand side of Figure Chapter 4 . Molecular orbital theory: the ligand group orbital approach and application to triatomic molecules 107
4.11.†Effectively, we are transforming the description of the bonding in XH2 from one in which the basis sets are the atomic orbitals of atoms X and H, into one in which the basis sets are the atomic orbitals of atom X and the ligand group orbitals of an H H fragment. This is a valuable approach for polyatomic molecules.
The number of ligand group orbitals formed¼ the number of atomic orbitals used.
In constructing an MO diagram for XH2(Figure 4.11), we consider the interactions of the valence atomic orbitals of X with the ligand group orbitals of the H H fragment.
Ligand group orbital LGO(1) has the correct symmetry to interact with the 2s atomic orbital of X, giving an MO with HXH -bonding character. The symmetry of LGO(2) is matched to that of the 2pz atomic orbital of X.
The resultant bonding MOs and their antibonding counter-parts are shown in Figure 4.12, and the MO diagram in Figure 4.11 shows the corresponding orbital interactions.
The 2pxand 2pyatomic orbitals of X become non-bonding orbitals in XH2. The final step in the construction of the MO diagram is to place the available electrons in the MOs according to the aufbau principle (see Section 1.9). An important result of the MO treatment of the bonding in Fig. 4.11 Application of the ligand group orbital (LGO) approach to construct a qualitative MO diagram for the formation of a linear XH2molecule from the interactions of the valence orbitals of X (2s and 2p atomic orbitals) and an H H fragment. For clarity, the lines marking the 2p orbital energies are drawn apart, although these atomic orbitals are actually degenerate.
Fig. 4.12 The lower diagrams are schematic representations of the MOs in linear XH2. The wavefunction labels correspond to those in Figure 4.11. The upper diagrams are more realistic representations of the MOs and have been generated computationally using Spartan ’04, #Wavefunction Inc. 2003.
†In Figure 4.11, the energies of the two ligand group orbitals are close together because the H nuclei are far apart; compare this with the situation in the H2molecule (Figure 1.18). Similarly, in Figure 4.17, the LGOs for the H3 fragment form two sets (all in-phase, and the degenerate pair of orbitals) but their respective energies are close because of the large H H separations.
XH2 is that the -bonding character in orbitals 1 and
2 is spread over all three atoms, indicating that the bonding character is delocalized over the HXH frame-work. Delocalized bonding is a general result within MO theory.
MO approach to bonding in linear XH
2: working from molecular symmetry
The method shown above for generating a bonding descrip-tion for linear XH2 cannot easily be extended to larger molecules. A more rigorous method is to start by identifying the point group of linear XH2 as D1h (Figure 4.13a). The D1hcharacter table is used to assign symmetries to the orbi-tals on atom X, and to the ligand group orbiorbi-tals. The MO diagram is then constructed by allowing interactions between orbitals of the same symmetry. Only ligand group orbitals that can be classified within the point group of the whole molecule are allowed.
Unfortunately, although a linear XH2molecule is structu-rally simple, the D1hcharacter table is not. This, therefore, makes a poor first example of the use of group theory in orbital analysis. We can, however, draw an analogy between the symmetries of orbitals in linear XH2and those in homo-nuclear diatomics (also D1h). Figure 4.13b is a repeat of Figure 4.11, but this time the symmetries of the orbitals on atom X and the two ligand group orbitals are given. Com-pare these symmetry labels with those in Figures 1.19 and 1.20. The construction of the MO diagram in Figure 4.13b follows by allowing interactions (bonding or antibonding) between orbitals on atom X and ligand group orbitals with the same symmetry labels.
A bent triatomic: H
2O
The H2O molecule has C2v symmetry (Figure 3.3) and we now show how to use this information to develop an MO picture of the bonding in H2O. Part of the C2v character table is shown below:
C2v E C2 vðxzÞ v’ðyzÞ
A1 1 1 1 1
A2 1 1 1 1
B1 1 1 1 1
B2 1 1 1 1
The inclusion of the xz and yz terms in the last two columns of the character table specifies that the H2O molecule is taken to lie in the yz plane, i.e. the z axis coincides with the principal axis (Figure 4.14). The character table has several important features.
