Δ-Machine Learning to Elevate DFT-Based Potentials and a Force Field to the CCSD(T) Level Illustrated for Ethanol.

[en] Progress in machine learning has facilitated the development of potentials that offer both the accuracy of first-principles techniques and vast increases in the speed of evaluation. Recently, Δ-machine learning has been used to elevate the quality of a potential energy surface (PES) based on low-level, e.g., density functional theory (DFT) energies and gradients to close to the gold-standard coupled cluster level of accuracy. We have demonstrated the success of this approach for molecules, ranging in size from H3O+ to 15-atom acetyl-acetone and tropolone. These were all done using the B3LYP functional. Here, we investigate the generality of this approach for the PBE, M06, M06-2X, and PBE0 + MBD functionals, using ethanol as the example molecule. Linear regression with permutationally invariant polynomials is used to fit both low-level and correction PESs. These PESs are employed for standard RMSE analysis for training and test data sets, and then general fidelity tests such as energetics of stationary points, normal-mode frequencies, and torsional potentials are examined. We achieve similar improvements in all cases. Interestingly, we obtained significant improvement over DFT gradients where coupled cluster gradients were not used to correct the low-level PES. Finally, we present some results for correcting a recent molecular mechanics force field for ethanol and comment on the possible generality of this approach.

Disciplines :

Chemistry

Author, co-author :

NANDI, Apurba ; University of Luxembourg

Pandey, Priyanka ; Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States

Houston, Paul L ; Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, United States ; Department of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332, United States

Qu, Chen ; Independent Researcher, Toronto, Ontario M9B0E3, Canada

Yu, Qi ; Department of Chemistry, Fudan University, Shanghai 200438, P. R. China

Conte, Riccardo ; Dipartimento di Chimica, Università degli Studi di Milano, via Golgi 19, 20133 Milano, Italy

TKATCHENKO, Alexandre ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Physics and Materials Science (DPHYMS)

Bowman, Joel M ; Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, United States

External co-authors :

yes

Language :

English

Title :

Δ-Machine Learning to Elevate DFT-Based Potentials and a Force Field to the CCSD(T) Level Illustrated for Ethanol.

Publication date :

22 October 2024

Journal title :

Journal of Chemical Theory and Computation

ISSN :

1549-9618

eISSN :

1549-9626

Publisher :

American Chemical Society, United States

Volume :

Issue :

Pages :

8807 - 8819

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://pubs.acs.org/doi/pdf/10.1021/acs.jctc.4c00977

Funders :

Universit? degli Studi di Milano
Fonds National de la Recherche Luxembourg
National Aeronautics and Space Administration

Funding text :

A.N. and A.T. acknowledge support from PHANTASTIC grant INTER/MERA22/16521502/PHANTASTIC. J.M.B. and P.P. acknowledge support from NASA grant 80NSSC22K1167. R.C. acknowledges support from Universita\u0300 degli Studi di Milano under grant PSR2022_DIP_005_PI_RCONT.

