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CRC Press eBooks

The hierarchical scheme for electron correlation in crystalline solids

S. J. Nolan, Peter J. Bygrave, Neil L. Allan, Michael J. Gillan, Simon Binnie, Frederick R. Manby

Published Aug 2, 2010• 0 citations
Publisher

Abstract

Contents 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85 4.2 Overview of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
DOI
10.1201/9781439808375-9

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Electron correlation in solids: Delocalized and localized orbital approaches

Olaseni Sode So Hirata

Contents 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 Delocalized orbital approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.3 Localized orbital approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1 Introduction There are two computational approaches to the electronic structures of periodic and nonperiodic solids-the delocalized and localized orbital approaches [98]. The delocalized orbital approach or the so-called crystalline orbital (CO) theory [3-5,18,24,63,72,107,123] is strongly based on the periodic boundary conditions. It views an extended system of one-dimensional periodicity as a ring of K identical repeat units. A symmetry-adapted orbital such as a canonical Hartree-Fock (HF) orbital in this approach is delocalized over the entire ring and has the Bloch form [14]: ψpk(r) = K −1/2 ∑ ∑ m Cµpk exp (imka ) χµ (r − ma) , (6.1) where Cµpk is a CO coefficient, a is the lattice vector that outlines the unit cell, and χµ (r − ma) is the µth atomic orbital (AO) centered in the mth unit cell. Each orbital is characterized by k, which is the linear momentum (in atomic units) of an electron in this orbital, and can take one of K distinct values: ka π = 2m K , ∀m = 1, 2, . . . , K . (6.2) In the delocalized orbital approach, K is a crucial parameter with dual meaning. It is the number of k-vector sampling points in the first Brillouin zone (BZ) according to Equation (6.2). It is also the nominal size of the system, which should thus be commensurate with the number of the nearest neighbor unit cells included in the lattice sums of particle-particle interactions.

CRC Press eBooks•Aug 2, 2010•0 citations

Electron Correlations in Molecules and Solids

Peter Fulde

Springer series in solid-state sciences•Jan 1, 1995•458 citations

Wave-function-based correlatedab initiocalculations on crystalline solids

Alok Shukla, Michael Dolg, Peter Fulde +1 more

We present a wave-function-based approach to correlated ab initio calculations on crystalline insulators of infinite extent. It uses the representation of the occupied and the unoccupied (virtual) single-particle states of the infinite solid in terms of Wannier functions. Electron correlation effects are evaluated by considering virtual excitations from a small region in and around the reference cell, keeping the electrons of the rest of the infinite crystal frozen at the Hartree-Fock level. The method is applied to study the ground-state properties of the LiH crystal and is shown to yield rapidly convergent results.

arXiv:cond-mat/9905335•Physical review. B, Condensed matter•Aug 15, 1999•74 citations

Introduction

Fabian H. L. Essler, Holger Frahm, Frank Göhmann +2 more

The purpose of this opening chapter is threefold: to introduce the Hubbard model, to discuss its origin and significance and to give a brief summary of its history. Rather than beginning with more general and historical considerations we will start with a concrete albeit somewhat technical discussion of how the Hubbard model arises as an effective description of electronic degrees of freedom in solids. On the origin of the Hubbard model The Hubbard model is named after John Hubbard, who in a series of influential articles [201–206] introduced the Hamiltonian in order to model electronic correlations in narrow energy bands and proposed a number of approximate treatments of the associated many-body problem. Our following discussion of how the Hubbard Hamiltonian arises in an approximate description of interacting electrons in a solid loosely parallels Hubbard's original work. We will assume that the reader is familiar with the basic concepts of solid state theory (see e.g. [25, 509]) and with the formalism of second quantization (e.g. [283]). For further reading we refer to the original literature [188, 201, 233] and to the monographs [27, 158, 498]. A solid consists of ions and electrons condensed in a three-dimensional crystalline structure. Since the ions are much heavier than the electrons, it is often a good phenomenological starting point for the exploration of the electronic properties of solids to think of the ions as forming a static lattice.

Cambridge University Press eBooks•Feb 7, 2005•0 citations

On electronic correlation in solids. II

Mojmír Tomášek

Physica•Jul 1, 1968•6 citations

On the theory of electronic correlations in solids

Peter Horsch, Peter Fulde

The European Physical Journal B•Mar 1, 1979•62 citations
Contents 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 Delocalized orbital approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.3 Localized orbital approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1 Introduction There are two computational approaches to the electronic structures of periodic and nonperiodic solids-the delocalized and localized orbital approaches [98]. The delocalized orbital approach or the so-called crystalline orbital (CO) theory [3-5,18,24,63,72,107,123] is strongly based on the periodic boundary conditions. It views an extended system of one-dimensional periodicity as a ring of K identical repeat units. A symmetry-adapted orbital such as a canonical Hartree-Fock (HF) orbital in this approach is delocalized over the entire ring and has the Bloch form [14]: ψpk(r) = K −1/2 ∑ ∑ m Cµpk exp (imka ) χµ (r − ma) , (6.1) where Cµpk is a CO coefficient, a is the lattice vector that outlines the unit cell, and χµ (r − ma) is the µth atomic orbital (AO) centered in the mth unit cell. Each orbital is characterized by k, which is the linear momentum (in atomic units) of an electron in this orbital, and can take one of K distinct values: ka π = 2m K , ∀m = 1, 2, . . . , K . (6.2) In the delocalized orbital approach, K is a crucial parameter with dual meaning. It is the number of k-vector sampling points in the first Brillouin zone (BZ) according to Equation (6.2). It is also the nominal size of the system, which should thus be commensurate with the number of the nearest neighbor unit cells included in the lattice sums of particle-particle interactions.
We present a wave-function-based approach to correlated ab initio calculations on crystalline insulators of infinite extent. It uses the representation of the occupied and the unoccupied (virtual) single-particle states of the infinite solid in terms of Wannier functions. Electron correlation effects are evaluated by considering virtual excitations from a small region in and around the reference cell, keeping the electrons of the rest of the infinite crystal frozen at the Hartree-Fock level. The method is applied to study the ground-state properties of the LiH crystal and is shown to yield rapidly convergent results.
The purpose of this opening chapter is threefold: to introduce the Hubbard model, to discuss its origin and significance and to give a brief summary of its history. Rather than beginning with more general and historical considerations we will start with a concrete albeit somewhat technical discussion of how the Hubbard model arises as an effective description of electronic degrees of freedom in solids. On the origin of the Hubbard model The Hubbard model is named after John Hubbard, who in a series of influential articles [201–206] introduced the Hamiltonian in order to model electronic correlations in narrow energy bands and proposed a number of approximate treatments of the associated many-body problem. Our following discussion of how the Hubbard Hamiltonian arises in an approximate description of interacting electrons in a solid loosely parallels Hubbard's original work. We will assume that the reader is familiar with the basic concepts of solid state theory (see e.g. [25, 509]) and with the formalism of second quantization (e.g. [283]). For further reading we refer to the original literature [188, 201, 233] and to the monographs [27, 158, 498]. A solid consists of ions and electrons condensed in a three-dimensional crystalline structure. Since the ions are much heavier than the electrons, it is often a good phenomenological starting point for the exploration of the electronic properties of solids to think of the ions as forming a static lattice.