(123) Ceriotti, M.; Markland, T. E. Efficient methods and practical guidelines for simulating isotope effects. *J. Chem. Phys.* **2013**, *138*, 014112.

(124) Geerlings, P.; De Proft, F. D.; Langenaeker, W. Conceptual Density Functional Theory. *Chem. Rev.* **2003**, *103*, 1793.

(125) Yang, W.; Zhang, Y.; Ayers, P. W. Degenerate ground states and fractional number of electrons in density and density reduced matrix functional theory. *Phys. Rev. Lett.* **2000**, *84*, 5172.

(126) Zeng, X.; Hu, H.; Hu, X.; Cohen, A. J.; Yang, W. Ab initio quantum mechanical/molecular mechanical simulation of electron transfer process: Fractional electron approach. *J. Chem. Phys.* **2008**, *128*, 124510.

(127) Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Discontinuous nature of the exchange-correlation functional in strongly correlated systems. *Phys. Rev. Lett.* **2009**, *102*, 066403.

(128) Schneider, G. Virtual screening: an endless staircase? *Nat. Rev. Drug Discovery* **2010**, *9*, 273.

(129) Bultinck, P.; Gironés, X.; Carbo-Dorcaz, R. Molecular quantum similarity: theory and applications. *Rev. Comput. Chem.* **2005**, *21*, 127.

(130) Kaji, M. Mendeleev's Discovery of the Periodic Law: The Origin and the Reception. *Found. Chem.* **2003**, *5*, 189.

(131) Pauling, L.; Yost, D. M. The additivity of the energies of normal covalent bonds. *Proc. Natl. Acad. Sci. U. S. A.* **1932**, *18*, 414.

(132) Pettifor, D. G. A chemical scale for crystal-structure maps. *Solid State Commun.* **1984**, *51*, 31–34.

(133) Glawe, H.; Sanna, A.; Gross, E. K. U.; Marques, M. A. L. The optimal one dimensional periodic table: a modified Pettifor chemical scale from data mining. *New J. Phys.* **2016**, *18*, 093011.

(134) Glawe, H.; Sanna, A.; Gross, E.; Marques, M. A. The optimal one dimensional periodic table: a modified Pettifor chemical scale from data mining. *New J. Phys.* **2016**, *18*, 093011.

(135) Allahyari, Z.; Oganov, A. R. Nonempirical definition of the Mendeleev numbers: Organizing the chemical space. *J. Phys. Chem. C* **2020**, *124*, 23867.

(136) Glushkovsky, A. AI Discovering a Coordinate System of Chemical Elements: Dual Representation by Variational Autoencoders. *arXiv* **2020**, arXiv:2011.12090

(137) Huang, B.; Zhuang, L.; Xiao, L.; Lu, J. Bond-energy decoupling: principle and application to heterogeneous catalysis. *Chem. Sci.* **2013**, *4*, 606–611.

(138) Wells, P. R. Linear Free Energy Relationships. *Chem. Rev.* **1963**, *63*, 171–219.

(139) Hammett, L. P. The Effect of Structure upon the Reactions of Organic Compounds. Benzene Derivatives. *J. Am. Chem. Soc.* **1937**, *59*, 96–103.

(140) Hansch, C.; Leo, A.; Taft, R. W. A survey of Hammett substituent constants and resonance and field parameters. *Chem. Rev.* **1991**, *91*, 165–195.

(141) Hammett, L. P. Some Relations between Reaction Rates and Equilibrium Constants. *Chem. Rev.* **1935**, *17*, 125–136.

(142) Hammond, G. S. A Correlation of Reaction Rates. *J. Am. Chem. Soc.* **1955**, *77*, 334–338.

(143) Bell, R. P. The theory of reactions involving proton transfers. *Proc. R. Soc. Lond., A* **1936**, *154*, 414–429.

(144) Evans, M. G.; Polanyi, M. Further considerations on the thermodynamics of chemical equilibria and reaction rates. *Trans. Faraday Soc.* **1936**, *32*, 1333–1360.

(145) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. *Phys. Rev.* **1965**, *140*, A1133.

(146) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. *Phys. Rev.* **1964**, *136*, B864.

(147) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. *Phys. Rev. Lett.* **1996**, *77*, 3865.

(148) Ernzerhof, M.; Scuseria, G. E. Assessment of the Perdew-Burke-Ernzerhof exchange-correlation functional. *J. Chem. Phys.* **1999**, *110*, 5029.

(149) Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. *J. Chem. Phys.* **1999**, *110*, 6158–6170.

(150) Hammer, B.; Nørskov, J. K. Electronic factors determining the reactivity of metal surfaces. *Surf. Sci.* **1995**, *343*, 211–220.

(151) Hammer, B.; Nørskov, J. K. Why gold is the noblest of all the metals. *Nature* **1995**, *376*, 238–240.

(152) Calle-Vallejo, F.; Martínez, J. I.; García-Lastra, J. M.; Sautet, P.; Loffreda, D. Fast Prediction of Adsorption Properties for Platinum Nanocatalysts with Generalized Coordination Numbers. *Angew. Chem., Int. Ed.* **2014**, *53*, 8316–8319 _eprint_:

(153) Calle-Vallejo, F.; Tymoczko, J.; Colic, V.; Vu, Q. H.; Pohl, M. D.; Morgenstern, K.; Loffreda, D.; Sautet, P.; Schuhmann, W.; Bandarenka, A. S. Finding optimal surface sites on heterogeneous catalysts by counting nearest neighbors. *Science* **2015**, *350*, 185–189.

(154) Huang, B.; Xiao, L.; Lu, J.; Zhuang, L. Spatially Resolved Quantification of the Surface Reactivity of Solid Catalysts. *Angew. Chem.* **2016**, *128*, 6347–6351 _eprint_:

(155) van Santen, R. A.; Neurock, M.; Shetty, S. G. Reactivity Theory of Transition Metal Surfaces: A Bronsted-Evans-Polanyi Linear Activation Energy Free Energy Analysis. *Chem. Rev.* **2010**, *110*, 2005–2048.

(156) Abild-Pedersen, F.; Greeley, J.; Studt, F.; Rossmeisl, J.; Munter, T. R.; Moses, P. G.; Skúlason, E.; Bligaard, T.; Nørskov, J. K. Scaling Properties of Adsorption Energies for Hydrogen-Containing Molecules on Transition-Metal Surfaces. *Phys. Rev. Lett.* **2007**, *99*, 016105.

(157) Fernández, E. M.; Moses, P. G.; Toftelund, A.; Hansen, H. A.; Martínez, J. I.; Abild-Pedersen, F.; Kleis, J.; Hinnemann, B.; Rossmeisl, J.; Bligaard, T.; Nørskov, J. K. Scaling Relationships for Adsorption Energies on Transition Metal Oxide, Sulfide, and Nitride Surfaces. *Angew. Chem., Int. Ed.* **2008**, *47*, 4683–4686 _eprint_:

(158) Calle-Vallejo, F.; Martínez, J. I.; García-Lastra, J. M.; Rossmeisl, J.; Koper, M. T. M. Physical and Chemical Nature of the Scaling Relations between Adsorption Energies of Atoms on Metal Surfaces. *Phys. Rev. Lett.* **2012**, *108*, 116103.

(159) Nørskov, J. K.; Bligaard, T.; Rossmeisl, J.; Christensen, C. H. Towards the computational design of solid catalysts. *Nat. Chem.* **2009**, *1*, 37.

(160) Parr, R. G.; Yang, W. *Density Functional Theory of Atoms and Molecules*; Oxford Science Publications, 1989.

(161) Geerlings, P.; De Proft, F.; Langenaeker, W. Conceptual Density Functional Theory. *Chem. Rev.* **2003**, *103*, 1793–1874.

(162) Geerlings, P.; Fias, S.; Boisdenghien, Z.; DE Proft, F. Conceptual DFT: chemistry from the linear response function. *Chem. Soc. Rev.* **2014**, *43*, 4989–5008.

(163) Huang, B.; von Lilienfeld, O. A. Communication: Understanding molecular representations in machine learning: The role of uniqueness and target similarity. *J. Chem. Phys.* **2016**, *145*, 161102.

(164) Faber, F. A.; Lindmaa, A.; von Lilienfeld, O. A.; Armiento, R. Machine Learning Energies of 2 Million Elpasolite (ABC₂D₆) Crystals. *Phys. Rev. Lett.* **2016**, *117*, 135502.

(165) Ong, S. P.; Jain, A.; Hautier, G.; Kocher, M.; Cholia, S.; Gunter, D.; Bailey, D.; Skinner, D.; Persson, K. A.; Ceder, G. The Materials Project. 2011; http://materialsproject.org/.

