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The bohmion method in nonadiabatic molecular dynamics

Darryl D. Holm, Jonathan Rawlinson, Cesare Tronci

Published Dec 7, 2020• 0 citations
OpenAlex

Abstract

Starting with the exact factorization of the molecular wavefunction, this paper presents the results from the numerical implementation in ab-initio nonadiabatic dynamics of the recently proposed bohmion method. Within the context of quantum hydrodynamics, the regularized nuclear Bohm potential in the bohmion method admits solutions comprising a train of $\delta$-functions which serve as a finite-dimensional sampling of the hydrodynamic flow paths. In addition, the nonlocal structure of the regularized Bohm potential admits nuclear quantum tunneling events. After reviewing the general theory, the bohmion method is applied to the well-known Tully models, which are used here as benchmark problems for the comparison of the new bohmion method with previous mixed quantum-classical methods.

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The bohmion method in nonadiabatic quantum hydrodynamics

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Abstract Starting with the exact factorization of the molecular wavefunction, this paper presents the results from the numerical implementation in nonadiabatic molecular dynamics of the recently proposed bohmion method. Within the context of quantum hydrodynamics, we introduce a regularized nuclear Bohm potential admitting solutions comprising a train of δ-functions which provide a finite-dimensional sampling of the hydrodynamic flow paths. The bohmion method inherits all the basic conservation laws from its underlying variational structure and captures electronic decoherence. After reviewing the general theory, the method is applied to the well-known Tully models, which are used here as benchmark problems. In the present case of study, we show that the new method accurately reproduces both electronic decoherence and nuclear population dynamics.

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The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether’s conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called ‘collective’. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite ħ $\hbar $ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ħ ≠ 0 $\hbar \ne 0$ dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called ‘Bohmions’, which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.

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Nonadiabatic dynamics via the classical limit Schrödinger equation

John C. Burant, John C. Tully

The coupled Schrödinger equations that describe nonadiabatic dynamics are recast using the Bohm formulation of quantum mechanics. The resulting coupled Bohm equations are solved numerically for two scattering models, giving results that are essentially identical to wave-packet solution of the original coupled Schrödinger equations. The classical limit of the set of coupled Bohm equations is then described, producing a mixed quantum-classical theory incorporating classical-like motion on each potential energy surface accompanied by quantum transitions between the quantum states. Numerical tests of the mixed quantum-classical method are in excellent agreement with the accurate full-quantum results for the model problems. The method is contrasted with the related surface-hopping approach. It is shown that computing the dynamics of a distribution of classical particles is more consistent and more accurate than computing the motion of independent point particles as with surface hopping.

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Werner Bauer, Paul Bergold, François Gay–Balmaz +1 more

Abstract. In order to alleviate the computational costs of fully quantum nonadiabatic dynamics, we present a mixed quantum-classical (MQC) particle method based on the theory of Koopman wavefunctions. Although conventional MQC models often suffer from consistency issues such as the violation of Heisenberg’s principle, we overcame these difficulties by blending Koopman’s classical mechanics on Hilbert spaces with methods in symplectic geometry. The resulting continuum model enjoys both a variational and a Hamiltonian structure, while its nonlinear character calls for suitable closures. Benefiting from the underlying action principle, here we apply a regularization technique previously developed within our team. This step allows for a singular solution ansatz that introduces the trajectories of computational particles—the koopmons—sampling the Lagrangian classical paths in phase space. In the case of Tully’s nonadiabatic problems, the method reproduces the results of fully quantum simulations with levels of accuracy that are not achieved by standard MQC Ehrenfest simulations. In addition, the koopmon method is computationally advantageous over similar fully quantum approaches, which are also considered in our study. As a further step, we probe the limits of the method by considering the Rabi problem in both the ultrastrong and the deep strong coupling regimes, where MQC treatments appear hardly applicable. In this case, the method succeeds in reproducing parts of the fully quantum results.

