Helium clusters and droplets: microscopic superfluidity and other quantum effects†

Ab Initio Potentials for the Ground S0 and the First Electronically Excited Singlet S1 States of Benzene-Helium with Application to Tunneling Intermolecular Vibrational States

We present new ab initio intermolecular potential energy surfaces for the benzene-helium complex in its ground (S0) and first excited (S1) states. The coupled-cluster level of theory with single, double, and perturbative triple excitations, CCSD(T), was used to calculate the ground state potential. The excited state potential was obtained by adding the excitation energies S0 → S1 of the complex, calculated using the equation of motion approach EOM-CCSD, to the ground state potential interaction energies. Analytical potentials are constructed and applied to study the structural and vibrational dynamics of benzene-helium. The binding energies and equilibrium distances of the ground and excited states were found to be 89 cm-1, 3.14 Å and 77 cm-1, 3.20 Å, respectively. The calculated vibrational energy levels exhibit tunneling of He through the benzene plane and are in reasonable agreement with recently reported experimental values for both the ground and excited states [Hayashi, M.; Ohshima, Y. J. Phys. Chem. Lett. 2020, 11, 9745]. Prospects for the theoretical study of complexes with large aromatic molecules and He are also discussed.

Unusual Dynamics and Vibrational Fingerprints of van der Waals Dimers Formed by Linear Molecules and Rare-Gas Atoms

Detailed structural, dynamical, and vibrational analyses have been performed for systems composed of linear triatomic molecules solvated by a single rare-gas atom, He, Ne, or Ar. Among the chromophores of these van der Waals (vdW) dimers, there are four neutral molecules (CO2, CS2, N2O, and OCS) and six molecular cations (HHe2+, HNe2+, HAr2+, HHeNe+, HHeAr+, and HNeAr+), both of apolar and polar nature. Following the exploration of bonding preferences, high-level four-dimensional (4D) potential energy surfaces (PESs) have been developed for 24 vdW dimers, keeping the two intramonomer bond lengths fixed. For these 24 complexes, over 1500 bound vibrational states have been obtained via quasi-variational nuclear-motion computations, employing exact kinetic-energy operators together with the accurate 4D PESs and their 2D/3D cuts. The reduced-dimensional (2D to 4D) dimer models have been compared with full-dimensional (6D) ones in the cases of the neutral CO2·Ar and charged HHe2+·He dimers, corroborating the high accuracy of the 2D to 4D vibrational energies. The reduced-dimensional models suggest that (a) while the equilibrium structures are T-shaped and planar, the effective ground-state structures are nonplanar, (b) certain bound states belong to collinear molecular structures, even when they are not minima, (c) the vdW vibrations are heavily mixed and many states have amplitudes corresponding to both the T-shaped and collinear structures, (d) there are a few dimers, for which even some of the vdW fundamentals lie above the first dissociation limit, and (e) the vdW vibrations are almost fully decoupled from the intramonomer bending motion.

Sizes of pure and doped helium droplets from single shot x-ray imaging

Advancements in x-ray free-electron lasers on producing ultrashort, ultrabright, and coherent x-ray pulses enable single-shot imaging of fragile nanostructures, such as superfluid helium droplets. This imaging technique gives unique access to the sizes and shapes of individual droplets. In the past, such droplet characteristics have only been indirectly inferred by ensemble averaging techniques. Here, we report on the size distributions of both pure and doped droplets collected from single-shot x-ray imaging and produced from the free-jet expansion of helium through a 5 μm diameter nozzle at 20 bars and nozzle temperatures ranging from 4.2 to 9 K. This work extends the measurement of large helium nanodroplets containing 10⁹-10¹¹ atoms, which are shown to follow an exponential size distribution. Additionally, we demonstrate that the size distributions of the doped droplets follow those of the pure droplets at the same stagnation condition but with smaller average sizes.

Hydrogen atom quantum diffusion in solid parahydrogen: The H + N2O → cis-HNNO → trans-HNNO reaction

The diffusion and reactivity of hydrogen atoms in solid parahydrogen at temperatures between 1.5 K and 4.3 K are investigated by high-resolution infrared spectroscopy. Hydrogen atoms are produced within solid parahydrogen as the by-products of the 193 nm in situ photolysis of N₂O, which induces a two-step tunneling reaction, H + N₂O → cis-HNNO → trans-HNNO. The second-order rate constant for the first step to form cis-HNNO is found to be inversely proportional to the N₂O concentration after photolysis, indicating that the hydrogen atoms move through solid parahydrogen via quantum diffusion. This reaction only readily occurs at temperatures below 2.8 K, not due to an increased rate constant for the first reaction step at low temperatures but rather due to an increased selectivity to the reaction. The rate constant for the second step of the reaction mechanism involving unimolecular isomerization is shown to be independent of the N₂O concentration as expected. The inverse concentration dependence of the rate constant for the reaction step that involves the hydrogen atom demonstrates clearly that quantum diffusion influences the reactivity of the hydrogen atoms in solid parahydrogen, which does not have an analogy in classical reaction kinetics.

