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Unified quantum framework for electrons and ions: The self-consistent harmonic approximation on a neural network curved manifold

Lorenzo Monacelli1,2,*, Antonio Siciliano2, and Nicola Marzari1

  • *Contact author: lorenzo.monacelli@uniroma1.it
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Phys. Rev. Research 7, 043277 – Published 9 December, 2025

DOI: https://doi.org/10.1103/7d6s-zz54

Abstract

The numerical solution of the many-body problem, which involves interacting electrons and ions, is a key challenge in condensed matter physics, chemistry, and materials science. Traditional methods to solve the multicomponent quantum Hamiltonian are usually specialized for one kind of particles—electrons or ions—and can suffer from a methodological gap when applied to the other ones. This work extends the self-consistent harmonic approximation, a proven successful technique for simulating quantum ions at finite temperatures in anharmonic crystals, to electrons. The approach minimizes the total free energy by optimizing an ansatz density matrix, solving a fermionic self-consistent harmonic Hamiltonian on a curved manifold parameterized through a neural network. This approach preserves an analytical expression for entropy, enabling the direct computation of free energies and phase diagrams of materials. By benchmarking this technique across several prototypical cases—a double-well potential, the hydrogen atom, and the H2 dissociation—we demonstrate that it can address both the ground- and excited-state properties of electronic systems, capture quantum tunneling and static electronic correlations, and thereby provide a unified quantum framework of electrons and atomic nuclei.

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References (59)

