Royal Society Publishing

Infrared spectroscopy of small-molecule endofullerenes

T. Rõõm, L. Peedu, Min Ge, D. Hüvonen, U. Nagel, Shufeng Ye, Minzhong Xu, Z. Bačić, S. Mamone, M. H. Levitt, M. Carravetta, J.Y.-C. Chen, Xuegong Lei, N. J. Turro, Y. Murata, K. Komatsu

Abstract

Hydrogen is one of the few molecules that has been incarcerated in the molecular cage of C60 to form the endohedral supramolecular complex H2@C60. In this confinement, hydrogen acquires new properties. Its translation motion, within the C60 cavity, becomes quantized, is correlated with its rotation and breaks inversion symmetry that induces infrared (IR) activity of H2. We apply IR spectroscopy to study the dynamics of hydrogen isotopologues H2, D2 and HD incarcerated in C60. The translation and rotation modes appear as side bands to the hydrogen vibration mode in the mid-IR part of the absorption spectrum. Because of the large mass difference of hydrogen and C60 and the high symmetry of C60 the problem is almost identical to a vibrating rotor moving in a three-dimensional spherical potential. We derive potential, rotation, vibration and dipole moment parameters from the analysis of the IR absorption spectra. Our results were used to derive the parameters of a pairwise additive five-dimensional potential energy surface for H2@C60. The same parameters were used to predict H2 energies inside C70. We compare the predicted energies and the low-temperature IR absorption spectra of H2@C70.

1. Introduction

A small cavity inside the fullerene cage is a potential site for trapping atoms and has attracted the attention of scientists from the moment of discovery of C60 [1]. The demonstration of formation of La@C60 after laser bombardment of La-impregnated graphite was immediate [2]. Since then the field of studies of endohedral fullerenes has been expanding. Endohedral fullerenes with noble gas (He and Ne [3]; Ar, Kr, Xe [4]), nitrogen [5] or phosphorus [6] atoms and with metal clusters [7] are made under extreme conditions using arc discharge, ion bombardment or high-pressure and high-temperature treatment.

Extreme methods are not suitable for encapsulation of small molecules. A different approach, ‘chemical surgery’, was applied by Rubin [8] when he made the first open-cage fullerene with an orifice large enough to load it with 3He or H2 using less extreme temperature and pressure [9]. Soon Murata et al. [10] synthesized another open-cage derivative of C60 and achieved 100% yield in filling with H2. Subsequently, the generation of a closed-cage H2@C60 was observed in the process of matrix-assisted laser desorption/ionization time-of-flight mass spectrometry analysis of this open-cage complex. Chemical methods were developed to close open-cage fullerenes and H2@C60 was produced in milligram quantities [11,12]. To accommodate two hydrogen molecules a cavity larger than C60 was needed. Two H2 were trapped in open-cage C70 with a yield of 3 : 97 in favour of species with one H2 per cage [13]. The restoration of the closed cage retains approximately the same ratio of (H2)2@C70 to H2@C70 [14]. Molecules other than hydrogen trapped in open-cage fullerenes are carbon monoxide [15], water [16,17], ammonia [18] and methane [19]. Recently, Kurotobi & Murata [20] succeeded in closing one of them and making the first closed-cage endohedral complex with a trapped polar molecule, H2O@C60. The rotational modes of endohedral water were observed by inelastic neutron scattering (INS), far infrared (far-IR) and nuclear magnetic resonance (NMR) at cryogenic temperatures [21].

To date, H2@C60 has been the most studied small-molecule endofullerene. The inhomogeneous distribution of interaction parameters is expected to be small, mainly because of the crystal field effects in solid H2@C60. The H2@C60 is a stable complex and can survive a short period of heating up to 500°C under vacuum [12]. These properties make H2@C60 appealing for spectroscopic and theoretical investigations of interactions between the molecular hydrogen and carbon nanosurfaces.

Three spectroscopic techniques—NMR, INS and infrared (IR)—have been used to study endohedral hydrogen. NMR studies cover spin lattice relaxation rates of H2@C60 in organic solvents [22,23], and in the presence of paramagnetic relaxants [24,25]. NMR was used to follow the orthopara conversion in H2@C60 in the presence of molecular oxygen at 77 K [26] or upon photoexcitation of a C70 triplet state [27]. NMR study of micro-crystalline H2@C60 samples at cryogenic temperatures shows splitting of the J=1 rotational state [28,29], a sign of the symmetry reduction from the icosahedral symmetry in the solid phase. Similarly, splitting of the ground ortho state was deduced from the heat capacity measurements [30].

An overview of the low-temperature NMR, INS and IR work on H2@C60 is given by Mamone et al. [31]. The first IR study of H2@C60 was limited to 6 K [32]. The translational and rotational transitions appeared as sidebands to the hydrogen molecule bond-stretching vibrational transition, Embedded Image, in the mid-IR spectral range. The direct translational and rotational transitions were not observed in the far-IR below 200 cm−1 [33]. The extension of IR studies to higher temperature made it possible to probe the hydrogen–C60 interaction potential in the ground v=0 and first excited v=1 vibrational states and a whole range of hydrogen isotopologues H2, D2 and HD were studied [33,34]. The isotope effects and translation–rotation coupling were also studied by INS in H2@C60 and HD@C60 [35]. The translational and rotational energies of H2@C60 and HD@C60 in the v=0 state determined by IR spectroscopy are consistent with the low-temperature INS results [35]. There are no Raman data on H2@C60, except a report on H2 inside an open-cage fullerene [36].

In this paper, we will review the IR studies of hydrogen isotopologues in C60 and present the analysis of IR low-temperature spectra of H2@C70. The far-IR properties of H2O@C60 will not be reviewed here [21].

2. Theory

Quantum statistics plays an important role in the dihydrogen wave function symmetry and has a pronounced effect on the rotation of the hydrogen molecule [37]. The symmetry relative to the interchange of two protons dictates that there are two forms of molecular hydrogen, called para- and ortho-H2. The two proton spins (Embedded Image) are in the antisymmetric I=0 total nuclear spin state in para-H2 and in the symmetric I=1 state in ortho-H2. Even rotational quantum numbers J are allowed for para-H2 and odd J for ortho-H2. The nucleus of D is a boson, nuclear spin Id=1. Thus, the rotational state with an even quantum number J has D2 in the state where the total nuclear spin of D2 is either zero or two, I=0,2. This is called ortho-D2, while para-D2 has the total nuclear spin I=1 and odd J values.

