# Exact Factorization Adventures: A Promising Approach for Non-Bound States

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The XF Approach in a Nutshell

#### 2.1. XF Adventures in Different Realms

## 3. Exact Potentials Driving Dissociation and Ionization

## 4. XF-Based Mixed Quantum-Classical Methods

#### 4.1. Computation of the Quantum Momentum

## 5. Computing the Quantum Momentum

#### 5.1. ECR Model ${k}_{0}=30$ a.u.

#### 5.2. ECR Model ${k}_{0}=10$ a.u.

## 6. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Crespo-Otero, R.; Barbatti, M. Recent Advances and Perspectives on Nonadiabatic Mixed Quantum–Classical Dynamics. Chem. Rev.
**2018**, 118, 7026–7068. [Google Scholar] [CrossRef] [PubMed][Green Version] - Agostini, F.; Curchod, B.F.E. Different flavors of nonadiabatic molecular dynamics. Wiley Interdiscip. Rev. Comput. Mol. Sci.
**2019**, 9, e1417. [Google Scholar] [CrossRef] - Thachuk, M.; Ivanov, M.Y.; Wardlaw, D.M. A semiclassical approach to intense-field above-threshold dissociation in the long wavelength limit. J. Chem. Phys.
**1996**, 105, 4094–4104. [Google Scholar] [CrossRef] - Thachuk, M.; Ivanov, M.Y.; Wardlaw, D.M. A semiclassical approach to intense-field above-threshold dissociation in the long wavelength limit. II. Conservation principles and coherence in surface hopping. J. Chem. Phys.
**1998**, 109, 5747–5760. [Google Scholar] [CrossRef] - Bajo, J.J.; González-Vázquez, J.; Sola, I.; Santamaria, J.; Richer, M.; Marquetand, P.; González, L. Mixed quantum-classical dynamics in the adiabatic representation to simulate molecules driven by strong laser pulses. J. Phys. Chem. A
**2012**, 116, 2800. [Google Scholar] [CrossRef] - Horenko, I.; Schmidt, B.; Schütte, C. A theoretical model for molecules interacting with intense laser pulses: The Floquet-based quantum-classical Liouville equation. J. Chem. Phys.
**2001**, 115, 5733–5743. [Google Scholar] [CrossRef][Green Version] - Fiedlschuster, T.; Handt, J.; Schmidt, R. Floquet surface hopping: Laser-driven dissociation and ionization dynamics of H
_{2}^{+}. Phys. Rev. A**2016**, 93, 053409. [Google Scholar] [CrossRef] - Schirò, M.; Eich, F.G.; Agostini, F. Quantum-classical nonadiabatic dynamics of Floquet driven systems. J. Chem. Phys.
**2021**, 154, 114101. [Google Scholar] [CrossRef] - Lépine, F.; Ivanov, M.Y.; Vrakking, M.J.J. Attosecond molecular dynamics: Fact or fiction? Nat. Photonics
**2014**, 8, 195. [Google Scholar] [CrossRef] - Hunter, G. Conditional probability amplitudes in wave mechanics. Int. J. Quantum Chem.
**1975**, 9, 237. [Google Scholar] [CrossRef] - Hunter, G. Ionization potentials and conditional amplitudes. Int. J. Quantum Chem.
**1975**, 9, 311. [Google Scholar] [CrossRef] - Hunter, G. Nodeless wave function quantum theory. Int. J. Quantum Chem.
**1980**, 9, 133. [Google Scholar] [CrossRef] - Hunter, G. Nodeless wave functions and spiky potentials. Int. J. Quantum Chem.
**1981**, 19, 755. [Google Scholar] [CrossRef] - Hunter, G.; Tai, C.C. Variational marginal amplitudes. Int. J. Quantum Chem.
**1982**, 21, 1041. [Google Scholar] [CrossRef] - Gidopoulos, N.I.; Gross, E.K.U. Electronic non-adiabatic states: Towards a density functional theory beyond the Born–Oppenheimer approximation. Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci.
**2014**, 372, 20130059. [Google Scholar] [CrossRef][Green Version] - Abedi, A.; Maitra, N.T.; Gross, E.K.U. Exact Factorization of the Time-Dependent Electron-Nuclear Wave Function. Phys. Rev. Lett.
**2010**, 105, 123002. [Google Scholar] [CrossRef][Green Version] - Abedi, A.; Maitra, N.T.; Gross, E.K.U. Correlated electron-nuclear dynamics: Exact factorization of the molecular wavefunction. J. Chem. Phys.
**2012**, 137, 22A530. [Google Scholar] [CrossRef][Green Version] - Agostini, F.; Gross, E.K.U. Ultrafast dynamics with the exact factorization. Eur. Phys. J. B
**2021**, 94, 179. [Google Scholar] [CrossRef] - Suzuki, Y.; Abedi, A.; Maitra, N.T.; Yamashita, K.; Gross, E.K.U. Electronic Schrödinger equation with nonclassical nuclei. Phys. Rev. A
**2014**, 89, 040501(R). [Google Scholar] [CrossRef] - Khosravi, E.; Abedi, A.; Maitra, N.T. Exact Potential Driving the Electron Dynamics in Enhanced Ionization of H
_{2}^{+}. Phys. Rev. Lett.**2015**, 115, 263002. [Google Scholar] [CrossRef][Green Version] - Ha, J.K.; Lee, I.S.; Min, S.K. Surface Hopping Dynamics beyond Nonadiabatic Couplings for Quantum Coherence. J. Phys. Chem. Lett.
