The transition between the gas-, supercritical-, and liquid-phase behavior is a fascinating topic, which still lacks molecular-level understanding. Recent ultrafast two-dimensional infrared spectroscopy experiments suggested that the vibrational spectroscopy of N2O embedded in xenon and SF6 as solvents provides an avenue to characterize the transitions between different phases as the concentration (or density) of the solvent increases. The present work demonstrates that classical molecular dynamics (MD) simulations together with accurate interaction potentials allows us to (semi-)quantitatively describe the transition in rotational vibrational infrared spectra from the P-/R-branch line shape for the stretch vibrations of N2O at low solvent densities to the Q-branch-like line shapes at high densities. The results are interpreted within the classical theory of rigid-body rotation in more/less constraining environments at high/low solvent densities or based on phenomenological models for the orientational relaxation of rotational motion. It is concluded that classical MD simulations provide a powerful approach to characterize and interpret the ultrafast motion of solutes in low to high density solvents at a molecular level.
INTRODUCTION
Solvent–solute interactions and the coupled dynamics between solute molecules embedded in a solvent are central for understanding processes ranging from rotational and vibrational energy relaxation to chemical reactivity in solution.1,2 Vibrational spectroscopy is a particularly suitable technique for following the structural dynamics of “hot” solutes in their electronic ground state interacting with a solvent environment. Experimentally, 1D- and 2D-infrared (IR) spectroscopies provide measures of the strength and time scales of solvent–solute interactions that couple to the resonant rovibrational excitation.3 These effects can also be probed directly by molecular dynamics (MD) simulations.4 The comparison of computations and experiments is a sensitive test for the quality of the intermolecular interaction potentials and provides a molecular-level understanding of energy transfer mechanisms. Furthermore, the entire structural dynamics at molecular-level detail is contained in the time-dependent motion of all solution species involved from the MD trajectories and is available for further analysis.
The understanding of solute equilibration in solvents including high-density gases and supercritical fluids (SCFs) is also important from a more practical perspective. For example, controlling the outcome of combustion reactions that are often performed in the supercritical regime requires knowledge of energy transfer rates and mechanisms in high temperature and pressure solutions.5,6 For reactions, it has been recognized that characterizing and eventually tuning the physicochemical properties of the solvent can be as important as determining the best catalyst.6 Hence, gaining molecular-level insight into the properties of solute–solvent dynamics is central not only for understanding fundamental solution interactions but also for industrial processes where solvents such as ionic, supercritical, or eutectic liquids are employed. On the other hand, the possibility to specifically manipulate the dynamical properties in a solvent system has been already successfully exploited in a wide range of applications.7,8
Linear and nonlinear optical and, in particular, infrared (IR) spectroscopy is a powerful means to characterize the structural and dynamical properties of condensed-phase systems. As an example, recent spectroscopic experiments in the terahertz and infrared region combined with the MD simulations provided an atomistically resolved picture of the structural properties of a eutectic mixture.9 More broadly speaking, the structural and dynamical properties of solutions can and have been studied at a molecular level using two-dimensional IR (2DIR) spectroscopy through vibrational energy transfer.10–18 The energy transfer rate for exchange of vibrational energy is expected to follow a sixth-power, distance dependent law,17 similar to Nuclear Overhauser Enhancement Spectroscopy (NOESY) in nuclear magnetic resonance (NMR) spectroscopy,19,20 or Förster energy transfer between electronic chromophores.21 As an example, the presence of cross-peaks in a 2D spectrum has been directly related to the formation of aggregated structures.11 However, interpretation and more in-depth understanding of the solvent structure requires additional information that can be obtained, for example, from MD simulations.
Recent 2DIR experiments on gas and supercritical phase solutions have shown how rotational energy returns to equilibrium following rovibrational excitation.22–24 The dynamics of N2O as the spectroscopic probe surrounded by xenon atoms and SF6 molecules has provided valuable information about the density-dependent change in the solvent structure and energy transfer dynamics as the fluid approaches and passes through the near-critical point density region. The 1D and 2D spectra of the N2O asymmetric stretch (ν3) in Xe and SF6 exhibit a significant dependence on solvent density.23 At low density, corresponding to SF6 and Xe in the gas phase, the FTIR (1D) band shape of the asymmetric stretch vibration is that of gas-phase N2O with clearly resolved P- and R-band structures, whereas at high solvent density of the liquid Xe or SF6 solvent, only a Q-branch-like absorption feature peaked at the pure vibrational transition frequency was observed. Because the solvent density ρ can be changed in a continuous fashion between gas and liquid densities, along a near-critical isotherm (T ∼ 1.01Tc), it is also possible to probe the intermediate, supercritical regime of the solvent. 2DIR spectral shapes as a function of waiting time and solvent density exhibit perfectly anti-correlated features that report on the J–scrambling or rotational energy relaxation rates.22–24
Given the molecular-level detail provided by quantitative MD simulations, the present work focuses on changes in the IR spectroscopy of N2O embedded in SF6 and xenon and their interpretation at a structural level. For this, an accurate kernel-based representation of high-level electronic structure calculations for the spectroscopic reporter (N2O) is combined with state-of-the-art treatment of electrostatic interactions employing the minimally distributed charge model (MDCM),25–27 which captures multipolar interactions. Extensive MD simulations for densities corresponding to those used in the recent 1D and 2D experiments were carried out.
The current work is structured as follows: first, the methods are presented. Next, the quality of the interaction potentials is discussed. Then, the density-dependent FTIR spectra are compared with the experiments, the vibrational frequency fluctuation correlation functions are characterized, and the organization of the solvent is analyzed for xenon and SF6 as the solvents. Finally, the findings are discussed in a broader context.
