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Nonadiabatic Ehrenfest dynamics with norm-conserving and ultrasoft pseudopotentials with nuclear velocity corrections on the atomic orbitals within the framework of the projector augmented wave method

Paolo Fachin1, Francesco Macheda1,2, Paolo Barone1,3, and Francesco Mauri1

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Phys. Rev. B 114, 134302 – Published 8 September, 2026

DOI: https://doi.org/10.1103/8fjf-zthz

Abstract

We derive the first-principles Ehrenfest molecular dynamics describing nonadiabatic processes with the inclusion of the nuclear-velocity-dependent phases (also known as electron-translation factors) on the atomic orbital basis. These phases, appearing when nuclei are treated dynamically, affect effective Hamiltonians constructed from localized orbitals. In this work, we focus on the effects on the first-principles pseudopotential Hamiltonian for both the norm-conserving and ultrasoft cases, derived within the projector augmented wave method framework. Peierls-like phases depending on the nuclear velocities appear in the nonlocal part of the potential, while additional nuclear velocity and acceleration-dependent corrections appear in the ultrasoft pseudopotential case. The use of the velocity-including atomic orbital basis enables a Galilean-invariant description of the nonadiabatic Ehrenfest molecular dynamics, removing spurious nonadiabatic couplings that arise from neglecting the nuclear velocity phases in the atomic orbitals.

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References (71)

