Dichroism of plasmonic chiral nanoalloys by rational design.

Time-dependent density functional theory (TDDFT) simulations are conducted on a series of chiral gold/silver alloy nanowires to explore whether silver doping can produce an enhancement of circular dichroism at the plasmon resonance in these systems, and to identify the quantum-mechanical origin of the observed effects. We find a strong plasmonic dichroism when one or two helixes of gold atoms are substituted by silver in a linear chiral nanotube, whose pure gold counterpart does not display any plasmonic dichroism, and we rationalize this finding in terms of "decoupling" the destructive interference of excitations in the pure gold nanotube via alloying. However, further attempts to increase the plasmonic dichroism by considering multi-shell gold nanowires in which one entire shell is doped with silver did not produce the desired effect, but rather a decrease in circular dichroism. We show that this latter result is due to a more severe destructive interference in the dipole excitation contributions, and suggest that further amplification should be possible in principle by properly tuning simultaneously the nanowire structure and chemical ordering.

Combined experimental and computational study of the photoabsorption of the monodoped and nondoped nanoclusters Au24Pt(SR)18, Ag24Pt(SR)18, and Ag25(SR)18.

Assessing the accuracy of first-principles computational approaches is instrumental to predict electronic excitations in metal nanoclusters with quantitative confidence. Here we describe a validation study on the optical response of a set of monolayer-protected clusters (MPC). The photoabsorption spectra of Ag25(DMBT)18-, Ag24Pt(DMBT)182- and Au24Pt(SC4H9)18, where DMBT is 2,4-dimethylbenzenethiolate and SC4H9 is n-butylthiolate, have been obtained at low temperature and compared with accurate TDDFT calculations. An excellent match between theory and experiment, with typical deviations of less than 0.1 eV, was obtained, thereby validating the accuracy and reliability of the proposed computational framework. Moreover, an analysis of the TDDFT simulations allowed us to ascribe all relevant spectral features to specific transitions between occupied/virtual orbital pairs. The doping effect of Pt on the optical response of these ultrasmall MPC systems was identified and discussed.

Reaction Mechanisms, Kinetics, and Improved Catalysts for Ammonia Synthesis from Hierarchical High Throughput Catalyst Design.

The Haber-Bosch (HB) process is the primary chemical synthesis technique for industrial production of ammonia (NH3) for manufacturing nitrate-based fertilizer and as a potential hydrogen carrier. The HB process alone is responsible for over 2% of all global energy usage to produce more than 160 million tons of NH3 annually. Iron catalysts are utilized to accelerate the reaction, but high temperatures and pressures of atmospheric nitrogen gas (N2) and hydrogen gas (H2) are required. A great deal of research has aimed at increased performance over the last century, but the rate of progress has been slow. This Account focuses on determining the atomic-level reaction mechanism for HB synthesis of NH3 on the Fe catalysts used in industry and how to use this knowledge to suggest greatly improved catalysts via a novel paradigm of catalyst rational design.We determined the full reaction mechanism on the two most active surfaces for the HB process, Fe(111) and Fe(211)R. We used density functional theory (DFT) to predict the free-energy barriers for all 12 important reactions and the 34 most important 2 × 2 surface configurations. Then we incorporated the mechanism into kinetic Monte Carlo (kMC) simulations run for several hours of real time to predict turnover frequencies (TOFs). The predicted TOFs are within experimental error, indicating that the predicted barriers are within 0.04 eV of experiment.With this level of accuracy, we are poised to use DFT to improve the catalyst. Rather than forming bulk alloys with uniform concentration, we aimed at finding additives that strongly prefer near-surface sites so that minor amounts of the additive might lead to dramatic improvements. However, even for a single additive, the combinations of surface species and reactions multiplies significantly, with ∼48 reaction steps to examine and nearly 100 surface configurations per 2 × 2 site. To make it practical to examine tens of dopant candidates, we developed the hierarchical high-throughput catalysis screening (HHTCS) approach, which we applied to both the Fe(111) and Fe(211) surfaces. For HHTCS, we identified the most important 4 reaction steps out of 12 for the two surfaces to examine >50 dopant cases, where we required performance at each step no worse than for pure Fe. With HHTCS, the computational cost is about 1% of that for doing the full reaction mechanism, allowing us to do ≈50 cases in about 1/2 the time it took to do pure Fe(111). The new leads identified with HHTCS are then validated with full mechanistic studies.For Fe(111), we predict three high-performance dopants that strongly prefer the second layer: Co with a rate 8 times higher, Ni with a rate 16 times higher, and Si with a rate 43 times higher, at 400 °C and 20 atm. We also found four dopants that strongly prefer the top layer and improve performance: Pt or Rh 3 times faster and Pd or Cu 2 times faster. For Fe(211), the best dopant was found to be second-layer Co with a rate 3 times faster than that for the undoped surface.The DFT/kMC data were used to predict reshaping of the catalyst particles under reaction conditions and how to tune dopant content so as to maximize catalytic area and thus activity. Finally, we show how to validate our mechanistic modeling via a comparison between theoretical and experimental operando spectroscopic signatures.

