Real-time observation of the diffusion-limited formation of a cation-molecule complex

Domain: physics
Published: 2025-12-29
Editorial outcome: major_revision

Abstract
For two molecules or atoms to react, they must first move towards each other and then, upon meeting, form new chemical bonds. Ultrafast spectroscopy and diffraction techniques have illuminated the bond-formation step for example by triggering reactions from weakly-bound precursors but not the initial approach, typically attributed to diffusion, although this is often the step that determines the reaction rate. Here, we measure and control the diffusion time for the reaction where a Li⁺ ion forms a complex with a benzene dimer, a textbook cation-π system, inside a liquid helium nanodroplet. Using femtosecond-timed Coulomb explosion, we find that Li⁺, initially at the droplet surface over 30 Å from the dimer, first solvates, then diffuses ballistically at 43 m/s and finally reacts. These results, rationalized by ring-polymer molecular dynamics simulations, pave the way for real-time imaging of stereodynamics in ion-molecule reactions.

Introduction
Unimolecular processes such as bond-breaking and isomerization start from the well-defined geometry of a single parent molecule. Therefore, their kinetics and even the changing molecular structure are, in principle, straightforward to explore on the natural atomic timescale using femtosecond (fs) pump-probe methods. This is not the case for a bimolecular reaction between two atomic or molecular reactants A and B, because such a process requires that A and B first meet. In gases and liquids, this happens via diffusion, and since the distance between reactants varies, so does the moment they meet for any defined starting time. Consequently, diffusion prevents the femtosecond time resolution needed to follow the part of the reaction where chemical bonds actually change. In certain cases, this timing problem may be overcome if a weakly-bound, non-reactive A-B complex can be created and if a fs pump pulse can initiate the bimolecular reaction by photoexciting A or B 1–3. Such a precursor complex eliminates diffusion and essentially turns the bimolecular reaction into a unimolecular process. Examples in gas phase and solvents have been reported, including proton transfer, vibrational dynamics and bond shortening 4–8. Another approach, demonstrated for radical-molecule and photo-redox reactions in a solvent, is to increase the reactant concentration so much that the diffusion time becomes almost negligible 9,10. It has also been shown that the intrinsic rate constant of e.g. bimolecular electron transfer reactions in solution can be determined when theoretical modelling of diffusion is applied to fs time-resolved experimental signals 11.

Under most synthetic or natural conditions, the two reactants are, however, not in a close-proximity, precursor geometry. Rather, they are brought together via diffusion, which for many reaction types determines the overall reaction rate and may involve stereodynamical effects as the reactants approach each other. Insight into the progression of the entire bimolecular reaction therefore requires time-resolved techniques that also capture and possibly control the diffusion stage, a so far unmet challenge. Here, we introduce an experimental method that enables us to prepare two reactants, an ion and a molecule, at a controllable initial distance, large enough that diffusion must occur before they meet and react. The separation of the reactants is obtained using helium nanodroplets doped with an alkali atom, here Li, at its surface and a molecular system, here the benzene dimer Bz2, near its center. A fs pump pulse initiates the reaction at a precise time through selective ionization of the Li atom. Coulomb explosion, triggered by ionizing Bz2 with a delayed fs probe pulse, provides time-resolved information about both the initial solvation of the Li+ ion, its diffusion through the droplet and the final Li+Bz2 complex formation. Ring-polymer molecular dynamics (RPMD) simulations serve to interpret the experimental findings and benchmark the experimentally determined diffusion velocity.

Results
Atomistic modelling of the Li+ + Bz2 → Li+Bz2 reaction in a helium nanodroplet
First, we provide a theoretical reference for the reaction:

Li+ + Bz2 → Li+Bz2 (1)

obtained through RPMD simulations 12,13, details given in Methods and Supplementary Information. We consider a helium droplet containing 3600 atoms corresponding to a radius of 33 Å with Bz2 near the center and a Li atom residing ∼2 Å above the surface 14. At time t=0, the Li atom is instantly ionized. The ensuing dynamics, including the local environment of the Li+ ion, its distance to Bz2 and the spatial orientation of Bz2, are determined in real time. Figure 1a, illustrating the droplet and dopants at four selected times, show that Li+ gradually moves towards the droplet center and eventually reaches Bz2 to form a Li+Bz2 complex. A movie of the RPMD trajectory is available as Supplementary Video 1.

The full curve in Fig. 1b, depicting the Li+–Bz2 distance, r(Li+–Bz2), for a single trajectory, as a function of time, shows that Li+ initially accelerates into the droplet, then decelerates and at t∼10 ps settles at a constant velocity. These early dynamics, caused by the polarizability interaction between Li+ and the He atoms, see ref. 13. and Supplementary Information, are independent of the presence of Bz2 as seen by comparing results without Bz2 in Fig. 1e to those with Bz2. The latter represent the average over 10 trajectories corresponding to different initial orientations of Bz2, i.e., a situation more accurately reflecting the experimental conditions than the single trajectory in Fig. 1b. At longer times, Fig. 1b shows that the Li+ velocity remains constant until t∼45 ps, where the ion accelerates and r(Li+–Bz2) drops from 13 Å to 3.5 Å at t=52 ps. Thereafter, r(Li+–Bz2) does not change. We ascribe this final step, only observed when Bz2 is present, to the short-range Li+–Bz2 attraction and the ensuing formation of the Li+Bz2 complex, where Li+ binds to the outside of the dimer (Fig. 1a).

Figure 1c shows that the distance between the two Bz molecules, r(Bz–Bz), remains constant at ∼3.9 Å, the calculated equilibrium distance of Bz2, until t=51 ps, where it drops to 3.2 Å within a few ps. This drop, highlighted by the grey shaded band, coincides with when Li+ makes the final approach to Bz2 and is another manifestation of the Li+Bz2 complex formation. Finally, Fig. 1d shows the coordination number for Li+, Nc(Li+), which is a measure of the number of He atoms closely bound to the ion. During the first ∼4 ps, Nc increases from 0 to 6, an effect ascribed to the solvation of Li+ 13,14. Thereafter, Nc increases slowly to about 8 and then, from t=51 to 2 ps, it drops sharply to below 2 and subsequently recovers at 3. We interpret the sharp drop, occurring simultaneously with r(Bz–Bz) and r(Li+–Bz2) reaching their final values, as part of the solvation shell of Li+ being broken during its bonding to Bz2. The asymptotic value of 3 for Nc is the number of He atoms attached to the side of Li+ facing away from Bz2.

