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Structures, Stabilities and Spectra Properties of Fused CH Endohedral Water Cage (CH) (HO)n Clusters From DFT-D Methods 4

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Lingli Tang, Ruili Shi, Yan Su, and Jijun Zhao J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b08073 • Publication Date (Web): 15 Oct 2015 Downloaded from http://pubs.acs.org on October 20, 2015

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The Journal of Physical Chemistry

Structures, Stabilities and Spectra Properties of Fused CH4 Endohedral Water Cage (CH4)m(H2O)n Clusters From DFT-D Methods Lingli Tanga, Ruili Shib,c, Yan Sub,c, Jijun Zhaob,c,d* a

b

School of Science, Dalian Nationalities University, Dalian 116600, China

College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China

c

Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian 116024,China d

Beijing Computational Science Research Center, Beijing 100089, China

*

Corresponding author. Email: [email protected]; Telephone: 086 0411-84709748; Fax: 086 0411-84706100 1

ACS Paragon Plus Environment


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Structures, Stabilities and Spectra Properties of Fused CH4 Endohedral Water Cage (CH4)m(H2O)n Clusters from DFT-D Methods

Abstract In order to understand the cage fusion behavior during the nucleation processes of methane hydrate (MH), methane-encapsulated double-cage clusters (CH4)2(H2O)n (n = 30−43) and several multi-cage structures with three or more cages were studied employing DFT-D methods. We find that almost all the lowest-energy double-cage structures can be constructed by merging the most stale structures of the mono-cage clusters CH4(H2O)n (n= 18−24). Double-cage structures can achieve higher stability through sharing a hexagon than a pentagon, which may be applicable to larger fused cage clusters. The preference of hexagons during cage fusion should be favorable for the appearance of the cages including hexagons such as the 51262, 51264 cages during MH nucleation process. The symmetric C−H stretching modes of methane molecules in the double-cage structures show clear trend of redshift with increasing size of the composing mono-cages. Compared with the case of mono-cages, the stretching frequencies of methane molecules in double-cage structures shift slightly, indicating variation of mono-cage configuration when cage fusion occurs. The larger multi-cage structures are found to possess higher fusion energies through sharing more polygons. Their thermodynamic stabilities do not simply increase with the number of fused mono-cages and are affected by the spatial arrangement of the building cages.

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1. Introduction Clathrate gas hydrates are crystalline compounds in which guest molecules, such as methane, carbon dioxide, hydrogen, ethane, are trapped in polyhedral water cages.1, 2 They have aroused intriguing interests because of their importance in energy resources, environment, chemical industries and geological disaster prevention.1, 3−6 Among the gas hydrates, methane hydrate (MH) is the most common either in nature or in oil and gas processing. As is well known, hydrates have three main clathrate structures, sI, sII, and sH.1, 2 The most thermodynamically stable structure for MH is the sI type with two 512 cages and six 51262 cages per unit cell (5n6m denotes a water cage consisting of n pentagons and m hexagons).1, 2 To date, molecular dynamic (MD) simulations have been widely conducted to explore the nucleation and growth mechanisms of MH.7−17 In spite of the detailed differences perhaps originated from different driving forces, potentials and system sizes adopted, most previous studies support the so-called “two-step” mechanism.7,

9−17

That is, amorphous hydrate-like structures first form rapidly and they subsequently