. The labels in the first column (under the point group symbol) tell us the symmetry types of orbitals that are permitted within the specified point group.
. The numbers in the column headed E (the identity operator) indicate the degeneracy of each type of orbital;
in the C2vpoint group, all orbitals have a degeneracy of 1, i.e. they are non-degenerate.
. Each row of numbers following a given symmetry label indicates how a particular orbital behaves when operated upon by each symmetry operation. A number 1 means that the orbital is unchanged by the operation, a 1 means the orbital changes sign, and a 0 means that the orbital changes in some other way.
Fig. 4.13 (a) A linear XH2molecule belongs to the D1hpoint group. Some of the symmetry operations are shown; the X atom lies on a centre of symmetry (inversion centre). (b) A qualitative MO diagram for the formation of linear XH2from atom X and two H atoms.
Chapter 4 . Molecular orbital theory: the ligand group orbital approach and application to triatomic molecules 109
To illustrate its use, let us consider the 2s atomic orbital of the O atom in water:
Apply each symmetry operation of the C2v point group in turn. Applying the E operator leaves the 2s atomic orbital unchanged; rotation about the C2 axis leaves the atomic orbital unchanged; reflections through the vand v’ planes leave the 2s atomic orbital unchanged. These results corre-spond to the following row of characters:
E C2 vðxzÞ v’ðyzÞ
1 1 1 1
and this matches those for the symmetry type A1 in the C2v
character table. We therefore label the 2s atomic orbital on the oxygen atom in water as an a1orbital. (Lower case letters are used for the orbital label, but upper case for the symmetry type in the character table.) The same test is now carried out on each atomic orbital of the O atom. The oxygen 2pxorbital is left unchanged by the E operator and by reflection through the vðxzÞ plane. Each of rotation about the C2 axis and reflection through the v’(yz) plane inverts the phase of the 2px orbital. This is summarized as follows:
E C2 vðxzÞ v’ðyzÞ
1 1 1 1
This matches the row of characters for symmetry type B1in the C2v character table, and the 2px orbital therefore pos-sesses b1 symmetry. The 2py orbital is left unchanged by the E operator and by reflection through the v’(yz) plane, but rotation about the C2 axis and reflection through the
v(xz) plane each inverts the phase of the orbital. This is summarized by the row of characters:
E C2 vðxzÞ v’ðyzÞ
1 1 1 1
This corresponds to symmetry type B2 in the C2vcharacter table, and the 2py orbital is labelled b2. The 2pz orbital is left unchanged by the E operator, by reflection through either of the v(xz) and v’(yz) planes, and by rotation about the C2 axis. Like the 2s orbital, the 2pzorbital there-fore has a1symmetry.
The next step is to work out the nature of the H H ligand group orbitals that are allowed within the C2vpoint group. Since we start with two H 1s orbitals, only two LGOs can be constructed. The symmetries of these LGOs are deduced as follows. By looking at Figure 4.14, you can see what happens to each of the two H 1s orbitals when each symmetry operation is performed: both 1s orbitals are left unchanged by the E operator and by reflection through the v’(yz) plane, but both are affected by rotation about the C2axis and by reflection through the v(xz) plane. This information is summarized in the following row of characters:
E C2 vðxzÞ v’ðyzÞ
2 0 0 2
in which a ‘2’ shows that ‘two orbitals are unchanged by the operation’, and a ‘0’ means that ‘no orbitals are unchanged by the operation’. Next, we note two facts: (i) we can con-struct only two ligand group orbitals, and (ii) the symmetry of each LGO must correspond to one of the symmetry types in the character table. We now compare the row of characters above with the sums of two rows of characters in the C2v character table. A match is found with the sum of the characters for the A1 and B2 representations. As a result, we can deduce that the two LGOs must possess a1 and b2 symmetries, respectively. In this case, it is relatively straightforward to use the a1 and b2 symmetry labels to sketch the LGOs shown in Figure 4.15, i.e. the a1orbital cor-responds to an in-phase combination of H 1s orbitals, while the b2orbital is the out-of-phase combination of H 1s orbi-tals. However, once their symmetries are known, the rigorous method of determining the nature of the orbitals is as follows.