Oxidation and de-alloying of PtMn particle models: a computational investigation.

We present a computational study of the energetics and mechanisms of oxidation of Pt-Mn systems. We use slab models and simulate the oxidation process over the most stable (111) facet at a given Pt2Mn composition to make the problem computationally affordable, and combine Density-Functional Theory (DFT) with neural network potentials and metadynamics simulations to accelerate the mechanistic search. We find, first, that Mn has a strong tendency to alloy with Pt. This tendency is optimally realized when Pt and Mn are mixed in the bulk, but, at a composition in which the Mn content is high enough such as for Pt2Mn, Mn atoms will also be found in the surface outmost layer. These surface Mn atoms can dissociate O2 and generate MnOx species, transforming the surface-alloyed Mn atoms into MnOx surface oxide structures supported on a metallic framework in which one or more vacancy sites are simultaneously created. The thus-formed vacancies promote the successive steps of the oxidation process: the vacancy sites can be filled by surface oxygen atoms, which can then interact with Mn atoms in deeper layers, or subsurface Mn atoms can intercalate into interstitial sites. Both these steps facilitate the extraction of further bulk Mn atoms into MnOx oxide surface structures, and thus the progress of the oxidation process, with typical rate-determining energy barriers in the range 0.9-1.0 eV.

A Resolution of Identity Technique to Speed up TDDFT with Hybrid Functionals: Implementation and Application to the Magic Cluster Series Au8n+4(SC6H5)4n+8 (n = 3-6).

The Resolution of Identity (RI) technique has been employed to speed up the use of hybrid exchange-correlation (xc) functionals at the TDDFT level using the Hybrid Diagonal Approximation. The RI has been implemented within the polTDDFT algorithm (a complex damped polarization method) in the AMS/ADF suite of programs. A speedup factor of 30 has been obtained with respect to a previous numerical implementation, albeit with the same level of accuracy. Comparison of TDDFT simulations with the experimental photoabsorption spectra of the cluster series Au8n+4(SR)4n+8(n = 3-6; R = C6H5) showed the excellent accuracy and efficiency of the method. Results were compared with those obtained via the more simplified and computationally cheaper TDDFT+TB and sTDDFT methods. The present method represents an accurate as well as computationally affordable approach to predict photoabsorption spectra of complex species, realizing an optimal compromise between accuracy and computational efficiency, and is suitable for applications to large metal clusters with sizes up to several hundreds of atoms.

Exploring the materials space in the smallest particle size range: from heterogeneous catalysis to electrocatalysis and photocatalysis.

Ultrasmall clusters of subnanometer size can possess unique and even unexpected physical and chemical propensities which make them interesting in various fields of basic science and for potential applications, such as catalysis, photocatalysis, electrocatalysis, and optical and chemical sensors, just to name a few examples. These small particles often offer the tunability of their performance in an atom-by-atom fashion and an economic atom-efficient use of the metal loading. In this paper we review recent progress in the characterization and theory of well-defined subnanometer clusters in catalytic processes, and discuss their optical properties and stability, along with the potential of the size-selected clusters for the understanding of catalytic processes and for the development of new classes of catalysts.

Coupling between Plasmonic and Molecular Excitations: TDDFT Investigation of an Ag-Nanorod/BODIPY-Dye Interaction.

