English

• 1. INTRODUCTION

  Hydrogen bond is a very important intermolecular force and widely exists in small molecular clusters, biological macromolecules and supramolecular systems. It plays an important role in many biological processes and chemical reactions, which has been paid much attention in both experimental and computational researches^[1]. For water molecules, a large number of hydrogen bonds are found in liquid water, ice and hydrates^[2]. To date, the exploration of clusters focusing on cluster's configuration, molecular kinetic behavior, hydrogen bonding networks and such physico-chemical properties has become a hot spot in the field of molecular cluster science^[3-10]. Dodecahedral 5¹² single cage cluster (H₂O)₂₀, consisting of 12 five-membered rings, is generally distributed in natural methane hydrates (MH) crystalline structure SI, SII and SH. Tetrakaidecahedral 5¹²6² single cage cluster (H₂O)₂₄ composed of 12 five-membered and 2 six-membered rings is generally distributed in natural methane hydrates (MH) crystalline structure SI. Dodecahedral 4³5⁶6³ single cage cluster (H₂O)₂₀, consisting of 3 four-membered rings, 6 five-membered rings and 3 six-membered rings, are generally distributed in natural methane hydrates (MH) crystalline structure SH^[11]. Dodecahedral cage of (H₂O)₂₀ and tetrakaidecahedral cage of (H₂O)₂₄ contain 30026 and 3043836 stereoisomeric configurations according to unique topologies, respectively^[12]. Because each water molecule in cage cluster shares donating or accepting hydrogen bonds with three neighbor waters, there are merely two types of water molecules hindered by neighbor combinations. One is that the water molecule provides a hydrogen bond OH and accepts two hydrogen bonds O…H, and the other is that the water molecule provides two hydrogen bonds OH and accepts one hydrogen bond O…H^{[13, 14]}. Anick defined them 'F components' water with a dangling OH covalent bond and 'L components' water based on Bernal-Fowler "ice rules". Therefore, four types of two-water-molecular pairs generally derived from finite size clusters containing only F waters and L waters are obtained, that is, L ⇐ F, F ⇐ F, L ⇐ L and F ⇐ L pairs. Previous studies have demonstrated that the energy of hydrogen bond in L ⇐ F pair is much stronger than any other pair^{[3, 4, 15]}. In the study of Znamenskiy and Green, the types of the two water molecules are represented as 12-type (F components) and 21-type (L components), respectively. The four types of hydrogen bonds are 1221, 2121, 1212 and 2112^[4]. Kirov et al. introduced the concept of 'strong' vs. 'weak' nearest-neighbor hydrogen bonds (SWB model) and divided the edge hydrogen bonds of convex polyhedral clusters into five groups as follows: t1d (equivalent to L ⇐ F), t1a, c2, c0 and c1a.^[15]. It is attractive that the F ⇐ L pair could be further refined as t1a or c1a by inherent stereoisomerism. Kuo et al. imported an auxiliary topological index ζ, which varies from 0 to 4, indicating the sum of F neighbor waters of the L central water^[16]. Furthermore, the derivative descriptor (t1P, t1Q) and (t2P, t2Q) were investigated for acceptor and donor water, respectively^{[14, 17-18]}. Considering the fact that only 3-coordinated water occurs in convex polyhedral water clusters, there are totally 49 sub-grouped hydrogen bonds based on his classification. Another topological four-digit list is also defined by Anick focusing on cluster isomer identification, e.g. list (4, 6, 0, 0) of 5¹² single cage cluster suggests such configuration which contains 4 L waters surrounded by totally 3 F neighbor waters, 6 L waters surrounded by 2 F neighbor waters and 1 L neighbor water. However, zero of the third digit implies there is no water surrounded by 1 F and 2 L neighbors, so does zero of the fourth digit, indicating the sum of water surrounded by merely 3 L neighbors. Iwata and coworkers used perturbation theory based on the locally-projected molecular orbitals (LPMO PT) to evaluate charge-transfer (CT) terms and dispersion interaction terms of every pair of the above 49 sub-grouped hydrogen bonds obtained from polyhedral (H₂O)₈, (H₂O)₂₀ and (H₂O)₂₄ stable isomers^[19-22]. The paper together with previous work revealed the specific contribution of the CT terms strongly correlated with hydrogen bond distances to the hydrogen bond energy in various sizes of (H₂O)_n cluster systems via Mulliken's charge-transfer theory and several rational assumptions^[20-22]. One of the assumptions was the simplified description of the next-nearest neighbor water of hydrogen bonding distribution, which made the descriptor of sub-grouped hydrogen bonds concise but ambiguous.

  In fact, there should be nearly 405 (5 × 3⁴) sub-grouped hydrogen bonds of six-water clusters completely considering all water molecules' steric distribution. Herein we also simplified the definition of sub-grouped hydrogen bonds considering the nearest and next-nearest neighbor water molecules. Enlightened by Znamenskiy and Kirov, we completely distinguished the central pair configuration and labeled all four environmental neighbors correspondingly. Finally, 74 sub-grouped hydrogen bonds were classified and calculated by a simpler mathematical iteration.

  2. METHODS AND CALCULATIONS

  The positions of oxygen atoms in 5¹², 4³5⁶6³, 5¹²6² and 5¹²6⁸ are determined by the crystal structures in SI, SII and SH. The O–O distance was set to 2.80 Å. Then the positions of the dangling hydrogens are randomly generated. Finally, the rest hydrogen atoms were randomly inserted into O–Os. The O–H covalent bond was set to 0.99 Å and the hydrogen bond length is set to 1.81 Å. ∠H–O–H covalent bond angle was set to 105.4°. A large number of hydrogen bonding isomers for cage structures about 5¹², 4³5⁶6³, 5¹²6² and 5¹²6⁸ are generated by program. Fig. 1 is the construction process of 5¹² cage hydrate cluster.

  Figure 1

  

  Figure 1.  Model construction of 5¹² cage water cluster

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  The sum of hydrogen bonding energy of every single cage cluster △E_BindE can be expressed as (1):

  $ \Delta E_{\text {BindE }}=E_{\text {cluster }}-E_{\text {loose }}$

  (1)

  E_cluster means the energy of single cage cluster, while E_loose implies the energy of loose cage structure, in which the bond length and bond angle of the water molecules are unchanging, and only the O…H distance is increased to 10 times of the original one. In this case, the hydrogen bond energy between the water molecules in the loose cage can be neglected. All single point energies were calculated at the DFT/B97D/6-311++G(2d, 2p) and DFT/M052X/6-311++G(2d, 2p) level in Gaussian09 package^[23-26].

