Theory of the fundamental vibration-rotation-translation spectrum of ${\mathrm{H}}_2$ in a ${\mathrm{C}}_{60}$ lattice

Calculations are presented for the fundamental vibration-rotation spectrum of ${\mathrm{H}}_2$ in fcc ${\mathrm{C}}_{60}$ (fullerite) lattices. The principal features are identified as lattice-shifted “vibration-rotation-translation” state absorption transitions. The level spacings of the ${\mathrm{H}}_2$ modes are calculated numerically for the potential function resulting from the summation of the individual $\mathrm{C}\text{−}{\mathrm{H}}_2$ potentials for all C atoms in the six nearest neighbor ${\mathrm{C}}_{60}$ molecules. The potential is approximately separable in Cartesian coordinates, giving a very good approximation to exactly calculated translational energies for the lower levels. The positions and relative strengths of the individual transitions are calculated from the eigenfunctions for this separable potential. The line shapes are assumed to be Lorentzian, and the widths are chosen so as to give good fits to the DRIFT spectrum of FitzGerald et al. [Phys. Rev. B 65, 140302(R) (2002)]. A theory of the $\mathrm{C}\text{−}{\mathrm{H}}_2$ induced dipole moment is developed with which to calculate intensities. In order to fit the observed DRIFTS transition frequencies it is found necessary to take the overlap part of the $\mathrm{C}\text{−}{\mathrm{H}}_2$ potential to be about 13% longer in range than the $\mathrm{C}\text{−}{\mathrm{H}}_2$ potential in graphene. Furthermore, differences in the theoretical spectra obtained with a near-optimal exp-6 potential and near-optimal Lennard-Jones 12-6 potential are clearly evident, with the exp-6 potential giving a better fit to observation than the Lennard-Jones potential. Similarly, Lorentzian line shapes assumed for the individual transitions yield better agreement with observation than Gaussian line shapes.

Figure 1

DRIFTS and our best theoretical spectrum for the fundamental band of ${\mathrm{H}}_2$ in a fullerite lattice as a function of the frequency $\omega$. The theoretical spectrum was calculated for the exp-6 potential for which $\sigma =3.22\mathrm{\AA }$ and $\epsilon =3.25\mathrm{meV}$. The principal component Lorentzian line shapes have HWHH of $24{\mathrm{cm}}^{-1}$. Other parameters are described in the text.

Figure 2

Theoretical spectrum using an exp-6 potential based on the known $\mathrm{C}\text{−}{\mathrm{H}}_2$ potential in graphene, for which $\sigma =2.85\mathrm{\AA }$ and $\epsilon =4.11\mathrm{meV}$; with DRIFTS.

Figure 3

Theoretical spectrum using the exp-6 form of the Wang et al.[15] potential ($\sigma =2.97\mathrm{\AA }$ and $\epsilon =3.69\mathrm{meV}$), with DRIFTS.

Figure 4

Theoretical spectrum using Gaussian line shapes. The width parameters are the same as for the Lorentzian fit of Fig. 1.

Figure 5

Theoretical spectrum using a Lennard-Jones interaction, with the same $\sigma$ and $\epsilon$ as used for the exp-6 potential in Fig. 1.