Explain why an electron in the bonding molecular orbital in the \(\mathrm{H}_{2}\) molecule has a lower energy than an electron in the 1 s atomic orbital of either of the separated hydrogen atoms.

The concept of a bonding molecular orbital is integral to understanding molecular structure and stability. In simple terms, when two atoms come close together, their respective atomic orbitals overlap to form molecular orbitals. Among these, the bonding molecular orbitals are of particular importance as they can accommodate pairs of electrons.

These orbitals form due to the constructive interference of the atomic orbital wave functions, which leads to an increase in electron density between the two atomic nuclei. This electron density acts like a 'glue' holding the nuclei together, which is essentially the chemical bond. Moreover, the electrons in these orbitals have a lower energy compared to those in individual atomic orbitals due to the stabilizing effect of this increased electron density.

Atomic orbitals serve as the foundation for understanding chemical bonding. They can be envisioned as the regions in space around an atom where there is a high probability of finding electrons.

Every atomic orbital has a distinct shape and energy level, for example, the spherical 1s orbitals of hydrogen. When considering molecular formation, atomic orbitals combine to yield molecular orbitals. This combination depends on the symmetry and energy compatibility of the atomic orbitals. One must appreciate that the way these orbitals merge has profound effects on the properties of the resulting molecule, including its shape, magnetism, and reactivity.

In the context of energy levels in molecules, electrons occupy molecular orbitals just as they occupy atomic orbitals, but with a notable difference in the energy landscape. The formation of molecular orbitals results in a set of energy levels that are distinct from those of separate atoms.

Splitting of Energy Levels

The amalgamation of atomic orbitals leads to the splitting of energy levels into bonding and antibonding molecular orbitals. Bonding orbitals have lower energy compared to the original atomic orbitals, while antibonding orbitals have higher energy. The arrangement of electrons into these new sorted energy levels is fundamental to the bond formation and the stability of molecules.

Electron stability is a key outcome of lower energy states in bonding molecular orbitals. An electron is more stable when it is in a lower energy state because a stable system is one that has minimized its potential energy.

In a bonding molecular orbital, electrons are shared between atoms and are attracted by more than one nucleus. This diffused electron cloud results in a lower potential energy for the system. The principle here is simple: lower potential energy equates to greater stability. Consequently, electrons in bonding molecular orbitals contribute to a more stable, lower-energy molecule compared to when those electrons are in separate, higher-energy atomic orbitals.