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We discuss an efficient Hierarchical Effective Mode (HEM) representation of a high-dimensional harmonic oscillator bath, which describes phonon-driven vibrational relaxation of an adsorbate-surface system, namely, deuterium adsorbed on Si(100). Starting from the original Hamiltonian of the adsorbate-surface system, the HEM representation is constructed via iterative orthogonal transformations, which are efficiently implemented with Householder matrices. The detailed description of the HEM representation and its construction are given in the second quantization representation. The hierarchical nature of this representation allows access to the exact quantum dynamics of the adsorbate-surface system over finite time intervals, controllable via the truncation order of the hierarchy. To study the convergence properties of the effective mode representation, we solve the time-dependent Schrodinger equation of the truncated system-bath HEM Hamiltonian, with the help of the multilayer extension of the Multiconfigurational Time-Dependent Hartree (ML-MCTDH) method. The results of the HEM representation are compared with those obtained with a quantum-mechanical tier-model. The convergence of the HEM representation with respect to the truncation order of the hierarchy is discussed for different initial conditions of the adsorbate-surface system. The combination of the HEM representation with the ML-MCTDH method provides information on the time evolution of the system (adsorbate) and multiple effective modes of the bath (surface). This permits insight into mechanisms of vibration-phonon coupling of the adsorbate-surface system, as well as inter-mode couplings of the effective bath.
We present a quantum-mechanical tier model for vibrational relaxation of low-lying excited states of an adsorbate vibrational mode (system), coupled to surface phonons (bath), at zero temperature. The tier model, widely used in studies of intramolecular vibrational energy redistribution in polyatomics, is adapted here to adsorbate-surface systems with the help of an embedded cluster approach, using orthogonal coordinates for the system and bath modes, and a phononic expansion of their interaction. The key idea of the model is to organize the system-bath zeroth-order vibrational space into a hierarchical structure of vibrational tiers and keep therein only vibrational states that are sequentially generated from the system-bath initial vibrational state. Each tier is generated from the previous one by means of a successor operator, derived from the system-bath interaction Hamiltonian. This sequential procedure leads to a drastic reduction of the dimension of the system-bath vibrational space. We notably show that for harmonic vibrational motion of the system and linear system-bath couplings in the system coordinate, the dimension of the tier-model vibrational basis scales as similar to N-lxv. Here, N is the number of bath modes, l is the highest-order of the phononic expansion, and l is the size of the system vibrational basis. This polynomial scaling is computationally far superior to the exponential scaling of the original zeroth-order vibrational basis, similar to M-N, with M being the number of basis functions per bath mode. In addition, since each tier is coupled only to its adjacent neighbors, the matrix representation of the system-bath Hamiltonian in this new vibrational basis has a symmetric block-tridiagonal form, with each block being very sparse. This favors the combination of the tier-model with iterative Krylov techniques, such as the Lanczos algorithm, to solve the time-dependent Schrodinger equation for the full Hamiltonian. To illustrate the method, we study vibrational relaxation of a D-Si bending mode, coupled via two-and (mainly) one-phonon interactions to a fully D-covered Si(100)-(2 x 1) surface, using a recent first-principles system-bath Hamiltonian. The results of the tier model are compared with those obtained by the Lindblad formalism of the reduced density matrix. We find that the tier model provides much more information and insight into mechanisms of vibration-phonon couplings at surfaces, and gives more reliable estimates of the adsorbate vibrational lifetimes. Moreover, the tier model might also serve as a benchmark for other approximate quantum-dynamics methods, such as multiconfiguration wavefunction approaches. Published under license by AIP Publishing.
We have measured differential cross sections (DCSs) for the reaction H + D-2 -> HD- (v' = 2, j' = 0,3,6,9) + D at center-of-mass collision energies E-coll of 1.25, 1.61, and 1.97 eV using the photoloc technique. The DCSs show a strong dependence on the product rotational quantum number. For the HD(v' = 2, j' = 0) product, the DCS is bimodal but becomes oscillatory as the collision energy is increased. For the other product states, they are dominated by a single peak, which shifts from back to sideward scattering as j' increases, and they are in general less sensitive to changes in the collision energy. The experimental results are compared to quantum mechanical calculations and show good, but not fully quantitative agreement.
