Rotor in a Cage: Infrared Spectroscopy of an Endohedral Hydrogen-Fullerene Complex
Abstract
We report the observation of quantized translational and rotational motion of molecular hydrogen inside the cages of C. Narrow infrared absorption lines at the temperature of 6 correspond to vibrational excitations in combination with translational and rotational excitations and show well-resolved splittings due to the coupling between translational and rotational modes of the endohedral H molecule. A theoretical model shows that H inside C is a three-dimensional quantum rotor moving in a nearly spherical potential. The theory provides both the frequencies and the intensities of the observed infrared transitions. Good agreement with the experimental results is obtained by fitting a small number of empirical parameters to describe the confining potential, as well as the ortho to para ratio.
Endohedral complexes of H molecules trapped inside fullerene cages have been synthesized recentlyRubin et al. (2001); Komatsu et al. (2005); Murata et al. (2006). Apart from their chemical interest and importance, these remarkable systems are ideal testbeds for the study of diatomic quantum rotors in a confined environment. H@C is different from quantum rotors studied so far, which were two-dimensional or showed hindered rotation, like H on a Cu surfaceSmith et al. (1996), H in intercalated graphiteBengtsson et al. (2000) or C in a metallofullereneKrause et al. (2004). The small mass and large rotational constant of H makes it the least sensitive of molecules to the corrugations of the potential surface. Also, C provides a nearly spherical bounding potential. Theoretically it has been shown that in C cages the quantum rotors COOlthof et al. (1996) and HCross (2001); Xu et al. (2008) should have a measurable translation-rotation coupling, a feature that has not been experimentally resolved to our knowledge. Moreover, there is very little experimental information on the quantum dynamics of H in fullerene cagesRafailov et al. (2005); Sartori et al. (2006); Carravetta et al. (2006, 2007).
In general, isolated homonuclear diatomics have no infrared (IR) activityHerzberg (1950). However, H does display IR activity in situations where there are intermolecular interactions present, such as in the solid and liquid phases Allin et al. (1955); Hare et al. (1955), in constrained environmentsHourahine and Jones (2003); FitzGerald et al. (2002, 2006); Herman and Lewis (2006), and in pressurized gassesKudian and Welsh (1971); McKellar and Welsh (1974). IR spectra of such systems are usually broad due to inhomogeneities in the system or due to random molecular collisions. As an exception, narrow lines are observed in semiconductor crystalsChen et al. (2002) and solid hydrogenOka (1993). Similarly we expect narrow lines in solid H@C, where the broadening of IR lines is suppressed by homogeneous distribution of trapping potentials provided by C molecules and by weak van der Waals interactions between the molecules.
In this Letter we study the dynamics of H in cages of C in the solid state with infrared spectroscopy. The observed spectra are described by a three-dimensional quantum rotor confined in a nearly spherical potential exhibiting translation-rotation coupling.
The H@C powder sample (10 mg) was prepared as described in Murata et al. (2006) and pressed into a mm thick pellet. IR transmission measurements were made with an interferometer Vertex 80v (Bruker), halogen lamp, and MCT detector with an apodized resolution 0.3 cm. The sample and the reference open hole were inside an optical cryostat with KBr windows. The absorption coefficient was calculated from the transmission through with the reflection coefficient . A frequency-independent index of refraction was assumed Homes et al. (1994).
The low temperature IR absorption peaks of H@C are located in four narrow spectral bands between 4060 and 4810, see Fig. 1. This region corresponds to the H stretching mode and its rotational/translational sidebands. Peaks in 4250, 4600 and 4800 regions (panels b, c, and d in Fig. 1) are assigned to vibrational excitations of H accompanied by translational and/or rotational excitations. It is the translation-rotation coupling that splits the line, shown in Fig. 1b, into three peaks. Weak transitions around 4070 (Fig. 1a) represent pure vibrational excitations of the H molecule and are forbidden in the approximate theory presented below.
