| Theory
The computational methodology implemented in the
program is the extension of the methodology developed and used
in the Prof. Bačić's group (New York University) to treat
the translation-rotation dynamics of hydrogen molecule confined
in different cages of clathrate hydrate (S. Liu et al.
JCP 103, 1829 (1995), M. Xu et al. JCP
128 244715 (2008)). However, the methane molecule
is a nonlinear molecule and its translational-rotational motion
must be described in the term of six coordinates (x, y, z,
&theta, &phi, &chi ). The first three coordinates represent
the Cartesian coordinates (x, y, z)
of the center of mass of the methane molecule, while the three
Euler angles (θ, φ, χ ) define the orientation
of the methane relative to the nanocage. To get the 6D translation-rotation
energy levels and wave functions one must solve the Schrödinger
equation with the following Hamiltonian:
All the details about the 6D bound state methodology implemented in the program
can be found here. In order to
calculate the eigenvalues and eigenfunctions of the Hamiltonian
above, a 6D basis was used for the matrix representation of the
Hamiltonian: a 3D direct product discrete variable representation
(DVR) for the x, y, z coordinates whileWigner D-functions
were used as the basis in the angular θ, φ, χ
coordinates. The program also includes the calculation of
the expectation values of the Cartesian coordinates and the calculation
of the 3D reduced probability densities in Cartesian and angular
coordinates. The former provide a measure of the wave function
delocalization in the x, y, z direction and are helpful
in making the quantum number assignment. The later are also used
for the visualization of the wave functions. To describe the interactions
between the methane and water molecules of the host we implemented
a specific force-field. For the general use of the program the
user should implement it's own force-field depending on the system
in hand. However, the implementation of any other force field
is quite strait forward and can be made by changing one subroutine
in the computer code.
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| Figure 2.
Methane hydrate has a sI crystal structure composed of two
nanocages formed by hydrogen-bonded water molecules: dodecahedral
(512) or a small cage formed by 20 water molecules
(left) and tetrakaidecahedral (51262)
or a large cage formed by 24 water molecules (right) |
Results
Potential
Methane hydrate has a sI crystal structure composed of two nanocages
formed by hydrogen-bonded water molecules: dodecahedral (512)
or a small cage formed by 20 water molecules and tetrakaidecahedral
(51262) or a large cage formed by 24 water
molecules (Figure 2.). All interactions between the methane molecule
and the water nanocages are assumed to be pairwise additive. Hence,
the interaction potential between the methane molecule and N
H2O molecules forming the cage (N =20 for
small cage and 24 for larger cage) was described with the equation:
where q = (x, y, z, θ, φ,
χ ) are the coordinates of methane molecule and the VCH4-H2O
is the interaction between a pair of CH4 and H2O
molecules defined bellow and the index w runs over the
water molecules of the cage, whose coordinates Qw
are fixed. The pair interaction between CH4 and H2O
is given by:
Three charges on the H2O molecule are taken from the
SPC/E effective pair potential model for water. Five point charges
are placed on the CH4 molecule. This charges together
with the C-H distance of 1.094 Å reproduce the calculated
octopole moment of CH4 (all the parameters for this
force field can be found in Alavi et. al. JCP 126,
124708 (2007)). The second term in the upper equation represents
the Lennard-Jones (LJ) potential chosen to describe the van der
Waals interactions between the O atom of H2O and the
center of mass of CH4. The LJ parameters εO-CH4
and σO-CH4 are determined following
the standard combination rules.
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|
Figure 3.
The 3D V (x, y, z) isosurfaces, at -1400,
-1200, -700, 700 and 1400 cm-1 for the CH4-cage
potential, obtained by minimizing the CH4 cage
interaction with respect to the three Euler angles of the
CH4 molecule, at every position of its center
of mass. The X, Y, Z axes coincide with the three principal
axes of the small dodecahedral cage. |
Figure 4.
Two dimensional (2D) cuts of the 3D V (θ, φ,
χ ) potential for the rotation of the methane molecule
in the center of mass of the small dodecahedral cage of methane
hydrate
Quantum 6D calculations
The T-R energy levels from the quantum 6D calculations
for CH 4 inside the small sI dodecahedral cage are
given in Table 1. The frequencies of all the levels are given
relative to the ground state T-R energy of the methane in the
small cage which was found to be -1339.9 cm-1 with
the Tse-Alavi potential. For one methane molecule inside the small
cage, the ground state energy has high negative value, implying
that the methane encapsulated in the cage represents truly bound
and stable state relative to the methane molecule at the large
distance outside the cage (zero energy).
| Table 1.
