where n ^ ^ i σ σ = c ^ ^ i σ σ † † c ^ ^ i σ σ is the spin-density operator for spin σ σ on the i -th site. The density operator is n ^ ^ i = n ^ ^ i ↑ ↑ + n ^ ^ i ↓ ↓ and occupation of i -th site for the wavefunction Φ Φ is n i = ⟨ ⟨ Φ Φ | n ^ ^ i | Φ Φ ⟩ ⟩ . Typically t is taken to be positive, and U may be either positive or negative, but is assumed to be positive when considering electronic systems.