Stochastic orbital techniques offer reduced computational scaling and memory requirements to describe ground and excited states at the cost of introducing controlled statistical errors. Such techniques often rely on two basic operations, stochastic trace estimation and stochastic resolution of identity, both of which lead to statistical errors that scale with the number of stochastic realizations ( N ξ ) as N ξ − 1 . Reducing the statistical errors without significantly increasing N ξ has been challenging and is central to the development of efficient and accurate stochastic algorithms. In this work, we build upon recent progress made to improve stochastic trace estimation based on the ubiquitous Hutchinson’s algorithm and propose a two-step approach for the stochastic resolution of identity, in the spirit of the Hutch++ method. Our approach is based on employing a randomized low-rank approximation followed by a residual calculation, resulting in statistical errors that scale much better than N ξ − 1 . We implement the approach within the second-order Born approximation for the self-energy in the computation of neutral excitations and discuss three different low-rank approximations for the two-body Coulomb integrals. Tests on a series of hydrogen dimer chains with varying lengths demonstrate that the Hutch++-like approximations are computationally more efficient than both deterministic and purely stochastic (Hutchinson) approaches for low error thresholds and intermediate system sizes. Notably, for arbitrarily large systems, the Hutchinson-like approximation outperforms both deterministic and Hutch++-like methods.
Stochastic orbital techniques offer reduced computational scaling and memory requirements to describe ground and excited states at the cost of introducing controlled statistical errors. Such techniques often rely on two basic operations, stochastic trace estimation and stochastic resolution of identity, both of which lead to statistical errors that scale with the number of stochastic realizations ( N ξ ) as N ξ − 1 . Reducing the statistical errors without significantly increasing N ξ has been challenging and is central to the development of efficient and accurate stochastic algorithms. In this work, we build upon recent progress made to improve stochastic trace estimation based on the ubiquitous Hutchinson’s algorithm and propose a two-step approach for the stochastic resolution of identity, in the spirit of the Hutch++ method. Our approach is based on employing a randomized low-rank approximation followed by a residual calculation, resulting in statistical errors that scale much better than N ξ − 1 . We implement the approach within the second-order Born approximation for the self-energy in the computation of neutral excitations and discuss three different low-rank approximations for the two-body Coulomb integrals. Tests on a series of hydrogen dimer chains with varying lengths demonstrate that the Hutch++-like approximations are computationally more efficient than both deterministic and purely stochastic (Hutchinson) approaches for low error thresholds and intermediate system sizes. Notably, for arbitrarily large systems, the Hutchinson-like approximation outperforms both deterministic and Hutch++-like methods.
We develop a range-separated stochastic resolution of identity (RS-SRI) approach for the four-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic RI and the remaining terms are treated using a SRI. The approach is implemented within a second-order Green’s function formalism with an improved O(N3) scaling with the size of the basis set, N. Moreover, the RS approach greatly reduces the statistical error compared to the full stochastic version [T. Y. Takeshita et al., J. Chem. Phys. 151, 044114 (2019)], resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains and water clusters.
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 present a real-time second-order Green’s function (GF) method for computing excited states in molecules and nanostructures, with a computational scaling of O ( N e 3 ), where N e is the number of electrons. The cubic scaling is achieved by adopting the stochastic resolution of the identity to decouple the 4-index electron repulsion integrals. To improve the time propagation and the spectral resolution, we adopt the dynamic mode decomposition technique and assess the accuracy and efficiency of the combined approach for a chain of hydrogen dimer molecules of different lengths. We find that the stochastic implementation accurately reproduces the deterministic results for the electronic dynamics and excitation energies. Furthermore, we provide a detailed analysis of the statistical errors, bias, and long-time extrapolation. Overall, the approach offers an efficient route to investigate excited states in extended systems with open or closed boundary conditions.
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.
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 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.