Infrared spectroscopy of an endohedral water in fullerene

Shugai, A.; Nagel, U.; Murata, Y.; Li, Yongjun; Mamone, S.; Krachmalnicoff, A.; Alom, S.; Whitby, R. J.; Levitt, M. H.; Rõõm, T.

doi:10.1063/5.0047350

An infrared absorption spectroscopy study of the endohedral water molecule in a solid mixture of H₂O@C₆₀ and C₆₀ was carried out at liquid helium temperature. From the evolution of the spectra during the ortho–para conversion process, the spectral lines were identified as para-H₂O and ortho-H₂O transitions. Eight vibrational transitions with rotational side peaks were observed in the mid-infrared: ω₁, ω₂, ω₃, 2ω₁, 2ω₂, ω₁ + ω₃, ω₂ + ω₃, and 2ω₂ + ω₃. The vibrational frequencies ω₂ and 2ω₂ are lower by 1.6% and the rest by 2.4%, as compared to those of free H₂O. A model consisting of a rovibrational Hamiltonian with the dipole and quadrupole moments of H₂O interacting with the crystal field was used to fit the infrared absorption spectra. The electric quadrupole interaction with the crystal field lifts the degeneracy of the rotational levels. The finite amplitudes of the pure v₁ and v₂ vibrational transitions are consistent with the interaction of the water molecule dipole moment with a lattice-induced electric field. The permanent dipole moment of encapsulated H₂O is found to be 0.50 ± 0.05 D as determined from the far-infrared rotational line intensities. The translational mode of the quantized center-of-mass motion of H₂O in the molecular cage of C₆₀ was observed at 110 cm⁻¹ (13.6 meV).

I. INTRODUCTION

Endohedral fullerenes consist of atoms or molecules fully encapsulated in closed carbon cages. The remarkable synthetic route known as “molecular surgery” has led to the synthesis of atomic endofullerenes He@C₆₀¹ and Ar@C₆₀² and several molecular endofullerene species, including H₂@C₆₀,³ H₂O@C₆₀,⁴ HF@C₆₀,⁵ CH₄@C₆₀,⁶ and their isotopologs.^7–9 It is well established that these endohedral molecules do not form chemical bonds with the carbon cage and rotate freely.^5,10,11 The rotation is further facilitated by the nearly spherical symmetry of the C₆₀ cage. The thermal and chemical stability of A@C₆₀ opens up the unique possibility of studying the dynamics and the interactions of a small molecule with carbon nano-surfaces. The trapping potential of dihydrogen H₂ has been described with high accuracy using infrared (IR) spectroscopy,^8,12 inelastic neutron scattering (INS),¹³ and theoretical calculations.¹⁴ The non-spherical shape of the small molecule and the quantized translational motion of its center of mass lead to coupled rotational and translational dynamics.^10,15,16

The water-endofullerene H₂O@C₆₀ is of particular interest. The encapsulated water molecule possesses rich spatial quantum dynamics. It is an asymmetric-top rotor, supports three vibrational modes, displays nuclear spin isomerism (para-water and ortho-water), and has electric dipole and quadrupole moments.

Low-temperature dielectric measurements¹⁷ on solid H₂O@C₆₀ show that the electric dipole moment of the encapsulated water is reduced to 0.51 ± 0.05 D from the free water value of 1.85 D. The C₆₀ carbon cage responds to the endohedral water molecule with a counteracting induced dipole, resulting in the lower total dipole moment.¹⁸

The dynamics of isolated or encapsulated single water molecules have been studied before in other environments, such as noble gas matrices, solid hydrogen, and liquid helium droplets. Although the trapping sites in these matrices have high symmetry and allow water rotation, these systems exist only at low temperature^19,20 or for a very short time.^21,22 Water has also been studied in crystalline environments with nano-size cavities. However, in this case, the interactions with the trapping sites inhibit the free rotation of the water molecules.^23–26

Several spectroscopic techniques have been used to study H₂O@C₆₀, including nuclear magnetic resonance (NMR), INS, and IR spectroscopy^4,11,27–29 and time-domain THz spectroscopy.³⁰ The low-lying rotational states of the encapsulated molecule are found to be very similar to those of an isolated water molecule, with the notable exception of a 0.6 meV splitting in the J = 1 rotational state.^11,31 This indicates that the local environment of the water molecule in H₂O@C₆₀ has a lower symmetry than the icosahedral point group of the encapsulating C₆₀ cage. The splitting has been attributed to the interaction between the electric quadrupole moment of H₂O and the electric field gradients generated by the electronic charge distribution of neighboring C₆₀ molecules.^32,33 The merohedral disorder present in solid C₆₀ leads to two C₆₀ sites with different quadrupolar interactions. This merohedral disorder also leads to splittings of the IR phonons in solid C₆₀.³⁴ There is also evidence from dielectric measurements that merohedral disorder leads to electric dipolar activity in solid C₆₀.³⁵

Confined molecules exhibit quantization of their translational motion (“particle in a box”), in addition to their quantized rotational and vibrational modes. Quantized translational modes have been observed at 60–70 cm⁻¹ for water in noble gas matrices.^19,36 However, comparatively, little is known about the center-of-mass translational mode of H₂O@C₆₀. The fundamental frequency of the water translation mode in H₂O@C₆₀ has been predicted to occur at ∼160 cm⁻¹.¹⁶ This relatively high frequency reflects the rather tight confinement of the water molecule in the C₆₀ cage.

The energy level separation between the ground para rotational state and the lowest ortho rotational state in H₂O@C₆₀ has been determined to be 2.6 meV (21 cm⁻¹) by INS.¹¹ This energy level separation corresponds to a temperature of 28 K. The full thermal equilibration of H₂O@C₆₀ at temperatures below 30 K, therefore, requires the conversion of ortho-water into para-water. This conversion process takes between tens of minutes and several hours below 20 K in H₂O@C₆₀.^11,17,30,37 The spin isomer conversion is much faster at ambient temperature, with a time constant of about 30 s reported for H₂O@C₆₀ dissolved in toluene.²⁸

In this paper, we report on a detailed low-temperature far-IR and mid-IR spectroscopic study of H₂O@C₆₀ and C₆₀ solid mixtures. The IR technique allows us to measure the frequencies of rotational, vibrational, and translational modes and, from the line intensities, to determine the dipole moment of encapsulated water. In addition, the IR spectra reveal the interaction of endohedral water with the electrostatic fields present in solid C₆₀.

The rest of this paper is organized as follows: Section II discusses the sample preparation, the recording procedures of the IR spectra, and the determination of the IR absorption cross sections for different filling factors, temperatures, and ortho–para ratios. The quantum mechanical vibrating rotor model for the encapsulated water molecules is introduced in Sec. III A. We include in this model the interactions of the H₂O electric dipole and quadrupole moments with the electrostatic fields present in solid C₆₀. The theory of the IR line intensities is presented in Sec. III B. Section IV presents the measured IR spectra and the fitting of these data by the quantum mechanical model. The results are discussed in Sec. V, followed by a summary in Sec. VI. Appendixes A–C contain a more detailed theory for the interaction of the water molecules with the electrostatic fields and the infrared radiation and more details on the fitting of the experimental data by the quantum mechanical model.

II. METHODS

A. Sample preparation

H₂O@C₆₀ was prepared by a multi-step synthetic route known as “molecular surgery.”^4,9 The H₂O-filled (number density N_•) and empty (number density N_○) C₆₀ were mixed and co-sublimed to produce small solvent-free crystals with a filling factor f = N_•/(N_• + N_○). Five samples with filling factors f = 0.014, 0.052, 0.10, 0.18, and 0.80 were studied. The powdered samples were pressed into pellets under vacuum. The diameter of sample pellets was 3 mm, and the thickness d varied from 0.2 mm to 2 mm. The thinner samples were used in the mid-IR because of light scattering in the powder sample. The samples were thicker for lower filling factors and thinner for higher filling factors to avoid the saturation of absorption lines in the far-IR.

B. Measurement techniques

The far-IR measurements were done with a Martin–Puplett type interferometer and ³He cooled bolometer from 5 to 200 cm⁻¹, as described in Ref. 11. The IR measurements between 600 and 12 000 cm⁻¹ were performed with an interferometer Vertex 80v (Bruker Optics), as described in Ref. 12.

Two methods were used to record the H₂O@C₆₀ absorption spectra.

1. Method 1

The intensity through the sample, I_s, was referenced to the intensity through a 3 mm diameter hole, I₀. The sample was allowed to reach ortho–para thermal equilibrium at a temperature of 30 or 45 K, and the temperature was rapidly reduced to 10 or to 5 K. The sample spectrum I_s(t = 0) was recorded immediately after the temperature jump. Since the ortho–para conversion process is slow, the ortho fraction was assumed to be preserved during the T jump, corresponding to the high temperature ortho fraction n_o ≈ 0.74. The absorption coefficient α was calculated from the ratio Tr = I_s(t = 0)/I₀ as α(t = 0) = −d⁻¹ ln[(1 − R)⁻² Tr] where the factor (1 − R)² with R = (η − 1)²(η + 1)⁻² corrects for the losses of radiation, one reflection from the sample front and one from the back face. The refraction index of solid C₆₀ was assumed to be given by η = 2.³⁸ To identify para-water and ortho-water absorption peaks, the difference of two spectra was calculated, Δα = α(t = 0) − α(Δt), where α(Δt) is the absorption spectrum measured after the waiting time Δt. Only the para-H₂O@C₆₀ and ortho-H₂O@C₆₀ peaks show up in the differential absorption spectra, with the para-H₂O@C₆₀ and ortho-H₂O@C₆₀ peak amplitudes having different signs, ortho positive and para negative. This method was used for the far-IR and the mid-IR part of the spectrum.

2. Method 2

The sample was allowed to reach ortho–para thermal equilibrium at a temperature of 30 or 45 K, leading to an ortho-rich state, as in the first method. The temperature was rapidly reduced to 10 or to 5 K, and a series of spectra were recorded at intervals of a few minutes starting immediately after the T jump, which continued until the ortho–para equilibrium was reached. The differential absorption Δα = −d⁻¹ ln[I_s(0)/I_s] was calculated, where I_s(0) is the spectrum recorded immediately after the T jump and I_s is the spectrum recorded when the low temperature equilibrium was reached. The equilibrium ortho fraction is ∼0.01 at 5 K. Method 2 was used for the far-IR part of the spectrum.

C. Line areas and absorption cross sections

The absorption line area $A_{j i}^{(k)}$ was determined by fitting the measured absorption $α_{j i}^{(k)} (ω)$ with Gaussian lineshape. $|i⟩$ and $|j⟩$ are the initial and final states of the transition, and k denotes para (k = p) or ortho (k = o) species. From these experimental line areas, $A_{j i}^{(k)}$ ⁠, a temperature and para (ortho) fraction independent line area $\bar{A_{j i}^{(k)}}$ was calculated,

\bar{A_{j i}^{(k)}} (f) = \frac{A_{j i}^{(k)} (f)}{n_{k} (p_{i}^{(k)} - p_{j}^{(k)})} .

(1)

The population difference of initial and final states, $p_{i}^{(k)} - p_{f}^{(k)}$ ⁠, is given by the sample temperature T, while the para (ortho) fraction n_k depends on the history of the sample because of the ortho–para conversion process. The normalized absorption line area $⟨ A_{j i}^{(k)} ⟩$ was determined from the linear fit of $\bar{A_{j i}^{(k)}} (f)$ for each absorption line,

\bar{A_{j i}^{(k)}} (f) = f ⟨ A_{j i}^{(k)} ⟩ .

(2)

Thus, the normalized absorption line area $⟨ A_{j i}^{(k)} ⟩$ is the absorption line area of a sample with a filling factor f = 1 and spin isomer fraction n_k = 1 where all the population is in the initial state, $p_{i}^{(k)} = 1$ ⁠. We used $⟨ A_{j i}^{(k)} ⟩$ to calculate a synthetic experimental spectrum for the spectral fit with the quantum mechanical model (Sec. IV B).

Furthermore, to compare the absorption cross sections of H₂O@C₆₀ in solid C₆₀ and free H₂O, a normalized absorption cross section $⟨ σ_{j i}^{(k)} ⟩$ was obtained as

\begin{matrix} 〈 σ_{j i}^{(k)} 〉 & = & {(\frac{3 \sqrt{η}}{η^{2} + 2})}^{2} \frac{σ_{j i}^{(k)}}{n_{k} (p_{i}^{(k)} - p_{j}^{(k)})} \\ = & {(\frac{3 \sqrt{η}}{η^{2} + 2})}^{2} \frac{〈 A_{j i}^{(k)} 〉}{N_{C_{60}}}, \end{matrix}

(3)

where η is the index of refraction of solid C₆₀ and $σ_{j i}^{(k)}$ is the absorption cross section of an endohedral water molecule [Eq. (20) in Sec. III B]. This H₂O@C₆₀ absorption cross section can be compared to the free water normalized cross section $⟨ σ_{j i}^{(k)} ⟩ = σ_{j i}^{(k)} n_{k}^{- 1} {(p_{i}^{(k)} - p_{j}^{(k)})}^{- 1}$ ⁠, where $p_{i}^{(k)} - p_{j}^{(k)}$ and n_k(T) are given by the temperature of the water vapor in the experiment reporting $σ_{j i}^{(k)}$ ⁠.

III. THEORY

A. Quantum mechanical model of H₂O@C₆₀: Confined vibrating rotor in an electrostatic field

We use the following Hamiltonian to model the endohedral water molecule in solid C₆₀:

H = H_{M} + H_{E S} + H_{T},

(4)

where $H_{M} = H_{v} + H_{r o t}$ is the free-molecule rovibrational Hamiltonian and $H_{E S}$ is the electrostatic interaction of H₂O with the surrounding electric charges. The translational Hamiltonian $H_{T}$ consists of water center-of-mass kinetic and potential energies in the molecular cavity of the C₆₀ molecule.

We neglect couplings between vibrational modes and between vibrational and rotational modes. In addition, the coupling between the translational motion and rotations is neglected. In $H_{E S}$ ⁠, terms describing the coupling of the solid C₆₀ crystal field to the electric dipole and quadrupole moments of H₂O are included.

The fitting of IR absorption spectra (Sec. IV B) is done with the Hamiltonian where the translational part is excluded,

H = H_{M} + H_{E S} .

(5)

We employ three coordinate frames. The space-fixed coordinate frame is denoted A. M = {x, y, z} is the molecule-fixed coordinate frame (Fig. 1). The Euler angles Ω_A→M^39,40 transform A to M. The crystal coordinate frame is C = {x′, y′, z′} with the z′ axis along the threefold symmetry axis of the S₆ point group, the symmetry group of the C₆₀ site in solid C₆₀. The Euler angles Ω_A→C transform A to C and Ω_C→M transform C to M. The coordinate systems A and C are used because the radiation interacting with the molecule is defined in the space-fixed coordinate frame A, while the local electrostatic fields are defined by the crystal coordinate frame C, which in a powder sample has a uniform distribution of orientations relative to the space-fixed frame A.

FIG. 1.

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(a) Molecule-fixed coordinate frame M = {x, y, z} and axes of principal moments of inertia, {a, b, c}. (b) Vibrations: v₁, symmetric stretch; v₂, symmetric bend; and v₃, asymmetric stretch. (c) para-water and ortho-water rotational energy levels in the ground and excited vibrational states (see Sec. III A 5) and rovibrational IR transitions (arrows) between the levels. The rotational states are labeled by $J_{K_{a} K_{c}}$ and p (para) and o (ortho). The IR transitions are between the para states or between the ortho states. 1, 3, and 5 are ortho-H₂O transitions, and 2 and 4 are para-H₂O transitions. Transitions 1 and 2 are forbidden for a free water molecule. The transitions where the one-quantum excitation of the asymmetric stretch vibration v₃ is involved are numbered 6 (para-H₂O), and 7 and 8 (ortho-H₂O transitions). The rotational and translational far-IR transitions in the ground vibrational states are shown in the inset of Fig. 2.

1. Vibrations

H₂O has three normal vibrations: the symmetric stretch of O–H bonds, denoted by the quantum number v₁, the bending motion of the H–O–H bond angle, v₂, and the asymmetric stretch of O–H bonds, v₃, ⁴⁰ as sketched in Fig. 1(b). The vibrational state is denoted $|v⟩$ ⁠, where the symbol v denotes the three vibrational quantum numbers, v ≡ v₁v₂v₃, each of which takes values v_i ∈ {0, 1, 2, …}. The vibrational energy for a harmonic vibrational potential is

E_{v} = \sum_{i = 1}^{3} ω_{i} (v_{i} + \frac{1}{2}), v_{i} = 0,1, 2, \dots,

(6)

where ω_i is the vibrational frequency of the ith vibration mode, i ∈ {1, 2, 3} and [ω_i] = cm⁻¹. Throughout this paper, the reported ω_i is the distance between the vibrational energy levels and is assumed to include anharmonic corrections.

2. Rotations

H₂O has the rotational properties of an asymmetric top with principal moments of inertia I_a < I_b < I_c.⁴⁰ The rotational states are indexed by three quantum numbers $J_{K_{a} K_{c}}$ ⁠, where J = 0, 1, 2, …, is the rotational angular momentum quantum number. K_a and K_c are the absolute values of the projection of J onto the a and c axes, in the limits of a prolate (I_a = I_b) and an oblate (I_b = I_c) top, respectively, K_a, K_c ≤ J.⁴⁰ Each rotational state $|J_{K_{a} K_{c}}, m⟩$ is (2J + 1)-fold degenerate, where m ∈ {−J, −J + 1, …, J} is the projection of J onto the z′ axis of the crystal coordinate frame C.

