# Classical Nuclear Motion: Comparison to Approaches with Quantum Mechanical Nuclear Motion

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results

#### 3.1. Wave Packets and Alternatives

#### 3.2. Multiple Spawning

#### 3.3. Path Integrals

#### 3.4. Full Treatment

## 4. Discussion

## Supplementary Materials

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AIMD | Ab initio molecular dynamics |

BOMD | Born–Oppenheimer molecular dynamics |

CPMD | Car–Parrinello molecular dynamics |

DFT | Density functional theory |

ESIPT | Excited-state intramolecular proton transfer |

HOMO | Highest occupied molecular orbital |

LUMO | Lowest unoccupied molecular orbital |

NEO | Nuclear-electronic orbital method |

QM/MM | Quantum mechanics/molecular mechanics |

ROKS | Restricted open-shell Kohn–Sham theory |

SCF | Self-consistent field theory |

TDDFT | Time-dependent DFT |

## References

- Frank, I.; Genuit, S.; Matz, F.; Oschinski, H. Ammonia, water, and hydrogen: Can nuclear motion be described classically? Int. J. Quantum Chem.
**2020**, 120, e26142. [Google Scholar] [CrossRef] - Frank, I. Classical motion of the nuclei in a molecules: A concept without alternatives. Chem. Sel.
**2020**, 5, 1872. [Google Scholar] [CrossRef] [Green Version] - Büchel, R.C.; Rudolph, D.A.; Frank, I. Deterministic quantum mechanics: The role of the Maxwell-Boltzmann distribution. Int. J. Quantum Chem.
**2021**, 121, e26555. [Google Scholar] [CrossRef] - Car, R.; Parrinello, M. Unified Approach for Molecular Dynamics and Density-Functional Theory. Phys. Rev. Lett.
**1985**, 55, 2471–2474. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hutter, J.; Hutter, J.; Alavi, A.; Deutsch, T.; Bernasconi, M.; Goedecker, S.; Marx, D.; Tuckerman, M.; Parrinello, M. Version 4.3, Copyright IBM Corp 1990–2015, Copyright MPI für Festkörperforschung Stuttgart 1997–2001. Available online: http://www.cpmd.org/ (accessed on 22 October 2022).
- Marx, D.; Parrinello, M. Ab initio path-integral molecular dynamics. Z. Phys. B
**1994**, 95, 143. [Google Scholar] [CrossRef] - Marx, D.; Parrinello, M. Structural quantum effects and three-centre two-electron bonding in CH
_{5}^{+}. Nature**1995**, 375, 216. [Google Scholar] [CrossRef] - Marx, D.; Parrinello, M. Ab initio path-integral molecular dynamics: Basic ideas. J. Chem. Phys.
**1996**, 104, 4077. [Google Scholar] [CrossRef] - Marx, D.; Parrinello, M. The effect of quantum and thermal fluctuations on the structure of the floppy molecule C
_{2}H_{3}^{+}. Science**1996**, 271, 179. [Google Scholar] [CrossRef] - Lan, Z.; Domcke, W.; Vallet, V.; Sobolewski, A.L.; Mahapatra, S. Time-dependent quantum wave-packet description of the
^{1}πσ^{*}photochemistry of phenol. J. Chem. Phys.**2005**, 122, 224315. [Google Scholar] [CrossRef] - Domcke, W.; Yarkony, D.R. Role of Conical Intersections in Molecular Spectroscopy and Photoinduced Chemical Dynamics. Annu. Rev. Phys. Chem.
**2012**, 63, 325. [Google Scholar] [CrossRef] - Ramesh, S.G.; Domcke, W. A multi-sheeted three-dimensional potential-energy surface for the H-atom photodissociation of phenol. Faraday Discuss.
**2013**, 163, 73. [Google Scholar] [CrossRef] - Xie, W.; Domcke, W. Accuracy of trajectory surface-hopping methods: Test for a two-dimensional model of the photodissociation of phenol. J. Chem. Phys.
**2017**, 147, 184114. [Google Scholar] [CrossRef] - Webb, S.P.; Iordanov, T.; Hammes-Schiffer, S. Multiconfigurational nuclear-electronic orbital approach: Incorporation of nuclear quantum effects in electronic structure calculations. J. Chem. Phys.
**2002**, 117, 4106. [Google Scholar] [CrossRef] [Green Version] - Ben-Nun, M.; Quenneville, J.; Martinez, T. Ab Initio Multiple Spawning: Photochemistry from First Principles Quantum Molecular Dynamics. J. Phys. Chem. A
**2000**, 104, 5161. [Google Scholar] [CrossRef] - Curchod, B.F.E.; Martinez, T.J. Ab Initio Nonadiabatic Quantum Molecular Dynamics. Chem. Rev.
