theoretical studies of co2(h2o)20,24,28 clusters: stabilization of cages in hydrates by co2 guest...

Theoretical studies of CO2(H2O)20,24,28 clusters: stabilization

of cages in hydrates by CO2 guest molecules

Arshad Khan*

Chemistry Department, Pennsylvania State University, DuBois, PA 15801, USA

Received 5 June 2003; revised 23 September 2003; accepted 23 September 2003

Abstract

Theoretical studies on dodecahedral (20-mer), tetrakaidecahedral (24-mer) and hexakaidecahedral (28-mer) water clusters

containing CO2 guest molecules are carried out by optimizing geometry at the Hartree–Fock (HF) level with 6-31G* basis set

followed by single point energy calculations with 6-311þþG** basis set and applying the Becke-3-parameter density

functional theory (DFT) and Lee-Yang-Parr correlation functional (B3LYP). While the filled tetrakaidecahedral and

hexakaidecahedral cage clusters are stabilized by 7.79 and 3.42 kcal/mol, respectively (relative to unfilled cage and separated

CO2 molecule, SEC), the filled dodecahedral cage shows no such stabilization. The largest SEC value for tetrakaidecahedral

cluster, resulting from two H-bonds between the guest and the host, explains the dominance of CO2-filled tetrakaidecahedral

structures in hydrates.

q 2003 Elsevier B.V. All rights reserved.

Keywords: CO2 in dodecahedral (20-mer); Tetrakaidecahedral (24-mer) and hexakaidecahedral (28-mer) water clustersIntroduction

1. Introduction

Hydrates are solid cage structures of water

molecules with various guest molecules in cavities

[1]. They can be found in natural gas pipelines, on the

ocean floor, in permafrost environments [2,3], deep

ice cores [4], rock inclusions [5], comets and certain

outer planets [6]. The hydrate structures occupied by

CO2 guest molecules have been known for many

years [7]. These hydrates are often considered to be

agents for global climate change and have received

significant attention from a large group of scientists

[1,8–10]. The most commonly observed cages in

various hydrates (especially hydrates I and II) are

dodecahedral (512), tetrakaidecahedral (51262) and

hexakaidecahedral (51264) clusters consisting of 20,

24 and 28 water molecules, respectively. Here,

Jeffrey’s notations [11,12] are used to represent

different cage types. For example, the notation 51262

represents a cage structure having 12 pentagonal and 2

hexagonal rings of water molecules.

It is believed that the CO2 molecules occupy most

of the large tetrakaidecahedral as well as a fraction of

smaller dodecahedral cages that exist in hydrate

structure I. Most of the studies reported so far are

confined to experimental methods of hydrate

preparation and characterization [1,10,13,14].

Because of large cluster size and number of electrons,

only a limited number of theoretical studies has so far

0166-1280/$ - see front matter q 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2003.09.010

Journal of Molecular Structure (Theochem) 664-665 (2003) 237–245

www.elsevier.com/locate/theochem

* Tel.: þ814-375-4744; fax: þ814-375-4784.

E-mail address: [email protected] (A. Khan).

been reported in this area [15–18]. These theoretical

studies primarily focus on cage structures occupied by

CH4 and N2 molecules. In one such study [16] it has

been noticed that the fused cage formation between

512 and 51262 (hydrate I) or between 512 and 51264

(hydrate II) provide significant stabilization. The guest

molecules like CH4 and N2 provide an additional

amount of stabilization and, thus, help in the formation

of above hydrates. Because of the importance of the

CO2-filled hydrate structures, similar studies are

undertaken so that one can understand the role of

CO2 guest molecules toward stabilization of cages that

are commonly found in hydrate structures I and II.

2. Method applied in calculation, definitions

of H-bonds and stabilization energies

The ab initio geometry optimization is done at the

Hartree–Fock (HF) level by using the 6-31G* basis

set on a number of assumed geometries. The

optimizations are followed by single point energy

calculations by using the 6-311þþG** basis set and

applying the Becke-3-parameter density functional

theory (DFT) and Lee-Yang-Parr correlation func-

tional (B3LYP) [19–22]. The GAUSSIAN 98 series of

programs [23] running parallel on Penn State’s IBM

RS/6000 AIX 5.1 cluster are used for these calcu-

lations. Each cluster node is a 4 CPU POWER-3 RISC

based system running at 375 MHz.