Quantum Tunneling of a He Atom Above and Below a Benzene Ring

Van der Waals (vdW) complexes with helium atoms have deserved much attention for their intriguing quantum nature relevant to microscopic superfluidity. However, tunneling splitting, the clear signature of quantum delocalization of He atoms, has rarely been identified in any of the He-containing complexes. Here, UV excitation spectra of benzene-He were extensively examined with almost full rotational resolution to identify two weak vibronic bands with vibrational excitation energies of only ∼13 and ∼16 cm^-1. Each of rotational transitions appears to be split into doublets in the higher-frequency band. This splitting is attributed to quantum tunneling due to the delocalization of He spread over two minimum locations below and above the benzene ring. The magnitude of the tunneling splitting as well as the vibrational frequencies of the two vdW modes are compared with the reported theoretical prediction to quantitatively assess the intermolecular potential energy surfaces so far derived.

Electron Diffraction of Pyrene Nanoclusters Embedded in Superfluid Helium Droplets

We report electron diffraction of pyrene nanoclusters embedded in superfluid helium droplets. Using a least-squares fitting procedure, we have been able to separate the contribution of helium from those of the pyrene nanoclusters and determine the most likely structures for dimers and trimers. We confirm that pyrene dimers form a parallel double-layer structure with an interlayer distance of 3.5 Å and suggest that pyrene trimers form a sandwich structure but that the molecular planes are not completely parallel. The relative contributions of the dimers and trimers are ∼6:1. This work is an extension of our effort of solving structures of biological molecules using serial single-molecule electron diffraction imaging. The success of electron diffraction from an all-light-atom sample embedded in helium droplets offers reassuring evidence of the feasibility of this approach.

Infrared Signatures of the HHen+ and DHen+ (n = 3-6) Complexes

Combination of a cryogenic ion-trap machine, operated at 4.7 K, with the free-electron-laser FELIX allows the first experimental characterization of the unusually bright antisymmetric stretch (ν₃) and π-bending (ν₂) fundamentals of the He-X⁺-He (X = H, D) chromophore of the in situ prepared HHe_n⁺ and DHe_n⁺ (n = 3-6) complexes. The band origins obtained are fully supported by first-principles quantum-chemical computations, performed at the MP2, the CCSD(T), and occasionally the CCSDTQ levels employing extended basis sets. Both the experiments and the computations are consistent with structures for the species with n = 3 and 6 being of T-shaped C_2v and of D_4h symmetry, respectively, while the species with n = 4 are suggested to exhibit interesting dynamical phenomena related to large-amplitude motions.

A method for constrained optimisation of the design of a scanning helium microscope

We describe a method for obtaining the optimal design of a normal incidence Scanning Helium Microscope (SHeM). Scanning helium microscopy is a recently developed technique that uses low energy neutral helium atoms as a probe to image the surface of a sample without causing damage. After estimating the variation of source brightness with nozzle size and pressure, we perform a constrained optimisation to determine the optimal geometry of the instrument (i.e. the geometry that maximises intensity) for a given target resolution. For an instrument using a pinhole to form the helium microprobe, the source and atom optics are separable and Lagrange multipliers are used to obtain an analytic expression for the optimal parameters. For an instrument using a zone plate as the focal element, the whole optical system must be considered and a numerical approach has been applied. Unlike previous numerical methods for optimisation, our approach provides insight into the effect and significance of each instrumental parameter, enabling an intuitive understanding of effect of the SHeM geometry. We show that for an instrument with a working distance of 1 mm, a zone plate with a minimum feature size of 25 nm becomes the advantageous focussing element if the desired beam standard deviation is below about 300 nm.

Propyne-water complexes hosted in helium droplets

A HElium Nanodroplet Isolation (HENDI) experiment was performed to explore the absorption spectrum of the propyne-water complex (CH3CCH⋯H2O). Two spectral regions were investigated, near the CH stretch v1 of the propyne moiety and near the asymmetric stretch v3 of the water moiety. Ab-initio calculations were performed at the MP2/aug-cc-pVTZ level to estimate the spectroscopic constants of the free complex. This provided the necessary parameters to simulate the absorption spectrum of the complex and thus facilitate the interpretation of the experiment. The observed spectrum is consistent with a structure of the complex where two H-bonds between water and propyne form a five member ring. The later was predicted by Lopes et al. [J. Mol. Struct. 834, 258 (2007)].

