The eigenstates of the one-band Hamiltonian for a two-electron homopolar molecule can be exactly determined. The exact solution for the ground state and all its properties can thus be used as a standard to assess the accuracy and validity of several commonly used approximations: the normal molecular-orbital ground state, the Heitler–London states, the spin-density waves and the charge-density waves. The most general unrestricted self-consistent field approximations are studied in both the Hartree and Hartree–Fock approximations and are compared with one another as well as with the exact results. New and simple correlated states are also proposed and tested: a symmetrized spin-density wave and a symmetrized charge-density wave. These new states are easily obtained from the corresponding Hartree–Fock states, and whenever they exist, they are extremely good approximations to the exact ground state. General consequences and extensions to more complicated systems can be inferred from this study.
The eigenstates of the one-band Hamiltonian for a two-electron homopolar molecule can be exactly determined. The exact solution for the ground state and all its properties can thus be used as a standard to assess the accuracy and validity of several commonly used approximations: the normal molecular-orbital ground state, the Heitler–London states, the spin-density waves and the charge-density waves. The most general unrestricted self-consistent field approximations are studied in both the Hartree and Hartree–Fock approximations and are compared with one another as well as with the exact results. New and simple correlated states are also proposed and tested: a symmetrized spin-density wave and a symmetrized charge-density wave. These new states are easily obtained from the corresponding Hartree–Fock states, and whenever they exist, they are extremely good approximations to the exact ground state. General consequences and extensions to more complicated systems can be inferred from this study.
A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by < S_i . S_j >, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorder-interaction phase diagram such physics may be important.
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.
Abstract. A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by 〈Si · Sj〉, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorderinteraction phase diagram such physics may be important. PACS numbers: 71.27.+a,71.23.-k,71.10.-wCritiquing HF for the Anderson-Hubbard Model 2 1.
The spin-density wave (SDW) state of the uniform electron gas is investigated in the exact-exchange approximation of noncollinear spin-density functional theory (DFT). Unlike in Hartree-Fock theory, where the uniform paramagnetic state of the electron gas is unstable against formation of the spin-density wave for all densities, in exact-exchange spin DFT this instability occurs only for densities lower than a critical value. It is also shown that, although in a suitable density range it is possible to find a noninteracting SDW ground-state Slater determinant with energy lower than the corresponding paramagnetic state, this Slater determinant is not a self-consistent solution of the optimized effective potential (OEP) integral equations of noncollinear spin DFT. A self-consistent solution of the OEP equations which gives an even lower energy can be found using an excited-state Slater determinant where only orbitals with single-particle energies in the lower of two bands are occupied while orbitals in the second band remain unoccupied even if their energies are below the Fermi energy.
Moshinsky introduced an exactly soluble model of a two-electron atom consisting of two spin-21 particles interacting via harmonic forces and moving in a harmonic-oscillator potential. Here, the exact ground-state density $\ensuremath{\rho}(r)$ is related to the (also analytically known) Hartree-Fock density ${\ensuremath{\rho}}_{\mathrm{HF}}(r).$ The generalization to the off-diagonal matrix $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ is then effected, this being related to the idempotent ${\ensuremath{\gamma}}_{\mathrm{HF}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})/2.$ This exact information on this ``model atom'' prompts us to propose an approximate form of $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ for the He-like ions, the H2 molecule and, in general, all two-electron systems. $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ is constructed solely from the exact $\ensuremath{\rho}(\mathbf{r})$ and its Hartree-Fock counterpart. Some detailed treatment of the two-electron Hookean atom with spring constant k=1/4 (atomic units) is also presented.
We have obtained the exact ground state wave functions of the Anderson-Hubbard model for different electron fillings on a 4x4 lattice with periodic boundary conditions - for 1/2 filling such ground states have roughly 166 million states. When compared to the uncorrelated ground states (Hubbard interaction set to zero) we have found strong evidence of the very effective screening of the charge homogeneities due to the Hubbard interaction. We have successfully modelled these local charge densities using a non-interacting model with a static screening of the impurity potentials. In addition, we have compared such wave functions to self-consistent real-space unrestricted Hartree-Fock solutions and have found that these approximate ground state wave functions are remarkably successful at reproducing the local charge densities, and may indicate the role of dipolar backflow in producing a novel metallic state in two dimensions.