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Grouptheorynotes.2.16.05
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Group Theory and the Molecular Orbitals of the water molecule.. . The bonding in the water
molecule can be approached from VSEPR as shown below, which suggests that it should have
two lone pairs of electrons, and an H-O-H angles of less than 109.5o because of the strong
repulsion of the lone pairs:. . . . The parent geometry for two bonds plus two electron pairs (n =
4) is tetrahedral. In this model, two of the coordination sites on the O-atom are occupied by Hatoms, and the other two are occupied by lone pairs:. . . . A simple qualitative MO view of this
bonding would have the 1s orbitals of the two H atoms overlapping with the sp3 hybrid orbitals
of the O-atom. This is shown on the next page in Figure 1. One sees that two of the sp3 hybrid
orbitals of the O-atom are involved in overlap with the 1s orbitals of the two H-atoms, while the
other two are left as n (non-bonding) orbitals, which are the lone pairs of the Lewis dot/VSEPR
structure above. . . This view of the water molecule is quite useful, but we can carry out a more
sophisticated analysis based on MO theory. We can use Hyperchem to calculate the energy
levels for the molecular orbitals for the H2O molecule, and then relate this quantitative MO
diagram to the character table for the C2v point group, to which H2O belongs.. . . Figure 1.
Simple qualitative MO diagram representing the overlap of the 1s orbitals with two sp3 hybrid
orbitals on oxygen , with the other two sp3 hybrid orbitals being the lone pairs on the oxygen
atom. These are indicated as n (non-bonding) orbitals.. . The MO diagram as calculated for the
H2O molecule using the PM3 semi-empirical approach is seen in Figure 2. Figure 2 shows six
orbitals, just as seen in the qualitative MO diagram for water in Fig. 1. However, we notice that
the six levels are given labels, which correspond to the symmetry labels in the character table for
the C2v point group (Figure 3). These labels derive from the effects of symmetry operations in
the character table for the C2v point group on the atomic orbitals on the O-atom (2s and 2px,
2py, 2pz) and the two H-atoms ( two 1s orbitals) that might participate significantly in the
bonding. The 1s orbital on the O-atom will participate very slightly, but we do not consider it
here.. . Use of the character Table for a molecule allows one to determine which atomic orbitals
will overlap significantly with each other to produce the bonding in the resulting molecule.
Significant overlap only occurs between atomic orbitals with the same symmetry labels. First, we
put all the atomic orbitals on oxygen and on the two H-atoms through the symmetry operations
indicated in the character Table. If the sign of the wavefunction of the orbitals does not change
when the symmetry operation is carried out, then this corresponds to a ‘1’ in the
representations in the column below that symmetry operation. If the sign changes, then this
corresponds to a ‘-1’ in the column of representations.. . . Figure 2. Quantitative energy
levels for the MO’s of the water molecule, calculated using the PM3 semi-empirical basis set
in Hyperchem.. . . Figure 3. Character table for the C2v point group.. . . . . Symmetry properties
of the atomic orbitals contributing to bonding in the H2O molecule:. . Consider the 2s orbital of
the oxygen atom. We ignore the E (identity) operation column, as this is always 1. It does
nothing, so it has no effect. The C2 operation, and the σv(xz) (Figure 4) and σv(yz) operations
(not shown) do not change the sign of the wavefunction on the 2s orbital:. . . . a). . . . . b). Figure
4. The effect on the 2s orbital (a) and the 2pz orbital (b) of oxygen of the symmetry operations
from the character Table for C2v. Note that these do not change the sign of the wavefunction of
the 2s or 2pz orbital, which shows that the symmetry of these orbitals is a1.. . . a). . . b). Figure 5.
