Rotational Superfluidity in Small Helium Droplets - American Chemical

Article pubs.acs.org/JPCA

Rotational Superfluidity in Small Helium Droplets David Mateo, Frisly Gonzalez, and Jussi Eloranta* Department of Chemistry and Biochemistry, California State University at Northridge, 18111 Nordhoff Street, Northridge, California 91330, United States S Supporting Information *

ABSTRACT: The first minimum appearing in molecular rotational constants as a function of helium droplet size has been previously associated with the onset of superfluidity in these finite systems. We investigate this relationship by bosonic density functional theory calculations of classical molecular rotors (OCS, N2O, CO, and HCN) interacting with the surrounding helium. The calculated rotational constants are in fair agreement with the existing experimental data, demonstrating the applicability of the theoretical model. Inspection of the spatial evolution of the global phase and density shows the increase in the rotational constant after the first minimum correlates with continuous coverage of the molecule by helium and the appearance of angular phase coherence rather than completion of the first solvent shell. We assign the observed phenomenon to quantum phase transition between a localized state and one-dimensional superfluid, which represents the onset of rotational superfluidity in small helium droplets.

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INTRODUCTION Helium droplets not only provide a unique matrix environment for high resolution spectroscopy and studying molecular solvation but also allow us to use guest molecules as probes of the surrounding quantum medium.1−3 After the initial discovery of the helium droplet technique for spectroscopic applications, attention quickly turned into characterizing the physical properties of the helium droplets themselves. The groundbreaking experiments by the Toennies group employed the glyoxal molecule as a probe to study the helium droplet response through optical absorption spectrum.4 These experiments revealed that the zero-phonon absorption lines were accompanied by a structured phonon wing, which was separated from the parent line by a gap. The two weak maxima occurring in the phonon wing were assigned to the turning points present in the superfluid helium dispersion relation (i.e., maxon and roton), hence demonstrating that the surrounding helium medium was indeed superfluid.4 The authors provided a frequency domain interpretation of the band maxima based on the density of states analysis and related it to the special shape of superfluid helium dispersion relation. Subsequent time domain theoretical calculations further demonstrated that the observed band maxima indeed corresponded to maxon and roton excitations localized near the probe molecule.5 These excitations have zero group velocity and provide a long time scale perturbation to the molecular electronic two-level system, which allows for their detection by frequency domain absorption spectroscopy. On the theoretical side, the gradual appearance of superfluidity in small helium droplets of sizes greater than ca. 50 atoms has been predicted by path-integral Monte Carlo and variational calculations,6,7 which is consistent with the experimental observation of superfluidity using the glyoxal probe in larger droplets (N ≈ 5500).4 The later experiments, employing various molecular probes, showed a © XXXX American Chemical Society

strong dependency of the phonon band structure to the molecule−helium interaction but many of the discrete lines appearing in the band gap region could be assigned to localized helium atom vibrations near the molecule.8,9 Another set of experiments was dedicated for studying the effect of helium on the rotational spectra of molecules embedded in helium droplets.10 For example, line intensity analysis of such spectra was used to determine the droplet base temperature (0.38 K) by assuming a thermodynamic equilibrium between the rotor and the rest of the droplet,11 and a comparison of oxygen carbon sulfide (OCS) rotational spectra in mixed 4He−3He droplets provided additional experimental evidence for the superfluidity of helium droplets.12 In the latter case, the lack of rotational structure in 3He and the presence of sharp rotational lines in 4He droplets was understood as a signature for the superfluidity of 4 He droplets. Note that the superfluid lambda transition of 3He occurs at much lower temperatures than for 4He, and therefore, 3 He is not superfluid at 0.38 K. The conclusion from these experiments was that superfluidity appears around N = 60, which is of the same order as path-integral Monte Carlo predictions for pure droplets.6,12,13 More recent experiments have attempted to extend this analogy for smaller droplets (N = 2, ..., 70) to demonstrate that even these are, in fact, also superfluid14−19 by determining the rotational constant of the probe molecule (e.g., OCS) as a function of the droplet size N. It was observed that the rotational constant initially decreases (or the effective moment of inertia increases) when N increases, Special Issue: Markku Räsänen Festschrift Received: June 9, 2014 Revised: August 6, 2014

