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)-() 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.
The rates Γ(1), Γ(2a), and Γ(2b) are, for T = 0, the same as those of Ref. 39, albeit written in a slightly different form.
Nf = N(N + 3)/2 results from summing the one-phonon excited states (N states) and the two kinds of two-phonon excited states, namely, same-oscillator doubly excited (N states) and two different oscillator singly excited ( states).
As evident from Fig. 1 and as described in Ref. 39, the small cluster is not a precise subsystem of the large one due to the presence of saturating H atoms at the cluster edges. As described in Ref. 39, these saturating H atoms are only used to minimize edge effects, and in the normal mode analysis, they are either fixed (small cluster) or associated with large masses (large cluster) such that they cannot couple to the bending mode of interest.
The agreement is in fact perfect for the selected H–Si mode and good but not perfect for the selected D–Si mode: For the latter, in Ref. 39, a lifetime of 155 fs was found, instead of 210 fs here. Reasons are (i) that in Ref. 39 two-phonon terms were included also for D–Si (which make some small contribution to the decay rate) and (ii) the orthogonalization of bath and system modes was accidentally not perfect for D–Si in Ref. 39, which caused a slightly too fast decay rate there.