{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T20:39:59Z","timestamp":1792010399537,"version":"4.4.0"},"reference-count":93,"publisher":"American Association for the Advancement of Science (AAAS)","issue":"6711","content-domain":{"domain":["www.science.org"],"crossmark-restriction":true},"short-container-title":["Science"],"published-print":{"date-parts":[[2024,8,23]]},"abstract":"<jats:p>We present an algorithm to estimate the excited states of a quantum system by variational Monte Carlo, which has no free parameters and requires no orthogonalization of the states, instead transforming the problem into that of finding the ground state of an expanded system. Arbitrary observables can be calculated, including off-diagonal expectations, such as the transition dipole moment. The method works particularly well with neural network ans\u00e4tze, and by combining this method with the FermiNet and Psiformer ans\u00e4tze, we can accurately recover excitation energies and oscillator strengths on a range of molecules. We achieve accurate vertical excitation energies on benzene-scale molecules, including challenging double excitations. Beyond the examples presented in this work, we expect that this technique will be of interest for atomic, nuclear, and condensed matter physics.<\/jats:p>","DOI":"10.1126\/science.adn0137","type":"journal-article","created":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T13:59:05Z","timestamp":1724335145000},"update-policy":"https:\/\/doi.org\/10.34133\/aaas_crossmark","source":"Crossref","is-referenced-by-count":74,"title":["Accurate computation of quantum excited states with neural networks"],"prefix":"10.1126","volume":"385","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0834-8653","authenticated-orcid":true,"given":"David","family":"Pfau","sequence":"first","affiliation":[{"name":"Google DeepMind, London N1C 4DJ, UK."},{"name":"Department of Physics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2936-0133","authenticated-orcid":true,"given":"Simon","family":"Axelrod","sequence":"additional","affiliation":[{"name":"Google DeepMind, London N1C 4DJ, UK."},{"name":"Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 01238, USA."},{"name":"Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 01239, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1635-1668","authenticated-orcid":true,"given":"Halvard","family":"Sutterud","sequence":"additional","affiliation":[{"name":"Department of Physics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-3102-196X","authenticated-orcid":true,"given":"Ingrid","family":"von Glehn","sequence":"additional","affiliation":[{"name":"Google DeepMind, London N1C 4DJ, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"James S.","family":"Spencer","sequence":"additional","affiliation":[{"name":"Google DeepMind, London N1C 4DJ, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"221","reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.447218"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature09346"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1021\/cr300503r"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature25483"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.chemrev.0c00749"},{"key":"e_1_3_2_7_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.aag2302"},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41570-023-00516-8"},{"key":"e_1_3_2_9_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.73.33"},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.87.1067"},{"key":"e_1_3_2_11_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.121.167204"},{"key":"e_1_3_2_12_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-022-35534-5"},{"key":"e_1_3_2_13_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.48.12037"},{"key":"e_1_3_2_14_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.77.115112"},{"key":"e_1_3_2_15_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.80.4558"},{"key":"e_1_3_2_16_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.2437215"},{"key":"e_1_3_2_17_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.123.036402"},{"key":"e_1_3_2_18_2","doi-asserted-by":"publisher","DOI":"10.1002\/wcms.1659"},{"key":"e_1_3_2_19_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.3220671"},{"key":"e_1_3_2_20_2","doi-asserted-by":"publisher","DOI":"10.1063\/5.0030949"},{"key":"e_1_3_2_21_2","doi-asserted-by":"publisher","DOI":"10.1088\/2516-1075\/ad38f8"},{"key":"e_1_3_2_22_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1752881"},{"key":"e_1_3_2_23_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