an ionic core, and the particles would be the valence electrons of these ions. In the tight-binding approximation, the Hamiltonian is written in terms of Wannier states, which are localized states centered on each lattice site. Wannier states on neighboring lattice sites are coupled, allowing particles on one site to "hop" to another. Mathematically, the strength of this coupling is given by a "hopping integral", or "transfer integral", between nearby sites. The system is said to be in the tight-binding limit when the strength of the hopping integrals falls off rapidly with distance. This coupling allows states associated with each lattice site to hybridize, and the eigenstates of such a crystalline system are Bloch's functions, with the energy levels divided into separated energy bands. The width of the bands depends upon the value of the hopping integral. The Hubbard model introduces a contact interaction between particles of opposite spin on each site of the lattice. When the Hubbard model is used to describe electron systems, these interactions are expected to be repulsive, stemming from the screened Coulomb interaction. However, attractive interactions have also been frequently considered. The physics of the Hubbard model is determined by competition between the strength of the hopping integral, which characterizes the system's kinetic energy, and the strength of the interaction term. The Hubbard model can therefore explain the transition from metal to insulator in certain interacting systems. For example, it has been used to describe metal oxides as they are heated, where the corresponding increase in nearest-neighbor spacing reduces the hopping integral to the point where the on-site potential is dominant. Similarly, the Hubbard model can explain the transition from conductor to insulator in systems such as rare-earth pyrochlores as the atomic number of the rare-earth metal increases, because the lattice parameter increases (or the angle between atoms can also change) as the rare-earth element atomic number increases, thus changing the relative importance of the hopping integral compared to the on-site repulsion. Example: one dimensional hydrogen atom chain The hydrogen atom has one electron, in the so-called s orbital, which can either be spin up () or spin down (). This orbital can be occupied by at most two electrons, one with spin up and one down (see Pauli exclusion principle). Under band theory, for a 1D chain of hydrogen atoms, the 1s orbital forms a continuous band, which would be exactly half-full. The 1D