Each geometry optimization is followed by a

nearest neighbor atom search. When two O atoms

are within a distance of 3.2 A, and an H atom in

between and makes an OHO angle of 1468 or larger,

the H atom is considered to be an H-bonding atom.

Based on these distance and angle criteria, the

numbers of H-bonds as well as dangling non-H-

bonding H (NHB H) atoms are determined.

Three different stabilization energy values are

discussed here. The magnitude of stabilization energy

value relative to separated constituent molecules (SE)

is calculated as follows for a n-mer water cluster

with a CO2 guest molecule in cavity, that is, for

the CO2[(H2O)n] cluster:

SEðCO2½ðH2OÞn�Þ ¼ EðCO2½ðH2OÞn�Þ2 EðnH2OÞ

2 EðCO2Þ

The SEP represents stabilization energy per mono-

mer, that is, SE/(n þ 1) for the above cluster with n

number of H2O and one CO2 molecules. The

stabilization energy, SEC, is calculated relative to

an empty cage and separated CO2 molecule as

follows:

SECðCO2½ðH2OÞn�Þ¼EðCO2½ðH2OÞn�Þ2Eð½ðH2OÞn�Þ

2EðCO2Þ

3. Reliability of the method

Even though the 6-31G* represents a moderately

sized basis set for HF optimizations, test results [24]

on water dimer suggest that the application of this

basis set provides an accurate dimer geometry. In

addition, the predicted structural features of larger

clusters are also in close agreement with those of

experiments. In large clusters, the size of cage

diameters is determined by both OO distances and

OOO angles. For irregular dodecahedral clusters [16]

the predicted radius is 4.1 A and the experimental

value obtained from hydrates is 4.1 A Ref. [1].

Similarly, the predicted radius for a dodecahedral

cage is around 4.1 A and the experimental value is

around 4.0 A. Hence, in general, one can say that the

HF/6-31G* optimization predicts reliable geometry

for both small and large water clusters. Since the

present study involves clusters of CO2 and H2O

molecules, additional tests were required. For this

Fig. 1. The HF/6-31G* optimized CO2(H2O)4 cluster is presented.

The HF/6-31G* optimized structure agrees closely with that

obtained by MP2/6-311þþG** optimization.

A. Khan / Journal of Molecular Structure (Theochem) 664-665 (2003) 237–245238

purpose geometry optimization of a relatively small

CO2(H2O)4 cluster (Fig. 1) was carried out at various

levels of ab initio theory to compare results with those

obtained by the method (B3LYP/6-311þþG**//HF/

6-31G*) applied in this study. Table 1 presents these

results on structural features and stabilization ener-

gies. A close agreement of structural features between

the HF/6-31G* and the MP2/6-311þþG** optimiz-

ation methods can be readily noticed. For example,

the OO, OH (non-H-bonded), CO (bonded) and C–O

(non-banded, nearest O atom) distances show dis-

crepancies of only 2.8, 1.4, 2.3 and 1.8%, respectively

from those calculated by the MP2/6-311þþG**

method. The bond angles like OOO angles (within the

ring) of around 908 and HOH (between covalently

bonded OH) angles of around 1068 also show a close

agreement with the MP2/6-311þþG** results.

The DFT calculations with a large basis set like 6-

311þþG** are known to provide reliable energy

values in comparison with experiments. An error

analysis of calculated energy values is presented in

Ref. [25], and shows only a small error for the

B3LYP method using the 6-311þþG** basis set. In

deed, for CO2(H2O)4 cluster the SE and SEP values

of 28.29 and 5.66 kcal/mol, calculated by the DFT

method, closely agree with the MP2/6-311þþG**

result of 30.95 and 6.19 kcal/mol, respectively

(Table 1).