Large amplitude motion within acetylene-rare gas complexes hosted in helium droplets

Near-infrared spectroscopy of the C2H2-Ar, Kr complexes was performed in the spectral region overlapping the ν3/ν2 + ν4 + ν5 Fermi-type resonance of C2H2. The experiment was conducted along the HElium NanoDroplet Isolation (HENDI) technique in order to study the coupling dynamics between a floppy molecular system (C2H2-Ar and C2H2-Kr) and a mesoscopic quantum liquid (the droplet). Calculations were performed using a spectral element based close-coupling program and state-of-the-art 2-dimensional potential energy surfaces to determine the bound states of the C2H2-Ar and C2H2-Kr complexes and simulate the observed spectra. This furnished a quantitative basis to unravel how the superfluid and non-superfluid components of the droplet affect the rotation and the deformation dynamics of the hosted complex.

Converged Colored Noise Path Integral Molecular Dynamics Study of the Zundel Cation Down to Ultralow Temperatures at Coupled Cluster Accuracy

For a long time, performing converged path integral simulations at ultralow but finite temperatures of a few Kelvin has been a nearly impossible task. However, recent developments in advanced colored noise thermostatting schemes for path integral simulations, namely, the Path Integral Generalized Langevin Equation Thermostat (PIGLET) and the Path Integral Quantum Thermal Bath (PIQTB), have been able to greatly reduce the computational cost of these simulations, thus making the ultralow temperature regime accessible in practice. In this work, we investigate the influence of these two thermostatting schemes on the description of hydrogen-bonded systems at temperatures down to a few Kelvin as encountered, for example, in helium nanodroplet isolation or tagging photodissociation spectroscopy experiments. For this purpose, we analyze the prototypical hydrogen bond in the Zundel cation (H₅O₂⁺) as a function of both oxygen-oxygen distance and temperature in order to elucidate how the anisotropic quantum delocalization and, thus, the shape of the shared proton adapts depending on the donor-acceptor distance. The underlying electronic structure of the Zundel cation is described in terms of Behler's Neural Network Potentials of essentially converged Coupled Cluster accuracy, CCSD(T*)-F12a/AVTZ. In addition, the performances of the PIQTB and PIGLET methods for energetic, structural, and quantum delocalization properties are assessed and directly compared. Overall, our results emphasize the validity and practical usefulness of these two modern thermostatting approaches for path integral simulations of hydrogen-bonded systems even at ultralow temperatures.

On the Existence of HeHe Bond in the Endohedral Fullerene Hе2 @C60

Twenty years have already been passed since the endohedral fullerene's void ceaselessly attracts attention of both, experimentalists and theoreticians, computational chemists and physicists in particular, who direct their efforts on computer simulations of encapsulating atoms and molecules into fullerene void and on unraveling the arising bonding patterns. We review recent developments on the endohedral He₂ @C₆₀ fullerene, on its experimental observation and on related computational works. The two latter are the main concerns in the present work: on the one hand, there experimentally exists the He dimer embedded into C₆₀ void. On the other, computational side, each He atom exhibits a negligible charge transfer to C₆₀ resulting in that altogether, the He dimer exists as a fractionally charged (He^+δ )₂ . Whether there exists a bond between these two helium atoms is the key question of the present work. Since a bond is a two-body creature, we assert that it suffices to define the bond on the basis of Löwdin's postulate of a molecule which we invoke to investigate such formation of the He dimer in a given C₆₀ void in terms of the HeHe potential energy well. It is analytically demonstrated that this well enables to maintain at least one bound (ground) state, and therefore, according to Löwdin's postulate which is naturally anticipated within quantum theory, we infer that (He^+δ )₂ is a molecule, a diatomic, where two heliums are bonded to each other. Using these arguments, we also propose to extend the concept of stability of endohedral fullerenes. © 2017 Wiley Periodicals, Inc.

A HElium NanoDroplet Isolation (HENDI) investigation of the weak hydrogen bonding in the propyne dimer (CH3CCH)2

A HElium Nanodroplet Isolation (HENDI) experiment was performed to explore the absorption spectra of the propyne monomer (CH₃CCH), dimer and (CH₃CCH)_≥3 multimers in the vicinity of the CH stretch region ν₁ of the monomer.

Dynamics of acetylene dimers hosted in helium droplets

The CH antisymmetric stretch of the C₂H₂ moieties in acetylene dimers was explored over the range 3270–3290 cm⁻¹ using the helium nanodroplet isolation (HENDI) technique.

