Density functional studies of fused dodecahedral and irregular-dodecahedral water cages

V. Shilpi, Surinder Pal Kaur and C. N. Ramachandran*
Department of Chemistry, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India-247667. E-mail: ramcnfcy@iitr.ac.in

Received 7th July 2015 , Accepted 17th August 2015

First published on 17th August 2015


Abstract

Fused cages with maximum number of t1d bonds are modelled by combining two dodecahedral cages (DD + DD), two irregular-dodecahedral cages (IDD + IDD) and a dodecahedral cage with an irregular-dodecahedral (DD + IDD) cage and are studied using the dispersion corrected density functional method B97-D in conjunction with the cc-pVTZ basis set. The stabilization energy per water molecule for the most stable fused cages followed the order fused dodecahedral (FDD) > fused dodecahedral-irregular-dodecahedral (FDI) > fused irregular-dodecahedral (FII), and is higher than that of the corresponding single cages from which they are constituted, showing an enhanced interaction between water molecules in the fused cages.


Introduction

Gas hydrates are crystalline solids that are formed when gas molecules are trapped inside the cavities of hydrogen-bonded water cages.1 Natural gas hydrates exist mainly in three forms, namely, sI, sII and sH hydrate structures. The combination of two dodecahedral (512) and six tetrakaidecahedral (51262) cages results in the formation of the sI hydrate. The sII type hydrate is formed by the combination of 16 dodecahedral (512) and eight hexakaidecahedral (51264) water cages, whereas the combination of three dodecahedral (512) cages, two irregular-dodecahedral (435663) cages and one icosahedral (51268) cage yields the sH hydrate.2,3

It is a well-known fact that the structure and the stability of a water cage depend largely on the hydrogen-bond topology.4–16 Among the studies on this subject, the investigations by Kirov et al.15,17 are unique and need special mention. They proposed two models, namely, the strong–weak bond model (SWB) and the strong–weak-effective-hydrogen bond model (SWEB), to explain the most stable networks in polyhedral water clusters. According to the SWB model, a hydrogen bond is considered to be strong when the hydrogen atom of the donor molecule, which is not involved in the hydrogen bond under consideration, is trans with respect to the bisector of ∠HOH of the acceptor water molecule. This model, however, has the drawback that it considers only the interactions between the nearest-neighbors (n-n). In the SWEB model, this drawback is rectified by considering the effect of the interactions between the next-nearest-neighbors (n-n-n). According to the SWEB model, the hydrogen bond with one dangling O–H bond on the donor water molecule, which is trans with respect to the bisector of ∠HOH of the acceptor water molecule, is designated as a t1d type and is considered to be the strongest hydrogen bond.17

Recently, we used the SWEB model to study the isomers of different families of (H2O)20 clusters.18 These families include (i) edge-sharing pentagonal prisms, (ii) face-sharing pentagonal prisms, (iii) fused cubes, (iv) dodecahedrons and (v) irregular dodecahedrons. Being the building blocks of gas hydrates, the dodecahedral and irregular-dodecahedral water cages have attracted special attention and are well discussed in the literature.19–27

Although several studies have been carried out in the past on the dodecahedral and irregular-dodecahedral water cages, not much is known about their fused cages. Using semi-empirical quantum mechanical calculations (ZINDO), Arshad Khan studied the fused cages formed by the various combinations of dodecahedral, irregular-dodecahedral and icosahedral cages.28 To the best of our knowledge, the relative stabilities of the isomers of fused cages formed from the same or different types of single cages have not yet been investigated. Because of the different possible orientations of O–H bonds, a large number of isomers are possible for a water cage and hence the modeling of these cages is tedious.29–32 Because the number of such isomers increases with increase in the number of water molecules, the modeling of fused water cages becomes more challenging.

Keeping this in mind, in the present study, the concept of the SWEB model was extended for investigating the structure and the stability of fused cages formed from dodecahedral and irregular-dodecahedral water cages. Being building blocks, the modeling of these fused cages is important to obtain a molecular level understanding of the gas hydrates.

