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arXiv:2609.32884

Reduced scaling implementation of the coupled cluster-based static embedding method, MPCC with density fitting

Avijit Shee, Fabian M. Faulstich, K. Birgitta Whaley, Lin Lin, Martin Head-Gordon

physics.chem-phquant-phSubmitted Sep 26, 2026• Announced Sep 29, 2026• 0 citations• Revised Oct 6, 2026 (v2)
v1 · Sep 26, 2026v2 · Oct 6, 2026
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Abstract

An improved algorithm to implement our previously proposed static quantum embedding method MPCC (J. Chem. Phys. 161, 164107 (2024)) is presented. It uses the density-fitting (DF) approximation for the electron-repulsion integrals (ERIs) across all layers of the embedding method: the low-level method, the screened interaction, and the high-level method. Additionally, an approximate factorized solution of the Sylvester equation is employed based on the Laplace transformation for the low-level method. To build the screened interaction, we have explored an approximate construction of it based on the distinguishable cluster approximation (DCA) by Kats and Manby (J. Chem. Phys. 141, 061101 (2014)), which amounts to neglecting certain higher-order exchange and ladder diagrams. Combining all these techniques, we have obtained an $\mathcal{O}(N^4)$ algorithm for the low-level method; a non-iterative $N_{\rm frag} \mathcal{O}(N^4)$ algorithm for the screened interaction; and an iterative $\mathcal{O}(N_{\rm frag}^6)$ algorithm for the fragment solver, where $N_{\rm frag}$ is the number of fragment orbitals and $N$ is the total number of orbitals. The resulting low-scaling implementation of the MPCC method is applied to trans polyacetylenes, that is, C$_{2n}$H$_{2n+2}$ molecules for chain lengths $n=1-8$, and the timing and accuracy of the method are compared to the original unfactorized implementation. Moreover, the approximations are further validated by applying the method to the potential energy surface of the \ce{O3} molecule. Finally, we apply the method to the selected molecules in the S22 dataset for which the performance of the MP2 method is known to be poor. The results demonstrate that the DF-based implementation of the MPCC method provides a significant reduction in computational cost while maintaining accuracy comparable to that of the original implementation.