The ground rotational state of a homonuclear diatomic molecule with the total nuclear spin I=1 is the J=1 state. This J=1 rotational state is 118 cm−1 for H2, and 58 cm−1 for D2, above the rotational ground state J=0 of even-I species. A thermal transition Embedded Image must be accompanied by a change in the total nuclear spin of the molecule, a process of very low probability. The time constant of thermal relaxation between the ortho and the para manifolds is very long and the room temperature orthopara ratio is maintained even at cryogenic temperatures. The equilibrium distribution of H2 nuclear spin isomers is no/np=3 and of D2 is no/np=2 at room temperature. To change the total nuclear spin of a molecule the two nuclei must experience different magnetic fields. The orthopara conversion can be activated by using a paramagnetic centre as a source of the magnetic field gradient. The equilibrium no/np=1.0 is reached at 77 K by dispersing H2@C60 on a zeolite surface and exposing it to molecular oxygen, which acts like a spin catalyst [38]. There are no ortho and para species for HD. All rotational levels of HD are in thermal equilibrium and there is one rotational ground state, J=0.

Quantum chemistry calculations are challenging for a hydrogen molecule in a weak van der Waals interaction with a large fullerene molecule. The availability of experimental data on endohedral H2 has stimulated theoretical work in this direction. Theoretical investigations currently cover the calculations of rotation–translation energies of hydrogen isotopologues in C60 [39,40] and H2 in C70 [41], and the stability of C60 or C70 with one or more incarcerated H2 [42,43]. Empirical parameters of the Morse potential between H–H and contact Dirac interaction between H–C were adjusted [44], and the density-fitting local Møller–Plesset theory was tested [45] using the experimental H2 vibrational frequency inside C60. Classical molecular dynamics and density functional theory have been combined to reproduce accurately the NMR chemical shift of 1H in H2@C60 [46].

Both the hydrogen molecule and fullerene have closed-shell electronic structures and therefore the interaction between them is the van der Waals interaction. This simplicity makes H2@C60 ideal for the studies of non-covalent bondings between H2 and carbon nanosurfaces, the knowledge needed for the design of carbon-based hydrogen storage materials. The high icosahedral symmetry of the C60 cavity is close to spherical; therefore, H2@C60 represents a textbook example of a body moving in a spherical potential well [47,48]. In addition, H2 rotates around its centre of mass. H2 is not spherical and therefore its interaction with the walls of the cavity depends on its orientation, which leads to the coupling between translational and rotational motion [49]. If the translational and the rotational motions are coupled then in the spherical potential the conserved angular momentum is the sum of translational and rotational angular momenta [50,39]. H2@C60 is a rare example in which the quantum dynamics of a diatomic rotor in a confined environment can be studied. Another, but with limited degrees of freedom, interesting example related to the fullerenes is the quantum rotor C2 in a metallofullerene C2@Sc2C84 [51,52]. The two scandium atoms limit the translational motion and fix the rotational axis of C2 relative to the fullerene cage. At low temperature, the rotation of C2 is hindered because it has a small rotational constant and is therefore more susceptible to the corrugations of the carbon surface. H2 provides examples of two-dimensional rotors, such as H2 on a Cu surface [53] or H2 in intercalated graphite [54].

High-pressure loading of solid C60 creates interstitial H2. Exohedral H2 has been studied by IR [55,56], INS [57,58], NMR [59,60] and Raman [61] spectroscopies. Hydrogen is trapped in an interstitial site of octahedral symmetry and theory predicts translation–rotation coupling [50,62]. However, broadening of experimental lines has prohibited accurate determination of the H2–C60 interaction potential.

The observed IR spectra of hydrogen encapsulated in C60 consist of several absorption lines. We construct a model Hamiltonian and a dipole moment operator with few adjustable parameters to describe accurately the position and intensity of such a multi-line spectrum.

(a) Diatomic molecule in a spherical potential well

To describe the motion of a hydrogen molecule inside C60, we use the following model. The C60 is considered to be rigid, its centre of mass is not moving and also does not rotate. We treat H2@C60 as an isolated molecular complex and approximate the true icosahedral symmetry of an isolated C60 with spherical symmetry. It means that, in this approximation, H2 moves in a rigid spherically symmetric bounding potential provided by the cavity of C60. Besides the translational movement inside C60 the hydrogen molecule has its internal degrees of freedom, vibration and rotation of two nuclei relative to its centre of mass. There are no coupling terms between ortho and para states in our model Hamiltonian.

The theoretical work of Olthof et al. [63] is a comprehensive description of the dynamics of a loosely bound molecule inside C60. Olthof et al. [63] model the intermolecular potential as a sum of atom–atom potentials and expand it in spherical harmonics. They determined the radial part of the wave function with a discrete variable representation method. The radial part of the wave function in our approach is given by algebraic functions, which are solutions of the three-dimensional spherical oscillator [47,48,64]. The advantage is that matrix elements are calculated in algebraic form, avoiding time-consuming numerical integration.

The position and orientation of the H2 molecule is given by spherical coordinates R={R,Ω}, Ω={Θ,Φ} and s={s,Ωs}, Ωs={θ,ϕ}, where R is the vector from the centre of the C60 cage to the centre of mass of H2 and s is the internuclear H–H vector. The centre of mass translational motion of H2 is given by eigenfunctions of the isotropic three-dimensional harmonic oscillator [48,64] Embedded Image2.1where Embedded Image is the radial wave function and Y LML is the spherical harmonic. The size of the H2 molecule depends on its vibrational state |v〉. Therefore both the bounding potential and Embedded Image depend on the vibrational quantum number v. The translational quantum numbers are N=0,1,2,…. The orbital angular momentum quantum number takes values L=N,N−2,…,1 or 0, depending on the parity of N, and the azimuthal quantum number is ML=−L,−L+1,…,L. The rotational wave functions, defined by the rotational quantum numbers J=0,1,… and MJ=−J,−J+1,…,J, are given by the spherical harmonics Y JMJ(θ,ϕ).

We use bipolar spherical harmonics with overall spherical rank Λ and component MΛ Embedded Image2.2where Embedded Image are the Clebsch–Gordan coefficients [64]. Then the full wave function describing the motion of the H2 molecule is Embedded Image2.3where Embedded Image is the vibrational wave function with the quantum number v.

