Reply to the 'Comment on "Perturbation theory of scattering for grazing-incidence fast-atom diffraction"' by G. A. Bocan, H. Breiss, S. Szilasi, A. Momeni, E. M. S. Casagrande, E. A. Sánchez, M. S. Gravielle and H. Khemliche, Phys. Chem. Chem. Phys., 2023, 25, DOI: 10.1039/D3CP02486E.

In this Reply, we show that criticisms of perturbation theory for grazing-incidence fast-atom diffraction (GIFAD) are ill-founded. We show explicitly that our formulation (W. Allison, S. Miret-Artés and E. Pollak, Phys. Chem. Chem. Phys., 2022, 24, 15851) provides a similar precision in describing the observed phenomena as ab initio potentials. Since that is the main criterion to distinguish between methods, it seems reasonable to conclude that the perturbation approach using a Morse-type potential reproduces the essential aspects of the dynamics correctly. In addition we expand on the historical context and summarize the physical insights provided by our methods.

The stochastic wave function method for diffusion of alkali atoms on metallic surfaces.

The stochastic wave function method is proposed to study the diffusion regimes of alkali atoms on metallic surfaces. The Lindblad approach, based on the microscopic Hamiltonian information in the Caldeira-Leggett model, is presented and numerical calculations of the dynamics are carried out to characterize surface diffusion for two different systems: Na-Cu(111) and Li-Cu(111). Calculations of the intermediate scattering function for an isolated adsorbate are compared, in the Brownian limit, with results deduced from helium spin-echo (HeSE) experiments after reducing them to single adsorbate dynamics. To illustrate the method we present the dependence on momentum transfer and the temperature dependency. Results show that the experiment can be described at a quantitative level by the 1-D quantum model (reduced dimensionality).

Atom-surface scattering in the classical multiphonon regime.

Many experiments that utilize beams of incident atoms colliding with surfaces as a probe of surface properties are carried out at large energies, high temperatures and with large mass atoms. Under these conditions the scattering process does not exhibit quantum mechanical properties such as diffraction or single-phonon excitation, but rather can be treated with classical physics. This is a review of work carried out by the authors over a span of several years to develop theoretical frameworks using classical physics for describing the scattering interactions of atom with surfaces.

Surface diffusion within the Caldeira-Leggett formalism.

Surface diffusion is described in terms of the intermediate scattering function in the time domain and reciprocal space. Two extreme time regimes are analyzed, ballistic (very short times) and Brownian or diffusive (very long times). This open dynamics is studied from the master equation for the reduced density matrix within the Caldeira-Leggett formalism. Several characteristic magnitudes in this decoherence process such as the coherence length, ensemble width and purity of the density matrix are analyzed. Furthermore, for flat surfaces, the surface diffusion is considered for the Schrödinger cat states and identical adsorbates or adparticles, bosons and fermions. The analytical results are compared with those issued from solving the Lindblad master equation through the stochastic wave function method. This numerical analysis is extended to be applied to corrugated surfaces.

Momentum-Space Decoherence of Distinguishable and Identical Particles in the Caldeira-Leggett Formalism.

In this work, momentum-space decoherence using minimum and nonminimum-uncertainty-product (stretched) Gaussian wave packets in the framework of Caldeira-Leggett formalism and under the presence of a linear potential is studied. As a dimensionless measure of decoherence, purity, a quantity appearing in the definition of the linear entropy, is studied taking into account the role of the stretching parameter. Special emphasis is on the open dynamics of the well-known cat states and bosons and fermions compared to distinguishable particles. For the cat state, while the stretching parameter speeds up the decoherence, the external linear potential strength does not affect the decoherence time; only the interference pattern is shifted. Furthermore, the interference pattern is not observed for minimum-uncertainty-product-Gaussian wave packets in the momentum space. Concerning bosons and fermions, the question we have addressed is how the symmetry of the wave functions of indistinguishable particles is manifested in the decoherence process, which is understood here as the loss of being indistinguishable due to the gradual emergence of classical statistics with time. We have observed that the initial bunching and anti-bunching character of bosons and fermions, respectively, in the momentum space are not preserved as a function of the environmental parameters, temperature, and damping constant. However, fermionic distributions are slightly broader than the distinguishable ones and these similar to the bosonic distributions. This general behavior could be interpreted as a residual reminder of the symmetry of the wave functions in the momentum space for this open dynamics.

Ag5-induced stabilization of multiple surface polarons on perfect and reduced TiO2 rutile (110).