The position and orientation of the H molecule is described using spherical coordinates and where is the vector from the center of the C cage to the center of mass of H and is the internuclear H-H vector. As first approximation we consider decoupled translational, rotational and vibrational movement of H. The translation of the confined molecule may be treated using the isotropic three-dimensional harmonic oscillator model. The appropriate translational quantum numbers are , the orbital angular momentum quantum number , which is an integer with the same parity as and the azimuthal quantum number . The radial part of the wavefunction depends both on and . The translational eigenfunctions are where the radial wave function and the spherical harmonics are defined in Flügge (1971). The rotational wavefunctions, defined by the rotational quantum numbers and , are given by the spherical harmonics . It is convenient to use bipolar spherical harmonics with overall spherical rank and component , defined as follows:
(1) |
where and are the Clebsch-Gordan coefficients Varshalovich et al. (1988). The full wavefunction describing the motion of the H molecule may be written as where is the vibrational wavefunction with a quantum number . The total nuclear spin of the H molecule determines whether it is either in a para state ( and even) or in an ortho state ( and odd). The Hamiltonian for the trapped molecule includes coupling terms between the vibrational, translational, and rotational motion. For simplicity, we neglect all matrix elements non-diagonal in and introduce a parametric dependence on :
(2) |
where is the vibration-rotation Hamiltonian, is the molecular momentum operator and is the molecular mass. The superscript prefix is used to indicate an implied dependence on the vibrational quantum number. is the potential energy of a molecule at a given position and orientation within the cavity, and includes terms that couple the rotational and translational motion. The vibrational-rotational Hamiltonian is diagonal in the basis set with eigenvalues given by , where is the fundamental vibration frequency; , where is an anharmonic correction to the rotational constant Herzberg (1950). Below 120, the thermally-activated rotational motion of the C cages is suppressed Tycko et al. (1991); Carravetta et al. (2007) and may be assumed to be time-independent. Expanded in multipoles it reads:
(3) |
where the functions are defined in Eq. 1 and takes even values. Terms with constitute a harmonic potential energy function, while terms with represent anharmonic perturbations. Translation-rotation coupling terms are terms with non-zero .
All odd- terms vanish for homonuclear diatomic molecules. For an icosahedral cavity, and assuming that longer-range intermolecular perturbations are negligible, all terms with odd values vanish, as well as the terms with and . We assume that all high-order terms starting from are small and express the potential energy as , where the isotropic harmonic term is given by and the perturbation due to translation-rotation and anharmonic coupling is given by . The unperturbed Hamiltonian eigenvalues in the basis are given by is the frequency for translational oscillations within the cavity. , where
The matrix elements of were evaluated analytically in the basis using 100 states with and for ortho-H, and 60 states with and for para-H. Matrix diagonalization leads to explicit but cumbersome expressions for the energy levels and eigenstates. A schematic energy level diagram is given in Fig.2. The ordering of the eigenvalues depends on the relative sign and magnitudes of the anharmonic term and the translation-rotation coupling term . The ordering in Fig.2 is consistent with the experimental results.
IR activity in H@C is due to a dipole moment, , induced by the constraining environment. The dipole moment operator can be expanded in multipoles depending on the instantaneous H configuration Poll and Hunt (1976):
(4) |
where denotes the spherical component and the coefficients describe the induced dipole moment. Since the dipole moment is a vector, there are restrictions on the allowed and values: (i) must be odd, (ii) from the triangle relation, (iii) for homonuclear molecules only even terms are allowed. These restrictions imply selection rules for IR spectroscopy of H@C. The selection rule for the total angular momentum is with the only allowed transitions having even values of and odd values of . In addition, because only the ground translational states () are populated at low temperature, the allowed transitions observed in the 6 K IR spectrum are to states (Fig. 2).