METHODS
Potential energy surfaces
The intramolecular potential energy surface (PES) of N2O in its electronic ground state (1A′) is provided by a machine-learned representation using the reproducing kernel Hilbert space (RKHS) method.28,29 Reference energies at the coupled-cluster level of theory CCSD(T)-F12/aug-cc-pVQZ including triple excitation perturbation were determined on a grid of Jacobi coordinates (R, r, θ) with r being the N–N separation, R being the distance between the center of mass of the diatom and the oxygen atom, and θ being the angle between the two distance vectors. The grid in r contained 15 points between 1.6 a0 and 3.1 a0, 15 points between 1.8 a0 and 4.75 a0 for R, and a ten-point Gauss–Legendre quadrature between 0° and 90° for θ. All calculations were carried out using the MOLPRO package.30
Intramolecular and intermolecular force field parameters for SF6 are those from the work of Dellis and Samios.31 Intermolecular interactions are based on Lennard-Jones potentials only, and the parameters are optimized such that the MD simulations of pure SF6 reproduce the experimentally observed PVT state points for liquid and gas SF6, as well as the states of liquid–vapor coexistence below and supercritical fluid above the critical temperature Tc, respectively. For xenon, the parameterization for a Lennard-Jones potential from Aziz and Slaman was used that reproduces dilute gas macroscopic properties such as virial coefficient, viscosity, and thermal conductivity over a wide temperature range but not specifically supercritical fluid properties.32 As discussed further below, the critical concentration for xenon as determined from the simulations (see Fig. S1) was found to be 5.19 M compared with the experimentally reported value33,34 of 8.45 M and variations for the critical temperature Tc for xenon compared with the experiments can also be anticipated. In comparison, the critical concentration of M for SF6 from the present work (see below) matches the experimental value of 5.06 M33,34 and is close to 5.12 M (obtained from the reported critical density of 0.74 g/ml) from the parameterization study.31
Electrostatic interactions are computed based on a minimally distributed charge model that correctly describes higher-order multipole moments of a molecule.25–27 For parameterization, a reference electrostatic potential (ESP) of N2O in the linear equilibrium conformation is computed at the CCSD/aug-cc-pVTZ level using the Gaussian program package.35 The optimized MDCM fit reproduces the ESP with a root mean squared error (RMSE) of 0.31 kcal/mol. For SF6 in its octahedral equilibrium conformation, the ESP is computed at the MP2/aug-cc-pVTZ level of theory using the Gaussian program, and the RMSE of the fitted ESP from MDCM is 0.11 kcal/mol. Recently,27 non-iterative polarization has also been included in MDCM, and this is also used here for N2O, SF6, and Xe. The polarizability of linear N2O computed at the CCSD/aug-cc-pVTZ level is 2.85 Å3 (with each atom contributing Å3 per atom), compared with 2.998 Å3 from the experiment.36 For Xe at the CCSD/aug-cc-pVTZ, the computed value of 2.96 Å3 compares with 4.005 Å3 from the experiment.36 For SF6, the experimentally measured polarizability of 4.49 Å3 was used and evenly distributed over the fluorine atoms (0.74 Å3 per F atom).37
The atomic van der Waals (vdW) parameters (ϵi, Rmin,i) of N2O were individually optimized by least-squares fitting using the trust region reflective algorithm38 to best describe the nonbonded interactions with Xe and SF6 from comparing with energies from CCSD/aug-cc-pVTZ calculations for a number of N2O–Xe and N2O–SF6 heterodimer structures. Lorenz–Berthelot combination rules of atomic vdW parameters between atom types i (solute) and j (solvent) ( at an atom–atom distance of Rmin,ij = Rmin,i/2 + Rmin,j/2) were used throughout. For the reference electronic structure calculations, the grid was defined by center of mass distances between N2O and the solvent with range r = [2.0, 10.0] Å, and angles α = [0, 180]° with step size of 30° between the N2O bond axis and the N2O-solvent center of mass direction. Solute–solvent dimer structures with interaction energies lower than 5 kcal/mol above the dimer minimum structure were considered, which led to 102 conformations for N2O–Xe, whereas for N2O–SF6, there were 203 structures. Reference interaction energies for fitting the vdW parameters were determined as follows: the total energy of the N2O–solvent (Xe or SF6) pair with the largest separation was considered to be the zero of energy, and energies for all other dimer structures were referred to this reference. Interaction energies for the equilibrium pair structure were found with −0.84 kcal/mol for N2O–Xe and −1.46 kcal/mol for N2O–SF6. Correcting for the basis set superposition error (BSSE) through counterpoise correction (CPC)39 reduces the interaction energies by up to to −0.36 and −0.85 kcal/mol, respectively. The alternative “chemical Hamiltonian approach”40 was found to yield similar results41 as the CPC, which, however, is not recommended for correlated wave function methods such as CCSD.42,43 In addition, the corrections due to BSSE are of a similar magnitude as the error in fitting the van der Waals parameters. Therefore, it was decided to not correct the interaction energies for BSSE. Then, the vdW parameters for each atom i of N2O were optimized to match the interaction energy predicted by the Chemistry at “Harvard Molecular Mechanics” (CHARMM) program with the reference interaction energies from the electronic structure calculations for the respective dimer conformations. The optimized vdW parameters are given in Table S2.
Molecular dynamics simulations
Molecular dynamics simulations were performed with the CHARMM program package.44 Each system (N2O in Xe and N2O in SF6 at given temperature and solvent concentration) was initially heated and equilibrated for 100 ps each, followed by the 10 ns production simulations in the NVT ensemble using a time step of 1 fs for the leapfrog integration scheme. The N2O/Xe systems were simulated at a temperature of 291.2 K, and for N2O/SF6, the temperature was 321.9 K, both of which are slightly above the experimental critical temperatures for condensation of xenon and SF6, respectively [Tc(Xe) = 289.74 K, Tc(SF6) = 318.76 K].23,33,34 A Langevin thermostat (coupling 0.1 ps−1) was used to maintain the temperature constant but was applied only to the solvent (Xe and SF6) atoms. Positions and velocities of snapshots of the simulations were stored every 1 fs for analysis. As the intermolecular vibrational energy transfer is slow,24 the structure of N2O was optimized, and new velocities from a Boltzmann distribution at the simulation temperature were assigned to N2O after the heating step. This ensures that the kinetic energies along the asymmetric, symmetric, and bending modes match the thermal energy with respect to the target simulation temperature.
The different simulation systems were prepared according to the conditions used in the experiments.22–24 Table S1 summarizes the concentration c(N2O) of N2O, molar volumes Vm, and critical density ratio ρ* = ρ/ρc. The experimentally determined critical densities are ρc = 1.11 g/ml for xenon and ρc = 0.74 g/ml for SF6 from which critical concentrations of 8.45 and 5.06 M for xenon and SF6 are obtained, respectively.33,34 In all setups, the simulation box contains one N2O molecule and 600 Xe atoms or 343 SF6 molecules, which corresponds to similar simulation box volumes for similar relative density ratios of the two solvents. In the original parameterization study, a simulation box containing 343 SF6 molecules was used to fit temperature–pressure properties.31
In the MD simulations for N2O in SF6, electrostatic and polarization interactions were only computed between the N2O solute and SF6 solvent. Electrostatic and polarization contributions to the SF6 solvent–solvent interactions were neglected. Such a procedure ensures that the pure (liquid, gas) properties of the solvent are unaltered. All force field parameters are listed in Table S2 in the supplementary material.