**2018**, 9, 1097–1104. [Google Scholar] [CrossRef] - Lee, I.S.; Ha, J.K.; Han, D.; Kim, T.I.; Moon, S.W.; Min, S.K. PyUNIxMD: A Python-based excited state molecular dynamics package. J. Comput. Chem.
**2021**, 42, 1755–1766. [Google Scholar] [CrossRef] [PubMed] - Pieroni, C.; Agostini, F. Nonadiabatic Dynamics with Coupled Trajectories. J. Chem. Theory Comput.
**2021**, 17, 5969–5991. [Google Scholar] [CrossRef] [PubMed] - Min, S.K.; Agostini, F.; Gross, E.K.U. Coupled-Trajectory Quantum-Classical Approach to Electronic Decoherence in Nonadiabatic Processes. Phys. Rev. Lett.
**2015**, 115, 073001. [Google Scholar] [CrossRef] [PubMed] - Agostini, F.; Min, S.K.; Abedi, A.; Gross, E.K.U. Quantum-Classical Nonadiabatic Dynamics: Coupled- vs Independent-Trajectory Methods. J. Chem. Theory Comput.
**2016**, 12, 2127–2143. [Google Scholar] [CrossRef][Green Version] - Min, S.K.; Agostini, F.; Tavernelli, I.; Gross, E.K.U. Ab Initio Nonadiabatic Dynamics with Coupled Trajectories: A Rigorous Approach to Quantum (De)Coherence. J. Phys. Chem. Lett.
**2017**, 8, 3048–3055. [Google Scholar] [CrossRef] - Khosravi, E.; Abedi, A.; Rubio, A.; Maitra, N.T. Electronic non-adiabatic dynamics in enhanced ionization of isotopologues of hydrogen molecular ions from the exact factorization perspective. Phys. Chem. Chem. Phys.
**2017**, 19, 8269–8281. [Google Scholar] [CrossRef] [PubMed][Green Version] - Abedi, A.; Maitra, N.T.; Gross, E.K.U. Response to “Comment on ‘Correlated electron-nuclear dynamics: Exact factorization of the molecular wavefunction’” [J. Chem. Phys. 139, 087101 (2013)]. J. Chem. Phys.
**2013**, 139, 087102. [Google Scholar] [CrossRef][Green Version] - Fiedlschuster, T.; Handt, J.; Gross, E.K.U.; Schmidt, R. Surface hopping in laser-driven molecular dynamics. Phys. Rev. A
**2017**, 95, 063424. [Google Scholar] [CrossRef][Green Version] - Suzuki, Y.; Abedi, A.; Maitra, N.T.; Gross, E.K.U. Laser-induced electron localization in H
_{2}^{+}: Mixed quantum-classical dynamics based on the exact time-dependent potential energy surface. Phys. Chem. Chem. Phys.**2015**, 17, 29271–29280. [Google Scholar] [CrossRef][Green Version] - Abedi, A.; Agostini, F.; Suzuki, Y.; Gross, E.K.U. Dynamical steps that bridge piecewise adiabatic shapes in the exact time-dependent potential energy surface. Phys. Rev. Lett.
**2013**, 110, 263001. [Google Scholar] [CrossRef] [PubMed][Green Version] - Agostini, F.; Abedi, A.; Suzuki, Y.; Gross, E.K.U. Mixed quantum-classical dynamics on the exact time-dependent potential energy surfaces: A novel perspective on non-adiabatic processes. Mol. Phys.
**2013**, 111, 3625. [Google Scholar] [CrossRef][Green Version] - Agostini, F.; Abedi, A.; Suzuki, Y.; Min, S.K.; Maitra, N.T.; Gross, E.K.U. The exact forces on classical nuclei in non-adiabatic charge transfer. J. Chem. Phys.
**2015**, 142, 084303. [Google Scholar] [CrossRef] [PubMed][Green Version] - Curchod, B.F.E.; Agostini, F. On the Dynamics through a Conical Intersection. J. Phys. Chem. Lett.
**2017**, 8, 831–837. [Google Scholar] [CrossRef][Green Version] - Curchod, B.F.E.; Agostini, F.; Gross, E.K.U. An exact factorization perspective on quantum interferences in nonadiabatic dynamics. J. Chem. Phys.
**2016**, 145, 034103. [Google Scholar] [CrossRef] [PubMed][Green Version] - Agostini, F.; Abedi, A.; Gross, E.K.U. Classical nuclear motion coupled to electronic non-adiabatic transitions. J. Chem. Phys.
**2014**, 141, 214101. [Google Scholar] [CrossRef] [PubMed][Green Version] - Abedi, A.; Agostini, F.; Gross, E.K.U. Mixed quantum-classical dynamics from the exact decomposition of electron-nuclear motion. Europhys. Lett.
**2014**, 106, 33001. [Google Scholar] [CrossRef][Green Version] - Davis, M.J.; Heller, E.J. Quantum dynamical tunneling in bound states. J. Chem. Phys.
**1981**, 75, 246. [Google Scholar] [CrossRef] - Hughes, K.H.; Parry, S.M.; Parlant, G.; Burghardt, I. A hybrid hydrodynamic-liouvillian approach to mixed quantum-classical dynamics: Application to tunneling in a double well. J. Phys. Chem. A
**2007**, 111, 10269–10283. [Google Scholar] [CrossRef] - Basire, M.; Borgis, D.; Vuilleumier, R. Computing Wigner distributions and time correlation functions using the quantum thermal bath method: Application to proton transfer spectroscopy. Phys. Chem. Chem. Phys.
**2013**, 15, 12591–12601. [Google Scholar] [CrossRef] - Litman, Y.; Behler, J.; Rossi, M. Temperature dependence of the vibrational spectrum of porphycene: A qualitative failure of classical-nuclei molecular dynamics. Faraday Discuss.