  The hydrogen bonding network of 5¹², 4³5⁶6³, 5¹²6² and 5¹²6⁸ single cage clusters have 30, 30, 36 and 54 independent hydrogen bonds, respectively, so the sum of hydrogen bonding energy of every single cage cluster △E_BindE can also be expressed as (2):

  $ \Delta E_{\text {BindE }}=\Sigma \xi_i \times E h_i $

  (2)

  ξ_i (i = 1, 2, …, 73, 74) represents the number of correlative sub-grouped hydrogen bonds appearing in any cluster isomer. Eh_i means the bond energy of various sub-grouped hydrogen bonds. Key information ξ_i and △E_BindE of each cluster was extracted to introduce the equation set and its coefficient matrix A satisfied:

  $ \mathrm{A}=\left(\xi_{i j}\right)_{n \times 74}$

  (4)

  Where j implied the j^th equation group and n was the number of equations contained in this system. The solution vector Eh was recorded as

  $ Eh = (Eh_{1}, Eh_{2}, …, Eh_{73}, Eh_{74})^{T}_{74 × 1} $

  (5)

  The single cage cluster energy vector △E_BindE was recorded as

  $ \begin{gathered} \Delta E_{\text {BindE }}=\left(\Delta E_{\text {BindE } 1}, \Delta E_{\text {BindE2 }}, \cdots, \right. \\ \left.\Delta E_{\text {BindEn-1 }}, \Delta E_{\text {BindEn }}\right)_{n \times 1}^T \end{gathered} $

  (6)

  The equation set could be expanded as

  $ \mathrm{A} \times E h=\Delta E_{\text {BindE }}$

  (7)

  To obtain the coefficient matrix convenient for Jacobi iterative method, we rewrote Eq. (7) and denoted column vector ξ_i (i = 1, 2, …, 73, 74) in A

  $ \mathrm{A}=\left(\xi_1, \xi_2, \cdots, \xi_{73}, \xi_{74}\right) $

  (8)

  $ \Sigma \xi_i \times E h_i=\Delta E_{\mathrm{BindE}} $

  (9)

  Pre-multiplied Eq. (9) by ξ_j^T

  $ \xi_j^{\mathrm{T}} \times\left(\Sigma \xi_i \times E h_i\right)=\xi_j^{\mathrm{T}} \times \Delta E_{\text {BindE }} $

  (10)

  Resulting in a new coefficient matrix B

  $ \mathrm{B}=\left(b_{i j}\right)_{74 \times 74} $

  (11)

  $ b_{i j}=\xi_j^{\mathrm{T}} \times \xi_i \quad i \neq j $

  (12-1)

  $ b_{i j}=0, i=j $

  (12-2)

  Thus the iterative solution of each element Eh_i of vector Eh

  $ E h_i^{(n+1)}=\left(\xi_i^{\mathrm{T}} \times \Delta E_{\text {BindE }}-\Sigma b_{i j} \times E h_j^{(n)}\right) /\left(\xi_i^{\mathrm{T}} \times \xi_i\right)$

  (13)

  Superscript n indicated the n^th iteration during calculation, and convergent solution was deemed to be got when the norm of difference between two adjacent iteration vectors is less than 0.1.

  3. RESULTS AND DISCUSSION

  3.1 Relationship between the stability of clusters and the type of central water molecule pairs

  The cage water cluster isomers of 602 5¹², 492 5¹²6², 501 4³5⁶6³ and 523 5¹²6⁸ were randomly generated. A few previous studies proposed that the hydrogen bonding energy of 1221 water pairs was higher than that of other water pairs, and we selected 1221 central hydrogen bonds as standard to classify isomers of diverse families defined by Anick to inspect the correlation between hydrogen bonding energies and the number of class 1221^{[3, 4, 15]}. The number of each cage is counted by n(1221) value, as listed in Table 1. In the four cage clusters, the n(1221) value in their isomers varies within isomer distribution area, and the most distributed number is in the middle of the region.

  Table 1

  

  Table 1.  The Number of Randomly Generated 5¹², 5¹²6², 4³5⁶6³ and 5¹²6⁸ Cage Water Clusters

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  Cage type	n(1221)
  	0	1	2	3	4	5	6	7	8	9	10	11	Sum
  512	6	51	150	196	128	54	16	1	-	-	-	-	602
  51262	2	27	74	144	130	78	29	7	1	-	-	-	492
  435663	2	51	104	156	124	53	10	1	-	-	-	-	501
  51268	-	2	7	42	81	158	104	77	38	11	2	1	523

  The energy of the 602 5¹², 492 5¹²6² and 501 4³5⁶6³ isomers was calculated by the B97D and M052X methods, as shown in Fig. 2. It was not difficult to find that the hydrogen bonding network of 5¹², 5¹²6² and 4³5⁶6³ single cage clusters always demonstrates the increasing tendency of bonding energy with the increase of n(1221).

  Figure 2

  

  Figure 2.  Binding energies. a) and b) are 5¹² cluster isomers, c) and d) are 5¹²6² cluster isomers, and e) and f) are 4³5⁶6³ cluster isomers. a), c) and e) are calculated by B97D. b), d) and f) are calculated by M052X

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  It should be noted that for the same value of n(1221), the bonding energies among various isomers always exhibit a huge difference. It could be easily seen from Fig. 2 that there is always a large region of energy overlap between the hydrogen bonding networks in different n values. Taking the 5¹² single cage by B97D method as an example, for n(1221) = 2, △E_bindE is between –778.25 and –684.88 kJ⋅mol^–1; for n(1221) = 3, △E_bindE is between –795.63 and –717.73 kJ⋅mol^–1, indicating that the △E_bindE of different n(1221) has a significant energy overlap scope. Therefore, it is unreasonable to distinguish the thermodynamic stability of different isomers only by the number of 1221 class hydrogen bonds in the cage cluster.

  3.2 Effect of neighbor water pair hydrogen bonds

  3.2.1   Rationality of the iterative solution

  Inheriting Znamenskiy's definition of water clusters, four-digit number index was utilized to characterize the two categories of hydrogen bonds consisting in the central water pair and neighbors. For instance, index 1221 indicates that the donor water oxygen of central water pair donates just one hydrogen bond (the first digit) and accepts two hydrogen bonds from neighbors (the second digit), which means that there is a dangling covalent O–H bond of donor water, while acceptor water oxygen donates two hydrogen bonds (the third digit) and accepts only one (the fourth digit). Hence, there is a 12-type water (equivalent to F water. Hereafter this definition is used in the following paper) performed as donor and a 21-type (equivalent to L) water as an acceptor of the central pair 1221.

  Taking into account that only 12-type water molecule and 21-type water molecule could occur in the single cage clusters, the candidate index of arbitrary water pair clusters could be 1212, 1221, 2112 or 2121. It's worth mentioning that there are isomeric clusters existing under certain indexes except 1221. These dissimilar clusters are shown in Fig. 3. We introduced superscript labeling digital index to identify isomers such as 1212^a, which corresponds to the cluster schemed in Fig. 3.

