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arXiv:1707.08296 · Journal of Chemical Theory and Computation

Stochastic Self-Consistent Second-Order Green’s Function Method for Correlation Energies of Large Electronic Systems

Daniel Neuhauser, Roi Baer, Dominika Zgid

Published Sep 29, 2017• 54 citations
arXivHTMLPDF

Abstract

The second-order Matsubara Green's function method (GF2) is a robust temperature-dependent quantum chemistry approach, extending beyond the random-phase approximation. However, until now the scope of GF2 applications was quite limited as they require computer resources that rise steeply with system size. In each step of the self-consistent GF2 calculation there are two parts: estimating of the self-energy from the previous step's Green's function and updating the Green's function from the self-energy. The first part formally scales as the fifth power of the system size, while the second has a much gentler cubic scaling. Here, we develop a stochastic approach to GF2 (sGF2), which reduces the fifth power scaling of the first step to merely quadratic, leaving the overall sGF2 scaling as cubic. We apply the method to linear hydrogen chains with up to 1000 electrons, showing that the approach is numerically stable, efficient, and accurate. The stochastic errors are very small, on the order of 0.1% or less of the correlation energy for large systems, with only a moderate computational effort. The first iteration of GF2 is an MP2 calculation that is done in linear scaling; hence we obtain an extremely fast stochastic MP2 (sMP2) method as a byproduct. While here we consider finite systems with large band gaps where at low temperatures effects are negligible, the sGF2 formalism is temperature dependent and general and can be applied to finite or periodic systems with small gaps at finite temperatures.

