ISSN 10637745, Crystallography Reports, 2010, Vol. 55, No. 3, pp. 353–361. © Pleiades Publishing, Inc., 2010. Original Russian Text © M.V. Kirov, 2010, published in Kristallografiya, 2010, Vol. 55, No. 3, pp. 389–397.
THEORY OF CRYSTAL STRUCTURES
Classification of Proton Configurations of Gas Hydrate Frameworks M. V. Kirov Earth Cryosphere Institute, Siberian Branch of the Russian Academy of Sciences, Tyumen, 625000 Russia email:
[email protected] Received January 26, 2009
Abstract—All molecular configurations differing in the position of hydrogen atoms (protons) in hydrogen bonds and satisfying periodic boundary conditions have been calculated for the unit cells of CSI, HSIII, and TSIV gas hydrate frameworks. The configurations obtained are ranged according to the number of stronger transconfigurations of hydrogenbound molecular pairs and according to the type of spatial sym metry. The configuration symmetry was analyzed taking into account the additional antisymmetry operation, which is related to the change in the direction of all hydrogen bonds. The strong dependence of the framework properties on the specific position of protons in H bonds is established. DOI: 10.1134/S1063774510030016
INTRODUCTION The properties of clathrate hydrates are of interest for developing modern technologies of gas extraction and transport from gas hydrate deposits [1]. The hydrogen storage and transport in the form of gas hydrates may be of great importance for hydrogen power engineering [2]. At the same time, gas hydrate frameworks (where all H bonds are mirrorsymmetric) are an interesting object for studying the structure and properties of water–ice systems. Singer and colleagues have developed a sequential statistical approach which makes it possible to relate the energy and other char acteristics of water clusters and various ice modifica tions to the hierarchy of structural invariants [3–5]. Discrete models of intermolecular interaction have been developed for polyhedral water clusters [6–8] and extended gas hydrate frameworks [9]; these mod els make it possible to directly estimate the energy and predict the most stable configurations. For many gas hydrate frameworks, despite the exponential proton disorder and a very large number of different configurations, one can calculate exactly and classify all defectfree configurations that obey the wellknown Bernal–Fowler rules: (i) there is one pro ton per H bond and (ii) two hydrogen atoms (protons) are located near each oxygen atom [10]. The proton configurations of cyclic and polyhedral water clusters were enumerated and classified (taking into account the approximate antisymmetry) in [11–15]. In this study we use periodic boundary conditions; they significantly limit the variety of proton configura tions. However, the exhaustive classification of the unitcell configurations with periodic boundary con ditions can be of great methodical and scientific inter est, especially for experts in computer simulation. For
example, configurations with the same type of actual symmetry (with allowance for the specific arrange ment of hydrogen atoms in H bonds) can be interest ing for studying the mechanical stability of gas hydrates. The nanostructural inhomogeneity, which is caused by the difference in the proton position in bonds, can be important for interpreting experimental data. In this paper the gas hydrate frameworks are con sidered according to the degree of increasing the num ber of molecules per unit cell: TSIV (12), HSIII (34), and CSI (46). A hypothetical framework con taining no pentagonal hydrogenbound cycles is also discussed. The stabilization energies of different pro ton configurations of the widespread HSIII and CSI frameworks (structures H and I [1]) are calculated using the TIP4P potential [16]. CALCULATION TECHNIQUE The framework structure was presented as an ori ented graph to enumerate all defectfree proton con figurations. The Hbond directions are indicated by arrows. According to the Bernal–Fowler rules, two arrows are incoming and two are outgoing for each graph vertex (oxygen atom). The problem was to enu merate all allowed configurations. Hydrogencoupled molecular pairs (with allowance for the periodic boundary conditions) were determined using a special computer program in which the initial data were the coordinates of oxygen atoms (vertices) in the unit cell. The program implementing the algorithm for enu merating and searching for defectfree configurations includes a large number of embedded cycles equal to the number of H bonds. Each cycle sets one of
