Calculations of excited states in the Green’s function formalism often invokes the diagonal approximation, in which the quasiparticle states are taken from a mean-field calculation. In this paper, we extend the stochastic approaches applied in the many-body perturbation theory and overcome this limitation for large systems in which we are interested in a small subset of states. We separate the problem into a core subspace whose coupling to the remainder of the system environment is stochastically sampled. This method is exemplified on computing hole injection energies into CO 2 on an extended gold surface with nearly 3000 electrons. We find that in the extended system the size of the problem can be compressed up to 95% using stochastic sampling. This result provides a way forward for self-consistent stochastic methods and determination of Dyson orbitals in large systems.
Calculations of excited states in the Green’s function formalism often invokes the diagonal approximation, in which the quasiparticle states are taken from a mean-field calculation. In this paper, we extend the stochastic approaches applied in the many-body perturbation theory and overcome this limitation for large systems in which we are interested in a small subset of states. We separate the problem into a core subspace whose coupling to the remainder of the system environment is stochastically sampled. This method is exemplified on computing hole injection energies into CO 2 on an extended gold surface with nearly 3000 electrons. We find that in the extended system the size of the problem can be compressed up to 95% using stochastic sampling. This result provides a way forward for self-consistent stochastic methods and determination of Dyson orbitals in large systems.
We develop a formalism to calculate the quasiparticle energy within the GW many-body perturbation correction to the density functional theory. The occupied and virtual orbitals of the Kohn-Sham Hamiltonian are replaced by stochastic orbitals used to evaluate the Green function G, the polarization potential W, and, thereby, the GW self-energy. The stochastic GW (sGW) formalism relies on novel theoretical concepts such as stochastic time-dependent Hartree propagation, stochastic matrix compression, and spatial or temporal stochastic decoupling techniques. Beyond the theoretical interest, the formalism enables linear scaling GW calculations breaking the theoretical scaling limit for GW as well as circumventing the need for energy cutoff approximations. We illustrate the method for silicon nanocrystals of varying sizes with N_{e}>3000 electrons.
Accurate many-body treatments of condensed-phase systems are challenging because correlated solvers such as the full configuration interaction (FCI) and the density matrix renormalization group (DMRG) scale exponentially with system size. Downfolding and embedding approaches mitigate this cost but typically require prior selection of a correlated subspace, which can be difficult to determine in heterogeneous or extended systems. Here, we introduce a stochastic cluster expansion framework for efficiently recovering the total correlation energy of large systems with near-DMRG accuracy and without the need to select an active space a priori . By combining correlation contributions from randomly sampled environment orbitals with an exactly treated subspace of interest, the method reproduces total energies for nonreacting and reactive systems while drastically reducing computational cost. The approach also provides a quantitative diagnostic for molecule–solvent correlation, guiding principled embedding decisions. This framework enables systematically improvable many-body calculations in extended systems, opening the door to high-accuracy studies of chemical processes in condensed phase environments.
This thesis develops stochastic many-body methodologies for the electronic structure description of chemically relevant systems with strong electron correlation and nonequilibrium dynamics. Many phenomena of interest in chemistry and materials science, including excited-state processes, charge transfer at catalytic interfaces, and plasmon-mediated photocatalysis, require an accurate treatment of interacting electrons beyond mean-field theory. However, direct solutions of the many-body Schrodinger equation remain computationally intractable for realistic systems due to the exponential growth of the configurational Hilbert space with system size. This challenge motivates the development of reduced representations and embedding strategies capable of retaining essential correlation effects at lower computational cost. The work presented here focuses on stochastic matrix compression and downfolding techniques for many-body electronic structure theory. In particular, new stochastic compression approaches are introduced for the efficient representation of many-body Hamiltonians and correlated wavefunctions within reduced active subspaces. These methods extend stochastic electronic structure techniques beyond their traditional applications in many-body perturbation theory and provide a framework for treating correlated subsystems embedded within large chemical environments. Applications of these approaches are demonstrated across several classes of problems. First, stochastic downfolding methodologies are developed for the computation of charged excitations and Dyson orbitals relevant to catalytic interfaces and molecular spectroscopy. These techniques enable the treatment of large orbital spaces while maintaining accurate descriptions of electron correlation in the active region. Second, stochastic compression methods are applied to the computation of total energies in large solvated and reactive systems, where conventional correlated approaches become prohibitively expensive. Finally, the thesis explores nonequilibrium electronic dynamics at driven catalytic interfaces, including plasmon-assisted charge transfer processes. In this context, time-dependent many-body simulations reveal that transient driving fields can dynamically modify molecular spectral structure, opening additional resonant pathways for hot-carrier injection that are absent in equilibrium descriptions. Taken together, the methods developed in this thesis provide new strategies for reducing the computational complexity of correlated electronic structure calculations while preserving the essential physics of many-body interactions. These results broaden the applicability of stochastic and embedding-based approaches to large-scale equilibrium and nonequilibrium problems in chemistry, condensed matter physics, and materials science.
