The efficiency of machine learning algorithms for electronically excited states is far behind ground-state applications. One of the underlying problems is the insufficient smoothness of the fitted potential energy surfaces and other properties in the vicinity of state crossings and conical intersections, which is a prerequisite for an efficient regression. Smooth surfaces can be obtained by switching to the diabatic basis. However, diabatization itself is still an outstanding problem. We overcome these limitations by solving both problems at once. We use a machine learning approach combining clustering and regression techniques to correct for the deficiencies of property-based diabatization which, in return, provides us with smooth surfaces that can be easily fitted. Our approach extends the applicability of property-based diabatization to multidimensional systems. We utilize the proposed diabatization scheme to achieve higher prediction accuracy for adiabatic states and we show its performance by reconstructing global potential energy surfaces of excited states of nitrosyl fluoride and formaldehyde. While the proposed methodology is independent of the specific property-based diabatization and regression algorithm, we show its performance for kernel ridge regression and a very simple diabatization based on transition multipoles. Compared to most other algorithms based on machine learning, our approach needs only a small amount of training data.

Department of Physical Chemistry University of Chemistry and Technology Technická 5 162 28 Prague Czech Republic

Departments of Chemistry Materials Science and Engineering and Physics University of Toronto St George Campus Toronto ON Canada

Institute of Theoretical Chemistry Faculty of Chemistry University of Vienna Währinger Str 17 1090 Wien Austria

Machine Learning Group Technische Universität Berlin and Institute for the Foundations of Learning and Data 10587 Berlin Germany

Vector Institute for Artificial Intelligence Toronto ON M5S 1M1 Canada

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Dral P. O. Quantum Chemistry in the Age of Machine Learning. J. Phys. Chem. Lett. 2020;11:2336–2347. PubMed

Westermayr J. Marquetand P. Machine Learning for Electronically Excited States of Molecules. Chem. Rev. 2021;121:9873–9926. doi: 10.1021/acs.chemrev.0c00749. PubMed DOI PMC

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. doi: 10.1063/1.4928757. PubMed DOI

Zhu X. Yarkony D. R. Fitting coupled potential energy surfaces for large systems: Method and construction of a 3-state representation for phenol photodissociation in the full 33 internal degrees of freedom using multireference configuration interaction determined data. J. Chem. Phys. 2014;140:024112. doi: 10.1063/1.4857335. PubMed DOI

Malbon C. L. Zhao B. Guo H. Yarkony D. R. On the nonadiabatic collisional quenching of OH(A) by H2: a four coupled quasi-diabatic state description. Phys. Chem. Chem. Phys. 2020;22:13516–13527. PubMed

Shen Y. Yarkony D. R. Construction of Quasi-diabatic Hamiltonians That Accurately Represent ab Initio Determined Adiabatic Electronic States Coupled by Conical Intersections for Systems on the Order of 15 Atoms. Application to Cyclopentoxide Photoelectron Detachment in the Ful. J. Phys. Chem. A. 2020;124:4539–4548. PubMed

Shu Y. Varga Z. Kanchanakungwankul S. Zhang L. Truhlar D. G. Diabatic States of Molecules. J. Phys. Chem. A. 2022;126:992–1018. doi: 10.1021/acs.jpca.1c10583. PubMed DOI

Abrol R. Kuppermann A. An optimal adiabatic-to-diabatic transformation of the 12A′ and 22A′ states of H3. J. Chem. Phys. 2002;116:1035–1062. doi: 10.1063/1.1419257. DOI

Köppel H. Regularized diabatic states and quantum dynamics on intersecting potential energy surfaces. Faraday Discuss. 2004;127:35–47. doi: 10.1039/B314471B. PubMed DOI

Zhu X. Yarkony D. R. Toward eliminating the electronic structure bottleneck in nonadiabatic dynamics on the fly: An algorithm to fit nonlocal, quasidiabatic, coupled electronic state Hamiltonians based on ab initio electronic structure data. J. Chem. Phys. 2010;132:104101. doi: 10.1063/1.3324982. PubMed DOI

Hoyer C. E. Parker K. Gagliardi L. Truhlar D. G. The DQ and DQΦ electronic structure diabatization methods: Validation for general applications. J. Chem. Phys. 2016;144:194101. doi: 10.1063/1.4948728. PubMed DOI

Varga Z. Parker K. A. Truhlar D. G. Direct diabatization based on nonadiabatic couplings: the N/D method. Phys. Chem. Chem. Phys. 2018;20:26643–26659. PubMed

Subotnik J. E. Yeganeh S. Cave R. J. Ratner M. A. Constructing diabatic states from adiabatic states: Extending generalized Mulliken-Hush to multiple charge centers with Boys localization. J. Chem. Phys. 2008;129:244101. PubMed

Pacher T. Köppel H. Cederbaum L. S. Quasidiabatic states from ab initio calculations by block diagonalization of the electronic Hamiltonian: Use of frozen orbitals. J. Chem. Phys. 1991;95:6668–6680.

