^{1}and Ivano Tavernelli

^{1,a)}

### Abstract

In this work, we present a complete derivation of the NonAdiabatic Bohmian DYnamics (NABDY) equations of motion. This approach naturally emerges from a transformation of the molecular time-dependent Schrödinger equation in the adiabatic representation of the electronic states. The numerical implementation of the method is discussed while simple nonadiabatic models are employed to address the accuracy of NABDY and to reveal its ability to capture nuclear quantum effects that are missed in trajectory surface hopping (TSH) due to the independent trajectory approximation. A careful comparison of the correlated, NABDY, and the uncorrelated, TSH, propagation is also given together with a description of the main approximations and assumptions underlying the “derivation” of a nonadiabatic molecular dynamics scheme based on classical trajectories.

We are grateful to Jiří Vaníček and Tomáš Zimmermann for providing us a version of their exact propagation code, and we thank Felipe Franco de Carvalho and Marine Bouduban for stimulating discussions. COST actions CM0702 and CM1204, Swiss National Science Foundation grants 200021-137717 and 200021-146396, and the NCCR-MUST interdisciplinary research program are acknowledged for funding and support.

I. INTRODUCTION

II. THEORY

A. Nonadiabatic Bohmian dynamics

B. Trajectory surface hopping from the nonadiabatic Bohmian dynamicsequations

III. NUMERICAL IMPLEMENTATION OF NABDY

A. Overview

B. Adiabatic representation

C. Computational details

IV. APPLICATIONS ON MODEL SYSTEMS

A. Nonadiabaticdynamics of NaI as a test system

B. Double arch model system

V. INSIGHTS ON TRAJECTORY SURFACE HOPPING FROM NONADIABATIC QUANTUM TRAJECTORIES

A. Notation

B. Single nonadiabatic crossing: NaI

C. Double arch model:Dynamics at the second nonadiabatic event

VI. CONCLUSIONS

### Key Topics

- Non adiabatic reactions
- 68.0
- Non adiabatic couplings
- 40.0
- Equations of motion
- 28.0
- Surface dynamics
- 21.0
- Trajectory models
- 14.0

## Figures

Cartoon representation of the different steps in NABDY propagation (GS = ground state and ES = excited state). The headings correspond to the different paragraphs in Sec. III A .

Cartoon representation of the different steps in NABDY propagation (GS = ground state and ES = excited state). The headings correspond to the different paragraphs in Sec. III A .

Schematic representation of the NABDY algorithm.

Schematic representation of the NABDY algorithm.

ALE frame adapted for nonadiabatic quantum trajectories (GS = ground state and ES = excited state). (a) Initial step in the Lagrangian frame for all electronic states involved, and (b) ALE step with a common grid.

ALE frame adapted for nonadiabatic quantum trajectories (GS = ground state and ES = excited state). (a) Initial step in the Lagrangian frame for all electronic states involved, and (b) ALE step with a common grid.

Nonadiabatic dynamics of photoexcited NaI. Final population in S1 (ionic state at large internuclear distances) is given for different initial momenta p 0. NABDY, TSH, and an exact solution of the time-dependent Schrödinger equation are compared. The dynamics starts in the first excited state (S1) which has a covalent character at equilibrium distance. TSH dynamics employs 1979 classical independent trajectories, whereas the maximum total number of ALE quantum trajectories used in NABDY is 136. Inset: Schematic representation of the NaI system. Potential energy curves are represented with black lines, nonadiabatic coupling with a red dashed line, and the initial nuclear wavepacket in grey.

Nonadiabatic dynamics of photoexcited NaI. Final population in S1 (ionic state at large internuclear distances) is given for different initial momenta p 0. NABDY, TSH, and an exact solution of the time-dependent Schrödinger equation are compared. The dynamics starts in the first excited state (S1) which has a covalent character at equilibrium distance. TSH dynamics employs 1979 classical independent trajectories, whereas the maximum total number of ALE quantum trajectories used in NABDY is 136. Inset: Schematic representation of the NaI system. Potential energy curves are represented with black lines, nonadiabatic coupling with a red dashed line, and the initial nuclear wavepacket in grey.

Nonadiabatic dynamics of photoexcited NaI. Convergence of the final population in the electronic state S1 in NABDY and TSH dynamics with respect to the number of trajectories employed. In both cases, the initial momentum is set to p 0 = 400 a.u.

Nonadiabatic dynamics of photoexcited NaI. Convergence of the final population in the electronic state S1 in NABDY and TSH dynamics with respect to the number of trajectories employed. In both cases, the initial momentum is set to p 0 = 400 a.u.