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Bibliography

Bowman, J. M.; Czakó, G.; Fu, B. High-dimensional ab initio potential energy surfaces for reaction dynamics calculations. Phys. Chem. Chem. Phys. 2011, 13, 8094- 8111, 10.1039/c0cp02722g
Qu, C.; Yu, Q.; Bowman, J. M. Permutationally invariant potential energy surfaces. Annu. Rev. Phys. Chem. 2018, 69, 151- 175, 10.1146/annurev-physchem-050317-021139
Fu, B.; Zhang, D. H. Ab initio potential energy surfaces and quantum dynamics for polyatomic bimolecular reactions. J. Chem. Theory Comput. 2018, 14, 2289- 2303, 10.1021/acs.jctc.8b00006
Jiang, B.; Li, J.; Guo, H. High-Fidelity Potential Energy Surfaces for Gas-Phase and Gas-Surface Scattering Processes from Machine Learning. J. Phys. Chem. Lett. 2020, 11, 5120- 5131, 10.1021/acs.jpclett.0c00989
Győri, T.; Czakó, G. Automating the Development of High-Dimensional Reactive Potential Energy Surfaces with the robosurfer Program System. J. Chem. Theory Comput. 2020, 16, 51- 66, 10.1021/acs.jctc.9b01006
Chmiela, S.; Sauceda, H. E.; Müller, K.-R.; Tkatchenko, A. Towards exact molecular dynamics simulations with machine-learned force fields. Nat. Commun. 2018, 9, 3887, 10.1038/s41467-018-06169-2
Sauceda, H. E.; Chmiela, S.; Poltavsky, I.; Müller, K. R.; Tkatchenko, A. Molecular force fields with gradient-domain machine learning: Construction and application to dynamics of small molecules with coupled cluster forces. J. Chem. Phys. 2019, 150, 114102, 10.1063/1.5078687
Houston, P. L.; Qu, C.; Nandi, A.; Conte, R.; Yu, Q.; Bowman, J. M. Permutationally invariant polynomial regression for energies and gradients, using reverse differentiation, achieves orders of magnitude speed-up with high precision compared to other machine learning methods. J. Chem. Phys. 2022, 156, 044120 10.1063/5.0080506
Bartók, A. P.; Csányi, G. Gaussian approximation potentials: A brief tutorial introduction. Int. J. Quantum Chem. 2015, 115, 1051- 1057, 10.1002/qua.24927
Smith, J. S.; Isayev, O.; Roitberg, A. E. ANI-1: an extensible neural network potential with DFT accuracy at force field computational cost. Chem. Sci. 2017, 8, 3192- 3203, 10.1039/C6SC05720A
Zhang, L.; Han, J.; Wang, H.; Car, R.; E, W. Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics. Phys. Rev. Lett. 2018, 120, 143001 10.1103/PhysRevLett.120.143001
Unke, O. T.; Meuwly, M. PhysNet: A Neural Network for Predicting Energies, Forces, Dipole Moments, and Partial Charges. J. Chem. Theory Comput. 2019, 15, 3678- 3693, 10.1021/acs.jctc.9b00181
Dral, P. O.; Owens, A.; Yurchenko, S. N.; Thiel, W. Structure-based sampling and self-correcting machine learning for accurate calculations of potential energy surfaces and vibrational levels. J. Chem. Phys. 2017, 146, 244108, 10.1063/1.4989536
Dral, P. O. MLatom: A program package for quantum chemical research assisted by machine learning. J. Comput. Chem. 2019, 40, 2339- 2347, 10.1002/jcc.26004
Christensen, A. S.; von Lilienfeld, O. A. On the role of gradients for machine learning of molecular energies and forces. Machine Learning: Science and Technology 2020, 1, 045018 10.1088/2632-2153/abba6f
Pinheiro, M.; Ge, F.; Ferré, N.; Dral, P. O.; Barbatti, M. Choosing the right molecular machine learning potential. Chem. Sci. 2021, 12, 14396- 14413, 10.1039/D1SC03564A