We discuss the effect of molecular symmetry on coherent tunneling in symmetric double-well potentials whose two molecular equilibrium configurations are interconverted by nuclear permutations. This is illustrated with vibrational tunneling in ammonia molecules, electronic tunneling in the dihydrogen cation, and laser-induced rotational tunneling of homonuclear diatomics. In this contribution, we reexamine the textbook picture of coherent tunneling in such potentials, which is depicted with a wavepacket shuttling back and forth between the two potential-wells. We show that the common application of this picture to the aforementioned molecules contravenes the principle of the indistinguishability of identical particles. This conflict originates from the sole consideration of the dynamics of the tunneling-mode, connecting the double-well energy minima, and complete omission of all the remaining molecular degrees of freedom. This gives rise to double-well wavepackets that are nonsymmetric under nuclear permutations. To obey quantum statistics, we show that the double-well eigenstates composing these wavepackets must be entangled with the wavefunctions that describe all the omitted molecular modes. These wavefunctions have compensating and opposite nuclear permutation symmetry. This in turn leads to complete quenching of interference effects behind localization in one potential-well or another. Indeed, we demonstrate that the reduced density of probability of the symmetrized molecular wavefunction, where all the molecular coordinates but the tunneling-mode are integrated out, is symmetrically distributed over the two potential-wells, at all times. This applies to any multilevel wavepacket of isotropic or fully aligned symmetric double-well achiral molecules. However, in the case of coherent electronic or vibrational tunneling, fully aligned molecules may exhibit dynamical localization in the space-fixed frame, where the tunneling-mode density shuttles between the opposite directions of the alignment axis. This dynamical spatial-localization results from linear combinations of molecular states that have opposite parity. In summary, this study shows that dynamical localization of the tunneling-mode density on either of the two indistinguishable molecular equilibrium configurations of symmetric double-well achiral molecules is forbidden by quantum statistics, whereas its dynamical localization in the space-fixed frame is allowed by parity. The subtle distinction between these two types of localization has far-reaching implications in the interpretation of many ultrafast molecular dynamics experiments.
The response of the hydrogen molecular ion, H-2(+), to few-cycle laser pulses of different intensities is simulated. To treat the coupled electron-nuclear motion, we use adiabatic potentials computed with Gaussian-type basis sets together with a heuristic ionization model for the electron and a grid representation for the nuclei. Using this mixed-basis approach, the time-dependent Schrodinger equation is solved, either within the Born-Oppenheimer approximation or with nonadiabatic couplings included. The dipole response spectra are compared to all-grid-based solutions for the three-body problem, which we take as a reference to benchmark the Gaussian-type basis set approaches. Also, calculations employing the fixed-nuclei approximation are performed, to quantify effects due to nuclear motion. For low intensities and small ionization probabilities, we get excellent agreement of the dynamics using Gaussian-type basis sets with the all-grid solutions. Our investigations suggest that high harmonic generation (HHG) and high-frequency response, in general, can be reliably modeled using Gaussian-type basis sets for the electrons for not too high harmonics. Further, nuclear motion destroys electronic coherences in the response spectra even on the time scale of about 30 fs and affects HHG intensities, which reflect the electron dynamics occurring on the attosecond time scale. For the present system, non-Born-Oppenheimer effects are small. The Gaussian-based, nonadiabatically coupled, time-dependent multisurface approach to treat quantum electron-nuclear motion beyond the non-Born-Oppenheimer approximation can be easily extended to approximate wavefunction methods, such as time-dependent configuration interaction singles (TD-CIS), for systems where no benchmarks are available.
