Research

  • 2017-present
  1. Moving boundary truncated grid method

    The moving boundary truncated grid method, developed by our group and further enhanced through collaboration with Dr. Chun-Yaung Lu at the Texas Advanced Computing Center, marks a significant advancement in computational quantum mechanics. This method dynamically allocates Eulerian grid points, which are activated or deactivated based on the evolving wave packet dynamics, without requiring prior knowledge. It has been effectively applied to various problems, including the time-dependent Schrodinger equations for photodissociation dynamics, barrier scattering problems, and the study of electronic nonadiabatic transitions in multidimensional quantum systems. In addition to these applications, it has also been employed for the time evolution of distribution functions in phase space, integrating the equations of motion like the Klein-Kramers, Wigner-Moyal, and modified Caldeira-Leggett equations, and accurately capturing negative basins in phase space resulting from interference effects. Demonstrating high accuracy and computational efficiency, this approach requires fewer grid points compared to traditional large space-fixed grid calculations, and is versatile enough to be used in complex applications ranging from the photodissociation of NOCl and barrier scattering problems to the dynamics of photoisomerization of retinal in rhodopsin.

    References:
    • Tsung-Yen Lee and Chia-Chun Chou, "Moving boundary truncated grid method for wave packet dynamics", J. Phys. Chem. A 122, 1451 (2018).
    • Chun-Yaung Lu, Tsung-Yen Lee, and Chia-Chun Chou, "Moving boundary truncated grid method: Multidimensional quantum dynamics", Int. J. Quantum Chem. 120, e26055 (2020).
    • Tsung-Yen Lee, Chun-Yaung Lu, and Chia-Chun Chou, "Moving boundary truncated grid method: Application to the time evolution of distribution functions in phase space", J. Phys. Chem. A 125, 476 (2021).
    • Chun-Yaung Lu, Tsung-Yen Lee, and Chia-Chun Chou, "Moving boundary truncated grid method for electronic nonadiabatic dynamics", J. Chem. Phys. 156, 044107 (2022).
  2. Adaptive moving grid methods for electronic nonadiabatic dynamics

    We developed a numerical method for solving the coupled complex quantum Hamilton-Jacobi equations in the diabatic representation for electronic nonadiabatic dynamics using adaptive moving grids. This approach, which transforms the coupled time-dependent Schrodinger equations into the coupled complex quantum Hamilton-Jacobi equations, involves the use of wave functions in exponential form and analytic equations for the spatial derivatives of complex action functions. The method allows grid points to move with the probability fluid, utilizing the arbitrary Lagrangian-Eulerian frame for diverse velocity options. We achieve a structured grid with uniform spacing by aligning boundary grid points with Lagrangian trajectories and maintaining equally spaced internal points, efficiently tracking wave packet evolution. Demonstrating accuracy in both one-dimensional systems like single and dual avoided crossings, and in three-dimensional nonadiabatic systems, this method yields accurate results for time-dependent wave packets and diabatic state populations, even under conditions like ultrashort laser pulse excitation. The adaptive grid method, verified against fixed grid calculations, offers a promising alternative for studying electronic nonadiabatic dynamics.

    Reference:
    • Chia-Chun Chou, "Complex-valued derivative propagation method with adaptive moving grids for electronic nonadiabatic dynamics", Ann. Phys. 445, 169084 (2022)
  3. Schrodinger-Langevin equation for wave packet dynamics

    We integrated the Schrodinger-Langevin equation on a fixed grid to analyze frictional effects in wave packet dynamics, using a hydrodynamic approach that highlights how environmental coupling impedes wave packet propagation and reduces its spreading. Our study encompassed scenarios like Eckart barrier scattering, metastable state decay, and the long-term behavior of decaying quantum systems, offering new insights through dissipative Bohmian trajectories. Additionally, we developed a two-state version of the Schrodinger-Langevin equation for electronic nonadiabatic dynamics, incorporating dissipative potentials into the coupled time-dependent Schrodinger equations in the diabatic representation.  This led to the derivation of hydrodynamic equations for dissipative Bohmian trajectories on each potential surface. We explored frictional effects on nonadiabatic transitions in two model systems, analyzing probability density evolution, quantum trajectory dynamics, and surface population transfers. This research demonstrates that the two-state Schrodinger-Langevin equation effectively models dissipative electronic nonadiabatic dynamics.

