Exchange Forces and Exchange Energy

The exchange integral is one of the most heavily used quantities in computational physics and chemistry. Every Hartree–Fock and density-functional calculation carries an exchange term — indeed, the accuracy of a DFT functional is judged largely by how well it approximates exchange (and correlation) energy, and the search for better exchange functionals is an active field. In magnetism, first-principles calculation of the exchange constants $\mathcal J_{ij}$ from electronic structure predicts Curie and Néel temperatures and designs new magnetic materials, including the interfacial Dzyaloshinskii–Moriya exchange that stabilizes skyrmions for spintronic memory. In quantum information, the exchange interaction between electron spins in coupled quantum dots is the physical mechanism behind two-qubit gates in semiconductor spin qubits — the same $2K$ singlet–triplet splitting of this article, tuned electrically in real time. And in cold-atom laboratories, exchange interactions between neutral atoms in optical lattices are engineered to simulate quantum magnetism directly, letting experimentalists watch the Heisenberg model of this article play out in a controllable setting.

Two identical particles do not carry two independent amplitudes — they share one, and that amplitude is either symmetric or antisymmetric under a swap. Where the amplitude is symmetric, the "particle 1 here, 2 there" contribution and its swapped partner add constructively near coincidence, so the particles are found closer together. Where it is antisymmetric, the spatial wavefunction vanishes when the two particles coincide, reducing the probability of finding them close together. That geometry is fixed before any interaction is switched on — it is the content of Symmetric and Antisymmetric States, which proved the mean-square separation shifts by $\mp 2|X_{ab}|^2$ purely from symmetrization. Now switch on a repulsive interaction. The state where electrons huddle together (symmetric space) pays more repulsion energy and sits higher; the state where they hold each other at arm's length (antisymmetric space) pays less and sits lower. The interaction has not become spin-dependent — but through the antisymmetry requirement, which ties spatial symmetry to spin, the energy has. This is the whole trick: an electrostatic, spin-blind Coulomb interaction ends up favoring one spin arrangement over another, and the energy scale is electrostatic (electron-volts), not magnetic (micro-electron-volts).

The figure is the thesis. A single spin-independent Coulomb term, evaluated in the two allowed symmetries, produces two energies separated by $2K$ — and because $K>0$ for the Coulomb kernel, the triplet is always the lower one. Historically this splitting was dramatic enough that the two families of helium lines, ortho and para, were mistaken for two distinct substances. There is one helium; the exchange integral just refuses to let its singlet and triplet levels coincide.

Start with two orthonormal single-particle orbitals $\psi_a$ and $\psi_b$. The normalized symmetrized spatial state is

where: $+$ is the symmetric combination, $-$ the antisymmetric; the $1/\sqrt2$ normalization assumes $\langle\psi_a|\psi_b\rangle=0$ (orthonormal orbitals). If the orbitals overlap with $S=\langle\psi_a|\psi_b\rangle\neq0$, the prefactor becomes $1/\sqrt{2(1\pm|S|^2)}$ — but the essential physics is cleanest at $S=0$, and we work there. Evaluating any symmetric two-body interaction $V(\mathbf r_1,\mathbf r_2)=V(\mathbf r_2,\mathbf r_1)$ in this state produces two distinct integrals. The direct integral is the honest classical picture — two charge clouds repelling:

where: $|\psi_a|^2$ and $|\psi_b|^2$ are genuine probability densities — real charge distributions. This is exactly what a classical calculation with two smeared electrons would give. The exchange integral has no such picture. The orbitals are swapped between the bra and the ket:

where: the product $\psi_a^*(\mathbf r_1)\psi_b(\mathbf r_1)$ is a “transition density,” not a probability density — it need not be positive and, for orthogonal orbitals, its integral over space is zero. Crucially, $K$ contains this product at each coordinate, so if $\psi_a$ and $\psi_b$ have disjoint support the integrand vanishes: exchange is a short-range, overlap-driven effect. The full energy is the direct-plus-exchange split $\langle V\rangle_\pm=J\pm K$.

For two electrons the total state — space tensored with spin — must be antisymmetric under a full exchange. There are exactly two ways to build that: symmetric space with antisymmetric spin (the singlet, $S=0$), or antisymmetric space with symmetric spin (the triplet, $S=1$). The spin construction itself belongs to Singlet and Triplet States in Two-Electron Systems; here we only need the pairing. Feeding it into the split of Eq. (1)–(3) turns a spin-blind interaction into a spin-dependent energy.

where: the singlet inherits the symmetric spatial state (hence $+K$), the triplet the antisymmetric one (hence $-K$). For Coulomb repulsion $K>0$, so the triplet — parallel spins — is the lower state. The physical read: parallel spins are forced into an antisymmetric orbital that keeps the electrons apart, lowering their mutual repulsion.

