add ref
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@ -288,3 +288,248 @@ swh:1:dir:6d82ae7ac757c78d7720dd89dfa52d7a453d2f68;origin=https://github.com/Qua
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publisher = {IOP Publishing},
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doi = {10.1088/2516-1075/ad2eb0}
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}
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@article{cizek_1966,
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author = {{\ifmmode\check{C}\else\v{C}\fi}{\ifmmode\acute{\imath}\else\'{\i}\fi}{\ifmmode\check{z}\else\v{z}\fi}ek, Ji{\ifmmode\check{r}\else\v{r}\fi}{\ifmmode\acute{\imath}\else\'{\i}\fi}},
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doi = {10.1063/1.1727484},
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issn = {0021-9606},
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journal = {J. Chem. Phys.},
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month = dec,
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number = {11},
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pages = {4256--4266},
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publisher = {American Institute of Physics},
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title = {{On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell{-}Type Expansion Using Quantum{-}Field Theoretical Methods}},
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volume = {45},
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year = {1966},
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bdsk-url-1 = {https://doi.org/10.1063/1.1727484}}
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@incollection{cizek_1969,
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address = {Chichester, England, UK},
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author = {{\ifmmode\check{C}\else\v{C}\fi}{\ifmmode\acute{\imath}\else\'{\i}\fi}{\ifmmode\check{z}\else\v{z}\fi}ek, Ji{\ifmmode\check{r}\else\v{r}\fi}{\ifmmode\acute{\imath}\else\'{\i}\fi}},
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booktitle = {{Advances in Chemical Physics}},
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doi = {10.1002/9780470143599.ch2},
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isbn = {978-0-47014359-9},
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journal = {Wiley Online Library},
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month = jan,
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pages = {35--89},
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publisher = {John Wiley {\&} Sons, Ltd},
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title = {{On the Use of the Cluster Expansion and the Technique of Diagrams in Calculations of Correlation Effects in Atoms and Molecules}},
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year = {1969},
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bdsk-url-1 = {https://doi.org/10.1002/9780470143599.ch2}}
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@incollection{paldus_1992,
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address = {Boston, MA, USA},
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author = {Paldus, Josef},
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booktitle = {{Methods in Computational Molecular Physics}},
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doi = {10.1007/978-1-4615-7419-4_7},
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journal = {SpringerLink},
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pages = {99--194},
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publisher = {Springer, Boston, MA},
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title = {{Coupled Cluster Theory}},
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year = {1992},
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bdsk-url-1 = {https://doi.org/10.1007/978-1-4615-7419-4_7}}
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@incollection{crawford_2000,
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address = {Chichester, England, UK},
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author = {Crawford, T. Daniel and Schaefer, Henry F.},
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booktitle = {{Reviews in Computational Chemistry}},
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doi = {10.1002/9780470125915.ch2},
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isbn = {978-0-47012591-5},
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journal = {Wiley Online Library},
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month = jan,
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pages = {33--136},
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publisher = {John Wiley {\&} Sons, Ltd},
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title = {{An Introduction to Coupled Cluster Theory for Computational Chemists}},
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year = {2000},
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bdsk-url-1 = {https://doi.org/10.1002/9780470125915.ch2}}
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@article{bartlett_2007,
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author = {Bartlett, Rodney J. and Musia{\l}, Monika},
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doi = {10.1103/RevModPhys.79.291},
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issn = {1539-0756},
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journal = {Rev. Mod. Phys.},
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month = feb,
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number = {1},
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pages = {291--352},
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publisher = {American Physical Society},
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title = {{Coupled-cluster theory in quantum chemistry}},
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volume = {79},
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year = {2007},
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bdsk-url-1 = {https://doi.org/10.1103/RevModPhys.79.291}}
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@book{shavitt_2009,
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address = {Cambridge, England, UK},
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author = {Shavitt, Isaiah and Bartlett, Rodney J.},
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doi = {10.1017/CBO9780511596834},
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isbn = {978-0-52181832-2},
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journal = {Cambridge Core},
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month = aug,
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publisher = {Cambridge University Press},
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title = {{Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory}},
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year = {2009},
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bdsk-url-1 = {https://doi.org/10.1017/CBO9780511596834}}
