# A Path Integral Molecular Dynamics Simulation of a Harpoon-Type Redox Reaction in a Helium Nanodroplet

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

**2016**, 145, 181101), this spatially quenched reaction was characterized as a harpoon-type or long-range electron transfer in a previous high-level ab initio study (de Lara-Castells et al., J. Phys. Chem. Lett.

**2017**, 8, 4284). To go beyond the static approach, classical and quantum PIMD simulations are performed at 2 K, slightly below the critical temperature for helium superfluidity (2.172 K). Calculations are executed in the NVT ensemble as well as the NVE ensemble to provide insights into real-time dynamics. A droplet size of 2090 atoms is assumed to study the impact of spatial hindrance on reactivity. By changing the number of beads in the PIMD simulations, the impact of quantization can be studied in greater detail and without an implicit assumption of superfluidity. We find that the reaction probability increases with higher levels of quantization. Our findings confirm earlier, static predictions of a rotational motion of the Cs${}_{2}$ dimer upon reacting with the fullerene, involving a substantial displacement of helium. However, it also raises the new question of whether the interacting species are driven out-of-equilibrium after impurity uptake, since reactivity is strongly quenched if a full thermal equilibration is assumed. More generally, our work points towards a novel mechanism for long-range electron transfer through an interplay between nuclear quantum delocalization within the confining medium and delocalized electronic dispersion forces acting on the two reactants.

## 1. Introduction

## 2. Potential Energy Surfaces and Hopping Neutral/Ionic Probabilities

#### 2.1. Analytic Potential Energy Surfaces

#### 2.2. Hopping Neutral/Ionic Probabilities

## 3. Computational Method and Details

#### 3.1. Initial Arrangement

#### 3.2. PIMD Simulation

## 4. Results

#### Real-Time Simulations in the NVE Ensemble

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**Diabatic potential energy curves characterizing the neutral Cs${}_{2}$-C${}_{60}$ and ionic Cs${}_{2}^{+}$-C${}_{60}^{-}$ pair interactions as function of their relative distance R at collinear and T-shaped orientations. Points: ab initio calculations; solid lines: analytic fits, see text.

**Figure 2.**Analytic potential energy surfaces characterizing neutral Cs${}_{2}$-C${}_{60}$ and ionic Cs${}_{2}^{+}$-C${}_{60}^{-}$ pair interactions as a function of their relative distance R and orientation $\theta $.

**Figure 3.**Computed (points) and fitted (lines) of probability for electron hopping obtained using the Landau–Zener model at collinear and T-shaped orientations as function of relative velocity [42]. Simple analytic extension at whatever be the orientation is also shown for an intermediate orientation $\theta =\pi /4$.

**Figure 4.**Initial structure of Cs${}_{2}$-C${}_{60}$@He${}_{2090}$ complex obtained by a combined evolutionary algorithm and geometric extension, see text.

**Figure 5.**Time evolution of the total energy of the Cs${}_{2}$-C${}_{60}$@He${}_{2090}$ system at $T=2$ K, obtained in simulations performed at the indicated number of beads M. Solid lines correspond to initial relative velocities of the Cs${}_{2}$-C${}_{60}$ interacting pair (${v}_{0}$) indicated in the figure, while dashed lines, maintaining the relation color/M, correspond to a common initial velocity ${v}_{0}=25.4$ m/s.

**Figure 6.**Evolution of relative Cs${}_{2}$-C${}_{60}$ distance (blue solid lines, right vertical axis) and velocity (red solid lines, left vertical axis) as a function of time. It was obtained through PIMD simulations for number M of beads considered, with common initial relative velocity being ${v}_{0}=25.4$ m/s. Constant dashed lines indicate the critical Landau velocity (shown in red) as well as the crossing distance between neutral and ionic potential energy curves at initial T-shaped orientation (shown in blue).

**Figure 7.**Evolution of relative Cs${}_{2}$-C${}_{60}$ distance (blue solid lines) and velocity (red solid lines), plotted as a function of time, and obtained through simulations for M beads ($M=1$, 5, and 10) but with different initial relative velocities of ${v}_{0}=129$ m/s ($M=1$), $169.4$ m/s ($M=5$), and $239.8$ m/s ($M=10$). Landau critical velocity and crossing distance between neutral and ionic potential energy curves at T-shaped initial orientation are also shown with dashed red and blue lines, respectively. Arrows indicate relative velocity of reactants when they get crossing distance between neutral and ionic potential energy curves.

**Figure 8.**For $M=10$, evolution of relative Cs${}_{2}$-C${}_{60}$ distance, R and orientation, $\theta $, as function of time. Dashed lines correspond to an initial velocity ${v}_{0}=25.4$ m/s, while solid lines show simulation starting with ${v}_{0}=239.8$ m/s. Blue lines: relative distance between two reactants; red lines: relative orientation; black lines: distance from C${}_{60}$ to its original position at time 0.

