---
title: Bonding Mechanisms
module: Molecules and Chemical Bonding
moduleNumber: 1
lessonNumber: 1
order: 101
summary: >
  A molecule forms when the total energy of two atoms drops below the energy of
  the separated pair. This lesson works through the four mechanisms that produce
  that minimum: the ionic bond from charge transfer, the covalent bond from
  shared electron wave functions, the metallic bond, and the weak dipole-dipole
  and hydrogen bonds, computing bond lengths and dissociation energies for NaCl,
  H₂, and H₂⁺.
topics: [Molecules and Chemical Bonding]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 9 — Molecular Structure and Spectra; §9-1 The Ionic Bond, §9-2 The Covalent Bond, §9-3 Other Bonding Mechanisms"
  - book: Kittel
    ref: "Ch. 3 — Crystal Binding and Elastic Constants"
---

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](/condensed-matter/crystal-structure/structure-of-solids) 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

$$
\Delta E_{\text{ion}} = 4.34\ \text{eV} - 3.62\ \text{eV} = 0.72\ \text{eV}.
$$

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 $-ke^2/r$, 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](/atomic-physics/many-electron-atoms/periodic-table-atomic-spectra) 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

$$
U(r) = -\frac{ke^2}{r} + \frac{A}{r^n} + \Delta E_{\text{ion}}.
$$

> **Definition (Dissociation energy and bond length).** The **dissociation
> energy** $E_d$ is the energy required to separate a molecule into its
> constituent neutral atoms; equivalently, the depth of the potential-energy
> minimum measured from the separated-atom energy. The **equilibrium
> separation** $r_0$ is the value of $r$ at which $U(r)$ is minimum, so that
> $\d U/\d r = 0$ there.

$$
% caption: The ionic potential energy is the sum of a steep exclusion-principle
% repulsion and a long-range Coulomb attraction; their sum has a minimum at the
% bond length r_0 (KCl: r_0 near 0.27 nm, E_d near 4.4 eV).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (0,-2.6) -- (0,3.0) node[left] {$U(r)$};
  \draw[->, black] (0,0) -- (7.2,0) node[below] {$r$};
  % repulsion A/r^n (steep, positive)
  \draw[acc, thick, domain=0.85:2.2, samples=60, variable=\x]
    plot ({\x},{2.4/(\x*\x*\x*\x*\x)});
  \node[acc, anchor=west] at (1.15,2.4) {exclusion repulsion};
  % Coulomb attraction -k/r
  \draw[black, thick, dashed, domain=1.0:6.8, samples=60, variable=\x]
    plot ({\x},{-2.6/\x});
  \node[black, anchor=west] at (4.4,-0.75) {Coulomb attraction};
  % total energy: sum with a well
  \draw[black, very thick, domain=0.92:6.8, samples=120, variable=\x]
    plot ({\x},{2.4/(\x*\x*\x*\x*\x) - 2.6/\x});
  \node[black, anchor=south west] at (3.5,0.35) {total energy};
  % equilibrium marker
  \fill[black] (1.72,-1.28) circle (2pt);
  \draw[black, dashed] (1.72,-1.28) -- (1.72,0) node[above right] {$r_0$};
  \draw[black, dashed] (1.72,-1.28) -- (0,-1.28) node[left] {$E_d$};
\end{tikzpicture}
$$

**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

$$
-\frac{ke^2}{r_0} = -\frac{(8.99\times10^9\ \text{N·m}^2/\text{C}^2)(1.60\times10^{-19}\ \text{C})^2}{1.93\times10^{-10}\ \text{m}} = -1.19\times10^{-18}\ \text{J} = -7.45\ \text{eV}.
$$

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 $r_0$.

