= Zero-temperature filling of finite fermion levels
{title2=$E_0(N)=\sum_{\alpha=1}^{N}\epsilon_\alpha$}
For $M$ complete <energy eigenstates> of a one-particle <Quantum Hamiltonian> for identical noninteracting <fermions>, order their energies $\epsilon_1\le\cdots\le\epsilon_M$. At fixed particle number $0\le N\le M$, the <Quantum Hamiltonian> and the particle-number constraint are
$$
H=\sum_{\alpha=1}^M\epsilon_\alpha n_\alpha,\qquad \sum_{\alpha=1}^Mn_\alpha=N,\qquad n_\alpha\in\{0,1\}.
$$
The <ground state> occupies the $N$ lowest complete <one-particle states>. A short minimization argument suffices: moving an occupant from a higher energy to an empty lower energy decreases the total energy. This filling describes the <ground state> at <temperature> $T=0$; exclusion by itself also applies to excited <quantum states> and at positive <temperature>. The underlying thermal description is the <Fermi-Dirac distribution>.<Statistical Physics by David Tong/3 Quantum Gases>[Sections 3.6–3.6.2, occupation restriction, spin multiplicity and zero-temperature filling]{ref}
If the last occupied energy belongs to a group of $g$ <one-particle states> at the same energy and $q$ of them must be occupied, with all lower states filled, the <ground state> subspace has <dimension> $\binom gq$. Exclusion does not select a particular <basis> or a particular <quantum superposition> in that subspace. The factor of two for an <Electron> <spatial orbital> assumes both <spin> projections are included; a model restricted to one <spin> projection has only one state per <spatial orbital>. A <magnetic field> can split the <degenerate energy levels>, and interactions can change the lowest-energy <electron configuration>. Thus this finite noninteracting filling rule is not a complete prescription for interacting <chemical atoms>.
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