The recent advent of cutting-edge experimental techniques allows for a precise synthesis of subnanometer metal clusters composed of just a few atoms, opening new possibilities for subnanometer science. In this work, via first-principles modeling, we show how the decoration of perfect and reduced TiO2 surfaces with Ag5 atomic clusters enables the stabilization of multiple surface polarons. Moreover, we predict that Ag5 clusters are capable of promoting defect-induced polarons transfer from the subsurface to the surface sites of reduced TiO2 samples. For both planar and pyramidal Ag5 clusters, and considering four different positions of bridging oxygen vacancies, we model up to 14 polaronic structures, leading to 134 polaronic states. About 71% of these configurations encompass coexisting surface polarons. The most stable states are associated with large inter-polaron distances (>7.5 Å on average), not only due to the repulsive interaction between trapped Ti3+ 3d1 electrons, but also due to the interference between their corresponding electronic polarization clouds [P. López-Caballero et al., J. Mater. Chem. A 8, 6842-6853 (2020)]. As a result, the most stable ferromagnetic and anti-ferromagnetic arrangements are energetically quasi-degenerate. However, as the average inter-polarons distance decreases, most (≥70%) of the polaronic configurations become ferromagnetic. The optical excitation of the midgap polaronic states with photon energy at the end of the visible region causes the enlargement of the polaronic wave function over the surface layer. The ability of Ag5 atomic clusters to stabilize multiple surface polarons and extend the optical response of TiO2 surfaces toward the visible region bears importance in improving their (photo-)catalytic properties and illustrates the potential of this new generation of subnanometer-sized materials.

Surface Structure of Bi(111) from Helium Atom Scattering Measurements. Inelastic Close-Coupling Formalism.

Elastic and inelastic close-coupling (CC) calculations have been used to extract information about the corrugation amplitude and the surface vibrational atomic displacement by fitting to several experimental diffraction patterns. To model the three-dimensional interaction between the He atom and the Bi(111) surface under investigation, a corrugated Morse potential has been assumed. Two different types of calculations are used to obtain theoretical diffraction intensities at three surface temperatures along the two symmetry directions. Type one consists of solving the elastic CC (eCC) and attenuating the corresponding diffraction intensities by a global Debye-Waller (DW) factor. The second one, within a unitary theory, is derived from merely solving the inelastic CC (iCC) equations, where no DW factor is necessary to include. While both methods arrive at similar predictions for the peak-to-peak corrugation value, the variance of the value obtained by the iCC method is much better. Furthermore, the more extensive calculation is better suited to model the temperature induced signal asymmetries and renders the inclusion for a second Debye temperature for the diffraction peaks futile.

Communication: Semiclassical perturbation theory for the quantum diffractive scattering of atoms on thermal surfaces.

Inspired by the semiclassical perturbation theory of Hubbard and Miller [J. Chem. Phys. 80, 5827 (1984)], we derive explicit expressions for the angular distribution of particles scattered from thermal surfaces. At very low surface temperature, the observed experimental background scattering is proportional to the spectral density of the phonons. The angular distribution is a sum of diffraction peaks and a broad background reflecting the spectral density. The theory is applied to measured angular distributions of Ne, Ar, and Kr scattered from a Cu(111) surface.

Dissipative geometric phase and decoherence in parity-violating chiral molecules.

Within a generalized Langevin framework for open quantum systems, the cyclic evolution of a two-level system is analyzed in terms of the geometric phase extended to dissipative systems for Ohmic friction. This proposal is applied to the dynamics of chiral molecules where the tunneling and parity violating effects are competing. The effect of different system-bath coupling functions in the dissipated energy is shown to be crucial to understand the behavior of the geometric phase as well as the decoherence displayed by the corresponding interference patterns.

Communication: Quantum Zeno-based control mechanism for molecular fragmentation.

A quantum control mechanism is proposed for molecular fragmentation processes within a scenario grounded on the quantum Zeno effect. In particular, we focus on the van der Waals Ne-Br(2) complex, which displays two competing dissociation channels via vibrational and electronic predissociation. Accordingly, realistic three-dimensional wave packet simulations are carried out by using ab initio interaction potentials recently obtained to reproduce available experimental data. Two numerical models to simulate the repeated measurements are reported and analyzed. It is found that the otherwise fast vibrational predissociation is slowed down in favor of the slow electronic (double fragmentation) predissociation, which is enhanced by several orders of magnitude. Based on these theoretical predictions, some hints to experimentalists to confirm their validity are also proposed.

Quantum Zeno and anti-Zeno effects in surface diffusion of interacting adsorbates.

Surface diffusion of interacting adsorbates is here analyzed within the context of two fundamental phenomena of quantum dynamics, namely the quantum Zeno effect and the anti-Zeno effect. The physical implications of these effects are introduced here in a rather simple and general manner within the framework of non-selective measurements and for two (surface) temperature regimes: high and very low (including zero temperature). The quantum intermediate scattering function describing the adsorbate diffusion process is then evaluated for flat surfaces, since it is fully analytical in this case. Finally, a generalization to corrugated surfaces is also discussed. In this regard, it is found that, considering a Markovian framework and high surface temperatures, the anti-Zeno effect has already been observed, though not recognized as such.