The energies and the wavefunctions, $|J_{K_{a} K_{c}}, m⟩$ ⁠, of the free rotor Hamiltonian $H_{r o t}$ depend on the rotational constants of an asymmetric top, A_v > B_v > C_v,⁴⁰

H_{r o t} = A_{v} Ĵ_{a}^{2} + B_{v} Ĵ_{b}^{2} + C_{v} Ĵ_{c}^{2},

(7)

where $Ĵ_{i}$ are the components of the angular momentum operator ${\hat{J}}^{2}$ along the principal directions a, b, and c. The index v labels the rotational constants in the vibrational state $|v⟩$ ⁠.

A molecule-fixed coordinate system M (axes x, y, and z, Fig. 1) with its origin at the nuclear center of mass is defined with the following orientations relative to the principal axes of the inertial tensor: x ∥ b, y ∥ c, and z ∥ a, where y is perpendicular to the H–O–H plane and x points toward the oxygen atom. The rotational wavefunctions of an asymmetric top in that basis are^39,40

\begin{gathered} |J, k, m, \pm⟩ = (|J, k, m⟩ \pm |J, - k, m⟩) / \sqrt{2}, \\ |J, 0, m, \pm⟩ \equiv |J, 0, m⟩ i f k = 0, \end{gathered}

(8)

where k is the projection of J on the a axis, axis z of M; for k > 0, k ∈ {1, 2, …, J} and each state is doubly degenerate. Here, $|J, k, m⟩$ denotes a normalized rotational function,

|J, k, m⟩ = \sqrt{\frac{2 J + 1}{8 π^{2}}} {[D_{m k}^{J} (Ω_{C \to M})]}^{*},

(9)

where the Euler angles Ω_C→M transform the crystal-fixed coordinate frame C into the molecule-fixed coordinate frame M and $D_{m k}^{J} (Ω)$ is the Wigner rotation matrix element or the Wigner D-function.^39,41

The correspondence between the asymmetric top wavefunctions $|J, k, m, \pm⟩$ [Eq. (8)] and the asymmetric top wavefunctions $|J_{K_{a} K_{c}}, m⟩$ is given in Ref. 40. The latter notation of wavefunctions is useful for the symmetry analysis and can be used to relate the wavefunction to para and ortho states of the water molecule (see Sec. III A 5) and to define the electric dipole transition selection rules [Eqs. (26) and (27)].

3. Electrostatic interactions

We assume two contributions to the electrostatic interaction,

H_{E S} = H_{Q} + H_{e d},

(10)

denoting the coupling of the quadrupole and dipole moments of H₂O to the corresponding multipole fields created by the surrounding charges.

a. Quadrupolar interaction.

It was shown by Felker et al.³² that C₆₀ molecules in neighboring lattice sites generate an electric field gradient at the center of a given C₆₀ molecule. In H₂O@C₆₀, the electric field gradient couples to the electric quadrupole moment of the water molecule, lifting the threefold degeneracy of the J = 1 ortho-H₂O rotational ground state.³²

The quadrupolar Hamiltonian may be expanded in rank-2 spherical tensors as follows: ^39,42

H_{Q} = \sum_{m = - 2}^{2} {(- 1)}^{m} V_{- m}^{(2)} Q_{m}^{C},

(11)

where ${V_{m}^{(2)}}$ are the spherical components of the electric field gradient tensor and ${Q_{m}^{C}}$ are the quadrupole moments of the water molecule, both expressed in the crystal-fixed coordinate frame C.

The experimental value of the H₂O quadrupole moment in the molecule-fixed coordinate frame M is given by {Q_xx, Q_yy, Q_zz} = {−0.13, −2.50, 2.63} esu × cm².⁴³ Since |Q_xx| ≪ |Q_yy|, |Q_zz|, it holds that Q_zz ≈ −Q_yy, and we may approximate the water quadrupole moment in spherical coordinates as follows:

\begin{matrix} {Q_{m}^{M}} & = & \{\frac{1}{2} (Q_{x x} - Q_{y y}), 0, \sqrt{\frac{3}{2}} Q_{z z}, 0, \frac{1}{2} (Q_{x x} - Q_{y y})\} \\ \approx & Q_{z z} \{\frac{1}{2}, 0, \sqrt{\frac{3}{2}}, 0, \frac{1}{2}\}, m = - 2, \dots, + 2 . \end{matrix}

(12)

The site symmetry of the C₆₀ molecule in solid C₆₀ is S₆, with the threefold symmetry axis along the cubic [111] axis, which is chosen here to be the z′ axis of the crystal coordinate frame C. The spherical tensor component in frame C is

{V_{m}^{(2) C}} = V_{Q} {0,0,1,0,0},

(13)

where m ∈ {−2, −1, 0, 1, 2} transforms like the fully symmetric A_g irreducible representation of the point group S₆.⁴⁴ After the transformation of the quadrupole moment (12) from the H₂O-fixed molecular frame M to the crystal frame C [see Eq. (A5)], the quadrupolar Hamiltonian (11) is given by

\begin{align} H_{Q} = & V_{Q} Q_{z z} [\sqrt{\frac{3}{2}} {[D_{00}^{2} (Ω_{C \to M})]}^{*}) \\ + (\frac{1}{2} {[D_{0, - 2}^{2} (Ω_{C \to M}) + D_{02}^{2} (Ω_{C \to M})]}^{*}] . \end{align}

(14)

b. Dipolar interaction.

Dielectric measurements of solid C₆₀ have provided evidence for the existence of electric dipoles in solid C₆₀.³⁵ We propose that these electric dipoles are the source of finite electric fields at the C₆₀ cage centers.

Consider a crystal electric field $E$ with spherical coordinates ${E, ϕ_{E}, θ_{E}}$ in the crystal-fixed frame C; see Appendix A. For simplicity, we assume a homogeneous crystal field with uniform orientation in the crystal-fixed frame. The interaction of the electric dipole moment with the electric field is given by

\begin{align} H_{e d} = & - \sum_{m = - 1}^{1} {(- 1)}^{m} E_{- m}^{E} μ_{m}^{E} \\ = & - \sum_{m^{'} = - 1}^{1} E {[D_{0 m^{'}}^{1} (Ω_{E \to C})]}^{*} \\ \times [\sum^{m^{″} = - 11} {[D_{m^{'} m^{″}}^{1} (Ω_{C \to M})]}^{*} μ_{m^{″}}^{M}], \end{align}

(15)

where the dipole moment in the molecule-fixed frame M is given by

{μ_{m}^{M}} = \frac{μ_{x}}{\sqrt{2}} {- 1,0,1}, m \in {- 1,0,1},

(16)

where μ_x is the permanent dipole moment of water in the Cartesian coordinates of frame M [Fig. 1(a)]. Since there are no other anisotropies than the axially symmetric electric field gradient tensor, the angle ϕ_E is arbitrary and we choose ϕ_E = 0.

The Hamiltonian (5) is diagonalized using the basis (8) up to J ≤ 4 for the ground vibrational state $|000⟩$ and for the three excited vibrational states $|100⟩$ ⁠, $|010⟩$ ⁠, and $|001⟩$ ⁠. The ground state and the three excited vibrational states are assumed to have independent rotational constants A_v, B_v, and C_v, where v = 000, 100, 010, or 001.

After separation of coordinates (see Appendix B 2), the quadrupole and dipole moments in Eqs. (14) and (15) are replaced by their expectation values, $⟨v| Q_{0}^{M} |v⟩$ ⁠, $⟨v| Q_{\pm 2}^{M} |v⟩$ ⁠, and $⟨v| μ_{\pm 1}^{M} |v⟩$ ⁠, in the ground and in the three excited vibrational states. We assume for simplicity that the dipole and quadrupole moments of H₂O are independent of the vibrational state $|v⟩$ ⁠.

4. Confined water translations: Spherical oscillator

The translational motion is the center-of-mass motion and is quantized for a confined molecule. The high icosahedral symmetry of the C₆₀ cavity is close to spherical symmetry, and therefore, the translational motion of the trapped molecule can be described by the three-dimensional isotropic spherical oscillator model.⁴⁵ For simplicity, we write the potential in the harmonic approximation as⁴²

V (R) = V_{2} R^{2},

(17)

where R is the displacement of the H₂O center of mass from the C₆₀ cage center. The C₆₀ cage is assumed rigid, and its center of mass is fixed.

The frequency of the spherical harmonic oscillator is

ω_{t}^{0} = \sqrt{2 V_{2} / m},

(18)

where m is the mass of a molecule moving in the potential and $[ω_{t}^{0}] = r a d s^{- 1}$ ⁠. The energy of the spherical harmonic oscillator is quantized,

E_{N} = ℏ ω_{t}^{0} (N + \frac{3}{2}),

(19)

where N is the translational quantum number, N ∈ {0, 1, 2, …}. The orbital quantum number L takes values L = N, N − 2, …, 1(0) for odd (even) N. The energy of the harmonic spherical oscillator does not depend on L, and in the isotropic approximation, there is an additional degeneracy of each E_N level in quantum number M_L, taking 2L + 1 values, M_L ∈ {−L, −L + 1, …, L}.

5. Nuclear spin isomers: para-water and ortho-water

The Pauli principle requires that the total quantum state is antisymmetric with respect to exchange of the two protons in water, which as spin-1/2 particles are fermions. This constraint leads to the existence of two nuclear spin isomers, with total nuclear spins I = 0 (para-H₂O) and I = 1 (ortho-H₂O), and different sets of rovibrational states. The antisymmetric nature of the quantum state has consequences on the IR spectra: only para to para and ortho to ortho transitions are allowed.

The ortho-H₂O states have odd values of K_a + K_c, while the para-H₂O states have even values of K_a + K_c in the ground vibrational state $|000⟩$ ⁠; see Fig. 1(c). The allowed rotational transitions are depicted in the inset of Fig. 2(a). The same rule applies to the excited vibrational states $|100⟩$ and $|010⟩$ [Fig. 1(c), upper left part]. However, the rules are inverted for the states $|001⟩$ ⁠, $|101⟩$ ⁠, and $|011⟩$ ⁠, which involve one-quantum excitation of the asymmetric stretch mode v₃. In these cases, ⁴⁰ para-H₂O has odd values of K_a + K_c, while ortho-H₂O has even values of K_a + K_c [Fig. 1(c), upper right part].

FIG. 2.

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Far-IR absorption spectra of H₂O@C₆₀ at 5 K. (a) Spectrum α(0) was measured after the temperature jump from 30 to 5 K (black), and the difference Δα = α(0) − α(Δt) was measured Δt = 44 h later (blue). The sample filling factor f = 0.05. Water rotational transitions corresponding to the absorption lines numbered 3 and 5 (ortho-water) and 4 (para-water) are shown in the inset. (b) Spectrum α(0) was measured after the temperature jump from 30 to 5 K (black), and the difference Δα = α(0) − α(Δt) was measured Δt = 5 h later (blue). The sample filling factor f = 0.18. The translational transitions N = 0 → N = 1 for para (T_p) and ortho (T_o1, T_o2) H₂O@C₆₀ are shown in the inset of panel (a). N is the quantum number of the spherical harmonic oscillator [Eq. (19)].

The energy difference between the lowest para rotational state $|0_{00}⟩$ and the lowest ortho rotational state $|1_{01}⟩$ is 2.6 meV (28 K).¹¹ Above 30 K, the ratio of ortho and para molecules is n_o/n_p ≈ 3. Hence, if the sample is cooled rapidly to 4 K, the number of para molecules slowly grows in the subsequent time interval, while the number of ortho molecules slowly decreases to the thermal equilibrium value n_o ≈ 0. The full conversion takes several hours.^11,17,30,37

B. Absorption cross section of H₂O@C₆₀

The strengths of the transitions between the rotational states of a polar molecule are determined by the permanent electric dipole moment of the molecule and by the electric field of the infrared radiation, corrected by the polarizability of the medium. In principle, the polarizability χ of the solid depends on the fraction f of C₆₀ cages that contain a water molecule, $χ (f) = χ_{C_{60}} + f χ_{H_{2} O @ C_{60}}$ ⁠. However, we found that, within the studied range of filling factors, f = 0.1–0.8, the absorption cross section of H₂O@C₆₀ was independent of f. Hence, only the polarizability of solid C₆₀ is relevant, $χ \approx χ_{C_{60}}$ ⁠, and the problem is similar to the optical absorption of an isolated impurity atom in a crystal.⁴⁶

Following Ref. 46, the electric field in the molecule embedded into the medium with an index of refraction η is $E_{e ff} = E (η^{2} + 2) / 3$ ⁠, where $E$ is the electric field of radiation in the vacuum. The refractive index of solid C₆₀ is η = 2, ⁴⁷ and hence, $E_{e ff} = 2 E$ ⁠.

In the following discussion, we use the index k ∈ {o, p} to indicate the ortho or para nuclear spin isomers. The absorption cross section⁴⁰ for a given nuclear spin isomer k, including the effective field correction, is given by

\begin{matrix} σ_{j i}^{(k)} & = & N_{k}^{- 1} \int_{L i n e} α_{j i}^{(k)} (ω) d ω \\ = & \frac{2 π^{2}}{h ε_{0} c_{0} η} {(\frac{η^{2} + 2}{3})}^{2} ω_{j i}^{(k)} (p_{i}^{(k)} - p_{j}^{(k)}) S_{j i}^{(k)}, \end{matrix}

(20)

where c₀ is the speed of light in the vacuum and ε₀ is the permittivity of vacuum; SI units are used, and the frequency ω is in number of waves per meter, [ω] = m⁻¹. The integral in (20) is the area of the absorption line of the transition from the state $|i⟩$ to $|j⟩$ ⁠.

The square of the electric dipole matrix element is given by

S_{j i}^{(k)} = \frac{1}{3} \sum_{σ = - 1}^{1} | ⟨j| μ_{σ}^{C} |i⟩ |^{2},

(21)

where $|i⟩$ and $|j⟩$ are the eigenstates with corresponding energies $E_{i}^{(k)}$ and $E_{j}^{(k)}$ ⁠. The symbol $μ_{σ}^{C}$ denotes the dipole moment components of a water molecule in the crystal-fixed frame C. This form of $S_{j i}^{(k)}$ is valid for random orientation of crystals in the powder sample and does not depend on the polarization of light; see Appendix B.

The absorption cross section $σ_{j i}^{(k)}$ was evaluated separately for para-water and ortho-water. The concentration of molecules is $N_{k} = f N_{C_{60}} n_{k}$ ⁠, where n_k is the ortho (or para) fraction and n_p + n_o = 1. The filling factor is denoted by f, and the number density of C₆₀ molecules in solid C₆₀ is given by $N_{C_{60}} = 1.419 \times 1 0^{21} {c m}^{- 3}$ ⁠.⁴⁸

$p_{i}^{(k)}$ and $p_{j}^{(k)}$ are the probabilities that the initial and final states are thermally populated,

p_{n}^{(k)} = {(Z^{(k)})}^{- 1} \exp (- \frac{E_{n}^{(k)} - E_{0}^{(k)}}{k_{B} T}),

(22)

where the statistical sum is

Z^{(k)} = \sum_{n} \exp (- \frac{E_{n}^{(k)} - E_{0}^{(k)}}{k_{B} T})

(23)

and $E_{0}^{(k)}$ is the ground state energy of para (ortho) molecules. At the low temperatures considered in this work, only the vibrational ground state is significantly populated. Thus, the significantly thermally populated states are rotational states in the ground vibrational states, which can be written as linear combinations of basis states in Eq. (8). Since we expect that the water molecule is not in a spherically symmetric environment in H₂O@C₆₀, the degeneracy in quantum number m is lifted, in general. Therefore, $p_{n}^{(k)}$ is the thermal population of a non-degenerate rotational state, and the eigenstates $|i⟩$ and $|j⟩$ in Eq. (21) include all possible m values for a given J.

When thermal equilibrium is reached between para-water and ortho-water, the fraction of the nuclear spin isomer k is

n_{k} (T) = \frac{g^{(k)} Z^{(k)} \exp (\frac{E_{0}^{(p)} - E_{0}^{(k)}}{k_{B} T})}{g^{(p)} Z^{(p)} + g^{(o)} Z^{(o)} \exp (\frac{E_{0}^{(p)} - E_{0}^{(o)}}{k_{B} T})} .

(24)

For the spin isomer k, the nuclear spin degeneracy g^(k) = 2I + 1, with I = 0 for para and I = 1 for ortho.

The absorption line areas are calculated from Eq. (20), where the matrix elements in Eq. (21) are between the eigenstates $|Φ_{r o t}^{v}⟩$ of the Hamiltonian given by Eq. (5). After separation of coordinates (see Appendix B 2), the matrix elements in the crystal-fixed coordinate frame C are

⟨j| μ_{σ}^{C} |i⟩ = \sum_{σ^{'} = - 1}^{1} ⟨Φ_{r o t}^{v^{'}}| {D_{σ σ^{'}}^{1} (Ω_{C \to M})}^{*} |Φ_{r o t}^{000}⟩ ⟨v^{'}| μ_{σ^{'}}^{M} |000⟩,

(25)

where the initial state is $|i⟩ = |000⟩ |Φ_{r o t}^{000}⟩$ and $|Φ_{r o t}^{v}⟩$ are the linear combinations of states (8). The dipole moments $⟨v^{'}| μ_{σ^{'}}^{M} |000⟩$ are given by Eqs. (B13) and (B14) in Cartesian coordinates.

