Graph theoretical enumeration of topology-distinct structures for ...

Graph theoretical enumeration of topology-distinct structures for hydrogen fluoride clusters (HF) n (n ≤ 6) Mahmutjan Jelil and Alimjan Abaydulla Citation: The Journal of Chemical Physics 143, 044301 (2015); doi: 10.1063/1.4926939 View online: http://dx.doi.org/10.1063/1.4926939 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/143/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in First principle structural determination of (B2O3) n (n = 1–6) clusters: From planar to cage J. Chem. Phys. 138, 094312 (2013); 10.1063/1.4793707 Hydrogen bond cooperativity and electron delocalization in hydrogen fluoride clusters J. Chem. Phys. 114, 5552 (2001); 10.1063/1.1351878 A theoretical study of Si 4 H 2 cluster with ab initio and density functional theory methods J. Chem. Phys. 114, 1278 (2001); 10.1063/1.1316032 Structure and vibrational spectra of H + (H 2 O) 8 : Is the excess proton in a symmetrical hydrogen bond? J. Chem. Phys. 113, 5321 (2000); 10.1063/1.1288918 Ionization energies of hyperlithiated and electronically segregated isomers of Li n (OH) n−1 (n=2–5) clusters J. Chem. Phys. 113, 1821 (2000); 10.1063/1.481986

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THE JOURNAL OF CHEMICAL PHYSICS 143, 044301 (2015)

Graph theoretical enumeration of topology-distinct structures for hydrogen fluoride clusters (HF)n (n ≤ 6) Mahmutjan Jelila) and Alimjan Abaydulla Department of Chemistry and Environmental Science, Kashgar University, 029, Xueyuan Road, Kashgar, Xinjiang 844008, China

(Received 28 April 2015; accepted 6 July 2015; published online 22 July 2015) A graph theoretical procedure to generate all the possible topology-distinct structures for hydrogen fluoride (HF) clusters is presented in this work. The hydrogen bond matrix is defined and used to enumerate the topology-distinct structures of hydrogen fluoride (HF)n (n = 2–8) clusters. From close investigation of the structural patterns obtained, several restrictions that should be satisfied for a structure of the HF clusters to be stable are found. The corresponding digraphs of generated hydrogen bond matrices are used as the theoretical framework to obtain all the topology-distinct local minima for (HF)n (n ≤ 6), at the level of MP2/6-31G**(d, p) of ab initio MO method and B3LYP/6-31G**(d, p) of density functional theory method. For HF clusters up to tetramers, the local minimum structures that we generated are same as those in the literature. For HF pentamers and hexamers, we found some new local minima structures which had not been obtained previously. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4926939]

I. INTRODUCTION

Hydrogen fluoride (HF) is a small molecule with a large tendency to self-associate due to hydrogen bonding in the gas phase. The chemical and biological importance of hydrogen bonded systems has led to a considerable experimental and theoretical interest in this phenomenon for many years.1,2 In these investigations, HF is a key molecule due to its small size and the strength of its hydrogen bond. From 1968 till now, (HF)n clusters have been the subject of study for more than four decades.3 A major reason for this interest is the fact that HF is perhaps the simplest strongly hydrogen bonding system. Though hydrogen bonds are weaker than covalent bonds, they can form long-lived structures for HF clusters. Erenow there have been a lot of theoretical4–14,20,22,26,33–36 and experimental studies15–19 to search on the determination of the potential energy surface,8–14,20 stable isomers, global minima, vibrational spectra, dissociation energy,3,15–20 and other thermodynamics properties.7,14,20 Most of the recent theoretical studies have used Monte Carlo sampling or Monte Carlo-based simulated annealing6–8,14 procedures. Although various aspects of HF clusters have been studied thoroughly in previous works, none of those studies have been devoted to elucidating all of the possible structures systematically. It is not trivial to present all of the possible structures nor to claim that a structure is indeed the global minimum. A graph is a mathematical structure which is related to the topology of a given molecule or molecular cluster. Graph theory has been used successfully and extensively to represent various properties of molecules, such as thermodynamic properties of alkenes21 and π-electron energies of aromatic hydrocarbons.22 A graph theoretical technique has been introduced to generate neutral and H2O cubes and dodecahedral clusters for (H2O)8, (H2O)20, H+(H2O)8, and a)Electronic mail: [email protected]