Doping of Green Fluorescent Protein into Superfluid Helium Droplets: Size and Velocity of Doped Droplets

We report doping of green fluorescent protein from an electrospray ionization (ESI) source into superfluid helium droplets. From analyses of the time profiles of the doped droplets, we identify two distinct groups of droplets. The faster group has a smaller average size, on the order of 106 helium atoms/droplet, and the slower group is much larger, by at least an order of magnitude. The relative populations of these two groups depend on the temperature of the droplet source: from 11 to 5 K, the signal intensity of the slower droplet group gradually increases, from near the detection limit to comparable to that of the faster group. We postulate that the smaller droplets are formed via condensation of gaseous helium upon expansion from the pulsed valve, while the larger droplets develop from fragmentation of ejected liquid helium. Our results on the size and velocity of the condensation peak at higher source temperatures (>7 K) agree with previous reports, but those at lower temperatures (<7 K) seem to be off. We attribute this discrepancy to the masking effect of the exceedingly large droplets from the fragmentation peak in previous measurements of droplet sizes. Within the temperature range of our investigation, although the expansion condition changes from subcritical to supercritical, there is no abrupt change in either the velocity distribution or the size distribution of the condensation peak, and the most salient effect is in the increasing intensity of the fragmentation peak. The absolute doping efficiency, as expressed by the ratio of ion-doped droplets over the total number of ions from the ESI source, is on the order of 10-4, while only hundreds of doped ions have been detected. Further improvements in the ESI source are key to extending the technology for future experiments. On the other hand, the separation of the two groups of droplets in velocity is beneficial for size selection of only the smaller droplets for future experiments of electron diffraction.

Photoexcited Ag ejection from a low-temperature He cluster: a simulation study by nonadiabatic Ehrenfest ring-polymer molecular dynamics

Nonadiabatic ring-polymer molecular dynamics simulations were performed to understand the photoexcitation dynamics of a low-temperature Ag·He₅₀₀ cluster.

Large amplitude motion of the acetylene molecule within acetylene-neon complexes hosted in helium droplets

Superfluid helium droplets provide an ideal environment for spectroscopic studies with rotational resolution. Nevertheless, the molecular rotation is hindered because the embedded molecules are surrounded by a non-superfluid component. The present work explores the dynamical role of this component in the hindered rotation of C2H2 within the C2H2-Ne complex. A HENDI experiment was built and near-infrared spectroscopy of C2H2-Ne and C2H2 was performed in the spectral region overlapping the ν3/ν2 + ν4 + ν5 Fermi-type resonance of C2H2. The comparison between measured and simulated spectra helped to address the above issue.

Lieb-Liniger-like model of quantum solvation in CO-(4)HeN clusters

Small (4)He clusters doped with various molecules allow for the study of "quantum solvation" as a function of cluster size. A peculiarity of quantum solvation is that, as the number of (4)He atoms is increased from N = 1, the solvent appears to decouple from the molecule which, in turn, appears to undergo free rotation. This is generally taken to signify the onset of "microscopic superfluidity." Currently, little is known about the quantum mechanics of the decoupling mechanism, mainly because the system is a quantum (N + 1)-body problem in three dimensions which makes computations difficult. Here, a one-dimensional model is studied in which the (4)He atoms are confined to revolve on a ring and encircle a rotating CO molecule. The Lanczos algorithm is used to investigate the eigenvalue spectrum as the number of (4)He atoms is varied. Substantial solvent decoupling is observed for as few as N = 5 (4)He atoms. Examination of the Hamiltonian matrix, which has an almost block diagonal structure, reveals increasingly weak inter-block (solvent-molecule) coupling as the number of (4)He atoms is increased. In the absence of a dopant molecule the system is similar to a Lieb-Liniger (LL) gas and we find a relatively rapid transition to the LL limit as N is increased. In essence, the molecule initially-for very small N-provides a central, if relatively weak, attraction to organize the cluster; as more (4)He atoms are added, the repulsive interactions between the identical bosons start to dominate as the solvation ring (shell) becomes more crowded which causes the molecule to start to decouple. For low N, the molecule pins the atoms in place relative to itself; as N increases the atom-atom repulsion starts to dominate the Hamiltonian and the molecule decouples. We conclude that, while the notion of superfluidity is a useful and correct description of the decoupling process, a molecular viewpoint provides complementary insights into the quantum mechanism of the transition from a molecular cluster to a quantum solvated molecule.