transform to crystalline solid. The types and numbers of cage emerged during MH nucleation and growth have been identified.7, 9−12, 14, 16 The 512 cage is the most abundant cage type and appears first. Two or more 512 cages will fuse and form face sharing 512 multi-cage structures, which were observed frequently in the initial process of MH nucleation.7, 9, 10, 14, 16 The next abundant cage is the 51262 cage. While the 51264 cage appears rarely, the 51263 cage was reported to form at the interface between sI and sII hydrates.18, 19 Apart from the standard cages (512, 51262, 51264, 51268 and 435663) which exist in the hydrate crystalline structures, many nonstandard cages (which do not exist in any hydrate crystalline structures) are observed during MD simulations. Walsh et al. reported that a set of seven cage types, in which there are three nonstandard cages (4151062, 4151063 and 4151064), comprises approximately 95% of all cages formed in the amorphous solid during nucleation.11 Guo et al. employed FSICA (face-saturated incomplete cage analysis) method to search for face-saturated cage types in the amorphous hydrate.12 They found thousands of cage types including both complete and incomplete cages (Here a cage is called a complete cage if its every vertex is shared with at least three edges and its every edge is shared with exactly two faces. Otherwise, the cage is an incomplete cage).12 It is noteworthy that a nonstandard cage 4151062 ranks second among the ten most abundant complete cages. Vatamanu and Kusalik also identified nonstandard cages in their initial formed disordered solid-like structures. These cages contain mostly tetragon, pentagon and hexagon and typically consist of 20−24 water molecules. A certain subset of them with higher symmetry can even survive the process of annealing and are observed in the annealed crystalline solid.13 Experimentally, it is challenging to explore the mechanism of MH nucleation on the molecular level due to the limited resolution in probing such small-scale and instantaneous events. However, some information about water cages during MH formation can still be obtained by various experimental techniques. Time-resolved Raman measurements suggested that the formation of 512 cages is earlier and faster than 51262 cages.20, 21 Schicks used both in-situ Raman spectroscopy and X-ray diffraction (XRD) to 3



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investigate the initial steps of MH formation.22 They found that the first X-ray signal for a hydrate structure delayed in time compared to Raman signals (i.e., no hydrate structures were detected although Raman spectra showed the appearance of 512 cages).22 This indicates the probability that an amorphous phase forms prior to the MH crystalline structure. The amorphous structure, as reported by MD simulations, may include nonstandard cages in addition to the standard cages. Neutron diffraction approach was employed to investigate the water structure during MH formation.23, 24 Once MH starts to form, the water shell of methane is slightly less ordered compared to methane solution. This implies that the hydrate cages around methane at the early stages of MH formation may be different from those in the aqueous solution, and are different again from those in the hydrate crystal structure23. Thompson showed that during MH formation, the O−O−O angel of water displays an enhancement in the trigonal peak at 53º and the tetrahedral peak shifts from 97º to 101º, which is a signature for the reordering of water molecules into pentagonal/hexagonal rings. 24 Ab initio methods, which provide more reliable description of intermolecular and intramolecular interactions than the empirical potentials used in MD simulations and are able to simulate the spectroscopic characteristics, have been utilized to study the CH4 endohedral water cage clusters. To examine how the methane molecules can stabilize the water cages, the interaction energies between methane molecules and the water cages were computed.25−31 It was found that the appearance of CH4 molecules can indeed stabilize the water cages, but the computed interaction energies varied with different methods.25−31 Based on the computed interaction energies, the occupancies of CH4 molecules in the 512, 435363, 51262, 51264 and 51268 cages were estimated by our group:28 the maximum occupancy is one, one, two, three and seven, respectively, and the optimum occupancy is one, one, one, two and four, respectively.28 Pérez et al. also studied the CH4 occupancy in the water cages.30, 31 But the interaction energies in this work were calculated using a periodic model (sI and sH crystalline structure) rather than the cluster model of water cages. They reported that 512, 435363 and 51262 cages can accommodate at most one CH4 molecule and the storage and optimum capacity of the 51268 cage is five and three, respectively.30, 31

The C−H stretching modes of methane molecules in the water cages (512, 435663, 51262, 51263, 51264, 51268)

were computed employing the density functional theory with dispersion correction (DFT-D).28, 32, 33 It was suggested that as the size of the water cavity increases, both symmetric and antisymmetric stretching frequencies first decrease from 512 to 51264 cage and then increase from 51264 to 51268 cage,28,

32, 33

consistent with the “loose cage−tight cage” model.34 The O−H stretching modes of H2O molecules in the cages were shown to shift towards lower frequencies compared to gas-phase H2O molecules due to hydrogen bonding and van der Waals interaction with the encapsulated CH4 molecules.33 Besides mono-cages, Khan et al. investigated some fused cage structures composed of 512, 435363, 51262 and 51268 empty cages utilizing the semiempirical ZINDO method. They found that, in most cases, stabilities of cage structures were significantly improved through cage fusion.25,