**2020**, 221, 526–546. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lawrence, J.E.; Manolopoulos, D.E. Path integral methods for reaction rates in complex systems. Faraday Discuss.
**2020**, 221, 9–29. [Google Scholar] [CrossRef] [PubMed] - Ghosh, S.; Giannini, S.; Lively, K.; Blumberger, J. Nonadiabatic dynamics with quantum nuclei: Simulating charge transfer with ring polymer surface hopping. Faraday Discuss.
**2020**, 221, 501–525. [Google Scholar] [CrossRef] [PubMed] - Gu, B.; Franco, I. Partial hydrodynamic representation of quantum molecular dynamics. J. Chem. Phys.
**2017**, 146, 194104. [Google Scholar] [CrossRef][Green Version] - Shushkov, P.; Li, R.; Tully, J.C. Ring polymer molecular dynamics with surface hopping. J. Chem. Phys.
**2012**, 137, 22A549. [Google Scholar] [CrossRef] - Dupuy, L.; Lauvergnat, D.; Scribano, Y. Smolyak representations with absorbing boundary conditions for reaction path Hamiltonian model of reactive scattering. Chem. Phys. Lett.
**2022**, 787, 139241. [Google Scholar] [CrossRef] - Suzuki, Y.; Watanabe, K. Bohmian mechanics in the exact factorization of electron-nuclear wave functions. Phys. Rev. A
**2016**, 94, 032517. [Google Scholar] [CrossRef][Green Version] - Talotta, F.; Agostini, F.; Ciccotti, G. Quantum trajectories for the dynamics in the exact factorization framework: A proof-of-principle test. J. Phys. Chem. A
**2020**, 124, 6764–6777. [Google Scholar] [CrossRef] - Agostini, F.; Tavernelli, I.; Ciccotti, G. Nuclear Quantum Effects in Electronic (Non)Adiabatic Dynamics. Eur. Phys. J. B
**2018**, 91, 139. [Google Scholar] [CrossRef] - Lopreore, C.L.; Wyatt, R.E. Quantum Wave Packet Dynamics with Trajectories. Phys. Rev. Lett.
**1999**, 82, 5190. [Google Scholar] [CrossRef][Green Version] - Wyatt, R.E. Quantum wavepacket dynamics with trajectories: Wavefunction synthesis along quantum paths. Chem. Phys. Lett.
**1999**, 313, 189–197. [Google Scholar] [CrossRef] - Wyatt, R.E.; Na, K. Quantum trajectory analysis of multimode subsystem-bath dynamics. Phys. Rev. E
**2001**, 65, 016702. [Google Scholar] [CrossRef] [PubMed] - Wyatt, R.E. Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics; Interdisciplinary Applied Mathematics; Springer: New York, NY, USA, 2005. [Google Scholar]
- Garashchuk, S.; Rassolov, V. Quantum Trajectory Dynamics Based on Local Approximations to the Quantum Potential and Force. J. Chem. Theory Comput.
**2019**, 15, 3906–3916. [Google Scholar] [CrossRef] [PubMed] - Garashchuk, S.; Vazhappilly, T. Multidimensional Quantum Trajectory Dynamics in Imaginary Time with Approximate Quantum Potential. J. Phys. Chem. C
**2010**, 114, 20595–20602. [Google Scholar] [CrossRef] - Wyatt, R.E.; Bittner, E.R. Quantum wave packet dynamics with trajectories: Implementation with adaptive Lagrangian grids. J. Chem. Phys.
**2000**, 119, 8898. [Google Scholar] [CrossRef] - Hughes, K.H.; Wyatt, R.E. Wavepacket dynamics on dynamically adapting grids: Application of the equidistribution principle. Chem. Phys. Lett.
**2002**, 366, 336–342. [Google Scholar] [CrossRef] - Kendrick, B.K. A new method for solving the quantum hydrodynamic equations of motion. J. Chem. Phys.
**2003**, 119, 5805. [Google Scholar] [CrossRef] - Trahan, C.J.; Wyatt, R.E. An arbitrary Lagrangian-Eulerian approach to solving the quantum hydrodynamic equations of motion: Equidistribution with “smart” springs. J. Chem. Phys.
**2003**, 4784, 336–342. [Google Scholar] [CrossRef] - Schild, A. Electronic quantum trajectories with quantum nuclei. arXiv
**2021**, arXiv:2109.13632v1. [Google Scholar] - Tavernelli, I.; Curchod, B.F.E.; Rothlisberger, U. Nonadiabatic molecular dynamics with solvent effects: A LR-TDDFT QM/MM study of ruthenium (II) tris (bipyridine) in water. Chem. Phys.
**2011**, 391, 101–109. [Google Scholar] [CrossRef] - Talotta, F.; Boggio-Pasqua, M.; González, L. Early relaxation dynamics in the photoswitchable trans-[RuCl(NO)(py)
_{4}]^{2+}. Chem.: Eur. J.**2020**, 26, 11522–11528. [Google Scholar] [CrossRef] [PubMed] - Atkins, A.J.; Talotta, F.; Freitag, L.; Boggio-Pasqua, M.; González, L. Assessing Excited State Energy Gaps with Time-Dependent Density Functional Theory on Ru(II) Complexes. J. Chem. Theory Comput.