In Figure 4.14, let the two H 1s orbitals be designated as
1 and 2. We now look at the effect of each symmetry operation of the C2v point group on 1. The E operator and reflection through the v’(yz) plane (Figure 4.14) leave
1 unchanged, but a C2rotation and reflection through the
v(xz) plane each transforms 1 into 2. The results are Fig. 4.14 The H2O molecule possesses a C2axis and two v
planes and belongs to the C2v point group.
written down as a row of characters:
E C2 vðxzÞ v’ðyzÞ
1 2 2 1
To determine the composition of the a1 LGO of the H H fragment in H2O, we multiply each character in the above row by the corresponding character for the A1 representation in the C2vcharacter table, i.e.
C2v E C2 vðxzÞ v’ðyzÞ
A1 1 1 1 1
The result of the multiplication is shown in equation 4.10 and gives the unnormalized wavefunction for the a1orbital.
ða1Þ ¼ ð1 1Þ þ ð1 2Þ þ ð1 2Þ þ ð1 1Þ
¼ 2 1þ 2 2 ð4:10Þ
This can be simplified by dividing by 2 and, after normali-zation (see Section 1.12), gives the final equation for the wavefunction (equation 4.11).
ða1Þ ¼ 1 ffiffiffi2
p ð 1þ 2Þ in-phase combination ð4:11Þ Similarly, by using the B2representation in the C2v charac-ter table, we can write down equation 4.12. Equation 4.13 gives the equation for the normalized wavefunction.
ðb2Þ ¼ ð1 1Þ ð1 2Þ ð1 2Þ þ ð1 1Þ
¼ 2 1 2 2 ð4:12Þ
ðb2Þ ¼ 1 ffiffiffi2
p ð 1 2Þ out-of-phase combination ð4:13Þ The MO diagram shown in Figure 4.15 is constructed as follows. Each of the 2s and 2pzorbitals of the O atom pos-sesses the correct symmetry (a1) to interact with the a1orbital of the H H fragment. These orbital interactions must lead to three MOs: two bonding MOs with a1symmetry and one antibonding (a1) MO. On symmetry grounds, the lower energy a1 MO could also include 2pz character, but 2s character dominates because of the energy separation of the 2s and 2pz atomic orbitals. The interaction between the 2pyatomic orbital and the LGO with b2 symmetry leads to two MOs which possess HOH bonding and antibonding Fig. 4.15 A qualitative MO diagram for the formation of H2O using the ligand group orbital approach. The two H atoms in the H2fragment are out of bonding range with each other, their positions being analogous to those in H2O. For clarity, the lines marking the oxygen 2p orbital energies are drawn apart, despite their being degenerate. Representations of the occupied MOs are shown at the right-hand side of the figure. For the a1and b2MOs, the H2O molecule is in the plane of the paper; for the b1MO, the plane containing the molecule is perpendicular to the plane of the paper.
Chapter 4 . Molecular orbital theory: the ligand group orbital approach and application to triatomic molecules 111
character respectively. The oxygen 2px orbital has b1 sym-metry and there is no symsym-metry match with a ligand group orbital. Thus, the oxygen 2pxorbital is non-bonding in H2O.
The eight valence electrons in H2O occupy the MOs according to the aufbau principle, and this gives rise to two occupied HOH bonding MOs and two occupied MOs with mainly oxygen character. (To appreciate this fully, see end of chapter problem 4.12.) Although this bonding model for H2O is approximate, it is qualitatively adequate for most descriptive purposes.