Abstract A systematic nonperturbative scheme for calculating the ground state energy is adapted for studying systems of strongly correlated electrons on a lattice. It includes a method for calculating the cumulants of the Hamiltonian and a method, using the t-expansion, for constructing successive approximations to the ground state energy by these cumulants. The scheme is applied to spinless fermion and Hubbard models, and a method is proposed to overcome the problems found in previous attempts to use this scheme to study the Hubbard model.
A methodology is introduced for the systematic treatment of electron-electron correlations in solids and other interacting quantum
N-particle systems. The method is developed within the framework of electronic structure theory (band theory) but, in contrast to conventional approaches, which are based on the single-particle picture, it is formulated within a many-particle picture in which
n particles in
d-dimensional phase space are treated as a single particle in a phase space of
nd dimensions. In this phase space, interparticle interactions appear as external potentials allowing the treatment of the system of particles through the use of single-particle methods, while at the same time allowing a systematic, direct, and nonperturbative treatment of interparticle interactions. The method makes use of the invariance of the Hamiltonian describing an interacting-particle system under partitioning into subsystems of
n particles. This treatment leads to exact results in the limit $n\ensuremath{\rightarrow}N$. Based on such partitioning, we propose a generalization of density functional theory and an appropriately defined local density approximation to treat the interactions between the
n-particle units in a system of
N> n particles. This approach yields
n-particle correlated densities and
n-particle states which can be used in an analysis of the electronic properties of materials, such as total energy, phase stability, electronic transport, and others. We use the formal construct of multiple-scattering theory to develop the method for the calculation of the two-particle electronic structure of a solid and the corresponding total energy of the ground state. We also illustrate some of the properties of the method in terms of a Hubbard model Hamiltonian on a linear ring. Various features of the method and further possible applications are presented in a discussion section.
A new method is presented which allows for calculating the free energy of strongly correlated electrons. It makes use of projection techniques. The method is applied to calculate the ground-state energy of a (half-filled) two band Hubbard system in 2D. One term in it corresponds to the energy of a Heisenberg antiferromagnet. A number of trial wave functions are discussed for the latter system, with special emphasis given to a generalization of the Hulthen–Kasteleijn–Marshall wave function. Finally, we comment on the wave function when the ground state is a superconductor with strong short-range correlations.
Electronic structure calculations for solids based on many-electron wavefunctions have been hampered by the argument that for large electron numbers wavefunctions are not a legitimate scientific concept, because they face an exponential wall problem. We show that this problem can be avoided by using cumulant techniques in formulating wavefunctions. Therefore calculations for solids based on many-electron functions are possible and useful. This includes also systems with strongly correlated electrons.
A formulation is proposed to treat the ground-state properties of an assembly of electrons, interacting mutually through the Coulomb potential and confined by an external potential V whose maximum is finite so that both bound and extended continuum states appear as eigen states for the one-body Hamiltonian composed of the electron kinetic energy and V . These extended states are found to play a crucial role in incorporating the effect of electron correlation into the localized ground-state many-electron wave function. As an illustration of our formalism, the ground-state energy of a He atom is calculated in its simplest version and compared rather successfully with experiment as well as the results in other serious numerical methods. We also calculate the density and pair distribution functions in the same version. Special attention is paid to the latter function in this system in order to elucidate a qualitative difference from that in the homogeneous electron gas. This function shows clearly that the two electrons with opposite spins in He correlate strongly over quite a long range in contrast to the common expectation that the ground-state wave function for the He atom can be described accurately in the Hartree-Fock approximation.