(166) Faber, F. A.; Christensen, A. S.; Huang, B.; von Lilienfeld, O. A. Alchemical and structural distribution based representation for universal quantum machine learning. *J. Chem. Phys.* **2018**, *148*, 241717.

(167) Ye, W.; Chen, C.; Wang, Z.; Chu, I.-H.; Ong, S. P. Deep neural networks for accurate predictions of crystal stability. *Nat. Commun.* **2018**, *9*, 3800.

(168) Schmidt, J.; Shi, J.; Borlido, P.; Chen, L.; Botti, S.; Marques, M. A. Predicting the thermodynamic stability of solids combining density functional theory and machine learning. *Chem. Mater.* **2017**, *29*, 5090–5103.

(169) Legrain, F.; Carrete, J.; van Roekeghem, A.; Curtarolo, S.; Mingo, N. How Chemical Composition Alone Can Predict Vibra-
tional Free Energies and Entropies of Solids. *Chem. Mater.* **2017**, *29*, 6220–6227.

(170) Hansen, K.; Biegler, F.; von Lilienfeld, O. A.; Müller, K.-R.; Tkatchenko, A.; et al. Interaction potentials in molecules and non-local information in chemical space. *Phys. Rev. Lett* **2012**, *108*, 058301.

(171) Ramakrishnan, R.; Dral, P. D.; Rupp, M.; von Lilienfeld, O. A. Quantum chemistry structures and properties of 134 kilo molecules. *Sci. Data* **2014**, 1, 140022.

(172) Koch, W.; Holthausen, M. C. *A Chemist's Guide to Density Functional Theory*; Wiley-VCH, 2002.

(173) Huang, B.; von Lilienfeld, O. A. The "DNA" of chemistry: Scalable quantum machine learning with "amons". *arXiv* 2017, arXiv:1707.04146, submitted to *Nature*.

(174) Braun, J.; Kerber, A.; Meringer, M.; Rücker, C. Similarity of molecular descriptors: The equivalence of Zagreb indices and walk counts. *MATCH Commun. Math. Comput. Chem.* **2005**, *54*, 163–176.

(175) Visco, J.; Pophale, R. S.; Rintoul, M. D.; Faulon, J. L. Developing a methodology for an inverse quantitative structure activity relationship using the signature molecular descriptor. *J. Mol. Graphics Modell.* **2002**, *20*, 429–438.

(176) Faulon, J.-L.; Visco, D. P., Jr.; Pophale, R. S. The Signature Molecular Descriptor. 1. Using Extended Valence Sequences in QSAR and QSPR Studies. *J. Chem. Inf. Comp. Sci.* **2003**, *43*, 707.

(177) Martin, S.; Roe, D.; Faulon, J.-L. Predicting protein-protein interactions using signature products. *Bioinformatics* **2005**, *21*, 218–226.

(178) Rogers, D.; Hahn, M. Extended-Connectivity Fingerprints. *J. Chem. Inf. Model.* **2010**, *50*, 742–754.

(179) Lounkine, E.; Keiser, M. J.; Whitebread, S.; Mikhailov, D.; Hamon, J.; Jenkins, J. L.; Lavan, P.; Weber, E.; Doak, A. K.; Côté, S.; Shoichet, B. K.; Urban, L. Large-scale prediction and testing of drug activity on side-effect targets. *Nature* **2012**, *486*, 361.

(180) Besnard, J.; et al. Automated design of ligands to polypharmacological profiles. *Nature* **2012**, *492*, 215.

(181) Ruddigkeit, L.; van Deursen, R.; Blum, L. C.; Reymond, J.-L. Enumeration of 166 Billion Organic Small Molecules in the Chemical Universe Database GDB-17. *J. Chem. Inf. Model.* **2012**, *52*, 2864–2875.

(182) Fink, T.; Bruggesser, H.; Reymond, J.-L. Virtual Exploration of the Small-Molecule Chemical Universe below 160 Da. *Angew. Chem., Int. Ed.* **2005**, *44*, 1504–1508.

(183) Faber, F. A.; Hutchison, L.; Huang, B.; Gilmer, J.; Schoenholz, S. S.; Dahl, G. E.; Vinyals, O.; Kearnes, S.; Riley, P. F.; von Lilienfeld, O. A. Prediction errors of molecular machine learning models lower than hybrid DFT error. *J. Chem. Theory Comput.* **2017**, *13*, 5255–5264.

(184) Schwaller, P.; Gaudin, T.; Lányi, D.; Bekas, C.; Laino, T. Found in Translation": predicting outcomes of complex organic chemistry reactions using neural sequence-to-sequence models. *Chem. Sci.* **2018**, *9*, 6091–6098.

(185) Schwaller, P.; Laino, T.; Gaudin, T.; Bolgar, P.; Hunter, C. A.; Bekas, C.; Lee, A. A. Molecular Transformer: A Model for Uncertainty-Calibrated Chemical Reaction Prediction. *ACS Cent. Sci.* **2019**, *5*, 1572–1583.

(186) Unsleber, J. P.; Reiher, M. The Exploration of Chemical Reaction Networks. *Annu. Rev. Phys. Chem.* **2020**, *71*, 121–142 eprint.

(187) Gómez-Bombarelli, R.; Wei, J. N.; Duvenaud, D.; Hernández-Lobato, J. M.; Sánchez-Lengeling, B.; Sheberla, D.; Aguilera-Iparraguirre, J.; Hirzel, T. D.; Adams, R. P.; Aspuru-Guzik, A. Automatic Chemical Design Using a Data-Driven Continuous Representation of Molecules. *ACS Cent. Sci.* **2018**, *4*, 268–276.

(188) Yang, Q.; Bassyouni, A.; Butler, C. R.; Hou, X.; Jenkinson, S.; Price, D. A.; et al. Ligand biological activity predicted by cleaning positive and negative chemical correlations. *Proc. Natl. Acad. Sci. U. S. A.* **2019**, *116*, 3373–3378.

(189) Krenn, M.; Häse, F.; Nigam, A.; Friederich, P.; Aspuru-Guzik, A. Self-Referencing Embedded Strings (SELFIES): A 100% robust

molecular string representation. *Mach. Learn.: Sci. Technol.* **2020**, 1, 045024.

(190) Hoffmann, C.; Menichetti, R.; Kanekal, K. H.; Bereau, T. Controlled exploration of chemical space by machine learning of coarse-grained representations. *Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top.* **2019**, *100*, 033302.

(191) John, S. T.; Csányi, G. Many-Body Coarse-Grained Interactions Using Gaussian Approximation Potentials. *J. Phys. Chem. B* **2017**, *121*, 10934–10949.

(192) Wang, J.; Olsson, S.; Wehmeyer, C.; Pérez, A.; Charron, N. E.; de Fabritiis, G.; Noé, F.; Clementi, C. Machine Learning of Coarse-Grained Molecular Dynamics Force Fields. *ACS Cent. Sci.* **2019**, *5*, 755–767.

(193) Karthikeyan, M.; Glen, R. C.; Bender, A. General Melting Point Prediction Based on a Diverse Compound Data Set and Artificial Neural Networks. *J. Chem. Inf. Model.* **2005**, *45*, 581–590.

(194) Ghiringhelli, L. M.; Vybiral, J.; Levchenko, S. V.; Draxl, C.; Scheffler, M. Big data of materials science: Critical role of the descriptor. *Phys. Rev. Lett.* **2015**, *114*, 105503.

(195) Rupp, M.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning. *Phys. Rev. Lett.* **2012**, *108*, 058301.

(196) Rasmussen, C. E.; Williams, C. K. I. Gaussian Processes for Machine Learning; Dietterich, T., Ed.; MIT Press: Cambridge, 2006; www.GaussianProcess.org.

(197) Schmidt, J.; Marques, M. R. G.; Botti, S.; Marques, M. A. L. Recent advances and applications of machine learning in solid-state materials science. *NPJ Comput. Mater.* **2019**, *5*, 83.

(198) Vapnik, V. *The Nature of Statistical Learning Theory*; Springer Science & Business Media, 2013.

(199) Brown, A.; Braams, B. J.; Christoffel, K.; Jin, Z.; Bowman, J. M. Classical and quasiclassical spectral analysis of CHS+ using an ab initio potential energy surface. *J. Chem. Phys.* **2003**, *119*, 8790.

(200) Neal, R. M. *Bayesian Learning for Neural Networks*; Springer, 1996; pp 29–53.

(201) čaylak, O.; von Lilienfeld, O. A.; Baumeier, B. Wasserstein metric for improved quantum machine learning with adjacency matrix representations. *Mach. Learn.: Sci. Technol.* **2020**, 1, 03LT01.