The following selection rules hold for the electric dipole moment along the z axis (the transition dipole moment of asymmetric stretch v₃) and along the x axis (the permanent dipole moment and the transition dipole moments of v₁ and v₂):⁴⁰

Δ K_{a} = e v e n, Δ K_{c} = o d d i f μ_{z}^{M},

(26)

Δ K_{a} = o d d, Δ K_{c} = o d d i f μ_{x}^{M},

(27)

where the convention of molecular axes is as shown in Fig. 1(a) and the definition of dipole moments is given by Eqs. (B13) and (B14). Corresponding transitions are sketched in Figs. 1(c) and 2.

IV. RESULTS AND INTERPRETATION OF SPECTRA

A. Spectra

The water IR absorption lines were identified unambiguously by taking advantage of the slow ortho–para conversion at low temperature. After rapid cooling from 30 K to 5 K, the slow ortho–para conversion causes the intensity of the para lines to slowly increase, while the intensity of the ortho lines decreases. The water absorption lines are readily identified and assigned to one of the two spin isomers, by taking the difference between spectra acquired shortly after cooling and spectra acquired after an equilibration time at the lower temperature.

A group of H₂O lines, numbered 3, 4, and 5, is seen below 60 cm⁻¹ [Fig. 2(a)]. These far-IR absorption lines have been reported earlier and correspond to the rotational transitions of H₂O in the C₆₀ cage.¹¹ The rotational energy levels involved are shown in the inset of Fig. 2(a). Lines 3 and 5 are ortho-water rotational transitions starting from the ortho-water ground state $|1_{01}⟩$ ⁠. Line 4 is the para-water transition from the ground rotational state $|0_{00}⟩$ ⁠. No other rotational transitions were observed at 5 K, which is consistent with the selection rules for the electric dipole allowed rotational transitions from states $|0_{00}⟩$ and $|1_{01}⟩$ ⁠.⁴⁰

Further H₂O lines are observed around 110 cm⁻¹ [Fig. 2(b)] and in six spectral regions above 600 cm⁻¹, as shown in Figs. 3–6. Below, we address each wavenumber range separately and assign the spectral lines to the transitions shown in the energy schemes of Figs. 1(c) and 2(a). The absorption lines associated with transitions of free water, labeled 1–8, are listed in Table I. Line assignments are supported by the results of spectral fitting using the model of a vibrating rotor in a crystal field.

FIG. 3.

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Absorption spectra of H₂O@C₆₀ vibrational, rovibrational, and vibration–translational transitions of the bending vibration v₂ at 5 K. The spectrum α(0) was measured after the temperature jump from 30 to 5 K (black), and the difference Δα = α(0) − α(Δt) was measured Δt = 3 h later (blue). The sample filling factor f = 0.1. Lines numbered 1 and 2 are pure vibrational transitions and 3–5 are rovibrational transitions [Fig. 1(c)]. The left inset shows the para, line 1, and ortho, line 2, components of the pure vibrational transition difference spectrum with time delay 1 h of the f = 0.8 sample. The differential spectrum of v₂ vibration–translational transitions T_p, T_o1, and T_o2, at 110 cm⁻¹ higher frequency from 1 to 2, is shown in the right inset.

FIG. 4.

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Absorption spectra of H₂O@C₆₀ vibrational, rovibrational, and vibration–translational transitions of symmetric stretching, v₁, and anti-symmetric stretching, v₃, vibrations at 5 K. The spectrum α(0) was measured after the temperature jump from 30 to 5 K (black), and the difference Δα = α(0) − α(Δt) was measured Δt = 3 h later (blue dashed line multiplied by 4 and blue line). The sample filling factor f = 0.1. Lines numbered 1 and 2 are pure vibrational transitions and 3–5 are rovibrational transitions of mode v₁, and lines 6–8 are rovibrational transitions of v₃ [Fig. 1(c)]. (*) marks the v₃ rovibrational transition at 3654 cm⁻¹ from the thermally excited rotational state $|000⟩ |1_{10}⟩$ to the rovibrational state $|001⟩ |1_{11}⟩$ ⁠. The inset shows the para, line 1, and ortho, line 2, components of the pure vibrational transition of mode v₁ of the f = 0.8 sample as the difference spectrum with time delay 1 h.

FIG. 5.

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Absorption spectra of H₂O@C₆₀ at 10 K. (a) Overtone, 2ω₂, and (b) combination, 2ω₂ + ω₃, rovibrational transitions. The spectrum α(0) was measured after the temperature jump from 45 to 10 K (black), and the difference Δα = α(0) − α(Δt) was measured Δt = 2.75 h later (blue). Numbers 3–8 label the transitions shown in Fig. 1(c). The sample filling factor f = 0.8.

FIG. 6.

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Absorption spectra of H₂O@C₆₀ at 5 K. Rovibrational combination (a) ω₂ + ω₃ and (b) ω₁ + ω₃ transitions. The spectrum α(0) was measured after the temperature jump from 30 to 5 K (black), and the difference Δα = α(0) − α(Δt) was measured Δt = 0.83 h later (blue). Numbers 6–8 label the transitions shown in Fig. 1(c). (*) in (a) at 5202 cm⁻¹ marks the ω₂ + ω₃ rovibrational transition from the thermally excited rovibrational state $|000⟩ |1_{10}⟩$ to $|011⟩ |1_{11}⟩$ ⁠. The 2ω₁ rovibrational transition at 7059 cm⁻¹, marked (+) in (b), is from $|000⟩ |0_{00}⟩$ to $|200⟩ |1_{11}⟩$ ⁠. The sample filling factor f = 0.8.

TABLE I.

Rotational and rovibrational transition frequencies ω_ji (unit: cm⁻¹) and normalized absorption cross sections ⟨σ_ji⟩ (unit: cm per molecule) [Eq. (3)] from the ground para, $|0_{00}⟩$ ⁠, and ground ortho, $|1_{01}⟩$ ⁠, rotational states of H₂O@C₆₀ and of free H₂O. The initial vibrational state is $|000⟩$ for all transitions. The spectral lines are labeled by Nos. 1–8, each number associated with one pair of initial and final rotational states. The frequency $ω_{j i}^{C_{60}}$ of lines 3–8 is the intensity-weighted average of line sub-component frequencies. $ω_{j i}^{g a s}$ and $⟨ σ_{j i}^{g a s} ⟩$ are from Ref. 49. The para–ortho index k in $⟨ σ_{j i}^{(k)} ⟩$ has been dropped to simplify the notation.

${\|v_{1} v_{2} v_{3}⟩}_{j}$	No.	${\|J_{K_{a} K_{c}}⟩}_{i}$	${\|J_{K_{a} K_{c}}⟩}_{j}$	$ω_{j i}^{g a s}$	$ω_{j i}^{C_{60}}$	$⟨ σ_{j i}^{g a s} ⟩$	$⟨ σ_{j i}^{C_{60}} ⟩$	$⟨ σ_{j i}^{g a s} ⟩ / ⟨ σ_{j i}^{C_{60}} ⟩$
000	3	1₀₁	1₁₀	18.6	16.8	(3.97 ± 0.08)10⁻¹⁷	(3.24 ± 0.26)10⁻¹⁸	12.1
	4	0₀₀	1₁₁	37.1	33.6	(5.30 ± 0.11)10⁻¹⁷	(3.02 ± 0.19)10⁻¹⁸	17.3
	5	1₀₁	2₁₂	55.7	51.1	(1.191 ± 0.024)10⁻¹⁶	(7.5 ± 0.4)10⁻¹⁸	16.0
100	1	0₀₀	0₀₀	3657.1	3573.2		(3.4 ± 0.6)10⁻²⁰
	2	1₀₁	1₀₁
	3	1₀₁	1₁₀	3674.7	3589.1	(5.03 ± 0.05)10⁻¹⁹	(2.89 ± 0.03)10⁻¹⁹	1.74
	4	0₀₀	1₁₁	3693.3	3606.5	(3.01 ± 0.03)10⁻¹⁹	(1.93 ± 0.14)10⁻¹⁹	1.56
	5	1₀₁	2₁₂	3711.1	3623.5	(3.97 ± 0.04)10⁻¹⁹	(1.57 ± 0.12)10⁻¹⁹	2.52
010	1	0₀₀	0₀₀	1594.8	1569.3		(2.3 ± 0.6)10⁻¹⁹
	2	1₀₁	1₀₁
	3	1₀₁	1₁₀	1616.7	1588.5	(1.66 ± 0.03)10⁻¹⁷	(1.03 ± 0.11)10⁻¹⁸	16.1
	4	0₀₀	1₁₁	1635.0	1605.3	(1.132 ± 0.023)10⁻¹⁷	(6.79 ± 0.23)10⁻¹⁹	16.7
	5	1₀₁	2₁₂	1653.3	1623.4	(1.73 ± 0.04)10⁻¹⁷	(1.26 ± 0.09)10⁻¹⁸	13.7
001	6	0₀₀	1₀₁	3779.5	3682.1	(8.08 ± 0.09)10⁻¹⁸	(1.73 ± 0.21)10⁻¹⁸	4.66
	7	1₀₁	0₀₀	3732.1	3637.4	(8.64 ± 0.09)10⁻¹⁸	(1.31 ± 0.12)10⁻¹⁸	6.57
	8	1₀₁	2₀₂	3801.4	3703.7	(1.526 ± 0.015)10⁻¹⁷	(2.18 ± 0.10)10⁻¹⁸	6.99
011	6	0₀₀	1₀₁	5354.9	5228.0	(8.75 ± 0.09)10⁻¹⁹	(1.15 ± 0.06)10⁻¹⁹	7.63
	7	1₀₁	0₀₀	5307.5	5183.6	(8.97 ± 0.09)10⁻¹⁹	(8.22 ± 0.04)10⁻²⁰	10.9
	8	1₀₁	2₀₂	5376.9	5249.2	(1.703 ± 0.017)10⁻¹⁸	(1.34 ± 0.06)10⁻¹⁹	12.7
101	6	0₀₀	1₀₁	7273.0	7088.4	(5.5 ± 0.5)10⁻¹⁹	(1.261 ± 0.021)10⁻²⁰	43.4
	7	1₀₁	0₀₀	7226.0	7043.5	(5.7 ± 0.6)10⁻¹⁹	(1.044 ± 0.012)10⁻²⁰	54.6
	8	1₀₁	2₀₂	7294.1	7109.3	(1.07 ± 0.11)10⁻¹⁸	(1.85 ± 0.05)10⁻²⁰	57.7
020	4	0₀₀	1₁₁	3196.1	3142.0	(7.72 ± 0.07)10⁻²⁰	(5.5 ± 0.4)10⁻²¹	14.1

${\|v_{1} v_{2} v_{3}⟩}_{j}$	No.	${\|J_{K_{a} K_{c}}⟩}_{i}$	${\|J_{K_{a} K_{c}}⟩}_{j}$	$ω_{j i}^{g a s}$	$ω_{j i}^{C_{60}}$	$⟨ σ_{j i}^{g a s} ⟩$	$⟨ σ_{j i}^{C_{60}} ⟩$	$⟨ σ_{j i}^{g a s} ⟩ / ⟨ σ_{j i}^{C_{60}} ⟩$
000	3	1₀₁	1₁₀	18.6	16.8	(3.97 ± 0.08)10⁻¹⁷	(3.24 ± 0.26)10⁻¹⁸	12.1
	4	0₀₀	1₁₁	37.1	33.6	(5.30 ± 0.11)10⁻¹⁷	(3.02 ± 0.19)10⁻¹⁸	17.3
	5	1₀₁	2₁₂	55.7	51.1	(1.191 ± 0.024)10⁻¹⁶	(7.5 ± 0.4)10⁻¹⁸	16.0
100	1	0₀₀	0₀₀	3657.1	3573.2		(3.4 ± 0.6)10⁻²⁰
	2	1₀₁	1₀₁
	3	1₀₁	1₁₀	3674.7	3589.1	(5.03 ± 0.05)10⁻¹⁹	(2.89 ± 0.03)10⁻¹⁹	1.74
	4	0₀₀	1₁₁	3693.3	3606.5	(3.01 ± 0.03)10⁻¹⁹	(1.93 ± 0.14)10⁻¹⁹	1.56
	5	1₀₁	2₁₂	3711.1	3623.5	(3.97 ± 0.04)10⁻¹⁹	(1.57 ± 0.12)10⁻¹⁹	2.52
010	1	0₀₀	0₀₀	1594.8	1569.3		(2.3 ± 0.6)10⁻¹⁹
	2	1₀₁	1₀₁
	3	1₀₁	1₁₀	1616.7	1588.5	(1.66 ± 0.03)10⁻¹⁷	(1.03 ± 0.11)10⁻¹⁸	16.1
	4	0₀₀	1₁₁	1635.0	1605.3	(1.132 ± 0.023)10⁻¹⁷	(6.79 ± 0.23)10⁻¹⁹	16.7
	5	1₀₁	2₁₂	1653.3	1623.4	(1.73 ± 0.04)10⁻¹⁷	(1.26 ± 0.09)10⁻¹⁸	13.7
001	6	0₀₀	1₀₁	3779.5	3682.1	(8.08 ± 0.09)10⁻¹⁸	(1.73 ± 0.21)10⁻¹⁸	4.66
	7	1₀₁	0₀₀	3732.1	3637.4	(8.64 ± 0.09)10⁻¹⁸	(1.31 ± 0.12)10⁻¹⁸	6.57
	8	1₀₁	2₀₂	3801.4	3703.7	(1.526 ± 0.015)10⁻¹⁷	(2.18 ± 0.10)10⁻¹⁸	6.99
011	6	0₀₀	1₀₁	5354.9	5228.0	(8.75 ± 0.09)10⁻¹⁹	(1.15 ± 0.06)10⁻¹⁹	7.63
	7	1₀₁	0₀₀	5307.5	5183.6	(8.97 ± 0.09)10⁻¹⁹	(8.22 ± 0.04)10⁻²⁰	10.9
	8	1₀₁	2₀₂	5376.9	5249.2	(1.703 ± 0.017)10⁻¹⁸	(1.34 ± 0.06)10⁻¹⁹	12.7
101	6	0₀₀	1₀₁	7273.0	7088.4	(5.5 ± 0.5)10⁻¹⁹	(1.261 ± 0.021)10⁻²⁰	43.4
	7	1₀₁	0₀₀	7226.0	7043.5	(5.7 ± 0.6)10⁻¹⁹	(1.044 ± 0.012)10⁻²⁰	54.6
	8	1₀₁	2₀₂	7294.1	7109.3	(1.07 ± 0.11)10⁻¹⁸	(1.85 ± 0.05)10⁻²⁰	57.7
020	4	0₀₀	1₁₁	3196.1	3142.0	(7.72 ± 0.07)10⁻²⁰	(5.5 ± 0.4)10⁻²¹	14.1

1. Translational transitions

A group of absorption lines around 110 cm⁻¹ is shown in Fig. 2(b). These lines do not correspond to any known water rotational transitions. We assign these peaks to the translational transitions (N = 0 → N = 1) of para-H₂O and ortho-H₂O, corresponding to the quantized center-of-mass vibrational motions of the water molecules in the encapsulating C₆₀ cages. Here, N denotes the quantum number of a spherical harmonic oscillator.⁴²

The assignment of these peaks to water center-of-mass translational oscillations is supported by the presence of lines at 1680 cm⁻¹, visible in the difference spectrum shown in the right-hand side inset of Fig. 3. These lines are 110 cm⁻¹ higher than the vibrational transitions 1 and 2 of the v₂ mode and correspond to the simultaneous excitation of the v₂ vibration and the translational modes. A similar combination has been observed in H₂@C₆₀ where a group of lines between 4240 and 4270 cm⁻¹ is the translational sideband to the H₂ stretching vibration.¹⁰

The translational side peak of the v₁ vibrational mode is expected at about 3683 cm⁻¹. However, this frequency coincides with a strong rovibrational absorption line 6 of the v₃ mode (Fig. 4), which probably obscures the 3683 cm⁻¹ translational side peak of the v₁ vibration.

The translational ortho transitions display a splitting of 2.9 cm⁻¹ in the ground vibrational state; see Fig. 2(b), and 2.7 cm⁻¹ in the excited vibrational state $|010⟩$ ⁠; see Fig. 4. These splittings may be attributed to the coupling between the water translation and rotation, associated with the interaction of the non-spherical rotating water molecule with the interior of the C₆₀ cage. The spectral structure of this type has been analyzed in detail for the case of H₂@C₆₀.^8,10,12 The simplified theoretical model used here does not include translation–rotation coupling and cannot explain these splittings. A theoretical analysis of the translational peaks will be given in a later paper.

2. Vibrational and rovibrational transitions

The vibrational and rovibrational transitions are shown in Fig. 3. The three major features that distinguish the spectrum of H₂O@C₆₀ from the spectrum of free water are as follows.

a. Pure vibrational transitions.

Absorptions corresponding to pure vibrational transitions, i.e., without simultaneous rotational excitation, are present around ω₁ = 3573 cm⁻¹ and ω₂ = 1570 cm⁻¹. Both features are split into two components, labeled 1 and 2, identified from the difference spectra (Figs. 3 and 4) as para (1) and ortho (2) transitions.

Transition 1 is a transition from the ground vibrational state to the excited vibrational state without a change in the rotational state $|0_{00}⟩$ ⁠. Transition 2 is a vibrational excitation without a change in the rotational state $|1_{01}⟩$ ⁠. The corresponding transitions are forbidden because they violate the selection rule ΔK_a = odd and ΔK_c = odd [Eq. (27)]. As will be discussed in Sec. V E, the pure vibrational transitions 1 and 2 can be activated by the crystal electric field of solid H₂O@C₆₀.

The ortho–para splitting, i.e., the separation of lines 1 and 2, is 0.5 cm⁻¹ for the v₁ vibrational mode and 1.8 cm⁻¹ for the v₂ vibrational mode.

b. Spectral splittings.