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dx.doi.org/10.1021/jp5057286 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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the contribution of single excitations to the reference wave function was not excessive.31 The purpose of this calculation was to assess how sensitive the calculated rotational constants are to the proper basis set convergence in the helium−molecule potential. For the other molecules considered in this work, we used the pair-potential data available in the literature. The N2O−He potential was taken from ref 32, HCN−He from ref 33, and CO−He from ref 34. Bosonic Density Functional Theory Calculations. Helium clusters were modeled by the Orsay-Trento DFT (OT-DFT) and the interaction with the guest molecule was included through an external potential.35 To compute the effective moment of inertia of the molecule−helium complex, we include an additional energy term of the form −ωLz10 and compute the “rotating” groundstate energy by minimizing

which is the expected behavior when helium atoms are trapped by the attractive parts of the molecule−helium potential and become part of the rotating complex. After a certain critical value of N, the rotational constant begins to increase suddenly, often exhibiting a probe-dependent oscillatory behavior before leveling off to the bulk value. For example, for the OCS molecule this turning point is observed already at N = 8 after which the effective moment of inertia begins to decrease.16 By analogy to the well-known Andronikashvili experiment,20 this has been assigned to the onset of superfluidity due to the reduced friction between the molecular rotor and the surrounding liquid.14,16 The following path integral Monte Carlo calculations showed the presence of local nonsuperfluid density near the probe molecule, which contributed to the effective moment of inertia by adiabatically following the rotor.21−24 Note that such partial adiabatic following effect may also appear in classical systems, where some of the solvent atoms become part of the rotating complex, but a decrease in the moment of inertia cannot occur as such an effect must clearly have quantum mechanical origin. Recently, the experimentally observed skewed rotational line shapes have been linked to the dynamical changes in the adiabatic coupling.25 At first glance, the superfluidity of small helium droplets, as observed through the turnover point of the rotational constant, appears to be incompatible with the earlier theoretical predictions for pure helium droplets where the characteristic superfluid features were shown to appear gradually when N > 40−70.6,7,13 Furthermore, the additional geometrical confinement provided by the probe molecule should only act to reduce the collective quantum effects. The idea of the turnover point marking the onset of superfluidity is difficult to reconcile with the facts that its position depends on the probe molecule and that additional local minima appear in the rotational constant as more helium is added.16,17 In this paper, we investigate the mechanism responsible for the nonmonotonic behavior of the rotational constant as a function of helium droplet size. We use bosonic density functional theory (DFT) calculations of OCS, N2O, CO, and HCN probe molecules solvated in helium droplets and bulk superfluid helium to identify the ingredients that play the essential role in the decrease of moment of inertia. The rotational constant as a function of the droplet size is compared with the existing experimental data and the turnover points occurring at certain droplet sizes are characterized. Finally, the relationship between superfluidity and the turnover point is discussed.

E[Ψ,ω] =

⎧

2

∫ ⎨⎩ 2ℏm |∇Ψ|2 + ϵOT[Ψ] + VX−He|Ψ|2 ⎫ − ωΨ*Lz Ψ⎬ ⎭

(1)

with respect to the liquid helium complex order parameter Ψ (“effective wavefunction”) normalized to the number of particles N, where m is the helium atom mass, ϵOT represents the Orsay−Trento energy density functional,35 VX−He is the molecule−helium interaction (molecule oriented along the xaxis), ω is a fixed parameter representing the rotation frequency, and Lz is the angular momentum operator about the z axis going through a given point r0⃗ . Due to the strongly bound nature of the OCS−He pair potential, an additional term to account for possible solidification of helium was included in OT-DFT.36 This term has no effect for the other molecules discussed in this work due to their smaller binding energy with helium. The kinetic correlation and backflow terms35 of OTDFT are computationally more demanding than the others and may lead to numerical instabilities in some cases. Inclusion of these terms did not provide a systematic improvement in the accuracy, and therefore, we left these terms out. Furthermore, we used the original OT-DFT spherical average radius rather than the slightly larger value suggested to be used with the high density correction term.36 With the energy for several values of ω computed, the contribution to the moment of inertia from helium can be obtained as Iadd =