(202) Hansen, K.; Montavon, G.; Biegler, F.; Fazli, S.; Rupp, M.; Scheffler, M.; von Lilienfeld, O. A.; Tkatchenko, A.; Müller, K.-R. Assessment and Validation of Machine Learning Methods for Predicting Molecular Atomization Energies. *J. Chem. Theory Comput.* **2013**, *9*, 3404–3419.

(203) Rupp, M. Machine learning for quantum mechanics in a nutshell. *Int. J. Quantum Chem.* **2015**, *115*, 1058–1073.

(204) Cortes, C.; Jackel, L. D.; Solla, S. A.; Vapnik, V.; Denker, J. S. Learning curves: Asymptotic values and rate of convergence. *Adv. Neur. Inform. Process. Syst.* **1994**, *6*, 327–334.

(205) Müller, K. R.; Finke, M.; Murata, N.; Schulten, K.; Amari, S. A numerical study on learning curves in stochastic multilayer feedforward networks. *Neural Comp.* **1996**, *8*, 1085.

(206) Christensen, A. S.; Faber, F. A.; von Lilienfeld, O. A. Operators in quantum machine learning: Response properties in chemical space. *J. Chem. Phys.* **2019**, *150*, 064105.

(207) Christensen, A. S.; von Lilienfeld, O. A. Operator quantum machine learning: Navigating the chemical space of response properties. *Chimia* **2019**, *73*, 1028–1031.

(208) Christensen, A. S.; Bratholm, L.; Faber, F. A.; von Lilienfeld, O. A. FCHL revisited: Faster and more accurate quantum machine learning. *J. Chem. Phys.* **2020**, *152*, 044107.

(209) Gastegger, M.; Schütt, K. T.; Müller, K.-R. Machine learning of solvent effects on molecular spectra and reactions. *arXiv* **2020**, arXiv:2010.14942

(210) Christensen, A. S.; von Lilienfeld, O. A. On the role of gradients for machine learning of molecular energies and forces. *Mach. Learn.: Sci. Technol.* **2020**, 1, 045018.

(211) Hammett, L. P. The Effect of Structure upon the Reactions of Organic Compounds. Benzene Derivatives. *J. Am. Chem. Soc.* **1937**, *59*, 96.
(212) Bragato, M.; von Rudorff, G. F.; von Lilienfeld, O. A. Data enhanced Hammett Equation: Reaction Barriers in Chemical Space. *Chem. Sci.* **2020**, *11*, 11859.

(213) Todeschini, R.; Consonni, V. *Handbook of Molecular Descriptors*; Wiley-VCH: Weinheim, 2009.

(214) Moussa, J. E. Comment on "Fast and Accurate Modeling of Molecular Energies with Machine Learning. *Phys. Rev. Lett.* **2012**, *109*, 059801.

(215) von Lilienfeld, O. A.; Ramakrishnan, R.; Rupp, M.; Knoll, A. Fourier series of atomic radial distribution functions: A molecular fingerprint for machine learning models of quantum chemical properties. *Int. J. Quantum Chem.* **2015**, *115*, 1084.

(216) Pozdnyakov, S. N.; Willatt, M. J.; Bartók, A. P.; Ortner, C.; Csányi, G.; Ceriotti, M. Incompleteness of Atomic Structure Representations. *Phys. Rev. Lett.* **2020**, *125*, 166001.

(217) Parsaeifard, B.; De, D. S.; Christensen, A. S.; Faber, F. A.; Kocer, E.; De, S.; Behler, J.; von Lilienfeld, A.; Goedecker, S. An assessment of the structural resolution of various fingerprints commonly used in machine learning. *Mach. Learn.: Sci. Technol.* **2021**, *2*, 015018.

(218) Ramakrishnan, R.; von Lilienfeld, O. A. Many Molecular Properties from One Kernel in Chemical Space. *Chimia* **2015**, *69*, 182.

(219) Montavon, G.; Rupp, M.; Gobre, V.; Vazquez-Mayagoitia, A.; Hansen, K.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Machine learning of molecular electronic properties in chemical compound space. *New J. Phys.* **2013**, *15*, 095003.

(220) Gilmer, J.; Schoenholz, S. S.; Riley, P. F.; Vinyals, O.; Dahl, G. E. Neural Message Passing for Quantum Chemistry. *Proceedings of the 34th International Conference on Machine Learning*, ICML, 2017.

(221) Eickenberg, M.; Exarchakis, G.; Hirn, M.; Mallat, S.; Thiry, L. Solid harmonic wavelet scattering for predictions of molecule properties. *J. Chem. Phys.* **2018**, *148*, 241732.

(222) Gubaev, K.; Podryabinkin, E. V.; Shapeev, A. V. Machine learning of molecular properties: Locality and active learning. *J. Chem. Phys.* **2018**, *148*, 241727.

(223) Welborn, M.; Cheng, L.; Miller III, T. F. Transferability in machine learning for electronic structure via the molecular orbital basis. *J. Chem. Theory Comput.* **2018**, *14*, 4772–4779.

(224) Langer, M. F.; Goebmann, A.; Rupp, M. Representations of molecules and materials for interpolation of quantum-mechanical simulations via machine learning. *arXiv:2020, arXiv:2003.12081v2*.

(225) Englert, B.-G. *Semiclassical Theory of Atoms*; Springer, 1988.

(226) Rupp, M.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Reply to Comment on "Fast and Accurate Modeling of Molecular Energies with Machine Learning. *Phys. Rev. Lett.* **2012**, *109*, 059802.

(227) Pilania, G.; Wang, C.; Jiang, X.; Rajasekaran, S.; Ramprasad, R. Accelerating materials property predictions using machine learning. *Sci. Rep.* **2013**, *3*, 2810.

(228) Schütt, K. T.; Glawe, H.; Brockherde, F.; Sanna, A.; Müller, K. R.; Gross, E. K. U. How to represent crystal structures for machine learning: Towards fast prediction of electronic properties. *Phys. Rev. B: Condens. Matter Mater. Phys.* **2014**, *89*, 205118.

(229) Rappé, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A., III; Skiff, W. M. UFF, a Full Periodic Table Force Field for Molecular Mechanics and Molecular Dynamics Simulations. *J. Am. Chem. Soc.* **1992**, *114*, 10024.

(230) Pronobis, W.; Tkatchenko, A.; Müller, K.-R. Many-Body Descriptors for Predicting Molecular Properties with Machine Learning: Analysis of Pairwise and Three-Body Interactions in Molecules. *J. Chem. Theory Comput.* **2018**, *14*, 2991.

(231) Faber, F.; Lindmaa, A.; von Lilienfeld, O. A.; Armiento, R. Crystal Structure Representations for Machine Learning Models of Formation Energies. *Int. J. Quantum Chem.* **2015**, *115*, 1094.

(232) Ramakrishnan, R.; Hartmann, M.; Tapavicza, E.; von Lilienfeld, O. A. Electronic Spectra from TDDFT and Machine Learning in Chemical Space. *J. Chem. Phys.* **2015**, *143*, 084111.

(233) Faber, F. A.; Hutchison, L.; Huang, B.; Gilmer, J.; Schoenholz, S. S.; Dahl, G. E.; Vinyals, O.; Kearnes, S.; Riley, P. F.; von Lilienfeld, O. A. Prediction errors of molecular machine learning models lower than hybrid DFT error. *J. Chem. Theory Comput.* **2017**, *13*, 5255–5264.

(234) Stuke, A.; Todorović, M.; Rupp, M.; Kunkel, C.; Ghosh, K.; Himanen, L.; Rinke, P. Chemical diversity in molecular orbital energy predictions with kernel ridge regression. *J. Chem. Phys.* **2019**, *150*, 204121.

(235) Ghosh, K.; Stuke, A.; Todorović, M.; Jørgensen, P. B.; Schmidt, M. N.; Vehtari, A.; Rinke, P. Deep learning spectroscopy: neural networks for molecular excitation spectra. *Adv. Sci.* **2019**, *6*, 1801367.

(236) Zhu, L.; Amsler, M.; Fuhrer, T.; Schaefer, B.; Faraji, S.; Rostami, S.; Ghasemi, S. A.; Sadeghi, A.; Grauzinyte, M.; Wolverton, C.; et al. A fingerprint based metric for measuring similarities of crystalline structures. *J. Chem. Phys.* **2016**, *144*, 034203.

(237) Schütt, O.; VandeVondele, J. Machine learning adaptive basis sets for efficient large scale density functional theory simulation. *J. Chem. Theory Comput.* **2018**, *14*, 4168–4175.

(238) Babaei, M.; Azar, Y. T.; Sadeghi, A. Locality meets machine learning: Excited and ground-state energy surfaces of large systems at the cost of small ones. *Phys. Rev. B: Condens. Matter Mater. Phys.* **2020**, *101*, 115132.

