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Surface hopping methods for nonadiabatic dynamics in extended systems

Advanced Review

Linjun Wang, Jing Qiu, Xin Bai, Jiabo Xu

Published Online: Jul 29 2019

DOI: 10.1002/wcms.1435

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Abstract The review describes recent method developments toward application of the trajectory surface hopping approach for nonadiabatic dynamics simulations of extended systems. Due to the ease of implementation and good balance between efficiency and reliability, surface hopping has become one of the most widely used mixed quantum‐classical methods for studying general charge and exciton dynamics. In extended systems (e.g., aggregates, polymers, surfaces, interfaces, and solids), however, surface hopping suffers from the difficulty to treat complex surface crossings in the adiabatic representation, and thus the relevant applications have been limited in the past years. The latest studies have allowed us to make a systematic classification of the surface crossings and identify their different influence mechanisms on the traditional surface hopping machinery, including problems related to the phase uncertainty correction of adiabatic states, the wave function propagation, the calculation of hopping probabilities, the velocity adjustment after surface hops, and the artificial long‐range population transfer amplified by decoherence corrections. Elegant solutions to each of these problems have enabled us to get fast time step convergence and size independence even for very large systems with different strengths of electron–phonon couplings. Thereby, the recent theoretical progresses have opened the door to simulate the real‐time and real‐space dynamics (e.g., charge separation, recombination, relaxation, and diffusion) in realistic extended systems, and will generate comprehensive understanding to promote the development of many research fields in chemistry, physics, biology, and material sciences in the near future. This article is categorized under: Structure and Mechanism > Computational Materials Science Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods

Schematic representation of (a) type 1, (b) type 2, (c) type 3, and (d) type 4 surface crossings during a time step. The adiabatic potential energy surfaces are represented by solid lines with different colors and the adiabatic states at both time t and t + dt are shown as indicated. The Roman numerals represent trivial crossings with the active or nonactive surfaces. Only representative trivial crossings are shown to guide the eyes. Reprinted with permission from Reference

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Distribution of the minimum energy gap between all adiabatic states for Holstein models (see Equation (10)) with different system sizes. The electronic coupling is set to be (a) 5 meV and (b) 50 meV, respectively. The other parameters are: m = 100 amu, ℏω = 0.01 eV, λ = 0.1 eV, and T = 300 K. Reprinted with permission from Reference

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Representative adiabatic potential energy surfaces of the spin‐boson model in Equation (7) for (a) an avoided crossing with V = 0.5 a.u. and (b) a trivial crossing with V = 0.01 a.u. The dashed blue lines represent the nonadiabatic couplings. The other parameters are chosen as K = α = 2.0 a.u.

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Distribution of the four types of surface crossings in (a) full space CC‐FSSH and (b) subspace CC‐FSSH for different system size N with dt = 1 fs. The Holstein model with V = 10 cm⁻¹ is adopted. MSD at 2 ps as a function of N by (c) full space CC‐FSSH (semi‐transparent solid squares) and CC‐GFSH (open squares), and (d) subspace CC‐FSSH (semi‐transparent solid circles) and CC‐GFSH (open circles) for V = 10 (purple), 50 (red), 200 (green) and 800 (blue) cm⁻¹ with dt = 1 fs. For each V, a horizontal dashed line is shown to guide the eyes. Reprinted with permission from Reference

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The MSD at 2 ps as a function of the time step size, dt, for (a) V = 10 cm⁻¹ with N = 9, (b) V = 50 cm⁻¹ with N = 17, (c) V = 200 cm⁻¹ with N = 45, and (d) V = 800 cm⁻¹ with N = 101 by SC‐FSSH, mSC‐FSSH, CC‐FSSH, SC‐FSSH‐RD and mSC‐FSSH‐RD. Reprinted with permission from Reference

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Time evolution of the charge population on each molecular site in representative trajectories obtained by different surface hopping strategies in the Holstein model with V = 200 cm⁻¹ at 300 K: (a) SC‐FSSH with dt = 0.1 fs, (b) SC‐FSSH with dt = 0.001 fs, and (c) SC‐FSSH‐RD with dt = 0.1 fs and P_c = 0.0001. The system consists of 45 molecular sites, and the initial charge is localized on the central molecule. Reprinted with permission from Reference

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Transmission/reflection population on the upper/lower surface for (a) and (b) the simple avoided crossing, (c) and (d) the dual avoided crossing, (e) and (f) the extended coupling with reflection models. Reprinted with permission from Reference

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