Efficient geometric integrators for nonadiabatic quantum dynamics. II. The diabatic representation

Roulet, Julien; Choi, Seonghoon; Vaníček, Jiří

doi:10.1063/1.5094046

Cited by 27 publications

(30 citation statements)

References 67 publications

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“…10,11,57 For additional details about the properties and numerical implementation of the higher-order split-operator algorithms, we refer the reader to Ref. 12.…”

Section: A Split-operator Algorithm and Dynamic Fourier Methodsmentioning

confidence: 99%

“…In addition, because of its symmetry, this algorithm can be composed by various symmetric composition schemes to obtain higher-order integrators. 11,12,[48][49][50][51] As well as having favorable geometric properties, the proposed algorithm is also very simple to implement because its implementation does not depend on the form of the potential energy arXiv:1909.06412v2 [quant-ph] 8 Nov 2019 function and because the grid adaptation requires no adjustable parameters.…”

Section: Introductionmentioning

confidence: 99%

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A time-reversible integrator for the time-dependent Schrödinger equation on an adaptive grid

Choi

Vaníček

2019

The Journal of Chemical Physics

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One of the most accurate methods for solving the time-dependent Schrödinger equation uses a combination of the dynamic Fourier method with the split-operator algorithm on a tensor-product grid. To reduce the number of required grid points, we let the grid move together with the wavepacket, but find that the naïve algorithm based on an alternate evolution of the wavefunction and grid destroys the time reversibility of the exact evolution. Yet, we show that the time reversibility is recovered if the wavefunction and grid are evolved simultaneously during each kinetic or potential step; this is achieved by using the Ehrenfest theorem together with the splitting method. The proposed algorithm is conditionally stable, symmetric, time-reversible, and conserves the norm of the wavefunction. The preservation of these geometric properties is shown analytically and demonstrated numerically on a three-dimensional harmonic model and collinear model of He-H 2 scattering. We also show that the proposed algorithm can be symmetrically composed to obtain time-reversible integrators of an arbitrary even order. We observed 10000-fold speedup by using the tenth-instead of the second-order method to obtain a solution with a time discretization error below 10 −9 . Moreover, using the adaptive grid instead of the fixed grid resulted in a 64-fold reduction in the required number of grid points in the harmonic system and made it possible to simulate the He-H 2 scattering for six times longer, while maintaining reasonable accuracy. Applicability of the algorithm to high-dimensional quantum dynamics is demonstrated using the strongly anharmonic eight-dimensional Hénon-Heiles model.

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“…10,11,57 For additional details about the properties and numerical implementation of the higher-order split-operator algorithms, we refer the reader to Ref. 12.…”

Section: A Split-operator Algorithm and Dynamic Fourier Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A time-reversible integrator for the time-dependent Schrödinger equation on an adaptive grid

Choi

Vaníček

2019

The Journal of Chemical Physics

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“…Therefore, they can be further composed using symmetric composition methods 4,34,[45][46][47][48][50][51][52][53] in order to obtain integrators of arbitrary even orders of convergence. To this end, starting from an integrator Ûp of even order p, an integrator Ûp+2 of order p + 2 is generated using the symmetric composition…”

Section: B Recovery Of Geometric Properties and Increasing Accuracy B...mentioning

confidence: 99%

An implicit split-operator algorithm for the nonlinear time-dependent Schrödinger equation

Roulet,

Vaníček

2021

Preprint

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The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schrödinger equations. However, when applied to certain nonlinear time-dependent Schrödinger equations, this algorithm loses time reversibility and second-order accuracy, which makes it very inefficient. Here, we propose to overcome the limitations of the explicit split-operator algorithm by abandoning its explicit nature. We describe a family of high-order implicit split-operator algorithms that are norm-conserving, time-reversible, and very efficient. The geometric properties of the integrators are proven analytically and demonstrated numerically on the local control of a two-dimensional model of retinal. Although they are only applicable to separable Hamiltonians, the implicit split-operator algorithms are, in this setting, more efficient than the recently proposed integrators based on the implicit midpoint method.

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“…As in the split-operator algorithm 58,82,83 for the TDSE or in the Verlet algorithm 59 for Hamilton's equations of motion, we can obtain a symmetric potential-kinetic-potential (VTV) algorithm of the second order in the time step ∆t by using the Strang splitting 56 and performing, in sequence, potential propagation for time ∆t/2, kinetic propagation for ∆t, and potential propagation for ∆t/2. The second-order kinetic-potential-kinetic (TVT) algorithm is obtained similarly, by exchanging the potential and kinetic propagations.…”

Section: Geometric Integratorsmentioning

confidence: 99%

High-order geometric integrators for representation-free Ehrenfest dynamics

Choi,

Vaníček

2021

Preprint

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Ehrenfest dynamics is a useful approximation for ab initio mixed quantum-classical molecular dynamics that can treat electronically nonadiabatic effects. Although a severe approximation to the exact solution of the molecular time-dependent Schrödinger equation, Ehrenfest dynamics is symplectic, time-reversible, and conserves exactly the total molecular energy as well as the norm of the electronic wavefunction. Here, we surpass apparent complications due to the coupling of classical nuclear and quantum electronic motions and present efficient geometric integrators for Ehrenfest dynamics, which are of arbitrary even orders of accuracy in the time step. These numerical integrators, obtained by symmetrically composing the second-order splitting method and exactly solving the kinetic and potential propagation steps, are norm-conserving, symplectic, and time-reversible regardless of the time step used. Using a nonadiabatic simulation in the region of a conical intersection as an example, we demonstrate that these integrators preserve the geometric properties exactly and, if highly accurate solutions are desired, can be even more efficient than the most popular non-geometric integrators.

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Efficient geometric integrators for nonadiabatic quantum dynamics. II. The diabatic representation

Abstract: Exact nonadiabatic quantum evolution preserves many geometric properties of the molecular Hilbert space. In a companion paper [S. Choi and J. Vaníček, 2019], arXiv:1903.04946v3 [physics.chem-ph]

Cited by 27 publications

References 67 publications

A time-reversible integrator for the time-dependent Schrödinger equation on an adaptive grid

A time-reversible integrator for the time-dependent Schrödinger equation on an adaptive grid

An implicit split-operator algorithm for the nonlinear time-dependent Schrödinger equation

High-order geometric integrators for representation-free Ehrenfest dynamics

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