Bonding Mechanisms

Two atoms bond when their combined energy, as a function of the separation $r$ of their nuclei, has a minimum below the energy of the two atoms infinitely far apart. The depth of that minimum is the dissociation energy $E_d$, the energy needed to pull the molecule apart; the location of the minimum is the equilibrium separation $r_0$, the bond length. Every bonding mechanism is a different physical route to the same shape: a long-range attraction that draws the atoms together and a short-range repulsion that stops them from collapsing. Four mechanisms cover almost every molecule and, as the next lesson shows, almost every solid.

The ionic bond

The cleanest case is the transfer of one electron from a metal atom to a halogen atom. Consider KCl. Removing the $4s$ electron from potassium costs its ionization energy $E_{\text{ion}}(\text{K})=4.34\text{eV}$; attaching that electron to chlorine returns its electron affinity $3.62\text{eV}$. The net cost of creating the ion pair ${\text{K}}^{+}+{\text{Cl}}^{-}$ from neutral atoms is

If that were the whole story KCl would not form. What pays it back is the Coulomb attraction of the two ions. At separation $r$ the electrostatic energy is $-k{e}^2/r$, which becomes more negative than $+0.72\text{eV}$ once $r$ drops below about $2\text{nm}$. From then on the ions accelerate toward each other.

Equilibrium requires a repulsion. As the closed-shell cores approach, their occupied orbitals begin to overlap, and the exclusion principle forces some core electrons into higher-energy states. This exclusion-principle repulsion rises steeply at small $r$ and is the dominant repulsion in nearly every molecule. Modeling it as a power law $A/r^n$, the total potential energy of the ion pair is

Worked example — bonding in NaF. Sodium has ionization energy $5.14\text{eV}$; fluorine has electron affinity $3.40\text{eV}$; the equilibrium separation of NaF is $r_0=0.193\text{nm}$. The net ionization cost is $\Delta E_{\text{ion}}=5.14-3.40=1.74\text{eV}$. The Coulomb energy at $r_0$ is

Adding the ionization cost, the net Coulomb-plus-ionization energy at $r_0$ is $-7.45+1.74=-5.71\text{eV}$. The measured dissociation energy is $4.99\text{eV}$, so the exclusion-principle repulsion contributes $-5.71+4.99=0.72\text{eV}$ at equilibrium. Setting the net force to zero at $r_0$ then fixes the exponent $n\approx 10$ and the constant $A$.

The ionic bond is a ledger of four terms. Ionizing the metal atom costs energy; attaching the electron to the halogen returns the electron affinity; the Coulomb attraction of the resulting ions is the large negative term; and the exclusion-principle overlap adds a small positive correction at $r_0$.

The degree of ionic character is read off the electric dipole moment. A purely ionic bond would place charge $+e$ and $-e$ a distance $r_0$ apart, giving $p_{\text{ionic}}=e{r}_0$. For NaCl, $p_{\text{ionic}}=3.78\times 10^{-29}\text{C⋅m}$, while the measured moment is $3.00\times 10^{-29}\text{C⋅m}$. The ratio $3.00/3.78=0.79$ means the NaCl bond is about 79% ionic and 21% covalent — no real bond is purely one type.

The covalent bond

Ionic bonding fails for ${\text{H}}_2$: forming ${\text{H}}^{+}+{\text{H}}^{-}$ costs more than $12\text{eV}$, and no separation makes the total energy negative. The attraction between two hydrogen atoms is purely quantum-mechanical, arising from the sharing of electrons and the symmetry of their wave functions.

Start with one electron shared between two identical square wells. When the wells are far apart the electron's energy is the same whether its wave function is symmetric (${\psi }_S$) or antisymmetric (${\psi }_A$) about the midpoint. As the wells approach, the two parts of the wave function overlap and the two cases diverge: ${\psi }_S$ piles probability density between the wells, while ${\psi }_A$ has a node there and keeps density away.

