A Stochastic Cluster Expansion for Electronic Correlation in Large Systems

Abstract

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.

Figures (4)

• Figure 1: Schematic: the full cluster expansion is used to describe the FCS while a lower order truncation treats the stochastically sampled region around the FCS.
• Figure 2: The stochastic cluster expansion method captures the total correlation energy independently of the FCS partitioning of the solvated sodium metaphosphate system. (A) For each FCS size in panel B, the stochastic convergence scales as $1/\sqrt{N_{\zeta}}$. (B) For 25 samples, the total correlation energy for Metaphosphate and 7 waters is repeated for 5 different FCS sizes. We include the same 10 unoccupied single-particle states in all FCS subspaces but add progressively lower energy occupied single-particle states.
• Figure 3: The stochastic cluster expansion approach captures the total correlation energy for the Menshutkin reaction $\mathrm{H_3N + CH_3Cl \;\rightarrow\; CH_3NH_3^{+} + Cl^{-}}$. For an FCS subspace of five occupied orbitals and all eight bound unoccupied orbitals, the total correlation energy is computed for the reactants, transition state, and products of the reaction of $\rm CH_3Cl$ and $\rm NH_3$. Error bars indicate the SEM over 25 stochastic samples.
• Figure 4: (Left) Two-body correlation terms for single-particle states localized on solvent molecules at varying distances from the metaphosphate. The distance in Bohr corresponds to the cutoff used to select water molecules. Symmetric error bars indicate the SEM over 25 stochastic samples. (Right) Panels A–D show the density of single-particle states sampled for each point in the left-hand plot.