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THE COMPUTATIONAL APPROACH Electronic Structure Calculations. The electronic structure calculations of OCS−He interaction were carried out at the CCSD(T) level of theory with an augmented correlation consistent basis set, aug-CC-pVQZ (AVQZ).26−28 The equilibrium geometry for OCS was obtained by the standard geometry optimization module as implemented in the Molpro code.29 The potential energy surface calculation for OCS−He included the basis set superposition error correction through the counterpoise method of Boys and Bernardi.30 The complete potential energy surface was generated from 17 different angular cuts distributed evenly between 0° and 180° by using interpolation for the full potential energy surface (raw numerical data given as Supporting Information). The standard T1-norm test was carried out in each calculation to ensure that

d2E dω 2

ω= 0

(2)

which, in practice, we found to be equivalent to computing the added moment of inertia as Iadd = ⟨Lz⟩/ω for sufficiently small values of ω (typically less than 1 GHz). The effective rotational constant is defined as

B=

ℏ 4πc(IX + Iadd)

(3)

where IX is the moment of inertia of the isolated molecule and c is the speed of light. This procedure accounts to treating the molecule as a classical rotor, an approximation that is more justified for the heavier molecules such as OCS and N2O than for the lighter molecules such as CO and HCN. This may somewhat impact the accuracy of the results obtained, but a quantum rotor cannot be directly coupled to OT-DFT in a B



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straightforward manner. Regarding the accuracy of this procedure, it is worth noting that while OT-DFT was originally designed to deal with bulk phenomena and is able to reproduce the static and dynamical properties of helium droplets with several hundreds of atoms, its accuracy is not guaranteed for small clusters. Despite these limitations, we have chosen to use the OT-DFT approach because it allows us to explore the ingredients that are essential to reproduce the qualitative behavior of rotational constants in helium droplets, even at the cost of quantitative accuracy. One peculiarity of a rotating molecule−helium complex is that the ground state solution must be obtained with the proper axis of rotation that minimizes the total energy. For a rigid solid this is always an axis passing through the center of mass of the system, but in a liquid-like environment, where parts of the system may respond differently to rotation, this criterion is not necessarily valid and the axis of rotation must be corrected selfconsistently during the minimization. Minimization of the energy in eq 1 with respect to the axis of rotation r0⃗ gives

r0⃗ = rX⃗ +

ω⃗ × ⟨p ⃗ ⟩ mX ω 2

Figure 1. Contour plot of OCS−He pair potential. Equipotential lines are separated by 5 cm−1, with the lowest level corresponding to −45 cm−1. Red solid lines show the potential from ref 40 (CCSD(T)/ AVTZ), and green dashed lines, the potential computed in this work (CCSD(T)/AVQZ).

helium density distributions have been discussed in previous works.40−42 The first five He atoms are localized in the highly attractive annular region between the O and C atoms. As the sixth atom is added, the helium starts to fill the O end of the molecule and, due to the zero-point motion and He−He interaction, the density in the annular well decreases slightly. At N = 8 both the minimum and the O end are filled, and the ninth atom starts to fill the opposite end thus rendering the molecule fully covered by He. The next helium atoms are spread along the molecular surface until the capacity of the first solvation layer is reached at around 20 atoms. Then, a second shell of helium starts to form and after ca. 35 atoms the cluster starts to have an approximately spherical shape. These structural density changes were also predicted by earlier Quantum Monte Carlo (QMC) calculations for the same number of atoms.42 A pictorial representation where each atom contributes as more He is added to the complex is presented in Figure 2 for selected values of N. This figure shows the

(4)

where rX⃗ is the center of mass of the molecule, mX is the mass of the molecule, and ⟨p⃗⟩ is the expectation value of the momentum operator. This correction has a small overall effect on the results; for example, for OCS−HeN clusters the axis is typically displaced from the center of mass of the free molecule by less than 0.5 Å, which implies an increase in the rotational constant by 0.0006 cm−1 at most. We have solved the nonlinear Schrödinger-type equation arising from the minimization of eq 1 by means of imaginary time propagation.37 The numerical treatment of the helium OT-DFT problem is described elsewhere.38 To be able to accommodate the angular momentum term in the Hamiltonian, we have used the Crank−Nicolson method39 for propagating the kinetic energy rather than the standard Fourier transform technique. To minimize the boundary condition artifacts arising from using a finite sized simulation box, a large grid consisting of 128 × 128 × 128 points with a spatial grid step of 0.25 Å was used in the calculations. An imaginary time step of 1 fs was used in minimizing the total energy of the system. To verify that there was no time step bias in the obtained solution, shorter time steps down to 0.01 fs were executed at the end of each run. Typically 104 imaginary time iterations were required for full convergence.

