Abstract
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.
Highlights
The celebrated Born–Oppenheimer approximation1,2 assumes the separability of the nuclear and electronic motions in a molecule and provides an appealing picture of independent electronic potential energy surfaces
As for the molecular Hamiltonian used in nonadiabatic simulations, the ab initio electronic structure methods typically yield the adiabatic potential energy surfaces, which are nonadiabatically scitation.org/journal/jcp coupled via momentum couplings
Exact diabatization is only possible in systems with two electronic states and one nuclear degree of freedom,29 there exist more general, approximate diabatization procedures,30–32 starting with the vibronic coupling Hamiltonian model.33. Another benefit of the diabatic representation is that it separates the Hamiltonian into a sum of kinetic energy, depending only on nuclear momenta, and potential energy, depending only on nuclear coordinates, which makes it possible to propagate the molecular wavefunction with the split-operator (SO) algorithm
Summary
The celebrated Born–Oppenheimer approximation assumes the separability of the nuclear and electronic motions in a molecule and provides an appealing picture of independent electronic potential energy surfaces. Exact diabatization is only possible in systems with two electronic states and one nuclear degree of freedom, there exist more general, approximate diabatization procedures, starting with the vibronic coupling Hamiltonian model.33 Another benefit of the diabatic representation is that it separates the Hamiltonian into a sum of kinetic energy, depending only on nuclear momenta, and potential energy, depending only on nuclear coordinates, which makes it possible to propagate the molecular wavefunction with the split-operator (SO) algorithm.. While the recursive compositions permit an automated generation of integrators of arbitrary even order in the time step, the efficiency of higher-order algorithms is sometimes questioned because the number of splitting steps grows exponentially with the order of accuracy, and so does the computational cost of a single time step.
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