Analysis
The response of the solute on the solvent structure and dynamics was evaluated by determining the frequencies of the quenched normal modes (QNMs) of N2O for frames every 5 fs of the simulation. For the QNM, a steepest descent geometry optimization for N2O, within a frozen solvent conformation for either 100 steps or until a gradient root mean square of kcal/mol/Å was reached, was carried out. The 9 × 9 mass-weighted Hessian matrix of the N2O solute in the solvent with regard to the solute atom displacements in Cartesian coordinates was determined and diagonalized to obtain time series of all nine normal modes including the effect of solvent.46,47 Instantaneous normal mode (INM) analyses were also applied on the solute by diagonalizing its mass-weighted Hessian matrix without prior geometry optimization on N2O to obtain insights on the impact of the solvent structure on the translational and rotational modes of the N2O solute.46
The normalized vibrational frequency–frequency correlation function (FFCF) / was computed for the time series of the asymmetric stretch frequency νas gathered from the QNM analyses. The amplitude A, lifetimes τi, and offset Δ of a bi-exponential function were optimized to fit the normalized FFCF for t ∈ [0.1, 2.0] ps.
RESULTS
Validation of the interaction potentials
First, the quality of the intramolecular PES for N2O is discussed, followed by a description of the van der Waals parameters for the N2O solute fit to the ab initio reference calculations.
The RKHS model provides a full-dimensional, intramolecular PES for N2O, which was originally developed for investigating the N + NO collision reaction dynamics.29 The Pearson coefficient R2 of the RKHS representation and the full set of reference values is 0.999 83, and the root mean squared error (RMSE) between the RKHS and reference energies up to 20 kcal/mol above the equilibrium structure (78 reference energies) is 0.13 kcal/mol. To establish the spectroscopic quality of the PES, the evaluation by the discrete variable representation (DVR) method using the DVR3D48 package yields a fundamental asymmetric stretch frequency of N2O of 2229 cm−1 compared with 2224 cm−1 from the experiments in the gas phase.49–51 The bending and symmetric stretch frequencies obtained from the DVR3D calculations are 598 and 1291 cm−1, respectively, and the overtone of the bending mode lies at 1184 cm−1. Experimentally, the bending and symmetric stretches are found at 589 and 1285 cm−1, respectively, while the overtone of the bending frequency is at 1168 cm−1.49–51
Using MDCMs for the electrostatics of N2O in the two solvent environments (Xe and SF6) requires a readjustment of the solute vdW parameters. To keep the solvent–solvent interaction unchanged, only the vdW parameters of N2O were optimized with respect to interaction energies from reference electronic structure calculations.
Figure 1 reports the correlation between the reference and model interaction energies with optimized vdW parameters for N2O for the N2O–Xe [panel (A)] and N2O–SF6 [panel (B)] complexes. The reference interaction energies ΔECCSD = Epair − Esolute − Esolvent are the differences between the respective total ab initio energy of the pair and the energies of the solute Esolute and solvent Esolvent fragment in a fixed minimum energy conformation. The force field energies ΔEFF = Eelec + ELJ include electrostatic and polarization contributions and the Lennard-Jones potential between the solute and solvent at the same conformation as in the reference dataset. The energies sample attractive (negative energy) and repulsive (positive energy) ranges of the N2O–solvent energies, and the comparison is restricted to energies within 5 kcal/mol of the separated molecules, which is the zero of energy. For N2O–Xe, the RMSE is 0.25 kcal/mol, and for N2O–SF6, it is 0.29 kcal/mol. The most stable structure is stabilized by −1.41 kcal/mol (−1.37 kcal/mol from the fitted energy function) for N2O–SF6 and −0.80 kcal/mol (−0.75 kcal/mol) for N2O–Xe.
Correlation between the CCSD/aug-cc-pVTZ reference interaction energies for the (A) N2O–Xe pair for 102 different conformations and (B) N2O–SF6 pair for 203 different conformations with interaction energies lower than 5 kcal/mol above the minimum.
Correlation between the CCSD/aug-cc-pVTZ reference interaction energies for the (A) N2O–Xe pair for 102 different conformations and (B) N2O–SF6 pair for 203 different conformations with interaction energies lower than 5 kcal/mol above the minimum.
The critical concentration at which transition to a supercritical fluid occurs is another relevant property for the present work. Earlier work showed that this transition can be correlated with a pronounced increase in the local solvent reorganization lifetime τρ.52,53 The quantity τρ is the integral over the local-density autocorrelation function and characterizes the time required for the local environment around a reference particle, e.g., the solute N2O in the present case, to change substantially. Hence, τρ can also be considered a local-density reorganization time.53
Figures S1 and S2 report the local solvent reorganization lifetimes τρ for pure xenon and SF6, respectively, from MD simulations. The solvent environments are defined by cutoff radii, which correspond approximately to the first and second minima of the solvent–solvent RDF; see Fig. S4. For SF6, the force field was parameterized31 to reproduce the experimentally measured critical density with corresponding concentration ccrit(SF6) = 5.06 M at the critical temperature.33,34 The present simulations yield a peak for τρ at c(SF6) = 5.02M, in close agreement with the experimental critical concentration at the critical temperature and the concentration with the local peak in τ2 in Fig. 4(D). For xenon, however, the parameterization of the PES32 did not include phase transition properties to the supercritical regime. Figure S1 shows that the maximum in τρ occurs at c(Xe) = 5.19 M. This compares with a critical concentration of 8.45 M at Tc from the experiment.33,34
Infrared spectroscopy
The computed IR spectra for the N2O asymmetric (νas) stretch in xenon and SF6 solution as a function of solvent density for near-critical isotherms allow for comparison between the experimental and simulation results. The present work focuses mainly on the change of the IR line shape at different solvent concentrations especially the P-, Q-, and R-branch structures; see Figs. 2 and 3. The P- and R-branches are the IR spectral features at lower and higher wavenumber from the pure vibrational transition frequency for the excitation of mode ν. The νas, νs, and νδ band shapes arise due to conservation of angular momentum during a vibrational excitation upon photon absorption.49 The Q-branch is the absorption band feature at the vibrational transition frequency. In addition to the asymmetric stretch νas ( cm−1, symmetry A1/Σ+), the symmetric stretch νs ( cm−1, A1/Σ+) and the bending vibration νδ ( cm−1, E1/Π) fundamentals from the MD simulations are reported and analyzed as a function of solvent density.
IR spectra of N2O in xenon at different solvent concentrations for frequency ranges of (A) 580–650 cm−1, (B) 1180–1350 cm−1, and (C) 2120–2310 cm−1 at 291.2 K. To make all spectra comparable, the amplitudes are scaled by the factor indicated at the center left for each range and density. In panel (C), the line shape frequency bands are shifted in frequency ω to maximize the overlap with the experimental IR signal (dashed black lines, at 291 K for gas, SCF, and at 287 K for liquid xenon) at the corresponding density for the N2O asymmetric stretch vibration. The frequency shift is also given in the bottom left corner, and an average shift is applied for densities without experimental reference spectra.
