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DISSERTATION

Presented in Partial Fulfillment of the Requirements for the

Degree Doctor of Philosophy in the Graduate School of The

Ohio State University

By

Michael David Tissandier, B.S.

*****

The Ohio State University 1997

Dissertation Committee: Approved by

Professor James V Coe, Advisor

Professor Terry Gustafson Advis

Professor Sherwin Singer Department of Chemistry TJMX Nmnber: 9813363

UMI Microform 9813363 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Titie 17, United States Code.

UMI 300 North Zeeb Road Ann Arbor, MI 48103 it can not be expressed in figures, it is not science; it is an opinion, ”

- Robert Heinlein

i ABSTRACT

The nature of water in combination with itself and solvated ions has been investigated in

terms o f structure and energetics. A PM3 investigation o f neutral water clusters from the well

known linear hydrogen-bonded dimer structure towards bulk has found several structural

families that contribute to the global m in im u m energy clusters. At the trim er (n=3), structures

consisting o f a single nearly planar ring form the most stable family. A t n=6, PM3 finds the

“cage” structure determined byab initio methods and spectroscopic experiments. Structures

consisting of stacked 4-, 5-, and 6-membered tings become the most stable from n=7 to an

undetermined size less than na=145, where ice-like crystal structures overtake the stacked ring

families. Use o f the extrapolation o f the ice-Ih structures to bulk on an excess molecular

binding energy per molecule basis has given a good calibration o f PM3 total atomic binding

energy values over the entire cluster range. PM3 finds energies that are 75% of the

experimental excess molecular binding energy across the entire cluster size regime on a per

molecule basis.

A new value for the bulk single-ion solvation enthalpy and free energy for the proton in water from a method forced to yield m axim u m consistency with bulk has been determined

from cluster data, -1148.7(2.1) kj/m ol and -1101(0.3) kj/m ol (e.s.d) respectively. A comparison with other reported values is included. The emission o f infrared light from coUisionally activated clusters in a fast ion beam has been detected and a low resolution spectrum obtained. The spectrum through the

O — H stretching region is in good agreement with spectroscopic data. The application of an electrostatic particle guide (EPG) to a fast ion beam is explored with the ultimate goal o f achieving greater photon collection efficiency and longer times in front of the detector. A total undispersed emission o f ~ lp W has been found with an EPG apparatus.

in F or Joanie

IV J ACKNOWLEDGMENTS

Some people have been invaluable in the time spent to produce this work. First and

foremost is my wife who, although not necessarily understanding how the progress of an

experiment can alter m oods, worked hard to keep my sanity intact. The three advisors I have

worked under, Jim Coe and Alan Marshall here at tOSU, and Howard Knachel at the University

of Dayton have each provided a different perspective on the addressing of difficulties that has

help crystallize in my mind methods o f problem solving which has served (and will continue to

serve) in good stead.

My coworkers past and present have had much to do with my success in this endeavor.

The Marshall group, during my short stint before the relocation to Florida, got my head on

straight and started me on the right track for graduate success, particularly the efforts o f Chuck

Ross and Troy Wood.

Once I became a refugee, the welcoming hospitality of the “original” Coe group members

helped immensely. Ken Cowen, who demonstrated the time that must be spent to accomplish a difficult experiment. Bob Plastridge, who taught me the ways o f the machine shop for late night creations, Deron Wood, who refused to let me sink into depressions during the tough times, and Cindy Barckholtz (nee Capp) who provided stability in a sometimes outrageous group, showed me the very different ways of a physical chemistry research group, “Basic

Research on a Dollar a Day.” The “new guys” in the group, who never cease to remind me that theoretical work was not in my original “job description,” had almost as much to do with the success o f this research as anyone. Alan Earhart introduced me to theoretical methods as much as I showed him the byways of experimental work. Michael Cohen shared the frustrations that only a malcontent ion beam can inflict. Ellen Gundlach, heiress to the title “Flying Dutchman of Projects,” reined us in at the right times merely by her quiet presence.

The variety o f custom equipment needed for the experimental work would have been infinitely more difficult to produce without the gang in the machine shop. Jerry Hoff, John

Herlinger, Kevin Tewell, Mike Skaggs, and Larry Rucker provided their knowledge and skill to make the many “back-of-the-envelope” designs a reality and provided an everyday perspective to some of the things we were trying to accomplish.

During my stay at the University o f Dayton Research Institute, three people first set me onto the instrumentation path, and I could not be happier. I owe a huge thank you to Wayne

Rubey, Rich Striebich, and Debbie Tirey for introducing me to the real world o f analytical chemistry when I really was not sure if I had the knack for any portion o f the field.

Earlier in my life, several influences put me onto this track and I will always be thankful.

My parents and immediate family began by pushing me to read, and to think about what I had read, on my own without being overbearing. They also put up with some o f the more outlandish ideas a young Itid ever spouted with only a gleam in their eye and a tolerant smile. I do not tell them often enough how much I appreciated all of it and I can only hope I do as well with any children o f my own. Later into school, several teachers influenced me, as they always do although often without even a simple thank you. Jim Malott, Ibby Henlein, Dave Brenner,

VI Gary Berlinger, Ron Mathis, Chuck Wolfe, Rose Wells, and Roger Courts were especially

influential at that point.

O f the teachers I had growing up, however, two must be mentioned separately. Linda

Lawwül, 4* grade teacher, science fair promoter, and gracious aunt, often bore the brunt o f my

silliness and responded with nothing but support. Even to this day, her opinion is one of the

first I seek in nearly any field, and more often than not she will provide perspectives that had

not even begun to occur to me. O f my teen years, the teacher who did the most for me was

Lanny McManus, basketball coach, biology, physics, chemistry teacher, and fiiend. Even after

high school, during one of our more recent, though all too infrequent, meetings, I would be

conversing while holding a mental image of Socrates instructing a young pupil, a holdover from

the days o f the “Physics Class Debating Team.”

VII J VITA

May 28, 1969...... Born - Georgetown, Ohio

1989-1991...... Intern, University of Dayton Research Institute, Dayton, Ohio

1991...... B.S, University of Dayton, Dayton, Ohio

1991-1997...... Graduate Teaching Assistant, The Ohio State University

PUBLICATIONS

M. H. Cohen, T. Schwope, M. D. Tissandier, and J. V. Coe, “The Eflfect of Clustering Water Molecules on the Rate and Mechanism of the H 3O (HzO)n + OHTCHaO)™ Reaction ', Dissociative Recombination: Theory, Experiment, and Applications III, D. 2^jfinan, J. B. A. MitcheU, D. Schwalm, and B. R. Rowe, Eds. (World Scientific, Singapore, 1996), p. 216.

W. A. Rubey, R. C. Striebich, M. D. Tissandier, D. A. Tirey, and S. D. Anderson, “Gas Chromatographic Measurement of Trace Oxygen and Other Dissolved Gases in Thermally Stressed Jet Fuel”, J. Chromât. Sci. 33, 1995, 433-437.

FIELD OF STUDY

Major Field: Chemistry

VIII TABLE OF CONTENTS

Dedication ...... iv Acknowledgments ...... v V ita...... viii List of Figures ...... xi List of Tables ...... xix

Chapter L Water Cluster Families and Their Trends Toward Bulk Structures ...... 1 1.01 PM3 Calculations...... 3 1.02 Plots of Excess Cluster Binding Energy vs. n ...... 4 1.03 General Fit Expression ...... 6 1.04 Water Dimer ...... 7 1.05 Chain Structures...... 8 1.06 Ring Structures ...... 9 1.07 The Flexamer Case ...... 10 1.08 Stacked Cube Structures ...... 12 1.09 Other Stacked Structures ...... 14 1.10 Hexagonal Ice Structures ...... 15 1.11 Cubic Ice Structures...... 18 1.12 Empty Clathrate C ages ...... 22 1.13 A .b Initio Results...... 23 1.14 Discussion ...... 25 1.15 Conclusions ...... 31 1.16 References...... 32

Chapter 2. Single Ion Bulk Solvation Enthalpies and Free Energies of Selected Halide Ions, Hydroxide, and Alkali Metal Ions from Cluster Studies ...... 61 2.01 Thermodynamic Conventions for the Electron ...... 63 2.02 Cluster Viewpoint of Bulk Solvation Enthalpies ...... 66 2.03 Previous Bulk Single-Ion Solvation Methods ...... 72 2.04 Coe’s Difference-Over-Sum Cluster Method ...... 74 2.05 Linear AH,„, ae vs. AH„,j M ethod...... 76 2.06 The Y’Method ...... 80 2.07 Discussion ...... 84 ix 2.08 Acknowledgments ...... 88 2.09 References ...... 89

Chapter 3. Studies of Intense Ion Beams in an Electrostatic Particle G u id e ...... 99 3.01 Current System ...... 100 3.02 Simien Results ...... 103 3.03 Acceptance Angle ...... 105 3.04 Energy and Mass D ependence ...... 106 3.05 Experimental Results ...... 108 3.06 Discussion ...... 109 3.07 Conclusions ...... 110 3.08 References ...... I l l

Chapter 4. Infrared Emission Spectra of H^O Clusters in Fast Ion Beams ...... 124 4.01 Multiple Collision Analysis ...... 125 4.02 Previous Results...... 128 4.03 Circular Variable Filter...... 129 4.04 Discussion ...... 133 4.05 Conclusions ...... 134 4.06 References ...... 136

A ppendices ...... 143 Appendix A - MathCAD Templates ...... 143 Appendix B - Fortran Programs ...... 161 Appendix C - Periodic Box Values in Hyperchem ...... 176 Appendix D - Calculation o f Cartesian Coordinates for Ordered Ice-Ih and Ice-Ic Crystal Unit Cells...... 179 Appendix E - Selected Hyperchem Files ...... 186 Appendix F - PM3 Calculated Binding Energies and Excess Binding Energies per Water Molecule for Selected Clusters ...... 192 List of References ...... 200

À List of Figures

1.1 The minima for the water dimer, (H ,0 ),. The linear hydrogen bonded structure is the experimentally determined global m in im u m ...... 38 1.2 The alternating chain structure. Two orientations are shown for the hydrogen atoms not involved in hydrogen-bonding. The second structure was used by Suhai as his model, projected to infinite length, for ice-Ih...... 39 1.3 The bridging chain structure. Both the alternating and bridging chains are present in the ice structures, but in differing degrees ...... 39 1.4 Excess binding energy plot for the chain families. The convergence o f the curves at n=4 is due to distortion o f both families towards a sim ilar structure...... 40

1.5 The ring family structures. These are the minimum energy structures for the rings found by PM3. In each case, the individual water molecules donate one hydrogen bond and accept one hydrogen bond to form the “backbone” o f the ring...... 41 1.6 Several structural isomers o f the tetramer. The cluster marked with the star is the minimum energy structure ...... 42 1.7 Excess binding energy plot for the ring structures. Ring structures for n=3...9 have been calculated and plotted on the same scale as the chain structures presented previously. The upward bend to the plot at n>6 is due to the “collapse” of the ring towards three-dimensional structures...... 43 1.8 Comparison o f four low energy hexamer water clusters. Four o f the low energy configurations for (H^O)^ are shown with the PM3 calculated binding energy. Structure a, the “ring,” was originally predicted to be the global minimum, but recent calculations have shown the other three structures to be more stable. It is generally believed that the “cage” structure is the global minimum, but all three remaining structures are nearly equivalent in energy and the ordering becomes dependent on the calculation method...... 44

XI 1.9 Electron attachment mass spectrum of water clusters by Bowen, et a i For an electron to attach to a cluster, the electron must either be solvated or the cluster have a large dipole moment. The continuum of peaks at n>l 1 is due to solvation of the electron. The isolated peaks at 2, 6, and 7 indicate the structures for the corresponding clusters must have a large dipole moment. The cyclic ( H ,0 ) g structure has a sm all dipole moment, while the other, more three-dimensional, structures have PM3 calculated dipoles >1.7 debye...... 45 1.10 Example stacked cube structures. The highly strained hydrogen bond angles are consistent throughout the structure which allows the fitting routine to be employed ...... 46 1.11 Excess binding energy plot for the stacked cube structures with the fitted curve. The stacked cube structures for n=8,12,...36, and 92 have been calculated and added to the previous plot of chains and stacked cubes. The extrapolation to bulk (n '^^=0) is found to be -7.51 kcal/mol ...... 47 1.12 Excess binding energy plot for the stacked hexagons and pentagons. The clusters and fitted expressions for the stacked hexagons and pentagons has been placed on a plot with the stacked cubes shown previously. The pentagons become most stable at n«28 and the hexagons become the minimum at n»70. The extrapolated bulk values for the pentagons and hexagons is -7.56 and -7.60 kcal/mol respectively ...... 48 1.13 Comparison of the stacked pentagon and stacked cube decamers. The stacked pentagon decamer is more stable at n=10 despite the fact that the stacked cube family is more stable at that cluster size. The global minimum for water clusters overall does not follow one family in particular as the value o f n is incremented, but in general follows the trends shown previously ...... 49 1.14 The crystal structure of ice-Ih. Normal hexagonal ice is comprised of sheets o f fused hexagonal rings (top). The stacking o f the layers form puckered hexagons in the conformational equivalent o f the cjxlohexane “boat” configuration (bottom). The c-crystal axis is unique in normal ice...... 50 1.15 The minimum ice-Ih structure. The cluster shown is the minimum structure with the ice-Ih bonding pattern that does not contain waters with only one hydrogen bond. Note that all the hexagonal rings present are in the “boat” configuration...... 51 1.16 Several ice-lh structures. The shown structures of normal ice were used to determine the values o f E^, Ej, and E, for the family. These clusters also represent the smallest o f the structures with values of n^, n,, and n, predicted by Eqs. (13), (14), and (15)...... 52

XII 1.17 Excess binding energy plot for hexagonal ice strucmres. The ice-Ih structures and the fitted extrapolation to bulk is added to the previous plot for comparison. The PM3 predicted OK. bulk value for ice-Ih is - 8.36 kcal/mol (compared to the experimental -11.3 kcal/mol) ...... 53 1.18 The minimum ice-Ic structure. The cluster shown is the minimum size structure for ice-Ic that does not co n rain molecules with only one hydrogen bond. Note that, unlike the normal ice structure, which contained all “boat” configuration hexagonal rings, all those present in this cluster are in the “chair” configuration...... 54

1.19 The fitted ice-Ic structures. These clusters follow the general shape, that o f the octahedron, o f materials with the cubic-crystal type at bulk, cubic ice, diamond, and silicon. These were used for the determination of the values of E^, Ej, and E, and were used to determine Eqs. (16), (17), and (18) ...... 1 ...... 55

1.20 Excess binding energy plot for cubic ice. The values and extrapolated curve for cubic ice has been plotted with the previous families. N ote that the ice-Ic clusters are everywhere more stable than the hexagonal ice structures. The curve determines a bulk value o f -9.09 kcal/mol ...... 56

1.21 The minimum energy (H,0)2i) clathrate structure. This structure is a representative o f the empty clathrate cages that form another family for analysis...... 57 1.22 Excess binding energy plot with the (H^O)?,, clathrate structure ...... 58 1.23 Excess binding energy plot for several levels o f ab initio calculation. The ring and early stacked cube structures were calculated and plotted at several levels of theory. The lower levels are shown to severely over­ estimate the value o f the hydrogen bond as many o f the clusters have calculated b in d in g energies per molecule greater than bulk. The MP2/aug-cc-pVDZ curve firom the data of Xantheas^^ is the closest to the predicted values firom the scaled PM3 and could become better if BSSE counterpoise techniques are employed ...... 59 1.24 Scaled excess binding energy plot. The values for the clusters and fitted curves were scaled globally by the value o f ap^o=1.35. The hexagonal ice curve is forced to extrapolate to the experimental value o f -11.3 kcal/mol. Cubic ice is predicted to be 0.97 kcal/mol more stable at OK ...... 60

XIII 2.1 Plot o f the bulk single-ion solvation enthalpies determined with Morris’ absolute value for the proton (-1103 kJ/mol) vs. the cluster solvation enthalpies for clusters with 5 water molecules. The positive and negative ions are grouped separately and display a different slope illustrating the two problems that must be overcome to achieve consistency of a particular absolute value o f the proton with the cluster data set...... 91 2.2 Plot of the bulk single-ion solvation enthalpies for clusters with Klots’ absolute value for the proton (-1136.4 kJ/mol) vs. the cluster solvation enthalpies for clusters with 5 water molecules. Klots’ chosen set of ions gives a fitted line successfully grouping the positive ions and selected negative ions onto a common line (albeit with a slightly increased slope over that of the positive ions alone). The strategy of eliminating OH' and F firom the process has not dealt successfully with the problem of a different slope for the negative ions, leaving the O H and P species as deviant without explanation ...... 92 2.3 Plot of the bulk single-ion solvation enthalpies determined with the Coe method’s absolute value for the proton (-1152.5 kJ/mol) vs. the cluster solvation enthalpies for clusters with 5 water molecules. The large halides were left out o f the fit because they are surface states at n=5 and will not exhibit the same functional dependence o f stepwise hydration energetics as do the internalized ions. The included ions all fit very well to a common line with a slope verj”^ close (less than a 0.5% difference) to that exhibited by the positive ions alone ...... 93 2.4 The correlation of the Y’ approximation and the bulk difference in ion volume for various pairs o f alkali and halide ions. The value o f Y’ (-1152.1±1.8 kJ/mol) when the difference in bulk ion volumes is zero represents the best value of the proton’s hydration enthalpy upon invoking the Bom concept that ions of the same volume have the same bulk solvation enthalpies...... 94 2.5 The correlation of the Y’ approximation and the bulk difference in ion solvation enthalpy. This correlation is very similar to that shown in Figure 2.4 regarding the position o f ion pairs. The intercept value o f (-1151.0±1.6 kJ/mol) is consistent with that determined by invoking the Bom concept showing that, in general, ions o f the same volume have the same cluster solvation enthalpies as well as bulk solvation enthalpies. The problem o f the proton’s bulk hydration enthalpy can be equivalently approached with the difference in bulk ion volumes or cluster solvation enthalpies...... 95

XIV 2.6 The correlarion o f the Y’ approximation and the difference in cluster ion solvation enthalpy vs. cluster size. This plot contains the same data as in Figure 2.5 except that the cluster size has been specified. If the Y’ approximation is good at only one particular pair o f ions, then each data set at a particular cluster size can be expected to share the good value in common. At each cluster size the data have been fit to a line by the method o f least squares. The spread in these fitted lines vs. the difference in cluster ion solvation enthalpy was used to determine a most common value of -1148.7±2.1 kJ/mol which represents our best determination of the absolute proton hydration value without invoking the Bom concept. This value is consistent with the values determined previously that had invoked the B om concept ...... 96 2.7 The correlation of the firee energy or X’ approximation with the difference in cluster ion solvation free energy vs. cluster size. These data show an even more commonly shared point of -1101.8+0.3 kJ/mol which represents our best determination o f the proton’s free energy of hydration without invoking the B om concept. This value compares favorably to the center-ofiweight o f previous determinations providing confidence in the enthalpic results o f Figure 2.6 ...... 97

2.8 The dependence of the bulk constants, -V 2 [k(A+)+k(B-)], on the difference in cluster solvation enthalpy vs. cluster size. The cluster approximation employed in the Y’ approximation involves only the first term o f Eq. (48) which is essentially zero at the common point, so that the proton’s absolute value is determined almost entirely by the behavior of the bulk constants from Table 2.3 vs. the difference in cluster ion solvation enthalpy. Notice the linearity o f the data sets at particular cluster sizes and the shared com m on point o f -1149.1 ±2.0 kJ/m ol which is consistent with the previous determinations for the proton’s absolute hydration enthalpy ...... 98

3.1 Diagram o f the Kingdon Trap. Ions were generated intemally and trapped into orbits about the central wire. The end caps restricted the ions axially ...... 112 3.2 Diagram of the current EPG system. Note the angle of the ion beam relative to the EPG...... 112 3.3 Diagram of the vacuum chamber. Two 1500 L/s diffusion pumps, one water baffled, one l-Nj, provided a base pressure of —5x10 Torr ...... 113 3.4 Gas rack plumbing diagram ...... 114 3.5 Kimble Physics pattem of holes for the elements of the ion beam. The lightly shaded holes were used for the primary support. Each hole is 0.3” center-to-center...... 114

XV 3.6 Typical Mass Spectrum obtained from the glow discharge source The x- axis is measured in volts as the spectrum is from a Wein velocity filter this also accounts for the non-linearity in mass ...... 115 3.7 Diagram o f the source circuit. The voltage applied by the beam power supply determines the beam voltage and the discharge voltage determines the source brighmess. The ballast resistance improves the steadiness of the beam output. The back plate is kept more negative than the front for maximum cluster products. Reversing this polarity will give electron bombardment ions ...... 116 3.8 Diagram of the ion optical element circuit. The series of resistors acts as a voltage divider for the beam voltage (VJ, adjustable by varying the position of the potentiometer inputs. This arrangement maintains a constant ratio between the voltage applied on the ion optical element and the beam voltage, despite any changes in V y ...... 116 3.9 Simion grid coordinates. The upper portion of the figure shows the potential surface of a cross-section of the EPG. The coordinate system and general layout o f the Simion grid is shown in the lower portion. Note the exaggerated size of the central wire (2mm diameter in the simulation vs. ~0.04 mm for the actual device). This constraint was imposed for ion trajectory purposes as an ion is considered to have struck a surface and been neutralized only if the grid points immediately surrounding the ion all are contained by an electrode. The 2 mm diameter was the smallest available that would guarantee the correct ion behavior near the central electrode...... 117 3.10 Potential plot inside the guide tube. The points are samples taken from the Simion simulation and were plotted as a function of radius. The solid curve is the predicted potential from Eq. (2) ...... 118 3.11 Spot picture. The initial trajectories for the ions in the simulation were arranged to imitate the real size of the ion source. Each o f the 5 points contained 10 rays varying in both the xy- and xz-plane ...... 118 3.12 Example trajectories from the simulation. The upper portion of the figure shows the typical spiral nature o f the ion spot trajectories for the initial portion o f the EPG. The lower plots are single ion trajectories viewed along the x-axis. The notation on each plot refers to the azimuth angle o f each initial ion condition. Note that as the angle increases, the eccentricity o f the ion “orbit” decreases...... 119 3.13 Azimuth and elevation angle plot. A surface involving flight times as a function o f azimuth and elevation angles for the initial ion trajectory for a given starting position. Note the narrow range where the ions are successfully transmitted down the length of the guide tube. The symmetry o f the plot confirms that the direction o f the spiral is unimportant...... 120

XVI 3.14 Time o f flight plot. Several single ion trajectories involving ions of different masses are shown. The linearity shows that the EPG is functioning in the same manner as a time-of-flight mass spectrometer, despite its inherent curved trajectories. The ions followed the same path and struck the Faraday tube in the same spot regardless o f the mass...... 121 3.15 Transmission window. Single ion trajectories were performed as a function o f radial position and ion energy for a given set o f input angles. The upper surface plot shows the flight time throughout the range studied. N ote that the flight time dramatically increases as the wire is approached. The lower plot shows the same plot as a single-line contour around the base of the surface. Three failure modes have been identified for ion transmission. The “Wire Shadow” is caused by the ions coming into contact with the central wire. This is exaggerated by the unphysical size o f the simulation wire. The “Low Vy” failure involves the ions not possessing sufficient energy to overcome the fidnge field o f the guide mbe and being repelled. “High refers to the ion energy being too great for the tube voltage to bend the ions into an orbit before striking the outer wall...... 122 3.16 Oscilloscope plot o f the modulated ion current. A trace o f the experimental modulated ion current at the Faraday tube o f the EPG. The modulation square wave was set to 100 Hz and a IkQ input impedance was used resulting in the y-axis being 1 m V equal to 1 mA o f ion current. The arrows denote the points where the modulation voltage changed. The delay was measured as —25 (xs ...... 123

4.1 Multiple collision analysis o f H 3 0 ^(H,0 ). The data points and the fit that produced the value for r,, the fraction of the original ion beam that underwent one glancing collision, did not dissociate, and reached the end of the beam path. The solid P* curve is the total fraction of the ion beam, at a given attenuation, that has underwent a collision and absorbed excess energy. The dash-dot curve labeled 1 is the contribution from ions that underwent a single collision, and the dash- dot curve labeled 2 is the result of 2 collisions and similarly for the CID curves (dashed). This analysis can be read as, on average, one collision does not dissociate the ion while 2 will cause dissociation. As the experimental dissociation energy for H 3 0 *(H, 0 ) is 1.37 eV, a collision energy of —I eV can be concluded ...... 137 4.2 Original apparatus for finding total emission o f infirared light. The ion beam was directed at through a set o f 4 grounded rods towards a CaF, lens. After a short time (< 1 second), the lens charges and repels the beam as it exits the region bounded by the rods. The fact that the beam is “pointed” at the detector gives long times (—5|xs) in the detector field o f view ...... 138

XVII 4.3 The undispersed infrared light detected from the emission o f water clusters. When the ion source was turned o ff (labeled “beam off”), litde light was observed on the detector. Turning the source on without adding collision gas (“gas off”), a light level of ~3 pW was detected. Addition o f collision gas (“gas on”), although decreasing the total ion current in the photon collection area, raised the detected light to ~ 13 pW. Note that a large time constant, as evidenced by the long tail between “gas on” and “gas off,” was applied to reduce the noise...... 139 4.4 Diagram of the spectrometer. Ion clusters are extracted from the ion source and undergo g la n cin g collisions with the argon introduced by a tube. The beam is amplitude modulated by a square wave sufficient to prevent to ions from entering the photon collection region during the “high” portion of the wave. The emission from the ions, also modulated with respect to the photon collection region, passes through the circular variable filter and is focused on the InSb detector. The signal is then demodulated through a lock-in amplifier and the signal averaged over long periods of time ...... 140

4.5 Low resolution infrared emission spectrum o f Hj0^(H20)„. The background scan was produced using Ar as the source gas. The downward feature at 3000 cm * common to both scans is an artifact o f the circular variable filter. The excess light above 3300 cm ' is consistent with absorption measurements by Lee et al. Peak power is ~33 fW...... 141 4.6 Typical mass spectrum from the ion source. The dispersive element is a Wein filter...... 142

D .l MathCAD generated positions o f atoms in the unit cell. Note that the atoms marked H14 and H16 must be translated -1 unit cell lengths along the x-dimension to achieve a real bond distance ...... 184 D.2 Hyperchem graphic of the Ice-Ih unit cell ...... 185

XVIII

1 List of Tables

1.1 Low lying local minima o f water dimer. The molecular binding energy for three minima was calculated with three semi-empirical and one ab initio method and compared to the experimental global minima, the linear hydrogen bonded structure ...... 8

1 .2 The P M 3 calculated excess molecular b in d in g energy per water molecule (E ^ for the alternating and bridging chain structures ...... 9 1.3 The PM3 calculated excess molecular binding energy per water molecule for the ting structures ...... 10 1.4 The numerical comparison o f the PM3 calculated excess molecular binding energy per water molecule and the predicted value from the fit o f the stacked cube structures...... 12 1.5 The numerical comparison o f the PM3 calculated excess molecular binding energy per water molecule (E^^) &nd the predicted value from the fit o f the stacked pentagon structures ...... 13 1.6 The numerical comparison o f the PM3 calculated excess molecular binding energy per water molecule (E ^ and the predicted value from the fit o f the stacked hexagon structures ...... 15 1.7 The progression o f water molecule types in ice-Ih. The fitting expression requires a mathematical expression for the number o f water molecules in a particular structure that have 2, 3, or 4

hydrogen bonds (n 2 , n^, and n^ respectively) as the clusters grow towards bulk. The clusters used were constructed and the numbers in each category tabulated as a function o f the “growth index,” or number o f shells, S ...... 16 1.8 The numerical comparison o f the PM3 calculated excess molecular binding energy per water molecule (E^^) and the predicted value from the fit o f the hexagonal ice structures ...... 18 1.9 The progression o f water molecule types in ice-lc. The fitting expression requires a mathematical expression for the number o f water molecules in a particular structure that have 2, 3, or 4 hydrogen bonds (nj, n,, and respectively) as the clusters grow towards bulk. The clusters used were constructed and the numbers in each category tabulated as a fiinction o f the “growth index,” or number o f shells, S ...... 21 xix

i) 1.10 The numerical comparison of the PM3 calculated excess molecular binding energy per water molecule (E ^ and the predicted value &om the fit of the cubic ice structures...... 22 1.11 Numerical results firom PM3 and two STO-KG (K=3,4) ab initio levels with MP2 corrections ...... 23 1.12 Numerical results firom PM3 and several Gaussian-type orbital ab initio levels with MP2 corrections ...... 24 1.13 Numerical results firom PM3 and two basis sets of Dunning with MP2 corrections ...... 25

2.1 Solvation fi^ee energy, AG”, ,^ (kJ/mol), and solvation enthalpy, (kf/mol), to place a pair o f oppositely charged, gas phase ions into water at 25 °C firom tabulated firee energies and enthalpies of formation (ref. 4). The values in parenthesis were not directly available firom data in ref. 4. There were determined from the differences (Table 2-2) o f values that were available...... 64 2.2 Bulk average differences (first ion-second ion) in the single ion bulk solvation firee energies o f similarly charged ions, AG,^,(ion 1)- AG,„|(ion 2), and enthalpies, AH,^,(ion 1)- AH„,(ion 2), Uncertainties are not available for the values based on firee energies and heats o f formation (ref. 4) because this data has apparendy been made intemally consistent. Alternative cycles (ref. 21) can be employed for the enthalpies (values are given in the second row of each enthalpy set with 1“ standard deviations in parenthesis) suggesting that a range o f uncertainties of 3-10 kJ/mol is reasonable for all of these values...... 67 2.3 The bulk solvation firee energy and enthalpy o f various ions relative to ...... 68 2.4 Cluster ion solvation firee energy and enthalpy partial sums o f various ions vs. cluster size. Ion values have been averaged firom the shown references. Entries in parenthesis are estimated ft-om the expected trend to the known bulk value ...... 71 2.5 Ion radii as reported by Gourary and Adrian ...... 83 2.6 Slopes and intercepts of Y’ (or X”) vs. the difference in cluster solvation enthalpy (or firee energy) for different cluster sizes, n. Estimated standard deviations are given in parenthesis...... 84 2.7 Single ion bulk solvation firee energies and enthalpies firom several different sources. Each data set has been used to generate data as in Table 2.1. The standard deviation of the comparison with the bulk experimental data in Table 2-1 is given as O’ at the bottom ...... 86

XX C.l Edge length to achieve a specified neutral water cluster size, n ...... 177 C.2 Edge length to achieve a specified X(H20)„ cluster size where X is a halide ion, alkali ion, OH', or HjO""...... 178

D.l Positions of unique atoms within the unit cell o f ordered Ice-Ih as reported by Leadbetter et al. Units are firactions o f unit cell edge length in each respective dimension ...... 181 D.2 Symmetry operations for the space group, C m c2,...... 181 D.3 Generated positions of atoms in the unit cell. Shaded atoms are in redundant positions...... 182 D.4 Final generated atomic positions in Angstroms ...... 183 D.5 Crystallographic properties of ice polymorphs. Note the difference in space group for Ice-Ih. This is the result of the hydrogen positions determined in ref. (1) ...... 183

F.l PM3 values for the monomer, alternating chains, and bridging chain structures studied in Chapter 1...... 193 F.2 Rings and stacked cube structures...... 194 F.3 Stacked pentagon and hexagon structures ...... 195 F.4 Partial stacked cube structures...... 196 F.5 Hexagonal ice structures ...... 197 F.6 Cubic ice structures...... 198 F.7 Other structures...... 199

XXI CHAPTER 1

WATER CLUSTER FAMILIES AND THEIR TRENDS TOWARD BULK STRUCTURES

The nature of bulk water is difficult to describe.^ This has resulted in a large body o f work in the realm of clusters. A simple electronic search of periodicals firom 1968-1996 found 473 references to water clusters, 421 of which are in English. The goal of this work is to perform calculations on the neutral water clusters which are valid for the whole cluster size range &om dimer to bulk. This “global view” o f the effect o f size is important as technology extends into the nanoparticle-sized region.-’ ^

Early models o f liquid water were divided into two broad categories. The continuum model was first expressed by Pople^ in 1951 and followed by the related “random network model” of Bemal^ in 1964. In both of these models, a continuous network o f molecules with tetrahedral coordination occurring to the first neighbor is present. Beyond that neighbor, flexible hydrogen bonds deform to create a continuous distribution of O O distances. Orentlicher^ used a similar approach, but allowed the hydrogen bonds to break rather than bend. Weres^ also used this basic idea to develop a thermodynamic lattice gas model in 1972. Ohmine® used networks of hydrogen bonds to describe collective morions in both liquid water and clusters.