26

Till now, no work about the fused

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endohedral CH4(H2O)n cages has been reported. In our previous study, CH4(H2O)n (n = 16−24) cage clusters with four-, five-, six- and seven-membered rings were systematically studied employing DFT-D methods.35 Several metastable nonstandard cages were revealed in addition to the standard 512 and 51262 cages. These low-energy cages were found to have great structural similarity and can transform one another through inserting water dimers or Stone-Wales arrangement. The C−H symmetric stretching modes of methane molecules in the low energy cage isomers were shown to have a clear redshift trend with the increase of cage size, which agreed with the results reported by others.28, 32, 33 As mentioned above, previous experimental observations and computer simulations indicate formation of amorphous structures in the first step of MH nucleation. Both standard and nonstandard cages may exist in the amorphous structures. Most importantly, these cages merge in a way which differs from that in the conventional MH crystalline structure. Therefore, it is crucial to investigate the fusion behaviors of these cages to gain new atomistic insight into the MH nucleation mechanism. Starting from the low-energy mono-cage structures (including both standard and nonstandard cages) obtained in our previously work,35 structures of a large number of double-cage clusters are generated and studied using ab initio methods in this work. The structures, stabilities and Raman spectroscopic properties of them are compared with the mono-cage clusters. Additionally, some multi-cage structures with three or more building cages are considered to elucidate how the thermodynamic stabilities of fused cage structures vary with number of building mono-cages.

2. Theoretical methods Double-cage (CH4)2(H2O)n (n = 30−43) clusters, were generated by merging two mono-cage structures of CH4(H2O)m (m =18−24) clusters.35 The key idea to merge two mono-cages is to “glue” two faces on the mono-cages together. The detailed operation includes the following steps. First, for each mono-cage, choose a face (a pentagon or a hexagon) randomly as the shared face. Second, put the two chosen faces in the same place by properly rotating and moving the two mono-cages. Here we place both faces in the xOy-plane and set the center of them at the origin. Third, remove one chosen face (i.e. the water molecules on the chosen face) from one of the mono-cages, and adjust the orientation of the hydrogen bonds to realize the connection of the two mono-cages. This technique can also be used to construct fused structures with three or more monocages. For each mono-cage size m, the lowest four to six isomers were chosen for the mergence. Two mono-cages were fused through sharing a pentagon or a hexagon. Totally 471 double-cages were generated in which 252 of them share a pentagon and the rest share a hexagon. Table 1 lists the number of generated double-cages and the size of the composed mono-cages for every (CH4)2(H2O)n cluster.

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Table 1. Number of generated isomers and sizes (n and m) of the two composing mono-cages for every double-cage cluster.

Double-cages sharing a pentagon Double-cage cluster

Isomer number

n

m

(CH4)2(H2O)31

21

18

18

(CH4)2(H2O)33

30

18

20 a

20 (22)

(CH4)2(H2O)35

51

20 (18)

(CH4)2(H2O)37

60

20 (18)

22 (24)

(CH4)2(H2O)39

46

20 (22)

24 (22)

(CH4)2(H2O)41

29

22

24

(CH4)2(H2O)43

15

24

24

Double-cages sharing a hexagon

a

Double-cage cluster

Isomer number

n

m

(CH4)2(H2O)30

20

18

18

(CH4)2(H2O)32

24

18

20

(CH4)2(H2O)34

46

20 (18)

20 (22)

(CH4)2(H2O)36

54

20 (18)

22 (24)

(CH4)2(H2O)38

35

22 (20)

22 (24)

(CH4)2(H2O)40

25

22

24

(CH4)2(H2O)42

15

24

24

Numbers in parentheses represent another combination for the corresponding double-cage cluster. For example, the

(CH4)2(H2O)35 cluster can be obtained by merging two CH4(H2O)20 cages or one CH4(H2O)18 and one CH4(H2O)22 cage.

Because quadrangles exist in the nonstandard cages, it is possible to construct double-cage structures sharing a quadrangle. However, quadrangles are not favored for water cages as indicated by the cages appeared during the process of MH nucleation. 512 is always the most abundant cage type.7, 9-16 Most nonstandard cages which were observed frequently contain only one quadrangle. 11,12 As quadrangles are rare in the water cages with high abundance, the appearance of double-cages sharing a quadrangle has a small probability. Additionally, in Ahan’s study, although most fused water cages were significantly stabilized by sharing pentagons or hexagons, few extra stabilizations were achieved when two 435363 cages were fused together through sharing a quadrangle

25, 26

. This indicates that double cages sharing a

quadrangle may not be energetically favored. Because of these results, here we do not consider double cage structures sharing a quadrangle. A less computational demanding method, PBE-TS/TNP36 was first employed to optimize all the 6