The Hamiltonian Embedded Image for the trapped molecule includes coupling terms between vibrational, translational and rotational motion. For simplicity, we neglect all matrix elements non-diagonal in v and introduce a parametric dependence on v, Embedded Image2.4where Embedded Image is the vibration–rotation Hamiltonian, p is the molecular momentum operator and m is the molecular mass of the diatomic molecule. vV =〈v|V (R,s,Ω,Ωs)|v〉 is the potential energy of the hydrogen molecule in the vibrational state ψvibv(s). The vibration–rotation Hamiltonian Embedded Image is diagonal in the basis set |vJNLΛMΛ〉 with eigenvalues given by Embedded Image2.5Embedded Image, where ω0 is the fundamental vibrational frequency, αe is the anharmonic correction and De is the centrifugal correction to the rotational constant Be [65,66].

We start from the general expansion of the potential in multi-poles, Embedded Image2.6and expand the radial part of potential Embedded Image in powers of R Embedded Image2.7where nl and the parities of l and n are the same.

V (R,s) is a scalar and transforms under fully symmetric representation Ag of the symmetry group Ih. The spherical harmonics λ=0,6,10,… transform as Ag of the symmetry group Ih [67]. We use fully spherical approximation of the potential, λ=0. Because λ=0 and λ=|lj|,|lj|+1,…,l+j it must be that l=j.

The total potential is Embedded Image2.8if we limit our expansion to j=l=2 and n=4. The odd-j terms are not allowed by symmetry for H2 and D2 and thus the coefficients Embedded Image and Embedded Image are zero for homonuclear diatomic molecules.

If we set the constant off-set Embedded Image to zero and write the perturbation part as Embedded Image2.9and the isotropic harmonic term as Embedded Image, then the total Hamiltonian reads Embedded Image2.10

The unperturbed Hamiltonian eigenvalues in the basis |vJNLΛMΛ〉 are Embedded Image2.11where Embedded Image is the frequency for translational oscillations within the cavity.

The meaning of different parts of the perturbation is explained by their influence on the energy levels of a harmonic three-dimensional spherical oscillator (table 1). The translation–rotation coupling term Embedded Image splits the energy of the |vJNLΛ〉 state into levels with different Λ, where Λ=|LJ|,|LJ|+1,…,L+J. For example, the N=L=J=1 state is split into three levels with different total angular momentum Λ=0,1,2. The ordering of levels depends on the sign of Embedded Image. The anharmonic correction to translation–rotation coupling is Embedded Image. If isotropic anharmonic correction Embedded Image is positive, the distance between energy levels increases with N and this correction is different for the levels with same N but different L. For example, for positive Embedded Image the N=2,L=0 level has higher energy than the N=2,L=2 level.

View this table:
Table 1.

Translation–rotation energies of a three-dimensional spherical oscillator Embedded Image for a perturbation given by equation (2.9) and Embedded Image by equation (2.10) for a few lower states. Here, Embedded Image, Embedded Image, Embedded Image and for vβ see equation (2.12).

The length scale Embedded Image (dimension m−2) of the radial part of a three-dimensional spherical oscillator wave function is related to the expectation value of the centre of mass amplitude in state |N〉 as [47] Embedded Image2.12

Terms described by the translation–rotation coupling coefficients Embedded Image and Embedded Image do not appear in table 1 because the first-order correction to energies vanishes as the matrix element of Embedded Image is zero if diagonal in L or J. These terms mix states with different N and J. For example, the first excited rotational state J=1, N=0 (expectation value of HD centre of mass is on the cage centre) has the state J=0, N=1 (expectation value of HD centre of mass is off the cage centre) mixed in [34]. The effect is that HD is forced to rotate about its geometric centre instead of its centre of mass.

It was found by the five-dimensional quantum mechanical calculation that the rotational quantum number J is almost a good quantum number for the homonuclear H2@C60 and D2@C60 and not for the heteronuclear HD@C60 [40]. Indeed, Embedded Image mixes states with different J for homonuclear species as well, but the effect is reduced compared with the effect of Embedded Image. In the former case, J±2 and L±2 are mixed, while in the latter case the J±1 and L±1 states that have a smaller energy separation are mixed.

The states with different Λ are not mixed in the spherical approximation, i.e. the total angular momentum Λ=L+J is conserved and Λ is a good quantum number. The other consequence of the spherical symmetry is that the energy does not depend on MΛ. Therefore, it is practical to use a reduced basis and reduced matrix elements [68] which are independent of MΛ. This reduces the number of states by factor 2Λ+1 for each Λ.

(b) Model Hamiltonian of H2@C70

A spherical approximation of the potential of a molecule trapped in C70 would be an oversimplification because of the elongated shape of C70. The symmetry of C70 is D5h, the distance between the centres of two capping pentagons (z-direction) is 7.906 Å. The diameter of the equatorial xy plane is 7.180 Å[69], similar to the diameter of the icosahedral sphere of C60, 7.113 Å[70]. The anisotropy of the potential of H2 inside C70 is supported by the five-dimensional quantum mechanical calculation [41] that shows that the lowest translational excitation in the z-direction is 54 cm−1 and in the xy plane is 132 cm−1, while in C60 it is 180 cm−1 and isotropic [33]. We derive from the IR spectra (see below) that the xy plane excitation energy is 151 cm−1, somewhat larger than theoretically predicted.

Although the z-axis translational energy in C70 is three times less than in the icosahedral C60, the effect of the C70 potential on the rotational motion is moderate. The splitting of the J=1 state is 7 cm−1, which is relatively small compared with the rotational energy 120 cm−1 in this state [41].

To analyse the IR spectra of H2@C70, we use a simplified Hamiltonian in which the translational energy is represented in the form of the sum of two oscillators, a one-dimensional linear and a two-dimensional circular oscillator, and we do not consider anharmonic corrections and the translation–rotation coupling.

The vibration–rotation energy Embedded Image2.13is the same as for H2@C60 except the last term, which accounts for the axial symmetry of the C70 potential with the rotational anisotropy parameter vκ [71]. For example, the three-fold degenerate J=1 rotational state in Ih symmetry is split in D5h symmetry and if vκ>0 the Jz=0 state is 3vκ below the twice degenerate Jz=±1 rotational state.

The translational part is added to the vibration–rotation Hamiltonian, equation (2.13), and the total energy becomes Embedded Image2.14Here, the translational energy is written as a sum of two oscillators, a linear oscillator along the z-axis with translational quantum Embedded Image and a two-dimensional (circular) oscillator [48,64] in the xy plane with translational quantum Embedded Image. Quantum numbers nz and n are positive integers including zero and l=n,n−2,…,−n+2,−n.