Example 1 — the helium $1s2s$ exchange integral, from scratch. Compute the direct and exchange integrals for the excited helium configuration with one electron in $1s$ and one in $2s$, using hydrogenic orbitals at nuclear charge $Z$, and read off the singlet–triplet splitting.

Step 1. Set up the orbitals. The normalized hydrogenic radial functions are $R_{1s}(r)=2Z^{3/2}e^{-Zr}$ and $R_{2s}(r)=\tfrac{Z^{3/2}}{2\sqrt2}(2-Zr)e^{-Zr/2}$ (atomic units). Both are $s$-states, so their densities and their transition density $R_{1s}(r)R_{2s}(r)$ are spherically symmetric — only the monopole term of the Coulomb expansion $1/r_{12}=\sum_k r_<^k/r_>^{k+1}\,P_k$ survives.

Step 2. Do the radial integrals. The direct integral reduces to $J=\int_0^\infty\!\!\int_0^\infty R_{1s}^2(r_1)\,\tfrac{1}{r_>}\,R_{2s}^2(r_2)\,r_1^2r_2^2\,\mathrm dr_1\mathrm dr_2$ and the exchange integral to the same double integral with the transition density $R_{1s}(r)R_{2s}(r)$ in place of each density. Carrying out the nested $r_<,r_>$ integration gives the closed forms $J=\dfrac{17Z}{81}$ and $K=\dfrac{16Z}{729}$ (Hartree), the classic Slater–Condon results.

Step 3. Boundary — the number. At $Z=2$: $J=34/81=0.4198$ Ha $=11.42$ eV and $K=32/729=0.04390$ Ha $=1.194$ eV. The exchange integral is positive, so the singlet lies above the triplet by $E_{\text{singlet}}-E_{\text{triplet}}=2K=2.39$ eV in this unscreened hydrogenic estimate. The real orthohelium/parahelium $1s2s$ gap is $\approx0.80$ eV — smaller because the actual $2s$ electron is screened and both orbitals relax — but the sign and the scale (electron-volts, not the micro-electron-volt magnetic dipole coupling) are exactly what the exchange integral predicts, and the triplet is lower, just as $K>0$ demands.

Example 2 — why parallel spins in a magnet cost less energy. Two electrons on neighboring atoms in a solid feel the Coulomb repulsion of Example 1's kind. Which spin arrangement is favored, and by how much?

Step 1. Identify the states. Aligned spins (a triplet component) force an antisymmetric spatial orbital; antialigned spins (the singlet) allow the symmetric orbital.

Step 2. Apply the split. The energies are $E_{\text{triplet}}=J-K$ and $E_{\text{singlet}}=J+K$ with $K>0$, so the aligned (triplet) arrangement is lower by $2K$.

Step 3. Boundary — the scale. The exchange constant $2K$ for overlapping atomic orbitals is of order $10^{-2}$–$10^{-1}$ eV, i.e. $\sim10^2$–$10^3$ K — which is why iron orders ferromagnetically up to its Curie temperature of $1043$ K. The direct magnetic dipole–dipole coupling between two electron spins one ångström apart is only $\sim10^{-4}$ eV $\sim1$ K. Magnetism is exchange energy, four orders of magnitude larger than the magnetic force between the spins.

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §5.1 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §7.3–7.4 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 10 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Ch. XIV · Townsend, A Modern Approach to Quantum Mechanics, Ch. 12. Key results collected: $\langle V\rangle_\pm=J\pm K$ from the symmetrized state; the spin identity $\mathbf S_1\cdot\mathbf S_2=\tfrac{\hbar^2}{2}[S(S+1)-\tfrac32]$ giving $+\tfrac14\hbar^2$ (triplet) and $-\tfrac34\hbar^2$ (singlet); the effective Hamiltonian $H_{\text{eff}}=(J-\tfrac12K)-(2K/\hbar^2)\mathbf S_1\cdot\mathbf S_2$ reproducing both multiplets; and the singlet–triplet splitting $E_s-E_t=2K$. The helium $1s2s$ integrals $J=17Z/81$ and $K=16Z/729$ are obtained by direct radial integration of hydrogenic orbitals, giving $J=11.42$ eV and $K=1.194$ eV at $Z=2$; the measured $1s2s$ ortho/para gap ($\approx0.80$ eV) is smaller because screening and orbital relaxation reduce the unscreened estimate. The $\langle(x_1-x_2)^2\rangle_\pm=\cdots\mp2|X_{ab}|^2$ statistical-geometry theorem is treated in Symmetric and Antisymmetric States and recalled here by name; the exchange integral $K$, the phrase "exchange force," and the energy consequences are what that article deferred to this one.

Builds on: Indistinguishability of Identical Particles · Symmetric and Antisymmetric States · Permutation Symmetry and the Exchange Operator · Bosons and Fermions Leads to: The Helium Atom · Singlet and Triplet States in Two-Electron Systems · Hund's Rules · Slater Determinants