Valence Bond Theory and Hybrid Atomic Orbitals

Valence bond theory describes a covalent bond as the overlap of half-filled atomic orbitals, resulting in a pair of electrons shared between the two bonded atoms.
Sigma ( $\sigma$ ) bonds are formed by end-to-end overlap and are stronger than pi ( $\pi$ ) bonds, which are formed by side-by-side overlap.
Single bonds are always sigma bonds, while multiple bonds consist of one sigma bond and one or two pi bonds.
VB theory explains molecular shapes using the arrangement of hybrid orbitals that maximize overlap in covalent bonds.
Resonance occurs when multiple valid Lewis structures can be drawn for a molecule; the actual electronic structure is an average of those shown by the individual Lewis structures.

Hybridization is introduced to explain the geometry of bonding orbitals in valence bond theory.
Hybrid orbitals are atomic orbitals formed by mathematically combining two or more valence atomic orbitals on an atom.
The number of hybrid orbitals formed equals the number of atomic orbitals combined.
Use the following method to define hybridization:
1. Draw the Lewis structure of the molecule or ion.
2. Determine the electron-group geometry around the central atom.
3. Specify the hybrid orbitals needed to accommodate this electron-group geometry.

Linear electron-group geometry requires two hybrid orbitals, achieved by mixing one s and one p orbital to form two sp hybrid orbitals.
These sp orbitals are oriented 180 degrees apart.
- $s + p \rightarrow 2 \space sp \space orbitals$
Examples include beryllium chloride ( $\text{BeCl}2$ ) and carbon dioxide ( $\text{CO}2$ ).

Trigonal planar electron-group geometry requires three hybrid orbitals, formed by mixing one s and two p orbitals to create three sp2 hybrid orbitals.
These sp2 orbitals point towards the corners of an equilateral triangle, 120 degrees apart.
- $s + 2p \rightarrow 3 \space sp^2 \space orbitals$
An example is boron trifluoride ( $\text{BF}_3$ ).

Tetrahedral electron-group geometry requires four hybrid orbitals, made by mixing one s and three p orbitals to generate four sp3 hybrid orbitals.
These sp3 orbitals are directed towards the corners of a tetrahedron, approximately 109.5 degrees apart.
- $s + 3p \rightarrow 4 \space sp^3 \space orbitals$
Examples include methane ( $\text{CH}4$ ) and water ( $\text{H}2\text{O}$ ).

Trigonal bipyramidal electron-group geometry requires five hybrid orbitals, formed by mixing one s, three p, and one d orbital, resulting in five sp3d hybrid orbitals.
- $s + 3p + d \rightarrow 5 \space sp^3d \space orbitals$
An example is phosphorus pentachloride ( $\text{PCl}_5$ ).

Octahedral electron-group geometry requires six hybrid orbitals, formed by mixing one s, three p, and two d orbitals, resulting in six sp3d2 hybrid orbitals.
- $s + 3p + 2d \rightarrow 6 \space sp^3d^2 \space orbitals$
An example is sulfur hexafluoride ( $\text{SF}_6$ ).

Multiple bonds (double and triple bonds) consist of sigma ( $\sigma$ ) and pi ( $\pi$ ) bonds.
A double bond consists of one sigma bond and one pi bond.
A triple bond consists of one sigma bond and two pi bonds.
Rotation around sigma bonds is allowed, but rotation around pi bonds is restricted, giving rise to geometric isomerism.
To describe multiple bonds using hybrid orbitals:
1. Assign hybridization using the method described earlier.
2. Describe the $\sigma$ bonds using hybrid orbitals and the $\pi$ bonds using unhybridized p orbitals.
For example, in ethene ( $\text{H}2\text{C=CH}2$ ):
- Each carbon is sp2 hybridized, forming sigma bonds with two hydrogens and the other carbon.
- The unhybridized p orbitals on each carbon overlap side-by-side to form the pi bond.
In ethyne ( $\text{H−C≡C−H}$ ):
- Each carbon is sp hybridized, forming sigma bonds with one hydrogen and the other carbon.
- Two unhybridized p orbitals on each carbon overlap side-by-side to form the two pi bonds.

Molecular orbital (MO) theory describes the electronic structure of molecules in terms of molecular orbitals, which are delocalized over the entire molecule.
Molecular orbitals are formed by combining atomic orbitals.
Bonding orbitals are lower in energy than the original atomic orbitals, while antibonding orbitals are higher in energy.
The bond order is a measure of the stability of a molecule and is calculated as:
- $\text{Bond Order} = \frac{\text{Number of Bonding Electrons} - \text{Number of Antibonding Electrons}}{2}$
A higher bond order indicates a more stable molecule.
MO diagrams show the relative energies and occupancy of molecular orbitals.
For diatomic molecules, MO diagrams can be constructed by combining the atomic orbitals of the constituent atoms.
For the second-row diatomic molecules (Li2 to Ne2), the order of MOs is typically:
- \sigma{2s} < \sigma{2s}^* < \sigma{2p} < \pi{2p} < \pi{2p}^* < \sigma{2p}^*
However, for molecules like O2 and F2, the order is slightly different:
- \sigma{2s} < \sigma{2s}^* < \pi{2p} < \sigma{2p} < \pi{2p}^* < \sigma{2p}^*
MO theory can also be used to explain the magnetic properties of molecules. For example, O2 is paramagnetic because it has unpaired electrons in its $\pi_{2p}^*$ orbitals.
Delocalized molecular orbitals are molecular orbitals that extend over more than two atoms, and are often found in molecules with resonance structures.

Hydrogen (H2) has a bond order of 1: (2 bonding electrons - 0 antibonding electrons)/2 = 1.
Helium (He2) has a bond order of 0: (2 bonding electrons - 2 antibonding electrons)/2 = 0, indicating that He2 does not exist.
Diatomic molecules of the second period like N2 and O2 can be analyzed using MO theory to describe their bonding characteristics and magnetic properties.
In summary, Molecular Orbital Theory provides a more complete description of chemical bonding that can explain properties that Valence Bond Theory cannot.