We will show below that the frequencies of z and xy translational modes, Embedded Image and Embedded Image, can be determined from the experimental data even though the translation–rotation coupling is not known. We take advantage of translation–rotation coupling being zero in the J=0 rotational state. The complication arises from the fact that the potential is different in the initial and final states of the IR transitions, v=0 and 1. However, this complication could be resolved if the energy of the fundamental vibrational transition Embedded Image (without change of n and nz) is known.

The ΔJ=0 transition from the para-H2 ground state leads to two excitation energies in the IR spectrum, first for the z mode and second for the xy mode Embedded Image2.15and Embedded Image2.16Defining the fundamental para transition energy as Embedded Image, we may rewrite equations (2.15) and (2.16) as Embedded Image2.17From these equations, translational frequencies in the excited v=1 state, Embedded Image and Embedded Image, can be determined without knowing the translation–rotation coupling.

The classification of energy levels up to J=1 and n=nz=1 by irreducible representations Γi of the symmetry group D5h is given in table 2. We get the irreducible representations Γj: Embedded Image and Embedded Image by subducting the translational states represented by spherical harmonics Y LML from the full rotational group O(3) to the symmetry group D5h. A1′ is the para-H2 ground state, n=l=nz=0. The first excited state of the z mode is nz=1 and A2′′. The first excited state of the xy mode n=l=1 is doubly degenerate E1′. The full symmetry when translations and rotations are taken into account is Γi=ΓjΓ(J). For example, the ortho-H2 ground state, J=1 and (nlnz)=(000), is split into Jz=0 (A2′′) and Jz=±1 (E1′) (table 2).

View this table:
Table 2.

Classification of energy levels of H2 inside the cage of C70 up to J=1 and n=nz=1 by irreducible representations Γi of the symmetry group D5h.

(c) Induced dipole moment of hydrogen in a spherical environment

IR light is not absorbed by vibrations and rotations of isolated homonuclear diatomic molecules [65]. IR activity of H2 is induced in the presence of intermolecular interactions, such as in the solid and liquid phases [72,73], in constrained environments [74,55,56,62], and in pressurized gases [75,76]. IR spectra of such systems are usually broad because of inhomogeneities in the system or because of random molecular collisions. As an exception, narrow lines are observed in semiconductor crystals [77] and solid hydrogen [78].

An overview of collision-induced dipoles in gases and gas mixtures is given in a book by Frommhold [79]. The confinement of the endohedral H2 introduces two differences when compared with H2 in the gas. First, the translational energy of H2 is quantized. In the gas phase, it is a continuum starting from zero energy. Second, the variation of the distance between H2 and the carbon atom is limited to the translational amplitude of H2 in the confining potential. In the gas phase, the distance varies from infinity to the minimal distance given by the collision radius. The selection rule ΔN=±1 for the endohedral H2 follows from these two conditions, as shown below.

Quantum mechanical calculations of induced dipoles are available for simple binary systems such as H2–He, H2–Ar and H2–H2. An extensive set of theoretical results for the H2–He system associated with the roto-translation electric dipole transitions, both in the vibrational ground state v=0 and accompanying the Embedded Image transition of the H2 molecule, can be found in [8084]. Related to the fullerene studies are calculations of the dipole moment of CO@C60 [63] and exohedral H2 in solid C60 [62].

We express the induced part of the dipole moment as an interaction between a hydrogen molecule and C60. Another approach was used by Ge et al. [33], who carried out a summation over 60 pair-wise induced dipole moments between H2 and carbon atoms. The relation between two sets of parameters was given [34].

We write the expansion of the dipole moment from the vibrational state v to v′ in bipolar spherical harmonics and in power series of R as Embedded Image2.18This is similar to the expansion of the potential discussed above, except the dipole moment is a polar vector whereas the potential is a scalar. The dipole moment transforms according to the irreducible representation T1u of the symmetry group Ih. The spherical harmonics of the order λ=1,5,7,… transform according to T1u of the symmetry group Ih [67]. We use λ=1 and are interested in Embedded Image transitions. In spherical symmetry, it is sufficient to calculate one component of the dipole moment vector, mλ=0, and if we drop the explicit dependence of Embedded Image on v,v′ and of Embedded Image on λ,mλ, the mλ=0 component of the dipole moment reads Embedded Image2.19

As λ=|lj|,|lj|+1,…,l+j and λ=1 it must be that l=|j±1|. The possible combinations are (lj)∈{(01),(10),(12),(21),(23),…}. If we restrict the expansion up to n=l=1, we get Embedded Image

A010 is zero for homonuclear diatomic molecules H2 and D2. It is the sum of the HD permanent rotational dipole moment in the gas phase and the induced rotational dipole moment inside C60. The selection rule is ΔN=0,ΔJ=0,±1, but Embedded Image is forbidden.

The expansion coefficients A101 and A121 are allowed by symmetry for both homo- and hetero-nuclear diatomic molecules. The expansion coefficient A101 describes the induced dipole moment that is independent of the orientation of the diatomic molecular axis s, selection rule ΔN=0,±1,ΔJ=0, but Embedded Image is forbidden. A121 describes the induced dipole moment that depends on the orientation of s, ΔN=0,±1, ΔJ=0,±2, but Embedded Image and Embedded Image are both forbidden. All terms in equation (2.20) satisfy the selection rule ΔΛ=0,±1, but Embedded Image is forbidden.

Although HD has a permanent rotational dipole moment, the induced dipole moment dominates inside C60 [34].

IR absorption line intensity [33] is proportional to the thermal population of the initial state. If the thermal relaxation between para-H2 and ortho-H2 is very slow we can define temperature-independent fractional ortho and para populations nk of the total number of molecules Embedded Image, where k=o,p selects ortho- or para-H2. Then the probability that the initial state |viJiNiLiΛiMΛi〉 is populated is Embedded Image2.20where Ei is the energy of the initial state measured from the ground state v=N=0 and j runs over all para- (or ortho-) H2 states in the basis used. gj=2Λj+1 is the degeneracy of the energy level Ej. Please note that gi does not appear in the numerator because pi is the population of an individual state |viJiNiLiΛiMΛi〉, although the reduced basis |vJNLΛ〉 is used. Hetero-nuclear HD has no para or ortho species and the coefficient nk in equation (2.20) must be set to one, nk=1.