A time-dependent density functional theory (TDDFT) computational approach is employed to study the optical coupling between a plasmonic system (a Ag50 nanorod) and a fluorescent dye (BODIPY). It is found that the BODIPY dye can interact with a plasmonic system in a rather different and selective way according to the mutual orientation of the fragments. Indeed, (i) the plasmon excitation turns out to be sensitive to the presence of the BODIPY transition and (ii) this can lead to amplify or suppress the resonance accordingly to the relative orientation of the corresponding transition dipoles. To understand the coupling mechanism, we analyze the shape of the induced density in real space and the Individual Component Map of the Oscillator Strength (ICM-OS) plots and achieve a simple rationalization and insight on the origin and features of the coupling. The resulting possibility of understanding plasmon/fluorophore interactions by simple qualitative arguments opens the way to a rational design of hybrid (plasmon + dye) systems with the desired optical behavior.

Reaction intermediates during operando electrocatalysis identified from full solvent quantum mechanics molecular dynamics.

Electrocatalysis provides a powerful means to selectively transform molecules, but a serious impediment in making rapid progress is the lack of a molecular-based understanding of the reactive mechanisms or intermediates at the electrode-electrolyte interface (EEI). Recent experimental techniques have been developed for operando identification of reaction intermediates using surface infrared (IR) and Raman spectroscopy. However, large noises in the experimental spectrum pose great challenges in resolving the atomistic structures of reactive intermediates. To provide an interpretation of these experimental studies and target for additional studies, we report the results from quantum mechanics molecular dynamics (QM-MD) with explicit consideration of solvent, electrode-electrolyte interface, and applied potential at 298 K, which conceptually resemble the operando experimental condition, leading to a prototype of operando QM-MD (o-QM-MD). With o-QM-MD, we characterize 22 possible reactive intermediates in carbon dioxide reduction reactions ([Formula: see text]RRs). Furthermore, we report the vibrational density of states (v-DoSs) of these intermediates from two-phase thermodynamic (2PT) analysis. Accordingly, we identify important intermediates such as chemisorbed [Formula: see text] ([Formula: see text]), *HOC-COH, *C-CH, and *C-COH in our o-QM-MD likely to explain the experimental spectrum. Indeed, we assign the experimental peak at 1,191 cm-1 to the mode of C-O stretch in *HOC-COH predicted at 1,189 cm-1 and the experimental peak at 1,584 cm-1 to the mode of C-C stretch in *C-COD predicted at 1,581 cm-1 Interestingly, we find that surface ketene (*C=C=O), arising from *HOC-COH dehydration, also shows signals at around 1,584 cm-1, which indicates a nonelectrochemical pathway of hydrocarbon formation at low overpotential and high pH conditions.

Understanding Reaction Networks through Controlled Approach to Equilibrium Experiments Using Transient Methods.

We report a combined experimental/theoretical approach to studying heterogeneous gas/solid catalytic processes using low-pressure pulse response experiments achieving a controlled approach to equilibrium that combined with quantum mechanics (QM)-based computational analysis provides information needed to reconstruct the role of the different surface reaction steps. We demonstrate this approach using model catalysts for ammonia synthesis/decomposition. Polycrystalline iron and cobalt are studied via low-pressure TAP (temporal analysis of products) pulse response, with the results interpreted through reaction free energies calculated using QM on Fe-BCC(110), Fe-BCC(111), and Co-FCC(111) facets. In TAP experiments, simultaneous pulsing of ammonia and deuterium creates a condition where the participation of reactants and products can be distinguished in both forward and reverse reaction steps. This establishes a balance between competitive reactions for D* surface species that is used to observe the influence of steps leading to nitrogen formation as the nitrogen product remains far from equilibrium. The approach to equilibrium is further controlled by introducing delay timing between NH3 and D2 which allows time for surface reactions to evolve before being driven in the reverse direction from the gas phase. The resulting isotopic product distributions for NH2D, NHD2, and HD at different temperatures and delay times and NH3/D2 pulsing order reveal the role of the N2 formation barrier in controlling the surface concentration of NHx* species, as well as providing information on the surface lifetimes of key reaction intermediates. Conclusions derived for monometallic materials are used to interpret experimental results on a more complex and active CoFe bimetallic catalyst.

Correction: 2D oxides on metal materials: concepts, status, and perspectives.

Correction for '2D oxides on metal materials: concepts, status, and perspectives' by Giovanni Barcaro et al., Phys. Chem. Chem. Phys., 2019, 21, 11510-11536, DOI: 10.1039/C9CP00972H.