  Figure 3

  

  Figure 3.  Nine classes of the hydrogen bonds considering the nearest neighbor water molecules. The water molecules labeled are 21- or 12-type water molecules, 21-type water molecules and 12-type water molecules

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  The number of all possible 1212 or 2121 topological configurations is two, compared with four configurations for 2112 clusters. To describe the sub-structure of hydrogen bonds of water molecules in different environments, the following improvements were made:

  $ 1212^{\mathrm{a}}\left(* * 12, * * 12, * * 12, 12^{* *}\right)$

  (20)

  Similar to previous definition, 1212^a characterization considers stereoisomeric central water molecule pairs, and the four sets of arrays in parentheses indicate four next-nearest-neighbor hydrogen bonds adjacent to the central water molecule pairs. **means that the environmental water molecule can be 12- or 21-type water molecule. These four sub-grouped indexes are defined by the following order: the two water oxygen atoms in center water pair are directed as vector from donor to acceptor water. The pairs of environmental water molecules adjacent to the donor water molecules counterclockwise are defined as the first and second sub groups, while acceptor water molecules adjacent to the environmental water molecules on the counterclockwise are defined as the third and fourth sub groups. The above improved definition method is aimed to analyze the pairs of water molecules in single-cage clusters, and any pair of water molecules appearing in the cluster can be uniquely represented in considering the sub hydrogen bonding type in adjacent environment. As shown in Fig. 4, since each environmental water pair contains one central molecule of identified configuration, the total number of sub-grouped hydrogen bonds in known central hydrogen bond type is limited to 16 (2 × 2 × 2 × 2). Based on the above classification, there are total 144 sub-grouped hydrogen bonds divided into 9 classes of the central water dimer configura-tions demonstrated respectively.

  Figure 4

  

  Figure 4.  Energy comparison of hydrogen bonds between B97D and M052X methods. (a) 5¹² cluster isomers, (b) 5¹²6² cluster isomers and (c) 4³5⁶6³ cluster isomers. The polyline with the square indicates the energies of 74 sub-grouped hydrogen bonds in the B97D method, and that with circle indicates the energies of 74 sub-grouped hydrogen bonds in M052X method

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  In Fig. 4, the index included environmental water pairs of class 1212^a is required to be (**12, **12, **12, 12**). Class 1212^b ought to be (**12, **12, 12**, **12), which represents that the dipole moment orientation of acceptor water molecule of the central hydrogen bond is different compared with class 1212^a. Class 1221 ought to be (**12, **12, 21**, 21**) uniquely. Class 2112^a should be (21**, **21, **12, 12**), while its mirror symmetric isomer class 2112^d is (**21, 21**, 12**, **12). Class 2112^b ought to be (**21, 21**, **12, 12**), and its mirror symmetric isomer class 2112^c is (21**, **21, 12**, **12). Class 2121^a defers to (21**, **21, 21**, 21**), while its mirror symmetric isomer class 2121^b is required to be (**21, 21**, 21**, 21**).

  It should be noted that the classification over refines some peculiar configurations from different classes, which share identical digit index, such as sub-grouped hydrogen bonds of classes 1212^a and 1212^b. For each hydrogen bond of class 1212^a, the acceptor water molecule of the central pair acts as an acceptor of the third group environmental water pair and donor of the fourth group environmental water pair, while 1212^b hydrogen bonds are exactly opposite: the central pair acceptor acts as donor of the third environmental group and acceptor of the fourth. Regardless of 1212^a or 1212^b hydrogen bonds, the central donor waters always act as acceptors for both the first and second environmental groups. To simplify this classification, we treat two sub-grouped hydrogen bonds as specular isomer configurations, which comply the following rules: their central pairs are mirror symmetric isomers and correlative environmental hydrogen bonds are mirror-isomeric of different positions separated by the central donors or acceptors. For instance, 1212^a (2112, 1212, 2112, 1221) and 1212^b (1212, 2112, 1221, 2112) (Fig. 4), two sub-grouped hydrogen bond configurations with the same hydrogen bonding connection pattern, are approximated to be specular isomerism. Considering that the cooperative effect of hydrogen bonds which have been discussed is mainly influenced by the connection pattern of hydrogen bonding net in the cluster, the hydrogen bonding energy of the water pair considering the next-nearest neighbors should be approximately equal in the case of mirror isomerism^[27-29].

  In this context, these specular symmetric sub-grouped hydrogen bonds should be combined into just one item. Classes 1212^a and 1212^b are merged into 1212, 2112^a and 2112^d into 2112^a, 2112^b and 2112^c into 2112^b, and 2121^a and 2121^b into 2121, thus a total of 64 (4 × 16) sub-grouped hydrogen bonds are acquired. Because of the central water pair's high symmetry of class 1221 hydrogen bonds, such as 1221 (1212, 2112, 2112, 2121) and 1221 (2112, 1212, 2121, 2112), which are categorized as specular isomers according to the rules above, total 6 sub-grouped hydrogen bonds from original class 1221 are selected to be merged into other congeneric sub-grouped hydrogen bonds. In summary, we finally get 5 classes, and 74 (16 + 16 + 16 + 16 + 10) diverse sub-grouped hydrogen bonds.

  Using Jacob iterative method, we calculated the energies of 74 sub-grouped hydrogen bonds. As shown in Fig. 4, the M052X method has greater energy values than the B97D method, and their changing tendency of energy values is almost unanimous.

  In order to test the iterative solution set, the energy values of 74 sub-grouped hydrogen bonds were used to estimate the hydrogen bond network energies of selected clusters. There are 602, 492 and 501 testing cages for corresponding 5¹², 5¹²6² and 4³5⁶6³ clusters. As shown in Fig. 5, the red curves indicate the fitting result of the clusters, the black curves show the original hydrogen bond network energy △E_bindE of each cluster, and they were arranged according to the number of 1221 water pairs. It can be clearly seen that the 5¹² single cage clusters have the best matching values compared with the original value, while 4³5⁶6³ cage clusters are worse for both B97D and M052X methods. For example, in B97D method, the maximum absolute values of the errors between the fitting and calculated energy values of 5¹², 5¹²6² and 4³5⁶6³ cage clusters are 3.71, 11.97 and 13.59 kJ⋅mol^-1, respectively, and the average absolute errors are 0.84, 2.81 and 3.87 kJ⋅mol^-1, respectively, which means the iterative solution set is correct.

