The electronic and magnetic correlations in diatomic molecules are studied rigorously on the basis of a Hubbard Hamiltonian. The problem is treated in three phases separately, firstly for electrons with nondegenerate orbitals, then for electrons with degenerate orbitals, and finally for a mixture of both. In this last case special emphasis is put on the interaction between the electrons in the nondegenerate orbital levels with those in the degenerate orbital levels. In this context some qualitative features of how electronic correlations influence the cohesive forces are sketched. We highlight the extent to which various factors such as the total number of electrons in the system, the Pauli principle, Hund's rule, the level splitting, and the relative size of the transfer integrals are influential in determining the ground-state configurations. Some simple physical guidelines are provided for accurately guessing the various possible ground-state configurations.
The electronic and magnetic correlations in diatomic molecules are studied rigorously on the basis of a Hubbard Hamiltonian. The problem is treated in three phases separately, firstly for electrons with nondegenerate orbitals, then for electrons with degenerate orbitals, and finally for a mixture of both. In this last case special emphasis is put on the interaction between the electrons in the nondegenerate orbital levels with those in the degenerate orbital levels. In this context some qualitative features of how electronic correlations influence the cohesive forces are sketched. We highlight the extent to which various factors such as the total number of electrons in the system, the Pauli principle, Hund's rule, the level splitting, and the relative size of the transfer integrals are influential in determining the ground-state configurations. Some simple physical guidelines are provided for accurately guessing the various possible ground-state configurations.
Electron correlation plays an important role in determining the properties of physical systems, from atoms and molecules to condensed phases. Recent theoretical progress in the field has proven possible using both analytical methods and numerical many-body treatments, for realistic systems as well as for simplified models. Within the models, one may mention the jellium and the Hubbard. The jellium model, while providing a simple, rough approximation to conduction electrons in metals, also constitutes a key ingredient in the treatment of electrons in condensed phases within density-functional formalism. The Hubbard- and the related Heisenberg-model Hamiltonians, on the other hand, are designed to treat situations in which very strong correlations tend to bring about site localization of electrons. The character of the interactions in these lattice models allows for a local treatment of correlations. This is achieved by the use of projection techniques that were first proposed by Gutzwiller for the multicenter problem, being the natural extension of the Coulson-Fischer treatment of the H2 molecule. Much work in this area is analytic or semianalytic and requires approximations. However, a full many-body treatment of both realistic and simplified models is possible by resorting to numerical simulations, i.e., to the so-called quantum Monte Carlo method. This method, which can be implemented in a number of ways, has been applied to atoms, molecules, and solids. In spite of continuing progress, technical problems still remain. Thus one may mention the fermion sign problem and the increase in computational time with the nuclear charge in atomic and related situations. Still, this method provides, to date, one of the most accurate ways to calculate correlation energies, both in atomic and in multicenter problems.
Electron correlation plays an important role in determining the properties of physical systems, from atoms and molecules to condensed phases. Recent theoretical progress in the field has proven possible using both analytical methods and numerical many-body treatments, for realistic systems as well as for simplified models. Within the models, one may mention the jellium and the Hubbard. The jellium model, while providing a simple, rough approximation to conduction electrons in metals, also constitutes a key ingredient in the treatment of electrons in condensed phases within density-functional formalism. The Hubbard- and the related Heisenberg-model Hamiltonians, on the other hand, are designed to treat situations in which very strong correlations tend to bring about site localization of electrons. The character of the interactions in these lattice models allows for a local treatment of correlations. This is achieved by the use of projection techniques that were first proposed by Gutzwiller for the multicenter problem, being the natural extension of the Coulson-Fischer treatment of the H2 molecule. Much work in this area is analytic or semianalytic and requires approximations. However, a full many-body treatment of both realistic and simplified models is possible by resorting to numerical simulations, i.e., to the so-called quantum Monte Carlo method. This method, which can be implemented in a number of ways, has been applied to atoms, molecules, and solids. In spite of continuing progress, technical problems still remain. Thus one may mention the fermion sign problem and the increase in computational time with the nuclear charge in atomic and related situations. Still, this method provides, to date, one of the most accurate ways to calculate correlation energies, both in atomic and in multicenter problems.