An augmented (multi-descriptor) grouping algorithm to optimize chemical ordering in nanoalloys.

We propose the Augmented Grouping Approach (AugGA) and its deployment in the Augmented Grouping GO (AugGGO) scheme, for an efficient exploration of the chemical ordering (or compositional structure) of multi-component (alloyed) nanoparticles. The approach is based on a 'grouping' strategy (previously proposed for high-symmetry structures) by which the number of compositional degrees of freedom of the system is decreased by defining sets of atoms (groups, or orbits, or shells) that are constrained to be populated by the same element. Three fundamental advances are here included with respect to previous proposals: (i) groups are defined on the basis of descriptors (no point-group symmetry is assumed), (ii) bulk groups can exploit general chemical ordering patterns taken from databases, and (iii) sub-grouping is realized via a multi-descriptor strategy (here using two basic descriptors: the atomic energy and a few types of geometry patterns). The AugGGO approach is applied to two prototypical examples of binary nanoalloys: Pd-Pt and Ag-Cu, with a size between ≈500 and ≈1300 atoms, in different configurations, and the convex hull of the mixing energy as a function of composition is derived. It is shown how the three advances here proposed decisively extend the power and scope of the grouping approach: (i) making it applicable to any generic structural framework, (ii) achieving a thorough sampling of the core regions of nanoparticles, and (iii) catching exotic/unexpected chemical ordering arrangements, at a computational cost which is 1-2 orders of magnitude smaller than that of traditional Monte Carlo single-exchange techniques.

Predictive optical photoabsorption of Ag24Au(DMBT)18 - via efficient TDDFT simulations.

We report a computational study via time-dependent density-functional theory (TDDFT) methods of the photo-absorption spectrum of an atomically precise monolayer-protected cluster (MPC), the Ag24Au(DMBT)18 single negative anion, where DMBT is the 2,4-dimethylbenzenethiolate ligand. The use of efficient simulation algorithms, i.e., the complex polarizability polTDDFT approach and the hybrid-diagonal approximation, allows us to employ a variety of exchange-correlation (xc-) functionals at an affordable computational cost. We are thus able to show, first, how the optical response of this prototypical compound, especially but not exclusively in the absorption threshold (low-energy) region, is sensitive to (1) the choice of the xc-functionals employed in the Kohn-Sham equations and the TDDFT kernel and (2) the choice of the MPC geometry. By comparing simulated spectra with precise experimental photoabsorption data obtained from room temperature down to low temperatures, we then demonstrate how a hybrid xc-functional in both the Kohn-Sham equations and the diagonal TDDFT kernel at the crystallographically determined experimental geometry is able to provide a consistent agreement between simulated and measured spectra across the entire optical region. Single-particle decomposition analysis tools finally allow us to understand the physical reason for the failure of non-hybrid approaches.

Optical Activity of Metal Nanoclusters Deposited on Regular and Doped Oxide Supports from First-Principles Simulations.

We report a computational study and analysis of the optical absorption processes of Ag20 and Au20 clusters deposited on the magnesium oxide (100) facet, both regular and including point defects. Ag20 and Au20 are taken as models of metal nanoparticles and their plasmonic response, MgO as a model of a simple oxide support. We consider oxide defects both on the oxygen anion framework (i.e., a neutral oxygen vacancy) and in the magnesium cation framework (i.e., replacing Mg++ with a transition metal: Cu++ or Co++). We relax the clusters' geometries via Density-Functional Theory (DFT) and calculate the photo-absorption spectra via Time-Dependent DFT (TDDFT) simulations on the relaxed geometries. We find that the substrate/cluster interaction induces a broadening and a red-shift of the excited states of the clusters, phenomena that are enhanced by the presence of an oxygen vacancy and its localized excitations. The presence of a transition-metal dopant does not qualitatively affect the spectral profile. However, when it lies next to an oxygen vacancy for Ag20, it can strongly enhance the component of the cluster excitations perpendicular to the surface, thus favoring charge injection.

Plasmonic Circular Dichroism in Chiral Gold Nanowire Dimers.