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two Hbond directions. Logical conditions are imposed on the cycles, which stop cycling when the Bernal–Fowler rules are violated by specified bonds (previous cycles). The existence of configurations entirely composed of stronger types of H bonds in cubic ice was proven using a similar algorithm con taining 220 embedded cycles [17]. The structure of each configuration was retained as a sequence of num bers 1 and –1, which correspond to the Hbond direc tions. Furthermore, this structure representation was used to calculate the coordinates of hydrogen atoms and perform calculations with the TIP4P potential. The specific residual entropy of 3D frameworks, ln(3/2) [18], is much lower than that of individual polyhedral water clusters (ln(3/ 2 ) [19]). This cir cumstance significantly facilitates the enumeration of configurations (provided that the number of mole cules is the same). Periodic boundary conditions addi tionally limit the number of allowed configurations. These circumstances made it possible to enumerate all configurations even for the frameworks with a fairly large number of molecules per unit cell. All calcula tions were performed on a conventional personal com puter. When selecting symmetrically independent (noni somorphic) proton configurations, we checked all symmetry operations [20] corresponding to the frame work space group. Proton configurations can be sepa rated into two classes: conventional and antisymmet ric. The concept of antisymmetry of water–ice sys tems, which is related to the change in the direction of all H bonds, was proposed by us in [11]. In the case of antisymmetric configurations, a change in the direc tion of all bonds leads to a symmetrically equivalent configuration related to the initial one by a spatial symmetry operation. The conventional proton configurations in the list of symmetrically different structures form pairs, which are related by an antisymmetry operation [11, 13]. Proton configurations in these pairs can be chosen so as to make them differ by only the direction of all H bonds. These configurations (antipodes) have some what different properties; therefore, the antisymmetry of water–ice systems is approximate [11–13]. The concept of antisymmetry makes it possible, in particu lar, to reduce the sets of symmetrically different defectfree proton configurations by a factor of almost 2 for large systems, because it is quite sufficient to retain only one antopode from each pair of conven tional configurations (which are dominant in such sys tems). The symmetry groups of proton configurations are subgroups of the symmetry group of Hbond frame work. The number of symmetrically equivalent config urations for each proton configuration is equal to the quotient of the number of symmetry operations of the framework group, νf, and the number of symmetry operations of the proton configuration, νс. Here, we
mean the symmetry operations without translations, which are generally given in crystallographic tables [20]. The numbers νf and νс are the orders of the cor responding crystalline classes. For individual clusters they are the orders of the symmetry groups of the Hbond framework and proton configuration, respec tively. The total number of proton configurations, N, and the number of symmetrically different configura tions, n, are related as n
n
νf 1 , (1) = νf N = ( ) (i) ν ν i i =1 c i =1 c where summation is over all symmetrically different configurations. Formula (1) is aimed at verifying the classification results and obtaining approximate esti mates. In sufficiently large systems, the fraction of proton configurations having any symmetry is negligi ble. Assuming that νс(i) = 1 in (1), we obtain a rather exact lower bound of the number of different proton configurations for such systems: (2) n ≈ N. νf Formulas (1) and (2) are also valid for individual structural classes, for example, for configurations with the same number of strong H bonds. To determine the antisymmetry of proton configurations, we checked the presence of all elements of the extended group, which contains, along with νf conventional symmetry elements, νf antisymmetry elements, which change the direction of all H bonds. Therefore, when verifying the balance with respect to antisymmetry, one has to use the doubled value of νf. Note that formulas (1) and (2) can be directly used for only the unit cells of extended icelike systems and isolated clusters.
