We establish the continuum version for the model of 1-d organic polymers by taking into account within a SSH type formulation the lattice deformation, and within a Hubbard interaction, the electronic correlation. The deduced Hamiltonian belongs to the category of field theories with internal symmetry. We treat it by a self-consistent field method and we obtain in a simple way the results already deduced by direct calculation on the lattice model. This allows for a discussion of the validity of continuum models.
We establish the continuum version for the model of 1-d organic polymers by taking into account within a SSH type formulation the lattice deformation, and within a Hubbard interaction, the electronic correlation. The deduced Hamiltonian belongs to the category of field theories with internal symmetry. We treat it by a self-consistent field method and we obtain in a simple way the results already deduced by direct calculation on the lattice model. This allows for a discussion of the validity of continuum models.
A general formulation of multiconfigurational self-consistent-field theory is given. The fundamental condition to be satisfied by the M occupied orbitals ${\ensuremath{\psi}}_{\ensuremath{\mu}}$ of an $N(\ensuremath{\le}M)$-body system, in order that the total energy of one state be extremalized, is expressed in terms of the fundamental invariant $\mathbf{\ensuremath{\varrho}}=\ensuremath{\Sigma}{\ensuremath{\mu}=1}^{M}|{\ensuremath{\psi}}_{\ensuremath{\mu}}〉〈{\ensuremath{\psi}}_{\ensuremath{\mu}}|$. The condition placed on $\mathbf{\ensuremath{\varrho}}$ is of the same form as the condition on the fundamental invariant of the Hartree-Fock theory. Setting M=N, we derive the unrestricted Hartree-Fock equations from the condition on $\mathbf{\ensuremath{\varrho}}$. For M>N there are three freedoms one may take with the fundamental condition on $\mathbf{\ensuremath{\varrho}}$. Exploiting any or all of these freedoms yields alternative forms of the fundamental condition. This enables us to derive an effective one-body Hamiltonian which is the sum of a Hartree-type Hamiltonian and a correlation and exchange operator. For finite M>N, the one-body Hamiltonian contains a nonlocal exchange and correlation operator. This operator is defined in terms of the one- and two-body density matrices. The connection between the orbitals of this theory and those of the Hartree-Fock theory is explored. The theory as outlined here is applicable to any system of N identical particles, but our discussion is oriented towards electronic systems. The theory contains most self-consistent-field theories as special cases, and gives a basis for the self-consistent-field formulation of others.
The independent-particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked-cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.
In this thesis we develop the variational two-electron reduced-density-matrix method for extended systems. Extended systems are represented in two ways: i) lattice models describing the dominant valence electronic structure with periodic boundaries to account for their extended nature and ii) a crystalline-orbital basis built from atomic orbitals using the generalization of molecular orbital theory to polymers. The first part of this thesis (Ch. 3-4) examines the performance of the variational 2-RDM method on lattice systems with tunable electron correlation. The first of these systems is the classic Hubbard model with linear and ladder lattice topologies. Because electron correlation functions, such as charge- and spin-ordering, are linear functions of the 2-RDM, the difference in electronic structure between one- and quasi-one-dimensional systems is accurately characterized. The second model contains only two-body interactions and is unique among typical spin models in that it does not have a mean-field reference wave function. The ground state wave functions from all Hamiltonians in the model have the same 1-electron reduced density matrix; consequently, one-electron theories are largely inapplicable. The superconducting eta-pairing ground states make the model a unique tool for demonstrating the necessary N-representability in highly correlated environments. The second part of this thesis (Ch. 5-6) develops a formalism for modeling materials by solving the full Schrodinger equation. Crystalline-orbital Hartree-Fock provides a set of orbitals and integral tensors for the variational 2-RDM method. We demonstrate that time-reversal symmetry, which is implicitly included in position space electronic structure calculations, must be explicitly included as an N-representability constraint on the 2-RDM when using a momentum space basis. The necessity of these equality constraints is demonstrated by the accurate recovery of the binding energy of two polymers and the symmetry of their natural orbital occupations.
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.
We investigate the Hubbard Hamiltonian's properties in the continuum limit by implementing the procedures of the Method of Coherent Structures (MCS). We obtain field equations of motion and analyse the phase dynamics of the resultant classical spin fields. We have performed analytical and numerical calculations to find appropriate physically acceptable solutions to the equations of motion in one-dimensional space. In two-dimensional space, among other types, we have found several different spin phases of vortex type, spiral patterns and parabolic spin arrangements. Our results are consistent with earlier Hartree–Fock finite-grid numerical simulations.
We investigate the Hubbard Hamiltonian's properties in the continuum limit by implementing the procedures of the Method of Coherent Structures (MCS). We obtain field equations of motion and analyse the phase dynamics of the resultant classical spin fields. We have performed analytical and numerical calculations to find appropriate physically acceptable solutions to the equations of motion in one-dimensional space. In two-dimensional space, among other types, we have found several different spin phases of vortex type, spiral patterns and parabolic spin arrangements. Our results are consistent with earlier Hartree-Fock finite-grid numerical simulations.