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Molecular orbitals represent the quantum states of electrons in molecules that extend over the entire molecule. They are vital for understanding how atoms bond to form molecules. In the molecular orbital model, we use the principles of quantum mechanics to describe the behavior of electrons within a molecule. Unlike atomic orbitals, which are centered around a single atom, molecular orbitals arise from the combination or overlap of atomic orbitals belonging to the atoms in a molecule.

This results in the creation of two types of orbitals - bonding orbitals, which have lower energy and ant-bonding orbitals with higher energy. Electrons will fill the lower energy molecular orbitals first, following the principles of the Aufbau principle, Hund's rule, and Pauli's exclusion principle, just as they do with atomic orbitals. The understanding of molecular orbitals allows chemists to explain the bond order, magnetism, and other properties of molecules.

In the context of molecular orbitals, the electron gain likelihood refers to how likely a molecule is to accept an additional electron. This is an important factor in chemical reactions and stability. When assessing the likelihood of electron gain, the focus is on the energy of the molecule's lowest unoccupied molecular orbital (LUMO).

A lower-energy LUMO means that less energy is required for an electron to be added to the molecule, increasing the likelihood of electron gain. Factors that can affect the energy of the LUMO include the type of atomic orbitals that combine to form the molecular orbitals, the electronegativity of the atoms involved, and the overall molecular geometry.

Comparing the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is crucial for predicting the reactivity of molecules. The HOMO represents the highest energy level occupied by an electron within a molecule under normal conditions, while the LUMO represents the next available energy level that can accept an electron.

The energy gap between these orbitals is known as the HOMO-LUMO gap. A smaller gap often signifies higher chemical reactivity and a greater likelihood of a molecule to participate in reactions, as it suggests that less energy is required to remove an electron from the HOMO or add an electron to the LUMO. By understanding the relative energy levels of the HOMO and LUMO, chemists can better predict which molecular species are likely to gain electrons. In the exercise, CN and NO molecules were compared. CN, with a lower energy LUMO, is more likely to gain an electron than NO. Similarly, \(\mathrm{N}_{2}^{2+}\) has a lower energy LUMO compared to \(\mathrm{O}_{2}^{2+}\), due in part to a higher effective nuclear charge, making it more electron-accepting.