∑
∑
ENUMERATION OF PROTON CONFIGURATIONS TSIV Framework In the tetragonal TSIV framework (of the TSIV type according to [21, 22]), there are only 12 mole cules per unit cell. A fragment of the TSIV framework structure is shown in Fig. 1. One characteristic feature of this framework is the presence of only one type of cavities. The direction of only 20 H bonds is indicated for each configuration. The direction of the other H bonds can be determined using the Bernal–Fowler rules and periodic boundary conditions. The proton configurations in Fig. 1 are ordered according to the number of stronger hydrogenbound transdimers nt (strong H bonds [6–8]). The maximum value of nt (configuration 1) is 18, which is 75% of the total num ber of bonds. For the fivepoint models of water mol ecule (ST2 [23] and TIP5P [24] potentials), the energy advantage of transdimers over cisdimers is due to the stabilizing effect of the effective charges involved in the formation of adjacent H bonds (Fig. 1, top). How
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trans 1
2
3
2/m 9
nnm
10
19
2 26 nn2 33
28 2
22
2
2
nnm 31
2
32
1
nn2
2
41 21 40
38
2 24
2
30
2 16
23
2 39
42 c
21/c
21
8
15
21212
37
7
2/m
2
29
cis
nnm
14
21
21 36
34
nn2
m
2
6
13
20
27 21212 35
2
2
2/m
2
5
12 2
21212 18
25
mn21
11
2 17
4
trans
355
c
Fig. 1. Proton configurations of the TSIV framework unit cell. From above: framework fragment, individual cavity, transdimer, and classification of H bonds on the polyhedron surface [6]. Configurations 1–32 are antisymmetric; what follows gives one rep resentative from each pair of conventional antisymmetrybound configurations.
ever, there is no explicit direct dependence between the stability of the system and the number of strong H bonds, nt, for 3D gas hydrate frameworks. The reason is the competition of interactions between the nearest hydrogenbound neighbors and more remote ones [9]. The total number of configurations in the unit cell of this framework (under periodic boundary condi tions) is 612. There are 52 symmetrically different configurations, 32 of which are antisymmetric. The other 20 configurations form 10 pairs of antipode con figurations; i.e., with allowance for the antisymmetry, there are 42 different configurations (Fig. 1). Periodic boundary conditions, which are conventional in com puter simulation, exclude antitranslations. Antisym metry groups are denoted similarly to conventional ones, but they have primes or are underlined if the cor responding element is an antielement [25]. The tetrag onal framework TSIV has the P42/mnm symmetry [21, 22]; the number of symmetry operations is νf = 16. The lower bound (2) for the number of nonisomor phic configurations is 612/16 ≈ 38. There are 52 differ ent configurations due to the presence of symmetric structures. The symbol “P,” which denotes the lattice type, is omitted in the designations of symmetry groups in Fig. 1. To check the balance in antisymmetry using formula (1), it is necessary to take into account CRYSTALLOGRAPHY REPORTS
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that νf = 32 and νс is 8, 4, 2, and 1 for 3, 11, 24, and 4 configurations, respectively. The easiest way to make sure of the presence of inversion and antiinversion in Fig. 1 is to compare the direction of opposite H bonds in the framework cavity. “Even” CSIV Framework We also considered a hypothetical gas hydrate framework of cubic symmetry, I m 3 m , which contains no pentagonal cycles [22]. All the cavities of this even framework (Fig. 2) are shaped like the Fedorov trun cated octahedron 4668. The use of this CSIV frame work (of the CSIV type according to [22]) is one of the best known ways for filling space with identical polyhedra [26]. Like in the TSIV framework, there are 12 molecules per unit cell. It was established that the total number of proton configurations with peri odic boundary conditions is 546. There are only 19 dif ferent configurations due to the high spatial symmetry of the framework (νf = 96). Only one configuration pair is not antisymmetric. Due to the absence of odd pentagonal cycles, this framework is of special theoretical interest. It was pre viously established that at any orientation of molecules
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bonds have conventional inversion. The number of antisymmetric configurations is much larger. The overwhelming majority (1296) are configurations with a twofold antisymmetry axis, and the other 55 have antiinversion symmetry. With allowance for the anti symmetry, the total number of different proton config urations is smaller by a factor of almost 2: 116 806.