  1. N. Marzari, A. Ferretti, and C. Wolverton, Electronic-structure methods for materials design, Nat. Mater. 20, 736 (2021).
  2. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
  3. K. Burke and L. O. Wagner, DFT in a nutshell, Int. J. Quantum Chem. 113, 96 (2012).
  4. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
  5. L. Monacelli, I. Errea, M. Calandra, and F. Mauri, Black metal hydrogen above 360 GPa driven by proton quantum fluctuations, Nat. Phys. 17, 63 (2021).
  6. L. Monacelli, M. Casula, K. Nakano, S. Sorella, and F. Mauri, Quantum phase diagram of high-pressure hydrogen, Nat. Phys. 19, 845 (2023).
  7. I. Errea, M. Calandra, C. J. Pickard, J. R. Nelson, R. J. Needs, Y. Li, H. Liu, Y. Zhang, Y. Ma, and F. Mauri, Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfide system, Nature (London) 532, 81 (2016).
  8. I. Errea, F. Belli, L. Monacelli, A. Sanna, T. Koretsune, T. Tadano, R. Bianco, M. Calandra, R. Arita, F. Mauri, and J. A. Flores-Livas, Quantum crystal structure in the 250-Kelvin superconducting lanthanum hydride, Nature (London) 578, 66 (2020).
  9. J. A. Morrone and R. Car, Nuclear quantum effects in water, Phys. Rev. Lett. 101, 017801 (2008).
  10. M. Cherubini, L. Monacelli, and F. Mauri, The microscopic origin of the anomalous isotopic properties of ice relies on the strong quantum anharmonic regime of atomic vibration, J. Chem. Phys. 155, 184502 (2021).
  11. U. Ranieri, S. Di Cataldo, M. Rescigno, L. Monacelli, R. Gaal, M. Santoro, L. Andriambariarijaona, P. Parisiades, C. De Michele, and L. E. Bove, Observation of the most H2-dense filled ice under high pressure, Proc. Natl. Acad. Sci. USA 120, e2312665120 (2023).
  12. M. Cherubini, L. Monacelli, B. Yang, R. Car, M. Casula, and F. Mauri, Quantum effects in the H-bond symmetrization and in the thermodynamic properties of high pressure ice, Phys. Rev. B 110, 014112 (2024).
  13. L. Binci, P. Barone, and F. Mauri, First-principles theory of infrared vibrational spectroscopy of metals and semimetals: Application to graphite, Phys. Rev. B 103, 134304 (2021).
  14. G. Marchese, F. Macheda, L. Binci, M. Calandra, P. Barone, and F. Mauri, Born effective charges and vibrational spectra in superconducting and bad conducting metals, Nat. Phys. 20, 88 (2023).
  15. N. Girotto and D. Novko, Dynamical renormalization of electron-phonon coupling in conventional superconductors, Phys. Rev. B 107, 064310 (2023).
  16. C.-J. Tong, X. Cai, A.-Y. Zhu, L.-M. Liu, and O. V. Prezhdo, How hole injection accelerates both ion migration and nonradiative recombination in metal halide perovskites, J. Am. Chem. Soc. 144, 6604 (2022).
  17. Y. J. Uemura et al., Universal correlations between Tc and nsm* (carrier density over effective mass) in high-Tc cuprate superconductors, Phys. Rev. Lett. 62, 2317 (1989).
  18. J. C. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys. 93, 1061 (1990).
  19. P. Nijjar, J. Jankowska, and O. V. Prezhdo, Ehrenfest and classical path dynamics with decoherence and detailed balance, J. Chem. Phys. 150, 204124 (2019).
  20. C. F. Craig, W. R. Duncan, and O. V. Prezhdo, Trajectory surface hopping in the time-dependent Kohn-Sham approach for electron-nuclear dynamics, Phys. Rev. Lett. 95, 163001 (2005).
  21. L. Wang, A. Akimov, and O. V. Prezhdo, Recent progress in surface hopping: 2011-2015, J. Phys. Chem. Lett. 7, 2100 (2016).
  22. P. Shushkov, R. Li, and J. C. Tully, Ring polymer molecular dynamics with surface hopping, J. Chem. Phys. 137, 22A549 (2012).
  23. D. M. Ceperley, Path integrals in the theory of condensed helium, Rev. Mod. Phys. 67, 279 (1995).
  24. K. P. Driver and B. Militzer, All-electron path integral Monte Carlo simulations of warm dense matter: Application to water and carbon plasmas, Phys. Rev. Lett. 108, 115502 (2012).
  25. I. Errea, M. Calandra, and F. Mauri, Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmonic approximation: Application to platinum and palladium hydrides, Phys. Rev. B 89, 064302 (2014).
  26. L. Monacelli and F. Mauri, Time-dependent self-consistent harmonic approximation: Anharmonic nuclear quantum dynamics and time correlation functions, Phys. Rev. B 103, 104305 (2021).
  27. M. Miotto and L. Monacelli, Fast prediction of anharmonic vibrational spectra for complex organic molecules, npj Comput. Mater. 10, 240 (2024).
  28. M. Borinaga, P. Riego, A. Leonardo, M. Calandra, F. Mauri, A. Bergara, and I. Errea, Anharmonic enhancement of superconductivity in metallic molecular Cmca-4 hydrogen at high pressure: A first-principles study, J. Phys.: Condens. Matter 28, 494001 (2016).
  29. M. Borinaga, I. Errea, M. Calandra, F. Mauri, and A. Bergara, Anharmonic effects in atomic hydrogen: Superconductivity and lattice dynamical stability, Phys. Rev. B 93, 174308 (2016).
  30. L. Monacelli and N. Marzari, First-principles thermodynamics of CsSnI3, Chem. Mater. 35, 1702 (2023).
  31. J. S. Zhou, R. Bianco, L. Monacelli, I. Errea, F. Mauri, and M. Calandra, Theory of the thickness dependence of the charge density wave transition in 1t−TiTe2, 2D Mater. 7, 045032 (2020).
  32. R. Bianco, I. Errea, L. Monacelli, M. Calandra, and F. Mauri, Quantum enhancement of charge density wave in NbS2 in the two-dimensional limit, Nano Lett. 19, 3098 (2019).