Wittenbrink N. Venghaus F. Williams D. Eisfeld W. A new approach for the development of diabatic potential energy surfaces: Hybrid block-diagonalization and diabatization by ansatz. J. Chem. Phys. 2016;145:184108. PubMed

Atchity G. J. Ruedenberg K. Determination of diabatic states through enforcement of configurational uniformity. Theor. Chem. Acc. 1997;97:47–58.

Yang K. R. Xu X. Truhlar D. G. Direct diabatization of electronic states by the fourfold-way: Including dynamical correlation by multi-configuration quasidegenerate perturbation theory with complete active space self-consistent-field diabatic molecular orbitals. Chem. Phys. Lett. 2013;573:84–89. doi: 10.1016/j.cplett.2013.04.036. DOI

Shu Y. Truhlar D. G. Diabatization by Machine Intelligence. J. Chem. Theory Comput. 2020;16:6456–6464. PubMed

Shu Y. Varga Z. Sampaio De Oliveira-Filho A. G. Truhlar D. G. Permutationally Restrained Diabatization by Machine Intelligence. J. Chem. Theory Comput. 2021;17:1106–1116. doi: 10.1021/acs.jctc.0c01110. PubMed DOI

Guan Y. Zhang D. H. Guo H. Yarkony D. R. Representation of coupled adiabatic potential energy surfaces using neural network based quasi-diabatic Hamiltonians: 1,2 2A’ states of LiFH. Phys. Chem. Chem. Phys. 2019;21:14205–14213. doi: 10.1039/C8CP06598E. PubMed DOI

Williams D. M. Eisfeld W. Neural network diabatization: A new ansatz for accurate high-dimensional coupled potential energy surfaces. J. Chem. Phys. 2018;149:204106. doi: 10.1063/1.5053664. PubMed DOI

Axelrod S. Shakhnovich E. Gómez-Bombarelli R. Excited state non-adiabatic dynamics of large photoswitchable molecules using a chemically transferable machine learning potential. Nat. Commun. 2022;13:3440. doi: 10.1038/s41467-022-30999-w. PubMed DOI PMC

Wang T. Y. Neville S. P. Schuurman M. S. Machine Learning Seams of Conical Intersection: A Characteristic Polynomial Approach. J. Phys. Chem. Lett. 2023;14:7780–7786. doi: 10.1021/acs.jpclett.3c01649. PubMed DOI PMC

Shu Y. Kryven J. Sampaio de Oliveira-Filho A. G. Zhang L. Song G.-L. Li S. L. Meana-Pañeda R. Fu B. Bowman J. M. Truhlar D. G. Direct diabatization and analytic representation of coupled potential energy surfaces and couplings for the reactive quenching of the excited 2Σ+ state of OH by molecular hydrogen. J. Chem. Phys. 2019;151:104311. doi: 10.1063/1.5111547. PubMed DOI

Guan Y. Guo H. Yarkony D. R. Extending the Representation of Multistate Coupled Potential Energy Surfaces to Include Properties Operators Using Neural Networks: Application to the 1,21A States of Ammonia. J. Chem. Theory Comput. 2020;16:302–313. doi: 10.1021/acs.jctc.9b00898. PubMed DOI

Mazouin B. Schöpfer A. A. von Lilienfeld O. A. Selected machine learning of HOMO-LUMO gaps with improved data-efficiency. Mater. Adv. 2022;3:8306–8316. doi: 10.1039/D2MA00742H. PubMed DOI PMC

Yarkony D. R. Conical Intersections: The New Conventional Wisdom. J. Phys. Chem. A. 2001;105:6277–6293. doi: 10.1021/jp003731u. DOI

Smith F. T. Diabatic and Adiabatic Representations for Atomic Collision Problems. Phys. Rev. 1969;179:111–123. doi: 10.1103/PhysRev.179.111. DOI

Mead C. A. Truhlar D. G. Conditions for the definition of a strictly diabatic electronic basis for molecular systems. J. Chem. Phys. 1982;77:6090–6098. doi: 10.1063/1.443853. DOI

Hoyer C. E. Xu X. Ma D. Gagliardi L. Truhlar D. G. Diabatization based on the dipole and quadrupole: The DQ method. J. Chem. Phys. 2014;141:114104. doi: 10.1063/1.4894472. PubMed DOI