Accurate many-body treatments of condensed-phase systems are challenging because correlated solvers such as full configuration interaction (FCI) and the density matrix renormalization group (DMRG) scale exponentially with system size. Downfolding and embedding approaches mitigate this cost but typically require prior selection of a correlated subspace, which can be difficult to determine in heterogeneous or extended systems. Here, we introduce a stochastic cluster expansion framework for efficiently recovering the total correlation energy of large systems with near-DMRG accuracy, without the need to select an active space a priori. By combining correlation contributions from randomly sampled environment orbitals with an exactly treated subspace of interest, the method reproduces total energies for non-reacting and reactive systems while drastically reducing computational cost. The approach also provides a quantitative diagnostic for molecule-solvent correlation, guiding principled embedding decisions. This framework enables systematically improvable many-body calculations in extended systems, opening the door to high-accuracy studies of chemical processes in condensed phase environments.
We introduce the concept of sparse stochastic compression, an efficient stochastic sampling of any general function. The technique uses sparse stochastic orbitals (SSOs), short vectors that sample a small number of space points. As a first demonstration, SSOs are applied in conjunction with simple direct projection to accelerate our recent stochastic GW technique; the new developments enable accurate prediction of G0W0 quasiparticle energies and gaps for systems with up to Ne>10,000 electrons, with small statistical errors of $\ifmmode\pm\else\textpm\fi{}0.05\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ and using less than 2000 core CPU hours. Overall, stochastic GW scales now linearly (and often sublinearly) with Ne.
We fully develop the Monte Carlo many-body Green’s function (MC-GF) method with the following enhancements: (1) The truncation order of the perturbation expansion of the Dyson self-energy is raised from the second order (MC-GF2) to the third order (MC-GF3) with the aid of a computerized procedure to enumerate and transform all 84 third-order Goldstone diagrams into Monte Carlo integrable expressions and then into central processing unit (CPU)/graphical processing unit (GPU)-parallel computer codes. (2) An efficient algorithm is proposed that computes all off-diagonal and diagonal elements of the MC-GF2 and MC-GF3 self-energy matrices by common subexpression elimination. (3) The frequency-independent approximation is lifted by introducing a method that computes frequency derivatives of the MC-GF2 and MC-GF3 self-energies up to any arbitrarily high order at nearly no additional computational cost. (4) The imaginary-time integration in the Laplace-transformed expressions of the self-energy is carried out stochastically (instead of using a quadrature in the previous implementations), resulting in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker convergence acceleration scheme is analyzed numerically, and the guidelines are established to select an optimal number of walkers for maximal efficiency. When such an optimal number is used, the cost per sample is constant of molecular size on either many CPUs or many GPUs. (6) The computational cost to obtain a binding energy within a given statistical uncertainty is observed to increase as (tentatively) O( n 4 ) and O( n 5 ) of molecular size ( n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O( n 5 ) and O( n 6 ) of the corresponding deterministic algorithms. With this method applied to the electron binding energies of C 60, we show that the third-order corrections to the self-energies are much greater in electron binding energies than in ground-state energies. They display a sign of oscillatory convergence toward experimental results, not necessarily improving the agreement with increasing perturbation order, justifying MC-GF3 and motivating even higher-order methods.