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On trajectory-based nonadiabatic dynamics: Bohmian dynamics versus trajectory surface hopping
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10.1063/1.4803835
/content/aip/journal/jcp/138/18/10.1063/1.4803835
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/18/10.1063/1.4803835
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

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 .

Image of FIG. 2.
FIG. 2.

Schematic representation of the NABDY algorithm.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

Nonadiabatic dynamics of photoexcited NaI. Final population in S (ionic state at large internuclear distances) is given for different initial momenta . NABDY, TSH, and an exact solution of the time-dependent Schrödinger equation are compared. The dynamics starts in the first excited state (S) 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.

Image of FIG. 5.
FIG. 5.

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

Image of FIG. 6.
FIG. 6.

Nonadiabatic dynamics of photoexcited NaI. The effect of the threshold value ε on the final population of state S 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 S/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 = 400 a.u. The red dotted curve indicates the exact final population of state S for the same initial momentum. Inset: Representation of the wavepacket probability density |(, τ)| 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 | ()| is represented by a dashed red line.

Image of FIG. 7.
FIG. 7.

Nonadiabatic dynamics for the double arch system. Final population in S is given for different initial momenta . 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.

Image of FIG. 8.
FIG. 8.

NaI model – Time derivatives of the amplitudes (nonadiabatic contributions) as a function of the space coordinate taken at three different times (black = GS, blue = S). 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 S) 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.

Image of FIG. 9.
FIG. 9.

NaI model – Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate taken at three different times (black = GS, blue = S). 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 S) 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.

Image of FIG. 10.
FIG. 10.

Double arch model (second nonadiabatic region) – Time derivatives of the amplitudes (nonadiabatic contributions) as a function of the space coordinate taken at three different times (black = GS, blue = S). 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 S) 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.

Image of FIG. 11.
FIG. 11.

Double arch model (second nonadiabatic region) - Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate taken at three different times (black = GS, blue = S). 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 S) 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.

Image of FIG. 12.
FIG. 12.

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.

Image of FIG. 13.
FIG. 13.

Double arch model (second nonadiabatic region) – Time derivatives of the phases (nonadiabatic contributions) as a function of the space coordinate taken at three different times (black = GS, blue = S) for the case in which all trajectories are initiated from a Gaussian distribution of configurations with the same momentum (σ = 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.

Image of FIG. 14.
FIG. 14.

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 = 1930 a.u. when the pure QTM propagation breaks down due to a quasinode formation. Lower panel: wavepacket at = 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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/content/aip/journal/jcp/138/18/10.1063/1.4803835
2013-05-14
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: On trajectory-based nonadiabatic dynamics: Bohmian dynamics versus trajectory surface hopping
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/18/10.1063/1.4803835
10.1063/1.4803835
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