IR spectra of N2O in xenon at different solvent concentrations for frequency ranges of (A) 580–650 cm−1, (B) 1180–1350 cm−1, and (C) 2120–2310 cm−1 at 291.2 K. To make all spectra comparable, the amplitudes are scaled by the factor indicated at the center left for each range and density. In panel (C), the line shape frequency bands are shifted in frequency ω to maximize the overlap with the experimental IR signal (dashed black lines, at 291 K for gas, SCF, and at 287 K for liquid xenon) at the corresponding density for the N2O asymmetric stretch vibration. The frequency shift is also given in the bottom left corner, and an average shift is applied for densities without experimental reference spectra.
IR spectra of N2O in SF6 at different solvent concentrations frequency ranges of (A) 580–650 cm−1, (B) 1180–1350 cm−1, and (C) 2120–2310 cm−1 at 321.9 K. Further information is provided in the caption of Fig. 2. The experimental IR signal (at 322 K for gas, SCF, and at 293 K for liquid SF6) at the corresponding density for the N2O asymmetric stretch vibration is shown by the dashed black line.
IR spectra of N2O in SF6 at different solvent concentrations frequency ranges of (A) 580–650 cm−1, (B) 1180–1350 cm−1, and (C) 2120–2310 cm−1 at 321.9 K. Further information is provided in the caption of Fig. 2. The experimental IR signal (at 322 K for gas, SCF, and at 293 K for liquid SF6) at the corresponding density for the N2O asymmetric stretch vibration is shown by the dashed black line.
From a quantum mechanical perspective, the rotational structure of an IR-active vibrational mode for a linear molecule (such as N2O) in the gas phase leads to P- and R-branches. Selection rules dictate the change of the rotational and vibrational quantum states, j and ν, satisfying Δν = ±1 and Δj = ±1 for vibrational modes of A1/Σ+ symmetry and Δj = 0, ±1 for vibrational modes of E1/Π symmetry, respectively. For N2O, the asymmetric νas and symmetric stretch νs are vibrational modes with a parallel vibrational transition dipole moment to the bond axis and of A1/Σ+ representation and the bending mode νδ with a perpendicular vibrational transition dipole moment of E1/Π representation. These rotational selection rules are only strictly valid in the absence of perturbations (exact free rotor) and break down with increasing deviation from a free rotor model, which, for example, can be due to embedding the rotor into a solvent of different density, which is a means to tune the strength of the perturbation. As a consequence of the perturbation, a Q-branch-like spectral feature emerges for parallel bands and becomes dominant with a band maximum at the wavenumber of the corresponding pure vibrational transition energy at sufficiently high solvent density.
For N2O in xenon, the IR band structure of νas in Fig. 2(C) ranges from resolved P- and R-branches in gaseous xenon up to a single near-Lorentzian-shaped band in liquid xenon in these classical MD simulations. From the simulations, at xenon concentrations higher than 5.19 M or relative density of ρ* = ρ/ρc = 0.62, the xenon solvent is a supercritical fluid. In the simulated absorption spectra at this density and above, the νas P- and R-band structure is not resolved, and a centrally peaked band at the pure vibrational frequency dominates the band shape. The black dashed lines in Fig. 2(C) are the experimentally determined spectra at the given solvent concentration.22–24
The agreement between experimental and computed N2O in Xe absorption band shapes is good and captures the dramatic gas phase-to-condensed phase line shape change as a function of solvent density. However, the frequency position of the νas band needs to be blue-shifted by a small frequency shift ω to achieve the best overlap with the experimental IR line shape. The density-dependent shifts are indicated in the panel and range from +6 to +22 cm−1 with no discernible trend. The shift originates from different effects, including insufficient sampling of the amount and distribution of internal energy within the N2O solutes vibrational degrees of freedom, remaining small inaccuracies in the intermolecular interactions, neglecting many-body contributions and slightly underestimating the anharmonicity in the PES along the relevant coordinates.
Figure 2(B) shows the corresponding band shape for νs of N2O in Xe around 1305 cm−1 in comparison to experimentally measured 1291 cm−1 for N2O in the gas phase. Similar to νas, it also displays resolved P- and R-branches at low solvent concentrations and changes into a Q-branch-like dominated structure in supercritical and liquid xenon. The IR band of the N2O bending mode νδ appears at around 615 cm−1 shown in Fig. 2(A), compared with 589 cm−1 from the experiment in the gas phase.49 At low solvent concentrations, the band structure of νδ from the MD simulations is a sharp Q-branch with weaker P- and R-branch side bands consistent with the quantum mechanical selection rules. The intensity of these bands is no longer evident relative to the central bending feature at higher solvent concentration. The first overtone of the νδ vibration is also detected between 1220 and 1230 cm−1 in Fig. 2(B) but with low intensity. The computed bending overtone (2νb) exhibits the same P- and R-branches at low density and the change to a single peaked band at high solvent concentration as observed for the νs and νas band structures, see Figs. S5 and S6, in agreement with the experimental results.54
The IR band structure of νas of N2O in SF6 is shown in Fig. 3(C). At the lowest solvent concentrations 0.20 and 0.51 M (ρ* = {0.04, 0.10}), the simulated distinct P- and R-branch structure is evident but gradually disappears for [SF6] > 1.52 M (ρ* > 0.30) for corresponding experimental state points. As the solvent concentration of SF6 increases, the band shape changes into a single-peaked band in agreement with the experimental IR spectra for [SF6] = 7.65 and 9.47 M (ρ* = {1.51, 1.87}). The computed IR spectra for 0.81 ≤ [SF6] ≤ 5.93 M (0.16 ≤ ρ* ≤ 1.17) do not exhibit the double-peak structure or are narrower than the experimental absorption spectra. The frequency position of the νas band is also adjusted by a small frequency blue shift ω within the range of +8 to +27 cm−1 to maximize the overlap with the experimental IR line shape.
The line shapes for the νδ and νs vibrational modes are shown in Figs. 3(A) and 3(B) as a function of SF6 concentrations, respectively. The νs mode changes from P-/R-branches at low solvent concentrations into a single featured structure in supercritical and liquid SF6. The calculated IR band of νδ around 615 cm−1 exhibits a sharp Q-branch with resolvable weak P- and R-branch satellites only at low SF6 concentration. The first overtone of the perpendicularly polarized νδ mode is again detected between 1220 and 1230 cm−1 with low intensity and a band shape as observed for the parallel polarized fundamental νs and νas band structures.
