Quantum Mechanics Course

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← Part VII/Exchange Interaction

6. Exchange Interaction

Reading time: ~24 minutes | Pages: 6

Purely quantum effect: energy depends on spin alignment without direct spin-spin interaction.

Origins of Exchange

Key insight: Antisymmetrization of spatial wave function affects energy

  • No direct force between spins required
  • Comes from Pauli exclusion + Coulomb repulsion
  • Affects where electrons can be (correlation)
  • Different spatial distributions → different energies

Two-Electron System

Total wave function = spatial × spin:

$$\Psi(\vec{r}_1, \vec{r}_2) = \psi(\vec{r}_1, \vec{r}_2) \chi(s_1, s_2)$$

Singlet (S = 0, antisymmetric spin):

$$\chi_S = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)$$

Must have symmetric spatial part:

$$\psi_S(\vec{r}_1, \vec{r}_2) = \frac{1}{\sqrt{2}}[\psi_a(\vec{r}_1)\psi_b(\vec{r}_2) + \psi_b(\vec{r}_1)\psi_a(\vec{r}_2)]$$

Triplet (S = 1, symmetric spin):

$$\chi_T \in \{|\uparrow\uparrow\rangle, \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle), |\downarrow\downarrow\rangle\}$$

Must have antisymmetric spatial part:

$$\psi_A(\vec{r}_1, \vec{r}_2) = \frac{1}{\sqrt{2}}[\psi_a(\vec{r}_1)\psi_b(\vec{r}_2) - \psi_b(\vec{r}_1)\psi_a(\vec{r}_2)]$$

Energy Difference

Energies for singlet and triplet:

$$E_S = E_0 + J, \quad E_T = E_0 - J$$

where $E_0$ is direct Coulomb energy, $J$ is exchange integral

Exchange integral:

$$J = \int\int \psi_a^*(\vec{r}_1)\psi_b^*(\vec{r}_2)\frac{e^2}{4\pi\epsilon_0|\vec{r}_1-\vec{r}_2|}\psi_b(\vec{r}_1)\psi_a(\vec{r}_2)d^3r_1d^3r_2$$

Always positive for electrons

Physical Interpretation

Triplet (parallel spins, $E_T = E_0 - J$):

  • Antisymmetric spatial part vanishes when $\vec{r}_1 = \vec{r}_2$
  • Electrons avoid each other
  • Lower Coulomb repulsion
  • Lower energy (more stable)

Singlet (antiparallel spins, $E_S = E_0 + J$):

  • Symmetric spatial part largest when $\vec{r}_1 = \vec{r}_2$
  • Electrons can be close together
  • Higher Coulomb repulsion
  • Higher energy

Heisenberg Exchange Hamiltonian

Effective spin-dependent interaction:

$$\hat{H}_{ex} = -2J\vec{S}_1\cdot\vec{S}_2$$

Using $\vec{S}_1\cdot\vec{S}_2 = \frac{1}{2}[S(S+1) - 3/2]$:

  • Triplet ($S=1$): $\vec{S}_1\cdot\vec{S}_2 = 1/4$ → $E_T = -J/2$
  • Singlet ($S=0$): $\vec{S}_1\cdot\vec{S}_2 = -3/4$ → $E_S = 3J/2$

Energy difference: $E_S - E_T = 2J$

Hund's First Rule Explained

Observation: Electrons in degenerate orbitals prefer parallel spins

Reason: Exchange interaction lowers energy

For $N$ electrons with parallel spins:

$$E_{ex} = -\sum_{i<j}J_{ij}$$

More parallel spins → more exchange energy lowering

Example: Carbon (2p²) has $^3P$ ground state, not $^1D$ or $^1S$

Ferromagnetism

Exchange interaction drives magnetic ordering:

  • J > 0 (ferromagnetic): Parallel spins favored → Fe, Co, Ni
  • J < 0 (antiferromagnetic): Antiparallel spins favored → MnO, FeO

Curie/Néel temperature:

$$k_B T_C \sim J$$

Above $T_C$: thermal energy overcomes exchange → paramagnetic

Molecular Bonding

Hydrogen molecule (H₂):

  • Singlet state: electrons with opposite spins can share space between nuclei
  • Forms covalent bond (bonding orbital)
  • Lower energy than two separate atoms

Triplet state:

  • Parallel spins → antisymmetric spatial part
  • Vanishes between nuclei (antibonding)
  • Higher energy → no bond formation

Exchange Hole

Reduced probability of finding two parallel-spin electrons near each other

Pair correlation function:

$$g(\vec{r}_1, \vec{r}_2) = |\psi(\vec{r}_1, \vec{r}_2)|^2$$

For antisymmetric state:

$$g(\vec{r}, \vec{r}) = 0$$

"Hole" in probability density around each electron

Density Functional Theory

Modern computational approach includes exchange:

$$E[n] = T[n] + V_{ext}[n] + J[n] + E_{xc}[n]$$

where $E_{xc}$ is exchange-correlation energy

  • Local density approximation (LDA)
  • Generalized gradient approximation (GGA)
  • Hybrid functionals (B3LYP, PBE0)

Key Takeaways

  • No classical analog: Purely quantum mechanical effect
  • Not a real force: Consequence of antisymmetrization
  • Spin-dependent: Energy depends on relative spin orientation
  • Ubiquitous: Affects chemistry, magnetism, bonding
  • Hund's rules: Explains why atoms prefer maximum spin
  • Covalent bonds: Singlet states allow electrons to share space
  • Magnetism: Exchange drives ferromagnetic ordering
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