  Figure 5

  

  Figure 5.  Hydrogen bonding network fitting curve. (a) and (b) are 5¹², (c) and (d) are 5¹²6² , (e) and (f) are 4³5⁶6³. The energies of (a), (c) and (e) are calculated by B97D method. The energies of (b), (d) and (f) are calculated by M052X method. The black curve indicates the calculated energies of selected clusters, while the red one shows the fitting energies using the solution of corresponding sub-grouped hydrogen bonds' energies

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  On the other side, to inspect the ability to estimate other cage clusters, the average values of the same sub-grouped hydrogen bonds in three different clusters are used since there is deviation between different clusters (Fig. 6). Both estimated and computed results are shown in Fig. 7, and they have the same variation tendency of hydrogen bond network energies, although the estimate energies almost become a little weaker as a whole. So, the values of the iterative solution can reflect how the next-nearest-neighbor hydrogen bond affects the central water molecule pairs.

  Figure 6

  

  Figure 6.  Energy comparison of hydrogen bonds between 5¹², 5¹²6² and 4³5⁶6³ single cage clusters. (a) B97D and (b) M052X methods. The polyline with the square indicates the energies of 74 sub-grouped hydrogen bonds of 5¹², the polyline with the circle is the energies of 74 sub-grouped hydrogen bonds of 5¹²6², and the polygonal line with the triangle shows the energies of 74 sub-grouped hydrogen bonds of 4³5⁶6³

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  Figure 7

  

  Figure 7.  Hydrogen bonding network fitting of 5¹²6⁸ cluster isomers. The energies of (a) are calculated by B97D and those of (b) by M052X. The black curve indicates the original calculated energies of selected clusters, while the red curve shows the fitting energies using the solution of corresponding sub-grouped hydrogen bonds' energies

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  3.2.2   A new sub-grouped hydrogen bond index and its regularity of cooperative effect

  It must point out that quoting the conclusion of Znamenskiy for small water clusters and the description generalized from a single cage clusters is still flawed. For example, to illustrate sub-grouped hydrogen bonds consisting of isomeric central water pair, nearest and next-nearest neighbors, we need to use twenty-digit index to describe five correlative water pairs in specific order, which may be too fussy to comprehend.

  Based on the existence of only two types of water molecu-les, 21- and 12-type water in 5¹² single cage clusters, we could describe a water molecule only by the number of donor hydrogen bonds shared with adjacent environmental water molecules. For example, digit "1" indicates 12-type water and digit "2" indicates 21-type water. This labeling method can simplify the complexity of characterizing the hydrogen bond network in a single cage cluster. In the above discussion utilizing twenty-digit index, over half of terms defined are actually repeated, that is, the center water molecules are also involved in the composition of environmental water pairs when the nearest neighbors are considered, so do the water molecules of environmental water pairs. Thus, we can only use a special method to describe each water molecule separately in the sub-grouped hydrogen bond configuration rather than explain the hydrogen bond pairs.

  As shown in Fig. 8, the water molecules in the primary fragment cluster are classified. The donor molecule of central water pair acts as the first digit, and the adjacent environmental water molecules are the second, and the third digit is counterclockwise. The acceptor molecule of central water pair acts as the sixth digit, and its adjacent environmental water molecules as the fourth, and the fifth digit is counterclockwise. The number of donor hydrogen bonds formed by defined subject water molecule and the environment is recorded on each digit. With this improvement, we can use the six-digit index to describe the sub-grouped hydrogen bond cluster consisting of the central pair and the nearest, next-nearest neighbors in detail. In order to consider the heterogeneity of the central water pair, we introduce a sign in the first or the sixth digit of the index. The oxygen atom of the donor molecule in central water pair is directed to the acceptor oxygen atom as the vector a. When the first digit is 2, the other donor hydrogen bond of central donor water molecule is directed as vector b, starting from donor oxygen atom to the acceptor one. If the rotation angle starting from a to b counterclockwise is less than 180°, it could be recorded as +2 (The positive sign can be omitted). Otherwise, it is recorded as –2. When the sixth digit is 1, the only donor hydrogen bond of central acceptor water molecule is directed as vector c. If the rotation angle starting from a to c counterclockwise is less than 180°, it could be recorded as +1 (The positive sign can be omitted). Otherwise, it's recorded as –1. In other case, there is only one distribution of central water molecular orientation. As for environmental water molecules, we just temporarily disregard their diverging hydrogen bond orientations.

  Figure 8

  

  Figure 8.  Improved index descriptor of sub-grouped hydrogen bonds. The numbers labeled in the figure explain the order of this six-digit index. The donor molecule of central water pair acts as the first digit, and the adjacent environmental water molecules are the second and third digit widdershins. The acceptor molecule of central water pair acts as the sixth digit, and its adjacent environmental water molecules as the fourth, and the fifth digit is counterclockwise.

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  According to this method, we could rewrite the modality of original array index of class 1212 hydrogen bonds as 1ddda1 and 1ddad-1, where letter 'd' indicates the ordered nearest neighbor water molecule acts as donor water pair consisting of environmental water and its nearest central water, and 'a' indicates the ordered nearest neighbor water molecule acts as acceptor of water pair consisting of the environmental water and its nearest central one. Since all class 1ddad-1 sub-grouped hydrogen bonds could find the mirror isomers in class 1ddda1 sub-grouped hydrogen bonds, we could identify all 32 kinds of class 1212 sub-grouped hydrogen bonds as 1ddda1. The same characterization could be applied to other four classes, so 1221 sub-grouped hydrogen bonds can be rewritten as 1ddaa2, 2112^a as 2adda1, 2112^b as 2adad-1, and 2121 as 2adaa2, respectively. The energies of 74 sub-grouped hydrogen bonds labeled by improving six-digit index are also relisted in Table 2.

  Table 2

  

  Table 2.  Numeric Sequence of 74 Subtype Hydrogen Bonds and Energies of Sub-grouped Hydrogen Bonds Labeled by Improved Index