Qu, C.; Bowman, J. M. An ab initio potential energy surface for the formic acid dimer: Zero-point energy, selected anharmonic fundamental energies, and ground-state tunneling splitting calculated in relaxed 1-4-mode subspaces. Phys. Chem. Chem. Phys. 2016, 18, 24835- 24840, 10.1039/C6CP03073D
Rasheeda, D. S.; Santa Daría, A. M.; Schröder, B.; Mátyus, E.; Behler, J. High-dimensional neural network potentials for accurate vibrational frequencies: the formic acid dimer benchmark. Phys. Chem. Chem. Phys. 2022, 24, 29381- 29392, 10.1039/D2CP03893E
Fu, Y.-L.; Lu, X.; Han, Y.-C.; Fu, B.; Zhang, D. H.; Bowman, J. M. Collision-induced and complex-mediated roaming dynamics in the H + C2H4 → H2 + C2H3 reaction. Chem. Sci. 2020, 11, 2148- 2154, 10.1039/C9SC05951B
Lu, D.; Behler, J.; Li, J. Accurate Global Potential Energy Surfaces for the H + CH3OH Reaction by Neural Network Fitting with Permutation Invariance. J. Phys. Chem. A 2020, 124, 5737- 5745, 10.1021/acs.jpca.0c04182
Papp, D.; Tajti, V.; Győri, T.; Czakó, G. Theory Finally Agrees with Experiment for the Dynamics of the Cl + C2H6 Reaction. J. Phys. Chem. Lett. 2020, 11, 4762- 4767, 10.1021/acs.jpclett.0c01263
Czakó, G.; Gruber, B.; Papp, D.; Tajti, V.; Tasi, D. A.; Yin, C. First-principles mode-specific reaction dynamics. Phys. Chem. Chem. Phys. 2024, 26, 15818- 15830, 10.1039/D4CP00417E
Ramakrishnan, R.; Dral, P. O.; Rupp, M.; von Lilienfeld, O. A. Big Data Meets Quantum Chemistry Approximations: The Δ-Machine Learning Approach. J. Chem. Theory Comput. 2015, 11, 2087- 2096, 10.1021/acs.jctc.5b00099
Zaspel, P.; Huang, B.; Harbrecht, H.; von Lilienfeld, O. A. Boosting Quantum Machine Learning Models with a Multilevel Combination Technique: Pople Diagrams Revisited. J. Chem. Theory and Comput. 2019, 15, 1546- 1559, 10.1021/acs.jctc.8b00832
Stöhr, M.; Medrano Sandonas, L.; Tkatchenko, A. Accurate Many-Body Repulsive Potentials for Density-Functional Tight Binding from Deep Tensor Neural Networks. J. Phys. Chem. Lett. 2020, 11, 6835- 6843, 10.1021/acs.jpclett.0c01307
Dral, P. O.; Owens, A.; Dral, A.; Csányi, G. Hierarchical machine learning of potential energy surfaces. J. Chem. Phys. 2020, 152, 204110, 10.1063/5.0006498
Fu, B.; Xu, X.; Zhang, D. H. A hierarchical construction scheme for accurate potential energy surface generation: An application to the F+H2 reaction. J. Chem. Phys. 2008, 129, 011103 10.1063/1.2955729
Nandi, A.; Qu, C.; Houston, P. L.; Conte, R.; Bowman, J. M. Δ-machine learning for potential energy surfaces: A PIP approach to bring a DFT-based PES to CCSD(T) level of theory. J. Chem. Phys. 2021, 154, 051102 10.1063/5.0038301
Qu, C.; Houston, P. L.; Conte, R.; Nandi, A.; Bowman, J. M. Breaking the Coupled Cluster Barrier for Machine-Learned Potentials of Large Molecules: The Case of 15-Atom Acetylacetone. J. Phys. Chem. Lett. 2021, 12, 4902- 4909, 10.1021/acs.jpclett.1c01142
Khire, S. S.; Gurav, N. D.; Nandi, A.; Gadre, S. R. Enabling Rapid and Accurate Construction of CCSD(T)-Level Potential Energy Surface of Large Molecules Using Molecular Tailoring Approach. J. Phys. Chem. A 2022, 126, 1458- 1464, 10.1021/acs.jpca.2c00025