(d) Infrared absorption by H2@C70

The infrared absorption in H2@C70 is given by electric dipole operators d0 and d±1, which transform as A2′′ and E1′ irreducible representations of D5h. At low temperature the symmetry-allowed transitions from the ground para state are Embedded Image and Embedded Image. Two ortho states, A2′′ and E1′, are populated at low temperature (figure 1). The transitions are Embedded Image and Embedded Image. The symmetry-allowed transitions from the E1′ state are Embedded Image and Embedded Image.

Figure 1.

Boltzmann population of the H2@C70 para ground translational state (000)(solid line) and the first thermally excited (dotted line) state (001) and of the two ortho states A2′′ (dashed line) and E1′ (dashed-dotted line) in the ground translational state (000) calculated using energy levels of the five-dimensional quantum mechanical calculation [41] and equation (2.20) with nk=1; irreducible representations are from table 2. (Online version in colour.)

The IR absorption line area is proportional to the Boltzmann population of the initial state (equation (2.20)). The degeneracy gj of the energy level Ej is one or two in the case of H2@C70. Figure 1 shows the Boltzmann population of the few first para and ortho levels in the v=0 vibrational state. The para state (000) population (solid line) starts to drop above 20 K as the first excited state (001) (dotted line) at 54 cm−1 above the ground state becomes populated and drains the population from the (000) state. The abrupt change of the population of ortho ground states above 2 K is because of the transfer of population from the non-degenerate A2′′ ground level (dashed line) to the doubly degenerate E1′ (dashed-dotted line) 7 cm−1 higher. The populations of the A2′′ and E1′ states decrease above 30 K as the first excited translational (001) ortho state becomes more populated.

3. Experiment

The endohedral complexes were prepared by ‘molecular surgery’ as described in [11,12]; H2@C60, D2@C60 and H2@C70 at Kyoto University, Kyoto, Japan, and HD@C60 at Kyoto University and Columbia University, New York, NY. The HD@C60 sample was a mixture of the hydrogen isotopologues H2:HD:D2 with the ratio 45 : 45 : 10. Since all C60 cages are filled, the filling factor for HD is ρ=0.45. The content of the C70 sample was empty : H2 : (H2)2=28:70:2 and the filling factor ρ=0.7. Experimental absorption spectra were corrected for the filling factor.

The para-enriched sample was made at Columbia University using molecular oxygen as a spin catalyst for the orthopara conversion [26]. Briefly, the H2@C60 adsorbed on the external surface of NaY zeolite was immersed in liquid oxygen at 77 K for 30 min, thereby converting the incarcerated H2 spin isomers to the equilibrium distribution at 77 K, no/np=1.0. The liquid oxygen was pumped away and the endofullerene–NaY complex was brought back rapidly to room temperature. The para-enriched H2@C60 was extracted from the zeolite with CS2 and the solvent was evaporated by argon. The powder sample under an argon atmosphere and on dry ice arrived at the National Institute of Chemical Physics and Biophysics, Tallinn, Estonia, 4 days after the preparation.

Powdered samples were pressed under vacuum into 3 mm diameter pellets for IR transmission measurements. Typical sample thickness was 0.3 mm. Two identical vacuum tight chambers with Mylar windows were employed in the IR measurements. The chambers were put inside an optical cold finger-type cryostat with KBr windows. During the measurements, the chamber containing the pellet for analysis was filled with He exchange gas while the empty chamber served as a reference. Transmission spectra were obtained using a Bruker interferometer Vertex 80 v with a halogen lamp and a HgCdTe or an InSb detector. The apodized resolution was typically 0.3 cm−1 or better.

Transmission Tr(ω) was measured as the light intensity transmitted by the sample divided by the light intensity transmitted by the reference. The absorption coefficient α(ω) was calculated from the transmission Tr(ω) through Embedded Image, with the reflection coefficient R=[(η−1)/(η+1)]2 calculated assuming a frequency-independent index of refraction [85], η=2. Absorption spectra were cut into shorter pieces around groups of H2 lines and a baseline correction was performed before fitting the H2 lines with Gaussians.

4. Results and discussion

(a) H2, D2 and HD in C60

The IR absorption spectra of H2@C60 at 6 K together with the simulated spectra are shown in figure 2 and the diagram of the energy levels involved is shown in figure 3a. The simulated spectra are calculated using Hamiltonian, dipole moment and the orthopara ratio parameters (table 3) obtained from the fit of 200 K spectra [33]. Temperature does not affect these parameters and the line intensities follow the Boltzmann population of initial states [33]. The intensity of three lines, Q(0)0 and Q(1)1 in figure 2a and S(1)3 in figure 2c, cannot be simulated because the induced dipole moment theory does not describe ΔN=0 transitions. However, the position of these three lines was used to fit the Hamiltonian parameters [33]. The intensity of the ortho state N=J=1 into Λ=0,1 and 2 by translation–rotation coupling is seen in figure 2b. The para N=1,J=2 and ortho N=1,J=3 states are split into three sublevels as well. However, because of the selection rule ΔΛ=±1 only one para and one ortho S-transition is IR active.

View this table:
Table 3.

Values of the fitted parameters for H2@C60 [33] and for HD and D2@C60 [34]. The vibrational and rotational constants of a free molecule of the three hydrogen isotopologues are shown for comparison; gas-phase ω0 is calculated including all terms Embedded Image up to k=3 [86]. The parameter Embedded Image is set to zero for HD@C60 and D2@C60. Parameter errors are given in [34].

Figure 2.

IR absorption spectra of H2@C60 at 6 K are shown by the black line. The green line in (a) is the Gaussian fit of fundamental para and ortho transitions, ΔN=ΔJ=0. The red line in (bd) is the simulated spectrum with parameters taken from the model fit of 200 K spectra [33]. The S(1)3 labelled green line in (c) is the Gaussian fit of the ΔN=0,ΔJ=2 forbidden ortho transition, experimentally observed at 4612.5 cm−1 and predicted to be at 4613.1 cm−1 by our model [33]. Line labels are the same as in figure 3a; the superscript to the line label shows the final state Λ.

Figure 3.

Diagrams of the observed low-temperature IR transitions in hydrogen isotopologues (a) H2, D2 and (b) HD incarcerated in C60. The initial states surrounded by a dashed box have the vibrational quantum number v=0; all the final states are in the excited vibrational state v=1. Dashed lines in (a) are forbidden transitions (ΔN=0) that are observed in H2@C60 but not in D2@C60. The S(1) line (dotted) with ΔN=1 is not observed in D2@C60. The line label Q(Ji) is used for ΔJ=0, R(Ji) is used for ΔJ=+1 and S(Ji) for ΔJ=+2 transitions where Ji is the initial state J.