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RESULTS Ab Initio OCS−He Potential. A comparison between the computed OCS−He pair potential (CCSD(T)/AVQZ) and the results of Paesani et al.40 (CCSD(T)/AVTZ) is presented in Figure 1, which shows only a small change in the van der Waals binding energy that can be attributed to the larger basis set used in this work. Our calculation gives an energy of −47.78 cm−1 at the global minimum, which is slightly wider and shallower than Paesani’s and the local minimum at the sulfur end appears slightly less bound. In practice, no significant differences in the computed rotational constants were found between these two potentials, which eliminates the OCS−He pair potential, a significant source of error. Heavy Rotors (OCS and N2O). We have computed the density profiles for rotating OCS−HeN complexes with N = 2, ..., 70 and the bulk liquid limit. The general aspects of the

Figure 2. Helium density differences between OCS−HeN and OCS− HeN−1 clusters. Red (dark) regions have a significant increase in density (difference at least 25% of the largest difference), and green (light) regions have a significant reduction of density (same threshold value as red regions with opposite sign) when one He atom is added.

difference in densities between the HeN and the HeN−1 clusters; regions where that difference is significantly positive, i.e., increase in He density, are displayed in red, and regions where it is significantly negative, i.e., decrease in He density, are shown in green. It is important to emphasize that this does not account for where each particle is; DFT describes a collection of indistinguishable bosons and does not contain information about the two-body correlation function nor any other measure of the relative positions of the atoms. This representation is meant to roughly highlight the changes in the density C



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for the observed changes in the rotational constant. The present method cannot quite provide quantitative accuracy as can be observed in Figure 4. For example, the minimum value of B at N = 9 is overestimated by a 20%, which is comparable to the accuracy obtained in the latest ab initio QMC calculations.42 We have also found a similar agreement with experimental data for N2O (heavy rotor), HCN, and CO (light rotors). A comparison between the computed rotational constants and the experimental data is presented in Figure 5.

distribution as the number of atoms is increased because, as will be discussed later, it has an important effect on the effective rotational constant of the molecule. We have also considered OCS solvated in the bulk liquid to mimic the behavior of large droplets. The resulting liquid distribution is presented in Figure 3 together with the current

Figure 3. Bulk liquid density (colors) and current field (arrows) around OCS molecule rotating counterclockwise in the xy-plane, as seen in the laboratory frame. The dashed white line follows the zero equipotential line of the OCS−He interaction.

field developed in response to the rotational term −ωLz. As long as one remains in the adiabatic following regime (ω ≲ 1 GHz), the density profile is insensitive to the value of ω chosen and the current field scales linearly with it. We have computed the rotational constant of OCS−HeN for N = 2−70 within the classical rotor approximation. The values obtained are presented in Figure 4 along with the

Figure 5. Overview of rotational constants (B in logarithmic scale) as a function of N for OCS, N2O, CO, and HCN molecules. Experimental reference data were taken from ref 17.

Light Rotors (HCN and CO). We have employed the same methodology to compute the rotational constants for light rotors CO and HCN solvated in small helium clusters. These molecules interact with helium through a pair potential that has a considerably simpler geometry. Although the OCS potential has a deep localized annular global minimum and two other wide local minima at each end of the molecule, HCN has only one minimum located at the nitrogen end and a smooth angular variation. This geometry produces a simpler helium distribution along the molecule and results in a straightforward filling scheme: for both CO and HCN, the first helium atoms go into the minimum of the potential well and when it fills up they start to spread angularly, gradually coating the molecule. When the molecule is completely coated, the complex starts to grow radially and the subsequent solvation layers become filled. The rotational constants computed for HCN and CO, shown in Figure 5, also exhibit similar shapes as measured experimentally. The agreement is worse than for the heavier rotors (OCS and N2O), presumably due to the gradual break down of the adiabatic approximation.43 In addition to the previously mentioned limitations of DFT used to describe small He clusters, approximating the molecule as a classical rotor works best for heavy molecules. Despite these shortcomings, light rotors present an interesting case because their simpler geometry makes them good candidates for inferring the contribution of the filling scheme on the effective rotational constant.