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  Subtype hydrogen bond array	Digital sequence	EH/kJ·mol-1
  		B97D	M052X
  1212(1212, 1212, 1212, 1212)	111111w	–10.33	–11.97
  1212(1212, 1212, 1212, 1221)	111121	–24.34	–26.41
  1212(1212, 1212, 2112, 1212)	111211w	–19.94	–21.87
  1212(1212, 1212, 2112, 1221)	111221	–25.37	–27.83
  1212(1212, 2112, 1212, 1212)	112111w	–16.43	–17.58
  1212(1212, 2112, 1212, 1221)	112121	–25.90	–27.92
  1212(1212, 2112, 2112, 1212)	112211	–23.25	–24.73
  1212(1212, 2112, 2112, 1221)	112221	–28.19	–30.00
  1212(2112, 1212, 1212, 1212)	121111w	–18.87	–20.01
  1212(2112, 1212, 1212, 1221)	121121s	–29.13	–30.76
  1212(2112, 1212, 2112, 1212)	121211	–23.71	–25.03
  1212(2112, 1212, 2112, 1221)	121221s	–29.22	–30.76
  1212(2112, 2112, 1212, 1212)	122111	–22.16	–23.58
  1212(2112, 2112, 1212, 1221)	122121	–28.96	–30.94
  1212(2112, 2112, 2112, 1212)	122211	–25.13	–26.74
  1212(2112, 2112, 2112, 1221)	122221	–26.68	–28.38
  1221(1212, 1212, 2112, 2112)	111112	–23.78	–25.36
  1221(1212, 1212, 2112, 2121)	111122	–27.40	–28.83
  1221(1212, 1212, 2121, 2121)	111222	–27.03	–28.76
  1221(1212, 2112, 2112, 2112)	112112s	–29.82	–31.95
  1221(1212, 2112, 2112, 2121)	112122s	–29.72	–31.75
  1221(1212, 2112, 2121, 2112)	112212s	–31.05	–33.07
  1221(1212, 2112, 2121, 2121)	112222	–25.90	–28.08
  1221(2112, 2112, 2112, 2112)	122112s	–29.12	–30.98
  1221(2112, 2112, 2112, 2121)	122122	–25.93	–27.66
  1221(2112, 2112, 2121, 2121)	122222w	–18.46	–20.23
  2112a(2112, 1221, 1212, 1212)	211111w	–19.68	–21.48
  2112a(2112, 1221, 1212, 1221)	211121s	–28.81	–30.39
  2112a(2112, 1221, 2112, 1212)	211211	–24.06	–25.82
  2112a(2112, 1221, 2112, 1221)	211221s	–31.39	–33.30
  2112a(2112, 2121, 1212, 1212)	212111w	–20.54	–22.41
  2112a(2112, 2121, 1212, 1221)	212121	–26.83	–28.62
  2112a(2112, 2121, 2112, 1212)	212211	–23.36	–25.04
  2112a(2112, 2121, 2112, 1221)	212221	–26.94	–28.85
  2112a(2121, 1221, 1212, 1212)	221111	–26.99	–28.81
  2112a(2121, 1221, 1212, 1221)	221121s	–31.91	–33.30
  2112a(2121, 1221, 2112, 1212)	221211s	–27.77	–29.34
  2112a(2121, 1221, 2112, 1221)	221221s	–30.51	–32.07
  2112a(2121, 2121, 1212, 1212)	222111	–23.35	–25.33
  2112a(2121, 2121, 1212, 1221)	222121	–25.56	–27.32
  2112a(2121, 2121, 2112, 1212)	222211	–22.69	–24.47
  2112a(2121, 2121, 2112, 1221)	222221	–21.83	–23.55
  2112b(1221, 2112, 1212, 1212)	–211111w	–19.00	–20.75
  2112b(1221, 2112, 1212, 1221)	–211121	–28.18	–29.99
  2112b(1221, 2112, 2112, 1212)	–211211	–24.55	–26.13
  2112b(1221, 2112, 2112, 1221)	–211221s	–31.28	–33.18
  2112b(1221, 2121, 1212, 1212)	–212111	–24.61	–26.66
  2112b(1221, 2121, 1212, 1221)	–212121s	–29.69	–31.64
  2112b(1221, 2121, 2112, 1212)	–212211	–26.60	–28.23
  2112b(1221, 2121, 2112, 1221)	–212221s	–29.69	–31.74
  2112b(2121, 2112, 1212, 1212)	–221111	–23.06	–24.94
  2112b(2121, 2112, 1212, 1221)	–221121s	–29.26	–30.95
  2112b(2121, 2112, 2112, 1212)	–221211	–24.80	–26.39
  2112b(2121, 2112, 2112, 1221)	–221221	–28.27	–30.09
  2112b(2121, 2121, 1212, 1212)	–222111	–23.58	–25.58
  2112b(2121, 2121, 1212, 1221)	–222121	–25.98	–27.74
  2112b(2121, 2121, 2112, 1212)	–222211	–22.39	–24.04
  2112b(2121, 2121, 2112, 1221)	–222221	–21.95	–23.71
  2121(2112, 1221, 2112, 2112)	211112	–27.01	–28.92
  2121(2112, 1221, 2112, 2121)	211122s	–31.41	–33.24
  2121(2112, 1221, 2121, 2112)	211212s	–30.65	–32.58
  2121(2112, 1221, 2121, 2121)	211222	–27.45	–29.25
  2121(2112, 2121, 2112, 2112)	212112	–24.63	–26.40
  2121(2112, 2121, 2112, 2121)	212122	–25.73	–27.51
  2121(2112, 2121, 2121, 2112)	212212	–25.50	–27.28
  2121(2112, 2121, 2121, 2121)	212222w	–18.96	–20.64
  2121(2121, 1221, 2112, 2112)	221112s	–31.32	–33.20
  2121(2121, 1221, 2112, 2121)	221122s	–30.93	–32.81
  2121(2121, 1221, 2121, 2112)	221212s	–30.60	–32.37
  2121(2121, 1221, 2121, 2121)	221222	–23.71	–25.49
  2121(2121, 2121, 2112, 2112)	222112	–25.27	–27.03
  2121(2121, 2121, 2112, 2121)	222122	–22.00	–23.72
  2121(2121, 2121, 2121, 2112)	222212	–22.11	–23.94
  2121(2121, 2121, 2121, 2121)	222222w	–12.37	–14.06

  It could be found that the values of hydrogen bonding energies are distributed between –7.22 and –35.00 kJ·mol^–1 in B97D and –9.98 and –36.67 kJ·mol^–1 in M052X. In addition, we use two significant thresholds to distinguish those values, that is, –21.00 and –29.00 kJ⋅mol^–1 in the B97D, and –22.00 and –30.00 kJ·mol^–1 in the M052X. We assume that all 74 sub-grouped hydrogen bonds are divided into three classes: weak (labeled superscript 'w'), normal and strong (labeled superscript 's'). To describe concepts in detail, we define the value of bonding energy of each strong hydrogen bond is smaller than –29.00 kJ·mol^–1 in B97D and –30.00 kJ·mol^–1 in the M052X, weak hydrogen bond is larger than –21.00 kJ·mol^–1 in B97D and –22.00 kJ·mol^–1 in M052X and others are normal hydrogen bonds. Then what we can conclude by using the classification from the above is that there are 20 strong, 44 normal and 10 weak hydrogen bonds.