The rovibrational transitions 3–8 are split into two or more components (see Figs. 3 and 4). These splittings, as in the case of the rotational transitions, are absent for a water molecule in the gas phase. Moreover, transitions 3–5 have the same splitting pattern as the rotational transitions in the ground vibrational state, also labeled 3–5; see Fig. 2. Transitions 6 and 7 are special since they are between the $|0_{00}⟩$ and $|1_{01}⟩$ rotational states and, thus, reflect directly the splitting of the triply degenerate $|1_{01}⟩$ state either in the ground vibrational state, transition 7, or in the excited vibrational state, transition 6 (Fig. 4).

We assign a weak ortho line at 3654 cm⁻¹, marked by ^* in Fig. 4, to the v₃ rovibrational transition from the thermally excited rovibrational state $|000⟩ |1_{10}⟩$ to $|001⟩ |1_{11}⟩$ ⁠. This assignment is further confirmed by calculating the transition frequency³⁹ with the parameters from Table III: $ω_{3} + E_{1_{11}} - E_{1_{10}} = ω_{3} + A_{001} + C_{001} - (A_{000} + B_{000}) = 3655 {c m}^{- 1}$ ⁠.

c. Red shifts.

The frequencies of vibrations are red-shifted relative to free H₂O. The stretching mode frequencies are red-shifted by about 2.4%, while the bending mode frequencies are red-shifted by about 1.6%; see Table II.

TABLE II.

Frequencies of symmetric stretching (ω₁), symmetric bending (ω₂), and asymmetric stretching (ω₃) modes and of their combinations 2ω₂, ω₂ + ω₃, 2ω₂ + ω₃, 2ω₁, and ω₁ + ω₃ measured from the ground vibrational state $|000⟩$ for H₂O@C₆₀ (this work) and for free H₂O.⁵⁰ ortho and para components of pure vibrational transitions v₁ and v₂ are indicated by superscripts o and p. The frequencies of transitions involving v₃ in H₂O@C₆₀ are the average of frequencies of transitions 6 and 7; see Figs. 4, 5(b), and 6. The overtone 2ω₂ is estimated from the frequency of line 4 [Fig. 5(a)], $2 ω_{2} + E_{1_{11}} = 3141 {c m}^{- 1}$ ⁠, where $E_{1_{11}} = A + C$ is the energy of the rotational state $|1_{11}⟩$ ⁠,³⁹ in which A = A₀₁₀ and C = C₀₁₀ from Table III. The 2ω₁ overtone frequency is estimated from the frequency of the para line $2 ω_{1} + E_{1_{11}} = 7059 {c m}^{- 1}$ [Fig. 6(b)], where $E_{1_{11}} = A_{100} + C_{100}$ ⁠. The shift $Δ ω = ω_{C_{60}} - ω_{g a s}$ ⁠.

$\|v_{1} v_{2} v_{3}⟩$	H₂O@C₆₀ cm⁻¹	Free H₂O cm⁻¹	Δω cm⁻¹	Δω/ω_gas
$\|010⟩$	1569.2^o	1594.8	−25.6	−0.016
	1571.0^p
$\|100⟩$	3573.2^o	3657.1	−83.9	−0.023
	3573.7^p
$\|001⟩$	3659.6	3755.3	−95.7	−0.025
$\|020⟩$	3105.5	3151.6	−46.1	−0.015
$\|011⟩$	5205.2	5331.3	−126.1	−0.024
$\|021⟩$	6716.6	6871.5	−154.9	−0.023
$\|200⟩$	7027.0	7201.5	−173.5	−0.024
$\|101⟩$	7065.8	7249.8	−184.0	−0.025

$\|v_{1} v_{2} v_{3}⟩$	H₂O@C₆₀ cm⁻¹	Free H₂O cm⁻¹	Δω cm⁻¹	Δω/ω_gas
$\|010⟩$	1569.2^o	1594.8	−25.6	−0.016
	1571.0^p
$\|100⟩$	3573.2^o	3657.1	−83.9	−0.023
	3573.7^p
$\|001⟩$	3659.6	3755.3	−95.7	−0.025
$\|020⟩$	3105.5	3151.6	−46.1	−0.015
$\|011⟩$	5205.2	5331.3	−126.1	−0.024
$\|021⟩$	6716.6	6871.5	−154.9	−0.023
$\|200⟩$	7027.0	7201.5	−173.5	−0.024
$\|101⟩$	7065.8	7249.8	−184.0	−0.025

3. Overtone and combination rovibrational transitions

Overtone and combination vibrational transitions where two vibrational quanta are excited are presented for the 2ω₂ transition in Fig. 5(a) and for the ω₂ + ω₃ and ω₁ + ω₃ transitions in Figs. 6(a) and 6(b). A three-quantum transition, 2ω₂ + ω₃, is shown in Fig. 5(b). Rotational levels involved are sketched in Fig. 1(c). Again, the splitting pattern of each higher order rovibrational transition is similar to the splitting of the rotational transition with Δv_i = 0 (Fig. 2) and rovibrational transitions with Δv_i = +1 (Figs. 3 and 4).

We assign a line marked by “+” at 7059 cm⁻¹ to 2ω₁ plus para-H₂O rotational transition, 0₀₀ → 1₁₁ [Fig. 6(b)]. Another two rotational side peaks of 2ω₁ are ortho transitions 3 and 5 expected at $2 ω_{1} + E_{1_{10}} - E_{1_{01}} = 7042 {c m}^{- 1}$ and at $2 ω_{1} + E_{2_{12}} - E_{1_{01}} = 7076 {c m}^{- 1}$ ⁠, where $E_{1_{10}} - E_{1_{01}} = A - C$ and $E_{2_{12}} - E_{1_{01}} = A + 3 C$ ⁠,³⁹ with the approximation A = A₂₀₀ ≈ A₁₀₀, C = C₂₀₀ ≈ C₁₀₀. The numerical values of A₁₀₀ and C₁₀₀ are taken from Table III. The first line overlaps with line 7, rovibrational transition of ω₁ + ω₃. The second line is not observed, but this could be due to the low intensity of ortho line 5 relative to para line 4; see, for example, Fig. 5(a).

TABLE III.

Parameters obtained from the quantum mechanical model fit of IR spectra of H₂O@C₆₀ at T = 5 K (Fig. 7). The vibration frequencies ω_i, rotational constants A_v, B_v, C_v, and quadrupolar energy V_QQ_zz are in units of cm⁻¹; electric field $E$ in 10⁶ V m⁻¹; θ_E in radians; and dipole moment μ^x (B13) and transition dipole moments $μ_{0 i}^{x, z}$ (B14) in D. It is assumed that μ^x and Q_zz do not depend on the vibrational state. The parameters with zero error were not fitted.

Parameter	Value	Error
n_o	0.7	0
f	1	0
ω₁	3574.1	0.3
ω₂	1569.2	0.3
ω₃	3659.9	0.9
A₀₀₀	24.15	0.17
B₀₀₀	15.3	0.8
C₀₀₀	8.48	0.07
A₁₀₀	23.1	0.3
B₁₀₀	14.3	0.8
C₁₀₀	8.50	0.09
A₀₁₀	26.7	0.3
B₀₁₀	14.6	0.9
C₀₁₀	8.81	0.07
A₀₀₁	26	6
B₀₀₁	15	3
C₀₀₁	8.2	1.8
μ^x	0.474	0.008
$μ_{01}^{x}$	1.031 × 10⁻²	0.021 × 10⁻²
$μ_{02}^{x}$	3.40 × 10⁻²	0.05 × 10⁻²
$μ_{03}^{z}$	2.83 × 10⁻²	0.07 × 10⁻²
$E$	110	5
θ_E	1.4	1.0
V_QQ_zz	−5.0	0.5

Parameter	Value	Error
n_o	0.7	0
f	1	0
ω₁	3574.1	0.3
ω₂	1569.2	0.3
ω₃	3659.9	0.9
A₀₀₀	24.15	0.17
B₀₀₀	15.3	0.8
C₀₀₀	8.48	0.07
A₁₀₀	23.1	0.3
B₁₀₀	14.3	0.8
C₁₀₀	8.50	0.09
A₀₁₀	26.7	0.3
B₀₁₀	14.6	0.9
C₀₁₀	8.81	0.07
A₀₀₁	26	6
B₀₀₁	15	3
C₀₀₁	8.2	1.8
μ^x	0.474	0.008
$μ_{01}^{x}$	1.031 × 10⁻²	0.021 × 10⁻²
$μ_{02}^{x}$	3.40 × 10⁻²	0.05 × 10⁻²
$μ_{03}^{z}$	2.83 × 10⁻²	0.07 × 10⁻²
$E$	110	5
θ_E	1.4	1.0
V_QQ_zz	−5.0	0.5

All two-quantum and three-quantum vibrational transitions are red-shifted approximately by 2.4% except 2ω₂ where the red shift is 1.5%; see Table II.

B. Spectral fitting with a quantum mechanical model

A synthetic spectrum consisting of Gaussian lines with full width at half maximum 1.5 cm⁻¹ was calculated from the experimental normalized line areas $⟨ A_{j i}^{(k)} ⟩$ [Eq. (2)] using f = 1, n_o = 0.7, n_p = 0.3, and T = 5 K. The parameters of the model Hamiltonian (from Sec. III A) and the transition dipole moments (from Sec. III B) were determined with a non-linear least squares method by minimizing the difference in synthetic and modeled spectra squared, Appendix C. The reported parameter error is the average of errors calculated with +δa_ν and −δa_ν in Eq. (C9), where the parameter variation of the νth parameter at its best value $a_{m i n}^{ν}$ is $| δ a_{ν} | = 0.005 a_{m i n}^{ν}$ ⁠. The fit was applied to the rotational transitions in the ground vibrational state $|000⟩$ and to the rovibrational transitions from the ground state to the vibrational states $|010⟩$ ⁠, $|100⟩$ ⁠, and $|001⟩$ ⁠. In total, 45 absorption lines were fitted.

The synthetic experimental spectra and the best fit spectra are shown in Fig. 7, with the best fit parameters given in Table III. The result of the fit overlaps well with the synthetic spectrum, except for transition 5 as seen in the first three panels of Fig. 7. While, for other transitions, one or two Gaussian components were sufficient, the experimental transition lineshape required three components to get a reliable fit of its line area. In addition, transitions 6 and 7 were represented by two components in the synthetic spectrum, although four peaks are seen in the experimental spectrum (Fig. 4). The additional structure of experimental peaks may originate from the merohedral disorder as discussed in Sec. V E.

FIG. 7.

$FIG. 7. Synthetic experimental spectra (black solid line) and spectra calculated with the best fit parameters from Table III (blue dashed line) with the filling factor f = 1 and ortho fraction no = 0.7 and at a temperature of 5 K. (a) Rotational transitions in the ground vibrational state. Vibrational (1 and 2) and rovibrational (3–5) transitions of (b) v2 and (c) v1. (d) Rovibrational transitions (6–8) of v3.$

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Synthetic experimental spectra (black solid line) and spectra calculated with the best fit parameters from Table III (blue dashed line) with the filling factor f = 1 and ortho fraction n_o = 0.7 and at a temperature of 5 K. (a) Rotational transitions in the ground vibrational state. Vibrational (1 and 2) and rovibrational (3–5) transitions of (b) v₂ and (c) v₁. (d) Rovibrational transitions (6–8) of v₃.

Tables IV and V list energies and the main components of rotational states in the ground vibrational state. The rotational energies are in qualitative agreement with recent computational estimates.⁵¹ The 2J + 1 degeneracy of rotational states is fully removed by the electrostatic field interacting with dipole and quadrupole moments of H₂O.

TABLE IV.

para-H₂O@C₆₀ rotational energies and wavefunctions in the ground vibrational state calculated with the best fit parameters from Table III. Wavefunction components with the absolute value of the amplitude less than 0.1 are omitted; $|J, - k, - m⟩ \equiv |J, \bar{k}, \bar{m}⟩$ ⁠.

$J_{K_{a} K_{c}}$	Energy (cm⁻¹)	Wavefunction in the symmetric top basis $\|J, k, m⟩$
0₀₀	0	−0.99 $\|0,0,0⟩$
1₁₁	33.07	0.49 $(\|1, \bar{1}, \bar{1}⟩ + \|1, \bar{1}, 1⟩ - \|1,1, \bar{1}⟩ - \|1,1,1⟩)$
	33.19	−0.69 $(\|1, \bar{1}, 0⟩ - \|1,1,0⟩)$
	33.75	0.48 $(\|1, \bar{1}, \bar{1}⟩ - \|1, \bar{1}, 1⟩ - \|1,1, \bar{1}⟩ + \|1,1,1⟩)$
2₀₂	67.04	−0.96 $\|2,0,0⟩ + 0.15 (\|2, \bar{2}, 0⟩ + \|2,2,0⟩)$
	67.73	0.68 $(\|2,0, \bar{1}⟩ + \|2,0,1⟩)$
	68.35	0.68 $(\|2,0, \bar{1}⟩ - \|2,0,1⟩)$
	71.27	0.68 $(\|2,0, \bar{2}⟩ - \|2,0,2⟩)$
	71.30	−0.68 $(\|2,0, \bar{2}⟩ + \|2,0,2⟩)$
2₁₁	94.65	−0.4 $(\|2, \bar{1}, 0⟩ + \|2,1,0⟩) + 0.41 (\|2, \bar{1}, \bar{2}⟩ + \|2, \bar{1}, 2⟩ + \|2,1, \bar{2}⟩ + \|2,1,2⟩)$
	94.72	−0.49 $(\|2, \bar{1}, \bar{2}⟩ - \|2, \bar{1}, 2⟩ + \|2,1, \bar{2}⟩ - \|2,1,2⟩)$
	94.85	−0.53 $(\|2, \bar{1}, 0⟩ + \|2,1,0⟩) - 0.24 (\|2, \bar{1}, \bar{2}⟩ + \|2, \bar{1}, 2⟩ + \|2,1, \bar{2}⟩ + \|2,1,2⟩)$
		$- 0.21 (- \|2, \bar{1}, \bar{1}⟩ + \|2, \bar{1}, 1⟩ - \|2,1, \bar{1}⟩ + \|2,1,1⟩)$
	94.91	0.45 $(- \|2, \bar{1}, \bar{1}⟩ + \|2, \bar{1}, 1⟩ - \|2,1, \bar{1}⟩ + \|2,1,1⟩) - 0.21 (\|2, \bar{1}, 0⟩ + \|2,1,0⟩)$
		$- 0.15 (\|2, \bar{1}, \bar{2}⟩ + \|2, \bar{1}, 2⟩ + \|2,1, \bar{2}⟩ + \|2,1,2⟩)$
	95.01	0.49 $(\|2, \bar{1}, \bar{1}⟩ + \|2, \bar{1}, 1⟩ + \|2,1, \bar{1}⟩ + \|2,1,1⟩)$

$J_{K_{a} K_{c}}$	Energy (cm⁻¹)	Wavefunction in the symmetric top basis $\|J, k, m⟩$
0₀₀	0	−0.99 $\|0,0,0⟩$
1₁₁	33.07	0.49 $(\|1, \bar{1}, \bar{1}⟩ + \|1, \bar{1}, 1⟩ - \|1,1, \bar{1}⟩ - \|1,1,1⟩)$
	33.19	−0.69 $(\|1, \bar{1}, 0⟩ - \|1,1,0⟩)$
	33.75	0.48 $(\|1, \bar{1}, \bar{1}⟩ - \|1, \bar{1}, 1⟩ - \|1,1, \bar{1}⟩ + \|1,1,1⟩)$
2₀₂	67.04	−0.96 $\|2,0,0⟩ + 0.15 (\|2, \bar{2}, 0⟩ + \|2,2,0⟩)$
	67.73	0.68 $(\|2,0, \bar{1}⟩ + \|2,0,1⟩)$
	68.35	0.68 $(\|2,0, \bar{1}⟩ - \|2,0,1⟩)$
	71.27	0.68 $(\|2,0, \bar{2}⟩ - \|2,0,2⟩)$
	71.30	−0.68 $(\|2,0, \bar{2}⟩ + \|2,0,2⟩)$
2₁₁	94.65	−0.4 $(\|2, \bar{1}, 0⟩ + \|2,1,0⟩) + 0.41 (\|2, \bar{1}, \bar{2}⟩ + \|2, \bar{1}, 2⟩ + \|2,1, \bar{2}⟩ + \|2,1,2⟩)$
	94.72	−0.49 $(\|2, \bar{1}, \bar{2}⟩ - \|2, \bar{1}, 2⟩ + \|2,1, \bar{2}⟩ - \|2,1,2⟩)$
	94.85	−0.53 $(\|2, \bar{1}, 0⟩ + \|2,1,0⟩) - 0.24 (\|2, \bar{1}, \bar{2}⟩ + \|2, \bar{1}, 2⟩ + \|2,1, \bar{2}⟩ + \|2,1,2⟩)$
		$- 0.21 (- \|2, \bar{1}, \bar{1}⟩ + \|2, \bar{1}, 1⟩ - \|2,1, \bar{1}⟩ + \|2,1,1⟩)$
	94.91	0.45 $(- \|2, \bar{1}, \bar{1}⟩ + \|2, \bar{1}, 1⟩ - \|2,1, \bar{1}⟩ + \|2,1,1⟩) - 0.21 (\|2, \bar{1}, 0⟩ + \|2,1,0⟩)$
		$- 0.15 (\|2, \bar{1}, \bar{2}⟩ + \|2, \bar{1}, 2⟩ + \|2,1, \bar{2}⟩ + \|2,1,2⟩)$
	95.01	0.49 $(\|2, \bar{1}, \bar{1}⟩ + \|2, \bar{1}, 1⟩ + \|2,1, \bar{1}⟩ + \|2,1,1⟩)$

TABLE V.

ortho-H₂O@C₆₀ rotational energies, counted from the para ground state $|0_{00}⟩$ ⁠, and wavefunctions in the ground vibrational state calculated with the best fit parameters from Table III. Wavefunction components with the absolute value of the amplitude less than 0.1 are omitted; $|J, - k, - m⟩ \equiv |J, \bar{k}, \bar{m}⟩$ ⁠.