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The Quantum Hydrodynamic Description of Tunneling

Brian K. Kendrick

The quantum hydrodynamic approach is based on the de Broglie-Bohm formulation of quantum mechanics. The resulting fluid-like equations of motion describe the flow of probability and an accurate solution to these equations is equivalent to solving the time-dependent Schroedinger equation. Furthermore, the hydrodynamic approach provides new insight into the mechanisms as well as an alternative computational approach for treating tunneling phenomena. New concepts include well-defined 'quantum trajectories', 'quantum potential', and 'quantum force' all of which have classical analogues. The quantum potential and its associated force give rise to all quantum mechanical effects such as zero point energy, tunneling, and interference. A new numerical approach called the Iterative Finite Difference Method (IFDM) will be discussed. The IFDM is used to solve the set of non-linear coupled hydrodynamic equations. It is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a 'smart' Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is also capable of treating anharmonic potentials. Example calculations using the IFDM will be presented which include: a one-dimensional Gaussian wave packet tunneling through an Eckart barrier, a one-dimensional bound-state Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. Approximate treatments of the quantum hydrodynamic equations will also be discussed which could allow scaling of the calculations to hundreds of degrees of freedom which is important for treating tunneling phenomena in condensed phase systems.

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Two‐dimensional wavepacket dynamics with quantum hydrodynamics

Daisuke Matsumoto, K. Hayashi, Tomonori Ida +3 more

Abstract Wyatt and coworkers introduced a new approach (Wyatt, Quantum Dynamics with Trajectories Introduction to Quantum Hydrodynamics, Springer, New York, 2005; Lopreore and Wyatt, Phys Rev Lett, 1999, 82, 5190; Wyatt et al., J Chem Phys, 2001, 114, 5113; Lopreore and Wyatt, J Chem Phys, 2002, 116, 1228; Trahan and Wyatt, J Chem Phys, 2003, 118, 4784; Trahan et al., J Chem Phys, 2005, 122, 164104; Babyuk and Wyatt, J Chem Phys, 2006, 125, 64112) which is based on quantum hydrodynamics to the quantum wavepacket dynamics and applied to several systems including tunneling and nonadiabatic transition. This approach is expected to overcome some disadvantages of the conventional wavepacket dynamics, such as order‐N problem and use of absorbing potential. We applied this method to two‐dimensional model of quantum tunneling system and compared the new approach with the conventional wavepacket simulation since the validity of this method to multidimensional systems is not well known. In the simulation, we adopted two different model potentials (Model I and Model II) and calculated time dependence of the probability density and the tunneling probability in both models, where we showed explicitly that the results of the quantum hydrodynamics are identical with those of the conventional method. Using the moving grid and the regridding algorithm in the quantum hydrodynamics, we can effectively avoid order‐N problem and use of the absorbing potential. In addition to these computational advantages, the flux vector is easily obtained by the quantum hydrodynamics and these vectors field helps us to intuitively understand detailed tunneling dynamics. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

