---
title: The Hydrogen Molecule, Exchange, and Hybridization
module: Molecules and Chemical Bonding
moduleNumber: 1
lessonNumber: 3
order: 103
summary: >
  Adding the second electron turns the one-electron ion into the two-electron
  hydrogen molecule, where electron-electron repulsion and the Pauli principle
  govern the bond. This lesson contrasts the Heitler-London valence-bond and
  molecular-orbital wave functions, derives the singlet-triplet splitting as an
  exchange energy, shows why naive molecular orbitals fail at dissociation, and
  builds the sp, sp², and sp³ hybrids that fix the directed geometry of covalent
  bonds.
topics: [Molecules and Chemical Bonding]
draft: false
sources:
  - book: Hook & Hall
    ref: "Ch. 1 — The Physics of Bonding"
  - book: Ashcroft & Mermin
    ref: "Ch. 32 — Electron Interactions and Magnetic Structure"
  - book: Kittel
    ref: "Ch. 11 — Diamagnetism and Paramagnetism; exchange interaction"
---

The [molecular-orbital
treatment](/condensed-matter/molecules-and-bonding/molecular-orbitals-and-h2-plus)
of $\text{H}_2^+$ used one electron. The neutral hydrogen molecule $\text{H}_2$
has two, and their mutual Coulomb repulsion together with the exclusion principle
turns the bond into a genuine many-body problem. Two approximate wave functions
compete: the molecular-orbital function, which places both electrons in the same
$\sigma_g$ orbital, and the Heitler-London valence-bond function, which keeps one
electron near each nucleus. Comparing them isolates the **exchange energy** that
splits spin singlet from spin triplet, and it exposes the electron correlation
that neither simple function captures.

## The two-electron Hamiltonian

With electrons labeled $1$ and $2$ and protons $A,B$ separated by $R$,

$$
\hat H = \hat h(1) + \hat h(2) + \frac{ke^2}{r_{12}} + \frac{ke^2}{R},
\qquad
\hat h(i) = -\frac{\hbar^2}{2m}\nabla_i^2 - \frac{ke^2}{r_{iA}} - \frac{ke^2}{r_{iB}}.
$$

The one-electron parts $\hat h(i)$ are the $\text{H}_2^+$ Hamiltonians already
solved. The new term $ke^2/r_{12}$ couples the electrons and prevents an exact
product solution. The total wave function must be antisymmetric under exchange of
the two electrons, including spin.

> **Definition (Spin-space factorization).** For two electrons the antisymmetric
> total state factors into a spatial part and a spin part of opposite symmetry:
> a symmetric spatial function pairs with the antisymmetric **singlet** spin
> state ($S=0$), and an antisymmetric spatial function pairs with the symmetric
> **triplet** ($S=1$). The Pauli principle thus ties the spatial distribution of
> the electrons to their total spin.

## The Heitler-London valence-bond function

The valence-bond ansatz builds the spatial state from products in which one
electron sits on each atom,

$$
\Psi_\pm(\vec r_1,\vec r_2) = N_\pm\left[\phi_A(\vec r_1)\phi_B(\vec r_2) \pm \phi_A(\vec r_2)\phi_B(\vec r_1)\right],
$$

symmetric ($+$, singlet) or antisymmetric ($-$, triplet) under $1\leftrightarrow
2$, with normalization $N_\pm^2 = 1/[2(1\pm S^2)]$ and $S = \langle\phi_A|\phi_B
\rangle$. Evaluating $\langle\Psi_\pm|\hat H|\Psi_\pm\rangle$ gives

$$
E_\pm(R) = 2E_{1s} + \frac{ke^2}{R} + \frac{C \pm X}{1 \pm S^2},
$$

where two two-electron integrals appear.

- **Coulomb energy** $C$ — the classical electrostatic interaction of the two
  charge clouds, $\phi_A^2$ around $A$ and $\phi_B^2$ around $B$, including their
  attraction to the opposite nucleus and their mutual repulsion. It is what a
  purely classical model of two overlapping charge distributions would give.
- **Exchange energy** $X$ — the integral over the interference density
  $\phi_A(\vec r_1)\phi_B(\vec r_1)\,\phi_A(\vec r_2)\phi_B(\vec r_2)$, a
  consequence of the indistinguishability of the electrons with no classical
  counterpart. For $\text{H}_2$ near the bond length $X < 0$.