Early versions of the mixture, or cluster, model, proposed by Frank and Wen,^ and

Frank and Quist,^® dealt with “flickering clusters,” where an amorphous globular sample of water at any instant contained water monomers and short-lived hydrogen-bonded clusters at room temperature. Nemethy and Scheraga provided a variation o f this in 1962.^^ They proposed a mixture o f species with two-, three- and four-coordination. Giguère^-"^^ and

Dannenberg^^ have also suggested that there is a significant firacrion o f species, possibly intermediates, where three-centered “bridging” hydrogen bonds are present as a defect, similar to the defect proposed by Dunitz to account for electrical conductivity in ice,^^ in order to explain some molecular diffusion and rotation observations.

Symons^® has proposed interpretation o f the properties o f water in terms o f concentrations of broken and unbroken hydrogen bonds. A statistical treatment by

Hagler^^ et al. is based on a distribution o f water clusters centered at n = ll. In a more recent development, Benson and Siebert^^ have put forward the idea that the heat capacity data for water can only be modeled with discrete water clusters, in their theory, hydrogen- bound octamers. Support for this theory can be found in the in&ared data of Libnau.-^

Water clusters are also becoming more prominent in several more recent problems.

The unexpected absorption of sunlight by clouds,— the “seeding” for water droplet growth,^^ and some formation pathways for acid rain formarion^^’ ^5 have also been attributed to the contribution of small clusters of water. Naturally occurring airborne ice crystals are believed to be the cause o f “haloes” seen around the sun at 22°, 46° (hexagonal ice),-^ and 28° (cubic ice).-^ These are examples of cases where knowledge o f water clusters over a broad size range is of direct importance.

PM3 Calculations

An approach similar to this line o f research, using PM3 for calculation o f small water clusters, was undertaken by Vasilyev,-® although this fact was unknown at the start o f this work. The present results correspond to the reported global minimum except at n=6. The n= l-16 results presented here that matched the earlier work o f Vasilyev were o riginally calculated by Muhammed-^ although vibrational frequencies were included in the later work.

The advent of affordable, high speed personal computers and commercial software packages has opened an avenue of research to scientists interested in comparing their results with those obtained by theoretical/computational chemistry. For clusters of even moderate size, however, ab initio methods become very time consuming and give artificially large molecular binding energies at even modest levels of theory, i.e. MP2/6-31G**, if more than one molecule is involved.^® It has been shown that, in order to get a reasonable answer for water clusters, large basis sets are required, greatly multiplying the computational time required.^ ^ Semi-empirical methods, such as AM l^-, and are better able to handle large cluster sizes, but some questions have been raised about their ability to calculate accurately the molecular binding energy o f clusters.^^’ A comparison o f these and several ab initio methods for strong and moderate hydrogen bonds, albeit hvdrocarbon-

water and molecular ion-water interactions, is reported by Zheng and Merz.^^

The PM3 calculations in this work were performed on several PI 66 computers with the

Hyperchem"^ molecular modeling program. This program provides a graphical interface

and a convenient method for manipulation of the positions of the constituent atoms or

molecules in a cluster.

Plots of Excess Cluster Binding Energy vs. n 1/3

The PM3 calculation o f all the clusters in this study, being semi-empirical and only

dealing with valence electrons, does not calculate a total electronic energy. The value

reported by the calculation is the atomic binding energy. This number is not in and of itself

useful for comparison to other methods, but calculation of the excess molecular binding

energy per water molecule, E^, via the expression

E ^ = E.. nEw (1). n

where E^ is the total atomic binding energy determined by the calculation, E^ is the

binding energy of a single water molecule from the same calculation method (-217.22

kcal/mol for PM3), and n is the number of waters in the cluster, allows any ab initio or semi-

empirical calculation to be direcdy compared to these results."*^

The choice of n*’^^ as the ordinate is from the relation of the radius of a sphere to the volume. Consider a property of a spherical cluster with n molecules that varies with the radius, R, such as the Laplace equation for the additional pressure inside a spherical cluster or drop,**^-

pO>_p(ü (2) R. where y is the bulk surface tension, and each constituent of the cluster occupies a given, constant, volume, Vg. Note that the interstitial volume is counted as part o f the occupied volume of the constituents. The volume o f the cluster can be approximated as

Yciuaer ~ *^^S (^)-

Substitution o f the volume expression for a sphere yields.

yttR^ = n y :tr^ (4),

where r, is the effective radius o f the molecule.'*^^ Simplification o f Eq.(4) gives the expression,

1 R = (5).

Rearrangement gives.

n ' L i (6). If the radius o f the cluster, R, is equal to r„ one molecule is present and n is unitv. If

the cluster gets very large, i.e. R » r , and bulk is approached, n approaches zero. Clusters

do not necessarily have a spherical shape, especially at small n, and, as such, will not always

resemble uniform spheres. This approach is an approximation only, but represents a

convenient method for extrapolation of a property, such as internal pressure, to bulk.

General Fit Expression

If the total atomic binding energy of a cluster is considered to be the result of

contributions of the water molecules with one, two, three, and four hydrogen bonds and all

the molecules in each category are considered equivalent throughout the family, the general

equation^ to which all the clusters are fit is given by

A

^ab.Faitiily ^(.Family i=l where is the calculated PM3 energy of a cluster in the designated family with n, (i=l-

4) water molecules, each with i hydrogen bonds, and E^^^y is the fitted value for the energy o f one water molecule in the cluster with i hydrogen bonds in that family. The quality o f the fit is given by calculating the predicted E^ for each cluster in the fit and calculating the estimated standard deviation

^ points ^ parameieis where a is the error in the fit, Np^^is the number o f clusters used in the fit, ...... is the number o f parameters being fit (in this instance, it is numerically equal to the number o f categories of water molecules present. Le. 1-4), is the PM3 energy of the j-th cluster, and

Eg, is the predicted energy o f the j-th cluster.

Water Dimer

The water dimer is one of the most studied clusters. The existence of associated water molecules in the gas phase was first put forward by Pimentel^^ firom PVT observations o f vapor. Many theoretical studies 46-66 experiments^^"^^ have been performed to determine the structure and binding energy o f this species. The dimer has been found to have three distinct minima, shown in Figure 1.1, o f which experimental evidence^^’ has shown the linear hydrogen bonded structure to be the global m in im u m - Table 1.1 shows the minimized molecular binding energy relative to the global minimum for three semi- empirical methods (MNDO, AMI and PM3) and a moderate level ab initio calculation

(MP2/6-31G**). One o f the primary reasons for the choice o f PM3 as the method for this work is the fact that it predicts the linear hydrogen bonded structure to be most stable, in agreement with ab initio and experimental results. Zheng and Merz^^ found good correlation between PM3 and ab initio to be the case over a wide range of compounds and ions with a single water molecule. Ene^y above ground state (eV) M P 2/ Structure MNDO AMI PM3 WIG** bifurcated -0.004 -0.074 0.067 0.005 dipole opposed 0.017 -0.087 0.065 0.092

linear hydrogen bonded 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0

Table 1.1. Low lying local minima o f water dimer. The m olecular binding energy for three minima o f the dimer was calculated with three semi-empirical and one ab initio method and compared to the experimental global minima, the linear hydrogen bonded structure.

Chain Structures

Chains o f hydrogen-bound water molecules represent the lowest stability, or highest energy, family o f small clusters that contain a global m in im u m structure at a given cluster size, n, the water dimer, (HjO),. Some chain structures have been explored by ab initio methods as they grow by Ojamae and Hermansson.^^

Two distinct repeating structures exist for the chains. The first, more stable, system, denoted as the “alternating chain,” is one where each water molecule in the chain contributes one hydrogen to the chain structure (Figure 1.2). Each water in this family

(excepting the terminal molecules) donates one hydrogen and accepts one hydrogen to the

“backbone” of the chain. The second structure, the “bridging chain,” involves 50% o f the waters contributing one hydrogen to the chain, 25%, two hydrogen atoms, and 25%, zero, in the following pattern: ...1,2,1,0,1,2,1,0... (Figure 1.3). A mixture o f these two extremes is possible, but these chains can be broken down into segments of these base structures in much the same way a vector can be given by the sum of orthogonal vectors along the axes of a coordinate system. 1 PM3 Excess Molecular Binding Ene%y n Alternating Chain Bridging Chain 3 -2.72 -2.62 4 -2.98 -2.77 5 -3.02 -2.92 6 -3.14 -2.83 7 -3.12 -3.00 8 1 -3.05 -2.92

Table 1.2. The PM3 calculated excess molecular binding energy per water molecule (E^ for the alternating and bridging chain structures.

The PM3 results for these structures are given in Figure 1.4 and Table 1.2. The

“convergence” of the 2 families at n=4 is due to similar distortions o f both chains toward a ring-hke configuration.

Ring Structures

Cyclic water structures, being easily accessible due to their small size, are among the most computationally studied clusters.^^^^ The infirared spectroscopy experiments of

Saykally et al. have confirmed^®'®^ the theoretical prediction^^’ 81-90 that the ring system o f structures becomes the global minimum at n=3. These minimum energy structures (Figure 1.5 and Table 1.3) consist o f a ring where each water contributes one covalently bound hydrogen to the backbone. The orientation of the hydrogen atoms not hydrogen-bonded is the determinant between several energetically near-equivalent structures

(Figure 1.6).

The general fitting routine breaks down in this case. While each water molecule present

is “2 -bonded,” the hydrogen bonds are not consistent firom one structure to another. n PM3 Eri, 3 -3.36 4 -4.59 5 -4.77 6 -4.91 7 -4.87 8 -4.86 9 -4.77

Table 1.3. The PM3 calculated excess molecular binding energy per water molecule for the ring structures.

There exists in these structures (up to n= 6 ) varying O O O angles and, as a result,

varying degrees o f strain in the hydrogen bonds, introducing an angular dependence in the energy not covered by the general expression.

The ring excess binding energy curve, shown in Figure 1.7, also bends towards lower stability, higher energy, beyond n = 6 . Closer examination o f the PM3-minimized structures reveal that these large rings tend to “buckle” towards more three-dimensional shapes.

The Hexamer Case

It has been shown that the global minimum structure for n=3-5 is a ring and n>7 form more three dimensional structures. O ne o f the issues in the cluster community is the structure o f the hexamer, Theoretical studies, separately by

Kim^- and Campbell®^, have predicted the cyclic symmetry structure (Figure 1.8a) to be the global minimum. Others®-*®^’ have predicted more three-dimensional, non- cyclic, structures to be m ost stable. Jordan^®, in his work utilizing TIP4P potentials, identified 137 distinct minimum energy structures, which were further grouped into 10

10 classes determined by the numbers of three-, four-, five-, and six-membered rings (based on the oxygen atoms) present in the structure. The calculations o f this work, though not as exhaustive for this one case, generally agree with the relative ordering determined by Jordan.

Both methods identify the “cage” structure (Figure 1.8b) as the most stable, in agreement with experimental results by Saykally.^^ The slighdy less stable structures differ between the two methods. Jordan^® reported a “prism” structure (Figure 1.8c) while PM3 finds the “book” structure (Figure 1.8 d) to be the next more stable structure. The differences between all 3 o f these structures is small (<1 kcal/mol in total atomic binding energy) and slight variations in the basis sets makes an absolute computational prediction difficult. In this case, it is known that TIP4P overestimates the strength of a hydrogen bond (6.4 kcal/mol^^ compared to an experimental 5.4 ± 0.7^^ and 5.4 ± 0.2^^ for the dimer binding energy) and the greater number o f such interactions in the “prism” structure could lead to an over-estimation o f the binding energy despite the greater “strain” o f the hydrogen bonds (average O —H O angle is —148° for the “prism” versus —154° for the

“cage”) in the structure. While none o f these structures are either “ring”- or “stacked

cube”-Hke, the “book” is a member of the stacked cube family (a cube of8 water molecules with an edge removed). The experimental evidence that the ring structures are not the

global m in im u m at n = 6 is the dipole moment and infirared spectroscopy. Bowen, in electron attachment experiments, found strong mass spectrometric peaks of (HnO)^ at n= 2 ,

6 , and 7, and the complete series at n > ll as shown in Figure 1.9.^^ For a cluster to bind an excess electron, a large dipole must be present, or the electron must be internalized, as is the case for the larger clusters. The exact dipole value required is an open question and is

11 probably a function o f cluster size. For the hexamer, the cyclic structure has a PM3 dipole

moment o f only 0.21 debye, compared to values of 1.92, 2.61, and 1.71 debye for the

“book”, “prism”, and “cage” structures respectively. This would seem to preclude the ring

as the dominant form experimentally, but the large dipole o f the rem a in in g structures makes

a conclusion about the actual structure difficult to make firom this data alone. In recent

vibration-rotation-tunneling work, Saykally^^ has found the spectrum for the hexamer to be

most consistent with the “cage” form.

Stacked Cube Structures

Stacked cube structures, rings o f 4 water molecules aligned with each other (with related structures having less than 4 waters in the last rin^, become the global m in im u m structure

at n= 7. The complete cube at n= 8 , in particular, as been the subject o f extensive studies.^^"^^^ Examples o f the calculated family o f structures are shown in Figure 1.10.

n 1 PM3 Prediction A

8 -6.074 --

1 2 -6.503 -6.498 -0.005 16 -6.765 -6.768 0.003

2 0 -6.928 -6.929 0 . 0 0 1

24 -7.036 -7.036 0

28 -7.113 -7.113 0

32 -7.171 -7.171 0

36 -7.216 -7.216 0

Table 1 .4 . The numerical comparison of the P M 3 calculated excess m o le c u lar binding energy per water molecule (E ^ and the predicted value from the fit o f the stacked cube structures. The octamer is not included in the fit because it lacks the highly “strained” internal rings and is more stable on a per molecule basis than the rest o f the family.

12 n PM3 Prediction A

1 0 -6.227 -- 15 -6.603 -6.599 -0.004

2 0 -6 . 8 6 6 -6.862 -0.004

25 -7.030 -7.031 0 . 0 0 1 30 -7.141 -7.139 -0.003

Table 1.5. The numerical comparison of the PM3 calculated excess binding energy per water molecule and the predicted value from the fit of the stacked pentagon structures. The decamer was excluded from the fit because it lacks the higher “strain” o f internal rings and is more stable on a per molecule basis than the rest o f the family.

The progression of the number of water molecules with 3 or 4 hydrogen bonds is a simple expression. The terminal rings each contain 4 molecules participating in 3 hydrogen bonds ( 8 water molecules total). The remaining, “internal,” molecules in the cluster (n- 8 molecules) all have 4 “strained” hydrogen bonds. This gives the expression for the growth of stacked cubes as

^abjsc ~ 8 )E 4 ^ (9 ), where (the energy contribution of water molecules with four hydrogen bonds) and

Ej^c (3 hydrogen bonds present) are -224.794(0.003) kcal/mol and -223.180 (0.002) kcal/mol respectively. These values contain the single water total atomic binding energy value of -217.22 kcal/mol as well as the fraction of the molecular binding energy that can be attributed to that type (n^ vs. n,) o f water molecule in the cluster. The overall estimated standard deviation (e.s.d.) for the stacked cube fit, CTso is 0.033 kcal/mol. The PM3 values for the family and the results o f this fit are shown graphically in Figure 1.11 and in numerical format in Table 1.4.

13 Other Stacked Structures

The stacked cube structures represent the global minimum structures from n»7 to an undetermined cluster size less than —145, where the cubic ice becomes more stable. Four- member rings are not the only possible stacked structure in this range. Five- and six- membered rings also form families o f similar binding energy on a per water molecule basis.

These structures are shown on an excess binding energy plot with the stacked cube structures presented previously (Figure 1.12).

The fitting expressions for the stacked pentagons and hexagons are similar to those for the stacked cubes. For the pentagons, the equation becomes,

^ab.sp = +(ri—10)E4^p ( 10).

with E 4 3 P and Ej^p -224.899 (0.009) kcal/mol and -223.279 (0.006) kcal/mol respectively, and a bulk value o f -7.68 kcal/mol. The overall error in the fit,

These results are presented in Table 1.5. Similarly, the stacked hexagons are,

^abXH ~ “1 ^)^4.SH (H)» with E^^H—-224.883 (0.002) kcal/mol, E3^„=-223.479 (0.002) kcal/mol (see Table 1.6) and a value o f -7.67 kcal/m ol at bulk. The error in this fit, CTsp(=0.016 kcal/mol (e.s.d.).

14 n H PM3 Prediction A

1 2 -6.406 -- 18 -6.728 -6.727 -0.003

24 -6.961 -6.961 0

30 -7.102 -7.102 0

Table 1.6. The numerical comparison o f the PM3 calculated excess binding energy per water molecule and the predicted value from the fit of the stacked hexagon structures. The dodecamer was omitted because it lacked the highly “strained” internal rings o f the rest o f the family.

The stacked pentagons do not, as a fam ily, overtake the stacked cubes in binding energy

per water molecule until n«28 (according to their fit expressions), but at n = 1 0 , the pentagonal structure is more stable (Figure 1.13). The stacked hexagons also overtake the stacked cube family in stability at n«45, although it does not become more stable than the pentagon structures. This is due to the fact that the internal angle o f a planar pentagon is closer to the tetrahedral 109.5° angle. It is interesting to note that a stack o f the other

“ting-like” hexamer structure, the “book” form (see the hexamer case section) is 5-6 kcal/mol less stable at n = 1 2 .

Hexagonal Ice Structures

Normal, or hexagonal ice (sometimes denoted ice-I or, in this work, ice-Ih), is the dominant form of bulk, solid water from ~-120°C to 0°C. This is the most common form o f solid water found in nature and is final form o f ice prior to melting (at atmospheric pressure). Small atmospheric ice crystals account for the observance o f haloes-^ about the

15 sun at 22° and 46° and may be responsible for larger than expected atmospheric absorption o f sunlight.—

Most o f its properties have been interpreted in terms o f its crystal structure and the

forces present in the crystal lattice.The basic structure, first proposed by Bragg,^®^ is well established for ice-Ih just below the melting poin t,and recendy, with a low

temperature annealed crystal firom neutron scattering experimentsTable D - 6 (see

Appendix D) lists the properties of ice-Ih and several other polymorphs.

In the bulk ice-Ih structure, each water molecule is hydrogen-bonded to the four nearest neighbors. The O — H bonds directed towards the lone pairs of two neighbors (forming

O — H O' hydrogen bonds) and the remaining neighbors reciprocate by directing an

O — H bond towards the lone pairs of the oxygen under scrutiny (O H — O' hydrogen bonds). The arrangement allows for 2 orientations, the hexagonal and cubic forms. The hexagonal arrangement, shown graphically in Figure 1.14, is an open lattice where the layers perpendicular to the z-axis (or c-crystal axis) consist o f fused hexagonal rings in a conformation similar to the chair form o f cyclohexane. The combined layers also result in hexagonal rings perpendicular to the z-axis, but these have the “boat” conformation. The structures utilized in this study were created from a large crystal based on the unit cell (See

Appendix D). Excess molecules were then removed until a likely structure candidate was present. One boundary condition was used, however. A water molecule with only 1 hydrogen bond is very easily distorted firom its “crystal” position and, as a result, none o f the structures in this study contained such a water molecule. The smallest structure possible for this arrangement and boundary condition is a local minimum octamer, (HzO),, shown in

16 Figure 1.15. At one point, this structure was purported^®^ to be the global minimum,

although later work has since shown this not to be the case.

The sub-family o f structures for determination of the constants E^, Ej, and E, are

shown in Figure 1.16. The structures are indexed by S, the n u m b e r o f “shells” of water

present around a central water pair and the energy was fit to the expression.

^To«aI.Ice-Ih ~ "^4^4 + IlsEs.Ice-Ih ^^îE^Jce-Ih (12).

The progression o f n^, n„ and n^ to bulk was determined firom the smallest clusters in the

sub-family. Table 1.7 shows the total n, and the number of waters in each cluster which

participated in either 2, 3, or 4 hydrogen bonds respectively. These values were then used

to obtain mathematical expressions, as a function of size (indexed by the number o f “shells’

in the cluster), for this subfamily:

s Total * * 2 «3 ®4

-- 8 6 2 0

1 34 1 2 14 8 2 152 30 50 72 3 402 60 98 244 4 832 1 0 2 158 572 5 1490 156 230 1104 6 2424 2 2 2 314 1888

Table 1.7. The progression of water molecule types in ice-Ih. The fitting expression requires a mathematical expression for the number o f water molecules in a particular structure that have 2, 3, or 4 hydrogen bonds (nj, n,, and n^ respectively) as the clusters grow towards bulk. The clusters used were constructed and the numbers in each category tabulated as a function o f the “growth index,” or number o f shells, S

17 J n 1 PM3 Prediction A 26 -5.903 -5.943 0.040 34 -6.032 -6.070 0.038 48 -6.350 -6.290 -0.060 70 -6.565 -6.574 0.009

Table 1.8. The aumehcal comparison of the PM3 calculated excess binding energy per water molecule and the predicted value from the fit of the hexagonal ice structures.

n4,Icc-lh(S) = 2 3 6 + £ 32 + X24(i-I) (13) j=3 i=3

s-i n„^D,(S) = 50+2y;6(i+2) (14) i=2

s-i n,.,^n.(S) = 30+2j;3(2i + l) (15).

The results o f this fit are shown in Table 1.8 and Figure 1.17. The values for

E 3j„.,h and Ey„.,H were found to be -225.578 (0.13), -223.468 (0.09), and -221.390 (0.19) kcal/mol respectively (recall the single water value is -217.22 kcal/mol). The error in this fit,

CT,„ ,h, was found to be 1.4 kcal/mol (e.s.d.) and an extrapolated bulk value, -8.36 kcal/mol. The slope in n *''^ space was found to be 11.7 kcal/mol, which can be considered a PM3 estimate of the surface tension for the sub-family.

Cubic Ice Structures

Cubic ice, often designated ice-Ic, is a low temperature, standard pressure, polymorph of solid water. It can be formed by warming amorphous ice (believed to be a glassy form

18 o f water at low temperature, <133 PC, from the deposition of water vapor). This

crystallization has been reported to occur at ~144 with an accompanying energy

release o f 0.2-0.3 kcal/mol,^®^» but this transition has recendy been reported to occur

anywhere from 143 to 203 dependent on the thermal history o f the sample. It

can also be formed by condensation of water vapor onto a surface held between 133K and

153 or by warming any of the quenched high-pressure ices.^^^-> In any case,

further heating causes the ice-Ic sample to transform irreversibly to ice-Ih.

There is some speculation about the presence o f cubic ice in nature. One o f the earliest

observations, in 1629, that has been attributed to natural occurrence of this ice polymorph

is the rare “Schemer’s halo” at an angle of —28° from the sun or moon.-^ Halos due to ice

crystals in the upper atmosphere are not uncommon, but the hexagonal prism structure o f

Ice-Ih gives the common halo angles-^’ of —22° on the hexagon sides and —46° if the

ray passes through the 90° terminal face o f the crystal (see Figures. 1-3 o f reference 27).

The index of refraction of cubic ice has not been ascertained, but if it is similar to that o f

ice-Ih, an angle near 28° is expected if the light passes through two sides o f the crystal

sharing an apex.-^ The rarity o f Schemer’s halo can be attributed to the relatively narrow

temperature range where laboratory experiments have found cubic ice to form on a surface, but the fact that it is present for a very short time, on the order of a minute, could also contribute to the lack of observations. The other tangible evidence comes from the structure of some dendritic snowflakes, where the c-crystal axes of the arms are often at

70° to one another.^ While the physical properties o f this crystal are that o f ice-lh, the

19 angle is atypical One speculation^ is that the core o f the flake is an octahedral crystal o f

cubic ice with the hexagonal ice growing from the surface o f that internal structure.

Theoretical calculations have confrrmed that this is a low energy interface.^

The structure o f cubic ice has been studied with X-ray^^® and electron^

diffraction techniques. The distribution of oxygen atoms is sim ila r to that o f ice-Ih

discussed previously. Each water molecule is tetrahedrally hydrogen bonded to its four

nearest neighbors. Like the hexagonal structure, the molecules are arranged in layers. Also

like the previous structure, the layers form hexagons perpendicular to the layers. These

rings, however, are in the “chair” form in a lattice where the oxygen atoms are isostructural with the diamond lattice, unlike the ice-Ih structure. The smallest structure possible, with

the same boundary condition o f eliminating structures with water molecules participating in a single hydrogen bond, is a local minimum decamer, (H,0),o, shown in Figure 1.18.

The minimized atomic binding energies for a sub-family o f cubic ice were calculated and

fit to the general expression given previously in Eq. (12) whereby the constants can be compared to the value for a single water (-217.22 kcal/mol). The structures used in this fit, shown in Figure 1.19, are the typical octagonal structure observed for a cubic crystal.-^ The series of structures were generated by adding a shell to one of the “pyramid-like” sides and presented in Table 1.9. The number of molecules in each of the bonding categories is a

function of the shell index, S and the number of molecules in each category (the number of waters with 2, 3, or 4 hydrogen bonds) was found to progress mathematically to bulk by the following equations:

2 0 s 1 Total * » 2 a . n.

-- 1 0 6 4 0

1 35 6 24 5 2 84 6 52 26 3 165 6 8 8 71 4 286 6 132 148 5 455 6 184 265 6 680 6 244 430

Table 1.9. The progression o f water molecule types in ice-Ic. The fitting expression requires a mathematical expression for the number o f water molecules in a particular structure that have 2, 3, or 4 hydrogen bonds (n?, n^, and n^ respectively) as the clusters grow towards bulk. The clusters used were constructed and the numbers in each category tabulated as a function o f the “growth index,” or number o f shells, S.

n 4.lce-lc (S)=S- Ê 4 (j- l) É 4 ( 2 i - 1 ) (16) Li=> i= l

ni,.^,c(S) = 4(S + I) + X *<' + ') (17) i=I

•^2 ,Ice-rc(S) — 6 (18).

The constant value for n, as a function of size is due to the fact that doubly hydrogen- bonded waters are only present at the comers of the octahedron.

The parameters, and Eyc^^ were fit to Eq.(7) and yielded values of-

226.31 (0.13) kcal/mol, -223.64 (0.07) kcal/mol, and -221.22 (0.12) kcal/mol respectively

(n,=0). The overall error in the fit, is 0.65 kcal/mol (e.s.d.) and the extrapolated bulk value, Ecaj„-ici is -9.09 kcal/mol. The tabulated results are shown in Table 1.10 and displayed in Figure 1.20.

21 n PM3 Prediction A

1 0 -4.916 -4.970 0.054 18 -5.658 -5.627 -0.031

35 -6.374 -6.385 0 . 0 1 1 53 -6.709 -6.704 -0.005

84 -7.071 -7.072 0 . 0 0 1

Table 1.10. The numerical comparison o f the PM3 calculated excess binding energy per water molecule (E,[J and the predicted value firom the fit o f the cubic ice structures.

Empty Clathrate Cages

Clathrates structures, particularly the X (H ,0 )2 „ species (X is an associated atom,

molecule, or ion) is o f interest in a number o f fields. The discovery o f the extraordinar}’^

stabilityof H (hTO)?: has been explained^— as an HjO* ion inside a clathrate cage

with 20 water molecules. The empty cage structures have become increasingly of interest

and have begun to receive attention from the theoretical community.

Graph theoretical work by Singer et aL has found 30,026 unique structures for the

(H^O),,, cage (Figure 1.21) based solely on hydrogen bonding topologies which satisfy the

Bemal-Fowler “ice rules.” To date, the energy o f 193 o f the structures have been calculated

and they span an estimated 53 kcal/mol range (total atomic binding energy) from least to

most stable.A sim ilar analysis would yield a family o f empty cages that could be

analyzed in much the same way as the earlier families. While the empty cages are not

necessarily of interest in and of themselves, knowledge o f the cage can help sort out the

factors contributing to the excess binding energy of a species associated with the cage. The

addition o f this clathrate cage to the excess binding energy plot is shown in Figure 1.22.

3L A b I n itio Results

Calculation of the excess binding energy per water molecule was performed with several

levels of theory using ab initio methods. This is described in detail elsewhere.

