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Graph Invariants – a Tool to Analyze Hydrogen Bonding in Ice and Water Clusters

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Graph Invariants – a Tool to Analyze Hydrogen Bonding in Ice and Water Clusters Graph Invariants – A Tool to Analyze Hydrogen Bonding in Ice and Water Clusters DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Jer-Lai Kuo, B.S., M.S. * * * * * The Ohio State University 2003 Dissertation Committee: Approved by Prof. Sherwin J. Singer, Adviser Prof. Anne B. McCoy Adviser Prof. James V. Coe Chemical Physics Program Prof. Eric Herbst ABSTRACT We have studied a wide range of aqueous systems, from the order/disorder in the hy- drogen bond network of ordinary ice, a fundamental problem in ice physics for over 70 years, to properties of water clusters. A common theoretical difficulty in these systems is the enormous number of hydrogen bond arrangements that must be considered in these systems, and for which no systematic treatment has been available. Our analytical method based on graph theory and the introduction of graph invariants provides a means for effi- cient and reliable analysis. Recent investigations of water clusters have demonstrated that graph invariants provide a powerful tool for capturing very complex behavior using only a small number of parameters, and some of our findings have uncovered unknown aspects of the behavior of water clusters. Applications of this analytic method provide insight into the nature of hydrogen bond disorder in ice, and impact our understanding of chemical reactions in water clusters important to environmental chemistry. ii To My Parents iii ACKNOWLEDGMENTS I would like to thank my advisor, Prof. Sherwin J. Singer, for his guidance and support through my graduate study. I also wish to thank Prof. Anne B. McCoy, Prof. James V. Coe and Prof. Eric Herbst for kindly serving on my dissertation examination committee. I owe a debt of thanks to Dr. Cristian V. Ciobanu and Prof. Lars P. Ojamae¨ and Prof. Isa- iah Shavitt have all in some way contribute to this thesis. I would like to acknowledge the Graduate School of the Ohio State University for the award of Presidential Fellowship. All the friends in Ohio State make my life in Columbus wonderful and memorable. My deepest thank goes to my family, through my life they have supported me with their love and encouragement. iv VITA January 12, 1973 . Born - Kinmen, Taiwan 1991-1995 . .B.S. Department of Physics, National Taiwan University, Taiwan 1995-1997 . .M.S. Department of Physics, National Taiwan University, Taiwan 1997-2001 . .Graduate Teaching and Research Associate, The Ohio State University. 2001-present . .Presidential Fellowship, The Ohio State University. PUBLICATIONS Research Publications Jer-Lai Kuo, Jame V. Coe, Sherwin Singer, Yehuda B. Band and Lars Ojamae¨ “On the Use of Graph Invariants for Efficiently Generating Hydrogen Bond Topologies and Predicting Physical Properties of Water Clusters and Ice”. J. Chem. Phys. 114, 2527-2540 (2001) FIELDS OF STUDY Major Field: Chemical Physics v TABLE OF CONTENTS Page Abstract . ii Dedication . iii Acknowledgments . iv Vita . v List of Tables . viii List of Figures . ix Chapters: 1. Introduction . 1 2. The Use of Graph Invariants for Efficiently Generating Hydrogen Bond Topolo- gies and Predicting Physical Properties of Water Clusters . 9 2.1 Graph invariants . 14 2.1.1 Generation of graph invariants . 18 2.1.2 Physical interpretation of graph invariants . 21 2.2 Graph invariants as a tool for enumerating H-bond topologies . 24 2.2.1 Sorting strategy . 26 2.2.2 Performation of sorting algorithm for realistic calculations . 27 2.3 Correlation and prediction of physical properties from H-Bond topology using graphical invariants . 31 ¡ £¦¥¨§ £ 2.3.1 ¢¡¤£¦¥¨§ © and . 31 2.3.2 Using invariants to calculate phase transitions: ¢¡£¥§ £ dodec- ahedral as a dry run . 37 vi 2.4 Discussion . 42 2.5 Appendix: Necessary and sufficient condition for a graphical invariant to be identically zero . 43 3. Graph Invariants for Periodic Systems: Predicting Physical Properties from the Hydrogen Bond Topology of Ice . 45 3.1 A gentle introduction to oriented graphs and graph invariants . 48 3.2 Graph invariants for periodic systems . 55 3.2.1 Graph invariants and space groups . 57 3.2.2 Invariants for arbitrary unit cell choice . 58 3.2.3 An illustration for square ice . 63 3.3 Graph invariants and graph enumeration for Ice-Ih . 69 3.3.1 Invariants for the 8-water orthorhombic unit cell . 69 3.3.2 Enumeration of H-bond arrangements in ice-Ih . 72 3.3.3 Analysis of enumeration results . 76 3.4 Conclusion . 80 3.5 Appendix: Graph invariants of the Orth § cell . 84 4. Effects of H-Bond Topology on Energetics, Structure and Chemistry on water clusters . 