0021-9606/2015/143(4)/044301/6/$30.00

H+(H2O)20.23 Some other previous works have used graph theory successfully to enumerate hydrogen bonding patterns of neutral water clusters and protonated water clusters, which correspond to all the topology-distinct structures of neutral water clusters24–26 and protonated water clusters.27,28 In this study, we developed an algorithm to enumerate all possible topology-distinct structures of HF clusters. For a rather small HF cluster, we can show by “intuition” all possible structures in each of which the topology is different, i.e., the hydrogen bonding network is different. It is difficult, however, to use “intuition” for a medium-sized HF cluster. We show here an algorithm to get the topology-distinct structures of HF clusters by a computer. The set of the topology-distinct structures for HF clusters serves as the basic theoretical framework for locating stationary structures. Possible local minima for the HF clusters up to (HF)6 are presented, the geometries of which are optimized by means of ab initio MO method and density functional theory (DFT) calculation.

II. THEORY AND COMPUTATIONAL DETAILS A. Graph and adjacency matrix

A graph is a set of vertices and edges. A graph has the corresponding matrix representation. For a graph with n vertices, the adjacency matrix A is the nth order square matrix, whose element ai j is equal to 1 for a pair of vertices i and j which are connected by an edge, and 0 otherwise.

B. Digraph and hydrogen bond (H-B) matrix equivalent of HF cluster

The important feature of the hydrogen bond is that it possesses the direction. When a direction is assigned to every edge of a graph, a graph is termed a directed graph (digraph);

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J. Chem. Phys. 143, 044301 (2015)

FIG. 2. A structure which we do not regard as hydrogen bonded. FIG. 1. Structure of a hydrogen-bonded hydrogen fluoride cluster with the equivalent rooted digraph and the corresponding hydrogen-bond matrix.

i.e., a digraph is a set of vertices and arrows. To represent the feature of hydrogen bonding of a HF cluster, we use a directed graph, or digraph. Here, vertices correspond to HF, and arrows correspond to hydrogen bonds from proton-donor to proton-acceptor (Fig. 1). A digraph has the corresponding matrix representation. For a digraph with n vertices, the matrix H is the nth order square matrix, whose element hi j is equal to 1 for an arrow directed from vertex i to vertex j, and 0 otherwise. We call this matrix the H-B matrix. There are several conditions for a digraph to be equivalent to a HF cluster. In a HF cluster, one HF can accept three protons from other hydrogen fluoride molecules, and can share one proton with other two hydrogen fluoride molecules. In addition, in a HF cluster, all of the neutral HF molecules should be connected by hydrogen bonds.

Step 1-3. Any vertex which is found to be connected to the first vertex is marked “connected.” After all the vertices connected to the first vertex are examined, the first vertex is marked “examined.” These two operations are repeated until all the “connected” vertices turn “examined.” Then, if the graph contains any “not-examined” vertex, it is not a connected graph. Any adjacency matrix which does not fulfill condition G-2 is removed here. Step 1-4. If a different numbering is used for the vertices in a graph, a different adjacency matrix can be written which corresponds to the same graph. Here, we define the unique adjacency matrix for a graph, by means of an n(n − 1)/2-figure binary number which is introduced in Step 1-1. We adopt the largest binary number as the representative of all other numbers which correspond to the same graph. The adjacency matrix, which corresponds to the largest n(n − 1)/2-figure binary number, is called the representation matrix here. 2. Counting up digraphs (DG)

C. Counting up the H-B matrices

All possible structures that are topology-distinct can be obtained by means of H-B matrix, i.e., by counting up all possible digraphs with the above-mentioned conditions. To this end, we take the following two steps. First, we neglect the direction of the arrow and count up all non-directed graphs; and second, we generate the digraphs from the graphs. 1. Counting up graphs

We count up graphs that fulfill the following two conditions, which come from the above-mentioned conditions for HF clusters. G-1: The maximum number of the edges connected to a vertex is 5. G-2: The graph is connected. First, we generate all the different adjacency matrices (Step 1-1); second, we remove those which do not fulfill G-1 (Step 1-2); third, we remove those which do not fulfill G-2 (Step 1-3); and finally, we remove duplications (Step 1-4). The detailed procedure here follows the one for counting up graphs for neutral water clusters.24 Step 1-1. In an adjacency matrix of a non directed graph, an element ai j is equivalent to a j i . For graphs with n vertices, 2n(n−1)/2 different matrices exist. To generate a complete set of the adjacency matrices, we generate n(n − 1)/2-figure binary numbers and assign ai j (1 or 0) to each place in order. Step 1-2. The sum of all the elements of the ith row of an adjacent matrix gives the number of the edges connected to the ith vertex. Any adjacency matrix which contains at least one row with the elements whose sum is larger than five does not fulfill condition G-1, and is removed.

