Supporting Information
for
Molecular Vibrations of Methane Molecules in the Structure I Clathrate Hydrate from Ab Initio
Molecular Dynamics Simulation
Masaki Hiratsuka, Ryo Ohmura, Amadeu K. Sum, and Kenji Yasuoka*
Vibrational modes of methane molecule
There are nine vibrational modes (15 degrees of freedom – 3 translations – 3 rotations), which are
classified into four kinds of vibrational types, such as three bending, two rocking, one symmetric
stretching, and three asymmetric stretching vibrational types. The small scattered vibrational spectra in
bending vibrations are possibly resulting from the three complex types of degenerated motions.
The animated images and frequencies shown here to illustrate the type of vibrational mode were
calculated in harmonic approximation using simulation package CPMD. (These schemes were not
calculated from molecular dynamics simulations). The calculation was based on the Density
Functional Theory. The basis set and treatment of the exchange correlation terms were the same as the
computation of the clathrate hydrate in the article. Only one methane molecule was in the box and the
lattice length was 11.83 Å. The actual vibrational frequencies and spectra for the clathrate hydrate in
article were generated from the ab initio MD simulations.
1
Bending: three degenerated motions (1260 cm-1).
Rocking: two degenerated motions (1502 cm-1).
Symmetric stretching: no degenerated motion (2962 cm-1).
Asymmetric stretching: three degenerated motion (3075 cm-1).
2
Autocorrelation functions
The autocorrelation function CAA is calculated by following equation
The physical value A is defined for five autocorrelation functions: velocity of hydrogen atoms, change
in the C-H bond length, change in the H-C-H angle, change in the C-H vector direction, and center of
mass velocity of the methane molecule. The mathematical expression and the computed
autocorrelation functions for guest molecules in small and large cages are shown as following. The red
and green lines show autocorrelation functions in the small and large cages, respectively.
Velocity of hydrogen atoms
3
C-H bond length
H-C-H angle:
Angle made by two C-H bonds, bond1 and bond2 in following equation, in a methane molecule.
4
C-H direction
Direction is defined as the angle between standard direction rCH(0) vector and rCH(t) vector.
Velocity of center of molecule
The values of i in the following equation are the atoms in the methane molecule: four hydrogen atoms
and one carbon atom.
5
Animation of the methane molecule in the cages
The animations of the methane molecule in the small and large cages are extracted from trajectory of
the Car-Parrinello molecular dynamics simulation from 5 ps to 9 ps.
6