Why is bonding in \(\mathrm{H}_{2}^{+}\) favorable? Express your answer in terms of the symmetry of the electron wave function.

H2+ is a molecular ion containing two hydrogen atoms and one electron. It is the simplest heteronuclear diatomic molecule with an odd number of electrons. Our task is to explain why bonding in this system is favorable using the symmetry of the electron wave function.

To understand the bonding in H2+, we need to consider the molecular orbitals (MOs) that describe the distribution of the single electron in the molecule. These molecular orbitals are formed through the Linear Combination of Atomic Orbitals (LCAO), which is a method to approximate the molecular orbitals by combining the atomic orbitals of the individual atoms in the molecule. In H2+, the two 1s atomic orbitals from each hydrogen atom form the molecular orbitals.

With LCAO, the two 1s atomic orbitals from the hydrogen atoms can combine in two ways to form molecular orbitals: 1. Constructive interference: When the two 1s atomic orbitals are in-phase (both positive lobes or both negative lobes), the constructive interference creates a bonding molecular orbital between the two hydrogen nuclei, which is lower in energy than the individual atomic orbitals. This bonding orbital is called the σ (sigma) molecular orbital. 2. Destructive interference: When the two 1s atomic orbitals are out-of-phase (one positive lobe and one negative lobe), the destructive interference creates an antibonding molecular orbital that is higher in energy than the individual atomic orbitals. This antibonding orbital is called the σ* (sigma-star) molecular orbital.

In H2+, there is only one electron available to occupy the molecular orbitals. Since the σ molecular orbital is lower in energy, this electron will occupy the σ bonding orbital. This leads to a decrease in energy and creates a stable bond between the two hydrogen nuclei, making bonding in H2+ favorable.

The configuration of the single electron in the bonding σ molecular orbital is symmetric with respect to the two nuclei, indicating that the electron density is higher between the nuclei. This increased electron density results in an attractive force between the hydrogen nuclei and stabilizes the molecule. In summary, bonding in H2+ is favorable because the electron occupies a symmetric bonding molecular orbital (σ) formed by the constructive interference of the hydrogen atoms' 1s orbitals, resulting in a stable bond between the two nuclei.