Translation-rotation energy levels of methane molecule in
the small cage of sI clathrate hydrate. The excitation energies
ΔE are relative to the ground-state energy
E0=-1344.91 cm-1. Also are
shown the root-mean-square (rms) amplitudes Δx,
Δy, Δz in bohr. The columns
labeled j=0, j=1, j=2 and j=3
represent the contributions of the corresponding rotational
basis functions to the 6D wave functions. Symbols (t) and
(d) assign the triplet and doublet states. |
|
For each T-R state Table 1. also displays the root-mean-square
(rms) amplitudes Δx, Δy, Δz
which provide a measure of the wave function delocalization in
the x-, y- and z- directions. Listed
next are the contributions of the rotational basis functions with
the same j quantum number to each 6D wave function. In
the last column the assignment of the 6D wave functions in the
translational (Cartesian) quantum numbers is given. The Cartesian
quantum numbers are assigned by inspecting the rms differences
and 3D reduced probability densities calculated as:
The Table 1. clearly shows that the fundamental excitation of
any one of the translational mode in the x, y,
or z direction increases its rms amplitude, by ≈0.1
bohr, relative to the (0,0,0) ground state. By subsequently adding
more quantum excitation to the x, y, or z
direction, the increase in corresponding rms amplitudes is smaller.
This reflects the fact that the space available for the translational
motion of the CH4 in the small cage is very restricted.
Namely, at the energy of interest, below -1400 cm-1,
the cage walls are only ≈1.2 bohr away from the center of
the cage. The 3D reduced probabilities for some 6D functions are
displayed in Figure 5. As can bee seen reduced densities of many
translational states are distorted and tilted with respect to
the Cartesian axes which is an evidence of a strong translation
mode coupling. However, we could assign the x-mode fundamental
(1,0,0), the y-mode fundamental (0,1,0) and the z-mode
fundamental (0,0,1) as 73.9 cm-1, 74.8 cm-1
and 76.4 cm-1. Corresponding fundamental excitations
in a pure 3D translational spectra, where the rotational effect
is included only by averaging the potential over the Euler angles,
are found to be 71.5 cm-1, 71.5 cm-1 and
71.6 cm-1. Compared to the 3D translational results,
translational fundamentals in a fully coupled T-R spectrum are
spread over 2.5 cm-1. The frequency spread can thus
be attributed not only to the non-symmetric arrangement of the
H atoms in the cage but also to the large coupling of the translational
and rotational modes. One additional strong evidence of the coupling
of the translational and rotational degrees of freedom can be
found by inspecting the contributions of the rotational basis
functions of the same j quantum number to the T-R eigenstates.
T-R eigenstates in a ground translational state (0,0,0) are almost
pure (93-99%) rotational states with j=1-5. Based on
this, one could conclude that the j is a good quantum
number. However, excitation of any one of the translational mode
in the x, y or z direction induces
strong mixing of the j=0 state with the j=3
rotational state. All excited translational states are 40-76%
j=0 states and 21-56% j=3 states.
Lets us now turn to the rotational excitations. For a free CH4
molecule, the j rotational level is (2j+1)2
degenerate. However, Table 1. shows that the confinement to the
cage changes this degeneracy, due to the anisotropy of the environment.
Breaking of the degeneracy of the rotational levels was observed
also for the hydrogen molecule inside the cages of sII clathrate
hydrate. For the ground translational state (0,0,0) first rotationally
excited state (j=1) consists of three triplets at 9.6
cm-1, 9.8 cm-1 and 10.0 cm-1.
Splitting in the j=1 state is thus only 0.4 cm-1.
Note also that the energy difference between the ground state
(j=0) and the j=1 state, is decreased to 9.8
cm-1 compared to the corresponding difference for the
freely rotating methane molecule 2B=10.5 cm-1.
