Show how a hydrogen 1\(s\) atomic orbital and a fluorine 2\(p\) atomic orbital overlap to form bonding and antibonding molecular orbitals in the hydrogen fluoride molecule. Are these molecular orbitals \(\sigma\) or \(\pi\) molecular orbitals?

Molecular orbitals are formed by the interaction of atomic orbitals when two atoms combine to form a chemical bond. The atomic orbitals combine to form new molecular orbitals that belong to the system as a whole. These molecular orbitals can either be bonding (lower energy, more stable) or antibonding (higher energy, less stable). The bonding molecular orbital is formed by constructive interference between the wave functions of atomic orbitals, while the antibonding molecular orbital is formed by destructive interference between their wave functions.

There are two types of molecular orbitals, namely \(\sigma\) and \(\pi\) orbitals: - \(\sigma\) orbitals: These molecular orbitals are cylindrically symmetrical around the bond axis (i.e., the line connecting the nuclei of the two atoms). They are formed by the head-on overlap (end-to-end) of the atomic orbitals. - \(\pi\) orbitals: These molecular orbitals are not cylindrically symmetrical around the bond axis. They are formed by the side-on overlap (side-by-side) of the atomic orbitals.

In the hydrogen fluoride (HF) molecule, the hydrogen atom contributes with its 1\(s\) atomic orbital, while the fluorine atom participates with one of its three 2\(p\) atomic orbitals (either \(2p_x\), \(2p_y\), or \(2p_z\)). Since the 1\(s\) atomic orbital is spherically symmetrical, and the 2\(p\) atomic orbitals are dumbbell-shaped, the overlap between these orbitals will be head-on, leading to the formation of \(\sigma\) molecular orbitals.

When the hydrogen 1\(s\) and fluorine 2\(p\) atomic orbitals overlap in a constructive manner, a bonding \(\sigma\) molecular orbital is formed. This bonding molecular orbital has lower energy and is more stable than the parent atomic orbitals. The electron density in this orbital is concentrated between the two nuclei, leading to the stabilization of the HF molecule. When the hydrogen 1\(s\) and fluorine 2\(p\) atomic orbitals overlap in a destructive manner, an antibonding \(\sigma^*\) molecular orbital is formed. This antibonding molecular orbital has higher energy and is less stable than the parent atomic orbitals. The electron density in this orbital has a node between the two nuclei, leading to repulsion and destabilization of the HF molecule if electrons were to occupy it. However, in the ground state of the HF molecule, this antibonding orbital remains unoccupied. In summary, the hydrogen 1\(s\) and fluorine 2\(p\) atomic orbital overlap in the hydrogen fluoride (HF) molecule produces both bonding and antibonding \(\sigma\) molecular orbitals. The bonding \(\sigma\) molecular orbital is responsible for the chemical bond between hydrogen and fluorine, while the antibonding \(\sigma^*\) molecular orbital remains unoccupied in the ground state.