The Hydrogen Molecule, Exchange, and Hybridization

The molecular-orbital treatment of $H_{2}^{+}$ used one electron. The neutral hydrogen molecule $H_{2}$ 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 $σ_{g}$ 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{k e ^{2}}{r _{12}} + \frac{k e ^{2}}{R}, \hat{h} (i) = - \frac{ℏ ^{2}}{2 m} \nabla_{i}^{2} - \frac{k e ^{2}}{r _{i A}} - \frac{k e ^{2}}{r _{i B}} .

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

The Heitler-London valence-bond function

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

Ψ_{\pm} (r_{1}, r_{2}) = N_{\pm} [ϕ_{A} (r_{1}) ϕ_{B} (r_{2}) \pm ϕ_{A} (r_{2}) ϕ_{B} (r_{1})],

symmetric ( $+$ , singlet) or antisymmetric ( $-$ , triplet) under $1 \leftrightarrow 2$ , with normalization $N_{\pm}^{2} = 1/ [2 (1 \pm S^{2})]$ and $S = ⟨ ϕ_{A} ∣ ϕ_{B} ⟩$ . Evaluating $⟨ Ψ_{\pm} ∣ \hat{H} ∣ Ψ_{\pm} ⟩$ gives

E_{\pm} (R) = 2 E_{1 s} + \frac{k e ^{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, $ϕ_{A}^{2}$ around $A$ and $ϕ_{B}^{2}$ 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 $ϕ_{A} (r_{1}) ϕ_{B} (r_{1}) ϕ_{A} (r_{2}) ϕ_{B} (r_{2})$ , a consequence of the indistinguishability of the electrons with no classical counterpart. For $H_{2}$ near the bond length $X < 0$ .

Because $X$ is negative, the symmetric spatial state $Ψ_{+}$ 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.

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.

The magnitude of the singlet-triplet gap defines an effective spin coupling. Mapping the two levels onto the spin Hamiltonian $\hat{H}_{spin} = - 2 J S_{1} \cdot S_{2}$ , whose eigenvalues are $- \frac{3}{4} \cdot (- 2 J)$ for the singlet and $+ \frac{1}{4} \cdot (- 2 J)$ for the triplet, gives

J = \frac{1}{2} (E_{singlet} - E_{triplet}) .

For $H_{2}$ the singlet lies below the triplet, so $J < 0$ : the exchange coupling is antiferromagnetic, favoring antiparallel spins. The same $- 2 J S_{1} \cdot S_{2}$ form, with the opposite sign, becomes the Heisenberg exchange that drives 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 $σ_{g} \propto ϕ_{A} + ϕ_{B}$ ,

Ψ_{MO} = σ_{g} (r_{1}) σ_{g} (r_{2}) \propto [ϕ_{A} (r_{1}) + ϕ_{B} (r_{1})] [ϕ_{A} (r_{2}) + ϕ_{B} (r_{2})] .

Expanding the product exposes the difference from Heitler-London,

Ψ_{MO} \propto covalent ϕ_{A} (1) ϕ_{B} (2) + ϕ_{A} (2) ϕ_{B} (1) + ionic ϕ_{A} (1) ϕ_{A} (2) + ϕ_{B} (1) ϕ_{B} (2) .

The molecular-orbital function weights the ionic configurations — both electrons on the same proton, that is $H^{-} + H^{+}$ — 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.

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.

Quantitatively, with a fixed hydrogen $1 s$ orbital: experiment gives $D_{e} = 4.75 eV$ at $R_{e} = 0.074 nm$ ; Heitler-London gives $D_{e} \approx 3.14 eV$ at $R_{e} \approx 0.087 nm$ ; the simple molecular orbital gives $D_{e} \approx 2.7 eV$ . 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 $1 s$ orbital, so its molecule has no shape beyond a bond length. Carbon, nitrogen, and oxygen bond through $2 s$ and $2 p$ 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 $2 s$ and $2 p$ orbitals into equivalent directed hybrid orbitals before bonding.

sp hybrids — one $s$ mixes with one $p$ , giving two hybrids at $180$ degrees. The remaining two $p$ orbitals form $π$ bonds. Linear molecules such as acetylene and carbon dioxide use $sp$ 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 $sp^{2}$ ; the perpendicular $p$ orbitals build the delocalized $π$ 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 $sp^{3}$ .

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

ψ_{1, 2, 3, 4} = \frac{1}{2} (ϕ_{2 s} \pm ϕ_{2 p_{x}} \pm ϕ_{2 p_{y}} \pm ϕ_{2 p_{z}}),

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

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.

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).

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 $sp^{3}$ bonds of carbon are what build the diamond and zincblende structures of the semiconductors, and the planar $sp^{2}$ network with its perpendicular $π$ orbitals is the electronic origin of graphene.

Summary

The two-electron Hamiltonian adds electron-electron repulsion $k e^{2} / r_{12}$ to two $H_{2}^{+}$ problems; the antisymmetry of the total state ties spatial symmetry to total spin (symmetric spatial ↔ singlet, antisymmetric ↔ triplet).
The Heitler-London energy $E_{\pm} = 2 E_{1 s} + k e^{2} / R + (C \pm X) / (1 \pm S^{2})$ splits singlet from triplet by the exchange energy $X$ ; since $X < 0$ for $H_{2}$ the singlet binds, and the mapping to $- 2 J S_{1} \cdot S_{2}$ gives $J = \frac{1}{2} (E_{singlet} - E_{triplet}) < 0$ .
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 $sp$ (linear, $18 0^{\circ}$ ), $sp^{2}$ (trigonal, $12 0^{\circ}$ ), and $sp^{3}$ (tetrahedral, $109. 5^{\circ}$ ) hybrids that set the directed geometry of covalent bonds.