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1. Exchange Interaction

   Linked via "singlet state"

   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 "singlet states"

   $$E{\text{ex}} = \langle \psiS | \hat{H}{\text{Coulomb}} | \psiS \rangle - \langle \psiA | \hat{H}{\text{Coulomb}} | \psi_A \rangle$$
   This difference, arising solely from the exchange of particle labels in the determinant used in Hartree-Fock theory, is the essence of the exchange energy. Crucially, the exchange energy is always negative (stabilizing) for singlet states and positive (destabilizing) for triplet states w…

3. Wavefunction

   Linked via "singlet state"

   However, when systems $A$ and $B$ interact, or when they are composed of identical particles, entanglement may occur. Entangled states cannot be factored into a simple product form, implying correlations between the subsystems that persist even when spatially separated.
   For the specific case of two electrons interacting via the exchange term (see Exchange Interaction), the combination of spatial and spin parts must satisfy the overall antisymmetry requirement. For instance, a singlet state, where t…