(239) Collins, C. R.; Gordon, G. J.; von Lilienfeld, O. A.; Yaron, D. J. Constant size descriptors for accurate machine learning models of molecular properties. *J. Chem. Phys.* **2018**, *148*, 241718.

(240) Drautz, R. Atomic cluster expansion for accurate and transferable interatomic potentials. *Phys. Rev. B: Condens. Matter Mater. Phys.* **2019**, *99*, 014104.

(241) Behler, J. Atom-centered symmetry functions for constructing high-dimensional neural networks potentials. *J. Chem. Phys.* **2011**, *134*, 074106.

(242) Huang, B.; von Lilienfeld, O. A. Quantum machine learning using atom-in-molecule-based fragments selected on the fly. *Nat. Chem.* **2020**, *12*, 945–951.

(243) Bartók, A. P.; De, S.; Poelking, C.; Bernstein, N.; Kermode, J. R.; Csányi, G.; Ceriotti, M. Machine learning unifies the modeling of materials and molecules. *Sci. Adv.* **2017**, *3*, e1701816.

(244) Gubaev, K.; Podryabinkin, E. V.; Shapeev, A. V. Machine learning of molecular properties: Locality and active learning. *J. Chem. Phys.* **2018**, *148*, 241727.

(245) Willatt, M. J.; Musil, F.; Ceriotti, M. Atom-density representations for machine learning. *J. Chem. Phys.* **2019**, *150*, 154110.

(246) Taylor, M. E.; Stone, P. Transfer learning for reinforcement learning domains: A survey. *J. Mach. Learn. Res.* **2009**, *10*, 1633.

(247) 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.

(248) Hobday, S.; Smith, R.; Belbruno, J. Applications of neural networks to fitting interatomic potential functions. *Modell. Simul. Mater. Sci. Eng.* **1999**, *7*, 397.

(249) Bereau, T.; Andrienko, D.; von Lilienfeld, O. A. Transferable atomic multipole machine learning models for small organic molecules. *J. Chem. Theory Comput.* **2015**, *11*, 3225.

(250) Ghasemi, S. A.; Hofstetter, A.; Saha, S.; Goedecker, S. Interatomic potentials for ionic systems with density functional accuracy based on charge densities obtained by a neural network. *Phys. Rev. B: Condens. Matter Mater. Phys.* **2015**, *92*, 045131.

(251) Sifain, A. E.; Lubbers, N.; Nebgen, B. T.; Smith, J. S.; Lokhov, A. Y.; Isayev, O.; Roitberg, A. E.; Barros, K.; Tretiak, S. Discovering a transferable charge assignment model using machine learning. *J. Phys. Chem. Lett.* **2018**, *9*, 4495–4501.

(252) Nebgen, B.; Lubbers, N.; Smith, J. S.; Sifain, A. E.; Lokhov, A.; Isayev, O.; Roitberg, A. E.; Barros, K.; Tretiak, S. Transferable Dynamic Molecular Charge Assignment Using Deep Neural Networks. *J. Chem. Theory Comput.* **2018**, *14*, 4687–4698.
(253) Dral, P. O.; von Lilienfeld, O. A.; Thiel, W. Machine Learning of Parameters for Accurate Semiempirical Quantum Chemical Calculations. *J. Chem. Theory Comput.* **2015**, *11*, 2120–2125 PMID: 26146493.

(254) Kranz, J. J.; Kubillus, M.; Ramakrishnan, R.; von Lilienfeld, O. A.; Elstner, M. Generalized density-functional tight-binding repulsive potentials from unsupervised machine learning. *J. Chem. Theory Comput.* **2018**, *14*, 2341–2352.

(255) 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.

(256) Zubatyuk, T.; Nebgen, B.; Lubbers, N.; Smith, J. S.; Zubatyuk, R.; Zhou, G.; Koh, C.; Barros, K.; Isayev, O.; Tretiak, S. Machine Learned Hückel Theory: Interfacing Physics and Deep Neural Networks. *arXiv* **2019**, arXiv:1909.12963

(257) Ramakrishnan, R.; Dral, P.; Rupp, M.; von Lilienfeld, O. A. Big Data meets Quantum Chemistry Approximations: The A-Machine Learning Approach. *J. Chem. Theory Comput.* **2015**, *11*, 2087–2096.

(258) Mezei, P. D.; von Lilienfeld, O. A. Noncovalent Quantum Machine Learning Corrections to Density Functionals. *J. Chem. Theory Comput.* **2020**, *16*, 2647–2653.

(259) Griego, C. D.; Zhao, L.; Saravanan, K.; Keith, J. A. Machine Learning Corrected Alchemical Perturbation Density Functional Theory for Catalysis Applications. *AIChE J.* **2020**, *66*, No. e17041.

(260) Bogojeski, M.; Vogt-Maranto, L.; Tuckerman, M. E.; Müller, K.-R.; Burke, K. Quantum chemical accuracy from density functional approximations via machine learning. *Nat. Commun.* **2020**, *11*, 5223.

(261) Townsend, J.; Vogiatzis, K. D. Transferable MP2-Based Machine Learning for Accurate Coupled-Cluster Energies. *J. Chem. Theory Comput.* **2020**, *16*, 7453.

(262) Nandi, A.; Qu, C.; Houston, P.; Conte, R.; Bowman, J. M. Delta-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.

(263) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 theory for molecular energies of first-and second-row compounds. *J. Chem. Phys.* **1991**, *94*, 7221–7230.

(264) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K.; Rassolov, V.; Pople, J. A. Gaussian-3 theory using reduced Mo/ller-Plesset order. *J. Chem. Phys.* **1999**, *110*, 4703–4709.

(265) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K. Gaussian-4 theory. *J. Chem. Phys.* **2007**, *126*, 084108.

(266) Zaspel, P.; Huang, B.; Harbrecht, H.; von Lilienfeld, O. A. Boosting quantum machine learning models with multi-level combination technique: Pople diagrams revisited. *J. Chem. Theory Comput.* **2019**, *15*, 1546.

(267) Le Gratiet, L.; Garnier, J. Recursive co-kriging model for design of computer experiments with multiple levels of fidelity. *Int. J. Uncertain. Quan.* **2014**, *4*, 365.

(268) Kennedy, M. C.; O'Hagan, A. Predicting the output from a complex computer code when fast approximations are available. *Biometrika* **2000**, *87*, 1–13.

(269) Cui, J.; Krems, R. V. Gaussian Process Model for Collision Dynamics of Complex Molecules. *Phys. Rev. Lett.* **2015**, *115*, 073202.

(270) Pilania, G.; Gubernatis, J. E.; Lookman, T. Multi-fidelity machine learning models for accurate bandgap predictions of solids. *Comput. Mater. Sci.* **2017**, *129*, 156–163.

(271) Batra, R.; Pilania, G.; Uberuaga, B. P.; Ramprasad, R. Multifidelity Information Fusion with Machine Learning: A Case Study of Dopant Formation Energies in Hafnia. *ACS Appl. Mater. Interfaces* **2019**, *11*, 24906.

(272) Wiens, A. E.; Copan, A. V.; Schaefer, H. F. Multi-fidelity Gaussian process modeling for chemical energy surfaces. *Chem. Phys. Lett.: X* **2019**, *3*, 100022.

(273) Egorova, O.; Hafizi, R.; Woods, D. C.; Day, G. M. Multifidelity Statistical Machine Learning for Molecular Crystal Structure Prediction. *J. Phys. Chem. A* **2020**, *124*, 8065–8078.

(274) Garcke, J.; Griebel, M.; Thess, M. Data Mining with Sparse Grids. *Computing* **2001**, *67*, 225–253.

(275) Delvos, F. J. d-Variate Boolean interpolation. *J. Approximation Theory* **1982**, *34*, 99–114.

(276) Pan, S. J.; Yang, Q. A Survey on Transfer Learning. *IEEE Trans. Knowl. Data Eng.* **2010**, *22*, 1345–1359 Conference Name: IEEE Transactions on Knowledge and Data Engineering.

(277) 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.

(278) Olivas, E. S.; Guerrero, J. D. M.; Sober, M. M.; Benedito, J. R. M.; Lopez, A. J. S. Handbook Of Research On Machine Learning Applications and Trends: Algorithms, Methods and Techniques; IGI Publishing: Hershey, PA, 2009.

(279) Iovanac, N. C.; Savoie, B. M. Simpler is Better: How Linear Prediction Tasks Improve Transfer Learning in Chemical Autoencoders. *J. Phys. Chem. A* **2020**, *124*, 3679–3685 PMID: 32267698.