**Figure 9.**For $M=10$ and ${v}_{0}=25.4$ m/s, and using cylindric coordinates with z-axis in direction of line joining Cs${}_{2}$ and C${}_{60}$ mass centers, final He density distribution ($\times {10}^{-3}$ Å${}^{-2}$). Positions of Cs${}_{2}$ and C${}_{60}$ impurities are also indicated.

**Figure 10.**Final He density distribution ($\times {10}^{-3}$ Å${}^{-2}$) in cylindric coordinates for $M=10$ when considering an initial relative velocity ${v}_{0}=239.8$ m/s. Positions of Cs${}_{2}$ and C${}_{60}$ impurities are also indicated.

**Figure 11.**Structure of Cs${}_{2}$-C${}_{60}$@He${}_{2090}$ system at 40 ps from a simulation with $M=10$ and ${v}_{0}=239.8$ m/s.

**Table 1.**Parameters defining the analytical description of the Cs${}_{2}$-C${}_{60}$ and Cs${}_{2}^{+}$-C${}_{60}^{-}$ interactions at collinear (C) and T-shaped (T) arrangements, Equation (6), as well as the He-C${}_{60}$ interaction, Equation (4). Energies and distances are given in cm${}^{-1}$ and Å, respectively. These global units determine those corresponding to the different parameters. For instance, the unit of the ${C}_{6}$ coefficient parameter is cm${}^{-1}$Å${}^{6}$.

Cs${}_{2}$-C${}_{60}$ | Cs${}_{2}^{+}$-C${}_{60}^{-}$ | He-C${}_{60}$ | |||||
---|---|---|---|---|---|---|---|

C | T | C | T | ||||

${A}_{2}$ | −2.22575d05 | 0 | ${B}_{0}$ | 5.89589d03 | 4.52665d03 | ||

${A}_{4}$ | 9.66771d07 | 0 | ${B}_{1}$ | −1.37653d05 | −1.03277d05 | ||

${A}_{6}$ | −1.60255d10 | 9.598560d7 | ${B}_{6}$ | −3.91681d08 | −3.39629d8 | ${F}_{6}$ | −1.90652d07 |

${A}_{8}$ | 1.18367d12 | −1.37271d11 | ${B}_{12}$ | 6.40779d12 | 8.58655d12 | ${F}_{8}$ | 3.76378d09 |

${A}_{10}$ | −4.10087d13 | 1.26854d13 | ${D}_{0}$ | 5.14755d03 | 5.14755d03 | ${F}_{10}$ | −2.79654d11 |

${A}_{12}$ | 5.48817d14 | −4.37303d14 | ${D}_{1}$ | 1.16141d05 | 1.16141d05 | ${F}_{12}$ | 6.20350d12 |

${A}_{14}$ | 0 | 5.43005d15 | ${q}_{0}$,${q}_{1}$ | 0 | 0 | ||

${C}_{6}$ | −2.71882d08 | −2.71882d08 | ${q}_{6}$,${q}_{12}$ | −3.9 | 0 | ||

${R}^{\left(n\right)}$ | 12.5 | 12.5 | ${R}^{\left(i\right)}$ | 28.0 | 20.0 |

**Table 2.**Parameters defining analytical description of electron hopping probability at the collinear (C) and T-shaped (T) orientations, Equation (8). Units of a and b are m${}^{2}$/s${}^{2}$, while those of c and d are m${}^{4}$/s${}^{4}$.

C | T | |
---|---|---|

a | 7047.479 | 1343.188 |

b | 10,489.841 | 2226.989 |

c | 469,477.985 | 13,407.221 |

d | 2,797,586.355 | 43,037.030 |

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Castillo-García, A.; Hauser, A.W.; de Lara-Castells, M.P.; Villarreal, P.
A Path Integral Molecular Dynamics Simulation of a Harpoon-Type Redox Reaction in a Helium Nanodroplet. *Molecules* **2021**, *26*, 5783.
https://doi.org/10.3390/molecules26195783

**AMA Style**

Castillo-García A, Hauser AW, de Lara-Castells MP, Villarreal P.
A Path Integral Molecular Dynamics Simulation of a Harpoon-Type Redox Reaction in a Helium Nanodroplet. *Molecules*. 2021; 26(19):5783.
https://doi.org/10.3390/molecules26195783

**Chicago/Turabian Style**

Castillo-García, Alvaro, Andreas W. Hauser, María Pilar de Lara-Castells, and Pablo Villarreal.
2021. "A Path Integral Molecular Dynamics Simulation of a Harpoon-Type Redox Reaction in a Helium Nanodroplet" *Molecules* 26, no. 19: 5783.
https://doi.org/10.3390/molecules26195783