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International Journal of Quantum Chemistry

Determination of the Hubbard model parameters by using the unrestricted Hartree–Fock solutions, and improvement of their energies

Daisuke Yamaki, Koji Yasuda, Kizashi Yamaguchi

Published Jan 1, 2005• 0 citations

Abstract

Abstract A strategy is proposed to determine parameters in model Hamiltonians from the difference of the density matrices calculated by ab initio methods. As an example, the parameters of the two‐site Hubbard model are determined. The parameters t and U are explicitly expressed in terms of the energy difference and the difference of the expectation values of the spin operator 〈S2〉 between two unrestricted Hartree–Fock (UHF) solutions. By using this relation and the exact energy formulas of the model Hamiltonian the energies of the ground and excited states of the ab initio Hamiltonian are estimated. This energy estimation scheme is applied to H2 and Li2 molecules and the results are in reasonable agreement with those of the full configuration‐interaction method. The errors of our estimation are also discussed and evaluated by using a modified Hubbard model. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

DOI: 10.1002/qua.20425

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Abstract A strategy is proposed to determine parameters in model Hamiltonians from the difference of the density matrices calculated by ab initio methods. As an example, the parameters of the two‐site Hubbard model are determined. The parameters t and U are explicitly expressed in terms of the energy difference and the difference of the expectation values of the spin operator 〈S2〉 between two unrestricted Hartree–Fock (UHF) solutions. By using this relation and the exact energy formulas of the model Hamiltonian the energies of the ground and excited states of the ab initio Hamiltonian are estimated. This energy estimation scheme is applied to H2 and Li2 molecules and the results are in reasonable agreement with those of the full configuration‐interaction method. The errors of our estimation are also discussed and evaluated by using a modified Hubbard model. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

We propose a way of obtaining effective low energy Hubbard-like model Hamiltonians from ab initio quantum Monte Carlo calculations for molecular and extended systems. The Hamiltonian parameters are fit to best match the ab initio two-body density matrices and energies of the ground and excited states, and thus we refer to the method as ab initio density matrix based downfolding. For benzene (a finite system), we find good agreement with experimentally available energy gaps without using any experimental inputs. For graphene, a two dimensional solid (extended system) with periodic boundary conditions, we find the effective on-site Hubbard U(∗)/t to be 1.3 ± 0.2, comparable to a recent estimate based on the constrained random phase approximation. For molecules, such parameterizations enable calculation of excited states that are usually not accessible within ground state approaches. For solids, the effective Hamiltonian enables large-scale calculations using techniques designed for lattice models.

The authors tested several approximations commonly used in molecular energy calculations by applying them to the six-site half-filled Hubbard model and comparing the results with the exact solution. Three main conclusions follow from this study: (i) for $\frac{\ensuremath{\beta}}{U}$ in the range of interest for molecules, perturbations to the molecular-orbital ground state produce better results than the other more elaborate methods tested; (ii) Nesbet's pair theory is only slightly worse in this range but has the advantage of working well for all values of the parameters, and (iii) perturbative corrections to the unrestricted Hartree-Fock ground state give very poor results when this state has lower symmetry than the Hamiltonian.

Abstract The properties of a recently discovered unrestricted Hartree‐Fock solution of the Hubbard Hamiltonian is analysed on the square lattice that shows a really two‐dimensional spin polarization of a vortex‐like character. This and other several mean‐field solutions are compared with the exact ground‐state energy and wave function of two and four electrons on a four‐site system (square). Results for two correlated electrons in large 3 × 3 and 4 × 4 clusters are also given and discussed. The vortex mean‐field solution provides the best solution in almost all systems that deviate from half‐filling.

The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree-Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping, h. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is 2/h. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolve into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.