Albert Polman

Mathieu Kociak

F Javier García De Abajo

Albert Orcid

Polman

Javier García De Abajo

Mathieu

Electron Beam Spectroscopy for Nanophotonics

Progress in electron-beam spectroscopies has recently enabled the study of optical excitations with combined space, energy and time resolution in the nanometer, millielectronvolt and femtosecond domain, thus providing unique access into nanophotonic structures and their detailed optical responses. These techniques rely on ∼1-300 keV electron beams focused at the sample down to subnanometer spots, temporally compressed in wavepackets a few femtoseconds long, and in some cases controlled by ultrafast light pulses. The electrons undergo energy losses and gains, also giving rise to cathodoluminescence light emission, which are recorded to reveal the optical landscape along the beam path. This review portraits these advances, with a focus on coherent excitations, emphasizing the increasing level of control over the electron wave functions and ensuing applications in the study and technological use of optically resonant modes and polaritons in nanoparticles, 2D materials and engineered nanostructures.

Introduction: fundamentals of EELS and CL

Cathodoluminescence (CL) and electron energy-loss spectroscopy (EELS) have advanced in recent decades to arguably provide the best combination of space, energy and time resolutions for the structural and optical characterization of materials. In these techniques, an energetic electron beam is raster-scanned over the specimen using either a transmission electron microscope (TEM, featuring 30-300 keV electron beams and equipped with an electron analyzer for acquisition of EELS spectra, and optionally with an optical spectrometer for CL) or a scanning electron microscope (SEM, 1-30 keV for CL). The acquired CL/EELS infrared-to-UV spectral data are then correlated with morphological, and structural information derived from secondary electron (SE) images (mostly in SEMs) or the high-angle annular dark field (HAADF) signal (in TEMs), all taken on the same sample during the same measurement. Optionally, energy-dispersive X-ray spectroscopy (EDX), electron backscatter diffraction (EBSD) and high-energy core-level spectroscopy (in EELS) can simultaneously provide compositional and atomic-structure correlations. [START_REF] Egerton | Electron Energy-Loss Spectroscopy in the Electron Microscope[END_REF] Beyond traditional materials science applications, the last decade has witnessed the emergence of both EELS and CL as unique tools in research on the behavior of light at the nanoscale. Due to the unsurpassed spatial resolution offered by electron microscopes down to the atomic (TEM) or nanometer (SEM) scale, pixel-by-pixel comparisons can be made between CL/EELS images and compositional as well as morphological information at length scales that are small compared with the light wavelength. These nanoscale correlations are at the heart of the success of spatially-resolved EELS and CL in nanophotonics research.

Figure 1(a-d) shows schematics of electron-light-matter interactions in different consolidated and emerging forms of electron beam spectroscopies. In conventional CL/EELS (Fig. 1a), two-dimensional (2D) CL/EELS maps are acquired by raster-scanning the beam over the specimen. A 2D map is constructed for each emitted light wavelength (CL) or electron energy loss (EELS) in what has been termed hyperspectral imaging. A key aspect of the CL and EELS excitation mechanism is that the specimen is polarized by time-varying electric fields produced by the moving electron, similar to the effect of an optical pulse. The spatial extent of the radial electric field around the electron trajectory is shown in Fig. 1e in reduced units. The field decays evanescently at large distances, as described by the modified Bessel functions K0 and K1. As an example, Fig. 1f shows the time evolution of the radial and axial fields for a 30 keV electron at a position 5 nm away from the trajectory. The electron creates a single electromagnetic field cycle within a few hundred attoseconds. The corresponding frequency spectrum is shown in Fig. 1g and has energies with significant weight in the 0-30 eV spectral region, the precise range depending on the electron kinetic energy. [START_REF] De Abajo | Optical excitations in electron microscopy[END_REF][START_REF] Brenny | Femtosecond plasmon and photon wave packets excited by a high-energy electron on a metal or dielectric surface[END_REF] The electron thus acts as a broadband source of optical excitation (i.e., its electromagnetic field covers a wide spectral range), with a spatial resolution limited by the extent of its evanescent field (∼0.5-10 nm, depending on electron energy and detection frequency). The EELS spectra are determined by the work done by each of the frequency components of the electron field acting on the polarized material, and probed through the ensuing losses experienced by the electron. Part of this work transforms into radiation emission (i.e., CL). An intuitive relation between EELS/CL and optical extinction/scattering can be rigorously established. [START_REF] Losquin | Unveiling nanometer scale extinction and scattering phenomena through combined electron energy loss spectroscopy and cathodoluminescence measurements[END_REF] Importantly, CL and EELS involve incident (produced by the electron) and scattered (induced by interaction with the sample) electromagnetic fields with well-defined phase relations, as well as broadband, ultrafast, nanometer-precision attributes that are currently being unfolded through advances in these spectroscopies. Additionally, the cascade decay of excitations triggered by the primary electron can produce incoherent CL emission, for example by recombination of electron-hole pairs in semiconductor (nano)structures or the radiative decay of excited color centers in insulators. [START_REF] Yacobi | Cathodoluminescence Microscopy of Inorganic Solids[END_REF][START_REF] Coenen | Cathodoluminescence for the 21st century: Learning more from light[END_REF][START_REF] Kociak | Cathodoluminescence in the scanning transmission electron microscope[END_REF] Figure 1  Electron-light-matter interactions. a Conventional electron spectroscopy. The electron energyloss spectrum (EELS) or the electron-induced (angle-and polarization-resolved) light emission (CL) is analyzed. b,c Ultrafast electron microscopy. Electron pulses are generated by an electrostatic beam blanker (b) or by photoemission from a pulsed-laser-driven cathode (c). Pulsed electron beams enable time-resolved imaging. When the specimen is optically excited, dynamic energy loss/gain (EELS/EEGS) and CL spectra can be acquired, enabling pump-probe spectroscopy to be performed by tuning the delay between laser and electron pulses on the sample (c). d Tailoring the electron wave function. Phase plates, microwave/THz cavities and intense optical near fields control the electron wave function 𝜓𝜓(𝑥𝑥, 𝑦𝑦, 𝑧𝑧, 𝑡𝑡). The shaped electron wave functions create unique ways to control coherent electron-matter excitations in advanced CL and EELS/EEGS. e. Spatial extent of the radial electric field around the electron trajectory in reduced units (R: radius, ω: angular frequency, γ: Lorentz contraction factor, v: electron velocity). The top horizontal scale shows the distance R for 30 keV electrons at a free-space wavelength of 800 nm. f Time evolution of the radial and axial electric fields for a 30 keV electron at a position 5 nm away from the trajectory. g Corresponding frequency/energy spectrum of the field intensity.

In the past few years, coherent CL and EELS spectroscopies have undergone revolutionary advances. Angle-and polarization-resolved CL have been introduced to fully characterize the state of the emitted CL light (Fig. 1a). Ultrashort electron pulses have been created through pulsed-laser photoemission in the electron cathode, laying the foundations of ultrafast electron microscopy (UEM) and time-resolved CL microscopy (Fig. 1b,c). As a consequence of this development, synchronized light and electron beam excitation of the specimen permits pump-probe CL or EELS spectroscopy to be performed and electron energy-gain processes can now be observed under pulsed sample excitation. Furthermore, several new approaches have been developed that shape the single-electron wave function in the spatial domain using elements placed along the electron beam path, or in the time domain using the interaction of the electron beam with suitably shaped optical fields (Fig. 1d).

In Section 2 of this review we briefly summarize the state-of-the art in the theory of CL, EELS and light-assisted UEM. We focus on coherent interactions related to the primary electron beam and do not discuss the incoherent processes that results from relaxation processes in the secondary electron cascade In Section 3 we describe highlights in CL/EELS nano-optics research enabled by technical developments in recent years. We focus on work that has led to new fundamental insights in coherent optical excitation of plasmons, plasmon polaritons, dielectric resonances and phonons in nanomaterials, as well as in the excitation of single-photon emitters. We also provide a comparison of the technical requirements and specifications of EELS and CL for nanophotonics research. In Section 4 we describe recent developments in ultrafast EELS and CL microscopy. In Section 5 we highlight recent advances in the shaping of electron wave functions in the spatial and time domains for time/spacestructured electron-matter interactions. We conclude in Section 6 by providing an outlook on novel fundamental studies that are enabled by the new developments in coherent EELS/CL spectroscopy. We intend this review to stimulate further developments in these highly exciting novel areas of research, further exploiting the potential of CL, EELS and UEM for the investigation of nanoscale optical phenomena.

Theoretical description of coherent electron interactions with nanoscale optical excitations

EELS and CL spectroscopies

Fundamentally, both EELS and CL are excitation spectroscopies, commonly applied to reveal the strength of individual optical modes, as excited by the field displayed by the electron. Importantly, this field is coherent in the sense that it generates an induced field that is phase-locked to it, and for example the CL light emission in metals or photonic structures maintains phase coherence with the electron-generated field. In fact, the EELS probability is intimately related to the local density of photonic states (LDOS), [START_REF] De Abajo | Probing the photonic local density of states with electron energy loss spectroscopy[END_REF] which is defined as the combined electric-field intensity of all normalized photonic modes as a function of light frequency and position in space, and in a more practical way, it is proportional to the decay rate of an optical emitter placed at that position. [START_REF] Hohenester | Electron-energy-loss spectra of plasmonic nanoparticles[END_REF] In fact, it has been shown that one can reconstruct the LDOS from tomographic EELS measurements. [START_REF] Horl | Tomographic imaging of the photonic environment of plasmonic nanoparticles[END_REF] In a similar way, the CL intensity is related the radiative component of the LDOS (i.e., the component that is proportional to the radiative emission rate from the noted emitter). Interestingly, the resonance line shape can differ between CL and EELS spectra, essentially as a result of different frequency-dependent weighting functions accompanying the Lorentzian resonance profile, depending on whether one probes it in the near-field (through EELS or light absorption) or far-field (through CL or elastic light scattering). [START_REF] Losquin | Unveiling nanometer scale extinction and scattering phenomena through combined electron energy loss spectroscopy and cathodoluminescence measurements[END_REF][START_REF] Zuloaga | On the energy shift between near-field and far-field peak intensities in localized plasmon systems[END_REF] Intuitive insight into the coherent interaction of electron beams with photonic structures is gained by describing each moving electron as a classical point charge, whose electromagnetic field is interacting with the sample, as a result of which an induced field is produced. The latter encompasses far-field components, which represent scattering into emitted light (CL), and evanescent waves that act back on the electron producing energy loss (EELS). In this classical description, the electron supplies an external current density, and because we are interested in performing spectroscopy, it is useful to Fourier-transform it in time and decompose it in frequency components as 𝐣𝐣(𝐫𝐫, 𝜔𝜔) for each sample position 𝐫𝐫.