87 4.1 Cluster stability and H-bond topology . 89 4.2 Self-dissociation and zwitterionic structures . 91 4.3 Short H-bonds . 95 4.4 Discussion . 99 Bibliography . 100 vii LIST OF TABLES Table Page 2.1 Group of vertex permutations for the triangle graph shown in Fig. 2.1, and the induced group of signed permutations on the bonds of the trian- gle graph. For the vertex permutations, we denote the permutation taking § vertices 1,2 and 3 to and as . It is also common to indicate per- mutations in terms of independent cycles, in terms of which, for example, § "!$#%§ the permutation would be written as . Our notation for signed bond permutations follows that for vertices. 17 2.2 First and second order invariants for the ¢¡&£¦¥¨§ © cage structure shown in Fig. 2.2. The invariants are calculated using a permutation symmetry group £( on the vertices isomorphic to the ' point group. In the text we refer to the invariants by any of the bond products that generate the invariant by application of a projection operator. For example, )+*,-) ./,102030 is at the £ £ 5 03030 top of the list of 1st order invariants, while )%4564,7) heads the list of 89,:)3;<5 8=,>03030 2nd order invariants and )3*5 is at the bottom of the list. 23 3.1 Value of the bond variables and graph invariants associated with each of the graphs depicted in Fig. 3.2. 51 viii LIST OF FIGURES Figure Page 1.1 Examples of the H-bond network. (a) The simplest H-bonded system. (b) An example of how the H-bond network can be summarized by an oriented graph. By convention H-bonds point from H-bond donor to the H-bond acceptor. An H-bond arrangement of ¡&?@ ¡A£¦¥¨§ B is shown on the left and the direction of the H-bonds are summarized by the oriented graph in the right. 2 1.2 Three H-bond isomers of dodecahedral ¡&£¥§ £ . They have similar oxygen # positions and differ only in the direction of H-bonds. Among DC%E%EFE$E%E H-bond isomers, many of them are related by a symmetry operation and # ! C there exist EE symmetry-unique representatives. 3 1.3 Trans and cis configurations in water dimer. From a pure electrostatic point of view, the trans configuration is energetically more stable than cis because the reduced repulsion between the non-H-bonded hydrogen atoms. 6 2.1 A simple example of an oriented graph, which might represent the config- ¡ £¦¥§ uration of a * cluster, is shown on the left. The direction shown on the edges indicate the orientation of the edges if all the bond variables GIH were taken equal to J , canonical orientations chosen arbitrarily for each ¡K£L¥¨§ bond. Two different H-bond topologies for * are shown on the right, along with the value of the bond variables, as referenced to the canonical orientations of the graph on the left. 16 2.2 The cage structure of ¢¡K£¦¥¨§ © . One of the 27 possible symmetry-distinct H-bonding arrangements for the cage structure is shown. The arrows and bond labels indicate the directions of the bonds when the bond variables are equal to J . 22 ix 2.3 Enumeration of all symmetry-distinct H-bond topologies for a dodecahe- dral ¡ £¥§ £ was performed by considering a sequence of structures con- taining fewer bonds than the full dodecahedron. Additional H-bonds were added to the structures after all symmetry-related duplicates were elimi- nated. This process furnishes data on the computational cost of eliminating symmetry-related structures as a function of the number of graphs. This data shown is for the calculation as performed in Ref. [1], without the use of the sorting method introduced in this work. The computational cost per £ graph edge is plotted as a function of the number of graphs M before sym- £ metry comparisons were made. Least squares fits of CPU time to MNM * M and M clearly show that the computational cost scales as or worse without the sorting method. 29 2.4 Data for the same calculation as in Fig. 2.3, this time employing the sorting method introduced in this work. CPU time per graph edge for sorting the graphs is plotted against the number of graphs M in the bottom panel. The total CPU time for symmetry comparisons within groups of size OQP7R$E%E is shown in the top panel. Least square fits clearly show that the compu- £ MTSUM tational cost scales as either M or in each case, and definitely not like M as in Fig. 2.3. On the basis of arguments presented in section 2.2 we expect MTSUM scaling in the bottom panel and linear scaling in the top panel. 30 2.5 In the upper panel, we test the degree to which the energies of the 27 iso- mers of the ¡ £L¥¨§ © cage are correlated with H-bond topology, and the ef- fectiveness of graphical invariants in capturing that trend. The V -coordinate is the energy of the isomers using the PM3 semi-empirical theory.
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