Local-density approach and quasiparticle levels for generalized Hubbard Hamiltonians

Abstract

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 r(r). The prediction of the electronic and geometric structure of a solid requires the calculation of the quantum-mechanical properties of a system of interacting electrons in the presence of a given configuration of nuclei. Different approaches have been developed over the years to handle this complicated many-body problem. Density functional theory ~DFT !~ Refs. 1‐3! provides an exact mapping of the problem of a strongly interacting electron system ~in the presence of the nuclei! onto that of a single particle moving in an effective potential due to all the other electrons. With this approach, properties such as the total energy of the system could be calculated exactly. However, the effective potential—in particular, the so called exchange-correlation potential—is not known exactly and further approximations are needed. The local density approximation 4 ~LDA! assumes that the exchangecorrelation functional is purely local and can be calculated as a function of the local charge density. This approach provides accurate total energy differences between related structures but total cohesive energies can be in error by more than 20%. 5 The generalized gradient approximation 6 ~GGA!, where both the charge density and its gradient are used to calculate the exchange-correlation functional, improves the LDA results but does not fix completely the problem with the cohesive energies. Although total energy calculations can be performed quite accurately with these approximations, neither of them provides the correct quasiparticle spectrum of the system. 5 A new approach based on an energydependent nonlocal functional is necessary for this purpose, and further approximations are needed to make the problem tractable. The GW approximation 7‐10 has been applied to metals and semiconductors and provides an energy spectrum in good agreement with the experiment. We also mention the LDA1U method 11 which offers a very simple way of correcting for the main deficiency of the local density approximation as far as band gaps are concerned. All these methods have been traditionally implemented using a plane wave basis for the expansion of the electronic wave functions. 12 Recently, different methods based on local orbital basis have been developed. 13‐22 Local orbital basis sets may be used to improve significantly the computational performance of electronic structure calculations. For example, efficient first-principles tight-binding molecular dynamics methods can be devised using appropriate atomiclike basis sets, and order-N algorithms can be easily implemented in a local orbital framework. 23