Abstract
Deep learning methods outperform human capabilities in pattern recognition and data processing problems and now have an increasingly important role in scientific discovery. A key application of machine learning in molecular science is to learn potential energy surfaces or force fields from ab initio solutions of the electronic Schrödinger equation using data sets obtained with density functional theory, coupled cluster or other quantum chemistry (QC) methods. In this Review, we discuss a complementary approach using machine learning to aid the direct solution of QC problems from first principles. Specifically, we focus on quantum Monte Carlo methods that use neural-network ansatzes to solve the electronic Schrödinger equation, in first and second quantization, computing ground and excited states and generalizing over multiple nuclear configurations. Although still at their infancy, these methods can already generate virtually exact solutions of the electronic Schrödinger equation for small systems and rival advanced conventional QC methods for systems with up to a few dozen electrons.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
$119.00 per year
only $9.92 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Carleo, G. et al. Machine learning and the physical sciences. Rev. Mod. Phys. 91, 045002 (2019).
Jumper, J. et al. Highly accurate protein structure prediction with AlphaFold. Nature 596, 583–589 (2021).
Deringer, V. L. et al. Origins of structural and electronic transitions in disordered silicon. Nature 589, 59–64 (2021).
Noé, F., Olsson, S., Köhler, J. & Wu, H. Boltzmann generators: sampling equilibrium states of many-body systems with deep learning. Science 365, eaaw1147 (2019).
Huang, B. & von Lilienfeld, O. A. Quantum machine learning using atom-in-molecule-based fragments selected on the fly. Nat. Chem. 12, 945–951 (2020).
Tkatchenko, A. Machine learning for chemical discovery. Nat. Commun. 11, 4125 (2020).
von Lilienfeld, O. A. & Burke, K. Retrospective on a decade of machine learning for chemical discovery. Nat. Commun. 11, 4895 (2020).
Noé, F., Tkatchenko, A., Müller, K.-R. & Clementi, C. Machine learning for molecular simulation. Annu. Rev. Phys. Chem. 71, 361–390 (2020).
Dral, P. O. Quantum chemistry in the age of machine learning. J. Phys. Chem. Lett. 11, 2336–2347 (2020).
Unke, O. T. et al. Machine learning force fields. Chem. Rev. 121, 10142–10186 (2021).
von Lilienfeld, O. A., Müller, K.-R. & Tkatchenko, A. Exploring chemical compound space with quantum-based machine learning. Nat. Rev. Chem. 4, 347–358 (2020).
Bian, Y. & Xie, X.-Q. Generative chemistry: drug discovery with deep learning generative models. J. Mol. Model. 27, 71 (2021).
Jones, R. O. Density functional theory: its origins, rise to prominence, and future. Rev. Mod. Phys. 87, 897–923 (2015).
Bartlett, R. J. & Musiał, M. Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 79, 291–352 (2007).
Deringer, V. L. et al. Gaussian process regression for materials and molecules. Chem. Rev. 121, 10073–10141 (2021).
Behler, J. Four generations of high-dimensional neural network potentials. Chem. Rev. 121, 10037–10072 (2021).
Musil, F. et al. Physics-inspired structural representations for molecules and materials. Chem. Rev. 121, 9759–9815 (2021).
Li, H., Collins, C., Tanha, M., Gordon, G. J. & Yaron, D. J. A density functional tight binding layer for deep learning of chemical Hamiltonians. J. Chem. Theory Comput. 14, 5764–5776 (2018).
Schütt, K. T., 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. 10, 5024 (2019).
Kirkpatrick, J. et al. Pushing the frontiers of density functionals by solving the fractional electron problem. Science 374, 1385–1389 (2021).
Chandrasekaran, A. et al. Solving the electronic structure problem with machine learning. Comput. Mater. 5, 1–7 (2019).
Welborn, M., Cheng, L. & Miller III, T. F. Transferability in machine learning for electronic structure via the molecular orbital basis. J. Chem. Theory Comput. 14, 4772–4779 (2018).