(280) Cai, C.; Wang, S.; Xu, Y.; Zhang, W.; Tang, K.; Ouyang, Q.; Lai, L.; Pei, J. Transfer Learning for Drug Discovery. *J. Med. Chem.* **2020**, *63*, 8683–8694.

(281) Browning, N. J.; Ramakrishnan, R.; von Lilienfeld, O. A.; Roethlisberger, U. Genetic Optimization of Training Sets for Improved Machine Learning Models of Molecular Properties. *J. Phys. Chem. Lett.* **2017**, *8*, 1351.

(282) Smith, J. S.; Nebgen, B.; Lubbers, N.; Isayev, O.; Roitberg, A. E. Less is more: Sampling chemical space with active learning. *J. Chem. Phys.* **2018**, *148*, 241733.

(283) Wang, T.; Zhu, J.-Y.; Torralba, A.; Efros, A. A. Dataset Distillation. *arXiv* **2020**, arXiv:1811.10959.

(284) Sucholutsky, I.; Schonlau, M. "Less Than One'-Shot Learning: Learning N Classes From M < N Samples. *arXiv* **2020**, arXiv:2009.08449

(285) Cerqueira, T. F.; Sarmiento-Pérez, R.; Amsler, M.; Nogueira, F.; Botti, S.; Marques, M. A. Materials design on-the-fly. *J. Chem. Theory Comput.* **2015**, *11*, 3955–3960.

(286) Sanchez-Lengeling, B.; Aspuru-Guzik, A. Inverse molecular design using machine learning: Generative models for matter engineering. *Science* **2018**, *361*, 360–365.

(287) Vilhelmsen, L. B.; Hammer, B. Systematic Study of Au₆ to Au₁₂ Gold Clusters on MgO100. *F Centers Using Density-Functional Theory. Phys. Rev. Lett.* **2012**, *108*, 126101.

(288) Janet, J. P.; Kulik, H. J. Predicting electronic structure properties of transition metal complexes with neural networks. *Chem. Sci.* **2017**, *8*, 5137–5152.

(289) Snyder, J. C.; Rupp, M.; Hansen, K.; Müller, K.-R.; Burke, K. Finding Density Functionals with Machine Learning. *Phys. Rev. Lett.* **2012**, *108*, 253002.

(290) Simm, G. N.; Reiher, M. Error-Controlled Exploration of Chemical Reaction Networks with Gaussian Processes. *J. Chem. Theory Comput.* **2018**, *14*, 5238–5248.

(291) Proppe, J.; Gugler, S.; Reiher, M. Gaussian Process-Based Refinement of Dispersion Corrections. *J. Chem. Theory Comput.* **2019**, *15*, 6046–6060.

(292) Goreinov, S. A.; Oseledets, I. V.; Savostyanov, D. V.; Tyrtyshnikov, E. E.; Zamarashkin, N. L. *Matrix Methods: Theory, Algorithms and Applications: Dedicated to the Memory of Gene Golub*; World Scientific, 2010; pp 247–256.

(293) Guo, C.; Pleiss, G.; Sun, Y.; Weinberger, K. Q. On calibration of modern neural networks. *Proceedings of the 34th International Conference on Machine Learning* Sydney, NSW, Australia, 2017; Vol. 70, pp 1321–1330.

(294) Skafte, N.; Jørgensen, M.; Hauberg, S. Reliable training and estimation of variance networks. *Adv. Neur. Inform. Process. Syst.* **2019**, 6326–6336.

(295) Peterson, A. A.; Christensen, R.; Khorshidi, A. Addressing uncertainty in atomistic machine learning. *Phys. Chem. Chem. Phys.* **2017**, *19*, 10978–10985.
(296) Cortés-Ciriano, I.; Bender, A. Deep Confidence: A Computationally Efficient Framework for Calculating Reliable Prediction Errors for Deep Neural Networks. *J. Chem. Inf. Model.* **2019**, *59*, 1269–1281.

(297) Musil, F.; Willatt, M. J.; Langovoy, M. A.; Ceriotti, M. Fast and Accurate Uncertainty Estimation in Chemical Machine Learning. *J. Chem. Theory Comput.* **2019**, *15*, 906–915.

(298) Gal, Y.; Ghahramani, Z. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. *International Conference on Machine Learning*, 2016; pp 1050–1059.

(299) Janet, J. P.; Duan, C.; Yang, T.; Nandy, A.; Kulik, H. J. A quantitative uncertainty metric controls error in neural network-driven chemical discovery. *Chem. Sci.* **2019**, *10*, 7913–7922.

(300) Prodan, E.; Kohn, W. Nearsightedness of electronic matter. *Proc. Natl. Acad. Sci. U. S. A.* **2005**, *102*, 11635–11638.

(301) Fias, S.; Heidar-Zadeh, F.; Geerlings, P.; Ayers, P. W. Chemical transferability of functional groups follows from the nearsightedness of electronic matter. *Proc. Natl. Acad. Sci. U. S. A.* **2017**, *114*, 11633–11638.

(302) Rappe, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard, W. A., III; Skiff, W. M. UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. *J. Am. Chem. Soc.* **1992**, *114*, 10024–10035.

(303) Misra, M.; Andrienko, D.; Baumeier, B.; Faulon, J.-L.; von Lilienfeld, O. A. Toward Quantitative Structure-Property Relationships for Charge Transfer Rates of Polycyclic Aromatic Hydrocarbons. *J. Chem. Theory Comput.* **2011**, *7*, 2549.

(304) Schütt, K. T.; Arbabzadah, F.; Chmiela, S.; Müller, K. R.; Tkatchenko, A. Quantum-chemical insights from deep tensor neural networks. *Nat. Commun.* **2017**, *8*, 13890.

(305) Heinen, S.; Schwilk, M.; von Rudorff, G. F.; von Lilienfeld, O. A. Machine learning the computational cost of quantum chemistry. *Mach. Learn.: Sci. Technol.* **2020**, *1*, 025002.

(306) 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 PMID: 31042390.

(307) McDonagh, J. L.; Silva, A. F.; Vincent, M. A.; Popelier, P. L. Machine learning of dynamic electron correlation energies from topological atoms. *J. Chem. Theory Comput.* **2018**, *14*, 216–224.

(308) Schütt, K. T.; Arbabzadah, F.; Chmiela, S.; Müller, K. R.; Tkatchenko, A. Quantum-chemical insights from deep tensor neural networks. *Nat. Commun.* **2017**, *8*, 13890.

(309) Chen, X.; Jørgensen, M. S.; Li, J.; Hammer, B. Atomic energies from a convolutional neural network. *J. Chem. Theory Comput.* **2018**, *14*, 3933.

(310) Wilkins, D. M.; Grisafi, A.; Yang, Y.; Lao, K. U.; DiStasio, R. A.; Ceriotti, M. Accurate molecular polarizabilities with coupled cluster theory and machine learning. *Proc. Natl. Acad. Sci. U. S. A.* **2019**, *116*, 3401–3406.

(311) Grisafi, A.; Nigam, J.; Ceriotti, M. Multi-scale approach for the prediction of atomic scale properties. *arXiv* **2021**, *12*, 2078.

(312) Paruzzo, F. M.; Hofstetter, A.; Musil, F.; De, S.; Ceriotti, M.; Emsley, L. Chemical shifts in molecular solids by machine learning. *Nat. Commun.* **2018**, *9*, 4501.

(313) Engel, E. A.; Anelli, A.; Hofstetter, A.; Paruzzo, F.; Emsley, L.; Ceriotti, M. A Bayesian approach to NMR crystal structure determination. *Phys. Chem. Chem. Phys.* **2019**, *21*, 23385–23400.

(314) Li, J.; Bennett, K. C.; Liu, Y.; Martin, M. V.; Head-Gordon, T. Accurate prediction of chemical shifts for aqueous protein structure on "Real World" data. *Chem. Sci.* **2020**, *11*, 3180–3191.

(315) Navarro-Vázquez, A. A DFT/machine-learning hybrid method for the prediction of 3JHCCH couplings. *Magn. Reson. Chem.* **2021**, *59*, 414.

(316) Bratholm, L. A.; Gerrard, W.; Anderson, B.; Bai, S.; Choi, S.; Dang, L.; Hanchar, P.; Howard, A.; Huard, G.; Kim, S., et al. A community-powered search of machine learning strategy space to find NMR property prediction models. *arXiv* **2020**, arXiv:2008.05994

(317) Gupta, A.; Chakraborty, S.; Ramakrishnan, R. Revving up ¹³C NMR shielding predictions across chemical space: Benchmarks for

atoms-in-molecules kernel machine learning with new data for 134 kilo molecules. *Mach. Learn.: Sci. Technol.* **2021**, *2*, 035010.

(318) 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.

