Stochastic Self-Energy in a Self-Consistent Second-Order Green's Function Scheme

Abstract

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.