Nagai, R., Akashi, R. & Sugino, O. Completing density functional theory by machine learning hidden messages from molecules. Comput. Mater. 6, 1–8 (2020).
Gómez-Bombarelli, R. et al. Automatic chemical design using a data-driven continuous representation of molecules. ACS Cent. Sci. 4, 268–276 (2018).
Hoogeboom, E., Satorras, V. G., Vignac, C. & Welling, M. Equivariant diffusion for molecule generation in 3D. Preprint at http://arxiv.org/abs/2203.17003 (2022).
Torlai, G. et al. Neural-network quantum state tomography. Nat. Phys. 14, 447–450 (2018).
Sutton, R. S. & Barto, A. G. Reinforcement Learning: An Introduction (MIT Press, 2018).
Tesauro, G. TD-Gammon, a self-teaching backgammon program, achieves master-level play. Neural Comput. 6, 215–219 (1994).
Mnih, V. et al. Human-level control through deep reinforcement learning. Nature 518, 529–533 (2015).
Silver, D. et al. Mastering the game of Go with deep neural networks and tree search. Nature 529, 484–489 (2016).
Degrave, J. et al. Magnetic control of tokamak plasmas through deep reinforcement learning. Nature 602, 414–419 (2022).
Heinrich, J., Lanctot, M. & Silver, D. Fictitious self-play in extensive-form games. In Proceedings of the 32nd International Conference on Machine Learning 805–813 (PMLR, 2015).
Silver, D. et al. A general reinforcement learning algorithm that masters chess, shogi, and Go through self-play. Science 362, 1140–1144 (2018).
Battaglia, S. Machine learning wavefunction. In Quantum Chemistry in the Age of Machine Learning 577–616 (Elsevier, 2023).
Manzhos, S. Machine learning for the solution of the Schrödinger equation. Mach. Learn. Sci. Techn. 1, 013002 (2020).
Piela, L. Ideas of Quantum Chemistry 2nd edn (Elsevier, 2014).
Foulkes, W. M. C., Mitas, L., Needs, R. J. & Rajagopal, G. Quantum Monte Carlo simulations of solids. Rev. Mod. Phys. 73, 33–83 (2001).
Bajdich, M., Mitas, L., Drobný, G., Wagner, L. K. & Schmidt, K. E. Pfaffian pairing wave functions in electronic-structure quantum Monte Carlo Simulations. Phys. Rev. Lett. 96, 130201 (2006).
Han, J., Zhang, L. & Weinan, E. Solving many-electron Schrödinger equation using deep neural networks. J. Comput. Phys. 399, 108929 (2019).
Acevedo, A., Curry, M., Joshi, S. H., Leroux, B. & Malaya, N. Vandermonde wave function ansatz for improved variational Monte Carlo. In 2020 IEEE/ACM Fourth Workshop on Deep Learning on Supercomputers (DLS) 40–47 (IEEE, 2020).
Szabo, A. & Ostlund, N. S. Modern Quantum Chemistry (Dover Publications, 1996).
Becca, F. & Sorella, S. Quantum Monte Carlo Approaches for Correlated Systems 1st edn (Cambridge Univ. Press, 2017).
Teale, A. M. et al. DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science. Phys. Chem. Chem. Phys. 24, 28700–28781 (2022).
Chan, G. K.-L. & Sharma, S. The density matrix renormalization group in quantum chemistry. Annu. Rev. Phys. Chem. 62, 465 (2011).
Huron, B., Malrieu, J. P. & Rancurel, P. Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions. J. Chem. Phys. 58, 5745–5759 (1973).
Booth, G. H., Thom, A. J. W. & Alavi, A. Fermion Monte Carlo without fixed nodes: a game of life, death, and annihilation in Slater determinant space. J. Chem. Phys. 131, 054106 (2009).
Olsen, J. The CASSCF method: a perspective and commentary: CASSCF Method. Int. J. Quantum Chem. 111, 3267–3272 (2011).
Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–606 (2017).