Recent theoretical investigations claim that tailored laser pulses may selectively steer benzene's aromatic ground state to localized non-aromatic excited states. For instance, it has been shown that electronic wavepackets, involving the two lowest electronic eigenstates, exhibit subfemtosecond charge oscillation between equivalent Kekule resonance structures. In this contribution, we show that such dynamical electron-localization in the molecule-fixed frame contravenes the principle of the indistinguishability of identical particles. This breach stems from a total omission of the nuclear degrees of freedom, giving rise to nonsymmetric electronic wavepackets under nuclear permutations. Enforcement of the latter leads to entanglement between the electronic and nuclear states. To obey quantum statistics, the entangled molecular states should involve compensating nuclear-permutation symmetries. This in turn engenders complete quenching of dynamical electron-localization in the molecule-fixed frame. Indeed, for the (six-fold) equilibrium geometry of benzene, group-theoretic analysis reveals that any electronic wavepacket exhibits a (D-6h) totally symmetric electronic density, at all times. Thus, our results clearly show that the six carbon atoms, and the six C-C bonds, always have equal Mulliken charges, and equal bond orders, respectively. However, electronic wavepackets may display dynamical localization of the electronic density in the space-fixed frame, whenever they involve both even and odd space-inversion (parity) or permutation-inversion symmetry. Dynamical spatial-localization can be probed experimentally in the laboratory frame, but it should not be deemed equivalent to charge oscillation between benzene's identical electronic substructures, such as Kekule resonance structures.
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.
Vibrational dynamics of adsorbates near surfaces plays both an important role for applied surface science and as a model lab for studying fundamental problems of open quantum systems. We employ a previously developed model for the relaxation of a D-Si-Si bending mode at a D:Si(100)-(2 x 1) surface, induced by a "bath " of more than 2000 phonon modes [Lorenz and P. Saalfrank, Chem. Phys. 482, 69 (2017)], to extend previous work along various directions. First, we use a Hierarchical Effective Mode (HEM) model [Fischer et al., J. Chem. Phys. 153, 064704 (2020)] to study relaxation of higher excited vibrational states than hitherto done by solving a high-dimensional system-bath time-dependent Schrodinger equation (TDSE). In the HEM approach, (many) real bath modes are replaced by (much less) effective bath modes. Accordingly, we are able to examine scaling laws for vibrational relaxation lifetimes for a realistic surface science problem. Second, we compare the performance of the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) approach with that of the recently developed coherent-state-based multi-Davydov-D2 Ansatz [Zhou et al., J. Chem. Phys. 143, 014113 (2015)]. Both approaches work well, with some computational advantages for the latter in the presented context. Third, we apply open-system density matrix theory in comparison with basically "exact " solutions of the multi-mode TDSEs. Specifically, we use an open-system Liouville-von Neumann (LvN) equation treating vibration-phonon coupling as Markovian dissipation in Lindblad form to quantify effects beyond the Born-Markov approximation. Published under an exclusive license by AIP Publishing.
We investigate the vibrational relaxation of a Morse oscillator, nonlinearly coupled to a finite-dimensional bath of harmonic oscillators at zero temperature, using two different approaches: Reduced dynamics with the help of the Lindblad formalism of reduced density matrix theory in combination with Fermi's Golden Rule, and exact dynamics (within the chosen model). with the multiconfiguration time-dependent Hartree (MCTDH) method. Two different models have been constructed, the situation where the bath spectrum is exactly resonant with the anharmonic oscillator transition frequencies, and the case for which the subsystem is slightly off-resonant with the environment. At short times, reduced dynamics calculations describe the relaxation process qualitatively well but fail to reproduce recurrences observed with MCTDH for longer times. Lifetimes of all the vibrational levels of the Morse oscillator have been calculated, and both Lindblad and MCTDH. results show the same dependence of the lifetimes on the initial vibrational state quantum number. A prediction, which should be generic for adsorbate systems is a striking, sharp increase of lifetimes of the subsystem vibrational levels close to the dissociation This is contradictory with harmonic/linear extrapolation laws, which predict a monotonic decrease of the lifetime with initial vibrational quantum number.
When a hydrogen (H) atom approaches a deuterium (D-2) molecule, the minimum-energy path is for the three nuclei to line up. Consequently, nearly collinear collisions cause HD reaction products to be backscattered with low rotational excitation, whereas more glancing collisions yield sideways-scattered HD products with higher rotational excitation. Here we report that measured cross sections for the H + D-2 -> HD(v' = 4, j') + D reaction at a collision energy of 1.97 electron volts contradict this behavior. The anomalous angular distributions match closely fully quantum mechanical calculations, and for the most part quasiclassical trajectory calculations. As the energy available in product recoil is reduced, a rotational barrier to reaction cuts off contributions from glancing collisions, causing high-j' HD products to become backward scattered.