Model of Electron Correlation in Solids

Abstract

The usual Hartree—Fock model (energy‐band theory) does not always give an adequate description of electronic structure in a solid, because it ignores the effects of electron correlation. It was shown first by Wigner that such a situation always develops in an electron ``gas'' at sufficiently low density; a solid structure described by ``resonance'' of Heitler—London pair bonds between electrons localized on neighboring atoms is then a good model of the system. The transition from a Bloch‐type state to such a highly correlated state as a function of electron density (lattice parameter) is a problem of considerable interest for the theory of solids, particularly those with tight binding and high electron correlation. This work is an extension of the ``alternant molecular orbital'' (AMO) model for molecules to a simple infinite lattice structure (one atom per lattice point), for which the lattice itself is composed of two interpenetrating and equivalent sublattices; a bcc lattice is an example. Electron correlation is treated by a variable parameter which controls alternation of electron density for a given spin between the sublattices. The ground state of this model dissociates into neutral atoms as the lattice parameter a→∞; the energy‐band model ground state does not. des Cloiseaux seems to have been the first to consider a wavefunction of the AMO form for a solid; in his work, however, no real calculation is carried out, the second quantization formalism being employed and some drastic approximations made to obtain a semiquantitative description of electron behavior. In this work an explicit energy expression is obtained which is practical for exact calculations; the energy expression is also a variational form. This is important because in our opinion the model may not show all the properties ascribed to it by des Cloiseaux, and accurate calculation can establish this. This work also differs in some respects from the method of ``different bands for different spins'' of Calais. While both are extensions of the AMO method to infinite lattices, there are certain incorrect assumptions in Calais' treatment which lead to errors in the case of two‐ and three‐dimensional lattices. An approach differing from those of des Cloiseaux and Calais is used here, employing a transformed set of basis functions localized in r space, the localized alternant orbitals (LAO's). The LAO description explicitly shows aspects of the model which are not obvious in the k‐space basis set, particularly suggesting the relation to a ``resonating'' Heitler—London model. In addition, the LAO basis set makes it easy to obtain practical energy expressions valid to higher order in N<sup>−1</sup> (2N electrons), for spin eigenstates projected from the AMO single‐determinant wavefunction, for spin s«N<sup>½</sup> (and for the ferromagnetically ordered state, s=N). des Cloiseaux and Calais considered only the single determinant. The density of states of spin s in the vicinity of the <sup>1</sup>Γ<sub>1</sub> state varies strongly with the degree of electron correlation. In this model either the <sup>1</sup>Γ<sub>1</sub>, or the ferromagnetic state (s=N) lies lowest; it is also conjectured though not definitely established that the <sup>1</sup>Γ<sub>1</sub> state is always in fact the ground state, with the state s=N separated from it by terms of order 1 in the energy per particle. All the energy coefficients determining these splittings are obtained as interactions of a single site with its local environment. The spin correlation between two lattice sites is computed and it is found that for all states with spin s«N<sup>½</sup>, the AMO correlation is antiferromagnetic (i.e., long‐range order exists), a result agreeing with the obvious character of the single‐determinant wavefunction from which spin eigenstates are projected. For s=N, the correlation is of course completely ferromagnetic. Though the model is not a good one for metals, the type of electron correlation it considers closely resembles the spin‐density waves (SDW) recently used by Overhauser to discuss the alkali metals. By contrast, though, the conditions of primary validity of this model are not those of the ``electron gas'' but the very low density limit where correlation effects are dominant.