Saito, H. Solving the Bose–Hubbard model with machine learning. J. Phys. Soc. Jpn. 86, 093001 (2017).
Nomura, Y., Darmawan, A. S., Yamaji, Y. & Imada, M. Restricted Boltzmann machine learning for solving strongly correlated quantum systems. Phys. Rev. B 96, 205152 (2017).
Adams, C., Carleo, G., Lovato, A. & Rocco, N. Variational Monte Carlo calculations of A≤4 nuclei with an artificial neural-network correlator ansatz. Phys. Rev. Lett. 127, 022502 (2021).
Astrakhantsev, N. et al. Broken-symmetry ground states of the Heisenberg model on the pyrochlore lattice. Phys. Rev. X 11, 041021 (2021).
Perronnin, F., Liu, Y., Sánchez, J. & Poirier, H. Large-scale image retrieval with compressed fisher vectors. In 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition 3384–3391 (IEEE, 2010).
Krizhevsky, A., Sutskever, I. & Hinton, G. E. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems Vol. 25, 1097–1105 (Curran Associates, 2012).
Schmidhuber, J. Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015).
LeCun, Y., Bengio, Y. & Hinton, G. Deep learning. Nature 521, 436–444 (2015).
McCulloch, W. S. & Pitts, W. A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 5, 115–133 (1943).
Rosenblatt, F. The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev. 65, 386 (1958).
Hornik, K., Stinchcombe, M. & White, H. Multilayer feedforward networks are universal approximators. Neural Netw. 2, 359–366 (1989).
Werbos, P. Beyond regression: new tools for prediction and analysis in the behavioral sciences. PhD thesis. Harvard Univ. (1974).
Linnainmaa, S. The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors. Master’s thesis (in Finnish), Univ. Helsinki (1970).
Linnainmaa, S. Taylor expansion of the accumulated rounding error. BIT Numer. Math. 16, 146–160 (1976).
Rumelhart, D. E., Hinton, G. E. & Williams, R. J. Learning representations by back-propagating errors. Nature 323, 533–536 (1986).
Glorot, X. & Bengio, Y. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the 13th International Conference on Artificial Intelligence and Statistics 249–256 (PMLR, 2010).
Hooker, S. The hardware lottery. Commun. ACM 64, 58–65 (2021).
Dauphin, Y. et al. Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. In Advances in Neural Information Processing Systems Vol. 27, 2933–2941 (Curran Associates, 2014).
Choromanska, A., Henaff, M., Mathieu, M., Arous, G. B. & LeCun, Y. The loss surfaces of multilayer networks. In Proceedings of the 18th International Conference on Artificial Intelligence and Statistics 192–204 (PMLR, 2015).
Bottou, L. & Bousquet, O. Learning using large datasets. In Mining Massive Data Sets for Security 15–26 (IOS Press, 2008).
Bottou, L. & Bousquet, O. The tradeoffs of large-scale learning. In Optimization for Machine Learning, 351–368 (MIT Press, 2011).
Russakovsky, O. et al. ImageNet large scale visual recognition challenge. Int. J. Comput. Vis. 115, 211–252 (2015).
Abadi, M. et al. TensorFlow: a system for large-scale machine learning. In Proceedings of the 12th USENIX Symposium on Operating Systems Design and Implementation 265–283 (USENIX Association, 2016).
Paszke, A. et al. Automatic differentiation in PyTorch. In NIPS Workshop on Automatic Differentiation (2017).
Bradbury, J. et al. JAX: Composable Transformations of Python+ NumPy Programs (GitHub, 2018); https://github.com/google/jax.
Goodfellow, I. J., Shlens, J. & Szegedy, C. Explaining and harnessing adversarial examples. Third International Conference on Learning Representations (ICLR) (ICLR, 2015).
LeCun, Y., Bottou, L., Bengio, Y. & Haffner, P. Gradient-based learning applied to document recognition. Proc. IEEE 86, 2278–2324 (1998).