By solving Maxwell's equation in the presence of structured materials, which enter through their frequency-dependent and spatially-dependent complex dielectric functions, the electron current, treated as a classical external source, permits us to obtain the resulting induced electromagnetic field 𝐄𝐄 ind (𝐫𝐫, 𝜔𝜔). For CL spectroscopy, the induced far-field directly yields the emitted energy spectrum (through the Poynting vector), or equivalently, upon dividing by the photon energy ℏ𝜔𝜔, the CL photon emission probability Γ CL (𝜔𝜔). In other words, the far electromagnetic field produced upon interaction of a moving charge (the electron) with the sample, which is obtained as the solution of the classical Maxwell equations, is time-Fourier transformed and its spectral decomposition interpreted as the probability of emission of photons as a function of their energy. While this interpretation emphasizes the quantum nature of the emitted photons, which is not really necessary to understand CL, it becomes essential to understand EELS. Indeed, for EELS, we can obtain the probability Γ EELS (𝜔𝜔) from the work done by the electron current, -Re�∫ 𝑑𝑑𝐫𝐫 𝐣𝐣(𝐫𝐫, -𝜔𝜔) • 𝐄𝐄 ind (𝐫𝐫, 𝜔𝜔)�, divided again by ℏ𝜔𝜔, and treating each frequency component separately. We note the quantum nature of the obtained spectra, emphasized by the introduction of Planck's constant ℏ: both CL and EELS are quantum-mechanical processes, in which individual photons or energy loss events are detected with a well-defined final electron energy, rather than an average classical far field or slowing down of the electron. Nonetheless, the prescription of treating each frequency component of the electron current separately, and ultimately dividing the emitted or absorbed energy by ℏ𝜔𝜔, produces results in agreement with full quantum mechanical descriptions. [START_REF] De Abajo | Optical excitations in electron microscopy[END_REF] It is remarkable that this type of semi-classical analysis permits writing the EELS and CL probabilities in terms of the macroscopic electromagnetic response of the sample, and this in turn as a simple, yet generally accurate approximation as a function of the local material permittivities; one can also readily incorporate nonlocal dispersion effects in terms of momentum-dependent response functions along directions of translational symmetry (e.g., in the bulk of a material), while inclusion of these effects in arbitrary geometries lacking translational symmetry requires a more involved description of the constitutive relations, such as provided by first-principles simulations.

Following the above classical prescription, closed-form expressions of the EELS and CL probabilities have been derived for several simple geometries using analytical methods. [START_REF] Ritchie | Plasma Losses by Fast Electrons in Thin Films[END_REF][START_REF] Zabala | Energy loss of electrons travelling through cylindrical holes[END_REF][START_REF] De Abajo | Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam[END_REF] In particular, it is instructive to consider a simple polarizable point particle of polarizability 𝛼𝛼(𝜔𝜔), for which, using CGS units, the EELS and CL probabilities per unit frequency reduce to: 2

Γ EELS (𝜔𝜔) Γ CL (𝜔𝜔) � = 4𝑒𝑒 2 𝜔𝜔 2 𝜋𝜋ℏ 2 𝑣𝑣 4 𝛾𝛾 2 � 1 𝛾𝛾 2 𝐾𝐾 0 2 � 𝜔𝜔𝜔𝜔 𝑣𝑣𝛾𝛾 � + 𝐾𝐾 1 2 � 𝜔𝜔𝜔𝜔 𝑣𝑣𝛾𝛾 �� × � Im{𝛼𝛼(𝜔𝜔)} (2𝜔𝜔 3 /3𝑐𝑐 3 )|𝛼𝛼(𝜔𝜔)| 2 (1)
where the Lorentz factor 𝛾𝛾 = 1/�1 -𝑣𝑣 2 /𝑐𝑐 2 accounts for relativistic effects at the electron velocity 𝑣𝑣 (= 0.33𝑐𝑐 and 0.70𝑐𝑐 at typical SEM and TEM beam energies of 30 keV and 200 keV, respectively).

Note that the dependence on beam-particle distance 𝑅𝑅 is described by the modified Bessel functions 𝐾𝐾 𝑚𝑚 , and that 𝑅𝑅 is in fact normalized to the characteristic distance 𝑣𝑣 𝜔𝜔𝛾𝛾 ⁄ , which determines the extension of the evanescent field associated with the passing electron, and in turn, the spatial resolution when imaging the excitation modes of the particle. [START_REF] Egerton | Electron Energy-Loss Spectroscopy in the Electron Microscope[END_REF][START_REF] Muller | Delocalization in inelastic scattering[END_REF] More precisely, the above probabilities decay as 𝐾𝐾 𝑚𝑚 2 ∝ 𝑒𝑒 -2𝜔𝜔𝜔𝜔/𝑣𝑣 √𝑅𝑅 ⁄ at large separations and diverge at small distances as 𝐾𝐾 0 ∝ log 𝑅𝑅 and 𝐾𝐾 1 ∝ 1 𝑅𝑅 ⁄ . This scaling with 𝑅𝑅 therefore limits spatial resolution by 𝑣𝑣/𝜔𝜔 in the regime of exponential decay, while only recoil, the physical width of the focused electron beam, and quantummechanical interactions impose a limit to spatial resolution at short distances; an accuracy as small as 3 nm is experimentally found for CL. [START_REF] Schefold | Spatial Resolution of Coherent Cathodoluminescence Super-Resolution Microscopy[END_REF] Additionally, these expressions reveal the same dependence of EELS and CL on the particle polarizability as the optical extinction and scattering cross sections, respectively, thus supporting the intuitive concept that EELS accounts for all loss channels of electronsample energy transfer, whereas CL corresponds only to losses that result in the emission of radiation.

As expected, the optical theorem [START_REF] Van De Hulst | Light Scattering by Small Particles[END_REF]

(Im{-1/𝛼𝛼(𝜔𝜔)} ≥ 2𝜔𝜔 3 /3𝑐𝑐 3 |𝛼𝛼(𝜔𝜔)| 2 ) directly implies Γ EELS ≥ Γ CL .
The relation between EELS/CL and extinction/scattering can be similarly extended to arbitrarily shaped nanoparticles in the quasi-static regime. [START_REF] Losquin | Unveiling nanometer scale extinction and scattering phenomena through combined electron energy loss spectroscopy and cathodoluminescence measurements[END_REF] In general, samples in actual experiments require a numerical solution of Maxwell's equations, for which various approaches have been developed, including the boundary-element method (BEM) [START_REF] García De Abajo | Retarded field calculation of electron energy loss in inhomogeneous dielectrics[END_REF] and the useful MNPBEM implementation, [START_REF] Waxenegger | Plasmonics simulations with the MNPBEM toolbox: Consideration of substrates and layer structures[END_REF] the discontinuous Galerkin time-domain method, [START_REF] Matyssek | Computing electron energy loss spectra with the Discontinuous Galerkin Time-Domain method[END_REF] the discrete dipole approximation, [START_REF] Geuquet | EELS and optical response of a noble metal nanoparticle in the frame of a discrete dipole approximation[END_REF][START_REF] Bigelow | Characterization of the electronand photon-driven plasmonic excitations of metal nanorods[END_REF] the finite-difference in the time-domain method (FDTD), [START_REF] Das | Probing higher order surface plasmon modes on individual truncated tetrahedral gold nanoparticle using cathodoluminescence imaging and spectroscopy combined with FDTD simulations[END_REF][START_REF] Cao | Electron energy-loss spectroscopy calculation in finite-difference time-domain package[END_REF] generalized Mie theory, [START_REF] García De Abajo | Relativistic description of valence energy losses in the interaction of fast electrons with clusters of dielectrics: multiple-scattering approach[END_REF][START_REF] Thomas | Application of Generalized Mie Theory to EELS Calculations as a Tool for Optimization of Plasmonic Structures[END_REF] and innovative approaches coupling these equations to the quantum electron-wave-function dynamics. [START_REF] Talebi | Electron-Light Interactions beyond the Adiabatic Approximation: Recoil Engineering and Spectral Interferometry[END_REF] In a complementary theoretical effort, advanced spectral-image processing techniques have been recently applied to obtain tomographic reconstructions of the spatial extent of localized optical modes [START_REF] Horl | Tomographic imaging of the photonic environment of plasmonic nanoparticles[END_REF][START_REF] Hörl | Tomography of particle plasmon fields from electron energy loss spectroscopy[END_REF] (see Section 3).

The above analytical and numerical methods rely on a dielectric description of the sample to obtain EELS and CL probabilities, an approach that has general applicability and can be used in combination with local, frequency-dependent dielectric functions to cope with plasmons, phononpolaritons, and excitons in most samples. In particular, surface phonon polaritons, which have been recently the subject of intense research because of their potential as mid-infrared modes with long lifetimes, [START_REF] Caldwell | Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons[END_REF] can now be probed by EELS thanks to recent advances in TEM instrumentation (see below), and the local dielectric approach generally leads to sufficiently accurate calculations of the obtained loss spectra. [START_REF] Lourenço-Martins | Vibrational surface electron-energy-loss spectroscopy probes confined surface-phonon modes[END_REF] A quantum-mechanical treatment of the fast electron further permits simulating lateral momentum transfers (i.e., inelastic electron distributions as a function of their deflection angle) and the reshaping of the lateral electron wave function upon interaction with excitations in the specimen. [START_REF] De Abajo | Optical excitations in electron microscopy[END_REF][START_REF] Asenjo-Garcia | Dichroism in the interaction between vortex electron beams, plasmons, and molecules[END_REF] It should be noted that when the sample is structured at length scales comparable to the Fermi wavelength of the involved materials, ranging from less than 1 nm in noble metals to 10s of nm in highly doped graphene, spatial dispersion and quantum finite-size effects become important and provide substantial corrections beyond local response, thus demanding the use of more sophisticated first-principle-based methods, that lie beyond the scope of the present review.

