^{1,2,a)}, Minzhong Xu

^{2}, Jules W. Moskowitz

^{2}, Juergen Eckert

^{3}and Zlatko Bačić

^{2,b)}

### Abstract

We report fully coupled quantum six-dimensional (6D) calculations of the translation-rotation (T-R) energy levels of molecule inside the small dodecahedral and large tetracaidecahedral cages of the structure I clathrate hydrate. The quantum dynamics of the three translational and three rotational degrees of freedom of are treated rigorously, while the guest molecule and the host cavities are taken to be rigid. The matrix of the full 6D T-R Hamiltonian is diagonalized in the product basis of contracted translational and angular basis functions, generated by solving two reduced-dimension (3D) eigenvalue problems. A pairwise additive -cage 6D potential energy surface (PES) is employed, constructed using the anisotropic pair potential which was utilized previously in the molecular dynamics simulations of methane hydrate. Our calculations elucidate the key features of the T-R energy level structure of the nanoconfined . The rotational levels of methane exhibit an elaborate pattern of splittings caused by the angular anisotropy of the environment; the splitting patterns are identical for both types of cages. Translationally excited T-R states in the small cage are assigned in terms of the quantum numbers and of the 3D isotropic harmonic oscillator and those in the large cage using the Cartesian quantum numbers. Extensive comparison is made with the data from the inelastic neutron scattering studies of methane hydrate, allowing an assessment of the accuracy of the 6D PES employed.

I.M. is grateful to the Unity through Knowledge Fund of Croatia for funding this research through Gaining experience Grant No. 12. Z.B. is grateful to the National Science Foundation for partial support of this research through Grant No. CHE- 0315508. The computational resources used in this work were funded in part by the NSF MRI Grant No. CHE-0420870. Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund for partial support of this research.

I. INTRODUCTION

II. THEORY

A. Quantum 6D calculation of the translation-rotation eigenstates of a nanoconfined methane molecule

B. Geometry of the cages

C. Potential energy surfaces

D. Computational details

III. RESULTS AND DISCUSSION

A. Quantum 3D results: Translational excitations in the small and large cages

B. Quantum 3D results: Rotational excitations in the small and large cages

C. Quantum 6D translation-rotation energy levels in the small and large cages

1. Rotational excitations

2. Translational excitations

D. Comparison with spectroscopic measurements

IV. CONCLUSIONS

### Key Topics

- Methane
- 45.0
- Excited states
- 36.0
- Eigenvalues
- 18.0
- Clathrate hydrates
- 17.0
- Excitation energies
- 15.0

## Figures

Geometries of the small cage (top) and the large cage (bottom) of structure I clathrate hydrate. The Cartesian axes shown coincide with the principal axes of the cages.

Geometries of the small cage (top) and the large cage (bottom) of structure I clathrate hydrate. The Cartesian axes shown coincide with the principal axes of the cages.

One-dimensional cuts through the 6D intermolecular PESs of in the small (top) and large cage (bottom), along the Cartesian axes , which coincide with the principal axes of the cages. The potential profiles are obtained by minimizing the -cage interaction with respect to the three Euler angles of at every position of its cm.

One-dimensional cuts through the 6D intermolecular PESs of in the small (top) and large cage (bottom), along the Cartesian axes , which coincide with the principal axes of the cages. The potential profiles are obtained by minimizing the -cage interaction with respect to the three Euler angles of at every position of its cm.

Top: 3D isosurfaces at −1400, −1100, −700, 700, and for the 6D intermolecular PES of in the small cage. Bottom: 3D isosurfaces at −1300, −1000, −700, 700, and for the 6D intermolecular PES of in the large cage. The isosurfaces are obtained by minimizing the -cage interaction with respect to the three Euler angles of at every position of its cm.

Top: 3D isosurfaces at −1400, −1100, −700, 700, and for the 6D intermolecular PES of in the small cage. Bottom: 3D isosurfaces at −1300, −1000, −700, 700, and for the 6D intermolecular PES of in the large cage. The isosurfaces are obtained by minimizing the -cage interaction with respect to the three Euler angles of at every position of its cm.

3D isosurfaces of the RPD in the Cartesian coordinates of the 6D T-R states of in the large cage corresponding to the three translational fundamentals: (a) (1,0,0), (b) (0,1,0), and (c) (0,0,1). The isosurfaces are drawn at 20% of the maximum value of the density. The excitations energies shown are relative to the ground state, from Table VI.

3D isosurfaces of the RPD in the Cartesian coordinates of the 6D T-R states of in the large cage corresponding to the three translational fundamentals: (a) (1,0,0), (b) (0,1,0), and (c) (0,0,1). The isosurfaces are drawn at 20% of the maximum value of the density. The excitations energies shown are relative to the ground state, from Table VI.