Gaussian94^^® was utilized for all the calculations on three Pentium 166MHz personal

computers and a RISC workstation.

Slater-type orbitals, STO-KG (K=3-4) basis sets,^^^"^^^ at the Moller-Plesset (MP2)

level give values shown in Table 1.11. It has been shown that the larger expansion (K=4) is

more suitable^^ when long range interactions become important, specifically the case of

hydrogen-bonded complexes, such as neutral water clusters.

Ring Structures n PM3 scaled PM3 STO-3G STO-4G

3 -3.36 -4.47 -8 . 0 2 -8.57 4 -4.59 - 6 . 1 0 -14.53 -15.64 5 -4.77 -6.34 -15.62 -16.73 6 -4.91 -6.53 -15.82 -16.94 7 -4.87 -6.48 -16.83

Stacked Cube Structures

6 -5.04 -6.70 -16.20 -17.13 7 -5.27 -7.01 -16.97 -17.98 8 -6.04 -8.03 -18.84 -20.32

Table 1.11. Numerical results from PM3 and 2 STO-KG (K=3,4) ab initio levels with MP2 corrections.

23 Ring Structures n PM3 scaled PM3 3-21G 6-31G 6-31G* 6-31G** 3 -3.36 -4.47 -14.27 -9.13 -7.88 -7.59 4 -4.59 - 6 . 1 0 -17.96 -11.84 -9.86 -9.51 5 -4.77 -6.34 -18.58 -12.53 -10.09 6 -4.91 -6.53 -18.95 -12.96 -10.27 -9.62 7 -4.87 -6.48 -18.81 -12.95

Stacked Cube Structures

6 -5.04 -6.70 -19.91 -12.77 -10.59 7 -5.27 -7.01 -2 1 . 1 0 -13.87 8 -6.04 -8.03 -23.60 -15.26

Table 1.12. Nvunerical results firom PM3 and several Gaussian-type orbital ab initio levels with MP2 corrections.

Gaussian-type orbital basis sets, 3-21G^^^"^^^ and 6-31G,^^® were also used to calculate the excess binding energy per water molecule at the MP2 level (Table 1.12).

Polarization functions for the higher level, both 6-31G* and 6-31G**,^'^^’ were included in the study. These results are also tabulated in Table 1.6. Each succeeding level gives results that are h^her energy (less stable) than the previous, approaching the PM3 and experimental values.

Finally, two basis sets o f Dunning were used (for comparison to the results o f Xantheas and Dunning. The cc-pVDZ and aug-cc-pVDZ sets,^^^^^® were utilized at the MP2 level to calculate the excess binding energy per water molecule (Table 1.13) and compared with the values derived firom the binding energies reported by Xantheas.Table 1.6 lists the literature results reported as well as the point firom this study that coincides with Xantheas.

24 Ring Structures aug-cc- n PM3 scaled PM3 cc-pVDZ pV D Z Ref. 95 3 -3.36 -4.47 -8.00 -5.45 -5.521 4 -4.59 -6.10 -9.99 -7.281 5 -4.77 -6.34

6 -4.91 -6.53 7 -4.87 -6.48

Table 1.13. Numerical results from PM3 and two o f basis sets o f Dunning with MP2 corrections.

The E,y values determined by these calculations are presented graphically in Figure 1.23.

The curve labeled “Scaled PM3” is discussed below.

Discussion

The intermediate region o f the cluster regime is difficult to access with computational methods. The large size and number of conformers and structural isomers make ab initio calculations, even with minimal basis sets, rim e - and computationally-intensive. While early versions of semi-empirical methods gave unacceptably large errors^^ in bulk determinations

of certain thermochemical properties, namely heats o f formation and heat capacity, more

recent methods, such as PM3, AMI, and some o f the specialized water-water potentials give

useful results and are able to extend the practical cluster size range available to computational analysis.

The PM3 results for n ^ o f this study yielded no surprising results. Vasilyev-® reported many o f the same results in 1993. The inclusion of the chain structures may at first seem superfluous as only the d im er is a global m in im u m . However, Suhai^^*^ used an alternating

25 structure has been used, when projected to infinite length, as a model for ice. The d im er

“chain” gave way to cyclic structures at n=3. The extensive studies o f the dimer structure

32,47-49, 64, 65, 151 energetics^^’ and the ready accessibility o f the trimer to setni-

empirical^®’ 81-87 initio studies^^’ 88-90 means this transition is well defined.

When the cluster size reached n= 6 , the rings no longer represent the most stable, lowest

energy, structural family. The stacked cubes, pentagons, and hexagons (all nearly equivalent

in stability) becom e the global minimum at n=7, with a curve that slowly flattens as the

effect o f the three-bonded water molecules o f the terminal rings becomes diluted in an

ever-increasing number of four-bonded waters, extrapolating to bulk values o f -7.57, -7.68,

and -7.66 kcal/m ol respectively.

At n«145 and 1000, the ice structures, cubic and hexagonal respectively, overtake the

stacked ring series in overall stability on a per water molecule basis. There exist a large

number of structural sub-families that fit in each class of ice, making determination of the

most stable sub-family difficult at best. The fitted sub-families represent one possible

solution.

A problem arises if the transition to ice structures is from the stacked ring series. At

such a large n, the stacked ring structures are not very compact, resembling in many ways a

two-dimensional chain structure more than a hulk-like three-dimensional entity. The

probability that another cluster family exists between the size region where stacked tings are

the global minimum and the ice structures dominate and is more stable is an open question.

The determination o f this family, if it exists, is beyond the scope o f this work, but the assignment of energies for two-three- and four-hydrogen bonded waters provides an avenue

26 o f research. A topological mapping routine is possible, with appropriate boundar}' conditions and based on these fitted energies, and can explore the number of each category o f water molecules to give clues to the structure of such a family.

The fact that ice-Ih is extrapolated to bulk is important. A bulk value for the binding energy o f hexagonal ice at 0 K. is available.The ratio o f the bulk value, -11.3 kcal/mol, and the extrapolated value is found firom the curve fit. Substitution of the expansion of

Ec£ (7) for in Eq. (1) yields

In the limit as n—>oo, n 4 wnTooi, n^, nj, and n, become negligible and Eq. (19) becomes

^Ext.Ice-Di ~ E^^(bulk) = E 4 — E, (20), and provides a way o f calibrating the results o f PM3 calculations. This ratio.

«PM3 = (21), Ext,Ice-Ih applies to the entire range o f cluster size and was determined to be 1.35. To some degree,

“absolute” values for the excess binding energy per water molecule for any optimized water cluster structure found by PM3 are possible via,

^xb.Scaled ~ ®^PM3^xb,Calc (22),

27 though the result is more credible at large n, near where the ratio was determined. The

scaled plot o f the families determined in this study are shown in Figure 1.24.

Application o f this calibration factor to the ring structures, early stacked cube structures

(n^ ), and the “book” form of the hexamer determined values for these structures for

comparison to ah initio values calculated at several levels of theory. The plots shown in

Figure 1.23 clearly show the level o f theory required to approach the answer given by the

scaled PM3 curve. Slater-type orbitals, even if the empty orbitals at K =4 are included,

clearly overestimate the value o f the hydrogen bond and is worse as the cluster gets larger.

It is clearly an unphysical result to have a value o f Ej, for 3 < n ^ be greater than that o f the

bulk hexagonal ice values. The other Gaussian-type orbitals fare little better until the

polarization functions are included, as represented by the MP2/6-31G* and NrP2/6-3lG**

curves. While neither plot actually exceeds the bulk E^y at n ^ , both are suffidendy close to

the value that larger structures would be approximately the same energy on a per molecule

basis, resulting in litde energetic reason for larger clusters to form.

The results from the HF/aug-cc-pVDZ and MP2/aug-cc-pVDZ basis sets, as reported

by Xantheas,^^^ actually bracket the predicted values for the cyclic structures from the

scaled PM3. It is possible that small corrections to these basis sets, such as basis set

superposition error (BSSE) counterpoise techniques, 153-156 may come closer to the

predicted values.

The comparison of the two ice polymorphs leads to several interesting conclusions.

The PM3 calculated structures and energies are all 0 K values. This method predicts the

2 8

Jt" cubic form of ice to be more stable over the entire size range, and the difference predicted at bulk is 0.9 kcal/mol. It is not surprising that the cubic form is more stable at low temperature as it is formed in the range discussed earlier. It would probably form lower than the specified range except for the severely slowed molecular motions that contribute to rearrangement, resulting in the amorphous form.

Since the hexagonal form is the dominant one atmosphere polymorph firom ~-120°C to the melting point of 0°C, the difference must either lie in the temperature response o f the crystal structure or the method o f calculation. The latter explanation is a possibility if an incorrect sub-family were used for either of the polymorphs. The ice-lc form is the expected macroscopic octahedral structure and is a reasonable choice. The sub-family of hexagonal ice was chosen due to the resemblance to the cubic structure and could represent a problem. As was discussed in the cubic ice section, the macroscopic crystal of ice-lh is a hexagonal prism shape, but the structure grows quickly as a function of “shells,” and is not as easily calculated or, as a result, extrapolated to bulk.

If the temperature effects are the reason for the difference, several lines o f reasoning found in the literature have a bearing. The cubic form will spontaneously undergo a phase transition to the hexagonal form upon heating. The energy change was found experimentally^®®’ to be —0.037 kcal/mol. Much o f this energy, however, has been attributed to the release o f surface energy when small ice-lc crystals transform into larger ice-lh crystals and the actual energy difference in the crystal lattice may be <0.005 kcal/mol, further complicating predictions.

29 One possible explanation for the change in global minimum from 0 K to room

temperature is hydrogen ordering in the structure. Bjerrum, using a point-charge model for

the water molecule, predicted that at room temperatures, the hydrogen atoms in hexagonal ice are more ordered than in ice-lc, up to the melting point. He attributed this to electrostatic interactions o f the molecules along the unique c-crystal axis. This is not consistent with later work by Pitzer^^® who showed that, if more interactions are included, the Bjerrum model breaks down. However, Pitzer did show that if a small amount o f ordering is present, hexagonal ice is the more stable form.

All the ice structures presented in this work were derived from unit cells, and the hydrogen atom positions were ordered to prevent an additional variable from entering the progression of the family structures. One structure was selected to test the effect of this ordering and the hydrogen positions were randomized. Optimization of this cluster gave a total PM3 energy <0.3 kcal/mol different from the ordered structure. This work does not prove nor disprove the notion that hydrogen ordering is responsible for the room temperature bulk dominance of the hexagonal form, but it does lend credence to the idea that there are discernible differences in the large cluster regime. In this vein, Beaumont^ explored the different reported unit cells for cubic ice. The one reported by Leadbetter^®^ used KOH as a dopant to help order the cubic structure, and the resulting crystal, referred to as the “C-structure,” had a definite dipole moment. Beaumont used the expansion of high-pressure ices instead o f KOH and (bund the “P-Structure” which did not have a large dipole, but turned out to be energetically equal rather than more stable. In a recent paper,

Dunning and Payneused a “supercell,” where 2 unit cells were combined and disordered

30 in their interior, to perform some ab initio studies on ice at a large (aug-cc-pVDZ) basis set

with periodic boundary conditions to avoid this problem.

The general fit expression proved to be quite sensitive to the family o f structures used.

In a separate series o f hexagonal ice structures, the error in the fit was quite small, 0.5

kcal/mol. Addition of an ice-lh cluster firom a different, but similar, sub-family, however,

increased the error to 5 kcal/moL It has been suggested"^ that the fit values for E; (i=l-4)

can be used to determine a general continuum fit firom dimer to bulk for global m in im u m

structures, although the details are not clear at this time.

Conclusions

The series o f water clusters in several fam ilie s has been optimized with PM3, a semi- empirical method. These families have been fit to a general expression to give bulk values

and crossing points. The calculated PM3 bulk value of hexagonal ice was found to be -8.36

kcal/mol. This value, when compared to the experimental bulk value o f -11.3 kcal/mol,

gave a global scaling factor, otp^^, o f 1.35. This value was applied to the n < 8 clusters and compared to several levels o f ab initio calculations. It has been determined that the

MP2/aug-cc-p VDZ basis set is the minimum level o f theory required to approach the predicted value o f these clusters derived from the scaled PM3, but at a much greater cost in computational capacity and time.

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35

3L 116. Lee, C.W. J M eteonlSoc]pn 50, 516 (1972). 117. Kobayashi, T., Furukawa, Y. & Takahashi, T . /C^st Growth 35, 262-8 (1976). 118. Shallcross, F.V. & Carpenter, G .B . J Chem P fys 26, 782-4 (1957). 119. Blackman, M. & Lisgarten, N .D . Proc R/^ Soc A 239, 93-107 (1957). 120. Shmaoka, K. J Pfys SocJpn 1 5 ,106 (1960). 121. Lin, S.S. Rev Set Instrum 44, 516 (1973). 122. Searcy, J.Q. & Fenn, J.B.J Chem Pfys 61, 5282 (1974). 123. Wei, S., Shi, Z. & Castieman, A.W., Jr. /Chem Pfys 94, 3268-70 (1991). 124. Nagashima, U., Shinohara, H., Nobuyuki, N. & Tanaka, H. / Chem Pfys 84, 209-14 (1986). 125. Yang, X. & Castieman, A.W., J r ./y lwChem Soc 111, 6845-6 (1989). 126. Holland, P.M. & Castieman, A.W., Jr./ Oe/wPfysl2, 5984-90 (1980). 127. FChan, A. Chem Pfys Lett 2SS, 574 (1996). 128. Khan, A. Chem Pfys Lett 253, 299 (1996). 129. Earhart, A.E. M.S. Thesis, The Ohio State University, (in preparation). 130. Frisch, M.J., Trucks, G.W., Schlegel, H.B., Gill, P.M.W., Johnson, B.G., Robb, M.A., Cheeseman, J.R., Keith, T , Petersson, G A.., Montgomery, J A., Raghavachari, K., Al- Laham, M A ., Zakrzerski, V.G., Ortiz, J.V., Foresman, J.B., Peng, C.Y., Ayala, P.Y., Chen, W., Wong, M.W., Andres, J.L., Replogle, E.S., Gomperts, R., Martin, R.L., Fox, D.J., Binkley, J.S., Defrees, D.J., Baker, J., Stewart, J.P., Head-Gordon, M., Gonzalez, C. & Pople, J A.Gaussian 94, Revision B.2. Gaussian, Inc., Pittsburgh, PA, (1995). 131. Hehre, W.J., Stewart, R.F. & Pople, J A .Chem / PfyshX, 2657-64 (1969). 132. Hehre, W.J., Ditchfieid, R., Stewart, R.F. & Pople, J A .Chem / Pfys 52, 2769-73 (1970). 133. Pietro, W.J., Levi, B A ., Hehre, W.J. & Stewart, R.F.Inorg Chem 19, 2225 (1980). 134. Pietro, W.J. & Hehre, W.J. /Comput Chem 4, 241 (1983). 135. Binkley, J.S., Pople, J A. & Hehre, W .J./v4wChem Soc VS2, 939 (1980). 136. Gordon, M.S., Binkley, J.S., Pople, J.A., Pietro, W.J. & Hehre, W.J.Am / Chem Soc 104, 2797 (1982). 137. Dobbs, K.D. & Hehre, W.J.J Comput Chem 7, 359 (1986). 138. Dobbs, K.D. & Hehre, W.J. / Comput Chem 8, 861 (1987). 139. Dobbs, K.D. & Hehre, W.J. / Comput Chem 8, 880 (1987). 140. Hehre, W.J., Ditchfieid, R. &Pople, J J Chem P fys 56, 2257-61 (1972). 141. Francl, M.M., Pietro, W.J., Hehre, W.J., Binkley, J.S., Gordon, M.S., DeFrees, D.J. & Pople, J A . /Chem P fys 77, 3654-65 (1982).

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36 143. Hariharan, P.C. & Pople, J.A.Theoretica Chimca A rta 2&y 213 (1973).

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37 Bifiircated

Dipole Opposed

Linear Hydrogen Bond

Figure 1.1 The minima for the water dimer, (H20)2- The linear hydrogen bonded structure is the experimentally determined global minimum.

38 ^ Y ^ Y Y

Figure 1.2. The alternating chain structure. A similar structure was used bv Suhai as his model, projected to infinite length, for ice-lh.

Figure 1.3. The bridging chain structure. Both the alternating and bridging chains are present in the ice structures, but in differing degrees.

39 Excess Binding Energy per Water Moiecuie - Chain Structures 0 1

-1 Bridging Chain -2 -|

-3

-4 - « -5 - \ u Alternating Chain -6 -

0 > -7 O e -8 - lU -9 -

-10 - Eiqserimental Ice-lh -11 : Bulk Value atO K

-12 0.0 0.1 0.2 0.3 0.4 0 .5 0.6 0 .7 0.8 0.9 1.0 .1 /3

Figure 1.4. Excess binding energy plot for the chain fam ilies. The convergence of the curves at n=4 is due to distortion o f both families towards a similar structure.

40 Figure 1.5. The ting family structures. These are the minimum energy structures for the rings found by PM3. In each case, the individual water molecules donate one hydrogen bond and accept one hydrogen bond to form the “backbone” of the ring.

41 -887.25 kcal/mol -886.50 kcal/mol

oft -886.32 kcal/mol -885.10 kcal/mol

Figure 1.6. The four unique isomers of the tetramer with each water as a single­ donor/ single acceptor.

42

1 Excess Binding Energy per Water Moiecuie - Ring Structures

-3 -

-4 -

ce

Ring Structures >» ? co lU

-10 - Experimental Ice-lh -11 Bulk Value atO K

-12 0.0 0.1 0.2 0 .3 0 .4 0.5 0.6 0.7 0.8 0.9 1.0 n ■1/3

Figure 1.7. Excess binding energy plot for the ring structures. Ring structures for n=3...9 have been calculated and plotted on the same scale as the chain structures presented previously. The upward bend to the plot at n>6 is due to the “collapse” of the ring towards three-dimensional structures.

43 - O ^ A

a - “Ring’ b - “Cage” -1332.76 kcal/mol -1333.74 kcal/mol

f M ^ o i

c - “Prism” d - “Book” 1333.19 kcaPmol -1333.57 kcal/mol

Figure 1.8. Comparison o f four low energy hexamer water clusters. Four o f the low energy configurations for (HiO)^ are shown with the PM3 calculated binding energy. Structure a, the “ring,” was originally predicted to be the global minimum, but recent calculations have shown the other three structures to be more stable. It is generally believed that the “cage” structure is the globalm in im u m , but all three rem a in in g structures are nearly equivalent in energy and the ordering becomes dependent on the calculation method.

44 -1.0% HjO

n=2

15

1 1 &Z.0% HjO

c Ë

mass

Figure 1.9. Electron attachment mass spectrum of water clusters by Bowen, et alP~^ For an electron to attach to a cluster, the electron must either be solvated or the cluster have a large dipole moment. The continuum of peaks at n> ll is due to solvation of the electron. The isolated peaks at 2, 6, and 7 indicate the structures for the corresponding clusters must have a large dipole moment. The cyclic (H,0)g structure has a small dipole moment, while the other, more condensed, structures have PM3 calculated dipole moments >1.7 debye.

45

i j . U o « - o

( H j O ) g

- u 3 3 >

( ^ 2 0 ) 2 8

Figure 1.10. Example stacked cube structures. The highly strained hydrogen bond angles are consistent throughout the structure which allows the fitting routine to be employed.

46 Excess Binding Energy per Water Molecule - Stacked Cubes

-2 -

I::

-7 - stacked Cubes ui

-10 - Experimental Ice-lh Bulk Value at 0 K

-12 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 n -1/3

Figure 1.11. Excess binding energy plot for the stacked cube structures with the fitted curve. The stacked cube structures for n=8,12,...36 have been calculated and added to the previous plot of chains and stacked cubes. The extrapolation to bulk (n '^^=0) is found to be -7.56 kcal/mol.

47 Excess Binding Energy per Water Moiecuie - 6.0 Stacked Pentagons and Hexagons

Stacked Pentagons -^-6 5 I Stacked Hexagons « &-7.0 Stacked Cubes I UJ -7.5

- 8.0 — I— 0.0 0.1 0.2 0.3 0.4 0.5 .-1/3

Figure 1.12. Excess binding energy plot for the stacked hexagons and pentagons. The clusters and fitted expressions for the stacked hexagons and pentagons has been placed on a plot with the stacked cubes shown previously..

48 f

Stacked Pentagon Stacked Cube -2234.33 kcal/mol -2223.54 kcal/mol

Figure 1.13. Comparison o f the stacked pentagon and stacked cube decamers. The stacked pentagon decamer is more stable at n=10. This is despite the fact that the stacked cube family is more stable in this range. The global minimum for water clusters overall does not follow one family in particular as the value o f n is incremented, but in general follows the trends shown previously.

49 ( ® 2 ® ) 7 2

X 0 -

x . _ ^ x * a r r

— a

W - v L c ® — ► c

u - T X ^ ^

W - . < ' w cF

Figure 1.14. The crystal structure of ice-lh. Normal hexagonal ice is comprised o f sheets of fused hexagonal rings (top). The stacking of the layers form puckered hexagons in the conformational equivalent of the cyclohexane “boat” configuration (bottom). The c- crystal axis is unique in normal ice.

50 Q ^ -

L

Figure 1.15. The minimum ice-Ih structure. The cluster shown is the m in im u m structure with the ice-Ih bonding pattern that does not contain waters with only one hydrogen bond. Note that all the hexagonal rings present are in the “boat” configuration.

51 >C5 ~^

V: 2

L j =

( 1 ^ 2 0 ) 7 8

•r"S^

> . . , A , 4 s ç > W - i - t . > General ( 1 ^ 2 0 ) 1 5 2 Shape

Figure 1.16. Several ice-Ih structures. The shown structures o f normal ice were used to determine the values o f E^, E,, and E; for the family. These clusters also represent the smallest o f the structures with values o f n^, nj, and n, predicted by Eqs. (13), (14), and (15).

52 Excess Binding Energy per 0 Water Molecule - Hexagonal Ice

-1

-2

-3

-4 Ice-Ih I -5 - u -6

2 -7 - § -8 ■ lU -9 - n=1200

-10 Experimental Ice-Ih -11 ; Bulk Value at 0 K

-12 —I— 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 *-1/3

Figure 1.17. Excess binding energy plot for hexagonal ice structures. The ice-Ih structures and the fitted extrapolation to bulk is added to the previous plot for comparison. The PM3 predicted OK bulk value for ice-Ih is -8.36 kcal/mol (compared to the experimental -11.3 kcal/mol).

53 Figure 1.18. The minimum ice-Ic structure. The cluster shown is the minimum size structure for ice-Ic that does not contain molecules with only one hydrogen bond. N ote that, unlike the normal ice structure, which contained all “boat” configuration hexagonal rings, aU those present in this cluster are in the “chair” configuration.

54 ( 1 ^ 2 0 ) 3 5

General ( ^ 2 ^ ) 8 4 Shape

Figure 1.19. The fitted ice-Ic structures. These clusters follow the general shape, that o f the octahedron, o f materials with the cubic-crystal type at bulk, cubic ice, diamond, and silicon. These were used for the determination o f the values of E^, E,, and Ej and were used to determine Eqs. (16), (17), and (18).

55

JL Excess Binding Energy per Water Molecule - Cubic Ice

-2 -

-3 - I Ice-Ic -5 -

I -6 -

2 -7 - O C -8 - lU n»165

-10 - Experimental Ice-Ih Bulk Value at 0 K

-12 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 .-1/3

Figure 1.20. Excess binding energy plot for cubic ice. The values and extrapolated curve for cubic ice has been plotted with the previous families. N ote that the ice-Ic clusters are everywhere more stable than the hexagonal ice structures. The curve determines a bulk value of -9.09 kcal/mol.

56 Figure 1.21. The minimum energy (H,O ) 2 0 clathrate structure. This structure is a representative o f the empty clathrate cages that form another family for analysis.

57 Excess Binding Energy per Water Molecule - (HgO);^ Clathrate

I CO

«2 (HzO)2o Empty C -8 - LU Clathrate Cage

-10 - Experimental Ice-Ih -11 : Bulk Value at 0 K

-12 0.0 0.1 0 .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -1A3

Figure 1.22. Excess binding energy plot with the clathrate structure (HjO)^)-

58 Excess Binding Energy per Water Moiecule - ab initio

-2 - MP2/au9-œ-pVDZ’ -4 - PM3

-6 - O MP2/aufl-cc-pVDZ E MP2/6-31G~ -10 -

S -12 -

P -14 - Experimental Bulk MP2A8-31G UJ= -16 - Value at OK -18 - MP2/STO-3G

-20 -

-22 - MP2/STO-4G

-24 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 „-1/3 'S.S. Xantheas. J. Chetn. Phys.. 100 1994. pp 7523-7534.

Figure 1.23. Excess binding energy plot for several levels o i ab initio calculation. The ring and early stacked cube structures were calculated and plotted at several levels o f theory. The lower levels are shown to severely over-estimate the value of the hydrogen bond as many of the clusteirs have calculated binding energies per molecule greater than bulk. The MP2/aug-cc-pVDZ curve from the data of Xantheas is the closest to the predicted values from the scaled PM3 and could become better if BSSE counterpoise techniques are employed.

59

il Scaled Excess Binding Energy per Water Molecule

0 - Bridging Chains HF/aug-cc-pVDZ^ -1 - Alternating Chains y 2 - » -3 - (HgOzo cage. MPZ/aug-cc-pVDZ^

MP2/6-31G** Ice-Ih ' MP2/6-31G*

lU -10 Stacked Cubes Experimental Ice-Ih MP2/6-31G Value at OK

Figure 1.24. Scaled excess binding energy plot. The values for the clusters and fitted curves were scaled globally by the value o f Gtp\o=l.35. The hexagonal ice curve is forced to extrapolate to the experimental value of -11.3 kcal/mol. Cubic ice is predicted to be 0.97 kcal/mol more stable at OK. The included al^ initio results are for comparison at small values o f n.

60 CHAPTER 2

SINGLE ION BULK SOLVATION ENTHALPIES AND FREE ENERGIES OF SELECTED HALIDE IONS, HYDROXIDE AND .ALKALI METAL IONS FROM CLUSTER STUDIES

With the advent of large cluster experiments and nanosynthetic methods,^’ “ it is currendy technologically and scientifically important to characterize the evolution o f chemical and physical properties o f clusters from small gas phase monomers to bulk. It is important to establish in what ways cluster properties are both similar and different from their bulk counterparts. The first step, in connecting the properties o f small clusters to bulk, is to obtain the best bulk parameters available- The issue o f absolute values for single ion solvation properties is a long standing problem in electrochemistry.^ Traditional bulk measurements are performed on inherently neutral, dilute, electrolytic solutions and basically characterize the thermochemistry of a pair of oppositely charged ions. Cluster ion experiments circumvent the bulk limitation by examining the properties of isolated single ions.

61 For the uninitiated, the problem of absolute, single ion solvation thermochemistry can be illustrated by considering the enthalpy to place a gas phase chloride ion into water. The

solvation enthalpy is just the difference between the heats o f formation of Cl'(aq)and Cl (^:

C r (g) ^ Cl (aq); AH„,,JCr ] (1),

= AH?[Cl-(aq)]-AH?[Cl(g)] (2)

Using values o f AH °[C I~(g)] = -246 kj/mol and AH °[CI~(aq)]= -167 kj/mol from the

CRC’s NBS tables,^ one gets ^^[CI”] = +79kJ/mol which has the wrong sign and magnitude for this obviously favorable process (the value determined in this work is

AH 5oi,aj[Cl”]= -308.4 kj/mol). This contrived example simultaneously employs both the

AH °[H *(aq)] = 0 and AH fjH jCg)] = 0 conventions producing erroneous results. This problem is generally avoided by constructing thermochemical cycles with pairs of oppositely charged ions (as it is pairs that exist in the neutral bulk solutions on which conventional experiments are employed) effectively canceling the unknown calibration offsets for the conversion o f heats of formation of the aqueous ions to an absolute scale. Although electrochemists may employ reference electrodes, this problem can no longer be avoided by those experimenting on large size cluster ions.

In this and other cluster-ion-based work, the adopted value for the absolute solvation free energy or enthalpy of the proton is required to be consistent with both the body of cluster ion data and bulk thermochemistry. There have been a number o f observations regarding the connection o f stepwise cluster ion thermochemistry to single-ion bulk

62 solvation thermochemistry.^"^ This work builds on these observations, but avoids some o f the more drastic assumptions which may have prevented general acceptance o f the cluster ion constrained results. In the following section, the consequences are examined with regard to the cluster ion solvation data o f three different literature values for the absolute solvation enthalpy o f the proton.

Thetmodynamic Conventions for the Electron

G ood values o f bulk solvation free energies and enthalpies are available for pairs o f oppositely charged ions as given in Table 2.1 using heats and free energies o f formation.'^ A thermochemical cycle can be constructed with the following reactions:

A(s) + l/2B,(g) -> AB(aq) = A*(aq) + B~(aq) ; AH°[AB(aq)] (3),

A"(g)->A(s); -AH°[A"(g)] (4),

B-(g)^ 1/2B,; -AH°[B-(g)] (5), such that

AH„,^(AB) = AH“[AB(aq)]-AH°[A"(g)]-AH?[B-(g)] (6).

Care must be taken when considering the gas phase ion formation enthalpies. Two conventions are present when a cycle includes gas phase ions. Consider the ionization reaction,

A^A*+e-;AH^ ( 7 ) ,

63 (kJ/mol) N egative ion Positive ion OHF Cl Br I -1530.6 -1522.9 -1404.0 -1377.7 -1340.6 Li^ (-955.9) (-948.2) -829.3 -803.0 -765.9 Na^ (-850.1) -842.4 -723.5 -697.2 -660.3 K* -778.3 (-770.6) -651.7 -625.4 (-588.5) Rb^ (-755.7) -748.0 (-629.1) -602.8 -565.8

(kj/mol) Negative ion Positive ion OH F Cl Br I -1622.6 -1613.8 -1470.0 -1439.0 -1396.8 U* -1050.7 -1041.8 -897.9 -866.9 -828.8 Na" -936.1 -927.4 -783.4 -752.4 -710.3 K* -852.7 -844.0 -700.0 -669.0 -627.0 R b' -827.1 -818.9 -675.0 -644.0 -601.9

Table 2.1. Solvarion free energy and enthalpy to place a pair o f oppositely-charged gas-phase ions into water at 25°C from the free energies and enthalpies of formation o f the gas phase neutral atoms and the corresponding salt. The values in parenthesis were not directly available from data in ref 4. They were determined from the differences (Table 2.2) that were available.

at OK and 298K. In the OK case, is the adiabatic ionization energy, IE„ o f A. At

298K, AH^ is the enthalpy of the reaction for the ionization, AH,^, and is related to Eq. (7)

by.