Because $X$ is negative, the symmetric spatial state $\Psi_+$ lies lower: the
singlet is the bonding ground state, and the triplet is repulsive. The exchange
energy, not the classical Coulomb energy, decides which spin state binds.

$$
% caption: The two spatial symmetries split by twice the exchange energy: the
% symmetric singlet binds while the antisymmetric triplet is repulsive, so the
% ground state of H2 has paired, antiparallel spins.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % reference level
  \draw[black, thick] (0.3,0) -- (2.1,0);
  \node[black, anchor=east] at (0.25,0) {two atoms};
  % triplet up
  \draw[black, thick, dashed] (4.4,1.25) -- (6.2,1.25);
  \node[black, anchor=west] at (6.25,1.25) {triplet, repulsive};
  % singlet down
  \draw[acc, thick] (4.4,-1.15) -- (6.2,-1.15);
  \node[acc, anchor=west] at (6.25,-1.15) {singlet, bonding};
  % connectors
  \draw[black] (2.1,0) -- (4.4,1.25);
  \draw[black] (2.1,0) -- (4.4,-1.15);
  % splitting bracket
  \draw[black, <->] (4.0,-1.15) -- (4.0,1.25);
  \node[black, anchor=east] at (3.95,0.05) {split};
\end{tikzpicture}
$$

The magnitude of the singlet-triplet gap defines an effective spin coupling.
Mapping the two levels onto the spin Hamiltonian $\hat H_{\text{spin}} = -2J\,
\vec S_1\cdot\vec S_2$, whose eigenvalues are $-\tfrac34\cdot(-2J)$ for the
singlet and $+\tfrac14\cdot(-2J)$ for the triplet, gives

$$
J = \tfrac12\left(E_{\text{singlet}} - E_{\text{triplet}}\right).
$$

For $\text{H}_2$ the singlet lies below the triplet, so $J < 0$: the exchange
coupling is antiferromagnetic, favoring antiparallel spins. The same
$-2J\,\vec S_1\cdot\vec S_2$ form, with the opposite sign, becomes the Heisenberg
exchange that drives [ferromagnetism](/condensed-matter/magnetism/exchange-and-ferromagnetism)
when parallel spins are favored. Exchange is a Coulomb effect wearing the disguise
of a spin interaction; no magnetic force enters.

## Molecular orbital versus valence bond

The molecular-orbital function places both electrons in the bonding orbital
$\sigma_g \propto \phi_A + \phi_B$,

$$
\Psi_{\text{MO}} = \sigma_g(\vec r_1)\,\sigma_g(\vec r_2)
\propto \big[\phi_A(\vec r_1)+\phi_B(\vec r_1)\big]\big[\phi_A(\vec r_2)+\phi_B(\vec r_2)\big].
$$

Expanding the product exposes the difference from Heitler-London,

$$
\Psi_{\text{MO}} \propto \underbrace{\phi_A(1)\phi_B(2) + \phi_A(2)\phi_B(1)}_{\text{covalent}}
+ \underbrace{\phi_A(1)\phi_A(2) + \phi_B(1)\phi_B(2)}_{\text{ionic}}.
$$

The molecular-orbital function weights the **ionic** configurations — both
electrons on the same proton, that is $\text{H}^- + \text{H}^+$ — equally with
the covalent ones. Heitler-London keeps only the covalent terms. Near the bond
length the truth includes some ionic character, so the valence-bond function
underbinds slightly; but as $R\to\infty$ the molecular-orbital function is
qualitatively wrong.