**2017**, 13, 4123–4145. [Google Scholar] [CrossRef] [PubMed][Green Version] - Garcáa, J.S.; Talotta, F.; Alary, F.; Dixon, I.M.; Heully, J.L.; Boggio-Pasqua, M. A Theoretical Study of the N to O Linkage Photoisomerization Efficiency in a Series of Ruthenium Mononitrosyl Complexes. Molecules
**2017**, 22, 1667. [Google Scholar] - Ando, H.; Iuchi, S.; Sato, H. Theoretical study on ultrafast intersystem crossing of chromium(III) acetylacetonate. Chem. Phys. Lett.
**2012**, 535, 177–181. [Google Scholar] [CrossRef][Green Version] - Brahim, H.; Daniel, C. Structural and spectroscopic properties of Ir(III) complexes with phenylpyridine ligands: Absorption spectra without and with spin-orbit-coupling. Comput. Theo. Chem.
**2014**, 1040–1041, 219–229. [Google Scholar] [CrossRef] - Hu, W.; Lendvay, G.; Maiti, B.; Schatz, G.C. Trajectory Surface Hopping Study of the O(
^{3}P) + Ethylene Reaction Dynamics. J. Phys. Chem. A**2008**, 112, 2093–2103. [Google Scholar] [CrossRef] - Fu, B.; Han, Y.C.; Bowman, J.M.; Angelucci, L.; Balucani, N.; Leonori, F.; Casavecchia, P. Intersystem crossing and dynamics in O(
^{3}P)+C_{2}H_{4}multichannel reaction: Experiment validates theory. Proc. Natl. Acad. Sci. USA**2012**, 109, 9733–9738. [Google Scholar] [CrossRef][Green Version] - Hu, W.; Lendvay, G.; Maiti, B.; Schatz, G.C. Electronic Structure and Excited States of the Collision Reaction O(
^{3}P)+C_{2}H_{4}: A Multiconfigurational Perspective. J. Phys. Chem. A**2021**, 125, 6075–6088. [Google Scholar] - Talotta, F.; Morisset, S.; Rougeau, N.; Lauvergnat, D.; Agostini, F. Spin-Orbit Interactions in Ultrafast Molecular Processes. Phys. Rev. Lett.
**2020**, 124, 033001. [Google Scholar] [CrossRef] - Talotta, F.; Morisset, S.; Rougeau, N.; Lauvergnat, D.; Agostini, F. Internal Conversion and Intersystem Crossing with the Exact Factorization. J. Chem. Theory Comput.
**2020**, 16, 4833–4848. [Google Scholar] [CrossRef] - Min, S.K.; Abedi, A.; Kim, K.S.; Gross, E.K.U. Is the molecular Berry phase an artefact of the Born-Oppenheimer approximation? Phys. Rev. Lett.
**2014**, 113, 263004. [Google Scholar] [CrossRef] [PubMed] - Requist, R.; Tandetzky, F.; Gross, E.K.U. Molecular geometric phase from the exact electron-nuclear factorization. Phys. Rev. A
**2016**, 93, 042108. [Google Scholar] [CrossRef][Green Version] - Requist, R.; Proetto, C.R.; Gross, E.K.U. Asymptotic analysis of the Berry curvature in the E ⊗ e Jahn-Teller model. Phys. Rev. A
**2017**, 96, 062503. [Google Scholar] [CrossRef][Green Version] - Agostini, F.; Curchod, B.F.E. When the exact factorization meets conical intersections. Eur. Phys. J. B
**2018**, 91, 141. [Google Scholar] [CrossRef] - Ibele, L.M.; Curchod, B.F.E.; Agostini, F. A photochemical reaction in different theoretical representations. J. Phys. Chem. A
**2022**, 126, 1263–1281. [Google Scholar] [CrossRef] - Hader, K.; Albert, J.; Gross, E.K.U.; Engel, V. Electron-nuclear wave-packet dynamics through a conical intersection. J. Chem. Phys.
**2017**, 146, 074304. [Google Scholar] [CrossRef] - Requist, R.; Gross, E.K.U. Exact Factorization-Based Density Functional Theory of Electrons and Nuclei. Phys. Rev. Lett.
**2016**, 117, 193001. [Google Scholar] [CrossRef][Green Version] - Li, C.; Requist, R.; Gross, E.K.U. Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF. J. Chem. Phys.
**2018**, 148, 084110. [Google Scholar] [CrossRef][Green Version] - Schild, A.; Gross, E.K.U. Exact Single-Electron Approach to the Dynamics of Molecules in Strong Laser Fields. Phys. Rev. Lett.
**2017**, 118, 163202. [Google Scholar] [CrossRef][Green Version] - Kocák, J.; Schild, A. Many-electron effects of strong-field ionization described in an exact one-electron theory. Phys. Rev. Res.
**2020**, 2, 043365. [Google Scholar] [CrossRef] - Gonze, X.; Zhou, J.S.; Reining, L. Variations on the “exact factorization” theme. Eur. Phys. J. B
**2018**, 91, 224. [Google Scholar] [CrossRef] - Lacombe, L.; Maitra, N.T. Embedding via the Exact Factorization Approach. Phys. Rev. Lett.
**2020**, 124, 206401. [Google Scholar] [CrossRef] [PubMed] - Requist, R.; Gross, E.K.U. Fock-Space Embedding Theory: Application to Strongly Correlated Topological Phases. Phys. Rev. Lett.
**2021**, 127, 116401. [Google Scholar] [CrossRef] [PubMed] - Salas, L.D.; Arce, J.C. Potential energy surfaces in atomic structure: The role of Coulomb correlation in the ground state of helium. Phys. Rev. A
**2017**, 95, 022502. [Google Scholar] [CrossRef][Green Version] - Salas, L.D.; Zamora-Yusti, B.; Arce, J.C. Characterization of the continuous transition from atomic to molecular shape in the three-body Coulomb system. Phys. Rev. A
**2022**, 105, 012808. [Google Scholar] [CrossRef] - Eich, F.G.; Agostini, F. The adiabatic limit of the exact factorization of the electron-nuclear wave function. J. Chem. Phys.