Kleier D. A. Halgren T. A. Hall J. H. Lipscomb W. N. Localized molecular orbitals for polyatomic molecules. I. a comparison of the Edmiston-Ruedenberg and Boys localization methods. J. Chem. Phys. 1974;61:3905–3919. doi: 10.1063/1.1681683. DOI

Werner H. J. Meyer W. MCSCF study of the avoided curve crossing of the two lowest 1Σ+ states of LiF. J. Chem. Phys. 1981;74:5802–5807. doi: 10.1063/1.440893. DOI

Karman T. Van Der Avoird A. Groenenboom G. C. Communication: Multiple-property-based diabatization for open-shell van der Waals molecules. J. Chem. Phys. 2016;144:121101. doi: 10.1063/1.4944744. PubMed DOI

Sun Q. Berkelbach T. C. Blunt N. S. Booth G. H. Guo S. Li Z. Liu J. McClain J. D. Sayfutyarova E. R. Sharma S. Wouters S. Chan G. K. PySCF: the Python-based simulations of chemistry framework. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2018;8:e1340.

Sun Q. Zhang X. Banerjee S. Bao P. Barbry M. Blunt N. S. Bogdanov N. A. Booth G. H. Chen J. Cui Z. H. Eriksen J. J. Gao Y. Guo S. Hermann J. Hermes M. R. Koh K. Koval P. Lehtola S. Li Z. Liu J. Mardirossian N. McClain J. D. Motta M. Mussard B. Pham H. Q. Pulkin A. Purwanto W. Robinson P. J. Ronca E. Sayfutyarova E. R. Scheurer M. Schurkus H. F. Smith J. E. Sun C. Sun S. N. Upadhyay S. Wagner L. K. Wang X. White A. Whitfield J. D. Williamson M. J. Wouters S. Yang J. Yu J. M. Zhu T. Berkelbach T. C. Sharma S. Sokolov A. Y. Chan G. K. L. Recent developments in the PySCF program package. J. Chem. Phys. 2020;153:024109. doi: 10.1063/5.0006074. PubMed DOI

Crouse D. F. On implementing 2D rectangular assignment algorithms. IEEE Trans. Aerosp. Electron. Syst. 2016;52:1679–1696.

Virtanen P. Gommers R. Oliphant T. E. Haberland M. Reddy T. Cournapeau D. Burovski E. Peterson P. Weckesser W. Bright J. van der Walt S. J. Brett M. Wilson J. Millman K. J. Mayorov N. Nelson A. R. J. Jones E. Kern R. Larson E. Carey C. J. Polat I. Feng Y. Moore E. W. VanderPlas J. Laxalde D. Perktold J. Cimrman R. Henriksen I. Quintero E. A. Harris C. R. Archibald A. M. Ribeiro A. H. Pedregosa F. van Mulbregt P. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat. Methods. 2020;17:261–272. doi: 10.1038/s41592-019-0686-2. PubMed DOI PMC

Tan P.-N., Steinbach M. and Kumar V., Introduction to Data Mining, Pearson Education Limited, Harlow, 2014

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. doi: 10.1021/acs.jpclett.0c00527. PubMed DOI PMC

Löwdin P. O. Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction. Phys. Rev. 1955;97:1474. doi: 10.1103/PhysRev.97.1474. DOI

Plasser F. Ruckenbauer M. Mai S. Oppel M. Marquetand P. González L. Efficient and Flexible Computation of Many-Electron Wave Function Overlaps. J. Chem. Theory Comput. 2016;12:1207–1219. doi: 10.1021/acs.jctc.5b01148. PubMed DOI PMC

Rupp M. Machine learning for quantum mechanics in a nutshell. Int. J. Quantum Chem. 2015;115:1058–1073. doi: 10.1002/qua.24954. DOI

Rupp M. von Lilienfeld O. A. Burke K. Guest Editorial: Special Topic on Data-Enabled Theoretical Chemistry. J. Chem. Phys. 2018;148:241401. doi: 10.1063/1.5043213. PubMed DOI

Schölkopf B. and Smola A. J., Learning with kernels: support vector machines, regularization, optimization, and beyond, MIT Press, Cambridge, 2002

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. doi: 10.1021/ct400195d. PubMed DOI

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. doi: 10.1021/acs.jctc.7b00577. PubMed DOI

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. doi: 10.1063/1.5020710. PubMed DOI

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;115:1084–1093. doi: 10.1002/qua.24912. DOI

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. doi: 10.1063/1.4964627. PubMed DOI

Longuet-Higgins H. C. The symmetry groups of non-rigid molecules. Mol. Phys. 1963;6:445–460. doi: 10.1080/00268976300100501. DOI

Keating S. P. Mead C. A. Conical intersections in a system of four identical nuclei. J. Chem. Phys. 1985;82:5102–5117. doi: 10.1063/1.448633. DOI

