Supplementary Material for: Free Energies of Carbon Dioxide Sequestration and
Methane Recovery in Clathrate Hydrates
Peter Dornan, Saman Alavi, and Tom K. Woo
Department of Chemistry, University of Ottawa, Ottawa, Ontario K1N 6N5
Computational Methods:
Notes on the specific procedures used for each replacement reaction.
For the Ψ → X′ conversion processes given in Eq. (9), different combinations of
atom annihilations Eq. (11) and potential parameter mutations, Eq. (12) are possible for
the coupling parameter λ3. Additionally, conversions involving N2 needed to be split into
two steps. Note that processes of modifying the guest van der Waals potentials are
considered in the direction of the products to reactants. The calculated values for each
component are listed in Table 4 of the main text.
R1: clathrate[8CH4(S+L)] + 8CO2(f) → sI_clathrate[8CO2 (S+L)] + 8CH4(f)
The mutation formula Eq. (11) was used to transform LJ potential
parameters of the CO2 carbon atom to the parameters of the CH4 carbon atom. The
annihilation formula, Eq. (10) was used to simultaneously annihilate the pair potentials of
the oxygen atoms of CO2. A plot of configurational energy versus λ for each of the three
steps is shown in Figure 1.
Figure 1. Configurational energy as a function of λi for (R1):
clathrate[8CH4(S+L)] + 8CO2(f) → clathrate[8CO2(S+L)] + 8CH4(f)
● CH4 (charged, λ=1) → X (uncharged, λ=0)
■ CO2 (charged, λ=1) → CO2 (uncharged, λ=0)
 CO2 (uncharged, λ=1) → X′ (uncharged, λ=0)
R2: clathrate[8CH4(S+L)] + 6CO2(f) → clathrate[2CH4(S)+6CO2(L)] + 6CH4(f)
Charge was removed from all guest molecules in the ΔGelec,i stages, even though
only the large cages guests are substituted. This was done to make use of the decharging
of the sI clathrate already calculated in (R1). A procedure similar to that of (R1) was used
to convert the CO2 guests in the large cages to X′ guests. A plot of configurational energy
versus λ for each of the three steps is shown in Figure 2.
Figure 2. Configurational energy as a function of λi for (R2):
clathrate[8CH4(S+L)] +6CO2(f)→ clathrate[2CH4(S)+6CO2(L)] +6CH4(f)
● CH4 (charged, λ=1) → X (uncharged, λ=0)
■ CH4/CO2 (charged, λ=1) → CH4/CO2 (uncharged, λ=0)
 CH4/CO2 (uncharged, λ=1) → X/X′ (uncharged, λ=0)
R3: clathrate[2CH4(S)+6CO2(L)]+2N2(f) → clathrate[2N2(S)+6CO2(L)] + 2CH4(f)
The process of converting uncharged N2 into uncharged CH4 was split into two
steps. First the LJ parameters of CH4 carbon were added to the center of mass of the N2
molecule using the modification formula Eq. (11), creating N2*, which is an uncharged
species with methane potential on the center of mass and N2 potential on the N atoms.
Next, the LJ parameters on the N atoms were removed, using the annihilation formula
Eq. (10), to give the X’ molecule. We observed that for a one step transformation of N2 to
X’, for some intermediate values of λ, the guests escaped their cages due to inadequate
intermolecular interactions. A plot of configurational energy versus λ for each of the four
steps is shown in Figure 3.
Figure 3. Configurational energy as a function of λi for (R3):
clathrate[2CH4(S)+6CO2(L)] + 2N2(f) → clathrate[2N2(S)+6CO2(L)] + 2CH4(f)
●
■

▲
CH4/CO2 (charged, λ=1) → X/CO2 (uncharged, λ=0)
N2/CO2 (charged, λ=1) → N2/CO2 (uncharged, λ=0)
N2/CO2 (uncharged, λ=1) → N2*/CO2 (uncharged, λ=0)
N2*/CO2 (uncharged, λ=1) → X′/CO2 (uncharged, λ=0)
R4: clathrate[8CH4(S+L)] + 2N2(f) → clathrate[2N2(S) + 6CH4 (L)] + 2CH4(f)
The procedure similar to (R3) was used. A plot of configurational energy versus λ
for each of the four steps is shown in Figure 4.