  33. R. Bianco, L. Monacelli, M. Calandra, F. Mauri, and I. Errea, Weak dimensionality dependence and dominant role of ionic fluctuations in the charge-density-wave transition of NbSe2, Phys. Rev. Lett. 125, 106101 (2020).
  34. J. Diego, A. H. Said, S. K. Mahatha, R. Bianco, L. Monacelli, M. Calandra, F. Mauri, K. Rossnagel, I. Errea, and S. Blanco-Canosa, Van der Waals driven anharmonic melting of the 3D charge density wave in VSe2, Nat. Commun. 12, 598 (2021).
  35. A. Siciliano, L. Monacelli, and F. Mauri, Beyond Gaussian fluctuations of quantum anharmonic nuclei, Phys. Rev. B 110, 134111 (2024).
  36. A. Siciliano, L. Monacelli, and F. Mauri, Beyond Gaussian fluctuations of quantum anharmonic nuclei: The case of rotational degrees of freedom, Phys. Rev. B 110, 144101 (2024).
  37. L. Monacelli, R. Bianco, M. Cherubini, M. Calandra, I. Errea, and F. Mauri, The stochastic self-consistent harmonic approximation: Calculating vibrational properties of materials with full quantum and anharmonic effects, J. Phys.: Condens. Matter 33, 363001 (2021).
  38. W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Quantum Monte Carlo simulations of solids, Rev. Mod. Phys. 73, 33 (2001).
  39. U. Aseginolaza, R. Bianco, L. Monacelli, L. Paulatto, M. Calandra, F. Mauri, A. Bergara, and I. Errea, Phonon collapse and second-order phase transition in thermoelectric SnSe, Phys. Rev. Lett. 122, 075901 (2019).
  40. C. Verdi, L. Ranalli, C. Franchini, and G. Kresse, Quantum paraelectricity and structural phase transitions in strontium titanate beyond density functional theory, Phys. Rev. Mater. 7, L030801 (2023).
  41. D. Romanin, L. Monacelli, R. Bianco, I. Errea, F. Mauri, and M. Calandra, Dominant role of quantum anharmonicity in the stability and optical properties of infinite linear acetylenic carbon chains, J. Phys. Chem. Lett. 12, 10339 (2021).
  42. L. Ranalli, C. Verdi, L. Monacelli, G. Kresse, M. Calandra, and C. Franchini, Temperature-dependent anharmonic phonons in quantum paraelectric KTaO3 by first principles and machine-learned force fields, Adv. Quantum Technol. 6, 2200131 (2023).
  43. A. Pedrielli, P. E. Trevisanutto, L. Monacelli, G. Garberoglio, N. M. Pugno, and S. Taioli, Understanding anharmonic effects on hydrogen desorption characteristics of MgnH2 nanoclusters by ab initio trained deep neural network, Nanoscale 14, 5589 (2022).
  44. F. Gygi, Adaptive Riemannian metric for plane-wave electronic-structure calculations, Europhys. Lett. 19, 617 (1992).
  45. D. R. Hamann, Application of adaptive curvilinear coordinates to the electronic structure of solids, Phys. Rev. B 51, 7337 (1995).
  46. N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1 ed. (Brooks Cole, California, US, 1976).
  47. D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, arXiv:1412.6980.
  48. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Commun. Pure Appl. Math. 44, 375 (1991).
  49. A. Cuzzocrea, A. Scemama, W. J. Briels, S. Moroni, and C. Filippi, Variational principles in quantum Monte Carlo: The troubled story of variance minimization, J. Chem. Theory Comput. 16, 4203 (2020).
  50. M. Dash, J. Feldt, S. Moroni, A. Scemama, and C. Filippi, Excited states with selected configuration interaction-quantum Monte Carlo: Chemically accurate excitation energies and geometries, J. Chem. Theory Comput. 15, 4896 (2019).
  51. R. B. Lehouc, D. C. Sorensen, and C. Yang, ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods (SIAM, Philadelphia, PA, 1998).
  52. P. Virtanen et al., and SciPy 1.0 Contributors, SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  53. T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics, Commun. Pure Appl. Math. 10, 151 (1957).
  54. T. Olsen and K. S. Thygesen, Static correlation beyond the random phase approximation: Dissociating H2 with the Bethe-Salpeter equation and time-dependent GW, J. Chem. Phys. 140, 164116 (2014).
  55. K. J. H. Giesbertz, A.-M. Uimonen, and R. van Leeuwen, Approximate energy functionals for one-body reduced density matrix functional theory from many-body perturbation theory, Eur. Phys. J. B 91, 282 (2018).
  56. R. L. Hall, N. Saad, K. D. Sen, and H. Ciftci, Energies and wave functions for a soft-core Coulomb potential, Phys. Rev. A 80, 032507 (2009).
  57. M. Iñarrea, V. Lanchares, J. F. Palacián, A. I. Pascual, J. P. Salas, and P. Yanguas, Effects of a soft-core Coulomb potential on the dynamics of a hydrogen atom near a metal surface, Commun. Nonlinear Sci. Numer. Simul. 68, 94 (2019).
  58. R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, in 32nd Conference on Neural Information Processing Systems (NeurIPS), Montréal, Canada (2018).
  59. Q. Zhang, X. Wang, R. Shi, X. Ren, H. Wang, and L. Wang, Neural canonical transformations for quantum anharmonic solids of lithium, Phys. Rev. Lett. 134, 246101 (2025).

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Phys. Rev. Research 7, 043277– Published 9 December, 2025
  • Received 24 October 2024
  • Accepted 29 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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