Methodology

All the geometries of the water cages reported in the present study were optimized at the B97-D/cc-pVTZ level of theory using the electronic structure program NWChem.33,34 The choice of the functional B97-D and the basis set cc-pVTZ was based on our previous studies on single water cages.18 The abovementioned method and basis set have also been reported to be ideal for similar systems in par with MP2 results.35–37 We also carried out single point energy calculations for the most stable cages obtained at the B97-D/cc-pVTZ level using various functionals of the M06 family and the results are reported in the ESI.

The stabilization energy (SE) was calculated using the supermolecular approach as

SE = EclusternEwater
where SE, Ecluster, Ewater and n are the stabilization energy, the energy of the cluster, the energy of one water molecule and the number of water molecules, respectively. The stabilization energies were corrected for the basis set super position (BSSE) error using the counterpoise method.38 Frequency calculations for the most stable geometries were carried out at the same level of theory to ascertain that the reported geometries belong to the minima of the respective potential energy surface. The stabilization energies were also corrected for zero point energy.

Because the present study involves the clusters of different numbers of water molecules, the stabilization energy per water molecule (SEP) was calculated using the equation

image file: c5ra13268a-t1.tif
where n is the number of water molecules forming the cage.

Results and discussion

(a) Modeling of the fused cages

The dodecahedral water cage occurs in all types of gas hydrates and its structure comprises twelve pentagonal rings, whereas the irregular-dodecahedral cage is constituted by three four-membered rings, six five-membered rings and three six-membered rings of water molecules. Three different types of fused water cages were considered, namely, (i) fused cages formed by the combination of two dodecahedral cages (FDD), (ii) two irregular-dodecahedral cages (FII) and (iii) a dodecahedral cage with an irregular-dodecahedral cage (FDI). The most stable dodecahedral (DD) and irregular-dodecahedral (IDD) cages with the maximum number of t1d hydrogen bonds reported in our previous study18 were used for this purpose.

To model the FDD cage, the most stable dodecahedral cage (DD) with seven t1d hydrogen bonds was used. There are three types of five-membered rings present in such a dodecahedral cage, each with zero, one and two t1d hydrogen bonds, as illustrated in Fig. 1 of the ESI. The fusion of two such dodecahedral cages by sharing a pentagon with no t1d hydrogen bonds leads to the formation of a fused dodecahedral cage. On fusion, two t1d bonds of the shared ring are transformed to t0 bonds (trans without any dangling hydrogen), thereby giving a fused cage with 12 t1d bonds; this is the maximum number possible (7 + 7 − 2), as shown in Fig. 2 of the ESI.


image file: c5ra13268a-f1.tif
Fig. 1 The most stable water cages formed by the fusion of two dodecahedral cages (FDD), a dodecahedral cage with an irregular-dodecahedral (FDI) cage and two irregular-dodecahedral cages (FII). The t1d hydrogen bonds are highlighted in green color. The numbers in parentheses correspond to the number of t1d hydrogen bonds.

The number of t1d bonds present in a fused cage (N) can be obtained from the expression N = x + yn; where x and y are the t1d bonds present in the respective single cages and n is the number of t1d bonds converted to t0 bonds on fusion.

The most stable dodecahedral (DD) and irregular-dodecahedral (IDD) cages have 7 and 8 t1d hydrogen bonds, respectively. Fusion of the abovementioned water cages is possible by sharing a five-membered ring. The maximum number of t1d hydrogen bonds for the fused dodecahedral-irregular-dodecahedral cage can be obtained by sharing a five-membered ring with zero t1d hydrogen bonds, similar to the previous case. The maximum number of t1d hydrogen bonds possible for the fused dodecahedral-irregular-dodecahedral water cage (FDI) is 13.

As it is known that fusion is not energetically feasible between two irregular-dodecahedral water cages via five-membered rings,28 we considered the fusion of the abovementioned two cages by sharing a four-membered ring. The fusion of two irregular-dodecahedral (IDD) cages sharing a four-membered ring, which has zero t1d hydrogen bonds, gives rise to 14 t1d hydrogen bonds; it is the maximum possible number.

Fig. 1 depicts the most stable geometry for each of the abovementioned combinations. The stabilization energy (SE), the stabilization energy per water molecule (SEP), the number of t1d hydrogen bonds and the number of four-membered rings for these fused cages are listed in Table 1. The optimized geometries of the higher energy isomers of each type are given in the ESI.

