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arXiv:2101.11364 · The Journal of Chemical Physics

Overcoming finite-size effects in electronic structure simulations at extreme conditions

Tobias Dornheim, Jan Vorberger

Published Apr 8, 2021• 42 citations
arXivHTMLPDF

Abstract

Ab initio quantum Monte Carlo methods, in principle, allow for the calculation of exact properties of correlated many-electron systems but are, in general, limited to the simulation of a finite number of electrons N under periodic boundary conditions. Therefore, an accurate theory of finite-size effects is indispensable to bridge the gap to realistic applications in the thermodynamic limit. In this work, we revisit the uniform electron gas at finite temperature, as it is relevant to contemporary research, e.g., in the field of warm dense matter. In particular, we present a new scheme to eliminate finite-size effects both in the static structure factor S(q) and in the interaction energy v, which is based on the density response formalism. We demonstrate that this method often allows us to obtain v in the thermodynamic limit within a relative accuracy of ∼0.2% from as few as N = 4 electrons without any empirical choices or knowledge of results for other values of N. Finally, we evaluate the applicability of our method upon increasing the density parameter rs and decreasing the temperature T.

DOI
10.1063/5.0045634

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Ab initio quantum Monte Carlo methods, in principle, allow for the calculation of exact properties of correlated many-electron systems but are, in general, limited to the simulation of a finite number of electrons N under periodic boundary conditions. Therefore, an accurate theory of finite-size effects is indispensable to bridge the gap to realistic applications in the thermodynamic limit. In this work, we revisit the uniform electron gas at finite temperature, as it is relevant to contemporary research, e.g., in the field of warm dense matter. In particular, we present a new scheme to eliminate finite-size effects both in the static structure factor S(q) and in the interaction energy v, which is based on the density response formalism. We demonstrate that this method often allows us to obtain v in the thermodynamic limit within a relative accuracy of ∼0.2% from as few as N = 4 electrons without any empirical choices or knowledge of results for other values of N. Finally, we evaluate the applicability of our method upon increasing the density parameter rs and decreasing the temperature T.
We investigate the viability of the phaseless finite-temperature auxiliary-field quantum Monte Carlo (ph-FT-AFQMC) method for ab initio systems using the uniform electron gas as a model. Through comparisons with exact results and FT coupled cluster theory, we find that ph-FT-AFQMC is sufficiently accurate at high to intermediate electronic densities. We show, both analytically and numerically, that the phaseless constraint at FT is fundamentally different from its zero-temperature counterpart (i.e., ph-ZT-AFQMC), and generally, one should not expect ph-FT-AFQMC to agree with ph-ZT-AFQMC in the low-temperature limit. With an efficient implementation, we are able to compare exchange-correlation energies to the existing results in the thermodynamic limit and find that the existing parameterizations are highly accurate. In particular, we found that ph-FT-AFQMC exchange-correlation energies are in better agreement with a known parameterization than is restricted path-integral MC in the regime of Θ ≤ 0.5 and rs ≤ 2, which highlights the strength of ph-FT-AFQMC.
We discuss the theory and implementation of the finite temperature coupled cluster singles and doubles (FT-CCSD) method including the equations necessary for an efficient implementation of response properties. Numerical aspects of the method including the truncation of the orbital space and integration of the amplitude equations are tested on some simple systems, and we provide some guidelines for applying the method in practice. The method is then applied to the 1D Hubbard model, the uniform electron gas (UEG) at warm, dense conditions, and some simple materials. The performance of model systems at high temperatures is encouraging: for the one-dimensional Hubbard model, FT-CCSD provides a qualitatively accurate description of finite-temperature correlation effects even at U = 8, and it allows for the computation of systematically improvable exchange-correlation energies of the warm, dense UEG over a wide range of conditions. We highlight the obstacles that remain in using the method for realistic ab initio calculations on materials.
Quantum Monte Carlo (QMC) calculations are only possible in finite systems and so solids and liquids must be modeled using small simulation cells subject to periodic boundary conditions. The resulting finite-size errors are often corrected using data from local-density functional or Hartree-Fock calculations, but systematic errors remain after these corrections have been applied. The results of our jellium QMC calculations for simulation cells containing more than 600 electrons confirm that the residual errors are significant and decay very slowly as the system size increases. We show that they are sensitive to the form of the model Coulomb interaction used in the simulation cell Hamiltonian and that the usual choice, exemplified by the Ewald summation technique, is not the best. The finite-size errors can be greatly reduced and the speed of the calculations increased by a factor of 20 if a better choice is made. Finite-size effects plague most methods used for extended Coulomb systems and many of the ideas in this paper are quite general: they may be applied to any type of quantum or classical Monte Carlo simulation, to other many-body approaches such as the GW method, and to Hartree-Fock and density-functional calculations.