**2016**, 145, 054110. [Google Scholar] [CrossRef][Green Version] - Schild, A.; Agostini, F.; Gross, E.K.U. Electronic Flux Density beyond the Born-Oppenheimer Approximation. J. Phys. Chem. A
**2016**, 120, 3316. [Google Scholar] [CrossRef] - Scherrer, A.; Agostini, F.; Sebastiani, D.; Gross, E.K.U.; Vuilleumier, R. On the mass of atoms in molecules: Beyond the Born-Oppenheimer approximation. Phys. Rev. X
**2017**, 7, 031035. [Google Scholar] [CrossRef] - Scherrer, A.; Agostini, F.; Sebastiani, D.; Gross, E.K.U.; Vuilleumier, R. Nuclear velocity perturbation theory for vibrational circular dichroism: An approach based on the exact factorization of the electron-nuclear wave function. J. Chem. Phys.
**2015**, 143, 074106. [Google Scholar] [CrossRef][Green Version] - Requist, R.; Proetto, C.R.; Gross, E.K.U. Exact factorization-based density functional theory of electron-phonon systems. Phys. Rev. B
**2019**, 99, 165136. [Google Scholar] [CrossRef][Green Version] - Gossel, G.H.; Lacombe, L.; Maitra, N.T. On the numerical solution of the exact factorization equations. J. Chem. Phys.
**2019**, 150, 154112. [Google Scholar] [CrossRef] [PubMed] - Lorin, E. Numerical analysis of the exact factorization of molecular time-dependent Schrödinger wavefunctions. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 95, 105627. [Google Scholar] [CrossRef] - Hoffmann, N.M.; Appel, H.; Rubio, A.; Maitra, N.T. Light-matter interactions via the exact factorization approach. Eur. Phys. J. B
**2018**, 91, 180. [Google Scholar] [CrossRef] - Yuen-Zhou, J.; Xiong, W.; Shegai, T. Polariton chemistry: Molecules in cavities and plasmonic media. J. Chem. Phys.
**2022**, 156, 030401. [Google Scholar] [CrossRef] [PubMed] - Galego, J.; Garcia-Vidal, F.J.; Feist, J. Cavity-Induced Modifications of Molecular Structure in the Strong-Coupling Regime. Phys. Rev. X
**2015**, 5, 041022. [Google Scholar] [CrossRef][Green Version] - Lacombe, L.; Hoffmann, N.M.; Maitra, N.T. Exact Potential Energy Surface for Molecules in Cavities. Phys. Rev. Lett.
**2019**, 123, 083201. [Google Scholar] [CrossRef][Green Version] - Martinez, P.; Rosenzweig, B.; Hoffmann, N.M.; Lacombe, L.; Maitra, N.T. Case studies of the time-dependent potential energy surface for dynamics in cavities. J. Chem. Phys.
**2021**, 154, 014102. [Google Scholar] [CrossRef] - Hoffmann, N.M.; Schäfer, C.; Rubio, A.; Kelly, A.; Appel, H. Capturing vacuum fluctuations and photon correlations in cavity quantum electrodynamics with multitrajectory Ehrenfest dynamics. Phys. Rev. A
**2019**, 99, 063819. [Google Scholar] [CrossRef][Green Version] - Rosenzweig, B.; Hoffmann, N.M.; Lacombe, L.; Maitra, N.T. Analysis of the classical trajectory treatment of photon dynamics for polaritonic phenomena. J. Chem. Phys.
**2022**, 156, 054101. [Google Scholar] [CrossRef] - Flick, J.; Ruggenthaler, M.; Appel, H.; Rubio, A. Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry. Proc. Natl. Acad. Sci. USA
**2017**, 114, 3026–3034. [Google Scholar] [CrossRef][Green Version] - Cederbaum, L.S. The exact wavefunction of interacting N degrees of freedom as a product of N single-degree-of-freedom wavefunctions. Chem. Phys.
**2015**, 457, 129. [Google Scholar] [CrossRef] - Scherrer, A.; Vuilleumier, R.; Sebastiani, D. Vibrational circular dichroism from ab initio molecular dynamics and nuclear velocity perturbation theory in the liquid phase. J. Chem. Phys.
**2016**, 145, 084101. [Google Scholar] [CrossRef] [PubMed] - Scherrer, A.; Vuilleumier, R.; Sebastiani, D. Nuclear Velocity Perturbation Theory of Vibrational Circular Dichroism. J. Chem. Theory Comput.