Shawe-Taylor, J. Building symmetries into feedforward networks. In 1989 First IEE International Conference on Artificial Neural Networks 158–162 (IET, 1989).
Vaswani, A. et al. Attention is all you need. In Advances in Neural Information Processing Systems Vol. 30, 5998–6008 (Curran Associates, 2017).
Schütt, K. T., Sauceda, H. E., Kindermans, P.-J., Tkatchenko, A. & Müller, K.-R. SchNet — a deep learning architecture for molecules and materials. J. Chem. Phys. 148, 241722 (2018).
Bronstein, M. M., Bruna, J., Cohen, T. & Veličković, P. Geometric deep learning: grids, groups, graphs, geodesics, and gauges. Preprint at http://arxiv.org/abs/2104.13478 (2021).
Hinton, G. E. & Salakhutdinov, R. R. Reducing the dimensionality of data with neural networks. Science 313, 504–507 (2006).
Kingma, D. P. & Welling, M. Auto-encoding variational Bayes. Preprint at http://arxiv.org/abs/1312.6114 (2013).
Goodfellow, I. et al. Generative adversarial nets. In Advances in Neural Information Processing Systems Vol. 27, 2672–2680 (Curran Associates, 2014).
Rezende, D. & Mohamed, S. Variational inference with normalizing flows. In Proceedings of the 32nd International Conference on Machine Learning 1530–1538 (PMLR, 2015).
van den Oord, A. et al. WaveNet: a generative model for raw audio. Preprint at http://arxiv.org/abs/1609.3499 (2016).
van den Oord, A. et al. Conditional image generation with PixelCNN decoders. In Advances in Neural Information Processing Systems Vol. 29, 4797–4805 (Curran Associates, 2016).
Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N. & Ganguli, S. Deep unsupervised learning using nonequilibrium thermodynamics. In Proceedings of the 32nd International Conference on Machine Learning 2256–2265 (PMLR, 2015).
Sharir, O., Levine, Y., Wies, N., Carleo, G. & Shashua, A. Deep autoregressive models for the efficient variational simulation of many-body quantum systems. Phys. Rev. Lett. 124, 020503 (2020).
Choo, K., Neupert, T. & Carleo, G. Two-dimensional frustrated J1-J2 model studied with neural network quantum states. Phys. Rev. B 100, 125124 (2019).
Hibat-Allah, M., Ganahl, M., Hayward, L. E., Melko, R. G. & Carrasquilla, J. Recurrent neural network wave functions. Phys. Rev. Res. 2, 023358 (2020).
Xie, H., null, L. Z. & Wang, L. Ab-initio study of interacting fermions at finite temperature with neural canonical transformation. J. Mach. Learn. 1, 38 (2022).
Behler, J. & Parrinello, M. Generalized neural-network representation of high-dimensional potential-energy surfaces. Phys. Rev. Lett. 98, 146401 (2007).
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. 108, 058301 (2012).
Bartók, A. P., Kondor, R. & Csányi, G. On representing chemical environments. Phys. Rev. B 87, 184115 (2013).
Schütt, K. T., Arbabzadah, F., Chmiela, S., Müller, K. R. & Tkatchenko, A. Quantum-chemical insights from deep tensor neural networks. Nat. Commun. 8, 13890 (2017).
Thomas, N. et al. Tensor field networks: rotation- and translation-equivariant neural networks for 3D point clouds. Preprint at https://arxiv.org/abs/1802.08219 (2018).
Schütt, K. T., Unke, O. T. & Gastegger, M. Equivariant message passing for the prediction of tensorial properties and molecular spectra. In Proceedings of the 38th International Conference on Machine Learning 9377–9388 (PMLR, 2021).
Miller, B. K., Geiger, M., Smidt, T. E. & Noé, F. Relevance of rotationally equivariant convolutions for predicting molecular properties. Preprint at https://arxiv.org/abs/2008.08461 (2020).
Geiger, M. & Smidt, T. e3nn: Euclidean neural networks. Preprint at http://arxiv.org/abs/2207.09453 (2022).