4. Results and discussions

Each of the optimized dodecahedral, tetrakaide-

cahedral and hexakaidecahedral structures with a

CO2 guest molecule in cavity shows similar struc-

tural features. For example, the average OH length

(Table 2) in each cluster is about the same with a

value of around 0.955 A and standard deviation (SD)

of around 0.005 A. Similarly, the average HOH angle

between two O–H bonds is around 1068 with a SD

value of around 18 in each cluster. The other

structural features (Table 2) and energy values

(Table 3) that change from one cluster type to

another are discussed below. Before examining the

filled clusters, optimizations and energy calculations

are carried out on unfilled cages like dodecahedron,

tetrakaidecahedron and hexakaidecahedron. The

dodecahedral structures are examined more

thoroughly to understand how the SE values change

when non-H-bonding H (NHB H) atoms are

rearranged on the surface.

Table 1

Structural features and energies of CO2(H2O)4 calculated at different levels of theory. The distances are expressed in A, energy values ðEÞ in

Hartree, stabilization energies SE and SEP in kcal/mol. The standard deviations from the average are shown in parentheses. The values for CO2

and H2O are given below the E values of the cluster. The results from the DFT/HF method applied in this study show a close agreement with

those calculated by the MP2 method of optimization with a basis set of 6-311þþG**

Method of calculation applied O–O

(A)

O–H

(cov.)

(A)

C–O

(non-banded)

(cov) (A)

C–O

(A)

E

(Hartree)

SE

(kcal/mol)

SEP

(kcal/mol)

B3LYP/6-311þþG**//HF/6-31G* 2.855 (0.047) 0.951 (0.005) 1.143 2.949 2494.523031 28.29 5.66

2188.645510 (CO2)

276.458106 (H2O)

MP2/6-311þþG** 2.778 (0.032) 0.965 (0.008) 1.170 3.002 2493.355198 30.95 6.19

2188.206200 (CO2)

276.274920 (H2O)

MP2/6-311G** 2.746 (0.030) 0.963 (0.008) 1.169 2.936 2493.311614 35.79 7.16

2188.198687 (CO2)

276.263972 (H2O)

MP2/6-31G* 2.774 0.974 1.180 2.876 2492.954474 37.23 7.45

2188.107747 (CO2)

276.196848 (H2O)

A. Khan / Journal of Molecular Structure (Theochem) 664-665 (2003) 237–245 239

5. (H2O)20 dodecahedral cluster: Different

arrangement of NHB H atoms

Fig. 2 shows 8 isomers (S1, S2, a – f) of

dodecahedral cluster with different arrangement of

NHB H atoms on the cluster surface. Each of this

structure has 10 NHB H atoms and 30 HB H atoms

along 30 edges of dodecahedron. The S1 represents

the most stable isomer with three sets (one set

separated from the other) of 2 NHB H atoms in

adjacent locations and rest 4 arranged in locations

without any adjacent NHB H atom. The isomer S2 is

similar to S1, except that one of its NHB H atoms is

turned inward (shown with an open circle) causing

Table 3

Energy, stabilization energy (SE, SEC) and SE/monomer (SEP) values presented. The SE values are calculated relative to separated constituent

molecules and SEC values relative to unfilled cage plus CO2 molecule. The energy values are calculated at the B3LYP/6-311þþG** level after

geometry optimization at the HF/6-31G* level. Energy values in parentheses are those obtained by geometry optimization and energy

calculation at the mp2/6-311þþG** level. Water molecules that form a cage are shown in brackets and the molecule in cavity is shown outside

of the brackets. For dodecahedral cluster, a number of isomers are examined that differ in the arrangement of 10 dangling H atoms on the surface

Clusters, molecules Energy values (Hartree) B3LYP/6-311þþG** Stabilization energy (kcal/mol)

SEC SE SEP

CO2[(H2O)20], Fig. 3 21718.128435 21.05 201 9.57

CO2 þ (H2O)20 21718.130113

CO2[(H2O)24], Fig. 4 22024.006746 7.79 230 9.20

CO2 þ (H2O)24 22023.994336

CO2[(H2O)28 22329.913647 3.42 277 9.55

CO2 þ (H2O)28 22329.908190

[(H2O)20], Fig. 2 dodecahedron 21529.495868 (S1) 209 10.45

21529.490742 (S2) 206 10.30

21529.484603 (2a) 202 10.10

21529.474909 (2b) 196 9.80

21529.473962 (2c) 196 9.80

21529.473458 (2d) 195 9.75

21529.470081 (2e) 193 9.65

21529.456324 (2f) 185 9.25

[(H2O)24] tetrakaidecahedron 21835.348826 222 9.25

[(H2O)28] hexakaidecahedron 22141.262680 273 9.75

CO2(H2O) 2265.110453 (2264.486835) 4.29 (3.59)