  1. G. Grosso and G. Pastori Parravicini, Solid State Physics, 2nd ed. (Academic, New York, 2013).
  2. D. R. Hamann, M. Schlüter, and C. Chiang, Norm-conserving pseudopotentials, Phys. Rev. Lett. 43, 1494 (1979).
  3. G. B. Bachelet, D. R. Hamann, and M. Schlüter, Pseudopotentials that work: From H to Pu, Phys. Rev. B 26, 4199 (1982).
  4. N. Troullier and J. L. Martins, Efficient pseudopotentials for plane-wave calculations, Phys. Rev. B 43, 1993 (1991).
  5. D. Vanderbilt, Soft self-consistent pseudopotentials in a generalized eigenvalue formalism, Phys. Rev. B 41, 7892 (1990).
  6. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  7. D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Phys. Rev. B 88, 085117 (2013).
  8. L. Kleinman and D. M. Bylander, Efficacious form for model pseudopotentials, Phys. Rev. Lett. 48, 1425 (1982).
  9. M. van Setten, M. Giantomassi, E. Bousquet, M. Verstraete, D. Hamann, X. Gonze, and G.-M. Rignanese, The PseudoDojo: Training and grading a 85 element optimized norm-conserving pseudopotential table, Comput. Phys. Commun. 226, 39 (2018).
  10. S. Fatehi, E. Alguire, Y. Shao, and J. E. Subotnik, Analytic derivative couplings between configuration-interaction-singles states with built-in electron-translation factors for translational invariance, J. Chem. Phys. 135, 234105 (2011).
  11. S. Fatehi and J. E. Subotnik, Derivative couplings with built-in electron-translation factors: Application to benzene, J. Phys. Chem. Lett. 3, 2039 (2012).
  12. D. A. Micha, Time-dependent many-electron treatment of electronic energy and charge transfer in atomic collisions, J. Phys. Chem. A 103, 7562 (1999).
  13. K. Runge and D. A. Micha, Time-dependent many-electron approach to slow ion-atom collisions for systems with several active electrons, Phys. Rev. A 62, 022703 (2000).
  14. M. Stengel, M. Royo, and E. Artacho, Rototranslational sum rules for nuclear dynamics via traveling pseudopotentials, Phys. Rev. Lett. 136, 196401 (2026).
  15. N. Santervás-Arranz, M. Stengel, and E. Artacho, Excess energy and countercurrents after a quantum kick, Phys. Rev. Res. 7, 033292 (2025).
  16. V. Athavale, X. Bian, Z. Tao, Y. Wu, T. Qiu, J. Rawlinson, R. G. Littlejohn, and J. E. Subotnik, Surface hopping, electron translation factors, electron rotation factors, momentum conservation, and size consistency, J. Chem. Phys. 159, 114120 (2023).
  17. D. R. Bates and R. McCarroll, Electron capture in slow collisions, Proc. R. Soc. A 245, 175 (1958).
  18. L. F. Errea, C. Harel, H. Jouini, L. Mendez, B. Pons, and A. Riera, Common translation factor method, J. Phys. B 27, 3603 (1994).
  19. L. F. Errea, L. Méndez, A. Riera, C. Harel, H. Jouin, and B. Pons, Common-translation-factor method with an atomic basis, Phys. Rev. A 50, 418 (1994).
  20. M. Kimura and N. F. Lane, Traveling-molecular-orbital-expansion studies of electron capture in collisions of fully stripped ions (Z = 6–9) with H and H2, Phys. Rev. A 35, 70 (1987).
  21. P. M. M. Gabás, L. F. Errea, L. Méndez, and I. Rabadán, Ab initio treatment of charge transfer in ion-molecule collisions based on one-electron wave functions, Phys. Rev. A 85, 012702 (2012).
  22. R. J. Allan and J. Hanssen, Quasimolecular treatment of Na-Na+, Li-Li+, Li-Na+ and Na-Li+ collisions with a common translation factor, J. Phys. B 18, 1981 (1985).
  23. R. J. Allan, A. Bahring, and J. Hanssen, Alignment and orientation of atomic orbitals in Na(3p)+Na+ interactions, J. Phys. B 18, 1999 (1985).
  24. W. R. Thorson and J. B. Delos, Theory of near-adiabatic collisions. I. Electron translation factor method, Phys. Rev. A 18, 117 (1978).
  25. A. K. Belyaev, Revised Born-Oppenheimer approach and a reprojection method for inelastic collisions, Phys. Rev. A 82, 060701 (2010).
  26. J. B. Delos, Theory of near-adiabatic collisions. III. Coupled equations arising from expansions involving single-center states, Phys. Rev. A 23, 2301 (1981).
  27. C. Illescas and A. Riera, Classical outlook on the electron translation factor problem, Phys. Rev. Lett. 80, 3029 (1998).