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generated double-cages. This method had been utilized to study the mono-cage clusters in our previous study and was shown to be appropriate to describe the noncovalent interaction in the present systems.35 The cutoff radius for the TNP basis set was set as 6.0 Ǻ. Geometry optimization was performed with a convergence criterion of 2.0 × 10-5 Hartree on maximum energy gradient and 0.005 Ǻ on maximum displacement for each atom. Self-consistent field calculations were carried out with a convergence criterion of 10-5 Hartree on the total energy. Because the nucleation progress of MH happened in water, it is necessary to consider the effect of water environment on the structures and stabilities of the water cage clusters, which has been taken into account by the conductor-like screening model (COSMO).37 All PBE-TS/TNP calculations were conducted using the DMol3 package.38, 39 To achieve more reliable geometry parameters and binding energies, the lowest-energy structure (plus a few low-energy isomers) for each (CH4)2(H2O)n cluster obtained from PBE-TS/TNP calculations36 was reoptimized employing B97-D/6-311++G(2d, 2p) method40 as implemented in Gaussian 09 program.41 Liu et al. assessed performances of a variety of ab initio methods on describing the noncovalent interactions in MH and this method was recommended as an appropriate comprise of computational cost and precision.42 The conductor-like solvation model,43, 44 as developed in the frame work of the polarizable continuum model was utilized to perform the geometry optimization in water solution. Frequency analyses were conducted for the lowest-energy structures to simulate the Raman spectra of various cage clusters. In addition to the double-cage clusters, some multi-cage clusters composed of three or more 512 and 51262 cages were constructed and optimized by PBE-TS/TNP method36 using DMol3 package.38,

39

However, due to large number of possible configurations, it is hard to perform a complete exploration of all of them. Instead, we considered a few representative ones. Most of selected clusters appear in sI or sII hydrate crystalline structures. For multi-cage clusters composed of three mono-cages, two arrangement ways for the building cages were considered: a linear or triangle shape. In addition to the clusters including three mono-cages, two clusters consisting of four and five mono-cages respectively were also considered in order to elucidate how the number of building cages affects the stability of a multi-cage cluster.

3. Results and Discussion 3.1 The lowest energy structures of the double-cage clusters The lowest-energy structures of the double-cage clusters (CH4)2(H2O)n (n = 30−43) are given in Fig. 2 (sharing a pentagon) and Fig. 3 (sharing a hexagon), respectively. For clarity, these structures are represented by their two composing mono-cages (depicted in Fig. 1) and the shared polygon. Taking the (CH4)2(H2O)34 cluster for example, its lowest-energy structure is formed by two 20a cages through sharing a pentagon, and this structure is named by 20a−20a−f5.

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Fig. 1 Mono-cage structures composed of the lowest-energy structures of the double-cage clusters (CH4)2(H2O)n (n = 30−43). The 18a, 20a, 22a and 24a structures are the ground state structures of the mono-cage clusters of CH4(H2O)m (m = 18−24).35 Both 20a (512) and 22a (51262) cages are basic building blocks of sI MH hydrate structure. The 22a structure (4151062) was reported to be the most abundant nonstandard cage type appearing during MH nucleation processes.12 The 20b and 22b cages are the second lowest-energy structures of the CH4(H2O)20 and CH4(H2O)22 clusters and are less stable than the 20a and 22a structures by 2.59 kcal/mol and 0.75 kcal/mol, respectively.35

For the lowest-energy double-cage structures sharing a pentagon (Fig. 2), all of their constructing mono-cages are also the ground state structures of the mono-cage CH4(H2O)m clusters. The 20a (512) and 24a (51262) cages, which are also the basic building units of the sI MH crystalline structure, are main constituents of these double-cage structures. Because the 20a cage does not contain hexagons, it cannot appear in the double-cage structures sharing a hexagon. As displayed in Fig. 3, the 20b cage, which is the second lowest-energy structure of the mono-cage CH4(H2O)20 cluster, takes over the 20a cage and becomes one of main constituents of the lowest-energy double-cage structures sharing a hexagon. Indeed, if we replace the 20a cage by the 20b structure, the composing mono-cages for the double-cage structures sharing a hexagon are almost the same as those for structures sharing a pentagon with the exception of the (CH4)2(H2O)36 cluster. The lowest-energy structure of the (CH4)2(H2O)36 cluster is 20b−22b−f6 instead of 20b−22a−f6. The latter is less stable than the former by 0.44 kcal/mol. This may arise from the better connection between the 20b and 22b cages (see below). Overall speaking, the lowest-energy double-cage structures tend to be originated from the ground-state structures of the mono-cage clusters.