Initial solvation of the Li+ ion
The starting point for the experiments is also helium nanodroplets, each containing one Li atom, at the surface, and one benzene dimer, expected to be located near the droplet center. For experimental reasons, many of the droplets will, however, contain a Li atom and only a single benzene (Bz) molecule (see Methods). First, the doped droplets are irradiated by a 50-fs pump laser pulse that multiphoton ionizes Li and leaves Bz2 (or Bz) intact. This prompt creation of a Li+ ion sets a precise starting time for the reaction, Eq. (1), at a well-defined initial separation of the reactants determined by the radius of the droplets.

Next, the 60-fs probe pulse, sent at time t, ionizes Bz2 (or Bz), which leads to ejection of different ion species, notably Li+, Bz2+ and Li+Bz2, from the droplet. We detect the time-dependent velocity-resolved yields of these ions to obtain information about different stages of the reaction path. First, we focus on the Li+ ion yield. Figure 2a-f shows the covariance maps 15,16 between the distributions of v(Li+) and v(Bz2+) at six selected times, v(Li+) and v(Bz2+) denoting the velocities of Li+ and Bz2+, respectively.

The pronounced covariance signal centred at (3.0 km/s, 0.2 km/s), observed at t=0.2 ps, shows that a Li+ ion with v around 3.0 km/s, corresponding to a kinetic energy, Ekin, of 0.33 eV, is correlated with a Bz2+ ion with v around 0.2 km/s (Ekin ∼ 0.02 eV). This correlated ion-pair stems from the Coulomb repulsion between a Li+ and a Bz2+ ion originating from the same droplet with Bz2+ produced by dissociative ionization of Bz2 or by ionization of Bz2. In either case, if Bz2 or Bz is located in the center of the droplets, with an expected average radius 〈R〉 of 35 Å, there will be a Coulomb energy of 0.37 eV between Li+ and Bz2+, the moment Bz2+ is created. Their repulsion causes the ejection of both ions from the droplet and momentum conservation gives the Li+ ion a final kinetic energy of 0.32 eV. The match with the measured Ekin corroborates that the initial Li+–Bz2-separation is to a good approximation given by the droplet radius. For details, see Supplementary Information.

At t=0.6 ps, Fig. 2b, the covariance signal at (3.0 km/s, 0.2 km/s) is weaker and at t=3.0 ps, Fig. 2c, it has essentially disappeared. Figure 2g shows the Li+ signal, coincidence-filtered with Bz2+ ions 17, as a function of time. The signal drops exponentially to zero within the first three ps, which we ascribe to Li+ solvating in the droplet, similar to recent observations of solvation of a single Na+ ion in a He droplet 18,19. Details of the solvation dynamics of the Li+ ion are reported elsewhere 20.

Diffusion and bond formation
Our main interest here is what happens at longer times. Figure 2g shows that after the initial decrease, the Li+ signal recovers and rises gradually until it reaches a plateau around 200 ps. We believe the reappearance of the Li+ signal occurs when the solvated Li+ ion moves sufficiently closely to Bz2 (or Bz) that the probe-pulse-induced Coulomb repulsion strips off all He atoms in the solvation shell. The significantly higher v(Bz2+) detected at longer times, Fig. 2d-f, compared to v(Bz2+) for t<1 ps, Fig. 2a-b, corroborates that Li+ has moved closer to Bz2 (or Bz). Thus, the results in Fig. 2 qualitatively show that Li+ gradually approaches and probably reacts with Bz2 (or Bz), but it is difficult to quantitatively determine when the Li+Bz2 complex forms.

Therefore, we turn to another experimental observable, namely Li+Bz2 ions detected in coincidence with Bz2+ ions. This observable, termed Co(Li+Bz2, Bz2+), has two decisive advantages. Firstly, unlike the signals shown in Fig. 2, it only originates from droplets doped with a Li atom and a Bz dimer. Secondly, Li+Bz2 is only formed when Li+ has moved closely to Bz2. Although this appears intuitively clear, we modelled the probe process and showed that formation of a Li+Bz2–Bz2+ ion pair only happens when r(Li+–Bz2) < 8 Å (Supplementary Information). According to Fig. 1b, r(Li+–Bz2) decreases from 8 Å to 3.5 Å, the value of the final Li+Bz2 complex, within ∼1 ps. Thus, we have an experimental observable that is nonzero only when Li+Bz2 has formed – or in the last ps before this happens.

Figure 3a displays the velocity distribution of the Li+Bz2 ions detected in coincidence with a Bz2+ ion as a function of time. The figure shows that these ions first appear after about 5 ps and the distribution centres around 0.6 km/s, which we interpret as the final mean velocity of the Li+Bz2 ions after Coulomb repulsion from a Bz2+ ion. Next, the time dependence of Co(Li+Bz2, Bz2+) is obtained by integrating the velocity distributions in Fig. 3a at each time step (Methods). The result, black circles in Fig. 3b, shows that the yield of Li+Bz2 complexes gradually increases from t∼5 ps until t∼100 ps where the signal flattens. This curve represents our measurement of the time-dependent yield of the reaction in Eq. (1).

To model and understand the experimental data, we note that according to the simulations, the total time, ttotal, to form Li+Bz2 from the Li+ and Bz2 reactants at their initial separated positions, i.e., the time we measure, has three contributions: 1) The solvation time, tsolvation, of Li+ including the initial acceleration and deceleration, i.e., the first ∼10 ps in Fig. 1b; 2) The diffusion time, tdiffusion, where Li+ moves with constant velocity, vdiffusion, i.e., ∼10–45 ps in Fig. 1b; 3) The bond formation time, tbond, where Li+ makes the final acceleration towards Bz2 and forms Li+Bz2, i.e., ∼45–52 ps in Fig. 1b. Thus,

ttotal = tsolvation + tdiffusion + tbond (2)

Equation (2) is based on the simulation for a droplet with R=33 Å, whereas the experimental results concern droplets with 33 Å ≤ 〈R〉 ≤ 49 Å. Currently, it is not computationally feasible to simulate larger droplets, but we expect that for a limited increase of R, tsolvation, tbond and vdiffusion remain essentially unchanged. Therefore, we rewrite Eq. (2) as

ttotal(R) = R/vdiffusion + tconstant (3)

and experimentally test its validity as described below.

The droplets are formed with a size distribution, well described by a log-normal distribution 21. According to Eq. (3), we therefore expect a log-normal distribution of ttotal. However, Co(Li+Bz2, Bz2+) represents integration over ttotal, because the smallest droplets give a contribution at the earliest times and then the contributions from larger droplets gradually add at longer times. Thus, Co(Li+Bz2, Bz2+) should be given as the cumulative distribution function of ttotal, denoted CDFLN (see Methods). The best fit of CDFLN to the data points, blue curve in Fig. 3b, closely matches the experimental results. We repeated the measurements for 14 other droplet size distributions and the results for three of them are displayed in Fig. 3b–e. Again, CDFLN provides excellent fits to the data points, and it is seen that the reaction time increases as the droplet size increases consistent with a diffusion-limited reaction.