**2013**, 9, 5305–5312. [Google Scholar] [CrossRef] - Diestler, D.J.; Kenfack, A.; Manz, J.; Paulus, B.; Pérez-Torres, J.F.; Pohl, V. Computation of the Electronic Flux Density in the Born-Oppenheimer Approximation. J. Phys. Chem. A
**2013**, 117, 8519–8527. [Google Scholar] [CrossRef] - Diestler, D.J. Beyond the Born-Oppenheimer Approximation: A Treatment of Electronic Flux Density in Electronically Adiabatic Molecular Processes. J. Phys. Chem. A
**2013**, 117, 4698–4708. [Google Scholar] [CrossRef] [PubMed] - Arce, J.C. Unification of the conditional probability and semiclassical interpretations for the problem of time in quantum theory. Phys. Rev. A
**2012**, 85, 042108. [Google Scholar] [CrossRef][Green Version] - Schild, A. Time in quantum mechanics: A fresh look at the continuity equation. Phys. Rev. A
**2018**, 98, 052113. [Google Scholar] [CrossRef][Green Version] - Kulander, K.C.; Mies, F.H.; Schafer, K.J. Model for studies of laser-induced nonlinear processes in molecules. Phys. Rev. A
**1996**, 53, 2562–2570. [Google Scholar] [CrossRef] - Chelkowski, S.; Foisy, C.; Bandrauk, A.D. Electron-nuclear dynamics of multiphoton H
_{2}^{+}dissociative ionization in intense laser fields. Phys. Rev. A**1998**, 57, 1176–1185. [Google Scholar] [CrossRef] - Walsh, T.D.G.; Ilkov, F.A.; Chin, S.L.; Châteauneuf, F.; Nguyen-Dang, T.T.; Chelkowski, S.; Bandrauk, A.D.; Atabek, O. Laser-induced processes during the Coulomb explosion of H
_{2}in a Ti-sapphire laser pulse. Phys. Rev. A**1998**, 58, 3922–3933. [Google Scholar] [CrossRef] - Lein, M.; Kreibich, T.; Gross, E.K.U.; Engel, V. Strong-field ionization dynamics of a model H
_{2}molecule. Phys. Rev. A**2002**, 65, 033403. [Google Scholar] [CrossRef] - McLachlan, A.D. A variational solution of the time-dependent Schrodinger equation. Mol. Phys.
**1964**, 8, 39–44. [Google Scholar] [CrossRef] - Tully, J.C. Mixed quantum-classical dynamics. Faraday Discuss.
**1998**, 110, 407. [Google Scholar] [CrossRef] - Kelkensberg, F.; Sansone, G.; Ivanov, M.Y.; Vrakking, M. A semi-classical model of attosecond electron localization in dissociative ionization of hydrogen. Phys. Chem. Chem. Phys.
**2011**, 13, 8647–8652. [Google Scholar] [CrossRef] - Sansone, G.; Kelkensberg, F.; Pérez-Torres, J.F.; Morales, F.; Kling, M.F.; Siu, W.; Ghafur, O.; Johnsson, P.; Swoboda, M.; Benedetti, E.; et al. Electron localization following attosecond molecular photoionization. Nature
**2010**, 465, 763–766. [Google Scholar] [CrossRef][Green Version] - He, F.; Ruiz, C.; Becker, A. Control of Electron Excitation and Localization in the Dissociation of H
_{2}^{+}and Its Isotopes Using Two Sequential Ultrashort Laser Pulses. Phys. Rev. Lett.**2007**, 99, 083002. [Google Scholar] [CrossRef][Green Version] - Zuo, T.; Bandrauk, A.D. Charge-resonance-enhanced ionization of diatomic molecular ions by intense lasers. Phys. Rev. A
**1995**, 52, R2511. [Google Scholar] [CrossRef] - Chelkowski, S.; Bandrauk, A.D. Two-step Coulomb explosions of diatoms in intense laser fields. J. Phys. B: At. Mol. Opt. Phys.
**1995**, 28, L723–L731. [Google Scholar] [CrossRef] - Chelkowski, S.; Zuo, T.; Atabek, O.; Bandrauk, A.D. Dissociation, ionization, and Coulomb explosion of H
_{2}^{+}in an intense laser field by numerical integration of the time-dependent Schrödinger equation. Phys. Rev. A**1995**, 52, 2977. [Google Scholar] [CrossRef] - Seideman, T.; Ivanov, M.Y.; Corkum, P.B. Role of Electron Localization in Intense-Field Molecular Ionization. Phys. Rev. Lett.
**1995**, 75, 2819–2822. [Google Scholar] [CrossRef] - Bandrauk, A.D.; Légaré, F. Enhanced Ionization of Molecules in Intense Laser Fields. In Progress in Ultrafast Intense Laser Science VIII; Yamanouchi, K., Nisoli, M., Hill, W.T., Eds.; Springer: Berlin/Heidelberg, Germany, 2012; pp. 29–46. [Google Scholar] [CrossRef][Green Version]
- Zuo, T.; Chelkowski, S.; Bandrauk, A.D. Harmonic generation by the H
_{2}^{+}molecular ion in intense laser fields. Phys. Rev. A**1993**, 48, 3837–3844. [Google Scholar] [CrossRef] [PubMed] - Takemoto, N.; Becker, A. Multiple Ionization Bursts in Laser-Driven Hydrogen Molecular Ion. Phys. Rev. Lett.
**2010**, 105, 203004. [Google Scholar] [CrossRef] [PubMed][Green Version] - Takemoto, N.; Becker, A. Time-resolved view on charge-resonance-enhanced ionization. Phys. Rev. A
**2011**, 84, 023401. [Google Scholar] [CrossRef][Green Version] - Beylerian, C.; Saugout, S.; Cornaggia, C. Non-sequential double ionization of H
_{2}using ultrashort 10 fs laser pulses. J. Phys. B: At. Mol. Opt. Phys.**2006**, 39, L105–L112. [Google Scholar] [CrossRef] - Bocharova, I.; Karimi, R.; Penka, E.F.; Brichta, J.P.; Lassonde, P.; Fu, X.; Kieffer, J.C.; Bandrauk, A.D.; Litvinyuk, I.; Sanderson, J.; et al. Charge Resonance Enhanced Ionization of CO
_{2}Probed by Laser Coulomb Explosion Imaging. Phys. Rev. Lett.**2011**, 107, 063201. [Google Scholar] [CrossRef][Green Version] - Légaré, F.; Litvinyuk, I.V.; Dooley, P.W.; Quéré, F.; Bandrauk, A.D.; Villeneuve, D.M.; Corkum, P.B. Time-Resolved Double Ionization with Few Cycle Laser Pulses. Phys. Rev. Lett.