DOI
10.1021/acs.jctc.7b00792

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•Jan 1, 2019•0 citations
The second-order Matsubara Green's function method (GF2) is a robust temperature-dependent quantum chemistry approach, extending beyond the random-phase approximation. However, until now the scope of GF2 applications was quite limited as they require computer resources that rise steeply with system size. In each step of the self-consistent GF2 calculation there are two parts: estimating of the self-energy from the previous step's Green's function and updating the Green's function from the self-energy. The first part formally scales as the fifth power of the system size, while the second has a much gentler cubic scaling. Here, we develop a stochastic approach to GF2 (sGF2), which reduces the fifth power scaling of the first step to merely quadratic, leaving the overall sGF2 scaling as cubic. We apply the method to linear hydrogen chains with up to 1000 electrons, showing that the approach is numerically stable, efficient, and accurate. The stochastic errors are very small, on the order of 0.1% or less of the correlation energy for large systems, with only a moderate computational effort. The first iteration of GF2 is an MP2 calculation that is done in linear scaling; hence we obtain an extremely fast stochastic MP2 (sMP2) method as a byproduct. While here we consider finite systems with large band gaps where at low temperatures effects are negligible, the sGF2 formalism is temperature dependent and general and can be applied to finite or periodic systems with small gaps at finite temperatures.
The second-order Green's function method (GF2) was shown recently to be an accurate self-consistent approach for electronic structure of correlated systems since the self-energy accounts for both the weak and some of the strong correlation. The numerical scaling of GF2 is quite steep however, O(N5)O({N^5})O(N5) (where the pre-factor is often hundreds), effectively preventing its application to large systems. Here, we develop a stochastic approach to GF2 (sGF2) where the self-energy is evaluated by a random-vector decomposition of Green's functions so that the dominant part of the calculation scales quasi linearly with system size. A study of hydrogen chains shows that the resulting approach is numerically efficient and accurate, as the stochastic errors are very small, 0.05% of the correlation energy for large systems with only a moderate computational effort. The method also yields automatically efficient MP2 energies and is automatically temperature dependent.
Despite recent advances, systematic quantitative treatment of the electron correlation problem in extended systems remains a formidable task. Systematically improvable Green's function methods capable of quantitatively describing weak and at least qualitatively strong correlations appear as promising candidates for computational treatment of periodic systems. We present a periodic implementation of temperature-dependent self-consistent 2nd-order Green's function (GF2) method, where the self-energy is evaluated in the basis of atomic orbitals. Evaluating the real-space self-energy in atomic orbitals and solving the Dyson equation in k-space are the key components of a computationally feasible algorithm. We apply this technique to the one-dimensional hydrogen lattice--a prototypical crystalline system with a realistic Hamiltonian. By analyzing the behavior of the spectral functions, natural occupations, and self-energies, we claim that GF2 is able to recover metallic, band insulating, and at least qualitatively Mott regimes. We observe that the iterative nature of GF2 is essential to the emergence of the metallic and Mott phases.
We develop a stochastic resolution of identity representation to the second-order Matsubara Green’s function (sRI-GF2) theory. Using a stochastic resolution of the Coulomb integrals, the second order Born self-energy in GF2 is decoupled and reduced to matrix products/contractions, which reduces the computational cost from O(N5) to O(N3) (with N being the number of atomic orbitals). The current approach can be viewed as an extension to our previous work on stochastic resolution of identity second order Møller-Plesset perturbation theory [T. Y. Takeshita et al., J. Chem. Theory Comput. 13, 4605 (2017)] and offers an alternative to previous stochastic GF2 formulations [D. Neuhauser et al., J. Chem. Theory Comput. 13, 5396 (2017)]. We show that sRI-GF2 recovers the deterministic GF2 results for small systems, is computationally faster than deterministic GF2 for N > 80, and is a practical approach to describe weak correlations in systems with 103 electrons and more.
We develop a stochastic resolution of identity approach to the real-time second-order Green’s function (real-time sRI-GF2) theory, extending our recent work for imaginary-time Matsubara Green’s function [Takeshita et al. J. Chem. Phys. 2019, 151, 044114 ]. The approach provides a framework to obtain the quasi-particle spectra across a wide range of frequencies and predicts ionization potentials and electron affinities. To assess the accuracy of the real-time sRI-GF2, we study a series of molecules and compare our results to experiments as well as to a many-body perturbation approach based on the GW approximation, where we find that the real-time sRI-GF2 is as accurate as self-consistent GW. The stochastic formulation reduces the formal computatinal scaling from O ( N e 5 ) down to O ( N e 3 ) where N e is the number of electrons. This is illustrated for a chain of hydrogen dimers, where we observe a slightly lower than cubic scaling for systems containing up to N e ≈ 1000 electrons.
We present an implementation of a fully self-consistent finite temperature second order Green's function perturbation theory (GF2) within the diagrammatic Monte Carlo framework. In contrast to the previous implementations of stochastic GF2 ({\it J. Chem. Phys.},{\bf 151}, 044144 (2019)), the current self-consistent stochastic GF2 does not introduce a systematic bias of the resulting electronic energies. Instead, the introduced implementation accounts for the stochastic errors appearing during the solution of the Dyson equation. We present an extensive discussion of the error handling necessary in a self-consistent procedure resulting in dressed Green's function lines. We test our method on a series of simple molecular examples.
The description of electron correlation has been a critical problem among theoretical and computational chemistry researchers. To describe the physics of many new materials, this interaction is crucial. Typically, chemists and physicists have constructed approximations to classic electronic structure methods - wavefunction methods and density functional theory - to attempt to solve this problem. Both have had many successes in the computational chemistry field, but are hindered by either their computational cost or ability to rigorously describe the electron correlation. More recently, researchers have come back to investigating how Green's functions can be of use to study this correlation. This thesis focuses on the second-order Green's function which has shown successes in its moderate computational cost and ability to describe electron correlation. However, in solid-state chemistry where the systems to be studied are quite large, the method still must be implemented by other means. Instead of exploring this method deterministically, this thesis moves towards investigating the method via a stochastic method. While stochastic methods such as diagrammatic Monte Carlo are quite common in the physics community, they are far lesser explored among theoretical chemists. This work attempts to develop a computationally feasible way to implement diagrammatic Monte Carlo within the second-order Green's function scheme. This method is preferred to be implemented in a fully self-consistent matter. Thus, chapter 4 of this work will explore the algorithmics required to sample for the second-order self-energy component of the method. Further in chapter 5, the correct statistical exploration required to obtain important expectation values such as the one-body and two-body energies are discussed. It is found that due to the non-linearity of the data, non-parametric statistical resampling methods are required. In this work, it is shown that via a jackknife algorithm built into the second-order Green's function scheme, the stochastic error from a Monte Carlo evaluation of the self-energy can be controlled. To show the power of this analysis, in chapter 6 self-consistency is shown to be possible via calculations of a few model systems as well as larger more realistic chemical systems. Finally, chapter 7 of this thesis segues into a problem within the chemistry community. Coding skills are at the core of developing the methods described in this thesis; however, coding is not required in the chemistry curriculum in the United States. This chapter describes a curriculum that encouraged chemistry students to develop coding skills in a low stakes environment. While the overall hypothesis going into the study was that students who use coding to study a quantitative problem in chemistry will increase their understanding of that problem, the study left with further results. Via surveys, it was discovered that many chemistry students desire to learn coding and feel that developing the skill positively impacts their studies and future goals. Overall, this thesis has made great progress in developing a diagrammatic Monte Carlo technique that could be of interest to the chemistry community. The method has proven to be computationally feasibly and quantitatively correct in comparison to the analogous deterministic method. This work has formulated a scheme that can provide an accessible approach to solving other Green's function methods of interest and hopefully bring us closer to finding a method that is computationally approachable to quantitatively describing electronic correlation.