From each fit, we obtain the mean value of the log-normal distribution of ttotal. This value, denoted 〈ttotal〉, represents the average reaction time at the particular droplet size distribution. Figure 3f displays 〈ttotal〉 as a function of 〈R〉. It appears that for 33 Å ≤ 〈R〉 ≤ 42 Å, 〈ttotal〉 scales linearly with 〈R〉. The linear dependence is quantified by the dashed, blue line representing the best linear fit to the points in the 33–42 Å range. The linearity validates Eq. (3) and allows a determination of the average velocity, vexp_diffusion, of Li+ in the diffusion region, as the inverse of the slope of the fitted line. We find vexp_diffusion = 43 ± 5 m/s (see Supplementary Information), i.e., larger than the simulated vdiffusion = 14 m/s, Fig. 1e.

Our simulations show that the linear motion results from the combined effect of Li+ being accelerated by the attractive interaction with the He atoms and Bz2 and decelerated by friction from the droplet (Supplementary Information). The simulations do, however, not account for the superfluidity that characterizes helium nanodroplets 22,23 and therefore vdiffusion is expected to underestimate the real diffusion velocity. The found vexp_diffusion is close to the critical Landau velocity measured to be 56 m/s for Ag atoms in helium droplets 23, indicating that the onset of friction on Li+ occurs when its speed is slightly below the Landau velocity. Similar to the findings in Ref. 23, we believe this is caused by heating of the droplet – in our case due to the ion solvation process. Finally, Fig. 3f shows that for the larger droplets, the reaction time gradually increases compared to the extrapolated linear behaviour. This could be caused by increased delocalization of Bz2 away from the droplet center 24, meaning Li+ not always moves straight towards Bz2, i.e., it may have to travel farther than the droplet radius to encounter the dimer. This effect is expected to be more pronounced for larger droplets in agreement with the observed behaviour of 〈ttotal〉. The excellent agreement between the fits of CDFLN and the experimental points suggest that it is unlikely that the delocalization becomes so pronounced that the Li+ ion does not meet Bz2 in the center – such an event would give a deviation from the log-normal distribution.

Methods
Experimental details
An illustration of the experimental setup is shown in Supplementary Information. More details can be found in Ref. 19. A beam of liquid helium droplets is formed by subcritical expansion of 99.9999% purity 4He gas into vacuum through a 5 µm diameter orifice. The backing pressure of the He gas was 50 bar in all experiments, while the temperature of the expansion nozzle was varied between 13.5 K–20.5 K. The resulting droplet distributions had an average number of helium atoms between 3200 and 14600. The formed helium droplet beam passes through two doping cells. The first contains a vapor of benzene molecules, let into the cell through a leak valve. The second contains a gas of Li atoms obtained by heating a sample of solid Li in an oven. The set-point of the PID controller was set to 390 °C. The opening of the leak valve and the temperature of the Li metal were selected to maximize the likelihood of doubly doping with benzene and singly doping with Li. The doping process is Poissonian 22, so other combinations of dopants are unavoidable. The case where a droplet is doped with one Bz molecule and one Li atom is addressed in the main text, and while other systems like LiBz3, Li2Bz2, Li2Bz etc. are possibly formed, they are present to a much lesser extent and do not contribute noticeably to the signal.

The doped droplet beam enters the target vacuum chamber, that contains a velocity map imaging (VMI) spectrometer 51,52. The VMI spectrometer is a three-plate setup, where the repeller (6000 V), extractor (3790 V) and ground (0 V) plate voltages are chosen to achieve velocity focusing. The droplets are irradiated by two pulsed, linearly polarized laser beams in the center of the VMI spectrometer. The two beams are spatially overlapped, and their linear polarization is in the plane of the detector at the end of the VMI spectrometer. The laser pulses are produced by a Coherent Astrella HE laser system (wavelength (nominal) λ0 = 800 nm, temporal FWHM τ = 35 fs, pulse energy Ep = 2 mJ, repetition rate frep = 5 kHz). The focused pump pulses used to ionize Li (τ = 50 fs, λ0 = 800 nm, Ep = 16 µJ, beam waist w0 = 41 µm) have a peak intensity of I0 = 1.1 × 10^13 W/cm^2, while the probe pulses used to ionize Bz (τ = 50 fs, λ0 = 400 nm, Ep = 18 µJ, w0 = 28 µm) have a peak intensity of I0 = 2.7 × 10^13 W/cm^2. The laser parameters were chosen to selectively ionize Li with the pump pulse and efficiently ionize Bz with the probe pulse. The probe pulse was delayed by time t using a linear delay stage.

The ions produced are projected by the VMI spectrometer onto a two-dimensional imaging detector. The detector consists of two micro-channel plates (MCP) and a phosphor screen (P47). The first MCP is a funnel-type plate, which increases the detection efficiency 53. The front of the MCP stack was grounded, while the back was set to 2300 V. The middle of the stack was kept at 1150 V using a shim-ring and the phosphor screen was biased to 4900 V. The light produced by the phosphor screen was imaged by a TPX3Cam detector 54–56. The TPX3Cam is event-based and records both the position and time-of-arrival of each lit-up pixel independently, which allows us to measure VMI images of multiple, different ions. The pixel data stream is time-walk corrected 57,58, clustered and centroided using the DBSCAN algorithm, and time-walk corrected again to retrieve the time-of-flight (ToF) from the time-of-arrival. The x- and y-hit coordinates are calibrated to give the projected velocities, vx and vy, using velocities of Li+ ions coming from the Coulomb explosion of the 3Σu+ state of the Li2 dimer 59. Mass-over-charge values (m/q) were obtained by calibrating the ToF-spectrum using known background ions (H2O+ and effusive Bz+). The final data stream then consists of tuples of values pertaining to each ion, including the laser shot number, (ls, m/q, vx, vy). The experiment was recorded for 15 different nozzle temperatures with a set of time delays between 1 ps and 1000 ps, where a trace was repeated between five and 20 times to achieve a reasonable signal-to-noise ratio.

Data analysis for extraction of Co(Li+Bz2, Bz2+)
The main observable in the experiment is the yield of Li+Bz2 ions detected in coincidence with Bz2+ ions, denoted Co(Li+Bz2, Bz2+). The experimental data acquired in the experiment consists of tuples of values from the TPX3Cam containing the shot number, mass-over-charge, velocity x-component and velocity y-component, (ls, m/q, vx, vy). The data analysis employed to extract Co(Li+Bz2, Bz2+) from these tuples is shown in Supplementary Fig. 10, for a dataset recorded with 〈R〉 = 35 Å. First, Li+Bz2 ions and Bz2+ ions are assigned in the m/q-spectrum, Supplementary Fig. 10a. VMI images are then constructed by calculating the 2D (vx, vy) histogram of the selected ion hits across all laser shots. A VMI image of Li+Bz2 is shown in Supplementary Fig. 10b for a time-resolved experiment, where images from each time delay are summed. The image shows a strong circular feature with a radius of around ∼ 0.4 km/s. There is also a dark line in the center of the image, which is attributed to a burn-in on the MCP detector. The VMI image shows that Li+Bz2 ions indeed are formed in the experiment and the majority of these have very low kinetic energy.