Batzner, S. E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nat. Commun. 13, 2453 (2022).
Batatia, I., Kovács, D. P., Simm, G. N., Ortner, C. & Csányi, G. MACE: higher order equivariant message passing neural networks for fast and accurate force fields. In Advances in Neural Information Processing Systems Vol. 35, 11423–11436 (Curran Associates, 2022).
Huang, X., Braams, B. J. & Bowman, J. M. Ab initio potential energy and dipole moment surfaces for H5O2+. J. Chem. Phys. 122, 44308 (2005).
Drautz, R. Atomic cluster expansion for accurate and transferable interatomic potentials. Phys. Rev. B 99, 014104 (2019).
Allen, A. E. A., Dusson, G., Ortner, C. & Csányi, G. Atomic permutationally invariant polynomials for fitting molecular force fields. Mach. Learn.: Sci. Tech. 2, 025017 (2021).
Benali, A. et al. Toward a systematic improvement of the fixed-node approximation in diffusion Monte Carlo for solids — A case study in diamond. J. Chem. Phys. 153, 184111 (2020).
Stokes, J., Izaac, J., Killoran, N. & Carleo, G. Quantum natural gradient. Quantum 4, 269 (2020).
Feynman, R. P. & Cohen, M. Energy spectrum of the excitations in liquid helium. Phys. Rev. 102, 1189–1204 (1956).
Kwon, Y., Ceperley, D. M. & Martin, R. M. Effects of three-body and backflow correlations in the two-dimensional electron gas. Phys. Rev. B 48, 12037–12046 (1993).
Tocchio, L. F., Becca, F., Parola, A. & Sorella, S. Role of backflow correlations for the nonmagnetic phase of the t–t’ Hubbard model. Phys. Rev. B 78, 041101 (2008).
Luo, D. & Clark, B. K. Backflow transformations via neural networks for quantum many-body wave functions. Phys. Rev. Lett. 122, 226401 (2019).
Robledo Moreno, J., Carleo, G., Georges, A. & Stokes, J. Fermionic wave functions from neural-network constrained hidden states. Proc. Natl Acad. Sci. USA 119, e2122059119 (2022).
Lovato, A., Adams, C., Carleo, G. & Rocco, N. Hidden-nucleons neural-network quantum states for the nuclear many-body problem. Phys. Rev. Res. 4, 043178 (2022).
Yang, Y. & Zhao, P. Deep-neural-network approach to solving the ab initio nuclear structure problem. Phys. Rev. C 107, 034320 (2023).
Taddei, M., Ruggeri, M., Moroni, S. & Holzmann, M. Iterative backflow renormalization procedure for many-body ground-state wave functions of strongly interacting normal Fermi liquids. Phys. Rev. B 91, 115106 (2015).
Ruggeri, M., Moroni, S. & Holzmann, M. Nonlinear network description for many-body quantum systems in continuous space. Phys. Rev. Lett. 120, 205302 (2018).
Chakravorty, S. J., Gwaltney, S. R. & Davidson, E. R. Ground-state correlation energies for atomic ions with to 18 electrons. Phys. Rev. A 44, 7071 (1991).
Hermann, J., Schätzle, Z. & Noé, F. Deep-neural-network solution of the electronic Schrödinger equation. Nat. Chem. 12, 891–897 (2020).
Ma, A., Towler, M. D., Drummond, N. D. & Needs, R. J. Scheme for adding electron–nucleus cusps to Gaussian orbitals. J. Chem. Phys. 122, 224322 (2005).
Schätzle, Z., Hermann, J. & Noé, F. Convergence to the fixed-node limit in deep variational Monte Carlo. J. Chem. Phys. 154, 124108 (2021).
Pfau, D., Spencer, J. S., Matthews, A. G. D. G. & Foulkes, W. M. C. Ab initio solution of the many-electron Schrödinger equation with deep neural networks. Phys. Rev. Res. 2, 033429 (2020).
Kingma, D. P. & Ba, J. Adam: a method for stochastic optimization. Preprint at http://arxiv.org/abs/1412.6980 (2015).