H2O 276.458106 (276.274920)

CO2 2188.645510 (2188.2061999)

Table 2

Optimized (HF/6-31G*) bond lengths and angles (average) for different clusters. The standard deviations (SD) from the mean are shown in

parentheses. The HOH angles are due to three covalently bonded atoms. The average O–H length represents the distance due to both NHB H

(non-H-bonded H atom) and HB H atoms. Water molecules that form a cage are shown in brackets and the guest molecule in cavity is shown

outside of brackets

Cluster type Lengths (SD) (A) Angles (SD) (deg)

O–H Av. (SD) OO Av. (SD) C–O Min. cage diam. (SD) OOO Av. (SD) H–O–H Av. (SD)

CO2[(H2O)20], Fig. 3 0.955 (0.005) 2.877 3.157 (0.0875) 8.07 (0.35) 108 (8) 106 (1.13)

CO2[(H2O)24], Fig. 4 0.955 (0.006) 2.917 2.925 (0.148) 9.65 (0.15) 102 (20) 106 (1.10)

CO2[(H2O)28] 0.955 (0.006) 2.922 2.894 (0.107) 9.37 (0.38) 104 (21) 106 (0.93)

[(H2O)20], Fig. 2a dodecahedron 0.956 (0.006) 2.877 (0.092) 8.12 (0.06) 108 (4) 106 (0.83)

[(H2O)24] tetrakaidecahedron 0.955 (0.005) 2.897 (0.100) 9.31 (0.59) 104 (17) 106 (1.06)

[(H2O)28] hexakaidecahedron 0.955 (0.005) 2.889 (0.089) 9.44 (0.52) 104 (21) 106 (0.86)


a distortion to the cage with the formation of a weak

H-bond (OHO angle of 1468) within the cavity. In

isomer 2a there are two sets of 3 NHB H atoms, one

set of 2 NHB H atoms, and two NHB H atoms by

themselves without having any adjacent NHB H atom.

The least stable isomer, f, on the other hand, has all

the 10 NHB H atoms in adjacent locations. Hence, one

can conclude that by spreading out the NHB H-atoms

on the cage surface with fewest of them in adjacent

positions, a cluster with high stability can be obtained.

The unfilled dodecahedral isomers are arranged

according to decreasing energy (S1 being the most

stable isomer with the SE value of 209 kcal/mol) in

Table 3. A number of other isomers (not shown here)

are also examined with one of the NHB H atoms

directed toward cavity. These structures have almost

the same SE values (0–3 kcal/mol difference) as those

with all the NHB H atoms directed outward.

6. CO2(H2O)20 cluster

Fig. 3 represents an optimized dodecahedral cluster

with a CO2 guest molecule in the cavity. The O and H

atoms are shown in the figure with black and light

spheres. The optimizations are carried out by selecting

two (2a and 2f) dodecahedral clusters each with a CO2

molecule in its cavity. The SE values of these clusters

are almost the same as their empty cage (1 kcal/mol

lower). The average cavity diameter of this cluster is

around 8.07 A and represents almost the same size as

that of an unfilled dodecahedron (8.12 A). The OO

distance in this cluster ranges from around 2.721 to

3.027 A with an average value of around 2.877 A

(SD ¼ 0.0875). This average OO distance is the same

as that of an unfilled dodecahedral cluster. The OOO

angle ranges from 87 to 1258 with an average value of

1088 (SD ¼ 88). The bonded C–O lengths are 1.143

and 1.144 A and OCO bond angle is around 1778 in

the CO2 guest molecule, and these values represent

almost an undistorted CO2 molecule within the cage

cavity. The nearest C–O distance (non-banded) from

Fig. 2. Unfilled dodecahedral cage with different arrangement of

NHB H atoms. The optimized structures (S1, S2, a–f) are shown in

order of decreasing stabilization energy (SE). The structure S1 is the

most stable and f is the least stable isomer among the eight isomers.