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@article{raghavachari_1989,
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author = {Raghavachari, Krishnan and Trucks, Gary W. and Pople, John A. and Head-Gordon, Martin},
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title = {{A fifth-order perturbation comparison of electron correlation theories}},
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journal = {Chem. Phys. Lett.},
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volume = {157},
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pages = {479--483},
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issn = {0009-2614},
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publisher = {North-Holland},
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doi = {10.1016/S0009-2614(89)87395-6}}
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@article{gyevi_2020,
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author = {Gyevi-Nagy, L{\ifmmode\acute{a}\else\'{a}\fi}szl{\ifmmode\acute{o}\else\'{o}\fi} and K{\ifmmode\acute{a}\else\'{a}\fi}llay, Mih{\ifmmode\acute{a}\else\'{a}\fi}ly and Nagy, P{\ifmmode\acute{e}\else\'{e}\fi}ter R.},
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title = {{Integral-Direct and Parallel Implementation of the CCSD(T) Method: Algorithmic Developments and Large-Scale Applications}},
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journal = {J. Chem. Theory Comput.},
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volume = {2020},
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number = {, 16},
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pages = {,1},
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year = {2020},
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month = jan,
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publisher = {American Chemical Society},
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doi = {10.1021/acs.jctc.9b00957}}
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@article{jiang_2023,
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author = {Jiang, Andy and Turney, Justin M. and Henry F. Schaefer, Iii},
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title = {{Tensor Hypercontraction Form of the Perturbative Triples Energy in Coupled-Cluster Theory}},
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journal = {J. Chem. Theory Comput.},
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volume = {2023},
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number = {, 19},
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pages = {,5},
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year = {2023},
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month = mar,
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publisher = {American Chemical Society},
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doi = {10.1021/acs.jctc.2c00996}}
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@article{janowski_2008,
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author = {Janowski, Tomasz and Pulay, Peter},
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title = {{Efficient Parallel Implementation of the CCSD External Exchange Operator and the Perturbative Triples (T) Energy Calculation}},
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journal = {J. Chem. Theory Comput.},
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volume = {4},
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number = {10},
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pages = {1585--1592},
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year = {2008},
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month = oct,
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issn = {1549-9618},
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publisher = {American Chemical Society},
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doi = {10.1021/ct800142f}}
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@article{anisimov_2014,
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author = {Anisimov, Victor M. and Bauer, Gregory H. and Chadalavada, Kalyana and Olson, Ryan M. and Glenski, Joseph W. and Kramer, William T. C. and Apr{\ifmmode\grave{a}\else\`{a}\fi}, Edoardo and Kowalski, Karol},
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title = {{Optimization of the Coupled Cluster Implementation in NWChem on Petascale Parallel Architectures}},
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journal = {J. Chem. Theory Comput.},
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volume = {10},
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number = {10},
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issn = {1549-9618},
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publisher = {American Chemical Society},
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doi = {10.1021/ct500404c}}
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@article{deumens_2011,
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author = {Deumens, Erik and Lotrich, Victor F. and Perera, Ajith and Ponton, Mark J. and Sanders, Beverly A. and Bartlett, Rodney J.},
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title = {{Software design of ACES III with the super instruction architecture}},
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journal = {WIREs Comput. Mol. Sci.},
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issn = {1759-0876},
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publisher = {John Wiley {\&} Sons, Ltd},
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doi = {10.1002/wcms.77}}
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@article{datta_2021,
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author = {Datta, Dipayan and Gordon, Mark S.},
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title = {{A Massively Parallel Implementation of the CCSD(T) Method Using the Resolution-of-the-Identity Approximation and a Hybrid Distributed/Shared Memory Parallelization Model}},
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journal = {J. Chem. Theory Comput.},
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volume = {17},
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year = {2021},
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month = aug,
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issn = {1549-9618},
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publisher = {American Chemical Society},
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doi = {10.1021/acs.jctc.1c00389}}
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@article{gyevi_2021,
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author = {Gyevi-Nagy, L{\ifmmode\acute{a}\else\'{a}\fi}szl{\ifmmode\acute{o}\else\'{o}\fi} and K{\ifmmode\acute{a}\else\'{a}\fi}llay, Mih{\ifmmode\acute{a}\else\'{a}\fi}ly and Nagy, P{\ifmmode\acute{e}\else\'{e}\fi}ter R.},
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title = {{Accurate Reduced-Cost CCSD(T) Energies: Parallel Implementation, Benchmarks, and Large-Scale Applications}},