Interaction with optical fields and ultrafast microscopy

The development of UEM has demanded a quantum treatment of the electron to explain the acquired EELS spectra. [START_REF] De Abajo | Multiphoton absorption and emission by interaction of swift electrons with evanescent light fields[END_REF][START_REF] Park | Photon-induced near-field electron microscopy (PINEM): Theoretical and experimental[END_REF] In fact, the interaction between short electron and laser pulses interacting through the mediation of the sample provides a handle to manipulate the electron wave function along the beam direction. The use of laterally extended beams adds intriguing effects resulting from the interplay between lateral and parallel wave function components, as recently shown through the demonstration of orbital angular momentum transfer between light and electrons. [START_REF] Vanacore | Generation and control of an ultrafast electron vortex beam via chiral plasmonic near-fields[END_REF][START_REF] Cai | Efficient orbital angular momentum transfer between plasmons and free electrons[END_REF] Although energy-momentum mismatch severely limits the free-space interaction between electrons and light, electrons can efficiently couple to evanescent optical fields that are produced by optical excitation of a material structure, as these can produce additional momentum transfer that break the noted mismatch. A semiclassical treatment of these interactions has been successfully used to explain recent experiments, [START_REF] De Abajo | Multiphoton absorption and emission by interaction of swift electrons with evanescent light fields[END_REF][START_REF] Barwick | Photon-induced near-field electron microscopy[END_REF] in which the quantum-mechanic electron wave function evolves in the presence of a classical external light field 2Re�𝐄𝐄 ext (𝐫𝐫) 𝑒𝑒 -𝑖𝑖𝜔𝜔𝑖𝑖 �. We note that this semiclassical description where the electron dynamics is described quantum mechanically and the light field classically accounts for electron-light interaction when the light is indeed classical (e.g., supplied through the coherent states of a pump laser), but it cannot be extended to describe the self-interaction of the electron through its own induced field (i.e., for EELS and CL), which requires a quantum description of the electron-sample interaction. [START_REF] De Abajo | Optical excitations in electron microscopy[END_REF] For monochromatic light of frequency 𝜔𝜔, the incident

electron wave function 𝜓𝜓 inc (𝐫𝐫, 𝑡𝑡) picks up inelastic components 𝜓𝜓 ℓ (𝐫𝐫, 𝑡𝑡) ≈ 𝜓𝜓 inc (𝐫𝐫, 𝑡𝑡) 𝐽𝐽 ℓ (2|𝛽𝛽|)𝑒𝑒 𝑖𝑖ℓ arg{-𝛽𝛽}+𝑖𝑖ℓ𝜔𝜔(𝑧𝑧-𝑣𝑣𝑖𝑖)
, where 𝐽𝐽 ℓ is the Bessel function of the first kind and corresponding to the absorption (ℓ > 0) or emission (ℓ < 0) of ℓ photons by the electron, where the coupling integral 37

𝛽𝛽(𝑥𝑥, 𝑦𝑦) = 𝑒𝑒 ℏ𝜔𝜔 ∫ 𝑑𝑑𝑧𝑧 𝐸𝐸 𝑧𝑧 ext (𝐫𝐫)𝑒𝑒 -𝑖𝑖𝜔𝜔𝑧𝑧/𝑣𝑣 , (2)
presented here for an electron moving along the 𝑧𝑧 direction, captures the interaction with the optical field as a function of lateral position (𝑥𝑥, 𝑦𝑦). A simple extension of this description has been formulated

for pulsed electrons and light, [START_REF] De Abajo | Multiphoton absorption and emission by interaction of swift electrons with evanescent light fields[END_REF][START_REF] Vanacore | Attosecond coherent control of free-electron wave functions using semiinfinite light fields[END_REF] with the interaction still governed essentially by 𝛽𝛽(𝑥𝑥, 𝑦𝑦). In its simplicity, Eq. ( 2) captures the essence of the interaction between swift electron beams and light in laser-illuminated samples, and consequently, it is widely used in the explanation of many experiments. [START_REF] Vanacore | Generation and control of an ultrafast electron vortex beam via chiral plasmonic near-fields[END_REF][START_REF] Feist | Quantum coherent optical phase modulation in an ultrafast transmission electron microscope[END_REF] We stress that the coupling only involves the electric field component along the electron trajectory, and conversely, the electron can mainly sample this component of the optical field.

This integral directly explains the vanishing interaction between electrons and free-space light, as 𝜔𝜔/𝑣𝑣 exceeds any light wave-vector component arising from 𝐸𝐸 𝑧𝑧 ext , therefore rendering 𝛽𝛽 = 0 in the absence of a material structure. It is reassuring to note that the CL probability of Eq. ( 1) can be directly obtained from Eq. ( 2) when applying it to a point particle and setting the incident light intensity to the LDOS times the field per vacuum photon mode. [START_REF] García De Abajo | Electron energy-gain spectroscopy[END_REF]

Coherent electron-matter interactions

Exciting localized plasmons

Noble metal nanoparticles possess strong localized plasmon resonances that render them as ideal building blocks in nanometer-scale photonic architectures, triggering different research areas within the field of nanophotonics during the last decade. The relatively strong coupling between energetic electrons and plasmons has been exploited in numerous EELS and CL studies, taking advantage of the unparalleled spatial resolution of these techniques. While the first EELS experiments on nanometersized metal (and semiconductor) nanoparticles were carried out several decades ago, [START_REF] Batson | Surface Plasmon Coupling in Clusters of Small Spheres[END_REF][START_REF] Batson | A new surface plasmon resonance in clusters of small aluminum spheres[END_REF][START_REF] Ouyang | Quantum size effects in the surface-plasmon excitation of small metallic particles by electron-energy-loss spectroscopy[END_REF][START_REF] Ugarte | Surface-and interface-plasmon modes on small semiconducting spheres[END_REF] it took until 2007 for improvements in energy and spatial resolution of EELS to enable direct visualization of plasmonic modes in Ag and Au nanoparticles (Fig. 2a). [START_REF] Bosman | Mapping surface plasmons at the nanometre scale with an electron beam[END_REF][START_REF] Nelayah | Mapping surface plasmons on a single metallic nanoparticle[END_REF] A very high spatial imaging resolution of ∼λ/40 was achieved, far below the optical diffraction limit. Following these pioneering papers, EELS has been extensively used to identify and map plasmonic modes in a wide variety of resonant plasmonic nanostructures at high spectral resolution (see Ref. [START_REF] Colliex | Electron Energy Loss Spectroscopy imaging of surface plasmons at the nanometer scale[END_REF] and references therein). Importantly, EELS probes electron losses associated with both radiative and non-radiative processes, so in contrast to far-field optics, it can be used to map not only electric dipole modes, but also quadrupoles and higher-order modes that do not, or only weakly, couple to far-field radiation.

CL spectra and images of plasmonic modes on Ag nanoparticles were first reported using 200 keV electrons in a TEM. [START_REF] Yamamoto | Photon emission from silver particles induced by a high-energy electron beam[END_REF] Several years later it was found that 30 keV electrons in a SEM also create efficient CL signals on plasmonic structures, and spatial maps of plasmons in Au nanowires were presented. [START_REF] Vesseur | Direct observation of plasmonic modes in Au nanowires using high-resolution cathodoluminescence spectroscopy[END_REF][START_REF] Coenen | Combined electron energy-loss and cathodoluminescence spectroscopy on individual and composite plasmonic nanostructures[END_REF] . Over the past decade, a number of CL studies have identified plasmonic modes in a wide variety of resonant nanostructures, nearly all carried out using SEM-CL systems (Fig. 2b). Plasmon resonances studied by CL range from the ultraviolet (e.g., in Ga and Al) to the visible and near-infrared (e.g., in Ag, Cu and Au) spectral domains (see Refs. [START_REF] Coenen | Cathodoluminescence for the 21st century: Learning more from light[END_REF] , and [START_REF] Kociak | Cathodoluminescence in the scanning transmission electron microscope[END_REF] and references therein), while recent advances in EELS spectrometers and monochromators also enable the study of mid-infrared modes. [START_REF] Rossouw | Plasmonic response of bent silver nanowires for nanophotonic subwavelength waveguiding[END_REF] Therefore, EELS and CL are key techniques to tackle novel emerging research areas in plasmonics, including transdimensional materials (between 2D and 3D), [START_REF] Boltasseva | Transdimensional Photonics[END_REF] aluminum plasmonics, [START_REF] Knight | Aluminum plasmonic nanoantennas[END_REF][START_REF] Lourenço-Martins | Vibrational surface electron-energy-loss spectroscopy probes confined surface-phonon modes[END_REF][START_REF] Pomarico | Resolution in Laser-Assisted Energy-Filtered Transmission Electron Microscopy[END_REF][START_REF] Martin | High-resolution imaging and spectroscopy of multipolar plasmonic resonances in aluminum nanoantennas[END_REF] doped metal oxides [START_REF] Agrawal | Resonant Coupling between Molecular Vibrations and Localized Surface Plasmon Resonance of Faceted Metal Oxide Nanocrystals[END_REF] or refractory transition metal nitrides. [START_REF] Herzing | Electron energy loss spectroscopy of plasmon resonances in titanium nitride thin films[END_REF]

Angle-and polarization-resolved CL

In CL, the angle dependence of the emitted light intensity can be measured over a wide angular range. Measurements are performed by projecting the emitted light that is collected by a parabolic mirror onto a CCD camera using a suitably designed optical path. Such angular measurements explore an important degree of freedom to probe details of localized plasmon resonances. [START_REF] Suzuki | Cathodoluminescent spectroscopic imaging of surface plasmon polaritons in a 1-dimensional plasmonic crystal[END_REF][START_REF] Coenen | Angle-resolved cathodoluminescence spectroscopy[END_REF][START_REF] Yamamoto | Gap and Mie plasmons in individual silver nanospheres near a silver surface[END_REF] For example, in CL experiments on a 150-nm-diameter Au disk, the excitation of multiple radiative resonant modes leads to interference in the far field, resulting in distinctly shaped angular emission profiles. [START_REF] Coenen | Directional emission from a single plasmonic scatterer[END_REF] Differences in the angular distribution of the CL emission can also yield information on plasmon mode symmetry. [START_REF] Myroshnychenko | Unveiling and Imaging Degenerate States in Plasmonic Nanoparticles with Nanometer Resolution[END_REF] This interference is a direct demonstration of the coherent nature of electron-beam excitation of multiple resonant plasmonic modes by a single electron. Furthermore, in CL polarimetry the four Stokes parameters can be derived from six independent polarization-filtered measurements, enabling spatial and angular mapping of the full polarization state of CL. [START_REF] Coenen | Polarization-sensitive cathodoluminescence Fourier microscopy[END_REF] Exploiting this effect, the localized electron excitation of specially designed plasmonic geometries can be used to create tailored angular and polarization states of light in the far field. [START_REF] Osorio | Angle-Resolved Cathodoluminescence Imaging Polarimetry[END_REF] Vice versa, angle-resolved imaging and polarimetry enable partitioning of the CL spectra in incoherent and coherent components. [START_REF] Osorio | Angle-Resolved Cathodoluminescence Imaging Polarimetry[END_REF][START_REF] Brenny | Quantifying coherent and incoherent cathodoluminescence in semiconductors and metals[END_REF] Plasmon CL/EELS tomography Three-dimensional information on the sampled specimen can be acquired from tomographic reconstructions using multiple CL or EELS measurements performed under different electron incidence angles. Using this concept, the three-dimensional (3D) plasmonic modes were determined using EELS in Ag nanocubes and nanoparticles by taking a series of measurements under different tilt angles [START_REF] Nicoletti | Three-dimensional imaging of localized surface plasmon resonances of metal nanoparticles[END_REF][START_REF] Collins | Eigenmode Tomography of Surface Charge Oscillations of Plasmonic Nanoparticles by Electron Energy Loss Spectroscopy[END_REF] (Fig. 2c). A similar technique was used to retrieve the LDOS in plasmonic Ag dimers. [START_REF] Horl | Tomographic imaging of the photonic environment of plasmonic nanoparticles[END_REF][START_REF] Hörl | Tomography of particle plasmon fields from electron energy loss spectroscopy[END_REF] Complementarily, 3D CL tomography was shown to retrieve the 3D distribution of the resonant modes of metallo-dielectric nanoparticles through the analysis of a large number of CL measurements on identical particles at different angles (Fig. 2d). [START_REF] Atre | Nanoscale optical tomography with cathodoluminescence spectroscopy[END_REF]

Exciting surface plasmon polaritons

In parallel to the broad research area of noble metal nanoparticles that possess localized plasmon resonances, the study of surface-plasmon polaritons (SPPs) that propagate at the interface between a metal and a dielectric has gained great interest. SPPs are highly confined waves with wavelengths that can be much smaller than free space waves at the same frequency. High-energy electrons serve as ideal point sources for SPPs on a planar metal-dielectric interface .68,69 They create femtosecond plasmon wavepackets that propagate at the interface with a gradually decaying intensity due to Ohmic dissipation in the metal. [START_REF] Brenny | Femtosecond plasmon and photon wave packets excited by a high-energy electron on a metal or dielectric surface[END_REF] Pioneering angle-resolved EELS experiments revealed the dispersion of SPPs on thin Al films. [START_REF] Pettit | Measurement of surface-plasmon dispersion in oxidized aluminum films[END_REF] Angle-resolved EELS has been used in general for systems in which translational invariance along one or several directions makes momentum rather than spatial resolution relevant.