International Journal of Quantum Chemistry•Jan 1, 2007•6 citations
Starting with the exact factorization of the molecular wavefunction, this paper presents the results from the numerical implementation in ab-initio nonadiabatic dynamics of the recently proposed bohmion method. Within the context of quantum hydrodynamics, the regularized nuclear Bohm potential in the bohmion method admits solutions comprising a train of δ\deltaδ-functions which serve as a finite-dimensional sampling of the hydrodynamic flow paths. In addition, the nonlocal structure of the regularized Bohm potential admits nuclear quantum tunneling events. After reviewing the general theory, the bohmion method is applied to the well-known Tully models, which are used here as benchmark problems for the comparison of the new bohmion method with previous mixed quantum-classical methods.
Abstract Starting with the exact factorization of the molecular wavefunction, this paper presents the results from the numerical implementation in nonadiabatic molecular dynamics of the recently proposed bohmion method. Within the context of quantum hydrodynamics, we introduce a regularized nuclear Bohm potential admitting solutions comprising a train of δ-functions which provide a finite-dimensional sampling of the hydrodynamic flow paths. The bohmion method inherits all the basic conservation laws from its underlying variational structure and captures electronic decoherence. After reviewing the general theory, the method is applied to the well-known Tully models, which are used here as benchmark problems. In the present case of study, we show that the new method accurately reproduces both electronic decoherence and nuclear population dynamics.
The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether’s conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called ‘collective’. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite ħ $\hbar $ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ħ ≠ 0 ℏ≠0\hbar \ne 0ℏ=0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called ‘Bohmions’, which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.
The coupled Schrödinger equations that describe nonadiabatic dynamics are recast using the Bohm formulation of quantum mechanics. The resulting coupled Bohm equations are solved numerically for two scattering models, giving results that are essentially identical to wave-packet solution of the original coupled Schrödinger equations. The classical limit of the set of coupled Bohm equations is then described, producing a mixed quantum-classical theory incorporating classical-like motion on each potential energy surface accompanied by quantum transitions between the quantum states. Numerical tests of the mixed quantum-classical method are in excellent agreement with the accurate full-quantum results for the model problems. The method is contrasted with the related surface-hopping approach. It is shown that computing the dynamics of a distribution of classical particles is more consistent and more accurate than computing the motion of independent point particles as with surface hopping.
Abstract. In order to alleviate the computational costs of fully quantum nonadiabatic dynamics, we present a mixed quantum-classical (MQC) particle method based on the theory of Koopman wavefunctions. Although conventional MQC models often suffer from consistency issues such as the violation of Heisenberg’s principle, we overcame these difficulties by blending Koopman’s classical mechanics on Hilbert spaces with methods in symplectic geometry. The resulting continuum model enjoys both a variational and a Hamiltonian structure, while its nonlinear character calls for suitable closures. Benefiting from the underlying action principle, here we apply a regularization technique previously developed within our team. This step allows for a singular solution ansatz that introduces the trajectories of computational particles—the koopmons—sampling the Lagrangian classical paths in phase space. In the case of Tully’s nonadiabatic problems, the method reproduces the results of fully quantum simulations with levels of accuracy that are not achieved by standard MQC Ehrenfest simulations. In addition, the koopmon method is computationally advantageous over similar fully quantum approaches, which are also considered in our study. As a further step, we probe the limits of the method by considering the Rabi problem in both the ultrastrong and the deep strong coupling regimes, where MQC treatments appear hardly applicable. In this case, the method succeeds in reproducing parts of the fully quantum results.
The quantum hydrodynamic approach is based on the de Broglie-Bohm formulation of quantum mechanics. The resulting fluid-like equations of motion describe the flow of probability and an accurate solution to these equations is equivalent to solving the time-dependent Schroedinger equation. Furthermore, the hydrodynamic approach provides new insight into the mechanisms as well as an alternative computational approach for treating tunneling phenomena. New concepts include well-defined 'quantum trajectories', 'quantum potential', and 'quantum force' all of which have classical analogues. The quantum potential and its associated force give rise to all quantum mechanical effects such as zero point energy, tunneling, and interference. A new numerical approach called the Iterative Finite Difference Method (IFDM) will be discussed. The IFDM is used to solve the set of non-linear coupled hydrodynamic equations. It is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a 'smart' Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is also capable of treating anharmonic potentials. Example calculations using the IFDM will be presented which include: a one-dimensional Gaussian wave packet tunneling through an Eckart barrier, a one-dimensional bound-state Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. Approximate treatments of the quantum hydrodynamic equations will also be discussed which could allow scaling of the calculations to hundreds of degrees of freedom which is important for treating tunneling phenomena in condensed phase systems.
Abstract Wyatt and coworkers introduced a new approach (Wyatt, Quantum Dynamics with Trajectories Introduction to Quantum Hydrodynamics, Springer, New York, 2005; Lopreore and Wyatt, Phys Rev Lett, 1999, 82, 5190; Wyatt et al., J Chem Phys, 2001, 114, 5113; Lopreore and Wyatt, J Chem Phys, 2002, 116, 1228; Trahan and Wyatt, J Chem Phys, 2003, 118, 4784; Trahan et al., J Chem Phys, 2005, 122, 164104; Babyuk and Wyatt, J Chem Phys, 2006, 125, 64112) which is based on quantum hydrodynamics to the quantum wavepacket dynamics and applied to several systems including tunneling and nonadiabatic transition. This approach is expected to overcome some disadvantages of the conventional wavepacket dynamics, such as order‐N problem and use of absorbing potential. We applied this method to two‐dimensional model of quantum tunneling system and compared the new approach with the conventional wavepacket simulation since the validity of this method to multidimensional systems is not well known. In the simulation, we adopted two different model potentials (Model I and Model II) and calculated time dependence of the probability density and the tunneling probability in both models, where we showed explicitly that the results of the quantum hydrodynamics are identical with those of the conventional method. Using the moving grid and the regridding algorithm in the quantum hydrodynamics, we can effectively avoid order‐N problem and use of the absorbing potential. In addition to these computational advantages, the flux vector is easily obtained by the quantum hydrodynamics and these vectors field helps us to intuitively understand detailed tunneling dynamics. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007