$J_{K_{a} K_{c}}$	Energy (cm⁻¹)	Wavefunction in the symmetric top basis $\|J, k, m⟩$
1₀₁	20.89	0.97 $\|1,0,0⟩$
	24.58	−0.69 $(\|1,0, \bar{1}⟩ + \|1,0,1⟩) + 0.16 (\|1, \bar{1}, 0⟩ + \|1,1,0⟩)$
	25.51	−0.7 $(\|1,0, \bar{1}⟩ - \|1,0,1⟩)$
1₁₀	38.88	0.49 $(\|1, \bar{1}, \bar{1}⟩ - \|1, \bar{1}, 1⟩ + \|1,1, \bar{1}⟩ - \|1,1,1⟩)$
	40.03	0.48 $(\|1, \bar{1}, \bar{1}⟩ + \|1, \bar{1}, 1⟩ + \|1,1, \bar{1}⟩ + \|1,1,1⟩) + 0.24 \|1,0,0⟩$
	43.62	0.69 $(\|1, \bar{1}, 0⟩ + \|1,1,0⟩)$
2₁₂	72.29	0.71 $(\|2, \bar{1}, 0⟩ - \|2,1,0⟩)$
	73.12	−0.49 $(\|2, \bar{1}, \bar{1}⟩ + \|2, \bar{1}, 1⟩ - \|2,1, \bar{1}⟩ - \|2,1,1⟩)$
	73.36	−0.49 $(\|2, \bar{1}, \bar{1}⟩ - \|2, \bar{1}, 1⟩ - \|2,1, \bar{1}⟩ + \|2,1,1⟩)$
	75.87	−0.49 $(\|2, \bar{1}, \bar{2}⟩ - \|2, \bar{1}, 2⟩ - \|2,1, \bar{2}⟩ + \|2,1,2⟩)$
	75.88	−0.49 $(\|2, \bar{1}, \bar{2}⟩ + \|2, \bar{1}, 2⟩ - \|2,1, \bar{2}⟩ - \|2,1,2⟩)$
2₂₁	119.5	−0.49 $(\|2, \bar{2}, \bar{2}⟩ + \|2, \bar{2}, 2⟩ - \|2,2, \bar{2}⟩ - \|2,2,2⟩)$
	119.5	0.49 $(\|2, \bar{2}, \bar{2}⟩ - \|2, \bar{2}, 2⟩ - \|2,2, \bar{2}⟩ + \|2,2,2⟩)$
	122.2	0.49 $(\|2, \bar{2}, \bar{1}⟩ - \|2, \bar{2}, 1⟩ - \|2,2, \bar{1}⟩ + \|2,2,1⟩)$
	122.2	0.49 $(\|2, \bar{2}, \bar{1}⟩ + \|2, \bar{2}, 1⟩ - \|2,2, \bar{1}⟩ - \|2,2,1⟩)$
	123.1	0.71 $(\|2, \bar{2}, 0⟩ - \|2,2,0⟩)$

$J_{K_{a} K_{c}}$	Energy (cm⁻¹)	Wavefunction in the symmetric top basis $\|J, k, m⟩$
1₀₁	20.89	0.97 $\|1,0,0⟩$
	24.58	−0.69 $(\|1,0, \bar{1}⟩ + \|1,0,1⟩) + 0.16 (\|1, \bar{1}, 0⟩ + \|1,1,0⟩)$
	25.51	−0.7 $(\|1,0, \bar{1}⟩ - \|1,0,1⟩)$
1₁₀	38.88	0.49 $(\|1, \bar{1}, \bar{1}⟩ - \|1, \bar{1}, 1⟩ + \|1,1, \bar{1}⟩ - \|1,1,1⟩)$
	40.03	0.48 $(\|1, \bar{1}, \bar{1}⟩ + \|1, \bar{1}, 1⟩ + \|1,1, \bar{1}⟩ + \|1,1,1⟩) + 0.24 \|1,0,0⟩$
	43.62	0.69 $(\|1, \bar{1}, 0⟩ + \|1,1,0⟩)$
2₁₂	72.29	0.71 $(\|2, \bar{1}, 0⟩ - \|2,1,0⟩)$
	73.12	−0.49 $(\|2, \bar{1}, \bar{1}⟩ + \|2, \bar{1}, 1⟩ - \|2,1, \bar{1}⟩ - \|2,1,1⟩)$
	73.36	−0.49 $(\|2, \bar{1}, \bar{1}⟩ - \|2, \bar{1}, 1⟩ - \|2,1, \bar{1}⟩ + \|2,1,1⟩)$
	75.87	−0.49 $(\|2, \bar{1}, \bar{2}⟩ - \|2, \bar{1}, 2⟩ - \|2,1, \bar{2}⟩ + \|2,1,2⟩)$
	75.88	−0.49 $(\|2, \bar{1}, \bar{2}⟩ + \|2, \bar{1}, 2⟩ - \|2,1, \bar{2}⟩ - \|2,1,2⟩)$
2₂₁	119.5	−0.49 $(\|2, \bar{2}, \bar{2}⟩ + \|2, \bar{2}, 2⟩ - \|2,2, \bar{2}⟩ - \|2,2,2⟩)$
	119.5	0.49 $(\|2, \bar{2}, \bar{2}⟩ - \|2, \bar{2}, 2⟩ - \|2,2, \bar{2}⟩ + \|2,2,2⟩)$
	122.2	0.49 $(\|2, \bar{2}, \bar{1}⟩ - \|2, \bar{2}, 1⟩ - \|2,2, \bar{1}⟩ + \|2,2,1⟩)$
	122.2	0.49 $(\|2, \bar{2}, \bar{1}⟩ + \|2, \bar{2}, 1⟩ - \|2,2, \bar{1}⟩ - \|2,2,1⟩)$
	123.1	0.71 $(\|2, \bar{2}, 0⟩ - \|2,2,0⟩)$

From our fit, the permanent dipole moment μ^x of the encapsulated water is given by the absorption cross section of the IR rotational transitions 3–5 in the ground vibrational state. With the value of μ^x in hand and by using the intensities of transitions 1 and 2, we were able to determine the internal static electric field in solid C₆₀. The interaction of μ^x with the crystal electric field mixes rotational states within ground and excited vibrational states. For example, in case of ortho-water, the components of state $|1_{10}⟩$ are mixed into the ground state $|1_{01}⟩$ ⁠, Table V. This mixing gives the oscillator strength to the pure vibrational transitions 1 and 2. As shown in Table III, the fitted value of the electric field at the C₆₀ cage centers is (110 ± 5)10⁶ V m⁻¹.

A splitting of 4 cm⁻¹ is observed for transition 7 and is due to the splitting of the ortho ground state $|1_{01}⟩$ [Fig. 1(c)]. In principle, a splitting could be caused by the interaction of the water electric dipole with an electric field, or by the interaction of the water electric quadrupole moment with an electric field gradient. The electric field 110 × 10⁶ V m⁻¹ is too small to cause splitting of this magnitude. This electric field lifts the degeneracy of m = ±1 levels of $|1_{01}⟩$ ⁠, but the gap between m = 0 and m = ±1 levels is due to the quadrupolar interaction. As the splitting is determined by the product of V_Q and Q_zz, it is not possible to have an estimate of how much is the water quadrupole moment Q_zz screened in C₆₀.

V. DISCUSSION

A. Vibrations of confined H₂O

All eight frequencies of the encapsulated H₂O vibrations found in this work are red-shifted relative to those of free water; see Table II. The red shift of the vibrational frequency has been observed for other endofullerenes, H₂@C₆₀, ^10,12 HD and D₂@C₆₀,⁸ and HF@C₆₀.⁵ Six water modes have a relative shift between −2.3% and −2.5%. Two frequencies, namely, the bond-bending mode frequency ω₂ and its overtone frequency 2ω₂, are shifted by −1.6% and −1.5%, respectively.

The observed red shifts of the stretching mode frequencies ω₁ and ω₃ are only partially consistent with previous density functional theory (DFT) calculations. The DFT-based calculations published by Varadwaj et al.⁵² do predict vibrational red shifts, while some of the calculations reported by Farimani et al.⁵³ predict blue shifts rather than red shifts.

The calculation predicts a blue shift of the bending mode ω₂ although ten times less in absolute value than the predicted shift of stretching modes.⁵² The experimental shift of ω₂ is less than that of stretching modes, but it is still red-shifted. The other method, fully coupled nine-dimensional calculation, predicts blue shifts for all three vibrational modes.⁵¹

B. Translations of H₂O

Table VI lists the measured translational energies from the ground to the first excited state, ω_t, of small-molecule endofullerenes. It is known that the potential of di-hydrogen in C₆₀ is anharmonic, ^8,12 while the degree of anharmonicity of HF@C₆₀ and H₂O@C₆₀ potentials is not known. For simplicity, we assume that the potential is harmonic for the current case of endohedral molecules, $ω_{t} \approx ω_{t}^{0}$ ⁠, and show its scaling relative to H₂ in Table VI. In this approximation, the harmonic potential parameter is similar among the hydrogen isotopologs, but for HF and H₂O, V₂ is larger by a factor of 1.9 and 3.4, respectively. The steeper translational potential for HF and H₂O, relative to dihydrogen, is consistent with the larger size of these molecules and, hence, their tighter confinement. The last line of Table VI is the frequency and the harmonic potential of H₂O@C₆₀ derived by Bacic and co-workers^16,51 using Lennard-Jones potentials. The calculated potential is steeper than the experimentally determined potential.

TABLE VI.

Translational energies from the ground to the first excited state, ω_t, of small-molecule endofullerenes and scaling of the harmonic spherical potential V₂ for the translational motion of an endohedral molecule, of mass m_i, relative to the H₂@C₆₀ potential $V_{2}^{H_{2}}$ ⁠. The anharmonicity is neglected, $ω_{t} \approx ω_{t}^{0}$ ⁠, where $ω_{t}^{0}$ is the frequency of an harmonic oscillator [Eq. (18)].

Molecule	m_i	ω_t (cm⁻¹)	$V_{2} / V_{2}^{H_{2}}$	References
H₂	2	179.5	1	12
HD	3	157.7	1.16	8
D₂	4	125.9	0.98	8
HF	20	78.6	1.92	5
H₂O	18	110	3.4	This work
H₂O	18	162	7.3	Theory^16,51

Molecule	m_i	ω_t (cm⁻¹)	$V_{2} / V_{2}^{H_{2}}$	References
H₂	2	179.5	1	12
HD	3	157.7	1.16	8
D₂	4	125.9	0.98	8
HF	20	78.6	1.92	5
H₂O	18	110	3.4	This work
H₂O	18	162	7.3	Theory^16,51

As seen in Fig. 2(b), the absorption line of the ortho-H₂O translational mode is split by 2.9 cm⁻¹. This splitting may be attributed to the coupling between the endohedral molecule translation and rotation, associated with the interaction of the non-spherical rotating molecule with the interior of the C₆₀ cage, as seen in H₂@C₆₀.^8,10,12 For the particular transition, shown in Fig. 2(b), it is the coupling between the translational state with N = 1, L = 1 and the rotational state $|1_{01}⟩$ of ortho-water. Within spherical symmetry, a good quantum number is Λ = J + L. Thus, the translation–rotation coupled translational state L = 1 and rotational state J = 1 form three states with Λ-values 0, 1, and 2. The calculated energy difference of ortho-water Λ = 0 and Λ = 1 states is 8 cm⁻¹¹⁶ as compared to the experimental value 2.9 cm⁻¹.

C. Rotations of H₂O

There are two possibilities why the rotational constants of water change when it is encapsulated. The first is that the bond length and angles of H₂O change. The second is that H₂O, because of confinement and being non-centrosymmetric, is forced to rotate about the “center of interaction,” which does not coincide with its nuclear center of mass.⁵⁴

The rotational constant relates to the moment of inertia I_aa as $A = h {(8 π^{2} c_{0} I_{a a})}^{- 1}$ ⁠, where c₀ is the speed of light. Similarly, $B = h {(8 π^{2} c_{0} I_{b b})}^{- 1}$ for the b-axis rotation and $C = h {(8 π^{2} c_{0} I_{c c})}^{- 1}$ for the c-axis rotation ([A] = m⁻¹ in SI units and 0.01[A] = cm⁻¹). The moments of inertia are $I_{α α} = \sum_{i} m_{i} (β_{i}^{2} + γ_{i}^{2})$ ⁠, where {α_i, β_i, γ_i} are the Cartesian coordinates of the ith nucleus with mass m_i with the origin at the nuclear center of mass.

For non-centrosymmetric molecules, the translation–rotation coupling shifts the rotational energy levels. In quantum mechanical terms, the shift of rotational states is caused by the mixing of rotational and translational states by translation–rotation coupling, for example, HD@C₆₀.^8,55 Translation–rotation coupling was not included in our quantum mechanical model of H₂O@C₆₀. The rotational constants of the model were free parameters to capture the effect of translation–rotation coupling and the change in the H₂O molecule geometry caused by the C₆₀ cage. The rotational constants of free water are A₀ = 27.88 cm⁻¹, B₀ = 14.52 cm⁻¹, and C₀ = 9.28 cm⁻¹ in the ground vibrational state.⁵⁶ The rotational constants of endohedral water have relative shifts −13%, 5.5%, −8.7% for A₀, B₀, and C₀, in the ground vibrational state, Table III. In the following discussion, we use classical arguments to assess whether the shift of rotational levels is caused by the change in water molecule geometry or by the translation–rotation coupling.

From the symmetry of the H₂O molecule, it is seen that the nuclear center of mass is on the b axis (Fig. 1). The shift of H₂O center of rotation in the negative direction of b axis decreases A and C, while it does not affect B. If B changes, it must be due to the change in H–O–H bond angle and O–H bond length. The calculation predicts the lengthening of the H–O bond by 0.0026 Å and decrease in the H–O–H bond angle by 0.87° of caged water with respect to free water.⁵² With these parameters, the relative change in rotational constants A, B, and C is −2.5%, 0.65%, and −0.46%, an order of magnitude smaller than those derived from the IR spectra of H₂O@C₆₀. However, shifting the rotation center by −0.07 Å in the b direction (further away from oxygen) gives relative changes −14%, 0.65%, and −5.2%. This relative change in A₀₀₀ and C₀₀₀ is not very different from the values derived from the IR spectra, while the relative change in B₀₀₀ is within the error limits, Table III. Thus, it is likely that the dominant contribution to the observed changes in rotational constants in the ground translational state comes not from the change in H₂O molecule bond lengths and bond angle but from shifting its center of rotation away from the nuclear center of mass of the H₂O molecule.

D. Permanent and transition dipole moments

A comparison of the normalized absorption cross sections ⟨σ_ji⟩ [Eq. (3)] for H₂O@C₆₀ and for free water is given in the last column of Table I. In general, for all observed transitions, the absorption cross section of endohedral water is smaller than that of free water. The v₁ mode has the smallest relative change, while the largest relative change is for the combination mode ω₁ + ω₃.

The comparison of H₂O@C₆₀ and free H₂O absorption cross sections (Table I) enables us to estimate independently from the spectral fit the permanent and transition dipole moments of encapsulated H₂O,

μ_{j i}^{C_{60}} = μ_{j i}^{g a s} \sqrt{\frac{〈 σ_{j i}^{C_{60}} 〉 ω_{j i}^{g a s}}{〈 σ_{j i}^{g a s} 〉 ω_{j i}^{C_{60}}}} .

(28)

The results are collected together with the dipole moments obtained from the fit of IR spectra in Table VII. The permanent dipole moment of encapsulated water is nearly four times smaller than that of free H₂O. The results of the IR spectroscopy study are consistent with the dipole moments determined by the capacitance method, 0.51 ± 0.05 D.¹⁷ The reduction of the fullerene-encapsulated water dipole moment has been predicted by several theoretical calculations.^18,52,60,61

TABLE VII.

Absolute values of the dipole moment, unit D, matrix elements of rotational [μ^x, Eq. (B13)] and rovibrational [ $μ_{01}^{x}, μ_{02}^{x}$ and $μ_{03}^{z}$ ⁠, Eq. (B14)] transitions of gaseous H₂O, as published, and of H₂O@C₆₀ calculated with Eq. (28) from the rotational and rovibrational absorption cross sections, Table I, and determined by the fit of IR spectra, Table III. The cross section-derived dipole moments are the cross section-error weighted averages of the three transitions, 3, 4, and 5 or 6, 7, and 8 in Table I.