**2003**, 91, 093002. [Google Scholar] [CrossRef] - Curchod, B.F.E.; Agostini, F.; Tavernelli, I. CT-MQC—A Coupled-Trajectory Mixed Quantum/Classical method including nonadiabatic quantum coherence effects. Eur. Phys. J. B
**2018**, 91, 168. [Google Scholar] [CrossRef][Green Version] - Marsili, E.; Olivucci, M.; Lauvergnat, D.; Agostini, F. Quantum and Quantum-Classical Studies of the Photoisomerization of a Retinal Chromophore Model. J. Chem. Theory Comput.
**2020**, 16, 6032–6048. [Google Scholar] [CrossRef] - Filatov, M.; Min, S.K.; Kim, K.S. Non-adiabatic dynamics of ring opening in cyclohexa-1,3-diene described by an ensemble density-functional theory method. Mol. Phys.
**2019**, 117, 1128–1141. [Google Scholar] [CrossRef] - Filatov, M.; Paolino, M.; Min, S.K.; Kim, K.S. Fulgides as Light-Driven Molecular Rotary Motors: Computational Design of a Prototype Compound. J. Phys. Chem. Lett.
**2018**, 9, 4995–5001. [Google Scholar] [CrossRef] - Filatov, M.; Paolino, M.; Min, S.K.; Choi, C.H. Design and photoisomerization dynamics of a new family of synthetic 2-stroke light driven molecular rotary motors. Chem. Commun.
**2019**, 55, 5247–5250. [Google Scholar] [CrossRef] [PubMed] - Filatov, M.; Min, S.K.; Choi, C.H. Theoretical modelling of the dynamics of primary photoprocess of cyclopropanone. Phys. Chem. Chem. Phys.
**2019**, 21, 2489–2498. [Google Scholar] [CrossRef] [PubMed] - Vindel-Zandbergen, P.; Ibele, L.M.; Ha, J.K.; Min, S.K.; Curchod, B.F.E.; Maitra, N.T. Study of the Decoherence Correction Derived from the Exact Factorization Approach for Nonadiabatic Dynamics. J. Chem. Theory Comput.
**2021**, 17, 3852–3862. [Google Scholar] [CrossRef] [PubMed] - Vindel-Zandbergen, P.; Matsika, S.; Maitra, N.T. Exact-Factorization-Based Surface Hopping for Multistate Dynamics. J. Phys. Chem. Lett.
**2022**, 13, 1785–1790. [Google Scholar] [CrossRef] - Tully, J.C. Molecular dynamics with electronic transitions. J. Chem. Phys.
**1990**, 93, 1061. [Google Scholar] [CrossRef] - Lu, J.; Zhoud, Z. Frozen Gaussian Approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms. Math. Comp.
**2018**, 87, 2189–2232. [Google Scholar] [CrossRef] - Wang, L.; Akimov, A.; Prezhdo, O.V. Recent Progress in Surface Hopping: 2011–2015. J. Phys. Chem. Lett.
**2016**, 7, 2100. [Google Scholar] [CrossRef] - Subotnik, J.E.; Jain, A.; Landry, B.; Petit, A.; Ouyang, W.; Bellonzi, N. Understanding the Surface Hopping View of Electronic Transitions and Decoherence. Ann. Rev. Phys. Chem.
**2016**, 67, 387–417. [Google Scholar] [CrossRef] - Gossel, G.H.; Agostini, F.; Maitra, N.T. Coupled-Trajectory Mixed Quantum-Classical Algorithm: A Deconstruction. J. Chem. Theory Comput.
**2018**, 14, 4513–4529. [Google Scholar] [CrossRef] - Agostini, F. An exact-factorization perspective on quantum-classical approaches to excited-state dynamics. Eur. Phys. J. B
**2018**, 91, 143. [Google Scholar] [CrossRef] - Agostini, F.; Marsili, E.; Talotta, F. G-CTMQC. 2021. Available online: https://gitlab.com/agostini.work/g-ctmqc (accessed on 13 May 2022).
- Lauvergnat, D. ModelLib. 2018. Available online: https://github.com/lauvergn/QuantumModelLib/tree/OOP_branch (accessed on 13 May 2022).
- Kim, T.I.; Ha, J.K.; Min, S.K. Coupled- and Independent-Trajectory Approaches Based on the Exact Factorization Using the PyUNIxMD Package. Top. Curr. Chem.
**2022**, 380, 8. [Google Scholar] [CrossRef] [PubMed] - Ha, J.K.; Min, S.K. Independent Trajectory Mixed Quantum-Classical Approaches Based on the Exact Factorization. J. Chem. Phys.
**2022**, 156, 174109. [Google Scholar] [CrossRef] [PubMed] - Barbatti, M. Velocity Adjustment in Surface Hopping: Ethylene as a Case Study of the Maximum Error Caused by Direction Choice. J. Chem. Theor. Comput.
**2021**, 17, 3010–3018. [Google Scholar] [CrossRef] [PubMed] - Carof, A.; Giannini, S.; Blumberger, J. Detailed balance, internal consistency, and energy conservation in fragment orbital-based surface hopping. J. Chem. Phys.
**2017**, 147, 214113. [Google Scholar] [CrossRef] - Tang, D.; Shen, L.; Fang, W.h. Evaluation of Mixed Quantum-Classical Molecular Dynamics on cis-Azobenzene Photoisomerization. Phys. Chem. Chem. Phys.
**2021**, 23, 13951–13964. [Google Scholar] [CrossRef] - Hammes-Schiffer, S.; Tully, J.C. Proton transfer in solution: Molecular dynamics with quantum transitions. J. Chem. Phys.
**1994**, 101, 4657–4667. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**1D model of laser-driven H${}_{2}^{+}$ dissociation: (

