Show how two \(2 p\) atomic orbitals can combine to form a \(\sigma\) or a \(\pi\) molecular orbital.

2p orbitals have two lobes on opposite sides of the nucleus, with a nodal plane passing through the nucleus where the electron probability is zero. There are three 2p orbitals, one along each of the x, y, and z axes: 2px, 2py, and 2pz. In forming molecular orbitals, the orbitals must have the same energy and proper orientation.

Sigma (σ) molecular orbitals are formed by the head-to-head (or end-to-end) overlap of two atomic orbitals along their internuclear axis. The electron density is symmetric along this axis. For forming σ molecular orbitals using 2p orbitals, we will consider the overlap of two 2pz orbitals: 1. Consider two atoms A and B, each with a 2pz orbital. 2. Align the internuclear axis (the axis connecting the nuclei of both atoms) along the z-axis. 3. Overlap the 2pz orbital of atom A with the 2pz orbital of atom B along the z-axis in a head-to-head fashion. 4. The resulting electron density will be symmetric around the internuclear axis, forming a sigma (σ) molecular orbital.

Pi (π) molecular orbitals are formed by the side-to-side overlap of two atomic orbitals, leading to electron density above and below (or in front and behind) the internuclear axis. For forming π molecular orbitals using 2p orbitals, we will consider the overlap of two 2px and two 2py orbitals: 1. Consider two atoms A and B, each with a set of 2px and 2py orbitals. 2. Align the internuclear axis along the z-axis (we will demonstrate with 2px orbitals, but the process is the same for 2py orbitals). 3. Overlap the 2px orbital of atom A with the 2px orbital of atom B in a side-to-side fashion, maintaining their orientation perpendicular to the internuclear axis. 4. With this side-to-side overlap, electron density will be above and below the internuclear axis, forming a pi (π) molecular orbital. The process for 2py orbitals is the same, with electron density accumulating in front and behind the internuclear axis in the resulting π molecular orbital. In summary, two 2p atomic orbitals can combine to form a σ molecular orbital by head-to-head overlap along the internuclear axis, or a π molecular orbital by side-to-side overlap perpendicular to the internuclear axis.