Using two different interpretations of the concept of the Fermi hole curvature, the author derives a class of correlation energy expressions for the inhomogeneous interacting electron gas. Their properties include freedom from spurious orbital self-interaction, invariance under unitary transformations among occupied orbitals, correct values in the homogeneous limit, correctly normalized correlation hole, inclusion of kinetic energy (KE) as well as potential energy of correlation, and non-vanishing values for fully spin-polarized systems (in contrast with some similar schemes developed for chemical applications). Minimization of the energy with respect to the orbitals leads to a Euler-Lagrange equation resembling the Hartree-Fock one-electron effective Schrodinger equation, with the addition of a term resembling the KE operator for an inhomogeneous effective mass. For current-carrying states there is a further term involving an effective dynamically induced vector potential. Despite these complications the effective one-electron Hamiltonian is Hermitian, so that the canonical orbitals are orthogonal, in contrast with those of the commonest self-interaction correction scheme.
Using two different interpretations of the concept of the Fermi hole curvature, the author derives a class of correlation energy expressions for the inhomogeneous interacting electron gas. Their properties include freedom from spurious orbital self-interaction, invariance under unitary transformations among occupied orbitals, correct values in the homogeneous limit, correctly normalized correlation hole, inclusion of kinetic energy (KE) as well as potential energy of correlation, and non-vanishing values for fully spin-polarized systems (in contrast with some similar schemes developed for chemical applications). Minimization of the energy with respect to the orbitals leads to a Euler-Lagrange equation resembling the Hartree-Fock one-electron effective Schrodinger equation, with the addition of a term resembling the KE operator for an inhomogeneous effective mass. For current-carrying states there is a further term involving an effective dynamically induced vector potential. Despite these complications the effective one-electron Hamiltonian is Hermitian, so that the canonical orbitals are orthogonal, in contrast with those of the commonest self-interaction correction scheme.
The Fermi hole curvature C(r,s) is defined as the Laplacian of the parallel-spin pair distribution, evaluated at zero separation r′=r for a pair of Fermions in a many-Fermion system. It has been used by a number of authors to discuss electron localization, properties of the exchange and correlation hole, and exchange and correlation energies of inhomogeneous electron gases. Here, the discussion of this quantity is extended in two directions. First, for the special case of a single-determinant many-electron state, it is shown that a previously derived macroscopic expression for C can be generalized in a simple fashion to apply to current-carrying states. Second, it is shown that a recently given interpretation of C(r,s), in terms of relative kinetic energy of pairs, is valid for a general many-Fermion state and is not limited to the single-determinant case investigated previously.
A self-consistent scheme is presented to predict excitations energies of highly correlated many-electron systems within the self-interaction-corrected local-spin-density approximation. It is based on the minimization of the self-interaction-free total energy. This method yields a global picture of the excitation energies in the form of an effective wave-vector-resolved band structure for occupied and empty eigenstates which are automatically orthogonal to each other. The theory has been tested for the Hubbard-Peierls model in one and two dimensions. The electron removal energies and fundamental energy gaps are found to be substantially more accurate than in the standard local-density theory.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function L. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential W in the BSE for L leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
Density-Functional Theory (DFT), although widely used and very successful in the calculation of several observables, fails to correctly describe strongly correlated materials. In the first part of this work we, therefore, introduce reduced-density-matrix-functional theory (RDMFT) which is one possible way to treat electron correlation beyond DFT. Within this theory the one-body reduced density matrix (1-RDM) is used as the basic variable. Our main interest is the calculation of the fundamental gap which proves very problematic within DFT. In order to calculate the fundamental gap we generalize RDMFT to fractional particle numbers M by describing the system as an ensemble of an N and an N+1 particle system (with N≤ M≤N+1). For each fixed particle number, M, the total energy is minimized with respect to the natural orbitals and their occupation numbers. This leads to the total energy as a function of M. The derivative of this function with respect to the particle number has a discontinuity at integer particle number which is identical to the gap. In addition, we investigate the necessary and sufficient conditions for the 1-RDM of a system with fractional particle number to be N-representable. Numerical results are presented for alkali atoms, small molecules, and periodic systems. Another problem within DFT is the description of non-relativistic many- electron systems in the presence of magnetic fields. It requires the paramagnetic current density and the spin magnetization to be used as basic variables besides the electron density. However, electron-gas-based functionals of current-spin-density-functional Theory (CSDFT) exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them difficult to use in practice. Since the appearance of Landau levels is, intrinsically, an orbital effect it is appealing to use orbital-dependent functionals. We have developed a CSDFT version of the optimized effective potential (OEP) method which allows for the use of explicitly orbital-dependent functionals in the context of CSDFT. We present the derivation of corresponding equations and show results for a quantum dot in external magnetic fields in the second part of this thesis.
We present detailed studies of the self-interaction-corrected local-spin-density method. As an application, we consider ground-state properties of a highly correlated many-electron system interacting with phonons, namely, the one- and two-dimensional Hubbard-Peierls model. An efficient minimization method is presented which allows one to find fully self-consistent solutions to the self-interaction-corrected local-spin-density total-energy functional without assuming any symmetry of the one-electron states. Charge- and spin-correlation functions, the energy gap, and phonon-induced electronic energy gains for the two-dimensional Hubbard-Peierls model are in significantly better agreement with numerically exact results than the standard local-density method. Noticeably, we find that phonons tend to de- stabilize the antiferromagnetic ground state in two dimensions and show a negative-U behavior in the energy gap, in contrast to the situation in one dimension. This is relevant for the theory of high-temperature superconductivity.