To gain further information about the time-resolved dynamics, we employ covariance map analysis 15,16,60,61 of the velocities and recoil directions of Li+Bz2 and Bz2+ ions. Covariance maps between the radial-velocity distributions, v(Li+Bz2) and v(Bz2+), of the corresponding VMI images are shown in Supplementary Fig. 10c, similarly summed across a time-resolved experiment. The covariance map shows that Li+Bz2 ions can be observed in coincidence with Bz2+ ions and that these ions have a well-defined kinetic energy that is larger than the dominating background signal observed in the VMI image in Supplementary Fig. 10b. To further corroborate this analysis, we also consider the covariance map between the angular distributions, θLi+Bz2 and θBz2+, shown in Supplementary Fig. 10d. The covariance map shows a 180° angular relation between the two ions, which is characteristic of a Coulomb explosion between two charges 62. The angular covariance map is very wide compared to gas-phase experiments 62, which is attributed to non-axial recoil effects coming from collisions with He atoms inside the droplet as the ions leave. The joint analysis of the velocities and recoil directions in Supplementary Fig. 10c-d shows that Li+Bz2 ions with a relation to Bz2+ ions representative of a Coulomb explosion process can be observed and that these ions have a kinetic energy that is larger than what can be seen in the VMI image of Li+Bz2, see Supplementary Fig. 10b. The covariance maps are then used to coincidence filter the Li+Bz2 ions 17. The filtering parameters used are:

|120° < |θLi+Bz2 − θBz2+| < 180°| < 120° (4)

|v(Li+Bz2) − 0.50v(Bz2+)| < 0.8 km/s (5)

The tolerances are chosen to fully cover the covariance maps shown in Supplementary Fig. 10c-d. The relation between the two velocities, v(Li+Bz2) and v(Bz2+), is not defined by the mass ratio between the species but is found empirically. This discrepancy is attributed to the ions having different interactions with the helium droplet as they leave the droplet. The filtered VMI image is shown in Supplementary Fig. 10e for Li+Bz2. There is still a significant signal in the center of the image, but a second channel now appears in the image with a radius of ∼ 0.8 km/s. To analyze this new feature, we extract the full velocity distribution, P(vr). The velocities reported so far have been the projected velocities in the plane of the detector, vx and vy, but the full velocity distribution can be reconstructed by Abel inverting the image, here using the MEVIR algorithm 63. The normalized P(vr) distributions for the raw and filtered VMI images are shown in Supplementary Fig. 10f. By applying coincidence filtering, a broad peak in the velocity distribution is revealed, which is suppressed by background signal in the raw image. This peak is attributed to the Coulomb explosion between Li+Bz2 and Bz2+ ions inside the droplet. Co(Li+Bz2, Bz2+) is then extracted by radially integrating the velocity distributions. In the experiment, this analysis is performed for each time delay. The resulting Co(Li+Bz2, Bz2+) trace for 〈R〉 = 35 Å is shown in Supplementary Fig. 10g and is identical to Fig. 3b. The panel also includes a fit of the cumulative-distribution function of a log-normal distribution, CDFLN. The cumulative-distribution function is defined as:

CDFLN(t) = A/2 [1 + erf( (ln(t) − μt) / (σt * sqrt(2)) )] + c (6)

where A, c, μt and σt are fitted to Co(Li+Bz2, Bz2+). A and c are parameters used to scale and offset the distribution. 〈ttotal〉 can then be calculated as the mean of the log-normal distribution:

〈ttotal〉 = exp(μt + σt^2 / 2) (7)

Theoretical details of ring-polymer molecular dynamics
Simulations of the ionization, diffusion and reaction processes were conducted at the atomistic level of detail, using the same approach previously implemented for the alkali-xenon pair of dopants inside helium nanodroplets 19. The system is prepared in the neutral state with the alkali atom sitting outside the droplet and the molecular dopant Bz2 sitting at the center of the droplet. Thermostatted path-integral molecular dynamics (PIMD) were carried out before sudden ionization of the lithium atom takes place, at a temperature of 2 K and using time step of 0.5 fs and a Trotter discretization number of 32, which is sufficient to ensure a liquid droplet. After 10 ps of equilibration, ring-polymer molecular dynamics (RPMD) simulations 20 were subsequently integrated up to 50 ps, using the same time step of 0.5 fs but without thermostatting, assuming the potential energy surface now corresponds to Li+. The details of the interaction potentials for the neutral and cationic systems are described in the Supplementary Information.

Different initial conditions for the trajectories were generated by supplementing the potential used in the equilibrated PIMD simulations with a simple biasing potential, detailed in the Supplementary Information. The dynamics of the Li+–Bz2 complex formation process was characterized using several descriptors that were monitored as a function of time, namely the distance r(Li+−Bz2) between the lithium cation and the Bz2 dimer, instantaneously measured at the dimers center of mass, the distance r(Bz−Bz) between the two benzene molecules, instantaneously measured as the minimum distance between the two molecules, and the coordination number Nc(Li+) of the two lithium atoms, defined by summing a cut-off function over all Li+–He pairs with value 1 for distances below 3.8 Å, 0 for distances above 4.2 Å, and smoothly interpolated between these values using a fifth-order polynomial. These values were chosen to yield a strictly vanishing coordination number for the neutral lithium atom sitting above the droplet. All the structural properties above were further averaged over the individual real-space beads in the path-integral description.