	Gas	References	Cross section	Fit
μ^x	1.855	57	0.50 ± 0.05	0.474 ± 0.008
$μ_{01}^{x}$	0.0153	58	(1.23 ± 0.13)10⁻²	(1.031 ± 0.021)10⁻²
$μ_{02}^{x}$	0.1269	59	(3.38 ± 0.19)10⁻²	(3.40 ± 0.05)10⁻²
$μ_{03}^{z}$	0.0684	58	(3.0 ± 0.3)10⁻²	(2.83 ± 0.07)10⁻²

	Gas	References	Cross section	Fit
μ^x	1.855	57	0.50 ± 0.05	0.474 ± 0.008
$μ_{01}^{x}$	0.0153	58	(1.23 ± 0.13)10⁻²	(1.031 ± 0.021)10⁻²
$μ_{02}^{x}$	0.1269	59	(3.38 ± 0.19)10⁻²	(3.40 ± 0.05)10⁻²
$μ_{03}^{z}$	0.0684	58	(3.0 ± 0.3)10⁻²	(2.83 ± 0.07)10⁻²

E. Effect of solid C₆₀ crystal field

The electric dipolar and quadrupolar interactions of the H₂O@C₆₀ molecule with the electrostatic field in solid C₆₀ explain the oscillator strength of the pure v₁ and v₂ vibrational transitions and the splitting of the rotational states with J > 0.

The theoretical work of Felker et al.³² shows that the source of the quadrupole crystal field is the orientation of electron-rich double bonds of 12 nearest neighbors of C₆₀ relative to the central H₂O@C₆₀. When the solid C₆₀ is cooled below 90 K, a merohedral disorder is frozen where ∼85% of C₆₀ have the electron-rich 6:6 bond (bond between the two hexagonal rings) facing pentagonal rings of a neighboring cage, the P-orientation.⁶² The rest are H-oriented where the 6:6 bond faces neighboring cage’s hexagonal rings. The calculated quadrupolar interaction for the P-oriented molecules is ten times bigger than that for the H-oriented molecules.³² The electric field gradient couples to the quadrupole moment of water and splits the $|1_{01}⟩$ rotational state by 4.2 cm⁻¹, where the m = ±1 doublet is above the m = 0 level.³² This theoretically predicted splitting of the $|1_{01}⟩$ state for the P-oriented molecules is remarkably close to the observed experimental value, seen as a 4 cm⁻¹ splitting of line 7, transition starting from the ortho ground state (Fig. 4). It is not possible to determine the crystal electric field gradient tensor and the encapsulated water quadrupole moment separately from our IR spectra.

Further splitting is possible if the symmetry is lower than S₆, but the maximum number of components for J = 1 remains three. However, we see that line 7 consists of four components instead of three (Fig. 4). This suggests that there are two sites with different local electrostatic fields. Anomalous splitting of triply degenerate phonons into quartets has been seen in solid C₆₀ by IR spectroscopy³⁴ and was attributed to the merohedral disorder. Thus, our work and the IR study of phonons³⁴ clearly show that there are two different quadrupolar interactions in solid C₆₀. As was proposed by Felker et al.,³² the crystal field has a different magnitude for P-oriented sites and for H-oriented sites. The small population of H-oriented sites (about 15%) justifies our spectral fitting with a single quadrupolar interaction.

We assumed that there is an internal electric field in C₆₀, and this field is a possible reason why the pure vibrational transitions 1 and 2 [see Fig. 1(c)] become visible in the IR spectrum. It is also plausible that 1 and 2 gain intensity through the translation–rotation coupling from the induced dipole moment of translational motion. However, there is evidence that local electric fields exist in solid C₆₀ as a result of merohedral disorder C₆₀.³⁵ The estimate of un-balanced charge by Alers et al.³⁵ was q = 6 × 10⁻³e assuming a dipole moment μ = qd₀, where e is the electron charge and d₀ = 0.7 nm is the diameter of a C₆₀ molecule. Our estimate is that the electric field $E = 1.1 \times 1 0^{8}$ V m⁻¹ at the center of the C₆₀ cage is created by the dipole moment with charge q = 4.7 × 10⁻³e. These two estimates are very close.

C₆₀ has six nearest-neighbor equatorial cages and three axial cages above and three axial cages below the equatorial plane, following the notation of Ref. 32. The z′ axis of the crystal field coordinate frame is normal to the equatorial plane. As our fit shows, the electric field is rotated away from the z′ axis by θ_E ≈ 80°, Table III, almost into the equatorial plane. It is possible that one of the six nearest-neighbors in the equatorial plane does not have P-orientation and this mis-oriented cage is the source of the electric field. θ_E has a large relative error consistent with the probability to have the mis-oriented cage in the equatorial or in the axial position.

VI. SUMMARY

The infrared absorption spectra of solid H₂O@C₆₀ samples were measured close to liquid He temperature, and rotational, vibrational, rovibrational, overtone, and combination rovibrational transitions of H₂O were seen. The spectral lines were identified as para-water and ortho-water transitions by following the para–ortho conversion process over the timescale of hours. The vibrational frequencies are shifted by −2.4% relative to free water, except bending mode frequency ω₂ and its overtone 2ω₂, where the shift is −1.6%. An absorption mode due to the quantized center-of-mass motion of H₂O in the molecular cage of C₆₀ was observed at 110 cm⁻¹. The dipole moment of encapsulated water is 0.50 ± 0.05 D, which is approximately four times less than that of free water and agrees with previous estimates.¹⁷

The rotational and rovibrational spectra were fitted with a quantum mechanical model of a vibrating rotor in the electrostatic field with dipolar and quadrupolar interactions. The quadrupolar interaction splits the J ≥ 1 rotational states of H₂O. The source of quadrupolar interaction is the relative orientation of electron-rich chemical bonds relative to pentagonal and hexagonal motifs of C₆₀ and its 12 nearest neighbors.³² Further IR study by using pressure to change the ratio of P-oriented and H-oriented motifs⁶³ would provide more information on the quadrupolar crystal fields of these motifs. The finite oscillator strength of the fundamental vibrational transitions is attributed to a finite electric field at the centers of C₆₀ cages due to merohedral disorder, as has been postulated in different contexts.³⁵ Our results are consistent with an internal electric field of 10⁸ V m⁻¹. However, it is also plausible that the fundamental vibrational transitions gain intensity through the translation–rotation coupling from the dipole moment induced by the translational motion, something that can be addressed in further theoretical studies.

To conclude, H₂O in the molecular cavity of C₆₀ behaves as a vibrating asymmetric top, its dipole moment is reduced, and the translational motion is quantized. The splitting of rotational levels is caused by the quadrupolar interaction with the crystal field of solid C₆₀. Evidence is found for the existence of a finite electric field at the centers of C₆₀ cages in water-endofullerene due to merohedral disorder.

Two out of three components necessary for a rigorous, comprehensive description of the water translations, rotations, and vibrations inside the C₆₀ molecular cage are now in place: first, the infrared spectroscopy data reported here and, second, the nine-dimensional quantum bound-state methodology⁵¹ plus the theory of symmetry breaking in solid C₆₀.^32,33 What is missing is a high-quality ab initio nine-dimensional potential energy surface for this system.

ACKNOWLEDGMENTS

We thank Professor Zlatko Bačić for useful discussions. This research was supported by the Estonian Ministry of Education and Research through institutional research funding under Grant No. IUT23-3 and personal research funding under Grant No. PRG736 and the European Regional Development Fund under Project No. TK134. We acknowledge the EPSRC (UK) for support under Grant Nos. EP/P009980/1 and EP/T004320/1.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX A: INTERACTION OF DIPOLE MOMENT WITH LOCAL ELECTRIC FIELD

The dipole moment in spherical components is⁴⁰

\begin{gathered} μ_{+ 1} = - \frac{1}{\sqrt{2}} (μ_{x} + i μ_{y}), \\ μ_{- 1} = \frac{1}{\sqrt{2}} (μ_{x} - i μ_{y}), \\ μ_{0} = μ_{z} . \end{gathered}

(A1)

The dipole moment of water in the Cartesian molecule coordinate frame, as shown in Fig. 1, is μ^M = {−μ_x, 0, 0}, where we use a convention that the dipole moment is directed from the negative charge to the positive charge. Then, from Eq. (A1), the dipole moment in spherical components is

{μ_{m}^{M}} = \frac{μ_{x}}{\sqrt{2}} {- 1,0,1}, m = - 1,0, + 1 .

(A2)

We consider the coupling of the H₂O dipole moment to the local electric field, $E$ ⁠, with spherical coordinates ${E, ϕ_{E}, θ_{E}}$ in the crystal frame C (frame where the electric field gradient tensor is defined) and in the coordinate frame E of electric field ${E_{m}^{E}} = E {0,1,0}$ ⁠, i.e., along the z_E axis. Corresponding Euler angles are Ω_C→E = {ϕ_E, θ_E, 0} and Ω_E→C = {π, θ_E, −π − ϕ_E}. The dipole moment of the molecule in frame E is

μ_{m}^{E} = \sum_{m^{'} = - 1}^{1} {[D_{m m^{'}}^{1} (Ω_{E \to C})]}^{*} μ_{m^{'}}^{C} .

(A3)

Here, we used Wigner D-functions relating the components of a spherical rank j irreducible tensor T_jm in coordinate frames A and B,³⁹

T_{j m}^{B} = \sum_{m^{'} = - j}^{j} D_{m^{'} m}^{j} (Ω_{A \to B}) T_{j m^{'}}^{A}

(A4)

and

T_{j m}^{A} = \sum_{m^{'} = - j}^{j} {[D_{m m^{'}}^{j} (Ω_{A \to B})]}^{*} T_{j m^{'}}^{B},

(A5)

where Ω_A→B = {ϕ, θ, χ} are Euler angles transforming coordinate frame A into frame B. The angles for the inverse transformation are Ω_B→A = {π − χ, θ, −π − ϕ}.⁴¹

The interaction of the molecular dipole moment $μ_{m}^{M}$ with the electric field is

\begin{array}{l} H_{e d} = & - \sum_{m = - 1}^{1} {(- 1)}^{m} E_{- m}^{E} μ_{m}^{E} \\ = & - \sum_{m = - 1}^{1} {(- 1)}^{m} E_{- m}^{E} [\sum_{m^{'} = - 1}^{1} {[D_{m m^{'}}^{1} (Ω_{E \to C})]}^{*} μ_{m^{'}}^{C}] \\ = & - \sum_{m, m^{'} = - 1}^{1} {(- 1)}^{m} E_{- m}^{E} {[D_{m m^{'}}^{1} (Ω_{E \to C})]}^{*} μ_{m^{'}}^{C} \\ = & - \sum_{m, m^{'} = - 1}^{1} {(- 1)}^{m} E_{- m}^{E} {[D_{m m^{'}}^{1} (Ω_{E \to C})]}^{*} \\ \times [\sum^{m^{″} = - 11} {[D_{m^{'} m^{″}}^{1} (Ω_{C \to M})]}^{*} μ_{m^{″}}^{M}] . \end{array}

The minus sign in front of sum in the formula above is consistent with $H_{e d} = - E \cdot μ$ if the spherical components of vectors $E$ and μ are defined as in (A1).

APPENDIX B: INTERACTION OF DIPOLE MOMENT WITH ELECTRIC FIELD OF RADIATION

Here, we derive the electric dipole transition matrix elements of a molecule, part (21) of the absorption cross section (20). The derivation of (21) starts from

S_{f i} = \frac{1}{e^{2}} | ⟨f| H_{e d} |i⟩ |^{2},

(B1)

where

H_{e d} = - \sum_{m = - 1}^{1} {(- 1)}^{m} e_{- m}^{A} μ_{m}^{A},

(B2)

where the electric field of radiation, $e = {e_{m}^{A}}$ ⁠, and the dipole moment of a molecule, $μ_{m}^{A}$ ⁠, are in the space-fixed frame A; $e^{2} = \sum_{m = - 1}^{1} {(e_{m}^{A})}^{2}$ ⁠.

The Euler angles of transformation of the crystal frame C to the space-fixed frame A, and vice versa, are Ω_C→A = {ϕ_R, θ_R, 0} and Ω_A→C = {π, θ_R, −π − ϕ_R}.

The dipole moment in frame A is

μ_{m}^{A} = \sum_{m^{'} = - 1}^{1} {[D_{m m^{'}}^{1} (Ω_{A \to C})]}^{*} μ_{m^{'}}^{C},

(B3)

where $μ_{m}^{C}$ is the dipole moment in the crystal frame C.

The absolute value of matrix element squared is

\begin{align} S_{f i} = & e^{- 2} {|⟨j| \sum_{m = - 1}^{1} {(- 1)}^{m} e_{- m}^{A} μ_{m}^{A} |i⟩|}^{2} \\ = & e^{- 2} {|\sum_{m, m^{'} = - 1}^{1} {(- 1)}^{m} e_{- m}^{A} {[D_{m m^{'}}^{1} (Ω_{A \to C})]}^{*} ⟨j| μ_{m^{'}}^{C} |i⟩|}^{2} \\ = & e^{- 2} \sum_{m_{1}, m_{1}^{'}, m_{2}, m_{2}^{'} = - 1}^{1} {(- 1)}^{m_{1} + m_{2}} e_{- m_{1}}^{A} {[e_{- m_{2}}^{A}]}^{*} \\ \times {[D_{m_{1} m_{1}^{'}}^{1} (Ω_{A \to C})]}^{*} D_{m_{2} m_{2}^{'}}^{1} (Ω_{A \to C}) \\ \times ⟨j| μ_{m_{1}^{'}}^{C} |i⟩ ⟨j| μ_{m_{2}^{'}}^{C} {|i⟩}^{*} . \end{align}

(B4)

1. Random orientation of crystals

We assume that all the crystals are identical and there are no other static fields outside the crystal that could break the directional isotropy. We average (B4) over the random orientation of crystal coordinate frames with respect to the space-fixed frame, Ω_A→C, and get

\begin{align} {⟨S_{f i}⟩}_{Ω_{A \to C}} = & \frac{1}{3 e^{2}} \sum_{m, m^{'} = - 1}^{1} {|e_{- m}^{A}|}^{2} {|⟨j| μ_{m^{'}}^{C} |i⟩|}^{2} \\ = & \frac{1}{3 e^{2}} (\sum_{m = - 1}^{1} {|e_{- m}^{A}|}^{2}) (\sum_{m^{'} = - 1}^{1} {|⟨j| μ_{m^{'}}^{C} |i⟩|}^{2}) \\ = & \frac{1}{3} (\sum_{m^{'} = - 1}^{1} {|⟨j| μ_{m^{'}}^{C} |i⟩|}^{2}), \end{align}

(B5)

where we used the property of rotation matrices,³⁹

\begin{matrix} \int {[D_{m_{1} m_{1}^{'}}^{j_{1}} (Ω)]}^{*} D_{m_{2} m_{2}^{'}}^{j_{2}} (Ω) d Ω \\ = \frac{8 π^{2}}{2 j_{1} + 1} δ_{j_{1} j_{2}} δ_{m_{1} m_{2}} δ_{m_{1}^{'} m_{2}^{'}} \end{matrix}

(B6)

and

\int_{0}^{π} \sin θ_{R} d θ_{R} \int_{0}^{2 π} d ϕ_{R} \int_{0}^{2 π} d χ_{R} = 8 π^{2} .

(B7)

If the sample is in the powder form, then it follows from Eq. (B5) that the absorption is independent of light polarization.

2. Transition matrix element and separation of coordinates

The absorption of radiation by a molecule [Eq. (20)] depends on the matrix elements of an electric dipole moment between the initial and final states,

\sum_{σ = - 1}^{1} {|⟨Φ^{'}| μ_{σ}^{A} |Φ⟩|}^{2},

(B8)

where $μ_{σ}^{A}$ is the molecule dipole moment in the space-fixed coordinate frame. The molecule wavefunction consists of the nuclear spin wavefunction $|I m_{I}⟩$ ⁠, electron wavefunction $|Φ^{e}⟩$ ⁠, electron spin wavefunction $|S m_{S}⟩$ ⁠, vibration wavefunction $|Φ^{v}⟩$ ⁠, and rotation wavefunction $|Φ^{r}⟩$ ⁠,⁴⁰

|Φ⟩ = |I m_{I}⟩ |S m_{S}⟩ |Φ^{e} Φ^{v} Φ^{r}⟩ .

(B9)

Using these separable wavefunctions, the matrix element (B8) is⁴⁰

\begin{matrix} ⟨Φ^{'}| μ_{σ}^{A} |Φ⟩ & = & ⟨I^{'} m_{I}^{'}| (I m_{I}⟩ ⟨S^{'} m_{S}^{'}| (S m_{S}⟩ \\ \times \sum_{σ^{'} = - 1}^{1} ⟨Φ^{r^{'}}| {D_{σ σ^{'}}^{1} (Ω_{A \to M})}^{*} |Φ^{r}⟩ ⟨Φ^{v^{'}}| μ_{σ^{'}}^{M (e)} |Φ^{v}⟩, \end{matrix}

(B10)

where

μ_{σ}^{M (e)} = ⟨Φ^{e^{'}}| μ_{σ}^{M} |Φ^{e}⟩,

(B11)

and we have taken into account that the electric dipole moment does not depend on nuclear and electron spin coordinates. Furthermore, if the energies of initial and final states of the transition are independent of spin projections m_I and m_s, the summation over initial and final states in the transition probability leads to degeneracy factors g_I = 2I + 1 and g_S = 2S + 1 in (20). The electron spin is zero in the ground electronic state of H₂O, and thus, g_S = 1. Degeneracy of para molecules (I = 0) is $g_{I}^{(p)} = 1$ ⁠, and that of ortho molecules (I = 1) is $g_{I}^{(o)} = 3$ ⁠.

If the electronic orbital does not change in the transition, then (B8) is the molecule electric dipole moment in the ground electronic state $|Φ^{e}⟩$ ⁠,

μ_{σ}^{M (e g)} \equiv ⟨Φ^{e}| μ_{σ}^{M} |Φ^{e}⟩ .

(B12)

For the rest of the discussion, we use a shorthand notation $μ_{σ}^{M}$ for the molecule dipole moment in the ground electronic state, $μ_{σ}^{M (e g)}$ ⁠.

