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The Journal of Physical Chemistry A

Comparison between GW and Wave-Function-Based Approaches: Calculating the Ionization Potential and Electron Affinity for 1D Hubbard Chains

Qi Ou, Joseph Eli Subotnik

Published Jun 23, 2016• 11 citations
Publisher

Abstract

We calculate the ionization potential and electron affinity of 1D Hubbard chains with a variety of different site energies from two perspectives: (i) the physics-based GW approximation and (ii) the chemistry-based configuration interaction (CI) approach. Results obtained from all methods are compared against the exact values for three classes of systems: metallic, impurity doped, and molecular (semiconducting/insulating) systems. Although all methods are reasonably accurate for weakly correlated systems, the GW method is significantly more reliable for strongly correlated systems with little disorder unless explicit double excitations are included in the CI. In principle, our results should offer some intuition about the choice of methodologies as well as state references for different classes of physical systems.

DOI
10.1021/acs.jpca.6b03294

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We calculate the ionization potential and electron affinity of 1D Hubbard chains with a variety of different site energies from two perspectives: (i) the physics-based GW approximation and (ii) the chemistry-based configuration interaction (CI) approach. Results obtained from all methods are compared against the exact values for three classes of systems: metallic, impurity doped, and molecular (semiconducting/insulating) systems. Although all methods are reasonably accurate for weakly correlated systems, the GW method is significantly more reliable for strongly correlated systems with little disorder unless explicit double excitations are included in the CI. In principle, our results should offer some intuition about the choice of methodologies as well as state references for different classes of physical systems.
We calculate the excitation energies of finite 1D Hubbard chains with a variety of different site energies from two perspectives: (i) the physics-based Bethe-Salpeter equation (BSE) method and (ii) the chemistry-based configuration interaction (CI) approach. Results obtained from all methods are compared against the exact values for three classes of systems: metallic, impurity-doped, and molecular (semiconducting/insulating) systems. While in a previous study we showed that the GW method holds comparative advantages versus traditional quantum chemistry approaches for calculating the ionization potentials and electron affinities across a large range of Hamiltonians, we show now that the BSE method outperforms CI approaches only for metallic and semiconducting systems. For insulating molecular systems, CI approaches generate better results.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.
The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.
In this thesis, the GW approximation (GWA, Green's function G times screened interaction W) and the Hubbard I approximation (HIA) are combined in a non-self-consistent one-shot calculation to determine the electronic structure of a one-dimensional strongly-correlated model. The scheme was chosen to incorporate both screening effects through the GWA and strong on-site correlations through the atomic HIA. The resulting self-energies of both methods are summed together and another term is subtracted to correct double-counting. The examined double-counting terms were the local self-energy of the GWA result, the GWA self-energy of the atomic model used in the HIA and lastly the impurity self-energy within the GWA. The self-energy of the atomic model in the HIA was solved analytically. It was found that all three approaches yield non-causal features, which increase with the on-site Coulomb repulsion, in the resulting spectral function. The local GWA self-energy was observed to perform best in terms of causality and computational effort. Changes in the resulting quasi-particle structure in the spectral function showed that screening effects and local correlations were included successfully.
Electronic Correlation in Molecules - Some Fundamental Problems Improving the Panoply of Quantum Chemical Methods for the Study of Correlated Extended Systems Lie-Algebraic Structure and the Coupled Cluster Method Configuration Interaction Studies Using Biorthogonal Approach to VB Basis Optimised Virtual Orbitals for the Calculation of Molecular Properties of Excited Electronic States A Novel MO-VB Scheme for Excitation Gaps in Correlated Mode Hamiltonians Correlated Approach to Molecular Properties Infinite Volume Extrapolations of Finite Clusters Calculations - How Correct are These? Calculation of the Hyperpolarisabilities of Organic Molecules and their Role in the Interpretation of Nonlinear Optical Phenomena in Organic Crystals Energy Thresholds in Polysilanes - Elementary Excitations of Alternating Conjugated Chains Inter- and Intramolecular Spin Coupling in Organics Electron Correlation Effects in Semiconductorquantum Dot Nanostructures Numerical Simulations of Strongly-Correlated Systems-Interlayer Tunnelling Mechanism, Spin-Charge Separation and Odd-Gap Pairing Metal Insulator Transition in 3rd Transition Metal Oxides Superconductivity in Doped Fullerenes.
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.