5.4: Larger (Polyatomic) Molecules

We can extend the method we used for diatomic molecules to draw the molecular orbitals of more complicated, polyatomic molecules (molecules with more than two atoms). To combine several different atoms in a molecular orbital diagram, we will group orbitals from different atoms into sets that match the symmetry of a central atom. These group orbitals are also referred to as symmetry adapted linear combinations (SALCs). We'll use a stepwise approach to do this as summarized below.

Symmetry adapted linear combinations (SALCs)

We need SALCs (aka group orbitals) to draw molecular orbital (MO) diagrams of polyatomic molecules. SALCs are groups of orbitals on pendant atoms. These groups of orbitals much match the symmetry of valence orbitals on the central atom in order to create a productive interaction. When we combine SALCs with the atomic orbitals on the central atom, we can generate an MO diagram that gives us information about the molecule’s bonding and electronic states.

Below is a set of steps you can follow to find SALCs and draw an MO Diagram. Each step will be illustrated in detail through the examples in the subsections that follow this page.

1. Find the point group of the molecule and assign Cartesian coordinates so that \(z\) is the principal axis.
   Sometimes we can simplify things by looking only at the point group of the relevant orbitals.* Only simplify when instructed to do so.
2. Identify and count the pendant atoms' valence orbitals.
   Is there more than 1 type for each atom (ex. just s, or s and p?) We expect 1 SALC for each ligand orbital.
3. Generate a reducible representation (\(\Gamma\)) for the group of pendant atom orbitals using the appropriate character table.
   You have to do this for each set of orbital types. If you only have s orbitals for each ligand, you only need to generate 1 reducible representation. If you have \(p\) orbitals, you would generate additional \(\Gamma\)'s for \(p_x\), \(p_y\), and \(p_z\).
4. Break the \(\Gamma\) into its component irreducible representations from the character table. Note the symmetry of each irreducible representation, its associated orbitals, and their degeneracy.
5. If you are asked to sketch the shapes of SALCs, determine what they “look like” using one of the following strategies.
   (a) Shortcut: if the reducible representations are listed with s, p, or d orbitals in the character table, just draw them (… and skip steps b-c)
   (b) Systematic (Projection Operator) method: Draw an expanded character table and draw your molecule with each ligand identified by letters or numbers (a,b,c... or i,j,k... or 1,2,3...) Determine where each pendant atom/orbital ends up under each of the operations of your expanded table. Project the values for each irreducible representation onto the chart you created and then add up the values. Positive and negative values are opposite signed orbitals.
6. Draw the MO diagram by combining SALCs with AO’s of like symmetry. When drawing SALC energy levels, remember that the more nodal planes in your SALC orbital drawing, the higher the energy for that SALC orbital.

* When we are focused on orbitals, as in the case for finding group orbitals and drawing molecular orbital diagrams, the symmetry of the orbitals is what we are interested in. In the case of high-symmetry point groups, like \(D_{\infty h}\) and \(C_{\infty v}\), even though the molecule may have a \(C_\infty\) axis, the orbitals do not necessarily retain this symmetry element. For example, the \(p_x\) orbital would not have a \(C_\infty\) axis, but rather a \(C_2\) axis. It is sufficient and useful to substitute \(D_{2h}\) for \(D_{\infty h}\) and \(C_{2v}\) for \(C_{\infty v}\) to simplify the problem.

• 5.4.1: Bifluoride anion

• 5.4.2: Carbon Dioxide

  Carbon dioxide is another linear molecule. This example is slightly more complex than the previous example of the bifluoride anion. While bifluoride had only one valence orbital to consider in its central H atom (the 1s orbital), carbon dioxide has a larger central atom, and thus more valence orbitals that will interact with SALCs.

• 5.4.3: H₂O

  Water is a bent molecule, and so it is important to remember that interactions of pendant ligands are dependent on their positions in space. You should consider the positions of the three atoms in water to be essentially fixed in relation to each other. The process for constructing the molecular orbital diagram for a non-linear molecule, like water, is similar to the process for linear molecules. We will walk through the steps below to construct the molecular orbital diagram of water.

• 5.4.4: NH₃

• 5.4.5: CO₂ (Revisted with Projection Operators)

• 5.4.6: BF₃

  BF₃ is more complex than previous examples because it is the first case in which there are multiple types of valence orbitals on the pendant atoms. BF₃ possesses s and p orbitals on both the central atom and all of the pendant atoms. We can follow the same steps that we have previously to derive other molecular orbital diagrams; however, there is one important difference: we will treat each type of pendant orbital as an individual set of SALCs.