Using Cartesian coordinates, the dipole moment in the ground vibrational state is

μ^{x} = ⟨000| μ_{x}^{M} |000⟩ .

(B13)

In the quantum mechanical model of H₂O@C₆₀ that we used to fit the IR spectra, we set the dipole moment equal to μ^x in three excited vibrational states $|100⟩$ ⁠, $|010⟩$ ⁠, and $|001⟩$ ⁠.

The vibrational transition dipole moments are

\begin{gathered} μ_{01}^{x} = ⟨100| μ_{x}^{M} |000⟩, \\ μ_{02}^{x} = ⟨010| μ_{x}^{M} |000⟩, \\ μ_{03}^{z} = ⟨001| μ_{z}^{M} |000⟩ . \end{gathered}

(B14)

The relation between spherical and Cartesian dipole moment components is given by Eq. (A1).

APPENDIX C: FITTING OF SYNTHETIC SPECTRA WITH QUANTUM MECHANICAL MODEL AND MODEL PARAMETER ERROR ESTIMATION

We determined the Hamiltonian parameters and the dipole moments, parameter set a = {a₁, …, a_ν, …, a_M}, by finding the parameter set a_min that gives the minimum value, $χ_{m i n}^{2}$ ⁠, of function

χ^{2} = \sum_{i = 1}^{N} {[S (ν_{i}) - f (ν_{i}; a)]}^{2},

(C1)

where S(ν_i) is the synthetic spectrum, with argument frequency ν_i, generated from the fit of the experimental spectra and f(ν_i; a) is the spectrum calculated from the model with M parameters a = {a₁, …, a_ν, …, a_M}; N is the number of points in the spectrum. The goal is to minimize χ² over the parameter set a. The result is $χ_{m i n}^{2}$ and a_min.

Let us define the matrix

F_{i ν} = \frac{\partial f (ν_{i}; a)}{\partial a_{ν}}

(C2)

and the dispersion matrix $\bar{V}$ [Eq. (12) of Ref. 64],

V_{μ ν} = \sum_{i = 1}^{N} F_{μ i}^{T} F_{i ν} .

(C3)

The covariance matrix [Eq. (11) of Ref. 64]

\bar{Θ} = σ^{2} {\bar{V}}^{- 1},

(C4)

where $σ^{2} = χ_{m i n}^{2} / (N - M)$ and ${\bar{V}}^{- 1}$ is the inverse matrix of $\bar{V}$ ⁠, ${\bar{V}}^{- 1} \bar{V} = \bar{1}$ ⁠.

The estimated variance of parameter a_ν is

Δ a_{ν} = \sqrt{Θ_{ν ν}} = \sqrt{\frac{χ_{m i n}^{2} {({\bar{V}}^{- 1})}_{ν ν}}{N - M}} .

(C5)

The correlation matrix $\bar{C}$ is

C_{ν μ} = \frac{Θ_{ν μ}}{\sqrt{Θ_{ν ν} Θ_{μ μ}}} .

(C6)

The element of $\bar{F}$ [Eq. (C2)] is

F_{i ν} = \frac{f (ν_{i}; a_{m i n} + δ a_{ν}) - f (ν_{i}; a_{m i n})}{δ a_{ν}},

(C7)

where a_min minimizes χ² and δa_ν is a small variation of the parameter a_ν.

We change the sum for an integral over ν in (C1),

χ^{2} = \sum_{k} \int_{ν_{k 1}}^{ν_{k 2}} A_{k}^{2} {[S (ν) - f (ν; a)]}^{2} d ν,

(C8)

where the experimental spectrum is available in several spectral ranges {ν_k1, ν_k2} indexed by k. A_k is the weight factor for each spectral range k. A_k is chosen so that the strongest lines for each range k are equal. The spectrum S(ν) is calculated with constant linewidth using line areas and frequencies from the fits of the experimental spectra.

By inserting (C7) into (C3) and using the continuum limit, we get the matrix $\bar{V}$ ⁠,

\begin{matrix} V_{μ ν} & = & \frac{1}{δ a_{μ} δ a_{ν}} \sum_{k} A_{k}^{2} \int_{ν_{k 1}}^{ν_{k 2}} d ν \\ \times [f (ν; a_{m i n} + δ a_{μ}) f (ν; a_{m i n} + δ a_{ν})) \\ - f (ν; a_{m i n} + δ a_{μ}) f (ν; a_{m i n}) \\ - f (ν; a_{m i n}) f (ν; a_{m i n} + δ a_{ν}) \\ + (f {(ν; a_{m i n})}^{2}] . \end{matrix}

(C9)

We define the number of experimental points N as the number of lines fitted in the experimental spectra multiplied by two as each line has two parameters, area and frequency. We estimate the parameter error with Eq. (C5) using (C8) and (C9).

REFERENCES

1.

Y.

Morinaka

,

F.

Tanabe

,

M.

Murata

,

Y.

Murata

, and

K.

Komatsu

, “

Rational synthesis, enrichment, and ¹³C NMR spectra of endohedral C₆₀ and C₇₀ encapsulating a helium atom

,”

Chem. Commun.

46

,

4532

–

4534

(

2010

).

https://doi.org/10.1039/c0cc00113a

Google Scholar

Crossref

2.

S.

Bloodworth

,

G.

Hoffman

,

M. C.

Walkey

,

G. R.

Bacanu

,

J. M.

Herniman

,

M. H.

Levitt

, and

R. J.

Whitby

, “

Synthesis of Ar@C₆₀ using molecular surgery

,”

Chem. Commun.

56

,

10521

–

10524

(

2020

).

https://doi.org/10.1039/d0cc04201c

Google Scholar

Crossref

3.

K.

Komatsu

,

M.

Murata

, and

Y.

Murata

, “

Encapsulation of molecular hydrogen in fullerene C₆₀ by organic synthesis

,”

Science

307

,

238

–

240

(

2005

).

https://doi.org/10.1126/science.1106185

Google Scholar

Crossref

PubMed

4.

K.

Kurotobi

and

Y.

Murata

, “

A single molecule of water encapsulated in fullerene C₆₀

,”

Science

333

,

613

–

616

(

2011

).

https://doi.org/10.1126/science.1206376

Google Scholar

Crossref

PubMed

5.

A.

Krachmalnicoff

,

R.

Bounds

,

S.

Mamone

,

S.

Alom

,

M.

Concistrè

,

B.

Meier

,

K.

Kouřil

,

M. E.

Light

,

M. R.

Johnson

,

S.

Rols

,

A. J.

Horsewill

,

A.

Shugai

,

U.

Nagel

,

T.

Rõõm

,

M.

Carravetta

,

M. H.

Levitt

, and

R. J.

Whitby

, “

The dipolar endofullerene HF@C₆₀

,”

Nat. Chem.

8

,

953

–

957

(

2016

).

https://doi.org/10.1038/nchem.2563

Google Scholar

Crossref

PubMed

6.

S.

Bloodworth

,

G.

Sitinova

,

S.

Alom

,

S.

Vidal

,

G. R.

Bacanu

,

S. J.

Elliott

,

M. E.

Light

,

J. M.

Herniman

,

G. J.

Langley

,

M. H.

Levitt

, and

R. J.

Whitby

, “

First synthesis and characterization of CH₄@C₆₀

,”

Angew. Chem., Int. Ed.

58

,

5038

(

2019

).

https://doi.org/10.1002/anie.201900983

Google Scholar

Crossref

7.

A. J.

Horsewill

,

S.

Rols

,

M. R.

Johnson

,

Y.

Murata

,

M.

Murata

,

K.

Komatsu

,

M.

Carravetta

,

S.

Mamone

,

M. H.

Levitt

,

J. Y.-C.

Chen

,

J. A.

Johnson

,

X.

Lei

, and

N. J.

Turro

, “

Inelastic neutron scattering of a quantum translator-rotator encapsulated in a closed fullerene cage: Isotope effects and translation-rotation coupling in H₂@C₆₀ and HD@C₆₀

,”

Phys. Rev. B

82

,

081410

(

2010

).

https://doi.org/10.1103/physrevb.82.081410

Google Scholar

Crossref

8.

M.

Ge

,

U.

Nagel

,

D.

Hüvonen

,

T.

Rõõm

,

S.

Mamone

,

M. H.

Levitt

,

M.

Carravetta

,

Y.

Murata

,

K.

Komatsu

,

X.

Lei

, and

N. J.

Turro

, “

Infrared spectroscopy of endohedral HD and D₂ in C₆₀

,”

J. Chem. Phys.

135

,

114511

(

2011

).

https://doi.org/10.1063/1.3637948

Google Scholar

Crossref

PubMed

9.

A.

Krachmalnicoff

,

M. H.

Levitt

, and

R. J.

Whitby

, “

An optimised scalable synthesis of H₂O@C₆₀ and a new synthesis of H₂@C₆₀

,”

Chem. Commun.

50

,

13037

–

13040

(

2014

).

https://doi.org/10.1039/c4cc06198e

Google Scholar

Crossref

10.

S.

Mamone

,

M.

Ge

,

D.

Hüvonen

,

U.

Nagel

,

A.

Danquigny

,

F.

Cuda

,

M. C.

Grossel

,

Y.

Murata

,

K.

Komatsu

,

M. H.

Levitt

,

T.

Rõõm

, and

M.

Carravetta

, “

Rotor in a cage: Infrared spectroscopy of an endohedral hydrogen-fullerene complex

,”

J. Chem. Phys.

130

,

081103

(

2009

).

https://doi.org/10.1063/1.3080163

Google Scholar

Crossref

PubMed

11.

C.

Beduz

,

M.

Carravetta

,

J. Y. C.

Chen

,

M.

Concistré

,

M.

Denning

,

M.

Frunzi

,

A. J.

Horsewill

,

O. G.

Johannessen

,

R.

Lawler

,

X.

Lei

,

M. H.

Levitt

,

Y.

Li

,

S.

Mamone

,

Y.

Murata

,

U.

Nagel

,

T.

Nishida

,

J.

Ollivier

,

S.

Rols

,

T.

Rõõm

,

R.

Sarkar

,

N. J.

Turro

, and

Y.

Yang

, “

Quantum rotation of ortho and para-water encapsulated in a fullerene cage

,”

Proc. Natl. Acad. Sci. U. S. A.

109

,

12894

–

12898

(

2012

).

https://doi.org/10.1073/pnas.1210790109

Google Scholar

Crossref

PubMed

12.

M.

Ge

,

U.

Nagel

,

D.

Hüvonen

,

T.

Rõõm

,

S.

Mamone

,

M. H.

Levitt

,

M.

Carravetta

,

Y.

Murata

,

K.

Komatsu

,

J. Y.-C.

Chen

, and

N. J.

Turro

, “

Interaction potential and infrared absorption of endohedral H₂ in C₆₀

,”

J. Chem. Phys.

134

,

054507

(

2011

).

https://doi.org/10.1063/1.3535598

Google Scholar

Crossref

PubMed

13.

A. J.

Horsewill

,

K. S.

Panesar

,

S.

Rols

,

J.

Ollivier

,

M. R.

Johnson

,

M.

Carravetta

,

S.

Mamone

,

M. H.

Levitt

,

Y.

Murata

,

K.

Komatsu

,

J. Y.-C.

Chen

,

J. A.

Johnson

,

X.

Lei

, and

N. J.

Turro

, “

Inelastic neutron scattering investigations of the quantum molecular dynamics of a H₂ molecule entrapped inside a fullerene cage

,”

Phys. Rev. B

85

,

205440

(

2012

).

https://doi.org/10.1103/physrevb.85.205440

Google Scholar

Crossref

14.

M.

Xu

,

F.

Sebastianelli

,

B. R.

Gibbons

,

Z.

Bačić

,

R.

Lawler

, and

N. J.

Turro

, “

Coupled translation-rotation eigenstates of H₂ in C₆₀ and C₇₀ on the spectroscopically optimized interaction potential: Effects of cage anisotropy on the energy level structure and assignments

,”

J. Chem. Phys.

130

,

224306

(

2009

).

https://doi.org/10.1063/1.3152574

Google Scholar

Crossref

PubMed

15.

M.

Xu

,

F.

Sebastianelli

,

Z.

Bačić

,

R.

Lawler

, and

N. J.

Turro

, “

Quantum dynamics of coupled translational and rotational motions of H₂ inside C₆₀

,”

J. Chem. Phys.

128

,

011101

(

2008

).

https://doi.org/10.1063/1.2828556

Google Scholar

Crossref

PubMed

16.

P. M.

Felker

and

Z.

Bačić

, “

Communication: Quantum six-dimensional calculations of the coupled translation-rotation eigenstates of H₂O@C₆₀

,”

J. Chem. Phys.

144

,

201101

(

2016

).

https://doi.org/10.1063/1.4953180

Google Scholar

Crossref

PubMed

17.

B.

Meier

,

S.

Mamone

,

M.

Concistrè

,

J.

Alonso-Valdesueiro

,

A.

Krachmalnicoff

,

R. J.

Whitby

, and

M. H.

Levitt

, “

Electrical detection of ortho–para conversion in fullerene-encapsulated water

,”

Nat. Commun.

6

,

8112

(

2015

).

https://doi.org/10.1038/ncomms9112

Google Scholar

Crossref

PubMed

18.

B.

Ensing

,

F.

Costanzo

, and

P. L.

Silvestrelli

, “

On the polarity of buckminsterfullerene with a water molecule inside

,”

J. Phys. Chem. A

116

,

12184

–

12188

(

2012

).

https://doi.org/10.1021/jp311161q

Google Scholar

Crossref

PubMed

19.

J.

Ceponkus

,

P.

Uvdal

, and

B.

Nelander

, “

The coupling between translation and rotation for monomeric water in noble gas matrices

,”

J. Chem. Phys.

138

,

244305

(

2013

).

https://doi.org/10.1063/1.4810753

Google Scholar

Crossref

PubMed

20.

M. E.

Fajardo

,

S.

Tam

, and

M. E.

DeRose

, “

Matrix isolation spectroscopy of H₂O, D₂O, and HDO in solid parahydrogen

,”

J. Mol. Struct.

695-696

,

111

–

127

(

2004

), Winnewisser Special Issue.

https://doi.org/10.1016/j.molstruc.2003.11.043

Google Scholar

Crossref

21.

C. M.

Lindsay

,

G. E.

Douberly

, and

R. E.

Miller

, “

Rotational and vibrational dynamics of H₂O and HDO in helium nanodroplets

,”

J. Mol. Struct.

786

,

96

–

104

(

2006

).

https://doi.org/10.1016/j.molstruc.2005.09.025

Google Scholar

Crossref

22.

K. E.

Kuyanov

,

M. N.

Slipchenko

, and

A. F.

Vilesov

, “

Spectra of the ν₁ and ν₃ bands of water molecules in helium droplets

,”

Chem. Phys. Lett.

427

,

5

–

9

(

2006

).

https://doi.org/10.1016/j.cplett.2006.05.134

Google Scholar

Crossref

23.

B. P.

Gorshunov

,

E. S.

Zhukova

,

V. I.

Torgashev

,

V. V.

Lebedev

,

G. S.

Shakurov

,

R. K.

Kremer

,

E. V.

Pestrjakov

,

V. G.

Thomas

,

D. A.

Fursenko

, and

M.

Dressel

, “

Quantum behavior of water molecules confined to nanocavities in gemstones

,”

J. Phys. Chem. Lett.

4

,

2015

–

2020

(

2013

).

https://doi.org/10.1021/jz400782j

Google Scholar

Crossref

PubMed

24.

E. S.

Zhukova

,

V. I.

Torgashev

,

B. P.

Gorshunov

,

V. V.

Lebedev

,

G. S.

Shakurov

,

R. K.

Kremer

,

E. V.

Pestrjakov

,

V. G.

Thomas

,

D. A.

Fursenko

,

A. S.

Prokhorov

, and

M.

Dressel

, “

Vibrational states of a water molecule in a nano-cavity of beryl crystal lattice

,”

J. Chem. Phys.

140

,

224317

(

2014

).

https://doi.org/10.1063/1.4882062

Google Scholar

Crossref

PubMed

25.

A. I.

Kolesnikov

,

G. F.

Reiter

,

N.

Choudhury

,

T. R.

Prisk

,

E.

Mamontov

,

A.

Podlesnyak

,

G.

Ehlers

,

A. G.

Seel

,

D. J.

Wesolowski

, and

L. M.

Anovitz

, “

Quantum tunneling of water in beryl: A new state of the water molecule

,”

Phys. Rev. Lett.

116

,

167802

(

2016

).

https://doi.org/10.1103/physrevlett.116.167802

Google Scholar

Crossref

PubMed

26.

M. A.

Belyanchikov

,

M.

Savinov

,

Z. V.

Bedran

,

P.

Bednyakov

,

P.

Proschek

,

J.

Prokleska

,

V. A.

Abalmasov

,

J.

Petzelt

,

E. S.

Zhukova

,

V. G.

Thomas

,

A.

Dudka

,

A.

Zhugayevych

,

A. S.

Prokhorov

,

V. B.

Anzin

,

R. K.

Kremer

,

J. K. H.

Fischer

,

P.

Lunkenheimer

,

A.

Loidl

,

E.

Uykur

,

M.

Dressel

, and

B.

Gorshunov

, “

Dielectric ordering of water molecules arranged in a dipolar lattice

,”

Nat. Commun.

11

,

3927

(

2020

).

https://doi.org/10.1038/s41467-020-17832-y

Google Scholar

Crossref

PubMed

27.

S.

Mamone

,

J. Y.-C.

Chen

,

R.