> **Result (Dissociation failure of the naive molecular orbital).** As the nuclei
> separate, $\Psi_{\text{MO}}$ retains fifty percent ionic character, so it
> dissociates to an unphysical equal mixture of $\text{H}^- + \text{H}^+$ and
> $\text{H} + \text{H}$. Its energy therefore approaches the average of the ionic
> and neutral limits rather than the correct energy of two neutral atoms. The
> valence-bond function dissociates correctly. Neither is right at all $R$; mixing
> the ground and doubly-excited molecular configurations (configuration
> interaction) interpolates between them and is the molecular form of **electron
> correlation**.

$$
% caption: Potential-energy curves for H2: the Heitler-London valence bond and
% the simple molecular orbital both underbind relative to experiment, and the
% molecular orbital rises to an unphysical limit at large separation.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-2.6) -- (0,1.2) node[left] {$E(R)$};
  \draw[->, black] (0,0) -- (7.2,0) node[below] {$R$};
  \draw[black, dashed] (0,0) -- (7.2,0);
  % experiment: deepest well
  \draw[black, very thick, domain=0.9:6.8, samples=140, variable=\x]
    plot ({\x},{9.5/(\x*\x) - 7.4/\x});
  \node[black, anchor=west] at (4.6,-0.35) {experiment};
  % Heitler-London: shallower well, correct flat limit
  \draw[acc, very thick, domain=0.95:6.8, samples=140, variable=\x]
    plot ({\x},{8.2/(\x*\x) - 5.6/\x});
  \node[acc, anchor=north] at (2.5,-1.05) {Heitler-London};
  % MO: shallower, rises at large R
  \draw[black, very thick, dashed, domain=1.0:6.8, samples=140, variable=\x]
    plot ({\x},{7.6/(\x*\x) - 5.0/\x + 0.11*\x});
  \node[black, anchor=south west] at (3.4,0.15) {molecular orbital};
\end{tikzpicture}
$$

Quantitatively, with a fixed hydrogen $1s$ orbital: experiment gives
$D_e = 4.75\ \text{eV}$ at $R_e = 0.074\ \text{nm}$; Heitler-London gives
$D_e \approx 3.14\ \text{eV}$ at $R_e \approx 0.087\ \text{nm}$; the simple
molecular orbital gives $D_e \approx 2.7\ \text{eV}$. Optimizing the orbital
exponent and adding configuration interaction closes the gap to experiment, at
the cost of the one-orbital simplicity.

## Hybridization and directed bonds

Hydrogen bonds with a spherical $1s$ orbital, so its molecule has no shape beyond
a bond length. Carbon, nitrogen, and oxygen bond through $2s$ and $2p$ orbitals,
and the observed geometries — the tetrahedral $109.5$-degree angles of methane
and diamond, the $120$-degree trigonal planar angles of graphene and ethylene,
the $180$-degree linear geometry of acetylene — do not match the $90$-degree
angles between bare $p$ orbitals. The resolution is that the atom mixes its $2s$
and $2p$ orbitals into equivalent directed **hybrid** orbitals before bonding.

> **Definition (Hybrid orbital).** A hybrid orbital is a normalized linear
> combination of the $s$ and $p$ atomic orbitals on one atom, oriented to
> maximize overlap with a neighbor. Mixing $s$ with $n$ of the three $p$ orbitals
> produces $n+1$ equivalent $\mathrm{sp}^n$ hybrids, each pointing along a bond
> direction.

- **sp hybrids** — one $s$ mixes with one $p$, giving two hybrids at $180$
  degrees. The remaining two $p$ orbitals form $\pi$ bonds. Linear molecules such
  as acetylene and carbon dioxide use $\mathrm{sp}$ carbon.
- **sp² hybrids** — one $s$ with two $p$, giving three coplanar hybrids at $120$
  degrees, with the leftover $p$ perpendicular to the plane. Graphene, benzene,
  and ethylene are $\mathrm{sp}^2$; the perpendicular $p$ orbitals build the
  delocalized $\pi$ system.
- **sp³ hybrids** — one $s$ with all three $p$, giving four hybrids toward the
  corners of a tetrahedron at $109.5$ degrees. Methane, and the diamond lattice
  of carbon, are $\mathrm{sp}^3$.

