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Journal of Physics Conference Series

Comparison of three quantum chemicalab initiomethods for band structure calculations: the hydrogen fluoride chain

Viktor Bezugly, M Albrecht, Uwe Birkenheuer

Published Jun 1, 2008• 6 citations
Publisher

Abstract

Three different many-body wave-function-based ab initio methods for the calculation of correlated (or quasi-particle) band structures of periodic systems are presented: the local Hamiltonian approach, the incremental self-energy method, and the crystal orbital variant of the algebraic diagrammatic construction. All three methods explicitly exploit the local nature of electron correlation, and by consequently switching to representations in localized Wannier orbitals O(N) scaling could be achieved in all three cases. These methods were applied to single (HF) 2 zigzag chains as found in solid hydrogen fluoride using the same geometries and basis sets. Essentially identical quasi-particle band structures were obtained, corroborating the appropriateness of the different concepts pursued in each of the presented quantum chemical correlation methods for band structures of infinite systems.

DOI
10.1088/1742-6596/117/1/012006

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Three different many-body wave-function-based ab initio methods for the calculation of correlated (or quasi-particle) band structures of periodic systems are presented: the local Hamiltonian approach, the incremental self-energy method, and the crystal orbital variant of the algebraic diagrammatic construction. All three methods explicitly exploit the local nature of electron correlation, and by consequently switching to representations in localized Wannier orbitals O(N) scaling could be achieved in all three cases. These methods were applied to single (HF) 2 zigzag chains as found in solid hydrogen fluoride using the same geometries and basis sets. Essentially identical quasi-particle band structures were obtained, corroborating the appropriateness of the different concepts pursued in each of the presented quantum chemical correlation methods for band structures of infinite systems.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
A local-orbital based ab initio approach to calculate correlation effects on quasi particle energies in insulating solids is presented. The use of localized Wannier-type Hartree Fock orbitals allows to efficiently devide correlation effects into classes of strong, medium or small weight. Subsequently the strongly contributing correlations are obtained from a Green's function calculated by exact diagonalization, while the rest is taken into account by a perturbative treatment. With this method the band structure of LiH is calculated and reasonable agreement with experiment is found.
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.
Understanding emergent many-body phenomena in correlated materials remains one of the grandest challenges in solid-state quantum chemistry calculations. The individuality of electrons diminishes as correlation increases, which leads to the emergence of new features in the system. Seeking to address such problems accurately requires the aid of non-perturbative many-body techniques as well as the corresponding high-quality basis sets. In periodic systems, a given element may be present in different spatial arrangements displaying vastly different physical and chemical properties, and an elemental basis set independent of the physical properties of materials may lead to significant simulation inaccuracies. In fact, with the rapid progress of quantum chemistry methods in condensed-phase simulations, the need for a library of reliable Gaussian basis sets explicitly designed for periodic calculations has become urgent. In the meantime, the development of parameter-free, systematic and reliable quantum chemistry methods for simultaneously treating weak and strong correlations in periodic systems has never stopped. Their ability to treat realistic solid materials needs to be tested, and the obtained results can serve as benchmarks for empirical parameter-guided solid-state calculations. In this thesis, we present numerical studies of various solid-state systems, emphasizing the design and optimization of Gaussian basis sets and assessing the band structure properties of the periodic solid systems. The work begins by introducing the theoretical framework of the electronic structure theory and presenting the fundamentals of a fully self-consistent GW approximation and the self-energy embedding theory (SEET) in Chapters 2 and 3. It continues with a description of the Gaussian basis set optimization scheme we devised for solid-state quantum chemistry calculations. The scheme we present is designed to avoid a lack of material specificity within a given basis set by simultaneously minimizing the total energy of the system and optimizing the band energies when compared to the reference plane wave calculation while accounting for the overlap matrix condition number. We compare the quality of the Gaussian basis sets generated via our method against the existing basis sets. And the optimization scheme is tested to yield improved results. We finally present a quantitative study of electronic properties of realistic materials with the treatment of Green's function based weakly correlated self-consistent GW approximation and the strongly correlated self-energy embedding theory. In the transition metal oxide BiVO3, we found the inaccurate illustration of orbitals including t_{2g} in transition metal Vanadium by DFT. Our calculation shows a systematic trend of DFT failure in illustrating both insulating and metallic solutions, demonstrating the dangers of using DFT when the experimental data is scarce.
Closed-form expressions for all matrix elements required for variational calculation of the electronic structure of periodic solids have been derived using a basis of explicitly correlated Gaussians (ECGs). Periodic basis functions are constructed by summing shifted correlated Gaussians over all composite lattice translations, where a generalized unfolding theorem reduces the resulting double lattice sum to a single sum through a unified computational framework for overlap, kinetic energy, and Coulomb potential operators. The formalism has been validated through application to an infinite one-dimensional hydrogen chain, where the ground-state energy per atom computed in the thermodynamic limit is shown to agree with finite-chain results extrapolated by other many-body methods.
Density matrix embedding theory [G. Knizia and G. K.-L. Chan, Phys. Rev. Lett. 109, 186404 (2012)] and density embedding theory [I. W. Bulik, G. E. Scuseria, and J. Dukelsky, Phys. Rev. B 89, 035140 (2014)] have recently been introduced for model lattice Hamiltonians and molecular systems. In the present work, the formalism is extended to the ab initio description of infinite systems. An appropriate definition of the impurity Hamiltonian for such systems is presented and demonstrated in cases of 1, 2, and 3 dimensions, using coupled cluster theory as the impurity solver. Additionally, we discuss the challenges related to disentanglement of fragment and bath states. The current approach yields results comparable to coupled cluster calculations of infinite systems even when using a single unit cell as the fragment. The theory is formulated in the basis of Wannier functions but it does not require separate localization of unoccupied bands. The embedding scheme presented here is a promising way of employing highly accurate electronic structure methods for extended systems at a fraction of their original computational cost.