Martens, J. & Grosse, R. Optimizing neural networks with kronecker-factored approximate curvature. In Proceedings of the 32nd International Conference on Machine Learning 2408–2417 (PMLR, 2015).
Gerard, L., Scherbela, M., Marquetand, P. & Grohs, P. Gold-standard solutions to the Schrödinger equation using deep learning: how much physics do we need? In Advances in Neural Information Processing Systems, Vol. 35, 10282–10294 (Curran Associates, 2022).
von Glehn, I., Spencer, J. S. & Pfau, D. A self-attention ansatz for ab-initio quantum chemistry. In International Conference on Learning Representations (ICLR 2023) (OpenReview, 2023).
Scherbela, M., Reisenhofer, R., Gerard, L., Marquetand, P. & Grohs, P. Solving the electronic Schrödinger equation for multiple nuclear geometries with weight-sharing deep neural networks. Nat. Comput. Sci. 2, 331 (2022).
Yang, L., Hu, W. & Li, L. Scalable variational Monte Carlo with graph neural ansatz. Preprint at http://arxiv.org/abs/2011.12453 (2020).
Gao, N. & Günnemann, S. Ab-initio potential energy surfaces by pairing GNNs with neural wave functions. In International Conference on Learning Representations (ICLR 2022) (OpenReview, 2022).
Gao, N. & Günnemann, S. Sampling-free inference for ab-initio potential energy surface networks. In International Conference on Learning Representations (ICLR 2023) (OpenReview, 2023).
Feynman, R. P. Forces in molecules. Phys. Rev. 56, 340 (1939).
Assaraf, R. & Caffarel, M. Zero-variance zero-bias principle for observables in quantum Monte Carlo: application to forces. J. Chem. Phys. 119, 10536–10552 (2003).
Umrigar, C. Two aspects of quantum Monte Carlo: determination of accurate wavefunctions and determination of potential energy surfaces of molecules. Int. J. Quantum Chem. 36, 217–230 (1989).
Qian, Y., Fu, W., Ren, W. & Chen, J. Interatomic force from neural network based variational quantum Monte Carlo. J. Chem. Phys. 157, 164104 (2022).
Pescia, G., Han, J., Lovato, A., Lu, J. & Carleo, G. Neural-network quantum states for periodic systems in continuous space. Phys. Rev. Res. 4, 023138 (2022).
Wilson, M. et al. Neural network ansatz for periodic wave functions and the homogeneous electron gas. Phys. Rev. B 107, 235139 (2023).
Cassella, G. et al. Discovering quantum phase transitions with fermionic neural networks. Phys. Rev. Lett. 130, 036401 (2023).
Li, X., Li, Z. & Chen, J. Ab initio calculation of real solids via neural network ansatz. Nat. Commun. 13, 7895 (2022).
Li, X., Fan, C., Ren, W. & Chen, J. Fermionic neural network with effective core potential. Phys. Rev. Res. 4, 013021 (2022).
Needs, R. J., Towler, M. D., Drummond, N. D., López Ríos, P. & Trail, J. R. Variational and diffusion quantum Monte Carlo calculations with the CASINO code. J. Chem. Phys. 152, 154106 (2020).
Shi, H. & Zhang, S. Some recent developments in auxiliary-field quantum Monte Carlo for real materials. J. Chem. Phys. 154, 024107 (2021).
Wilson, M., Gao, N., Wudarski, F., Rieffel, E. & Tubman, N. M. Simulations of state-of-the-art fermionic neural network wave functions with diffusion Monte Carlo. Preprint at http://arxiv.org/abs/2103.12570 (2021).
Ren, W., Fu, W. & Chen, J. Towards the ground state of molecules via diffusion Monte Carlo on neural networks. Nat. Commun. 14, 1860 (2023).
Schautz, F. & Filippi, C. Optimized Jastrow–Slater wave functions for ground and excited states: application to the lowest states of ethene. J. Chem. Phys. 120, 10931 (2004).