We report a computational study at the time-dependent density functional theory (TDDFT) level of the chiro-optical spectra of chiral gold nanowires coupled in dimers. Our goal is to explore whether it is possible to overcome destructive interference in single nanowires that damp chiral response in these systems and to achieve intense plasmonic circular dichroism (CD) through a coupling between the nanostructures. We predict a huge enhancement of circular dichroism at the plasmon resonance when two chiral nanowires are intimately coupled in an achiral relative arrangement. Such an effect is even more pronounced when two chiral nanowires are coupled in a chiral relative arrangement. Individual component maps of rotator strength, partial contributions according to the magnetic dipole component, and induced densities allow us to fully rationalize these findings, thus opening the way to the field of plasmonic CD and its rational design.

Si-Doped Fe Catalyst for Ammonia Synthesis at Dramatically Decreased Pressures and Temperatures.

The Haber-Bosch (HB) process combining nitrogen (N2) and hydrogen (H2) into ammonia (NH3) gas plays an essential role in the synthesis of fertilizers for food production and many other commodities. However, HB requires enormous energy resources (2% of world energy production), and the high pressures and temperatures make NH3 production facilities very expensive. Recent advances in improving HB catalysts have been incremental and slow. To accelerate the development of improved HB catalysts, we developed a hierarchical high-throughput catalyst screening (HHTCS) approach based on the recently developed complete reaction mechanism to identify non-transition-metal (NTM) elements from a total set of 18 candidates that can significantly improve the efficiency of the most active Fe surface, Fe-bcc(111), through surface and subsurface doping. Surprisingly, we found a very promising subsurface dopant, Si, that had not been identified or suggested previously, showing the importance of the subsurface Fe atoms in N2 reduction reactions. Then we derived the full reaction path of the HB process for the Si doped Fe-bcc(111) from QM simulations, which we combined with kinetic Monte Carlo (kMC) simulations to predict a ∼13-fold increase in turnover frequency (TOF) under typical extreme HB conditions (200 atm reactant pressure and 500 °C) and a ∼43-fold increase in TOF under ideal HB conditions (20 atm reactant pressure and 400 °C) for the Si-doped Fe catalyst, in comparison to pure Fe catalyst. Importantly, the Si-doped Fe catalyst can achieve the same TOF of pure Fe at 200 atm/500 °C under much milder conditions, e.g. at a much decreased reactant pressure of 20 atm at 500 °C, or alternatively at temperature and reactant pressure decreased to 400 °C and 60 atm, respectively. Production plants using the new catalysts that operate under such milder conditions could be much less expensive, allowing production at local sites needing fertilizer.

The Missing Link: Au191(SPh-tBu)66 Janus Nanoparticle with Molecular and Bulk-Metal-like Properties.

Understanding the evolution of the structure and properties in metals from molecule-like to bulk-like has been a long sought fundamental question in science, since Faraday's 1857 work. We report the discovery of a Janus nanomolecule, Au191(SPh-tBu)66 having both molecular and metallic characteristics, explored crystallographically and optically and modeled theoretically. Au191 has an anisotropic, singly twinned structure with an Au155 core protected by a ligand shell made of 24 monomeric [-S-Au-S-] and 6 dimeric [-S-Au-S-Au-S-] staples. The Au155 core is composed of an 89-atom inner core and 66 surface atoms, arranged as [Au3@Au23@Au63]@Au66 concentric shells of atoms. The inner core has a monotwinned/stacking-faulted face-centered-cubic (fcc) structure. Structural evolution in metal nanoparticles has been known to progress from multiply twinned, icosahedral, structures in smaller molecular sizes to untwinned bulk-like fcc monocrystalline nanostructures in larger nanoparticles. The monotwinned inner core structure of the ligand capped Au191 nanomolecule provides the critical missing link, and bridges the size-evolution gap between the molecular multiple-twinning regime and the bulk-metal-like particles with untwinned fcc structure. The Janus nature of the nanoparticle is demonstrated by its optical and electronic properties, with metal-like electron-phonon relaxation and molecule-like long-lived excited states. First-principles theoretical explorations of the electronic structure uncovered electronic stabilization through the opening of a shell-closing gap at the top of the occupied manifold of the delocalized electronic superatom spectrum of the inner core. The electronic stabilization together with the inner core geometric stability and the optimally stapled ligand-capping anchor and secure the stability of the entire nanomolecule.

An efficient hybrid scheme for time dependent density functional theory.