Estimation of mean droplet radius 〈R〉
A crucial parameter used in the analysis to determine the diffusion velocity of Li+ in liquid helium droplets is the mean droplet radius, 〈R〉. Direct measurements of the helium droplet size distributions typically require a dedicated experimental setup 21,64 and is therefore not currently possible in our experiment. Experimentally recorded droplet sizes have, however, been reported for conditions similar to ours also using a 5 µm orifice and similar stagnation pressures and temperatures 22. These measurements form the basis for estimating the size of the droplets used in our experiment. To this aim, we first consider the droplet size distribution characterized by a log-normal distribution 65:

P(N) = 1 / (N * δN * sqrt(2 * π)) * exp( −(ln(N) − μN)^2 / (2 * δN^2) ) (8)

where N is the number of helium atoms and δN and μN are parameters that characterize the distribution. For the expansion conditions in our experiment, the two parameters are given as 65:

δN = 0.55 (9)

μN = ln(〈N〉) − δN^2 / 2 (10)

which are uniquely determined by the mean number of helium atoms, 〈N〉. The mean radius, 〈R〉, can then be found assuming that the droplets are hard spheres with uniform density as:

〈R〉 = 2.22 Å * 〈N〉^(1/3) * exp( δN^2 / 18 ) (11)

Estimated values for 〈N〉 and 〈R〉, based on previously reported droplet sizes 22, are shown in Supplementary Table 2 for the nozzle temperatures used in the experiment. The droplets of interest in our experiment, which give rise to our observable Co(Li+Bz2, Bz2+), are those that are doubly doped with benzene molecules and singly doped with lithium atoms. To estimate the size distributions of these droplets, we must then also include the doping probability of the molecules and atoms. The doped droplet size distribution is determined as the renormalized product of the two distributions:

P(N; k) = P(N) * P(k; N, pd) / Σ [P(N0) * P(k; N0, pd)] (12)

where P(k; N, pd) is the probability of doping k particles at pressure pd. This probability is described by a Poissonian distribution:

P(k; N, pd) = (βN * pd)^k / k! * exp(−βN * pd) (13)

where k is the number of doped particles, pd is the doping pressure, N is the number of He atoms and βN is a collection of factors:

βN = Fa0 * (σD(N) * Ld) / (vc * kB * Td) (14)

Here, Ld is the length of the doping cell, σD(N) is the cross-section of the droplets, kB is the Boltzmann constant and Td is the dopant temperature. The correction factor Fa0 accounts for the movement of the droplets and dopant particles, and includes the most probable speed of the dopants (cvd) and the speed of the droplets (vD):

Fa0(vD, cvd) = sqrt(π) * (vD / cvd + 1 / (2 * vD / cvd)) * erf(vD / cvd) + exp(−(vD / cvd)^2) (15)

with vD = sqrt(5 * Tnozzle * kB / mHe), cvd = sqrt(2 * Td * kB / md) and erf(x) the Gaussian error function. The temperatures of the dopants were T Bz = 298 K and T Li = 628 K and the doping cells had lengths of LBz = 7.7 cm and LLi = 1.7 cm. Based on the experimental data given in reference 65, the droplet cross section can be written as:

σD(N) = π * (2.22 Å)^2 * N^(2/3) * (1 + 6.51 Å^2 / (2.22 Å * N^(1/3))^2) (16)