    References:
    • Chia-Chun Chou, "Hydrodynamic analysis of the Schrodinger-Langevin equation for wave packet dynamics", Phys. Lett. A 381, 3384 (2017).
    • Ching-Hwa Ho and Chia-Chun Chou, "Dissipative electronic nonadiabatic dynamics within the framework of the Schrodinger-Langevin equation", Eur. Phys. J. Plus 136, 973 (2021).
  4. Trajectory-based analysis of physical systems: From classical and quantum to superquantum regimes

    We employed the transition trajectory formalism previously developed by our group to study the quantum-classical transition in wave packet dynamics, focusing on the scattering from the Eckart barrier and the decay of a metastable state. This approach maintains conservation of total probability and energy across varying degrees of quantumness, transitioning to classical dynamics as the quantum contribution diminishes. Computational analyses demonstrated that wave packet evolution and transmission probabilities converge to classical results as quantumness decreases, with discrepancies arising in regions where single-valued wave functions prevent trajectory crossing. Additionally, using a scaled Schrodinger-Langevin equation, we examined the transition of dissipative systems from quantum to classical regimes, finding that energy dissipation rates remain constant across this spectrum. This research also explored 'superquantum' phenomena by extending the degree of quantumness beyond one. In this regime, simulations revealed enhanced transmission probabilities and quantum tunneling effects in scenarios like Eckart barrier scattering and metastable state decay. Thus, our work provides novel insights into the transition between quantum and classical behaviors and the implications of superquantum effects on physical systems.

    References:
    • Chia-Chun Chou, "Trajectory-based understanding of the quantum-classical transition for barrier scattering", Ann. Phys. 393, 167 (2018).
    • Chia-Chun Chou, "Quantum-classical transition of the dissipative wave packet dynamics for barrier scattering", Int. J. Quantum Chem. 119, e25812 (2019).
    • Chia-Chun Chou, "Superquantum effects on physical systems from a hydrodynamic perspective", Ann. Phys. 461, 169592 (2024).

  • 2012-2017
  1. Complex quantum Hamilton-Jacobi equation with Bohmian trajectories

    We have developed a novel synthetic trajectory approach to wave packet dynamics (the CQHJE-BT method). In this method, wave packets are evolved through the time integration of the complex quantum Hamilton-Jacobi equation (CQHJE) by propagating an ensemble of real-valued Bohmian trajectories (BT), and the complex action and the real Bohmian trajectories are computed concurrently on the fly. This trajectory approach has been applied to quantum dynamical problems such as chemical reactive scattering and photodissociation. For the two-dimensional model reactive scattering problem, time-dependent transmission probabilities are evaluated by the time integration of the probability flux. Excellent computational results can be obtained through the time integration of the CQHJE by propagating an ensemble of trajectories. In addition, the CQHJE-BT method has been applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. The energy-resolved absorption spectrum calculated by the Fourier transform of the autocorrelation function indicates that the broad spectrum corresponds to fast and direct dissociation. The main peak position is in good agreement with the exact full three-dimensional calculations. Moreover, the CQHJE is approximated solved by propagating individual quantum trajectories. Excellent transmitted wave packets and transmission probabilities can be obtained using transmitted trajectories.

    References:
    • Chia-Chun Chou, "Complex quantum Hamilton-Jacobi equation with Bohmian trajectories for wave packet dynamics", Chem. Phys. Lett. 591, 203 (2014).
    • Chia-Chun Chou, "Complex quantum Hamilton-Jacobi equation with Bohmian trajectories: Application to the photodissociation dynamics of NOCl", J. Chem. Phys. 140, 104307 (2014).
    • Chia-Chun Chou, "Two-dimensional reactive scattering with transmitted quantum trajectories", Int. J. Quantum Chem. 115, 419 (2015).
    • Chia-Chun Chou, "Computation of transmission probabilities for thin potential barriers with transmitted quantum trajectories",  Phys. Lett. A 379, 2174 (2015).
    • Chia-Chun Chou, "Complex-valued derivative propagation method with approximate Bohmian trajectories for quantum barrier scattering", Chem. Phys. 457, 160 (2015).
  2. Schrodinger-Langevin equation for the ground state of quantum systems

    The Schrodinger-Langevin equation with linear dissipation is integrated by propagating an ensemble of Bohmian trajectories for the ground state of quantum systems. Excellent ground state energies and wave functions have been obtained for the harmonic oscillator, the double well potential, and the ground vibrational state of methyl iodide. In addition, we also employ the derivative propagation method (DPM) to approximately solve the Schrodinger-Langevin equation for the ground state energy of quantum systems. The ground state energy is calculated from the amplitude of the wave function by propagating only one single trajectory. As computational demonstrations, excellent ground state energies have been obtained for the Morse potential, the strongly anharmonic potential, the coupled Morse oscillator-harmonic oscillator system, and the ground vibrational state of methyl iodide. Furthermore, we have developed dissipative quantum trajectories in complex space in the framework of the logarithmic nonlinear Schrodinger equation. Extending all the relevant functions into complex space, we show that dissipative complex quantum trajectories satisfy a quantum Newtonian equation of motion including a friction force.  As time progresses, the friction term causes the fluid elements in complex space to lose kinetic energy. Analogous to the Schrodinger-Langevin equation, the complex-extended logarithmic nonlinear Schrodinger equation provides a theoretical framework for dissipative quantum trajectories in complex space.