This has been applied to bulk structures, for example for band gap determination, [START_REF] Gu | Band-gap measurements of direct and indirect semiconductors using monochromated electrons[END_REF] 2D systems such as plasmonic surfaces [START_REF] Shekhar | Momentum-Resolved Electron Energy Loss Spectroscopy for Mapping the Photonic Density of States[END_REF] and 1D systems such as carbon nanotubes. [START_REF] Hage | Momentum-and spaceresolved high-resolution electron energy loss spectroscopy of individual single-wall carbon nanotubes[END_REF] In resonant nanostructures, CL measurements of the plasmonic standing waves provide a unique way to probe the SPP wave vector at a given frequency, so that from a range of measurements over a broad frequency band the dispersion relation can be determined. [START_REF] Vesseur | Direct observation of plasmonic modes in Au nanowires using high-resolution cathodoluminescence spectroscopy[END_REF][START_REF] Vesseur | Experimental verification of n=0 structures for visible light[END_REF][START_REF] Talebi | Excitation of Mesoscopic Plasmonic Tapers by Relativistic Electrons: Phase Matching versus Eigenmode Resonances[END_REF][START_REF] Schröder | Real-space imaging of nanotip plasmons using electron energy loss spectroscopy[END_REF] Complementarily, angle-resolved CL measurements probe the SPP wave vector in periodic plasmonic crystal structures, from which the plasmonic band structure can be derived (Fig. 2e). [START_REF] Takeuchi | Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence[END_REF] EELS is also used to study optical properties of anisotropic materials. It particular, it was employed to study plasmons in graphite [START_REF] Chen | Detection of optical surface guided modes in thin graphite films by high-energy electron scattering[END_REF] or hexagonal boron nitride [START_REF] Terauchi | Electron energy-loss spectroscopy study of the electronic structure of boron nitride nanotubes[END_REF] and their curved forms, such as onion fullerenes and nanotubes, [START_REF] Kociak | Plasmons in layered nanospheres and nanotubes investigated by spatially resolved electron energy-loss spectroscopy[END_REF] including those formed with transition metals dichalcogenides. [START_REF] Kociak | Experimental Evidence of Surface-Plasmon Coupling in Anisotropic Hollow Nanoparticles[END_REF] With successive advances in energy resolution, finer lowenergy details, such as band gaps [START_REF] Kociak | Electron Energy Loss Spectroscopy Measurement of the Optical Gaps on Individual Boron Nitride Single-Walled and Multiwalled Nanotubes[END_REF] or excitonic lines could be measured in nanotubes. [START_REF] Rossouw | Metallic and Semiconducting Single-Walled Carbon Nanotubes : Differentiating Individual SWCNTs by Their Carbon 1s Spectra[END_REF] The anisotropy of these systems further allow to detect different forms of optical excitations such as the Dyakonov mode. 84 From Ref. [START_REF] Coenen | Combined electron energy-loss and cathodoluminescence spectroscopy on individual and composite plasmonic nanostructures[END_REF] . c 3D tomographic reconstruction using EELS (300 keV) of the field distributions of five different resonant plasmonic modes indicated by different colors (energy 2.2-3.5 eV) in a 100-nm Ag nanocube. From Ref. [START_REF] Nicoletti | Three-dimensional imaging of localized surface plasmon resonances of metal nanoparticles[END_REF] . d 3D tomographic reconstruction using CL (30 keV) of the plasmonic field distributions for different light wavelengths in a 230-nm-diameter polystyrene/Au-nanoshell geometry. After Ref. [START_REF] Atre | Nanoscale optical tomography with cathodoluminescence spectroscopy[END_REF] . e (top) Dispersion diagram derived from angle-resolved CL data on a square array of holes (pitch 600 nm) drilled in a Ag film. The CL intensity is plotted along the Γ-X direction for p-polarized emission. Green lines indicate band-folded calculated SPP dispersion curves without the holes. (bottom) SEM image of the hole array. From Ref. [START_REF] Takeuchi | Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence[END_REF] .

Exciting resonant dielectric nanostructures

Electron beam excitation provides a unique way to probe resonant modes of dielectric (nonplasmonic) nanoparticles. Here, the fast electrons directly couple to the localized polarizable molecular bonds inside the dielectric, setting up displacement polarization currents governed by resonant electric and magnetic Mie modes. Using this principle, EELS was used in an aloof geometry (i.e., with the beam passing just outside the material) to identify Mie modes in silica spheres in the far-ultraviolet spectral range. [START_REF] Hyun | Measuring far-ultraviolet whispering gallery modes with high energy electrons[END_REF] Further EELS measurements on dielectrics are scarce, partly due to limitations imposed by broadening of the zero-loss peak in thick specimens. In contrast, CL is ideally suited to identify the Mie modes of silicon nanodisks in the visible-to-near-infrared spectral range, and their spatial modal distributions were imaged at a resolution far below the optical diffraction limit. [START_REF] Coenen | Resonant modes of single silicon nanocavities excited by electron irradiation[END_REF] CL has enabled imaging of the cavity modes in dielectric photonic crystals, for which the angular CL emission profiles reflect their photonic band structure. [START_REF] Sapienza | Deep-subwavelength imaging of the modal dispersion of light[END_REF] Electron beams also strongly couple to travelling optical waves:

infrared CL measurements have probed the modal field distributions of TE and TM polarized modes in Si photonic crystal waveguides, [START_REF] Brenny | Near-Infrared Spectroscopic Cathodoluminescence Imaging Polarimetry on Silicon Photonic Crystal Waveguides[END_REF] and most recently, topological Si photonic crystals. [START_REF] Peng | Probing the band structure of topological silicon photonic lattices in the visible spectrum[END_REF] In a related context, EELS was used to detect photonic Bloch modes in porous Al2O3 membranes 90 and Si photonic structures. [START_REF] Cha | Mapping local optical densities of states in silicon photonic structures with nanoscale electron spectroscopy[END_REF]

Imaging phonons

A key advance in EELS is the development of microscope systems with an energy resolution down to <10 meV, bringing within reach an entirely new field of vibrational electron spectroscopy. [START_REF] Krivanek | Vibrational spectroscopy in the electron microscope[END_REF] This is the result of the development of bright cold field-emission electron sources, new efficient electron monochromator designs, spectrometers, improved aberration-corrected beam optics and improved resolution in the electron energy detectors. Taking advantage of this high energy resolution it has recently been demonstrated that surface and bulk phonons in MgO cubes can be mapped at the nanoscale using their characteristic EELS peaks (Fig. 3). [START_REF] Lagos | Mapping vibrational surface and bulk modes in a single nanocube[END_REF] Interestingly, the low-energy vibrational states can be excited using aloof excitation, as the relative contribution of low frequencies in the electron excitation increases for larger distances away from the beam path. [START_REF] Lagos | Excitation of long-wavelength surface optical vibrational modes in films, cubes and film/cube composite system using an atom-sized electron beam[END_REF] Among them, excitations such as surface phonons are of prime interest in IR nanophotonics, playing a similar role as surface plasmons in nanophotonics at higher frequencies. Surface phonons, in addition, may present competetive advantages compared to plasmons because, for a proper choice of materials, they can exhibit quality factors much larger than those predicted for plasmons. [START_REF] Caldwell | Low-Loss, Extreme Subdiffraction Photon Confinement via Silicon Carbide Localized Surface Phonon Polariton Resonators[END_REF] The interaction of fast electrons with phonons under aloof excitation can be described in similar terms as the excitation of surface plasmons, [START_REF] Lourenço-Martins | Vibrational surface electron-energy-loss spectroscopy probes confined surface-phonon modes[END_REF] where EELS closely map the photonic density of states.

The description for penetrating trajectories, where bulk phonons are excited becomes more involved. [START_REF] Lagos | Mapping vibrational surface and bulk modes in a single nanocube[END_REF][START_REF] Forbes | Modeling energy-loss spectra due to phonon excitation[END_REF] . Nevertheless, upon reciprocal-space removal of the surface phonon signals, atomicallyresolved maps of bulk phonons can be resolved. [START_REF] Hage | Phonon Spectroscopy at Atomic Resolution[END_REF] Interestingly, the vibrational EELS spectra measured on MgO showed both energy loss and gain components, reflecting both the creation and annihilation of phonons by the electron beam, with the EELS gain/loss peaks described by a Boltzmann distribution, confirming the inelastic nature of the electron scattering by phonons in thermal equilibrium. [START_REF] Lagos | Thermometry with Subnanometer Resolution in the Electron Microscope Using the Principle of Detailed Balancing[END_REF][START_REF] Idrobo | Temperature Measurement by a Nanoscale Electron Probe Using Energy Gain and Loss Spectroscopy[END_REF] Figure 3  EELS phonon microscopy. a EELS spectra taken on a single 100-nm MgO cube. b EELS maps showing the spatial distribution of corner (left, 70 meV) and face (right, 77 meV) vibrational modes on the edge of the MgO cube. From Ref. [START_REF] Lagos | Mapping vibrational surface and bulk modes in a single nanocube[END_REF] .

CL photon statistics

The unique way in which electrons excite optical materials creates special fingerprints in the photon statistics of the CL light that is emitted. When single-photon emitters, such as for example nitrogen-vacancy (N-V) centers in diamond, are excited by energetic electrons, the emission itself has no phase relation with the incoming electron, as intermediate energy relaxation processes precede the excitation of emitter. However, TEM-CL measurements of the g [START_REF] De Abajo | Optical excitations in electron microscopy[END_REF] second-order autocorrelation function for CL photon emission from individual N-V centers in diamond 100 or point defects in hexagonal BN 101 show strong anti-bunching, corresponding to the excitation of a single photon emitter (Fig. 4a).