  From these 74 sub-grouped hydrogen bonds labeled by new index, it is obviously rational to find that both donor and acceptor fragments share identical value of three independent digits called same-digital-array, usually perform poor cooperative effect among weak and normal hydrogen bonds, e.g. 111111, 222222 or 111211, which owns the lowest bonding energy compared with others. Therefore, we could declaim that the existence of donor or acceptor fragment expressed by same-digital-array is the necessary condition to judge poor cooperative effect. On the other hand, there are none same-digital-array fragments existing in any strong hydrogen bond. We could also make the conclusion that the existence of donor or acceptor fragment expressed by different-digital-array is the necessary condition to judge strong cooperative effect. Furthermore, the beginning 16 digital sequences listed in Table 2 belong to the central hydrogen bond 1212, and their average value can be regarded as the hydrogen bond energy of 1212. Similarly, the energies of other central hydrogen bonds can also be got, and those for 1212, 1221, 2112^a, 2112^b, 2121 are 23.60, 26.82, 25.76, 25.81 and 25.60 kJ·mol^-1 gained by B97D, 25.28, 28.67, 27.51, 27.61 and 27.40 kJ·mol^-1 by M052X. Thus, the central hydrogen bond of 1221 is indeed stronger than the others, but the 74 sub-grouped hydrogen bonds labeled by new index can perfectly distinguish the effect coming from the environment even within the same central hydrogen bond.

  4. CONCLUSION

  In this paper, a series of cage water clusters was randomly generated. For the same cage water cluster, when n(1221) values are different, the energy intervals of the hydrogen bond networks overlap, so the thermodynamic stability of isomers could not be distinguished only by the value of n(1221).

  In order to obtain bonding energy of each sub-type, cluster conformation was got from selected random samples to establish equation group. After a few iterations, the hydrogen bonding energy solution of 74 sub-grouped hydrogen bonds was got and passed matching test. Besides, we also tested 5¹²6⁸ samples to confirm the generality of the solution. In order to relate bonding energy with six water molecular cluster modules, a simpler six-digit index based on preceding Znamenskiy's work was imported to classify the 74 sub-grouped hydrogen bonds. This improvement effectively explains some regularity contained in hydrogen binding energy distribution, e.g. same-digital-array donor or acceptor fragment often represents poor cooperative effect, while different-digital-array donor or acceptor fragment indicates strong cooperative effect. The above simplified method helps us not only get rid of complex theoretical calculation analysis but also focus on specific cooperative effect of different sub-grouped hydrogen bonds. What's more, 74 sub-grouped hydrogen bonds are classified into 1212, 1221, 2112 and 2121 hydrogen bond types, and it is found that the hydrogen bond type with the strongest energy is 1221, and the sub-grouped hydrogen bond with the strongest energy is 221121. Generally speaking, our study could be applied to more changeable polyhedral cluster hydrogen bonding networks and the influence determined by complete next-nearest neighbor configuration will be discussed in future.

  
  
• 1. [1]

     Desiraju, G. R.; Steiner, T. The Weak Hydrogen Bond in Structural Chemistry and Biology. Oxford University Press, Oxford 1999.

  2. [2]

     Steiner, T. Hydrogen bonds from water molecules to aromatic acceptors in very high-resolution protein crystal structures. Biophy. Chem 2002, 95, 195−201. doi: 10.1016/S0301-4622(01)00256-3

  3. [3]

     Xantheas, S. S. Cooperativity and hydrogen bonding network in water clusters. Chem. Phys. 2000, 258, 225−231. doi: 10.1016/S0301-0104(00)00189-0

  4. [4]

     Znamenskiy, V. S.; Green, M. E. Quantum calculations on hydrogen bonds in certain water clusters show cooperative effects. J. Chem. Theo. Comput. 2007, 3, 103−114. doi: 10.1021/ct600139d

  5. [5]

     Yoo, S.; Aprà, E.; Zeng, X. C.; Xantheas, S. S. High-level ab initio electronic structure calculations of water clusters (H₂O)₁₆ and (H₂O)₁₇: a new global minimum for (H₂O)₁₆. J. Phys. Chem. Lett. 2014, 1, 3122−3127.

  6. [6]

     Shields, R. M.; Temelso, B.; Archer, K. A.; Morrell, T. E.; Shields, G. C. Accurate predictions of water cluster formation, (H₂O)_n (n = 2~10). J. Phys. Chem. A 2010, 114, 11725−11737.

  7. [7]

     Satya, B.; Soohaeng, Y.; Aprà, E.; Xantheas, S. S.; Zeng, X. C. Lowest-energy structures of water clusters (H₂O)₁₁ and (H₂O)₁₃. J. Phys. Chem. A 2006, 110, 11781−11784. doi: 10.1021/jp0655726

  8. [8]

     Temelso, B.; Archer, K. A.; Shields, G. C. Benchmark structures and binding energies of small water clusters with anharmonicity corrections. J. Phys. Chem. A 2011, 115, 12034−12046. doi: 10.1021/jp2069489

  9. [9]

     Hodges, M. P.; Stone, A. J.; Xantheas, S. S. Contribution of many-body terms to the energy for small water clusters: a comparison of ab initio calculations and accurate model potentials. J. Phys. Chem. A 1997, 101, 9163−9168. doi: 10.1021/jp9716851

  10. [10]

      Akase, D.; Aida, M. Distribution of topologically distinct isomers of water clusters and dipole moments of constituent water molecules at finite atmospheric temperatures. J. Phys. Chem. A 2014, 118, 7911−7924. doi: 10.1021/jp504854f

  11. [11]

      Sloan, D. E. Fundamental principles and applications of natural gas hydrates. Nature 2003, 426, 353−363. doi: 10.1038/nature02135

  12. [12]

      Kuo, J. L.; Coe, J. V.; Singer, S. J.; Band, Y. B.; Ojamäe, L. On the use of graph invariants for efficiently generating hydrogen bond topologies and predicting physical properties of water clusters and ice. J. Chem. Phys. 2001, 114, 2527−2540. doi: 10.1063/1.1336804

  13. [13]

      Anick, D. J. Polyhedral water clusters, Ⅰ: formal consequences of the ice rules. J. Mol. Struc-Theochem. 2002, 587, 87−96. doi: 10.1016/S0166-1280(02)00101-X

  14. [14]

      Anick, D. J. Polyhedral water clusters, Ⅱ: correlations of connectivity parameters with electronic energy and hydrogen bond lengths. J. Mol. Struc-Theochem. 2002, 587, 97−110. doi: 10.1016/S0166-1280(02)00100-8

  15. [15]

      Kirov, M. V.; Fanourgakis, G. S.; Xantheas, S. S. Identifying the most stable networks in polyhedral water clusters. Chem. Phys. Lett. 2008, 461, 180−188. doi: 10.1016/j.cplett.2008.04.079

  16. [16]