Vibrational frequency–frequency correlation function
The vibrational frequency fluctuation correlation function (FFCF), which probes the coupling of the solute vibrational modes to the solvent environment, can be determined from 2DIR experiments. Of particular interest are the time scales of the lifetimes τi and amplitudes Ai with which the FFCF decays, which are shown in Fig. S7. In the condensed phase, the FFCF is also related to the change of the center-line slope at different 2DIR waiting times.3
Figures 4(A) and 4(B) report the FFCFs for the asymmetric stretch νas mode of N2O in xenon and SF6 from QNM, respectively.1,55 The fit of the FFCFs using a bi-exponential decay yields two times: a rapid inertial component (τ1 ∼ 0.05 ps) and a longer spectral diffusion time scale (τ2 ∼ 0.2 … 0.5 ps). It is not possible to directly compare the computed vibrational FFCFs from INM with the experimentally measured 2DIR spectra as they are overwhelmingly dominated by the contribution of rotational energy relaxation dynamics.22–24 Nevertheless, these time scales can be compared with measured and computed time scales [τ1, τ2, τ3] for CN− ([0.2, 2.9, n.a.] and [0.04, 0.87, 9.2] ps) and for ([n.a., n.a., ∼1.2] and [0.04, 0.23, 1.2] ps) in H2O.55–58 Here, “n.a.” refers to the time scales that were not determined or available from the experiments but clearly appeared in the analysis of the simulations. A similar behavior of the vibrational FFCF as the one found in the present work—a first rapid relaxation time on the 0.05–0.1 ps time scale, followed by a slower time scale of 0.5–1.0 ps—was reported in an earlier MD simulation study for liquid HCl.59
FFCF of the asymmetric stretch frequencies νas of N2O from the QNM analysis in (A) xenon and (C) SF6 solvent at different concentrations. Asymptotically, all FFCFs decay close to zero (Δ < 0.01). An offset of 0.01 was added to the normalized FFCF to avoid overlap, and the lifetimes τ from the fitted bi-exponential function are shown for the (B) xenon and (D) SF6 solvent depending on solvent concentration. The critical concentration ccrit determined from experiments and its estimation from simulations at Tc of xenon and SF6 are marked by the vertical dashed and dotted lines, respectively. It should be noted that the Xe–Xe interactions were not optimized to reproduce the experimentally known Tc for pure Xe, whereas the model for SF6 does.
FFCF of the asymmetric stretch frequencies νas of N2O from the QNM analysis in (A) xenon and (C) SF6 solvent at different concentrations. Asymptotically, all FFCFs decay close to zero (Δ < 0.01). An offset of 0.01 was added to the normalized FFCF to avoid overlap, and the lifetimes τ from the fitted bi-exponential function are shown for the (B) xenon and (D) SF6 solvent depending on solvent concentration. The critical concentration ccrit determined from experiments and its estimation from simulations at Tc of xenon and SF6 are marked by the vertical dashed and dotted lines, respectively. It should be noted that the Xe–Xe interactions were not optimized to reproduce the experimentally known Tc for pure Xe, whereas the model for SF6 does.
It is important to note that experimentally decay times on the several 10 fs time scale are difficult to determine with confidence from vibrational FFCFs. Therefore, only those longer than that are considered in the following. It is noted that for both ions (CN− and ) in water, mentioned above, the longer decay times are 1 ps or longer compared with ps for N2O in SF6. This is consistent with the fact that ion–water interactions are considerably stronger than N2O–SF6 interactions. Furthermore, the N2O–Xe interaction is weakest among all those considered here, which leads to the rather short τ2 value for this system, even relative to SF6. Given that for the present system the intermolecular interactions are weak and the decay times are rapid, it is anticipated that only τ2 for N2O in SF6 would be amenable to the 2DIR experiments.
Figures 4(B) and 4(D) show the dependence of the short (τ1) and long (τ2) time scales as a function of the xenon and SF6 solvents concentrations, respectively. In both solvents, the longer lifetime slows down with increasing concentration, and a peak in the lifetime at 5.19 M in Xe and 5.02 M in SF6 is observed. The pronounced increase in τ2 is likely to be related to approaching the critical point as the local peak in τ2 at a SF6 solvent concentration of 5.02 M matches the critical concentration for pure SF6. Similarly, for N2O in Xe, the peak at 5.19 M is consistent with a lengthening of the τ2 timescale in this solution at the critical concentration for pure xenon which was obtained from the local solvent reorganization lifetimes τρ; see Fig. S1. Thus, both solvents exhibit critical slowing (lengthening of τ2) at the critical concentration of the respective pure solvent.
It is also noted that the slope for τ2(c) for N2O in SF6 is considerably steeper than for Xe as the solvent. This is most likely also due to the increased solvent–solute interaction strength between N2O and SF6 compared with Xe. The results for τ2 indicate that interesting dynamical effects, such as critical slowing, can be expected to develop around the critical point of the solvent. It is worthwhile to note that at the critical concentration c(Xe) = 5.19 M for the transition to a SCF, the spectroscopy of the asymmetric stretch changes from P-/R-branches to Q-branch-like (Fig. 2), and the decay time τ2 from the FFCF of the asymmetric stretch [see Fig. 4(B)] features a pronounced maximum. The fact that for xenon the critical density from the simulations is underestimated (5.19 vs 8.45 M from experiments), but the concentration-dependence of the spectroscopy agrees with the experiment suggests that orientational relaxation and inhomogeneous effects play the dominant role for the density dependence of the dipole correlation function.
A direct comparison of the vibrational FFCFs from INM with the prior experimental FFCFs is not meaningful at present because of the overwhelming contribution to the reported FFCF due to the rotational energy relaxation dynamics. The 2DIR spectral features corresponding to the vibrational FFCF overlap the rotational energy relaxation but may be accessible in the future experimental studies that employ higher spectral resolution detection than in Refs. 22–24. Furthermore, unlike typical 2DIR spectra of condensed phase systems, the available experimental FFCFs report only on the quasi-free rotor members of the ensemble contrary to the computed vibrational FFCFs.
Radial distribution functions and size of the solvent shells
A structural characterization of the solvent environment is afforded by considering the radial distribution functions g(r) between the center of mass of N2O and the central sulfur atom and the Xe atom, which are shown in Figs. S3A and S3C, respectively. They show a wide first peak corresponding to the first solvation shell around the N2O solute within Å [panel (A)] for the xenon and Å [panel (C)] for the SF6 solvent, followed by considerably weaker features around 8 Å and beyond depending on the density.