The intensity of other lines in figure 2bd is described accurately by two dipole moment parameters, A101 and A121, and the orthopara ratio no/np=2.89±0.045, which is very close to the statistical value three (table 3). We confirmed the assignment of spectral lines to para- and ortho-H2 by measuring the spectrum of a para-enriched H2@C60 sample [33]. Low-temperature spectra in the region of Q lines are shown in figure 4. The time delay between para enrichment and the first IR measurement was 4 days. The 4255 cm−1 para line is stronger and the other three ortho lines are weaker in the para-enriched sample than in the non-enriched sample.

Figure 4.

IR absorption spectra in the region of Q lines of a para-enriched H2@C60 (dashed line) and non-enriched sample (solid line) at low temperature (T). The thermal equilibrium orthopara ratio at 77 K where the sample was para enriched is no/np=1.0. (Online version in colour.)

Figure 4 deserves some attention. Lines Q(0)1 and Q(1)0 have a clear multi-component structure and it is different for the two samples. This is not possible in the approximation of spherical symmetry—not even by considering the true icosahedral symmetry of C60 because the lowest Λ value for the state which is split by the icosahedral symmetry is three while the initial and final states of the optical transitions under discussion only have Λ=0,1. Later measurements on the relaxed but initially para converted sample showed that the difference between the line shapes of the normal and para converted sample are not due to the different orthopara ratio. Thus, it is likely that the difference in the line splitting is due to a different impurity content of the two samples. However, it is not completely excluded that the crystal field or the distortion of the C60 cage is responsible for part of this splitting. Note that the Q(1)0 line in D2@C60 (figure 5) bears a similar splitting pattern to that in H2@C60.

Figure 5.

IR absorption spectra of D2@C60 at 5 K. The simulated spectrum is shown by the red line (para), the blue line (ortho) and the thin black line (sum of para and ortho spectra). The parameters for the simulated spectrum are taken from the model fit of the 90 K spectrum [34]. Line labels are the same as in figure 3a; superscript to the line label shows the final state Λ.

The D2@C60 spectrum (figure 5) is shifted to a lower frequency than that of H2@C60 because of the heavier mass of D2. The spectrum has fewer lines than the H2@C60 spectrum. The missing transitions in D2@C60 (table 4) belong to the J-odd species, which are in the minority for D2.

View this table:
Table 4.

Experimental IR absorption line parameters of endohedral dihydrogen isotopologues in C60 at 5 K. ω is the transition frequency in wavenumber units, cm−1, and S is the integrated absorption line area in cm−2 units. JNLΛ are the quantum numbers of the main component with the weight |ξ1|2 in the final state v=1. The |ξ0|2 of the main component JNLΛ in the initial v=0 state for H2 is 0.97 of 0000 and 0.97 of 1001, for D2 is 0.99 of 0000 and 0.99 of 1001, and for HD is 0.95 of 0000. Integrated absorption cross-section σ (unit: cm per molecule) is used by some authors, Embedded Image, where Embedded Image is the number of hydrogen molecules in the volume V [87]; Embedded Image cm−3 in solid C60.

The splitting of N=J=1 into Λ=0,1 and 2 states is similar in D2 and H2. The magnitude of the splitting, Embedded Image (table 1), is less in D2 because Embedded Image is smaller, although Embedded Image and Embedded Image are similar for the two isotopologues (table 3). Line Q(1)2 overlaps partially with the Q(0)1 line (figure 5). However, this is not because the translation–rotation splitting is different for the two isotopologues but because of a smaller anharmonic correction αe of the rotational constant for D2 (table 3).

The spectrum of HD@C60 is more simple (figure 6) because there are no para and ortho species and therefore only one state is populated at low temperature. There is one spectral line, not present in homonuclear dihydrogen, labelled R(0) in figures 3b and 6. HD has no inversion symmetry and the ban on ΔJ=1 transitions is lifted. The classification of this transition as ΔJ=1 is arbitrary, first, because the weight of JNLΛ=1201 is only 0.5 in this state [34] and, second, the change of translational state is ΔN=2, which is not allowed by any of the dipole operator components in our model. The next component in the final state is JNLΛ=2111 with weight 0.29, which makes this transition A121-active, Embedded Image, and therefore it would be more appropriate to classify it as an S line.

Figure 6.

IR absorption spectra of HD@C60 at 5 K, black line. The parameters for the simulated spectrum (red dashed line) are taken from the model fit of 90 K spectra [34]. The broad lines indicated by arrows are not due to HD. Line labels are the same as in figure 3b; superscript to the line label shows the final state Λ. (Online version in colour.)

Another interesting feature of the HD spectrum is the absence of the ΔN=0,ΔJ=1 (Embedded Image) transition, although the induced dipole moment coefficient A010 gives a two orders of magnitude larger dipole moment than the permanent dipole moment of free HD, as was discussed in [34]. This ΔJ=1 transition in HD@C60 is suppressed because there is an interference of two dipole terms, A010 and A101, which have opposite signs. The final state consists of 80% of the pure rotational state J=1,N=0 and 18% of the pure translational state N=1,J=0. This mixed final state has matrix elements from the ground state for both A coefficients, A010 and A101, which nearly cancel each other.

The observed low-temperature IR transitions of H2, D2 and HD@C60 are collected in table 4. The content of the unperturbed |JNLΛ〉 state in the final state is above |ξ1|2=0.9 in homonuclear species in most cases, while for HD it varies and could be as low as 0.5. The mixing of states ξv is proportional to the energy separation of mixed states, EjEi in the first-order perturbation theory. Embedded Image couples states where Li=Lj and Ji=Jj. These states are far from each other as Li=Lj only if Nj=Ni±2. The other term Embedded Image mixes Lj=Li±2 and Jj=Ji±2, which are even further from each other for small N. It was found by Xu et al. [41] that J is almost a good quantum number in H2 and D2@C60 but not in HD@C60. Indeed, the distance between mixed states decreases for HD as Embedded Image mixes Lj=Li±1 and Jj=Ji±1 and for this Nj=Ni±1.