The last parameters required to estimate the doped droplet distribution are the vapor pressures of the samples. The optimal vapor pressures of benzene and lithium were found in the following way. We performed a pump-probe experiment at t=500 ps, where Co(Li+Bz2, Bz2+) was maximized by varying the leak valve opening for Tnozzle = 17 K, see Supplementary Fig. 11. This leak valve position was then used throughout the experiment for other nozzle temperatures. The maximum of Co(Li+Bz2, Bz2+) must coincide with the vapor pressure of benzene where the doping of two molecules is most likely. This vapor pressure was 1.0 * 10^−5 mbar, which corresponds to a temperature of T = 298 K 66. The Li sample was similarly estimated as the pressure which optimizes the likelihood of doping one lithium atom for the 17 K droplet size distribution, pLi = 1.99 * 10^−5 mbar, which corresponds to a temperature of T = 628 K 66. Note that this is not entirely consistent with the vapor pressure of Li metal at the set temperature of the oven, which indicates that the temperature measured on the outside of the oven is not exactly the same as the temperature of the sample. The final doped droplet distribution was then found by first estimating the undoped droplet distribution, using values in Supplementary Table 2, finding the doped droplet distribution for k=2 with benzene and then finding the doubly doped droplet distribution for k=1 with lithium. The resulting droplet size distribution was then transformed from N to R and the numerical average 〈R〉 was computed. The effect of including the doping probabilities on 〈R〉 is shown in Supplementary Fig. 12 for the experimental conditions along with the undoped distributions.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper \"Real-time observation of the diffusion-limited formation of a cation-molecule complex\" by J.K. Christensen et al. addresses bond formation dynamics, using Li⁺ and the benzene dimer as a model system. The authors take advantage of the differing solubilities of Li atoms and benzene in helium droplets to design their initial conditions, allowing them to tune the initial separation distance by varying the droplet diameter. This is a clever approach, although it remains limited to reactions involving alkali ions and molecules.\n\nThe results combine experimental measurements with RPMD calculations. The dynamics are triggered by a femtosecond laser that ionizes the metal atom, and the distance between the reaction partners is determined using Coulomb explosion imaging following molecular ionization, detected in coincidence with a state-of-the-art detector. Using a dimer is a smart way to observe ion-molecule contact, as the recoil occurs then between Li(Bz)⁺ and Bz⁺.\n\nThe authors derive a diffusion coefficient from their measurements that differs from the calculated value.\n\nThis paper is clearly written, elegantly tackles an interesting problem, and is accessible to a broad audience. It fits well with Nature Communications, and I am happy to recommend its publication after minor revisions:\n• In the introduction, the authors could mention the extensive work on the stereodynamics of bimolecular reactions using crossed molecular beams.\n• The discrepancy in the diffusion coefficient could also be explained by the fact that Bz₂ may not be centered in the cluster. Instead, it likely roams around, meaning the first observed coincidences may involve the smallest droplets or the closest dimers to Li. The experimental measurements assume that systems containing both Li and Bz₂ are 100% reactive. However, as seen in the videos, some systems may remain for a long time in the \"c\" step of the dynamics (invisible to the probe). Why not for ever? especially if Li and Bz₂ are initially far apart. This point could be discussed further.", "reviewer_2": "General Comment: This article addresses a highly relevant topic by introducing an experimental method to explore, in real time, a bimolecular reaction, complementing it with ring-polymer molecular dynamics simulations, and paving the the way for real-time imaging of stereo-dynamics in ion-molecule reactions. It is nicely shown how a Li+ ion, initially located at the surface of a helium nanodroplet, diffuses towards a benzene dimer, initially located inside the liquid drop, after reaching the regime of a ballistic collision, leading to the Li+ –dimer complex formation. The use of the helium nanodroplet allows to control the initial arrangement of the Li+ ion and the benzene dimer. This work complements previous research on a harpoon-type electron transfer between heliophobic Cs and Cs2 species, initially located at the droplet surface, and a heliophilic fullerene molecule, fully embedded in its interior (DOI: 10.1021/acs.jpclett.7b01910 and DOI:10.3390/molecules26195783), with the results having explained experimental measurements using electron ionization mass spectroscopy (DOI: 10.1063/1.4967405). In spite of the high relevance of the present work, the manuscript needs a major revision in order to to make it more complete, reliable, and interesting. I have included a list of suggestions for improvement below.\n\n1) Motivation. The introduction could be extended to better motivate the work in a broader perspective of helium nanodroplet-mediated bimolecular collisions. Besides the mentioned research above (DOI: 10.1021/acs.jpclett.7b01910 and DOI:10.3390/molecules26195783), it could be interesting to mention on the bimolecular collision between superfluid helium nanodroplets with embedded metal atoms and ions and different surfaces such as titanium dioxide-based surfaces and graphene sheets (see, e.g., DOI: 10.1063/1.4916955 and DOI: DOI: 10.1039/D3CP01303K).\n\n2) Intermolecular interactions.\nAs an addition, it would be recommendable to dissect the physical contributions of the different pair interactions using Symmetry Adapted Perturbation Theory, which is implemented in open-access quantum-chemistry (QC) packages. In particular, it would allow to establish a correlation between the different steps of the ion-dimer formation, and particularly, between the regime of diffusion dynamics and the dominance of the dispersion contribution to the interaction.\n\nAs presented in the supplementary information, the Li-dimer interaction is taken as a sum of pair potentials of the 6-12 Lennard-Jones (LJ)-type, with the parameters having been fitted to QC calculations using the coupled cluster approach with single, double, and perturbative triples excitation (i.e., the CCSD(T) method). Although the CCSD(T) method is an accurate approach for this type of interaction, the used electronic basis set, the augmented polarized correlation-consistent double-ζ basis set, is not large enough to describe it precisely, specially at long-range. It would be thus convenient that the authors extend their computations with the triple-ζ basis set, including an extrapolation to the complete basis set limit (see, e.g., DOI: 10.1063/1.1415078) in order to explain how much the basis se enlargement affects the fitting parameters.\n\nAs very well explained in the supplementary information, the interaction between the Li+ ion and the benzene dimer, it is much more complex, involving a partial charge transfer upon the cation-π complex formation. The inter-atomic pairwise functional form is correct although it would be advisable to use more accurate potential functions that have been previously applied to complexes formation implying charge transfer and other strong attractive energy contributions beyond van-der-Waals forces (see, e.g., DOI: 10.1039/D4RA05401F). It would be also desirable to apply higher levels of ab initio theory either to fit the potential parameters or to validate the performance of the Density Functional Theory approaches that have been considered. If the application of the CCSD(T) approach is computationally too demanding, the the domain-based pair natural orbital coupled-cluster approach DLPNO-CCSD(T) could be a feasible option (see, e.g., DOI: 10.1063/1.4773581). A practical option could also be the application of the coupled Möller Plesset second-order perturbation theory method (see, e.g., DOI: 10.1063/1.2905808, and DOI: 10.1021/ct9005882).\n\n3) Diffusion dynamics and friction.\nIn page 7 of the main manuscript (line 25), it can be read ‘Our simulations show that the linear motion results from the combined effect of Li+ being accelerated by the attractive interaction with the He atoms and Bz2 and decelerated by friction from the droplet’. This is a very engaging observation that can be connected with findings in the previous research on the spatial quenching of a bimolecular collision due the reaction barrier stemming from the extrusion of the helium droplet upon approach of the two reactants (DOI:10.1039/C6CP06858H). As a future prospect, it would be instructive to use the same approach as a complement to the ring-polymer molecular dynamics simulations since it accounts for the superfluid nature of the quantum fluid through the application of helium density functional theory.\n\nMinor Comment. Please include the coordinates of the most relevant complexes at the lowest-energy configurations for the sake of reproducibility."}, "author_response": "Reply to Reviewer 1\nReviewer:\nThe paper \"Real-time observation of the diffusion-limited formation of a cation-molecule complex\"\nby J.K. Christensen et al. addresses bond formation dynamics, using Li⁺ and the benzene dimer as a\nmodel system. The authors take advantage of the differing solubilities of Li atoms and benzene in\nhelium droplets to design their initial conditions, allowing them to tune the initial separation\ndistance by varying the droplet diameter. This is a clever approach, although it remains limited to\nreactions involving alkali ions and molecules.