Fig. 2. Configuration of an individual cavity in which all H bonds of the 3D CSIV framework form trans dimers (strong bonds). The white and black colors of the faces correspond to the sequential and alternate orientations of H bonds.
in pentagonal and other odd cycles at least one of the H bonds is formed by a weaker cisdimer [6, 11]. Pro ton configurations, which are completely formed by Hbond transconformations, exist in the CSIV framework due to the absence of such frustrated cycles. The position of protons in the H bonds in such configurations can be characterized by using the blackandwhite coloring of polyhedra faces like a soccer ball [6]. The white and black colors correspond, respectively, to sequential (homodromic cycles) and alternate orientations of H bonds (Fig. 2). Under peri odic boundary conditions, only one (antisymmetric) configuration (which is much more stable among two such configurations, where all hexagonal cycles are homodromic) is implemented in the unit cell (Fig. 2). Note that, according to the results of ab initio calcula tions, this configuration of truncated octahedron (H2O)24 has the lowest energy of all polyhedral water clusters [12, 27]. HSIII Framework The proton configurations of the hexagonal HSIII framework are listed in Table 1. The proton configura tions were separately analyzed for isomorphism for each number of transdimers (nt). This framework has P6/mmm symmetry [1] with 34 molecules per unit cell. The numbers of conventional symmetry operations and generalized symmetry operations are 24 and 48, respectively. The total number of proton configura tions is 5 568 720. The lower bound for the number of different configurations yielded by formula (2) (5568720/24 = 232 030) for this fairly large unit cell only slightly differs from the actual value of 232 261. Due to the larger size of the unit cell (when com pared with the TSIV and CSIV frameworks), the fraction of symmetric and antisymmetric configura tions is much smaller. The symmetry group of the HS III framework is symmorphic: it contains neither screw axes nor glidereflection planes; therefore, the number of symmetry types is limited. Only 231 pairs of antipode configurations with opposite directions of H
CSI Framework The proton configurations of the CSI framework are classified in Table 2. This framework has the Pm 3 n symmetry, with 46 molecules per unit cell [1]. The numbers of symmetry operations and generalized symmetry operations are 48 and 96, respectively. The total number of proton configurations is 822 823 440. The exhaustive classification in Table 2 was obtained to a great extent using the fact that most configurations in large unit cells are asymmetric. When searching for nonequivalent configurations for each nt, we retained and compared only symmetric and antisymmetric structures. For absolutely asym metric configurations, formula (2) is exact. After determining the symmetry type of each generally sym metric configuration and the total number of configu rations that are equivalent for them, the number of dif ferent asymmetric configurations was found by sub traction. For example, for nt = 20 (Table 2), the number of symmetric antipode pairs is 18 + 304 = 322. Each such configuration has 48/2 = 24 symmetrically equivalent ones. In addition, each of the 19 antisym metric configurations in the last six columns of Table 2 contains one conventional symmetry element. The 2/с, 21/с and 2/с, 21/с symmetry groups contain anti inversion –1 (or 1 in the other designation) and inver sion –1 ( 1 ), respectively. Therefore, the number of absolutely asymmetric configurations is 24 × 2 × 322 – 24 × 19)/48 = 128 791, and the total number of sym metrically different structures is 128791 + 2 × 322 + 19 = 129 454. The numbers of different (taking into account the antisymmetry) configurations were calcu lated similarly for each nt. When checking the configurations for isomor phism, we retained 8910 symmetric (having one anti pode each) and 23 702 antisymmetric configurations, i.e., only 32 612 configurations in total or 0.38% of the total number of different (taking into account the anti symmetry) configurations: 8 587 446. The above described method for enumerating nonisomorphic proton configurations has a computational complexity close to linear, because the first (most important stage) retains only generally symmetric configurations. PROTON CONFIGURATION ENERGIES The most energetically favorable configurations of molecular systems are very often symmetric [14, 28]. The reason is as follows: the configurations that are