Guan Y. Xie C. Guo H. Yarkony D. R. Neural Network Based Quasi-diabatic Representation for S0 and S1 States of Formaldehyde. J. Phys. Chem. A. 2020;124:10132–10142. PubMed

Williams D. M. Eisfeld W. Complete Nuclear Permutation Inversion Invariant Artificial Neural Network (CNPI-ANN) Diabatization for the Accurate Treatment of Vibronic Coupling Problems. J. Phys. Chem. A. 2020;124:7608–7621. doi: 10.1021/acs.jpca.0c05991. PubMed DOI

Yin Z. Braams B. J. Guan Y. Fu B. Zhang D. H. A fundamental invariant-neural network representation of quasi-diabatic Hamiltonians for the two lowest states of H3. Phys. Chem. Chem. Phys. 2021;23:1082–1091. doi: 10.1039/D0CP05047D. PubMed DOI

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. doi: 10.1063/1.4989536. PubMed DOI

Dral P. O. MLatom: A program package for quantum chemical research assisted by machine learning. J. Comput. Chem. 2019;40:2339–2347. doi: 10.1002/jcc.26004. PubMed DOI

Guan Y. Xie C. Guo H. Yarkony D. R. Enabling a unified description of both internal conversion and intersystem crossing in formaldehyde: A global coupled Quasi-Diabatic hamiltonian for its S0, S1, and T1 states. J. Chem. Theory Comput. 2021;17:4157–4168. doi: 10.1021/acs.jctc.1c00370. PubMed DOI

Bartók A. P. Csányi G. Gaussian approximation potentials: A brief tutorial introduction. Int. J. Quantum Chem. 2015;115:1051–1057. doi: 10.1002/qua.24927. DOI

Frisch M. J., Trucks G. W., Schlegel H. B., Scuseria G. E., Robb M. A., Cheeseman J. R., Scalmani G., Barone V., Mennucci B., Petersson G. A., Nakatsuji H., Caricato M., Li X., Hratchian H. P., Izmaylov A. F., Bloino J., Zheng G., Sonnenberg J. L., Hada M., Ehara M., Toyota K., Fukuda R., Hasegawa J., Ishida M., Nakajima T., Honda Y., Kitao O., Nakai H., Vreven T., Montgomery Jr. J. A., Peralta J. E., Ogliaro F., Bearpark M., Heyd J. J., Brothers E., Kudin K. N., Staroverov V. N., Keith T., Kobayashi R., Normand J., Raghavachari K., Rendell A., Burant J. C., Iyengar S. S., Tomasi J., Cossi M., Rega N., Millam J. M., Klene M., Knox J. E., Cross J. B., Bakken V., Adamo C., Jaramillo J., Gomperts R., Stratmann R. E., Yazyev O., Austin A. J., Cammi R., Pomelli C., Ochterski J. W., Martin R. L., Morokuma K., Zakrzewski V. G., Voth G. A., Salvador P., Dannenberg J. J., Dapprich S., Daniels A. D., Farkas O., Foresman J. B., Ortiz J. V., Cioslowski J. and Fox D. J., Gaussian 09, Revision D.01, 2013

Hillery M. O'Connell R. Scully M. Wigner E. Distribution functions in physics: Fundamentals. Phys. Rep. 1984;106:121–167. doi: 10.1016/0370-1573(84)90160-1. DOI

Zobel J. P. Nogueira J. J. González L. Finite-temperature Wigner phase-space sampling and temperature effects on the excited-state dynamics of 2-nitronaphthalene. Phys. Chem. Chem. Phys. 2019;21:13906–13915. doi: 10.1039/C8CP03273D. PubMed DOI

Mead A. Review of the Development of Multidimensional Scaling Methods. J. R. Stat. Soc. Ser. D. 1992;41:27–39. doi: 10.2307/2348634. DOI

Sršeň Š. Sita J. Slavíček P. Ladányi V. Heger D. Limits of the Nuclear Ensemble Method for Electronic Spectra Simulations: Temperature Dependence of the (E)-Azobenzene Spectrum. J. Chem. Theory Comput. 2020;16:6428–6438. doi: 10.1021/acs.jctc.0c00579. PubMed DOI

Crespo-Otero R. Barbatti M. Spectrum simulation and decomposition with nuclear ensemble: formal derivation and application to benzene, furan and 2-phenylfuran. Theor. Chem. Acc. 2012;131:1237.

Xue B. X. Barbatti M. Dral P. O. Machine Learning for Absorption Cross Sections. J. Phys. Chem. A. 2020;124:7199–7210. doi: 10.1021/acs.jpca.0c05310. PubMed DOI PMC