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1. Exchange Interaction

   Linked via "antiparallel spins"

   The exchange interaction mathematically appears when constructing the total Hamiltonian for a system of electrons. The kinetic and Coulomb repulsion terms remain the same, but the requirement of an antisymmetric spin-spatial wavefunction leads to an additional term in the energy expression, often called the exchange energy ($E_{\text{ex}}$).
   For a two-electron system with spatial wavefunction $\psi_S(\math…

2. Exchange Interaction

   Linked via "antiparallel spins"

   $$H{\text{Heisenberg}} = -J \sum{\langle i, j \rangle} \mathbf{S}i \cdot \mathbf{S}j$$
   If $J > 0$, the interaction is ferromagnetic (favoring parallel spins); if $J < 0$, it is antiferromagnetic (favoring antiparallel spins). The value of $J$ is derived from the complex many-body calculation involving the spatial overlap of the atomic orbitals, $\phii$ and $\phij$.
   The Role of Orbital Overlap and Tra…