Nandi, A.; Nagy, P. R. Combining state-of-the-art quantum chemistry and machine learning make gold standard potential energy surfaces accessible for medium-sized molecules. Artif. Intell. Chem. 2024, 2, 100036 10.1016/j.aichem.2023.100036
Nandi, A.; Laude, G.; Khire, S. S.; Gurav, N. D.; Qu, C.; Conte, R.; Yu, Q.; Li, S.; Houston, P. L.; Gadre, S. R.; Richardson, J. O.; Evangelista, F. A.; Bowman, J. M. Ring-Polymer Instanton Tunneling Splittings of Tropolone and Isotopomers using a Δ-Machine Learned CCSD(T) Potential: Theory and Experiment Shake Hands. J. Am. Chem. Soc. 2023, 145, 9655- 9664, 10.1021/jacs.3c00769
Houston, P. L.; Qu, C.; Yu, Q.; Pandey, P.; Conte, R.; Nandi, A.; Bowman, J. M.; Kukolich, S. G. Formic Acid-Ammonia Heterodimer: A New Δ-Machine Learning CCSD(T)-Level Potential Energy Surface Allows Investigation of the Double Proton Transfer. J. Chem. Theor. Comp. 2024, 20, 1821- 1828, 10.1021/acs.jctc.3c01273
Bowman, J. M.; Qu, C.; Conte, R.; Nandi, A.; Houston, P. L.; Yu, Q. Δ-Machine Learned Potential Energy Surfaces and Force Fields. J. Chem. Theory Comput. 2023, 19, 1- 17, 10.1021/acs.jctc.2c01034
Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988, 37, 785- 789, 10.1103/PhysRevB.37.785
Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648- 5652, 10.1063/1.464913
Devereux, M.; Boittier, E. D.; Meuwly, M. Systematic improvement of empirical energy functions in the era of machine learning. J. Comput. Chem. 2024, 45, 1899- 1913, 10.1002/jcc.27367
Unke, O. T.; Stöhr, M.; Ganscha, S.; Unterthiner, T.; Maennel, H.; Kashubin, S.; Ahlin, D.; Gastegger, M.; Medrano Sandonas, L.; Berryman, J. T.; Tkatchenko, A.; Müller, K. R. Biomolecular dynamics with machine-learned quantum-mechanical force fields trained on diverse chemical fragments. Sci. Adv. 2024, 10, eadn4397 10.1126/sciadv.adn4397
Plé, T.; Lagardère, L.; Piquemal, J.-P. Force-field-enhanced neural network interactions: from local equivariant embedding to atom-in-molecule properties and long-range effects. Chem. Sci. 2023, 14, 12554- 12569, 10.1039/D3SC02581K
Song, K.; Käser, S.; Töpfer, K.; Vazquez-Salazar, L. I.; Meuwly, M. PhysNet meets CHARMM: A framework for routine machine learning/molecular mechanics simulations. J. Chem. Phys. 2023, 159, 024125 10.1063/5.0155992
Rogers, T. R.; Wang, F. Accurate MP2-based force fields predict hydration free energies for simple alkanes and alcohols in good agreement with experiments. J. Chem. Phys. 2020, 153, 244505, 10.1063/5.0035032
Zhao, Y.; Truhlar, D. G. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor. Chem. Acc. 2008, 120, 215- 241, 10.1007/s00214-007-0310-x
Walker, M.; Harvey, A. J. A.; Sen, A.; Dessent, C. E. H. Performance of M06, M06-2X, and M06-HF Density Functionals for Conformationally Flexible Anionic Clusters: M06 Functionals Perform Better than B3LYP for a Model System with Dispersion and Ionic Hydrogen-Bonding Interactions. J. Phys. Chem. A 2013, 117, 12590- 12600, 10.1021/jp408166m
Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865- 3868, 10.1103/PhysRevLett.77.3865, Erratum Phys. Rev. Lett. 1997, 78, 1396.
Perdew, J. P.; Ernzerhof, M.; Burke, K. Rationale for mixing exact exchange with density functional approximations. J. Chem. Phys. 1996, 105, 9982- 9985, 10.1063/1.472933
Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158- 6170, 10.1063/1.478522
