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

Atomic orbitals are regions around a nucleus where electrons with specific energy levels and angular momentum are likely to be found. The shape and orientation of atomic orbitals depend on their energy levels (principal quantum number, n) and the type of orbital (angular momentum quantum number, l). In this case, we are interested in the 2\(p\) atomic orbitals, which means n=2 and l=1. For \(p\) orbitals, there are three orientations possible: \(p_x\), \(p_y\), and \(p_z\). These orbitals are dumbbell-shaped and are oriented along their respective axes.

Molecular orbitals are formed by the combination of atomic orbitals from individual atoms in a molecule. They can accommodate the same number of electrons as all the atomic orbitals combined. Linear combination of atomic orbitals (LCAO) is a method used to describe the formation of molecular orbitals from atomic orbitals. In this process, atomic orbitals combine in such a way that results in constructive or destructive interference of their wave functions, leading to bonding or anti-bonding molecular orbitals.

A \(\sigma\) bond is formed by the head-on overlap of atomic orbitals. In the case of 2\(p\) orbitals, it's the overlap of two \(p_z\) orbitals (oriented along the z-axis) that results in a \(\sigma\) bond. The wave functions of the atomic orbitals constructively interfere, forming areas of high electron density (called a bonding orbital) between the two nuclei. This increased electron density leads to a stronger attraction between the positively charged nuclei and the electrons, resulting in a bond.

A \(\pi\) bond is formed by the side-on overlap of atomic orbitals. In the case of 2\(p\) orbitals, it's the overlap of two 2\(p_x\) or 2\(p_y\) orbitals (oriented along the x- and y-axis, respectively). The wave functions of the atomic orbitals constructively interfere, forming areas of high electron density (called a bonding orbital) above and below the nuclei along the internuclear axis. The electron density is not as concentrated between the nuclei as in the \(\sigma\) bond, resulting in a weaker bond compared to \(\sigma\) bonding. By combining two 2\(p\) atomic orbitals through the LCAO method, we have demonstrated the formation of both \(\sigma\) and \(\pi\) molecular orbitals. The head-on overlap of 2\(p_z\) orbitals results in a \(\sigma\) bond, while the side-on overlap of 2\(p_x\) or 2\(p_y\) orbitals results in a \(\pi\) bond. These linear combinations of atomic orbitals help to explain the various bonding types observed in molecules.