The IR absorption amplitude at frequency is Herzberg (1950):
(5) |
where = or selects ortho- or para-H. At the sample temperature , the fractional population of the initial state is given by the Boltzmann distribution for ortho and para manifolds separately. Since the spin isomer interconversion is negligible for the endohedral complex Carravetta et al. (2007), the number of ortho and para molecules, and , is not in general governed by the Boltzmann distribution and must be determined empirically. Since all the observed transitions are from to , only terms with and are to be considered in the dipole expansion. This implies that only transitions are observable. The dipole matrix elements for the observed transitions in H@C between states and can be expressed as
(6) |
where and . The matrix elements reduce into a sum of products over radial integrals and known angular integrals.
Three potential parameters , , , the rotational constant , and , , were fitted to match the experimental frequencies and intensities. The potential parameters in the first vibrational state are . The low- data is insufficient to derive accurately the anharmonic correction , which is poorly defined from the separation of and levels only. This affects the value of , because and are correlated. However, the translation-rotation coupling term is well-defined. and describe the potential within approximately 0.5Å from the C cage center, which is the root square of the average square displacement for the translational state. The fitted rotational constant is while is obtained directly from the difference in the fundamental vibrational frequencies for the ortho- and para-H. The ratio between the induced dipole moment parameters is . The ortho to para ratio is consistent with the equilibration at any temperature warmer than 120K and suggests that there has been negligible spin isomer interconversion since the molecules were synthesized. The data and the best fit results are displayed in Fig. 1 and summarized in Table 1. and
Experimental | Fitted | |||||
---|---|---|---|---|---|---|
initial | final | (cm) | (cm) | (cm) | (cm) | |
4065.44 | ||||||
4071.39 | ||||||
4244.5 | 4244.1 | 4.5 | ||||
4250.7 | 4250.7 | 20.0 | ||||
4261.0 | 4261.0 | 10.0 | ||||
4255.0 | 4255.5 | 11.2 | ||||
4591.5 | 4590.7 | 2.9 | ||||
4802.5 | 4803.0 | 5.1 |
The frequencies of the pure vibrational transitions (Fig. 1a) are shifted by from the free H value. The reduction in both the vibrational frequency and the rotational constant Bragg et al. (1982) are consistent with a predominantly attractive C-H interaction that slightly stretches the H-H bond.
The results described here are in qualitative agreement with previous theoretical studies. The quantum-chemical calculation by CrossCross (2001) gave a factor of two smaller translation-rotation coupling potential term and larger values for both harmonic and anharmonic potential terms, leading to significant discrepancies in the IR line positions if used to reproduce the experimental data. Xu et al.Xu et al. (2008) used pair-wise Lennard-Jones potentials in their quantum-mechanical calculation. The ordering of the sublevels is consistent with our results, even though the splittings are different. Xu et al. used two different potential energy surfaces, which gave different , one smaller and the other larger than the experimentally observed . The splitting among the energy levels, that is the measure of , remained larger than the experimentally observed value for both potential energy surfaces. Although our measurements refer to the excited state, while the numerical calculations are for the ground state, it seems that the theory using a pairwise C-H potential is not accurate enough to describe the dynamics of H@C. We believe that our results are a reference for theories modeling the interaction between H and curved carbon nano-surfaces.
Deformation of the cage, crystal field, or carbon isotopomers in the cage may lower the symmetry and split the 4255 and 4261 lines (Fig. 1b) and cause the IR activity of the weak fundamental transitions (Fig. 1a).
In summary, the IR spectrum of endohedral H displays a rich structure due to the coupled translational and rotational modes of the confined quantum rotor. Line positions and intensities are described by a theory involving multipole expansion of the confining potential and fitting of a small number of parameters. The next targets will be analyzing higher- data to extract information about anharmonic corrections and the vibrational ground state, studying lower symmetry cages and different dihydrogen isotopomers. The accurate determination of the energy levels in the vibrational ground state may play a key role in understanding fully the low- NMR-behavior of endohedral H in different fullerenesCarravetta et al. (2007).
The support by the EPSRC, the EstSF grants 6138 and 7011, and the University Research Fellowship (Royal Society) is acknowledged. S.M. thanks Dr. G. Pileio for useful discussions.
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