(319) Gastegger, M.; Behler, J.; Marquetand, P. Machine learning molecular dynamics for the simulation of infrared spectra. *Chem. Sci.* **2017**, *8*, 6924–6935.

(320) Lopez-Bezanilla, A.; von Lilienfeld, O. A. Modeling electronic quantum transport with machine learning. *Phys. Rev. B: Condens. Matter Mater. Phys.* **2014**, *89*, 235411.

(321) Arsenault, L.-F.; Lopez-Bezanilla, A.; von Lilienfeld, O. A.; Millis, A. J. Machine learning for Many-Body Physics: the case of the Anderson impurity model. *Phys. Rev. B: Condens. Matter Mater. Phys.* **2014**, *90*, 155136.

(322) Arsenault, L.-F.; von Lilienfeld, O. A.; Millis, A. J. Machine learning for many-body physics: efficient solution of dynamical mean-field theory. *arXiv*2015, http://arxiv.org/abs/1506.08858.

(323) Dral, P. O.; Barbatti, M.; Thiel, W. Nonadiabatic excited-state dynamics with machine learning. *J. Phys. Chem. Lett.* **2018**, *9*, 5660–5663.

(324) Westermayr, J.; Faber, F. A.; Christensen, A. S.; von Lilienfeld, O. A.; Marquetand, P. Neural networks and kernel ridge regression for excited states dynamics of CH2NH: From single-state to multi-state representations and multi-property machine learning models. *Mach. Learn.: Sci. Technol.* **2020**, *1*, 025009.

(325) Westermayr, J.; Gastegger, M.; Marquetand, P. Combining SchNet and SHARC: The SchNarc machine learning approach for excited-state dynamics. *J. Phys. Chem. Lett.* **2020**, *11*, 3828–3834.

(326) Richter, M.; Marquetand, P.; González-Vázquez, J.; Sola, I.; González, L. SHARC: ab initio molecular dynamics with surface hopping in the adiabatic representation including arbitrary couplings. *J. Chem. Theory Comput.* **2011**, *7*, 1253–1258.

(327) Westermayr, J.; Marquetand, P. Deep learning for UV absorption spectra with SchNarc: First steps toward transferability in chemical compound space. *J. Chem. Phys.* **2020**, *153*, 154112.

(328) Westermayr, J.; Marquetand, P. Machine learning for electronically excited states of molecules. *Chem. Rev.* **2020**, DOI: 10.1021/acs.chemrev.0c00749

(329) Westermayr, J.; Marquetand, P. Machine learning for nonadiabatic molecular dynamics. *Machine Learning in Chemistry* **2020**, *17*, 76.

(330) Westermayr, J.; Marquetand, P. Machine learning and excited-state molecular dynamics. *Mach. Learn.: Sci. Technol.* **2020**, *1*, 043001.

(331) Zubatyuk, R.; Smith, J.; Nebgen, B. T.; Tretiak, S.; Isayev, O. Teaching a neural network to attach and detach electrons from molecules. *ChemRxiv*2020, DOI: 10.26434/chemrxiv.12725276.v2

(332) Zhang, Y.; Ye, S.; Zhang, J.; Hu, C.; Jiang, J.; Jiang, B. Efficient and Accurate Simulations of Vibrational and Electronic Spectra with Symmetry-Preserving Neural Network Models for Tensorial Properties. *J. Phys. Chem. B* **2020**, *124*, 7284–7290.

(333) Ward, L.; Liu, R.; Krishna, A.; Hegde, V. I.; Agrawal, A.; Choudhary, A.; Wolverton, C. Including crystal structure attributes in machine learning models of formation energies via Voronoi tessellations. *Phys. Rev. B: Condens. Matter Mater. Phys.* **2017**, *96*, 024104.

(334) Yao, K.; Herr, J. E.; Brown, S. N.; Parkhill, J. Intrinsic Bond Energies from a Bonds-in-Molecules Neural Network. *J. Phys. Chem. Lett.* **2017**, *8*, 2689.

(335) Janet, J. P.; Kulik, H. J. Predicting electronic structure properties of transition metal complexes with neural networks. *Chem. Sci.* **2017**, *8*, 5137.

(336) Grisafi, A.; Wilkins, D. M.; Csányi, G.; Ceriotti, M. Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems. *Phys. Rev. Lett.* **2018**, *120*, 036002.

(337) Xie, T.; Grossman, J. C. Crystal Graph Convolutional Neural Networks for an Accurate and Interpretable Prediction of Material Properties. *Phys. Rev. Lett.* **2018**, *120*, 145301.
(338) Rossi, K.; Juraskova, V.; Wischert, R.; Garel, L.; Corminboeuf, C.; Ceriotti, M. Simulating Solvation and Acidity in Complex Mixtures with First-Principles Accuracy: The Case of CH3SO3H and H2O2 in Phenol. *J. Chem. Theory Comput.* **2020**, *16*, 5139.

(339) Moosavi, S. M.; Nandy, A.; Jablonka, K. M.; Ongari, D.; Janet, J. P.; Boyd, P. G.; Lee, Y.; Smit, B.; Kulik, H. J. Understanding the diversity of the metal-organic framework ecosystem. *Nat. Commun.* **2020**, *11*, 4068.

(340) Jablonka, K. M.; Ongari, D.; Moosavi, S. M.; Smit, B. Big-Data Science in Porous Materials: Materials Genomics and Machine Learning. *Chem. Rev.* **2020**, *120*, 8066.

(341) Ulissi, Z. W.; Singh, A. R.; Tsai, C.; Nørskov, J. K. Automated Discovery and Construction of Surface Phase Diagrams Using Machine Learning. *J. Phys. Chem. Lett.* **2016**, *7*, 3931–3935.

(342) Ulissi, Z. W.; Tang, M. T.; Xiao, J.; Liu, X.; Torelli, D. A.; Karamad, M.; Cummins, K.; Hahn, C.; Lewis, N. S.; Jaramillo, T. F.; et al. Machine-learning methods enable exhaustive searches for active bimetallic facets and reveal active site motifs for CO2 reduction. *ACS Catal.* **2017**, *7*, 6600–6608.

(343) Ulissi, Z. W.; Medford, A. J.; Bligaard, T.; Nørskov, J. K. To address surface reaction network complexity using scaling relations machine learning and DFT calculations. *Nat. Commun.* **2017**, *8*, 14621.

(344) Singh, A. R.; Rohr, B. A.; Gauthier, J. A.; Nørskov, J. K. Predicting chemical reaction barriers with a machine learning model. *Catal. Lett.* **2019**, *149*, 2347–2354.

(345) Meyer, B.; Sawatlon, B.; Heinen, S.; von Lilienfeld, O. A.; Corminboeuf, C. Machine learning meets volcano plots: computational discovery of cross-coupling catalysts. *Chem. Sci.* **2018**, *9*, 7069–7077.

(346) Heinen, S.; von Rudorff, G. F.; von Lilienfeld, O. A. Quantum based machine learning of competing chemical reaction profiles. *arXiv* **2020**, arXiv:2009.13429

(347) von Rudorff, G. F.; Heinen, S. N.; Bragato, M.; von Lilienfeld, O. A. Thousands of reactants and transition states for competing E2 and S2 reactions. *Mach. Learn.: Sci. Technol.* **2020**, *1*, 045026.

(348) Flores, R. A.; Paolucci, C.; Winther, K. T.; Jain, A.; Torres, J. A. G.; Aykol, M.; Montoya, J.; Nørskov, J. K.; Bajdich, M.; Bligaard, T. Active Learning Accelerated Discovery of Stable Iridium Oxide Polymorphs for the Oxygen Evolution Reaction. *Chem. Mater.* **2020**, *32*, 5854–5863.

(349) Mamun, O.; Winther, K. T.; Boes, J. R.; Bligaard, T. A. Bayesian framework for adsorption energy prediction on bimetallic alloy catalysts. *NPJ Comput. Mater.* **2020**, *6*, 177.

(350) Garijo del Río, E.; Kaappa, S.; Garrido Torres, J. A.; Bligaard, T.; Jacobsen, K. W. Machine Learning with bond information for local structure optimizations in surface science. *J. Chem. Phys.* **2020**, *153*, 234116.

(351) Groenenboom, M. C.; Anderson, R. M.; Wollmershauser, J. A.; Horton, D. J.; Policastro, S. A.; Keith, J. A. Combined Neural Network Potential and Density Functional Theory Study of TiAl2O5 Surface Morphology and Oxygen Reduction Reaction Overpotentials. *J. Phys. Chem. C* **2020**, *124*, 15171–15179.

(352) Schlexer Lamoureux, P.; Winther, K. T.; Garrido Torres, J. A.; Streibel, V.; Zhao, M.; Bajdich, M.; Abild-Pedersen, F.; Bligaard, T. Machine learning for computational heterogeneous catalysis. *ChemCatChem* **2019**, *11*, 3581.

