Open-system Density Matrix Theory Research Articles

Non-Markovian vibrational relaxation dynamics at surfaces.

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 × 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 Schrödinger 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.

Isotopic effects in vibrational relaxation dynamics of H on a Si(100) surface

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.

Current versus temperature-induced switching of a single molecule: Open-system density matrix theory for 1,5-cyclooctadiene on Si(100)

The switching of single cyclooctadiene molecules chemisorbed on a Si(100) surface between two stable conformations, can be achieved with a scanning tunneling microscope [Nacci et al., Phys. Rev. B 77, 121405(R) (2008)]. Recently, it was shown by quantum chemical and quantum dynamical simulations that major experimental facts can be explained by a single-mode model with switching enforced by inelastic electron tunneling (IET) excitations and perturbed by vibrational relaxation [Nacci et al., Nano Lett. 9, 2997 (2009)]. In the present paper, we extend the previous theoretical work in several respects: (1) The model is generalized to a two-mode description in which two C(2)H(4) units of COD can move independently; (2) contributions of dipole and, in addition, (cation and anion) resonance-IET rates are considered; (3) the harmonic-linear vibrational relaxation model used previously is generalized to anharmonic vibrations. While the present models highlight generic aspects of IET-switching between two potential minima, they also rationalize specific experimental findings for COD/Si(100): (1) A single-electron excitation mechanism with a linear dependence of the switching rate on tunneling current I, (2) the capability to switch both at negative and positive sample biases, and (3) a crossover temperature around ~60 K from an IET-driven, T-independent atom tunneling regime, to classical over-the-barrier isomerization with exponential T-dependence at higher temperatures for a bias voltage of +1.5 V and an average tunneling current of 0.73 nA.

Role of electronic friction during the scattering of vibrationally excited nitric oxide molecules from Au(111)

Some time ago, it has been observed that vibrationally highly excited $\text{NO}(v)$ molecules (with typical vibrational quantum numbers $v\ensuremath{\approx}15$) lose substantial amounts of vibrational energy when scattering off a Au(111) surface [H. Huang, C. Rettner, D. Auerbach, and A. Wodtke, Science 290, 111 (2000)]. This has been interpreted as a sign for the breakdown of the Born-Oppenheimer approximation due to vibration-electron coupling. It has been argued that this process cannot be understood on the basis of single-quantum transitions which are typical for ``electronic friction'' models based on a perturbative treatment of weak vibration-electron couplings. Rather, multiple-quanta transitions characteristic for strong nonadiabatic effects are needed according to recent classical surface hopping calculations involving multiple potential-energy surfaces and model Hamiltonians [N. Shenvi, S. Roy, and J. C. Tully, Science 326, 829 (2009)]. Here we address the importance and magnitude of electronic friction for $\text{NO}@\text{Au}(111)$ by using fully quantum-mechanical, parameter-free first-principles theories in reduced dimensions. Periodic density-functional theory calculations are performed to obtain a ground-state potential-energy surface along the desorption and NO-vibration coordinates, and coordinate-resolved, finite NO vibrational lifetimes due to vibration-electron coupling. Using this input, the scattering event is modeled by open-system density-matrix theory in the frame of the coupled-channel-density-matrix method, which allows for the inclusion of energy relaxation of the scattering NO molecules. It is found that within this model at least, electronic friction accounts for the observed vibrational deactivation of NO scattering from gold.

Femtosecond-Laser Desorption of H2 (D2) from Ru(0001): Quantum and Classical Approaches

The femtosecond-laser-induced, substrate-mediated associative desorption of molecular hydrogen and deuterium from a Ru(0001) surface in the so-called DIMET limit is studied theoretically. Two widely used models, a nonadiabatic approach and a adiabatic one are employed and compared to each other. The quantum model is realized by the Monte Carlo wave packet (MCWP) method in the framework of open-system density matrix theory. The classical approach is realized with the help of (frictional) Langevin dynamics with stochastic forces. For both models the same ground-state potential energy surface is used and the same two-temperature model adopted to describe the hot-electron-driven desorption dynamics. Apart from these common features both models are different. Still, both account well for the main experimental findings (Wagner et al. Phys. Rev. B 2005, 72, 205404). In particular, an isotope effect in desorption probabilities, the energy content of the desorbing molecules, and the scaling of these observables with laser fluence are reproduced and explained. The similarity of the results obtained with both models is traced back to the fact that, in the present case, the photodynamics takes place dominantly in the ground electronic state because the electronically excited state is rapidly quenched. The short lifetime of the excited state has also the effect that photoreaction cross sections are typically very small. An IR+vis hybrid scheme, by which the adsorbate is vibrationally excited by IR photons prior to the heating of metal electrons with the vis pulse, is shown to successfully promote the reaction even for strongly coupled adsorbate―surface systems.