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Fig. 2 The Lowest-energy structures of double-cage clusters (CH4)2(H2O)n (n = 31−43) sharing a pentagon. The structures are named by the composed mono-cages and the shared polygon.

Fig. 3 The Lowest-energy structures of double-cage clusters (CH4)2(H2O)n (n = 30−42) sharing a hexagon. The structures are named by the composed mono-cages and the shared polygon.

As listed in Table 1, six double-cage clusters have two combination ways as for the sizes of their composing mono-cages. For example, the (CH4)2(H2O)35 or (CH4)2(H2O)34 clusters can be obtained by merging two CH4(H2O)20 cages or one CH4(H2O)18 and one CH4(H2O)22 cage. Therefore, two combination ways will compete for the lowest-energy structure for each cluster. As displayed in Fig. 2 and Fig. 3, the CH4(H2O)18 cage does not show up in the lowest-energy structures of the (CH4)2(H2O)35, (CH4)2(H2O)34, (CH4)2(H2O)37, and (CH4)2(H2O)36 clusters. The most favorable combinations of the (CH4)2(H2O)39 and CH4)2(H2O)38 clusters are 20a−24a−f5 and 20b−24a−f6, respectively, rather than two CH4(H2O)22 cages.

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3.2 Stabilities of the lowest-energy double-cage structures The thermodynamic stabilities of the lowest-energy structures for the double-cage (CH4)2(H2O)n clusters can be characterized by stabilization energy (SE). The stabilization energy per H2O molecule (SEP) allows us to compare the stabilities of these structures with different n. For double-cage structures, they are defined as

SE = 2E(CH4) + nE(H2O) −E((CH4)2(H2O)n)

(1)

SEP = SE/n.

(2)

and

Here n is the number of water molecules in the double-cages; E((CH4)2(H2O)n), E(CH4), E(H2O) represent the total energies of the double-cage (CH4)2(H2O)n cluster, the methane and water monomer, respectively. To estimate the enhanced stabilities in a double-cage structures gained by fusing the isolated mono-cages, the cage fusion energy (FE), which was introduced first by Khan,25, 26 as well as the fusion energy per H2O molecule (FEP) were also calculated for these lowest-energy structures. The definitions of FE and FEP for double-cage clusters were

FE = SE (double cage) − [SE (cage 1) + SE (cage 2) − shared ring size × SEP(cage 1 or 2, which is lower)],

(3)

and

FEP = FE/n.

(4)

By simple derivation, we found that if a double-cage structure is constructed from two same mono-cages (such as 20a−20a−f5), then

FEP = SEP (double cage) − SEP (mono cage).

(5)

Otherwise, the FEP value will fall in between

$%&' SEP (double cage) − SEP (cage1), SEP (double cage)− SEP (cage2)(

(6)

and

$)*' SEP (double cage) − SEP (cage1), SEP (double cage)− SEP (cage2)(. (7) Thus, bigger positive FEP (FE) value implies higher stability derived by double-cage structures from the isolated mono-cages. Negative FEP (FE) value indicates that the formed double-cage structure is less stable than the isolated mono-cages. The calculated values of SE, SEP, FE and FEP for the lowest-energy double-cage structures are listed in Table 2 and Table 3.

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Table 2. Stabilization energies (SE), stabilization energies per H2O molecule (SEP), fusion energies (FE) and fusion energies per H2O molecule (FEP) for the lowest-energy structures of the double-cage clusters sharing a pentagon.

a

Structures

SE (kcal/mol)a

SEP (kcal/mol)a

FE (kcal/mol)a

FEP (kcal/mol)a

18a−18a−f5

238.4

7.69

14.76

0.48

18a−20a−f5

257.22

7.79

15.72

0.48

20a−20a−f5

275.81

7.88

17.31

0.49

20a−22a−f5

288.38

7.79

16.81

0.45

20a−24a−f5

303.74

7.79

16.84

0.43

22a−24a−f5

317.49

7.74

17.82

0.43

24a−24a−f5

331.91

7.72

16.91

0.39

The energies are obtained by B97-d/6-311++G(2d,2p) method.