A(jk ^ ■^298K ’ IHC(A) (8).

IHC(A*) (9 ),

e;'OK ®298K ’ IHC(e ) (10),

64

I where IFIC is the integrated heat capacities of the species. In both conventions, AH°(e~) is

zero at OK. As a result,

AH?(A*W = AH°(AU (11),

the energy difference between the molecule and the ion is simply the adiabatic ionization

potential.

When the temperature is raised to 298K, Eq. (11) becomes,

^Hf(A )29gK = AHf(A)29g^ — AHf(e )298K (1-),

where

= IE, + IHC(e") + IHC(A" ) - IHC(A) (13).

A typical assumption is that the integrated heat capacities for a molecule and its ion are

approximately equal, i.e. IHC(A)«IHC(A^, and Eq. (12) becomes,

AHXA1298K = AH ?(A) 2 9 gK - A H ?(e-) 2 9 gK +IE, +IHC(e-) (14).

In the “Electron Convention,” or EC, the electron is treated as a chemical element in its standard state, i.e. A H ° ( e ~ ) is zero at all temperatures. In this convention, Eq. ( 1 4 ) is reduces to

AH “(A*)298k = A H ?(A ) 2 9 gK +IE, +IHC(e-) (15).

65 'ITie “Ion Convention,” or IC, used in most o f the body o f literature on the thermochemistry of ions, has been stated several ways, but essentially assumes the integrated heat capacity o f the electron is equal to enthalpy of formation of the electron at any temperature. This reduces Eq. (14) to

+ IE , (16).

Usually, when the “Electron Convention” is used, the integrated heat capacity for the electron is estimated to be an ideal gas following Boltzmann statistics, 5/2RT. At 298K, this gives the relation between the two conventions as

^ ? ( A ^ ) .p8k(IC) = AH°(A"),,8 k(EC)- 6.197 kJ/mol (17).

A similar series of arguments exist for negative ions and electron affinities, resulting in the relation

AH"(B') 2 9gK(IC) = A H °(B ') 2 , 8 k(EC) + 6.197 kJ/mol (18).

Note the change in sign for the Boltzmarm factor.

Cluster Viewpoint of Bulk Solvation Enthalpies

The merits and difficulties associated with a particular value for the absolute solvation enthalpy o f the proton can be illustrated with regard to cluster work by plotting the absolute bulk solvation enthalpies o f a variety o f ions (that correspond to the absolute proton value of interest) vs. the cluster ion solvation enthalpy at a certain cluster size (n=5 in this case).

In the following paragraph the procedure to obtain absolute bulk single-ion solvation

66 AG^.„(ion 1) - AG%,(ioa 2) (k[/mol) Positive ions H \ Na" Li", Na" K", Na" Rb", Na" -680.5 -105.8 71.8 94.4

Negative ions F , OH Cl , OH Br , OH I , OH 7.7 126.6 152.9 189.9

AH%,(ion 1) - AH“„,,^on 2) (kf/m ol) Positive ions H", Na" Li", Na" K", Na" Rb", Na" -686.6 -115.3 83.3 108.5

Negative ions F , OH Cl , OH Br , OH I , OH 8.6 152.6 183.6 224.9

Table 2.2. Bulk average differences (first ion - second ion) in the single ion bulk solvation free energies and enthalpies. Uncertainties are not available for the thermochemical cycle used to find these differences because the original data (ref. 4) has apparendy been made internally consistent. Alternative cycles can be employed for the enthalpies (ref. 21) suggesting that a range o f uncertainties o f 3-10 kJ/mol is reasonable for all o f these values.

enthalpies given an absolute value for the proton is described. Following that is the procedure for obtaining the corresponding cluster ion solvation enthalpy.

It is unclear how each entry in Table 2.1 is distributed between the two contributing ions of opposite charge. 'Ihe entries in Table 2.1 reveal accurate differences between ions o f similar charge as summarized in Table 2.2 with positive ions relative to Na* and negative ions relative to OH'; but there is no bulk constraints which fix the positive ions relative to the negative ions. The bulk situation is such that a determination of the absolute value for one ion effectively determines all the rest. Table 2.3 makes use of the data in Tables 2.1 and

67 ion AG„l« (kJ/moI) / moi j H' X Y Li' 574.7 + X 571.2 + Y Na' 680.5 + X 6 8 6 . 6 + Y K' 752.3 + X 769.9 + Y Rb' 774.9 + X 795.1 + Y OH -(1530.6 + X) (1622.6 + Y) F -(1522.9 + X) -(1614.0 + Y) Cl -(1404.0 + X) -(1470.0 + Y) Br -(1377.7 + X) (1439.0 + Y) I -(1340.7 + X) -(1397.8 + Y)

Table 2.3. The bulk solvation free energy and enthalpy o f various ions relative to H *.

2.2 to express the absolute solvation free energy (X) and enthalpy (Y) o f the proton. In the examples to follow, different sets of single-ion solvation enthalpies, generated using different absolute proton values, share a common set of bulk constraints.

In order to compare bulk solvarion values to cluster data, the extensive literature o f stepwise ion solvarion thermochemistry must be converted from stepwise values to solvarion. Throughout this paper, we employ enthalpies to illustrate the concepts, however it should be understood that for each enthalpic equation or relation, there is a similar equation or relation for free energies. The stepwise enthalpy for addition of water to an aqueous cluster ion is given by

Z(H,0),_, + H,0 ^ Z(H,0). ; AH._,, (Z) (19).

A set o f equations like Eq. (19) from i= l to n, will build up a cluster ion o f size n,

Z + H , 0 ^ Z ( H , 0 ) ; AHo.,(Z) (20)

Z (H ,0 ) + H^OZCH.O), ; A H ,,(Z ) (21)

68 Z(H,0), + H,0 Z(H,0)3 ; AH, 3(Z) (22)

Z(H,0)„_, + H,0 Z(H,0)„ ; AH„.,„(Z) (23)

Z + nH,0^Z(H30)„; ^AH._,,(Z) (24). i=I

This is only one part o f solvating an ion in a cluster o f size, n. A set o f equations for the

production o f a neutral water cluster must also be considered,

(H;0)^, + H,0->.(H,0),; AH._,„(H,0) (25).

Note that in this index, AH„ ,(H ,0) = 0, or, stated in terms of a chemical expression,

H,0^H,0; AHo,,(H,0) (26).

Subtraction of the equations for the stepwise construction o f the neutral water cluster, Eq.

(25), from the set of equations for the ion-molecule cluster, Eq. (19), gives the solvation relation:

Z + (H30)„ Z (H ,0). ; AH„,.„(Z) (27),

where,

AH„,.(Z) = X (Z) - è (H.O) (28). i=l i=i

The bulk single ion solvation enthalpy appears in the limit as n goes to infinity. 'ITie determination o f the solvation enthalpy o f an ion, Z, requires knowledge o f the stepwise

69 enthalpies o f neutral water clusters in addition to the stepwise enthalpies of the aqueous cluster ion. The summations of stepwise enthalpies for solute and solvent that appear in

Eq. (28) are called partial sums and are accumulated from literaturevaluesin Table 2.4.

The solvation enthalpy for each ion is simply the value entered in Table 2.4 for the ion of interest minus the value in Table 2.4 for water. The water value has been determined ^ ^ from PM3 water cluster calculations that have been corrected for zero point energy and calibrated for the excess energy o f hydrogen bonding by comparison to high level ab initio

(HF/aug-cc-pVDZ and MP2/aug-cc-pVDZ) calculations^^’ on small water clusters.

70 2AG,_„(kJ/mol)6-9 i=l n Species 1 2 3 4 5 U" -113.8 -192.9 -248.5 -279.9 -298.7 HjO" -100.8 -148.5 -185.7 -208.3 -224.6 Na" -78.7 -133.9 -171.6 -196.3 -211.4 K" -49.4 -86.6 -113.0 -131.4 -144.8 Rb" -41.0 -70.3 -91.2 -107.1 -118.8

OH -74.9 -121.8 -154.0 -177.0 -194.6 F -75.7 -125.1 -159.0 -183.7 -202.5 Cl -40.0 -67.6 -88.5 -103.6 -112.0 Br -33.5 -57.3 -75.3 -88.7 -98.5 I -22.6 -39.8 -53.2 -62.8 -69.3

H ,0 0 6.7 11.3 14.6 (16.9)

1=1 n Species 1 2 3 4 5 Li" -142.3 -250.2 -336.8 -405.4 -463.6 HjO" -144.0 -230.5 -302.0 -356.8 -407.0 Na" -104.6 -187.4 -251.8 -307.0 -355.1 K" -75.4 -143.1 -198.3 -247.7 -292.5 Rb" -66.9 -123.8 -174.8 -221.7 -265.6

OH -109.6 -188.3 -257.7 -317.1 -376.1 F -97.5 -172.4 -233.1 -290.4 -344.0 Cl -60.2 -113.8 -162.8 -208.4 -248.1 Br -54.4 -104.6 -152.3 -198.3 -243.4 I -43.9 -84.6 -123.5 -162.0 -199.7

H ,0 0.0 -15.7 -29.1 -45.7 (-66.4)

Table 2.4. Cluster ion solvation free energy and enthalpy partial sums o f various ions vs. cluster size. Ion values have been averaged from the shown references. Entries in parenthesis are estimated from the expected trend to the known bulk value.

71 Previous Bulk Single-Ion Solvadon Methods

Two sets of single-ion solvation values often quoted are those of Motris^^ (Y=-1103

kJ/moI) and Klots-^ (Y=-l 136.4 kJ/moI) The bulk solvarion enthalpies for various ions of

both Morris and Klots are plotted against the cluster solvarion enthalpies at n=5 in Figures

2.1 and 2.2 respectively. The ions have been separately fit to lines by the method o f least

squares.

Examination o f Figure 2.1 shows two disturbing features: 1) the positive ions are

grouped separately from the negative ions and 2) the negative ions display a different slope

(1.399) than the positive ions (1.109). The observations are disturbing because it is not

expected that there should be any generalized differences between the negative ions and positive ions on such a plot.-^ It hardly needs to be mentioned that there would be a large

standard deviation in AFI^» (or a low correlation coefficient) if the positive and negative ions were fit to a common line. Both of these problems cannot be fixed simply by changing the absolute proton value.

Klots presented a set o f absolute single-ion solvarion enthalpies and free energies based on cluster data. The approach assumes that all o f the stepwise additions of a water molecule to the cluster ions of and B are equivalent after n=5, i.e.

g a H ,_ ,X A 3 = g A H ,_ ,,(B -) (29). i=6 i=6

72 Inspection of a plot— of cluster solvation enthalpy vs. n shows that strongly solvated ions (like Li*) and weakly solvated ions (like Cl) have different stepwise solvation energies at n=6. It is not necessary to make such a drastic assumption, since the difference in the cluster ion partial sums could be taken as a relative or proportional indication o f the difference that will develop at bulk.

Figure 2.2 shows the plot o f the bulk single-ion solvation enthalpies based on the proton value reported by Klots o f -1136.4 kJ/mol against the cluster solvation enthalpies at n=5. Klots chose to work with a set of ions, Li*, Na*, K*, Rb*, Cl , and Br (but not OH and F). This set exhibits a common linear relation for the positive ons and chosen negative ions with a slope increased a bit (6%) over that of the positive ions alone and a standard deviation in AH,„,œ of 13 kJ/mol which has doubled from that of the positive ions alone.

By eliminating OH and F and adjusting the absolute solvation enthalpy of the proton, the method of Klots succeeds in finding a common fitting line for the positive and selected negative ions addressing the first problem with Figure 2.1, Le. the issue of why the negative ions, as a group, exhibit a different slope than the positive ions. It is now reasonably well established-^'-^ that large halide ions prefer to reside on the surface o f small water clusters, gradually working their way into the interior o f clusters upon addition o f 60 or so solvating water molecules.-^ The stepwise energetics associated with the hydration o f a surface state can be expected to be different than the stepwise energetics of the corresponding internalized ion at a given n. Klots noted that OH fails to conform to the pattern, but in retrospect it would seem that the large halides (he used only chloride and bromide) are the deviant species.

73 Coe*s Di£fetence Over Sum Cluster M ethod

The cluster ion method of C oe^ solves both of the above-mentioned problems of

Figure 2.1 and avoids the drastic assumptions inherent in the method o f Klots. The

method of Coe assumes only that both the positive and negative intemalit^ed cluster ions

have a common functional dependence in stepwise hydration, i.e. there is a c o m m o n linear

relation for internally solvated positive and negative cluster ions (that gets better as n gets

larger) between the cluster solvation enthalpy and the bulk solvation enthalpy. There is no

generalized energetic difference in the way negative and positive cluster ions progress to

bulk; all differences result from the differences in the most immediate interactions between

the ion and its nearby solvating water molecules. The cluster data indicate bulk differences

between ions of opposite charge in a relative way (the differences are not fully developed by

n=5); so the cluster relative differences were normalized to corresponding sums and

projected to bulk with the parameter l/s as follows:

■ a h „,„(a *)- a h „,„(B-)' "AH ^,.(A1-AH ^.(B-)' (30). AH„,„(AO-h AH„,„(B')AH„,„(A*)-h AH„,„(B-)

The normalized differences (difference over sum ratios) are projected to bulk without the

assumption of equivalency for A* and B an n>5 by using a scaling parameter, s, which is

common to all pairs of ions. The normalization of differences by sums makes this method

equivalent to minimizing the deviations from a common line for positive and negative ions

on the AH„|,0 vs. plots. The value o f s for any particular set o f bulk single ion

values does not need to be fitted and can be determined by dividing the sum of the

74

a. magnitudes o f the cluster difference-over-sum ration for all pairs of oppositely charged ions

by the same type o f sum for the magnitudes of the bulk difference-over-sum rados as

follows

A* B" s = (31). ZZ A ' B‘

The optimal set o f absolute single-ion values is the set that m in im ise s the differences between the left and right sides o f Eq. (30) over all pairs of oppositely charged internalized ions. In recalculating the results from ref. 21, two m in o r errors were discovered: 1) the partial sum for water at n=6 was used instead of n=5 giving a high s value but not affecting the absolute proton value, and 2) there was a small data entry error (the numbers in the table are correct). The final results should have been reported as AH^, = -1147.7±6.0 kJ/mol and s = 1.573. This value is 4.8 kJ/m ol smaller in magnitude than that reported in ref. 21, but is still within the error bars (±lcr).

The same type of vs. AH^, g plot as presented in Figures 2.1 and 2.2 is now presented in Figure 2.3 using the absolute proton solvarion enthalpy determined by the Coe difference-over-sum method (-1152.5 kJ/mol, i.e. as given in ref. 21). Only internalized cluster ions at n=5 were desired, so OH' and F were used and the large halides were left out. The subset o f internalized ions, Li^, Na*, K*, Rb*, OH', and F , all fit very well to a common line [ AH^,^(kJ / mol) =1.103 AH^, ^ -162.9]. The standard deviation (5.8 kJ/mol)

75 and the slope (1.103) are quite close to the values exhibited by the positive ions themselves

(as in Figure 2.2). This observation supports the fundamental assertion that the stabilization o f sufficiently-sized, internalized cluster ions by further clustering is energetically insensitive to the sign of the charge being solvated. The halides do not fall on the common internalized AH ^,vs. line because they are surface states at n=5 and progress to bulk with a different stepwise functional dependence. So the difference-over-sum ratio method also succeeds in grouping the positive ions together (without doubling the standard deviation of the positive ions alone as with the Klots method), although the group now consists only of internalized ions. The difference-over-sum ratio method further identifies the physical reason for the difference in the negative ion group slope, the surface vs. internal ion issue. In practice, the most important difference o f this method firom the Klots method is the use of the OH and F internalized cluster ions (excluded as deviant by Klots) in place of the surface state cluster ions (at n=5) of Cl and Br which have a different stepwise solvation functional dependence.

The Linear AH^^= vs. AH„,^ Method

The basic assumptions and contrasts of the Coe difference-over-sum and Klots methods are much more apparent when presented in terms o f AH^, „ vs. AH^,^ plots. In fact, the Coe difference-over-sum method criteria (choosing an absolute proton solvation enthalpy which minimizes the comparison o f the left and right sides of Eq. (30) over all the pairs o f oppositely charged ions) is mathematically equivalent to the plot with m in im um

76 deviations. This latter approach is less complicated mathematically, easier to explain, and

easier to visualize; so we present the following alternative to the Coe method.

The method starts by choosing a value o f Y, the absolute bulk solvation enthalpy o f the

proton. Once a value of Y is chosen, absolute, single-ion, bulk values are determined for aU

o f the ions by virtue o f the data in Table 2.3. This set o f bulk values is used with the

corresponding cluster solvation enthalpies at n=5 to fit to the following linear equation

where i is an index for the N internally solvated ions (both positive and negative) at cluster

size n=5. The slope (m) and intercept (b) are given by the standard least squares

expressions

(33)

b = V t V I (34).

i \ i

Note that m and b are functions o f Y. For any particular choice o f Y, the standard

deviation (cr) o f the fit is evaluated using

77 J -V------^ ------(35)-

The method proceeds by varying the proton solvation (Y) until

This AHjoi.a, vs. AH^,^ method is simpler but essentially equivalent to the Coe difference-over-sum method. To demonstrate this, the scaling parameter s o f Eq. (30) can

be related to the slope (m) and intercept (b) o f the AH„, „ vs. AH^,5 plot as

, 2 b S= l4----- (36), m

or expanded as

I 2 b 1 y ______1______n?)

m npairs A-.'Tpairs ) 4- AH„,^(B- )

where is the number of pairs of oppositely charged ions and the summation is over the pairs. Use of the values of m and b determined from both the bulk data and absolute proton solvation enthalpy firom ref. 21 gives s=1.578 which is essentially the same value determined by the difference-over-sum method as given above (s=1.573). The optimized

absolute proton solvation enthalpy using the AH^,, „ vs. method with the ref. 2 1 data set is -1151.3+6.5 kj/mol which can be compared to the difference-over-sum value of -

1152.5±6 kJ/moL The two methods have the same result within the random errors of the

78 procedures, although there may be some subde differences in the contribution of errors to each method.

Both the difference-over-sum and the vs. methods differ firom the Klots method in the use o f stepwise clustering energetics o f neutral water clusters. The difference-over-sum and AH^,^ vs. methods are not very sensitive to the value of the water partial sum (Table 2.4). This value can be changed by a factor o f more than two without any change in the optimal absolute value for the proton (although this does effect the value of s). Therefore, it is not essential for these methods to have the best value for the water partial sum. Most importantly, the use o f the neutral water cluster data allows the comparison of similar bulk and cluster properties providing the key to the simplidtv of this new alternative method.

As mentioned previously, that the water partial sum for enthalpies was obtained using semi-empirical PM3 calculations corrected for zero point energy with calibration o f the excess energy o f hydrogen bonding by comparison to high level ab initio methods.

The free energy partial sum was obtained using the equilibrium concentration measurements of Kell and McLaurin-^ on small water clusters in steam at temperatures from 425-725K.

These data (see Fig. 2 o f ref. 25) have been graphically extrapolated back to 300K in order to provide room temperature values for AG„.,j,[H,0] of 6.7, 4.6, and 3.8 kJ/mol for n=2, 3, and 4 respectively. We only require a value for n=5 which is obtained by estimating the general trend. The stepwise free energies of water will extend to a value"^ of -8.59 kJ/mol at

bulk ^.e. —AG ^ [H , 0 ], the negative of the vaporization free energy of water at room

79 temperature). The same functional form that was previously found to connect the

calculated stepwise enthalpies o f Buffey and Brown-^ (in an averaged way) to bulk was used

for the free energies. The following relations were obtained by fitting this functional form

to the cluster data of Kell and McLaurin-^ with the constant term fixed at its bulk value to

obtain (for n>l):

/ mol) = -8.59 + (26.6± OJ)[n-'^- (n-1)"'"] (38),

^n-i.„[H20](kJ/moI) = -44.0 + (57.6±8.1)[n=^"-(n-l)"'"] (39).

It is interesting that the stepwise free energies for adding a water cross over from

unfavorable to favorable at some intermediate cluster size (~n=10). Over the cluster size

range presendy being used, the neutral stepwise solvation free energies and enthalpies are

much smaller than the ion stepwise solvation energies, so again some lack of exact

knowledge of the neutral values can be tolerated by the method. Partial sums of the

stepwise free energy and enthalpy o f neutral water clusters have been accumulated in Table

2.4.

Y* M ethod

A correlation has been observed-® that provides further evidence for a value for the proton enthalpy greater than previously reported and achieves a greater consistency with bulk. The data in Table 2.3 can be written as

AH„,,„[A*] = Y + k(A*) (40),

8 0 AH„,„[B-] =-(Y+k(B-)) (41),

where k(A*) and k(B ) are the constants representing the bulk differences of the proton and

the positive and negative ion respectively. Solving Eq. (41) for Y gives

(42).

The difference between the bulk positive and negative ions in any pair is not known. The

solvation enthalpy for a cluster o f size n, however, can be related to the bulk single-ion

solvation enthalpy by

where c„(A^ and c„,(B ) are the fraction o f the bulk solvation enthalpy of A^ and B

respectively that a cluster o f size n or m has achieved. This remains an unknown quantity,

but introduces the cluster data into Eq. (42) giving

Y = - (45). 2 c„(A*) c^ (B -)

The values o f the cluster sizes n and m are not rigorously required to be equal.

As mentioned previously, the difference at bulk between the solvation enthalpies of A'^ and B are not known. The sums are known and are the bulk solvation enthalpies of Table

81 2.1. A new constant, c^(A ^3 ) can be defined from the sums of both the cluster and bulk

solvation enthalpies, which are known.

When the two ions in question are similar in single-ion solvation enthalpy at both cluster

size n and bulk, then

c^(A\B-) = c„(A*) s c„(B") (47).

This allows an approximation for Y, the proton single-ion solvation enthalpy, in Eq.

(45) as

^sol.n(A ) ^so,,n(B > _ - k(B') (48) C„.„(A*,B-)

where Y’ is the value of the proton single ion solvation enthalpy based on the cluster data,

AH,ui^(A^ and AH„^„(B ), which are known. Y’ is rigorously equal to Y when the fraction o f the bulk single-ion solvation enthalpy for the positive ion, c„(A*) is equal to the fraction for the negative ion, c„(B ).

According to the Bom concept for ionic solvation,-^ this will be true when the volume o f the ions being solvated is similar. Figure 2.4 shows the calculated value for Y’ with respect to the difference in ion volumes (Table 2.5) at different cluster sizes, as reported by

Gourary and Adrian.^® The cluster volumes were estimated by adding a constant value for

82 io n ra d iu s (A) ion radius (A) Li^ 0.94 F 1.16 Na" 1.17 Cl 1.64 1.49 Br 1.80 Rb^ 1.63 I 2-05

Table 2.5. Ion radii as reported by Gourary and Adrian^®

each water molecule in the cluster to the ion volume. The best fit line shows an intercept

(when Bomian ions would have equal solvation bulk solvation energies) of -1152.1±1.8 kj / mol.

A more informative plot would involve the differences in cluster ion solvation enthalpies instead o f volumes (Figure 2.5). Once again, a good correlation is seen, with an intercept of -1151.0+1.6 kj/mol. The observation of a similar intercept in both Figures 2.4 and 2.5 gives credence to the Bomian concept applying to cluster solvation enthalpies.

With this in mind, the cluster-based difference-over-sum method is given some justification as the cluster ordering is indicative o f the bulk values.

Figure 2.6 shows the same information at Figure 2.5 except each cluster size is individually fit. Table 2.6 gives the slopes and intercepts for these fitted lines. The common point in the cluster enthalpy differences in Figure 2-6 is near the intercept, i.e. the cluster calculated value for Y’ exhibits the least standard deviation (2.1 kJ/mol) at the

Bomian expectation value. The corresponding firee energy difference plot by cluster size

(Figure 2-7) shows an even greater consistency (X’=-1101.8±0.3 kJ/mol).

83 Enthalpies Free Ene%ies n V slope Y* intercept X* slope X* intercept

1 0.662(0.039) -1147.2(1.8) 1.451(0.074) -1101.2(3.2)

2 0.397(0.026) -1150.9(2.0) 0.723(0.036) -1101.3(2.6) 3 0.269(0.024) -1150.3(2.3) 0.516(0.026) -1101.8(2.4) 4 0.198(0.021) -1148.8(2.3) 0.424(0.021) - 1 1 0 2 .0 (2 .2 ) 5 0.169(0.019) -1146.0(2.3) 0.370(0.019) - 1 1 0 1 .2 (2 .0 )

6 0.101(0.048) -1151.7(6.2) 0.320(0.024) -1099.2(2.8)

Table 2.6. Slopes and intercepts o f Y’ (or X') vs. the difference in cluster solvation enthalpy (or free energy) for different cluster sizes, n. Estimated standard deviations are given in parenthesis.

The correlation shown by Eq. (48) can also be expressed by plotting its components against one another. A plot o f l/c ^ (A '’,B ) vs. AH,^,^(A^-AH,^|^(B ) is nearly constant

(although the constant differs for each n). As a result, the first term o f Eq. (48) is linear in

AH,„|^(A ^-AH,oij,(B ) and will not contribute to Y’ at the intercept; the latter terms, k(A*^ and k(B ), determine the best value for Y’. These constants are known bulk values and are not derived from cluster data. Figure 2.8 shows the plot of the latter terms (k(A*) and k(B ) vs AH,„|^(A^-AH,„,^(B ). The relations are linear and share a common point near the intercept at a value of -1148.7±2.0 kj/mol.

Discussion

The results o f the Y’ method are compared to other selected results in Table 2.7. The presently obtained value of AH^^, =-1148.7±2.1 kJ/mol is somewhat higher than many literature values (see Table 2.7) and constitutes an important reassessment o f this value. The presently obtained value o f =-ll0l±0.3 kJ/mol compares favorably

84 to the center of weight of determinations in the chronological survey of Friedman and

Krishnan (see Fig. 5 in Ref. 35, Chapter 1, p.25).

When the three different proton solvation enthalpies were examined in the early sections of this paper, they were compared using a common set o f bulk constraints (from

Table 2.3) in order to highlight the consequences o f these values relative to the cluster data.

O ne can also evaluate any set o f absolute single ion values for consistency to bulk. Since bulk measurements on inherendy neutral, dilute, electrolytic solutions basically characterize the thermochemistry of a pair o f oppositely charged ions, any set of single ion thermochemical properties can be compared to bulk by adding the values of oppositely charged pairs in the set. The standard deviation o f comparison, between the bulk values available in the NBS tables'^ as provided in Table 2.1 using all but the top row (avoiding the issue o f HVs HjO*) and each o f the selected data sets, is presented at the bottom o f the free energy and enthalpy sections in Table 2.7. For instance, Friedman and Krishnan’s^^ set o f single ion solvation free energies exhibits a first standard deviation o f 28.1 kJ/mol with respect to bulk, while the set of Gomer and Tryson^- shows a value of 40.6 kj/mol. If there is confidence in the bulk data, then this comparison can be used to evaluate the quality o f single ion thermochemical data sets. Consistency does not in itself guarantee quality (the examples in Figures 2.1 to 2.2 are equally consistent with bulk, but not with the cluster data), but lack of consistency is indicative o f problems.

85 AGV- (kJ/moÇ ion 1 this work Klots^O Conway^^ Friedman^ ^ Gomer^- -1101.8(0.3) -1097.9 -1066 -1059.4 U" -527.1 -518.8 -487.5 -516.7 -494.0

H 3O" -452.2 -432.8 Na^ -421.3 -418.4 387.6 -411.3 -379.2 OH -428.8 -425.9 -379.1 -395.6 F -421.1 -426.3 456.8 -434.3 -463.1 K* -349.5 -345.2 314.0 -338.1 -305.9 Rb^ -326.9 -323.8 292.7 -320.5 -284.4* Cl -302.2 -309.2 339.7 -317.1 -340.6 Br -275.9 -295.4 325.9 -303.1 -318.4 I- -238.9 -248.9 279.5 -256.9 -281.1*

Comparison to bulk CT (kJ/mol) 0.04 8 . 2 8 . 0 28.1 40.6

AH“«,„(kJ/mol) ion this work C oe^ Klots-® Randles^^ Friedman^ ^ FF -1148.7(2.1) -1152 -1136 (-1131) -1129 Li" -577.5 -576.1 -560 -552 -559

H 3O" (-502)' -513.4 -469 -432 Na" -462.1 -466.5 -451 -444 -444 OH -473.9 -479.5 -486.2 -485 -423 F -465.3 -456.9 -466.5 -473 -474 K" -378.8 -381.3 -367 -360 -360 Rb" -353.6 -357.9 -341 -335 -339 Cl -321.3 -316.3 -333 -326 -340 Br -290.3 -282.7 -318 -326 -326 I -249.1 -245.9 -261 -268 -268

Comparison to bulk CT (kj/mol) 0 . 6 9.9" 15.5 15.4 35.9

Table 2.7. Single ion bulk solvation free energies and enthalpies from several sources. Each data set has been used to generate data as in Table 2.1. The standard deviation o f the comparison with the bulk experimental data in Table 2.1 is given as CT at the bottom, a) calculated using the current web page value o f 691 kJ/m ol for the room temperature proton affinity o f water, b) this value was 2.4 kJ/m ol with the bulk set used in ref. 21.

86 The single ion sets o f Conway,^^ Klots-® and Coe— were all forced to be consistent

with their bulk data sets (Rosseinsky^^ for the Conway-®’ and Klots sets), but there is

considerable difference between that older set and the CRC’s NBS tables (17 kJ/mol for

enthalpies and 8.4 kJ/mol for free energies). The enthalpic single ion set of Coe exhibited a

standard deviation o f 2.4 kJ/m ol from its alternative bulk data (based on heats o f dilution,

gas phase bond energies, and lattice energies), but shows a standard deviation of 9.9 kJ/mol

relative to the presendy employed bulk constraints. However the differences in the bulk

data have had littie (if any) effect on the absolute proton solvation enthalpies determined

with the present method and from ref. 2 1 .

The overly consistent differences between similarly charged ions in Table 2.1 shows that

the NBS database"*" contains bulk values which have been optimized for internal consistency.