Figure 4. Rotational constant B for OCS−HeN as a function of the number of helium atoms N. The experimental data were taken from ref 16 and QMC data from ref 42. The thin dashed line shows the classical prediction assuming that all helium rotates with the molecule.

experimental16 and QMC42 data. Despite the approximations involved in the present model, the qualitative agreement with the experimental data is remarkable: the model correctly reproduces the position of the first turnover point and displays the oscillatory behavior for larger clusters that have been observed experimentally. The presence of these oscillations was already hinted in previous QMC calculations42 for N = 30, 40, 50, and 64, but the higher resolution of the present calculations makes the oscillation clearly evident. The figure also shows the classical rotational constant evaluated using the OT-DFT densities, which assumes that all helium rotates rigidly with the molecule. The absence of structure in this curve can be used to discard “classical” geometrical effects (i.e., particles moving closer or farther from the axis of rotation) as the main source D



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DISCUSSION

The first turnover point in rotational constants as a function of helium droplet size (Figure 4) has been previously proposed to be a consequence of the highly attractive region in the molecule−helium interaction that binds a small number of He atoms so strongly that they rotate rigidly with the molecule, whereas the additional atoms fall in the less bound regions and are not dragged by the rotating molecule. Such behavior has been interpreted to manifest a superfluid free-flow around the probe molecule.21 This model relates the position of the turnover point to the capacity of the potential well, which explains why it is expected to be molecule-dependent. However, the model cannot explain the additional oscillations in B after the first turnover point. The fact that DFT is capable of reproducing the experimental behavior albeit its implicit approximations suggests a new interpretation of the experimental data. The zero-temperature DFT model describes the whole liquid helium as a single scalar complex field, neglecting the individual degrees of freedom for each atom, imposing irrotationality and forcing effective Bose condensation in the whole system. Hence, a classical phase transition to a superfluid state cannot be responsible for the turnover in B. The calculations show that this point is always related to complete coating of the guest molecule by helium as opposed to complete filling of the first solvent shell. The secondary minima appearing in B are then loosely related to the coverage of the consequent solvation layers. The decrease in the moment of inertia correlates with the coating of the molecule because the irrotationality restriction can only oppose following of the molecule when the surrounding helium is connected. With an incomplete coating, even an irrotational liquid can follow the molecular rotation adiabatically without developing any vorticity. This is a consequence of the manner angular momentum is related to the circulation in inhomogeneous systems. As the first solvation shell starts filling up and the angular distribution of helium becomes more homogeneous, the irrotationality requirement begins to increasingly limit the amount of angular momentum allowed, effectively decoupling the liquid from molecular rotation. This point is illustrated in Figure 6, where the circulation (i.e., gradient of the global phase), angular momentum, and liquid density are plotted for CO−He5 and CO−He10. On the one hand, although in both cases the net circulation is zero, for N = 5 the negative circulation can “hide” in the low density region and not contribute to the angular momentum. On the other hand, when N = 10 there is no longer a region with negligible density and the negative circulation contributes with negative angular momentum, effectively reducing the overall moment of inertia (i.e., increase in B). We have observed the same behavior in all molecules studied here: when the first solvation layer becomes connected, the effective value of B begins to increase. Stating that there is angular phase coherence at N = 10 but not at N = 5 could be taken as abuse of language considering that OTDFT calculations always have phase coherence for the reasons exposed previously. However, when a region with a negligible density is present in the angular distribution of helium, the value of the phase there is irrelevant and the phase coherence is lost at finite temperatures. In practice, even the low temperature of these droplets, typically of 0.2 K,16 is enough to randomize the phase in these near zero density regions. To treat this aspect of the problem more accurately, path integral QMC or Bose−Hubbard-type models should be employed.6,44

Figure 6. Angular cuts for CO−HeN along the pair-potential minimum for helium density (filled area), circulation (solid line), and angular momentum (dashed line). The circulation and angular momentum are shown in arbitrary units, with the zero marked by the black horizontal line. The angle ϕ = 0° corresponds to the carbon end and ϕ = π to the oxygen end.