Table 3. Stabilization energies (SE), stabilization energies per H2O molecule (SEP), fusion energies (FE) and fusion energies per H2O molecule (FEP) for the lowest-energy structures of the double-cage clusters sharing a hexagon.

a

Structures

SE (kcal/mol)a

SEP (kcal/mol)a

FE (kcal/mol)a

FEP (kcal/mol)a

18a−18a−f6

232.44

7.75

16.02

0.53

18a−20b−f6

249.75

7.80

18.06

0.56

20b−20b−f6

265.01

7.79

18.29

0.54

20b−22b−f6

280.25

7.38

19.1

0.53

20b−24a−f6

296.83

7.81

19.42

0.51

22a−24a−f6

312.99

7.82

20.61

0.52

24a−24a−f6

328.99

7.83

21.31

0.51

The energies are obtained by B97-D/6-311++G(2d,2p) method.

From Table 2 and Table 3, the double-cage structures can gain stabilities from the isolated mono-cages by 0.4~0.56 kcal/mol per H2O molecule. Additionally, it is clear that the FEP values of the double-cages sharing a hexagon are bigger than those of the structures sharing a pentagon. This finding demonstrates that the double-cage structures can achieve more stability through sharing a hexagon than sharing a pentagon. It is natural to expect that this result is valid for not only double-cage clusters, but also larger fused cage clusters, especially when the mono-cages are arranged in a linear shape. As seen from the SEP values, the 18a−18a−f6, 22a−24a−f6 and 24a−24a−f6 structures are more stable than the 18a−18a−f5, 22a−24a−f5 and 24a−24a−f5 structures, respectively. As the composing mono-cages are same, the achieved more stability for structures sharing a hexagon arises from the obtained bigger fusion energies. The 20a−20a−f5 (face sharing 512 cage) structure gains the highest stability among all the lowest-energy double-cage structures although its fusion energy is smaller than structures sharing a hexagon. The high stability of this structure should originate from its composing mono-cage 20a, which is the most stable 11



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mono-cage structure.35 It seems that the stabilities of the double-cage structures are determined by two factors: the stabilities of the composing mono-cages and the fusion energies gained by connecting two isolated cages. As stated above, the lowest-energy structure of the (CH4)2(H2O)36 cluster 20b−22b−f6 is 0.44 kcal/mol more stable than the 20b−22a−f6 structure in spite of the less stability of 22b than 22a.35 We compared the fusion energies of the two structures and found that the FE of the 20b−22b−f6 structure is 1.2 kcal/mol higher than that of the 20b−22a–f6 structure. The present theoretical results may provide some useful insights into the MH nucleation and growth process. Water molecules prefer to form pentagons which largely preserve the tetrahedral structure of water.7, 20 As a result, the 20a (512) cage has high stability and tends to form in the initial process of MH nucleation, as demonstrated by many MD simulations and experimentally spectra measurements.7, 9−12, 14, 16, 20−22

At this time, the appearances of cages containing hexagons such as 51262, 51264 and 51268 structures

are rare and transitory. However, when cage fusion happens, since fused cage can achieve more stabilities through sharing hexagons than sharing pentagons, the relative stabilities of the hexagon-containing cages should be promoted and the life time of these cages may be prolonged, which is favorable during the formation of the MH crystalline structure and thus should be observable in experiment.

3.3 Raman spectra of the lowest-energy double-cage structures To provide useful reference for identifying these fused cages in future experiments, Raman spectra of symmetric C−H stretching modes of the encapsulated methane molecules in the lowest-energy double-cage structures were simulated and depicted in Fig. 4. For comparison, the corresponding spectra for the composing mono-cages were also given. Similar to the case of the mono-cages,35 the C−H stretching modes of methane molecules in the double-cage clusters also exhibit a clear trend of redshift with the increasing size of the composing mono-cage. Upon merging two isolated mono-cages into one double-cage, the C−H stretching modes of methane molecules in the corresponding cages change slightly. Both redshift and blueshift may happen, depending on the specific cage structures. For example, the stretching frequencies of methane molecules in the 20a cage shift to lower values when the 18a−20a−f5, 20a−22a−f5 and 20a−24a−f5 structures form; however, a very weak blueshift was observed in the 20a−20a−f5 structure. This indicates that the details of the cage configuration have varied upon suffering mergence and the cages will experience different structural change when they fused with different cages or the same cages in different ways (through sharing a hexagon or a pentagon). According to the “loose cage−tighe cage” model,34 the redshift of the C−H stretching modes implies the enlargement of the cage and the blueshift means that the cage shrinks. Thus from Fig. 4, when cage fusion occurs, the mono-cages may be either enlarged or shrunk, relying on the concrete double-cage structures formed.