**2018**, 118, 3305. [Google Scholar] [CrossRef] - Frank, I.; Hutter, J.; Marx, D.; Parrinello, M. Molecular dynamics in low-spin excited states. J. Chem. Phys.
**1998**, 108, 4060. [Google Scholar] [CrossRef] - Tavernelli, I.; Röhrig, U.; Roethlisberger, U. Molecular dynamics in electronically excited states using time-dependent density functional theory. Mol. Phys.
**2005**, 103, 963. [Google Scholar] [CrossRef] - Alonso, J.L.; Andrade, X.; Echenique, P.; Falceto, F.; Prada-Gracia, D.; Rubio, A. Efficient Formalism for Large-Scale Ab Initio Molecular Dynamics based on Time-Dependent Density Functional Theory. Phys. Rev. Lett.
**2008**, 101, 096403. [Google Scholar] [CrossRef] [Green Version] - Lopata, K.; Govind, N. Modelling Fast Electron Dynamics with Real-Time Time-Dependent Density Functional Theory: Application to Small Molecules and Chromophores. J. Chem. Theory Comput.
**2011**, 7, 1344. [Google Scholar] [CrossRef] - Lian, C.; Hu, S.Q.; Guan, M.X.; Meng, S. Momentum-resolved TDDFT algorithm in atomic basis for real time tracking of electronic excitation. J. Chem. Phys.
**2018**, 149, 154104. [Google Scholar] [CrossRef] - Lian, C.; Ali, Z.A.; Kwon, H.; Wong, B.M. Indirect but Efficient: Laser-Excited Electrons Can Drive Ultrafast Polarization Switching in Ferroelectroc Materials. J. Phys. Chem. Lett.
**2019**, 10, 3402. [Google Scholar] [CrossRef] [PubMed] - Marx, D.; Hutter, J. Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem.
**2006**, 27, 1787–1799. [Google Scholar] [CrossRef] [PubMed] - Troullier, N.; Martins, J.L. Efficient Pseudopotentials for Plane-Wave Calculations. Phys. Rev. B
**1991**, 43, 1993. [Google Scholar] [CrossRef] [PubMed] - Boero, M.; Parrinello, M.; Terakura, K.; Weiss, H. Car-Parrinello study of Ziegler-Natta heterogeneous catalysis: Stability and destabilization problems of the active site models. Mol. Phys.
**2002**, 100, 2935–2940. [Google Scholar] [CrossRef] - Bernardi, F.; Olivucci, M.; Robb, M.A. Potential energy surface crossings in organic photochemistry. Chem. Soc. Rev.
**1996**, 25, 321. [Google Scholar] [CrossRef] - Nonnenberg, C.; Grimm, S.; Frank, I. Restricted open-shell Kohn-Sham theory for π-π
^{*}transitions. II. Simulation of photochemical reactions. J. Chem. Phys.**2003**, 119, 11585. [Google Scholar] [CrossRef] - Frank, I.; Damianos, K. Restricted Open-Shell Kohn-Sham Theory: Simulation of the Pyrrole Photodissociation. J. Chem. Phys.
**2007**, 126, 125105. [Google Scholar] [CrossRef] - Shu, Y.; Truhlar, D.G. Diabatization by machine intelligence. J. Chem. Theory Comput.
**2020**, 16, 6456. [Google Scholar] [CrossRef] - Sobolewski, A.L.; Domcke, W. Photoinduced Electron and Proton Transfer in Phenol and Its Clusters with Water and Ammonia. J. Phys. Chem. A
**2001**, 105, 9275. [Google Scholar] [CrossRef] - Ben-Nun, M.; Molnar, F.; Lu, H.; Phillips, J.C.; Martinez, T.J.; Schulten, K. Quantum dynamics of the femtosecond photoisomerization of retinal in bacteriorhodopsin. Faraday Discuss.
**1998**, 110, 447. [Google Scholar] [CrossRef] - Röhrig, U.; Guidoni, L.; Laio, A.; Frank, I.; Röthlisberger, U. A molecular spring for vision. J. Am. Chem. Soc.
**2004**, 126, 15328. [Google Scholar] [CrossRef] - Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures on Physics, 2nd ed.; Pearson Education: London, UK; California Institute of Technology: Pasadena, CA, USA, 2006. [Google Scholar]
- Yu, Q.; Pavosevic, F.; Hammes-Schiffer, S. Development of nuclear basis sets for multicomponent quantum chemistry methods. J. Chem. Phys.
**2020**, 152, 244123. [Google Scholar] [CrossRef] - Tao, Z.; Roy, S.; Schneider, P.E.; Pavosevic, F.; Hammes-Schiffer, S. Analytical Gradients for Nuclear–Electronic Orbital Time-Dependent Density Functional Theory: Excited-State Geometry Optimizations and Adiabatic Excitation Energies. J. Chem. Theory Comput.
**2021**, 17, 5110. [Google Scholar] [CrossRef] - Rohloff, E.; Rudolph, D.A.; Strolka, O. Classical nuclear motion: Does it fail to explain reactions and spectra in certain cases? Int. J. Quantum Chem.
**2022**, 122, e26902. [Google Scholar] [CrossRef]