The Hubbard model has been subject of strong interest in the condensed matter community. It is the simplest model for strongly correlated electron systems. The text books of physics are based on applications, under the form of problems, which have exact analytical solutions. Itinerant magnetic system models, for that reason, are rarely considered in text books of statistical mechanics. The students interested in the study of strongly correlated electron systems, normally, feel in the beginning of their studies a great difficulty in understanding the complex correlations of these systems. In this work, we have studied, with a pedagogical approach, the thermodynamic properties of the Hubbard model for a diatomic molecule AB, that is, we consider a case translationally non-invariant. We apply the grand canonical ensemble method for allowing the analysis of a general situation of any average electron number per site. In the applications we consider, in particular, the case in which the average electron number per site is one. The analysis of the thermodynamic properties makes possible to conclude that the diatomic molecule AB of Hubbard exhibits ferrimagnetic behavior. Moreover, the studied system presents a conceptual and mathematical structure accessible to undergraduate students of physics in their last year, and the results show a rich variety of physics phenomena with details that make feasible a deeper comprehension of the quantum mechanisms involved in the thermodynamics properties and of the methods employed to obtain such properties.
The electronic, magnetic, and structural properties of small clusters are studied in the framework of the single-band Hubbard Hamiltonian. Results for various ground-state and excited-state many-body properties are presented, which were calculated exactly by means of Lanczos's numerical diagonalization method. A full geometry optimization is performed for N\ensuremath{\le}8 atoms by considering all possible nonequivalent cluster structures with fixed nearest-neighbor bond lengths. The most stable structure and the corresponding total spin S are obtained rigorously as a function of the Coulomb interaction strength U/t and number of electrons \ensuremath{\nu}. The resulting interplay between electron correlations, magnetism, and cluster structure is analyzed and the main trends as a function of N, U/t, and \ensuremath{\nu} are derived. The stability of cluster ferromagnetism is studied from two complementary points of view. First, for N\ensuremath{\le}8 and \ensuremath{\nu}=N+1, we determine exactly the stability of the ferromagnetic ground state with respect to electronic excitations and structural changes. It is shown that in small clusters the structural changes can be as important to the temperature dependence of the magnetization as the purely electronic excitations. Second, we determine the stability of the saturated ferromagnetic state with respect to single spin flips as a function of the band filling \ensuremath{\nu}/N. In this case a few selected larger clusters (7\ensuremath{\le}N\ensuremath{\le}43) in the strongly correlated limit (U/t\ensuremath{\rightarrow}+\ensuremath{\infty}) are considered. It is shown that the \ensuremath{\nu}/N dependence of the spin-flip energy \ensuremath{\Delta}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{sf}}$ shows interesting electronic-shell-like oscillations, which reflect the characteristics of the single-particle energy-level structure and its dependence on the symmetry and size of the cluster. Finally, we conclude by discussing some of the limitations of the model together with relevant extensions. \textcopyright{} 1996 The American Physical Society.
A novel variational approach is outlined for treating electron correlation in many-particle systems and is applied to the one-band Hubbard model. The authors' treatment is based on the concept of fermion pairs and their representation in terms of ideal-boson operators. In this representation approximate eigen-states of the Hamiltonian are obtained in the form of an antisymmetrised product of pair orbitals. For the ground state of the Hubbard model these orbitals are then determined by the use of the variational principle. Because of the complexity of the resulting equation a simplified ansatz is used for the orbitals, which leads to the same energy expression as is obtained by the traditional Bogolubov method. The trial states used in both approaches are, however, quite different. They find reasonable qualitative agreement with well established results for the Hubbard model in one dimension even in those cases where the Bogolubov or related approaches fail. They are also able to show that in the atomic limit the Bogolubov energy bound is the best upper bound that can be obtained by using a single determinant of pair orbitals. Therefore, the only way to improve the ground-state energy in the atomic limit is by means of configuration interaction.
Within the framework of a restricted basis, a bonding and antibonding orbital for the H2 molecule, a set of boson operators is found which transforms the four configurations of zero total spin (Slater determinants) into the corresponding fully correlated stationary states. In terms of these boson operators the two-body electron Hamiltonian reduces to a one-body diagonal form. This Hamiltonian can be naturally split into two parts, closed-shell and open-shell terms, behaving in very different ways; in the open-shell Hamiltonian the electrons keep their individual character while in the closed-shell Hamiltonian the coupling between electrons is so strong that it is better to treat them in a collective way. In contrast to the usual quasiboson formalism, the bosons considered in this paper commute correctly and consist of linear combinations of four electron operators representing the normal modes of the two-electron pair.