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Table 1. Classification of the proton configurations of the HSIII framework nt
M
S/2
n
a
m
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 Σ
72 240 240 1368 1584 2856 4344 7128 11928 18768 30000 40368 63600 93576 127296 173568 212448 262680 317712 345312 394944 400608 413136 394728 400176 368928 344808 277488 241512 181344 148704 97584 72480 43080 30336 18240 10944 6384 4152 1848 1152 456 456 48 96 5568720
0 0 0 0 0 0 0 0 1 0 0 0 1 0 6 0 20 0 22 0 29 0 22 0 25 0 26 0 19 0 17 0 12 0 5 0 16 0 3 0 5 0 0 0 2 231
3 10 10 57 66 119 181 297 498 782 1250 1682 2651 3899 5310 7232 8872 10945 13260 14388 16485 16692 17236 16447 16699 15372 14393 11562 10082 7556 6213 4066 3032 1795 1269 760 472 266 176 77 53 19 19 2 6 232261
3 2 6 7 12 11 11 13 12 20 28 32 33 43 54 56 60 59 64 78 63 84 76 73 59 58 47 44 42 42 27 30 22 25 17 12 8 8 4 1 3 1 1 0 0 1351
3 6 8 32 39 65 96 155 255 401 639 857 1342 1971 2682 3644 4466 5502 6662 7233 8274 8388 8656 8260 8379 7715 7220 5803 5062 3799 3120 2048 1527 910 643 386 240 137 90 39 28 10 10 1 3 116806
I 0 0 0 0 1 0 1 6 20 22 29 22 25 26 19 17 12 5 16 3 5 0 2 231
2
–1
2 2 5 7 10 11 11 13 11 20 23 32 28 43 42 56 60 59 57 78 59 84 74 73 51 58 44 44 39 42 27 30 21 25 17 12 8 8 4 1 3 1 1 0 0 1296
1 1 2 0 1 5 5 12 0 7 4 2 8 3 3 0 1 0 0 0 0 0 0 55
Note: nt is the number of transconfigurations of hydrogenbound pairs, M is the total number of proton configurations, S/2 is the number of antisymmetrybound pairs of the configurations possessing the conventional spatial symmetry, n is the number of symmetrically dif ferent (nonisomorphic) configurations, a is the number of antisymmetric configurations, and m is the number of nonisomorphic con figurations (with allowance for the antisymmetry). What follows is the statistics of spatial generalized symmetry. CRYSTALLOGRAPHY REPORTS
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Table 2. Classification of the proton configurations of the CSI framework nt
M
n
m
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
24 48 144 528 1056 3072 5376 10368 20400 34704 62928 117024 206808 340704 566232 862800 1360464 2022912 3022920 4370352 6197880 8565504 11563968 15078432 19440456 24026448 29356560 34169808 39654120 43928208 48260208 50500320 52725984 52596720 52220976 49571904 46788528 42052992 37664136 32013552 27319320 21949152 17710896 13416864 10244352
1 1 4 11 29 64 125 216 439 723 1342 2438 4377 7098 11891 17975 28470 42144 63208 91049 129454 178448 241354 314134 405543 500551 612205 711871 826868 915171 1006167 1052090 1099196 1095765 1088732 1032748 975511 876104 785340 666949 569734 457274 369416 279518 213771
1 1 2 7 17 35 70 116 231 372 690 1236 2225 3581 6006 9052 14333 21178 31762 45674 64967 89427 120962 157320 203175 250613 306544 356352 414015 458049 503658 526526 550240 548359 544957 516824 488305 438430 393184 333825 285325 228917 185067 139963 107154
–1
c
0
0
0
1
0
7
0
11
0
13
0
29
0
68
0
94
1
123
13
215
18
304
39
394
58
467
73
529
88
640
114
629
101
631
128
660
137
602
104
561
90
483
83
352
75
266
2
21
–1
c
2/c 21/c 2/c 21/c nc2 nc2
0 1 0 1 2 6 8 16 14 16 20 29 43 46 77 99 113 169 201 242 276 309 346 360 454 504 498 578 662 651 680 666 717 618 644 549 549 485 540 430 438 363 360 229 234
0 0 0 2 0 0 0 0 1 5 10 5 18 18 28 30 45 43 64 57 110 97 130 146 199 171 207 255 245 276 264 296 315 335 303 351 291 271 286 271 247 197 220 179 154
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
2
0
0
0
0
0
0
2
1
3
0
0
0
1
0
2
4
0
1
0
0
1
0
2
2
1
1
0
0
1
1
4
7
0
0
0
0
0
1
11
4
0
0
0
0
0
1
13
19
2
2
0
0
1
1
12
34
2
0
1
0
2
0
13
62
2
5
3
2
6
1
16
68
1
1
0
1
1
6
20
117
4
0
4
2
3
4
30
132
3
3
2
7
1
0
29
201
11
2
3
2
3
4
33
166
0
1
1
0
3
1
33
207
0
1
2
5
4
0
27
194
1
2
5
0
4
2
27
210
8
2
5
4
0
3