Complementarily, g [START_REF] De Abajo | Optical excitations in electron microscopy[END_REF] measurements on multiple centers in TEM-CL showed strong photon bunching, in strong contrast to the photoluminescence observations. This is due to the fact that a single electron can excite multiple energetic secondary electrons that can in turn create many optical excitations, leading to the emission of a bunch of photons within a time determined by the excited-state lifetime of the emitter. 102 This renders the measurement of g [START_REF] De Abajo | Optical excitations in electron microscopy[END_REF] as a powerful tool for the determination of lifetimes at high spatial resolution. 103 Photon correlation measurements were also carried out in SEM-CL (Fig. 4b), in which pulsed excitation using electrostatic blanking creates an additional degree of freedom to control the statistics of the excitation process. These measurements enabled the direct determination of the efficiency of materials excitation by the primary electron 104 and have also enabled unravelling the relative excitation and emission probabilities of semiconductor nanostructures by analyzing g [START_REF] De Abajo | Optical excitations in electron microscopy[END_REF] data with a statistical model. 105

EELS CL Instrument configuration (S)TEM with electron energy detector S(T)EM with CL light collection and analysis system Optical processes probed Sum of non-radiative and radiative processes

Radiative processes

Energy range > 0.03 eV (λ<30 µm) λ=200-900 nm using Si CCD detector λ=900-1700 nm using InGaAs detector Spectral resolution ∼10 meV (2 nm bandwidth at λ=500 nm) ∼0.1 nm (0.3 meV at 2 eV) Electron probe size TEM-EELS: < 0.1 nm TEM-CL: < 0.1 nm SEM-CL: ∼1-10 nm Specimen thickness <50 ∼ 100 nm No limit

Normalization of spectral intensity

Normalizing by the full spectral integral.

Using transition radiation as a reference, for example from a planar Al surface.

Ultrafast EELS and CL

Ultrafast electron microscopy

The low-energy gain effects observed in vibrational EELS spectroscopy (Fig. 3) result from the annihilation of thermal phonons. Much larger energy gains can be observed in EELS when the specimen is excited by spatially and temporally overlapping laser and electron pulses. This idea was proposed in Ref. 106 , theoretically elaborated, [START_REF] García De Abajo | Electron energy-gain spectroscopy[END_REF] and eventually demonstrated in experiments on carbon nanotubes and silver nanowires under visible light excitation (Fig. 5a). [START_REF] Barwick | Photon-induced near-field electron microscopy[END_REF] In these photon-induced near-field electron microscopy (PINEM) experiments, a ladder of energy loss and gain peaks was observed. [START_REF] Barwick | Photon-induced near-field electron microscopy[END_REF] The data represent the strong nonlinear coupling between the electron and photon fields, in which many energy quanta are exchanged for each electron in the beam. Key to the efficient electron-light interaction described here is the use of optical near fields, as plane waves and electrons cannot exchange energy because of lack of energy-momentum conservation, as was already mentioned in Section 2. In contrast, optical near-fields contain evanescent components that can fulfill the required conservation laws, thus making PINEM possible. Interestingly, by mapping the energy gain spectra across a specimen that is resonant with the incident light, the optical near-field of the resonance can be reconstructed, as recently shown by probing the optically excited SPP Fabry-Perot modes on a Ag nanowire, 107 and a subsequent demonstration of 20 meV resolution based on varying the light frequency around those modes. [START_REF] Pomarico | Resolution in Laser-Assisted Energy-Filtered Transmission Electron Microscopy[END_REF] A key element in the energy gain experiments is the synchronous excitation of a specimen by laser and electron pulses. 108 In these experiments a beam from a femtosecond pulsed laser is split in two parts: one to excite the specimen, and one to induce photoemission from the electron cathode, which leads to the generation of electron bunches. Typical electron pulse durations are around 1 ps and the average number of electrons per pulse can range between less than 1 and over 1000, depending on laser pulse fluence and cathode settings such as temperature and extraction voltage.

Pulsed-laser-driven cathodes were first used in electron diffraction to investigate, for example, transient phase transformations, 109 as well as structural relaxations. 110 Their use in time-resolved electron microscopy has rapidly grown in recent years. Recent detailed studies of the PINEM effect have shown that the quantized energy sidebands are populated in a quantum-coherent way according to the Rabi oscillations between the light and electron states. [START_REF] Feist | Quantum coherent optical phase modulation in an ultrafast transmission electron microscope[END_REF] These experiments confirmed theoretical predictions, [START_REF] De Abajo | Multiphoton absorption and emission by interaction of swift electrons with evanescent light fields[END_REF] and the quantized energy gain/loss transitions are so strong that they can lead to a near-complete depletion of the initial electron energy state (Fig. 5b). Importantly, the quantum superposition of the excited electron ladder states results in coherent shaping of the electron wavepacket in momentum space, creating a train of attosecond electron pulses in the time domain.

Tailoring of the electron wavefunction will be further discussed in Section 5.

Ultrafast CL

The first use of a pulsed photoemission microscope in CL spectroscopy was reported in Ref. 111 .

Later, using 200 fs laser pulses to generate 10-ps electron pulses (10 keV), the carrier dynamics in GaAs nanostructures was probed with a spatial resolution of 50 nm (Fig. 8). 112 The ultrafast pulsed geometry also enables pump-probe spectroscopy with laser and electron beams as pump and probe, or vice versa, opening up an entirely new research area of ultrafast excited matter spectroscopy using CL and EELS. [START_REF] Feist | Quantum coherent optical phase modulation in an ultrafast transmission electron microscope[END_REF] . c Streak images of CL emission from quantum dot, quantum wire and quantum well structures in a InGaAs/AlGaAs micropyramid (1.52-1.90 eV, 90 K) taken using a 10 keV pulsed electron beam generated by photoemission with a 200-fs laser pulse (λ=266 nm). d Schematic of sample geometry and SEM image. Tops of pyramids are separated by 5 µm. From Ref. 112 .

Tailoring the electron wave function

As is clear from Sections 3 and 4, there have been many advances in EELS, EEGS and CL spectroscopy in the past years that have led to new insights in multiple aspects of electron-matter-light interactions. These experiments have relied on state-of-the-art (pulsed) electron microscopes employing "conventional" electron beams that have phase fronts similar to plane waves, with the spatial and temporal coherence determined by the source and electron column geometries and settings. Recent exciting new developments concern tailoring the electron wave function itself, both in the spatial and time domains. A first example was already discussed above, where PINEM experiments on Au nanowires created electron pulses composed of a train of attosecond pulses. [START_REF] Feist | Quantum coherent optical phase modulation in an ultrafast transmission electron microscope[END_REF]

Time domain

It is well known that electrons can elastically scatter from light fields by the ponderomotive force, such as for example in the Kapitza-Dirac effect, 113,114 in which two counterpropagating optical waves configure a light grating that can diffract a passing electron wave. However, these elastic interactions are too weak for practical applications in beam shaping. The evanescent optical components produced when light interacts with material boundaries provide an efficient way to enhance such interactions.

For example, in the inelastic Smith-Purcell effect, electrons propagating above a grating interact with their induced electromagnetic surface waves, resulting in the generation of light in the visible spectral range and a concomitant loss in electron energy. 115,116 A similar diffraction effect is achieved when electrons interact with periodic field patterns of plasmonic standing waves excited by optical pumping. 117 Complementarily, in the "inverse Smith-Purcell effect" electromagnetic surface waves generated by optical pulses can accelerate electrons, as most recently demonstrated for near-infrared pulsed laser excitation. 118,119 Inelastic electron-near-field-light scattering effects provide a unique way to tailor the wave function of the (single) electron itself, enabling coherent control over electron-lightmatter interaction.

Additional approaches have been recently explored to tailor the electron wave function with farfield radiation pulses. For example, by placing a pulsed microwave source inside the electron column, single-electron pulses could be compressed and tailored. 120 The oscillating microwave cavity field pulse is synchronized with the electron pulse such that it acts differently on the front and rear parts of the electron pulse, effectively widening the energy spread and shrinking the spatial distribution (pulse width) of the electron pulse in the direction along the beam (Fig. 6a,b). In this way, the electron pulse can be made shorter than the original laser pulse generating it. Using this concept, a train of electron pulses compressed by THz pulses was used to follow the excitation and relaxation of THz-excited Ag butterfly antennas in the time domain at a time resolution of ∼5 fs, by using an ingenious scheme in which the time-varying electric near field of the antenna controlled the streaking of the electron pulse. 121,122

Spatial domain

Initial experiments in shaping the electron beam in the spatial domain used either rotated superimposed graphene sheets 123 or holographic gratings 124 to induce a singular spiraling phase to the electron wave, generating vortex electron beams that carry orbital angular momentum. Since then, numerous methods for producing various phase-shaped beams have been proposed (see Ref. 125 for a review). For instance, using advanced phase plates, electrons carrying quantized amounts of orbital angular momentum up to ±100ℏ per electron have been created (Fig. 6c). 126 An alternative technique to create electron vortex beams uses a magnetic needle placed in the electron beam path. 127 Recently, direct transfer of angular momentum from an optical beam to the electrons has been demonstrated. [START_REF] Vanacore | Generation and control of an ultrafast electron vortex beam via chiral plasmonic near-fields[END_REF] The research area of structured beams is now emerging further, and some initial applications have been demonstrated. Vortex beams can probe circular magnetic dichroism, similar to X-ray magnetic circular dichroism spectroscopy, 124 but now with the prospect of atomic resolution. Similarly, it has been predicted that the chirality of plasmonic systems could be determined using a vortex beam. 128,129 Aside from vortex beams, several other beam geometries have been realized using structured surfaces. 130,131,132 The first application of a phase-shaped beam in nanophotonics was in the measurement of the symmetry of the electric field amplitude of localized modes of a plasmonic nanorod, which is not possible in a direct way using EELS or CL, as these techniques probe the field intensity rather than the amplitude. 133 The same concept can be expanded to other symmetries, in order to reveal the symmetries of a broad range of coherent excitations. The use of ultrathin materials, such as for example graphene, which interact strongly with both electrons and light, can provide further control over electron-light-energy exchange processes. 134 More generally, phase information about the coupling to the nanophotonic system is encoded in the electron wave front after interaction with the sample; it can be unfolded by acquiring and analyzing the reciprocal (angular) space images. However, this information is generally lost in conventional EELS setups, in which collection up to high detection angles averages out phase variations, as early envisioned by Kohl 135 and Ritchie and Howie. 136 Ptychographic techniques (i.e., those reconstructing the phase by using information hidden in the diffraction pattern acquired for every point of a scan) constitute a generalization of the concepts of structured beams and permit retrieving phase information without requiring dedicated beam phase shapes adapted for each problem, as experimentally demonstrated in recent experiments. 137 In a recent exciting development, a programmable electron phase plate was demonstrated that is composed of an array of cylinders in which the electric field is individually manipulated (Fig. 6d). 138 This enables tunable control over the beam geometry that is reconstructed by interference after the electron is transmitted through the array. 138 This concept, which can be expanded further to large array sizes, holds great potential as a unique way to control beam shape in a detailed way. The creation of electron wave functions with tailored spatial, temporal and angular momentum distributions is strongly connected to developments in structured illumination and pulse shaping that have proven very powerful in (super-resolution) optical microscopy and spectroscopy. Similar applications in electron microscopy are opening up at a fast pace. In fact, one could imagine that temporally and spatially structured (multi-color) illumination sources interacting with electrons could provide even further control over the optical shaping of electron wavepackets.