  28. L. A. Nafie, Velocity‐gauge formalism in the theory of vibrational circular dichroism and infrared absorption, J. Chem. Phys. 96, 5687 (1992).
  29. L. A. Nafie, Adiabatic molecular properties beyond the Born–Oppenheimer approximation. Complete adiabatic wave functions and vibrationally induced electronic current density, J. Chem. Phys. 79, 4950 (1983).
  30. A. Scherrer, F. Agostini, D. Sebastiani, E. K. U. Gross, and R. Vuilleumier, Nuclear velocity perturbation theory for vibrational circular dichroism: An approach based on the exact factorization of the electron-nuclear wave function, J. Chem. Phys. 143, 074106 (2015).
  31. S. Patchkovskii, Electronic currents and Born-Oppenheimer molecular dynamics, J. Chem. Phys. 137, 084109 (2012).
  32. E. Ditler, J. Mattiat, and S. Luber, The position operator problem in periodic calculations with an emphasis on theoretical spectroscopy, Phys. Chem. Chem. Phys. 25, 14672 (2023).
  33. A. Scherrer, R. Vuilleumier, and D. Sebastiani, Nuclear velocity perturbation theory of vibrational circular dichroism, J. Chem. Theory Comput. 9, 5305 (2013).
  34. E. Ditler, T. Zimmermann, C. Kumar, and S. Luber, Implementation of nuclear velocity perturbation and magnetic field perturbation theory in CP2K and their application to vibrational circular dichroism, J. Chem. Theory Comput. 18, 2448 (2022).
  35. R. Kumar and S. Luber, Calculation of vibrational circular dichroism spectra employing nuclear velocity perturbation or magnetic field perturbation theory using an atomic-orbital-based linear response approach, J. Phys. Chem. A 129, 4325 (2025).
  36. A. Abedi, N. T. Maitra, and E. K. U. Gross, Exact factorization of the time-dependent electron-nuclear wave function, Phys. Rev. Lett. 105, 123002 (2010).
  37. A. Reyes, D. Micha, and K. Runge, First principles dynamics of Li–He collisional excitation using atomic core potentials, Chem. Phys. Lett. 363, 441 (2002).
  38. E. C. Alguire, Q. Ou, and J. E. Subotnik, Calculating derivative couplings between time-dependent Hartree–Fock excited states with pseudo-wavefunctions, J. Phys. Chem. B 119, 7140 (2015).
  39. Z. Pu, X. Wu, Y. Wang, C. Fan, W. Yan, Z. Zhou, Y. Q. Gao, and Q. Sun, Analytical excited-state gradients and derivative couplings in TDDFT with minimal auxiliary basis set approximation and GPU acceleration, J. Chem. Theory Comput. 22, 1793 (2026).
  40. Q. Ou, G. D. Bellchambers, F. Furche, and J. E. Subotnik, First-order derivative couplings between excited states from adiabatic TDDFT response theory, J. Chem. Phys. 142, 064114 (2015).
  41. A. Ojanperä, V. Havu, L. Lehtovaara, and M. Puska, Nonadiabatic Ehrenfest molecular dynamics within the projector augmented-wave method, J. Chem. Phys. 136, 144103 (2012).
  42. X. Qian, J. Li, X. Lin, and S. Yip, Time-dependent density functional theory with ultrasoft pseudopotentials: Real-time electron propagation across a molecular junction, Phys. Rev. B 73, 035408 (2006).
  43. J. J. Mortensen et al., gpaw: An open Python package for electronic structure calculations, J. Chem. Phys. 160, 092503 (2024).
  44. A. V. Akimov and O. V. Prezhdo, The PYXAID program for non-adiabatic molecular dynamics in condensed matter systems, J. Chem. Theory Comput. 9, 4959 (2013).
  45. W. Chu and O. V. Prezhdo, Concentric approximation for fast and accurate numerical evaluation of nonadiabatic coupling with projector augmented-wave pseudopotentials, J. Phys. Chem. Lett. 12, 3082 (2021).
  46. W. Chu, Q. Zheng, A. V. Akimov, J. Zhao, W. A. Saidi, and O. V. Prezhdo, Accurate computation of nonadiabatic coupling with projector augmented-wave pseudopotentials, J. Phys. Chem. Lett. 11, 10073 (2020).
  47. J.-R. Vogt, M. Schulz, R. Souza Mattos, M. Barbatti, M. Persico, G. Granucci, J. Hutter, and A. Hehn, A density functional theory and semiempirical framework for trajectory surface hopping on extended systems, J. Chem. Theory Comput. 21, 10474 (2025).
  48. J. Xu, R. Zhou, T. E. Li, S. Hammes-Schiffer, and Y. Kanai, Lagrangian formulation of nuclear–electronic orbital ehrenfest dynamics with real-time tddft for extended periodic systems, J. Chem. Phys. 161, 194109 (2024).