Alternatively, one can also view the decrease in moment of inertia as a consequence of the increased energy cost associated with the development of vorticity, i.e., the difference between the groundstate energy and the minimum energy compatible with one unit of circulation (2πℏ/m). This energy cost is a measure how efficiently the molecule can couple its rotation to helium and, consequently, increase the moment of inertia. When a region of negligible density is present, the circulation can be either 0 or 1 without affecting the current. On the one hand, because the energy depends on the current and not the phase, the energy gap for vortex nucleation is negligible in this situation. On the other hand, if more atoms are added and the molecule becomes fully coated, increasing the circulation by one unit increases the total current and thus the energy. This implies the existence of an energy gap between the ground state and the lowest vortex state, which decouples the rotational motion from the surrounding helium. The energy required for vortex nucleation around the OCS molecule is shown in Figure 7. This is practically zero up to N = 9, at which point it begins to increase logarithmically with the number of helium atoms, as one would expect on the basis of vortex hydrodynamic models.45 This value of N corresponds exactly to the first turnover point in B observed experimentally for OCS. Superfluidity of Small Helium Droplets. On the basis of the present calculations, the first turnover point in the rotational constant is not an indication of the onset of “regular” superfluidity in small systems, at least not in the common understanding of bulk superfluidity and its characteristic excitation spectrum. The energy gap for vortex creation, as described above, plays a similar role here as the roton gap in the appearance of translational superfluidity in bulk liquid helium (i.e., the Landau criterion for superfluidity). The rotational superfluidity in small helium droplets appears when the probe molecule becomes fully coated by helium. As will be shown in the following section, this decrease in inertia can also be understood as a quantum phase transition between a localized E



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to V0 at the opposite end of the molecule (ϕ = π). The Hamiltonian of this model has the typical rigid rotor form H=−

V ℏ2 ∂ 2 + 0 (1 − cos φ) 2 φ 2 2mR

(5)

The added moment of inertia introduced by helium, as defined in eq 2, can be computed using the second-order perturbation theory as Iadd = 2 ∑ k≠0

|⟨k|Lz|0⟩|2 E 0 − Ek

(6)

where |k⟩ and Ek are the kth eigenstate and eigenenergy of the system, respectively. For V0 ≪ ℏ2/2mR2, helium is spread over the whole ring and the quantized circulation forbids any net current, giving Iadd ≈ 0. For V0 ≫ ℏ2/2mR2, helium becomes trapped in the bound angular well and the net current can grow indefinitely while still maintaining zero circulation. In this regime the classical limit Iadd = mR2 is obtained. The moment of inertia as a function of the potential barrier height can be fitted to the following empirical expression

Figure 7. Energy required for vortex nucleation in OCS−HeN complexes as a function of the number of He atoms. The logarithmic fit is shown for reference (see text).

state and one-dimensional superfluid.44 The secondary local minima observed after the first turnover point are an attenuated repetition of this effect when additional solvation layers become covered. In contrast with the first solvation layer, whose capacity depends mostly on the molecule−He interaction, the details on how these additional layers are filled depend heavily on He−He correlations, which may not be fully captured by DFT. For that reason, the first turnover point is reproduced by DFT calculations with a higher accuracy than following secondary oscillations. Regarding the appearance of the rotationally resolved OCS spectrum when ca. 60 4He atoms surround the molecule,12 the present calculations do not predict any new phenomena that would appear at this particular value of N. This number of atoms already provides bulk-like solvation of OCS, and therefore, this observation is not directly related to the onset of rotational superfluidity, which, as discussed above, starts already at N = 9. This could be related to the onset of translational superfluidity; however, the pure helium droplet calculations7,13 predict only very weak roton-maxon resonances in the range N = 40−70 atoms, and furthermore, the presence of the OCS molecule can only act to reduce the number of helium atoms that contribute to the collective quantum behavior. The line broadening, which is the main observable in this experiment, depends on the dynamic response of the mixed 3He−4He droplet that takes place after rotational excitation of OCS. To model such a system theoretically, a real time propagation of the droplet is required to evaluate the relevant time correlation function for computing the line shape, which is beyond the scope of this work. It is interesting to note that previous variational calculations on mixed 3He/4He droplets have established that the 3He density begins to retract from the center of the droplet in a similar size regime (N ≈ 40−60),7 which could, consequently, restore Bose symmetry in the first solvation shell. One-Dimensional Superfluidity and Phase Coherence. To gather additional evidence that the first turnover point in the rotational constant is related to the coating of the molecule and the appearance of angular phase coherence, we have developed a simplified model by stripping out most of the complexity and leaving only the essential ingredients to reproduce the behavior of B. In this model, helium is confined to move on a circle with a fixed radius R, which corresponds to the minimum distance of the molecule−He pair potential. The system has only one periodic degree of freedom, ϕ ∈ [0, 2π). The potential V(ϕ) has a minimum at ϕ = 0 and increases up