Effective doping of low energy ions into superfluid helium droplets

We report a facile method of doping cations from an electrospray ionization (ESI) source into superfluid helium droplets. By decelerating and stopping the ion pulse of reserpine and substance P from an ESI source in the path of the droplet beam, about 10(4) ion-doped droplets (one ion per droplet) can be recorded, corresponding to a pickup efficiency of nearly 1 out of 1000 ions. We attribute the success of this simple approach to the long residence time of the cations in the droplet beam. The resulting size of the doped droplets, on the order of 10(5)/droplet, is measured using deflection and retardation methods. Our method does not require an ion trap in the doping region, which significantly simplifies the experimental setup and procedure for future spectroscopic and diffraction studies.

Formation of dimers of light noble atoms under encapsulation within fullerene's voids

Van der Waals (vdW) He2 diatomic trapped inside buckminsterfullerene's void and preserving its diatomic bonding is itself a controversial phenomenon due to the smallness of the void diameter comparing to the He-He equilibrium distance. We propound a computational approach, including smaller fullerenes, C20 and C28, to demonstrate that encapsulation of He2 inside the studied fullerenes exhibits an interesting quantum behavior resulting in a binding at shorter, non-vdW internuclear distances, and we develop a computational model to interpret these He-He bonding patterns in terms of Bader's atom-in-molecule theory. We also conjecture a computational existence of He2@C60 on a solid basis of its theoretical UV absorption spectrum and a comparison with that of C60.

Microwave spectroscopy of the seeded binary and ternary clusters CO-(pH2)2, CO-pH2-He, CO-HD, and CO-(oD2)(N=1,2)

We report the Fourier transform microwave spectra of the a-type J = 1-0 transitions of the binary and ternary CO-(pH2)2, CO-pH2-He, CO-HD, and CO-(oD2)N=1,2 clusters. In addition to the normal isotopologue of CO for all clusters, we observed the transitions of the minor isotopologues, (13)C(16)O, (12)C(18)O, and (13)C(18)O, for CO-(pH2)2 and CO-pH2-He. All transitions lie within 335 MHz of the experimentally or theoretically predicted values. In comparison to previously reported infrared spectra [Moroni et al., J. Chem. Phys. 122, 094314 (2005)], we are able to tentatively determine the vibrational shift for CO-pH2-He, in addition to its b-type J = 1-0 transition frequency. The a-type frequency of CO-pH2-He is similar to that of CO-He2 [Surin et al., Phys. Rev. Lett. 101, 233401 (2008)], suggesting that the pH2 molecule has a strong localizing effect on the He density. Perturbation theory analysis of CO-oD2 reveals that it is approximately T-shaped, with an anisotropy of the intermolecular potential amounting to ∼9 cm(-1).

The structure of the asymmetric helium trimer (3)He(4)He2

Despite their apparent simplicity, the properties of the two helium trimers, (4)He3 and (3)He(4)He2, are still not completely understood. In particular, the existence of a bound state of the asymmetric trimer (3)He(4)He2 was established many years ago, using different theoretical approaches, and later it was experimentally detected. However its structural properties have not been thoroughly investigated so far, probably because an accurate theoretical description of this very weakly bound system is computationally quite demanding. In this work we give for the first time an accurate and complete theoretical description of the geometrical structure of this fragile system using quantum Monte Carlo techniques employing the TTY potential and compare its properties with those of (4)He2 and (4)He3. We compute average values of distances and angles, along with the angle-angle distribution function: a two-dimensional probability distribution well suited to discuss the shape of a trimer. Our analysis shows that the lighter isotope is very diffuse and can be found at large distances from the other two atoms, but also close to the center of mass of the system in nearly linear configurations. For this system the concept of "equilibrium structure" is meaningless and all kinds of three-atom configurations must be taken into account in its description.

Microscopic superfluidity in 4He clusters stirred by a rotating impurity molecule

The effective moment of inertia of a CO impurity molecule in 4HeN and p-(H2)N solvent clusters initially increases with N but then commences a nonclassical decrease at N=4 (4He) or N=6 (p-H2). This suggests molecule-solvent decoupling and a transition to microscopic superfluidity. However, the quantum decoupling mechanism has not been elucidated. To understand the decoupling mechanism, a one-dimensional model is introduced in which the 4He atoms are confined to a ring. This model captures the physics and shows that decoupling happens primarily because of bosonic solvent-solvent repulsion. Quantum Monte Carlo and basis set calculations suggest that the system can be modeled as a stirred Tonks-Girardeau gas. This allows the N-particle time-dependent Schrödinger equation to be solved directly. Computations of the integrated particle current reveal a threshold for stirring and current generation, indicative of superfluidity.