Describe what happens to the electron density around two hydrogen atoms as they come together to form an \(\mathrm{H}_{2}\). What happens to the energy of two hydrogen atoms as they approach each other? What happens to the spins of the electrons?

When two hydrogen atoms come close to each other, a fascinating interplay begins between their respective electron clouds. These clouds are also known as 1s orbitals, each housing a single electron. As the distance between the hydrogen atoms decreases, the electron clouds start to merge, creating a zone of shared electron density. This zone represents an overlap of the atomic orbitals, and marks the birthplace of a molecular orbital.

This overlap is crucial because it essentially defines the space between the atoms where the probability of finding electrons becomes notably high. Think of it as a Venn diagram—the overlap area in the middle grows as the circles (or in this case, orbitals) draw nearer to each other. This area of overlap is the cornerstone of bond formation, as it creates the stage for electrons to dwell together between the nuclei, thus keeping the atoms bonded in a molecule of hydrogen H2.

The journey of two hydrogen atoms towards each other is not only a dance of electron clouds but also involves a critical change in energy. As they gravitate closer, the potential energy of the system declines. This decrease in energy is because, within the overlapping orbital space, the electrical forces exert a mutual attraction between the positively charged nuclei and the now-concentrated electron density.

The energy of the system hits a sweet spot, known as the minimum energy level, when the atoms find themselves at an arm's length—the perfect distance apart for the bond. This distance is scientifically termed the bond length. Here, the system is the most stable and the energy released during this process is often reflected as heat or light. The energy curve that represents this change is concave downward, showing energy decrease up to the bond length, and beyond that, an energy increase if the atoms were pushed unnaturally close.

Now let's discuss the tiny, yet phenomenal, world of electron spins in the formation of an H2 molecule. Each hydrogen atom has one electron, which behaves a bit like a tiny magnet due to its spin. There are two possibilities for an electron's spin—up or down. For a bond to form, these two electrons from separate hydrogen atoms have to agree to dance to the same tune, in opposite directions.

This movement is known as 'electron spin pairing', and it creates a paired-up, or singlet, state that is essential for the bond. Thanks to the Pauli exclusion principle, which states that no two electrons can have identical quantum numbers in the same atom, the paired electrons with opposite spins can coexist peacefully in the same molecular orbital. Without this harmonious pairing, the electrons would repel each other, and the bond simply couldn't form. This spin behavior is critical and indispensable to the stability and existence of the molecule.