A path-integral Monte Carlo study of a small cluster: The Ar trimer.

The Ar(3) system has been studied between T=0 K and T=40 K by means of a path-integral Monte Carlo (PIMC) method. The behavior of the average energy in terms of the temperature has been explained by comparison with results obtained with the thermal averaged rovibrational spectra estimated via: (i) a quantum mechanical method based on distributed Gaussian functions for the interparticle distances and (ii) an analytical model which precisely accounts for the participation of the dissociative continua Ar(2)+Ar and Ar+Ar+Ar. Beyond T approximately 20 K, the system explores floppier configurations than the rigid equilateral geometry, as linear and Ar-Ar(2)-like arrangements, and fragmentates around T approximately 40 K. A careful investigation of the specific heat in terms of a confining radius in the PIMC calculation seems to discard a proper phase transition as in larger clusters, in apparent contradiction with previous reports of precise values for a liquid-gas transition. The onset of this noticeable change in the dynamics of the trimer occurs, however, at a remarkably low value of the temperature in comparison with Ar(n) systems formed with more Ar atoms. Quantum mechanical effects are found of relevance at T

Two-bath model for activated surface diffusion of interacting adsorbates.

The diffusion and low vibrational motions of adsorbates on surfaces can be well described by a purely stochastic model, the so-called interacting single adsorbate model, for low-moderate coverages (theta < or approximately equal to 0.12). Within this model, the effects of thermal surface phonons and adsorbate-adsorbate collisions are accounted for by two uncorrelated noise functions, which arise in a natural way from a two-bath model based on a generalization of the one-bath Caldeira-Leggett Hamiltonian. As an illustration, the model is applied to the diffusion of Na atoms on a Cu(001) surface with different coverages.

He atom scattering from ZnO surfaces: calculation of diffraction peak intensities using the close-coupling approach.

Diffraction intensities of a molecular He beam scattered off the clean and water-covered ZnO(1010) surface have been simulated using a new potential model in conjunction with the close-coupling formalism. The effective corrugation functions for the systems He-H2O/ZnO(1010) and He-H2O/ZnO(1010) have been obtained from density functional theory calculations within the Esbjerg-Nørskov approximation. Using these data a potential model is constructed consisting of a corrugated Morse potential at small He-surface distances and a semiempiric attractive part at larger distances. The diffraction patterns obtained from close-coupling calculations agree with the experimental data within about 10%, which opens the possibility to simulate He diffraction from surfaces of any structural complexity and to verify surface and adsorbate structures proposed theoretically by employing this kind of analysis.

Phonon lineshapes in atom-surface scattering.

Phonon lineshapes in atom-surface scattering are obtained from a simple stochastic model based on the so-called Caldeira-Leggett Hamiltonian. In this single-bath model, the excited phonon resulting from a creation or annihilation event is coupled to a thermal bath consisting of an infinite number of harmonic oscillators, namely the bath phonons. The diagonalization of the corresponding Hamiltonian leads to a renormalization of the phonon frequencies in terms of the phonon friction or damping coefficient. Moreover, when there are adsorbates on the surface, this single-bath model can be extended to a two-bath model accounting for the effect induced by the adsorbates on the phonon lineshapes as well as their corresponding lineshapes.

A theoretical investigation on the spectrum of the Ar trimer for high rotational excitations.

A detailed study of the rovibrational spectrum of the Ar trimer is performed by means of an exact hyperspherical coordinate (HC) method and a variational approach based on distributed Gaussian functions (DGFs) to describe the interparticle distances. The good agreement observed between the energy levels obtained with both procedures for high values of the total angular momentum (J=15 and 20) reveals the quality of the DGF method to describe the rotation of the title system. Rotational constants for the lowest bound states, obtained as averages for each vibrational state, have been obtained and compared to previous results. A detailed analysis of density probability functions obtained by means of the HC approach for rovibrational states at J=0 and 20 shows close similitudes thus supporting the vibration-rotation separation adopted within the DGF scheme for the Ar(3) system.

Quantum Markovian activated surface diffusion of interacting adsorbates.

A quantum Markovian activated atom-surface diffusion model with interacting adsorbates is proposed for the intermediate scattering function, which is shown to be complex-valued and factorizable into a classical-like and a quantum-mechanical factor. Applications to the diffusion of Na atoms on flat (weakly corrugated) and corrugated-Cu(001) surfaces at different coverages and surface temperatures are analyzed. Quantum effects are relevant to diffusion at low surface temperatures and coverages even for relatively heavy particles, such as Na atoms, where transport by tunneling is absent.