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An improved algorithm to implement our previously proposed static quantum embedding method MPCC (J. Chem. Phys. 161, 164107 (2024)) is presented. It uses the density-fitting (DF) approximation for the electron-repulsion integrals (ERIs) across all layers of the embedding method: the low-level method, the screened interaction, and the high-level method. Additionally, an approximate factorized solution of the Sylvester equation is employed based on the Laplace transformation for the low-level method. To build the screened interaction, we have explored an approximate construction of it based on the distinguishable cluster approximation (DCA) by Kats and Manby (J. Chem. Phys. 141, 061101 (2014)), which amounts to neglecting certain higher-order exchange and ladder diagrams. Combining all these techniques, we have obtained an O(N4)\mathcal{O}(N^4)O(N4) algorithm for the low-level method; a non-iterative NfragO(N4)N_{\rm frag} \mathcal{O}(N^4)Nfrag​O(N4) algorithm for the screened interaction; and an iterative O(Nfrag6)\mathcal{O}(N_{\rm frag}^6)O(Nfrag6​) algorithm for the fragment solver, where NfragN_{\rm frag}Nfrag​ is the number of fragment orbitals and NNN is the total number of orbitals. The resulting low-scaling implementation of the MPCC method is applied to trans polyacetylenes, that is, C2n_{2n}2n​H2n+2_{2n+2}2n+2​ molecules for chain lengths n=1−8n=1-8n=1−8, and the timing and accuracy of the method are compared to the original unfactorized implementation. Moreover, the approximations are further validated by applying the method to the potential energy surface of the \ce{O3} molecule. Finally, we apply the method to the selected molecules in the S22 dataset for which the performance of the MP2 method is known to be poor. The results demonstrate that the DF-based implementation of the MPCC method provides a significant reduction in computational cost while maintaining accuracy comparable to that of the original implementation.
We incorporate a canonical polyadic decomposition (CPD) based low-level solver as a means to accelerate the environment-level solver for the recently developed MPCC embedding framework. Using CPD, we both factorize the three dominant order-three density-fitting two-electron integral (DF TEI) tensors and develop a novel formulation that reduces the storage complexity of the low-level solver from O(N3){O}(N^3)O(N3) to O(NR)O(NR)O(NR), where RRR is the CPD rank, and the computational scaling of the most time-consuming contractions from O(N4){O}(N^4)O(N4) to O(NR2){O}(NR^2)O(NR2). We provide benchmarks on representative chemical environments, namely water clusters (H2O)n\mathrm{(H_2O)_n}(H2​O)n​ with n=1n = 1n=1 to 666 and linear alkane chains CnH2n+2\mathrm{C_nH_{2n+2}}Cn​H2n+2​ with n=1n = 1n=1 to 666. For both test sets, using the CPD-compressed DF TEI tensors reproduces the DF reference convergence behavior of the low-level solver, the subsequent high-level step, and the fully self-consistent MPCC iterations, while introducing only small, rank-controlled shifts in absolute energies. At a fixed tolerance in the absolute MPCC energy, the CP ranks required for these tensor approximations increase linearly with system size. Chemically relevant energy differences are likewise preserved, as demonstrated for water-cluster dissociation energies and in a proof-of-concept embedding calculation of methane in a four-water cluster.
We develop a static quantum embedding scheme that utilizes different levels of approximations to coupled cluster (CC) theory for an active fragment region and its environment. To reduce the computational cost, we solve the local fragment problem using a high-level CC method and address the environment problem with a lower-level Møller-Plesset (MP) perturbative method. This embedding approach inherits many conceptual developments from the hybrid second-order Møller-Plesset (MP2) and CC works by Nooijen [J. Chem. Phys. 111, 10815 (1999)] and Bochevarov and Sherrill [J. Chem. Phys. 122, 234110 (2005)]. We go beyond those works here by primarily targeting a specific localized fragment of a molecule and also introducing an alternative mechanism to relax the environment within this framework. We will call this approach MP-CC. We demonstrate the effectiveness of MP-CC on several potential energy curves and a set of thermochemical reaction energies, using CC with singles and doubles as the fragment solver, and MP2-like treatments of the environment. The results are substantially improved by the inclusion of orbital relaxation in the environment. Using localized bonds as the active fragment, we also report results for N=N bond breaking in azomethane and for the central C-C bond torsion in butadiene. We find that when the fragment Hilbert space size remains fixed (e.g., when determined by an intrinsic atomic orbital approach), the method achieves comparable accuracy with both a small and a large basis set. Additionally, our results indicate that increasing the fragment Hilbert space size systematically enhances the accuracy of observables, approaching the precision of the full CC solver.