20
171
1
3
2
2
2
1
18
196
6
5
1
4
0
1
16
114
0
4
2
0
1
1
6
131
1
0
2
3
4
2
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Table 2. (Contd.) nt 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 Σ
M 7271136 5359704 3553104 2483064 1572864 1050624 628800 384672 218160 138144 64608 43800 15216 8952 2544 1416 384 96
n 151482 111947 74023 51907 32768 22009 13100 8067 4545 2928 1346 934 317 201 53 33 8 3
m
–1
75874 56169 37100 26091 16448 11080 6586 4086 2284 1492 681 482 162 109 28 20 4 3
c
63
219
32
140
28
90
14
37
18
31
7
14
4
9
0
3
0
0
2
21
171 185 124 125 88 64 51 52 19 31 11 14 6 8 2 4 0 0
95 115 53 79 40 50 21 24 4 11 5 10 1 2 1 0 0 0
822823440 17151190 8587446 1288 7622 14198 6653
optimal on the whole (not only with respect to energy) are most often sufficiently optimal locally, which sig nificantly limits their structural variety and thus increases the probability of higher order symmetry. This holds true for the antisymmetry of icelike sys tems. However, the symmetric and antisymmetric proton configurations form a representative set of con figurations in this case, which demonstrates the varia tion in the properties most clearly due to the proton disorder. Figure 3 shows the distributions of proton configu rations of the two most widespread frameworks: HSIII and CSI. The energies and total dipole moments cor respond to the configurations optimized using the TIP4P potential. The optimization was very approxi mate. However, it was established that further optimi zation does not change the general pattern. In the HS III framework (Fig. 3a), generally symmetric configu rations cover the entire range of parameter values almost completely. It is especially important that the extreme configurations (circles) belong to this repre sentative set. This gives grounds to state that the gen erally symmetric configurations of the CSI frame work (Fig. 3b) also fairly exactly characterize the gen eral distribution. The data of Tables 1 and 2 confirm the high repre sentativeness of the set of generally symmetric config urations, especially for the optimization problems. The fraction of generally symmetric configurations increases significantly when nt approaches the maxi CRYSTALLOGRAPHY REPORTS
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–1
c
2/c 21/c 2/c 21/c nc2 nc2
9
73
0
0
2
3
2
2
5
57
2
1
2
1
1
2
0
31
0
0
0
4
1
1
0
25
1
1
0
1
1
0
0
12
0
0
1
0
0
1
0
5
0
0
0
1
0
0
0
4
0
0
3
0
0
0
0
2
0
0
0
0
0
1
0
1
0
0
1
0
0
1
349
2252
49
35
42
42
44
38
mum and minimum values, whereas in the middle of the tables it is less than 1%. Note that the antipode configuration pairs only slightly differ in properties. The difference in the stabilization energies for them is less than 1%. That is why the calculations were per formed for only one antipode from each pair. The number of transdimers nt determines the framework energy, with only the interaction between the nearest neighbors taken into account [9]. The average fractions of strong H bonds in the HSIII and CSI frameworks are 0.355 and 0.356, respectively, and the maximum fractions are 0.677 and 0.674. The configurations with nt values far from maximum are most stable due to the competition of the contribu tions from the nearest and more remote neighbors [9]. According to our calculations, the configuration with the symmetry P21/c (nt = 44) is the most stable in the CSI framework; this configuration was predicted by us previously in the combinatorial optimization based on the discrete model of intermolecular inter action [9]. RESIDUAL ENTROPY According to the wellknown and rather exact Pauling formula, the residual entropy of icelike sys tems S = lnM/N = ln(3/2) = 0.4055 [18]. Here, M is the total number of proton configurations and N is the number of molecules. The statistics obtained by us demonstrate a tendency to this limit: S(TSIV) =
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p, Da HSIII
20
30
CSI
20 10 10 0 –57
–55
–53
–51
–49
0 –47 –57 U, kJ/mol
–55
–53
–51
–49
–47
Fig. 3. Distributions of framework proton configurations over energy U and total dipole moment р. The generally symmetric and asymmetric configurations are shown black and gray, respectively.