Fig. 3. CO2-filled optimized dodecahedral cage with 10 NHB H

atoms directed outward. The O atoms are shown in black spheres

and H atoms are shown with grayish-white spheres. The C atom is

shown within the cavity with a gray sphere.


the C atom of the guest molecule to an O atom of

water molecule on the cage surface is around 3.16 A,

and represents an off-centered location for the guest

molecule (cage radius is around 4.04 A). This seems

to be the case with most guest molecules within

different cages that have been studied so far.


The OO distance in this filled tetrakaidecahedral

cluster (Fig. 4) ranges from around 2.287 to 3.172 A

with an average value of around 2.917 A

(SD ¼ 0.148). On an average the OO distance in

this cluster is longer than that in the CO2(H2O)20

cluster (2.908 A) or the unfilled dodecahedral

(2.884 A) or tetrakaidecahedral (2.897 A) cluster.

The average cavity diameter is around 9.65 A and is

larger than that of the unfilled tetrakaidecahedral cage

(9.31 A, SD ¼ 0.59 A). The lengthening of the OO

distance is therefore caused by the enlargement of the

cage structure caused by the inclusion of the guest

molecule. The OOO angle in this cluster ranges from

42 to 1618 with an average value of 1028 (SD ¼ 208).

The unfilled tetrakaidecahedral structure has the OOO

angles ranging from 57 to 1538 with an average value

of 1048 (SD ¼ 178). These values suggest that the

cage structure is more distorted when the CO2

molecule occupies its cavity. The bonded C–O

lengths are 1.142 and 1.144 A and the OCO bond

angle is around 1798 in the CO2 guest molecule. These

values represent almost an undistorted CO2 molecule

within the cage cavity. The nearest C–O distance

(non-banded) from the C atom of the guest to an O

atom (of a H2O molecule) on the cage surface is

around 2.92 A and represents an off-centered location

for the guest molecule (cage radius is around 4.82 A).

In this cluster the CO2 guest molecule is held to the

cage cavity by two H-bonds contributed from two

H2O molecules on the cage surface. Each of these

H2O molecules is oriented in such a way so that one of

their H atoms is directed toward cavity for H-bonding

with the guest molecule. The OO distances from O

atoms of these water molecules to O atoms of the CO2

molecule are 2.979 and 3.002 A with the OHO (H

lying in between) angles of 174 and 1758. Interest-

ingly, the filled tetrakaidecahedral cluster is substan-

tially more stable (by 7.8 kcal/mol, Table 3) than the

unfilled cage and separated CO2 molecule (SEC,

Table 3).


The optimized hexakaidecahedral cluster with a

CO2 molecule in cavity (not shown here) has an

average diameter of around 9.37 A (SD ¼ 0.38 A)

and represents a slightly reduced cage size compared

to that of the unfilled hexakaidecahedral cage

(9.44 A). The OO distance in this cluster ranges

from around 2.729 to 3.170 A with an average value

of around 2.922 A (SD ¼ 0.107). The average OO

distance in this cluster is longer than that in the

unfilled hexakaidecahedral (2.889 A) cluster and the

OOO angle ranges from 43 to 1568 with an average

value of 1048 (SD ¼ 218). The unfilled hexakaideca-

hedral structure has the OOO angles ranging from 57

to 1488 with an average value of 1048 (SD ¼ 218).

These values suggest that the hexakaidecahedral cage

structure does not show much distortion when the CO2

molecule occupies its cavity. The bonded C–O

lengths are 1.142 and 1.144 A and the OCO bond

angle is around 1788 in the CO2 guest molecule. These

values, as in other clusters discussed above, represent

almost an undistorted CO2 molecule within the cage

cavity. Also, the nearest C–O distance from the C

atom of the guest to an O atom of a H2O molecule on

the cage surface is around 2.89 A and represents

Fig. 4. The optimized tetrakaidecahedral cage (24-mer) with a CO2

guest molecule within the cavity. The guest molecule is bonded by

two H-bonds.


an off-centered location for the guest molecule (cage

radius is around 4.68 A).