\nThe results combine experimental measurements with RPMD calculations. The dynamics are\ntriggered by a femtosecond laser that ionizes the metal atom, and the distance between the reaction\npartners is determined using Coulomb explosion imaging following molecular ionization, detected\nin coincidence with a state-of-the-art detector. Using a dimer is a smart way to observe ion-\nmolecule contact, as the recoil occurs then between Li(Bz)⁺ and Bz⁺.\nThe authors derive a diffusion coefficient from their measurements that differs from the calculated\nvalue. This paper is clearly written, elegantly tackles an interesting problem, and is accessible to a\nbroad audience. It fits well with Nature Communications, and I am happy to recommend its\npublication after minor revisions:\nAuthors response\nWe appreciate the positive comments very much.\nReviewer\nIn the introduction, the authors could mention the extensive work on the stereodynamics of\nbimolecular reactions using crossed molecular beams.\nAuthors response\nWe fully agree that past work on stereodynamics of bimolecular reactions and scattering using\ncrossed molecular beams should be mentioned. We believe, however, that it would fit better in the\nDiscussion section where future possibilities enabled by the current work are discussed. Therefore,\nwe have rewritten this part significantly and added a 149-word-long paragraph (‘In this connection .\n. . experimental method is time-resolved’) to mention and emphasize the importance of crossed\nmolecular beam studies of stereodynamics, and how our work open new opportunities for insight.\nReviewer\nThe discrepancy in the diffusion coefficient could also be explained by the fact that Bz₂ may not be\ncentered in the cluster. Instead, it likely roams around, meaning the first observed coincidences may\ninvolve the smallest droplets or the closest dimers to Li. The experimental measurements assume\nthat systems containing both Li and Bz₂ are 100% reactive. However, as seen in the videos, some\nsystems may remain for a long time in the \"c\" step of the dynamics (invisible to the probe). Why\nnot for ever? especially if Li and Bz₂ are initially far apart. This point could be discussed further.\nAuthors response\nWe agree that the benzene dimer may not initially be located in the center of the droplets. The\ndelocalization of neutral species has been addressed in theoretical work previously (K. K. Lehmann,\nRef. 25), where it was found that for larger droplets, the molecule tends to roam more (be more\nlocalized). We believe this also plays a role for our experimental results. As seen in Fig. 3f in the\nmanuscript, the larger droplets show a deviation in t from the linear diffusion regime. We\ntotal\nattribute this discrepancy to the delocalization of the Bz molecule; in the text (p. 7 l. 33-36). We\n2\nhave revised this sentence and added 64 words to fully cover this.\nRegarding the possibility of Li+ and Bz not reacting; we do not think this happens. If the dimer\n2\nhappens to be significantly away from the droplet center, the ion may miss it on the first passage of\nthe center but the attraction between Li+ and Bz will bring them together later and lead to\n2\nformation of the Li+Bz complex. If we assume that a substantial part of the droplet size\n2\ndistributions resulted in no reaction between Li+ and Bz , we would not expect the cumulative\n2\ndistribution function, CDF , to provide a good fit of the experimental signal Co(Li+Bz,Bz+). The\nLN\nfact that CDF , actually gives very good fit to the data points for all droplet sizes, see Fig. 3 b-e\nLN\nstrongly supports that non-reactive trajectories do not occur.\nReply to Reviewer 2\nReviewer\nGeneral Comment: This article addresses a highly relevant topic by introducing an experimen-\ntal method to explore, in real time, a bimolecular reaction, complementing it with ring-polymer\nmolecular dynamics simulations, and paving the the way for real-time imaging of stereo-dynamics\nin ion-molecule reactions. It is nicely shown how a Li+ ion, initially located at the surface of\na helium nanodroplet, diffuses towards a benzene dimer, initially located inside the liquid drop,\nafter reaching the regime of a ballistic collision, leading to the Li+ –dimer complex formation. The\nuse of the helium nanodroplet allows to control the initial arrangement of the Li+ ion and the\nbenzene dimer. This work complements previous research on a harpoon-type electron transfer\nbetween heliophobic Cs and Cs2 species, initially located at the droplet surface, and a helio-\nphilic fullerene molecule, fully embedded in its interior (DOI: 10.1021/acs.jpclett.7b01910 and\nDOI:10.3390/molecules26195783), with the results having explained experimental measurements\nusing electron ionization mass spectroscopy (DOI: 10.1063/1.4967405). In spite of the high rel-\nevance of the present work, the manuscript needs a major revision in order to to make it more\ncomplete, reliable, and interesting. I have included a list of suggestions for improvement below.\nAuthors response\nWe appreciate the positive comments. Also, we respect the criticism and suggestions for changes.\nWe have addressed all the points below.\nReviewer\n1) Motivation. The introduction could be extended to better motivate the work in a broader\nperspective of helium nanodroplet-mediated bimolecular collisions. Besides the mentioned re-\nsearch above (DOI: 10.1021/acs.jpclett.7b01910 and DOI:10.3390/molecules26195783), it could\nbe interesting to mention on the bimolecular collision between superfluid helium nanodroplets with\nembedded metal atoms and ions and different surfaces such as titanium dioxide-based surfaces\nand graphene sheets (see, e.g., DOI: 10.1063/1.4916955 and DOI: DOI: 10.1039/D3CP01303K).\nAuthors response\nFirst of all, we thank the reviewer for bringing our attention to the work on helium droplets co-\ndoped with an alkali atom (or dimer) at the surface and a fullerene in the interior. Those studies are\nhighly relevant for the results presented in our work and should obviously be mentioned. Like the\nwork on crossed beam bimolecular reactions, brought up by reviewer 1, we believe that the\nmentioning of the alkali-fullerene studies fits better in the end of the manuscript, where we discuss\nnovel opportunities opened by the current findings, rather than in the introduction. Therefore, we\nhave added new text (49 words) at the end of the Conclusion to mention the alkali-fullerene studies\nand that our new method may provide insight into e.g. electron transfer in such systems. The new\ntext includes the two references suggested by the reviewer, another one to Krupka, Hauser and de\nLara-Castells (2017) as well as to the experimental paper by Renzler et al. (2016).\nRegarding the work on deposition of metal-clusters on surfaces via beams of doped helium droplets\nwe respectfully disagree with the reviewer. That work – both the experimental and theoretical – is\ncertainly most interesting but in our opinion, it is topic-wise far away from the femtosecond time-\nresolved studies of reactions between individual ions and molecules explored in our work. The\nintroduction focuses on previous results on time-resolved studies of bimolecular reactions going\nback to the pioneering work of Ahmed Zewail in the 1980s. This is, we believe, the most pertinent\ncontext for the work presented in our manuscript.\nExtending the introduction to mentioning work on doped helium droplets colliding with surfaces or\nother complex formation or spectroscopic characterization of bimolecular reactions in helium\ndroplets would neither provide a more relevant background nor serve as any extra useful\ninformation for the readers in terms of understanding our research results. Therefore, we\nrespectfully keep the introduction as it is but, as said, extend the discussion of the future\nopportunities opened by our work – to mention the topics suggested by the two reviewers.\nReviewer\n2) Intermolecular interactions.\nAs an addition, it would be recommendable to dissect the physical contributions of the different pair\ninteractions using Symmetry Adapted Perturbation Theory, which is implemented in open-access\nquantum-chemistry (QC) packages. In particular, it would allow to establish a correlation between\nthe different steps of the ion-dimer formation, and particularly, between the regime of diffusion\ndynamics and the dominance of the dispersion contribution to the interaction.\nAuthors response\nThe regime where dispersion forces dominate is actually the one before the lithium atom is ionized,\ni.e. a regime where the reaction cannot take place because the two partners are separated from each\nother and barely interact. Once the lithium atom is ionized, it is polarization forces that cause the\ncation to enter the droplet, become solvated and then diffuse. This also happens without benzene\nmolecules in the center. The interaction with the benzene molecules is also important, as they will\nof course trigger the formation of the complex at some later stage. We agree with the reviewer that\nit is interesting to distinguish the different contributions to the interaction energy in this case and\nwhen revisiting the Li+-benzene interaction potential (see below), we have now quantified the\nmagnitude of the non-electrostatic contribution.