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# The divide–expand–consolidate coupled cluster scheme

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The Divide‐Expand‐Consolidate (DEC) scheme is a linear‐scaling and massively parallel framework for high accuracy coupled cluster (CC) calculations on large molecular systems. It is designed as a black‐box method, which ensures error control in the correlation energy and molecular properties. DEC is combined with a massively parallel implementation to fully utilize modern manycore architectures providing a fast time to solution. The implementation ensures performance portability and will straightforwardly benefit from new hardware developments. The DEC scheme has been applied to several levels of CC theory and extended the range of application of those methods. WIREs Comput Mol Sci 2017, 7:e1319. doi: 10.1002/wcms.1319

DEC‐RI‐MP2/cc‐pVDZ strong scaling plot for coarse‐grained parallelization: AAT 10 using 3738 to 18,400 nodes with strong scaling efficiency numbers along the curve. The blue line displays ideal strong scaling behavior.
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Schematic representation of the coarse and medium‐grained parallelization of the divide–expand–consolidate scheme for a 12 fragment calculation on nine nodes. Node 0 corresponds to the global master while each light green block corresponds to a fragment calculation. The red and gray lines denote local masters and slaves, respectively. Finally, the dashes inside the fragments correspond to idle time during the medium‐grained parallelization, while the wiggles denote idle time during the coarse‐grained parallelization.
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Overview of a typical divide–expand–consolidate calculation. The Hartree–Fock (HF) orbitals are localized and assigned to atomic sites. The atomic fragment optimization is then performed as described in Figure along with the calculation of pair energy estimates used for the pair screening procedure. The pairs that have not been screened away are then calculated (usually the time‐dominating part). Finally, the fragment energies are collected and added to obtain the total correlation energy.
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DEC‐RI‐MP2/cc‐pVDZ computational scaling with system size for 1‐aza‐adamantane‐trione (AAT) molecular wires of increasing lengths using 14,952 Titan nodes. The blue line displays the ideal linear‐scaling behavior. A molecular wire containing 10 AAT monomers and one of the localized molecular orbitals are also shown.
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Decay of the Møller–Plesset perturbation theory pair fragment interaction energies |ΔE PQ | with the pair distance R PQ for a reduced titin protein (392 atoms) using a cc‐pVDZ basis set (3772 basis functions). The expected $R P Q − 6$ pair decay is also plotted. The pair fragment interaction energies are separated into the calculated “final” pairs (in green) and the pairs screened away by the pair screening procedure (in blue).
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Benchmarking density functional theory functionals against divide–expand–consolidate and Møller–Plesset perturbation theory (MP2) electrostatic potential. (a) Difference between the MP2 and B3LYP electrostatic potential. (b) Difference between the MP2 and CAMB3LYP electrostatic potential. It was demonstrated that B3LYP incorrectly predicts partial electron transfer from anionic to cationic sites due to a combination of self‐interaction errors and an incorrect distance dependencies of the B3LYP functional. However, the range‐separated CAMB3LYP functional performs much better because of the elimination of self‐interaction errors at long distances.
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Illustration of the fragment optimization procedure, E P is the atomic fragment energy associated with atomic site P, P AOS denotes the space in which the CC amplitude equations are solved, and fragment optimization threshold (FOT) is a user‐defined threshold. The rectangles represent the complete orbital space where the blue color denotes orbitals that are prioritized (from left to right) according to their estimated contribution to the fragment energy (E P ). The red color denotes orbitals included in the fragment orbital space (P AOS), while discarded orbitals are represented in gray. Finally, the index n refers to the expanded fragment energy ($E P n$) which serves as a reference in the fragment reduction procedure.
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