A more concise comparison of how the solvent environment depends on solvent density can be obtained by considering the average coordination number ⟨N(r′)⟩ of the Xe and SF6 solvents within the first solvation shell [Figs. 5(A) and 5(B)] that are computed from the respective g(r), see Fig. S3. At the respective lowest sampled solvent concentration, ⟨N(r′)⟩ yields an average of 0.31 Xe atoms [panel (A) at 0.34 M] and 0.29 SF6 molecules [panel (B) at 0.20 M] within the first solvation shell. The coordination number rises monotonically with increasing solvent concentration but shows a decrease in the slope within the SCF regime as the solvation shell fills. It is interesting to note that both solvents exhibit two slopes for N(r) depending on c: a first, steeper one for the gas-phase-like solvent into the SCF regime, and a second, flatter one, leading into the liquid-like regime. The slopes for the gas-phase-like regime is steeper (1.40 M−1) for SF6 than for Xe (0.94 M−1), which may be related to the solvent–solute interaction strength.
Average coordination number ⟨N(r′)⟩ of (A) the Xe (r′ = 6.5 Å) and (B) SF6 solvent compounds (r′ = 7.5 Å) within the first solvation shell around N2O at cutoff distance r′. The red and blue lines are a linear fit of the first and last three data points, respectively. The vertical dotted lines mark the estimation of the critical concentration in the simulations at Tc of (A) xenon and (b) SF6, and the vertical dashed lines mark the approximate concentrations of the solvent phase transition.
Average coordination number ⟨N(r′)⟩ of (A) the Xe (r′ = 6.5 Å) and (B) SF6 solvent compounds (r′ = 7.5 Å) within the first solvation shell around N2O at cutoff distance r′. The red and blue lines are a linear fit of the first and last three data points, respectively. The vertical dotted lines mark the estimation of the critical concentration in the simulations at Tc of (A) xenon and (b) SF6, and the vertical dashed lines mark the approximate concentrations of the solvent phase transition.
It is also of potential interest to consider the present results in light of the independent binary collision (IBC) model60–62 although the model has also been criticized to be potentially oversimplified when applied to liquids.63,64 In the simplest implementation of the IBC model—where the contact distance is associated with the position of the first maximum of the solvent–solute radial distribution function, which only insignificantly changes with density—the rate for a given process, e.g., vibrational relaxation, depends linearly on (a) the collision rate and (b) a density independent probability that a given collision is effective; see Figs. S3A and S3C. As a consequence, rates adequately described by an IBC model depend linearly on density, which itself is proportional to the occupancy of the first solvation shell. This is what is found in the present work for low densities; see Fig. 5. However, toward higher densities, another linear dependence develops with lower slope for both solvents. The breakdown of the IBC model occurs at a slightly higher solvent concentration for Xe than for SF6 (c(Xe) ∼ 3.5 M vs c(SF6) ∼ 2.9 M, see Fig. 5) compared with values of M and 4.0 M from 2DIR experimentally determined rotational energy relaxation, respectively.24 Just as found for comparison of the simulated and experimental absorption spectra, further refinement of the N2O–SF6 interactions will provide even better quantitative agreement with the experiment, for example, within a PES morphing approach.65
DISCUSSION
Classical MD simulations have been previously used to determine and analyze rotational vibrational spectra of simple solutes, including CO or HCl in a solvent (argon)66,67 or for water.68 These calculations demonstrated that classical MD simulations are capable of realistically capturing P- and R-branch structures in moderate to high density gases and even in liquid water.68 Along similar lines, the present MD simulations of N2O in xenon or SF6 as the solvent reproduce well the experimentally observed splitting in P-, Q-, and R-branches and the overall line shapes in the IR spectra of N2O depending on the solvent concentration (see Figs. 2 and 3). The agreement in the computed line shapes of the FTIR νas spectrum of N2O in xenon does match with the experimental spectra as a function of solvent concentration. The separation between the maxima of the P- and R-branches is ∼[27, 25, 29, 22] cm−1 at [0.34, 0.84, 1.26, 3.10] M from the simulations, compared with ∼[25, 25, 22] cm−1 at [0.84, 1.26, 3.10] M from the experiments. The IR signals for the νas vibration of N2O in SF6 show a larger contribution of the Q-branch-like feature at lower ρ* than that observed in the experimental spectra. For simulations in both solvents, the computed spectra are shifted to the red compared with the results from experiments for νas.
The computed IR spectra from the classical MD simulations of N2O/Xe in Fig. 2 agree with the relative intensity and band width of the νas vibration signal at the corresponding solvent concentration. The IR spectra of N2O in SF6 in Fig. 3 deviate more from the corresponding experimental line shapes by overestimating solvent–solute interactions, leading to a somewhat stronger contribution of the Q-branch-like absorption character at lower solvent concentration. The experimental νas signal at the lowest measured solvent density of 0.81 M overlaps better with the computed signal at lower solvent densities of 0.51 and 0.20 M. However, the line shape in highly dense supercritical and liquid SF6 agrees well with the experiment where the Q-band-like feature dominates the spectral line shape and resolvable P- and R-branch contributions are not evident. At the same absolute density, the asymmetric stretch of N2O is indicative of a more liquid-like character of SF6 as the solvent compared with xenon; see Figs. 2 and 3(C).
Splitting of the IR spectrum into P- and R-branches correlates with free rotor properties of the N2O solute molecule. Based on this, the results in Fig. 3 indicate an overestimation of the solute–solvent interaction in SF6 because a merged line shape for νas arises between 0.81 and 1.52 M from the computations, whereas experimentally, this occurs for concentrations between 3.39 and 4.36 M. Consequently, the rotation of N2O in the simulations starts to become hindered at lower solvent densities compared to the experiment. This observation correlates with the long-range and stronger electrostatic interaction between N2O and polar SF6 rather than xenon as the solvent.
As a comparison, the electrostatic contribution to the interaction energy of a N2O/SF6 pair in its equilibrium conformation with −1.42 kcal/mol is considerably stronger then the polarization contribution in an N2O/Xe pair with −0.84 kcal/mol. As the electrostatic MDCM and vdW parameter fit are only applied to single molecules or molecule pairs, respectively, the energy function might insufficiently capture many-body interactions between the N2O solute and multiple SF6 solvent molecules.
An overtone of the νδ vibration (615 cm−1) of N2O is found at about twice the frequency between 1220 and 1230 cm−1 in both solvents. The splitting into P- and R-branches at low solvent densities originates from excitation of simultaneously two bending mode quanta (2νδ) and the change of the rotational state by ΔJ = ±1.69,70 Consequently, as the splitting of the 2νδ IR band depends on the free rotor properties such as for the νas and νs IR band structure, the band shape changes toward a single Q-branch peak at higher solvent concentrations. In other words, the rotational “selection rules” followed by the bending overtone are the same as for the asymmetric stretch vibration, both of which have the same symmetry. This quantum mechanical result is seen in the present classical MD simulations.