Time-dependent configuration-interaction calculations of laser-driven dynamics in presence of dissipation

Correlated, multielectron dynamics of "open" electronic systems within the fixed-nuclei approximation are treated here within explicitly time-dependent configuration-interaction schemes. Specifically, we present simulations of laser-pulse driven excitations of selected electronic states of LiCN in the presence of energy and phase relaxation. The evolution of the system is studied using open-system density matrix theory, which embeds naturally in the time-dependent configuration-interaction singles (doubles) formalism. Different models for dissipation based on the Lindblad semigroup formalism are presented. These models give rise to lifetimes for energy relaxation ranging from a few hundreds of femtoseconds to several nanoseconds. Pure dephasing is treated using a Kossakowski-like Gaussian model, proceeding on similar time scales. The pulse lengths employed range from very short (tens of femtoseconds) to very long (several nanoseconds). To make long-time propagations tractable, the quasiresonant approximation is used. The results show that despite the loss of efficiency, selective dipole switching can still be achieved in the presence of dissipation when using appropriately designed laser pulses.

Mode-selective excitation of hydrogen atoms on a Si surface: Non-Markovian and Markovian treatment of infrared laser driven dissipative quantum dynamics

Mode-selective excitation of adsorbates by shaped infrared laser pulses is investigated here theoretically, for the example of a H atom on a hydrogen-covered $\mathrm{Si}(100)\text{\ensuremath{-}}2\ifmmode\times\else\texttimes\fi{}1$ surface. The mode-selective excitation is perturbed by the intermode coupling within the system (bending and stretching modes) and by system-bath coupling to substrate phonons. Using a force-field based model, vibration-phonon coupling was found and predicted to lead to vibrational relaxation of the H-Si stretching mode on a ns timescale, and of the Si-Si-H bending mode on a ps timescale [I. Andrianov and P. Saalfrank, J. Chem. Phys. 124, 034710 (2006)]. To address the question as to whether in such a dissipative situation mode-selective control of adsorbate vibrational dynamics is still possible, a system-bath ansatz is used to derive an open-system density matrix theory in which the H vibrations are driven either by ${\mathrm{sin}}^{2}$, or by freely optimized infrared ps laser pulses. Both for the Si-H stretching and Si-Si-H bending vibrations mode-selective excitation is predicted to be possible. It is also found that the Markov approximation works well in most of the applications, and that simple ${\mathrm{sin}}^{2}$ are nearly as effective as pulses which were freely optimized by optimal control theory.

STM and laser-driven atom switch: An open-system density-matrix study of H/Si(100)

A recently proposed single atom switch, experimentally realized with a scanning tunneling microscope (STM) for H on a Si(100) surface [U.J. Quaade et al., Surf. Sci. 415, L1037 (1998)], is examined here theoretically. Using a two-state model and open-system density matrix theory which includes both electronic and vibrational relaxation of the adsorbate, we argue that the switching is dominated by tunneling in the ground state. The model accounts for observed isotope and, qualitatively, for temperature effects. We suggest that also a laser-driven H switch should be possible.

Open-system density matrix description of femtosecond laser desorption of electronically and vibrationally relaxing adsorbates: Single- and two-pulse scenarios

The femtosecond laser pulse induced desorption of a molecule (NO) from a metal surface (Pt) is investigated by means of open-system density matrix theory. Single- and two-pulse scenarios in the so-called DIMET limit (Desorption Induced by Multiple Electronic Transitions) are considered. Special emphasis is given (a) to a realistic modeling of the temperature profiles of substrate degrees of freedom in response to a laser pulse, (b) to the role of vibrational relaxation of the adsorbate-surface bond, and (c) to a critical comparison of our anharmonic “excitation–deexcitation” approach to the simpler Truncated Harmonic Oscillator model with electronic friction. For the single-pulse case, nonlinear scaling of the desorption yield with laser fluence is demonstrated and found to be in good agreement with experiment. For the two-pulse case, time-resolved correlation diagrams are calculated and interpreted.

