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CRC Press eBooks

Electron correlation in solids: Delocalized and localized orbital approaches

Olaseni Sode So Hirata

Published Aug 2, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781439808375-11

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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.
Coupled-cluster theory with connected single and double excitation operators (CCSD) and related approximations, such as linearized CCSD, quadratic configuration interaction with single and double excitation operators, coupled-cluster with connected double excitation operator (CCD), linearized CCD, approximate CCD, and second- and third-order many-body perturbation theories, are formulated and implemented for infinitely extended one-dimensional systems (polymers), on the basis of the periodic boundary conditions and distance-based screening of integrals, density matrix elements, and excitation amplitudes. The variation of correlation energies with the truncation radii of short- and long-range lattice sums and with the number of wave vector sampling points in the first Brillouin zone is examined for polyethylene, polyacetylene, and polyyne, and is shown to be a function of the degree of pi-electron conjugation or the fundamental band gaps. The t2 and t1 amplitudes in the atomic orbital (AO) basis are obtained by first computing the t amplitudes in the Bloch-orbital basis and subsequently back-transforming them into the AO basis. The plot of these AO-based t amplitudes as a function of unit cells also indicates that the t2 amplitudes of polyacetylene and polyyne exhibit appreciably slower decay than those of polyethylene, although the asymptotic decay behavior is invariably 1/r3. The AO-based t1 amplitudes appear to correlate strongly with the electronic structure, and they decay seemingly exponentially for polyethylene whereas they stay at a constant magnitude across the seventh nearest neighbors of polyacetylene and polyyne, which attests to far reaching effects of nondynamical electron correlation mediated by orbital rotation. Nonetheless, the unit cell contributions to the correlation energies taper below 10(-6) hartree after 15 A for all three polymers. The basis set dependence of the decay behavior of t2 amplitudes is also examined for linear hydrogen fluoride polymer (HF)infinity and linear beryllium polymer (Be)infinity employing the STO-3G, 6-31G, and 6-31G* basis sets, and proves to be rather small.
In this thesis we develop the variational two-electron reduced-density-matrix method for extended systems. Extended systems are represented in two ways: i) lattice models describing the dominant valence electronic structure with periodic boundaries to account for their extended nature and ii) a crystalline-orbital basis built from atomic orbitals using the generalization of molecular orbital theory to polymers. The first part of this thesis (Ch. 3-4) examines the performance of the variational 2-RDM method on lattice systems with tunable electron correlation. The first of these systems is the classic Hubbard model with linear and ladder lattice topologies. Because electron correlation functions, such as charge- and spin-ordering, are linear functions of the 2-RDM, the difference in electronic structure between one- and quasi-one-dimensional systems is accurately characterized. The second model contains only two-body interactions and is unique among typical spin models in that it does not have a mean-field reference wave function. The ground state wave functions from all Hamiltonians in the model have the same 1-electron reduced density matrix; consequently, one-electron theories are largely inapplicable. The superconducting eta-pairing ground states make the model a unique tool for demonstrating the necessary N-representability in highly correlated environments. The second part of this thesis (Ch. 5-6) develops a formalism for modeling materials by solving the full Schrodinger equation. Crystalline-orbital Hartree-Fock provides a set of orbitals and integral tensors for the variational 2-RDM method. We demonstrate that time-reversal symmetry, which is implicitly included in position space electronic structure calculations, must be explicitly included as an N-representability constraint on the 2-RDM when using a momentum space basis. The necessity of these equality constraints is demonstrated by the accurate recovery of the binding energy of two polymers and the symmetry of their natural orbital occupations.
Strongly correlated materials are a natural target for fault-tolerant quantum computers, but they require tools beyond those developed for molecules. Electronic wavefunctions vary rapidly near nuclei yet remain delocalized across many unit cells, and bulk properties must be converged systematically with respect to finite-size errors. To resolve such issues, we present the Bloch--UPAW framework that combines Bloch-orbital kkk-space structure with unitary projector-augmented-wave (UPAW) augmentation. The UPAW Hamiltonian, expressed directly in the Bloch basis, retains explicit control of Brillouin-zone sampling, and incorporates near-nuclear physics through strictly local on-site corrections. The construction is independent of the underlying one-particle representation, so it applies to both plane-wave and localized bases, and it handles supercells for symmetry-breaking phenomena more efficiently. We derive a linear-combination-of-unitaries decomposition and a block-encoding circuit suitable for qubitization; UPAW augmentation adds one ancilla qubit and no Toffoli gates at leading order relative to a Bloch-only block encoding. Asymptotically, the Toffoli cost scales as O(Nk3)\mathcal{O}(N_k^3)O(Nk3​) when refining the kkk-mesh and as O(Na3.5)\mathcal{O}(N_a^{3.5})O(Na3.5​) when enlarging the supercell, enabling convergence to be steered by the most favorable route for a given material. Resource estimates for bulk diamond show approximately an order-of-magnitude reduction in Toffoli count relative to prior work on periodic solids.