The double excited rotational state j=2 consists of five
doublets and five triplets located between 22.6 cm-1
and 42.3 cm-1. Splitting in the j=2 state,
≈20 cm-1 respectively, is much bigger than splitting
in a j=1 state, but comparable to the splitting in the
j=3 state. To conclude, the frequencies of the j=1
← j=0, j=2 ← j=0 and j=2
← j=1 transitions are calculated to be 9.8 cm-1,
31.8 cm-1 and 22.0 cm-1.
| Figure 5. 3D isosurfaces
of the translational components of the wave functions of the
j=0 states of methane molecule for (left to right
and up to down) n=85 (1,0,0), n=86 (0,1,0),
n=422 (0,1,1), n=423 (0,0,2), n=1340
(1,1,1) and n=1474 (0,0,3) |
Comparison with the experimental
data
Rotational-translational motion of methane encaged in clathrate
hydrates have been studied in the inelastic neutron scattering
experiments (INS). Rotational excitations of the confined methane
molecules are found in the low-energy part of the INS spectrum
of the sI methane hydrate, i.e. below ≈30 cm-1
(Gutt et al. Appl.Phys.A 74 S1299 (2002) and Gutt at
all. Europhys.Lett 48 269 (1999)). Three bands are observed
at T=2.1 K and are assigned to the following rotational
transitions: j=1 ← j=0 at 8.7 cm-1,
j=2 ← j=0 at 26.6 cm-1 and j=2
← j=1 at 17.7cm-1 . The corresponding
transitions in our study are calculated as bands centered at 9.8cm-1
, 30.9 cm-1 and 21.1 cm-1. As can be seen,
calculated values are slightly overestimated compared to to experimental
values due to the certain deficiencies of the force field used
in the computations. The quantum mechanical energy levels of the
methane are calculated exactly for the methane-cage potential
energy surface employed, taking into account all the mode couplings
and anharmonicities. Our results can thus serve as a true test
of the quality of the widely used methane-cage potentials. Our
calculations further show that j=1 and j=2 levels
are split into three and ten components, allowing for numerous
rotational transitions differing very little in energy, which
may be hard to resolve experimentally. This could account for
the apparent lack of the structure in the measured bands which
appear as broad peaks.
In the region between 30 cm-1 and 105 cm-1
broad inelastic excitations are observed in the INS spectra of
methane sI hydrate (Baumert et. al. Phys.Rev.B 68, 174301
(2003)). While the peak at ≈42 cm-1 could be
clearly identified, it was not very clear how many excitations
contribute to the broad shoulder which is observed
from 56-105 cm-1. The assignment is done on the basis
of previous molecular dynamic (MD) simulations in which three
excitations are favoured. Namely, inelastic excitations were fitted
with three Gaussian functions and the positions of the three peaks
are then found to be 43.6 cm-1, 61.3 cm-1
and 80.7 cm-1. However our calculations of T-R energy
levels of methane in the small and large cage (preliminary results
for the large cage) show that the translational fundamentals can
be separated in three groups, but they are not degenerate. As
discussed previously, the splittings in the translational transitions
are due to the coupling between the translation and rotational
motion of the methane molecule which can not been accounted in
MD simulations. Two lower frequency transitions (around 30 cm-1
and 50.5 cm-1 ) are attributed to a translational modes
of the methane in the large cage while the higher frequency transitions
centered at 75 cm-1 are attributed to the translational
excitations of the methane in the small cage. The fundamental
translational excitations at 30cm-1 consist of two
closely separated lines and correspond to the translational modes
along the two longer axes in the large cage while the excitation
of the translational mode along the shorter axis in the large
cage is responsible for the singlet at 50.5 cm-1. Due
to the cubic symmetry of the small cage, translation fundamentals
in a small cage appear as three closely separated lines at 73.9
cm-1, 74.8 cm-1 and 76.4 cm-1.
These findings correlate extremely well with the change of the
environment experienced by the methane molecules in the two cages:
small cage is essentially a spherical top, while large cage has
a lower symmetry of oblate symmetric top. Because of the small
rotational constant of the methane molecule, every translational
excitation is accompanied by a vast amount of rotational transitions.
Only in a lowest 200cm-1 around 1000 T-R transitions
are calculated. This can explain huge broadness of measured translational
bands. Obviously it will be very hard to resolve experimentally
the fine structure of the lines and we hope our work can provide
valuable help in this aspect as well.
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