      Kuo, J. L.; Ciobanu, C. V.; Ojamäe, L.; Shavitt, I.; Singer, S. J. Short H-bonds and spontaneous self-dissociation in (H₂O)₂₀: effects of h-bond topology. J. Chem. Phys. 2003, 118, 3583−3588. doi: 10.1063/1.1538240

  17. [17]

      Anick, D. J. Application of database methods to the prediction of B3LYP-optimized polyhedral water cluster geometries and electronic energies. J. Chem. Phys. 2003, 119, 12442−12456. doi: 10.1063/1.1625631

  18. [18]

      Anick, D. J. O–H stretch modes of dodecahedral water clusters: a statistical ab initio study. J. Phys. Chem. A 2006, 110, 5135−5143. doi: 10.1021/jp055632s

  19. [19]

      Iwata, S. Dispersion energy evaluated by using locally projected occupied and excited molecular orbitals for molecular interaction. J. Chem. Phys. 2011, 135, 094101−12. doi: 10.1063/1.3629777

  20. [20]

      Iwata, S.; Bandyopadhyay, P.; Xantheas, S. S. Cooperative roles of charge transfer and dispersionterms in hydrogen-bonded networks of (H₂O)_n, n = 6, 11, and 16. J. Phys. Chem. A 2013, 117, 6641−6651. doi: 10.1021/jp403837z

  21. [21]

      Iwata, S. Analysis of hydrogen bond energies and hydrogen bonded networks in water clusters (H₂O)₂₀ and (H₂O)₂₅ using the charge-transfer and dispersion terms. Phys. Chem. Chem. Phys. 2014, 16, 11310−11317. doi: 10.1039/C4CP01204F

  22. [22]

      Iwata, S.; Akase, D.; Aida, M.; Xantheas, S. S. Electronic origin of dependence of hydrogen bond strengths on next-nearest-neighbor hydrogen bonds in polyhedral water clusters (H₂O)_n; n = 8, 20 and 24. Phys. Chem. Chem. Phys. 2016, 18, 19746−19756. doi: 10.1039/C6CP02487D

  23. [23]

      Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. Comput. Chem. 2006, 27, 1787−1799. doi: 10.1002/jcc.20495

  24. [24]

      Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A. Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Keith, T.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, O.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Revision D. 01. Gaussian, Inc., Wallingford CT 2013.

  25. [25]

      Tang, L.; Shi, R.; Su, Y.; Zhao. J. J. Structures, stabilities and spectra properties of fused CH₄ endohedral water cage (CH₄)_m(H₂O)_n clusters from DFT-D methods. J. Phys. Chem. A 2015, 119, 10971−10979. doi: 10.1021/acs.jpca.5b08073

  26. [26]

      Yuan, D.; Li, Y.; Ni, Z.; Pulay, P.; Li, W.; Li, S. Benchmark relative energies for large water clusters with the generalized energy-based fragmentation method. J. Chem. Theory. Comput. 2017, 13, 2696−2704. doi: 10.1021/acs.jctc.7b00284

  27. [27]

      Mcdonald, S.; Ojamae, L.; Singer, S. J. Graph theoretical generation and analysis of hydrogen-bonded structures with applications to the neutral and protonated water cube and dodecahedral clusters. J. Phys. Chem. A 1998, 102, 2824−2832. doi: 10.1021/jp9803539

  28. [28]

      Kirov, M. V. Atlas of optimal proton configurations of water clusters in the form of gas hydrate cavities. J. Struct. Chem. 2002, 43, 790−797. doi: 10.1023/A:1022825324222

  29. [29]

      Yoo, S.; Kirov, M. V.; Xantheas, S. S. Low-energy networks of the T-cage (H₂O)₂₄, cluster and their use in constructing periodic unit cells of the structure I (SI) hydrate lattice. J. Am. Chem. Soc. 2009, 131, 7564−7566. doi: 10.1021/ja9011222

• 

  Figure 1  Model construction of 5¹² cage water cluster

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  Figure 2  Binding energies. a) and b) are 5¹² cluster isomers, c) and d) are 5¹²6² cluster isomers, and e) and f) are 4³5⁶6³ cluster isomers. a), c) and e) are calculated by B97D. b), d) and f) are calculated by M052X

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  Figure 3  Nine classes of the hydrogen bonds considering the nearest neighbor water molecules. The water molecules labeled are 21- or 12-type water molecules, 21-type water molecules and 12-type water molecules

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  Figure 4  Energy comparison of hydrogen bonds between B97D and M052X methods. (a) 5¹² cluster isomers, (b) 5¹²6² cluster isomers and (c) 4³5⁶6³ cluster isomers. The polyline with the square indicates the energies of 74 sub-grouped hydrogen bonds in the B97D method, and that with circle indicates the energies of 74 sub-grouped hydrogen bonds in M052X method

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  Figure 5  Hydrogen bonding network fitting curve. (a) and (b) are 5¹², (c) and (d) are 5¹²6² , (e) and (f) are 4³5⁶6³. The energies of (a), (c) and (e) are calculated by B97D method. The energies of (b), (d) and (f) are calculated by M052X method. The black curve indicates the calculated energies of selected clusters, while the red one shows the fitting energies using the solution of corresponding sub-grouped hydrogen bonds' energies

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  Figure 6  Energy comparison of hydrogen bonds between 5¹², 5¹²6² and 4³5⁶6³ single cage clusters. (a) B97D and (b) M052X methods. The polyline with the square indicates the energies of 74 sub-grouped hydrogen bonds of 5¹², the polyline with the circle is the energies of 74 sub-grouped hydrogen bonds of 5¹²6², and the polygonal line with the triangle shows the energies of 74 sub-grouped hydrogen bonds of 4³5⁶6³

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  Figure 7  Hydrogen bonding network fitting of 5¹²6⁸ cluster isomers. The energies of (a) are calculated by B97D and those of (b) by M052X. The black curve indicates the original calculated energies of selected clusters, while the red curve shows the fitting energies using the solution of corresponding sub-grouped hydrogen bonds' energies

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  Figure 8  Improved index descriptor of sub-grouped hydrogen bonds. The numbers labeled in the figure explain the order of this six-digit index. The donor molecule of central water pair acts as the first digit, and the adjacent environmental water molecules are the second and third digit widdershins. The acceptor molecule of central water pair acts as the sixth digit, and its adjacent environmental water molecules as the fourth, and the fifth digit is counterclockwise.