.2786997"},{"key":"e_1_3_2_24_2","doi-asserted-by":"publisher","DOI":"10.1021\/ct900227j"},{"key":"e_1_3_2_25_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.0c00147"},{"key":"e_1_3_2_26_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.1c00212"},{"key":"e_1_3_2_27_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.226401"},{"key":"e_1_3_2_28_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.033429"},{"key":"e_1_3_2_29_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41557-020-0544-y"},{"key":"e_1_3_2_30_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.2768360"},{"key":"e_1_3_2_31_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10910-008-9355-x"},{"key":"e_1_3_2_32_2","volume-title":"Advances in Neural Information Processing Systems","author":"Bottou L.","year":"2007","unstructured":"L. Bottou, O. Bousquet, in Advances in Neural Information Processing Systems, J. Platt, D. Koller, Y. Singer, S. Roweis, Eds. (Curran Associates, Inc., 2007)."},{"key":"e_1_3_2_33_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.455398"},{"key":"e_1_3_2_34_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.87.043401"},{"key":"e_1_3_2_35_2","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9474(84)90127-1"},{"key":"e_1_3_2_36_2","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729586"},{"key":"e_1_3_2_37_2","volume-title":"Eleventh International Conference on Learning Representations","author":"von Glehn I.","year":"2023","unstructured":"I. von Glehn, J. S. Spencer, D. Pfau, A Self-Attention Ansatz for Ab-initio Quantum Chemistry, Eleventh International Conference on Learning Representations (ICLR, 2023)."},{"key":"e_1_3_2_38_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1800011"},{"key":"e_1_3_2_39_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.0c01111"},{"key":"e_1_3_2_40_2","doi-asserted-by":"publisher","DOI":"10.1002\/wcms.1517"},{"key":"e_1_3_2_41_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.8b00406"},{"key":"e_1_3_2_42_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.8b01205"},{"key":"e_1_3_2_43_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.9b01216"},{"key":"e_1_3_2_44_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.0c00227"},{"key":"e_1_3_2_45_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.1c00226"},{"key":"e_1_3_2_46_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jpca.1c08524"},{"key":"e_1_3_2_47_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.2c00416"},{"key":"e_1_3_2_48_2","doi-asserted-by":"publisher","DOI":"10.1088\/0031-8949\/1984\/T8\/020"},{"key":"e_1_3_2_49_2","doi-asserted-by":"publisher","DOI":"10.1016\/1010-6030(92)80001-C"},{"key":"e_1_3_2_50_2","doi-asserted-by":"publisher","DOI":"10.1525\/9780520331556"},{"key":"e_1_3_2_51_2","doi-asserted-by":"publisher","DOI":"10.1093\/mnras\/stw1820"},{"key":"e_1_3_2_52_2","doi-asserted-by":"publisher","DOI":"10.1038\/nchem.1263"},{"key":"e_1_3_2_53_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.4998614"},{"key":"e_1_3_2_54_2","doi-asserted-by":"publisher","DOI":"10.1086\/147485"},{"key":"e_1_3_2_55_2","doi-asserted-by":"publisher","DOI":"10.1086\/147486"},{"key":"e_1_3_2_56_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.477764"},{"key":"e_1_3_2_57_2","doi-asserted-by":"crossref","unstructured":"M. Barbatti R. Crespo-Otero in Density-Functional Methods for Excited States N. Ferr\u00e9 M. Filatov M. Huix-Rotllant Eds. vol. 368 of Topics in Current Chemistry (Springer 2014) pp. 415\u2013444.","DOI":"10.1007\/128_2014_605"},{"key":"e_1_3_2_58_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1807378"},{"key":"e_1_3_2_59_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.0c00372"},{"key":"e_1_3_2_60_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.2c00769"},{"key":"e_1_3_2_61_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.4c00410"},{"key":"e_1_3_2_62_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1670062"},{"key":"e_1_3_2_63_2","doi-asserted-by":"publisher","DOI":"10.1021\/ct300591z"},{"key":"e_1_3_2_64_2","doi-asserted-by":"publisher","DOI":"10.1016\/0009-2614(92)85419-B"},{"key":"e_1_3_2_65_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.479571"},{"key":"e_1_3_2_66_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jctc.1c01162"},{"key":"e_1_3_2_67_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1751-1097.1977.tb06918.x"},{"key":"e_1_3_2_68_2","volume-title":"Tenth International Conference on Learning Representations","author":"Gao N.","year":"2022","unstructured":"N. Gao, S. G\u00fcnnemann, Ab-initio Potential Energy Surfaces by Pairing GNNs with Neural Wave Functions, Tenth International Conference on Learning Representations (ICLR, 2022)."