Figure 4. Configurational energy as a function of λi for (R4):
clathrate[8CH4(S+L)] + 2N2(f) → clathrate[2N2(S) + 6CH4 (L)] + 2CH4(f)
●
■

▲
CH4 (charged, λ=1) → X (uncharged, λ=0)
N2/CH4(charged, λ=1) → N2/CH4 (uncharged, λ=0)
N2/CH4 (uncharged, λ=1) → N2*/CH4 (uncharged, λ=0)
N2*/CH4 (uncharged, λ=1) → X′/CH4 (uncharged, λ=0)
R5: clathrate[2N2(S)+6CH4(L)] +6CO2(f) → clathrate[2N2(S)+6CO2(L)] +6CH4(f)
A procedure similar to (R2) was used. A plot of configurational energy versus λ
for each of the three steps is shown in Figure 5.
Figure 5. Configurational energy as a function of λi for (R5):
clathrate[2N2(S)+6CH4(L)] +6CO2(f)→ clathrate[2N2(S)+6CO2(L)] +6CH4(f)
● N2/CH4 (charged, λ=1) → N2/X (uncharged, λ=0)
■ N2/CO2 (charged, λ=1) → N2/CO2 (uncharged, λ=0)
 N2/CO2 (uncharged, λ=1) → N2/X′ (uncharged, λ=0)
R6: sII:clathrate[24CH4(S+L)] +24CO2(f) → sII:clathrate[24CO2(S+L)] +24CH4(f)
The mutation formula Eq. (11) was used both to transform the LJ pair potentials
of the CO2 carbon to the parameters for CH4 carbon, as well as simultaneously annihilate
the pair potentials involving the O atoms of CO2 (here the parameters σf and εf are both
taken as 0). Note that the annihilation formula Eq. (10) was not used to eliminate the
oxygen atoms because some of the simulations were unstable to this potential. A plot of
configurational energy versus λ for each of the three steps is shown in Figure 6.
Figure 6. Configurational energy as a function of λi for (R6):
sII:clathrate[24CH4(S+L)] + 24CO2(f) → clathrate[24CO2 (S+L)] +24CH4(f)
● CH4 (charged, λ=1) → X (uncharged, λ=0)
■ CO2 (charged, λ=1) → CO2 (uncharged, λ=0)
 CO2 (uncharged, λ=1) → X′ (uncharged, λ=0)
Free energy calculations for multiple occupancies.
The sequestering capacity sI or sII CO2 clathrates can be studied by determining
the free energy for CO2 insertion into the large cages, (R7 in the main text for sII), or
equivalently, and computationally more convenient, by the annihilation of a single CO2
guests from doubly occupied sI and sII clathrate large cages, the inverse reaction. For the
annihilation reaction, the free energy calculation is divided into a two steps. In the first
stage, the electrostatic potentials of one of the CO2 molecules in each large cage are
removed via a coupling parameter, λelec in a function of the form given in Eq. (6).
Subsequently the LJ potentials for these same molecules are removed with the
introduction of a coupling parameter for the LJ potential, λvdW, the form of which is given
in Eq. (10). For the first process, (removal of the electrostatic potential), λvdw =1 and
simulations are performed with λelec varying from 1 to 0 in increments of 0.1. The final
configuration of the simulation with λelec=0 is taken as the initial configuration for the
process whereby the van der Waals potentials are removed. The Gibbs free energy for the
overall process is:
0
G 21   delec
1
U vdw ,elec 
elec
0
NPT ,vdw 1
  dvdw
1
U vdw ,elec 
vdw
(S1)
NPT ,elec  0
The numerical integration is performed by fitting the curves of U vs λi to cubic
polynomials. A plot of configurational energy versus λ for each of the three steps is
shown in Figure 7.
Figure 7. Configurational energy as a function of λi for (R7):
sII:clathrate[24CO2]+ 8CO2(f) → clathrate[32CO2 ]
● CO2 (charged, λ=1) → CO2 (uncharged, λ=0)
■ CO2 (uncharged, λ=1) → CO2 (Ideal gas, λ=0)
A representation of the simulated clathrate structures is shown in Figure 8.
Figure 8. A 3×3×3 supercell of the sI framework and guests for the CO2(L)/CH4(S)
clathrate with O atoms as vertices and H atoms not shown. One small and
large cage are highlighted in red.