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arXiv:2105.10548 · The Journal of Chemical Physics

Cluster many-body expansion: A many-body expansion of the electron correlation energy about a cluster mean field reference

Vibin Abraham, Nicholas J. Mayhall

Published Aug 2, 2021• 18 citations
arXivHTMLPDF

Abstract

The many-body expansion (MBE) is an efficient tool that has a long history of use for calculating interaction energies, binding energies, lattice energies, and so on. In the past, applications of MBE to correlation energy have been unfeasible for large systems, but recent improvements to computing resources have sparked renewed interest in capturing the correlation energy using the generalized nth order Bethe–Goldstone equation. In this work, we extend this approach, originally proposed for a Slater determinant, to a tensor product state (TPS) based wavefunction. By partitioning the active space into smaller orbital clusters, our approach starts from a cluster mean field reference TPS configuration and includes the correlation contribution of the excited TPSs using the MBE. This method, named cluster MBE (cMBE), improves the convergence of MBE at lower orders compared to directly doing a block-based MBE from a RHF reference. We present numerical results for strongly correlated systems, such as the one- and two-dimensional Hubbard models and the chromium dimer. The performance of the cMBE method is also tested by partitioning the extended π space of several large π-conjugated systems, including a graphene nano-sheet with a very large active space of 114 electrons in 114 orbitals, which would require 1066 determinants for the exact FCI solution.

DOI
10.1063/5.0057752

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The many-body expansion (MBE) is an efficient tool that has a long history of use for calculating interaction energies, binding energies, lattice energies, and so on. In the past, applications of MBE to correlation energy have been unfeasible for large systems, but recent improvements to computing resources have sparked renewed interest in capturing the correlation energy using the generalized nth order Bethe–Goldstone equation. In this work, we extend this approach, originally proposed for a Slater determinant, to a tensor product state (TPS) based wavefunction. By partitioning the active space into smaller orbital clusters, our approach starts from a cluster mean field reference TPS configuration and includes the correlation contribution of the excited TPSs using the MBE. This method, named cluster MBE (cMBE), improves the convergence of MBE at lower orders compared to directly doing a block-based MBE from a RHF reference. We present numerical results for strongly correlated systems, such as the one- and two-dimensional Hubbard models and the chromium dimer. The performance of the cMBE method is also tested by partitioning the extended π space of several large π-conjugated systems, including a graphene nano-sheet with a very large active space of 114 electrons in 114 orbitals, which would require 1066 determinants for the exact FCI solution.
Facilitated by a rigorous partitioning of a molecular system's orbital basis into two fundamental subspaces-a reference and an expansion space, both with orbitals of unspecified occupancy-we generalize our recently introduced many-body expanded full configuration interaction (MBE-FCI) method to allow for electron-rich model and molecular systems dominated by both weak and strong correlation to be addressed. By employing minimal or even empty reference spaces, we show through calculations on the one-dimensional Hubbard model with up to 46 lattice sites, the chromium dimer, and the benzene molecule how near-exact results may be obtained in an entirely unbiased manner for chemical and physical problems of not only academic but also applied chemical interest. Given the massive parallelism and overall accuracy of the resulting method, we argue that generalized MBE-FCI theory possesses an immense potential to yield near-exact correlation energies for molecular systems of unprecedented size, composition, and complexity in the years to come.
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.
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.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of (Nε−1)O(log⁡Δ)(N\varepsilon^{-1})^{\mathcal{O}(\logΔ)}(Nε−1)O(logΔ), where NNN is the system size, ε\varepsilonε the desired precision, and ΔΔΔ the interaction-graph degree. This becomes particularly severe in long-range systems, where Δ≡NΔ\equiv NΔ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime O~(N4ε−3)\tilde{\mathcal{O}}(N^4\varepsilon^{-3})O~(N4ε−3). For quantum partition functions at temperatures β≤βsampling≡O(1)β\leqβ_{\mathrm{sampling}}\equiv\mathcal{O}(1)β≤βsampling​≡O(1), this improves to O~(N3+1/Dε−3)\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})O~(N3+1/Dε−3), where DDD is the lattice dimension. The framework directly applies to long-range quantum systems with interactions ∝1/rα\propto1/r^α∝1/rα, including 222-local models with α>2Dα>2Dα>2D.
We present a wide-reaching revamp of the generalized many-body expanded full configuration interaction (MBE-FCI) method. First, we outline how to automatize the selection of reference active spaces, whereby the inherent bias introduced through a manual identification is reduced, also within the context of traditional complete active space methods. Second, we allow for the use of compact orbital clusters as expansion objects, which works to circumvent the unfavorable scaling with the number of orbitals included in the space complementary to the reference orbitals. Finally, we present a new algorithm for ensuring that many-body expansions can be efficiently terminated while conservatively accounting for resulting errors. These developments are all tested on a variety of molecular systems and different orbital representations to illustrate the abilities of our algorithm to produce correlation energies within predetermined error bounds, significantly broadening the overall applicability of the MBE-FCI method.
Size extensivity, defined as the correct scaling of energy with system size, is a desirable property for any many-body method. Traditional configuration interaction (CI) methods are not size extensive, hence the error increases as the system gets larger. Coupled electron pair approximation (CEPA) methods can be constructed as simple extensions of a truncated CI that ensures size extensivity. One of the major issues with the CEPA and its variants is that singularities arise in the amplitude equations when the system starts to be strongly correlated. In this work, we extend the traditional Slater determinant based coupled electron pair approaches like CEPA-0, averaged coupled-pair functional, and average quadratic coupled-cluster to a new formulation based on tensor product states (TPS). We show that a TPS basis can often be chosen such that it removes the singularities that commonly destroy the accuracy of CEPA based methods. A suitable TPS representation can be formed by partitioning the system into separate disjoint clusters and forming the final wave function as the tensor product of the many body states of these clusters. We demonstrate the application of these methods on simple bond breaking systems such as CH 4 and F 2 where determinant based CEPA methods fail. We further apply the TPS-CEPA approach to stillbene isomerization and few planar π-conjugated systems. Overall, the results show that the TPS-CEPA method can remove the singularities and provide improved numerical results compared to common electronic structure methods.