(353) Fink, T.; Reymond, J.-L. Virtual Exploration of the Chemical Universe up to 11 Atoms of C, N, O, F: Assembly of 26.4 Million Structures (110.9 Million Stereoisomers) and Analysis for New Ring Systems, Stereochemistry, Physicochemical Properties, Compound Classes, and Drug Discovery. *J. Chem. Inf. Model.* **2007**, *47*, 342–353.

(354) Blum, L. C.; Reymond, J.-L. 970 Million Druglike Small Molecules for Virtual Screening in the Chemical Universe Database GDB-13. *J. Am. Chem. Soc.* **2009**, *131*, 8732–8733.

(355) Ruddigkeit, L.; van Deursen, R.; Blum, L. C.; Reymond, J.-L. Enumeration of 166 Billion Organic Small Molecules in the Chemical Universe Database GDB-17. *J. Chem. Inf. Model.* **2012**, *52*, 2864–2875.

(356) Fink, T.; Bruggesser, H.; Reymond, J.-L. Virtual Exploration of the Small-Molecule Chemical Universe below 160 Da. *Angew. Chem., Int. Ed.* **2005**, *44*, 1504–1508 _eprint_:

(357) Delarue Bizzini, L.; Müntener, T.; Häussinger, D.; Neuburger, M.; Mayor, M. Synthesis of trinorbornane. *Chem. Commun.* **2017**, *53*, 11399–11402.

(358) Montavon, G.; Rupp, M.; Gobre, V.; Vazquez-Mayagoitia, A.; Hansen, K.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Machine learning of molecular electronic properties in chemical compound space. *New J. Phys.* **2013**, *15*, 095003.

(359) Hoja, J.; Medrano Sandonas, L.; Ernst, B. G.; Vazquez-Mayagoitia, A.; DiStasio, R. A.; Tkatchenko, A. QM7-X: A comprehensive dataset of quantum-mechanical properties spanning the chemical space of small organic molecules. *Sci. Data* **2021**, *8*, 43.

(360) Blum, L. C.; Reymond, J.-L. 970 Million Druglike Small Molecules for Virtual Screening in the Chemical Universe Database GDB-13. *J. Am. Chem. Soc.* **2009**, *131*, 8732.

(361) Narayanan, B.; Redfern, P. C.; Assary, R. S.; Curtiss, L. A. Accurate quantum chemical energies for 133000 organic molecules. *Chem. Sci.* **2019**, *10*, 7449–7455.

(362) Chen, G.; Chen, P.; Hsieh, C.-Y.; Lee, C.-K.; Liao, B.; Liao, R.; Liu, W.; Qiu, J.; Sun, Q.; Tang, J.; Zemel, R.; Zhang, S. Alchemy: A Quantum Chemistry Dataset for Benchmarking AI Models. *arXiv* **2019**, arXiv:1906.09427

(363) Ramakrishnan, R.; Hartmann, M.; Tapavicza, E.; von Lilienfeld, O. A. Electronic spectra from TDDFT and machine learning in chemical space. *J. Chem. Phys.* **2015**, *143*, 084111.

(364) Irwin, J. J.; Shoichet, B. K. ZINC - A Free Database of Commercially Available Compounds for Virtual Screening. *J. Chem. Inf. Model.* **2005**, *45*, 177–182.

(365) Huang, B.; von Lilienfeld, O. A. Dictionary of 140k GDB and ZINC derived AMONs. *arXiv* **2020**, arXiv:2008.05260

(366) Smith, J. S.; Isayev, O.; Roitberg, A. E. ANI-1, A data set of 20 million calculated off-equilibrium conformations for organic molecules. *Sci. Data* **2017**, *4*, 170193.

(367) Smith, J. S.; Zubatyuk, R.; Nebgen, B.; Lubbers, N.; Barros, K.; Roitberg, A. E.; Isayev, O.; Tretiak, S. The ANI-1ccx and ANI-1x data sets, coupled-cluster and density functional theory properties for molecules. *Sci. Data* **2020**, *7*, 134.

(368) Tkatchenko, A.; Scheffler, M. Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data. *Phys. Rev. Lett.* **2009**, *102*, 073005.

(369) Chmiela, S.; Tkatchenko, A.; Sauceda, H. E.; Poltavsky, I.; Schütt, K. T.; Müller, K.-R. Machine learning of accurate energy-conserving molecular force fields. *Sci. Adv.* **2017**, *3*, e1603015.

(370) Dandu, N. K.; Ward, L.; Assary, R. S.; Redfern, P. C.; Narayanan, B.; Foster, I. T.; Curtiss, L. A. Quantum Chemically Informed Machine Learning: Prediction of Energies of Organic Molecules with 10 to 14 Non-Hydrogen Atoms. *J. Phys. Chem. A* **2020**, *124*, 5804.

(371) Kim, S.; Chen, J.; Cheng, T.; Gindulyte, A.; He, J.; He, S.; Li, Q.; Shoemaker, B. A.; Thiessen, P. A.; Yu, B.; Zaslavsky, L.; Zhang, J.; Bolton, E. E. PubChem 2019 update: improved access to chemical data. *Nucleic Acids Res.* **2019**, *47*, D1102–D1109.

(372) Nakata, M.; Shimazaki, T. PubChemQC Project: A Large-Scale First-Principles Electronic Structure Database for Data-Driven Chemistry. *J. Chem. Inf. Model.* **2017**, *57*, 1300–1308.

(373) Glavatskikh, M.; Leguy, J.; Hunault, G.; Cauchy, T.; Da Mota, B. Dataset's chemical diversity limits the generalizability of machine learning predictions. *J. Cheminf.* **2019**, *11*, 69.

(374) Allen, F. H. The Cambridge Structural Database: a quarter of a million crystal structures and rising. *Acta Crystallogr., Sect. B: Struct. Sci.* **2002**, *58*, 380–388.

(375) Stuke, A.; Kunkel, C.; Golze, D.; Todorović, M.; Margraf, J. T.; Reuter, K.; Rinke, P.; Oberhofer, H. Atomic structures and orbital energies of 61,489 crystal-forming organic molecules. *Sci. Data* **2020**, *7*, 58.
(376) Schober, C.; Reuter, K.; Oberhofer, H. Virtual Screening for High Carrier Mobility in Organic Semiconductors. *J. Phys. Chem. Lett.* **2016**, *7*, 3973–3977.

(377) Grambow, C. A.; Pattanaik, L.; Green, W. H. Reactants, products, and transition states of elementary chemical reactions based on quantum chemistry. *Sci. Data* **2020**, *7*, 137.

(378) Schwilk, M.; Tahchieva, D. N.; von Lilienfeld, O. A. Large yet bounded: Spin gap ranges in carbenes. *arXiv* **2020**, arXiv:2004.10600

(379) Heinen, S.; Schwilk, M.; von Rudorff, G. F.; von Lilienfeld, O. A. Machine learning the computational cost of quantum chemistry. *Mach. Learn.: Sci. Technol.* **2020**, *1*, 025002.

(380) Donchev, A. G.; Taube, A. G.; Decolvenaere, E.; Hargus, C.; McGibbon, R. T.; Law, K.-H.; Gregersen, B. A.; Li, J.-L.; Palmo, K.; Siva, K.; Bergdorf, M.; Klepeis, J. L.; Shaw, D. E. Quantum chemical benchmark databases of gold-standard dimer interaction energies. *Sci. Data* **2021**, *8*, 55.

(381) Korth, M.; Grimme, S. Mindless" DFT Benchmarking. *J. Chem. Theory Comput.* **2009**, *5*, 993–1003 PMID: 26609608.

(382) Balcells, D.; Skjelstad, B. B. tmQM Dataset—Quantum Geometries and Properties of 86k Transition Metal Complexes. *J. Chem. Inf. Model.* **2020**, *60*, 6135.

(383) Janet, J. P.; Ramesh, S.; Duan, C.; Kulik, H. J. Accurate Multiobjective Design in a Space of Millions of Transition Metal Complexes with Neural-Network-Driven Efficient Global Optimization. ACS Cent. Sci. **2020**, *6*, 513–524.

(384) Rosen, A. S.; Iyer, S. M.; Ray, D.; Yao, Z.; Aspuru-Guzik, A.; Gagliardi, L.; Notestein, J. M.; Snurr, R. Q. Machine learning the quantum-chemical properties of metal-organic frameworks for accelerated materials discovery. *Matter* **2021**, *4*, 1578.

(385) Rupp, M.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning. *Phys. Rev. Lett.* **2012**, *108*, 058301.

