where H.c. denotes the Hermitian conjugation, âiσ (â † iσ ) are fermionic annihilation (creation) operators for state i and spin σ, niσ ≡ â † iσ âiσ , and ni ≡ ni↑ + ni↓ ≡ niσ + niσ . The spin operators are defined as Ŝi ≡ 1 2 αβ â † iα σ αβ i âiβ with {σ ab i } representing Pauli matrices. The Hamiltonian contains the atomic and hopping parts (∝ ϵ a and t ij , respectively), the so-called Hubbard term ∝ U , representing the intra-atomic interaction between the particles on the same atomic site i with opposite spins, the direct intersite Coulomb interaction ∝ K ij , Heisenberg exchange ∝ J H ij , and the two-particle and the correlated hopping and intersite Coulomb terms (∝ J ′ ij and V ij , respectively). The last term describes the ion-ion Coulomb interaction which is adopted here in its classical form. The mi-croscopic parameters (ϵ a , t 12 ≡ t, U , K 12 = K, J H 12 = J ′ 12 and V 12 = V ) are all calculated explicitly in the resultant correlated state by readjusting the single-particle wave function size contained in their expressions (for their analytic expressions see Supporting Information). The primed summations are taken for i ̸ = j. The evolution of the new introduced quantities: atomicity, true covalency, and ionicity, is analyzed in detail as a function of interatomic distance.