Dissipation in anharmonic molecular systems: beyond the linear coupling limit

Open-system density matrix theory is a powerful method for treating the dynamics of an active quantum mechanical system embedded in a “bath” of less active modes. For the sake of economy and conceptual simplicity, the system and bath modes are often considered harmonic and the coupling between them bilinear in the system and bath coordinates. We present a systematic way of how to include nonlinear position dependence of the coupling strength between an anharmonic system and a bath. This extends the applicability of open-system density matrix theory considerably, while still retaining the advantages of the harmonic-linear model. To achieve this we suggest to interpret operators derived from supersymmetric quantum mechanics as approximate raising/lowering operators, and to use them for constructing a dissipative Liouvillian of Lindblad form. The properties of this Liouvillian are examined in some detail. As a concrete example, we apply the formalism to the vibrational relaxation and the inelastic scattering of a gas phase atom (argon) at a surface. The method will also be generalized to finite temperatures.

Open-system quantum dynamics for gas-surface scattering: Nonlinear dissipation and mapped Fourier grid methods

When a gas phase atom or molecule hits a nonrigid surface, both elastic and inelastic scattering and sticking can occur. We suggest treating the dynamics of these processes using open-system density matrix theory. For the “free→free” and “free→bound” events at hand, both fundamental and numerical problems arise. The fundamental problem is that the adsorbate “system” is anharmonic and the coupling between the system and the substrate “bath” has to be nonlinear at least in the system coordinates. Here we propose a new Lindblad-type open-system density matrix approach which accounts for system anharmonicity and nonlinearity of the system–bath coupling. The numerical problem is that for a dissipative scattering process large basis sets or grids are required, making the storage and direct propagation of a density matrix difficult. To overcome this problem we use a mapped Fourier method which reduces the grid size and hence the storage requirements significantly. We apply the new methods and techniques to a simple model resembling the simultaneous scattering and sticking of an O2 molecule at a metal surface.

Photodesorption of NO from a metal surface: quantum dynamical implications of a two-mode model

Open-system density matrix theory is used to study the dissipative quantum dynamics of photoinduced, hot-electron mediated desorption of NO from Pt(111). Two electronic states are considered and both the DIET (Desorption Induced by Electronic Transitions) and DIMET (Desorption Induced by Multiple Electronic Transitions) limits addressed. Besides the center-of-mass motion of the molecule perpendicular to the surface plane, the vibrational motion of the NO molecule is accounted for. In a first, methodological part the interrelations between various “direct” and “jumping wave packet” methods to solve the underlying open-system Liouville-von Neumann equations of motion, are emphasized. In a second, applied part we investigate (i) the relevance of the popular “negative ion resonance” model, (ii) the role of coordinate-dependent electronic relaxation, and (iii) the implications of multiple-electronic excitations.

Direct and indirect DIET and DIMET from semiconductor and metal surfaces: what can we learn from 'toy models'?