In order to evaluate the presendy determined set o f single ion solvation properties, we

compare it to the set determined with more rigorous error analysis ( 2 "‘* column, bottom

section of Table V. Ref. 21.). These two sets exhibit a standard deviation of 6.3 kJ/mol

which falls comfortably within the range o f individual errors reported by Coe, providing

confidence in the extension o f the method to free energies (where data for such checks are

not available).

The enthalpic data set o f Randles^"^ represents an experimental determination o f the

absolute single ion enthalpies. It exhibits a first standard deviation o f 15.4 kJ/mol relative

to the present bulk set which could be taken as an indication o f the random error o f the

experimental method. Each of Randles single ion enthalpy values has been used to

determine absolute solvation enthalpy using the data in Table 2.3. The average and standard

87

iJ deviation of this exercise (-1131±9 kJ/mol) is exhibited in brackets in Table 2.7 under

Notice that the uncertainty o f this procedure is in reasonable agreement with the

comparison to bulk. Comparison o f the present single ion values to those of Randles

shows a standard deviation o f 21 kj/m ol. Consequently, roughly half o f the difference may

be attributed to systematic differences between the cluster approach and the Randles

separation method which does not account for the surface potential of water (~13

kj/mol^^). The presendy determined set o f single ion values is: 1) consistent with experiment when surface potentials are considered, 2 ) completely consistent with bulk as only NBS heats and free energies of formation have been employed, and 3) maximally consistent with the cluster. It is hoped that these values will find utility in the increasingly important activity o f connecting cluster ion data to bulk.

88 References

1. Casdeman, A.W. & Bowen, jr. J P^s Chem 100, 12911-44 (1996). 2. Andres, R.P., Averback, R.S., Brown, W.L., Brus, L.E., Goddard III, W.A., Kaldor, A., Louie, S.G., Moscovitz, M., Peercy, P.S., Riley, S.J., Siegel, R.W., Saepen, F. & Wang, Y. / M ater Res 4, 704-36 (1989). 3. Bard, A.J., Parsons, R. & Jordan, j.Standard Potentials in Aqueous Solution (Marcel Dekker, New York, 1985). 4. CRC Handbook of Chemistiy and Posies (CRC Press, Boca Raton, 1985).

5. Cieplak, P., Lybrand, T.P. & KoUman, P.A. J Chem Phys 8 6 , 6393 (1987).

6 . Kebarle, P. in Modem Aspects o f Electrochemistiy (eds. Conway, B.E. & Bockris, J.O.M.) (Plenum Press, New York, 1974). 7. Keesee, R.G., Lee, N. & Casdeman, A.W., Jr. /Chem PhysTi, 2195-202 (1980).

8 . Keesee, R.G. & Casdeman, A.W., Jr.Chem Plys L^ett 74, 139 (1980). 9. Lee, N., Keesee, R.G. & Casdeman, A.W., Jr. / Interface ScilS, 555 (1980). 10. Cunningham, A.J., Payzant,J.D. & Kebarle, P. J A m Chem Soc9A, 7627 (1972).

1 1. Keesee, R.G. & Casdeman, A.W., Jr. /Plys Chem Ref Data 15, 1011 (1986). 12. Hiraoka, K., Mizuse, S. & Yamabe, S. / Pfys Chem 92, 3943-52 (1988). 13. Dalleska, N.F., Honma, K. & Armentrout, P.R. J A m Chem SocU5, 12125 (1993). 14. Magnera, T.F., David, D.E. & Michl, J.Chem Plys Lett 182, 363 (1991). 15. Shi, Z., Ford, J.V., Wei, S. & Casdeman, A.W., Jr. / P/ys99, 8009-15 (1993). 16. Tissandier, M.D., Earhart, A.D., Gundlach, E., Muhammed, S.I. & Coe, J.V. (in preparation). 17. Xantheas, S.S. & Dunning, T.H., Jr./ Chem P ly sV i, 8774-92 (1993). 18. Xantheas, S.S. & Dunning, T.H.,]r.J P ^ s Chem 98, 13489-97 (1994). 19. Morris, D.F.C. Electrochimica A cta “21, 1481-6 (1982). 20. Klots, C.E. / P lys Chem 85, 3585 (1981). 21. Earhart, A.E. M.S. Thesis, The Ohio State University, (in preparation). 22. Coe, J.V. Chem Plys Lett 229, 161-8 (1994). 23. Coe, J.V. /Plys Chem A 101, 2055-63 (1997). 24. Markovich, G., Pollack, S., Giniger, R. & Chesnovsky, O. / Chem P lys 101, 9344-53 (1994). 25. Perera, L. & Berkowitz, M.L./ Chem P lys 100, 3085-93 (1994). 26. Kell, G.S. & McLaurin, G.E. / Chem P lys 51, 4345 (1969).

89 27. Buffey, I.P. & Brown, W.B. Chem Phys Celt 109, 59 (1984). 28. Tissandier, M.D., Cowen, K.A., Feng, W.Y., Gundlach, E., Cohen, M.H., Earhart, A.D. 8c QoCj].Y. in preparation (1997). 29. Bom, M. Z PfysiJk 1, 45-8 (1920). 30. Gourary, B.S. & Adrian, F.J.So/idState Piysics 10, 127 (1960). 31. Ftiedman, H.L. & Krishnan, C.V. in Water: Ji. Comprehensive Treatise (ed. Franks, F.) (Plenum Press, New York, 1973). 32. Gomer, R. & Tryson, G. / Chem P lys 66, 4413-24 (1977). 33. Conway, B.R. J Soi Chem 1, 721-70 (1978). 34. Randles, E.B. Trans Faraday Soc 52, 1573 (1956). 35. Rosseinsky, D R. Chem Rev65, 467-90 (1965). 36. Trasatti, S. in Modem Aspects of Electrochemisty (eds. Conway, B.E. & Bockris) (Plenum Press, N ew York, 1979).

90 Bulk Solvation Enthalpy Against Cluster Data - Morris -200

R b^ -3 0 0 - o E B r Na cr -400 _8 o (0 I <

-5 0 0 -

OH

-600 — 1______-4 0 0 -300 -200 -100 AHgq, g (kJ/mol)

Figure 2.1. Plot o f the bulk single-ion solvation enthalpies determined with Morris’ absolute value for the proton (-1103 kJ/mol) vs. the cluster solvation enthalpies for clusters with 5 water molecules. The positive and negative ions are grouped separately and display a different slope illustrating the two problems that must be overcome to achieve consistency o f a particular absolute value o f the proton with the cluster data set.

91

1 Bulk Solvation Enthalpy Against

-200 Cluster Data - Klots

-300 Rb Br­ o E er -400 - Na

X” < 1

-500 - OH

-600 1 -400 -300 -200 -100 AHgQi g (kJ/mol)

Figure 2.2. Plot of the bulk single-ion solvation enthalpies determined with Klots’ absolute value for the proton (-1136.4 kJ/mol) vs. the cluster solvation enthalpies for clusters with 5 water molecules. Klots’ chosen set o f ions gives a fitted line successfully grouping the positive ions and selected negative ions onto a common line (albeit with a slighdy increased slope over that of the positive ions alone). The strategy of eliminating OH and P from the process has not dealt with the problem of a different slope for the negative ions, leaving the OH and F species as deviant without explanation.

92 Bulk Solvation Enthalpy Against

-200 Cluster Data - Coe

-3 0 0 Br o C l E

-4 0 0 - Na X

-5 0 0

-6 0 0 -4 0 0 -3 0 0 -200 -100 AHgqi g (kJ/m ol)

Figure 2.3. Plot o f the bulk single-ion solvation enthalpies determined with the Coe method’s absolute value for the proton (-1152.5 kJ/mol) vs. the cluster solvation enthalpies for clusters with 5 water molecules. The large halides were left out o f the fit because they are surface states at n=5 and wül not exhibit the same functional dependence of stepwise hydration energetics as do the internalized ions. The included ions fit very well to a common line with a slope very close (less than 0.5% difference) to that exhibited by the positive ions alone.

93 V vs Difference in Bulk ion Volumes

-110O -1110 -1120 -1130 i ^ - 1 1 4 0 E -1 1 5 0 2-1160 > - 1 1 7 0 -1180 intercept -1190 -1152.1(1.8) kJ/mol -1200 -1210 -1220

-40 -3 0 -20 -10 10 20

Figure 2.4. The correlation o f the Y’ approximation and the bulk difference in ion volume for various pairs o f alkali and halide ions. The value o f Y’ (-1152.1+1.8 kJ/mol) when the difference in bulk ion volumes is zero represents the best value of the proton’s hydration enthalpy upon invoking the Bom concept that ions o f the same volume have the same bulk solvation enthalpies.

94 Y' vs Difference in Cluster Solvation Enthalpy

-1100

- 1 1 1 0

-1120 -1130 ^ -1140 O E -1150 3 -1160 ^ -1170 -1180 -1190 intercept -1200 \T' -1151.0(1.6)kJ/mol -1210 -1220

-300 -200 -100 - 0 100 200

Figure 2.5. The correlation of the Y’ approximation and the difference in cluster ion solvation enthalpy. This correlation is very sim ilar to that shown in Figure 2.4 regarding the position o f ion pairs. The intercept value o f -1151.0+1.6 kJ/mol is consistent with that determined by invoking the Bom concept showing that, in general, ions o f the same volume have the same cluster solvation enthalpies as well as bulk solvation enthalpies. The problem of the proton's bulk hydration enthalpy can be equivalently approached with the difference in bulk ion volumes or cluster solvation enthalpies.

95 Y' vs Différence in Cluster Solvation Enthaipy

-1100 -

-1 1 1 0 -

-1120 - -1130 - 3 -1 1 4 0 - -1 1 5 0 - -1 1 6 0 - -1 1 7 0 - -1180 - -1 1 9 0 - common point -1148.7(2.1) kJ/mol -1200 -

-1210 -

-1220 -

-300 -200 -100 - 0 100 200 „(B ) (kJ/mol)

Figure 2.6. The correlation of the Y’ approximation and the difference in cluster ion solvation enthalpy vs. cluster size. This plot contains the same data as in Figure 2.5 except that the cluster size has been specified. If the Y’ approximation is good at only one particular pair o f ions, then each data set at a particular cluster size can be expected to share the good value in common. At each cluster size the data have been fit to a line by the method of least squares. The spread in these fitted lines vs. the difference in cluster ion solvation enthalpy was used to determine a most common value of -1148.7±2.1 kJ/mol which represents our best détermina non of the absolute proton hydration value without invoking the Bom concept. This value is consistent with the values determined previously that had invoked the Bom concept.

96 X' vs Différence in Cluster Solvation Free Energy

-1010 -1030 -1050 -1070 101.8(0.3) kJ/mol O -1090 3^ -1110 -1130 X -1150 -1170 -1190 -1210 -1230 -1250

-300 -200 -100 - 0 100 AG,oi.n(A*)-AG,„, „(B ) (kJ/mol)

Figure 2.7. The correlation of the free energy or X’ approximation with the difference in cluster ion solvation free energy vs. cluster size. These data show an even more commonly shared point of -1101.8+0.3 kJ/mol which represents our best determination o f the proton’s free energy of hydration without invoking the Bom concept. This value compares favorably to the center-ofrweight of previous determinations providing confidence in the enthalpic results o f Figure 2.6.

97 Experimental Solvation

-950 Enthalpy Différences

g -1050

m= ? ^-1150 $ » o

-1250 -3 0 0 -200 -100 100 200

Figure 2.8. The dependence o f the bulk constants, -! 6 [k(A^+k(B )], on the difference in cluster solvation enthalpy vs. cluster size. The cluster approximation employed in the Y’ approximation involves only the first term o f Eq. (48) which is essentially zero at the common point, so that the proton’s absolute value is determined almost entirely by the behavior of the bulk constants firom Table 2.3 vs. the difference in cluster solvation enthalpy. Notice the linearity o f the data sets at particular cluster sizes and the shared common point at -1149.1±2.0 kJ/mol which is consistent with the previous determinations for the proton’s absolute hydration enthalpy.

98 j CHAPTERS

STUDIES OF INTENSE ION BEAMS IN AN ELECTROSTATIC PARTICLE GUIDE

Transmission o f ion beams over long distances or through the fringe field of a magnet

has been extensively studied. Quadrupole, hexapole, and octopole guide rods,^’ - elaborate

lens configurations,^’ and stacked ring setups^ have been used to maintain ion beam

density or overcome the magnetic “mirror.” The simplest o f these methods is the

Electrostatic Particle Guide (EPG) based on the work o f H. H. Kingdon.

Kingdon introduced an ion trap to minimize space charge effects in 1923.^ His design consisted o f a tube with electrically connected end caps (Figure 3.1). A thin filam e n t wire was mounted concentrically and a negative voltage was applied. Positive ions generated by a discharge were confined by the field directed inward towards the wire. The work was based on the investigation by HuU^ into the motion o f an ion between concentric cylinders in electric and magnetic fields and expanded upon by Page.® The specific case of motion in the Kingdon trap was studied much later by Lewis.^

99 In 1964, Herb introduced the EPG as an avenue to better ion gauges. Termed the

“Orbitron,” Herb produced electrons firom a filament inside the outer tube and used the

extended path provided by the radial force to achieve the same level o f accuracy as other

contemporary gauges with ten times less emission current. He later extended this idea to

ion pumps. Morion in the “Orbitron” was described numerically by Hooverman. ^ -

More recently, the EPG has been applied to studies of radioactive recoil ions and other

secondary particles by time o f flight mass spectrometry (TOF-MS).^^’ The primary concern to these experiments is the collection efficiency, or entry angle, and distortion, if any, o f the TOF-MS peaks.Low energyand high-resolurion^® TOF-MS have also been conducted with some success.

Current System

A version of the EPG was applied to a glow discharge source with the ultimate goal of detecting an infrared emission spectrum from the cluster ions of the beam (See Chapter 4).

The goal of this line of investigation was to increase the transmitted current over a longer path, in turn increasing the average time each cluster was in a photon detection region.

A diagram o f the EPG as it is applied to our system is given in Figure 3.2. The entry into the EPG consisted o f a grounded stainless steel tube (2.5 cm ID and 1.2 cm in length).

Mounted on this by means o f Kimble Physics plates, and electrically isolated by Teflon spacers, is a stainless steel rod (1 mm diameter). This rod provides support for the central trichrome electrode (20 pm diameter). The repulsive drift tube was also constructed of stainless steel (2.5 cm ID and ~35 cm in length). The drift tube extended through the

1 0 0 Plexiglas flange and was sealed with a Cajon o-ring fitting (2.5 cm ID). third stainless steel tube (2.5 cm ID and 4.5 cm in length) acted as the Faraday cup separated by a Plexiglas spacer (2.3 cm ID) and sealed with epoxy. A stainless steel rod mounted in the Plexiglas supported the nichrome wire at the exit o f the drift tube. The spiral path predicted by theoretical models precluded the need to “cap” this mbe with a conducting material as all ion trajectories left the drift tube with sufficient radial velocities to reach the outer tube without exiting the end. The end of the Faraday “tube” was sealed for vacuum use by a second Plexiglas spacer (2.3 cm ID) on which was mounted a NaCl window sealed with epoxy.

The experiments were performed inside a stainless steel vacuum chamber diagrammed in Figure 3.3. The pumping system consisted of 2 5000 L /s oil diffusion pumps (Balzers

Diff 5000 with Dow Coming DP704 diffusion pump oil). The backing pumps are a Welch

1375 and a Welch 1398 mechanical pump, rated at 35.4 and 53.1 cfin respectively. An ion gauge (Kurt J. Lesker G075N Ion Gauge Tube, MKS Type 290 Ion Gauge Controller) monitored the chamber pressure throughout the experiment. Typical base pressure in the chamber was ~2 x 10 * Torr. Source gas routinely raised this value to ~5 x 10^ Torr.

Addition o f collision gas further raised the pressure from 7x10^ up to a maximum of 2 x

10 Torr. Beyond this point, the chamber pressure is high enough that discharges occur at other points in the apparatus. The foreline pressure, monitored by thermocouple gauges

(Varian Type 0531 Gauge, Varian Type 801 Controller) generally ranged from 5 mTorr to

15 mTorr.

Source and collision gases were introduced and regulated through use of a gas regulation system. A diagram o f the gas line arrangement is shown in Figure 3.4. A mechanical pump

101 (Welch 1397, 17.7 cfin) was used to evacuate the gas lines before the valves to the chamber

are opened. This pressure is monitored by a thermocouple gauge mounted on the ro u g h in g

line. Needle valves (Nupro) are used for precise control o f the gas flow to the apparatus.

All lines are polyethylene tubing ( 6 mm ID).

The ion beam used in these experiments utilized a glow discharge source reported in

earlier documentation^^. The source consisted o f 2 plates o f mild steel separated by a Pyrex

glass tube «4 cm in length. Viton O-dngs in protective grooves on the plates provide a

vacuum seal between the plates and the glass. The anodic fi^ont plate was a hexagon 3.5 cm

across and 6 mm thick with a tapered 2.5 mm aperture. Holes were drilled into the plate to

allow easy connection o f ion optical components with a Kimble Physics hole pattern

(Figure 3.5). The back plate was also a hexagon 18 mm thick. This plate had a port to allow introduction of the source gas to the discharge. A potential of ~2500V placed across the plates with a 500kQ ballast resistance in series will typically produce HjO* predominately with small amounts o f H j0*(H ,0)n clusters where n = l-4 (Figure 3.6). This relative cluster ratio could be adjusted by changing the source conditions.

The ion beam voltage was established by an isolation scheme which floated the ion source to the requested voltage. A diagram o f the circuit is shown in Figure 3.7. The discharge power supply determines the potential difference between the source plates and is isolated firom ground. Initially, this power supply consisted o f a bank o f 510V batteries

(Eveready #497) connected in series. This provided a steady voltage, but the cost o f replacement batteries soon became prohibitive. In later experiments, the batteries were replaced with a Fluke 41 OB power supply and an isolation transformer. The beam power

1 0 2 supply (Fluke 41 OB) was used to establish the beam voltage, V,,. This supply provides the

float voltage for the front plate of the source. It is important to note that the polarity of the power supplies is arranged in such a way that the back plate is at a more negative voltage than the front plate as given by

^baclcplate ^dischaige ^ftontplale ^discharge • (^)*

A source with the back plate positive relative to the front will give electron bombardment ions. Ion currents between 5 and 20 pA were routinely obtained throughout these experiments.

The voltage to the other elements was provided through the use o f a voltage divider circuit as shown in Figure 3.8. The beam voltage is divided by multiple 2W 160kQ resistors in series. The position o f the potentiometer is selected by a banana-plug arrangement.

Connection to a Plexiglas disk electrically isolates the potentiometer. The wiper of the potentiometer is connected to the element being controlled.

Sixnion Results

The potential o f a radial electric field, E(r), i.e. that generated by two coaxial cylinders is given^ by

103 where V„ and V, are the voltages at the outer and inner cylinders respectively, r„ is the inner diameter o f the outer cylinder, and r^ is the outer diameter o f the inner cylinder. Plac in g a positive potential on the outer cylinder (or a negative potential on the inner cylinder) yields a net force inward for positive ions. From a safety point of view, the potential should be applied to the inner cylinder, but the present physical arrangement o f the ion optics made this problematic.

The EPG tube assembly dimensions were entered into an ion optical simulation program, Simion.-® This program sets up a grid with cylindrical symmetry for calculating potentials. The grid used in this investigation and coordinate system are shown in Figure

3.9. A potential was placed on the drift mbe and the potential array refined. As a check, the potential at several points was obtained and plotted in Figure 3.10 against the theoretical curve obtained from Eq. (2) with V„=1000 and V,=0.

The injection of the ions presents 3 variables with respect to the EPG. The first is the position of the ion beam where it crosses the y-z plane at x=0. This represents the point of entry into the grounded end-cap o f the EPG. The other parameters are the angles that the ions enter the mbe relative to the coordinate system shown in Figure 3.9. The azimuth angle is defined at the angle firom the x-y plane towards positive z and the elevation angle is from the x-z plane towards positive y. Other items to be controlled are the ion mass and energy. For the purposes o f this investigation, the ion is assumed to be H^O (H^O)^ where n=0-3.

As the smdies were intended for real world applications, single ion trajectories would be insufficient for comparison. Glow discharges sources are not point sources and space

104 charge effects lead to beam spreading. As a result, a “spot” was simulated for the input of

the EPG (Figure 3.11). A 3 mm diameter circle was divided into 5 points, the center and 4

ordinate positions relative to the simulation coordinate system. From each o f these points,

a pencil of rays was initiated with a 5° spread in both the y- and z-axis. The program took

space charge effects into account by assigning a portion of the arbitrary 10 |xA ion current to each trajectory and integrating the ions as line charges.

Some example ion trajectories are shown in Figure 3.12. The different trajectories shown are the result of varying the injection angle from a fixed point.

Acceptance Angle

Angle o f injection is the single most important element in transmission through the

EPG. For a given ion mass, ion energy and drift tube voltage, only a narrow range of azimuth-elevation angle combinations permit passage through the EPG. Figure 3.13 shows the results o f simulations on HjO* at lOOOV and a drift tube voltage of 700V. The x and y- axes are the elevation and azimuth angles respectively. The elevation, z, on the plots is the average flight time of all ions entering the grounded entrance tube. O f those trajectories that successfully reached the end of the EPG, the average flight time ranged from ~5 to 14 fis. The straight line flight time for this device under those conditions, i.e. ^y^=0, and zero elevation angle was 3.2 fis, an increase of only ~30% on the low end, although experimentally some advantage is gained (see discussion). The closer the ion trajectory passed the central wire during the initial entry into the high field region, the longer the observed flight time. This effect reached an upper limit when the attraction o f the ion to

105 the center electrode increased to the point that the ion collides with the wire and is

neutralized.

As expected by symmetry, the direction o f the spiral is unimportant- This is shown by

the fact that the transmission band is “mirrored” about the center point o f the plot in

Figure 3.13.

Energy and Mass Dependence

The behavior o f the ion transmission through the EPG was explored in terms o f energy and mass dependence. The change in mass from one simulation to the next changed only in the time-of-flight o f the cluster yielding no change in the actual ion path, and the effect was equivalent to a field free linear time-of-flight experiment where the time can be anticipated merely by the velocity, whether initially determined or derived from starting

conditions, as evidenced by the linear time-vs. mass '''" plot of H^O 2 ,, in Figure

3.14. For the conditions used for the figure, the expression,

L = vt (3), where L is the ion path length in meters, v is the ion velocity and t is the time in seconds, can be expanded by substitution for v.

- J ® to give.

106 (5).

Rearrangement to a linear expression yields.

where t is the flight time, m is the ion mass, q is the fundamental charge, and is the beam voltage.

The energy dependence is more complex. The ratio of the energy of the ion to the potential difference of the cylinder-wire combination.

O I determines the curvature of the ion path. As a approaches 0, the ion passes a critical threshold where it no longer possesses the energy to overcome the fnnge fields of the EPG and is repulsed (for a given angle and starting position). A large a results in the internal field o f the EPG being too small to bend the ion into a spiral path and the ion colliding into the outer cylinder and being neutralized.

The “energy transmission window” that is produced is dependent on the initial position and angle o f the ion beam. Figure 3.15 shows the typical window produced at a given beam angle as a function o f radial position. The “wire shadow” portion o f the curve is exaggerated by the unrealistic size of the central wire in the simulation. The failure mode

107 shown as “Low Vy" is the result of reflection out of the tube by the large repulsive field of

the apparatus. The slant o f the “High Vy” area, a smaller window at large radius than at

smaller, is counterintuitive without a closer examination of the injection region. When the

radial initial position is large, the fringe field of the electrodes reduces the elevation angle o f

the trajectory, towards the straight line path. As the radius is reduced, the fring e field

encountered changes, causing the elevation angle to increase beyond the transmission angle

shown previously.

Experimental Results

The apparatus previously described (Figure 3.2) was intended to be used in experiments

dealing with infrared emission (see Chapter 4). The behavior o f the ion current was found

to coincide with the behavior expected from the simulations. Experimentally it is difficult

to decouple the position and angle dependence. A deflector mounted on the ion beam can

change the angle, but also changes the position where the ion beam crosses the y-z plane

and enters the first grounded tube o f the EPG.

A deflector modulation of ~250V was applied and the current on the “Faraday mbe”

was recorded on a digitizing oscilloscope (LeCroy Model 9350AM). An example of these

traces is shown in Figure 3.16. The broadcast from the modulation square wave is visible as

a artifact prior to the rise or fall o f the ion current. This allowed a direct measurement o f

the flight time between the modulation voltage going to zero and the arrival of the first

ions. Small changes in the entry angle/position (via deflector or lens voltage changes earlier

in the beam path) had profound effects on the profile observed. For maximum response

1 0 8

i from the lock-in amplifier, the square wave shape was optimized, even at the expense of

longer flight times.

D isc u ssio n

Throughout its history, the EPG has been used to increase the path length or the

collection angle for charged species. The behavior of the ion inside the field tube with no

perturbation by the ends has been dealt with in detail by Lewis,^ but starting in a field free

region and entering the EPG, through the fiinge fields, is a more difficult proposition.

While the increase in time over the “straight-line” flight time is not necessarily large enough

to warrant the added work, the real gain is in terms o f current. The current o f a similar

source as a function o f the beam path distance was investigated and the details are reported

elsewhere.-^ At times, the EPG s y s te m transmitted ~2Ü^iA over its length (—40 cm), far in

excess of the “open path” current o f ref. 2 1 .

Use of the EPG with an ion beam requires careful alignment of the beam angle to the

entrance tube and the point the beam crosses the entry plane. Some small corrections are

possible with electrostatic deflectors, but gross changes usually result in very low currents,

indicative of most of the ions being lost.

Often, the modulated current is adjusted to give the cleanest square wave possible for

use with a lock-in amplifier. This particular combination of deflector and lens voltages also

typically corresponds to the maximum transmitted current, although at the expense o f flight

time. An examination of Figure 3-15 demonstrates the reason for this phenomena. A

“clean” square wave will occur when the range o f flight times is small from a spot that

109

JLI contains a range o f both position and angle. This corresponds to the “flat” regions of the

transmission windows, well away from the maximum flight times near the wire. If flight

time is increased (Le. the spot is moved closer to the wire), the range of flight times increases, distorting the square wave, and often loss o f current as these longer times are near the edge of the window.

One enormous advantage to the stringent conditions necessary for ion transmission is in modulation. Since the transmission is very angle dependent, very littie voltage (~200 V) is required to completely eliminate the transmission, providing a clean modulation, despite beam energies in excess of 1 keV.

Conclusions

The application o f an electrostatic particle guide to the ion beams utilised in this research group is demonstrated theoretically and experimentally. The transmission for a representative ratio was explored in terms of entry angle, position, mass, and energy as a guide for coupling an ion beam to an EPG to be used in other experiments.

no R eferen ces

1. March, R.E. & Hughes, RJ.Quadrupok Storage Mass Spectrometjy (Wiley, New York, NY, 1989). 2. Wollnik, H. Optics o f Charged Partic/es (Academic Press, San Diego, CA, 1987). 3. Kofel, P. & McMahon, T.B. Int J Mass Spectrom Ion Proc (1990). 4. Alford, J.M., Williams, P.E., Trevor, D.J. & Smalley, R EInt . J Mass Spectrom Ion Proc 72, 33-51 (1986). 5. Guan, S. & Marshall, A .G . J Jim Soc Mass Spectrom 7, 101-6 (1996).

6 . Kingdon, K.H. Plys Rev 21, 408-18 (1923). 7. Hull, A.W. Pi^s Rev 18, 31-57 (1921).

8 . Page, L. PJ^s Rev 18, 59-61 (1921). 9. Lewis, R.R. J A p p i Pfp/s 53, 3975-80 (1982). 10. Mourad, W.G., Pauly, T. & Herb, R.G. Rev S ci Instrum 35, 661-5 (1964). 11. Douglas, R.A., Zabritski, J. & Herb, R.G. Rev Sci Instrum 36, 1-6 (1965). 12. Hooverman, R.H./ App/ Pfys 53, 3505-8 (1963). 13. Abbe, J.C., Amiel, S. & MacFarlane, R.D.NuciInst andMeth 102, 73-6 (1972). 14. Oakey, N.S. & MacFarlane, R.D. Hucl Inst and Meth 49, 220-8 (1967). 15. Kenigsberg, A., Leon, J. & Sha&ir, N.H. PIucl Inst and Meth 148, 605-12 (1978). 16. Igersheim, R., Abbe, R.C., Paulus, j.M. & Amiel, S. Nuci Inst and Meth 109, 301-4 (1973). 17. G eno, P.W. & MacFarlane, R.D. Int J Mass Spectrom Ion P r o d A, 43-57 (1986). 18. MacFarlane, R.D. & Torgerson, D.F. Int J Mass Spectrom Ion Phys 21, 81-92 (1976). 19. Cowen, K.A. Ph.D. Dissertation, The Ohio State University, (1994). 20. Dahl, D.A. va.43rdASM S Conference on Mass Spectrometry and Allied Topics 111 (1995). 21. Earhart, A.E. M.S. Thesis, The Ohio State Urtiversity, (in preparation).

Ill Wire Electrode Outer Trap Cylinder

I t

Figure 3.1. Diagram o f the Kingdon Trap. Ions were generated internally and trapped into orbits about the central wire. The end caps restricted the ions axiallv.

Corrective Collision Deflector Glow Discharge Source Terminator Faraday "Tube" Guide Tube

/ Collision Entrance Modulating Cell Tube Deflector

Figure 3.2. Diagram o f the current EPG system. Note the angle o f the ion beam relative to the EPG.

1 1 2 Wire K Harness ^

Gate Gate Valve V Valve - - - — 1 r —- . - — — — —r— —rrn— .. IICJ. N, Water ^ < Baffle Baffle ^ '

Diffusion Diffusion Pump —> <—Pump

Figure 3.3. Diagram of the vacuum chamber. Two 1500 L /s diffusion pumps, one water baffled, one l-Nj, provided a base pressure o f ~5xl0'

113 To Vacuum Chamber

G as DGas A Gas B GasC Gas DGas

Source Gas CutofT Valve

% f Needle Valve Pumpout Valve

Vacuum Cutoff Valve

Mechanical Pump

Figure 3.4. Gas rack plumbing diagram.

1 . 2 0 m

Figure 3.5. Kimble Physics pattern of holes for the elements o f the ion beam. The lightly shaded holes were used for the primary support. Each hole is 0.3” center-to-center.

