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The Journal of Chemical Physics

Spectral line shapes in dissipative systems: Molecules adsorbed on metal surfaces

Salvador Miret-Artés, David A. Micha, Dario Bekšić

Published Mar 8, 2001• 0 citations
Publisher

Abstract

The temporal evolution of rates and probabilities of transitions, induced by light between states of a molecule interacting with a dissipative medium, are derived for a Hamiltonian containing an anti-Hermitian term which describes dissipation. Our theoretical treatment introduces a biorthonormal basis set of decaying eigenstates to obtain the rates and probabilities from perturbation theory. The formalism is developed for excitation by continuous and pulsed light. As an example, it is applied to the excitation of molecules adsorbed on solid surfaces, taking into account the dissipation of energy into the solid as the molecule relaxes or is desorbed.

DOI
10.1063/1.1340672

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The temporal evolution of rates and probabilities of transitions, induced by light between states of a molecule interacting with a dissipative medium, are derived for a Hamiltonian containing an anti-Hermitian term which describes dissipation. Our theoretical treatment introduces a biorthonormal basis set of decaying eigenstates to obtain the rates and probabilities from perturbation theory. The formalism is developed for excitation by continuous and pulsed light. As an example, it is applied to the excitation of molecules adsorbed on solid surfaces, taking into account the dissipation of energy into the solid as the molecule relaxes or is desorbed.
Adsorbed species on a solid-state surface interact with the large number of substrate modes, which gives rise to thermal relaxation. Commonly, the temporal evolution of the quantum state of the adsorbate is described by a master equation for the level populations (vibrational bond, internal modes, etc). It is pointed out that this approach does not necessarily give a correct account of the coupling to the solid when the effective level widths become comparable to the level separations, or larger. It is shown that the evolution of the populations does not decouple any more from the time evolution of the coherences (off-diagonal matrix elements), which implies that a random-phase approximation cannot be justified, and that the density matrix of the adsorbate is not determined only by the golden-rule transition constants. Especially, the line profiles turn out to be very sensitive to the coherence-coherence couplings. Although the coherences vanish in thermal equilibrium, their time-regression operator, and hence their mutual couplings and their couplings to the populations, which determine the absorption profile, do not. This information is lost in a master-equation treatment of relaxation.
It has been postulated that the state function can be expressed as a linear superposition of eigenstates of Hermitian operators. When a system is subjected to an external impulse in the form of radiation, the probability of finding the system in different eigenstates of the state function changes. The total probability is always unity, which means the change in probabilities for different states occurs at the expense of each other. This change in probability of finding the system in different eigenstates brought about by system-radiation interaction is called the spectroscopic transition. Here, we use a two-level system to develop the details of a general spectroscopic transition, and work further to examine the effects of interruption in the coherent evolution of the state function. The physical meaning of absorption coefficient is discussed in detail. We derive the general forms of relaxation of the excited state, spectral line shape, and line broadening. The origin and the meaning of homogeneous and inhomogeneous line broadening are discussed. We describe the Doppler effect and its use in the interpretation of inhomogeneous line shape.
We investigate the effect of inter-mode coupling on the vibrational relaxation dynamics of molecules in weak dissipative environments. The simulations are performed within the reduced density matrix formalism in the Markovian regime, assuming a Lindblad form for the system-bath interaction. The prototypical two-dimensional model system representing two CO molecules approaching a Cu(100) surface is adapted from an ab initio potential, while the diatom-diatom vibrational coupling strength is systematically varied. In the weak system-bath coupling limit and at low temperatures, only first order non-adiabatic uni-modal coupling terms contribute to surface-mediated vibrational relaxation. Since dissipative dynamics is non-unitary, the choice of representation will affect the evolution of the reduced density matrix. Two alternative representations for computing the relaxation rates and the associated operators are thus compared: the fully coupled spectral basis, and a factorizable ansatz. The former is well-established