Future directions

As is clear from the many examples reviewed above, the EELS/CL community is very lively with new discoveries continuously being made. Based on the described developments we envision several notable trends.

Time resolution

Developments in ultrafast laser-driven cathodes are continuing and electron pulses as short as 200 fs were recently demonstrated, 139 bringing time-resolved studies of hot electron and electron-phonon relaxation in solids within reach. So far, the generation of single pulses short enough to directly probe plasmon relaxation (∼10 fs) has remained elusive. Trains of attosecond pulses have been realized, as described above. An exception is a recent point-projection technique, in which plasmon dynamics of the optically excited electron cathode itself was imaged with a spatial resolution of 20 nm and a time resolution of 25 fs, thus getting closer to the real-time characterization of plasmon dynamics. 140 An alternative proposal is to retrieve the sub-cycle dynamics of plasmons by using interference between the field of an specially designed plasmonic metamaterial lens excited by a fast electron and a plasmonic field of interest. 141,142 Recent developments with electron diffraction (without spatial resolution) have demonstrated sub-fs resolution in pulse trains, paving the way to sub-optical cycle temporal resolution in EELS/CL. 143 As a benchmark in this context, photoemission electron microscopy (PEEM) has been shown to render <10 nm spatial resolution through electron imaging, accompanied by ~10 fs resolution associated with pump/probe delay in two-photon photoemission (see below). 144,145 We note that temporal information on the electron excitation processes can also be derived from EELS data using Fourier analysis methods. 146 Ultrafast transient behavior may also be resolved by studying the effect of attosecond forces exerted on plasmonic nanoparticles induced by swift electrons. 147,148 We note that recent advances have been triggered by the development of high-brightness Schottky guns. We anticipate that the search for even brighter guns will play an important role in the years to come, while significant progress has been already made with the conception of the first timeresolved cold-field emission gun. 149 Finally, we note that recent work shows how electrostatic beam blankers placed in the electron column can now deliver electron pulses as short as 30-90 ps 150,151 thus enabling CL lifetime imaging of a broad range of materials at very high spatial resolution. The advantage of these blankers is that they can be easily integrated in the electron column. Another exciting new development is the use of an electrostatic beam blanker driven by a laser-driven photoconductive switch, which may produce electron pulses as short as 100 fs. 152

Energy resolution

A key new development in EELS instrumentation is the demonstration of energy resolution below 10 meV. This enables a wide range of high-resolution phonon/electron spectroscopy studies in bulk materials, as well as at surface and interfaces. [START_REF] Krivanek | Vibrational spectroscopy in the electron microscope[END_REF] An interesting parallel development is the use of microwave fields created by RF cavities that are integrated in the electron column to tailor the electron pulse and energy resolution. A new design shows that 200-fs electron pulses may be created with an energy width below 500 meV. 153 Using a sequence of GHz RF cavities even promises energy resolutions down to 20 meV, far better than the energy spread in the source itself. These designs come within a factor 10 of the Heisenberg limit of the variance in beam energy and position. In principle, such new designs could simplify the EELS instrumentation, as they require a much simpler microscope column, although the required (high) beam currents have not yet been realized. The use of pulsed electron beams in EELS opens up the use of time-of-flight energy loss analysis.

We note that electron energy-gain spectroscopy (EEGS) has been proposed [START_REF] García De Abajo | Electron energy-gain spectroscopy[END_REF]

Structured electron beams as quantum electron probes

As discussed in Sections 4 and 5, it has now become possible to tailor the electron wave function in both the spatial and temporal domains. Spatially structured beams provide a new degree of freedom in spatially-resolved excitation and will enable advanced studies on symmetries in plasmonic excitations. Temporally structured beams can provide coherent control over optical excitations and effectively perform pump-probe spectroscopy with both the pump and probe encoded in the same pulse, thus effectively granting access to sub-optical-cycle dynamics.

An exciting aspect of these developments is that they pursue to control the electron wave function itself. From a fundamental perspective, electron beams are used as quantum probes, with the electron microscope operating as a quantum instrument using well-prepared initial states that can be entangled with materials excitations. In particular, when a CL photon or a loss signal is recorded in EELS, one makes sure that one quantum of excitation has been produced on the sample. This enables exciting studies on correlations in time and space, as well as real-time studies of excited states by their spectral, spatial and diffraction signatures. A key element in these studies is to obtain optimum sensitivity, extracting as much information as possible from the smallest possible number of electrons, aiming to achieve unity detection quantum efficiency. A fundamental question arises, whether quantum information can be encoded in the electron spectrum in the form of coherent superpositions of optically excited states. This also raises the question how electrons can be used to probe quantum aspects in a specimen, how the collapse of the electron wave function to an observable eigenstate can be controlled, and whether this could be exploited in (scalable) quantum technology. Quantum measurements may also enable new forms of microscopy in which the electron-sample interaction is probed with one part of the electron wave function, while a weak entangled part is interacting with the specimen, keeping electron-induced degradation to a minimum. 154

Incoherent electron-matter interactions

Although this review focuses mostly on coherent light-matter interactions, it is important to note that several new developments are place that involve incoherent processes and that provide unique new insights into quantum optical phenomena and condensed-matter energy landscapes.

Recently, the very high spatial resolution of electron spectroscopy has been employed to investigate transition-metal dichalcogenides (TMDCs) with unique (semi-)metallic and semiconducting properties.

For example, using EELS the spin-orbit energy splitting in MoS2 and MoSe2 was measured 155 and lowenergy hyperbolic phonon polariton modes were observed in hexagonal BN. 156 Additionally, CL measurements have revealed sulfur impurity inhomogeneities in MoSe2 flakes. 157 By embedding a TMDC layer in a heterostructure geometry, the effective capture range of electron-induced excitations can be strongly enhanced, and the specimen stability improved, enabling the observation via CL of strain-induced lateral bandgap variations in WSe2 monolayers at length scales below the optical diffraction limit. 158

Complementary techniques

Aside from the focus on EELS and CL in this review, two other techniques involve (ultrafast) electron-matter-light interactions. In particular, PEEM, which renders images constructed from photoelectrons emitted from a specimen, can render a spatial resolution down to <10 nm, as determined by the electron optics. Field enhancement caused by, for example, plasmonic hotspots locally enhance the photoemission intensity, so that PEEM images directly probe plasmon fields. 159 The spectral and temporal resolution is determined by the exciting photon beam properties, allowing fewfemtosecond-resolution observation of plasmon interference dynamics when the exciting light is administered in a pump-probe configuration. 160 Also, the polarization and phase of the incoming light beam can be adapted for studying the plasmon symmetry. As a striking example, PEEM has been recently used to investigate the ultrafast dynamics of plasmonic vortices. 145 More details on PEEM and its relation with EELS, EEGS and CL can be found in Ref. 161 . We note that the related ultrafast pointprojection electron microscopy (see above) also presents a way to directly image the plasmon intensity distributions at high spatial and temporal resolution. 140 In scanning tunneling microscopy (STM), inelastic electron tunneling produces luminescence with spectral features that render information on the sampled materials and structures. Optical spectra are collected with the atomic spatial resolution offered by the STM, although the details of the tip morphology, which are usually unknown, add a certain degree of uncertainty on the imaged optical modes. Nevertheless, this technique has proven to be useful for the investigation of electro-optical molecules 162,163 and plasmonic fields. 164

Compact microscopes

The new fundamental developments also raise the question whether further improvements in electron microscope designs can be made by critically reconsidering features that have historically been developed. Obviously, the use of lower electron energies simplifies the design of the columns.

Introducing novel beam shaping concepts as described above can reduce the need for correctors, further simplifying the electron column. Altogether, the distinction between TEM and SEM geometries may vanish for some applications. Ultimately, compact ("table-top") microscope designs may become possible taking advantage of the new electron beam shaping concepts.

Conclusions

In summary, the research field of electron beam spectroscopy for nanophotonics has grown into The high degree of spatial and temporal control that can now be achieved over electron beams anticipates the development of exciting new applications in quantum coherent control. Electron pulses are quantum probes that can be potentially entangled with materials excitations, raising fundamental questions if they can carry quantum information to create entirely new forms of electron microscopy with entangled beams. Overall, the many new insights in electron beam spectroscopy for nanophotonics promises many exciting new discoveries in electron-light-matter interaction in the coming years.

Figure 2 

 2 Figure 2  EELS and CL spectroscopy of plasmonic nanostructures. a (top) EELS map at corner plasmon resonance energy (1.75 eV) for a triangular Ag particle probed with 100 keV electrons.(bottom) TEM image of the same structure. From Ref.[START_REF] Nelayah | Mapping surface plasmons on a single metallic nanoparticle[END_REF] . b. (top) CL images of transverse and longitudinal resonant plasmonic modes at 590 nm and 860 nm of a Au nanowire, acquired with 30 keV electrons. (bottom) TEM image of the same structure. From Ref.[START_REF] Coenen | Combined electron energy-loss and cathodoluminescence spectroscopy on individual and composite plasmonic nanostructures[END_REF] . c 3D tomographic reconstruction using EELS (300 keV) of the field distributions of five different resonant plasmonic modes indicated by different colors (energy 2.2-3.5 eV) in a 100-nm Ag nanocube. From Ref.[START_REF] Nicoletti | Three-dimensional imaging of localized surface plasmon resonances of metal nanoparticles[END_REF] . d 3D tomographic reconstruction using CL (30 keV) of the plasmonic field distributions for different light wavelengths in a 230-nm-diameter polystyrene/Au-nanoshell geometry. After Ref.[START_REF] Atre | Nanoscale optical tomography with cathodoluminescence spectroscopy[END_REF] . e (top) Dispersion diagram derived from angle-resolved CL data on a square array of holes (pitch 600 nm) drilled in a Ag film. The CL intensity is plotted along the Γ-X direction for p-polarized emission. Green lines indicate band-folded calculated SPP dispersion curves without the holes. (bottom) SEM image of the hole array. From Ref.[START_REF] Takeuchi | Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence[END_REF] .

Figure 4 

 4 Figure4 Photon bunching and anti-bunching in incoherent CL. Spectrum of the second-order autocorrelation function g[START_REF] De Abajo | Optical excitations in electron microscopy[END_REF] of CL emission. a TEM-CL on diamond nanoparticles with a single embedded N-V center (80 keV electrons, 570-720 nm spectral band) showing photon emission antibunching. From Ref.100 . b SEM-CL on InGaN quantum wells for different beam currents showing bunching. From Ref.104 .