As in tetrakaidecahedral cluster, the CO2 guest

molecule in this cluster is held to the cage cavity by

two H-bonds contributed from two H2O molecules on

the cage surface. Unlike the filled tetrakaidecahedral

cluster, the filled hexakaidecahedral cage is more

stable than its unfilled cage (þCO2) by only

3 kcal/mol (SEC). This energy is less than 1/2 the

value obtained for filled tetrakaidecahedral cluster

and requires further examination. Possible expla-

nation for these energy differences is given under

Section 9.

9. Comparison of Stabilization energies of different

filled clusters

Among the three types of filled clusters studied

here, the tetrakaidecahedral (Fig. 4) cluster shows the

maximum stabilization relative to unfilled cage (SEC,

around 8 kcal/mol) and the separated guest molecule.

As mentioned above, the filled hexakaidecahedral

structure shows significantly less stabilization (SEC

of around 3 kcal/mol) and the dodecahedral cluster

shows no stabilization at all. In order to examine

whether the H-bonding between the guest and the host

has anything to do with the above SEC values, the

following calculations are carried out on a CO2(H2O)

cluster bonded by a single H-bond. The geometry is

optimized at the HF/6-31 G* level followed by a

single point energy calculation at the B3LYP/6-

311þþG** level. The stabilization energy (SE) for

this cluster relative to separated CO2 and H2O

molecules is around 4 kcal/mol (Table 3) and,

hence, represents the H-bond strength without the

zero-point energy (ZPE) correction. Thus, the cage

stabilization (SEC) by 8 kcal/mol for filled tetrakai-

decahedral cluster can be explained on the basis of

two H-bonds that bind the guest molecule to the host

(Fig. 4). Interestingly, much lower amount of

stabilization (around 3 kcal/mol) is achieved for a

filled hexakaidecahedral cluster even when there are

two H-bonds connecting the guest molecule to the

host, and hence, requires further examination. The

analysis of OHO angles (H in between two O atoms)

that indicates the effectiveness of the H-bond is

performed on these clusters. When the OHO angle is

1808 the H atom lies along a line joining the two O

atoms, and is expected to provide an effective

H-bonding. On the other hand, as this angle gets

smaller the H-bond becomes increasingly less effec-

tive. The OHO angles in hexakaidecahedral cluster

responsible for H-bonds between the guest and the

host molecule are significantly smaller (162 and 1668)

than 1808. These H-bonds can be considered to be

much less effective than those in filled tetrakaideca-

hedral cluster in which OHO angles are 174 and 1758

even though the OO distance is almost the same in

each cluster type (around 3 A). The filled dodecahe-

dral structure does not have any such H-bond (Fig. 3)

and hence, there is no cage stabilization noticed when

the guest molecule fills the cage cavity. These results

are consistent with those observed experimentally;

that is, most of the tetrakaidecahedral cages are filled

with CO2 with only a limited number of filled

dodecahedral cage in the hydrate I structure. Even

though no stabilization is achieved by filling a

dodecahedral cage with a CO2 molecule, it is expected

that a guest in the cavity will prevent the collapse of

the cage under pressure, and thus, may provide

stability to hydrate structures under pressure.

The SE values (Table 3) relative to constituent

monomers increase from the filled dodecahedral to

hexakaidecahedral cluster with the values of 201, 230

and 277 kcal/mol. Similarly, for unfilled cages the SE

values increase from smaller dodecahedral to the

largest hexakaidecahedral clusters. These values

suggest that an increasingly larger amount of energy

is released when the cluster size is increased. The SEP

values indicate that each of the above clusters is stable

and the dissociation of a single monomer molecule

from the cluster requires about 9–10 kcal/mol.

10. Guest–host H-bonding and experimental

evidence

The present calculation suggests that a tetrakai- or

hexakaidecahedral cluster has two H atoms turned

toward cavity when it is filled with a CO2 molecule. As

mentioned above, the H-bond energy without the ZPE

correction is around 4 kcal/mol. By applying the ZPE

correction (mp2/6-311þþG*) this value reduces to

around 2 kcal/mol, and is significantly lower than that

in a water dimer (5.3 kcal/mol, after ZPE correction,


mp2/6-311þþG* calculation). It should be pointed

out that the hydrate structures are formed by the fusion

of different cage structures. Since an H-bond between

two H2O molecules is stronger (5 kcal/mol) than that

between a CO2 molecule and a H2O molecule

(2 kcal/mol), one would expect most of the inward

turned H atoms to be turned outward during a hydrate

formation so that fused structure can form with

maximum number of H-bonds between water mol-

ecules. This would suggest that a smaller number of

clusters could still exist with inward turned H atoms.

