3.4.5: Nonbonding Orbitals and Other Factors

The simplest case is when there is an even number of atomic orbitals that all combine to form strong bonding and antibonding orbitals. But what if there is an uneven number of atomic orbitals? Or what if there are some orbitals that don't meet the criteria for bonding? Or what if the bonding interactions are weak? In these cases, there will be molecular orbitals on the molecule that have non-bonding character.

It is important to note that the bonding, non-bonding, and antibonding nature of orbitals exist on a spectrum. Some bonding and anti-bonding orbitals may have some non-bonding character depending on where their energies lie with respect to the original atomic orbital energies. When molecular orbitals have energies similar to their original atomic orbitals, they will have some non-bonding character. The closer the energies of atomic and molecular orbitals, the more non-bonding the molecular orbitals.

Differences in symmetry

The MO picture for a molecule gets complicated when many valence AOs are involved. We can simplify the problem enormously by noting (without proof here) that orbitals of different symmetry do not interact. Here we will take a simple approach understanding the symmetry of orbitals based on their nodal symmetry. Note, a more rigorous approach to the symmetry classes of bonds aand orbitals is defined by group theory and is the topic of advanced inorganic courses. 

To form a molecular orbital, atomic orbitals must have the same nodal symmetry or their overlap is zero. This means that two orbitals must have the same symmetry with respect to the bonding axis in order to form a molecular orbital. Figure \(\PageIndes{1}\) shows a hydrogen 1s orbital and a chlorine 3p_y orbital. The H 1s orbital is symmetric about the bonding axis; the sign of the orbital is unchanged by a 180° rotation about the bond axis. However, the Cl 3p_y orbital is antisymmetric about the bonding axis; the same rotation inverts the sign of the Cl 3p_y wavefunction. Because these two orbitals have different symmetries, the Cl 3p_y orbital is nonbonding and doesn’t interact with the H 1s. The same is true of the Cl 3p_x orbital. The p_x and p_y orbitals have π symmetry (nodal plane containing the bonding axis) and are labeled π_nb in the MO energy level diagram shown in Figure \(\PageIndex{2}\). In contrast, the H 1s and Cl 3p_z orbitals both have σ symmetry so they can make the bonding and antibonding combinations.

Figure \(\PageIndex{1}\): Orbitals of σ and π symmetry do not interact.

The MO diagram of HCl that includes all the valence orbitals of the Cl atom is shown in Figure \(\PageIndex{2}\). Two of the Cl valence orbitals (3p_x and 3p_y) have the wrong symmetry to interact with the H 1s orbital. The Cl 3s orbital has the same σ symmetry as H 1s, but it is much lower in energy so there is little orbital interaction. The energy of the Cl 3s orbital is thus affected only slightly by forming the molecule. The pairs of electrons in the π_nb and σ_nb orbitals are therefore nonbonding.

Figure \(\PageIndex{2}\): Energy level diagram of the HCl molecule showing MOs derived from the valence AOs.

Note that the MO result in Figure \(\PageIndex{2}\) (1 bond and three pairs of nonbonding electrons) is the same as we would get from valence bond theory for HCl. The nonbonding orbitals are localized on the Cl atom, just as we would surmise from the valence bond picture. In order to differentiate it from the σ bonding orbital, the σ antibonding orbital, which is empty in this case, is designated with an asterisk.

Differences in energy

Combination of orbitals with different energies may lead to orbitals with non-bonding character. Atomic orbitals that have similar energies will have the strongest interactions, and result in bonding molecular orbitals with much lower energies than the component atomic orbitals. On the other hand, atomic orbitals with very unequal energies have a weaker interaction because the molecular orbitals are closer in energy to the atomic orbital energies, thus there is less energy benefit to putting electrons in the bonding molecular orbitals (Figure \(\PageIndex{3}\)). When bonding or antibonding orbitals are close to the energies of the contributing atomic orbitals, those molecular orbitals may have some non-bonding character.

Figure \(\PageIndex{3}\). This figure illustrates how relative energies of atomic orbitals result in molecular orbitals that are more or less favorable. The greater the energy difference between atomic orbitals and the bonding molecular orbital, the more eneretically favorable it is for bonding to occur. (CC-BY-NC-SA, Kathryn Haas)

Uneven number of atomic orbitals

In the case that there is an uneven number of atomic orbitals with compatible symmetry, orbitals with non-bonding character will form. For example, in the case where three atomic orbitals combine, the most common result is formation of a low-energy bonding orbital, a high energy antibonding orbital, and a non-bonding orbital of intermediate energy (Figure \(\PageIndex{4}\)).

Figure \(\PageIndex{4}\). This figure illustrates the most common case for three atomic orbitals that combine to form three molecular orbitals; formation of one low-energy bonding, one high-energy antibonding, and one intermediate energy non-bonding molecular orbital. (CC-BY-NC-SA, Kathryn Haas)

Problems

Exercise 1

What will most likely lead to the smallest covalent interaction?

a) Overlap of a small and a large orbital.

b) Overlap of two small orbitals.

c) Overlap of two large orbitals.

Answer

a) Overlap of a small and a large orbital.

Exercise 2

What will most likely lead to the largest covalent interaction?

a) orbital overlap in sigma-fashion

b) orbital overlap in pi-fashion

c) orbital overlap in delta-fashion

Answer

a) orbital overlap in sigma-fashion

Exercise 4

Decide by “inspection” which of the following combinations of orbitals have the “right” symmetries to form molecular orbitals.

a) The 2p_x orbital of the first N atom and the 2p_y orbital of the second N atom in the molecule N₂. The z axis is defined as the bond axis in N₂.

b) The F 2p_x and the H 1s orbital in the HF molecule. The z axis is defined as the bond axis.

c) The 2p_z orbital of F and the 1s orbital in the HF molecule. The z axis is defined as the bond axis.

Answer

a)

b)

c)