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journal = {J. Chem. Theory Comput.},
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volume = {17},
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number = {2},
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pages = {860--878},
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year = {2021},
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month = feb,
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issn = {1549-9618},
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publisher = {American Chemical Society},
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doi = {10.1021/acs.jctc.0c01077}}
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@article{deprince_2013,
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author = {Deprince, A. Eugene Iii and Sherrill, C. David},
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title = {{Accuracy and Efficiency of Coupled-Cluster Theory Using Density Fitting/Cholesky Decomposition, Frozen Natural Orbitals, and a t1-Transformed Hamiltonian}},
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journal = {J. Chem. Theory Comput.},
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volume = {9},
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number = {6},
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pages = {2687--2696},
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year = {2013},
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month = jun,
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issn = {1549-9618},
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publisher = {American Chemical Society},
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doi = {10.1021/ct400250u}}
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@article{peng_2019,
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author = {Peng, Chong and Calvin, Justus A. and Valeev, Edward F.},
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title = {{Coupled-cluster singles, doubles and perturbative triples with density fitting approximation for massively parallel heterogeneous platforms}},
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journal = {Int. J. Quantum Chem.},
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volume = {119},
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number = {12},
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pages = {e25894},
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year = {2019},
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month = jun,
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issn = {0020-7608},
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publisher = {John Wiley {\&} Sons, Ltd},
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doi = {10.1002/qua.25894}}
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@article{shen_2019,
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author = {Shen, Tonghao and Zhu, Zhenyu and Zhang, Igor Ying and Scheffler, Matthias},
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title = {{Massive-Parallel Implementation of the Resolution-of-Identity Coupled-Cluster Approaches in the Numeric Atom-Centered Orbital Framework for Molecular Systems}},
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journal = {J. Chem. Theory Comput.},
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volume = {15},
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number = {9},
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pages = {4721--4734},
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year = {2019},
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month = sep,
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issn = {1549-9618},
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publisher = {American Chemical Society},
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doi = {10.1021/acs.jctc.8b01294}}
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@article{pitonak_2011,
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author = {Pito{\ifmmode\check{n}\else\v{n}\fi}{\ifmmode\acute{a}\else\'{a}\fi}k, Michal and Aquilante, Francesco and Hobza, Pavel and Neogr{\ifmmode\acute{a}\else\'{a}\fi}dy, Pavel and Noga, Jozef and Urban, Miroslav},
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title = {{Parallelized implementation of the CCSD(T) method in MOLCAS using optimized virtual orbitals space and Cholesky decomposed two-electron integrals}},
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journal = {Collect. Czech. Chem. Commun.},
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volume = {76},
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number = {6},
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pages = {713--742},
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year = {2011},
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month = may,
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issn = {1212-6950},
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publisher = {Institute of Organic Chemistry and Biochemistry AS CR, v.v.i.},
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doi = {10.1135/cccc2011048}}
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@article{wang_2020,
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author = {Wang, Zhifan and Guo, Minggang and Wang, Fan},
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title = {{Single-precision open-shell CCSD and CCSD(T) calculations on graphics processing units}},
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journal = {Phys. Chem. Chem. Phys.},
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volume = {22},
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number = {43},
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pages = {25103--25111},
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year = {2020},
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publisher = {Royal Society of Chemistry},
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doi = {10.1039/D0CP03800H}}
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@ -25,11 +25,12 @@
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\toto}[1]{\textcolor{blue}{#1}}
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\newcommand{\joonho}[1]{\textcolor{purple}{#1}}
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\newcommand{\yann}[1]{\textcolor{purple}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\AS}[1]{\toto{(\underline{\bf AS}: #1)}}
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\newcommand{\JL}[1]{\joonho{(\underline{\bf JL}: #1)}}
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\newcommand{\YANN}[1]{\yann{(\underline{\bf YANN}: #1)}}
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\newcommand{\byann}[1]{\textcolor{purple}{\sout{#1}}}
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\usepackage{listings}
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\definecolor{codegreen}{rgb}{0.58,0.4,0.2}
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@ -121,23 +122,23 @@ that were previously computationally prohibitive.