Iadd(V0) = mR2

(V0/Vc)β 1 + (V0/Vc)β

(7)

where the optimized values are β = 2.3 and Vc = 1.3ℏ2/2mR2. To demonstrate the concept, we apply this model to HCN and CO molecules as their pair potential with He resembles closely that of eq 5. The only input required in this model is the height of the effective barrier V0, which depends on the number of He atoms present. We get the value for V0 = V0(N) from DFT calculations using not just the molecule−helium pair potential, but the whole OT-DFT self-consistent potential, i.e., the functional derivative of the relevant part of eq 1 with respect to the order parameter. The results are shown in Figure 8, and though it is clear that to reproduce the experimental values with precision one needs a richer model, this simple approach is sufficient to obtain the oscillatory behavior in Iadd (or, equivalently, B) and predict the turnover point with a qualitative accuracy. This is consistent with the fact that the oscillatory behavior is related to angular phase coherence.

Figure 8. Comparison of the moment of inertia obtained using DFT and the one-dimensional model with the experimental values17 for CO (top) and HCN (bottom). F



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this implies that rotation takes place in two dimensions. In the present calculations the possible coupling between the two degrees of freedom is neglected (i.e., two uncoupled onedimensional rotors), which may partially be responsible for the lack of quantitative reproduction of the rotational constants. For molecules with multiple different moments of inertia, it may be possible to observe the onset of one-dimensional superfluidity along each axis separately, provided that the coupling between them is small. This would produce complicated behavior in B as a function of N where multiple primary minima are predicted.

There is a distant analogy of this phenomenon with Mott insulator−superfluid quantum phase transition, which is more conveniently discussed in the coordinate frame of the rotating probe molecule. When the number of He atoms around the rotor is less than required for the first turnover point, they remain localized in the effective OT-DFT potential well (or potential wells depending on the probe molecule). In the rotating frame of reference and within the adiabatic following regime, the atoms remain stationary because the potential energy dominates over the kinetic energy. As discussed earlier, the finite temperature present in the experiments destroys the phase coherence in the low density regions outside the potential well, which also means that there would be no coherence between the atoms trapped in adjacent potential minima. The temperature must also be sufficiently low such that the trapped atoms cannot be thermally excited. Under this condition, there is no net current unless the value of angular velocity (ω) is increased above a threshold value, which could lead to the creation of “particle-hole”-type excitations. If only one potential minimum is present, such excitation would essentially correspond to a bound-to-free transition. The existence of a threshold angular velocity (i.e., the adiabatic regime) for creating excitations in the system is a direct consequence of the presence of an energy gap in the spectrum, which is characteristic for a Mott insulating state. Note that the coordinate perpendicular to the plane of rotation could induce phase coherence between certain symmetry related potential wells (e.g., the ring of five He atoms around OCS connects the two in-plane potential minima shown in Figure 3). We emphasize that our current OT-DFT model cannot deal with the loss of phase coherence, which is required for describing Mott insulator states, and therefore, there is no rigorous correspondence with the present case. After the turnover point in B, the He atoms become delocalized over the rotor surface, the kinetic energy dominates, and phase coherence develops even at finite temperatures. In the rotating frame of reference, these atoms flow around the molecule (i.e., nonzero net current) and constitute a onedimensional superfluid. The current depends now directly on the angular velocity ω and thus no energy gap is present. However, if ω is increased above some critical value (ωc), excitations with quantized vorticity will appear. In the rotating reference frame, any vorticity formed opposes the induced current and hence leads to reduction in superfluidity. Thus, ωc plays the same role for rotational superfluidity as the Landau critical velocity does for translational motion in bulk superfluid helium. We conclude that the first turnover point in B corresponds to quantum phase transition from a localized to one-dimensional superfluid state, where the surrounding helium current is driven by the rotating probe molecule (“stirbar”). The coupling to the surrounding helium is provided by the anisotropic probe molecule−helium potential, which also allows the detection of current through the changes in effective momentum of inertia. Analogously, in superconductors an electric potential difference can be used to drive such a current. In the present case, the appearance of the phase transition depends primarily on the probe molecule geometry and the number of surrounding He atoms (N) whereas ωc determines the maximum current that the one-dimensional superfluid state can sustain without creating vorticity. Finally, we note that even though we only consider one-dimensional superfluidity here, molecular rotation is quantized about each principal axis and for linear molecules