The effect on a) the 2py and b) the 2px orbital of oxygen of the symmetry operations from the
character Table for C2v. Note that for the 2py the C2 and σv(xz) operation changes the sign of
the wavefunction, so the symmetry is b2. For the 2px the C2 operation changes the sign, but the
σv(xz) does not, which shows that the symmetry of this orbital is b1.. The 1s orbitals of the two
H atoms may similarly be assigned symmetries from the character table. The two possibilities are
that the 1s orbitals from both H atoms will have the same sign of the wavefunction, or opposite
signs. In Figure 6 is shown the C2 symmetry operation on the two H 1s orbitals where they have
the same sign. The reader can work out that the σv(xz) operation will also leave the sign of the
wavefunctions unchanged. The symmetry for the situation where the sign of the two
wavefunctions of the two H-atoms have the same sign has a1 symmetry.. . Figure 6. Effect of the
C2 rotational operation on the signs of the wavefunctions of two H 1s orbitals having
wavefunctions of the same sign. The reader can work out that this pairing of H 1s orbitals has a1
symmetry.. . Where the two H 1s orbitals have opposite signs, the C2 operation (Figure 7)
changes the sign of their wavefunctions. The reader can work out that this pairing of H 1s
orbitals has b2 symmetry. . . Figure 7. Effect of the C2 rotational operation on the signs of the
wavefunctions of two H 1s orbitals having wavefunctions of the opposite sign. The reader can
work out that this pairing of H 1s orbitals has b2 symmetry.. . . . . . Construction of the MO
diagram for the water molecule:. . The important rule for constructing an MO diagram is that
only atomic orbitals of the same symmetry can overlap significantly. Thus, we place the atomic
orbitals for oxygen on the left of ten diagram, and the two sets of orbitals for the H-atoms, and
label them according to their symmetry:. . . Step 1. Place the four bonding atomic orbitals for
Oxygen on the left, and the two atomic 1s bonding orbitals for the two H atoms on the right. The
orbitals with a1 symmetry are colored red. . . Step 2. Now overlap all the orbitals that have a1
symmetry, highlighted in red. Note that three orbitals have a1 symmetry, so only three orbitals
having a1 symmetry can be produced, hence there is only one antibonding a1* level. The 1s (a1)
orbital is to some extent a non-bonding orbital.. . Step 3. Now construct a bonding b2 and an
antibonding b2* level from the two b2 symmetry atomic orbitals present, shown in blue. Note
that there are two b2 symmetry atomic orbitals, so there will be two b2 symmetry molecular
orbitals. . . . . Step 4. The 2px orbital has b1 symmetry, but none of the H 1s orbitals have b1
symmetry, so no overlap occurs. Place this lone b1 symmetry orbital as a non-bonding orbital
which is at the same energy as the original atomic orbital.. . . . Step 5. Feed the electrons (6 from
O and 2 from the 2 H atoms) into the molecular orbitals. The bonding derives mainly from the
overlap of the 2py and 2pz orbitals of O with the 1s orbitals of the two H atoms. The 2s (a1)
orbital of O is to some extent a non-bonding level, so that we have shown the overlaps of the 1s
orbital of O with other atomic orbitals with broken lines, indicating that the overlap is small.. .
The MO’s for water are shown in Figure 8. None of them correspond to the ‘bunny
ears’ lone pairs suggested by VSEPR. The only totally non-bonding orbital is the 2px orbital,
which is shown enlarged in Figure 9.. . Figure 8. The energy levels of the molecular orbitals of
the water molecule, together with drawing of the corresponding molecular orbitals.. . . Figure 9.
The non-bonding b1 MO of the O atom in water, which is essentially the 2px orbital.. . There are
no bunny ears on water in the MO model. When the water forms, for example, H-bonds, the
interaction is a largely ionic interaction between the partial positive charges on the H atoms, and
the partial negative charge on the 2px orbital of the O atom of a neighboring water, as shown in
Figure 10. One can measure the predicted H---O-H angles of the H-bond, and find that these are
about 110o, giving the appearance of a tetrahedral arrangement around the O-atom, comprised of
two H atoms bonded to it, and two other H-atoms from neighboring water molecules H-bonded
to it.. . Figure 10. PM3 generated structure of two water molecules H-bonded to a central water
molecule, showing how the H-bonds form at a spot suggesting the presence of ‘bunny ear’
lone pairs, as described in the text. . . . . 10. . . . .
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