A hybrid approach able to perform Time Dependent Density Functional Theory (TDDFT) simulations with the same accuracy as that of hybrid exchange-correlation (xc-) functionals but at a fraction of the computational cost is developed, implemented, and validated. The scheme, which we name Hybrid Diagonal Approximation (HDA), consists in employing in the response function a hybrid xc-functional (containing a fraction of the non-local Hartree-Fock exchange) only for the diagonal elements of the omega matrix, while the adiabatic local density approximation is employed for the off-diagonal terms. HDA is especially (but not exclusively) advantageous when using Slater type orbital basis sets and allows one to employ them in a uniquely efficient way, as we demonstrate here by implementing HDA in a local version of the Amsterdam Density Functional code. The new protocol is tested on NH3, C6H6, and the [Au25(SCH3)18]- cluster as prototypical cases ranging from small molecules to ligand-protected metal clusters, finding excellent agreement with respect to both full kernel TDDFT simulations and experimental data. Additionally, a specific comparison test between full kernel and HDA is considered at the Casida level on seven other molecular species, which further confirm the accuracy of the approach for all investigated systems. For the [Au25(SCH3)18]- cluster, a speedup by a factor of seven is obtained with respect to the full kernel. The HDA, therefore, promises to provide a quantitative description of the optical properties of medium-sized systems (nanoclusters) at an affordable cost, thanks to its computational efficiency, especially in combination with a complex polarization algorithm version of TDDFT.

2D oxides on metal materials: concepts, status, and perspectives.

Oxide materials at the two-dimensional limit, in particular in the form of ultrathin films of oxides (UTOx) grown on metal surfaces, represent promising materials in view of both fundamental science and technological applications. While the former aspect is widely recognized, these systems have not yet realized their full potential in terms of the latter (technological) aspect. In the present perspective, we review the field and its basic underlying concepts, and at the same time we provide an overview of the most promising future directions with a focus on their potential toward and relationships with real-world exploitation.

Reaction mechanism and kinetics for ammonia synthesis on the Fe(211) reconstructed surface.

To provide guidelines to accelerate the Haber-Bosch (HB) process for synthesis of ammonia from hydrogen and nitrogen, we used Quantum Mechanics (QM) to determine the reaction mechanism and free energy reaction barriers under experimental reaction conditions (400 °C and 20 atm) for all 10 important surface reactions on the Fe(211) reconstructed (Fe(211)R) surface. These conditions were then used in full kMC modeling for 30 minutes to attain steady state. We find that the stable surface under Haber-Bosch conditions is the missing row 2 × 1 reconstructed surface (211)R and that the Turn Over Frequency (TOF) is 18.7 s-1 per 2 × 2 surface site for 1.5 Torr NH3 pressure, but changes to 3.5 s-1 for 1 atm, values close (within 6%) to the ones on Fe(111). The experimental ratio between (211) and (111) rates at low (undisclosed) NH3 pressure was reported to be 0.75. The excellent agreement with experiment on two very different surfaces and reaction mechanisms is a testament of the accuracy of QM modeling. In addition, our kinetic analysis indicates that Fe(211)R is more active than Fe(111) at high pressure, close to HB industrial conditions, and that (211)R is more abundant than (111) via a steady-state Wulff construction under HB conditions. Thus, at variance with common thinking, we advocate the Fe(211)R surface as the catalytically active phase of pure iron ammonia synthesis catalyst under HB industrial conditions.

Pd doping, conformational, and charge effects on the dichroic response of a monolayer protected Au38(SR)24 nanocluster.

TDDFT simulations of the absorption and CD spectra of a Pd2Au36(SC2H4Ph)24 monolayer-protected cluster (MPC) are carried out with the aim of investigating the effects of doping, conformational degrees of freedom of the thiolates' end-groups, and charge states on the optical and dichroic response of a prototypical MPC species. Clear signatures of Pd doping in both absorption and CD spectra are found to be a consequence of the participation of Pd (4d) states in the ligand-based d-band and on the unoccupied MOs of lower energy. Exploration of conformational space points to a much greater sensitivity of optical rotation to the conformation of the end-groups of the organic monolayer compared to absorption. Finally, the effect of charge is mainly seen as a decreased dependence of the dichroic response on conformation. The agreement between the TDDFT predictions and the available experimental data is good, and enables an assignment of absorption and CD bands to specific classes of one-particle excitations.