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CONCLUSIONS The rotational constants of probe molecules solvated in small helium clusters present an incipient minimum as a function of the system size, which means that addition of more helium atoms after this point reduces the moment of inertia of the complex. This minimum appears between N = 3 and N = 9 depending on the probe molecule and has been previously associated with the onset of superfluidity in these finite systems. 46 This is somewhat surprising as theoretical calculations predict a gradual appearance of bulk-like superfluidity after N = 40−70 in pure helium droplets. Experiments for OCS and N2O, among other probe molecules, show additional oscillations in the rotational constant after the first turnover point. The DFT calculations can reproduce this behavior qualitatively, namely the number of atoms at which the turnover takes place and the subsequent oscillations, and offer insight into the mechanism responsible for the decrease in moment of inertia (i.e., reduction of “rotational friction” by accumulation of negative angular momentum). By inspecting the evolution of the global phase and the topological changes in the liquid distribution as more helium is added, we conclude that the minimum in the rotational constant is a signature of quantum phase transition that bears some similarities to Mott insulator to one-dimensional superfluid transition. This transition is triggered by the reduction in the effective potential barrier arising from the helium−molecule and helium−helium interactions. In that sense, the introduction of a rotating probe molecule does not lower the onset of bulk-like superfluidity in small helium droplets, as understood by the existence of the characteristic excitation spectrum and the associated translational critical Landau velocity, but rather introduces a new phenomenon related to the appearance of rotational superfluidity when the probe molecule becomes fully coated with helium. The number of atoms at which this transition takes places has more to do with the particular geometry of the probe molecule than the intrinsic properties of helium (excluding irrotationality). This phenomenon and its relationship to other phase transitions observed in helium are summarized in Figure 9. Notice especially the different size regimes where the rotational and translational superfluidity appear. Finally, we note that strong field alignment of molecules in helium droplets has been demonstrated recently to create rotational wavepackets,47 which can be made to mimic the behavior of classical rotors. Provided that the wavepacket is given a sufficient initial angular momentum, the following rotational dynamics could reveal important information about the possible differences between quantum and classical rotors in the presence of helium. In addition, such measurement might also be able to determine the critical rotational velocities required for vortex nucleation (ωc), which should depend on the probe molecule as well as the droplet size. G



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Figure 9. Overview of phase transitions observed in helium as a function of system size (N) and temperature. Classical phase transitions are denoted by CPT (red) and quantum phase transitions by QPT (green). ΔEv represents the energy required to create a singly quantized vortex and ΔEr denotes the roton energy gap. The bottom panel labeled by “Rotation” represents the rotational superfluid regime whereas “Translation” corresponds to translational superfluid. The orange arrows indicate the flux of helium in the reference frame of the molecule (rotating frame in the bottom panels and a frame moving at constant velocity in the top panel).

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ASSOCIATED CONTENT

S Supporting Information *

The raw numerical data for the calculated CCSD(T)/AVQZ potential energy surface is given as Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*J. Eloranta. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the National Science Foundation grants CHE-1262306 and DMR-1205734 and the Interdisciplinary Research Institute for the Sciences (IRIS) are gratefully acknowledged.

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REFERENCES

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Rotational Superfluidity in Small Helium Droplets - American Chemical

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