3D isosurfaces of the RPD in the Cartesian coordinates of several 6D T-R states of in the large cage, with two or more quanta of excitation: (a) (1,0,1), (b) (0,1,1), (c) (0,0,2), (d) (2,0,1), and (e) (0,2,1). The isosurfaces are drawn at 20% of the maximum value of the density. The excitations energies shown are relative to the ground state.

3D isosurfaces of the RPD in the Cartesian coordinates of several 6D T-R states of in the large cage, with two or more quanta of excitation: (a) (1,0,1), (b) (0,1,1), (c) (0,0,2), (d) (2,0,1), and (e) (0,2,1). The isosurfaces are drawn at 20% of the maximum value of the density. The excitations energies shown are relative to the ground state.

## Tables

Translational energy levels of in the small cavity of sI clathrate hydrate, from the quantum 3D calculations. The translational excitation energies are relative to the ground-state energy . The rms amplitudes , , and are in bohr. The quantum numbers and are those of the 3D isotropic HO.

Translational energy levels of in the small cavity of sI clathrate hydrate, from the quantum 3D calculations. The translational excitation energies are relative to the ground-state energy . The rms amplitudes , , and are in bohr. The quantum numbers and are those of the 3D isotropic HO.

Translational energy levels of in the large cavity of sI clathrate hydrate, from the quantum 3D calculations. The translational excitation energies are relative to the ground-state energy . The rms amplitudes , , and are in bohr. The energy levels are assigned in terms of the Cartesian quantum numbers whenever possible, or as , where is the total number of quanta in the and modes. For additional explanation see the text.

Translational energy levels of in the large cavity of sI clathrate hydrate, from the quantum 3D calculations. The translational excitation energies are relative to the ground-state energy . The rms amplitudes , , and are in bohr. The energy levels are assigned in terms of the Cartesian quantum numbers whenever possible, or as , where is the total number of quanta in the and modes. For additional explanation see the text.

Rotational energy levels of inside the small cavity of sI clathrate hydrate, from the quantum 3D calculations for the cm of kept fixed at the cm of the cage. The rotational excitation energies are relative to the ground-state energy , and denotes the degeneracy of the levels. The columns labeled show the contributions of the corresponding rotational basis functions to the eigenstates.

Rotational energy levels of inside the small cavity of sI clathrate hydrate, from the quantum 3D calculations for the cm of kept fixed at the cm of the cage. The rotational excitation energies are relative to the ground-state energy , and denotes the degeneracy of the levels. The columns labeled show the contributions of the corresponding rotational basis functions to the eigenstates.

Rotational energy levels of inside the large cavity of sI clathrate hydrate, from the quantum 3D calculations for the cm of kept fixed at the cm of the cage. The rotational excitation energies are relative to the ground-state energy , and denotes the degeneracy of the levels. The columns labeled show the contributions of the corresponding rotational basis functions to the eigenstates.

Rotational energy levels of inside the large cavity of sI clathrate hydrate, from the quantum 3D calculations for the cm of kept fixed at the cm of the cage. The rotational excitation energies are relative to the ground-state energy , and denotes the degeneracy of the levels. The columns labeled show the contributions of the corresponding rotational basis functions to the eigenstates.

Select T-R energy levels of in the small cavity of sI clathrate hydrate, from the quantum 6D calculations. The T-R excitation energies are relative to the ground-state energy . For the purely rotationally excited levels, those with up to , denotes their degeneracy. The rms amplitudes , , and are in bohr. The columns labeled show the contributions of the corresponding rotational basis functions to the eigenstates. The translational quantum numbers and are those of the 3D isotropic HO.

Select T-R energy levels of in the small cavity of sI clathrate hydrate, from the quantum 6D calculations. The T-R excitation energies are relative to the ground-state energy . For the purely rotationally excited levels, those with up to , denotes their degeneracy. The rms amplitudes , , and are in bohr. The columns labeled show the contributions of the corresponding rotational basis functions to the eigenstates. The translational quantum numbers and are those of the 3D isotropic HO.

Select T-R energy levels of in the large cavity of sI clathrate hydrate, from the quantum 6D calculations. The T-R excitation energies are relative to the ground-state energy . For the purely rotationally excited levels, denotes their degeneracy. The rms amplitudes , , and are in bohr. The columns labeled show the contributions of the corresponding rotational basis functions to the eigenstates. The translational excitations are assigned in terms of the Cartesian quantum numbers whenever possible, or as , where is the total number of quanta in the and modes. For additional explanation see the text.

Select T-R energy levels of in the large cavity of sI clathrate hydrate, from the quantum 6D calculations. The T-R excitation energies are relative to the ground-state energy . For the purely rotationally excited levels, denotes their degeneracy. The rms amplitudes , , and are in bohr. The columns labeled show the contributions of the corresponding rotational basis functions to the eigenstates. The translational excitations are assigned in terms of the Cartesian quantum numbers whenever possible, or as , where is the total number of quanta in the and modes. For additional explanation see the text.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content