114 Positive Ion Mass Spectrum Torr neat water vapor 900

800

700

600

< 500

e 400 Ë oa 300 c a 200 -

100

0 100 200 300 Deflector Potential (V)

Figure 3.6. Typical mass spectrum obtained from the glow discharge source. The x-axis is measured in volts as the spectrum is firom a Wein velocity filter. This also accounts for the non-linearity in mass.

115 Front

I Ballast Resistance

Discharge Power Power Supply Supply

Figure 3.7. Diagram o f the source circuit. The voltage applied by the beam power supply determines the beam voltage and the discharge voltage determines the source brightness. The ballast resistance improves the steadiness of the beam output. The back plate is kept more negative than the front for maximum cluster products. Reversing this polarity will give electron bombardment ions.

Ion OOptlcal Element

Figure 3.8. Diagram o f the ion optical element circuit. The series o f resistors acts as a voltage divider for the beam voltage (V(J, adjustable by varying the position o f the potentiometer inputs. This arrangement maintains a constant ratio between the voltage applied on the ion optical element and the beam voltage, despite any changes in Vy.

116

JÜ u

Figure 3.9. Simion grid and coordinates. The upper portion o f the figure shows the potential surface of a cross-section of the EPG. The coordinate system and general layout o f the Simion grid is shown in the lower portion. Note the exaggerated size o f the central wire (2 mm diameter in the simulation vs. ~0.04 mm for the actual device). This constraint was imposed for ion trajectory purposes as an ion is considered to have struck a surface and been neutralized only if the grid points immediately surrounding the ion all are contained by an electrode. The 2 mm diameter was the smallest available that would guarantee the correct ion behavior near the central electrode.

117 Radial Electric Potential

1000 w i r e > 8 0 0

400 Outer

200 C y lin d er

0 1 2 3 4 5 6 7 8 9 10 11 12 13 R adius (mm)

Figure 3.10. Potential plot inside the guide tube. The points are samples taken from the Simion simulation and were plotted as a function of radius. The solid curve is the predicted potential from Eq. (2).

Figure 3.11. Spot picture. The initial trajectories for the ions in the simulation were arranged to imitate the real size of the ion source. Each of the 5 points contained 10 rays varying both in the xy- and xz-plane.

118 Figure 3.12. Example trajectories firom the simulation. The upper portion o f the figure shows the typical spiral nature o f the ion spot trajectories for the initial portion o f the EPG. The lower plots are single ion trajectories viewed along the x-axis. The notation on each plot refers to the azimuth angle o f each initial ion condition. Note that as the angle increases, the eccentricity o f the ion “orbit” decreases.

119 Flight time profile as a function of input angle

I I:

Figure 3.13. Azimuth and elevation angle plot. A surface involving flight time as a function o f azimuth and elevation angles for the initial ion trajectory for a given starting position. Note the narrow range where the ions are successfully transmitted down the length o f the guide tube. The symmetry of the plot confirms that the direction of the spiral is unimportant.

120 Time of Flight (^2®)n=0-20

CO

20 O E

o u.

0 5 10 15 20 Mass 1/2

Figure 3.14. Time o f flight plot. Several single ion trajectories involving ions o f different masses are shown. The linearity shows that the EPG is functioning in the same manner as a time-of-flight mass spectrometer, despite its inherendy curved trajectories. The ions followed the same path and struck the Faraday tube in the same spot regardless o f mass.

121 Position Transmission Window

Wire Shadow

E 10 i: 5 7 I : 75 4 ^ 3 5 f LowV, 0 2000 4000 6000 8000 Energy (eV)

Figure 3.15. Transmission window. Single ion trajectories were performed as a function o f radial position and ion energy for a given set o f input angles. The upper surface plot shows the flight time throughout the range studied. Note that the flight time dramatically increases as the wire is approached. The lower plot shows the same plot as a single-line contour around the base of the surface. Three fltilure modes have been identified for ion transmission. The “Wire Shadow” is caused by the ions coming into contact with the central wire. This is exaggerated by the unphysical size o f the simulation wire. The “Low Vy” fidlure involves the ions not possessing sufficient energy to overcome the fiinge field of the guide tube and being repelled. “High V^” refers to the ion energy being too great for the tube voltage to bend the ions into an orbit before striking the outer wall.

1 2 2 . L J

1 1 t ? S, y V Time (0.2 ms/div)

Figure 3.16. Oscilloscope plot o f the modulated ion current. A trace o f the experimental modulated ion current at the Faraday tube o f the EPG. The modulation square wave was set to 100 Hz and a IkQ input impedance was used resulting in the y-axis being 1 mV equal to 1 pA o f ion current. The arrows denote the points where the modulation voltage changed. The delay was measured as —25 ps.

123 CHAPTER 4

INFRARED EMISSION SPECTRA OF Hj0^(H,0)„ CLUSTERS IN FAST ION BEAMS

Numerous methods exist to record vibrational spectra of molecular ions such as direct absorption in ion beams,^ photodissociation,-"^ photoinduced charge transfer cross-section change,^ and absorption/emission from discharges.^"^^ A general technique for obtaining variable resolution spectra over a wide range o f cluster ions and larger polyatomic ions is not presently available. Application o f ion beams to this problem offers two clear advantages. The emitting ion species can be identified and isolated with standard mass spectrometric methods such as a Wein filter, and spatial separation o f the photon collection region from the ion generation region where excited neutrals and radicals can interfere, complicating the resultant spectrum.

Infrared emission techniques require a high density o f vibrationally excited ions to obtain suitable spectra. The spectroscopic method of choice is often cool ions “pumped”

by a laser, but direct absorption in fast ion beams resulted in only 1 0 ^ excited ions. ^

124 Discharge techniques, however, often expend great amounts of effort^"^’ to minimize the

“hotbands” often observed.

Ion beams do present some challenges, however. Sufficient densities of like-charged emitting ions is limited by space charge effects and ion lifetimes in fast ion beams are considerably shorter (*5 (Xs) than the vibrational lifetimes o f the cluster ions (> 1 ms).

Unfortunately, optimization of one of these effects is at the expense of the other. Each apparatus presented here represents a different compromise.

Multiple Collision Anatysis

The application of M.S. Kim’s glancing collision definitions to the ion beam experiments performed in this research group has been previously discussed in detail

Attenuation measurements revealed that, even under multiple collision conditions, some fraction of the original ion beam current reached the end o f the beam path. Kim’s

definition for total collision cross-section, 0 4 - is given by

CT t = C T l + C T c a ^ ( 1 )

where is the cross-section resulting in ion loss and is the coUisional activation cross- section, where ion current is still detected at the end o f the beam path, either as the original parent ion, or as a fragment ion as a result o f collision induced dissociation (CID).

125 The value of <1 ^,^ can be thought o f as occurring in one of 2 pathways. The first is the

case where the energy transferred in the collision is greater than the weakest bond in the cluster. This is the CID path and can be shown as

+ B A * + n B . ( 2 )

The second case is one where the energy imparted to the ion cluster is less than the weakest bond, internal excitation. Multiple collisions of this type could be considered equivalent to the spectroscopic method o f multi-photon dissociation and can be written as

A*(B)„ ^ A*(B)n ->A"^(B)^ ->...^ A"^(B)^_,+B (3)

where energy is added until the total accumulated energy overcomes the weakest bond. The difference between the spectroscopic method and collisional activation is due to the nature of the excitation event. Photons with a narrow range of energies provide the excitation in spectroscopy, whereas the energy transfer due to collisions is a wider distribution.

These definitions, while useful for theoretical purposes, are not the parameters measured in the laboratory. Typical measurements include parent attenuation cross-section, dp, and the collision induced dissociation cross-section, dao- The relation of these 2 parameters to Kim’s definitions is easily seen. The ion loss cross-section, cTl, is the loss of total ion current and does not include the fragment ions that reach the end of the beam path. The parameter dp, however, does include CID products as parent loss, and d^ can be written as

126 CTl — CTp CTc id - (4 )

The relationship between CT^^ and the measurable parameters is also a simple one,

(5)

The parameter r, is the key to the technique. It represents the fraction o f the original ion

beam that underwent a single collision, did not dissociate, and reached the end of the beam

path. The expression in Eq. 5 yields the cross-section for ions that underwent one collision

and reached the end o f the beam path, regardless o f dissociation.

All of these parameters were determined by fitting attenuation data with derived

equations from Kim using r,. The values o f r, for H j0 * (H ,0 ) and Hj 0 ^(H2 0 ), were

determined to be 0.90 and 0.60, respectively. Once the value of r, is known for a species,

the values of for the fraction that underwent n collisions, did not dissociate and remained in

the ion beam, r„, are known. An inherent assumption in this method is that r„=0, i.e. no

unimolecular processes account for ion loss. This assumption is valid for the experiments

that determined these values for reasons given elsewhere.

The fact that the values o f r, for these clusters are non-zero means a fraction o f the

collisions result in internal excitation of the ion cluster without loss from the ion beam.

The cross-section that results in parent excitation, written as oj», is expressed as

CTp. = C T c a ~ ^ c id • ( ^ )

127 The results o f this detertnination are shown in Figure 4.1 along with the data points for

the fit o f r,. In the case o f H ,0 a single collision m ost often results in internal

excitation o f up to 6 % o f the original beam (solid line) while 2 collisions cause dissociation in the majority o f the cases (dashed line). This can be interpreted as transferring, on average, less than the bond energy o f the cluster, 1.37 eV, with 1 collision and greater than this amount with 2 collisions, ~ 1 eV.

The determination that a firaction o f the ion beam can undergo a collision, acquire energy, and remain in the ion beam lends itself to a number o f possibilities. One of the more attractive possibilities is detecting infrared photons emitted as the ion clusters relax.

Previous Results

Experiments performed earlier on a simple test apparatus (Figure 4.2) investigated total infrared emission from an ion beam o f water clusters through the use o f a collision gas to internally excite water clusters via glancing collisions. The water ion clusters were extracted and passed through a collision cell containing a higher pressure of argon. Next, parallel plate deflectors used for the modulation scheme and beam steering in one dimension were employed. The photon collection region consisted of a large ( 6 ”) diameter cylinder containing 4 grounded rods. The end of the cylinder was sealed with an alu m in u m flange. Centered on this flange was a biconvex CaF, lens (f=2.5 cm). The outer cylinder was allowed to “float” at the potential placed upon it by the charging o f the ion beam. The ions entered the tube and were kept nominally at a ground potential by the rods. When the ions left the “shielding” rods, they experienced a repulsive potential due to the “charged”

1 2 8 tube/lens flange and deflected away from the ion beam axis. The net effect o f this arrangement was a “long” time (»5 fis) spent by the ions in front of the collecting lens.

The fact that the emission from the ion clusters is several orders o f magnimde lower than the blackbody background required the use o f a modulation scheme in all the experiments performed. A 700-1000V square wave was applied to a set of parallel plate deflectors. The deflection was sufficient to move the ion beam out of the photon collection region. The net effect of this was an amplitude modulation of the transmitted ion current. All light measurements were taken with a liquid nitrogen cooled InSb detector

(Cincinnati Electronics Model SDD-1963-S1) which has a maximum response in the O-H stretching region and very little response in the visible. The output o f the detector was sent to a lock-in amplifier (Stanford Research SR510) and the signal was recorded.

Figure 4.3 shows the results o f this preliminary investigation. The discharge turned o ff represents the background signal level. The discharge on, with the collision gas, in this case

Ar, off, the light level increased slighdy. Addition o f the collision gas, despite the fact that the total ion current is reduced by 33%, increased the detected light by a factor of 4. The low response of the detector in the visible region lends credence to the identification o f this response as infrared radiation.

Circular Variable Filter

Initial attempts at dispersion o f the emission were accomplished by use of a Circular

Variable Filter (CVF) with a transmission range o f 2.5-4.S (im (Optical Coating Laboratory,

Inc.). The experimental apparatus is shown in Figure 4.4. Water ion clusters extracted at 1

129 keV Érom the glow discharge source and undergo g la n cin g collisions with Ar introduced

into the beam path via a 1/8” teflon tube, gaining on average 1 eV of internal energy. This

ion beam is then modulated by a 700-1000 V square wave applied to the deflector at 160

Hz, experimentally determined to be a quiet area in the noise spectrum. The excited

clusters then enter the photon collection cell where they “dwell” for approximately 0 . 1 jis.

A Faraday cup to monitor the ion current terminated the beam path. This short rime is

offset by obtaining the highest possible ion density, limited only by space charge. This

constraint required the ion beam path from the source to the photon collection cell to be as

short as possible.

A fraction of the light emitted from the clusters while in the photon collection cell

passes through the CVF and is focused on the InSb detector with a fast CaF; (f=2.5cm)

lens. The CVF was controlled by a stepping motor directed by a FORTRAN program,

Step6 .for (see Appendix C), which divided the filter into 50 steps (10 o f these steps at the

2.5 pm end were lost due to a crack that developed in the filter, which was subsequently

covered by a small piece o f stainless steel shim stock to prevent unfiltered light from

reaching the detector).

The advantage of this arrangement, the ion beam orthogonal to the detector, as opposed to in line as in the previous experiment, is one of background. The presence of the ion source itself in the field o f view o f the detector greatly increased the background light level present in the original experiment. While the source light is not directly modulated, some experiments revealed that the modulation applied to the deflectors did result in the plasma of the source also being modulated. Under certain conditions, with a

130 suffîciendy low frequency and high pressure, the discharge could be seen with the naked eve coming out of the source and modulated by the deflectors.

Detection of the emission perpendicular to the ion beam meant a fundamental tradeoff was present. To increase the time in front o f the detector, the beam voltage should be lowered. The velocity of ions in an ion beam is found from.

ymv- =qV^ (7),

or, rearrangement gives.

'loo - y (8 ).

Inside the photon collection cell, the electric potential is at ground, and is equal to V,,, the real voltage applied to the beam. It is obvious that the velocity is decreased with the beam voltage and the time to pass a given distance, or field o f view, is increased. As the voltage is decreased, however, space charge effects have an increased influence on the ion current (see Chapter 3). The general space charge equation for ions is given-^ as

(9)

where is the maximum current in |lA, q is the ion charge, m is the mass in a.m.u., is the accelerating voltage, D is the tube diameter, and L is the tube length. This equation

131 dearly shows the decrease in maximum ion current (for a transport tube of given diameter)

as or Vy at ground, is decreased.

The optimum condition for light detection is where both the number of ions and the

time in front of the detector is a maximum. This can be written as

where Ipc is the ion current in the Faraday cup, t is the time in front o f the detector and T is

the fluorescence lifetime. Use o f the velodty relation.

and substitution o f Eq. (7) into Eq. (10) yields.

(1 2 )

Collection o f the constants for a given species and apparatus gives

(13)

where k contains the instrumental and system factors and is expressed as.

132 Each system could be optimized by careful choice of such that Eq. 13 is a m a x i m u m .

The low resolution IR spectrum obtained and a background scan are shown in Figure

4.5. The blackbody background was produced with a discharge o f Ar in place o f the neat

water vapor. The feature centered at 3700 cm ' shows good correlation with various

experiments on HjO^(H 2 0 )„ clusters-^» ^ if the low resolution nature o f the experiment is

taken into account. A typical mass spectrum from a similar source is nominally 40% H,0^,

25% H ;0 (H2 O), 20% Hj0 ^(H2 0 )2 , 10% H 3 0 *(H 2 0 )j and 5% other species (Figure 4.6).

The central downward feature is an artifact o f the CVF.

D iscu ssio n

The low resolution IR emission spectrum presented above demonstrates the feasibility of application of fast ion beam techniques to the general problem. The “peak” nature of the feature argues against the light being bremsstrahlung in origin. Such a source would be expected to emit over a much larger range of wavelengths, although it could be present in the background curve. The good correlation to the absorption spectra reported by Yeh-^ and Begemann-- is also indicative of emissions from a molecular source.

Some fundamental problems are also shown, however. The peak power from the spectrum is ~40 fW. Each data point on the spectrum required 20 minutes of run time and a large time constant (10 sec typically) to acquire. This low power and long observation

133 time required means careful consideration is needed before attempting the experiment with

a higher resolution spectrometer.

The use o f a monochromator or other spatially dispersive method is difficult in the

extreme. Loss o f signal through the instrument makes alig n m e n t and subsequent scanning very time consuming. A Fourier transform approach, where the multiplex advantage could result in a greater “scan time” per point (for a fixed experiment time) and could conceivably give better results, but the interferometer would require a stepping action rather than a continuous sweep because o f the large time constant required.

Cursory examination of the system used for the CVF shows that it was not optimized for resolution o f the infirared peaks. Light that passes through a broad section o f the filter can reach the detector, spreading a peak over a wide range. This was a concession to the physical dimensions of the vacuum system utilized. A two lens system where the photon

collection region is imaged with a magnification of ~ 0 . 2 would gready improve the resolution of the spectrum. This technique was not employed for this “proof of concept” study.

Initial work with undispersed emission firom the EPG device has found ~1 pW of infirared light present from the unattenuated ion beam. Examination o f Fig. 4-3 reveals that the corresponding light level (unattenuated ion beam. Le. no collision gas) in the orthogonal setup was ~3 pW. The greater level observed was due to modulated background signal from ion-surface collisions in the modulation region. The EPG system does not have the modulation region in the field o f view o f the detector, eliminating this source of background

134 and lowering the amount of undispersed light recorded. This is encouraging as the ~1 pW

of power observed is wholly due to vibrational relaxation.

Conclusions

An infrared emission spectrum from a fast ion beam of unresolved HjO*(Fl 2 0 )„ clusters has been obtained through the crudest o f spectrometers, a circular variable filter. The spectrum shows good agreement with published high resolution spectra.-^’ — Despite the low power o f the signal, this method shows promise for general, low resolution spectra of cluster ions once optimization of the system (Le. the physical layout of the photon collection region, improved optical setup for spectrum resolution).

135 References

1. Coe, J.V., Owrutsky, J.C., Keim, E.R., Agman, N.V. & Savkally, R.J.J Chem 90, 3893-902 (1989). 2. Carrington, A. & Sarre, P.J.M ol Pl^s 33, 1495-7 (1977). 3. Draves, J.A., Luthey-Schulten, Z., Liu, W.L. & Lisy, J.M. /Chem Phys 93, 4589-602 (1990). 4. Winkel, J.F., Woodward, C.A., Jones, A.B. & Stace, A.J.J Chem Phys 103, 5177-93 (1995). 5. Choi, J.H., Kumata, K.T., Hass, B.M., Cao, Y., Johnson, M.S. & Okumura, M.J Chem 100, 7153-65 (1994).

6 . Wing, W.H., Ruff, GA.., Lamb, W.E. & Spezeski, J.J.Plys Rev Lett 36, 1488 (1976). 7. Oka, T. Plys Rev LeU 45, 531 (1980).

8 . Gudeman, C.S. & Saykally, R.J.^ntt Rev Pl^s Chem 35, 387 (1984). 9. Gudeman, C.S., Begemaim, M.H., Pfaff, J. & Saykally, RJ.Phys Rev Lett 50, 727 (1983). 10. Foster, S.C., McKellar, A.R.W. & Sears, T .].J Chem Plys% L 578-9 (1984). 11. Hirota, E. & Kawaguchi, K. yinn Rev Pfys Chem 36, 53-76 (1985). 12. Brault, J.W. & Davis, S.P.Pfys Rev Lett 65, 2535 (1990). 13. Martin, PA . & GuelachvOi, G. P fys Rev L ett 65, 2535 (1990). 14. Coe, J.V. Ph.D. Dissertation, The Johns Hopkins University, (1986). 15. Okumura, M. Ph.D., University o f California, Berkeley, (1986). 16. Nesbitt, D.J., Petek, H., Gudeman, C.S., Moore, C.B. & Saykally, R.J.J Chem Phys 81, 5281 (1984). 17. Bawendi, M.G., Rehfuss, B.D. & Oka, T . ] Chem P lys9 5 , 6200 (1990). 18. Siegel, M.W., Celotta, R.J., Hall, R.J., Levine, J. & Bennett, R APl^sical . Reviews A 6, 607 (1972). 19. Cowen, K A . Ph D . Dissertation, The Ohio State University, (1994). 20. Moore, J.H., Davis, C.C. & Coplan, M.A. Building Scient^c Apparatus (Addison Wesley, London, 1989). 21. Yeh, L.I., Okumura, M., Myers, J.D., Price, J.M. & Lee, Y.T.J Chem Plys 91, 7319-30 (1989). 22. Begemann, M.H. & Saykally, R.].J Chem PlysBT., 3570-9 (1985).

136 MuKiple Collision Analysis Curves for 0.07 P* total (no fragmentation)

0.03 - .CO total (l^cy fragntnts)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Figure 4.1. Multiple coUision analysis o f H^O*(H 2 C)). The data points and the fit that produced the value for r„ the fraction o f the original ion beam that underwent one glancing collision, did not dissociate, and reached the end of the beam path. The solid P* curve is the total fraction of the ion beam, at a given attenuation, that has undergone a collision and absorbed excess energy. The lower solid curve and the dashed cur^'^e are the fraction o f the original beam that dissociated. The

dash-dot curve labeled 1 is the contribution from ions that underwent a single collision, and the dash-dot curve labeled 2 is the result o f 2 collisions and similarly for the CID curves (dashecQ. This analysis can be

read as, on average, one collision does not dissociate the ion while 2 will cause dissociation. As the experimental dissociation energy for

H 3 0 ^(H,0 ) is 1.37 eV, a collWon energy of ~1 eV can be concluded.

137 Vacuum Wall CaFg Lens i Grounded Rods

Ion Beam

InSb Detector ------— '------— — — — - - -

Figure 4.2. Original apparatus for finding total emission o f infrared light. The ion beam was directed at through a set o f 4 grounded rods towards a CaF, lens. After a short time (< 1 second), the lens charges and repels the beam as it exits the region bounded by the rods. The fact that the beam is “pointed” at the detector gives long times (~5|xs) in the detector field o f view.

138 J Total IR Emission

16 gas on 1 gas on 14

12

10 IW Q) 8 I 6 K 4 gas off gas off

2 beam off 0 beam off

0 100 200 300 tim e (s)

Figure 4.3. The undispersed infrared light detected from the emission of water clusters. When the ion source was turned off (labeled “beam off”), little light was observed on the detector. Turning the source on without adding collision gas (“gas off”), a light level of ~3 pW was detected. Addition of collision gas (“gas on”), although decreasing the total ion current in the photon collection area, raised the detected light to ~13 pW. N ote that a large time constant, as evidenced by the long tail between “gas on” and “gas off,” was applied to reduce the noise.

139 Ar Ion Source Faraday : Cup ■A - t V, f

/ / \ / \ f ^ InSb

Figure 4.4. Diagram o f the spectrometer. Ion clusters are extracted from the ion source and undergo glancing collisions with the argon introduced by a tube. The beam is amplitude modulated by a square wave sufficient to prevent to ions from entering the photon collection region during the “high” portion of the wave. The emission from the ions, also modulated with respect to the photon collection region, passes through the circular variable filter and is focused on the InSb detector. The signal is then demodulated through a lock-in amplifier and the signal averaged over long periods o f time.

140 Infrared Emission 0.25

0.20

I0.15 w

I 0.10 Blackbody Background

0.05

0.00 2500 2750 3000 3250 3500 3750 4000 Frequency (cm''')

Figure 4.5. Low resolution in&ared emission spectrum o f H 3 0 *(H, 0 )„. The background scan was produced using Ar as the source gas. The downward feature at 3000 cm ' common to both scans is an artifact o f the circular variable filter. The excess light above 3300 cm '* is consistent with absorption measurements by Lee et al. Peak power is ~33 fW.

141

J Positive ion Mass Spectrum 900 -I "1 Torr neat water vapor 800 - 700 - < =L 600 - c 500 ■ s 400 • d 300 ■

o 200

100 -

0 1 0 0 200 300 Deflector Potential (V)

Figure 4.6. Typical mass spectrum firom the ion source. The dispersive element is a Wein filter.

142 A P P E N D IX A

MATHCAD TEMPLATES

Famify Eneigedcs II.mcd

This is the original calculation and plotting worksheet for the neutral water cluster

calculations studied in Chapter 1. Each data set is converted to the excess binding energy

per water molecule and plotted against n

143

£ Family Energetics II.mcd

Read energetics data for chains and set up counter, i

Chains - READPRN energetics > - O..rows(Chains) - 1

Display values for chain data Bridging Altematin n Chain Chain Chains. 0 Chains. Chains. 2 1.72 1.74 3 ' 2.72 2.98 4 2.98 2.97 5 3.02 33 6 3.14 3.48 7 3.12 3.67 8 3.05 3.78 9 3.06 3.89 10 3.12 3.96 II 3.2 4.04 12 3.16 4.09

Enter values for ring data and set up co Ring Energy ii 3 .9 Convert counter to n (Total) n ' - plot Ringji ii 3 661.74 n 4 887.25 5 1109.96 1332.79 6 Convert to binding energy per particle 7 1554.61 1776.66 8 Ring.. ii-( 2I7JZ2) 1997.90 9 Ring:,

144 Enter stacked cube data and set np counter, j j 0..9 n Stacked Cube Energy (Total) nsc. SCube. = JJ Convert counter to n-* ^ plot 1786.12 12 268424 16 3583.12 "Ji nsc. 20 4482.01 24 5380.96 Convert to binding energy per particle 28 6279.87 32 7178.81 SCube. nsc. (2 1 7 2 2 ) 36 8077.69 SCube. nsc. 60 1347129 92 20662.91

data and set up counter n Ice Energy Convert counter to p 0..5 (Total) n plot m = Ice p p 1770.41 nm m 30 668325 39 869723 Convert to binding energy per particle 57 12736.72 72 16101.54 Ice - m ( 21722) Ice 90 20140 m

Place point at bulk

nbulk 0 Bulk 112

145 Enter clathrate data and set up counter, me Clathrate Convert counter to 0.. 16 Energy (Total) n-' 5 plot me Clath = q To" 2232.54 nmc me 12 2674.81 ü 3123.45 Convert to binding energy per particle 15 3346.93 " Clath me •( 217.22) 16 3571.90 Clath ------9- q_ 18 4013.83 me 20 4459.63

22" 4897.48 24 5358.69 26 5804.17 2 ^ 6248.77 30 6692.72 32 7146 JO 36 8038.68 40 8923.62 " 50 11154.58 60 13388.22

Enter Stacked cube-1 data and set up counter, mm I Convert counter to pp 0.. 3 n Cube Energy (Total) n-I 3 plot mm I SCml PP pp nmml mml 7 1557.45 PP pp 1 1 2455.59 15 33^ J2 SCml mml -( 217J2) SCml _PP. PP______19 4252.94 PP mml pp Enter Stacked cube-2 data and set up counter, mm2 n Cube Energy Convert counter to ppp 0.. 3 (Total) n-I'3 plot mm2 SCm2 ppp ppp nmm2 mm2 6 1333.57 ppp ppp w 2233.54 SCm2 mm2 •( 217J2) 14 3131.79 SCm2 _PPP PPPJ______ppp mm2 18 4030.37 ppp

146 Enter Stacked cube-3 data and set np counter, mm2 Cube Energy Convert counter to pq 0..2 (Total) n-> 3 plot mm3. SCtn3 pq pq nmm3 = mm3 ~9 2008.03 pq pq 13^ 2907.50 17 3804.76 SCm3 mm3 -( 217J22) SCm3 =------®------»------mm3 pq

Enter cubic ice data and set up counter, m Ice Energy Convert counter to pc 0.. 17 (Total) n - ' 3 p l o t met IceC pc pc nmci - met 10 222135 pc pc 18 4011.93 26 5804 Convert to binding energy per particle 32 7154.6 34 7605.58 IceC mci •( 217.22) IceC pc 38 8499.63 pc met 42 939737 pc 43 9614 45 10066.01 48 10742.43 57 12762 52 11638 68 15240 90 20180 92 20635.77 113 25343.25 120 26921.46 136 30500.63

Enter Stacked cubes of 5 data and set up counter, pq5 n Cube Energy Convert counter to pqS - 0.. 2 (Total) n ' 3 plot nsc5 SC5 pq5 pq5 nsc5p nsc5 kT 2234J5 pqs pqS 15 3357.15 20 4481.27 SC5 nsc5 , ( 217.22) SC5 pqL pqlj______:: pqs nsc5 pqS

147 Molecular Binding Energy per Molecule

Bridging Chain

Alternating Chain , Rings I Clathrate J “6 Ice-Ih s Partial and Complete Ice-Ic Stacked Cubes

10

* * Bulk

0.20.3 0.4 0.5 0.7 09

148 Cubic Ice Octa Multifit.incd

Once the equations for “growing” a crystal o f octahedral cubic ice to bulk are determined

(see Chapter 1), the equations are used along with the calculated energies and the resulting fits for E 4 , E 3 , and E2 to place the large n curve on the excess binding energy plot. A similar template is created for the hexagonal ice structures.

149 Cubic ice Muitifit - Octahedral Structures

Input number of parameters to be fit (E^, E^, and E^) npar 3 input number of structures In fit npts - 5 Set up counters i 0.. npts 1 j 0.. npar I k - 0.. npar - I

Enter Structures by number of waters, n, and total atomic binding energy. E N. = E. - I 1

IÔ" "I2 2 1 J6 18 40H^.8J[ 35 7825.80 53 11868.25 84 18840.44

Input an array with the number of waters with 2, 3, and 4 hydrogen bonds for each structure

6 4 0 7 10 1 n 6 24 5 7 34 12 6 52 26 Convert the total atomic binding energy to excess binding energy per water molecule

E N - 217.22 Eb ----- !------■ N. I Set up A and H matrices for fit

" j Ainv =A ' i i Output A values

206 788 277 A = 788 4552 1890 277 1890 846

150 J Calculate a for each parameter -22IJ225 a ^Ainv-H a = 223.637 -22631

Calculate the fitted total atomic binding energy for each structure and output values

Ecal. - ^ a - n . Ecal. 1 . ' 1 1 EI j 2221.895 222136 4011351 4011.81 7826.183 7825.8 11867.947 1186835 18840.528 18840.44

Calculate the error in y

E Ecal. 2

o y ------oy = 0.65 npts npar

Calculate the individual errors

0.122 oa Ainv^ yoy oa = 0.072 0.13 Setup the counter for the cluster "shells"

S = 1.100 Input the equations for cluster growth

S n4g S- ^ ^ (4 (2 i - 1)) j = I i = I

n3g 4-(S - 1) - (8 (i - 1))

n 2 s 6

151 Get total number of waters

NCj n4g-n3g n2g

Calculate the total atomic binding energy from the fitting expression

Es a4s.cy n3s-a, nls'Oo

Convert to excess binding energy per water molecule Eg - Ncg 21722 Exbf Nc,

0 0.2 0.4 0.6 0.8

Ncs \ N.