The three conditions listed below must be fulfilled for a digraph to be equivalent to a HF cluster. DG-1: The maximum number of arrows directed toward the vertex is 3 and the maximum number of arrows directed from the vertex is 2. DG-2: When an arrow is directed from the vertex i toward the vertex j, an arrow directed from the vertex j toward the vertex i does not exist. DG-3: The digraph must be connected. Condition DG-1 indicates that in a HF cluster a HF can accept three protons from other HF molecules, and can share one proton with other two hydrogen fluoride molecules. Condition DG-2 indicates that no structure shown in Fig. 2 can exist in a HF cluster. We use these three conditions to count up digraphs by means of an adjacency (representation) matrix. First, we change all edges in a representation matrix to arrows (Step 2-1); second, we remove those which do not fulfill condition DG-1 (Step 2-2); and finally, we remove duplications (Step 2-3). Note that condition DG-2 is fulfilled in Step 2-1 as will be shown below, and that condition DG-3 is already fulfilled in the previous step of counting up the graphs. Step 2-1. There are 2m different ways to generate the directed adjacency matrices for a graph with m edges. We make use of the binary number of the representation matrix here to generate all of them. In this step, condition DG-2 is automatically fulfilled, since hi j is not equal to h j i . Step 2-2. The sum of the elements in the ith row of a directed adjacency matrix gives the number of arrows directed from the ith vertex. The sum of the elements in the jth column of a directed adjacency matrix gives the number of arrows directed to the jth vertex. Any matrix which does not fulfill condition DG-1 is removed here.

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J. Chem. Phys. 143, 044301 (2015)

TABLE I. Numbers of graphs, digraphs, and restrictive rooted digraphs up to 8 vertices and numbers of local minima of HF clusters (HF)n (n = 2–5). Vertexa

2

3

4

5

6

7

8

Graphb

1 1 1 1 1

2 5 2 2 5 2

7 27 2 6 22 2

21 247 4 21 161 9

112 3272 ... 78 1406 51

697 55 005 ... 353 14 241 320

6 386 1 104 449 ... 1 929 164 461 2 443

2

2

4

8

Digraphc Local minimad Restrictive graphe Restrictive digraph

(1) (1) (2)

Local minimaf a The

number of the vertices. number of the graphs generated. c The number of the digraphs generated. d The number of the local minima searched from the generated digraph. e (1) The number of restrictive graph and restrictive digraph (see Sec. II D 1) according to the restrictions RDG-1 and RDG-2 generated, (2) the number of restrictive digraph according to restrictions RDG-1, RDG-2, and RDG-3. f The number of the local minima searched from the generated restrictive rooted digraph. b The

Step 2-3. The directed adjacency matrices generated after the previous steps include duplications. To remove duplications, the same procedure as in the last step in the previous Step 1-4 is used here. Thus, we finally obtain the representative directed adjacency matrix, i.e., the H-B matrix. D. Restrictive digraph

According to the steps above, we enumerated the graphs and digraphs of HF clusters (HF)n (n = 2-8) (Table I). Various initial geometries for a HF cluster corresponding to these possible digraphs (for n = 2–5) were constructed and each of the trial geometries was optimized by means of ab initio MO method. As seen in Table I, although the number of possible topology-distinct digraphs increases rapidly with cluster size, the number of stable topology distinct structures is very limited; e.g., only 2 local minimum out of 5 digraphs for trimer and 2 local minimum out of 22 digraphs are found and tetramer, and only 4 local minimum out of 247 digraphs are found for pentamer, respectively. Making use of these results as well as other previous works,4–16,20 we extracted several structural rules and found the restrictions that stable structures should fulfill as below. We remove those H-B matrices of corresponding H-B patterns which do not fulfill these restrictions from all possible topology distinct H-B patterns and generate the restrictive digraphs. The corresponding H-B patterns of restrictive digraphs supply ideal initial structures for searching local minima of medium size (HF)n clusters (i.e., n = 5–8) systematically. 1. Counting up restrictive digraph (RDG)s

A digraph which fulfills the following four restrictions is equivalent to a restrictive HF cluster. RDG-1: The maximum number of arrows towards the vertex is 2. RDG-2: The number of arrows directed from the vertex is 1 or 2, not equal to zero.