The vibrational frequency ω0 is redshifted from its gas-phase value for H2 and D2 (table 3). The relative change in the frequency [ω0(gas)−ω0(C60)]ω0(gas) depends on the cage and is independent of the hydrogen isotopologue [88]. Based on our fit results (table 3), for H2 ω0(C60)/ω0(gas)=0.9763 and for HD ω0(C60)/ω0(gas)=0.9773. We used the average of these two ratios to calculate the D2@C60 fundamental vibrational frequency, ω0=2924  cm−1, that was necessary to fit the IR spectra to a model Hamiltonian [34]. At this point we cannot say how much of the redshift is caused by a change in the zero-point vibrational energy and how much is caused by the change of anharmonic corrections to the vibrational levels in the C60 as our dataset is limited to energy differences of v=0 and 1 levels only. Note that the vibrational frequency of H2@C60 is ω0=4062.4 cm−1 (table 3), while the para vibrational transition is shifted up by 9 cm−1, to ω0=4071.4 cm−1 (table 4). This is because the zero-point translational energies are different in the v=0 and 1 vibrational states [33].

The rotational constant Be, the vibrational correction αe, and the centrifugal correction De of the hydrogen inside C60 and in free space are compared in table 3. The smaller than the gas-phase value of Be may be interpreted as 0.8% (0.9%) stretching of the nucleus–nucleus distance s in H2@C60 (D2@C60), as Be∼〈1/s2〉. An attractive interaction between hydrogen atoms and the cage causes s to be longer. The elongation of the equilibrium proton–proton distance is consistent with the redshift of ω0 [89]. However, the anharmonic vibrational correction to the rotational constant, αe, is smaller inside the cage than in the gas phase. Here, the cage has the opposite, repulsive, effect and reduces the elongation of the proton–proton distance in the excited v states when compared with H2 in the free space. This is supported by the fact that within the error bars αe of D2@C60 is equal to αe of D2. The vibrational amplitude of D2 is less than that of the H2 and therefore the repulsive effect of the cage becomes important at v>1 for D2.

Among the rotational and vibrational constants of HD@C60 the centrifugal correction De to the rotational constant is anomalously different from its gas-phase value compared with the other two isotopologues (table 3). Positive De means that the faster the molecule rotates, the longer is the bond. We speculate that, since the rotation centre of HD inside the cage is further away from the deuteron, the centrifugal force on the deuteron increases and the bond is stretched more than in the free HD molecule.

A similar system to the one studied here is exohedral H2 in C60. There, H2 occupies the octahedral interstitial site in the C60 crystal. The prominent features in the exohedral H2 IR spectra [56] are the translational, ΔN=±1, sidebands to the fundamental transitions, Δv=1 and ΔJ=0,2. The redshift of the fundamental vibrational frequency is about 60 cm−1, which is less than in H2@C60, where it is 98.8 cm−1. Also, the separation of the translational N=0 and 1 levels, approximately 120 cm−1, is less than that in H2@C60, 184.4 cm−1. It is likely that the main contribution to the latter difference comes from the larger van der Waals volume available for H2 in the octahedral site than in the C60 cage.

(b) H2@C70

IR absorption spectra of H2@C70 measured at 5 K are shown in figure 7, and the scheme of energy levels with the low-temperature transitions indicated with arrows is shown in figure 8. The transitions where ΔJ=0 and the translational state changes by Δn=1 or Δnz=1 are labelled as Qxy(J) and Qz(J), where J is the rotational quantum number of the initial state. The transitions where ΔJ=2 are labelled by Sxy(J) and Sz(J). The z and xy translational modes have distinctly different energies. For the Q lines they are shown in different panels (figure 7a,b). The para and ortho S lines are well separated because of different rotational energies. The para S lines are shown in figure 7c, whereas ortho S lines are shown in figure 7d. The Q lines cannot be sorted out into para and ortho that easily. An exception is the Q(1) transition, inset to figure 7a. This is a pure vibrational transition, Embedded Image, without the change of translational or rotational states. The corresponding (fundamental) para transition Q(0) would be at 4069.3 cm−1 according to the model (table 5). We expect it to be much weaker than the ortho Q(1) transition, as in H2@C60, and for this reason it is not observed in the experiment.

Figure 7.

IR absorption spectra of H2@C70 at 5 K. The line labels (table 5) refer to the z (xy) mode transition where Qz(J) [Qxy(J)] is the Δnz= 1,ΔJ=0 [Δn=1,ΔJ=0] and Sz(J) [Sxy(J)] is the Δnz=1,ΔJ=2 [Δn=1,ΔJ=2] transition; J is the initial state rotational quantum number. Inset to (a) shows the pure vibrational ortho-H2 transition without the change of translational or rotational state.

Figure 8.

Energy levels and observed IR transitions of H2@C70 from the ground para and ortho translational states in the v=0 state to the excited vibrational state v=1. The irreducible representations of the symmetry group D5h (table 2) are given for a few lower states. The ordering of the states is based on [41]. Ortho-H2 transitions marked by a and b are Qz(1)a and Qz(1)b in table 5 and start from the thermally excited state, which is E1′ (000) according to Xu et al. [41]. There are three experimentally observed ortho-H2 transitions from the ground A2′′(000) state: Qxy(1), Qxy(1)c and Qxy(1)d.

View this table:
Table 5.

IR transitions of H2@C70 observed at 5 K from the vibrational state v=0 to the state v=1. J(nlnz) are the quantum numbers for the initial and final states and the labels indicate corresponding transitions in figures 7 and 8. The experimental spectra were fitted with Gaussians, Sexp is the experimental line area; the experimental line position ωexp could always be determined with a precision better than 0.1 cm−1. The model parameters, equation (2.14), were determined as described in the text by setting the model frequency ωmod equal to the experimental frequency ωexp for four lines indicated with error 0. The model parameters are vibrational frequency ω0=4069.1 cm−1, one-dimensional oscillator frequency along the z-axis Embedded Image cm−1, two-dimensional circular oscillator frequency Embedded Image cm−1 and the rotational anisotropy parameter vκ=3.1 cm−1. The rotational constant Be=59.865 cm−1 and its corrections, αe=2.974 cm−1 and De=0.04832 cm−1, were assumed to be the same as in H2@C60.

In figure 9, we plot the normalized line area and normalized Boltzmann population. Figure 9a shows the temperature dependence of para lines. The assignment of the 4125.6 cm−1 line to Qz(0) and the 4219.9 cm−1 line to Qxy(0) is supported by our model, which is summarized in table 5 and will be discussed below.

Figure 9.