and serves as a benchmark for the solution of Liouville-von Neumann equation. In the latter, a contracted grid basis of potential-optimized discrete variable representation is tailored to incorporate most of the inter-mode coupling, while the Lindblad operators are represented as tensor products of one-dimensional operators, for consistency. This procedure results in a marked reduction of the grid size and in a much more advantageous scaling of the computational cost with respect to the increase of the dimensionality of the system. The factorizable method is found to provide an accurate description of the dissipative quantum dynamics of the model system, specifically of the time evolution of the state populations and of the probability density distribution of the molecular wave packet. The influence of intra-molecular vibrational energy redistribution appears to be properly taken into account by the new model on the whole range of coupling strengths. It demontrates that most of the mode mixing during relaxation is due to the potential part of the Hamiltonian and not to the coupling among relaxation operators.
The dynamics of an adsorbed atom irradiated by an infrared laser in resonance with the vibrational levels of the adbond can be described by the reduced density matrix of the adbond. A nonperturbative approach for the laser-adbond interaction is presented. Relaxation against the substrate degrees of freedom is described by second-order perturbation theory. Both pulsed lasers and cw lasers are considered. Expressions for the transition probabilities are derived in the low-intensity and in the high-intensity limits. It appears that the time evolution of the occupation probabilities is coupled to those of the coherences, i.e., a master equation cannot be obtained. The time evolution of all the elements of the reduced density matrix has to be taken into account. These expressions act as a basis for the study of physical processes as resonant de sorption, resonant heating, and laser-induced thermal desorption. Resonant desorption is described by the addition of loss terms into the equations of motion of the occupation probabilities. Expressions for the rate of desorption as well as the rate of energy transfer to the substrate are given. In particular, a comparison is made for the effect of a cw laser vs a pulsed laser on the desorption and resonant heating.
Strong light-matter coupling between molecules and optical or plasmonic cavity modes has emerged as a promising platform for advancing photonics, materials science, and chemistry. However, optical cavities and plasmonic resonators in particular are inherently lossy systems characterized by finite photon lifetimes. Accurate theoretical descriptions of molecular dynamics under strong coupling therefore require a proper treatment of cavity losses. In this work, we compare three theoretical approaches for modeling dissipative molecule-cavity dynamics within a realistic parameter regime: the Lindblad master equation, the stochastic Schrödinger equation, and the non-Hermitian Schrödinger equation. As an example, we consider the two lowest energy state of Na2 molecule coupled to a cavity mode and analyze the time evolution of the excited-state population and the mean photon number. Our results demonstrate that the stochastic Schrödinger equation provides an accurate and computationally efficient alternative to the Lindblad master equation, while the non-Hermitian Schrödinger approach is found to be applicable only within a limited range of conditions. Furthermore, we show that inclusion of molecular rotation leads to rotational-vibrational-photonic coupling and gives rise to pronounced nonadiabatic dynamics through light-induced conical intersections. These findings highlight the importance of both dissipation and rotational degrees of freedom for a realistic description of molecular dynamics in strongly coupled molecule-cavity systems.
The decay rate of an excited molecule (atom) embedded in a dispersive and absorbing planar cavity is derived by using a recently obtained compact form of the Green's function for a multilayer. As a by-product, a hint is provided for a straightforward extension of the results obtained for lossless cavities of other shapes to the corresponding absorbing cavities. The decay rate in an absorbing cavity consists of the spontaneous emission rate and of the nonradiative rates caused by the near-field interaction of the molecule with the cavity medium and, for nearby molecules, with the cavity mirrors. Only the spontaneous emission rate is satisfactorily described in the macroscopic approach adopted. The theory is applied to an analysis of the effects of the weak cavity absorption on the decay rate in a dielectric microcavity formed by two metallic mirrors. As expected, dissipation in the cavity medium spoils the conditions for controlled spontaneuos emission and strongly suppresses the intensity of spontaneous emission. However, its effect on the spontaneous emission rate is much less pronounced.