Animated Solution for Chemistry - Chemical Bonding and Molecular Structure: In which of the following pairs of molecules/ions both the species are not likely to exist?

Imagine you are a molecular architect, trying to build molecules from individual atoms. But there's a catch! Not all combinations are allowed by the laws of quantum mechanics. In this problem, we are tasked with finding a pair of species that simply refuse to exist in nature.

To solve this mystery, we must consult the ultimate rulebook of chemical bonding: Molecular Orbital Theory (MOT). According to MOT, the stability of a molecule is determined by its Bond Order.

Here, $N_b$ represents the number of electrons in bonding molecular orbitals (which hold the atoms together), and $N_a$ represents the number of electrons in antibonding molecular orbitals (which push the atoms apart). If the bond order is zero or negative, the repulsive forces overwhelm or equal the attractive forces, and the molecule falls apart. It simply cannot exist!

Let's put our first suspect, the $H_2^{2+}$ ion, under the microscope. A standard hydrogen molecule ($H_2$) is formed by two hydrogen atoms, each contributing one electron, giving a total of two electrons.

However, the $+2$ charge on $H_2^{2+}$ indicates a dramatic event: both of these electrons have been stripped away!

With absolutely zero electrons left in the system, the molecular orbitals are completely vacant. There are no bonding electrons ($N_b=0$) and no antibonding electrons ($N_a=0$).

Since the bond order is exactly zero, there is no electron glue to hold the two bare hydrogen nuclei (protons) together. The immense electrostatic repulsion between the two positive protons will instantly blow them apart. Thus, $H_2^{2+}$ cannot exist.

Now, let's turn our attention to the hypothetical helium molecule, $H{e}_2$. Helium is a noble gas, notorious for its reluctance to form bonds. Let's see why MOT agrees with this behavior.

Each helium atom brings two electrons to the table, making a total of four electrons for the $H{e}_2$ molecule.

Following the Aufbau principle, we fill these electrons into the molecular orbitals starting from the lowest energy level. The first two electrons happily pair up in the bonding $\sigma 1s$ orbital, creating a stabilizing force.

But we still have two electrons left! These must go into the higher energy, destabilizing antibonding ${\sigma }^{\ast }1s$ orbital.

Now, let's calculate the bond order. We have two bonding electrons ($N_b=2$) and two antibonding electrons ($N_a=2$).

The stabilizing effect of the bonding electrons is perfectly canceled out by the destabilizing effect of the antibonding electrons. The net bonding force is zero. Consequently, the two helium atoms will just bounce off each other rather than forming a stable molecule. $H{e}_2$ does not exist!

We have successfully interrogated both suspects. Both $H_2^{2+}$ and $H{e}_2$ yield a bond order of zero, proving that neither species is viable.

Looking at our options, this perfectly aligns with option (c). The beauty of Molecular Orbital Theory lies in its ability to mathematically prove what we observe in nature—or in this case, what we don't observe!