2.4. Combining AOs to Build MOs

Molecular orbitals, which we will also later use as group orbitals, can be built from AOs in exactly the same way as MO-programs do, except that we can use the LCAO principle qualitatively to understand the AO-combination process. We will consider a simple example, methylene, CH₂, in order to illustrate the principles involved. We can then use the MOs obtained as generic orbitals for the fragment or group AH₂, where A can be any main group element, in order to explain the shapes of these molecules, and also as group orbitals in order to build the MOs of more complicated molecules like ethylene or cyclopropane.

2.4.1. Methylene, CH₂

The MOs of methylene are built from the AOs of one carbon and two hydrogens within the LCAO approximation. However, because the two hydrogens are symmetrically equivalent, their AOs cannot be considered separately, but must be combined to symmetry-adapted combinations. This is because the CH₂ molecule has C_2v-symmetry, for which the most relevant symmetry element in this discussion is the mirror plane shown in blue in Figure 2.9. An introduction to symmetry elements will be given in Sect. 5.1.

Figure 2.9 The horizontal mirror plane of the methylene molecule makes it necessary to combine the two individual hydrogen AOs (left) to give the symmetry-adapted combinations shown on the right.

The symmetry-adapted combinations of hydrogen s-orbitals that we use to build the methylene MOs must be either symmetric or antisymmetric with respect to reflection in this plane. The individual AOs do not fulfill this condition, but can be combined to give the two symmetry-adapted combinations shown in Figure 2.9. These combinations can then be used to build the MOs.

Once we have built symmetry-adapted combinations of AOs for all equivalent sets of atoms, we can begin to combine them to form the MOs of methylene. Symmetry is a great help in this process, allowing us to determine which AOs and symmetry-adapted combinations can interact with each other and which not. Orbitals that cannot interact with each other are said to be orthogonal. The horizontal mirror plane already used to obtain the symmetry-adapted combinations of hydrogen AOs and the one situated in the molecular plane suffice to distinguish all the different symmetry types involved in forming the methylene MOs from the AOs used here. Figure 2.10 shows the two different symmetry planes that we will use.

Figure 2.10 The two mirror planes used to characterise the symmetry of orbitals for methylene.

We can now place the two symmetry-adapted combinations of hydrogen s-orbitals and the four carbon AOs at their correct positions in methylene and classify them as to whether they are symmetric or antisymmetric with respect to each of the two mirror planes. This is shown in Figure 2.11.

Figure 2.11 The carbon AOs and hydrogen symmetry-adapted combinations of AOs used to build the MOs of methylene. The letters in parentheses denote symmetric (S) or antisymmetric (A) with respect to the s_xy and s_yz mirror planes, in that order.

Only one orbital, the carbon p_x, is antisymmetric with respect to reflection in the s_yz-plane. This AO is thus orthogonal to all the others and will be used unchanged in the methylene MOs. The carbon p_y-orbital is antisymmetric with respect to reflection in the s_xy-plane, as is the symmetry-adapted combination denoted Y'_HH. These two orbitals can thus interact with each other, as shown in Figure 2.12.

Figure 2.12 The interaction of the carbon p_y-AO with the antisymmetric combination of hydrogen s-orbitals to give the CH-bonding p_CH2-MO and the antibonding p*_CH2-MO. These MOs are designated p because of their nodal plane. They are not part of a conjugated p-system, but can be involved in hyperconjugation.

The resulting two MOs are bonding (p_CH2) and antibonding (p*_CH2) combinations of the carbon p-AO and the antisymmetric combination of hydrogen s-AOs. We will discuss the details of these MOs in more detail when we consider the quantitative aspects of bonding in methylene, but note that the carbon contribution is purely p, making it less favourable than an MO in which carbon uses its s-orbital. These two MOs are often denoted p because they are antisymmetric with respect to their nodal plane. This designation does not imply that the MO is part of a conjugated p-system. The shape of the orbitals does, however, let them interact very effectively with p-systems, so that they are important in hyperconjugation.

The three remaining orbitals, the symmetrical Y_HH, the carbon s and p_z, are all symmetrical with respect to both mirror planes. They can therefore all interact with each other. The resulting MOs are shown in Figure 2.13.

Figure 2.13 The interaction between the three symmetric MOs used to build the MOs of methylene. The s_CH-MO is CH-bonding and consists mainly of s-contributions and is therefore particularly stable. The n-MO is a carbon lone-pair (and therefore essentially non-bonding). The s*_CH-MO is the antibonding counterpart of the s_CH-MO.

Figure 2.14 shows the electron density due to each of the six MOs of methylene with their energy levels calculated at the AM1 semiempirical level of theory. Note that the mixing of the s- and p_z-AOs on carbon leads to the familiar orbital shapes described by hybrid orbitals in valence bond theory. The s_CH-orbital is composed almost exclusively of s-AOs, so that the negative lobe remains very small. The higher orbitals have more p-character on carbon.

Figure 2.14 AM1-calculated electron densities and energy levels for the valence molecular orbitals of singlet methylene.

H₂ and He₂ -The Simplest Examples
The Effect of Electronegativity
p-Orbitals and p-Overlap
Combining AOs to Build MOs
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The AH₂ Walsh Diagram