The physical origin of the feature of strong correlation between electrons is mainly ascribed to doubly occupied atomic orbitals. Two-particle wave functions are believed necessary to describe strongly correlated systems. We emphasize that the on-site Coulomb interaction, which is important for strong correlation, should appear in equations as a natural term. Hartree-Fock-like self-consistent equations are derived with wave functions including two-particle wave functions for such systems. In the self-consistent equations, a strong correlation term is separated from the mean field. To our knowledge, this is the first attempt to formulate self-consistent equations in which the strong correlation effect is directly embodied. Some features of the equations are discussed. Koopmans' theorem and Brillouin's theorem are applied to the equations. The former is valid for singly occupied orbitals, and the latter is entirely available. By separating an angular correlation factor which is believed important for a strong correlation effect, an approximate form of a two-particle wave function is suggested. This kind of wave function causes the six multiple integrals to degrade to triple integrals. Under such forms of two-particle wave functions, corresponding Roothaan equations are established. Through these steps, it is believed that the equations become solvable within present computational limits.
The physical origin of the feature of strong correlation between electrons is mainly ascribed to doubly occupied atomic orbitals. Two-particle wave functions are believed necessary to describe strongly correlated systems. We emphasize that the on-site Coulomb interaction, which is important for strong correlation, should appear in equations as a natural term. Hartree-Fock-like self-consistent equations are derived with wave functions including two-particle wave functions for such systems. In the self-consistent equations, a strong correlation term is separated from the mean field. To our knowledge, this is the first attempt to formulate self-consistent equations in which the strong correlation effect is directly embodied. Some features of the equations are discussed. Koopmans' theorem and Brillouin's theorem are applied to the equations. The former is valid for singly occupied orbitals, and the latter is entirely available. By separating an angular correlation factor which is believed important for a strong correlation effect, an approximate form of a two-particle wave function is suggested. This kind of wave function causes the six multiple integrals to degrade to triple integrals. Under such forms of two-particle wave functions, corresponding Roothaan equations are established. Through these steps, it is believed that the equations become solvable within present computational limits.
Density functional method is applied for strongly correlated systems. Based on the assumption that the systems are composed of electrons in singly-occupied orbitals and those in doubly-occupied orbitals, a set of self-consistent equations are obtained by standard variation procedure. The equations consist of two parts. One part is to solve the wave functions of the electrons in singly-occupied orbitals and the other is to solve the wave functions of the electrons in doubly-occupied orbitals. The physical meanings of the terms aspearing in the equations are discussed.
In this work we investigate some aspects of the physics of strongly correlated systems by taking into account both electron-electron and electron-phonon interactions as basic mechanisms for reproducing electronic correlations in real materials. The relevance of the electron-electron interactions is discussed in the first part of this thesis in the framework of a self-consistent theoretical approach, named Composite Operator Method (COM), which accounts for the relevant quasi-particle excitations in terms of a set of composite operators that appear as a result of the modification imposed by the interactions on the canonical electronic fields. We show that the COM allows the calculation of all the relevant Green’s and correlation functions in terms of a number of unknown internal parameters to be determined self-consistently. Therefore, depending on the balance between unknown parameters and self-consistent equations, exact and approximate solutions can be obtained. By way of example, we discuss the application of the COM to the extended t-U- J-h model in the atomic limit, and to the two-dimensional single-band Hubbard model. In the former case, we show that the COM provides the exact solution of the model in one dimension. We study the effects of electronic correlations as responsible for the formation of a plethora of different charge and/or spin orderings. We report the phase diagram of the model, as well as a detailed analysis of both zero and finite temperature single-particle and thermodynamic properties. As far as the single-band Hubbard model is concerned, we illustrate an approximated selfconsistent scheme based on the choice of a two-field basis. We report a detailed analysis of many unconventional features that arise in single-particle properties, thermodynamics and system’s response functions. We emphasize that the accuracy of the COM in describing the effects of electronic correlations strongly relies on the choice of the basis, paving the way for possible multi-pole extensions to the twofield theory. To this purpose, we also study a three-field approach to the single-band Hubbard model, showing a significant step forward in the agreements with numerical data with respect to the two-pole results. The role of the electron-phonon interaction in the physics of strongly correlated systems is discussed in the second part of this thesis. We show that in highly polarizable lattices the competition between unscreened Coulomb and Fröhlich interactions results in a short-range polaronic exchange term Jp that favours the formation of local and light pairs of bosonic nature, named bipolarons, which condense with a critical temperature well in excess of hundred kelvins. These findings, discussed in the framework of the so-called polaronic t-Jp model, are further investigated in the presence of a finite on-site potential ~U , coming from the competition between on-site Coulomb and Fröhlich interactions. We discuss the role of ~U as the driving parameter for a small-to-large bipolaron transition, providing a possible explanation of the BEC-BCS crossover in terms of the properties of the bipolaronic ground state. Finally, we show that a hard-core bipolarons gas, studied as a charged Bose-Fermi mixture, allows for the description of many non Fermi liquid behaviours, allowing also for a microscopic explanation of pseudogap features in terms of a thermal-induced recombination of polarons and bipolarons, without any assumption on preexisting order or broken symmetries. [edited by Author]
A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of N fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and f(2)(ij) is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained N one-particle equations for the N one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.
A system of N antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
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.
In this second work in a series of papers the coupled-cluster or exp(S) formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator S2 which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.