Normalized line area and Boltzmann population of para and ortho transitions of H2@C70 starting from the ground translational state. (a) para lines; (b) and (c) ortho lines. Solid line in (b) is the normalized intensity when the initial state is the ground state A2′′ (Jz=0) and in (c) when the initial state is E1′ (Jz=±1), 7 cm−1 above A2′′. This corresponds to vκ>0, equation (2.13). Dashed line in (b) and (c) is the reverse situation when the ground state is doubly degenerate E1′ and A2′′ is 7 cm−1 above it, i.e. vκ<0. The normalized intensities are calculated using the translational energy levels from the five-dimensional quantum mechanical calculation [41] and equation (2.20).

The uniaxial symmetry of C70 splits the ortho ground state J=1. This splitting is about 7 cm−1≈10 K and creates a sharp temperature dependence of the ortho line intensity at low temperature as is shown in figure 1. This feature could be used to determine which of the Q lines belong to the ortho species. Two lines, 4110.3 cm−1 and 4124.2 cm−1, have a temperature dependence consistent with the ortho transitions starting from the ground translational (000) state of E1′ symmetry (figure 9c). However, according to our model, there is no transition close to 4124.2 cm−1 from the thermally excited ortho E1′ state (table 5). This might be due to our simplified model, where the rotational anisotropy parameter vκ is assumed equal in the v=0 and 1 vibrational states. The rest of the ortho lines are shown in figure 9b. Several lines follow the same temperature dependence, the fundamental transition at 4063.2 cm−1, Qxy(1) lines 4213.4, 4224.2 and 4227.2 cm−1 and the Sz(1) line at 4678.8 cm−1. Lines Qz(1) at 4115.7 and Sxy(1) at 4761.3 cm−1 have distinctly different temperature dependence from the rest of the lines. Such deviation could be due to the overlap of transitions starting from A2′′and E1′ symmetry states. However, none of the ortho lines behaves like the theoretical prediction (black line) so the possibility that the ground ortho state is doubly degenerate E1′ and not the A2′′, i.e. 0κ<0, is not ruled out by our data. The dashed black line shows the ground-state population if the ground state is E1′ and the A2′′ state is 7 cm−1 above it. This temperature dependence is closer to the experimental situation than the solid black line, where 0κ>0. The change of sign of 0κ affects the temperature dependence of intensities only a little if the transitions start from the thermally excited state, as shown in figure 9c, where the red dashed line is the normalized Boltzmann population of the A2′′ state if this state is 7 cm−1 above the ground state E1′, i.e. if 0κ<0.

It is not unreasonable that 0κ<0 and the E1′ (Jz=±1) state is the ground rotational state of ortho-H2. It requires that there is an attraction instead of repulsion between H and C when H2 is in the centre of the C70 cage. The attraction is stronger for the H2 if its molecular axis is in the xy plane, Jz=±1, where the distance between H and C is less than if the axis is along the z plane, Jz=0, where the H–C distance is longer. Indeed, the attraction between C and H in H2@C60 was deduced from the redshift of vibrational energy and reduced rotational constant compared with H2 in the gas phase [33]. However, it was found by five-dimensional quantum mechanical calculation that A2′′ is the ground state [41]. A more elaborate model with dipole moment parameters that describe the IR line intensities together with the Boltzmann population could resolve the issue.

Table 5 summarizes the assignment of transitions observed in the experiment. Since our model described in §2b did not include translation–rotation coupling we used the transitions where both L and J are zero or one of them is zero in the initial and final states. The fundamental frequency ω0=4069.1 cm−1 was chosen to match the experimental fundamental ortho transition Q(1); the rotational constant Be=59.865 cm−1 and its corrections αe=2.974 cm−1, De=0.04832 cm−1 for H2@C60 were used. Next the two para transitions were matched, Qz(0) and Qxy(0), to obtain Embedded Image cm−1, Embedded Image cm−1. The 4220 cm−1 line was chosen as Qxy(0) because its temperature dependence (figure 9a) is similar to the Sz(0) and Sxy(0) line temperature dependence. The ortho ground-state splitting is 9.4 cm−1 (7.4 cm−1 from the five-dimensional calculation [41]) with vκ=3.1 cm−1, which was obtained by assuming that the Qz(1)a transition, Embedded Image in figure 8, is centred at 4110.3 cm−1. It was assumed that the anharmonic corrections to translational energy and to vκ are same for the v=0 and v=1 vibrational states.

The variation in the experimental frequencies compared with the model frequencies is within ±10 cm−1 (table 5). This is reasonable since our model did not include translation–rotation coupling, and the translation–rotation coupling in H2@C60 creates splittings of this magnitude. The vibrational frequency ω0 is larger in H2@C70 than in H2@C60. This difference should not be overinterpreted, because the correction owing to the difference in zero-point translational energies in two vibrational states was not taken into account for H2@C70, as was done for H2@C60 [33]. Again, a more elaborate model is needed for C70 than the two-oscillator model described in §2b.

5. Conclusions

IR absorption spectra of endohedral hydrogen isotopologues H2 [32,33], D2 [34] and HD [34] in C60 are informative, involving excitations of vibrations, rotations and translational motion of dihydrogen. The translational motion of the encapsulated molecule is quantized and coupled to its rotations because of the surrounding C60 cage. The vibrational frequency of dihydrogen is redshifted compared with the gas-phase value. Together with the smaller rotational constant it shows that the hydrogen bond is stretched inside the cage and there is an attraction between H (or D) and C. The heteronuclear HD does not rotate about its centre of mass because of the surrounding cage. Different rotational and translational states are mixed and rotational quantum number J is not a good quantum number for HD@C60.

Our study shows that the vibrations and rotations of C60 and the crystal field effects of solid C60 are not important on the energy scale of IR measurements. If these effects are important their contribution to the IR spectra is of the order of one wavenumber splitting of a few absorption lines. Such small splitting is consistent with the NMR [28,29] and heat capacity [30] results.

C70 has an ellipsoidal shape and this splits the translation–rotation states of H2. The translational frequency is about 180 cm−1 in H2@C60. In C70, this three-dimensional mode is split into a two-dimensional mode at 151 cm−1 and a one-dimensional mode at 56 cm−1. The five-dimensional quantum mechanical calculation [41] is in very good agreement with this experimental result.

Funding statement

This research was supported by the Estonian Ministry of Education and Research, grant SF0690029s09, and by the Estonian Science Foundation, grant nos. ETF8170, ETF8703 and JD187.

Footnotes

References

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