12


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Fig. 4 Simulated Raman spectra for the symmetric C−H stretching modes of methane molecules in the lowest-energy double-cage structures. For comparison, the corresponding spectra for the composed mono-cage clusters were also given. The short dashed lines and solid lines represent the spectra for the mono-cages and double cages, respectively.

3.4 Structure and stability of selected multi-cage clusters In order to further explore stabilities of fused cage structures when the number of fused mono-cages increases, some multi-cage (CH4)m(H2O)n structures with three or more mono-cages (most of them exist in the MH crystalline structures) are generated and optimized using PBE-TS/TNP method. The equilibrium structures are depicted in Fig. 5. The 20a−20a−20a−1, 20a−20a−20a−2 and 20a−20a−20a−20a structures consisting of adjacent 20a cages appear in the sII hydrate crystalline structure, while the structures containing adjacent 24a cages, i.e. the 24a−24a−24a−1, 24a−24a−24a−2 and 24a−24a−20a, can be found in the hydrate structure of sI phase. The 20a−20a−20a−20a−24a structure composed of five mono-cages does 13



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not exist in any hydrate crystalline structure. But this structure would appear in the sII hydrate structure if we replace its 24a cage by the standard 51264 cage which was reported to emerge infrequently during MH nucleation.7, 9−12, 14, 16

Fig. 5 Multi-cage structures optimized by PBE-TS/TNP method. The structures are named by their composed mono-cages.

We also calculated the SE, SEP, FE and FEP values for these multi-cage structures. SE and SEP for (CH4)m(H2O)n structures are defined as SE = m×E(CH, ) + n×E(H- O)−E((CH4)m(H2O)n)

(8)

and SEP = SE/n.

(9)

By generalizing the definition of FE for double-cage structures, we define FE for (CH4)m(H2O)n structures as FE = SE((CH4)m(H2O)n) −∑1 234 SE(CH, (H- O)/0 ) + LE

(10)

FEP = FE/n.

(11)

and

Here SE((CH4)m(H2O)n) is stabilization energy of the (CH4)m(H2O)n structure; n2 in the second term ∑1 234 SE(CH, (H- O)/0 ) stands for the number of water molecules in the ith mono-cages. Since we assume 14


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that each mono-cage is occupied by one methane molecule, there are m mono-cages in the multi-cage (CH4)m(H2O)n cluster, and ∑1 234 SE(CH, (H- O)/0 ) represents the sum of stabilization energies of m isolated mono-cages; the third term LE is the energies lost by the isolated mono-cages when H2O molecules are removed during cage fusion. The ways to calculate SE((CH4)m(H2O)n) and ∑1 234 SE(CH, (H- O)/0 ) are obvious. We will take the 24a−24a−20a structure (Fig. 5) as an example to illustrate how to calculate LE. When the three cages fuse together, fourteen H2O molecules need to be removed. Among the fourteen H2O molecules, six of them must be removed from the 24a cage and the rest can be removed either from the 24a cage or from the 20a cage. Since the 24a cage is less stable than the 20a cage,35 H2O molecules can be removed easier from the former than from the latter. Therefore, we can assume that all the fourteen H2O molecules are removed from the two 24a cages and the energies lost (LE) by the two 24a cages from the removed H2O molecules is calculated by: 14×SEP(24a) (i.e., LE = 14 × SEP(24a)). The calculated SE, SEP, FE and FEP values for the multi-cage structures (depicted in Fig. 5) are given in Table 4. It seems that the fused multi-cage structures can gain more fusion energies through sharing more water polygons. As seen in Table 4, the 20a−20a−20a−2, 24a−24a−24a−2 and 24a−24a−20a structures which have triangle shape and share three polygons possess about 0.1 kcal/mol higher FEP values than the linear structures of 20a−20a−20a−1 and 24a−24a−24a−1 sharing two polygons. The 20a−20a−20a−2 (24a−24a−24a−2) structure is indeed more stable than the 20a−20a−20a−1 (24a−24a−24a−1) structure as indicated by the SEP values. The 20a−20a−20a−20a structure has both linear and triangle character for the arrangement of the composing mono-cages. Interestingly, both its FEP and SEP values fall in between those of the linear 20a−20a−20a−1 structure and the triangle 20a−20a−20a−2 structure. The largest multi-cage structure 20a−20a−20a−20a−24a considered in this work shares seven polygons and has the biggest FEP value and the highest stability.