The four $\mathrm{sp}^3$ hybrids on carbon, written in terms of the atomic
orbitals, are

$$
\psi_{1,2,3,4} = \tfrac12\left(\phi_{2s} \pm \phi_{2p_x} \pm \phi_{2p_y} \pm \phi_{2p_z}\right),
$$

with the sign patterns $({+}{+}{+}),\ ({+}{-}{-}),\ ({-}{+}{-}),\ ({-}{-}{+})$
selecting the four tetrahedral directions.

$$
% caption: The four sp-cubed hybrid lobes point to the corners of a tetrahedron,
% the geometry of methane and of the diamond lattice, with equal interbond
% angles of 109.5 degrees.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % central atom
  \fill[black] (0,0) circle (2.6pt);
  \node[black, anchor=north east] at (-0.05,-0.05) {C};
  % four tetrahedral directions projected to 2D
  \draw[acc, very thick, ->] (0,0) -- (0,1.9);
  \draw[acc, very thick, ->] (0,0) -- (-1.7,-1.0);
  \draw[acc, very thick, ->] (0,0) -- (1.7,-1.0);
  \draw[acc, very thick, ->] (0,0) -- (0.55,-0.6);
  % small lobe blobs at tips
  \fill[acc!25] (0,1.9) circle (0.3);
  \fill[acc!25] (-1.7,-1.0) circle (0.3);
  \fill[acc!25] (1.7,-1.0) circle (0.3);
  \fill[acc!25] (0.55,-0.6) circle (0.22);
  \node[acc, anchor=south] at (0,2.2) {hybrid};
  \node[black, anchor=north] at (0,-1.5) {tetrahedral};
\end{tikzpicture}
$$

$$
% caption: The sp-squared hybrids lie in a plane at 120 degrees (trigonal, the
% graphene and ethylene geometry); the sp hybrids point along one axis at 180
% degrees (linear, the acetylene geometry).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % --- sp2 trigonal ---
  \begin{scope}
    \fill[black] (0,0) circle (2.4pt);
    \draw[acc, very thick, ->] (0,0) -- (0,1.7);
    \draw[acc, very thick, ->] (0,0) -- (-1.47,-0.85);
    \draw[acc, very thick, ->] (0,0) -- (1.47,-0.85);
    \node[acc, anchor=south] at (0,1.9) {trigonal};
    \node[black, anchor=north] at (0,-1.15) {three at 120 degrees};
  \end{scope}
  % --- sp linear ---
  \begin{scope}[xshift=6.4cm]
    \fill[black] (0,0) circle (2.4pt);
    \draw[acc, very thick, ->] (0,0) -- (1.9,0);
    \draw[acc, very thick, ->] (0,0) -- (-1.9,0);
    \node[acc, anchor=south] at (0,0.35) {linear};
    \node[black, anchor=north] at (0,-0.5) {two at 180 degrees};
  \end{scope}
\end{tikzpicture}
$$

Hybridization is a choice of basis, not new physics: any complete set of orbitals
spans the same space, and the hybrids are the combinations that make the bonding
picture look like localized, directed sticks. The directed $\mathrm{sp}^3$ bonds
of carbon are what build the diamond and [zincblende
structures](/condensed-matter/crystal-structure/bravais-lattices-and-crystal-systems)
of the semiconductors, and the planar $\mathrm{sp}^2$ network with its
perpendicular $\pi$ orbitals is the electronic origin of
[graphene](/condensed-matter/nanostructures/graphene-and-dirac-materials).

## Summary

- The two-electron Hamiltonian adds electron-electron repulsion $ke^2/r_{12}$ to
  two $\text{H}_2^+$ problems; the antisymmetry of the total state ties spatial
  symmetry to total spin (symmetric spatial ↔ singlet, antisymmetric ↔ triplet).
- The Heitler-London energy $E_\pm = 2E_{1s} + ke^2/R + (C\pm X)/(1\pm S^2)$
  splits singlet from triplet by the exchange energy $X$; since $X<0$ for
  $\text{H}_2$ the singlet binds, and the mapping to $-2J\,\vec S_1\cdot\vec S_2$
  gives $J = \tfrac12(E_{\text{singlet}} - E_{\text{triplet}}) < 0$.
- The naive molecular orbital weights ionic and covalent configurations equally
  and dissociates incorrectly; configuration interaction between the two
  functions is the molecular form of electron correlation.
- Mixing $s$ and $p$ orbitals gives the $\mathrm{sp}$ (linear, $180^\circ$),
  $\mathrm{sp}^2$ (trigonal, $120^\circ$), and $\mathrm{sp}^3$ (tetrahedral,
  $109.5^\circ$) hybrids that set the directed geometry of covalent bonds.