152 Yp calculation.mcd

This is the template that calculated the various values for the proton solvation enthalpy

from the Y’ approximation method.

153 J bulk sums 10472 1020.0 8792 845.7 816.6 We don't know the radius to 9382 911.0 770.2 736.8 707.4 use for OH- so we pick one Hbsum 855.8 828.6 687.8 6542 625.0 that fit's trend. 1.40 is Goldschmidt value, 1.1 fits 830.4 8032 662.4 628.9 599.8 well cluster index pos ion index neg ion index n 0.. 5 I 0..3 m 0.. 5 j 0..4 w angstroms angstroms n T’i m. 6 . 0 6.o~ .94 T.T I&9 1.17 1.16 41.9 1.44 1^4^ 67.1 L63 1.80 93.9 2.05

0 142.3 250.2 336.8 405.4 463.6 0 104.6 187.4 251.8 307.0 355.1 0 75.7 143.1 1982 247.7 292.5 Hp, w 0 66.9 123.8 174.8 221.7 265.6

0 109.6 1882 257.7 317.1 376.1 0 97.5 172.4 233.1 290.4 344.0 Hn 0 602 113.8 162.8 208.4 248.1 Hsn. Hn W 0 54.4 104.6 1522 1982 243.4 J.n J.n 0 43.9 84.6 123.5 162.0 199.7

kp. kn. replaced Br- value of 1423.2 with a J constants Rosseinsky scaled value of 1444, i.e. relative to 57&6T 16242 the relative spacing between CI-, Br-, and I- in Ross, was scaled to the 01-, I- gap in absolute proton 687.0 1597.5 value the NBS data, the best fit value is 769T 1456.7 1464.0. this would give Bromine a bigger 794.8 14232 bulk value than Chlorine by 6.3 kJ/mol, 1393.9 i.e. they are the same! since internalized fit better, maybe this procedure should only be done with the halides, with all nx 5 mx nx cluster ions, optimal Br- constant was found to be 1449.9 154 Hsp. ^ Hsn. this is the fraction of . V *^i.n j,m c(i,j,n,m) ------c(0,0,nx,mx) =0.6225 bulk obtained at n=5 Hbsum. . '.J

absolute proton value is Ytry Ytry - 1146_2 Hsp. Hsn. Coe results Hpb. - kp. - Ytry Hnb. - kn. - Ytry cp(i,n) ^ -----— c n ( j ,n ) ------J Ytry=-1150.4 ^ Hpb. Hnb Hpb; Hnb. best is -1148.3 J cp(i.nx) cn(j,nx) 5682 478.3: 0.6507 0.59 4592 4512 0.5688 0.5542 376.8 310.5 0.5271 0.4966 351.4 277 0.4886 0.5397 247.7 0.4271 define the difference from the approximating constant, 5p and Sn 5p(i,j,n,m) =cp(i,n) - c(i,j,n,m) 5n(i,j,n,m) = cn(j,m) - c(i,j,n,m)

constraints on cp and on values using only sign of cluster differences nn I..5 mm 1..5 Hsp. - Hsn. cdiflf( i, j, nn, mn ) = ------— ------^®Pi,nn cdiff( 1,1 ,nx,nx) = 1

All of the calculated proton values are either upper or lower limits as indicated by cdiff, this includes only cluster data because the c are undefined for the monomers

Hsn. I j.mm Ycalc(i,j,nn,mm) k p .- kn. 2 c(i,j,nn,mm) c(i,j,nn,mm)

1 Ycalc(3,0,nx,mx) = 1.1086-10^ rp, -rp . 0 .0 1.0 rpbCi,. rp. ^ - n-{ 1.928-.885)^ rPi -- rp. 0.15 1.19: m 0.00 -1.0 m. m. 0.0 -1.33: mbc. m. ^ - n-(1.928-.885)^ J.n

mbc. 5 n3 - 1-5 m3 - 1.. 5 rpbc; 5 rpbCj„ mbc-o 2.96891 1.1 2.9499 0.94 2.9776 i 1.16 29791^ TTf 3.0811: 3.0419 T 4 9 I 3.1302 hS 3.0783 1763^ J ^ 2 2 ' 295

155 4 3 3 4 3 AV(i,j,n3,m3) - % ! rpbc. ^ - mbc. ^ AVl(il,jI,nl,ml) - rpbc.^ - mbc.j

kl 0..499 yl,2 5 -i^ 2 5 j - 5 -(„3 -l)--(m 3 -l) ^ Ycalc(i,j,n3,in3)

’‘*i 2 5 i^ 2 5 j.S („ 3 -i).(m 3 -i) &V(i,j,n3,m3) intercept(xl,yl)= 1.1714*10^ ycalC|.| = slope(xl ,yl ) xl^, - interceptxI ,yl ) k l

il = 0 jl =4 nl -5 ml =nl

900

-1000

Ycalc( ij.n3.m 3) coo 1100 — YcalcC i . j . n l .m l ) OOO CO Ytry -1200 og ycalc.

1300

1400 150 100 -50 100

4V (i.j.n3.m 3).A V (i.j.nl.m l ).AV(i.j.n3.m 3).xl

0 .5

5p( i.j.nn.m m )

5n( i.j.nn.m m )

H p. - Hn. i.n n j.m m

Check for consistency of both monomer ions and clusters with Born theory Hv -44.01 Hpbc. - Hp. - Hpb. - n-Hv Hnbc. - Hn. - Hnb. - n Hv i . n ^^i.n ^ I j.n j.n j

156 Hpbc, 5 HnbCj J

324.65 322.25 324.15 327.35 304J5 282.45

30^85 ' 2 5 3 ^ "268.05

Born slope ib 0.. 1 bo 0 b. - 685.8 rmlg 0 rml, -1

\ 1 1 1 " 1 1

100 — V -

» P ‘^ 3 . n 200 - - » " '^ 2 . n OOO oOO'o ^ '^ o o 300 - - i OOO » o 8 I H pbc, 0 -400 O 'v \ HnbCj 0

500 " ib 5

-6 0 0

700 1 1 0 0 .2 0.4 0 .6 0.8 I 1 I I rpbcj^ mbcj „ rpbc, „ mbc, rpbc, „ mbc. „ mbc^ „ ib

kn 0.. 19 ynlg.j^j - Ycalc(i,j,l,I) xnl^, . ^ Hsp, , - Hsn. ,

ynlcalc,^ =slope(xnl ,ynl )-xnl,^^ i- intercept(xnl ,ynl )

slope(xnl ,ynl) =0.6379 intercepte xn 1 ,ynl) = 1.1427*10

157 y^Si^j = Ycalc(i,j,2,2) xn2j. J. Hsp. ^ Hsn. ^

yn2calC|^ - slope(xn2,yn2)-xn2^ - intercept(xn2,yn2)

sIope(xn2,yn2) =0.4144 intercept(xn2,yn2) = 1.1467-10^

y ^ 5 i-j " Vcalc(i,j,3.3) xn3j. . Hsp^^ Hsn.^

yn3calC|^ = slope( xn3, yn3 ) -xn3 - intercept( xn3, yn3 )

slope(xn3,yn3) =0J3I2 intercept(xn3,yn3) = 1.1462-10^

yn4j . . = Y calc(i,j,4,4) xn4^ . . = Hsp. ^ - Hsn. ^

yn4calC|^ = slope(xn4,yn4)-xn4|^ - interceptxn4,yn4)

slope(xn4,yn4) =0.2799 intercept(xn4,yn4) = 1.1447-10^

ynSj . . = Ycalc(i,j,5,5) xn5g . . Hsp. ^ Hsn. ^

ynScaiCj^ - s!ope( xnS, yn5 ) -xn5 ^ r intercept( xn5, ynS )

slope(xn5,yn5) =0.254 intercept(xn5,ynS) = 1.1422-10^

f I i n2 1.. 5

j, k2 0..99 2 5 - j - 5 -n 2 -I Ycalc(i,j,n2,n2) * 2 2 5 - i - 5 j - n 2 - i ^^'^('-j-i'2,n2)

y2calC|^ = slope(x2,y2)-x2|^ - interceptx2,y2intercept(x2,y2) = 1.148-10^ slope(x2,y2) = 1.7387

! "6 1..5 »*Pi.n6 »®"j.n6 I ...... :------...... :------,n3 y6calc^,kZ ------slope( x6, y2 ’) -x6,., Ic2 - intercept( x6, y2 ) intercept( x6, y2) = 1.1462-10 slope(x6,y2) =0.3088 ii=3 jj=2 nn =2 mm =nn

158 1100

1150 OOO

Ytry

Ycalc(ii,jjj.nn,mm) 1200 OOO

1250 40 -20 10 20

x2. , .x2. , ,x2. . dV( ii .Ü, nn. m m J

nn - 5 mm = nn

1100

y-k2 -1120 O OO yôcalcj^

1140 Ytry

Ycalc n n .m m ) OOO 1160 »o o yn5calCj 03 kn

yii4calC| 1180 kn

yn3calC| kn 1200 yn2caIC| kn

y n lcalc. kn

-1 2 4 0 -3 0 0 -2 5 0 -200 -1 5 0 100 50 50 100 150

x6^.x6^.x6^.H sp. r„ - Hsn. ^.xn5^.xn4^^.xn3^.xn2^.xnl^

159 fits with error in both x and y, k2 is index of points, y2 vs x2 and y2 vs x6 for bulk electron density volumes and cluster solvation enthalpies respectively

0x2 3 0x6 - 8 oy2 - 10

ib 0..20 ‘”^ib 1.9 , ib .01 bx6.|j = .25 - ib .005 1 '^ i b \ 1 (oy2)‘ - bx2 ^ ( 0 x2 / ' ^ ' ’ib 2 2 lb (oy2) - bx6.b^-(ax2)

X! '^ib- y^ici - bx2;b'x2^ Z ! '^^ib ■ y2^-bx6,^ x6^ k2 ax2;b ------ax6.lb ------10 ' wx6 . k2 k2

*”‘^ib'’^k2 ax6.lb bx6. lb rx6 '•"“ib __ , 2 2 2 k2 (oyZ) - bx2.^ '(0x2) k2 (oyz) ^ - bx6.^ ^-(ox6)^ ax2..lb %*2(b %*^ib 1146.73481 1.9 187.4719 1149.1006! 035 153.4376 1146.6581 1.91 1873989 1148.8562! 0355 149.7822 1146.5813 1.92 1873485 1148.61191 036 146.4364 1146.5046 1.931 1873204 11483676 03651 1433992 1146.4278 1.94! 1873145 1148.1232! 037 140.6694 1146J511 1.95! 1873305 1147.8789 i 0375 1383456 1146.2743 1.96 1873684 1147.6346 038 136.1265

1146.1976 1.97 187.4278 11473902 0385 1 1343109 1146.12081 1.981 18730881 1147.14591 039 132.7974 1146.0441 1.99! 187.611 1146.9016! 0395! 131.5844 1145.9673 2 187.7344 1146.6572 03 130.6705 1145.8906; 2.011 187.8787 1146.41291 0305 130.0541 1145.81381 2.02! 188.0438: 1146.16861 031 129.7338 1145.7371 2.03! 18832941 1145.92421 0315! 129.7078 1145.6603 2.041 188.43561 1145.67991 032 129.9746 1145.58361 2.05! 188.6619! 1145.4355! 0325! 130.53241 1145.5068! 2.06! 1188.90841 1145.1912! 033 ; 1313795 1145.4301 2.071 189.17481 ! 1144.9469! 0335! 132.5141 11453533! 2.081 ! 189.461 1144.70251 ! 034 : 133.9345 114537661 2.09; 189.7667! 1144.4582! 0345; 135.6387 1145.1998 2.1 190.09181 114431391 035 137.6248!

160 A P P E N D IX B

FORTRAN PROGRAMS

Step6.for

This program was designed to control the step motor circuit for the circular variable filter (CVF) and display the output o f the lock-in amplifier as read by a analog-digital converter (ADC) board d uring the later infirared emission experiments (Chapter 4).

c routine for controlling the step motor and dispersing IR c emission. STEP version 6.0 c —Michael D Tissandier c Alan Earhart c 8 /1 9 /9 5 c Variable List c y d a ta Array to hold averaged values from all scans c y si n g Array to hold values from single scan c xy Needed for graphics display-moveto c dummy Needed for graphics display-lineto, setpixel c f i n Filename for saving data c n r e a d Number of readings per point c n w a it Delay time before recording detector output c nav g Number of scans to average c x yz Dummy variable to slow loop c p o s i t Record of CVF step number c in p o s Integer form of posit c key Holder for proceed/stop inputs from keyboard c kk ey Placeholder for stopping a scan c npos New p o s i t i o n f o r CVF movement c s t e p s Counter for movement by position number c b a s e l Baseline for non-CVF scan c z l e v e l Array for summation of light levels 161 c zleveltot Final value for summation c i,j,k,kk,t,tt,z Loop counters c This list is now incomplete.... c c End Variable List c

INCLUDE 'FGRAPH.FI' INCLUDE 'FGRAPH.FD' dimension ydata(50) dimension ysing(50) dimension yy(700) dimension zlevel(7201) dimension zlevavg(2000) dimension zlevsd(2000) RECORD /xycoord/ xy INTEGER*2 dummy integer*2 iiter character*15 fin logical*l kkey

c address for A/D

data ioadl,ioadh,ibyte/768,769,0/

c s e t g a in

» call oup(0,779,ibyte)

! c set mux input channel to 0 (use 17 in first parameter to read j c from channel 1) j; I call oup(0,770,ibyte) | i c be sure CVF is in start position

5 c a l l e l s p o s it= 0 w rite(5,*)'Set CVF into starting (0) position' write(5,*)' press when ready' call inkey(key) if(key.eg.32)goto 100 g o to 5 c ***************************************************************** c display main menu

100 call els w rite (5,*) ' Step Progreim Main Menu' write(5,*)' ' w rite(5,*)'1-Scan CVF' w rite(5,*)'2-Reset CVF to starting point' w r i t e ( 5 , * ) ' 3-C hange p o s i t i o n o f CVF' w rite(5,*)'4-Move CVF to "open" position' w rite(5,*)'5-Move CVF to "blocked" position' w rite(5,*)'6-Manual reset of CVF to O' w rite(5,*)'7-Scan without CVF'

162 write (5,*) '8-Integrate light level' w rite ( 5,*) 'Q-Quit' write (5,*) ' ' w rite(5,*)'Enter Number of Choice:' call inkey(key)

Goto section specified

i f ( k e y . e q . 4 9 )g o to 10 if(key.eq.50)goto 20 if(key.eq.51)goto 30 if(key.eq.52)goto 40 if(key.eq.53)goto 50 if(key.eq.54)goto 60 if(key.eq.55)goto 70 if(key.eq.56)goto 80 if(key.eq.113)goto 999 g o to 100

c Scan CVF Section 0 *****************************************************************

10 call els

c set number of readings per point

w rite(5,*)'enter number of readings per point' read(5,*)nread I ! ! c set scan wait time f w rite(5,*)'enter scan wait time' read(5,*)nwait

c set number of scans to average

w rite(5,*)'enter number of scans' read(5,*)navg

c wait to begin scanning

110 write (5,*) 'press to begin scan' call inkey(key) if(key.eq.32)goto 120 g o to 110 120 continue

c initialize ydata

do 130 i=l,50 130 ydata(i)=0

c record scan loop

dummy= setvideomode ( $vresl6COLOR) dummy= s e t c o l o r (1) do 140 k=lfnavg

163 call moveto (2,230,xy) call oup(2,771,0)

c point gathering loop

c take single spectrum

do 150 i = l , 50 dummy= setp ix el(2,479)

c take multiple readings and average for data point

y= 0 . do 160 kk=l,nread call oup(iword,ioadl,ibyte) call inp(iadl,ioadl,ibyte) call inp(iadh,ioadh,ibyte) 160 y=y+16.*iadh+int(iadl/16. ) y=y/( nread*1.) ydata(i )=y+ydata(i ) y s i n g ( i ) = y

c take step for next reading

call oup(10,771,0) call oup(2,771,0) posit=posit+l

c wait loop

do 170 tt=l,nwait f x y z = l i 170 continue I I c plot on screen - single scan

i y = i n t ( ( y s i n g ( i ) *4 6 0 . ) /4 0 9 6 . ) dummy=setpixel(2,479) ix = i* 1 0 150 dummy=lineto(ix,iy)

c changing directions for reset t call oup (0,771,0) ' do 180 i=l,50 do 190 j=l,1000 190 xx= l call oup(8,771,0) call oup(0,771,0) posit=posit-l 180 continue 140 continue

c normalize signal average

do 200 i=l,50 200 ydata(i)=ydata(i)/ (navg*l.)

164 c display total average

duiratiy=setcolor ( 15 ) call moveto(2,230, xy) do 210 t=l,50 ty = in t{(ydata(t)*460. ) /4096.) dummy=setpixel(2,479) tx = t* 1 0 210 duinmy=lineto (tx, ty)

c ready for next scan?

220 write(5,*)'press to stop viewing' call inkey(key) if(key.eq.32)goto 230 g o to 220 c a l l e l s 230 continue dum m y=setvideom ode ($DEFAULTMODE)

c Save spectrum

240 write(5,*)'Would you like to save spectrum? (y/n)' call inkey(key) if(key.eq.110)goto 250 if(key.eq.121)goto 260 g o to 240 260 write(*,'(a\)') 'enter filename ’ read(*,' (a)')fin open(3,file=fIn) I: > do 270 i=l,50 270 write (3, *) i, ydata (i) [ c l o s e (3) I 250 continue

I c Scan again?

5 280 write(5,*)'Would you like to scan with the same settings? (y/n)' f call inkey(key) \ c a l l e l s ; if(key.eq.121)goto 110 I if(key.eq.110)goto 290 g o to 280 ; 290 write(5,*)'Scan with different settings? (y/n)' call inkey(key) if(key.eq.121)goto 10 [ if(key.eq.110)goto 100 i g o to 290 c ***************************************************************** c Reset CVF Section Q *****************************************************************

c Determine current position of CVF and set direction

20 if(posit.gt.0)goto 300 if(posit.I t.0)goto 310 if(posit.eq.0)goto 320

165 c Greater than 0 case

300 call oup(0,771,0) c n t = p o s i t do 330 z=l,cnt call oup(8,771,0) call oup(0,771,0) do 340 i= l,1000 340 xy z= l posit=posit-l 330 continue write(5,*)'Position reset to start' 350 write(5,*)'press to proceed' call inkey(key) if(key.eq.32)goto 100 g o to 350

c Less than 0 case

310 call oup(2,771,0) cnt=posit*(-1) do 360 z=l,cnt call oup(10,771,0) call oup(2,771,0) do 370 j=l,1000 370 xy z= l posit=posit+l 360 continue w rite(5, *)'Position reset to start' 380 write(5,*)'press to proceed' call inkey(key) if(key.eq.32)goto 100 g o to 380

c Equal to 0 case

320 write(5,*)'CVF already appears to be at the starting position' w rite(5,*)'Please make sure that this is the case.' write(5,*)'If not, manually set the filter to O' 390 write(5,*)'press to continue' call inkey(key) i f (key.eq.32)goto 100 g o to 390 c ***************************************************************** c Change Position Section c *****************************************************************

30 call els

c Manually or by position number?

w rite(5,*)'Change position menu' w rite(5,*) ' ' w rite(5,*)'1-Manual change using arrow keys' w rite(5,*)'2-By position number' w rite(5, * )'3-Return to Main Menu'

166

Jl? w r i t e (5,*) ' ' w rite(5,*)'Enter choice ' call inkey(key) c Goto section specified

i f (key.eq.49)goto 400 if(key.eq.50)goto 410 if(key.eq.51)goto 100 g o to 30 g o to 400 c Manual change section

400 call els w rite(5,*)'Right arrow moves clockwise' w rite(5,*)'Left arrow moves counterclockwise* write(5,*) ' ' w rite(5,*) ' returns to Position menu' call inkey(key) c Determine filter direction

i f (key.eq.54)goto 420 i f (key.eq.52)goto 430 i f (k e y . e q . 3 2 )g o to 30 c Go clockwise (right arrow)

420 call oup(2,771,0) call oup(10,771,0) call oup(2,771,0) do 440 z=l,500 440 x y z= l posit=posit+l g o to 400 c Go counterclockwise (left arrow)

430 call oup(0,771,0) call oup(8,771,0) call oup(0,771,0) do 450 z=l,500 450 x y z= l posit=posit-l g o to 400 c Change by position number

410 call els inpos=int(posit) w rite(5,*)'Present position number is *, inpos w rite(5,*) ' ' w rite(5,*)'Enter new position number ' r e a d ( 5 , * ) npos c Determine direction for movement

167 if{npos.It.posit)goto 460 if(npos.gt.posit)goto 470 if(npos.eq.posit)goto 480

c Less than case (move counterclockwise)

460 call els call oup(0,771,0) steps=posit-npos do 490 z=l,steps call oup(8,771,0) call oup(0,771,0) posit=posit-l do 500 i=l,1000 500 x y z = l 490 continue g o to 30

c Greater than case (move counterclockwise)

47 0 call els call oup(2, 771,0) steps=npos-posit do 510 z=l,steps call oup(10,771, 0) call oup(2,771,0) posit=posit+l do 520 1=1,1000 520 x y z = l 510 continue g o to 30

c Equal to case (no movement)

480 call els w rite(5,*)'It appears that the CVF is already in that position.' w rite(5,*)'If you do not believe that to be the case, you w ill' write (5,*)'probably want to run the "Manual reset" portion o f w rite(5,*)'the program.' w rite(5,*)' ' 530 write(5,*)'press to continue' call inkey(key) i f ( k e y . e q . 3 2 )g o to 30 g o to 530

c Move to open position Section c ******************★**********<

40 call els w rite(5,*)'Press to move CVF to the' w rite(5,*)'"open" (-50) position.' call inkey(key) if(key.eq.32)goto 600 g o to 40

c move CVF to position

1 6 8 J c determine direction for movement

600 if(posit.gt.-50)goto 610 if(posit.it.-50)goto 620 if(posit.eq.-50)goto 630

c Greater than case

610 steps=posit+50 call oup(0,771,0) do 640 z=l,steps call oup(8,771,0) call oup(0,771,0) do 650 i= l,1000 650 x y z= l 640 continue p o s it= - 5 0 w rite(5,*)'CVF in "open" position.' 660 write(5,*)'press to return to Main Menu' call inkey(key) if(key.eq.32)goto 100 g o to 660 c Lesser than case

620 steps=posit-50 call oup(2,771,0) do 670 z=l,steps call oup(10,771,0) call oup(2,771,0) do 680 i= l,1000 680 x y z= l 670 continue p o s it= - 5 0 w r i t e ( 5 , * ) 'CVF i n "open" p o s i t i o n . ' 690 w rite(5,*)'press to return to Main Menu' call inkey(key) i f (key.eq.32)goto 100 g o to 690 c Equal to case

630 call els w rite(5,*)'It appears that the CVF is already in that position.' w rite(5,*)'If you do not believe that to be the case, you w ill' w rite(5,*)'probably want to run the "Manual Reset" portion o f w rite(5,*)'the program. ' write(5,*)' ' w rite(5,*)'press to return to Main Menu' call inkey(key) if(key.eq.32)goto 100 g o to 690 c ***************************************************************** c Move to blocked position Section c *****************************************************************

169 50 c a l l e l s w rite(5,*)'Press to move CVF to the' w rite(5,*)*"blocked" (-20) position.' call inkey(key) if(key.eq.32)goto 700 g o to 50

c move CVF to position

c determine direction for movement

700 if(posit.gt.-20)goto 710 if(posit.It.-20)goto 720 if(posit.eq.-20)goto 730

c Greater than case

710 steps=posit+20 call oup(0,771,0) do 740 z=l,steps call oup(8,771,0) call oup(0,771,0) do 750 i=l,1000 750 x y z = l 740 continue p o s it= - 2 0 w rite(5,*)'CVF in "blocked" position.' 760 w rite(5,*)'press to return to Main Menu' call inkey(key) if(key.eq.32)goto 100 g o to 760

c Lesser than case

720 steps=((posit+20)*(-1)) w rite(5,*)'steps = ',steps call oup(2,771,0) do 770 z=l,steps c a l l o u p d O , 771, 0) call oup(2,771,0) do 780 i=l,1000 780 x y z= l 770 continue p o s it= - 2 0 w rite(5,*)'CVF in "blocked" position.' 790 w rite(5,*)'press to return to Main Menu' call inkey(key) if(key.eq.32)goto 100 g o to 790 c Equal to case

730 call els write (5,*)'It appears that the CVF is already in that position.' w rite(5,*)'If you do not believe that to be the case, you w ill' w rite(5,*)'probably want to run the "Manual Reset" portion o f w rite(5,*)'the program. '

170 write(5,*)' ' w rite(5,*)'press to return to Main Menu' call inkey(key) if(key.eq.32)goto 100 g o to 790 c ****************************************************: c Manual reset Section c ****************************************************:

60 call els w rite(5,*) This is to reset the position counter. Do this ' w rite(5,*) ONLY if you believe that the CVF has lost track o f w rite(5,*) its position.' w rite (5, *) w rite(5,*) Make sure the CVF is in the "start" position before' w rite (5, *) continuing.' w r i t e (5, *) 800 write(5,*) Do you wish to reset counter? (y/n)' call inkey(key) if(key.eq.121)goto 810 if(key.eq.110)goto 820 g o to 800 c Reset counter

810 posit=0 w rite(5, *) 'Counter re se t.' 830 w rite(5,*)'Press to return to Main Menu.' call inkey(key) if(key.eq.32)goto 100 g o to 830 c Do NOT r e s e t

820 write (5,*) 'Counter NOT reset.' 840 w rite(5,*)'Press to return to Main Menu' call inkey(key) if(key.eq.32)goto 100 g o to 840 c ***************************************************** c Scan Without CVF Section

70 call els w rite(5,*)'Enter number of readings per point read(5,*)nread 900 write(5,*)'press any key to begin' call inkey(key) if(key.eq.32)goto 905 g o to 905

905 dummy=setvideomode(SvreslSCOLOR) d u m m y = se tc o lo r(1) b a s e l= 2 4 0 call moveto(2,479,xy) do 910 i=l,638

171 y = 0 . do 920 kk=l,nread dunmy=setpixel(2,479) call oup(iword,ioadl,ibyte) call inp(iadl,ioadl,ibyte) call inp(iadh,ioadh,ibyte) y=y+16.*iadh+int(iadl/16.) 920 continue y=y/(1.*nread) y y ( i) = y iy= int((y-2048.)*460./2048. ) iy=basel-iy duinm y= setpixel (2, 479) duitimy=lineto (i, iy) 910 continue

c Check keyboard

930 if(key.eq-114)goto 940 call chkchr(kkey) if(kkey)goto 940 g o to 40

c Output to disk

940 call inkey(key) dum m y= setvideom ode($DEFAULTMODE) 950 w rite(5,*)'Would you like to save this scan? (y/n)' call inkey(key) if(key-eq.110)goto 960 if(key.eq.121)goto 970 g o to 950

970 write(*,'(a\)')'Enter filename ' read(* ,'(a)')fin open(2,file=fIn) do 980 i=l,638 980 write(3,*)i,yy(i) c l o s e (3)

960 write(5,*)'would you like to scan again? (y/n)' call inkey(key) if(key.eq.110)goto 100 if(key.eq.121)goto 990 g o to 960

990 write(5,*)'Use same number of readings per point? (y/n) call inkey(key) if(key.eq.110)goto 70 if(key.eq.121)goto 900 g o to 990

c Integration Section c ********************

80 call els

172 J c initialize variables

ziteravg=0. sdsum =0. zleverr=0. zlevtot=0. elapsed=0. i s e c s = 0 . do 85 jj=l,7201 85 zlevel(jj)=0. do 86 11=1,2000 zlevavg(11)=0. 86 zlevsd(ll)=0.

w rite(5,*) 'Enter time for Integration in seconds' w rite(5,*) 'Present maximum: 7200s (2 hours)' read(5,*)itime w rite(5,*) 'Enter number of iterations' w rite(5,*) 'Present maximum: 2000' read(5,*)iiter write(5,*) ' ' 88 write(5,*) 'Press to begin' call inkey(key) if (key.ne.32)goto 88

c number of iterations loop

c a l l e l s irow=7 ic o l= 4

call locate(irow -6,icol,1er) ztime=itime/60. w rite(5,*) 'Time for each iteration (in sec) ' ; itim e call locate(irow-5,icol,1er) w rite(5,*) 'Time for each iteration (in min) ' ; ztim e call locate(irow-4,icol,1er) write(5,*) ' Total number of iteration(s) ' ; i i t e r

do 9999 i= l,iiter

z le v to t= 0 e la p se d = 0 is e c s = 0

number of seconds loop

inelapsed=int(elapsed) call locate(irow-2,icol,1er) write(5,*) ' ' call locate(irow-2,icol,1er) write (5,*) 'Current iteration number : ',i call locate(irow-1,icol,1er) write(5,*) ' ' call locate(irow-1,icol, 1er) w rite(5,*) ' Time elapsed (in min) : ',inelapsed do 1000 kk=l,itime

173 J c 1 second of scanning loop

do 1010 jj= l,10000 call oup(iword, ioadl,ibyte) call inp(iadl,ioadl,ibyte) call inp(iadh,ioadh,ibyte) 1010 y=y+16.*iadh+int(iadl/16.)

y=y/10000. z l e v e l (klc) =y isecs=isecs+l

if(isecs.ne.60)goto 1000 is e c s = 0 elapsed=elapsed+l inelapsed=int(elapsed) c c a l l e l s call locate(irow,icol,1er) write(5,*) ' ' call locate(irow ,icol,1er) write(5,*) ' Time Elapsed (in min) : ',inelapsed 1000 continue

c total averaging loop

do 1020 jj=l, itime zlevtot=(zlevtot*l.)+(zlevel(j j)*1.) 1020 continue

zlevavg(i)=(zlevtot*!.)/(itime*1.)

c Display integration result

call locate(irow+i,icol,1er) write(5,*) 'Iteration : ' , i , ' : ', & ' Ave Light Value (in DAC units) = ',zlevavg(i) c write(5,*) 'Average Light Value (in DAC units) = ',zlevavg(i) write(5,*) ' '

9999 continue

c Display average and standard deviation

do 1099 zz=l,iiter ziteravg=ziteravg+zlevavg(zz) 1099 continue ziteravg=ziteravg/(iiter*l.)

do 1098 zy=l,iiter zlevsd(zy)=zlevavg(zy)-ziteravg zlevsd(zy)=zlevsd(zy)*zlevsd(zy) sdsum=sdsum+zlevsd(zy) 1098 continue

sdsum=sdsum/(iite r* 1.-1.) sd=sqrt(sdsum) zleverr=sd/sqrt(iiter) w rite(5,*) 'Average Light Value = ziteravg,' +/- ',zleverr

174

Â.' 1025 write(5,*) 'Would you like to save this data to a file? (y /n )' call inkey(key) if(key.eq.110)goto 1030 i f (key.eq.121)goto 1027 g o to 1025

E n te r file n a m e :1027 write(*,'(a\)') E n te r file n a m e :1027 read(*,' (a)')fin open(3, file=fln) do 1028 i= l,iiter 1028 write(3,*) i,zlevavg(i) w rite(3,*) ziteravg,zleverr c l o s e (3)

1030 write(5,*) 'Would you like to integrate again? (y/n) call inkey(key) if(key.eq.110)goto 100 if(key.eq.121)goto 80 g o to 1030 c c Q u it c

999 end

175 APPENDIX C

PERIODIC BOX VALUES IN HYPERCHEM

One o f the features o f Hyperchem is the ability to create a box o f specified size around a solute and fill it with randomly oriented water molecules. Use of this feature required a knowledge of the appropriate edge length of the box to obtain the desired cluster size.

