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

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.

High-order geometric integrators for representation-free Ehrenfest dynamics

Seonghoon Choi, Jiri Vanicek

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 "representation-free" Ehrenfest dynamics, which do not rely on a diabatic or adiabatic representation of electronic states and 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.

2021

Efficient geometric integrators for the linear and nonlinear time-dependent Schrödinger equation

Julien Roulet

Many physical and chemical reactions are driven by nonadiabatic processes, which imply the breakdown of the celebrated Born-Oppenheimer approximation. To understand these processes, experimentalists employ spectroscopic techniques. However, the obtained results are difficult to decipher, and accurate molecular quantum dynamics simulations are used to interpret the results.The second-order split-operator algorithm is one of the most popular numerical methods for simulating the nonadiabatic quantum dynamics because it is explicit, easy to implement, and it preserves many geometric properties of the exact solution. However, the second-order accuracy of this algorithm makes it unaffordable if very accurate results are needed, as tiny time steps are required. To remedy this lack of efficiency, we use composition methods to generate higher-order split-operator algorithms.Although compositions methods increase the accuracy of the standard split-operator algorithm to arbitrary even orders of convergence, the efficiency of the obtained algorithms is still questioned because the computational cost per time step increases drastically with the order of convergence. Therefore, using one- and three-dimensional models of NaI and pyrazine, respectively, we investigate the convergence, efficiency, and geometric properties of these high-order integrators and find that they are, for accurate simulations, more efficient than the standard split-operator algorithm while still preserving the same geometric properties. Besides employing these integrators for simulating the nonadiabatic quantum dynamics, we also explore quantum control and, more specifically, local control theory. This technique uses the instantaneous dynamics of the system to compute an electric field, which interacts with the system in order to drive the state in a desired direction. Because the electric field is obtained from the state itself, we demonstrate that this technique translates into a nonlinear time-dependent Schrödinger equation. Although it is geometric and second-order accurate for simple nonlinearities, the standard split-operator algorithm loses its time-reversal symmetry and second-order accuracy when employed for more complicated nonlinear time-dependent Schrödinger equations. One example of the latter is the one appearing in local control theory.We demonstrate that this lack of generality of the standard split-operator algorithm occurs due to its explicit nature. Thus, we propose two strategies to overcome this issue: First, we completely abandon the split-operator algorithm and present a numerical method based on the implicit midpoint method instead. Second, we make the standard split-operator algorithm implicit, which avoids abandoning the split-operator algorithm altogether. The accuracy and geometric properties of both strategies are then numerically verified using a two-dimensional model of retinal, a molecule whose photochemistry triggers the first event in the biological process of vision. The results demonstrate that both approaches yield second-order methods that preserve all the geometric properties of the exact solution. Because the developed integrators are symmetric, we further improve their accuracy and efficiency using composition methods.

EPFL2022

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

Seonghoon Choi, Jiri Vanicek

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.

2019

Efficient geometric integrators for the linear and nonlinear time-dependent Schrödinger equation

Julien Roulet

EPFL2022