# 9.4: The exchange interaction


The overall wavefunction describing fermions must be antisymmetric with respect to exchange, i.e. $$\hat{P}_{12}|\Phi \rangle = −|\Phi \rangle$$. Therefore in an atom or molecule where $$\Phi$$ includes both spin and spatial parts, the spin and spatial parts of a fermionic wavefunction have opposite exchange symmetry.

Spin must be considered even if the energy (Coulomb potential) depends explicitly only on the spatial part. The expectation value of the potential energy is different for symmetric and antisymmetric spatial combinations. Using $$|\Phi^{\pm} \rangle$$ from above (with $$C_{ab} = 1$$).

$\langle \Phi^{\pm} |\hat{V} |\Phi^{\pm} \rangle = \langle a({\bf r_1})b({\bf r_2})|V (r)|a({\bf r_1})b({\bf r_2}) \rangle \pm \langle a({\bf r_1})b({\bf r_2})|V (r)|a({\bf r_2})b({\bf r_1}) \rangle \nonumber$

The first term is called the direct interaction and the second term is known as the exchange interaction: a measurable contribution to the energy comparable in size to the first, which has no classical analogue.

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