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  Table 1.  The Number of Randomly Generated 5¹², 5¹²6², 4³5⁶6³ and 5¹²6⁸ Cage Water Clusters

  Cage type	n(1221)
  	0	1	2	3	4	5	6	7	8	9	10	11	Sum
  512	6	51	150	196	128	54	16	1	-	-	-	-	602
  51262	2	27	74	144	130	78	29	7	1	-	-	-	492
  435663	2	51	104	156	124	53	10	1	-	-	-	-	501
  51268	-	2	7	42	81	158	104	77	38	11	2	1	523

  下载: 导出CSV

  Table 2.  Numeric Sequence of 74 Subtype Hydrogen Bonds and Energies of Sub-grouped Hydrogen Bonds Labeled by Improved Index

  Subtype hydrogen bond array	Digital sequence	EH/kJ·mol-1
  		B97D	M052X
  1212(1212, 1212, 1212, 1212)	111111w	–10.33	–11.97
  1212(1212, 1212, 1212, 1221)	111121	–24.34	–26.41
  1212(1212, 1212, 2112, 1212)	111211w	–19.94	–21.87
  1212(1212, 1212, 2112, 1221)	111221	–25.37	–27.83
  1212(1212, 2112, 1212, 1212)	112111w	–16.43	–17.58
  1212(1212, 2112, 1212, 1221)	112121	–25.90	–27.92
  1212(1212, 2112, 2112, 1212)	112211	–23.25	–24.73
  1212(1212, 2112, 2112, 1221)	112221	–28.19	–30.00
  1212(2112, 1212, 1212, 1212)	121111w	–18.87	–20.01
  1212(2112, 1212, 1212, 1221)	121121s	–29.13	–30.76
  1212(2112, 1212, 2112, 1212)	121211	–23.71	–25.03
  1212(2112, 1212, 2112, 1221)	121221s	–29.22	–30.76
  1212(2112, 2112, 1212, 1212)	122111	–22.16	–23.58
  1212(2112, 2112, 1212, 1221)	122121	–28.96	–30.94
  1212(2112, 2112, 2112, 1212)	122211	–25.13	–26.74
  1212(2112, 2112, 2112, 1221)	122221	–26.68	–28.38
  1221(1212, 1212, 2112, 2112)	111112	–23.78	–25.36
  1221(1212, 1212, 2112, 2121)	111122	–27.40	–28.83
  1221(1212, 1212, 2121, 2121)	111222	–27.03	–28.76
  1221(1212, 2112, 2112, 2112)	112112s	–29.82	–31.95
  1221(1212, 2112, 2112, 2121)	112122s	–29.72	–31.75
  1221(1212, 2112, 2121, 2112)	112212s	–31.05	–33.07
  1221(1212, 2112, 2121, 2121)	112222	–25.90	–28.08
  1221(2112, 2112, 2112, 2112)	122112s	–29.12	–30.98
  1221(2112, 2112, 2112, 2121)	122122	–25.93	–27.66
  1221(2112, 2112, 2121, 2121)	122222w	–18.46	–20.23
  2112a(2112, 1221, 1212, 1212)	211111w	–19.68	–21.48
  2112a(2112, 1221, 1212, 1221)	211121s	–28.81	–30.39
  2112a(2112, 1221, 2112, 1212)	211211	–24.06	–25.82
  2112a(2112, 1221, 2112, 1221)	211221s	–31.39	–33.30
  2112a(2112, 2121, 1212, 1212)	212111w	–20.54	–22.41
  2112a(2112, 2121, 1212, 1221)	212121	–26.83	–28.62
  2112a(2112, 2121, 2112, 1212)	212211	–23.36	–25.04
  2112a(2112, 2121, 2112, 1221)	212221	–26.94	–28.85
  2112a(2121, 1221, 1212, 1212)	221111	–26.99	–28.81
  2112a(2121, 1221, 1212, 1221)	221121s	–31.91	–33.30
  2112a(2121, 1221, 2112, 1212)	221211s	–27.77	–29.34
  2112a(2121, 1221, 2112, 1221)	221221s	–30.51	–32.07
  2112a(2121, 2121, 1212, 1212)	222111	–23.35	–25.33
  2112a(2121, 2121, 1212, 1221)	222121	–25.56	–27.32
  2112a(2121, 2121, 2112, 1212)	222211	–22.69	–24.47
  2112a(2121, 2121, 2112, 1221)	222221	–21.83	–23.55
  2112b(1221, 2112, 1212, 1212)	–211111w	–19.00	–20.75
  2112b(1221, 2112, 1212, 1221)	–211121	–28.18	–29.99
  2112b(1221, 2112, 2112, 1212)	–211211	–24.55	–26.13
  2112b(1221, 2112, 2112, 1221)	–211221s	–31.28	–33.18
  2112b(1221, 2121, 1212, 1212)	–212111	–24.61	–26.66
  2112b(1221, 2121, 1212, 1221)	–212121s	–29.69	–31.64
  2112b(1221, 2121, 2112, 1212)	–212211	–26.60	–28.23
  2112b(1221, 2121, 2112, 1221)	–212221s	–29.69	–31.74
  2112b(2121, 2112, 1212, 1212)	–221111	–23.06	–24.94
  2112b(2121, 2112, 1212, 1221)	–221121s	–29.26	–30.95
  2112b(2121, 2112, 2112, 1212)	–221211	–24.80	–26.39
  2112b(2121, 2112, 2112, 1221)	–221221	–28.27	–30.09
  2112b(2121, 2121, 1212, 1212)	–222111	–23.58	–25.58
  2112b(2121, 2121, 1212, 1221)	–222121	–25.98	–27.74
  2112b(2121, 2121, 2112, 1212)	–222211	–22.39	–24.04
  2112b(2121, 2121, 2112, 1221)	–222221	–21.95	–23.71
  2121(2112, 1221, 2112, 2112)	211112	–27.01	–28.92
  2121(2112, 1221, 2112, 2121)	211122s	–31.41	–33.24
  2121(2112, 1221, 2121, 2112)	211212s	–30.65	–32.58
  2121(2112, 1221, 2121, 2121)	211222	–27.45	–29.25
  2121(2112, 2121, 2112, 2112)	212112	–24.63	–26.40
  2121(2112, 2121, 2112, 2121)	212122	–25.73	–27.51
  2121(2112, 2121, 2121, 2112)	212212	–25.50	–27.28
  2121(2112, 2121, 2121, 2121)	212222w	–18.96	–20.64
  2121(2121, 1221, 2112, 2112)	221112s	–31.32	–33.20
  2121(2121, 1221, 2112, 2121)	221122s	–30.93	–32.81
  2121(2121, 1221, 2121, 2112)	221212s	–30.60	–32.37
  2121(2121, 1221, 2121, 2121)	221222	–23.71	–25.49
  2121(2121, 2121, 2112, 2112)	222112	–25.27	–27.03
  2121(2121, 2121, 2112, 2121)	222122	–22.00	–23.72
  2121(2121, 2121, 2121, 2112)	222212	–22.11	–23.94
  2121(2121, 2121, 2121, 2121)	222222w	–12.37	–14.06

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