Table 4. Stabilization energies (SE), stabilization energies per H2O molecule (SEP), fusion energies (FE) and fusion energies per H2O molecule (FEP) for multi-cage structures depicted in Fig. 5.

a

Structures

SE (kcal/mol)a

SEP (kcal/mol)a

FE (kcal/mol)a

FEP (kcal/mol)a

20a−20a−20a−1

388.37

7.77

24.42

0.49

20a−20a−20a−2

371.33

7.9

29.23

0.62

24a−24a−24a−1

459.22

7.65

29.73

0.5

24a−24a−24a−2

451.93

7.79

36.76

0.63

24a−24a−20a

422.64

7.83

33.68

0.62

20a−20a−20a−20a

486.57

7.85

35.28

0.57

20a−20a−20a−20a−24a

604.67

8.06

60.33

0.8

The energies are obtained using PBE-TS/TNP method

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By comparing the SEP values of the 20a−20a−20a−1, 20a−20a−20a−2 and 20a−20a−20a−20a structures, it was found that the stabilities of the fused cage structures do not simply increase with the number of the fused mono-cages and are affected by the arrangement of the composing mono-cages, which is relevant to the fusion energies of the fused cage structures as stated above. According to our theoretical results for the multi-cage clusters, when cage fusion happens during MH nucleation and growth, the cages tend to fuse in such a way to maximize the shared water polygons; thus linear configurations are usually not favored. Even though nearly linear multi-cage structures were observed by Hawtin et al. during their MD simulations in early stages of MH nucleation,9 these structures only existed transitorily and may originate from the stochastic nature of MH nucleation. In contrast, the compact structures would survive for a longer time and act as seeds for further nucleation. As seen in previous studies, the nucleated structures obtained from MD simulations are compact and contain certain number of mono-cages.7, 10−14, 16

4. Conclusions Double-cage clusters (CH4)2(H2O)n (n = 30−43) generated by merging the low-energy mono-cages are systematically studied utilizing B97-D/6-311++G(2d, 2p)//PBE-TS/TNP method. It was found that most of mono-cages composed of the lowest energy double-cage structures are also the ground state structures of the mono-cage clusters CH4(H2O)n (n = 18−24). The 20a−20a−f5 structure (face sharing 512 cages) has the highest thermodynamic stability among all the 471 double-cage structures explored. Double-cage clusters can gain more stability from two isolated mono-cages through sharing a hexagon than sharing a pentagon. This trend is expected to be not only valid for the double-cage clusters but also for fused cages composed of more mono-cages. The preference of hexagons for cage fusion may have direct implication on the appearance of the cages containing hexagons during MH nucleation and growth process. The symmetric C−H stretching modes of methane molecules in double-cage structures change slightly compared with the case of isolated mono-cages. Both redshift and blueshift may happen, depending on the specific double-cages formed. In other words, the details of the mono-cage configuration have varied upon suffering fusion and the mono-cages will experience different structural change when different double-cage structures are formed. The stretching frequencies of methane molecules in the double-cages show a clear redshift trend with increasing sizes of the composing mono-cages. A few selected multi-cage structures composed of three or more mono-cages have been also studied using PBE-TS/TNP method. Generally speaking, the multi-cage structures can achieve more fusion energies through sharing more water polygons. However, their energetic stabilities do not simply increase with the number of fused mono-cages and are affected by the way the composing mono-cages are arranged.

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Acknowledgments This work was supported by the National Natural Science Foundation of China (11174045, 11404051, 11304030).

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The graphic for the TOC

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Fig. 1 338x190mm (300 x 300 DPI)


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Fig. 2 338x190mm (300 x 300 DPI)



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Fig. 3 338x190mm (300 x 300 DPI)


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Fig. 4 304x711mm (300 x 300 DPI)



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The graphic for the TOC 490x200mm (300 x 300 DPI)


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