(386) Janet, J. P.; Kulik, H. J. Resolving Transition Metal Chemical Space: Feature Selection for Machine Learning and Structure-Property Relationships. *J. Phys. Chem. A* **2017**, *121*, 8939–8954.

(387) Curtarolo, S.; Setyawan, W.; Hart, G. L.; Jahnatek, M.; Chepulskii, R. V.; Taylor, R. H.; Wang, S.; Xue, J.; Yang, K.; Levy, O.; Mehl, M. J.; Stokes, H. T.; Demchenko, D. O.; Morgan, D. AFLOW: An automatic framework for high-throughput materials discovery. *Comput. Mater. Sci.* **2012**, *58*, 218–226.

(388) Saal, J. E.; Kirklin, S.; Aykol, M.; Meredig, B.; Wolverton, C. Materials design and discovery with high-throughput density functional theory: the open quantum materials database (OQMD). *JOM* **2013**, *65*, 1501–1509.

(389) Chanussot, L.; Das, A.; Goyal, S.; Lavril, T.; Shuaibi, M.; Riviere, M.; Tran, K.; Heras-Domingo, J.; Ho, C.; Hu, W. The Open Catalyst 2020 (OC20) Dataset and Community Challenges. ACS Catal. **2021**, *11*, 6059.

(390) Talirz, L.; Kumbhar, S.; Passaro, E.; Yakutovich, A. V.; Granata, V.; Gargiulo, F.; Borelli, M.; Uhrin, M.; Huber, S. P.; Zoupanos, S.; et al. Materials Cloud, a platform for open computational science. *Sci. Data* **2020**, *7*, 299.

(391) Jinnouchi, R.; Miwa, K.; Karsai, F.; Kresse, G.; Asahi, R. On-the-Fly Active Learning of Interatomic Potentials for Large-Scale Atomistic Simulations. *J. Phys. Chem. Lett.* **2020**, *11*, 6946–6955.

(392) Garijo del Río, E.; Mortensen, J. J.; Jacobsen, K. W. Local Bayesian optimizer for atomic structures. *Phys. Rev. B: Condens. Matter Mater. Phys.* **2019**, *100*, 104103.

(393) Thompson, A.; Swiler, L.; Trott, C.; Foiles, S.; Tucker, G. Spectral neighbor analysis method for automated generation of quantum-accurate interatomic potentials. *J. Comput. Phys.* **2015**, *285*, 316–330.

(394) Christensen, A. S.; Faber, F. A.; Huang, B.; Bratholm, L. A.; Tkatchenko, A.; Müller, K.-R.; von Lilienfeld, O. A. *QML: A Python Toolkit for Quantum Machine Learning*, 2017; https://github.com/qmlcode/qml.

(395) Huang, B.; von Lilienfeld, O. A. *AQML: Amons-based Quantum Machine Learning Code for Quantum Chemistry*. 2020; https://github.com/binghuang2018/aqml.

(396) Bonomi, M. Promoting transparency and reproducibility in enhanced molecular simulations. *Nat. Methods* **2019**, *16* (8), 670–673.

(397) Yao, K.; Herr, J. E.; Toth, D. W.; Mckintyre, R.; Parkhill, J. The TensorMol-0.1 model chemistry: a neural network augmented with long-range physics. *Chem. Sci.* **2018**, *9*, 2261–2269.

(398) Gao, X.; Ramezanghorbani, F.; Isayev, O.; Smith, J.; Roitberg, A. TorchANI: A Free and Open Source PyTorch Based Deep Learning Implementation of the ANI Neural Network Potentials. *J. Chem. Inf. Model.* **2020**, *60*, 3408.

(399) Huber, S. P.; Zoupanos, S.; Uhrin, M.; Talirz, L.; Kahle, L.; Hauselmann, R.; Gresch, D.; Muller, T.; Yakutovich, A. V.; Andersen, C. W.; et al. AiiDA 1.0, a scalable computational infrastructure for automated reproducible workflows and data provenance. *Sci. Data* **2020**, *7* (1), 300.

(400) Gómez-Bombarelli, R.; Wei, J. N.; Duvenaud, D.; Hernández-Lobato, J. M.; Sánchez-Lengeling, B.; Sheberla, D.; Aguilera-Iparraguirre, J.; Hirzel, T. D.; Adams, R. P.; Aspuru-Guzik, A. Automatic chemical design using a data-driven continuous representation of molecules. *ACS Cent. Sci.* **2018**, *4*, 268–276.

(401) Carrasquilla, J.; Torlai, G.; Melko, R. G.; Aolita, L. Reconstructing quantum states with generative models. *Nat. Mach. Intell.* **2019**, *1*, 155–161.

(402) Nesterov, V.; Wieser, M.; Roth, V. 3DMolNet: A Generative Network for Molecular Structures. *arXiv* **2020**, arXiv:2010.06477

(403) Segler, M. H.; Preuss, M.; Waller, M. P. Planning chemical syntheses with deep neural networks and symbolic AI. *Nature* **2018**, *555*, 604.

(404) Schwaller, P.; Gaudin, T.; Lanyi, D.; Bekas, C.; Laino, T. Found in Translation*: predicting outcomes of complex organic chemistry reactions using neural sequence-to-sequence models. *Chem. Sci.* **2018**, *9*, 6091–6098.

(405) Nair, V. H.; Schwaller, P.; Laino, T. Data-driven Chemical Reaction Prediction and Retrosynthesis. *Chimia* **2019**, *73*, 997–1000.

(406) Schwaller, P.; Laino, T.; Gaudin, T.; Bolgar, P.; Hunter, C. A.; Bekas, C.; Lee, A. A. Molecular transformer: A model for uncertainty-calibrated chemical reaction prediction. *ACS Cent. Sci.* **2019**, *5*, 1572–1583.

(407) Schwaller, P.; Petraglia, R.; Zullo, V.; Nair, V. H.; Haeuselmann, R. A.; Pisoni, R.; Bekas, C.; Iuliano, A.; Laino, T. Predicting retrosynthetic pathways using transformer-based models and a hyper-graph exploration strategy. *Chem. Sci.* **2020**, *11*, 3316–3325.

(408) Pesciullesi, G.; Schwaller, P.; Laino, T.; Reymond, J.-L. Transfer learning enables the molecular transformer to predict region- and stereoselective reactions on carbohydrates. *Nat. Commun.* **2020**, *11*, 4874.

(409) Carrasquilla, J.; Melko, R. G. Machine learning phases of matter. *Nat. Phys.* **2017**, *13*, 431–434.

(410) Ch'Ng, K.; Carrasquilla, J.; Melko, R. G.; Khatami, E. Machine learning phases of strongly correlated fermions. *Phys. Rev. X* **2017**, *7*, 031038.

(411) Broecker, P.; Carrasquilla, J.; Melko, R. G.; Trebst, S. Machine learning quantum phases of matter beyond the fermion sign problem. *Sci. Rep.* **2017**, *7*, 8823.

(412) Vargas-Hernández, R. A.; Sous, J.; Berciu, M.; Krems, R. V. Extrapolating quantum observables with machine learning: inferring multiple phase transitions from properties of a single phase. *Phys. Rev. Lett.* **2018**, *121*, 255702.

(413) Cheng, B.; Engel, E. A.; Behler, J.; Dellago, C.; Ceriotti, M. Ab initio thermodynamics of liquid and solid water. *Proc. Natl. Acad. Sci. U. S. A.* **2019**, *116*, 1110–1115.

(414) Cheng, B.; Mazzola, G.; Pickard, C. J.; Ceriotti, M. Evidence for supercritical behaviour of high-pressure liquid hydrogen. *Nature* **2020**, *585*, 217–220.

(415) Carleo, G.; Troyer, M. Solving the quantum many-body problem with artificial neural networks. *Science* **2017**, *355*, 602–606.

(416) Schütt, K.; Gastegger, M.; Tkatchenko, A.; Müller, K.-R.; Maurer, R. J. Unifying machine learning and quantum chemistry with
a deep neural network for molecular wavefunctions. *Nat. Commun.* **2019**, *10*, 5024.

(417) Hermann, J.; Schätzle, Z.; Noé, F. Deep-neural-network solution of the electronic Schrödinger equation. *Nat. Chem.* **2020**, *12*, 891–897.

(418) Carrasquilla, J. Machine learning for quantum matter. *Adv. Phys. X* **2020**, *5*, 1797528.

(419) Raccuglia, P.; Elbert, K. C.; Adler, P. D.; Falk, C.; Wenny, M. B.; Mollo, A.; Zeller, M.; Friedler, S. A.; Schrier, J.; Norquist, A. J. Machine-learning-assisted materials discovery using failed experiments. *Nature* **2016**, *533*, 73–76.