**a**) Shows a sketch of the model molecule, and the electric field of the $\lambda =228$ nm laser applied to the system, with the shaded part indicating the optical cycle for which snapshots for two different intensities are shown in the other panels. (

**b**) The top panel shows $\langle R\rangle (t)$ as a measure of dissociation, for the stronger field intensity of ${I}_{1}$, and the lower panel shows ${I}_{p}=1-{\int}_{{\mathrm{box}}_{\mathrm{e}}}dz\int dR{|\Psi (z,R,t)|}^{2}$ which measures ionization through the number of electrons outside a box chosen of size $|z|\le 10$ a.u; the exact is shown in black, while predictions of the traditional classical Ehrenfest method is in red, the quantum time-dependent Hartree in blue, and a classical evolution on the exact TDPES (labelled as exact-Ehrenfest) as green. The lowest panel shows time-snapshots of the nuclear density and the TDPES, and the black dot shows the position and energy of a classical particle evolving under the TDPES. (

**c**) This shows the same quantities as in (

**b**) but for the weaker field intensity ${I}_{2}$. Reproduced from ref. [17] with the permission of AIP Publishing.

**Figure 2.**Analytical form of the electronic Hamiltonian on the

**left**, with the BO potential energy surfaces (red and black) and NAC (dashed green) shown on the

**right**for ECR model.

**Figure 3.**Upper panels: electronic populations (

**A**,

**B**), and of the excited state ${\rho}_{22}(t)$ (solid) and fraction of trajectories on the excited state ${\Pi}_{2}(t)={N}_{2}(t)/{N}_{traj}$ (dotted), with ${N}_{2}(t)$ being the number of trajectories running on the upper surface at a given time and ${N}_{traj}$ the total number of trajectories, are plotted for ECR model with ${k}_{0}=30$ a.u., as functions of time starting at 400 a.u. which is when the trajectories and the quantum wavepacket approach the NAC region. The label ${\mathrm{Q}}_{\mathrm{m}}$ indicates the use of Equation (27), i.e., the modified definition of the quantum momentum, while ${\mathrm{Q}}_{\mathrm{o}}$ indicates the use of Equation (22), i.e., the original definition. Lower panels: electronic coherences (

**C**,

**D**), $|{\rho}_{12}{|}^{2}={\sum}_{\alpha}^{{N}_{traj}}{|{C}_{1}^{\alpha *}(t){C}_{2}^{\alpha}(t)|}^{2}/{N}_{traj}$. The color code is the same as in the upper panels. The coupled-trajectory results are compared with exact results in the left panels, whereas auxiliary-trajectory results are shown in the right panels. For reference, surface-hopping (SH) results with no decoherence corrections are shown in the right panels.

**Figure 4.**Spatial distribution of trajectories for the ECR model with ${k}_{0}=30$ a.u.: CTSH with original definition of the quantum momentum using Equation (22) (panel

**A**), CTSH with the modified definition of the quantum momentum using Equation (27) (panel

**B**), SHXF with $\sigma ={\sigma}_{0}$ (panel

**C**) and SHXF $\sigma ={\sigma}_{0}$/10 (panel

**D**).

**Figure 5.**Upper panels: electronic populations (

**A**,

**B**) of the excited state (solid) and fraction of trajectories on the excited state (dotted) are plotted for ECR model with ${k}_{0}=10$ a.u. Lower panels: electronic coherences (

**C**,

**D**).

**Figure 6.**Top panels in each plot: Time snapshots of the electronic populations of the excited state for CTMQC (

**left**) and CTSH (

**right**), with the modified definition of the quantum momentum, at the positions of the trajectories (green dots) and running state (red dots). Middle panels in each plot: XF contribution to the time-variation of the electronic populations (blue dots), i.e., the second-term on the right-hand-side of Equation (24) but without the trajectory-sum. Lower panels in each plot: BO surfaces (thin red lines), NAC (thin dashed lines), and spatial distribution of the nuclear trajectories on the BO surfaces (black dots).

**Figure 7.**Same as in Figure 6 but using the original definition of the quantum momemtum.

**Figure 8.**Spatial distribution of trajectories for the ECR $k=10$ a.u. model. CTSH with original definition of the QM (panel

**A**), CTSH with the modified definition of the QM (panel

**B**), SHXF with $\sigma ={\sigma}_{0}$ (panel

**C**) and SHXF $\sigma ={\sigma}_{0}$/10 (panel

**D**).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Villaseco Arribas, E.; Agostini, F.; Maitra, N.T. Exact Factorization Adventures: A Promising Approach for Non-Bound States. *Molecules* **2022**, *27*, 4002.
https://doi.org/10.3390/molecules27134002

**AMA Style**

Villaseco Arribas E, Agostini F, Maitra NT. Exact Factorization Adventures: A Promising Approach for Non-Bound States. *Molecules*. 2022; 27(13):4002.
https://doi.org/10.3390/molecules27134002

**Chicago/Turabian Style**

Villaseco Arribas, Evaristo, Federica Agostini, and Neepa T. Maitra. 2022. "Exact Factorization Adventures: A Promising Approach for Non-Bound States" *Molecules* 27, no. 13: 4002.
https://doi.org/10.3390/molecules27134002