We incorporate a solver for the fragment problem with accuracy beyond coupled cluster singles and doubles (CCSD) into the previously proposed static embedding framework, MPCC. To this end, we employ a CCSDT solver for the fragment subsystem. For the environment subsystem, we construct a perturbative estimate of the triples amplitudes, explicitly accounting for feedback from all fragment amplitudes. The resulting approach is denoted MPCCSDT(pt). We further introduce a more complete formulation in which feedback from the environment amplitudes to the fragment amplitudes is also included. This scheme involves an iterative treatment of the environment triples amplitudes and is denoted MPCCSDT(it). In addition, we assess the accuracy of the previously proposed low-level method by introducing a modified low-level approach that incorporates a lowest-order treatment of selected long-range effects, including spin fluctuations and charge polarization. All resulting approaches may be viewed as post-CCSD(T) methods. We therefore consider test cases for which CCSD(T) exhibits substantial deviations from CCSDT. Our results demonstrate that inclusion of triples amplitudes at the fragment level alone is insufficient; a perturbative treatment of the environment triples amplitudes is required. For many energy-difference applications, feedback from the environment triples amplitudes to the fragment amplitudes, is not essential, but it does play a role in the very challenging molecules. A very interesting finding from our study is that in some challenging cases, we need an improved (second-order) perturbative method for the SD amplitudes, going beyond the first-order one used in our earlier work.
We incorporate a canonical polyadic decomposition (CPD)-based low-level solver as a means to accelerate the environment-level solver for the recently developed Møller-Plesset Coupled-Cluster (MPCC) embedding framework. Using CPD, we both factorize the three dominant order-three density-fitting two-electron integral (DF TEI) tensors and develop a novel formulation that reduces the storage complexity of the low-level solver from O(N3) to O(NR), where R is the CPD rank, and the computational scaling of the most time-consuming contractions from O(N4) to O(NR2). We provide benchmarks on representative chemical environments, namely water clusters (H2O)n with n = 1 to 6 and linear alkane chains CnH2n+2 with n = 1 to 6. For both test sets, using the CPD-compressed DF TEI tensors reproduces the DF reference convergence behavior of the low-level solver, the subsequent high-level step, and the fully self-consistent MPCC iterations, while introducing only small, rank-controlled shifts in absolute energies. At a fixed tolerance in the absolute MPCC energy, the CP ranks required for these tensor approximations increase linearly with system size. Chemically relevant energy differences are likewise preserved, as demonstrated for water-cluster dissociation energies and in a proof-of-concept embedding calculation of methane in a four-water cluster.
Recent developments in quantum embedding theories have provided attractive approaches to correlated calculations for large systems. In this work, we extend our previous work [ J. Chem. Theory Comput. 2019, 15, 4497–4506; J. Phys. Chem. Lett. 2019, 10, 6368–6374] on bootstrap embedding (BE) to enable correlated ab initio calculations at the coupled cluster with singles and doubles (CCSD) level for large molecules. We introduce several new algorithmic developments that significantly reduce the computational cost of BE, while maintaining its accuracy. The resulting implementation scales as O ( N 3 ) for the integral transform and O ( N ) for the CCSD calculation. Numerical results on a series of conjugated molecules suggest that BE with reasonably sized fragments can recover more than 99.5% of the total correlation energy of a full CCSD calculation, while the required computational resources (time and storage) compare favorably to one popular local correlation scheme: domain localized pair natural orbital (DLPNO). The largest BE calculation in this work involves ∼2900 basis functions and can be performed on a single node with 16 CPU cores and 64 GB of memory in a few days. We anticipate that these developments represent an important step toward the application of BE to solve practical problems.
We consider the rank-reduced coupled-cluster theory with single and double (RR-CCSD) excitations introduced recently [Parrish et al., J. Chem. Phys. 150, 164118 (2019)]. The main feature of this method is the decomposed form of doubly excited amplitudes, which are expanded in the basis of largest magnitude eigenvectors of MP2 or MP3 amplitudes. This approach enables a substantial compression of amplitudes with only minor loss of accuracy. However, the formal scaling of the computational costs with the system size (N) is unaffected in comparison with the conventional CCSD theory (∝N6) due to the presence of some terms quadratic in amplitudes, which do not naturally factorize to a simpler form even within the rank-reduced framework. We show how to solve this problem, exploiting the fact that their effective rank increases only linearly with the system size. We provide a systematic way to approximate the problematic terms using the singular value decomposition and reduce the scaling of the RR-CCSD iterations down to the level of N5. This is combined with an iterative method of finding dominant eigenpairs of MP2 or MP3 amplitudes, which eliminates the necessity to perform the complete diagonalization, making the cost of this step proportional to the fifth power of the system size, as well. Next, we consider the evaluation of perturbative corrections to CCSD energies resulting from triply excited configurations. The triply excited amplitudes present in the CCSD(T) method are decomposed to the Tucker-3 format using the higher-order orthogonal iteration procedure. This enables us to compute the energy correction due to triple excitations non-iteratively with N6 cost. The accuracy of the resulting rank-reduced CCSD(T) method is studied for both total and relative correlation energies of a diverse set of molecules. Accuracy levels better than 99.9% can be achieved with a substantial reduction of the computational costs. Concerning the computational timings, the break-even point between the rank-reduced and conventional CCSD implementations occurs for systems with about 30–40 active electrons.