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\section{Introduction}
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\label{sec:introduction}
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Coupled cluster (CC) theory is an accurate quantum mechanical approach widely used in computational chemistry and physics to describe the electronic structure of atoms, molecules, and materials.
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Coupled cluster (CC) theory is an accurate quantum mechanical approach widely used in computational chemistry and physics to describe the electronic structure of atoms, molecules, and materials.\cite{Cizek_1966,Cizek_1969,Paldus_1992}
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It offers a systematic and rigorous framework for accurate predictions of molecular properties and reactions by accounting for electron correlation effects beyond the mean-field approximation.
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CC theory starts with a parametrized wave function, typically referred to as the CC wave function, which is expressed as an exponential series of excitation operators acting on a reference:
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\begin{equation}
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\ket{\Psi_{\text{CC}}} = e^{\hat{T}} \ket{\Phi}
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\end{equation}
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where $\ket{\Phi}$ is the reference determinant, and $\hat{T}$ is the cluster operator representing single, double, triple, and higher excitations from the reference wave function.
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where $\ket{\Phi}$ is the reference determinant, and $\hat{T}$ is the cluster operator representing single, double, triple, and higher excitations from the reference wave function.\cite{crawford_2000,bartlett_2007,shavitt_2009}
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Coupled Cluster with Singles and Doubles (CCSD) includes single and double excitations and represents the most commonly used variant of CC theory due to its favorable balance between accuracy and computational cost.
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Coupled Cluster with Singles, Doubles, and perturbative Triples (CCSD(T)) incorporates a perturbative correction to the CCSD energy to account for some higher-order correlation effects, and stands as the gold standard of quantum chemistry.
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Coupled Cluster with Singles, Doubles, and perturbative Triples (CCSD(T)) incorporates a perturbative correction to the CCSD energy to account for some higher-order correlation effects, and stands as the gold standard of quantum chemistry.\cite{raghavachari_1989}
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CCSD(T) has demonstrated exceptional accuracy and reliability, making it one of the preferred choices for benchmark calculations and highly accurate predictions.
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It has found successful applications in a diverse range of areas, including spectroscopy,\cite{villa_2011,watson_2016,vilarrubias_2020} reaction kinetics,\cite{dontgen_2015,castaneda_2012} and materials design,\cite{zhang_2019} and has played a pivotal role in advancing our understanding of complex chemical phenomena.
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In the context of CC theory, the perturbative triples correction represents an important contribution to the accuracy of electronic structure calculations.\cite{stanton_1997}
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However, the computational cost associated with the calculation of this correction can be prohibitively high, especially for large systems.
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The inclusion of the perturbative triples in the CCSD(T) method leads to a computational scaling of $\order{N^7}$, where $N$ is proportional to the number of molecular orbitals.
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This scaling can rapidly become impractical, posing significant challenges in terms of computational resources and time requirements.
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This scaling can rapidly become impractical, posing significant challenges in terms of computational resources and time requirements.\cite{janowski_2008,deumens_2011,pitonak_2011,deprince_2013,anisimov_2014,peng_2019,shen_2019,gyevi_2020,wang_2020,datta_2021,gyevi_2021,jiang_2023}
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To address this computational bottleneck, our goal is to develop a novel semi-stochastic algorithm that brings back the computational time to a level smaller or comparable to that of the CCSD method, which has a scaling of $\order{N^6}$, while ensuring well-controlled approximations.
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Our algorithm strikes a balance between computational efficiency and
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@ -156,7 +157,7 @@ The perturbative triples correction,
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E_{(T)} = \sum_{ijk\,abc} E_{ijk}^{abc},
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\end{equation}
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is a sum of $N_{\text{o}}^3 \times N_{\text{v}}^3$ terms, where $N_{\text{o}}^3$ and $N_{\text{v}}^3$ denote the number of occupied and virtual molecular orbitals, respectively.
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Each individual term is expressed as
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\yann{For a closed-shell reference with canonical orbitals,} each individual term is expressed as\cite{rendell_1991}
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\begin{equation}
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E_{ijk}^{abc} = \frac{1}{3} \frac{(4 W_{ijk}^{abc} +
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W_{ijk}^{bca} + W_{ijk}^{cab})
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