Dash, M., Feldt, J., Moroni, S., Scemama, A. & Filippi, C. Excited states with selected configuration interaction-quantum Monte Carlo: chemically accurate excitation energies and geometries. J. Chem. Theory Comput. 15, 4896 (2019).
Zhao, L. & Neuscamman, E. An efficient variational principle for the direct optimization of excited states. J. Chem. Theory Comput. 12, 3436 (2016).
Pathak, S., Busemeyer, B., Rodrigues, J. N. B. & Wagner, L. K. Excited states in variational Monte Carlo using a penalty method. J. Chem. Phys. 154, 034101 (2021).
Entwistle, M., Schätzle, Z., Erdman, P. A., Hermann, J. & Noé, F. Electronic excited states in deep variational Monte Carlo. Nat. Commun. 14, 274 (2023).
Choo, K., Carleo, G., Regnault, N. & Neupert, T. Symmetries and many-body excitations with neural-network quantum states. Phys. Rev. Lett. 121, 167204 (2018).
Cuzzocrea, A., Scemama, A., Briels, W. J., Moroni, S. & Filippi, C. Variational principles in quantum Monte Carlo: the troubled story of variance minimization. J. Chem. Theory Comput. 16, 4203 (2020).
Jordan, P. & Wigner, E. über das Paulische Äquivalenzverbot. Z. Phys. 47, 631 (1928).
Bravyi, S. & Kitaev, A. Fermionic quantum computation. Ann. Phys. 298, 210–226 (2002).
Sorella, S. Green function Monte Carlo with stochastic reconfiguration. Phys. Rev. Lett. 80, 4558–4561 (1998).
Choo, K., Mezzacapo, A. & Carleo, G. Fermionic neural-network states for ab-initio electronic structure. Nat. Commun. 11, 2368 (2020).
Yang, P.-J., Sugiyama, M., Tsuda, K. & Yanai, T. Artificial neural networks applied as molecular wave function solvers. J. Chem. Theory Comput. 16, 3513–3529 (2020).
Torlai, G., Mazzola, G., Carleo, G. & Mezzacapo, A. Precise measurement of quantum observables with neural-network estimators. Phys. Rev. Res. 2, 022060 (2020).
Iouchtchenko, D., Gonthier, J. F., Perdomo-Ortiz, A. & Melko, R. G. Neural network enhanced measurement efficiency for molecular groundstates. Mach. Learn. Sci. Technol. 4, 015016 (2023).
Glielmo, A., Rath, Y., Csányi, G., De Vita, A. & Booth, G. H. Gaussian process states: a data-driven representation of quantum many-body physics. Phys. Rev. X 10, 041026 (2020).
Del Re, G., Ladik, J. & Biczó, G. Self-consistent-field tight-binding treatment of polymers. I. Infinite three-dimensional case. Phys. Rev. 155, 997–1003 (1967).
Yoshioka, N., Mizukami, W. & Nori, F. Solving quasiparticle band spectra of real solids using neural-network quantum states. Commun. Phys. 4, 1–8 (2021).
Barrett, T. D., Malyshev, A. & Lvovsky, A. I. Autoregressive neural-network wavefunctions for ab initio quantum chemistry. Nat. Mach. Intell. 4, 351 (2022).
Zhao, T., Stokes, J. & Veerapaneni, S. Scalable neural quantum states architecture for quantum chemistry. Mach. Learn. Sci. Technol. 4, 025034 (2023).
Giner, E., Scemama, A. & Caffarel, M. Using perturbatively selected configuration interaction in quantum Monte Carlo calculations. Can. J. Chem. 91, 879–885 (2013).
Holmes, A. A., Tubman, N. M. & Umrigar, C. Heat–bath configuration interaction: an efficient selected configuration interaction algorithm inspired by heat-bath sampling. J. Chem. Theory Comput. 12, 3674–3680 (2016).