Figure 5 

 5 Figure5 Ultrafast electron microscopy a Energy gain/loss spectroscopy and quantum coherent manipulation of electron energy distributions. a Energy spectrum of 200 keV electrons after interaction with carbon nanotubes that are simultaneously excited with a pulsed laser (200 fs, ℏ𝜔𝜔= 2.4 eV), showing three energy gain and three energy loss quanta. From Ref.[START_REF] Barwick | Photon-induced near-field electron microscopy[END_REF] . b Energy spectrum of 120 keV electrons after interaction with an optical near field around a Au needle (3.5 ps, ℏ𝜔𝜔= 1.6 eV). Data are shown for different pulse energies increasing from bottom to top. Energy gain/loss peaks are quantized by the photon energy and distributed according to Rabi oscillations coupled among the ladder states, creating an attosecond train of electron pulses. From Ref.[START_REF] Feist | Quantum coherent optical phase modulation in an ultrafast transmission electron microscope[END_REF] . c Streak

Figure 6 

 6 Figure 6  Tailoring the electron wave function in the temporal and spatial domain. a The timevarying electric field in a microwave pulse modifies the electron wavepacket to a desired shape. b Schematic of electron pulse in the velocity and time domain, before and after passing through a microwave cavity. From Ref. 120 . c Phase plate composed of a transmission grating with fork dislocations. Transmitted electron waves carry discrete orbital angular momenta with topological charges indicated at the bottom. From Ref. 126 . d Programmable tunable phase plate composed of a 2×2 array of cylindrical electrodes that individually control the phase of 4 electron beams that recombine in the far field to form a programmable interference pattern. From Ref. 138 .

  as a way to combine the excellent resolution in the frequency of the external laser with the atomic resolution provided by electron beams. The promised EELS mapping of optical excitations with sub-meV resolution still remains as an experimental challenge. A recent development has used this principle to map narrowly separated plasmon standing waves in the spectra of long silver wires with 20 meV resolution.[START_REF] Pomarico | Resolution in Laser-Assisted Energy-Filtered Transmission Electron Microscopy[END_REF]

  photon bunching and anti-bunching, providing further detailed insights into electron-matter interactions.

Table I Characteristics of state-of-the-art EELS and CL spectroscopy.

 I

Acknowledgements

We gratefully acknowledge the assistance of Sophie Meuret and Toon Coenen in preparing this review. Future visions described in this paper partly originate from presentations and a panel discussion

session at the workshop Electron Beam Spectroscopy for Nanophotonics (EBSN) held in Sitges, Spain, during October 25-27, 2017. We thank the workshop participants for providing their insights; in particular the discussion panelists Joanne Etheridge, Ido Kaminer, Claus Ropers, and Johan Verbeeck.


Bernardino Tirri

Ilaria Ciofini

Juan Carlos

Carlo Adamo
email: carlo.adamo@chimieparistech.psl.eu

Éric Brémond
email: eric.bremond@u-paris.fr

Computation of Covalent and Noncovalent Structural Parameters at Low Computational Cost: the Efficiency of the DH-SVPD Method

DH-SVPD is a tailored atomic basis set originally developed to enhance the domain of applicability of double-hybrid density functionals to large molecular systems in weak interactions. In combination with any density functional belonging to this approximation, it provides an accurate estimate of noncovalent interaction energies at the cost of a double-ζ basis set, without adding a posteriori an empirical dispersion correction. We show here that the accuracy/cost ratio observed previously for energy properties can be safely extended to the modeling of structural parameters of small-and medium-sized organic molecules. In particular, we demonstrate that in combination with the nonempirical PBE-QIDH double hybrid, DH-SVPD is competitive with very large quadruple-ζ basis sets when modeling both covalent and noncovalent structural parameters.

INTRODUCTION

Modern double-hybrid (DH) density functionals [1][2][3] are considered among the most advanced and accurate approximations to the exchange-correlation energy term within the Kohn-Sham approach to density-functional theory. 4,5 Developed from the middle of the 2000s, their original recipe merges wavefunction-and density-based approaches by mixing a fraction of nonlocal exact-like exchange (EXX) and second-order perturbation (PT2) correlation with a pure semilocal density-functional approximation (DFA). 6,7 Owing to these ingredients, they peak on the highest rung of the Perdew's Ladder 8 and become a clear improvement both in terms of sophistication and performance over hybrid and semilocal density functionals. 9 Some variants to the standard definition of a DH were recently developed to solve specific issues. Some of them focus on the orbital-optimization (OO) approach 10 to correct an artificial spin-symmetry breaking occurring in spin unrestricted systems. [11][12][13] Others are based on the range-separation exchange (RSX) scheme [START_REF] Savin | Recent Developments and Applications of Modern Density Functional Theory[END_REF] to (partially) cure the self-interaction and delocalization errors [START_REF] Cohen | [END_REF] (SIE and DE, respectively). [16][17][18][19] However, all of them conserve similar features, i.e. a fraction of EXX generally larger than 50% and a fraction of PT2 correlation ranging between 10 and 40%, which allow them to reach the chemical accuracy when modeling a large panel of properties ranging from ground-state energies 20 to structures 21,22 and extending to singlet-singlet vertical excitation energies. 23,24 Despite their undeniable robustness, DHs remain more computationally demanding than standard density functionals. The computation of their PT2 correlation energy term scales formally as O(n 5 ), n referring to the size of the basis set. This larger effort in comparison to global hybrids, can however be alleviated under certain conditions 25 by transforming the DHs into their spin-opposite-scaled (SOS) variants. [26][27][28] The resulting SOS-DHs get a final formal scaling in O(n 4 ), but in return, strongly loose in performance with respect to weakinteraction properties, 29 one of their main advantage with respect to other classes of density functionals. As a result, it is common practice to couple a SOS-DH with an empirical pairwise dispersion correction 30 to recover the missing correlation effects as an error compensation. 27 The resulting DHs corrected empirically for dispersion interactions are among the best performing DFT-like approaches. 20 Several energy benchmark tests show that the coupling is particularly optimal by considering semiempirical DHs, i.e. DHs whose parameters are trained to reproduce reference energy databases. It remains however questionable when it enrols nonempirical DHs since it often leads to overbinding situations when the size of the weakly interacting systems is not large enough. 31 To overcome these limitations while keeping homogeneous performances independently of the system-size, we recently proposed an alternative to empirical dispersion corrections based on the use of a tailored atomic basis set. 32 Like second-order Møller-Plesset (MP2) perturbation theory, DHs exhibit a larger binding behavior in weakly interacting systems when using smaller basis set (i.e. double-versus quadruple-ζ or larger, respectively). 33 From this observation, we specifically developed the so-called DH-SVPD, 32 a double-ζ basis set which takes advantage of an error compensation between the basis set incompleteness and superposition errors (BSIE and BSSE, respectively). DH-SVPD is derived under the Varandas' zeroth-order-exchange-energy optimization criterion 34 referred here to as a constraint:

min {α i } (E -E 0 ) -(J -K) (E -E 0 ) + (J -K) 2 (1)
where the quantity E -E 0 refers to the interaction energy between monomers in a molecular aggregate, and J and K the corresponding Coulomb and exchange energies of the dimer itself.

The optimization of the exponents of the more external Gaussian functions of the def2-SVPD atomic basis set, 35 denoted here {α i }, allows the expression of the interaction energy of the molecular aggregate within its one one-exchange level perturbation development.

It is worth to note that the satisfaction of the constraint imposed by Eq. 1 does not introduce any external parameterization. Moreover, in association with a DH, the DH-SVPD basis set allows the recovery of both energy and structural features of dispersion interactions in small to large systems 32,36 at the reduced computational cost of a modest double-ζ split-valence basis set.

We propose here to go further the energy benchmarking and evaluate the performance of the association between a DH density functional and the tailored DH-SVPD basis set to model intramolecular structural parameters of organic molecules. Structure optimizations are indeed often performed in small basis set due to the increasing computational time caused by the successive evaluations of energy and gradient up to the fulfillment of convergence criteria. Furthermore, we want to verify that the observed improvement on energy properties of weakly interacting molecules is not done at the expense of a basic property such as molecular geometry, but that the performance improvement on both properties stays in close correlation. 21,37 Our benchmarking investigation focuses on the CCse and B3se databases of accurate structural parameters introduced some years ago by Barone and co-workers. [38][39][40][41] The former gathers a collection of 21 small organic molecules whose reference covalent structural parameters are derived from a mixed coupled-cluster singles and doubles plus perturbative triples [CCSD(T)], and experimental protocol. The latter is an extension to the former containing 26 extra medium-sized organic molecules but derived using a lower (B3LYP) level of theory.

To illustrate the performance of the DH-SVPD basis set on weakly interacting systems, we also include to the previous collection of data, 4 extra molecular aggregate examples, each of them governed by H-bond, mixed electrostatics/dispersion or dispersion-dominated stacking interactions (Figure 1). Their reference structural parameters were recently derived with a mixed CCSD(T)/experimental protocol, case of the formamide dimer, 39 , or at CCSD(T) level of the theory at the complete basis set (CBS) limit, case of T-shaped ethyne aggregate and methane-methane aggregates. 42

COMPUTATIONAL DETAILS

All the computations were performed with the Revision B.01 of the Gaussian'16 software using a tight threshold as a convergence criteria for energy and structure optimization in addition to an ultrafine integration grid. [START_REF] Frisch | Gaussian˜16 Revision B.01[END_REF] The performance of the tailored DH-SVPD doubleζ basis set were benchmarked versus the CCse and B3se databases, 40,41 in addition to the formamide dimer, 39 T-shaped ethyne and methane-methane aggregates. 42 They were compared with those obtained with 4 Ahlrichs double-to quadruple-ζ basis sets (i.e., def2-SVP, def2-SVPD, def2-TZVPP and def2-QZVPP), 35 plus the aug-cc-pVTZ Dunning augmented triple-ζ basis set. [START_REF] Dunning | Encyclopedia of Computational Chemistry[END_REF] Since DH-SVPD is developed for the first-row atoms, we restricted our benchmarking investigation to subsets of the above-mentioned datasets containing 17 and 38 reference structures, respectively, that is to say a total of 217 references covalent bond lenghts. More details are provided in Figures S1 andS2 of the Supporting Information.

The impact of the coupling between the density functional and the basis set on the structure optimization property was evaluated by testing 10 different exchange-correlation approximations. We select 3 of them from the 06' edition of the Minnesota family which mix different percentage of EXX, i.e. M06-L (EXX = 0%), 45 M06 (27%) 46 and M06-2X (54%). 46 We complete this list with other commonly used DHs: PBE0-DH (EXX = 50%, PT2 = ∼12%), 47 B2-PLYP (53%, 27%), 7 mPW2-PLYP (55%, 25%), 48 PBE-QIDH 49 Going further into details, Figure 2 shows that the covalent structural parameters smoothly converge with the size of the basis set. Already, a triple-ζ basis set augmented or not with diffuse functions (def2-TZVPP or aug-cc-pVTZ, respectively) provides MADs in close agreement with the very large def2-QZVPP quadruple-ζ basis set. As an example, the MADs for the CH, CC, CN and CO bond lengths gathered in the CCse dataset are calculated to be 0.001, 0.006, 0.005 and 0.009 Å, respectively, with def2-TZVPP while they are estimated to 0.002, 0.007, 0.007 and 0.010 Å, respectively, at def2-QZVPP level of theory. A deviation not larger than 0.002 Å is therefore observed between the two levels of theory.