0.5347, S(CSIV) = 0.5252, S(HSIII) = 0.4568, and S(CSI) = 0.4463. Note that the similar results for the elementary orthorhombic and extended hexagonal (2 × 2 × 1) unit cells of hexagonal ice Ih give the fol lowing values: ln114/8 = 0.5920 and ln2404144962/48 = 0.4500 [5]. One may wonder why the residual entropy tends to the upper rather than to the lower bound. Indeed, the correlations imposed by artificial periodic boundary conditions limit the proton disorder. The point is that the Pauling formula determines the specific entropy of disordered infinite 3D systems. The entropy of the sys tems formed by the repetition of individual fragments is indeed lower. Specifically, it is zero because the number of configurations is finite. The periodic boundary conditions limit the number of proton con figurations for isolated fragments in the form of a unit cell. However, the residual entropy of such structures is much higher. For example, for polyhedral clusters, as was noted above, the analog of the Pauling formula is the expression S = ln(3/ 2 ) = 0.7520 [19]. In this context, the formulas for planar and tetrahe drally coordinated water cycles are illustrative [12]. For example, the number of proton configurations in a cycle composed of N molecules is 3N + 1. The resid ual entropy of cyclic fragments, ln(3N + 1)/N, also tends from above to the bound value of ln3. For an open finite chain composed of N molecules, the num ber of configurations is 2 × 3N, because it is 6 for an individual molecule; an additional molecule can be added to such structures in three ways (rotation by 120°).
fer significantly in properties, and each of them is unique. The properties of gashydrate formations may also depend strongly on the structure of the framework proton subsystem. When the sizes of a system increase infinitely, the distribution of random proton configurations in energy (and in other characteristics) in gas hydrate frameworks tends to a Δ function. This is especially pronounced on the set of configurations with a zero total dipole moment. Configurations with intermedi ate values of characteristics are achieved with close tounity probability in large systems. However, the actual variation in properties is determined by the very unlikely extreme configurations. Configurations with extreme properties can be found by successively screening the database of all proton configurations. In this case, the optimal unitcell configurations fairly exactly characterize the general variation in the prop erties of extended systems. Note that energetically optimal configurations can also be calculated by applying discrete models of the intermolecular inter action and special methods of combinatorial optimi zation [9]. For frameworks with large unit cells, such as CSII (136 water molecules per unit cell), represen tative sets of random proton configurations can be important [29]. ACKNOWLEDGMENTS This study was supported by the Russian Founda tion for Basic Research, project no. 080300338a. REFERENCES
CONCLUSIONS The proton configurations calculated in this study form a special crystallographic database, which can easily be used for investigating the properties of gas hydrates using the most diverse computer simulation techniques. The framework proton configurations dif
1. E. D. Sloan, Jr., Clathrate Hydrates of Natural Gases, 2nd ed. (Marcel Dekker, NewYork, 1998). 2. W. L. Mao, C. A. Koh, and E. D. Sloan, Phys. Today. 60, 42 (2007). 3. S. McDonald, L. Ojamé, and S. J. Singer, J. Phys. Chem. 102, 2824 (1998).
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Translated by Yu. Sin’kov