Bhattacharyya

,

M. H.

Levitt

,

R. G.

Lawler

,

A. J.

Horsewill

,

T.

Rõõm

,

Z.

Bačić

, and

N. J.

Turro

, “

Theory and spectroscopy of an incarcerated quantum rotor: The infrared spectroscopy, inelastic neutron scattering and nuclear magnetic resonance of H₂@C₆₀ at cryogenic temperature

,”

Coord. Chem. Rev.

255

,

938

–

948

(

2011

).

https://doi.org/10.1016/j.ccr.2010.12.029

Google Scholar

Crossref

28.

B.

Meier

,

K.

Kouřil

,

C.

Bengs

,

H.

Kouřilová

,

T. C.

Barker

,

S. J.

Elliott

,

S.

Alom

,

R. J.

Whitby

, and

M. H.

Levitt

, “

Spin-isomer conversion of water at room temperature and quantum-rotor-induced nuclear polarization in the water-endofullerene H₂O@C₆₀

,”

Phys. Rev. Lett.

120

,

266001

(

2018

).

https://doi.org/10.1103/physrevlett.120.266001

Google Scholar

Crossref

PubMed

29.

Y.

Li

,

J. Y.-C.

Chen

,

X.

Lei

,

R. G.

Lawler

,

Y.

Murata

,

K.

Komatsu

, and

N. J.

Turro

, “

Comparison of nuclear spin relaxation of H₂O@C₆₀ and H₂@C₆₀ and their nitroxide derivatives

,”

J. Phys. Chem. Lett.

3

,

1165

–

1168

(

2012

).

https://doi.org/10.1021/jz3002794

Google Scholar

Crossref

PubMed

30.

S. S.

Zhukov

,

V.

Balos

,

G.

Hoffman

,

S.

Alom

,

M.

Belyanchikov

,

M.

Nebioglu

,

S.

Roh

,

A.

Pronin

,

G. R.

Bacanu

,

P.

Abramov

,

M.

Wolf

,

M.

Dressel

,

M. H.

Levitt

,

R. J.

Whitby

,

B.

Gorshunov

, and

M.

Sajadi

, “

Rotational coherence of encapsulated ortho and para water in fullerene-C₆₀ revealed by time-domain terahertz spectroscopy

,”

Sci. Rep.

10

,

18329

(

2020

).

https://doi.org/10.1038/s41598-020-74972-3

Google Scholar

Crossref

PubMed

31.

K. S. K.

Goh

,

M.

Jiménez-Ruiz

,

M. R.

Johnson

,

S.

Rols

,

J.

Ollivier

,

M. S.

Denning

,

S.

Mamone

,

M. H.

Levitt

,

X.

Lei

,

Y.

Li

,

N. J.

Turro

,

Y.

Murata

, and

A. J.

Horsewill

, “

Symmetry-breaking in the endofullerene H₂O@C₆₀ revealed in the quantum dynamics of ortho and para-water: A neutron scattering investigation

,”

Phys. Chem. Chem. Phys.

16

,

21330

–

21339

(

2014

).

https://doi.org/10.1039/c4cp03272a

Google Scholar

Crossref

PubMed

32.

P. M.

Felker

,

V.

Vlček

,

I.

Hietanen

,

S.

FitzGerald

,

D.

Neuhauser

, and

Z.

Bačić

, “

Explaining the symmetry breaking observed in the endofullerenes H₂@C₆₀, HF@C₆₀, and H₂O@C₆₀

,”

Phys. Chem. Chem. Phys.

19

,

31274

–

31283

(

2017

).

https://doi.org/10.1039/c7cp06062a

Google Scholar

Crossref

PubMed

33.

Z.

Bačić

,

V.

Vlček

,

D.

Neuhauser

, and

P. M.

Felker

, “

Effects of symmetry breaking on the translation-rotation eigenstates of H₂, HF, and H₂O inside the fullerene C₆₀

,”

Discuss. Faraday Soc.

212

,

547

–

567

(

2018

).

https://doi.org/10.1039/C8FD00082D

Google Scholar

Crossref

34.

C. C.

Homes

,

P. J.

Horoyski

,

M. L. W.

Thewalt

, and

B. P.

Clayman

, “

Anomalous splitting of the F_1u(→ 3F_u) vibrations in single-crystal C₆₀ below the orientational-ordering transition

,”

Phys. Rev. B

49

,

7052

–

7055

(

1994

).

https://doi.org/10.1103/physrevb.49.7052

Google Scholar

Crossref

35.

G. B.

Alers

,

B.

Golding

,

A. R.

Kortan

,

R. C.

Haddon

, and

F. A.

Theil

, “

Existence of an orientational electric dipolar response in C₆₀ single crystals

,”

Science

257

,

511

(

1992

).

https://doi.org/10.1126/science.257.5069.511

Google Scholar

Crossref

PubMed

36.

L.

Abouaf-Marguin

,

A.-M.

Vasserot

,

C.

Pardanaud

, and

X.

Michaut

, “

Nuclear spin conversion of H₂O@C₆₀ trapped in solid xenon at 4.2 K: A new assignment of ν₂ rovibrational lines

,”

Chem. Phys. Lett.

480

,

82

–

85

(

2009

).

https://doi.org/10.1016/j.cplett.2009.08.071

Google Scholar

Crossref

37.

S.

Mamone

,

M.

Concistrè

,

E.

Carignani

,

B.

Meier

,

A.

Krachmalnicoff

,

O. G.

Johannessen

,

X.

Lei

,

Y.

Li

,

M.

Denning

,

M.

Carravetta

,

K.

Goh

,

A. J.

Horsewill

,

R. J.

Whitby

, and

M. H.

Levitt

, “

Nuclear spin conversion of water inside fullerene cages detected by low-temperature nuclear magnetic resonance

,”

J. Chem. Phys.

140

,

194306

(

2014

).

https://doi.org/10.1063/1.4873343

Google Scholar

Crossref

PubMed

38.

C. C.

Homes

,

P. J.

Horoyski

,

M. L. W.

Thewalt

,

B. P.

Clayman

, and

T. R.

Anthony

, “

Effect of isotopic disorder on the F_u modes in crystalline C₆₀

,”

Phys. Rev. B

52

,

16892

–

16900

(

1995

).

https://doi.org/10.1103/physrevb.52.16892

Google Scholar

Crossref

39.

R. N.

Zare

,

Angular Momentum

, Baker Lecture Series (

John Wiley & Sons, Inc.

,

1988

).

Google Scholar

40.

P. P.

Bunker

and

P.

Jensen

,

Molecular Symmetry and Spectroscopy

, 2nd ed. (

NRC of Canada

,

1998

), p.

747

.

Google Scholar

41.

D. A.

Varshalovich

,

A. N.

Moskalev

, and

V. K.

Khersonskii

,

Quantum Theory of Angular Momentum

(

World Scientific

,

1988

).

Google Scholar

Crossref

42.

C.

Cohen-Tannoudji

,

B.

Diu

, and

F.

Laloë

,

Quantum Mechanics

(

John Wiley & Sons

,

2005

).

Google Scholar

43.

J.

Verhoeven

and

A.

Dymanus

, “

Magnetic properties and molecular quadrupole tensor of the water molecule by beam-maser Zeeman spectroscopy

,”

J. Chem. Phys.

52

,

3222

–

3233

(

1970

).

https://doi.org/10.1063/1.1673462

Google Scholar

Crossref

44.

S. L.

Altmann

and

P.

Herzig

,

Point-Group Theory Tables

, 2nd ed. (

University of Vienna

,

Vienna

,

2011

).

Google Scholar

45.

W. H.

Shaffer

, “

Degenerate modes of vibration and perturbations in polyatomic molecules

,”

Rev. Mod. Phys.

16

,

245

–

259

(

1944

).

https://doi.org/10.1103/revmodphys.16.245

Google Scholar

Crossref

46.

D. L.

Dexter

, “

Absorption of light by atoms in solids

,”

Phys. Rev.

101

,

48

–

55

(

1956

).

https://doi.org/10.1103/physrev.101.48

Google Scholar

Crossref

47.

Solid C₆₀ has strong optical phonons between 500 and 1400 cm⁻¹. The index of refraction above 1400 cm⁻¹ is 2.0 (Ref. 38). Using the phonon data from Table II (Ref. 38) and Eq. (1) (Ref. 38), we calculate the index of refraction at 50 cm⁻¹, $η = \sqrt{R [ε]} = 2.01$ ⁠. Thus, η = 2.0 is a good approximation for the index of refraction in the frequency range of H₂O transitions reported here.

48.

S.

Aoyagi

,

N.

Hoshino

,

T.

Akutagawa

,

Y.

Sado

,

R.

Kitaura

,

H.

Shinohara

,

K.

Sugimoto

,

R.

Zhang

, and

Y.

Murata

, “

A cubic dipole lattice of water molecules trapped inside carbon cages

,”

Chem. Commun.

50

,

524

–

526

(

2014

).

https://doi.org/10.1039/c3cc46683c

Google Scholar

Crossref

49.

I. E.

Gordon

,

L. S.

Rothman

,

C.

Hill

,

R. V.

Kochanov

,

Y.

Tan

,

P. F.

Bernath

,

M.

Birk

,

V.

Boudon

,

A.

Campargue

,

K. V.

Chance

,

B. J.

Drouin

,

J.-M.

Flaud

,

R. R.

Gamache

,

J. T.

Hodges

,

D.

Jacquemart

,

V. I.

Perevalov

,

A.

Perrin

,

K. P.

Shine

,

M.-A. H.

Smith

,

J.

Tennyson

,

G. C.

Toon

,

H.

Tran

,

V. G.

Tyuterev

,

A.

Barbe

,

A. G.

Császár

,

V. M.

Devi

,

T.

Furtenbacher

,

J. J.

Harrison

,

J.-M.

Hartmann

,

A.

Jolly

,

T. J.

Johnson

,

T.

Karman

,

I.

Kleiner

,

A. A.

Kyuberis

,

J.

Loos

,

O. M.

Lyulin

,

S. T.

Massie

,

S. N.

Mikhailenko

,

N.

Moazzen-Ahmadi

,

H. S. P.

Müller

,

O. V.

Naumenko

,

A. V.

Nikitin

,

O. L.

Polyansky

,

M.

Rey

,

M.

Rotger

,

S. W.

Sharpe

,

K.

Sung

,

E.

Starikova

,

S. A.

Tashkun

,

J. V.

Auwera

,

G.

Wagner

,

J.

Wilzewski

,

P.

Wcisło

,

S.

Yu

, and

E. J.

Zak

, “

The HITRAN 2016 molecular spectroscopic database

,”

J. Quant. Spectrosc. Radiat. Transfer

203

,

3

–

69

(

2017

).

https://doi.org/10.1016/j.jqsrt.2017.06.038

Google Scholar

Crossref

50.

J.

Tennyson

,

N. F.

Zobov

,

R.

Williamson

,

O. L.

Polyansky

, and

P. F.

Bernath

, “

Experimental energy levels of the water molecule

,”

J. Phys. Chem. Ref. Data

30

,

735

(

2001

).

https://doi.org/10.1063/1.1364517

Google Scholar

Crossref

51.

P. M.

Felker

and

Z.

Bačić

, “

Flexible water molecule in C₆₀: Intramolecular vibrational frequencies and translation-rotation eigenstates from fully coupled nine-dimensional quantum calculations with small basis sets

,”

J. Chem. Phys.

152

,

014108

(

2020

).

https://doi.org/10.1063/1.5138992

Google Scholar

Crossref

PubMed

52.

A.

Varadwaj

and

P. R.

Varadwaj

, “

Can a single molecule of water be completely isolated within the subnano-space inside the fullerene C₆₀ cage? A quantum chemical prospective

,”

Chem. - Eur. J.

18

,

15345

–

15360

(

2012

).

https://doi.org/10.1002/chem.201200969

Google Scholar

Crossref

53.

A. B.

Farimani

,

Y.

Wu

, and

N. R.

Aluru

, “

Rotational motion of a single water molecule in a buckyball

,”

Phys. Chem. Chem. Phys.

15

,

17993

–

18000

(

2013

).

https://doi.org/10.1039/c3cp53277a

Google Scholar

Crossref

PubMed

54.

H.

Friedmann

and

S.

Kimel

, “

Theory of shifts of vibration–rotation lines of diatomic molecules in noble-gas matrices. Intermolecular forces in crystals

,”

J. Chem. Phys.

43

,

3925

–

3939

(

1965

).

https://doi.org/10.1063/1.1696622

Google Scholar

Crossref

55.

M.

Xu

,

F.

Sebastianelli

,

Z.

Bačić

,

R.

Lawler

, and

N. J.

Turro

, “

H₂, HD, and D₂ inside C₆₀: Coupled translation-rotation eigenstates of the endohedral molecules from quantum five-dimensional calculations

,”

J. Chem. Phys.

129

,

064313

(

2008

).

https://doi.org/10.1063/1.2967858

Google Scholar

Crossref

PubMed

56.

R. A.

Toth

, “

ν₂ band of H₂¹⁶O: Line strengths and transition frequencies

,”

J. Opt. Soc. Am. B

8

,

2236

–

2255

(

1991

).

https://doi.org/10.1364/josab.8.002236

Google Scholar

Crossref

57.

S. A.

Clough

,

Y.

Beers

,

G. P.

Klein

, and

L. S.

Rothman

, “

Dipole moment of water from Stark measurements of H₂O, HDO, and D₂O

,”

J. Chem. Phys.

59

,

2254

–

2259

(

1973

).

https://doi.org/10.1063/1.1680328

Google Scholar

Crossref

58.

J. M.

Flaud

and

C.

Camy-Peyret

, “

Vibration-rotation intensities in H₂O-type molecules application to the 2ν₂, ν₁, and ν₃ bands of H₂¹⁶O

,”

J. Mol. Spectrosc.

55

,

278

–

310

(

1975

).

https://doi.org/10.1016/0022-2852(75)90270-2

Google Scholar

Crossref

59.

C.

Camy-Peyret

and

J.-M.

Flaud

, “

Line positions and intensities in the ν₂ band of

H_{2}^{16}

O

,”

Mol. Phys.

32

,

523

–

537

(

1976

).

https://doi.org/10.1080/00268977600103261

Google Scholar

Crossref

60.

C. N.

Ramachandran

and

N.

Sathyamurthy

, “

Water clusters in a confined nonpolar environment

,”

Chem. Phys. Lett.

410

,

348

–

351

(

2005

).

https://doi.org/10.1016/j.cplett.2005.04.113

Google Scholar

Crossref

61.

K.

Yagi

and

D.

Watanabe

, “

Infrared spectra of water molecule encapsulated inside fullerene studied by instantaneous vibrational analysis

,”

Int. J. Quantum Chem.

109

,

2080

(

2009

).

https://doi.org/10.1002/qua.22066

Google Scholar

Crossref

62.

P. A.

Heiney

, “

Structure, dynamics and ordering transition of solid C₆₀

,”

J. Phys. Chem. Solids

53

,

1333

–

1352

(

1992

).

https://doi.org/10.1016/0022-3697(92)90231-2

Google Scholar

Crossref

63.

B.

Sundqvist

, “

Fullerenes under high pressures

,”

Adv. Phys.

48

,

1

–

134

(

1999

).

https://doi.org/10.1080/000187399243464

Google Scholar

Crossref

64.

D. L.

Albritton

,

A. L.

Schmeltekopf

, and

R. N.

Zare

, “

An introduction to the least-square fitting of spectroscopic data

,” in

Molecular Spectroscopy: Modern Research

(

Academic Press, Inc.

,

New York

,

1976

), pp.

1

–

67

.

Google Scholar

2021

Author(s)

Infrared spectroscopy of an endohedral water in fullerene Open Access

I. INTRODUCTION

II. METHODS

A. Sample preparation

B. Measurement techniques

1. Method 1

2. Method 2

C. Line areas and absorption cross sections

III. THEORY

A. Quantum mechanical model of H2O@C60: Confined vibrating rotor in an electrostatic field

1. Vibrations

2. Rotations

3. Electrostatic interactions

a. Quadrupolar interaction.

b. Dipolar interaction.

4. Confined water translations: Spherical oscillator

5. Nuclear spin isomers: para-water and ortho-water

B. Absorption cross section of H2O@C60

IV. RESULTS AND INTERPRETATION OF SPECTRA

A. Spectra

1. Translational transitions

2. Vibrational and rovibrational transitions

a. Pure vibrational transitions.

b. Spectral splittings.

c. Red shifts.

3. Overtone and combination rovibrational transitions

B. Spectral fitting with a quantum mechanical model

V. DISCUSSION

A. Vibrations of confined H2O

B. Translations of H2O

C. Rotations of H2O

D. Permanent and transition dipole moments

E. Effect of solid C60 crystal field

VI. SUMMARY

ACKNOWLEDGMENTS

DATA AVAILABILITY

APPENDIX A: INTERACTION OF DIPOLE MOMENT WITH LOCAL ELECTRIC FIELD

APPENDIX B: INTERACTION OF DIPOLE MOMENT WITH ELECTRIC FIELD OF RADIATION

1. Random orientation of crystals

2. Transition matrix element and separation of coordinates

APPENDIX C: FITTING OF SYNTHETIC SPECTRA WITH QUANTUM MECHANICAL MODEL AND MODEL PARAMETER ERROR ESTIMATION

REFERENCES

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Infrared spectroscopy of an endohedral water in fullerene

A. Quantum mechanical model of H₂O@C₆₀: Confined vibrating rotor in an electrostatic field

B. Absorption cross section of H₂O@C₆₀

A. Vibrations of confined H₂O

B. Translations of H₂O

C. Rotations of H₂O

E. Effect of solid C₆₀ crystal field