Concentrated negative charge between the two protons is what holds them together. Only ${\psi }_S$ produces a stable bond. Applying this to the hydrogen molecule ion ${\text{H}}_2^{+}$ — one electron, two protons — the electronic energy splits into two branches as $r$ decreases. As $r\to 0$ the symmetric state approaches the $1s$ ground state of a $Z=2$ (helium) nucleus, with energy $-54.4\text{eV}$; the antisymmetric state approaches the $2p$ first excited state, with energy $-13.6\text{eV}$. Adding the proton-proton repulsion $U_p=+k{e}^2/r$ gives the total energy.

Numerically, $U_p+E_S$ has a minimum at $r_0=0.106\text{nm}$ with binding energy $2.7\text{eV}$; $U_p+E_A$ has no minimum and cannot bind. The symmetric branch is the bonding orbital, the antisymmetric branch the antibonding orbital.

Adding a second electron gives the neutral ${\text{H}}_2$ molecule. Both electrons occupy the bonding orbital with antiparallel spins (a spin singlet, $S=0$). The extra charge between the protons binds them more tightly: the binding energy rises from $2.7\text{eV}$ to $4.5\text{eV}$ and the bond length shrinks to $r_0=0.074\text{nm}$.

The exclusion principle explains what does not bond. Three hydrogen atoms cannot form ${\text{H}}_3$: the third electron cannot join the two already filling the bonding orbital with opposite spins, so it must occupy the antibonding orbital, whose repulsion outweighs the attraction. The ${\text{H}}_2$ bond is saturated. For the same reason two helium atoms do not form ${\text{He}}_2$: with four electrons, two must occupy the antibonding orbital, canceling the bond.

Metallic, dipole-dipole, and hydrogen bonds

Two further mechanisms complete the catalog. Metallic bonding, treated in the free-electron gas lesson, has no single-molecule version: valence electrons detach entirely and move through the whole lattice of positive ions.

The weak bonds are electrostatic attractions between dipoles. The field of a permanent dipole $p_1$ falls off as $1/r^3$, so the force on a nearby dipole falls as $1/r^4$ — much shorter range than the $1/r^2$ Coulomb force. A second dipole $p_2$ aligns itself along the field lines of the first, lowering its energy $U=-p_2\cdot E_d$. Polar molecules such as ${\text{H}}_2\text{O}$ attract one another this way.

When the dipole-dipole bond involves hydrogen — as in water — it is a hydrogen bond, viewable as a proton shared between two electronegative atoms. Hydrogen bonds (about $0.5\text{eV}$ per bond in water) cross-link giant biological molecules; the two strands of DNA are held together by hydrogen bonds weak enough to unzip during replication yet strong enough to hold shape otherwise.

Even two nonpolar molecules attract. The average dipole moment of a nonpolar molecule is zero, but its instantaneous moment is not, because the electrons are in constant motion. Fluctuating instantaneous dipoles in neighboring molecules correlate to produce attraction, with potential energy $\propto 1/r^6$ and force $\propto 1/r^7$. This is the van der Waals (or London dispersion) force, which acts between all atoms and molecules. It is the only bond available to the noble gases, and its weakness is why helium boils at $4.2\text{K}$ and does not solidify at atmospheric pressure at any temperature.

The five mechanisms occupy distinct regions of a bond-length versus dissociation-energy plane: covalent and ionic bonds are both deep, the metallic bond intermediate, and the hydrogen and van der Waals bonds progressively shallower and longer-ranged.

The exclusion-principle repulsion is common to all five: whatever draws the atoms together, it is the overlap of filled cores that ultimately sets the bond length. With the ground-state bond in place, the next lessons refine the covalent picture. The molecular-orbital method makes the ${\text{H}}_2^{+}$ splitting quantitative through the variational secular equation, and the rotational and vibrational spectra excite the finished bond to measure $r_0$ and the force constant directly.