$$
% caption: Charge-transfer energy ledger for an alkali halide such as NaCl:
% ionization plus affinity is a small net cost, the Coulomb term is the large
% drop, and core overlap lifts the floor slightly to the bound state.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-3.6) -- (0,1.6) node[above, black] {energy};
  % neutral atoms at 0
  \draw[acc, very thick] (0.3,0) -- (2.2,0);
  \node[acc, anchor=south] at (1.2,0.05) {neutral atoms};
  % ion pair (net cost up)
  \draw[black, dashed] (2.2,0) -- (2.2,0.7);
  \draw[acc, very thick] (2.2,0.7) -- (4.0,0.7);
  \node[acc, anchor=south] at (3.1,0.75) {ion pair};
  % Coulomb drop
  \draw[black!70, very thick, ->] (4.0,0.7) -- (4.0,-3.2);
  \node[black!70, anchor=west] at (4.15,-1.3) {Coulomb attraction};
  % after Coulomb + repulsion floor
  \draw[acc, very thick] (4.0,-3.2) -- (5.6,-3.2);
  \draw[black, dashed] (5.6,-3.2) -- (5.6,-2.7);
  \draw[acc, very thick] (5.6,-2.7) -- (7.2,-2.7);
  \node[acc, anchor=north] at (6.4,-2.75) {bound molecule};
\end{tikzpicture}
$$

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, $p_{\text{ionic}} = 3.78\times10^{-29}\
\text{C·m}$, while the measured moment is $3.00\times10^{-29}\ \text{C·m}$. 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.

$$
% caption: For two wells brought close, the symmetric combination concentrates
% electron density in the region between the nuclei; the antisymmetric
% combination has a node there and leaves that region nearly empty.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- symmetric ---
  \begin{scope}
    \draw[black] (-2.4,0) -- (2.4,0);
    \fill[black] (-1.0,0) circle (2.4pt);
    \fill[black] (1.0,0) circle (2.4pt);
    \draw[acc, very thick, domain=-2.3:2.3, samples=90, variable=\x]
      plot ({\x},{1.5*exp(-(\x-1.0)*(\x-1.0)/0.6) + 1.5*exp(-(\x+1.0)*(\x+1.0)/0.6)});
    \node[acc, anchor=south] at (0,1.7) {symmetric};
    \node[black, anchor=north] at (0,-0.15) {large density between};
  \end{scope}
  % --- antisymmetric ---
  \begin{scope}[xshift=6.4cm]
    \draw[black] (-2.4,0) -- (2.4,0);
    \fill[black] (-1.0,0) circle (2.4pt);
    \fill[black] (1.0,0) circle (2.4pt);
    \draw[acc, very thick, domain=-2.3:2.3, samples=120, variable=\x]
      plot ({\x},{1.5*exp(-(\x-1.0)*(\x-1.0)/0.6) - 1.5*exp(-(\x+1.0)*(\x+1.0)/0.6)});
    \node[acc, anchor=south] at (0,1.7) {antisymmetric};
    \node[black, anchor=north] at (0,-0.15) {node between};
  \end{scope}
\end{tikzpicture}
$$

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
$U_p = +ke^2/r$ gives the total energy.

$$
% caption: For H₂⁺ only the symmetric branch U_p + E_S develops a minimum (bond
% length 0.106 nm, binding 2.7 eV); the antisymmetric branch U_p + E_A falls
% monotonically and never binds.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-2.2) -- (0,2.6) node[left] {energy};
  \draw[->, black] (0,0) -- (6.6,0) node[below] {$r$};
  % antibonding branch: monotonic decreasing to a positive tail then flat
  \draw[black, thick, dashed, domain=0.7:6.3, samples=90, variable=\x]
    plot ({\x},{2.4*exp(-\x/1.4)});
  \node[black, anchor=west] at (2.3,1.05) {$U_p + E_A$ antibonding};
  % bonding branch: well
  \draw[acc, very thick, domain=0.75:6.3, samples=120, variable=\x]
    plot ({\x},{6.0/(\x*\x) - 4.2/\x});
  \node[acc, anchor=north west] at (2.9,-0.55) {$U_p + E_S$ bonding};
  \fill[acc] (1.43,-0.735) circle (2pt);
  \draw[black, dashed] (1.43,-0.735) -- (1.43,0) node[above] {$r_0$};
\end{tikzpicture}
$$

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**.

> **Definition (Bonding and antibonding orbitals).** When two atomic orbitals
> overlap, they combine into a lower-energy **bonding orbital** (symmetric,
> concentrating charge between the nuclei) and a higher-energy **antibonding
> orbital** (antisymmetric, with a node between the nuclei). Like an atomic
> orbital, each holds at most two electrons of opposite spin.