},{"key":"e_1_3_2_69_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-44216-9"},{"key":"e_1_3_2_70_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.98.075122"},{"key":"e_1_3_2_71_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevC.107.034320"},{"key":"e_1_3_2_72_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-022-35627-1"},{"key":"e_1_3_2_73_2","unstructured":"D. Pfau S. Axelrod H. Sutterud I. von Glehn J. S. Spencer Code for \u201cAccurate Computation of Quantum Excited States with Neural Networks \u201d Zenodo (2024); https:\/\/doi.org\/10.5281\/zenodo.11937084.10.5281\/zenodo.11937084"},{"key":"e_1_3_2_74_2","doi-asserted-by":"publisher","DOI":"10.1002\/wcms.1364"},{"key":"e_1_3_2_75_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.102.205122"},{"key":"e_1_3_2_76_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.124.020503"},{"key":"e_1_3_2_77_2","first-page":"10282","volume-title":"Advances in Neural Information Processing Systems","author":"Gerard L.","year":"2022","unstructured":"L. Gerard, M. Scherbela, P. Marquetand, P. Grohs, in Advances in Neural Information Processing Systems, S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, A. Oh, Eds. (Curran Associates, Inc., 2022), pp. 10282\u201310294."},{"key":"e_1_3_2_78_2","unstructured":"N. Gao S. G\u00fcnnemann Generalizing Neural Wave Functions. arXiv:2302.04168 [cs.LG] (2023)."},{"key":"e_1_3_2_79_2","doi-asserted-by":"crossref","unstructured":"G. Pescia J. Nys J. Kim A. Lovato G. Carleo Message-Passing Neural Quantum States for the Homogeneous Electron Gas. arXiv:2305.07240 [quant-ph] (2023).","DOI":"10.1103\/PhysRevB.110.035108"},{"key":"e_1_3_2_80_2","doi-asserted-by":"publisher","DOI":"10.1038\/s42256-024-00794-x"},{"key":"e_1_3_2_81_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.46.509"},{"key":"e_1_3_2_82_2","unstructured":"J. S. Spencer D. Pfau A. Botev W. M. C. Foulkes Better Faster Fermionic Neural Networks. arXiv:2011.07125 [physics.comp-ph] (2020)."},{"key":"e_1_3_2_83_2","volume-title":"Advances in Neural Information Processing Systems","author":"Vaswani A.","year":"2017","unstructured":"A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, I. Polosukhin, in Advances in Neural Information Processing Systems, I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, R. Garnett, Eds. (Curran Associates, Inc., 2017)."},{"key":"e_1_3_2_84_2","unstructured":"J. L. Ba J. R. Kiros G. E. Hinton Layer Normalization. arXiv:1607.06450 [stat.ML] (2016)."},{"key":"e_1_3_2_85_2","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2016.90"},{"key":"e_1_3_2_86_2","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jpca.2c01338"},{"key":"e_1_3_2_87_2","doi-asserted-by":"publisher","DOI":"10.1002\/wcms.1340"},{"key":"e_1_3_2_88_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.42.3503"},{"key":"e_1_3_2_89_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.5038135"},{"key":"e_1_3_2_90_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.4.013021"},{"key":"e_1_3_2_91_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.97.1474"},{"key":"e_1_3_2_92_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.468585"},{"key":"e_1_3_2_93_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.37.4950"},{"key":"e_1_3_2_94_2","unstructured":"N. Gao J. K\u00f6hler A. Foster folx - Forward Laplacian for JAX version 0.2.5 Github (2023); https:\/\/github.com\/microsoft\/folx."}],"container-title":["Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.science.org\/doi\/pdf\/10.1126\/science.adn0137","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T13:59:16Z","timestamp":1724335156000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.science.org\/doi\/10.1126\/science.adn0137"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,23]]},"references-count":93,"journal-issue":{"issue":"6711","published-print":{"date-parts":[[2024,8,23]]}},"alternative-id":["10.1126\/science.adn0137"],"URL":"https:\/\/doi.org\/10.1126\/science.adn0137","relation":{},"ISSN":["0036-8075","1095-9203"],"issn-type":[{"value":"0036-8075","type":"print"},{"value":"1095-9203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,23]]},"assertion":[{"value":"2023-11-17","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-05-17","order":1,"name":"revised","label":"Revised","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-06-21","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2024-08-23","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"eadn0137"}}