Tkatchenko, A.; DiStasio, R. A.; Car, R.; Scheffler, M. Accurate and Efficient Method for Many-Body van der Waals Interactions. Phys. Rev. Lett. 2012, 108, 236402 10.1103/PhysRevLett.108.236402
Blum, V.; Gehrke, R.; Hanke, F.; Havu, P.; Havu, V.; Ren, X.; Reuter, K.; Scheffler, M. Ab initio molecular simulations with numeric atom-centered orbitals. Comput. Phys. Commun. 2009, 180, 2175- 2196, 10.1016/j.cpc.2009.06.022
Nandi, A.; Conte, R.; Qu, C.; Houston, P. L.; Yu, Q.; Bowman, J. M. Quantum calculations on a new CCSD (T) machine-learned potential energy surface reveal the leaky nature of gas-phase trans and gauche ethanol conformers. J. Chem. Theory Comput. 2022, 18, 5527- 5538, 10.1021/acs.jctc.2c00760
Braams, B. J.; Bowman, J. M. Permutationally invariant potential energy surfaces in high dimensionality. Int. Rev. Phys. Chem. 2009, 28, 577- 606, 10.1080/01442350903234923
Bowman, J. M.; Braams, B. J.; Carter, S.; Chen, C.; Czaḱo, G.; Fu, B.; Huang, X.; Kamarchik, E.; Sharma, A. R.; Shepler, B. C.; Wang, Y.; Xie, Z. Ab-initio-based potential energy surfaces for complex molecules and molecular complexes. J. Phys. Chem. Lett. 2010, 1, 1866- 1874, 10.1021/jz100626h
Xie, Z.; Bowman, J. M. Permutationally Invariant Polynomial Basis for Molecular Energy Surface Fitting via Monomial Symmetrization. J. Chem. Theory Comput. 2010, 6, 26- 34, 10.1021/ct9004917
Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M. MOLPRO, version 2015.1, a package of ab initio programs. 2015; see http://www.molpro.net.
Ren, X.; Rinke, P.; Blum, V.; Wieferink, J.; Tkatchenko, A.; Sanfilippo, A.; Reuter, K.; Scheffler, M. Resolution-of-identity approach to Hartree-Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions. New J. Phys. 2012, 14, 053020 10.1088/1367-2630/14/5/053020
MSA 2.0 Software with Gradients, 2019. https://github.com/szquchen/MSA-2.0; Accessed: 2019-01-20.
Video MSA 2.0 Software with Gradients, 2024. https://scholarblogs.emory.edu/bowman/msa/.
Quade, C. A Note on Internal Rotation-Rotation Interactions in Ethyl Alcohol. J. Mol. Spectrosc. 2000, 203, 200- 202, 10.1006/jmsp.2000.8156
Pearson, J. C.; Sastry, K. V. L. N.; Winnewisser, M.; Herbst, E.; De Lucia, F. C. The Millimeter- and Submillimeter-Wave Spectrum of trans-Ethyl Alcohol. J. Phys. Chem. Ref. Data 1995, 24, 1- 32, 10.1063/1.555972
Durig, J.; Larsen, R. Torsional vibrations and barriers to internal rotation for ethanol and 2,2,2-trifluoroethanol. J. Mol. Struct. 1990, 238, 195- 222, 10.1016/0022-2860(90)85015-B
Romanowski, H.; Bowman, J. M.; Harding, L. B. Vibrational energy levels of formaldehyde. J. Chem. Phys. 1985, 82, 4155- 4165, 10.1063/1.448858
Carter, S.; Culik, S. J.; Bowman, J. M. Vibrational self-consistent field method for many-mode systems: A new approach and application to the vibrations of CO adsorbed on Cu(100). J. Chem. Phys. 1997, 107, 10458- 10469, 10.1063/1.474210
Pflüger, K.; Paulus, M.; Jagiella, S.; Burkert, T.; Rauhut, G. Multi-level vibrational SCF calculations and FTIR measurements on furazan. Theor. Chem. Acc. 2005, 114, 327- 332, 10.1007/s00214-005-0678-4
Yagi, K.; Hirata, S.; Hirao, K. Multiresolution potential energy surfaces for vibrational state calculations. Theor. Chem. Acc. 2007, 118, 681- 691, 10.1007/s00214-007-0363-x