The perturbation of the rotational modes by the solvent is visualized by the INM in Figs. S8A and S8B for N2O in SF6 and xenon at different densities, respectively. The INM histogram for N2O in SF6 shows two peaks around the zero frequency line for low positive and imaginary frequencies (positive and negative second derivative of the potential projected along normal mode displacement, respectively). The emerging shoulder(s) in the INM histogram toward higher solvent densities (Fig. S8) are due to low-frequency modes involving solute–solvent interactions. This leads to larger deviations from a free rotor movement of the N2O, akin to motion in a density-dependent rotationally constraining solvent potential.23 Visual inspection of the normal modes identifies these as librational modes or frustrated rotations. The INM histogram for N2O in xenon in Fig. S8B shows broader peaks and lower shoulders for the gaseous solvent in comparison with the INM distribution of N2O in SF6 in Fig. S8A. The higher mode density around low frequencies for the INMs of N2O in xenon relative to SF6 also indicates lower perturbation of the free rotor character in gaseous and supercritical xenon solvent than SF6.
The change of the IR spectra for linear molecules in gaseous, supercritical up to liquid solvents can be compared with the experimental and theoretical work on hydrogen cyanide in an electric field.71,72 With increasing field strength, the molecules evolve from a free rotor to those trapped in pendular states, which is one way to control the population of rotational states in molecular beam experiments. The electric field lifts the degeneracy of the quantum number mj (the projection of the angular momentum on the electric field direction) for each rotational state j and leads to a manifold of different transitions that correspond to the IR spectra. Pendular states are linear combinations of field-free (j, mj) states covering a range of j–values but sharing the same mj quantum number, i.e., j–mixing occurs.71 With increasing electric field strength, the field-free selection rule Δj = ±1 breaks down. For excitation along dipole transition vectors parallel to the electric field, the P- and R-branches change into a single Lorentzian-shaped Q-branch at large electric field strength.71
The effects found for polar molecules in external electrical fields can be compared with the motion of N2O in a supercritical or liquid solvent. Here, the solvent molecules form a cavity around the solute where the direction of the angular momentum is not energetically degenerate, and N2O behaves like in a pendular state. In analogy to a molecule in an electric field, the P- and R-branches collapse into a single Q-branch-like band structure for vibrational modes with transition dipole moments perpendicular to the angular momentum of the molecule upon breakdown of the usual selection rules.
Figure 6 shows computed IR spectra from the Fourier transform of the dipole–dipole correlation function from the MD simulations of a single N2O molecule with different initial conditions. For all cases (A)–(D), intramolecular energy is distributed along all vibrational normal modes with respect to the thermal energy at 321.9 K. Without any rotational energy assigned, the IR spectra in Fig. 6(A) show no split into P- and R-branches. With rotational energy assigned around the rotational axis parallel to the transition dipole vector, the IR spectra in Fig. 6(B) show separation into P- and R-branches by 28 cm−1 for the νas stretch that correlates for a rotational constant of Bj=16 = 0.43 cm−1. The IR signal of the bending mode primarily consists of the Q-branch with two small P- and R-branch satellites separated by ± 14 cm−1 each from the Q-branch. For rotational energy assigned around a rotational axis perpendicular to both stretch and bending transition dipole moments, all IR signals in Fig. 6(C) split in P- and R-branches as well (Bj=16 = 0.42 cm−1), but no Q-branch is observed. The rotational energy corresponding to the rotational quantum state j = 16 is the state with the highest probability according to the Maxwell–Boltzmann distribution function Pj(T) for N2O at 321.9 K.
Model IR spectra from 100 ps MD simulations of a single N2O molecule with vibrational energy equivalent to T = 321.9 K and either (A) no rotational energy, rotation around axis (B) perpendicular or (C) parallel to the transition dipole moment of the bending vibration of N2O. Panel (D) shows a superposition of N2O IR spectra from MD simulation at different rotational states j scaled by the respective probability according to the Maxwell–Boltzmann distribution Pj at T = 321.9 K. The experimental spectra of N2O in gaseous SF6 (c = 1.26 M) are shown by the dashed line and shifted by −14.5 cm−1 to maximize the overlap with the computed line shape. An illustration of an angle N2O molecule with the principle axis of inertia is shown in the top right corner and the order of the moments of inertia I.
Model IR spectra from 100 ps MD simulations of a single N2O molecule with vibrational energy equivalent to T = 321.9 K and either (A) no rotational energy, rotation around axis (B) perpendicular or (C) parallel to the transition dipole moment of the bending vibration of N2O. Panel (D) shows a superposition of N2O IR spectra from MD simulation at different rotational states j scaled by the respective probability according to the Maxwell–Boltzmann distribution Pj at T = 321.9 K. The experimental spectra of N2O in gaseous SF6 (c = 1.26 M) are shown by the dashed line and shifted by −14.5 cm−1 to maximize the overlap with the computed line shape. An illustration of an angle N2O molecule with the principle axis of inertia is shown in the top right corner and the order of the moments of inertia I.
At lower solvent density, the appearance of P-, Q-, and R-branches in the model IR spectra in Fig. 6(B) can be explained by the mechanical instability of a rotation around the transition dipole vector of the bending mode ey. For bent N2O, the moment of inertia are ordered by Ix > Iy ≫ Iz. According to the tennis racket theorem,73 the rotation around the axis ey with the middle moment of inertia is unstable. Thus, the angle between rotation axis or angular momentum vector and the transition dipole vector of the bending mode varies in time and gives rise to P-, Q-, and R-branch signals.
A superposition of model IR spectra at varying rotational energy states and randomly sampled rotation axis but perpendicular to the N2O bond axis yields a line shape close to the computed spectra of N2O in the gaseous solvent at lower concentrations. The contribution of the single IR spectra at rotational state j to the model spectra in Fig. 6(D) iss scaled by the probability value according to Pj(T = 321.9 K). An average rotational constant B = 0.42 cm−1 is computed from the spectra with rotational states j = [2, 26], which is in perfect agreement with the experimentally determined rotational constant of Bexp = 0.419 cm−1.74 Hence, overall two regimes can be distinguished: for low densities, the rotational motion of the solute is oscillatory and guided by the mechanical instability around the middle rotational axis (tennis racket theorem), whereas at higher densities, the constraining effect of the solvent due to tighter packing leads to a more diffusive rotational motion of the solute.
