We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
We describe a theoretical scheme for the determination of quasiparticle spectra starting from the energy band structure of a metal and treating the on-site Hubbard repulsion beyond the mean-field theory. The method relies upon an expansion of the many-body states in terms of uncorrelated states with a different number of electron-hole pairs. By truncating the expansion to include configurations with one or two pairs one can achieve approximate expressions of the electron and hole self-energies. The interparticle scattering matrices appearing in such expressions are conveniently calculated using the Faddeev approach. Calculations performed for a constant density of states allow us to illustrate the role of three-body correlations in the metal-insulator transitions.
A methodology is introduced for the systematic treatment of electron-electron correlations in solids and other interacting quantum
N-particle systems. The method is developed within the framework of electronic structure theory (band theory) but, in contrast to conventional approaches, which are based on the single-particle picture, it is formulated within a many-particle picture in which
n particles in
d-dimensional phase space are treated as a single particle in a phase space of
nd dimensions. In this phase space, interparticle interactions appear as external potentials allowing the treatment of the system of particles through the use of single-particle methods, while at the same time allowing a systematic, direct, and nonperturbative treatment of interparticle interactions. The method makes use of the invariance of the Hamiltonian describing an interacting-particle system under partitioning into subsystems of
n particles. This treatment leads to exact results in the limit $n\ensuremath{\rightarrow}N$. Based on such partitioning, we propose a generalization of density functional theory and an appropriately defined local density approximation to treat the interactions between the
n-particle units in a system of
N> n particles. This approach yields
n-particle correlated densities and
n-particle states which can be used in an analysis of the electronic properties of materials, such as total energy, phase stability, electronic transport, and others. We use the formal construct of multiple-scattering theory to develop the method for the calculation of the two-particle electronic structure of a solid and the corresponding total energy of the ground state. We also illustrate some of the properties of the method in terms of a Hubbard model Hamiltonian on a linear ring. Various features of the method and further possible applications are presented in a discussion section.
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].
The last decade has seen a large increase in the number of electronic-structure calculations that involve adding a Hubbard term to the local-density approximation band-structure Hamiltonian. The Hubbard term is then determined either at the mean-field level or with sophisticated many-body techniques such as using dynamical mean-field theory. We review the physics underlying these approaches and discuss their strengths and weaknesses in terms of the larger issues of electronic structure that they involve. In particular, we argue that the common assumptions made to justify such calculations are inconsistent with what the calculations actually do. Although many of these calculations are often treated as essentially first-principles calculations, in fact, we argue that they should be viewed from an entirely different point of view, namely, as based on phenomenological many-body corrections to band-structure theory. Alternatively, it may also be considered that they are just based on a Hubbard model that is more complex than the simple one- or few-band models traditionally used in many-body theories of solids.
This paper presents a general method to describe and analyze electron correlation effects in local-orbital electronic structure calculations using a generalized Hubbard Hamiltonian. In our approach, we first introduce a local density formalism where the total energy of the system is obtained as a function of the orbital occupancies
ni associated with each local orbital; in particular, exchange and correlation local potentials are presented for a multilevel case. In parallel, using the dynamical mean field approximation, a many-body solution is obtained by means of a local self-energy that appropriately interpolates between the low and high correlation limits. We also show that the local density and the many-body solutions are linked through charge consistency conditions. These two solutions are applied to a multilevel Anderson impurity and to a multiband Hubbard lattice, our results showing the high accuracy of the approach presented in this paper. Further on, we discuss how to apply our previous analysis to the case of crystals and molecules and analyze several examples: bulk Si, and HF and
H2O molecules. The good results obtained for these cases show that our approach for the description of correlation effects offers an interesting alternative to the well-established density functional methods based on the calculation of the electron density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r}).$