General Chemistry: Chemical Bonding Advanced

Lewis structures and valence bond theory provide a crucial foundation for understanding chemical bonds, but they hit a wall when explaining the magnetic properties of oxygen or the conductivity of metals. Advanced bonding theory, specifically molecular orbital theory, overcomes these limitations by treating electrons as belonging to the entire molecule, not just individual atoms. This framework allows us to quantitatively predict bond strength, magnetic behavior, and the electronic structure of complex solids, making it indispensable for explaining real-world molecular properties.

The Core Idea: Molecular Orbital Theory

Molecular orbital theory (MOT) is a model that describes the behavior of electrons in a molecule by forming molecular orbitals from the mathematical combination of atomic orbitals. Unlike valence bond theory, which localizes electrons between two specific atoms, MOT delocalizes electrons over the entire molecule. The key principle is the linear combination of atomic orbitals (LCAO), where atomic wavefunctions from different atoms are added or subtracted to create new molecular wavefunctions.

This combination results in two types of molecular orbitals: bonding molecular orbitals and antibonding molecular orbitals. Bonding orbitals are formed by the in-phase (constructive) addition of atomic orbitals, resulting in electron density concentrated between the nuclei, which stabilizes the molecule. Antibonding orbitals are formed by out-of-phase (destructive) addition, marked with an asterisk (*), resulting in a node of zero electron density between the nuclei, which destabilizes the molecule. The energy of a bonding orbital is lower than that of the original atomic orbitals, while the antibonding orbital's energy is higher.

Constructing Molecular Orbital Diagrams

A molecular orbital diagram visually represents the relative energies of atomic and molecular orbitals and tracks the filling of electrons according to the Pauli exclusion principle and Hund's rule. For homonuclear diatomic molecules (e.g., $O_2$, $N_2$), the diagram is symmetrical. The construction follows a standard sequence: place the atomic orbitals of the separated atoms on the sides, draw the molecular orbitals in the center in order of increasing energy, and then fill the molecular orbitals with the total number of valence electrons from both atoms.

The order of orbital filling is critical. For molecules like $O_2$ and $F_2$, the molecular orbital energy order is: $${\sigma }_{1s}<{\sigma }_{1s}^{\ast }<{\sigma }_{2s}<{\sigma }_{2s}^{\ast }<{\sigma }_{2p_z}<{\pi }_{2p_x}={\pi }_{2p_y}<{\pi }_{2p_x}^{\ast }={\pi }_{2p_y}^{\ast }<{\sigma }_{2p_z}^{\ast }.$$ However, for $B_2$, $C_2$, and $N_2$, the ${\pi }_{2p}$ orbitals are lower in energy than the ${\sigma }_{2p_z}$ orbital. Accurately placing electrons in this framework allows for the calculation of key molecular properties.

Bond Order, Stability, and Magnetism

The power of MOT becomes clear when we calculate bond order, a direct indicator of bond strength and stability. The formula is: $$\text{Bond Order}=\frac{(\text{Number of electrons in bonding orbitals})-(\text{Number of electrons in antibonding orbitals})}{2}.$$ A bond order greater than zero suggests a stable bond. For example, $H{e}_2$ has a bond order of zero, correctly predicting its non-existence as a stable molecule, while $O_2$ has a bond order of 2.

Furthermore, MOT perfectly explains paramagnetism, the attraction of a molecule to an external magnetic field due to the presence of unpaired electrons. By examining the filled molecular orbital diagram for $O_2$, we see two unpaired electrons in the degenerate ${\pi }_{2p}^{\ast }$ orbitals. This prediction, which eludes Lewis structure theory (which shows $O_2$ with all electrons paired), is a major triumph of MOT. Diamagnetic substances, with all electrons paired, are weakly repelled by a magnetic field.

Delocalized Bonding and Resonance

Delocalized bonding describes a bonding situation where electrons are not confined to a single bond or between two atoms, but are shared over three or more atoms. This concept is the quantum mechanical explanation for resonance structures in Lewis theory. In molecules like benzene ($C_6{H}_6$) or the carbonate ion ($C{O}_3^{2-}$), atomic p-orbitals overlap side-by-side across multiple adjacent atoms to form a set of pi molecular orbitals that extend over the entire region.

These delocalized pi systems are more stable than localized double bonds would be, a stabilization known as resonance energy. In MOT terms, the electrons in these systems occupy bonding molecular orbitals that span several atoms. This delocalization is responsible for the unique properties of aromatic compounds and explains why all carbon-carbon bonds in benzene are of equal length, an intermediate between a single and double bond.

Metallic Bonding and Band Theory

Metallic bonding is an extreme case of electron delocalization, where valence electrons are shared among all the atoms in a metal lattice. Molecular orbital theory scales up to explain this through band theory. In a metal, the enormous number of atomic orbitals (e.g., the 3s orbitals in sodium) combine to form a nearly continuous set of molecular orbitals called a band.

The valence band is the highest occupied band, and the conduction band is the next available band of higher energy. In metals, these bands overlap, allowing electrons to move freely into empty orbitals with minimal energy input. This explains metallic properties: electrical conductivity (free-moving electrons), malleability (delocalized bonds allowing layers to slide), and luster (electrons absorbing and re-emitting light). In semiconductors and insulators, a band gap exists between the valence and conduction bands; the size of this gap determines the material's electrical properties.

Common Pitfalls

1. Misapplying the Diatomic MO Diagram Order: Using the $O_2/F_2$ orbital order for all molecules is a frequent error. Remember, for elements with atomic number less than 8 (i.e., $B_2$, $C_2$, $N_2$), the ${\pi }_{2p}$ orbitals are lower in energy than the ${\sigma }_{2p}$ orbital. Always check which sequence applies before filling the diagram.
2. Confusing Bond Order with Bond Length Alone: While bond order is the primary factor, bond length is also influenced by atomic size. A C-C single bond is shorter than a Si-Si single bond due to smaller atomic radius, even though both have a bond order of 1. Use bond order to compare bonds between the same two elements.
3. Overlooking Magnetism in Diagrams: When predicting paramagnetism, don't just count unpaired electrons in atomic orbitals. You must fill the molecular orbitals correctly. A molecule can have an even number of total electrons yet still be paramagnetic if the molecular orbital filling leaves unpaired electrons, as in $O_2$.
4. Equating Delocalization with "Floating" Electrons: Delocalized electrons are not randomly dispersed; they occupy specific, well-defined molecular orbitals that have a high probability density over several nuclei. The bonding is quantized, not arbitrary.

Summary

• Molecular orbital theory (MOT) provides a more powerful and quantitative model for bonding than Lewis or valence bond theories by describing electrons in molecules using delocalized molecular orbitals formed from atomic orbitals.
• Bond order, calculated from molecular orbital diagrams, directly predicts bond stability and strength, while the presence of unpaired electrons in these diagrams accurately explains paramagnetism.
• Delocalized bonding is the MOT explanation for resonance, where electrons occupy molecular orbitals spread over multiple atoms, leading to increased stability, as seen in benzene and polyatomic ions.
• Metallic bonding and band theory represent the large-scale application of MOT, where overlapping bands of molecular orbitals allow for electron delocalization throughout a solid, explaining conductivity, malleability, and other key metallic properties.