Efficient geometric integrators for nonadiabatic quantum dynamics. I. The adiabatic representation

Geometric integrators of the Schrödinger equation conserve exactly many invariants of the exact solution. Among these integrators, the split-operator algorithm is explicit and easy to implement but, unfortunately, is restricted to systems whose Hamiltonian is separable into kinetic and potential terms. Here, we describe several implicit geometric integrators applicable to both separable and nonseparable Hamiltonians and, in particular, to the nonadiabatic molecular Hamiltonian in the adiabatic representation. These integrators combine the dynamic Fourier method with the recursive symmetric composition of the trapezoidal rule or implicit midpoint method, which results in an arbitrary order of accuracy in the time step. Moreover, these integrators are exactly unitary, symplectic, symmetric, time-reversible, and stable and, in contrast to the split-operator algorithm, conserve energy exactly, regardless of the accuracy of the solution. The order of convergence and conservation of geometric properties are proven analytically and demonstrated numerically on a two-surface NaI model in the adiabatic representation. Although each step of the higher order integrators is more costly, these algorithms become the most efficient ones if higher accuracy is desired; a thousand-fold speedup compared to the second-order trapezoidal rule (the Crank-Nicolson method) was observed for a wavefunction convergence error of 10^−10. In a companion paper [J. Roulet, S. Choi, and J. Vaníček, J. Chem. Phys. 150, 204113 (2019)], we discuss analogous, arbitrary-order compositions of the split-operator algorithm and apply both types of geometric integrators to a higher-dimensional system in the diabatic representation.

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

Seonghoon Choi, Julien Roulet, Jiri Vanicek

Exact nonadiabatic quantum evolution preserves many geometric properties of the molecular Hilbert space. In the first paper of this series ["Paper I," S. Choi and J. Vaníček, J. Chem. Phys. 150, 204112 (2019)], we presented numerical integrators of arbitrary-order of accuracy that preserve these geometric properties exactly even in the adiabatic representation, in which the molecular Hamiltonian is not separable into kinetic and potential terms. Here, we focus on the separable Hamiltonian in diabatic representation, where the split-operator algorithm provides a popular alternative because it is explicit and easy to implement, while preserving most geometric invariants. Whereas the standard version has only second-order accuracy, we implemented, in an automated fashion, its recursive symmetric compositions, using the same schemes as in Paper I, and obtained integrators of arbitrary even order that still preserve the geometric properties exactly. Because the automatically generated splitting coefficients are redundant, we reduce the computational cost by pruning these coefficients and lower memory requirements by identifying unique coefficients. The order of convergence and preservation of geometric properties are justified analytically and confirmed numerically on a one-dimensional two-surface model of NaI and a three-dimensional three-surface model of pyrazine. As for efficiency, we find that to reach a convergence error of 10^−10, a 600-fold speedup in the case of NaI and a 900-fold speedup in the case of pyrazine are obtained with the higher-order compositions instead of the second-order split-operator algorithm. The pyrazine results suggest that the efficiency gain survives in higher dimensions.

2019

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

Julien Roulet, Jiri Vanicek

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.

2021

High-order geometric integrators for nonadiabatic molecular quantum dynamics and their applications to explore conical intersections

Seonghoon Choi

Accurate simulations of molecular quantum dynamics are crucial for understanding numerous natural processes and experimental results. Yet, such high-accuracy simulations are challenging even for relatively simple systems where the Born-Oppenheimer approximation is valid. Worse still, due to the ubiquity of conical intersections in molecules, it is often necessary to go beyond this celebrated approximation and employ even more sophisticated methods that can describe nonadiabatic processes involving multiple coupled electronic states.Geometric integrators provide a way to simulate nonadiabatic quantum dynamics with high accuracy while exactly conserving geometric properties, such as norm, energy, symplecticity, and time reversibility. However, the exact conservation of these properties typically leads to a higher computational cost when compared with non-geometric integrators of the same order of accuracy. We remedy this lack of efficiency by employing various composition methods to obtain high-order geometric integrators, competitive in efficiency even with some non-geometric integrators.To overcome the limited applicability of the popular split-operator algorithms, we present geometric integrators that can solve the time-dependent Schroedinger equation efficiently regardless of the form of the Hamiltonian. Employing these integrators, we systematically compare the different representations of the molecular Hamiltonian for their suitability for simulating nonadiabatic dynamics at a conical intersection and find that the rarely used exact quasidiabatic Hamiltonian, which includes the often-neglected residual nonadiabatic couplings, yields the most accurate results. Accordingly, we present a method for quantifying the validity of ignoring these residual couplings and show that depending on the system, initial state, and employed quasidiabatization scheme, neglecting the residual couplings can indeed result in inaccurate dynamics.The computational cost of exact quantum simulations quickly becomes prohibitively high as the system size increases. For high-dimensional simulations, one can either optimize the available basis set, e.g., by employing adaptive phase space grids or, alternatively, employ a more approximate approach, such as mixed quantum-classical Ehrenfest dynamics, that provides a useful qualitative picture. Although Ehrenfest dynamics and exact quantum dynamics simulated on an adaptive grid may appear unrelated, we show that, surprisingly, the high-order geometric integrators for these two problems can, in fact, be obtained by using the same splitting and composition methods.

EPFL2022

High-order geometric integrators for nonadiabatic molecular quantum dynamics and their applications to explore conical intersections

Seonghoon Choi

EPFL2022