2.2 Atomic and molecular orbitals

Atomic and Molecular Orbitals

Atomic and molecular orbitals describe where electrons are most likely found, first around individual atoms and then within molecules. In photochemistry, these orbitals matter because light absorption and emission involve electrons moving between them. This section covers the quantum mechanical description of atomic orbitals, how they combine into molecular orbitals via LCAO theory, and how to read molecular orbital diagrams to predict bond strength and magnetic behavior.

Atomic Orbitals

Atomic and Molecular Orbital Properties

An atomic orbital is a mathematical function that describes the probability of finding an electron in a region of space around a single atom. Each orbital is fully specified by four quantum numbers:

• $n$ (principal): determines the energy level and overall size of the orbital ($n = 1, 2, 3, \ldots$)
• $l$ (angular momentum): determines the orbital shape ($l = 0$ for s, $l = 1$ for p, $l = 2$ for d, $l = 3$ for f)
• $m_l$ (magnetic): determines the orbital's orientation in space ($m_l = -l, \ldots, 0, \ldots, +l$)
• $m_s$ (spin): specifies the electron's intrinsic spin ($+\tfrac{1}{2}$ or $-\tfrac{1}{2}$)

A molecular orbital extends over an entire molecule rather than a single atom. Molecular orbitals form when atomic orbitals on neighboring atoms combine, producing three possible types:

• Bonding orbitals concentrate electron density between nuclei, stabilizing the molecule.
• Antibonding orbitals (marked with an asterisk, e.g., $\sigma^*$) place a node between nuclei, destabilizing the molecule.
• Non-bonding orbitals remain localized on one atom and don't significantly affect bond strength.

Types of Atomic Orbitals

Each orbital type has a characteristic shape, a set number of orientations per energy level, and a specific count of angular nodes (equal to $l$):

• s orbitals ($l = 0$): spherical, one orbital per energy level, zero angular nodes. Examples: 1s, 2s, 3s. Each higher $n$ adds a radial node, but the overall shape stays spherical.
• p orbitals ($l = 1$): dumbbell-shaped, three orientations per level ($p_x, p_y, p_z$), one angular node each. These first appear at $n = 2$.
• d orbitals ($l = 2$): mostly cloverleaf-shaped (with one "doughnut-plus-lobes" exception, $d_{z^2}$), five orientations per level, two angular nodes. First appear at $n = 3$.
• f orbitals ($l = 3$): more complex multi-lobed shapes, seven orientations per level, three angular nodes. First appear at $n = 4$.

For photochemistry, the shapes and symmetry labels of these orbitals directly determine which electronic transitions are allowed when a molecule absorbs light.

Molecular Orbitals

Formation of Molecular Orbitals

Molecular orbitals are constructed using the Linear Combination of Atomic Orbitals (LCAO) method. The idea: take the wave functions of atomic orbitals on two (or more) atoms and add or subtract them to produce new wave functions that span the molecule.

1. Identify atomic orbitals of similar energy and compatible symmetry on neighboring atoms. Orbitals must belong to the same symmetry representation to combine effectively.
2. Add the wave functions (constructive interference). The resulting bonding orbital has increased electron density between the nuclei and sits lower in energy than the original atomic orbitals.
3. Subtract the wave functions (destructive interference). The resulting antibonding orbital has a node between the nuclei and sits higher in energy than the original atomic orbitals.

Two important symmetry types arise from how the orbitals overlap:

• $\sigma$ bonds: head-on overlap along the internuclear axis (e.g., two s orbitals, or two $p_z$ orbitals pointing at each other). Cylindrically symmetric about the bond axis.
• $\pi$ bonds: side-on overlap above and below (or in front of and behind) the internuclear axis (e.g., $p_x$ with $p_x$). These have a nodal plane containing the bond axis.

Each bonding combination produces a corresponding antibonding counterpart ($\sigma^*$, $\pi^*$). In photochemistry, $\pi \rightarrow \pi^*$ and $n \rightarrow \pi^*$ transitions are among the most common excitations triggered by UV-visible light.

Molecular Orbital Diagrams

An MO diagram is a visual map of the molecular orbital energy levels, showing which orbitals electrons occupy. Here's how to build and read one for a homonuclear diatomic:

1. Draw the atomic orbital energy levels for each atom on the left and right sides of the diagram.
2. Connect atomic orbitals of matching symmetry to the molecular orbitals they form in the center column. Place bonding orbitals below and antibonding orbitals above the parent atomic orbital energies.
3. Fill electrons from lowest energy up, following the Aufbau principle, Pauli exclusion (max two electrons per orbital, opposite spins), and Hund's rule (maximize unpaired spins in degenerate orbitals).

Energy level ordering has a subtlety worth knowing. For $\text{O}_2$, $\text{F}_2$, and $\text{Ne}_2$, the expected ordering holds: $\sigma_{2s} < \sigma_{2s}^* < \sigma_{2p} < \pi_{2p} < \pi_{2p}^* < \sigma_{2p}^*$. But for $\text{Li}_2$ through $\text{N}_2$, s-p mixing pushes the $\sigma_{2p}$ orbital above the $\pi_{2p}$ orbitals. Forgetting this switch is a common mistake.

Bond order quantifies bond strength:

$\text{Bond order} = \frac{(\text{bonding electrons}) - (\text{antibonding electrons})}{2}$

• $\text{N}_2$: 10 electrons total, 8 bonding, 2 antibonding → bond order = 3 (triple bond, very strong).
• $\text{O}_2$: 12 electrons total, 8 bonding, 4 antibonding → bond order = 2 (double bond).
• A bond order of 0 means the molecule won't form (e.g., $\text{He}_2$).

Higher bond order corresponds to shorter, stronger bonds.

Magnetic properties follow directly from the diagram:

• Paramagnetic: the molecule has unpaired electrons and is attracted to a magnetic field. $\text{O}_2$ is paramagnetic because its two highest-energy electrons occupy separate degenerate $\pi_{2p}^*$ orbitals with parallel spins.
• Diamagnetic: all electrons are paired, and the molecule is weakly repelled by a magnetic field. $\text{N}_2$ is diamagnetic.

The fact that MO theory correctly predicts $\text{O}_2$'s paramagnetism, while Lewis structures do not, is one of the strongest arguments for the MO approach. For photochemistry specifically, knowing which orbitals are occupied and which are empty tells you exactly which electronic transitions are possible when a photon arrives.