RDG-3: When two arrows are accepted from a neighbor vertex, a vertex which accepts two arrows from the vertex can donate two arrows to other vertex, otherwise it cannot donate two arrows to the other vertex. We have written our own program in FORTRAN to enumerate all the possible H-B matrices and restrictive H-B matrix of every size of HF clusters according to the algorithm described above. Then, generated all the possible digraphs and restrictive digraphs which are topologically distinguishable for each of (HF)n (n = 2–8) clusters from H-B matrices of every size of HF clusters by means of our novel Python program and free graph visualization software, GraphViz 2.38.29 E. Stable structure corresponding to a H-B matrix

We construct various trial initial geometries for a HF cluster with a hydrogen bonding topology corresponding to a digraph by use of a graphical tool of GaussView 5.030 and Spartan ’08.31 For each of the trial geometries, we perform the geometry optimization by means of the ab initio MO method at the HF/6-31G∗∗(d, p), MP2/6-31G∗∗(d, p), and DFT methods at the B3LYP/6-31G∗∗(d, p) level of theory. In this way, we obtain all the optimized structures which are topologically distinguishable for each of (HF)n (n = 2–6). The program package of Gaussian 0932 is used for the ab initio MO and DFT calculations.

III. RESULT AND DISCUSSION

Using the algorithm and designed program described above, we have enumerated the graphs and the digraphs containing up to 8 vertices, which correspond to the hydrogen fluoride clusters (HF)n (n = 2–8). The numbers of the generated graphs and the generated restricted digraphs according to our algorithm are summarized in Table I. It should be noted here that we dealt with only topology-distinct structures in the current work. We do not take account of any fine structures of HF clusters, such as the direction of a free F–H bond.14 All of the possible topology-distinct structures of the HF clusters for

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J. Chem. Phys. 143, 044301 (2015)

FIG. 3. All of the enumerated possible topology-distinct graphs and digraphs with n vertices for (HF)n (n = 2–4). The designated patterns (2A, 3A, 3B, 4A, and 4B) correspond to the stable structures.

(HF)n (n = 2–4) and the restrictive topology-distinct structures for (HF)n (n = 5–6) are shown in Figs. 3 and 4, respectively. The number of the graphs and the digraphs with 3 vertices were 2 and 5, the number of the graphs and the digraphs with 4 vertices were 7 and 27, respectively. We have found that 2 local minima out of 5 topology-distinct structures of hydrogen fluoride trimers and found that only 2 out of 27 correspond to the stable structures of HF tetramers. The graphs and the digraphs of HF trimers and tetramers are shown in Fig. 3, the optimized structures and the corresponding digraphs of HF

trimers and tetramers are shown in Fig. 5. For HF trimers, the most stable structure corresponded to 3B; the energy difference between 3A and 3B was 8.22 kcal mol−1. For the HF tetramer, the 4-membered cyclic cluster (4B in Fig. 5) is the most stable, and cluster 4A is less stable by 12.4 kcal/mol than 4B. The 4-membered cyclic structures (4B and 4A) are generated from the different identical non-directed graphs (see Fig. 4) and have 4 hydrogen bonds, though the combination in the directions of the hydrogen bonds is different. The obtained structures of HF trimer and tetramer are topologically same as

FIG. 4. Enumerated restrictive digraphs generated from H-B matrices by our own Python program and GraphViz 2.37 program package corresponding to HF pentamer and hexamer. The designated patterns (5A, 5B, 5C, 5D, 6A, 6B, 6C, 6D, 6E, 6F, and 6G) correspond to the stable structures.

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M. Jelil and A. Abaydulla

J. Chem. Phys. 143, 044301 (2015)

FIG. 5. The digraphs and the optimized geometries of HF clusters (HF)n (n = 2–6).

those found by using a Monte Carlo and molecular dynamic simulation,6 Monte Carlo simulated annealing procedure,7 ab intio MO,13,20,34,37 and DFT calculation.37 Compared with the water cluster tetramer, the enumerated numbers of possible topology-distinct hydrogen bonded pattern of HF cluster tetramer are more than that of water cluster tetramer,24,25 but the local minima structures are less than that of water cluster tetramer. From the HF pentamer, the number of the possible topology-distinct H-B patterns increased rapidly with the cluster size (Table I). It is difficult to construct various trial initial geometries and search local minima with these huge numbers of possible H-B patterns. Thus, according to the restrictions 9 restrictive digraphs were generated from 247 possible digraphs for HF pentamers (Fig. 4), and various initial geometries were constructed for a HF pentamer with a HB topology corresponding to each of the above restrictive digraphs, followed by the geometry optimization. We found 4 local minima out of 247 possible digraphs and 9 restrictive digraphs by means of both MP2/6-31G∗∗ and B3LYP/6-31G∗∗ levels of theory (Table I). The optimized structures with MP2/6-31G∗∗ and the corresponding digraphs of four local