Sharma, S., Holmes, A. A., Jeanmairet, G., Alavi, A. & Umrigar, C. J. Semistochastic heat–bath configuration interaction method: selected configuration interaction with semistochastic perturbation theory. J. Chem. Theory Comput. 13, 1595–1604 (2017).
Greer, J. Monte Carlo configuration interaction. J. Comput. Phys. 146, 181–202 (1998).
Coe, J. P. Machine learning configuration interaction. J. Chem. Theory Comput. 14, 5739–5749 (2018).
Goings, J. J., Hu, H., Yang, C. & Li, X. Reinforcement learning configuration interaction. J. Chem. Theory Comput. 17, 5482–5491 (2021).
Pineda Flores, S. D. Chembot: a machine learning approach to selective configuration interaction. J. Chem. Theory Comput. 17, 4028 (2021).
Nooijen, M., Shamasundar, K. & Mukherjee, D. Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory. Mol. Phys. 103, 2277–2298 (2005).
Hutter, M. On representing (anti)symmetric functions. Preprint at http://arxiv.org/abs/2007.15298 (2020).
Neuscamman, E. The Jastrow antisymmetric geminal power in Hilbert space: theory, benchmarking, and application to a novel transition state. J. Chem. Phys. 139, 194105 (2013).
Sabzevari, I. & Sharma, S. Improved speed and scaling in orbital space variational Monte Carlo. J. Chem. Theory Comput. 14, 6276–6286 (2018).
Rubenstein, B. Introduction to the variational monte carlo method in quantum chemistry and physics. In Variational Methods in Molecular Modeling 285–313 (Springer, 2017).
Toulouse, J., Assaraf, R. & Umrigar, C. J. Introduction to the variational and diffusion Monte Carlo methods. In Advances in Quantum Chemistry Vol. 73, 285–314 (Elsevier, 2016).
Amari, S. Natural gradient works efficiently in learning. Neural Comput. 10, 251–276 (1998).
Ay, N., Jost, J., Lê, H. V. & Schwachhöfer, L. Information Geometry. No. 64 in a Series of Modern Surveys in Mathematics (Springer, 2017).
Spencer, J. S., Pfau, D., Botev, A. & Foulkes, W. M. C. Better, faster fermionic neural networks. Preprint at http://arxiv.org/abs/2011.07125 (2020).
Acknowledgements
The authors acknowledge funding from the German Ministry for Education and Research (Berlin Institute for the Foundations of Learning and Data, BIFOLD), the Berlin Mathematics Research Center MATH+ (AA1-6 and AA2-8) and European Commission (ERC CoG 772230 ScaleCell). G.C. is supported by the Swiss National Science Foundation under Grant No. 200021_200336, and by the NCCR MARVEL, a National Centre of Competence in Research, under Grant No. 205602. The authors are grateful to N. Yoshioka for providing us with the raw data of ref. 157. W.M.C.F. and his co-workers gratefully acknowledge PRACE for awarding them access to the JUWELS Booster supercomputer (https://apps.fz-juelich.de/jsc/hps/juwels/booster-overview.html); the HPC RIVR Consortium (https://www.hpc-rivr.si) and EuroHPC JU for providing computing resources on the Vega HPC system at the Institute of Information Science (https://www.izum.si) and the UK Engineering and Physical Sciences Research Council for providing computing resources at the Baskerville Tier 2 HPC service (https://www.baskerville.ac.uk). Baskerville was funded by the EPSRC and UKRI through the World Class Labs scheme (EP/T022221/1) and the Digital Research Infrastructure programme (EP/W032244/1) and is operated by Advanced Research Computing at the University of Birmingham.
Author information
Authors and Affiliations
Contributions
J.H., J.S. and K.C. contributed equally and all authors were part of the discussion, reviewing and editing of the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Reviews Chemistry thanks Pavlo Dral and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hermann, J., Spencer, J., Choo, K. et al. Ab initio quantum chemistry with neural-network wavefunctions. Nat Rev Chem 7, 692–709 (2023). https://doi.org/10.1038/s41570-023-00516-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41570-023-00516-8