Listed below are the sizes required for a cube (a=b=c) to achieve a specified number o f solvating waters, n, for both neutral water (Table C-1) and with a solute, X, where X is a halide ion, alkali ion, OH , or HjO^ (Table C-2).

176 n Box n Box n Box Size Size Size 1 2 26 8.98 51 12 2 4 27 9 52 12.1 3 5 28 9.3 53 12.165 4 5.5 29 9.4 54 12.2 5 6 30 9.462 55 12.25 6 6.2 31 9.5 56 12.3 7 6.5 32 9.7 57 12.5 8 6.7 33 10 58 12.55 9 7 34 10.3 59 12.6 10 7.3 35 10.35 60 12.7 11 7.4 36 10.4 61 12.77 12 7.5 37 10.425 62 13 13 7.6 38 10.5 63 13.05 14 7.9 39 10.56 64 13.07 15 7.997 40 10.57 65 13.2 16 8 41 10.59 66 13.25 17 8.1 42 10.7 67 13.3 18 8.2 43 11 68 13.35 19 8.22 44 11.1 69 13.4 20 8.3 45 11.3 70 13.45 21 8.5 46 11.35 71 13.5 22 8.51 47 11.5 72 13.51 23 8.6 48 11.8 73 13.6 24 8.9 49 11.9 74 13.65 25 8.96 50 11.95 75 13.7

Table C-1. Edge length to achieve a specified neutral water cluster size, n.

177 i X ( H z O ) ,

n Box n Box n Box Size Size Size 1 5 26 9.2 51 12.165 2 5.5 27 9.4 52 12.2 3 6 28 9.462 53 12.25 4 6.3 29 9.5 54 12.3 5 6.5 30 9.7 55 12.5 6 6.7 31 10 56 12.55 7 7 32 10.3 57 12.6 8 7.3 33 10.35 58 12.7 9 7.4 34 10.4 59 12.77 10 7.5 35 10.425 60 13 11 7.7 36 10.5 61 13.05 12 7.9 37 10.56 62 13.07 13 7.997 38 10.57 63 13.2 14 8 39 10.59 64 13.25 15 8.1 40 10.7 65 13.3 16 8.2 41 11 66 13.35 17 8.22 42 11.2 67 13.4 18 8.4 43 11.3 68 13.45 19 8.5 44 11.35 69 13.5 20 8.52 45 11.5 70 13.51 21 8.7 46 11.8 71 13.6 22 8.9 47 11.9 72 13.65 23 8.97 48 11.95 73 13.7 24 8.98 49 12 74 13.73 25 9 50 12.1 75 13.75

Table C-1. Edge length to achieve a specified X(H20)„ cluster size where X is a halide ion, alkali ion, OH', or

178 A P P E N D IX D

CALCULATION OF CARTESIAN COORDINATES FOR THE ORDERED ICE-Ih CRYSTAL UNIT CELL

Leadbetter et aL obtained a neutron scattering pattern of annealed 0 ,0 at 5 K^. The a n n e aling process and low temperature obtained a highly ordered crystal for the subsequent experiment. The crystal was found to belong to the space group with the following dimensions for the unit cell: a=4.5019 A, b=7.7978 A, and c=7.3280 A. The five unique positions for atoms in the unit cell are given in Table D-1.

The Cwf2, space group has 4 symmetry operations as well as 2 positions for each operation. These are given in Table D-2.

Each of the 5 atoms given in Table D-1 are used to generate 7 additional atomic positions, in fractional coordinates, through use of the operations given in Table D-2. The generated results are given in Table D-3. N ote that the ratio o f oxygen to hydrogen is greater than 1:2. This is the result of the fact that not all of the generated coordinates are unique. Elimination of the shaded equivalent positions in Table D-3 yields the coordinates given in Table D-4. These coordinates are in Angstroms, obtained by multiplying the fractional coordinates by the appropriate unit cell dimension. Negative values for positions have been translated 1 unit cell length. The hydrogen atoms were paired with their

179 corresponding oxygen atoms by the distance between positions in the unit cell. Note that

two hydrogen atoms, labeled H14 and H16, are on the opposite side of the unit cell from their corresponding oxygen (Figure D - 1 ) . The position of these atoms has been translated

1 unit cell to allow a reasonable bond distance. The structure of the unit cell was then entered into a Hyperchem input file (.hin) format (Figure D - 2 ) . The graphical output is shown in Figure D - 3 . The rem a in in g structures in the ice studies were generated by addition o f one or more unit cells in the 3 dimensions.

R eferences

1. Leadbetter, A.J., Ward, R., Clark, J.W., Tucker, PL&., Matsuo, T. & Suga, H.J Chem Phys 82, 424-8 (1985). 2. Hahn, T. (D Reidel Publishing Co., Boston, 1983). 3. Lonsdale, K. Proc Roy Sac A 247, 424-34 (1958). 4. Eisenberg, D. & BCauzmann, W. T6e Structure and Properties of Water (Oxford University Press, New York, 1969).

1 8 0 9 o i 0 2 D1 D 2 D3 X 1 0.0000 0.5000 0.0000 0.0000 0.6766 y 1 0.6648 0.8255 0.6636 0.5363 -0.2252 z 1 0.0631 -0.0631 0.1963 0.0183 -0.0183

Table D-1. Positions of unique atoms within the unit cell o f ordered Ice-Ih as reported by Leadbetter et aL Units are fractions of unit cell edge length in each respective dimension.

(0, 0, 0) (+ ’/ 2 , +Vz, 0)

(x, Y, z) (-X, -y, Vz + z) (x, -y, Yz + z) (-x, y, z)

Table D-2. Symmetry operations for the space group, Cmc2,.-

181 X y z X y z O l 0.0000 0.6648 0.0631 HI 0.0000 0.6636 0.1963 0.0000 -0.6648 0.5631 0.0000 -0.6636 0.6963 KdddOf :0 # Q 0 : # 6 5 3 5 #W 03965- o .(x m ? o # # ? 0 : # 6 01965 0.5000 1.16^ 0.0631 0.5000 i.1636 0.1963 0.5000 -0.1648 0.5631 0.5000 -0.1636 0.6963 0.5000= '030# ; 0.6965# 0 # # ;03000 . 0:1963

02 0.5000 0.8255 -0.0631 H2 0.0000 0.5363 0.0183 -0.5000 -0.8255 0.4369 0.0000 -0.5363 0.5183 (kspoo : 0 . 0 # 4)3365; 03185 :0:&255: 0.0000 . 03565 0:0183 1.0000 1.3255 -0.0631 0.5000 1.0363 0.0183 0.0000 -0.3255 0.4369 0.5000 -0.0363 0.5183 r n 3 2 m 0 3 # -0.0365 03183 0.0000 13255; ^ -0.0(551 03000 10365 0.0183

H3 0.6766 -0.2252 -0.0183 -0.6766 0.2252 0.4817 -0.6766 0.2252 0.4817 0.6766 -0.2252 -0.0183 1.1766 0.2748 -0.0183 -0.1766 0.7252 0.4817 1.1766 0.7252 0.4817 -0.1766 0.2748 -0.0183

Table D-3. Generated positions o f atoms in the unit cell. Shaded atoms are in redundant positions.

182 X V z X y 2 o 0.000 5.184 0.4624 H 0.000 5.175 1.438 0.000 2.614 4.126 0.000 2.623 5.102 2.251 1.285 0.4624 2.251 1.276 1.438 2.251 6.513 4.126 2.251 6.522 5.102 2.251 6.437 6.866 0.000 4.182 0.134 2.251 1.361 3.202 0.000 3.616 3.798 0.000 2.538 6.866 2.251 0.283 0.134 0.000 5.260 3.202 2.251 7.515 3.798 3.046 6.042 7.194 1.456 1.756 3.530 3.046 1.756 3.530 1.456 6.042 7.194 0.795 2.143 7.194 3.707 5.655 3.530 0.795 5.655 3.530 3.707 2.143 7.194

Table D -4. Final generated atomic positions in Angstroms.

Ice Ih Ic II III V VI VII VIII Crystal Hexagonal Cubic Rhombohedral Tetragonal Monoclinic Tetragonal Cubic Cubic System Space P63 / mmc F43m R.3 P4,2,2 A2/a P4:/nmc Im3m Im3m Group Cell a 4.48 a 6 J 5 a 7.78 a 6.73 a 9 0 2 b 7 j 4 a 60 7 a 3.41 Dimensions C7.3I a 113.1° C6.83 c IO_35 3 1090° C5.79 No rtralecules / 4 8 12 12 28 10 2 2 unit cell Density at I atm 0.94 •• 1.17 1.14 103 101 1.50 and-l75°C

Table D-5. Crystallographic properties of ice polymorphs.^ Note the difference in space group for Ice-Ih. This is the result of the hydrogen positions determined in ref. (1).

183 M (4.13) 01 (P.46) 07(6.87) 08 0 JO)

- - ^ ^ 1 ^ 1 1 1 H5CB.13) V 1 1 1 1 { I H13(7.19 1/ j ^ H p 0 J 3 ) 1

1 O 0J3) j H12 (7.19) O I > I i 03 Cq.4(5)v J t S r ' 06 (3 JO) Æ 05(6.07) H3ni.44') H i 11 (3 J3 ) ! i ■ œ (T 19) '

i ! H16(7.19; o i » aJ 3 ) i ; i ! > 1 Î A

YpofjÜcn(b)

Figure D-1. MathCAD generated positions of atoms in the unit cell. Note that the atoms marked H14 and H16 must be translated -1 unit cell lengths along the x-dimension to achieve a real bond distance.

184 Figure D -Z Hyperchem graphie o f the Ice-Ih unit cell.

185

i l APPEN D IX E

SELECTED HYPERCHEM FILES

Ice-Ih Unit Cell

This is the unit cell derived from the calculations of Appendix D. As a first step for creation of larger clusters, this set of 8 molecules is combined with additional unit cells where combinations o f the x-, y-, and z- values have been added to an integer number of cell lengths to provide molecules in a different unit cell position in the same crystal. The values for the unit cell lengths are: a (or x): 4.5019, b (y): 7.7978, and c (z): 7.3280 A. This file is ilhuc.hin. forcefield opls sys 0 view 40 0.16869 55 15 100010001 -1.1255 -3.899 -58.664 seed -1111 mol 1 atom 1 - O ** - -0.3586454 0 5.184 0.4624 2 2 s 3 s atom 2 - H ** - 0.1793224 0 5.175 1.438 1 1 s atom 3 - H ** - 0.1793228 0 4.182 0.1341 1 1 s endmol 1 mol 2 atom 1 - O ** - -0.3586454 0 2.614 4.126 2 2 s 3 s atom 2 - H**- 0.1793224 0 2.623 5.102 1 1 s atom 3 - H ** - 0.1793228 0 3.616 3.798 1 1 s endmol 2 mol 3 186 atom 1 43 ** - -0.3586454 2.251 1.285 0.4624 2 2 s 3 s atom 2 - H - 0.1793224 2.251 1.276 1.438 1 1 s atom 3 - H * * - 0.1793228 2.251 0.2831 0.1341 1 1 s endmol 3 mol 4 atom 1 -O** - -0.3586454 2.251 6.513 4.126 2 2 s 3 s atom 2 - H - 0.1793224 2.251 7.515 3.798 I l s atom 3 - H ** - 0.1793228 2.251 6.522 5.102 1 1 s endmol 4 mol 5 atom 1 - O ** - -0.3586454 2.251 6.437 6.866 2 2 s 3 s atom 2 - H - 0.1793224 3.046 6.042 7.194 1 1 s atom 3 - H - 0.1793228 1.456 6.042 7.194 1 1 s endmol 5 mol 6 atom l -(0** - -0.3586454 2.251 1.361 3.202 2 2 s 3 s atom 2 - H ** - 0.1793224 1.456 1.756 3.53 1 1 s atom 3 - H ** - 0.1793228 3.046 1.756 3.53 1 1 s endmol 6 m ol 7 atom 1 - O - -0.3586454 0 2.538 6.866 2 2 s 3 s atom 2 - H ** - 0.1793224 0.795 2.143 7.194 1 1 s atom 3 - H ** - 0.1793228 -0.795 2.143 7.194 1 1 s endmol 7 mol 8 atom 1 -O ** - -0.3586454 0 5.26 3.202 2 2 s 3 s atom 2 - H ** - 0.1793224 0.795 5.655 3.53 1 1 s atom 3 - H - 0.1793228 -0.795 5.655 3.53 1 1 s endmol 8

187 Ice-Ic Unit Cell

This unit cell is also derived &om the calculations of Appendix D. This file is named ilcuc.hin.

forcefield opls sys 0 view 40 0.16056 55 15 -0.5683673 -0.6634484 0.4866156 0.04593136 -0.6160949 -0.7863316 0.8214919 -0.4245742 0.3806414 -0.53265 3.5352 -55.27 seed -1111 mol 1 atom 1 - O ** - -0.3586454 00022s3s atom 2 - H ** - 0.1793224 -0.57 -0.57 0.57 1 1 s atom 3 - H ** - 0.1793228 0.57 -0.57 -0.57 1 1 s endmol 1 mol 2 atom l-(3 **- -0.3586454 03322s3s atom 2 - H ** - 0.1793224 -0.57 3.57 2.43 1 1 s atom 3 - H ** - 0.1793228 0.57 3.57 3.57 1 1 s endmol 2 mol 3 atom 1 - O ** - -0.3586454 30322s3s atom 2 - H ** - 0.1793224 2.43 0.57 2.43 1 1 s atom 3 - H ** - 0.1793228 3.57 0.57 3.57 1 1 s endmol 3 mol 4 atom 1 - O ** - -0.3586454 33022s3s atom 2 - H ** - 0.1793224 2.43 2.43 0.57 1 1 s atom 3 - H ** - 0.1793228 3.57 2.43 -0.57 1 1 s endmol 4 mol 5 atom 1 - O ** - -0.3586454 1.5 1.5 1.5 2 2 s 3 s atom 2 - H - 0.1793224 0.93 0.93 0.93 1 1 s atom 3 - H ** - 0.1793228 0.93 2.07 2.07 1 1 s endmol 5 mol 6 atom 1 -<3 **- -0.3586454 1.5 4.5 4.5 2 2 s 3 s atom 2 - H ** - 0.1793224 2.07 3.93 5.07 1 1 s atom 3 - H ** - 0.1793228 2.07 5.07 3.93 1 1 s endmol 6 mol 7 atom 1 - O ** - -0.3586454 4.5 1.5 4.5 2 2 s 3 s atom 2 - H ** - 0.1793224 5.07 0.93 5.07 1 1 s atom 3 - H ** - 0.1793228 5.07 2.07 3.93 1 1 s

1 8 8 endmol 7 m ol 8 atom 1 - O ** - -0.3586454 4.5 4.5 1.5 2 2 s 3 s atom 2 - H - 0.1793224 3.93 3.93 0.93 1 1 s atom 3 - H**- 0.1793228 3.93 5.07 2.07 1 1 s endmol 8

189 Hexamer “Cage” Structure

This is the predicted global m in i m u m for the ( H iO ) ^ cluster. The file is named

wôcage.hin.

forcefield opls svs 0 view 40 0.25252 55 15 -0.6417714 0.03414106 -0.7661357 0.672842 0.5044283 -0.541143 0.3679853 -0.8627783 -0.3466992 -0.24365 -3.6632 -51.523 seed -1111 mol 1 atom 1 - O ** - -0.4493923 2.126599 2.323017 -1.497497 2 2 s 3 s atom 2 - H**- 0.218823 2.525375 2.981963 -0.9197923 1 1 s atom 3 - H**- 0.2217103 1.528313 2.785526 -2.095171 1 1 s endmol 1 mol 2 atom 1 - O ** - -0.4573765 1.128879 4.16688 -3.200283 2 2 s 3 s atom 2 - H ** - 0.2031428 0.7029992 4.005824 -4.034363 1 1 s atom 3 - H ** - 0.2575462 2.03613 4.475164 -3.363219 1 1 s endmol 2 mol 3 atom 1 - O ** - -0.4133043 0.4083072 5.518684 -0.9975004 2 2 s 3 s atom 2 - H**- 0.2324671 0.4652861 5.219837 -1.912589 1 1 s atom 3 - H ** - 0.1909525 -0.2147924 4.959805 -0.5452226 1 1 s endmol 3 mol 4 atom 1 - O ** - -0.4390173 2.94619 4.703122 -0.5362201 2 2 s 3 s atom 2 - H**- 0.1999322 3.407141 4.984723 0.2456926 1 1 s atom 3 - H**- 0.2454461 2.101593 5.173109 -0.5885701 1 1 s endmol 4 mol 5 atom 1- O** - -0.4307775 4.163649 2.056881 -3.2327 2 2 s 3 s atom 2 - H ** - 0.1871048 4.92021 1.621016 -2.858141 1 1 s atom 3 - H ** - 0.2352961 3.400439 1.884941 -2.6651 1 1 s endmol 5 mol 6 atom 1 - O ** - -0.4522634 3.759859 4.712348 -3.129035 2 2 s 3 s atom 2 - H**- 0.2181992 3.72924 4.807296 -2.171691 1 1 s atom 3 - H ** - 0.2315114 4.137646 3.84959 -3.333793 1 1 s endmol 6

190 Hexamer "Book^ Structure

This is the (HiO)^ structure related to the stacked cube family. The file is named w6book.hin. forcefield opls sys 0 view 40 0.27429 55 15 -0.4983553 -0.03145489 -0.8664022 0.6402284 0.6604965 -0.3922396 0.5845936 -0.75017 -0.3090237 0.31213 4.4924 -35.684 seed -1111 mol 1 atom 1 - O ** - -0.419446 16.85972 2.18321 7.683013 2 2 s 3 s atom 2 - H ** - 0.1889791 17.19156 1.300119 7.564598 1 1 s atom 3 - H ** - 0.228382 17.53305 2.800068 7.374084 1 1 s endmol 1 mol 2 atom 1 - O ** - -0.4120512 18.88703 3.968249 7.349269 2 2 s 3 s atom 2 - H ** - 0.2287725 18.99966 4.226315 8.270097 1 1 s atom 3 - H ** - 0.1891757 18.67931 4.749976 6.848832 1 1 s endmol 2 mol 3 atom 1 - O ** - -0.4309664 16.9319 6.900735 10.31988 2 2 s 3 s atom 2 - H ** - 0.2413985 16.19659 6.382643 10.67272 1 1 s atom 3 - H ** - 0.1925387 16.61993 7.361262 9.548277 1 1 s endmol 3 m ol 4 atom 1 - O ** - -0.4366894 14.97405 5.104637 10.80846 2 2 s 3 s atom 2 - H ** - 0.1914896 14.66856 4.940009 11.69374 1 1 s atom 3 - H ** - 0.242732 15.41553 4.304416 10.49283 1 1 s endmol 4 mol 5 atom 1 - O ** - -0.4631286 16.64746 3.060616 10.23047 2 2 s 3s atom 2 - H ** - 0.2319939 16.50785 2.610924 9.389964 1 1 s atom 3 - H ** - 0.2167262 17.47842 3.542399 10.17341 1 1 s endmol 5 mol 6 atom 1 -43 ** -0.4410105 18.68808 4.865365 9.911213 2 2 s 3s atom 2 - H ** - 0.2033096 19.38601 4.898227 10.55652 1 1s atom 3 - H ** - 0.2477954 18.15038 5.664542 9.997121 1 1 s endmol 6

191 APPENDIX F

PM3 CALCULATED BINDING ENERGIES AND EXCESS BINDING ENERGIES PER WATER MOLECULE FOR SELECTED CLUSTERS

The following tables, organized by f a m ily , contain the results o f the calculations on numerous water clusters using PM3. This is intended as a reference guide o f structures for future work.

192 Total Atomic B inding F ilenam e n Ene% y E.b (*.hin) Note 1 -217.22 0 monomer water monomer for determining

Bridging Chain Structures 3 -659.53 -2.62 bc-w3mn 4 -879.95 -2.77 bc-w4mn 5 -1100.70 -2.92 bc-w5mn 6 -1320.32 -2.83 bc-w6mn 7 -1541.56 -3.00 bc-w7mn 8 -1761.12 -2.92 bc-w8mn

Alternating Chain Structures 3 -659.82 -2.72 ac-w3mn 4 -880.79 -2.98 ac-w4mn 5 -1101.21 -3.02 ac-w5mn 6 -1322.13 -3.14 ac-w6mn 7 -1542.41 -3.12 ac-w7mn 8 -1762.18 -3.05 ac-w8mn

Table F.l. PM3 values for the monomer, alternating chains, and bridging chain structures studied in Chapter 1.

193 Total Atomic B in d in g Slenam e n E nc% y (*.hin) Notes Ring Structiues 3 -661.74 -3.36 ring-w3mn 4 -887.25 -4.59 ring-w4mn 5 -1109.96 -4.77 ring-w5nin 6 -1332.76 -4.91 ring-w6mn first structure not the global minimum 7 -1554.56 -4.86 ring-w7inn 8 -1776.22 -4.81 ring-w8tnn 9 -1997.90 -4.77 ring-w9mn 10 -2219.04 -4.68 ring-wlOtnn

Stacked Cube Structures Aligned Rings 8 -1786.12 -6.05 sc-w08mn 12 -2684.24 -6.47 sc-wl2mn 16 -3583.12 -6.73 sc-wl6mn 20 -4482.01 -6.88 sc-w20mn 24 -5380.96 -6.99 sc-w24tnn 28 -6279.87 -7.06 sc-w28tnn 32 -7178.81 -7.12 sc-w32mn 36 -8077.69 -7.16 sc-w36mn 92 -20662.91 -7.38 sc-w92mn

Opposing Rings 8 -1786.35 -6.07 sac-w08tnn global minimum octamer 12 -2684.67 -6.50 sac-wl2mn 16 -3583.76 -6.77 sac-wl6mn 20 -4482.95 -6.93 sac-w20mn 24 -5382.14 -7.04 sac-w24tnn 28 -6281.33 -7.11 sac-w28mn 32 -7180.50 -7.17 sac-w32mn 36 -8079.71 -7.22 sac-w36mn

Table F.2 Rings and stacked cube structures

194 J T o tal A tom ic B inding filenam e n E nergy E xb (*.hin) Notes Stacked Pentagons Aligned Rings 10 -2234.35 -6.22 sp-wlOmn 15 -3357.15 -6.59 sp-wl5mn 20 -4481.27 -6.84 sp-w20mn 25 -5605.56 -7.00 sp-w25tnn 30 -6729.24 -7.09 sp-w30mn 35 -7854.11 -7.18 sp-w35mn 40 sp-w40st not optimized 45 sp-w45st not optimized 50 -11227.29 -7.33 sp-w50mn

Opposing Rings 10 -2234.47 -6.23 spa-wlOmn 15 -3357.34 -6.60 spa-wl5mn 20 -4481.72 -6.87 spa-w20mn 25 -5606.24 -7.03 spa-w25mn 30 -6730.82 -7.14 spa-w30mn

Stacked Hexagons Aligned Rings 12 -2681.40 -6.23 sh-wl2mn 18 -4030.41 -6.69 sh-wl8mn 24 -5379.28 -6.92 sh-w24mn 30 -6728.18 -7.05 sh-w30mn 72 -16129.20 -6.80 sh-w72mn

Opposing Rings 12 -2683.51 -6.40 sha-wl2mn 18 -4031.06 -6.73 sha-wl8mn 24 -5380.34 -6.96 sha-w24mn 30 -6729.66 -7.10 sha-w30mn

Table F.3 Stacked pentagon and hexagon structures

195 Total Atomic B in d in g filenam e a E n erg y E.b (*.hin) Notes Partial Stacked Cubes Aligned Rings 6 -1333.57 -5.04 psc-w06mn 7 -1557.44 -5.27 psc-w07tnn 9 -2006.65 -5.74 psc-w09inn 10 -2233.54 -6.13 psc-wlOmn 11 -2455.59 -6.02 psc-wllmn 13 -2907.50 -6.43 psc-wl3m n 14 -3131.79 -6.48 psc-wl4m n 15 -3354.12 -6.39 psc-wl5tnn 17 -3805.10 -6.61 psc-wl7m n 18 -4030.37 -6.69 psc-wl8tnn 19 -4252.94 -6.62 psc-wl9tnn 21 -4704.10 -6.78 psc-w21mn 22 -4925.00 -6.64 psc-w22mn 23 -5151.57 -6.76 psc-w23mn 26 -5823.87 -6.78 psc-w26tnn 27 -6050.47 -6.87 psc-w27tnn

Table F.4 Partial stacked cube structures

196

SU T o ta l A tom ic B in d in g filenam e n E n erg y Ed, (* h in ) N otes Hexagonal Ice “Double Cone” Structures 8 -1770.41 -4.08 Ilh-dc08mn 26 -5801.21 -5.90 Ilh-dc26tnn 34 -7590.57 -6.03 Ilh-dc34m n 70 -15664.94 -6.56 Ilh-dc70m n 78 -17451.46 -6.52 Ilh-dc78m n 144 Ilh-dcl44st not optimized 152 Ilh-dcl52st not optimized 248 Ilh-dc248st nor optimized 256 Ilh-dc256st not optimized 394 Ilh-dc394st not optimized 402 Ilh-dc402st not optimized

“Partial Prism” Structures 30 -6683.40 -5.56 Ilh-pp30m n 39 -8697.39 -5.79 Ilh-pp39m n 48 -10713.98 -5.99 Ilh-pp48mn 57 -12736.65 -6.23 Ilh-pp57m n 72 -16101.54 -6.41 Ilh-pp72m n 90 -20140.00 -6.56 Ilh-pp90m n

Table F.5 Hexagonal ice structures

197 Total Atomic B inding filenam e n E ne% y (*.hin) Notes C ubic Ice “Octahedron” Structures 10 -2221.36 -4.92 Ilc-octalOmn 18 -4011.81 -5.66 Ilc-octal8m n 35 -7825.80 -6.37 Ilc-octa35tnn 53 -11868.25 -6.71 Ilc-octa53mn 84 -18840.44 -7.07 Ilc-octa84tnn

“Pyramid” Structures 26 -5803.98 -6.01 Ilc-pyr26mn 51 -11409.00 -6.49 Ilc-pyr51mn 136 -30500.72 -7.05 Ilc-pyrl36mn

indeterminate structures 14 -3115.99 -5.35 Ilc-idl4m n 18 -4012.11 -5.68 Ilc-idl8mn 26 -5804.02 -6.01 Ilc-id26Amn 26 -5800.30 -5.87 Ilc-id26Bmn 26 -5800.83 -5.89 Ilc-id26Cmn 30 -6699.54 -6.10 Ilc-id30mn 32 -7154.63 -6.34 Ilc-id32mn 34 -7605.58 -6.47 Ilc-id34mn 38 -8496.62 -6.38 Ilc-id38Amn 38 -8496.55 -6.37 Ilc-id38Bmn 38 -8495.39 -6.34 Ilc-id38Cmn 38 -8499.63 -6.45 Ilc-id38Dmn 42 -9397.27 -6.52 Ilc-id42mn 43 -9616.58 -6.42 Ilc-id43mn 45 -10066.01 -6.47 Ilc-id45mn 48 -10742.43 -6.58 Ilc-id48mn 57 -12762.32 -6.68 Ilc-id57mn * Ilc-borg4096 Cube o f 4096 waters for creating clusters

Table F.6 Cubic ice structures

198 Total Atomic B inding filenam e n. E nergy Ed, (*.hin) N o tes 2 -437.94 -1.75 LHB-dimer Linear Hydrogen Bonded dimer 2 -436.38 -0-97 BIF-dimer Bifurcated dimer 2 -436.20 -0.88 DPO-dimer Dipole Opposed dimer 4 -885.10 -4.06 alt-4uuuu tetramer ring with no opposite H’s 4 -885.10 -4.06 alt-4dddd “ 4 -886.32 -4.36 alt-4uuud one outer H on the opposite side 4 -886.33 -4.36 alt-4uudu 4 -886.33 -4.36 alt-4uduu 4 -886.33 -4.36 alt-4duuu 4 -886.33 -4.36 alt-4dddu “ 4 -886.33 -4.36 alt-4ddud 4 -886.33 -4.36 alt-4dudd “ 4 -886.32 -4.36 alt-4uddd “ 4 -886.50 -4.40 alt-4uudd two outer H’s on opposite side, together 4 -886.49 -4.40 alt-4uddu 4 -886.49 -4.40 alt-4dduu 4 -886.49 -4.40 alt-4duud 4 -887.25 -4.59 alt-4udud alternating side outer H’s 4 -887.21 -4.58 alt-4dudu 12 -2675.56 -5.74 s-bookl2tnn 2 stacked “book” structures

Table F.7 Other structures

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