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](/condensed-matter/free-electron-fermi-gas/free-electron-gas-and-conduction)
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.

$$
% caption: Permanent dipoles orient so that each charge sits nearest an opposite
% charge on its neighbor, giving a net attraction that falls off as 1/r^4.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \x in {0,2.0,4.0,6.0}{
    \draw[acc, thick, ->] (\x,-0.5) -- (\x,0.7);
    \node[acc, anchor=south] at (\x,0.72) {$p$};
    \fill[black] (\x,0.7) circle (1.6pt);
    \fill[black] (\x,-0.5) circle (1.6pt);
    \node[anchor=west, font=\scriptsize] at (\x+0.06,0.62) {pos};
    \node[anchor=west, font=\scriptsize] at (\x+0.06,-0.42) {neg};
  }
  \draw[black, <->] (0.35,-1.05) -- (1.65,-1.05) node[midway, below] {attraction};
\end{tikzpicture}
$$

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.

$$
% caption: In ice each water molecule hydrogen-bonds to four neighbors, the
% proton sitting between the donor oxygen and an acceptor lone pair.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % central oxygen
  \fill[black!70] (0,0) circle (5pt);
  \node[black!70, anchor=east] at (-0.2,0) {O};
  % two covalent O-H bonds
  \draw[black, thick] (0,0) -- (1.2,0.9);
  \draw[black, thick] (0,0) -- (1.2,-0.9);
  \fill[acc] (1.2,0.9) circle (3pt);
  \fill[acc] (1.2,-0.9) circle (3pt);
  \node[acc, anchor=south west] at (1.25,0.9) {H};
  \node[acc, anchor=north west] at (1.25,-0.9) {H};
  % neighbor oxygen accepting a hydrogen bond (dashed)
  \fill[black!70] (3.4,0.9) circle (5pt);
  \node[black!70, anchor=west] at (3.6,0.9) {O};
  \draw[black, thick, dashed] (1.55,0.9) -- (3.2,0.9);
  \node[black, anchor=south] at (2.45,1.25) {hydrogen bond};
\end{tikzpicture}
$$

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.

$$
% caption: The four bond types on one energy-length chart: dissociation energy
% (vertical) against bond length (horizontal), each point marking a typical
% molecule, spanning two orders of magnitude in binding energy.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black] {bond length};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black] {dissociation energy};
  % points (x = length, y = energy scaled)
  \fill[acc] (1.3,4.0) circle (2.6pt);
  \node[acc, anchor=south] at (1.3,4.1) {covalent};
  \fill[acc] (3.4,3.8) circle (2.6pt);
  \node[acc, anchor=south] at (3.4,3.9) {ionic};
  \fill[acc] (4.2,2.5) circle (2.6pt);
  \node[acc, anchor=west] at (4.35,2.5) {metallic};
  \fill[black] (4.0,1.0) circle (2.6pt);
  \node[black, anchor=west] at (4.15,1.0) {hydrogen};
  \fill[black] (5.6,0.4) circle (2.6pt);
  \node[black, anchor=west] at (5.75,0.4) {van der Waals};
\end{tikzpicture}
$$

| Bond type | Origin | Range | Typical energy |
| --- | --- | --- | --- |
| Ionic | charge transfer, Coulomb attraction | $1/r^2$ force | $3$–$5\ \text{eV}$ |
| Covalent | shared electron wave functions | short, saturable | $2$–$5\ \text{eV}$ |
| Metallic | delocalized valence electrons | lattice-wide | $1$–$4\ \text{eV}$ |
| Hydrogen | proton shared between polar molecules | $1/r^4$ force | $\sim 0.5\ \text{eV}$ |
| van der Waals | correlated instantaneous dipoles | $1/r^7$ force | $0.01$–$0.1\ \text{eV}$ |

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](/condensed-matter/molecules-and-bonding/molecular-orbitals-and-h2-plus)
makes the $\text{H}_2^+$ splitting quantitative through the variational secular
equation, and the [rotational and vibrational
spectra](/condensed-matter/molecular-spectra/rotational-vibrational-spectra)
excite the finished bond to measure $r_0$ and the force constant directly.