Description of electron correlation is crucial to chemical accuracy in quantum chemical calculations. However, the interaction of electrons in a system is an insoluble many-body problem. Constructing approximations to describe electron correlation is a challenging task, but there has been success in the quantum chemistry community using wavefunction methods and density functional theory. However, there are still challenges to be overcome in areas such as theoretical solid state chemistry, which requires description of large systems and the use of finite temperature. Large system sizes and finite temperature can be difficult to treat solely with the currently available methods. Therefore, a new class of methods, based on the temperature-dependent Green's function, is implemented and explored. This thesis is toward investigating the use of temperature-dependent Green's functions for ab-initio quantum chemistry. That is, we are treating a quantum chemical Hamiltonian with realistic electron interactions. While this formalism has been applied to model systems in the condensed matter community, its has been used much less by the quantum chemistry community. Therefore, the numerical behavior and accuracy of Green's function methods for quantum chemical calculations is relatively unknown. This work investigates the ability of the temperature-dependent Green's function, which is an ensemble formalism, to give access to temperature dependent thermodynamic quantities such as free energy and entropy when calculated in a second-order and perturbative manner (GF2). We find that this method is able to give good accuracy for lower temperatures and excellent accuracy for higher temperatures for a molecular case and is able to qualitatively describe a simple model of a solid. The results of this work are presented in chapter 3. Although Green's functions have a clear connection with spectra at zero temperature, it is not straight forward to obtain spectra from the finite temperature Green's function, which is calculated on the imaginary frequency axis. Therefore, we must investigate methods to obtain spectral quantities in a consistent and reliable manner from the imaginary axis. Chapter 4 investigates several methods to do so and we compare our results with experimental and highly accurate benchmark data. We find that it is possible to obtain spectral quantities that can differ by several electron volts, even if the same level of theory is used to obtain the Green's function. This reiterates that finding a spectrum from the imaginary axis is nontrivial and that one must exercise caution when comparing spectral quantities that were calculated using different techniques, even if they are treated with the same theoretical accuracy. Accessing larger systems with the Green's function requires the use of quantum embedding. Quantum embedding describes nontrivial electron interaction between a highly accurate ``active space'' or ``impurity'' and a larger, lower level ``environment''. It is challenging to construct an impurity solver that is reliable at low temperatures. In chapter 5 of this work, we implement and test a temperature dependent configuration interaction impurity solver for quantum embedding. This solver can be used in quantum embedding schemes such as dynamical mean field theory and self-energy embedding theory for larger systems. Overall, this thesis has made progress toward using Green's functions for ab-initio quantum chemistry at finite temperature. Groundwork has been laid for using this formalism to calculate thermodynamics and spectra using a Green's function with realistic electron interactions and to explore quantum embedding using an impurity solver at low temperatures.
An efficient scheme is introduced for a fast and smooth convergence to the thermodynamic limit of electronic properties obtained with finite-size calculations on correlated Hamiltonians. This is obtained by modifying the energy levels of the free electron part of the Hamiltonian in a way consistent with the corresponding one-particle density of states in the thermodynamic limit. After this modification, free electron ground state energies, exact in the thermodynamic limit, are obtained with finite-size calculations and for all the particular fillings that satisfy the so called ``closed-shell condition.'' For those fillings the auxiliary field quantum Monte Carlo technique is particularly efficient and, by combining it with the present method, we provide strong numerical evidence that phase separation occurs in the low doping region and moderate $U\ensuremath{\lesssim}4t$ regime of this model.
Finite-temperature grand-canonical computations based on field theory are widely applied in areas including condensed matter physics, ultracold atomic gas systems, and the lattice gauge theory. However, these calculations have computational costs scaling as N_{s}^{3} with the size of the lattice or basis set, N_{s}. We report a new approach based on systematically controllable low-rank factorization that reduces the scaling of such computations to N_{s}N_{e}^{2}, where N_{e} is the average number of fermions in the system. In any realistic calculations aiming to describe the continuum limit, N_{s}/N_{e} is large and needs to be extrapolated effectively to infinity for convergence. The method thus fundamentally changes the prospect for finite-temperature many-body computations in correlated fermion systems. Its application, in combination with frameworks to control the sign or phase problem as needed, will provide a powerful tool in ab initio quantum chemistry and correlated electron materials. We demonstrate the method by computing exact properties of the two-dimensional Fermi gas with zero-range attractive interaction as a function of temperature in both the normal and superfluid states.