\nThe results obtained by partitioning the interaction energy clearly show that the electrostatic\ncomponent (point charge-quadrupole) is essential in explaining why the cation-π complex is so\nstable as the cation approaches the benzene dimer along the reaction coordinate, perpendicular to\nthe dimer plane. Parallel interactions, on the other hand, barely exhibit any contribution of the\ncharge-quadrupole term, as expected.\nReviewer\nAs presented in the supplementary information, the Li-dimer interaction is taken as a sum of pair\npotentials of the 6-12 Lennard-Jones (LJ)-type, with the parameters having been fitted to QC\ncalculations using the coupled cluster approach with single, double, and perturbative triples\nexcitation (i.e., the CCSD(T) method). Although the CCSD(T) method is an accurate approach for\nthis type of interaction, the used electronic basis set, the augmented polarized correlation-consistent\ndouble-ζ basis set, is not large enough to describe it precisely, specially at long-range. It would be\nthus convenient that the authors extend their computations with the triple-ζ basis set, including an\nextrapolation to the complete basis set limit (see, e.g., DOI: 10.1063/1.1415078) in order to explain\nhow much the basis se enlargement affects the fitting parameters.\nAuthors response\nThe interaction between neutral Li and the benzene dimer is extremely weak at the long distances\nrelevant prior to ionization. The interaction at arbitrary ranges is more complex but not of interest in\nour work because we only consider the Li atom at the droplet surface where it is very far (tens of Å)\nfrom the benzene dimer. The Lennard-Jones form that we chose is consistent with the main\ninteraction in the systems, which are also of the LJ form (especially among helium atoms), but\nshould only be considered as a compromise between computational efficiency and chemical\naccuracy. The CCSD(T) calculations that we performed were limited to the long-range region as\nwell - that is, for distances above 10 Å. The few points selected along the Z and X axes were used\nto fit the parameters of the LJ potential between Li and C and H atoms of the benzene monomer, in\nthe region well above the equilibrium distance of a few Å.\nIt is true that the double zeta basis set remains relatively modest, however we could not achieve\nquadruple zeta quality due to excessive memory requirements. Two calculations at the triple zeta\nlevel could be successfully performed, for Li-benzene distances of 10 Å and 20 Å along the Z axis,\nwith the values given in the table below, in comparison with the double zeta basis set.\nZ (Å) aug-cc-pVDZ (meV) aug-cc-pVTZ (meV) AVTZ-AVDZ (meV)\n10 0.9265 0.7031 -0.2234\n20 0.00843 0.00843 <0.00001\nBased on these new calculations, and without attempting to extrapolate to the complete basis set\nlimit, we can see that for Li-benzene distances larger than 20 Å (which is always the case in the\nexperiments and simulations) the difference between the results for the two basis sets is negligible.\nThus, we can safely conclude that the Li-benzene interaction remains extremely weak and would\nnot affect the equilibrium properties of the droplet, prior to ionization. Actually, with magnitudes of\nthe interaction of a few micro-eV, we believe that neglecting the Li-benzene interactions altogether\nin these equilibrium simulations would not alter the subsequent results after ionization. In our\nrevision, we have further mentioned the additional calculations performed for Li-benzene in the\nSupplementary Information and commented on the relevance of the resulting interaction for\ndroplets containing 3600 helium atoms or more.\nReviewer\nAs very well explained in the supplementary information, the interaction between the Li+ ion and\nthe benzene dimer, it is much more complex, involving a partial charge transfer upon the cation-π\ncomplex formation. The inter-atomic pairwise functional form is correct although it would be\nadvisable to use more accurate potential functions that have been previously applied to complexes\nformation implying charge transfer and other strong attractive energy contributions beyond van-der-\nWaals forces (see, e.g., DOI: 10.1039/D4RA05401F). It would be also desirable to apply higher\nlevels of ab initio theory either to fit the potential parameters or to validate the performance of the\nDensity Functional Theory approaches that have been considered. If the application of the CCSD(T)\napproach is computationally too demanding, the the domain-based pair natural orbital coupled-\ncluster approach DLPNO-CCSD(T) could be a feasible option (see, e.g., DOI: 10.1063/1.4773581).\nA practical option could also be the application of the coupled Möller Plesset second-order\nperturbation theory method (see, e.g., DOI: 10.1063/1.2905808, and DOI: 10.1021/ct9005882).\nAuthors response\nThe discussion in the Supplementary Information refers to charge transfer in the dicationic case (a\nsecond ionization is performed to trigger Coulomb explosion and record the charged fragments).\nThe purpose of this calculation was to show that in the dicationic complex, one charge localized on\nthe benzene molecule furthest from Li+ and the second on the Li+Bz complex. The resulting\nCoulomb explosion would then yield Li+Bz and Bz+ ions, which is what we observe. However, we\nagree that we did not discuss at length the case of the cationic case (singly-charge), which is\nperhaps even more important as it is the one where the reaction takes place. For cation-π complexes,\nDFT turns out to capture the interaction rather well, as discussed in earlier papers of such\ncompounds [Ref. 46-47 of manuscript + ref. 5 of SI], and we did not consider coupled cluster\napproaches to such a problem. However, to support our statement, we have performed additional\nquantum chemistry calculations, using MP2 instead of DFT, but keeping the basis set as 6-\n311++G(d,p). The results, now included in Supplementary Fig. 2, are very close to the DFT data\nused to parametrize the potential, at least within the natural error associated with the simple\nfunctional form used for this potential. The resulting data are also shown below, with the\nelectrostatic contribution removed from the potential energy depicted with dashed lines:\nConcerning the specific charge transfer problem, also in relation to the potential itself, there is\nessentially no charge transfer between Li+ and the benzene monomer upon complex formation. The\nfigure below gives the variations of the effective charge on the lithium atom along the Z coordinate,\nas obtained from a natural bond orbital population analysis:\nThe residual electronic fraction transferred from the Li+ to the benzene molecule thus amounts to\nless than 0.05 electron at the equilibrium position, which is well accounted for by polarization\nforces.\nFinally, and while we agree with the reviewer that more sophisticated forms for the Li+-benzene\ninteraction models could always be designed (we added a reference to Krupka, Krezminska and de\nLara-Castells), they would likely not change the mechanisms observed in our numerical\nexperiments, nor the quantitative reaction times or diffusion velocities – but they would make the\nsimulations quite more expensive.\nReviewer\n3) Diffusion dynamics and friction.\nIn page 7 of the main manuscript (line 25), it can be read ‘Our simulations show that the linear motion results from the combined effect of Li+ being accelerated by the attractive interaction with the He atoms and Bz2 and decelerated by friction from the droplet’. This is a very engaging observation that can be connected with findings in the previous research on the spatial quenching of a bimolecular collision due the reaction barrier stemming from the extrusion of the helium droplet upon approach of the two reactants (DOI:10.1039/C6CP06858H). As a future prospect, it would be instructive to use the same approach as a complement to the ring-polymer molecular dynamics simulations since it accounts for the superfluid nature of the quantum fluid through the application of helium density functional theory.\nAuthors response\nWe agree with the reviewer that the He-(TD)DFT approach would be a natural step beyond the\npresent work. At the moment, it is unclear whether this approach can treat droplets of approximately\n4000 atoms, over simulation times of several tens of picosecond and with a minimum account of\nstatistics, but it is definitely an effort worth considering. In the revision, we have added a sentence\nin the concluding sections to motivate such an effort, together with the appropriate reference.\nReviewer\nMinor Comment. Please include the coordinates of the most relevant complexes at the lowest-\nenergy configurations for the sake of reproducibility.\nAuthors response\nWe now provide the equilibrium geometries of the Li+-benzene and Li+-benzene dimer (side and\nsandwich forms) cation-π complexes, as obtained from our DFT optimizations, in the\nSupplementary Information."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have answer the reviewers properly and changed the manuscript accordingly.\nI propose to accept the manuscript for publication.", "reviewer_2": "The authors carried out significant modifications in line with my comments, having provided satisfactory answers. I recommend its acceptance for publication in Nature Communications."}, "author_response": ""}]

Decision letter