For additional clarification for how the transition between P-/R-features and Q-branch-like line shapes occurs, two different models are briefly considered. The first one operates directly on the dipole–dipole correlation function of a single N2O molecule, whereas the second one uses a Langevin-type simulation of one N2O molecule in the gas phase. The influence of a solvent can be heuristically included by multiplication of the dipole–dipole correlation function ϕ(t), see Eq. (6), with an exponential damping, i.e., considering ϕ(t) × exp(−t/τ). The damping with characteristic time τ models all sources of the dipole correlation function decay, primarily the solvent density dependence on orientational relaxation due to collisions between N2O and the solvent in the present case. With increasing density of the solvent, for example, τ decreases, which leads to loss of knowledge about its rotational motion and broadening of the computed IR signals; see Fig. 7(A). Consequently, the P-/R-features collapse into a single Q-branch-like peak for relaxation decays 0.1 ≤ τ ≤ 0.2 ps, which are considerably shorter than the rotational periods.
(A) Model IR spectra from a Fourier transform of the dipole–dipole correlation function multiplied by an exponential damping factor with τ, which describes orientational (rotational) relaxation and inhomogeneous effects. The dipole–dipole correlation function is obtained from a 100 ps MD simulation of a single N2O molecule with vibrational energy equivalent to T = 321.9 K and rotation around the axis parallel to the transition dipole moment of the bending vibration of N2O. (B) Model IR spectra from 10 ns Langevin simulations of a single N2O molecule with initial vibrational and rotational energy equivalent to T = 321.9 K. A Langevin thermostat with different friction coefficients γi = {10, 1, 0.1, 0.01, 10−4} ps−1 (from top to bottom) was applied to the dynamics to a model energy exchange between N2O and the solvent.
(A) Model IR spectra from a Fourier transform of the dipole–dipole correlation function multiplied by an exponential damping factor with τ, which describes orientational (rotational) relaxation and inhomogeneous effects. The dipole–dipole correlation function is obtained from a 100 ps MD simulation of a single N2O molecule with vibrational energy equivalent to T = 321.9 K and rotation around the axis parallel to the transition dipole moment of the bending vibration of N2O. (B) Model IR spectra from 10 ns Langevin simulations of a single N2O molecule with initial vibrational and rotational energy equivalent to T = 321.9 K. A Langevin thermostat with different friction coefficients γi = {10, 1, 0.1, 0.01, 10−4} ps−1 (from top to bottom) was applied to the dynamics to a model energy exchange between N2O and the solvent.
Alternatively, interaction with a solvent can be modeled from the MD simulations with a stochastic thermostat, such as in Langevin dynamics, which applies random forces on the atoms to simulate orientational interactions with the heat bath. For this, 10 ns Langevin simulations with varying coupling strengths γ were run for a single N2O molecule in the gas phase using CHARMM, and the dipole–dipole correlation function was determined. Figure 7(B) shows the computed IR spectra. At low coupling strength γi = 10−4 ps−1, the IR spectra still yield a split of the asymmetric stretch into P- and R-branches (yellow trace). With increasing friction γi = 0.01 ps−1, still multiple peaks are visible in the asymmetric stretch absorption line shape, but the spectral Q-branch-like region is getting filled in (pink trace). At yet higher γi ≥ 0.1 ps−1 (green, red, and blue traces), the P-/R-features wash out entirely.
Finally, it is of some interest to discuss the results on the spectroscopy (Figs. 2 and 3) and the solvent structure (Fig. 5) from a broader perspective. The interaction between N2O and Xe is weaker by about 50% than that between N2O and SF6. The type of interaction also differs: pure van der Waals (in the present treatment) for Xe vs van der Waals and electrostatics for SF6. It is interesting to note that the transition between the P-/R- and the Q-band-like spectra in Xe occurs at ρ* ∼ 0.6 ( M) compared with ρ* ∼ 0.2 ( M) for SF6; see Figs. 2 and 3. This is consistent with the increased interaction strength for N2O–SF6 as fewer solvent molecules are required to sufficiently perturb the system to illicit the change in the spectroscopy. Similarly, the slope of N(r) vs concentration can also serve as a qualitative measure for the interaction strength. To stabilize a solvent shell of a certain size, a smaller number (lower concentration) of solvent molecules is required if the solvent–solute interaction is sufficiently strong. These qualitative relationships are also reflected in the longer τ2 values of the FFCFs for N2O in SF6 compared with Xe as the solvent; see Fig. 4. For large polyatomic molecules in highly compressible fluids, “attractive” solutes have been found to recruit increased numbers of solvent molecules, which leads to local density augmentation.75 While the amount of augmentation was found to correlate well with the free energy of solvation of the solute from simulations, computations and experiments were reported to follow different correlations.
CONCLUSION
In conclusion, the present work establishes that the rotational structure of the asymmetric stretch band is quantitatively described for N2O in xenon and is qualitatively captured in SF6 compared with the experiment. Specifically, the change from P-/R-branches in the gas phase environment changes to a Q-branch-like structure, eventually assuming a Lorentzian-like line shape once the solvent passes through the supercritical fluid to the liquid state. Transition between the gas- and SCF-like solvent is reflected in a steep increase in the longer correlation time τ2 in the FFCF for both solvents. As the density of the solvent increases, additional solvent shells appear in g(r). The present work suggests that atomistic simulations together with machine learned and accurate electrostatic interactions yield a quantitative understanding of the spectroscopy and structural dynamics4 from very low density to the liquid environment with the possibility to characterize transition to the SCF.
SUPPLEMENTARY MATERIAL
See the supplementary material for additional data and figures referred to in the article that further support the findings of this work.
ACKNOWLEDGMENTS
This work was supported by the Swiss National Science Foundation, Grant Nos. 200021-117810 and 200020-188724, the NCCR MUST, and the University of Basel (to M.M.) and by European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 801459—FP-RESOMUS, which is gratefully acknowledged. The support of the National Science Foundation, Grant No. CHE-2102427 (L.D.Z.), and the Boston University Photonics Center is gratefully acknowledged. We thank James L. Skinner for valuable correspondence on Ref. 68.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Kai Töpfer: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Debasish Koner: Investigation (supporting); Methodology (supporting); Software (supporting); Validation (supporting); Writing – review & editing (equal). Shyamsunder Erramilli: Conceptualization (equal); Methodology (equal); Writing – review & editing (equal). Lawrence D. Ziegler: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Markus Meuwly: Conceptualization (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (equal).
DATA AVAILABILITY
Raw data were generated at the University of Basel large scale facility. The derived data supporting the findings of this study are available from the corresponding author upon reasonable request. The data and source code that allow for reproducing the findings of this study are openly available at https://github.com/MMunibas/N2O_SF6_Xe. The original release of the N2O model potential is available at https://github.com/MMunibas/N2O-PESs.