Abstract
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.
Highlights
Separating electronic from nuclear degrees of freedom leads to the Born–Oppenheimer approximation1,2 and the intuitive picture of electronic potential energy surfaces
We have described geometric integrators for nonadiabatic quantum dynamics in the adiabatic representation in which the popular split-operator algorithms cannot be used due to nonseparability of the Hamiltonian into kinetic and potential terms
The proposed methods are based on the symmetric composition of the trapezoidal rule or implicit midpoint method and, as a result, are symmetric, stable, unitary, symplectic, and time-reversible, and, in addition, conserve the energy exactly
Summary
Separating electronic from nuclear degrees of freedom leads to the Born–Oppenheimer approximation and the intuitive picture of electronic potential energy surfaces. Another issue with the second-order differencing is that a much too small time step is required to obtain an accurate solution.42 This problem has been addressed by using the Chebyshev and short iterative Lanczos algorithms; both methods increase remarkably the efficiency of numerical integration by effectively approximating the exact evolution operator. These two methods are neither time-reversible nor symplectic, and the Chebyshev propagation scheme does not even conserve the norm. III, the convergence properties and conservation of geometric invariants by various methods are analyzed numerically on a two-surface NaI model in the adiabatic representation This system has a nonseparable Hamiltonian due to an avoided crossing between its potential energy surfaces and a corresponding region of large nonadiabatic momentum coupling.
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