minima of HF pentamers are given in Fig. 5. All of the local minima of HF pentamers are in Cs symmetry, except for 5A. The local minimum structure 5D (in Fig. 5), in which two three-membered molecular rings have common HF edges have not been reported in previous works,6,20,34,36,37 was found. The obtained global minimum structure (the five member ring, 5A in Fig. 5) is in C5h symmetry and it is topologically the same as previous work.6,20,34,36,37 The energy differences between 5A and 5B, 5C, 5D were 6.96, 16.70, and 11.95 kcal mol−1, respectively, on the MP2/6-31G∗∗(d, p) level of theory. The most weakly bound minimum was the 3-membered cyclic cluster with two “tails” (see Fig. 5, structure 5C).20 The number of enumerated possible topology-distinct hydrogen bonded pattern of HF cluster pentamer is more than water cluster pentamer,24,25 but number of topology-distinct local minima structures is less than water cluster pentamer.25 According to the restrictions (above section), 51 restrictive digraphs (Fig. 4) were generated for HF cluster hexamer. The corresponding restricted digraphs are shown in Figure 4. Various initial geometries were constructed for a HF hexamer with a HB topology corresponding to each of the above restrictive digraphs, followed by the geometry optimization

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by means of the ab initio MO method and DFT method. We found 8 local minima out of 51 restrictive digraphs at MP2/6-31G∗∗(d, p) and B3LYP/6-31G∗∗(d, p) levels of theory, respectively (Table I). The optimized geometries that are obtained with MP2/6-31G∗∗ and B3LYP/6-31G∗∗ and those corresponding restrictive digraphs are shown in Figure 5. Two new local minima (6G and 6H in Fig. 5) in which a square connected with a triangle structure (6G) by common HF edges or two triangle structure with a “tail” (6H) have common HF edges and with seven hydrogen bonds these had not been reported in previous works.6,20,36,37 The most stable structure was 6A, which is topologically same as the one found by means of Monte Carlo and molecular dynamics simulation by semi classical method,6 ab initio MO calculation,20,34 and DFT calculation.37 The energy differences between 6A, 6B, 6C 6D, 6E, 6F, 6G, and 6H were 5.45, 12.71, 10.00, 12.97, 22.61, 9.43, and 16.07 kcal mol−1, respectively, at MP2/6-31G∗∗(d, p) calculation level of theory. The most weakly bound minimum was 6F, which is 3-membered cyclic cluster with two “tails,” (Fig. 5, 6F) and formed with 6 hydrogen bonds. The number of enumerated possible topology-distinct hydrogen bonded pattern of HF cluster hexamer is three times more than water cluster hexamer,24,25 but number of topology-distinct local minima structures is only 8.

IV. CONCLUSION

The enumeration of digraphs is a useful method to present all the possible topology–distinct hydrogen bond patterns of HF clusters. We extracted the restrictions that a stable structure of a HF cluster should fulfill, and applied the restrictions to find all of the local minima for a HF pentamer and hexamer. In addition, we found some new HF pentamer and hexamer structures. The enumeration method in the frame of graph theory together with the extraction of the restrictions can be used to obtain all possible hydrogen-bonding patterns corresponding to HF clusters. Combination of graph theoretical enumerations with ab initio MO calculations and DFT calculations allows us to find all topology distinct stable structures for HF clusters. It should be emphasized that the numbers and the possible hydrogen-bonding patterns of HF clusters are valid. For HF cluster tetramer to hexamer, we have found that, after optimization, the initial chain structures collapse into cyclic ones. However, a chain like saddle-point structure (one imaginary frequency) is found in tetramer to hexamer cases. The graph theoretical enumeration method guarantees that there cannot be any other patterns. A digraph corresponding to a HF cluster contains the H–bond information that includes not only the two–body pair wise terms but also the three–body and higher terms. The H–bond pattern, as well as the number of the H–bonds, influences the stability of a HF cluster.

J. Chem. Phys. 143, 044301 (2015)

ACKNOWLEDGMENTS

The support of the National Natural Science Foundation of China (Grant No. 21363010) is gratefully acknowledged. We also gratefully acknowledge the support of the Kashgar Teachers College Research Key Program Foundation (Grant No.13-2457). 1Hydrogen

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