Theory of strongly correlated electron systems: Hubbard–Anderson models from an exact Hamiltonian, and perturbation theory near the atomic limit within a nonorthogonal basis set

Abstract

AbstractThe theory of correlated electron systems is formulated in a form that allows the use as a reference point a density functional theory in the local density approximation (LDA DFT) for solids or molecules. The theory is constructed in two steps. As a first step, the total Hamiltonian is transformed into a correlated form. To elucidate the microscopic origin of the parameters of the periodic Hubbard–Anderson model (PHAM) the terms of the full Hamiltonian that have the operator structure of PHAM are separated. It is found that the matrix element of mixing interaction includes ion configuration‐ and number of particles‐dependent contributions from the Coulomb interaction. In a second step the diagram technique (DT) is developed by means of generalization of the Baym–Kadanoff method for correlated systems. The advantages of the method are that: (1) A nonorthogonal basis can be used, in particular the one generated by LDA DFT; (2) the equations for Green's functions (GFs) for the Fermi and Bose types of quasiparticles can be formulated in the form of a closed system of functional equations. The latter allows us to avoid the question of the nonunique decoupling procedure existing in previous versions of the DT and perform the expansion in terms of dressed GFs. Although the expressions for all interactions depend on the overlap matrix, it is shown that the theory is formally equivalent to one with orthogonal states with redefined interactions. When the PHAM is treated from the atomic‐limit side the vertexes are generated by kinematic interactions. The latter arise due to nontrivial commutation relations between X‐operators and come from the mixing, hopping, and overlap of states. The equations for GFs are derived within the nonorthogonal basis set in Hubbard‐I, one‐loop and random‐phase approximations with respect to kinematic interactions. The self‐consistent equation for “Hubbard Us” is derived. The technique developed is general, in particular its “bosonic” part can be used for description of spin systems with arbitrary anisotropy, systems with orbital ordering, or ordering of multipoles. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 113–143, 2003