The DH-SVPD tailored double-ζ basis set slips from this convergence trend and provides here the best performance with deviations calculated to 0.004, 0.003, 0.002, 0.006, 0.002 and 0.004 Å for the CH, CC, CN, CO, NH and OH bond lengths, respectively, for the CCse benchmark set (Figure 2). In comparison, the original def2-SVPD basis set exhibits poorer performances determined as 0.008, 0.003, 0.003, 0.007, 0.002 and 0.001 Å, respectively. DH-SVPD tends indeed to shorten the covalent bond lengths. As a result, CH bonds, which are systematically modeled as too long with def2-SVPD, are better estimated with DH-SVPD. However, OH bonds, which are correctly predicted with def2-SVPD, tends to be too shorten with DH-SVPD. As found for energy properties, 32,36 the error balance ruled by the association between a DH density functional and the DH-SVPD basis set results here in an increase of the binding behavior from a structural point of view.

The latter conclusion regarding covalent bonds can be extended to all types of noncovalent interactions. Indeed, the combination of PBE-QIDH with DH-SVPD models H-bonds, mixed electrostatics/dispersion and dispersion-dominated stacking interactions in better agreement with the reference than with a large triple-or very large quadruple-ζ basis set like aug-cc-pVTZ or def2-QZVPP, respectively (Figure 2). As a first example, the r(O • • • H cis ) bond length of the formamide dimer is predicted as 1.816 Å (the reference being 1.836 Å, Figure 1) while aug-cc-pVTZ and def2-QZVPP model them as 1.818 and 1.823 Å, respectively (Figure 2). On the same line, the r(π • • • H) distance ruling the mixed electrostatics/dispersion interaction in the T-shaped ethyne aggregate is modeled as 2.715 Å with DH-SVPD, and 2.655 and 2.693 Å with aug-cc-pVTZ and def2-QZVPP, respectively, while the reference is set to 2.696 Å (Figure 2). The methane-methane aggregate which is governed by dispersiondominated stacking interaction is at the origin of a similar trend but with a larger error of about one order of magnitude. With respect to the reference r(C • • • C) bond length (3.638 Å), the error is about 0.080 Å with DH-SVPD while it is about 0.177 and 0.196 Å at augcc-pVTZ and def2-QZVPP level of theory, respectively (Figure 2). As a conclusion, the tailored DH-SVPD basis set in association with PBE-QIDH systematically shortens covalent and noncovalent bond lengths and provides a better estimate structural parameters.

To confirm the trend obtained with PBE-QIDH, we compare its results with the performance of 9 other popular density functionals belonging to the semilocal, hybrid and double-hybrid approximations and gathering different fraction of EXX and PT2 correlation (Figure 3). Concerning the 53 covalent bond lengths gathered into the CCse dataset, we observe a non-uniform influence of the basis set size with respect to the density functional (Figure 3). If a triple-ζ augmented or not with diffuse functions unanimously provides converged results, DH-SVPD can be view either as a compromise in between the double-and triple-ζ level of theory or as a real improvement with respect to all the investigated basis sets. Indeed, the combination of DH-SVPD with semilocal and hybrid density functionals provides a real improvement with respect to the original def2-SVPD basis set. The mean signed deviation (MSD) for M06-L, M06 and M06-2X are about 0.003, 0.002 and 0.002 Å, respectively, at DH-SVPD level and 0.005, 0.005 and 0.005 Å, respectively, at def2-SVPD level. The same trend is observed for the MAD of the B3se dataset (see Figures S3 to S5 within the Supporting Information). Results of an average quality are found for semiempirical DHs. DH-SVPD provides a slight improvement with respect to def2-SVPD and does not reach the performance obtained with def2-TZVPP. As an example, with B2-PLYP, the MSD is calculated as 0.008, 0.006 and 0.001 Å for def2-SVPD, DH-SVPD and def2-TZVPP, respectively. The addition of a pairwise dispersion correction does not significantly influence that much the deviations, the MSD change being lower than 0.001 Å for B2-PLYP-D and B2-PLYP-D3(bj). The real improvement with respect to other basis sets observed is found for the PBE0-DH and PBE-QIDH nonempirical DHs. Their MSD are part of those of the best approaches and are evaluated to less than 0.001 Å for the former and 0.001 Å for the latter. This association allows thus to obtain reliable results at the computational prize of a double-ζ basis set.

On the r(O • • • H cis ) H-bond length of the formamide dimer example (Figure 4), the DH-SVPD basis set illustrates well its ability to reinforce the binding behavior of weakly interacting systems by shortening weak bond lengths. In association with semiempirical DHs, DH-SVPD tends to improve the modeling of H-bond lengths with respect to larger basis sets since this type of density functionals are recognized to lack from dispersion interactions. Indeed, the error compensation between BSIE and BSSE in the DH-SVPD double-ζ basis set artificially enhances the binding behavior of the dimer while a very large basis set (less prone to the latter error) weakens its structure. The deviations are particularly small when the semiempiricals DHs are coupled with a pairwise dispersion correction like for B2-PLYP-D, B2-PLYP-D3(bj) and DSD-PBEP86-D3(bj). In association with a nonempirical DH like PBE0-DH or PBE-QIDH, the DH-SVPD tailored basis set emphasizes however the already excellent binding behavior of the dimer and thus overshortens the H-bond lengths.

The diverging behavior observed between empirical and nonempirical DHs is related to the semilocal approximation used by construction in the DH, the PBE 53 exchange density functional capturing better dispersion forces than other semilocal approximation. 31,54 The other weakly interacting aggregates ruled by mixed electrostatics/dispersion or dispersiondominated stacking interactions, case of the T-shaped ethyne or methane dimers, respectively, confirm the above mentioned trend (Figure 4). For all the basis set investigated, the errors are however about one order of magnitude larger with respect to those measured with H-bond interactions. It shows that the errors found are more related to the density functional than to the basis set, even if again, the error compensation between BSIE and BSSE in small basis set tends to improve the binding character of a system in weak interactions.

More than a question of accuracy, the use of a small basis set strongly alleviates the computational cost.

CONCLUSIONS

In this research article, we showed the ability of the coupling of the tailored and cost-effective DH-SVPD basis set with a DH density functional to model covalent structural parameters of small-and medium-sized organic molecules. In particular, the results obtained for the nonempirical PBE-QIDH density functional are competitive with those found when using a very large basis set (quadruple-ζ). Since the DH-SVPD basis set provides an accurate estimation of weak noncovalent interaction energies, the resulting level of theory allows the accurate modeling of both covalent and noncovalent structural parameters at a reasonable computational cost. These results further increase the possibility of applying DHs by enhancing their accuracy/cost ratio. At the same time, they strength the theoretical background of nonempirical DHs since they prove that the pairing with empirical dispersion corrections can be avoided by the used of tailored basis set. nuclear gradient computations, with a given basis set with respect to the DH-SVPD basis set on the methane dimer example. The computations are performed on Intel Xeon CPUs (E5-2690 v3 @2.60GHz) using 8 processors in shared memory.

(O • • • H cis ) r(π • • • H) r(C • • • C)

  (EXX = ∼69%, PT2 = ∼33%) and extend it with 3 extra DHs coupled with empirical dispersion corrections: DSD-PBEP86-D3(bj) (EXX = 69%, PT2 SS = 22%, PT2 OS = 52%), 50 B2-PLYP-D 51 and B2-PLYP-D3(bj).

52

 52

Figure 2

 2 Figure 2 reports the performance of the DH-SVPD basis set in association with the nonempirical PBE-QIDH double-hybrid density functional to model CH, CC, CN, CO, NH and OH bond lengths, and compares it with those obtained with the def2-SVP, def2-SVPD, def2-TZVPP, def2-QZVPP Ahlrichs and aug-cc-pVTZ Dunning basis sets. At first glance, the statistics derived from both CCse and B3se datasets are homogeneous, independently on the basis set. The mean absolute deviations (MADs) derived from both datasets are indeed in close agreement. The only exception are the absolute deviations calculated with respect to the CN bonds which differ because of the poor performance of PBE-QIDH with respect to aziridine, a molecule only included in B3se.

Figure 5

 5 illustrates it on the structure optimization of the methane dimer performed with the PBE-QIDH DH density functional. With respect to DH-SVPD, the CPU time to perform one SCF cycle is multiplied by 3, 12 and 36 with def2-TZVPP, aug-cc-pVTZ and def2-QZVPP, respectively. It reaches its maximum with 5, 35 and 94 times, respectively, during the PT2 computation, and lies around 2, 7 and 17 times, respectively, during the coupled-perturbated Kohn-Sham (CPKS) and gradient computations. As depicted above, the huge time gain observed is of course not as the expense of the accuracy of the approach.It opens thus a new route to the application of DHs to much larger molecular systems of chemical interest for theoretician and experimentalists.

Figure 1 : 5 • 10 - 3 1 • 10 - 2 1. 5 • 10 - 2 2 • 10 - 2 2. 5 • 10 - 2 3 • 10 - 2 3. 5 • 10 - 2 4 • 10 - 2 4. 5 • 10 - 2 5 • 10

 15103102510210251021025102102510210 Figure 1: From left to right, molecular structures and reference bond lengths ( Å) of the formamide dimer,39 , and the T-shaped ethyne and methane-methane aggregates.42

Figure 2 : 42 M 1 . 5 • 10 - 2 - 1 • 10 - 2 - 5 • 10 - 3 0 5 • 10 - 3 1 • 10 - 2 1

 24215102110251035103102 Figure 2: (left) Mean absolute deviations ( Å) computed with the PBE-QIDH double hybrid and different basis sets taking as reference the 53 (164) CCSD(T) (B3LYP) equilibrium bond length references gathered into the CCse (B3se) databases. 40,41 (right) Absolute bond length deviations ( Å) computed with the PBE-QIDH double hybrid and a selection of basis sets versus the r(O • • • H cis ) H-bond CCSD(T) equilibrium reference parameter of the formamide dimer, 39 the r(π • • • H) mixed electrostatics/dispersion CCSD(T)/CBS reference parameter of T-shaped ethyne aggregate, 42 and the r(C • • • C) dispersion-dominated stacking CCSD(T)/CBS reference parameter of the methane-methane aggregate. 42

Figure 3 :

 3 Figure 3: Mean signed deviations ( Å) computed over a selection density functionals and basis sets versus the 53 CCSD(T) equilibrium bond length references gathered into the CCse database.40,41

Figure 5 :

 5 Figure 5: Ratio of the CPU time spent to achieve the (red) self-consistent field (SCF) convergence, (blue) PT2, (orange) coupled-perturbated Kohn-Sham (CPKS) and (green)

The authors acknowledge the GENCI-CINES for HPC resources (Projects A0040810359 and A0060810359). C. A. and I. C. acknowledge ANR (grand E-Storic) for financial support. reference distance of the methane-methane aggregate. 42