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The non-Born-Oppenheimer quantum dynamics of pp mu and pd mu molecular ions excited by ultrashort, superintense VUV laser pulses polarized along the molecular axis (z) is studied by the numerical solution of the time-dependent Schrodinger equation within a three-dimensional (3D) model, including the internuclear distance R and muon coordinates z and rho, a transversal degree of freedom. It is shown that in both pp mu and pd mu, muons approximately follow the applied laser field out of phase. After the end of the laser pulse, expectation values < z >, < p >, and < R > demonstrate "post-laser-pulse" oscillations in both pp mu and pd mu. In the case of pd mu, the post-laser-pulse oscillations of < z > and < R > appear as shaped "echo pulses." Power spectra, which are related to high-order harmonic generation (HHG), generated due to muonic and nuclear motion are calculated in the acceleration form. For pd mu it is found that there exists a unique characteristic frequency omega(pd mu)(osc) representing both frequencies of post-laser-pulse muonic oscillations and the frequency of nuclear vibrations, which manifest themselves by very sharp maxima in the corresponding power spectra of pd mu. The homonuclear pp p. ion does not possess such a unique characteristic frequency. The "exact" dynamics and power, and HHG spectra of the 3D model are compared with a Born-Oppenheimer, fixed-nuclei model featuring interesting differences: postpulse oscillations are absent and HHG spectra are affected indirectly or directly by nuclear motion.
In a recent paper [U. Lorenz and P. Saalfrank, Chem. Phys. 482, 69 (2017)], we proposed a robust scheme to set up a system-bath model Hamiltonian, describing the coupling of adsorbate vibrations (system) to surface phonons (bath), from first principles. The method is based on an embedded cluster approach, using orthogonal coordinates for system and bath modes, and an anharmonic phononic expansion of the system-bath interaction up to second order. In this contribution, we use this model Hamiltonian to calculate vibrational relaxation rates of H–Si and D–Si bending modes, coupled to a fully H(D)-covered Si(100)-(2×1) surface, at zero temperature. The D–Si bending mode has an anharmonic frequency lying inside the bath frequency spectrum, whereas the H–Si bending mode frequency is outside the bath Debye band. Therefore, in the present calculations, we only take into account one-phonon system-bath couplings for the D–Si system and both one- and two-phonon interaction terms in the case of H–Si. The computation of vibrational lifetimes is performed with two different approaches, namely, Fermi’s golden rule, and a generalized Bixon-Jortner model built in a restricted vibrational space of the adsorbate-surface zeroth-order Hamiltonian. For D–Si, the Bixon-Jortner Hamiltonian can be solved by exact diagonalization, serving as a benchmark, whereas for H–Si, an iterative scheme based on the recursive residue generation method is applied, with excellent convergence properties. We found that the lifetimes obtained with perturbation theory, albeit having almost the same order of magnitude—a few hundred fs for D–Si and a couple of ps for H–Si—, are strongly dependent on the discretized numerical representation of the bath spectral density. On the other hand, the Bixon-Jortner model is free of such numerical deficiencies, therefore providing better estimates of vibrational relaxation rates, at a very low computational cost. The results obtained with this model clearly show a net exponential decay of the time-dependent survival probability for the H–Si initial vibrational state, allowing an easy extraction of the bending mode “lifetime.” This is in contrast with the D–Si system, whose survival probability exhibits a non-monotonic decay, making it difficult to define such a lifetime. This different behavior of the vibrational decay is rationalized in terms of the power spectrum of the adsorbate-surface system. In the case of D–Si, it consists of several, non-uniformly distributed peaks around the bending mode frequency, whereas the H–Si spectrum exhibits a single Lorentzian lineshape, whose width corresponds to the calculated lifetime. The present work gives some insight into mechanisms of vibration-phonon coupling at surfaces. It also serves as a benchmark for multidimensional system-bath quantum dynamics, for comparison with approximate schemes such as reduced, open-system density matrix theory (where the bath is traced out and a Liouville-von Neumann equation is solved) or approximate wavefunction methods to solve the combined system-bath Schrödinger equation.