  49. B. Barrera, D. P. Arovas, A. Chandran, and A. Polkovnikov, The moving Born–Oppenheimer approximation, Proc. Natl. Acad. Sci. USA 123, e2507816123 (2026),.
  50. X. Bian, T. Duston, N. Bradbury, Z. Tao, M. Bhati, T. Qiu, X. Wu, Y. Wu, and J. E. Subotnik, The phase-space way to electronic structure theory and subsequently chemical dynamics, Chem. Phys. Rev. 7, 011303 (2026).
  51. X. Bian, Y. Wu, T. Qiu, Z. Tao, and J. E. Subotnik, A semiclassical non-adiabatic phase-space approach to molecular translations and rotations: Surface hopping with electronic inertial effects, J. Chem. Phys. 161, 234114 (2024).
  52. T. Duston, Z. Tao, X. Bian, M. Bhati, J. Rawlinson, R. G. Littlejohn, Z. Pei, Y. Shao, and J. E. Subotnik, A phase-space electronic Hamiltonian for vibrational circular dichroism, J. Chem. Theory Comput. 20, 7904 (2024).
  53. Y. Wu, J. Rawlinson, R. G. Littlejohn, and J. E. Subotnik, Linear and angular momentum conservation in surface hopping methods, J. Chem. Phys. 160, 024119 (2024).
  54. Z. Tao, T. Qiu, M. Bhati, X. Bian, T. Duston, J. Rawlinson, R. G. Littlejohn, and J. E. Subotnik, Practical phase-space electronic Hamiltonians for ab-initio dynamics, J. Chem. Phys. 160, 124101 (2024).
  55. Z. Tao, X. Bian, Y. Wu, J. Rawlinson, R. G. Littlejohn, and J. E. Subotnik, Total angular momentum conservation in Ehrenfest dynamics with a truncated basis of adiabatic states, J. Chem. Phys. 160, 054104 (2024).
  56. T. N. Todorov, Time-dependent tight binding, J. Phys.: Condens. Matter 13, 10125 (2001).
  57. R. P. Woodard, Ostrogradsky's theorem on Hamiltonian instability, Scholarpedia 10, 32243 (2015).
  58. M. Ostrogradsky, Mémoires sur les équations différentielles, relatives au problème des isopérimètres, Mem. Acad. Imperiale Sci. St.-Petersbourg Ser. 6 4, 385 (1850).
  59. H. Motohashi and T. Suyama, Third order equations of motion and the Ostrogradsky instability, Phys. Rev. D 91, 085009 (2015).
  60. K. Takahashi, H. Motohashi, T. Suyama, and T. Kobayashi, General invertible transformation and physical degrees of freedom, Phys. Rev. D 95, 084053 (2017).
  61. A. Dal Corso, Projector augmented-wave method: Application to relativistic spin-density functional theory, Phys. Rev. B 82, 075116 (2010).
  62. G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  63. C. Rostgaard, The projector augmented-wave method, arXiv:0910.1921.
  64. K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, Car-Parrinello molecular dynamics with Vanderbilt ultrasoft pseudopotentials, Phys. Rev. B 47, 10142 (1993).
  65. A. Dal Corso, A. Pasquarello, and A. Baldereschi, Density-functional perturbation theory for lattice dynamics with ultrasoft pseudopotentials, Phys. Rev. B 56, R11369 (1997).
  66. A. Dal Corso, Density-functional perturbation theory with ultrasoft pseudopotentials, Phys. Rev. B 64, 235118 (2001).
  67. V. Zobač, M. Kuisma, A. H. Larsen, T. Rossi, and T. Susi, Ehrenfest dynamics with localized atomic-orbital basis sets within the projector augmented-wave method, J. Chem. Phys. 162, 124117 (2025).
  68. C. J. Pickard and F. Mauri, All-electron magnetic response with pseudopotentials: NMR chemical shifts, Phys. Rev. B 63, 245101 (2001).
  69. C. J. Pickard and F. Mauri, Nonlocal pseudopotentials and magnetic fields, Phys. Rev. Lett. 91, 196401 (2003).
  70. J. R. Yates, C. J. Pickard, and F. Mauri, Calculation of NMR chemical shifts for extended systems using ultrasoft pseudopotentials, Phys. Rev. B 76, 024401 (2007).
  71. P. Umari, X. Gonze, and A. Pasquarello, Density-functional perturbational theory for dielectric tensors in the ultrasoft pseudopotential scheme, Phys. Rev. B 69, 235102 (2004).

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Phys. Rev. B 114, 134302– Published 8 September, 2026
  • Received 4 June 2026
  • Revised 6 August 2026
  • Accepted 17 August 2026
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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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