Indeed, the dielectric and NMR studies suggest the

presence of a number of H-bonds between the host

water cluster and the CO2 guest molecule, and the

existence of Bjerrum defects into the lattice [10,26,27]

of CO2-hydrates. These defects result in certain OO

distances (within 3.2 A) without any H atom in

between, and thus, allow inward turning of certain H

atoms for the H-bond formation with oxygen atom of

the CO2 guest molecule. Such a weak guest–host H-

bond in CO2-filled cluster is vulnerable to cleavage

when temperature is increased allowing the guest

molecule to undergo translational and rotational

motion. Indeed, neutron diffraction study by Ikeda

et al. [27] suggest rotation of the CO2 molecule within

the tetrakaidecahedral cage cavity allowing the

formation of H-bonds at various locations of the host.

The rotation, as expected, becomes increasingly

restricted when the temperature is lowered. Thus, all

the known experimental results can be explained on the

basis of this study.

11. Selection of starting geometry: SE, SEP

and SEC values

Even though there is some degree of randomness in

the selection of a starting geometry (for example, a

CO2 filled cage with all NHB H atoms outward), the

optimized structure shows significant deviation from

the starting geometry with two H atoms reoriented

toward cavity of tetrakai and hexakaidecahedral

clusters. This observation suggests a relatively

stronger interaction between the guest and the host

in CO2-hydrate than in say methane-hydrate where

such reorientation of dangling NHB H atoms is not

noticed [15]. Since the reorientation of water

molecules on the cage surface is caused by

the presence of CO2 guest molecule in cavity, and

the weak H-bonds (2 kcal/mol, with ZPE correction)

exist between the guest and the host, a low activation

energy for the reorientation of water molecules is

expected. This, indeed, supports the experimental

findings [10,26].

Even though various locations of NHB H atoms

change the SE values, much smaller change in SEP

values (SE per monomer) is noticed. As discussed

earlier, the SEP value provides key information about

the stability of a cluster toward dissociation of a

monomer. Similarly, the SEC value, that gives

stabilization energy relative to an empty cage (and

separated guest), also shows only a small change from

one geometry to another with changed arrangement of

dangling H atoms. Even though this has been tested

for different dodecahedral cages, similar results are

expected for larger clusters.

From the above discussions one can say that the

major conclusions of this paper regarding the

stabilization energy relative to empty cage (SEC),

stabilization energy per monomer (SEP) and type of

interaction (H-bonding) between the host and the guest

remain unchanged even if the global minimum

structure has not been obtained in the study.

12. Concluding remarks

Even though the theoretical studies are done on

isolated cage structures that are present as fused cages

in various hydrates, one can still draw a conclusion

about the role of guest molecules on the basis of above

studies. As has been noticed in an earlier study [16],

the fused cage formation does not change the structure

of a cage in any substantial way for which the

predicted cage diameters and OO distances are close

to those of the experiments obtained from solid

hydrates. For example, the predicted hexakaidecahe-

dral diameter [17] with and without guest molecules

range from 9.29 to 9.55 A with an average value of

around 9.45 A and is close to the experimental

average value [1] of around 9.46 A.

Three stabilization energy values are discussed for

a filled cage cluster. While the SE value provides the

total binding energy for the cluster relative to

constituent molecules, the SEP value provides the

energy requirement for the dissociation of a monomer


from the cluster, and hence, provides its survival

possibility under collisions. The third stabilization

energy, SEC, allows one to determine whether a cage

is stabilized when it is filled with a guest molecule.

Acknowledgements

The author acknowledges help from Jeff Nucciar-

one and the Numerically Intensive Computing Group

at the Center for Academic Computing at Penn State

for generous computation time provided on the IBM

RS/6000 SP2 system.

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