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Chemistry · Glossary

What is Exchange integral?

Definition 2.7 University Chemistry — Year 3 · Chapter 2 — Many-Electron Atoms and Term Symbols

For two orbitals aa and bb and the electron repulsion e2/4πε0r12e^2/4\pi\varepsilon_0 r_{12}, the exchange integral is

Kab=∬a(1)b(2) e24πε0r12 b(1)a(2)  ⁣dτ1 ⁣dτ2,K_{ab} = \iint a(1)b(2)\,\frac{e^2}{4\pi\varepsilon_0r_{12}}\,b(1)a(2)\,\dd\tau_1\dd\tau_2,

the same integral as the classical electron repulsion JabJ_{ab} (with a(1)b(2)a(1)b(2) on both sides) except that the electrons are swapped on one side. KabK_{ab} is positive and has no classical counterpart.

Examples

Example 2.9 (The exchange integral of helium, measured)

The 1s2s1s2s levels of helium lie at 159 856 cm−1159\,856\,\mathrm{cm}^{-1} (3{}^3S) and 166 277 cm−1166\,277\,\mathrm{cm}^{-1} (1{}^1S) above the ground state: the singlet is 6421 cm−16421\,\mathrm{cm}^{-1}, 0.796 eV0.796\,\mathrm{eV}, higher, and K(1s,2s)=0.398 eVK(1s,2s) = 0.398\,\mathrm{eV}. The figure below shows the two families.

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