Table 1 The stabilization energy (SE), stabilization energy per water molecule (SEP), number of t1d hydrogen bonds and number of four-membered rings for the most stable geometries of fused cages. The BSSE corrected energy values are given in parenthesis. The values in bold correspond to the BSSE and ZPE corrected stabilization energies
Isomer SE (kcal mol−1) SEP (kcal mol−1) Number of hydrogen bonds Number of t1d hydrogen bonds Number of four membered rings
FDD −452.28 (−357.76), −264.17 −12.92 (−10.22), −7.55 55 12 0
FDI −446.98 (−353.04), −260.24 −12.77 (−10.08), −7.43 55 13 3
FII −452.74 (−357.88), −263.01 −12.56 (−9.94), −7.31 56 14 5


(b) Stability of fused cages

The FDD and FDI cages are made of 35 water molecules, whereas the FII cage is composed of 36 water molecules. Considering the difference in the number of water molecules, the fused cages were analyzed in terms of stabilization energy per water molecule (SEP). The stabilization energy per water molecule for FDD, FDI and FII cages is −7.55, −7.43 and −7.31 kcal mol−1, respectively. The stabilization energy per water molecule for the abovementioned fused cages is higher than that for a water dimer. This is understandable, because the hydrogen bond energy per water molecule increases as we move from dimer to trimer, tetramer etc. as a result of cooperativity in water clusters, which is greater in cyclic structures.16

The stabilization energy per water molecule for the dodecahedral and irregular-dodecahedral water cages is −6.99 and −6.93 kcal mol−1, respectively.18 The high value of stabilization energy per water molecule for the fused cages compared to that of the constituent single cages imply enhanced interactions between water molecules of the fused cages. These enhanced interactions can be considered as the driving force for the formation of fused cages.

The number of t1d hydrogen bonds in FDD, FDI and FII are 12, 13 and 14, respectively, which is in the reverse order of their SEP. As there is no correlation between the number of t1d hydrogen bonds and the value of SEP, it can be concluded that the relative stability of fused water cages cannot be explained on the basis of the SWEB model. However, the difference in SEP between the most stable cages of different types of fused cages can be correlated to the number of four-membered rings present in them. The fused cage FDD, which has the highest SEP, does not have any four-membered rings, whereas the fused cages FDI and FII possess three and five such rings, respectively. The presence of four-membered rings results in strain, thereby increasing the energy of the cage. Thus, the SEP of FII is lower, despite the fact that it has more t1d hydrogen bonds.

The relative stability of the isomers within the family of a fused water cage given in the ESI can be correlated to the number of t1d hydrogen bonds present in them. It was found that within a family of fused cage, the stabilization energy decreases with a decrease in the number of t1d hydrogen bonds, similar to our observation for (H2O)20 clusters.18 For the isomers with the same number of t1d hydrogen bonds, the energy difference can be attributed to the difference in the hydrogen bond parameters (O–H bond distance, O–H⋯O distance and O–H⋯O angle). The bond parameters for the isomers with the same number of t1d hydrogen bonds for the FDD cages are also given in the ESI.

Conclusion

Fused water cages with a maximum number of t1d hydrogen bonds were modeled using the lowest energy dodecahedral and irregular-dodecahedral water cages. The stabilization energy per water molecule for the fused cages followed the order FDD > FDI > FII, which is a reverse order with respect to the number of four-membered rings present. Due to the presence of five strained four-membered rings, FII has a higher energy than the FDI and FDD water cages. The calculated stabilization energy per water molecule for the fused cages were found to be more than that for the constituent single cages, showing enhanced interactions between the water molecules in the fused cages. The present study also showed that the SWEB model cannot be used to compare the relative stability of fused cages of different families. However, it can be used to eliminate a large number of higher energy isomers of a given family of a fused cage, thus reducing the computational cost in modeling such cages. The fused water cages proposed in the present study belong to the minimum energy isomers and can be further used in the studies of gas hydrates.

Acknowledgements

CNR is thankful to the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India for a research grant (No. SB/S1/PC-019/2013). The authors VS and SPK acknowledge MHRD for their fellowships. The infrastructure provided by IIT Roorkee is also greatly acknowledged.

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Footnotes

Dedicated to Professor Eli Ruckenstein on the occasion of his 90th birthday.
Electronic supplementary information (ESI) available: See DOI: 10.1039/c5ra13268a

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