You probably recall from chemistry that a covalent molecular bond, such as the bond between the two atoms in molecules such as $\text{H}{}_2$ and $\text{O}{}_2$, is a bond in which the electrons are shared between the atoms. The basic idea of covalent bonding can be understood with a one-dimensional quantum-mechanical model.

The simplest molecule, the hydrogen molecular ion ${\text{H}}_2{}^{+}$, consists of two protons and one electron. Although it seems surprising that such a system could be stable, the two protons form a molecular bond with one electron. This is the simplest covalent bond.

How can we model the ${\text{H}}_2{}^{+}$, ion? To begin, FIGURE 40.27a shows a one-dimensional model of a hydrogen atom in which the electron’s Coulomb potential energy, with its $1\text{/}r$ dependence, has been approximated by a finite potential well of width 0.10 nm $(\approx 2a_{\text{B}})$ and depth 24.2 eV. You learned in Chapter 38 that an electron in the ground state of the Bohr hydrogen atom orbits the proton with radius $r_1=a_{\text{B}}$ (the Bohr radius) and energy $E_1=-13.6\text{eV}$. A numerical solution of the Schrödinger equation finds that the ground-state energy of this finite potential well is $E_1=-13.6\text{eV}$. This model of a hydrogen atom is very oversimplified, but it does have the correct size and ground-state energy.

FIGURE 40.27 A molecule can be modeled as two closely spaced potential wells, one representing each atom.

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We can model ${\text{H}}_2{}^{+}$, by bringing two of these potential wells close together. The molecular bond length of ${\text{H}}_2{}^{+}$, is known to be $\approx 0.12\text{nm,}$ so FIGURE 40.27b shows potential wells with 0.12 nm between their centers. This is a model of ${\text{H}}_2{}^{+}$, not a complete $\text{H}{}_2$ molecule, because this is the potential energy of a single electron. (Modeling $\text{H}{}_2$ is more complex because we would need to consider the repulsion between the two electrons.)

FIGURE 40.28 shows the allowed energies, wave functions, and probability densities for an electron with this potential energy. The $n=1$ wave function has a high probability of being found within the classically forbidden region between the two protons. In other words, an electron in this quantum state really is “shared” by the protons and spends most of its time between them.

FIGURE 40.28 The wave functions and probability densities of the electron in ${\text{H}}_{\text{2}}{}^{+}$.

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In contrast, an electron in the $n=2$ energy level has zero probability of being found between the two protons because the $n=2$ wave function has a node at the center. The probability density shows that an $n=2$ electron is “owned” by one proton or the other rather than being shared.

To learn the consequences of these wave functions we need to calculate the total energy of the molecule: $E_{\text{mol}}=E_{\text{p}-\text{p}}+E_{\text{elec}}$. The $n=1$ and $n=2$ energies shown in Figure 40.28 are the energies $E_{\text{elec}}$ of the electron. At the same time, the protons repel each other and have electric potential energy $E_{\text{p}-\text{p}}$. It’s not hard to calculate that $E_{\text{p}-\text{p}}=12.0\text{eV}$ for two protons separated by 0.12 nm. Thus

The $n=1$ molecular energy is less than zero, showing that this is a bound state. The $n=1$ wave function is called a bonding molecular orbital. Although the protons repel each other, the shared electron provides sufficient “glue” to hold the system together. The $n=2$ molecular energy is positive, so this is not a bound state. The system would be more stable as a hydrogen atom and a distant proton. The $n=2$ wave function is called an antibonding molecular orbital.

Both $E_{\text{elec}}$ and $E_{\text{p}-\text{p}}$ depend on the separation between the protons, which we assumed to be 0.12 nm in this calculation. If we were to calculate and graph $E_{\text{mol}}$ for many different values of the proton separation, the graph would look like the molecular-bond energy curve shown in Figure 40.23. In other words, a molecular bond has an equilibrium length where the bond energy is a minimum because of the interplay between $E_{\text{p}-\text{p}}$ and $E_{\text{elec}}$.

Although real molecular wave functions are more complex than this one-dimensional model, the $n=1$ wave function captures the essential idea of a covalent bond. Notice that a “classical” molecule cannot have a covalent bond because the electron would not be able to exist in the classically forbidden region. Covalent bonds can be understood only within the context of quantum mechanics. In fact, the explanation of molecular bonds was one of the earliest successes of quantum mechanics.