**Figure 1.**HOMO of phenol during the photoreaction (ROKS simulation). Apart from a rapid change of sign between 12 and 24 fs, not much happens to the electronic structure. A hydrogen atom is expelled at the end of the simulation (at the right side of the molecule).

**Figure 2.**LUMO of phenol during the photoreaction (ROKS simulation). The motion starts from a ${\pi}^{*}$ orbital which has already an antibinding interaction concerning the O–H bond. The elongation of this bond leads to the formation of the 1s orbital of the dissociating hydrogen atom.

**Figure 3.**Photoreaction of [2,2${}^{\prime}$-bipyridyl]-3,3${}^{\prime}$-diol, ROKS simulation. Upper and lower panel: reaction of the two OH groups, respectively. The isomerization events, which are characterized by a crossing of the orange and red curves, are not exactly simultaneous, but follow closely one after the other. Color code of educt and product plots: white: hydrogen, black: carbon, green: nitrogen, red: oxygen.

**Figure 4.**HOMO and LUMO of [2,2′-bipyridyl]-3,3′-diol before and after the photoreaction. The central carbon–carbon bond is strengthened in the excited state and adopts double bond character.

**Figure 5.**(

**a**) LUMO of [2,2′-bipyridyl]-3-ol; (

**b**) LUMO of bipyridine. The LUMOs resemble the LUMO of bipyridyl-diol. Upon occupation of this orbital, the central carbon–carbon bond is strengthened. This prevents a rotation.

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**MDPI and ACS Style**

Frank, I.
Classical Nuclear Motion: Comparison to Approaches with Quantum Mechanical Nuclear Motion. *Hydrogen* **2023**, *4*, 11-21.
https://doi.org/10.3390/hydrogen4010002

**AMA Style**

Frank I.
Classical Nuclear Motion: Comparison to Approaches with Quantum Mechanical Nuclear Motion. *Hydrogen*. 2023; 4(1):11-21.
https://doi.org/10.3390/hydrogen4010002

**Chicago/Turabian Style**

Frank, Irmgard.
2023. "Classical Nuclear Motion: Comparison to Approaches with Quantum Mechanical Nuclear Motion" *Hydrogen* 4, no. 1: 11-21.
https://doi.org/10.3390/hydrogen4010002