Nonadiabatic dynamics of photoexcited NaI. The effect of the threshold value ε on the final population of state S1 is investigated by looking at the dependence on the delay time τ used to switch on the dynamics in GS once entered into the region of S1/GS coupling (ε(τ) is a monotonic growing function of τ until the maximum of the nonadiabatic coupling is reached). The initial momentum is set to p 0 = 400 a.u. The red dotted curve indicates the exact final population of state S1 for the same initial momentum. Inset: Representation of the wavepacket probability density |A(x, τ)|2 for given delay time τ (from the left: τ = 0.1, 41.2, 63.8, 105.7, 126.9, 196.5, 263.4, and 357.3 a.u.). The nonadiabatic coupling function |d 21(x)| is represented by a dashed red line.

Nonadiabatic dynamics of photoexcited NaI. The effect of the threshold value ε on the final population of state S1 is investigated by looking at the dependence on the delay time τ used to switch on the dynamics in GS once entered into the region of S1/GS coupling (ε(τ) is a monotonic growing function of τ until the maximum of the nonadiabatic coupling is reached). The initial momentum is set to p 0 = 400 a.u. The red dotted curve indicates the exact final population of state S1 for the same initial momentum. Inset: Representation of the wavepacket probability density |A(x, τ)|2 for given delay time τ (from the left: τ = 0.1, 41.2, 63.8, 105.7, 126.9, 196.5, 263.4, and 357.3 a.u.). The nonadiabatic coupling function |d 21(x)| is represented by a dashed red line.

Nonadiabatic dynamics for the double arch system. Final population in S1 is given for different initial momenta p 0. NABDY, TSH, and an exact solution of the time-dependent Schrödinger equation are compared. TSH labels trajectory surface hopping runs with 1500 trajectories and different initial conditions (“TSH”: initial conditions sampled from a Gaussian distribution for positions and momenta (see Sec. III C ); “TSH*”: same initial conditions (momentum and position) for all trajectories). The maximum total number of ALE quantum trajectories used in NABDY is 162. Inset: Schematic representation of the double-arch system. Potential energy curves are represented with black lines, nonadiabatic couplings with a red dashed line, and the initial nuclear wavepacket in grey.

Nonadiabatic dynamics for the double arch system. Final population in S1 is given for different initial momenta p 0. NABDY, TSH, and an exact solution of the time-dependent Schrödinger equation are compared. TSH labels trajectory surface hopping runs with 1500 trajectories and different initial conditions (“TSH”: initial conditions sampled from a Gaussian distribution for positions and momenta (see Sec. III C ); “TSH*”: same initial conditions (momentum and position) for all trajectories). The maximum total number of ALE quantum trajectories used in NABDY is 162. Inset: Schematic representation of the double-arch system. Potential energy curves are represented with black lines, nonadiabatic couplings with a red dashed line, and the initial nuclear wavepacket in grey.

NaI model – Time derivatives of the amplitudes (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1). Upper panel: NABDY. Different terms of Eq. (5.2) : (• − •, the dots locate the position of the grid in the ALE frame), (− − −), and (⋯·). The last row reports the probability densities of the two nuclear wavepackets (in GS and S1) and the corresponding first-order nonadiabatic coupling term. Lower panel: pTSH. The dots (•) correspond to term #1 of Eq. (5.4) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The grey lines represent the same quantities obtained for a single trajectory along the complete dynamics. In this case, all trajectories reproduces the same curve. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

NaI model – Time derivatives of the amplitudes (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1). Upper panel: NABDY. Different terms of Eq. (5.2) : (• − •, the dots locate the position of the grid in the ALE frame), (− − −), and (⋯·). The last row reports the probability densities of the two nuclear wavepackets (in GS and S1) and the corresponding first-order nonadiabatic coupling term. Lower panel: pTSH. The dots (•) correspond to term #1 of Eq. (5.4) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The grey lines represent the same quantities obtained for a single trajectory along the complete dynamics. In this case, all trajectories reproduces the same curve. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

NaI model – Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1). Upper panel: NABDY. Different terms of Eq. (5.1) : (• − •, the dots locate the position of the grid in the ALE frame), (− − −), (⋯·), and (-··-). The last row reports the probability densities of the two nuclear wavepackets (in GS and S1) and the corresponding first-order nonadiabatic coupling term. Lower panel: pTSH. The dots (•) correspond to term #1 of Eq. (5.3) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The grey lines represent the same quantities obtained for a single trajectory along the complete dynamics. In this case, all trajectories reproduces the same curve. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