    References:
    • Chia-Chun Chou, "Trajectory approach to the Schrodinger-Langevin equation with linear dissipation for ground states", Ann. Phys. 362, 57 (2015).
    • Chia-Chun Chou, "Ground state energy from the single trajectory propagation of the Schrodinger-Langevin equation", Chem. Phys. Lett. 633, 195 (2015).
    • Chia-Chun Chou, "Dissipative quantum trajectories in complex space: Damped harmonic oscillator", Ann. Phys. 373, 325 (2016).
  3. Schrodinger-Langevin equation for wave packet dynamics

    The Schrodinger-Langevin equation is integrated to study the wave packet dynamics of quantum systems subject to frictional effects by propagating an ensemble of quantum trajectories.  The equations of motion for the complex action and quantum trajectories are derived from the Schrodinger-Langevin equation. The moving least squares approach is used to evaluate the spatial derivatives of the complex action required for the integration of the equations of motion.   Computational results have been presented and analyzed for the evolution of a free Gaussian wave packet, a two-dimensional barrier model, and the photodissociation dynamics of NOCl. For the time evolution of a free Gaussian wave packet, frictional effects not only impede the propagation of the wave packet but also suppress the spreading of the wave packet. The center of the wave packet follows the classical trajectory of a particle subject to a linear damping force.  For the two-dimensional barrier model, the transmission probability is reduced as the magnitude of the friction coefficient increases. In addition, the Schrodinger-Langevin equation has been employed to simulate the photodissociation dynamics of NOCl in condensed environments. Because of frictional effects, the propagation of the wave packet into the NO+Cl dissociation channel is significantly impeded, and the maximum of the absorption spectrum displays a redshift. The computational result is in good qualitative agreement with the experimental results in the solution-phase photochemistry of NOCl.  Moreover, we have also employed the derivative propagation method (DPM) to approximately solve the Schrodinger-Langevin equation for barrier transmission problems.  The equations of motion are derived for the amplitude of the wave function and the spatial derivatives of the complex action, and the Schrodinger-Langevin equation can be
    approximately solved for the amplitude of the wave function by propagating individual quantum trajectories. It has been shown that frictional effects arising from the interaction between the system and the environment not only impede the propagation of the transmitted wave packet but also suppress the spreading of the transmitted wave packet.

    References:
    • Chia-Chun Chou, "Schrodinger-Langevin equation with quantum trajectories for photodissociation dynamics", Ann. Phys. 377, 22 (2017).
    • Chia-Chun Chou, "Approximate quantum trajectory approach to the Schrodinger-Langevin equation for barrier transmission", Chem. Phys. Lett. 669, 181 (2017).
  4. Trajectory description of the quantum-classical transition

    The quantum-classical transition for wave packet interference is investigated within the framework of the nonlinear transition equation using a hydrodynamic description. The transition equation provides a description of the continuous process for physical systems from purely quantum to purely classical regimes.  Based on the nonlinear transition equation, we have developed the transition trajectory formalism to provide a hydrodynamic description for the quantum-classical transition.  This formalism has been employed to analyze the free Gaussian wave packet, the head-on collision of two free Gaussian wave packets, the multiple-slit interference, and the dynamics of a wave packet bouncing off a wall. It has been shown that the interference process changes continuously from a diffraction-like to collision-like case when the degree of quantumness gradually decreases.  This study provides an insightful trajectory interpretation for the quantum-classical transition of wave packet interference.  In addition, we have also derived a quantum-classical transition equation in complex space (QCTE-CS) in the framework of the complex quantum Hamilton-Jacobi formalism.  The QCTE-CS describes the continuous quantum-classical transition of physical systems in complex space.  Because the complex transition trajectory formalism establishes the connection between complex quantum and classical trajectories, it provides an alternative way to study the quantum-classical correspondence.

    References:
    • Chia-Chun Chou, "Trajectory description of the quantum-classical transition for wave packet interference", Ann. Phys. 371, 437 (2016).
    • Chia-Chun Chou, "Quantum-classical transition equation with complex trajectories", Int. J. Quantum Chem. 116, 1752 (2016).
 

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