Desorption induced by electronic transitions (DIET) and its variant DIMET (M = 'Multiple'), are among the simplest possible "reactions" of ad-species involving ultra-short lived electronically excited states at surfaces. The non-adiabatic bond-cleavage can be enforced, for example, with laser irradiation or with electrons or holes emitted from the tip of a scanning tunnelling microscope (STM). The transient creation of excited intermediates can proceed directly (localised to the adsorbate-substrate complex), or indirectly (i.e., through the substrate). To understand the basic processes, simple one-mode two-state "toy models" such as the Menzel-Gomer-Redhead (MGR) or the Antoniewicz scenarios have proven very useful in the past. We adopt and extend MGR- and Antoniewicz-type models together with numerically exact open-system density matrix theory to address a few actual problems/experiments in DI(M)ET: (1) Direct, laser-induced desorption of H(D) from Si(100) surfaces which has been realised in the continuous-wave DIET regime only recently [T. Vondrak and X.-Y. Zhu, Phys. Rev. Lett., 1999, 82, 1967], is studied and compared to so-far hypothetical femtosecond laser desorption. The possibility of controlling the reaction by shaping the laser pulses is addressed. (2) For the same system, temperature effects are studied for electron- or hole-stimulated desorption with an STM [T. C. Shen, C. Wang, G. C. Abeln, T. R. Tucker, J. W. Lyding, Ph. Avouris and R. E. Walkup, Science, 1995, 268, 1590; C. Thirstrup, M. Sakurai, T. Nakayama and K. Stokbro, Surf. Sci., 1999, 424, L329]. A modified version of Gadzuk's "sudden transition and averaging" approach is adopted which accounts for temperature dependent excited state lifetimes. (3) For photodesorption of NO from Pt(111), based on quantum dynamical simulations possible experimental tests involving static electric fields are suggested to address the relevance of the recently challenged [F. M. Zimmermann, Surf. Sci., 1997. 390, 174], "negative ion resonance" model of the Antoniewicz type.

Quantum dynamics of laser‐ and field‐induced desorption of molecules from metal surfaces

The desorption of adspecies from surfaces by laser radiation or static electric fields is an important subject for the microstructuring of materials and a challenge to theory: Not only the excitation mechanism, but also the subsequent, ultrafast dynamics in a dissipative environment, needs to be properly described. After a brief review on previous work, in this work a few new issues concerning the laser- or field-induced desorption of molecules from metal surfaces [NO from Pt(111), and NH3 from Cu(111)] will be addressed with the help of open-system density matrix theory and wave packet propagation methods. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 210–219, 2000

Quantum dynamics of laser- and field-induced desorption of molecules from metal surfaces

The desorption of adspecies from surfaces by laser radiation or static electric fields is an important subject for the microstructuring of materials and a challenge to theory: Not only the excitation mechanism, but also the subsequent, ultrafast dynamics in a dissipative environment, needs to be properly described. After a brief review on previous work, in this work a few new issues concerning the laser- or field-induced desorption of molecules from metal surfaces [NO from Pt(111), and NH3 from Cu(111)] will be addressed with the help of open-system density matrix theory and wave packet propagation methods. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 210–219, 2000

Infrared-laser driven vibrational excitation of relaxing adsorbates: Quantum dynamical aspects

As a first step to the active manipulation of adsorbates by external, time-dependent electromagnetic fields, the infrared-laser driven selective excitation of molecular vibrations of adsorbates at metal surfaces is investigated here in the framework of time-dependent open-system density matrix theory. Special emphasis is given to the inclusion of vibrational damping, caused by the coupling of the adsorbate vibrations to possibly electronic substrate degrees of freedom. For the example system NH3/Cu, a non-Markovian, two-mode open-system Liouville–von Neumann model for the vibrational relaxation of an excited adsorbate is proposed. After studying the field-free decay of excited adsorbates, it is shown that even in rapidly relaxing environments optimal IR laser pulses in the picosecond domain can be designed which lead to temporarily high populations of selected target states of adsorbates at metal surfaces.

Vibrationally excited products after the photodesorption of NO from Pt(111): a two-mode open-system density matrix approach

The experimentally observed vibrational excitation of NO molecules photodesorbing from a Pt(111) surface is investigated numerically with the help of open-system density matrix theory. We extend Gadzuk's jumping wave packet model to treat DIET processes (desorption induced by electronic transitions, single-excitation limit) which is equivalent to a one-channel open system Liouville-von Neumann equation with coordinate-independent quenching, to cases when the electronic relaxation becomes coordinate-dependent. This allows for a more realistic but still economic and, hence, systematic study of DIET within a two-state, two-degrees of freedom model. Adopting “reasonable” Antoniewicz-type model potentials, we find that the experimental observations can semi-quantitatively be rationalized if an electronic quenching rate decreasing with increasing NO separation, is assumed. Preliminary two-mode DIMET simulations (desorption induced by multiple electronic transitions, multiple-excitation limit) within a stochastic wave packet approach are also presented.