NaI model – Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1). Upper panel: NABDY. Different terms of Eq. (5.1) : (• − •, the dots locate the position of the grid in the ALE frame), (− − −), (⋯·), and (-··-). The last row reports the probability densities of the two nuclear wavepackets (in GS and S1) and the corresponding first-order nonadiabatic coupling term. Lower panel: pTSH. The dots (•) correspond to term #1 of Eq. (5.3) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The grey lines represent the same quantities obtained for a single trajectory along the complete dynamics. In this case, all trajectories reproduces the same curve. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

Double arch model (second nonadiabatic region) – Time derivatives of the amplitudes (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1). Upper panel: NABDY. Different terms of Eq. (5.2) : (• − •, the dots locate the position of the grid in the ALE frame), (− − −), and (⋯·). The last row reports the probability densities of the two nuclear wavepackets (in GS and S1) and the corresponding first-order nonadiabatic coupling term. Lower panel: pTSH. The dots (•) correspond to term #1 of Eq. (5.4) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The grey lines represent the same quantities obtained for single trajectories along the complete dynamics. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

Double arch model (second nonadiabatic region) – Time derivatives of the amplitudes (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1). Upper panel: NABDY. Different terms of Eq. (5.2) : (• − •, the dots locate the position of the grid in the ALE frame), (− − −), and (⋯·). The last row reports the probability densities of the two nuclear wavepackets (in GS and S1) and the corresponding first-order nonadiabatic coupling term. Lower panel: pTSH. The dots (•) correspond to term #1 of Eq. (5.4) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The grey lines represent the same quantities obtained for single trajectories along the complete dynamics. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

Double arch model (second nonadiabatic region) - Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1). Upper panel: NABDY. Different terms of Eq. (5.1) : (• − •, the dots locate the position of the grid in the ALE frame), (− − −), (⋯·), and (-··-). The last row reports the probability densities of the two nuclear wavepackets (in GS and S1) and the corresponding first-order nonadiabatic coupling term. Lower panel: pTSH. The dots (•) correspond to term #1 of Eq. (5.3) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The grey lines represent the same quantities obtained for single trajectories along the complete dynamics. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

Double arch model (second nonadiabatic region) - Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1). Upper panel: NABDY. Different terms of Eq. (5.1) : (• − •, the dots locate the position of the grid in the ALE frame), (− − −), (⋯·), and (-··-). The last row reports the probability densities of the two nuclear wavepackets (in GS and S1) and the corresponding first-order nonadiabatic coupling term. Lower panel: pTSH. The dots (•) correspond to term #1 of Eq. (5.3) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The grey lines represent the same quantities obtained for single trajectories along the complete dynamics. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

Schematic representation of the time evolution of a FE in trajectory-based nonadiabatic dynamics. The local equal-time correlation in TSH is represented by vertical lines while the nonlocal equal-time correlations captured within NABDY is shown by a wavy line.

Schematic representation of the time evolution of a FE in trajectory-based nonadiabatic dynamics. The local equal-time correlation in TSH is represented by vertical lines while the nonlocal equal-time correlations captured within NABDY is shown by a wavy line.

Double arch model (second nonadiabatic region) – Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1) for the case in which all trajectories are initiated from a Gaussian distribution of configurations with the same momentum (σ p = 0). pTSH. The dots (•) correspond to term #1 of Eq. (5.3) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

Double arch model (second nonadiabatic region) – Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate x taken at three different times (black = GS, blue = S1) for the case in which all trajectories are initiated from a Gaussian distribution of configurations with the same momentum (σ p = 0). pTSH. The dots (•) correspond to term #1 of Eq. (5.3) (divided by 10) at the positions sampled by all 483 trajectories at the given time. The last row reports the histograms of the positions sampled by all classical trajectories in their corresponding driving state at the given times and the corresponding first-order nonadiabatic coupling term.

Adiabatic ALE quantum trajectories propagation scheme applied to the collision of a nuclear wavepacket with an Eckart potential (black solid curve). Upper panel: Initial conditions, the red dotted line corresponds to the initial value of the kinetic energy of the nuclear wavepacket. Middle panel: wavepacket at t = 1930 a.u. when the pure QTM propagation breaks down due to a quasinode formation. Lower panel: wavepacket at t = 5560 a.u. with the modified QTM scheme for different thresholds η (dotted lines indicate a × 10 zoom on the transmitted part of the nuclear wavepacket).

Adiabatic ALE quantum trajectories propagation scheme applied to the collision of a nuclear wavepacket with an Eckart potential (black solid curve). Upper panel: Initial conditions, the red dotted line corresponds to the initial value of the kinetic energy of the nuclear wavepacket. Middle panel: wavepacket at t = 1930 a.u. when the pure QTM propagation breaks down due to a quasinode formation. Lower panel: wavepacket at t = 5560 a.u. with the modified QTM scheme for different thresholds η (dotted lines indicate a × 10 zoom on the transmitted part of the nuclear wavepacket).

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