Sparta, M.; Ho̷yvik, I.-M.; Toffoli, D.; Christiansen, O. Potential Energy Surfaces for Vibrational Structure Calculations from a Multiresolution Adaptive Density-Guided Approach: Implementation and Test Calculations. J. Phys. Chem. A 2009, 113, 8712- 8723, 10.1021/jp9035315
Schröder, B.; Rauhut, G. From the Automated Calculation of Potential Energy Surfaces to Accurate Infrared Spectra. J. Phys. Chem. Lett. 2024, 15, 3159- 3169, 10.1021/acs.jpclett.4c00186
Ischtwan, J.; Collins, M. A. Molecular potential energy surfaces by interpolation. J. Chem. Phys. 1994, 100, 8080- 8088, 10.1063/1.466801
Nguyen, K. A.; Rossi, I.; Truhlar, D. G. A dual-level Shepard interpolation method for generating potential energy surfaces for dynamics calculations. J. Chem. Phys. 1995, 103, 5522- 5530, 10.1063/1.470536
Pan, S. J.; Yang, Q. A Survey on Transfer Learning. IEEE Trans. Knowl. Data Eng. 2010, 22, 1345- 1359, 10.1109/TKDE.2009.191
Smith, J. S.; Nebgen, B. T.; Zubatyuk, R.; Lubbers, N.; Devereux, C.; Barros, K.; Tretiak, S.; Isayev, O.; Roitberg, A. E. Approaching coupled cluster accuracy with a general-purpose neural network potential through transfer learning. Nat. Commun. 2019, 10, 2903- 2906, 10.1038/s41467-019-10827-4
Käser, S.; Boittier, E. D.; Upadhyay, M.; Meuwly, M. Transfer Learning to CCSD(T): Accurate Anharmonic Frequencies from Machine Learning Models. J. Chem. Theory Comput. 2021, 17, 3687- 3699, 10.1021/acs.jctc.1c00249
Käser, S.; Meuwly, M. Transfer learned potential energy surfaces: accurate anharmonic vibrational dynamics and dissociation energies for the formic acid monomer and dimer. Phys. Chem. Chem. Phys. 2022, 24, 5269- 5281, 10.1039/D1CP04393E
Käser, S.; Meuwly, M. Transfer-learned potential energy surfaces: Toward microsecond-scale molecular dynamics simulations in the gas phase at CCSD(T) quality. J. Chem. Phys. 2023, 158, 214301, 10.1063/5.0151266
Käser, S.; Richardson, J. O.; Meuwly, M. Transfer Learning for Affordable and High-Quality Tunneling Splittings from Instanton Calculations. J. Chem. Theory Comput. 2022, 18, 6840- 6850, 10.1021/acs.jctc.2c00790
Liu, Y.; Li, J. Permutation-Invariant-Polynomial Neural-Network-Based Δ-Machine Learning Approach: A Case for the HO2 Self-Reaction and Its Dynamics Study. J. Chem. Phys. Lett. 2022, 13, 4729- 4738, 10.1021/acs.jpclett.2c01064
Song, K.; Li, J. The neural network based Δ-machine learning approach efficiently brings the DFT potential energy surface to the CCSD(T) quality: a case for the OH + CH3OH reaction. Phys. Chem. Chem. Phys. 2023, 25, 11192- 11204, 10.1039/D3CP00665D
Jiang, B.; Guo, H. Permutation invariant polynomial neural network approach to fitting potential energy surfaces. J. Chem. Phys. 2013, 139, 054112 10.1063/1.4817187
Conte, R.; Nandi, A.; Qu, C.; Yu, Q.; Houston, P. L.; Bowman, J. M. Semiclassical and VSCF/VCI Calculations of the Vibrational Energies of trans- and gauche-Ethanol Using a CCSD(T) Potential Energy Surface. J. Phys. Chem. A 2022, 126, 7709- 7718, 10.1021/acs.jpca.2c06322
Slootman, E.; Poltavsky, I.; Shinde, R.; Cocomello, J.; Moroni, S.; Tkatchenko, A.; Filippi, C. Accurate quantum Monte Carlo forces for machine-learned force fields: Ethanol as a benchmark. J. Chem. Theory Comput. 2024, 20, 6020- 6027, 10.1021/acs.jctc.4c00498