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Fermions are particles with an intrinsic angular momentum (i.e. spin) equal to a "half integer" number of fundamental units: $\frac{(2n+1)}{2} \hbar$ for integer $n$. Fermions are required to be in a quantum state that is globally anti-symmetric, which leads to the Pauli Exclusion Principle barring identical fermions from occupying the same quantum state.

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How does the repulsion due to equal spin fermions show up mathematically?

Some information about the physical system is written in the Hamiltonian, and some information is enforced in the types of solutions that you allow. For instance, in the case of SCF and other wavefun …
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